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2412.02046v1	http://arxiv.org/abs/2412.02046v1	Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results	\documentclass[a4paper, 10pt, twoside, notitlepage]{amsart} \usepackage{amsmath,amscd} \usepackage{amssymb} \usepackage{amsthm} \usepackage{comment} \usepackage{graphicx, xcolor} \usepackage{mathrsfs} \usepackage[linktocpage,ocgcolorlinks, linkcolor=blue]{hyperref} \usepackage{bm} \usepackage{bbm} \usepackage{url} \usepackage[utf8]{inputenc} \usepackage{mathtools,amssymb} \usepackage{esint} \usepackage{tikz} \usepackage{dsfont} \usepackage{relsize} \usepackage{url} \urlstyle{same} \usepackage{xcolor} \usepackage{graphicx} \usepackage{mathrsfs} \usepackage[shortlabels]{enumitem} \usepackage{lineno} \usepackage{amsmath} \usepackage{enumitem} \usepackage{amsthm} \usepackage{verbatim} \usepackage{dsfont} \numberwithin{equation}{section} \renewcommand{\thefigure}{\thesection.\arabic{figure}} \DeclarePairedDelimiter\ceil{\lceil}{\rceil} \DeclarePairedDelimiter\floor{\lfloor}{\rfloor} \allowdisplaybreaks \newcommand{\para}[1]{\vspace{3mm} \noindent\textbf{#1.}} \mathtoolsset{showonlyrefs} \graphicspath{{images/}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{claim}[theorem]{Claim} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}{Question} \newtheorem{remark}[theorem]{Remark} \newtheorem{assumption}{Assumption} \newtheorem{observation}{Observation} \newtheorem{centertext}{} \title[Inverse problems for damped nonlocal wave equations]{Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results} \author[P. Zimmermann]{Philipp Zimmermann} \address{Departament de Matem\`atiques i Inform\`atica, Universitat de Barcelona, Barcelona, Spain} \email{[email protected]} \newcommand{\todo}[1]{\footnote{TODO: #1}} \newcommand{\C}{{\mathbb C}} \newcommand{\R}{{\mathbb R}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\N}{{\mathbb N}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\A}{{\mathcal A}} \newcommand{\Order}{{\mathcal O}} \newcommand{\order}{o} \newcommand{\eps}{\varepsilon} \newcommand{\der}{{\mathrm d}} \newcommand{\id}{\mathrm{Id}} \newcommand {\p} {\partial} \newcommand{\LC}{\left(} \newcommand{\RC}{\right)} \newcommand{\wt}{\widetilde} \newcommand{\Kelvin}{K}\newcommand{\riesz}{I_{\alpha}}\newcommand{\xrt}{X}\newcommand{\dplane}{R_d} \newcommand{\no}{N}\newcommand{\nod}{N_d} \newcommand{\schwartz}{\mathscr{S}} \newcommand{\cschwartz}{\mathscr{S}_0} \newcommand{\tempered}{\mathscr{S}^{\prime}} \newcommand{\rapidly}{\mathscr{O}_C^{\prime}} \newcommand{\slowly}{\mathscr{O}_M} \newcommand{\fraclaplace}{(-\Delta)^s} \newcommand{\fourier}{\mathcal{F}} \newcommand{\ifourier}{\mathcal{F}^{-1}} \newcommand{\vev}[1]{\left\langle#1\right\rangle} \newcommand{\pol}{\mathcal{O}_M} \newcommand{\borel}{\mathcal{M}} \newcommand{\Hcirc}{\overset{\hspace{-0.08cm}\circ}{H^s}} \newcommand{\test}{\mathscr{D}}\newcommand{\smooth}{\mathscr{E}}\newcommand{\cdistr}{\mathscr{E}'}\newcommand{\distr}{\mathscr{D}^{\prime}}\newcommand{\dimens}{n}\newcommand{\kernel}{h_{\alpha}} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\aabs}[1]{\left\lVert #1 \right\rVert}\newcommand{\ip}[2]{\left\langle #1,#2 \right\rangle}\newcommand{\im}{\mathsf{i}} \DeclareMathOperator{\spt}{spt}\DeclareMathOperator{\ch}{ch}\DeclareMathOperator{\Div}{div} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\loc}{loc} \newcommand{\radon}{\mathscr{M}}\newcommand{\weak}{\rightharpoonup}\newcommand{\weakstar}{\overset{\ast}{\rightharpoonup}} \newcommand{\Vareps}{\boldsymbol{\varepsilon}} \begin{document} \maketitle \begin{abstract} The main purpose of this article is to establish new uniqueness results for Calder\'on type inverse problems related to damped nonlocal wave equations. To achieve this goal we extend the theory of very weak solutions to our setting, which allows to deduce an optimal Runge approximation theorem. With this result at our disposal, we can prove simultaneous determination results in the linear and semilinear regime. \medskip \noindent{\bf Keywords.} Fractional Laplacian, wave equations, nonlinear PDEs, inverse problems, Runge approximation, very weak solutions. \noindent{\bf Mathematics Subject Classification (2020)}: Primary 35R30; secondary 26A33, 42B37 \end{abstract} \tableofcontents \section{Introduction} \label{sec: introduction} In recent years, inverse problems for nonlocal partial differential equations (PDEs) of elliptic, parabolic and hyperbolic type have been studied. This line of research was initiated by Ghosh, Salo and Uhlman \cite{GSU20}, in which they have considered the (partial data) Calder\'on problem related to the \emph{fractional Schr\"odinger equation} \begin{equation} \label{eq: fractional Schroedinger equation} \begin{cases} ((-\Delta)^s+q)u=0&\text{ in }\Omega,\\ u =\varphi & \text{ in } \Omega_e, \end{cases} \end{equation} where $\Omega\subset\R^n$ is a bounded domain, $\Omega_e=\R^n\setminus \overline{\Omega}$, $0<s<1$, $q$ is a suitable potential and $(-\Delta)^s$ is the \emph{fractional Laplacian} which is the operator with Fourier symbol $\|\xi\|^{2s}$. In this problem one asks whether the knowledge of the \emph{(partial) Dirichlet to Neumann (DN) map} \begin{equation} \label{eq: partial DN map Schroeding} \Lambda_q \varphi= (-\Delta)^s u_\varphi\|_{W_2}, \quad \varphi\in C_c^{\infty}(W_1), \end{equation} where $W_1,W_2\subset\Omega_e$ are given measurement sets (i.e.~nonempty open sets) and $u_\varphi$ denotes the unique solution to \eqref{eq: fractional Schroedinger equation}, uniquely determines the potential $q$. The overall strategy to establish unique determination results for the above Calder\'on problem is as follows (see \cite{GSU20,RS17,RZ-unbounded}): \begin{enumerate}[(i)] \item\label{item: integral identity} \emph{Integral identity:} Assume that the potentials $q_j$ are suitably regular, then one can write \begin{equation} \label{eq: integral identity schroeding} \langle (\Lambda_{q_1}-\Lambda_{q_2})\varphi_1,\varphi_2\rangle=\int_{\Omega} (q_1-q_2)(u_{\varphi_1}-\varphi_1),(u_{\varphi_2}-\varphi_2)\,dx, \end{equation} when the right hand side is interpreted accordingly. \item\label{item: Runge approx} Establish one of the following \emph{Runge approximation theorems}: \begin{enumerate}[(I)] \item\label{L2 Runge} $\mathcal{R}_W =\{u_f\|_{\Omega}\,;\,f\in C_c^{\infty}(W)\}$ is dense in $L^2(\Omega)$ (see \cite{GSU20} for $q\in L^{\infty}(\Omega)$). \item\label{Hs Runge} $\mathscr{R}_W =\{u_f-f\,;\,f\in C_c^{\infty}(W)\}$ is dense in $\widetilde{H}^s(\Omega)$ (see \cite{RS17} for Sobolev multipliers $q$ or \cite{RZ-unbounded} for local, bounded bilinear forms). \end{enumerate} \item\label{item: Conclusion} If the potentials $q_j$ for $j=1,2$ have suitable continuity properties, then $\Lambda_{q_1}=\Lambda_{q_2}$ together with \ref{item: Runge approx} ensure that there holds $q_1=q_2$ in $\Omega$. \end{enumerate} In \ref{Hs Runge}, the space $\widetilde{H}^s(\Omega)$ is the closure of $C_c^{\infty}(\Omega)$ in the energy space \[ H^s(\R^n)=\{u\in\tempered(\R^n)\,;\,\\|u\\|_{H^s(\R^n)}\vcentcolon = \\|\langle D\rangle^s u\\|_{L^2(\R^n)}<\infty\}, \] where $\langle D\rangle^s$ is the Bessel potential operator. Observe the similarity of the above strategy to the one of \cite{SU87} for showing unique determination for the classical Calder\'on problem, where instead of the Runge approximation theorem suitable geometric optics solutions are used. Moreover, the Runge approximation \ref{item: Runge approx} relies on a Hahn--Banach argument and the \emph{unique continuation property (UCP)} of the fractional Laplacian $(-\Delta)^s$. For more results on Calder\'on problems for elliptic nonlocal PDEs, we refer the interested reader to \cite{GLX,cekic2020calderon,CLL2017simultaneously,LL2020inverse,LL2022inverse, LZ2023unique,KLZ-2022,KLW2022,LRZ2022calder,Semilinear-nonlocal-wave-paper,LLU2023calder,CGRU2023reduction,LLU2023calder,RZ-unbounded,RZ2022LowReg,CRTZ-2022,LZ2024uniqueness,feizmohammadi2021fractional,feizmohammadi2021fractional_closed,FKU24,Trans-anisotropic-LNZ} and the references therein. \subsection{Mathematical model and main results} \label{subsec: mathematical model and main results} Recently, the above approach for solving elliptic nonlocal inverse problems has also been adapted to deduce uniqueness results for the Calder\'on problem of nonlocal hyperbolic equations. Let us next describe some of these results in more detail and for this purpose consider the problem \begin{equation} \label{eq: discussion existing results} \begin{cases} \partial_t^2u+\lambda (-\Delta)^s \partial_t u+(-\Delta)^s u +f(u)= 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega, \end{cases} \end{equation} where $\lambda\in\R$ and $f\colon \Omega\times\R\to\R$ is a possibly nonlinear function. If the problem \eqref{eq: discussion existing results} is well-posed in the energy class $H^s(\R^n)$, then for any two given measurement sets $W_1,W_2\subset\Omega_e$ we may introduce the DN map $\Lambda^{\lambda}_{f}$ via \[ \Lambda^{\lambda}_{f}\varphi=\left.(\lambda (-\Delta)^s\partial_t u_\varphi+(-\Delta)^s u_\varphi)\right\|_{(W_2)_T}, \] whenever $\varphi$ is supported in $(W_1)_T$ and $u_\varphi$ is the solution of \eqref{eq: discussion existing results}. The \emph{Calder\'on problem} for \eqref{eq: discussion existing results} reads as follows: \begin{question} \label{question: calderon discussion} Does the DN map $\Lambda^{\lambda}_{f}$ uniquely determine the function $f$? \end{question} A suitable class of nonlinearities are the so-called weak nonlinearities, which are defined next. \begin{definition}\label{main assumptions on nonlinearities} We call a Carath\'eodory function $f\colon \Omega\times \R\to\R$ \emph{weak nonlinearity}, if it satisfies the following conditions: \begin{enumerate}[(i)] \item\label{prop f} $f$ has partial derivative $\partial_{\tau}f$, which is a Carath\'eodory function, and there exists $a\in L^p(\Omega)$ such that \begin{equation} \label{eq: bound on derivative} \left\|\partial_\tau f(x,\tau)\right\|\lesssim a(x)+\|\tau\|^r \end{equation} for all $\tau\in\R$ and a.e. $x\in\Omega$. Here the exponents $p$ and $r$ satisfy the restrictions \begin{equation} \label{eq: restrictions on p} \begin{cases} n/s\leq p\leq \infty, &\, \text{if }\, 2s< n,\\ 2<p\leq \infty, &\, \text{if }\, 2s= n,\\ 2\leq p\leq \infty, &\, \text{if }\, 2s\geq n. \end{cases} \end{equation} and \begin{equation} \label{eq: cond on r} \begin{cases} 0\leq r<\infty, &\, \text{if }\, 2s\geq n,\\ 0\leq r\leq \frac{2s}{n-2s}, &\, \text{if }\, 2s< n, \end{cases} \end{equation} respectively. Moreover, $f$ fulfills the integrability condition $f(\cdot,0)\in L^2(\Omega)$. \item\label{prop F} The function $F\colon \Omega\times\R\to\R$ defined via \[ F(x,\tau)=\int_0^\tau f(x,\rho)\,d\rho \] satisfies $F(x,\tau)\geq 0$ for all $\tau\in\R$ and $x\in\Omega$. \end{enumerate} \end{definition} Let us note that $f(x,\tau)=q(x)\|\tau\|^r\tau$ with $r$ satisfying \eqref{eq: cond on r} and $0\leq q\in L^{\infty}(\Omega)$ is a weak nonlinearity. Using the above notions, we can now discuss some of the existing results. \begin{enumerate}[(a)] \item\label{item: basic nonlocal wave} The article \cite{KLW2022} gives a positive answer for $\lambda=0, f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(\Omega)$. Their proof relied on the observation that the related nonlocal wave equation \eqref{eq: discussion existing results} satisfies an $L^2(\Omega_T)$ Runge approximation theorem. \item\label{item: viscous nonlocal wave} The work \cite{zimmermann2024calderon} deals on the one hand with the linear case $\lambda=1$, $f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(0,T;L^p(\Omega))$, where $p$ satisfies the restrictions \eqref{eq: restrictions on p}, and $q$ is weakly continuous in $t$ and on the other hand with the nonlinear case $\lambda=1$ and $f$ is a $r+1$ homogeneous, weak nonlinearity. The uniqueness proofs use substantially that due to presence of the viscosity term $(-\Delta)^s\partial_t$ solutions $u$ to \eqref{eq: discussion existing results} satisfy $\partial_t u\in L^2(0,T;H^s(\R^n))$ and as a consequence the linearized equations have the Runge approximation property in $L^2(0,T;\widetilde{H}^s(\Omega))$. \item\label{item: semilinear nonlocal wave} In \cite{Semilinear-nonlocal-wave-eq}, uniqueness is proved in the case $\lambda=0$ and $f$ satisfies the same properties as in \ref{item: viscous nonlocal wave}, but with the additional restriction $r\leq 1$. This article only uses an $L^2(\Omega_T)$ Runge approximation result for the linearized nonlocal wave equation. \item\label{item: optimal runge} By establishing a theory for very weak solutions of linear nonlocal wave equations with $\lambda=0$, the authors of \cite{Optimal-Runge-nonlocal-wave} could deduce an optimal $L^2(0,T;\widetilde{H}^s(\Omega))$ Runge approximation theorem for these equations. This allowed to extend the results in \ref{item: semilinear nonlocal wave} to the cases $r>1$ and additionally showed that one can recover any linear perturbation $q\in L^p(\Omega)$ with $p$ satisfying the restrictions \eqref{eq: restrictions on p}. Furthermore, by this improved Runge approximation theorem the authors could also treat the case of serially or asymptotically polyhomogeneous nonlinearities (see \cite[Theorem 1.5]{Optimal-Runge-nonlocal-wave}). \end{enumerate} In this context, let us also mention the recent article \cite{fu2024wellposednessinverseproblemsnonlocal} which deals with the Calder\'on problem for a third order semilinear, nonlocal, viscous wave equation. The goal of this paper us to present an extension of the models described in \ref{item: viscous nonlocal wave} and \ref{item: semilinear nonlocal wave}, which we discuss next. Let $0<s<1$ and suppose that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^p(\Omega)$, where $0<s<\alpha\leq 1$ and $1\leq p\leq \infty$ satisfies the restrictions in \eqref{eq: restrictions on p}. Then we define the following \emph{damped, nonlocal wave operator} \begin{equation} \label{eq: damped, nonlocal wave operator} L_{\gamma}\vcentcolon = \partial_t^2+\gamma\partial_t +(-\Delta)^s \end{equation} and consider the problem \begin{equation} \label{eq: wave problem} \begin{cases} L_{\gamma}u +f(u)= F & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} where $f$ is a weak nonlinearity or $f(u)=q(x)u$. In fact, this is a possibly nonlinear generalization of the model (G2) with $s_1=0$ in \cite[Section 1.1]{zimmermann2024calderon}. By \cite[Proposition 3.7]{Semilinear-nonlocal-wave-eq} (see Section \ref{subsec: weak solutions} for the linear case), we know that the problem \eqref{eq: wave problem} is well-posed, whenever the source $F$, exterior condition $\varphi$ and initial conditions $u_0,u_1$ are sufficiently regular. Thus, we can introduce the related (partial) DN map via \begin{equation} \label{eq: formal DN map} \Lambda_{\gamma,f}\varphi\vcentcolon = \left.(-\Delta)^s u_{\varphi}\right\|_{(W_2)_T}, \end{equation} where $W_1,W_2\subset \Omega_e$ are some measurement sets, $\varphi$ is supported in $(W_1)_T$ and $u_\varphi$ is the unique solution of \eqref{eq: wave problem} with $u_0=u_1=0$. Then we ask the following question: \begin{question} \label{question: Caldeorn problem of this work} Does the partial DN map $\Lambda_{\gamma,f}$ uniquely determine the damping coefficient $\gamma$ and the function $f$? \end{question} In this work we establish the following affirmative answers to this question, whereas the first result discusses the linear case and the second one the semilinear perturbations. \begin{theorem}[Uniqueness for linear perturbations] \label{thm: uniqueness linear} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that for $j=1,2$ we have given coefficients $(\gamma_j,q_j)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$ and let $\Lambda_{\gamma_j,q_j}$ be the DN map associated to the problem \begin{equation} \label{eq: PDEs uniqueness theorem} \begin{cases} (L_{\gamma_j}+q_j)u = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} for $j=1,2$. If $W_1,W_2\subset\Omega_e$ are two measurement sets such that \begin{equation} \label{eq: equality of DN maps} \left.\Lambda_{\gamma_1,q_1}\varphi\right\|_{(W_2)_T}=\left.\Lambda_{\gamma_2,q_2}\varphi\right\|_{(W_2)_T} \end{equation} for all $\varphi\in C_c^{\infty}((W_1)_T)$, then there holds \begin{equation} \label{eq: equal coefficients} \gamma_1=\gamma_2\text{ and }q_1=q_2 \text{ in }\Omega. \end{equation} \end{theorem} \begin{theorem}[Uniqueness for semilinear perturbations] \label{thm: uniqueness semilinear} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$ and $0<s<\alpha\leq 1$. Assume that for $j=1,2$ we have given coefficients $\gamma_j\in C^{0,\alpha}(\R^n)$ and $r+1$ homogeneous, weak nonlinearities $f_j$, where $r>0$ satisfies \eqref{eq: cond on r}. Let $\Lambda_{\gamma_j,f_j}$ be the DN map associated to the problem \begin{equation} \label{eq: PDEs uniqueness theorem} \begin{cases} L_{\gamma_j}u +f_j(u) = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} for $j=1,2$. If $W_1,W_2\subset\Omega_e$ are two measurement sets such that \begin{equation} \label{eq: equality of DN maps semilinear} \left.\Lambda_{\gamma_1,f_1}\varphi\right\|_{(W_2)_T}=\left.\Lambda_{\gamma_2,f_2}\varphi\right\|_{(W_2)_T} \end{equation} for all $\varphi\in C_c^{\infty}((W_1)_T)$, then there holds \begin{equation} \label{eq: equal coefficients} \gamma_1=\gamma_2\text{ in }\Omega\text{ and }f_1=f_2 \text{ in }\Omega\times \R. \end{equation} \end{theorem} \begin{remark} For simplicity we restrict our attention to homogeneous nonlinearities $f$, but the unique determination remains valid in some polyhomogeneous cases as described in \cite{Optimal-Runge-nonlocal-wave} for $\gamma=0$. \end{remark} \subsection{Organization of the article} The rest of this article is structured as follows. In Section \ref{sec: Very weak solutions to damped, nonlocal wave equations}, we establish the existence of unique weak and very weak solutions to damped nonlocal wave equations. In Section \ref{sec: inverse problem} we then move on to the inverse problem part of this work. First, in Section \ref{sec: Runge approx} we establish the optimal Runge approximation theorem. Afterwards, in Section \ref{sec: integral identity} we prove a suitable integral identity that allows us to recover simultaneously the damping coefficient $\gamma$ and potential $q$ in Section \ref{sec: linear uniqueness}. Finally, Section \ref{sec: semilinear uniqueness} contains the proof of the simultaneous determination of the damping coefficient $\gamma$ and the homogeneous nonlinearity $f$. \section{Weak and very weak solutions to damped, nonlocal wave equations} \label{sec: Very weak solutions to damped, nonlocal wave equations} The main purpose of this section is to show existence of unique weak and very weak solutions to \emph{damped, nonlocal wave equations (DNWEQ)} \begin{equation} \label{eq: damped nonlocal wave equations well-posedness} \begin{cases} (L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} where $L_\gamma$ is given by \eqref{eq: damped, nonlocal wave operator} and only the case $\varphi=0$ is considered for very weak solutions. \subsection{Weak solutions} \label{subsec: weak solutions} This section deals with the well-posedness of \eqref{eq: damped nonlocal wave equations well-posedness} for regular sources, exterior conditions and initial data. We also prove well-posedness for the case when instead of initial values the values at $t=T$ are specified, which will be needed for the development of the theory of very weak solutions. \begin{theorem}[Weak solutions to homogeneous DNWEQ] \label{thm: weak sol to hom DNWEQ} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;\widetilde{L}^2(\Omega))$\footnote{Here and below we set $\widetilde{L}^2(\Omega)\vcentcolon =\widetilde{H}^0(\Omega)$.} and initial conditions $(u_0,u_1)\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, there exists a unique weak solution $u\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ of \begin{equation} \label{eq: damped nonlocal wave equations well-posedness weak sol} \begin{cases} (L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} which means that $(u(0),\partial_tu(0))=(u_0,u_1)$ in $\widetilde{H}^s(\Omega)\times \widetilde{L}^2(\Omega)$ and there holds \begin{equation} \label{eq: weak formulation damped nonlocal wave eq} \begin{split} &\frac{d}{dt} \langle \partial_t u,v\rangle_{L^2(\Omega)}+\langle \gamma\partial_t u,v\rangle_{L^2(\Omega)}+\langle (-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^2(\R^n)}+\langle qu,v\rangle_{L^2(\Omega)}\\ &=\langle F,v\rangle_{L^2(\Omega)} \end{split} \end{equation} for all $v\in \widetilde{H}^s(\Omega)$ in the sense of distributions on $(0,T)$. Moreover, the unique solution $u$ obeys the energy identity \begin{equation} \label{eq: energy identity} \begin{split} &\\|\partial_t u(t)\\|_{L^2(\Omega)}^2+\\|(-\Delta)^{s/2}u(t)\\|_{L^2(\R^n)}^2+2 \int_0^t \langle \gamma\partial_t u(\tau)+qu(\tau),\partial_t u(\tau)\rangle_{L^2(\Omega)}\,d\tau\\ &=\\|(-\Delta)^{s/2}u_0\\|_{L^2(\R^n)}^2+\\|u_1\\|_{L^2(\Omega)}^2+2 \int_0^t\langle F(\tau),\partial_tu (\tau)\rangle_{L^2(\Omega)}\,d\tau \end{split} \end{equation} which implies \begin{equation} \label{eq: energy estimate} \begin{split} &\\|\partial_t u(t)\\|_{L^2(\Omega)}+\\|(-\Delta)^{s/2}u(t)\\|_{L^2(\R^n)}\\ &\leq C(\\|u_1\\|_{L^2(\Omega)}+\\|(-\Delta)^{s/2}u_0\\|_{L^2(\R^n)}+\\|F\\|_{L^2(0,t;L^2(\Omega))}) \end{split} \end{equation} for all $0\leq t\leq T$ and some $C>0$ only depending on $\\|q\\|_{L^p(\Omega)}$, $\\|\gamma\\|_{L^{{\infty}(\Omega)}}$ and $T>0$. \end{theorem} \begin{proof} Throughout the proof, we endow $\widetilde{H}^s(\Omega)$ with the equivalent norm $\\|u\\|_{\widetilde{H}^s(\Omega)}=\\|(-\Delta)^{s/2}u\\|_{L^2(\R^n)}$ (see \cite[Lemma 2.3]{Semilinear-nonlocal-wave-eq}) and we introduce the following continuous sesquilinear forms \begin{equation} \label{eq: sesquilinear forms} a_0(u,v)=\langle (-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^2(\R^n)},\quad a_1(u,v)= \langle qu,v\rangle_{L^2(\Omega)} \end{equation} for $u,v\in \widetilde{H}^s(\Omega)$ and \begin{equation} b(u,v)=\langle \gamma u,v\rangle_{L^2(\Omega)} \end{equation} for $u,v\in \widetilde{L}^2(\Omega)$. Next, recall that by \cite[eq.~(3.7)]{Semilinear-nonlocal-wave-eq} one has \begin{equation} \label{eq: L2 estimate potential} \\|qu\\|_{L^2(\Omega)}\leq C\\|q\\|_{L^p(\Omega)}\\|u\\|_{\widetilde{H}^s(\Omega)} \end{equation} for all $u\in \widetilde{H}^s(\Omega)$. It is not hard to see that we can invoke the existence and uniqueness results \cite[Chapter XVIII, \S 5, Theorem~3 \& 4]{DautrayLionsVol5} (see \cite[p.~571]{DautrayLionsVol5}), which ensure the existence of a unique, real-valued solution $u\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ to \eqref{eq: damped nonlocal wave equations well-posedness weak sol}. Furthermore, by \cite[Chapter XVIII, \S 5, Lemma~7]{DautrayLionsVol5} the solution $ u$ satisfies the following energy identity \begin{equation} \label{eq: energy identity proof} \begin{split} &\\|\partial_t u(t)\\|_{L^2(\Omega)}^2+\\|(-\Delta)^{s/2}u(t)\\|_{L^2(\R^n)}^2+2 \int_0^t \langle \gamma\partial_t u(\tau)+qu(\tau),\partial_t u(\tau)\rangle_{L^2(\Omega)}\,d\tau\\ &=\\|(-\Delta)^{s/2}u_0\\|_{L^2(\R^n)}^2+\\|u_1\\|_{L^2(\Omega)}^2+2 \int_0^t\langle F(\tau),\partial_tu (\tau)\rangle_{L^2(\Omega)}\,d\tau \end{split} \end{equation} for $0\leq t\leq T$. Hence, we have shown the identity \eqref{eq: energy identity}. Let us define $\Psi \in C([0,T])$ by \[ \Psi(t)\vcentcolon = \\|\partial_t u(t)\\|_{L^2(\Omega)}^2+\\|(-\Delta)^{s/2}u(t)\\|_{L^2(\R^n)}^2 \] for $0\leq t\leq T$. Using \eqref{eq: L2 estimate potential}, \eqref{eq: energy identity proof} and $\gamma\in L^{\infty}(\Omega)$, we get \[ \begin{split} &\Psi(t)\leq \Psi(0)+\int_0^t \\|F(\tau)\\|_{L^2(\Omega)}^2\,d\tau+ C\int_0^t (1+\\|\gamma\\|_{L^{\infty}(\Omega)}+\\|q\\|^2_{L^p(\Omega)})\Psi(\tau)\,d\tau \end{split} \] and via Gronwall's inequality we deduce the energy estimate \begin{equation} \begin{split} &\Psi(t)\leq C(\Psi(0)+\\|F\\|_{L^2(0,t;L^2(\Omega))}^2) \end{split} \end{equation} for all $0\leq t\leq T$ and some $C>0$ only depending on $\\|q\\|_{L^p(\Omega)}$, $\\|\gamma\\|_{L^{\infty}(\Omega)}$ and $T>0$. This establishes the estimate \eqref{eq: energy estimate}. \end{proof} As a consequence we have the following result: \begin{proposition}[Weak solutions to inhomogeneous DNWEQ] \label{prop: Weak solutions to inhomogeneous DNWEQ} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;\widetilde{L}^2(\Omega))$, exterior condition $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and initial conditions $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(0)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(0)\in \widetilde{L}^2(\Omega)$, there exists a unique weak solution $u\in C([0,T];H^s(\R^n))\cap C^1([0,T];L^2(\R^n))$ of \begin{equation} \label{eq: damped nonlocal wave equations well-posedness weak sol inhom} \begin{cases} (L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} which means that $u$ satisfies \eqref{eq: weak formulation damped nonlocal wave eq}, the $(u(0),\partial_t u(0))=(u_0,u_1)$ in $H^s(\R^n)\times L^2(\R^n)$ and $u=\varphi$ in $(\Omega_e)_T$ means that $u(t)=\varphi(t)$ a.e. in $\Omega_e$ for any $0<t<T$. Furthermore, the following energy estimate holds \begin{equation} \label{eq: energy estimate inhomogeneous} \begin{split} &\\|\partial_t u(t)\\|_{L^2(\R^n)}+\\|(-\Delta)^{s/2}u(t)\\|_{L^2(\R^n)}\\ &\leq C(\\|u_1\\|_{L^2(\R^n)}+\\|(-\Delta)^{s/2}u_0\\|_{L^2(\R^n)}+\\|\varphi\\|_{C^2([0,t];H^{2s}(\R^n))}+\\|F\\|_{L^2(0,t;L^2(\Omega))}) \end{split} \end{equation} for any $0\leq t\leq T$. \end{proposition} \begin{proof} Observe, under the current regularity assumptions and compatibility conditions, that $u$ solves \eqref{eq: damped nonlocal wave equations well-posedness weak sol inhom} if and only if $w\vcentcolon = u-\varphi$ solves \begin{equation} \label{eq: Usual cauchy homogeneous} \begin{cases} (L_{\gamma}+q)w = F-(L_\gamma+q) \varphi & \text{ in } \Omega_T,\\ w =0 & \text{ on } (\Omega_e)_T,\\ w(0) = u_0-\varphi(0),\, \partial_{t}w(0) = u_1-\partial_t\varphi(0) & \text{ on } \Omega. \end{cases} \end{equation} The only fact to keep in mind is that if $u\in C([0,T];H^s(\R^n))$, then the condition $u(t)=\varphi(t)$ a.e. in $\Omega_e$ is equivalent to $u(t)-\varphi(t)\in \widetilde{H}^s(\Omega)$ as $\Omega\subset\R^n$ is a bounded Lipschitz domain. So, the assertions of Propsition \ref{prop: Weak solutions to inhomogeneous DNWEQ} follow immediately from Theorem \ref{thm: weak sol to hom DNWEQ}. \end{proof} Next, let us define for any $g\in L^1_{loc}(V_T)$, $V\subset\R^n$ open, its \emph{time-reversal} \begin{equation} \label{eq: time reversal} g^\star(x,t)=g(x,T-t). \end{equation} Then, we have the following lemma. \begin{lemma} \label{lemma: time reversal of solution} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Let $F\in L^2(0,T;\widetilde{L}^2(\Omega))$, $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(0)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(0)\in \widetilde{L}^2(\Omega)$. Then $u$ solves \begin{equation} \label{eq: Usual cauchy} \begin{cases} (L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} if and only if $u^\star$ solves \begin{equation} \label{eq: time reversed cauchy} \begin{cases} (L_{-\gamma}+q)v = F^\star & \text{ in } \Omega_T,\\ v =\varphi^\star & \text{ on } (\Omega_e)_T,\\ v(T) = u_0,\, \partial_{t}v(T) = u_1 & \text{ on } \Omega. \end{cases} \end{equation} In particular, for any $F\in L^2(\Omega_T)$, $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(T)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(T)\in \widetilde{L}^2(\Omega)$, there exists a unique solution $u^\star$ of \begin{equation} \label{eq: backwards damped, nonlocal wave equation} \begin{cases} (L_{-\gamma}+q)v = F & \text{ in } \Omega_T,\\ v =\varphi & \text{ on } (\Omega_e)_T,\\ v(T) = u_0,\, \partial_{t}v(T) = u_1 & \text{ on } \Omega. \end{cases} \end{equation} \end{lemma} \begin{proof} First, note that by the proof of Proposition \ref{prop: Weak solutions to inhomogeneous DNWEQ}, we can assume without loss of generality that $\varphi=0$. Secondly, one easily sees that $\partial_t u^\star=-(\partial_t u)^\star$ and thus a change of variables in \eqref{eq: weak formulation damped nonlocal wave eq} gives the asserted equivalence. The unique solvability of \eqref{eq: backwards damped, nonlocal wave equation} follows from the equivalence and Theorem \ref{thm: weak sol to hom DNWEQ}. \end{proof} \subsection{Very weak solutions} Let us start by making some simple observations. Suppose that $u$ and $v$ are smooth solutions of the problems \begin{equation} \label{eq: motivation very weak solutions 1} \begin{cases} (L_\gamma+q) u = F & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} and \begin{equation} \label{eq: motivation very weak solutions 2} \begin{cases} (L_{-\gamma}+q) v = G & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega, \end{cases} \end{equation} respectively. If we multiply the PDE \eqref{eq: motivation very weak solutions 1} by $v$ and integrate over $\Omega_T$, then we get \begin{equation} \label{eq: formal computation} \begin{split} &\int_{\Omega_T}Fv\,dxdt=\int_{\Omega_T}[(L_{\gamma}+q)u]v\,dxdt\\ &= \int_{\Omega}u_0\partial_t v(0)\,dx-\int_{\Omega}u_1v(0)\,dx-\int_\Omega \gamma u_0v(0)\,dx+\int_{\Omega_T}u(L_{-\gamma}+q)v\,dxdt\\ &= \int_{\Omega}u_0\partial_t v(0)\,dx-\int_{\Omega}u_1v(0)\,dx-\int_\Omega \gamma u_0v(0)\,dx+\int_{\Omega_T}Gu\,dxdt. \end{split} \end{equation} Notice that if $G\in L^2(0,T;\widetilde{L}^2(\Omega))$, $v\in L^2(0,T;\widetilde{H}^s(\Omega))$ and $(v(0),\partial_t v(0))\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, then one can make sense of the first integral and the last line in \eqref{eq: formal computation}, even in the case $F\in L^2(0,T;H^{-s}(\Omega))$, $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$ and $u\in L^2(0,T;\widetilde{L}^2(\Omega))$. Here, $H^{-s}(\Omega)\subset\distr(\Omega)$ is defined by \[ H^{-s}(\Omega)=\{u\|_\Omega\,;\,u\in H^{-s}(\R^n)\} \] and it can be identified with the dual space of $\widetilde{H}^s(\Omega)$, when $\Omega$ is Lipschitz. The previous computation motivates the following definition. \begin{definition}[Very weak solutions] \label{def: very weak solutions} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$, source $F\in L^2(0,T;H^{-s}(\Omega))$ and initial conditions $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$. Then we say that $u\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is a \emph{very weak solution} of \begin{equation} \label{eq: def very weak} \begin{cases} (L_\gamma+q) u = F & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} whenever there holds\footnote{Here and below we sometimes write $\langle \cdot,\cdot\rangle$ to denote the duality pairing between $H^{-s}(\Omega)\times \widetilde{H}^s(\Omega)$.} \begin{equation} \label{eq: weak formulation of very weak sols} \int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle F,v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v(0)\rangle \end{equation} for all $G\in L^2(0,T;\widetilde{L}^2(\Omega))$, where $v\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ is the unique weak solution of the adjoint equation \begin{equation} \label{eq: adjoint eq of def very weak} \begin{cases} (L_{-\gamma}+q) v = G & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega, \end{cases} \end{equation} (see Theorem \ref{thm: weak sol to hom DNWEQ}). \end{definition} Next, let us recall the following well-posedness result of very weak solutions. \begin{theorem}[{Very weak solutions for $\gamma=q=0$,\cite[Theorem 3.6]{Optimal-Runge-nonlocal-wave}}] \label{thm: well-posedness very weak without damping} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$ and $0<s< 1$. Then for any given source $F\in L^2(0,T;H^{-s}(\Omega))$ and initial conditions $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$, there exists a unique solution to \begin{equation} \label{eq: very weak without damping and potential} \begin{cases} (\partial_t^2+(-\Delta)^s) u = F & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega \end{cases} \end{equation} and it satisfies the following energy estimate \begin{equation} \label{eq: energy estimate very weak without damping and potential} \\|u(t)\\|_{L^2(\Omega)}+\\|\partial_t u(t)\\|_{H^{-s}(\Omega)}\leq C(\\|u_0\\|_{L^2(\Omega)}+\\|u_1\\|_{H^{-s}(\Omega)}+\\|F\\|_{L^2(0,t;H^{-s}(\Omega))}) \end{equation} for all $0\leq t\leq T$. \end{theorem} Hence, we have a well-defined solution map. \begin{proposition}[Solution map] \label{prop: solution map} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and let $X_s\vcentcolon =\widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$ be endowed with the usual product norm \[ \\|(u,w)\\|_{X_s}\vcentcolon = (\\|u\\|_{L^2(\Omega)}^2+\\|w\\|_{H^{-s}(\Omega)}^2)^{1/2}. \] Then the \emph{solution map} $S\colon L^2(0,T;H^{-s}(\Omega))\to C([0,T];X_s)$ defined by \begin{equation} \label{eq: solution map} S(F)\vcentcolon = (u,\partial_t u), \end{equation} where $u\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is the unique solution of \eqref{eq: very weak without damping and potential} with $(u_0,u_1)=0$. Moreover, the solution map is continuous and satisfies the estimate \begin{equation} \label{eq: continuity estimate solution map} \\|S(F)(t)\\|_{X_s}\leq C\\|F\\|_{L^2(0,t;H^{-s}(\Omega))} \end{equation} for any $0\leq t\leq T$. \end{proposition} \begin{proof} First of all note that the solution map $S$ is well-defined by Theorem \ref{thm: well-posedness very weak without damping}. The estimate \eqref{eq: continuity estimate solution map} follows from \eqref{eq: energy estimate very weak without damping and potential}, which together with the linearity of $S$ gives the continuity of $S$. Observe that the linearity of $S$ is a direct consequence of the unique solvability of \eqref{eq: very weak without damping and potential} and the fact that the PDE is linear. \end{proof} \begin{theorem} \label{thm: general well-posedness result of very weak solutions} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $\mathcal{F}\colon C([0,T];X_s)\to L^2(0,T;H^{-s}(\Omega))$ satisfies the Lipschitz estimate \begin{equation} \label{eq: Lipschitz estimate nonlinearity} \\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\\|_{H^{-s}(\Omega)}\leq C\\|U(t)-V(t)\\|_{X_s} \end{equation} for a.e.~$0\leq t\leq T$ and $U,V\in C([0,T];X_s)$. Then for all $(u_0,u_1)\in X_s$, there exists a unique solution $u$ of \begin{equation} \label{eq: well-posedness nonlinear very weak} \begin{cases} (\partial_t^2+(-\Delta)^s) u = \mathcal{F}(u,\partial_t u) & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} that is the formula \eqref{eq: weak formulation of very weak sols} holds with $F$ replaced by $\mathcal{F}(u,\partial_t u)$ in which we test against every weak solution $v$ of the adjoint equation \begin{equation} \label{eq: adjoint eq nonlinear} \begin{cases} (\partial_t^2+(-\Delta)^s) v = G & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega \end{cases} \end{equation} with $G\in L^2(0,T;\widetilde{L}^2(\Omega))$. \end{theorem} \begin{proof}[Proof of Theorem \ref{thm: general well-posedness result of very weak solutions}] Let $u_h\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ be the unique solution to \begin{equation} \label{eq: homogeneous part} \begin{cases} (\partial_t^2+(-\Delta)^s) u = 0 & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega \end{cases} \end{equation} and let us set $U_h\vcentcolon =(u_h,\partial_t u_h)\in C([0,T];X_s)$. Furthermore, we define the operator $\mathcal{T}\colon C([0,T];X_s)\to C([0,T];X_s)$ as \begin{equation} \mathcal{T}(U)\vcentcolon = U_h+S(\mathcal{F}(U)), \end{equation} which is well-defined by \eqref{prop: solution map} and the properties of $\mathcal{F}$. Next, we show that $\mathcal{T}$ has a unique fixed point $U=(U_1,U_2)$. \medskip \noindent\textit{Step 1.~Existence.} Let $U,V\in C([0,T];X_s)$, then by linearity of $S$, \eqref{eq: continuity estimate solution map} and \eqref{eq: Lipschitz estimate nonlinearity} we get \[ \begin{split} \\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\\|_{X_s}&=\\|S(\mathcal{F}(U))(t)-S(\mathcal{F}(V))(t)\\|_{X_s}\\ &=\\|S(\mathcal{F}(U)-\mathcal{F}(V))(t)\\|_{X_s}\\ &\leq C\\|\mathcal{F}(U)-\mathcal{F}(V)\\|_{L^2(0,t;H^{-s}(\Omega))}\\ &\leq C\\|U-V\\|_{L^2(0,t;X_s)}. \end{split} \] Next, let us define the following norm on $X_s$ \begin{equation} \label{eq: new norm on Xs} \\|U\\|_{\theta}\vcentcolon = \sup_{0\leq t\leq T}\left(e^{-\theta t}\\|U(t)\\|_{X_s}\right) \end{equation} for $\theta>0$, which will be fixed in a moment. Then we have the estimate \[ \\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\\|_{X_s}\leq C\left(\int_0^te^{2\theta \tau}\,d\tau\right)^{1/2}\\|U-V\\|_{\theta}\leq \frac{C}{(2\theta)^{1/2}}e^{\theta t}\\|U-V\\|_{\theta} \] and hence there holds \[ \\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\\|_{\theta}\leq \frac{C}{(2\theta)^{1/2}}\\|U-V\\|_{\theta}. \] Therefore, we deduce that $\mathcal{T}$ is a strict contraction from the complete metric space $(C([0,T];X_s),\\|\cdot\\|_{\theta})$ to itself, when $\theta>0$ is chosen such that $C/(2\theta)^{1/2}<1$. Now, we may invoke Banach's fixed point theorem to obtain a unique fixed point $U=(u,w)$ of $\mathcal{T}$. Next, observe that the definition of the solution map $S$ and $U=\mathcal{T}(U)=U_h+S(\mathcal{F}(U))$ imply \[ u=u_h+u_n \text{ and } w=\partial_t u, \] where $u_n$ solves \begin{equation} \label{eq: nonlinear part} \begin{cases} (\partial_t^2+(-\Delta)^s) v = \mathcal{F}(U) & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega. \end{cases} \end{equation} Going back to the definition of very weak solutions, we see this implies that $u$ solves \eqref{eq: well-posedness nonlinear very weak}. \medskip \noindent\textit{Step 2.~Uniqueness.} Suppose $\Tilde{u}\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is any other solution to \eqref{eq: well-posedness nonlinear very weak}, then $\bar{u}\vcentcolon = u-\widetilde{u}$ solves \begin{equation} \label{eq: uniquenes very weak nonlinear} \begin{cases} (\partial_t^2+(-\Delta)^s) v = \mathcal{F}(u,\partial_t u)-\mathcal{F}(\widetilde{u},\partial_t \widetilde{u}) & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega. \end{cases} \end{equation} Thus, applying the energy estimate \eqref{eq: energy estimate very weak without damping and potential} together with the Lipschitz assumption on $\mathcal{F}$, we see that \[ \begin{split} \\|U(t)-\widetilde{U}(t)\\|^2_{X_s}&\leq C\int_0^t\\|\mathcal{F}(U)(\tau)-\mathcal{F}(\widetilde{U})(\tau)\\|_{H^{-s}(\Omega)}^2\,d\tau\\ &\leq C\int_0^t\\|U(t)-\widetilde{U}(t)\\|^2_{X_s}\,d\tau, \end{split} \] where $U=(u,\partial_tu)$ and $\widetilde{U}=(\widetilde{u},\partial_t\widetilde{u})$. So, Gronwall's inequality shows that $u=\widetilde{u}$. This establishes the uniqueness assertion and we can conclude the proof. \end{proof} As an application of Theorem \ref{thm: general well-posedness result of very weak solutions}, we can show the unique solvability of \eqref{eq: damped nonlocal wave equations well-posedness} for rough source and initial data. \begin{theorem}[Very weak solutions to DNWEQ] \label{thm: very weak sol DNWEQ} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;H^{-s}(\Omega))$ and $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$, there exists a unique solution of \begin{equation} \label{eq: well-posedness very weak DNWEQ} \begin{cases} (L_\gamma+q) u = F & \text{ in } \Omega_T,\\ u =0 & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega. \end{cases} \end{equation} \end{theorem} \begin{proof} Let us define the mapping $\mathcal{F}\colon C([0,T];X_s)\to L^2(0,T;H^{-s}(\Omega))$ by \[ \mathcal{F}(U)(t)\vcentcolon =F-\gamma w(t)-qu(t), \] where $U=(u,w)\in C([0,T];X_s)$. On the one hand, using the estimate \eqref{eq: L2 estimate potential} we see that for any $u\in \widetilde{L}^2(\Omega)$ one has $qu\in H^{-s}(\Omega)$ and there holds \begin{equation} \label{eq: continuity estimate dual potential} \begin{split} \\|qu\\|_{H^{-s}(\Omega)} &=\sup_{\\|v\\|_{\widetilde{H}^s(\Omega)}\leq 1}\|\langle u,qv\rangle_{L^2(\Omega)}\|\\ &\leq C\\|q\\|_{L^p(\Omega)}\\|u\\|_{L^2(\Omega)}. \end{split} \end{equation} On the other hand, by applying \cite[Lemma 3.1]{Stability-fractional-conductivity} and $\partial\Omega\in C^0$ we deduce that for any $v\in \widetilde{H}^s(\Omega)$ one has $\gamma v\in \widetilde{H}^s(\Omega)$ and it obeys the estimate \begin{equation} \label{eq: multiplication by gamma} \\|\gamma v\\|_{\widetilde{H}^s(\Omega)}\leq C\\|\gamma\\|_{C^{0,\alpha}(\R^n)}\\|v\\|_{\widetilde{H}^s(\Omega)}. \end{equation} Thus, we can again infer from a duality argument that $H^{-s}(\Omega)\ni w\mapsto \gamma w\in H^{-s}(\Omega)$ is a continuous map satisfying \begin{equation} \label{eq: continuity estimate dual damping} \begin{split} \\|\gamma w\\|_{H^{-s}(\Omega)}&=\sup_{\\|v\\|_{\widetilde{H}^s(\Omega)}\leq 1}\|\langle \gamma w,v\rangle\|\\ &=\sup_{\\|v\\|_{\widetilde{H}^s(\Omega)}\leq 1}\|\langle w,\gamma v\rangle\|\\ &\leq C\\|\gamma\\|_{C^{0,\alpha}(\R^n)}\\|w\\|_{H^{-s}(\Omega)}. \end{split} \end{equation} From the estimates \eqref{eq: continuity estimate dual potential} and \eqref{eq: continuity estimate dual damping}, we easily deduce that $\mathcal{F}$ is well-defined and satisfies the Lipschitz estimate \begin{equation} \label{eq: Lipschitz estimate application} \\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\\|_{H^{-s}(\Omega)}\leq C(\\|\gamma\\|_{C^{0,\alpha}(\R^n)}+\\|q\\|_{L^p(\Omega)})\\|U(t)-V(t)\\|_{X_s} \end{equation} for all $U,V\in C([0,T];X_s)$. Thus, we can apply Theorem \ref{thm: general well-posedness result of very weak solutions} to get the existence of a unique solution to \eqref{eq: well-posedness very weak DNWEQ} in the sense that for any $G\in L^2(0,T;\widetilde{L}^2(\Omega))$ and corresponding solution $v$ of \eqref{eq: adjoint eq nonlinear}, there holds \begin{equation} \label{eq: prelim def of very weak sol} \int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle (F-\gamma \partial_t u-qu),v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}. \end{equation} It remains to verify that $u$ is indeed a solution of \eqref{eq: well-posedness very weak DNWEQ} in the sense of Definition \ref{def: very weak solutions}. For this purpose let $G\in L^2(0,T;\widetilde{L}^2(\Omega))$ and suppose that $v$ is the unique solution to \eqref{eq: adjoint eq of def very weak}. Hence, $v$ solves \[ \begin{cases} (\partial_t^2+(-\Delta)^s) v = \widetilde{G} & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega \end{cases} \] with $\widetilde{G}=G+\gamma \partial_t v-qv\in L^2(0,T;\widetilde{L}^2(\Omega))$ (see \eqref{eq: L2 estimate potential}). Next, we claim that there holds \begin{equation} \label{eq: integration by parts formula} \int_0^T \langle \gamma \partial_t u,v\rangle\,dt=-\int_0^T \langle \gamma\partial_t v,u\rangle_{L^2(\Omega)}\,dt-\langle \gamma u_0,v(0)\rangle. \end{equation} For this purpose, let us consider for $\eps>0$ the unique solution $v_\eps\in H^1(0,T;\widetilde{H}^s(\Omega))$ with $\partial_t^2 v_\eps\in L^2(0,T;H^{-s}(\Omega))$ to the following parabolically regularized problem \begin{equation} \label{eq: regularization for equation of w} \begin{cases} (\partial_t^2 -\eps (-\Delta)^s \partial_t +(-\Delta)^s)v = \widetilde{G}&\text{ in }\Omega_T,\\ v=0 &\text{ in }(\Omega_e)_T,\\ v(T)= \partial_t v(T)=0 &\text{ in }\Omega \end{cases} \end{equation} (see \cite[Chapter XVIII, Section 5.3.1]{DautrayLionsVol5}). By \cite[Chapter XVIII, Section 5.3.4]{DautrayLionsVol5} we know that there holds \begin{equation} \label{eq: convergence} \begin{split} v_{\eps}&\weakstar v \text{ in }L^{\infty}(0,T;\widetilde{H}^s(\Omega)),\\ \partial_t v_{\eps}&\weakstar \partial_t v \text{ in }L^{\infty}(0,T;\widetilde{L}^2(\Omega)),\\ v_\eps(t)&\to v(t) \text{ in } \widetilde{H}^s(\Omega)\text{ for all }0\leq t\leq T. \end{split} \end{equation} First, note that the conditions $u\in C^1([0,T];H^{-s}(\Omega))$ and $v_\eps\in C^1([0,T];\widetilde{L}^2(\Omega))$, where the latter follows from the Sobolev embedding, guarantee that $\langle \gamma u,v_\eps\rangle\in C^1([0,T])$ with \begin{equation} \label{eq: product rule} \partial_t \langle \gamma u,v_\eps\rangle=\langle \partial_t u,\gamma v_\eps\rangle+\langle u,\gamma \partial_t v_\eps\rangle_{L^2(\Omega)}. \end{equation} Thus, by the fundamental theorem of calculus we deduce that there holds \[ \langle \gamma u(T),v_\eps(T)\rangle-\langle \gamma u_0,v_\eps(0)\rangle=\int_0^T\langle \partial_t u,\gamma v_\eps\rangle+\langle u,\gamma \partial_t v_\eps\rangle_{L^2(\Omega)}\,dt. \] By the convergence assertions \eqref{eq: convergence} and $v_\eps (T)=0$, we get \[ -\langle \gamma u_0,v(0)\rangle=\int_0^T\langle \partial_t u,\gamma v\rangle+\langle u,\gamma \partial_t v\rangle_{L^2(\Omega)}\,dt. \] This proves \eqref{eq: integration by parts formula}. Hence, inserting this into \eqref{eq: prelim def of very weak sol}, we obtain \[ \begin{split} \int_0^T\langle\widetilde{G},u\rangle_{L^2(\Omega)}\,dt&=\int_0^T\langle (F-\gamma \partial_t u-qu),v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}\\ &=\int_0^T\langle F,v\rangle\,dt+\int_0^T \langle u,\gamma \partial_t v\rangle_{L^2(\Omega)}\,dt-\int_0^T\langle u,qv\rangle\,dt\\ &\quad+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v( 0)\rangle. \end{split} \] As $\widetilde{G}=G+\gamma \partial_t v-qv$ this gives \[ \begin{split} \int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt&=\int_0^T \langle F,v\rangle\, dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v(0)\rangle. \end{split} \] Hence, we observe that $u$ is indeed a solution of \eqref{eq: well-posedness very weak DNWEQ} in the sense of Definition \ref{def: very weak solutions}. By reversing the above arguments one can also observe that if $u$ is a solution in the sense of Definition \ref{def: very weak solutions}, then by \eqref{eq: integration by parts formula} it is a solution in the sense of \eqref{eq: prelim def of very weak sol} and thus the solution in the sense of Definition \ref{def: very weak solutions} is unique. \end{proof} \section{The inverse problem} \label{sec: inverse problem} After establishing the theory of very weak solutions to damped, nonlocal wave equations, we now turn our attention to the inverse problem part. First, in Section \ref{sec: Runge approx} we prove the optimal Runge approximation theorem (Theorem \ref{thm: Runge approx}) and in Section \ref{sec: integral identity} a suitable integral identity. Using these results, we then show in Section \ref{sec: linear uniqueness} our first main result dealing with linear perturbations (Theorem \ref{thm: uniqueness linear}). Finally, in Section \ref{sec: semilinear uniqueness} we prove Theorem \ref{thm: uniqueness semilinear} showing that the damping coefficient and the nonlinearity can be determined simultaneously. \subsection{Runge approximation} \label{sec: Runge approx} With the material from Section \ref{sec: Very weak solutions to damped, nonlocal wave equations} at our disposal, we can now show the following Runge approximation theorem, whose proof is very similar to the one of \cite[Theorem 1.2]{Optimal-Runge-nonlocal-wave}. \begin{theorem}[Runge approximation] \label{thm: Runge approx} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Then for any measurement set $W\subset\Omega_e$ and initial conditions $(u_0,u_1)\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, the \emph{Runge set} \begin{equation} \label{eq: Runge set} \mathcal{R}^{u_0,u_1}_W\vcentcolon = \{u_\varphi-\varphi\,;\,\varphi\in C_c^{\infty}(W_T)\} \end{equation} is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$, where $u_\varphi$ is the unique solution to \begin{equation} \label{eq: PDE Runge} \begin{cases} (L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega, \end{cases} \end{equation} (see Proposition \ref{prop: Weak solutions to inhomogeneous DNWEQ}). \end{theorem} \begin{proof} First of all note that it is enough to consider the case $(u_0,u_1)=0$. To see this assume that the density holds for $\mathcal{R}_W\vcentcolon =\mathcal{R}_W^{0,0}$ and let $f\in L^2(0,T;\widetilde{H}^s(\Omega))$. Let $v_0$ be the unique solution to \begin{equation} \begin{cases} (L_{\gamma}+q)v = 0 & \text{ in } \Omega_T,\\ v =0 & \text{ on } (\Omega_e)_T,\\ v(0) = u_0,\, \partial_{t}v(0) = u_1 & \text{ on } \Omega \end{cases} \end{equation} and define $\widetilde{f}\vcentcolon = f-v_0\in L^2(0,T;\widetilde{H}^s(\Omega))$. By assumption there exists $(\varphi_k)_{k\in\N}\subset C_c^{\infty}(W_T)$ such that $u_k-\varphi_k\to \widetilde{f}$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ as $k\to \infty$, where $u_k$ is the unique solution to \begin{equation} \label{eq: PDE Runge vanishing initial} \begin{cases} (L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} with $\varphi=\varphi_k$. Then $v_k\vcentcolon = u_k+v_0$ is the unique solution to \eqref{eq: PDE Runge} with $\varphi=\varphi_k$. The above convergence now implies $v_k-\varphi_k\to f$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ as $k\to\infty$ and we get that $\mathcal{R}_W^{u_0,u_1}$ is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$. Therefore, it remains to show that $\mathcal{R}_W$ is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$. As usual, we show this by a Hahn--Banach argument. Thus, suppose that $F\in L^2(0,T;H^{-s}(\Omega))$ vanishes on $\mathcal{R}_W$. Let us recall that if $\varphi\in C_c^{\infty}(W_T)$ and $u$ solves \eqref{eq: PDE Runge vanishing initial}, then by \eqref{eq: Usual cauchy homogeneous} and Lemma \ref{lemma: time reversal of solution} the function $v=(u-\varphi)^\star$ satisfies \begin{equation} \begin{cases} (L_{-\gamma}+q)v = -(-\Delta)^s\varphi^\star &\text{ in }\Omega_T,\\ v=0 &\text{ in }(\Omega_e)_T,\\ v(T)=\partial_t v(T)=0 &\text{ in }\Omega. \end{cases} \end{equation} Next, let $w$ be the unique solution to \begin{equation} \label{eq: solution to weak RHS} \begin{cases} (L_{\gamma}+q)w= F^\star &\text{ in }\Omega_T,\\ w=0 &\text{ in }(\Omega_e)_T,\\ w(0)= \partial_t w(0)=0 &\text{ in }\Omega \end{cases} \end{equation} (see Theorem \ref{thm: very weak sol DNWEQ}). By testing the equation for $w$ by $v$, we get \[ -\int_0^T \langle (-\Delta)^s\varphi^\star, w\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle F^\star,v\rangle\,dt=\int_0^T\langle F,u_\varphi-\varphi\rangle\,dt=0 \] for any $\varphi\in C_c^{\infty}(W_T)$. This ensures that there holds \[ (-\Delta)^s w=0\quad \text{ in }W_T. \] Furthermore, by construction $w$ vanishes in $(\Omega_e)_T$ and hence the unique continuation principle for the fractional Laplacian guarantees $w=0$ in $\R^n_T$ (see \cite{GSU20}). As very weak solutions are distributional solutions, we get \[ \int_0^T\langle F^\star,\Phi\rangle\,dt=\int_0^T \langle (L_{-\gamma}+q)\Phi,w\rangle_{L^2(\Omega)}\,dt=0 \] for all $\Phi\in C_c^{\infty}(\Omega_T)$. To see that very weak solutions are distributional solutions, one can simply take $G=\chi_\Omega(L_\gamma+q)\Phi$ with $\Phi\in C_c^{\infty}(\Omega\times [0,T))$ in Definition \ref{def: very weak solutions}, where $\chi_\Omega$ denotes the characteristic function of $\Omega$ (see also \cite[Proposition 3.8]{Optimal-Runge-nonlocal-wave}). By density of $C_c^{\infty}(\Omega_T)$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ we deduce that $F=0$. This concludes the proof. \end{proof} As a consequence we have the following lemma. \begin{lemma}[Convergence of time derivative] \label{lemma: convergence of time derivatives} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Let $\Phi,\Psi\in L^2(0,T;\widetilde{H}^s(\Omega))\cap H^1(0,T;H^{-s}(\Omega))$ and suppose $(\varphi_k)_{k\in\N}\subset C_c^{\infty}((\Omega_e)_T)$ is such that \begin{equation} \label{eq: convergence assertion for time derivative} u_k-\varphi_k\to \Phi\text{ in }L^2(0,T;\widetilde{H}^s(\Omega))\text{ as }k\to\infty, \end{equation} where $u_k$ solves \begin{equation} \label{eq: approx time derivative} \begin{cases} (L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\ u =\varphi_k & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} for $k\in\N$. If $\Phi,\Psi$ satisfy one of the conditions \begin{enumerate}[(a)] \item \label{item cond 1 first} $\Psi(T)=\Phi(0)=0$ \item \label{item cond 2 first} or $\Psi(T)=\Psi(0)=0$, \end{enumerate} then we have \begin{equation} \label{eq: first order limit} \lim_{k\to\infty}\int_0^T\langle \partial_t (u_k-\varphi_k),\Psi\rangle \,dt =\int_0^T\langle \partial_t\Phi,\Psi\rangle\,dt. \end{equation} \end{lemma} \begin{remark} Let us note that the same formula \eqref{eq: first order limit} holds for second order time derivatives under appropriate conditions. \end{remark} \begin{proof} Using the integration by parts formula, we may compute \[\small \begin{split} &\lim_{k\to\infty}\int_0^T\langle \partial_t (u_k-\varphi_k),\Psi\rangle \,dt \\ &= \lim_{k\to\infty}\left(\langle (u_k-\varphi_k)(T),\Psi(T)\rangle_{L^2(\Omega)}-\langle (u_k-\varphi_k)(0),\Psi(0)\rangle_{L^2(\Omega)}-\int_0^T\langle \partial_t\Psi,u_k-\varphi_k\rangle \,dt\right)\\ &=-\lim_{k\to\infty}\int_0^T\langle \partial_t\Psi,u_k-\varphi_k\rangle \,dt\\ &=-\int_0^T \langle \partial_t \Psi,\Phi\rangle \, dt\\ &=\langle \Phi(0),\Psi(0)\rangle_{L^2(\Omega)}-\langle \Phi(T),\Psi(T)\rangle_{L^2(\Omega)}+\int_0^T \langle \partial_t\Phi, \Psi\rangle \, dt\\ &=\int_0^T \langle \partial_t\Phi, \Psi\rangle \, dt. \end{split} \] In the first equality sign we used an integration by parts, in the second equality we used \eqref{eq: approx time derivative}, $\Psi(T)=0$ and \eqref{eq: approx time derivative}, in the third equality the convergence \eqref{eq: convergence assertion for time derivative}, in the fourth equality again an integration by parts and finally in the last equality the conditions \ref{item cond 1 first} or \ref{item cond 2 first}. \end{proof} \subsection{DN map and integral identities} \label{sec: integral identity} Next, we define the \emph{Dirichlet to Neumann (DN) map} $\Lambda_{\gamma,q}$ related to \begin{equation} \label{eq: PDE for integral identity} \begin{cases} (L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega, \end{cases} \end{equation} via \begin{equation} \label{eq: DN map} \langle \Lambda_{\gamma,q}\varphi,\psi\rangle=\int_{\R^n_T}(-\Delta)^{s/2} u_\varphi (-\Delta)^{s/2}\psi\,dx \end{equation} for all $\varphi,\psi\in C_c^{\infty}((\Omega_e)_T)$, where $u_\varphi$ is the unique solution to \eqref{eq: PDE for integral identity} with exterior condition $\varphi$. Using the above preparation, we now establish the following integral identity. \begin{proposition}[Integral identity for linear perturbations] \label{prop: integral identity} Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma_j,q_j)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$ for $j=1,2$. Let $\varphi_j\in C_c^{\infty}((\Omega_e)_T)$ and denote by $u_j$ the corresponding solution of \eqref{eq: PDE for integral identity} with $(\gamma,q)=(\gamma_j,q_j)$. Then there holds \begin{equation} \label{eq: integral identity} \begin{split} &\langle (\Lambda_{\gamma_1,q_1}-\Lambda_{\gamma_2,q_2})\varphi_1,\varphi_2^\star\rangle\\ &\quad =\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt. \end{split} \end{equation} \end{proposition} \begin{proof} Let $(\Gamma_j,Q_j)\in C^{0,\alpha}(\R^n)\times L^p(\Omega)$, $j=1,2$, and suppose $U_j$ is the unique solutions of \eqref{eq: PDE for integral identity} with $(\gamma,q)=(\Gamma_j,Q_j)$ and exterior condition $\varphi=\psi_j$. Then we may compute \begin{equation} \label{eq: calculation for integral identity} \begin{split} &\int_{\Omega_T}\{[(\Gamma_1-\Gamma_2)\partial_t+Q_1-Q_2](U_1-\psi_1)\}(U_2-\psi_2)^\star\,dxdt\\ & =\int_{\Omega_T}\{[\Gamma_1\partial_t+Q_1](U_1-\psi_1)\}(U_2-\psi_2)^\star\,dxdt\\ &\quad -\int_{\Omega_T}(U_1-\psi_1)[- \Gamma_2\partial_t+Q_2](U_2-\psi_2)^\star\,dxdt\\ &=\int_{0}^T\langle L_{\Gamma_1,Q_1}(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt\\ &\quad -\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt\\ &\quad -\int_{0}^T\langle L_{-\Gamma_2,Q_2}(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\ &\quad +\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\ &=-\int_{0}^T\langle (-\Delta)^s\psi_1,(U_2-\psi_2)^\star\rangle_{L^2(\Omega)}\,dt +\int_{0}^T\langle (-\Delta)^s \psi_2^\star,(U_1-\psi_1)\rangle_{L^2(\Omega)}\,dt\\ &=\int_{\R^n_T} ((-\Delta)^s\psi_2^\star) U_1\,dxdt-\int_{\R^n_T} ((-\Delta)^s\psi_1) U_2^\star\,dxdt\\ &=\langle \Lambda_{\Gamma_1,Q_1}\psi_1,\psi_2^\star\rangle-\langle \Lambda_{\Gamma_2,Q_2}\psi_2,\psi_1^\star\rangle. \end{split} \end{equation} In the first equality we used that $U_1-\psi_1$ has vanishing initial conditions, $(U_2-\psi_2)^\star$ has vanishing terminal conditions and an integration by parts. In the third equality we used that the PDEs for $U_1-\psi_1$ and $(U_2-\psi_2)^\star$ hold in the sense of $L^2(0,T;H^{-s}(\Omega))=(L^2(0,T;\widetilde{H}^s(\Omega))'$ (see Lemma \ref{lemma: time reversal of solution}). In the fourth equality, we used the PDEs for $U_1$ and $U_2$, Lemma \ref{lemma: time reversal of solution} and that there holds \[ \begin{split} &\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\ &=\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt, \end{split} \] which can be established similarly as \cite[Claim 4.2]{Semilinear-nonlocal-wave-eq} (see also the proof of Theorem \ref{thm: very weak sol DNWEQ}). In the last equality, we have made the change of variables $\tau=T-t$ for the second integral. On the one hand, using \eqref{eq: calculation for integral identity} with \[ \Gamma_1=\Gamma_2=\gamma_j\text{ and }Q_1=Q_2=q_j, \] we observe that \begin{equation} \label{eq: self-adjointness DN map} \langle \Lambda_{\gamma_j,q_j}\psi_1,\psi_2^\star\rangle=\langle \Lambda_{\gamma_j,q_j}\psi_2,\psi_1^\star\rangle \end{equation} for all $\psi_j\in C_c^{\infty}((\Omega_e)_T)$, $j=1,2$. On the other hand, choosing \[ \Gamma_j=\gamma_j,\,Q_j=q_j\text{ and }\psi_j=\varphi_j \] in \eqref{eq: calculation for integral identity} and taking into account the self-adjointness \eqref{eq: self-adjointness DN map}, we get \eqref{eq: integral identity}. \end{proof} \subsection{Simultaneous determination of damping coefficient and linear perturbations} \label{sec: linear uniqueness} \begin{proof}[Proof of Theorem \ref{thm: uniqueness linear}] First note that by the integral identity in Proposition \ref{prop: integral identity}, we may deduce from the condition \eqref{eq: equality of DN maps} that there holds \begin{equation} \label{eq: uniqueness proof help identity} \int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt=0 \end{equation} for all $\varphi_j\in C_c^{\infty}((W_j)_T)$, where $u_j$ denotes the unique solution to \begin{equation} \label{eq: PDE uniqueness proof} \begin{cases} (L_{\gamma_j}+q_j)u = 0 & \text{ in } \Omega_T,\\ u =\varphi_j & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega. \end{cases} \end{equation} Let $\omega\Subset \Omega$ and choose a cutoff function $\Phi_1\in C_c^{\infty}(\Omega)$ satisfying $\Phi_1=1$ on $\omega$. Moreover, let $\Phi_2\in C_c^{\infty}(\omega_T)$. By the Runge approximation (Theorem \ref{thm: Runge approx}), there exist sequences $(\varphi_j^k)_{k\in\N}\subset C_c^{\infty}((W_j)_T)$ with corresponding solutions $u_j^k$ of \eqref{eq: PDE uniqueness proof} with $\varphi_j=\varphi_j^k$ such that $u_j^k-\varphi_j^k\to \Phi_j$ in $L^2(0,T;\widetilde{H}^s(\Omega))$. Taking $\varphi_1=\varphi_1^k$ and $\varphi_2=\varphi_2^{\ell}$ in \eqref{eq: uniqueness proof help identity} gives \[ \int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u^k_1-\varphi^k_1)\}(u^{\ell}_2-\varphi^{\ell}_2)^\star\,dxdt=0 \] for all $k,\ell\in\N$. First, we let $\ell\to\infty$ to deduce \begin{equation} \label{eq: uniqueness proof help identity 2} \int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u^k_1-\varphi^k_1)\}\Phi_2^\star\,dxdt=0 \end{equation} for all $k\in\N$. As $\gamma_1-\gamma_2\in C^{0,\alpha}(\R^n)$ the estimate \eqref{eq: multiplication by gamma} ensures that we can apply Lemma \ref{lemma: convergence of time derivatives} under the condition \ref{item cond 2 first} and so $\partial_t\Phi_1=0$ shows that the first term in \eqref{eq: uniqueness proof help identity 2} goes to zero. So in the limit $k\to\infty$ what remains is \[ \int_{\Omega_T}(q_1-q_2)\Phi_2^\star \,dxdt=0, \] where we used $\Phi_1=1$ on $\omega$. This ensures that $q_1=q_2$ on $\omega$. As the set $\omega$ is arbitrary, we get $q_1=q_2$ in $\Omega$. Now, the identity \eqref{eq: uniqueness proof help identity} reduces to \[ \int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt=0 \] for all $\varphi_j\in C_c^{\infty}((W_j)_T)$. We choose $\eta\in C_c^{\infty}(\Omega_T)$, define \[ \Phi_1(x,t)=\int_0^t\eta(x,\tau)\,d\tau\in C_c^{\infty}(\Omega\times (0,T]) \] and take $\Phi_2\in C_c^{\infty}(\Omega_T)$. Then using $\partial_t \Phi_1=\eta$ and arguing as above via a Runge approximation and Lemma \ref{lemma: convergence of time derivatives}, we get from \eqref{eq: uniqueness proof help identity} the identity \[ \int_{\Omega_T}(\gamma_1-\gamma_2)\eta\Phi_2^\star\,dxdt=0. \] This again implies $\gamma_1=\gamma_2$ in $\Omega$. \end{proof} \subsection{Simultaneous determination of damping coefficient and nonlinearity} \label{sec: semilinear uniqueness} Before turning to the proof of our second main result, let us recall that the \emph{DN map} related to the problem \begin{equation} \label{eq: PDE for semilinear problem} \begin{cases} L_{\gamma}u+f(u) = 0 & \text{ in } \Omega_T,\\ u =\varphi & \text{ on } (\Omega_e)_T,\\ u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} is defined by \begin{equation} \langle \Lambda_{\gamma,f}\varphi,\psi\rangle\vcentcolon = \int_{\R^n_T}(-\Delta)^{s/2}u_\varphi(-\Delta)^{s/2}\psi\,dxdt, \end{equation} where $\varphi,\psi\in C_c^{\infty}((\Omega_e)_T)$ and $u_\varphi$ is the unique solution to \eqref{eq: PDE for semilinear problem} (see \cite[Proposition 3.7]{Semilinear-nonlocal-wave-eq}). \begin{proof}[Proof of Theorem \ref{thm: uniqueness semilinear}] Let $\eps>0$ and denote by $u^{(j)}_\eps$ the unique solutions to \eqref{eq: PDE for semilinear problem} with $f=f_j$, $\gamma=\gamma_j$ and $\varphi=\eps \eta$ for some fixed $\eta\in C_c^{\infty}((W_1)_T)$. Let us observe that the UCP for the fractional Laplacian and the condition \eqref{eq: equality of DN maps semilinear} imply that $u_\eps\vcentcolon = u^{(1)}_\eps=u^{(2)}_\eps$. Next, let us note that we can write \begin{equation} \label{eq: decomposition of u eps} u_\eps=\eps v_j+R^{(j)}_\eps \end{equation} for $j=1,2$, where $v_j$ and $R^{(j)}_\eps$ are the unique solutions of \begin{equation} \label{eq: linear part of u eps} \begin{cases} L_{\gamma_j}v = 0 & \text{ in } \Omega_T,\\ v =\eta & \text{ on } (\Omega_e)_T,\\ v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega \end{cases} \end{equation} and \begin{equation} \label{eq: nonlinear part of u eps} \begin{cases} L_{\gamma_j}R = -f_j(u_\eps) & \text{ in } \Omega_T,\\ R =0 & \text{ on } (\Omega_e)_T,\\ R(0) = 0,\, \partial_{t}R(0) = 0 & \text{ on } \Omega, \end{cases} \end{equation} respectively. This simply follows from the unique solvability of \eqref{eq: PDE for semilinear problem} and both functions $u_\eps$ and $\eps v_j+R^{(j)}_\eps$ are solutions. Furthermore, we notice that the energy estimate of \cite[Theorem 3.1]{Semilinear-nonlocal-wave-eq}, \cite[eq.~(3.18)]{Semilinear-nonlocal-wave-eq} and the $r+1$ homogeneity of $f_j$ ensure that $R^{(j)}_\eps$ satisfies \begin{equation} \label{eq: energy estimate remainder} \begin{split} \\|\partial_t R^{(j)}_\eps\\|_{L^{\infty}(0,T;L^2(\Omega))}+\\|R^{(j)}_\eps (t)\\|_{L^{\infty}(0,T;H^s(\R^n))}&\lesssim \\|f_j(u_\eps)\\|_{L^2(\Omega_T)}\\ &\lesssim \\|u_\eps\\|^{r+1}_{L^{\infty}(0,T;H^s(\R^n))}. \end{split} \end{equation} Moreover, we may estimate \begin{equation} \label{eq: estimate u eps} \begin{split} &\\|\partial_t u_\eps\\|_{L^{\infty}(0,T;L^2(\R^n))}+\\|u_\eps\\|_{L^{\infty}(0,T;H^s(\R^n))}\\ &\lesssim \\|\partial_t (u_\eps-\eps\eta)\\|_{L^{\infty}(0,T;L^2(\Omega))}+\\|u_\eps-\eps\eta\\|_{L^{\infty}(0,T;H^s(\R^n))}+\eps\\|\eta\\|_{W^{1,\infty}(0,T;H^{2s}(\R^n))}\\ &\lesssim \eps\\|\eta\\|_{W^{1,\infty}(0,T;H^{2s}(\R^n))}. \end{split} \end{equation} This follows from the following observations. If $u$ solves \eqref{eq: PDE for semilinear problem} for a damping coefficient $\gamma\in C^{0,\alpha}(\R^n)$, a weak nonlinearity $f$ and $\varphi\in C_c^{\infty}((\Omega_e)_T)$, then $v=u-\varphi$ solves \begin{equation} \begin{cases} L_{\gamma}v+f(v) = -(-\Delta)^s\varphi & \text{ in } \Omega_T,\\ v =0& \text{ on } (\Omega_e)_T,\\ v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega. \end{cases} \end{equation} Now, we may invoke \cite[eq.~(3.15)]{Semilinear-nonlocal-wave-eq} to find that there holds \[ \begin{split} &\\|\partial_t v(t)\\|_{L^2(\Omega)}^2+\\|v(t)\\|_{H^s(\R^n)}^2\\ &\lesssim \int_0^t\|\langle \gamma\partial_t v,\partial_t v\rangle_{L^2(\Omega)}\|\,d\tau +\int_0^t\|\langle (-\Delta)^{s}\varphi,\partial_t v\rangle_{L^2(\Omega)}\|\,d\tau\\ &\lesssim \\|(-\Delta)^{s}\varphi\\|_{L^2(0,t;L^2(\Omega))}^2+\int_0^t\\|\partial_t v\\|_{L^2(\Omega)}^2\,d\tau. \end{split} \] Thus, Gronwall's inequality gives \[ \\|\partial_t v(t)\\|_{L^2(\Omega)}+\\|v(t)\\|_{H^s(\R^n)}\lesssim \\|(-\Delta)^{s}\varphi\\|_{L^2(0,t;L^2(\Omega))}. \] This ensures the validity of the second estimate in \eqref{eq: estimate u eps}. Next, observe that by subtracting the PDEs for $u^{(1)}_\eps$ and $u^{(2)}_\eps$, we deduce that \begin{equation} \label{eq: condition for uniqueness semilinear} (\gamma_1-\gamma_2)\partial_t u_\eps=f_2(u_\eps)-f_1(u_\eps)\text{ in }\Omega_T. \end{equation} By \eqref{eq: decomposition of u eps}, we may write \begin{equation} \label{eq: condition for uniqueness semilinear 2} (\gamma_1-\gamma_2)(\eps\partial_tv_1+\partial_t R^{(1)}_\eps) =f_2(u_\eps)-f_1(u_\eps)\text{ in }\Omega_T. \end{equation} Combining \eqref{eq: energy estimate remainder} and \eqref{eq: estimate u eps}, we see that \begin{equation} \label{eq: decay estimate R eps} \\|\partial_t R^{(j)}_\eps\\|_{L^{\infty}(0,T;L^2(\Omega))}+\\|R^{(j)}_\eps (t)\\|_{L^{\infty}(0,T;H^s(\R^n))}\lesssim \eps^{r+1}. \end{equation} Multiplying \label{eq: condition for uniqueness semilinear 2} by $\eps^{-1}$ gives \begin{equation} \label{eq: condition for uniqueness semilinear 3} \begin{split} &(\gamma_1-\gamma_2)(\partial_tv_1+\eps^{-1}\partial_t R^{(1)}_\eps) =f_2(\eps^{-1/(r+1)}u_\eps)-f_1(\eps^{-1/(r+1)}u_\eps)\text{ in }\Omega_T. \end{split} \end{equation} Next, let us focus one the case $2s<n$ as the other one can be treated similarly. As $r>0$ we deduce from \eqref{eq: estimate u eps} that $\eps^{-1/(1+r)}u_\eps\to 0$ in $L^{\infty}(0,T;H^s(\R^n))$ and so by Sobolev's embedding in $L^q(0,T;L^{2_s^}(\Omega))$ for all $1\leq q\leq \infty$ and $2_s^=\frac{2n}{n-2s}$. Hence, by our assumptions on $f_j$ and \cite[Lemma 3.6]{zimmermann2024calderon}, we get \begin{equation} \label{eq: zero convergence nonlinearity} f_j(\eps^{-1/(r+1)}u_\eps)\to 0\text{ in }L^{q/(r+1)}(0,T;L^{2_s^/(r+1)}(\Omega)) \end{equation} for all $q\geq r+1$ as $\eps\to 0$. Additionally, using \eqref{eq: decay estimate R eps} we know that \begin{equation} \label{eq: zero convergence time derivative remainder} \eps^{-1}\partial_t R^{(j)}_\eps\to 0\text{ in }L^{\infty}(0,T;L^2(\Omega)). \end{equation} Therefore, from \eqref{eq: condition for uniqueness semilinear 3}, \eqref{eq: zero convergence nonlinearity} and \eqref{eq: zero convergence time derivative remainder}, we infer \[ (\gamma_1-\gamma_2)\partial_t v_1=0\text{ in }\Omega_T. \] In particular, this ensures that there holds \[ \int_{\Omega_T}(\gamma_1-\gamma_2)\partial_t (v_1-\eta)(w_2-\psi)^\star\,dxdt=0 \] for any $\psi \in C_c^{\infty}((W_2)_T)$, where $w_2$ is the unique solution of \begin{equation} \begin{cases} L_{\gamma_2}w = 0 & \text{ in } \Omega_T,\\ w =\psi & \text{ on } (\Omega_e)_T,\\ w(0) = 0,\, \partial_{t}w(0) = 0 & \text{ on } \Omega. \end{cases} \end{equation} Now, arguing as in the previous section, we get $\gamma_1=\gamma_2$ in $\Omega$. Hence, \eqref{eq: condition for uniqueness semilinear} reduces to \begin{equation} f_1(u_\eps)=f_2(u_\eps)\text{ in }\Omega_T. \end{equation} Multiplying this identity by $\eps^{-(r+1)}$ and arguing as before, we deduce that \[ f_1(v)=f_2(v)\text{ in }\Omega_T, \] where $v\vcentcolon =v_1=v_2$ as $\gamma_1=\gamma_2$. One can now show $f_1(x,\tau)=f_2(x,\tau)$ for all $x\in\Omega$ and $\tau\in\R$ exactly as described in \cite[p.~29]{Optimal-Runge-nonlocal-wave}. Hence, we can conclude the proof. \end{proof} \medskip \noindent\textbf{Acknowledgments.} P.~Zimmermann was supported by the Swiss National Science Foundation (SNSF), under grant number 214500. \section{Statements and Declarations} \subsection{Data availability statement} No datasets were generated or analyzed during the current study. \subsection*{Conflict of Interests} Hereby we declare there are no conflict of interests. \bibliography{refs} \bibliographystyle{alpha} \end{document}
2412.02042v1	http://arxiv.org/abs/2412.02042v1	Delta invariants of plumbed 3-manifolds	\documentclass[11pt,a4paper, reqno,dvipsnames]{amsart} \usepackage[utf8]{inputenc} \usepackage{amsfonts, amsmath,amsthm, amssymb} \usepackage{enumerate} \usepackage{rotating} \usepackage{tikz} \usetikzlibrary{decorations.pathreplacing} \usepackage{graphicx} \usepackage{mathtools} \usepackage{comment} \usepackage[giveninits=true]{biblatex} \renewbibmacro{in:}{} \addbibresource{bibliography.bib} \usepackage[mode=buildnew]{standalone} \usepackage{geometry} \geometry{left=35mm,right=35mm,top=42mm,bottom=44mm} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=Violet, filecolor=magenta, urlcolor=cyan, citecolor=Violet, pdftitle={Delta invariants of plumbed manifolds}, } \usepackage[capitalise]{cleveref} \usepackage{mymacros} \title{$\Delta$ invariants of plumbed manifolds} \author{Shimal Harichurn, Andr\'as N\'emethi, Josef Svoboda} \date{\today} \begin{document} \maketitle \begin{abstract} We study the minimal $q$-exponent $\Delta$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^c$-structure. We express $\Delta$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $\Delta$ for non-Seifert manifolds. Finally, we compare $\Delta$ invariants with correction terms in Heegaard--Floer homology. \end{abstract} \tableofcontents \section{Introduction} \(\Zhat_b(Y; q)\) is a $q$-series invariant of a negative definite plumbed 3-manifold \(Y\) equipped with a \(\spc\) structure \(b\) \cite{GPPV20}. It recovers Witten--Reshetikhin--Turaev $U_q(\mathfrak{sl}_2)$-invariants in radial limits to roots of unity as conjectured by \cite{GPPV20}, and recently proved in \cite{Mur23}. In this paper, we focus on the behavior of \(\Zhat_b(Y; q)\) near \(q=0\). In other words, we study the smallest \(q\)-exponents \(\Delta_b\) in \(\Zhat_b(Y,q)\): \[ \Zhat_b(Y,q) = q^{\Delta_b}(c_0+c_1 q + c_2 q^2 + \dots), \quad c_0 \neq 0. \] The rational numbers $\Delta_b$ were studied in \cite{GPP21} where their fractional part was related to various invariants of 3-manifolds. In this work, we focus on the actual value of $\Delta_b$. For plumbed manifolds it is natural to consider the `canonical' $\spc$ structure $can$. Related to it, there is a numerical topological invariant \(\gamma(Y) := k^2+s\) (see \ref{ss:spinc}). It appears, e.g., in the study of Seiberg--Witten invariants \cite{NeNi02} of the associated plumbed 3-manifold, and its use in topology goes back to Gompf \cite{Gompf98}. It also plays an important role in singularity theory, e.g. in Laufer's formula \cite{Laufer1977} for Milnor number of a Gorenstein normal surface singularity. As the main result of this paper, we prove that for Seifert manifolds, \(\Delta_{can}(Y)\) can be expressed using $\gamma(Y)$: \begin{theorem}\label{thm:seifert_hom} Let $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ be a Seifert manifold associated with negative definite plumbing graph. Let $\emph{can}$ be the canonical $\spc$ structure of $Y$. Then $\Delta_{can}$ satisfies \begin{equation} \Delta_{can} = -\frac{\gamma(Y)}{4} + \frac{1}{2}. \end{equation} If $Y$ is not a lens space, then $\Delta_{can}$ is minimal among all ${\Delta_b, b \in \spc(Y)}$. \end{theorem} In the special case of Seifert homology spheres $Y=\Sigma(a_1,a_2,\dots,a_n)$, we deduce that $\Delta_{can}(Y)$ grows polynomially with the leading term given by \[ \Delta_{can}(Y) \approx (n-2)a_1 a_2 \cdots a_n. \] The idea of the proof is to express $\Delta_{can}$ as a minimum of a quadratic form, given by the linking form of the plumbing graph, over certain integral vectors. The key is identifying this set over which we minimize (\cref{lm:delta_minimizer}). We show that in this set, there exists a unique vector (up to sign) that minimizes the quadratic form. Once we leave Seifert manifolds, the computation of \(\Delta_b\) invariants becomes much more complicated. We illustrate this on plumbing graphs with exactly two vertices of degree 3 and no vertices of degree $\geq 4$ in \cref{s:H_shaped}. In this case, \(\Delta_{can}\) is often smaller than $-\gamma(Y)/4 + 1/2$, because we have a larger freedom in finding minimizing vectors. To analyze $\Delta$ of these graphs, we use splice diagrams \cite{Sie80,NeumannWallBook86,NW05a} of plumbed manifolds, building on \cite{GKS23}. Surprisingly, $\Delta_{can}$ can also be larger than $-\gamma(Y)/4 + 1/2$, as a result of interesting cancellations in the formula for $\Zhat_b(Y)$, see \cref{ex:cancel_H}. Namely, $\Zhat_b(Y)$ can be expressed as a weighted sum over certain lattice points and those can sometimes be organized in pairs with weights of opposite signs. This can be avoided by refining the weights, as provided by the two-variable series $\Zhathat_b(q,t)$ defined in \cite{AJK23}. However, \cref{ex:t_cancel_0} shows that cancellations may occur even in $\Zhathat_b(q,t)$. An analogy between $\Delta_b(Y)$ invariants and the correction terms $d_b(Y)$ in Heegaard--Floer homology was proposed in \cite{GPP21}. In \cite{harichurn2023deltaa}, the first author demonstrated on Brieskorn spheres that unlike $d_b(Y)$, $\Delta_b(Y)$ are not cobordism invariants. Using the explicit formula for $\Delta_{can}$ of Seifert manifolds given by \cref{thm:seifert_hom}, we compare $\Delta_{can}$ and $d_{can}$ for some classes of Brieskorn spheres, where $d_{can}$ is explicitly known \cite{BorNem2011}. We find that $\Delta_{can}$ is generically much larger than $d_{can}$. The reason for this discrepancy is that although $\Delta_{can}$ and $d_{can}$ are both minima of certain closely related quadratic forms, for $\Delta_{can}$, the quadratic form is being minimized over a much smaller set of vectors than the one for $d_{can}$. \subsection{Acknowledgement} We are grateful to Sergei Gukov, Mrunmay Jagadale and Sunghyuk Park for useful discussions. A. N\'emethi was partially supported by ``\'Elvonal (Frontier)'' Grant KKP 144148. J. Svoboda was supported by the Simons Foundation Grant {\it New structures in low-dimensional topology}. S. Harichurn was supported by the 2020 FirstRand FNB Fund Education Scholarship Award. \section{Preliminaries} In this section, we collect the necessary material on plumbed manifolds and \(\spc\) structures on them. \subsection{Plumbed 3-manifolds} Let $\Gamma = (V,E,m)$ be a finite tree with the set of vertices $V$ and edges $E$, and a vector \(m \in \ZZ^{\lvert V \rvert}\) consisting of integer labels \(m_v\) for each vertex \(v \in V\). Let $s = \lvert V\rvert$ and let $\delta = (\delta_v)_{v \in V} \in \ZZ^s$ be the vector of the degrees (valencies) of the vertices. We often implicitly order vertices of \(V\), so that \(V = {v_1,v_2,\dots,v_s}\) and write the quantities associated to \(v_i\) with subscript \(i\). We can record \(\Gamma\) using $s\times s$ \emph{plumbing matrix} \(M=M(\Gamma)\), defined by \[ M_{vw} = \begin{cases} m_v & \text{if} \quad v = w \\ 1 & \text{if} \quad (v, w) \in E \\ 0 & \text{otherwise.} \end{cases} \] We always assume that \(M\) is a negative definite matrix. For a vector $l \in \Z^s$ we write \[ l^2 = l^T M^{-1}l. \] Note that the quadratic form \(l \mapsto -l^2\) is a positive definite. From \(\Gamma\), we can construct a closed oriented 3-manifold $Y := Y(\Gamma)$ by \emph{plumbing} \cite{Neu81}: For each vertex \(v\) we consider a circle bundle over \(S^2\) with Euler number \(m_v\). Then we glue the bundles together along tori corresponding to the edges in \(E\). Manifolds given by this construction, with \(M\) negative definite, are called \emph{negative definite plumbed manifolds}. From the construction above, it follows that \(Y\) is a \emph{rational homology sphere}, that is, \(H_1(Y) = H_1(Y,\ZZ)\) is finite. We have \(H_1(Y) \cong \ZZ^s/M \ZZ^s\) and $\lvert H_1(Y)\rvert=\det M$. The quadratic form $l \mapsto -l^2$ takes values in $\lvert H\rvert^{-1}\ZZ$ as the adjugate matrix $\adj M = (\det M) M^{-1}$ of $M$ has integer entries. \subsection{\texorpdfstring{\(\Spc\)}{Spinc} structures}\label{ss:spinc} The set $\spc(Y)$ of \(\spc\) structures on $Y$ admits a natural free and transitive action of \(H_1(Y)\), hence it is finite. It can be identified with the set $(2\ZZ^s + m)/2M\ZZ^s$ of \emph{characteristic vectors}. For us, it is convenient to use another identification, with \(\spc(Y) \cong (2\ZZ^s + \delta)/2M\ZZ^s\), obtained from the usual characteristic vectors via the map \(l \to l - Mu\), where \(u = (1, 1, \dots, 1)\). This is justified by the identity \( \delta + m = Mu\). For a vector $b \in 2\ZZ^s + \delta,$ we denote the corresponding $\spc$ structure as $[b] \in (2\ZZ^s + \delta)/2M\ZZ^s$. We often omit the brackets when it is clear from the context, e.g. we write $\Zhat_b$ for $\Zhat_{[b]}$. We consider the vector $2u - \delta$ and the corresponding `canonical' $\spc$ structure $can = [2u-\delta]$. Its characteristic vector is \(k = 2u-\delta + Mu = m + 2u\). The rational number \(\gamma(Y) := k^2+s = (2u-\delta +Mu)^2+s\) does not depend on the plumbing representation of $Y$, so it is a topological invariant of $Y$. Denote by $\Tr(M) = \sum_{v \in V}m_v$ the trace of the plumbing matrix $M$. Then $\gamma(Y)$ can be expressed as follows: \begin{proposition}[\cite{NeNi02}]\label{base-k2-s} Let $Y:=Y(\Gamma)$ be a negative definite plumbed manifold, which is a rational homology sphere. Then \begin{equation} \begin{split} \gamma(Y) &= 3s + \Tr(M) + 2 + \sum_{v,w \in V} (2-\delta_v)(2-\delta_w)M_{vw}^{-1}\\ &=3s + \Tr(M) + 2 + (2u-\delta)^2. \end{split} \end{equation} \end{proposition} \subsection{The \texorpdfstring{\(\Zhat_b\)}{Zhat} invariants} For the rest of the paper, let \(Y = Y(\Gamma)\) be a negative definite plumbed rational homology sphere with a chosen negative definite plumbing matrix \(M\) of $\Gamma$. \begin{definition} Let \(b \in 2\ZZ^s + \delta\) be a vector representing a $\spc$ structure $[b]$. For $\lvert q \rvert<1$, the GPPV $q$-series $\Zhat_b(Y;q)$ is defined by \begin{equation}\label{def:zhat} \Zhat_b(Y;q) = q^{\frac{-3s - \Tr(M)}{4}} \vp\oint\limits_{\lvert z_i \rvert = 1}\prod_{v_i\in V} \frac{dz_i}{2\pi i z_i} \left(z_i - \frac{1}{z_i}\right)^{2-\delta_i} \Theta_{b}(z), \end{equation} where \begin{equation}\label{eq:theta} \Theta_{b}(z) = \sum_{l \in 2M\ZZ^s + b} z^l q^{-\frac{l^2}{4}}, \end{equation} and $\vp$ denotes the principal part of the integral. \end{definition} We often omit the manifold $Y$ when it is clear from the context, e.g. we write $\Zhat_b(q)$ for $\Zhat_b(Y,q)$. Clearly, $\Zhat_b(q)$ does not depend on the choice of $b \in [b]$. The convergence of $\Zhat_b(Y)$ for $\lvert q \rvert<1$ follows from the negative definiteness of \(M\). \section{\texorpdfstring{$\Delta_b$}{Delta} invariants} \begin{definition} The \emph{Delta invariant} \(\Delta_b\) is the smallest $q$-exponent in \(\Zhat_b(q)\). If $\Zhat_b(q) = 0$, we set \(\Delta_b = \infty\). \end{definition} Using $\Delta_b$, we can write the $\Zhat_b(q)$ in the following form: \[ \Zhat_b(q) = 2^{-s} q^{\Delta_b}\sum_{i=0}^\infty r_iq^i = 2^{-s} q^{\Delta_b}(r_0 + r_1q^1 + r_2q^2 + \cdots) \] for some integers $r_i$ with $r_0 \neq 0$. This is justified by part (1) of the following Lemma (recall that $\lvert H\rvert$ is the order of $H_1(Y,\ZZ)$): \begin{lemma}\label{delta-h-mod-1} \begin{enumerate} \item The differences between the exponents of \(\Zhat_b(q)\) are integers. \item The fractional part of the exponents (and in particular of \(\Delta_b\)) is given by \[ \frac{-3s - \Tr(M) -b^2}{4} \Mod{1}. \] Consequently $4 \lvert H \rvert \Delta_b \in \ZZ \cup \{\infty\}$. \item $\lvert H\rvert\Delta_a \equiv \lvert H\rvert\Delta_{b} \Mod{1}$ for any two $\spc$ structures $[a],[b]$ on $Y$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item Consider two vectors \(l, l+2Mn \in [b]\), where $l,n \in \ZZ^s$. The difference of the corresponding exponents is \[ \frac{-l^2}{4}+\frac{(l+2Mn)^2}{4} = l^Tn + n^T Mn \in \ZZ. \] \item The exponent of a representative $b \in [b]$ reads \[ \frac{-3s - \Tr(M) - b^2}{4} \in \frac{1}{4\lvert H\rvert}\ZZ \] By (1), all the other exponents have the same fractional part. \item We have \[ \lvert H\rvert\Delta_b(Y) - \lvert H\rvert\Delta_{a}(Y) \equiv \frac{\lvert H\rvert(b^2 -a^2)}{4} \Mod{1}. \] By writing $b = 2l + \delta$ and $a = 2n+\delta$ with $l,n \in \ZZ$, we see that \[ \frac{1}{4}\lvert H\rvert (b^2 - a^2) = \lvert H\rvert (l^2 + l^TM^{-1}\delta - n^2 + n^TM^{-1}\delta) \in \ZZ. \] \end{enumerate} \end{proof} \begin{remark} The set of $\spc$ structures admits a natural involution, called conjugation, denoted by $b \mapsto \bar{b}$. It is known that \(\Zhat_b(q) = \Zhat_{\bar{b}}(q)\), hence we have \(\Delta_b =\Delta_{\bar{b}}\). \end{remark} \subsection{The exponents of \texorpdfstring{$\Zhat_b$}{Zhat}}\label{computation-via-set-c} The \(\Delta_b\) invariant is the minimal $q$-exponent in the series \(\Zhat_b(q)\). The exponents are, up to an overall shift, given by the quadratic form \(l \mapsto -l^2\). Here $l$ are lattice vectors that run over a certain subset \(\LC_b \subset \ZZ^s\). We need to identify this subset. We order the set of vertices \(V\) of the plumbing tree $\Gamma$ by their degree: \[ V = \{v_1, \dots, v_{s_1}, v_{s_{1}+1}, \dots, v_{s_{1}+s_{2}}, v_{s_{1}+s_{2}+1}, \dots, v_{s_{1}+s_{2} + s_{3} = s}\}. \] We have \(s_1\) leaves, \(s_2\) vertices of degree 2 and \(s_3\) vertices of degree \(\geq 3\). The integrand of \(\Zhat\) contains the rational function \begin{equation}\label{eq:rat_function} \prod_{i=1}^s(z_i- z_i^{-1})^{2-\delta_i} = \prod_{i=1}^{s_1} (z_i-z_i^{-1}) \prod_{i=s_1+s_2+1}^{s} \frac{1}{(z_i-z_i^{-1})^{\delta_i-2}}. \end{equation} The integration in \eqref{def:zhat} is equivalent to the following procedure: First, we expand each term of the product above using the \emph{symmetric expansion}---the average of Laurent expansions as $z_i \rightarrow 0$ and $z_i \rightarrow \infty$, and multiply these together, giving an element of $\Z[z_1^{\pm 1},\dots, z_s^{\pm 1}]$: \begin{equation}\label{eq:rat_function_expn} \prod_{i=1}^s \se (z_i- z_i^{-1})^{2-\delta_i} = \sum_{l \in \ZZ^s} \ct_l z^l = \sum_{l \in \ZZ^s} \ct_l z_1^{l_1}z_2^{l_2}\cdots z_s^{l_s}. \end{equation} Then we multiply the result with the theta function $\Theta_b(z)$ in \eqref{eq:theta}, and we extract the constant coefficient in variables $z_i$, giving the $q$-series $\Zhat_b(q)$. Let \(\LCt\) denote the set of all the vectors \(l \in \ZZ^s\) with nonvanishing coefficient $\ct_l \neq 0$ in \eqref{eq:rat_function_expn}. Similarly, define $\LCt_b = \LCt \cap -(2\ZZ^s+b)$. The reason for the sign is that we are pairing $l \in \LCt$ with $-l \in 2\ZZ^s+b$ when extracting the constant coefficient. Note that while \(\LCt\) is symmetric about the origin, $2\ZZ^s+b$ in general is not. \begin{lemma}\label{description-c} The set $\LCt$ consists of vectors whose components satisfy the following conditions: \[\LCt = \left\{(l_1,\dots,l_{s_1}, 0, \dots 0, m_1, \dots, m_{s_3}) \in \ZZ^s \;\middle\vert\; \begin{aligned} &l_i = \pm 1 \\ &m_i \equiv \delta_i \Mod{2} \\ &\lvert m_i \rvert \geq \delta_i-2 \end{aligned} \right\}. \] \end{lemma} \begin{proof} The expansion for 1-vertices is simply \(z_i-z_i^{-1}\), giving the entries \(l_i = \pm 1\). The variables corresponding to 2-vertices are absent in \ref{eq:rat_function_expn}. For a vertex \(v_i\) of degree \(\delta_{i} \geq 3\), put \(d := \delta_{i}-2 \geq 1\). We have the following symmetric expansion: \begin{align} 2 \cdot \se (z-z^{-1})^{-d} &= \underset{z \to \infty}{\expn} \frac{z^{-d}}{(1-z^{-2})^{d}} + \underset{z \to 0}{\expn} \frac{z^{d}}{(z^2-1)^{d}}\\ &= z^{-d} \sum_{k=0}^\infty \binom{k-1+d}{k}z^{-2k} + (-1)^d z^{d} \sum_{k=0}^\infty \binom{k-1+d}{k} z^{2k}\\ &= (z^{-d} + d z^{-d-2} + \dots) + (-1)^{d} (z^{d} + d z^{d+2} +\dots) \end{align} It follows that the corresponding entry \(m_i\) must have the same parity as \(\delta_i\) and \(\lvert m_i \rvert \geq d = \delta_i-2\). \end{proof} \subsection{Cancellations}\label{ss:cancel} By the previous section, the $q$-series \(\Zhat_b(Y,q)\) can then be expressed as a sum over $\LCt_b$: \begin{equation}\label{eq:before_cancel} \Zhat_b(Y,q) = q^{\frac{-3s - \Tr(M)}{4}} \sum_{l \in \LCt_b} \ct_l q^{\frac{-l^2}{4}}. \end{equation} The $q$-exponents of $\Zhat_b(q)$ are therefore given by $( -3s - \Tr(M) -l^2)/4$ for those $l \in \LCt_b$ whose contribution does not cancel out in the sum above, in other words: \begin{equation}\label{eq:cancel} c_l:= \sum_{\substack{l' \in \LCt_b\\l'^2 = l^2}} \ct_{l'} \neq 0. \end{equation} However, $c_l$ can vanish even if there are more nonzero terms $\tilde{c}_l$ is \eqref{eq:cancel}. We refer to this phenomenon as `cancellations' -- see Examples \ref{ex:cancel_H} and \ref{ex:cancel_seifert}. Motivated by this, we define \[ \LC_b := \{l \mid l \in \LCt_b; l \text{ satisfies \eqref{eq:cancel}} \} \subseteq \LCt_b. \] It follows that $\Delta_b$ is determined by minimizing $-l^2$ over the set $\LC_b$. We refer to the elements $l$ in $C_b$ for which $-l^2$ is minimal, as \emph{minimizing vectors}. \begin{lemma}\label{lm:delta_minimizer} \begin{equation} \Delta_b(Y) = \frac14 \left(-3s - \Tr(M) + \min_{l \in \LC_b} \{ -l^2 \}\right). \end{equation} \end{lemma} \section{$\Delta$ for Seifert Manifolds} In this section, we will prove \cref{thm:seifert} which gives an explicit formula for $\Delta_{can}(Y)$ of a Seifert manifold $Y$ equipped with the canonical $\spc$ structure $can$. \subsection{Seifert manifolds} Seifert manifold $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$, fibered over $S^2$, is given by an integer $b_0$ and tuples of integers $0<\omega_i<a_i$ for $1 \leq i \leq n$. It can be represented by a star-shaped plumbing as shown in \cref{fig:seifert}. The graph consists of a central vertex with label $-b_0$ and $n$ `strings'. The labels $-b_{j_1},-b_{j_2},\dots, -b_{j_{s_j}}$ on the $j$-th string are determined by Hirzebruch--Jung continued fraction \[\frac{a_j}{\omega_j} = [b_{j_1}, \dots, b_{j_{s_j}}] =b_{j_1} - \cfrac{1}{b_{j_2} - \cfrac{1}{\cdots - \cfrac{1}{b_{j_{s_j}}}}}. \] \begin{figure} \centering \includestandalone{graphics/seifert/new_seifert_plumbing} \caption{Plumbing graph of a Seifert manifold.} \label{fig:seifert} \end{figure} The intersection form $M$ is negative definite if and only if the orbifold Euler number $e = -b_0 + \sum_i \omega_i/a_i$ is negative. We have the following formula for $\gamma(Y)$ of a Seifert manifold $Y$ \cite[p. 296]{NeNi02}, a consequence of \cref{base-k2-s}: \begin{equation}\label{eq:formula_gamma} \gamma(Y) = \frac{1}{e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 + e + 5 + 12 \sum_{i=1}^n \ds(\omega_i, a_i). \end{equation} Here $\ds$ denotes the Dedekind sum. \subsection{\texorpdfstring{$\Delta_{can}$}{Delta} of Seifert manifolds} \begin{theorem}\label{thm:seifert} Let $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ be a negative definite Seifert manifold. Then $\Delta_{can}$ of the canonical $\spc$ structure satisfies \begin{equation}\label{eq:seifert} \Delta_{can} = -\frac{\gamma(Y)}{4} + \frac{1}{2}. \end{equation} If $Y$ is not a lens space, then $\Delta_{can}$ is minimal among all ${\Delta_b, b \in \spc(Y)}$. \end{theorem} \begin{proof} We will treat the lens spaces separately in \cref{ssec:lens}, so we may assume that $n \geq 3$ and \(a_i \geq 2\) for each \(i\). Denote $A = \prod_{i=1}^n a_i$. We will first show that over the set $\LCt$ from \cref{description-c}, \(-l^2\) is minimized by exactly be vectors $\pm(2u-\delta)$. From that, it will follow that $2u-\delta \in \LC_{can}$ is a true minimizing vector in the sense of \cref{ss:cancel}. The formula \eqref{eq:seifert} for $\Delta_{can}$ then follows from \cref{base-k2-s}. The set $\LCt$ consists of the vectors \[ l = (l_1, l_2, \dots, l_n, 0, \dots, 0, m) \] where $l_i = \pm 1$ and $m \equiv n \Mod{2}$ and $\lvert m \rvert \geq n-2$ by \cref{description-c}. The quadratic form \(l^2\) can be expressed using Seifert data as follows: \begin{equation}\label{eq:quadratic} -l^2 = m^2 A + \sum_{i=1}^n l_i m \frac{A}{a_i} + \sum_{i \neq j}^n l_i l_j \frac{A}{a_ia_j} - \sum_{i=1}^n M^{-1}_{ii}. \end{equation} By the symmetry \(l^2 = (-l)^2\), we may assume that $m \geq n-2$. Taking the derivative with respect to \(m\), we obtain \[ \frac{1}{A} \frac{\partial}{\partial m}\left(-l^2\right) = 2m + \sum_{i=1}^n \frac{l_i}{a_i} \geq 2(n-2) - \frac{n}{2} > 0, \] so the minimum is attained when \(\lvert m \rvert = n-2.\) Similarly, pick \(j \in 1,\dots, s_1.\) Let \(l^+, l^-\) be two vectors with \(m = n-2\) that only differ by having \(l_j^+=1\), \(l^-_j=-1\), respectively. We have \begin{align} l_+^2 - l_-^2 & = \frac{2A}{a_j} (n-2 + \sum_{i \neq j} \frac{l_i}{a_i}) \\ & \geq \frac{2A}{a_j} (n-2 - \frac{n-1}{2}) \geq 0. \end{align} It follows that \[ l = (-1,\dots, -1, 0, \dots, 0, n-2) = \delta - 2u. \] minimizes \eqref{eq:quadratic}. The argument also implies that the only other vector giving the same value of $l^2$ is $-l=2u-\delta$ which represents the canonical $\spc$ structure $can$. These two vectors have the same coefficients in the expansion of \eqref{eq:rat_function}: \[ c_l = c_{-l} = \frac12\operatorname{sgn}(n-2)^n \binom{\frac{n+\lvert n-2 \rvert}{2}-2}{n-3}. \] Therefore even in the case that $\pm l$ belong to the same $\spc$ structure, i.e. $can$ is $\spin$, their contributions do not cancel in $\Zhat_{can}(q)$ and consequently $-l \in \LC_{can}$. Thus the minimum of \(-l^2\) over \(\LC_{can}\) is given by $-(2u-\delta)^2$ and we have \begin{equation}\label{delta-cleaning-up} \Delta = \frac{-3s - \Tr(M)}{4} - \frac{(2u-\delta)^2}{4} = - \frac{\gamma(Y)}{4} + \frac{1}{2} \end{equation} The last equality follows the formula for \(\gamma(Y)\) in \cref{base-k2-s}. \end{proof} \subsection{Lens spaces}\label{ssec:lens} In this section, we compute $\Delta_b$ of lens spaces. Let \(Y= L(p,r) \) be a lens space with \(p>r>0\). Denote by \(g\) a generator of \(H_1(Y,\ZZ) \cong \ZZ/p\ZZ\) and $can$ the canonical \(\spc\) structure. We have the following formula for \(\Zhat_b(q)\) of lens spaces: \[ \sum_{i=0}^{p-1} \Zhat_{g^i can}(q)g^i = q^{3\ds(r,p)} \left((g^{-r-1}+1) q^{1/2p} - (g^{-r}+g^{-1}) q^{-1/2p}\right) \] Here \(\ds(r,p)\) denotes the Dedekind sum. From the formula, we read off the four finite \(\Delta_b\) invariants. \begin{equation}\label{eq:delta_lens} \Delta_{g^{-r-1}can} = \Delta_{can} = 3\ds(r,p) + \frac{1}{2p}, \quad \Delta_{g^{-r}can} = \Delta_{g^{-1}can} = 3\ds(r,p) - \frac{1}{2p}. \end{equation} \noindent \(\gamma(Y)\) can be described using Dedekind sums in the following manner \cite[p. 304]{NeNi02}: \[ \gamma(Y) = 2 - \frac{2}{p} - 12\ds(p,r). \] Comparing with \eqref{eq:delta_lens}, we obtain that \(\Delta_{can}\) satisfies the same formula as for the other Seifert manifolds: \[ \Delta_{can} = -\frac{\gamma(Y)}{4} + \frac12. \] Note that \eqref{eq:delta_lens} shows that for lens spaces, $\Delta_{can}$ is not minimal among all $\Delta_b$. See also \cref{ex:cancel_H}. \begin{remark} We originally proved \cref{thm:seifert} using the reduction theorem \cite[Thm. 4.2]{GKS23} which may be used to compute all $q$-exponents of $\Zhat_b(q)$ of Seifert manifolds. Later, we found a simpler argument presented here, which focuses on the smallest exponent. It also emphasizes the role of the vector $2u-\delta$, making the presence of the invariant $\gamma(Y)$ more transparent. \end{remark} \begin{remark} As we have seen above, \(\Delta_b\) invariants are often infinite. In \cite{GPPV20}, the authors conjectured, based on Physics considerations, the existence of a categorification of \(\Zhat_b(q)\), i.e. a doubly-graded cohomology theory \(\mathcal{H}_b^{i,j}(Y)\) whose graded Euler characteristic is \(\Zhat_b(q)\): \[ \Zhat_b(q) = 2^{-s} q^{\Delta_b} \sum_{i, j \in \ZZ} q^i (-1)^j \dim \mathcal{H}_b^{i,j}(Y). \] Smaller \(q\)-exponents than \(\Delta_b\) could appear in the corresponding two-variable generating series \(q^{\Delta_b}\sum_{i,j} q^i t^j \dim \mathcal{H}_b^{i,j}(Y)\). In \cite{GPV16}, a Poincar\'e series of this sort was defined for lens spaces \(L(p,1)\). Its minimal \(q\)-power is finite for all \(\spc\) structures, in contrast with \(\Delta_b\). \end{remark} \section{Beyond Seifert manifolds}\label{s:H_shaped} In the proof of \cref{thm:seifert} giving the formula for $\Delta_{can}$ of Seifert manifolds, we used a special form \eqref{eq:quadratic} of the quadratic from $l \mapsto -l^2$ following from the fact that Seifert manifolds admit a star-shaped graph. For more general graphs, we need a generalization of \eqref{eq:quadratic}. This is realized by some properties of \emph{splice diagrams}, which are certain weighted graphs built from plumbing graphs. We will illustrate this method on plumbing graphs with exactly two vertices of degree 3 and no vertices of degree 4 or more. The corresponding splice diagrams are ``H-shaped'' graphs with 6 vertices, as in \cref{fig:H_splice}. In general, there is no uniform choice of minimizing vector as was the vector $2u-\delta$ for $\Delta_{can}$ of Seifert manifolds. Therefore we cannot hope for a simple universal formula for $\Delta_{can}$ as in \cref{thm:seifert}. Nevertheless, the minimizing vectors keep a specific form in certain regions given by the relative size of the weights in splice diagram. On the boundaries of these regions, we may see multiple minimizing vectors. In principle, one can divide the study into those particular cases. Some of these can be effectively reduced to the case of Seifert manifolds, as we will illustrate in \cref{ss:computation_h_shaped}. The techniques described here can be used for more general plumbings, but the number of cases grows significantly with the complexity of the splice diagram. \subsection{Splice diagrams} Following \cite{NW05a} (see also \cite{SavelievBook}), given a plumbed manifold $Y$ with plumbing graph $\Gamma$, we construct a splice diagram $\Omega$ of $Y$ as follows: $\Omega$ is a tree obtained by replacing each maximal string in $\Gamma$ (a simple path in $\Gamma$ whose interior is open in $\Gamma$) by a single edge. Thus $\Omega$ is homeomorphic to $\Gamma$ but has no vertices of degree two. We identify the vertices of $\Omega$ with the corresponding vertices of $\Gamma$. At each vertex \(v\) of $\Omega$ of degree \(\geq 3\), we assign a weight $w_{v\varepsilon}$ on an incident edge $\varepsilon$ as follows. The edge $\varepsilon$ in $\Omega$ corresponds to a string in $\Gamma$ starting in $v$ with some edge $e$. Let $\Gamma_{ve}$ be the subgraph of $\Gamma$ cut off by the edge of $\Gamma$ at $v$ in the direction of $e$, as in the following picture. \begin{center} \includestandalone{graphics/H_shaped/explanation_subgraph} \end{center} The corresponding weight \(w_{v\varepsilon}\) is given by the determinant of \(-M(\Gamma_{ve})\), where \(M(\Gamma_{ve})\) denotes the intersection matrix of $\Gamma_{ve}$. We draw the weight \(w_{v \varepsilon}\) on the edge \(\varepsilon\) near \(v\). For example, the following is a plumbing graph and its splice diagram: \begin{center} \includestandalone{graphics/H_shaped/explanation_splice} \end{center} If $Y(\Gamma)$ is a homology sphere, the splice diagram uniquely determines the plumbing graph, see \cite{NW02}. Although $\Gamma$ and $\Omega$ are essentially equivalent in this case, $\Omega$ is much smaller. More importantly, it is the relative size of the weights of $\Omega$, which directly influences the minimizing vectors. For example, if one weights is significantly larger than the others, the minimizing vectors must be of some specific form, see \cref{ss:computation_h_shaped}. The values of the quadratic form $l \mapsto -l^2=-l^T M^{-1}l$ can be computed from the splice diagram as follows: For two vertices \(v\), \(v'\) of the splice diagram, consider the shortest path $P$ connecting them. Let \(N_{vv'}\) be the product of all weights adjacent to vertices of $P$, but not lying on $P$. \begin{center} \includestandalone{graphics/H_shaped/explanation_path} \end{center} \begin{theorem}[{\cite[Thm. 12.2]{NW05a}}]\label{theorem:neumann} With the notation above, we have \[ M^{-1}_{vv'} = -\frac{N_{vv'}}{\det M}. \] \end{theorem} Recall that the vectors $l \in \Z^s$ that contribute to $\Zhat_b(q)$ lie in the set $\LCt$ from \cref{description-c}. If $l \in \LCt$, we have $l_v=0$ if $v$ is of degree 2 and $l_v^2 = (\pm 1)^2=1$ if $v$ is of degree 1. We obtain the following expression for $-l^2$ generalizing \eqref{eq:quadratic}: \begin{equation}\label{eq:quadr_form_restricted} -l^2 = \sum_{\substack{v \neq v'\\ \delta_v,\delta_{v'} \neq 2}} M^{-1}_{vv'} l_v l_{v'} + \sum_{\delta_v \geq 3} M^{-1}_{vv} l_v^2 + \sum_{\delta_v =1} M^{-1}_{vv} \end{equation} Note that the coefficients in the first and second sum can be expressed using the weights of the splice diagram $\Omega$, up to the multiplication by $\det M$. On the other hand, the third sum is not expressed in terms of weights of $\Omega$ in a simple way, but it is independent of $l$ (whenever $l \in \LCt$), so it does not influence which vectors $l \in \LCt$ have the minimal value of $-l^2$. \subsection{\texorpdfstring{$\Delta$}{Delta} for homology spheres with $H$-shaped splice diagrams} \label{ss:computation_h_shaped} We now consider a plumbing graph $\Gamma$ with exactly two vertices of degree 3 and no vertices of higher degree, e.g. the graph in \cref{fig:plumbing_from_splice}. For simplicity, we assume that the associated plumbed manifold $Y$ is a homology sphere. The associated splice diagram $\Omega$ is an `H-shaped' graph with six vertices, see \cref{fig:H_splice}. Its six weights are denoted \(a_1,a_2,a_3\) and \(a'_1,a'_2,a'_3\). They are pairwise coprime integers, which we further assume to be $\geq 2$. In this case, $Y$ can be realized as splicing of Brieskorn spheres $\Sigma(a_1,a_2,a_3)$ and $\Sigma(a'_1,a'_2,a'_3)$ along their third singular fibers. We consider the projection $\ZZ^s \to \ZZ^6$, denoted by $l \mapsto \bar{l}$, which removes the components corresponding to the vertices of degree 2. We order the components of $\bar{l} = (x_1,x_2,x_3,x'_1,x'_2,x'_3)$ as in \cref{fig:H_splice} and keep the ordering throughout this section. \begin{figure}[ht] \centering \includestandalone{graphics/H_shaped/splice_H_shaped} \caption{H-shaped splice diagram $\Omega$} \label{fig:H_splice} \end{figure} The quadratic form $l \mapsto -l^2$, restricted to the set $\LCt$ from \cref{description-c}, can be expressed in terms of the weights of $\Omega$ as: \begin{equation}\label{eq:form_H_shaped} \begin{split} -l^2 &= x_3^2 a_1 a_2 a_3 + x_1 x_3 a_2 a_3 + x_2 x_3 a_1 a_3 + x_1 x_2 a_3 + (\dots)' + \\ & + x_1 x'_2 a_2 a'_1 + x_1' x_3 a_1 a_2 a'_2 + x'_2 x_3 a_1 a_2 a'_1 + (\dots)' + \\ & + x_1 x'_1 a_2 a'_2 + x_2 x'_2 a_1 a'_1 + x_3x'_3a_1a_2a'_1a'_2 + C. \end{split} \end{equation} Here $(\dots)'$ means that we repeat the terms on each line with the usual and dashed variables reversed. \(C\) is constant on the set $\LCt$ so it does not influence the minimizing vectors for $\Delta_{can}$. The general strategy to identify $\Delta_{can}$ can be described as follows: We know that $x_1,x_2,x'_1,x'_2 \in \{\pm1\}$ because $l \in \LCt$. For each of the $2^4$ possibilities of the signs, the form above reduces to a quadratic form in variables $x_3$, and $x_3'$. We can then minimize these forms over odd integers, using standard optimization methods. Finally, we must check that the resulting vectors do not cancel out, as in \cref{ss:cancel}, so they are true minimizing vectors. Clearly, the form of minimizing vectors depends on the relative size of the coefficients \(a_i, a'_i\). We describe in greater detail the case when \(a_3\) is very large compared to other \(a_i\) and \(a'_i\). This allows us to effectively reduce the minimizing problem to the Seifert case. If $a_3 \gg a_1,a_2,a'_1,a'_2,a'_3$, the substantial terms are those containing \(a_3\): \begin{equation}\label{eq:half_graph} x^2_3 a_1 a_2 a_3 + x_1 x_3 a_2 a_3 + x_2 x_3 a_1 a_3 + x_1 x_2 a_3 \end{equation} Any vector $l \in \LCt$ with the minimal value of $-l^2$ also minimizes the expression \eqref{eq:half_graph}. As this is (almost) a quadratic form of a star-shaped graph, we can repeat the first part of the argument in the proof of \cref{thm:seifert}. Namely, assuming that $a_i \geq 2$ and taking the $x_3$-derivative, we obtain that \(x_3=\pm 1\) in any minimizing vector, say $x_3=1$. The signs of $x_1$ and $x_2$ are easily determined by the relative size of $a_1$ and $a_2$. This consideration `freezes' the left-hand side of the graph and for each choice of $x'_1 = \pm 1$ and $x'_2 = \pm 1$, we are left with a quadratic function of a single variable $x'_3$. Explicitly, in the case $x_1=x_2=-1$, the minimum in the variable $x'_3$ (over $\R$) is given by \begin{equation} \begin{split} x_3' &= \frac{x'_1 a'_2 a'_3 + x'_2 a'_1 a'_3 - a_1 a_2 a'_2 + a_1 a_2 a'_1 a'_2 - a_1 a_2 a'_1}{2a'_1 a'_2 a'_3}\\ &= \frac12 \left(\frac{x_1'}{a'_1}+\frac{x_2'}{a'_2}-\frac{a_1 a_2}{a'_1 a'_3}+\frac{a_1 a_2}{a'_2 a'_3}-\frac{a_1 a_2}{a'_3}\right). \end{split} \end{equation} This computation illustrates that the value of $x'_3$ can be arbitrarily large. This shows that the minimizing vector is very different from the vector $2u-\delta$ which was minimizing in the case of Seifert manifolds. In a similar way, we could analyze other cases of the relative size of the weights, giving (at least approximate) `explicit formulas' for $\Delta_{can}$. However, in practice, it is easier to use a computer to search for the minimizing vectors. We illustrate the variability of possible minimizing vectors and possible values of $\Delta_{can}$ on several examples in this and the following section. \begin{example}\label{ex:original} Consider the integral homology sphere $Y$ associated with the splice diagram \begin{center} \includestandalone{graphics/H_shaped/example_splice_big} \end{center} Then $\Delta(Y) = \frac{3045}{1000}$, with minimizing vectors $\pm(-1,-1,3,1,1,-1).$ $\Delta(Y)$ is strictly smaller than $-\frac{\gamma(Y)}{4} + \frac{1}{2} = \frac{3885}{1000}$ given by the vector $2u-\delta=(-1,-1,1,-1,-1,1)$. The corresponding plumbing graph on $28$ vertices is shown in Figure \ref{fig:plumbing_from_splice}. \begin{figure}[ht] \centering \resizebox{\columnwidth}{!}{ \includestandalone{graphics/H_shaped/example_plumbing_from_splice} } \caption{The plumbing graph associated to the splice diagram in Example \ref{ex:original}} \label{fig:plumbing_from_splice} \end{figure} \end{example} \begin{example} Consider the manifold $Y$ given by the following plumbing: \begin{center} \includestandalone{graphics/H_shaped/example_H_shaped/plumbing_H} \end{center} Then $Y$ is an integral homology sphere with $\Delta(Y) = \frac{1}{2}$ and the minimizing vectors are $\{\pm v, \pm w \}$, where $v=(1, 1, -1, -1, 1, 1)$, $w = (1, 1, -1, 1, -1, 1)$. Again, $\Delta(Y) < - \frac{\gamma(Y)}{4} + \frac{1}{2} = \frac{5}{2}$. Note that this plumbing corresponds to the following splice diagram: \begin{center} \includestandalone{graphics/H_shaped/example_H_shaped/splice_H} \end{center} \end{example} \section{Upper Bounds and Cancellations} In this section, we focus on upper bounds for $\Delta_b$. In principle, bounding $\Delta_b$ from above should be easy---the (shifted) norm $(-3s - \Tr(M) -l^2)/4$ of any element $l \in \mathcal{C}_b$ gives an upper bound. However, when finding elements of $\LC_b$, we are facing two issues. First of all, $\LCt_b$ can be empty, e.g. for some $\spc$ structures on lens spaces, giving $\Zhat_b(q)=0$ and $\Delta_b = \infty$. Secondly, even if $\LCt_b$ is non-empty, there may be drastic cancellations of the coefficients, as we will illustrate on the Seifert manifold in \cref{ex:cancel_seifert}. This prevents us from establishing general results in this direction. In particular, the vector $2u-\delta$ does not always give an upper bound for $\Delta_{can}$, unlike for Seifert manifolds (where it was optimal), see \cref{ex:cancel_H}. We believe that those cancellations are rather special, being related to some additional symmetry of the plumbing graph. In particular, it would be rather surprising if they occurred for all $\spc$ structures at once. Therefore, we expect the following: \begin{conjecture}\label{conj:ineq} For a negative-definite plumbed manifold $Y$ we have \[\min_{b \in \spc{Y}} \Delta_b \leq -\frac{\gamma(Y)}{4}+\frac12.\] \end{conjecture} We now proceed with several examples of manifolds for which the cancellations occur. We also compare $\Zhat(q)$ with the two-variable extension $\Zhathat$ defined in \cite{AJK23}. The new variable \(t\) can distinguish vectors $l$ with the same value of $l^2$, removing some cancellations, but not all of them, see \cref{ex:t_cancel_0}. \begin{example}\label{ex:cancel_H} Consider the following plumbing: \begin{center} \includestandalone{graphics/H_shaped/example_cancel/plumbing_cancel} \end{center} The resulting plumbed manifold $Y$ admits $20$ $\spc$ structures. For the canonical $\spc$ structure $can$ we have $\Delta_{can} = \frac{33}{20}$ with the (sole) minimizing vector $(1, -1, 1, -1, -1, 1)$. This is strictly larger than $-\frac{\gamma(Y)}{4} + \frac12 = \frac{13}{20}$. The only vectors within $\LCt_{can}$ that satisfy $\frac{1}{4}(-3s - \operatorname{Tr}(M) - l^2) = \frac{13}{20}$ are $l_1 = (1, 1, -1, 1, 1, -1)$ and $l_2 = (-1, -1, 1, 1, 1, -3)$. However their coefficients are $c_{l_1} = \frac{1}{4} = -c_{l_2}$ and so they cancel out in $\Zhat_{can}(q)$. The above cancellation does not happen for $\Zhathat_{can}(q,t)$: \[ \Zhathat_{can}(q,t) = -\frac{1}{4}\left((t^{-1} -t)q^\frac{13}{20} +q^\frac{33}{20} -t^{-1}q^\frac{53}{20} + (t^{-2} + t^2)q^\frac{73}{20} - t^{-1}q^\frac{93}{20} +\cdots\right). \] \end{example} \begin{example}\label{ex:cancel_seifert} Consider Seifert manifold \(M(2;(3,1),(3,2),(3,2))\). It is described by the following negative definite plumbing: \begin{center} \includestandalone{graphics/seifert/plumbingM2_13_23_23} \end{center} Let \(b\) be the \(\spc\) structure with the associated vector \(\delta = (1,2,3,2,1,1) \in 2\ZZ^s+\delta\). The $q$-series $\Zhat_b(q)$ is a single monomial: \[ \Zhat_b(q) = \frac12 q^{-5/6} \] This is due to the following cancellation: All vectors but one in the set $\mathcal{C}$ can be split into pairs \(l_1,l_2\) satisfying \(l_1^2=l_2^2\) and \(c_{l_1} = -c_{l_2}\). Similarly to the previous example, $\Zhathat_b(q,t)$ removes the cancellations and it gives an infinite series: \[ \Zhathat_b(q,t) = \frac12 ( q^{-5/6} t^{-1} + q^{13/6} (1-t^{-2}) + q^{67/6} (t-t^{-3}) + q^{157/6} (t^2 - t^{-4}) \dots). \] \end{example} \begin{example}\label{ex:t_cancel_0} Consider the following plumbing: \begin{center} \includestandalone{graphics/H_shaped/example_cancel/plumbing_t_cancel} \end{center} The resulting plumbed manifold $Y$ admits $21$ $\spc$ structures. For the canonical $\spc$ structure $can$ represented by $2u - \delta = ( 1, 1, -1, 1, 1, -1)$ we have that $\Delta_{can}' = \frac{5}{2}$ and the minimizing vectors for the quadratic form that produces $\Delta'_{can}$ are given by $l_1 = (-1, -1, -3, 1, 1, 1)$, $l_2 = (-1, -1, 7, -1, -1, -1)$, $l_3 = (1, 1, -7, 1, 1, 1)$ and $l_4 = (1, 1, 3, -1, -1, -1)$ with coefficients $c_{l_1} = c_{l_3}= -\frac{1}{4}t^{-1}$ and $c_{l_2} = c_{l_4 }= -\frac{1}{4}t$. This is strictly larger than $-\frac{\gamma(Y)}{4} + \frac12 = \frac{1}{2}$. For this manifold, the only vectors within $\tilde{\mathcal{C}}_{can}$ which satisfy $\frac{1}{4}(-3s - \operatorname{Tr}(M) - l^2) = \frac{1}{2}$ are $l_1 = (-1, -1, 1, -1, -1, 1)$, $l_2 = (-1, -1, 3, 1, 1, -1)$, $l_3 = (1, 1, -1, 1, 1, -1)$ and $l_4 = (1, 1, -3, -1, -1, 1)$. However their coefficients are $c_{l_1} = \frac{1}{4}t^{-1} = -c_{l_4}$ and $c_{l_2} = -\frac{1}{4}t = -c_{l_3}$ and so they cancel out in $\Zhathat_{can}(q,t)$. The entire $(t, q)$ series is given by: \[ \Zhathat_{can}(q,t) = -\frac{1}{4}\left((2t^{-1} + 2t)q^\frac{5}{2} + (2t^{-3} +2t^3)q^\frac{9}{2} + (-t^{-4} + t^{-2} +t^2 -t^4)q^\frac{15}{2} + \cdots\right) \] This example shows that cancellations do occur even for $\Zhathat_{can}(q,t)$ and as a result $\Delta_{can}' > -\frac{\gamma(Y)}{4} + \frac{1}{2}$ where $\Delta_{can}'$ denotes the smallest $q$-exponent of $\Zhathat_{can}(q,t)$. \end{example} \section{Comparison with correction terms} In the last section, we compare $\Delta_b(Y)$ with correction terms $d_b(Y)=d(Y,[b])$ in Heegaard--Floer homology\footnote{Again, we omit the brackets for $\spc$ structures.}. We include this discussion because there were some expectations that $\Delta_b(Y)$ and $d_b(Y)$ might be related \cite{GPP21, AJK23}. In the Seifert case, we have an explicit formula for $\Delta_{can}(Y)$ in terms of the $\gamma(Y)$ invariant by \cref{thm:seifert}. We can then use elementary bounds for Dedekind sums to obtain estimates on $\Delta_{can}(Y)$. Finally, we compare $\Delta_{can}(Y)$ to $d_{can}(Y)$ for some classes of Brieskorn spheres, where $d_{can}(Y)$ is known, finding that they are very different. \subsection{Quadratic forms} Correction terms can be expressed as minimizers of a quadratic form over the characteristic vectors: \begin{theorem}[{\cite{Nem05}}]\label{thm:cor_terms} For an almost rational graph $\Gamma$, the correction terms are given by \[ d_k(Y) = \max_{k' \in [k]} \frac{(k')^2+s}{4} \] \end{theorem} Note that this formula holds in many other cases and was conjectured by the second author to be true for any negative definite rational homology sphere \cite{Nemethi_lattice08}. To compare with $\Delta$, we need to shift to our conventions on $\spc$ structures, so we set $l = k' - Mu$. Then we have \begin{align} \frac{(k')^2+s}{4} &= \frac{(l + Mu)^T M^{-1}(l + Mu) + s}{4}\\ &= \left(\frac{\Tr(M) +3s}{4} + \frac{l^2}{4} \right) + \frac{l^T u}{2} - \frac12 \end{align*} We see that the minimized quadratic forms for $d$ and $\Delta$ differ in the linear term $l^T u/2= (\sum_{i=1}^s l_i)/2$. \begin{remark} In \cite{GPP21}, the authors observed that in the setting of \cref{thm:cor_terms}, the difference between $d_b(Y)$ and $\Delta_b(Y)-\frac12$ is an integer. For Seifert manifold $Y$ with the canonical $\spc$ structure, this can be explained as follows: By \cite[Thm. 8.3.]{Nem08}, $d_{can}(Y)$ and $\gamma(Y)$ (and therefore $\Delta_{can}(Y)$) satisfy the following relation: \begin{equation} d_{can}(Y)=\frac{\gamma(Y)}{4} - 2 \chi_{can} \left( = -\Delta_{can} + \frac12 - 2 \chi_{can} \right). \end{equation} Here $\chi_{can}$ is the holomorphic Euler characteristic of a certain holomorphic line bundle on the corresponding quasihomogeneous singularity. In particular, it is an integer. \end{remark} \subsubsection{Lower Bound for Seifert Manifolds} We end this section with some elementary estimates considering the $\gamma(Y)$ invariant of Seifert manifolds, and hence of $\Delta_{can}(Y)$. The formula \eqref{eq:formula_gamma} for $\gamma(Y)$ contains Dedekind sums. We have the following well-known inequality for $p>0$, $a \in \ZZ$: \begin{equation} -\mathbf{s}(1, p) \leq \mathbf{s}(a, p) \leq \mathbf{s}(1, p) = \frac{p}{12} + \frac{1}{6p} - \frac{1}{4}. \end{equation} Combining this with \eqref{eq:formula_gamma}, we obtain the following result. \begin{proposition}\label{lower-bound-delta-general-seifert} For the Seifert manifold $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ and the canonical $\spc$ structure, we have \[ \Delta_{can} \geq - \frac{1}{4e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 - \frac{e+3(n+1)}{4} + \sum_{i=1}^n \frac{a_i}{4} + \frac{1}{2a_i} \] and \[ \Delta_{can} \leq -\frac{1}{4e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 -\frac{e+3(1-n)}{4} - \sum_{i=1}^n \frac{a_i}{4} + \frac{1}{2a_i} \] where $e$ is the orbifold Euler number of $Y$. \end{proposition} If $Y$ is a Brieskorn sphere $\Sigma(a_1, \dots, a_n)$, then $e = -\prod_{i=1}^n a_i^{-1}=-A^{-1}$. In particular, if $a_i \gg 1$, the leading term of $\Delta$ is given by $(n-2)^2A$. We obtain that the $\Delta$ invariant grows polynomially with $a_i$ for large values of $a_i$. \subsubsection{Brieskorn spheres} We illustrate the difference between $\Delta = \Delta_{can}$ and $d = d_{can}$ for some families of Brieskorn spheres, for which correction terms are explicitly known \cite{BorNem2011}. For $p, q > 0$ set $\rho = (p-1)(q-1)/2$. Then for $\Sigma(p, q, p q+1)= S^3_{-1}(T_{p, q})$ we have $\gamma(Y) = -4\rho(\rho-1)$. From \cref{thm:seifert} we immediately obtain the next result: \begin{corollary} For $Y = \Sigma(p, q, pq+1)$, \begin{equation}\label{eq:Delta-p-p+1} \Delta(Y) = \rho(\rho-1) + \frac{1}{2}. \end{equation} \end{corollary} In contrast to this, the correction term vanishes, so that $\Delta(Y) \gg d(Y) = 0$. For $Y = \Sigma(p, p+1, p(p+1)-1)$, we have \begin{equation}\label{eq:d} d(Y) = \bigg\lfloor \frac{p}{2} \bigg\rfloor\left(\bigg\lfloor \frac{p}{2} \bigg\rfloor+1\right).\end{equation} From \cref{lower-bound-delta-general-seifert} we obtain, after some manipulations, that for $p>2$ \begin{equation}\label{inequality-delta-d-comparison} \Delta(Y) \geq \frac{1}{4}\left(p^4 + 2p^3-5p \right)-3 > d(Y). \end{equation} The case of $p = 2$ correspond to Poincar\'e sphere $Y = \Sigma(2, 3, 5)$, where we have $\Delta(Y) = -\frac{3}{2} < d(Y) = 2$. This seems to be a boundary case and in general, we expect the following conjecture: \begin{conjecture} For all but finitely many Seifert manifolds $Y$ we have \[ d_{can}(Y) < \Delta_{can}(Y). \] \end{conjecture} \printbibliography \end{document} \documentclass[tikz]{standalone} \begin{document} \begin{tikzpicture} \begin{scope}[every node/.style={circle,fill,draw,inner sep=1.5pt}] \node (1) at (-2,0.5) {}; \node (2) at (-2,-0.5) {}; \node (3) at (-1,0) {}; \node (4) at (1,0) {}; \node (5) at (2,0.5) {}; \node (6) at (2,-0.5) {}; \end{scope} \node[xshift=-.3cm] at (2) {$v$}; \node[yshift=.3cm] at (4) {$v'$}; \node[xshift=.3cm, yshift=-.3cm] at (3) {$P$}; \begin{scope}[every edge/.style ={draw}, every node/.style ={fill=white,inner sep=0pt,scale=.9}] \draw (1) -- node[near end] {$2$} (3); \draw[thick] (2) -- (3); \draw[thick] (3) -- (4); \draw (4) -- node[near start] {$2$} (5); \draw (4) -- node[near start] {$2$} (6); \end{scope} \node[xshift=2cm] at (4) {$N_{vv'}=8$}; \end{tikzpicture} \end{document} \documentclass[tikz]{standalone} \begin{document} \begin{tikzpicture} \begin{scope}[every node/.style={circle,fill,draw,inner sep=1.5pt}] \node (1) at (-2,0.5) {}; 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2412.02123v3	http://arxiv.org/abs/2412.02123v3	Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios	\documentclass[11pt]{amsart} \usepackage{amsfonts,amsthm,amsmath,enumerate,amssymb,latexsym,color,tcolorbox,tikz,mathrsfs,bm,subfig,tcolorbox} \usetikzlibrary{patterns,patterns.meta,bending,angles,quotes,shapes.geometric} \usepackage[text={6in,9in},centering]{geometry} \usepackage[backref=page, colorlinks, linkcolor=blue, anchorcolor=red, citecolor=red]{hyperref} \renewcommand{\baselinestretch}{1.2} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{fact}{Fact} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \DeclareMathOperator{\dimh}{dim_H} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\inte}{int} \DeclareMathOperator{\por}{por} \DeclareMathOperator{\dimb}{dim_B} \DeclareMathOperator{\diam}{diam} \newcommand{\blue}{\color{blue}} \newcommand{\red}{\color{red}} \newcommand{\purple}{\color{purple}} \newcommand{\rd}{\,\mathrm{d}} \def\R{\mathbb R} \def\Z{\mathbb Z} \def\D{\mathcal D} \def\A{\mathcal A} \def\C{\mathcal C} \def\L{\mathcal L} \def\i{\bm i} \def\j{\bm j} \def\k{\bm k} \def\bi{\mathbf i} \def\bj{\mathbf j} \def\N{\mathbb N} \def\Q{\mathbb Q} \def\h{\mathcal H} \def\S{\mathcal S} \def\U{\mathcal U} \def\dh{d_{\textup{H}}} \date{\today} \begin{document} \title[Self-embeddings of Bedford-McMullen carpets with dependent ratios]{Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios} \author{Jian-Ci Xiao} \address{School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China} \email{[email protected]} \subjclass[2010]{Primary 28A80; Secondary 28A78} \keywords{Bedford-McMullen carpets, self-embedding, obliqueness, generalized Sierpi\'nski carpets, logarithmic commensurability, deleted-digit sets.} \begin{abstract} We prove that any non-degenerate Bedford-McMullen carpet does not allow oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of the principal axes. We also establish a logarithmic commensurability result on the contraction ratios of such embeddings. This completes a previous study by Algom and Hochman [Ergod. Th. \& Dynam. Sys. {\bf 39} (2019), 577--603] on Bedford-McMullen carpets generated by multiplicatively independent exponents, together with a new proof of their non-obliqueness statement. For the self-similar case, however, we construct a generalized Sierpi\'nski carpet that is symmetric with respect to an appropriate oblique line and hence allows a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpi\'nski carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$. \end{abstract} \maketitle \section{Introduction} One way to gain a better comprehension of the geometric structure of fractal sets is to delve into the study of their self-embedding mappings, especially affine mappings or even similitudes. Note that a similitude $f$ on $\R^d$ can be written as $f(x)=\lambda Ox+a$, where $\lambda\geq 0$ is the \emph{similarity ratio} (also \emph{contraction ratio} if $\lambda<1$), $O$ is an orthogonal transformation on $\R^d$ (the \emph{orthogonal part}) and $a\in\R^d$. Here a rough principle is that the imposition of additional restrictions on these sets usually corresponds to an increase in the constraints placed on their self-embedding mappings. A pioneering work in this direction is a logarithmic commensurability theorem established by Feng and Wang~\cite{FW09}. This theorem asserts that if $K\subset\R$ is a self-similar set generated by a homogeneous iterated function system (IFS) with the open set condition and $f$ is a similitude with $f(K)\subset K$, then the contraction ratio of $f$ must be a rational power of the common ratio of mappings in the IFS. This result was later extended to higher dimensional cases by Elekes, Keleti and M\'ath\'e~\cite{EKM10}, albeit with the open set condition being substituted by the strong separation condition. If in addition the mappings in the corresponding IFS share a common contraction ratio and orthogonal part, the author~\cite{Xiao24} proved the relative openness of the embedded image. In~\cite{Alg201}, Algom characerized all affine self-embeddings of self-similar sets with the strong separation condition. For self-affine sets, however, there are much fewer existing results. One reason for this disparity is that a natural rescaling approach, which works well in the self-similar case, usually fails in the self-affine case due to the inherent distortion. Nevertheless, the classic family of Bedford-McMullen carpets, with their simple lattice structure and delicate intrinsic properties, presents an optimal subject for investigation. A formal definition is as follows. \begin{definition} Let $n\geq m\geq 2$ be integers and let $\Lambda\subset \{0,1,\ldots,n-1\}\times\{0,1,\ldots,m-1\}$ be non-empty. Set $\varphi_{i,j}$ to be the affine map given by \begin{equation}\label{eq:varphiij} \varphi_{i,j}(x,y) = \Big( \frac{x+i}{n}, \frac{y+j}{m} \Big), \quad (i,j)\in\Lambda. \end{equation} The attractor $K(n,m,\Lambda)$ associated with the self-affine IFS $\{\varphi_{i,j}:(i,j)\in\Lambda\}$ is usually called a \emph{Bedford-McMullen carpet} when $n>m$, and a \emph{generalized Sierpi\'nski carpet} when $n=m$. More precisely, \[ K(n,m,\Lambda) = \Big\{ \Big( \sum_{k=1}^\infty \frac{i_k}{n^k}, \sum_{k=1}^\infty \frac{j_k}{m^k} \Big): (i_k,j_k)\in\Lambda \Big\}. \] To avoid trivial cases, in this paper we always assume that $1<\#\Lambda<mn$, where $\#$ denotes the cardinality. \end{definition} In~\cite{EKM10}, Elekes, Keleti and M\'ath\'e studied the intersections of Bedford-McMullen carpets with their translated copies. They observed a measure drop phenomenon occurring in these intersections, except in trivial cases. A systematic study on self-embedding similitudes of Bedford-McMullen carpets was conducted by Algom and Hochman~\cite{AH19}, who showed that if the horizontal and vertical ratios $n,m$ are multiplicatively independent, then the carpet permits only nearly trivial self-embedding similitudes. To state and extend their findings, let us introduce a convenient terminology. \begin{definition}[Obliqueness] \, \begin{enumerate} \item A line $L\subset\R^2$ is called \emph{oblique} if it is not parallel to one of the two principle coordinate axes. \item A similitude $f$ on $\R^2$ with positive similarity ratio is called \emph{oblique} if $f$ sends the $x$-axis to an oblique line. \end{enumerate} \end{definition} Algom and Hochman's main result is the following theorem. \begin{theorem}[\cite{AH19}]\label{thm:AH19} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet with $\frac{\log n}{\log m}\notin\Q$. Suppose $K$ is not supported on any line and is not a Cartesian product of the unit interval $[0,1]$ and some Cantor set. If $f$ is a similitude sending $K$ into itself, then $f$ is not oblique and has similarity ratio $1$. \end{theorem} The non-degenerate and non-product assumptions on $K$ are natural. For example, if $K=[0,1]\times\{0\}$ or if $K=[0,1]\times C_{1/3}$ (where $C_{1/3}$ denotes the middle third Cantor set), then $f$ clearly need not to be an isometry. Meanwhile, the assumption of the independence between $n$ and $m$ is a key factor preventing oblique and contracting self-embeddings of the carpet. In fact, it is well known that a Cantor $m$-set cannot be embedded into another Cantor $n$-set if $\frac{\log m}{\log n}\notin\Q$, see~\cite{FHR14}. Heuristically, this phenomenon rules out oblique embeddings of non-degenerate Bedford-McMullen carpets with independent ratios, and forces the contraction ratio of any embedding similitude to be dependent on $n$ and $m$ simultaneously (so isometry). Our main purpose is to study to what extent Theorem~\ref{thm:AH19} remains true for the dependent case (i.e., $\frac{\log n}{\log m}\in\Q$) or even the self-similar case (i.e., $n=m$). The presence of dependence inherently complicates the self-embedding problem. Fortunately, first we are able to extend the non-obliqueness statement to the dependent case, together with a new proof for the independent case without analyzing the tangent sets of $K$ as in~\cite{AH19}. \begin{theorem}\label{thm:main1} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet that is not supported on any line. If $f$ is a similitude sending $K$ into itself, then $f$ is not oblique. \end{theorem} We remark that the non-obliqueness statement is closely related to the slicing problem of Bedford-McMullen carpets. For example, writing $N$ to be the maximal number of rectangles that are selected in a row in the initial pattern (see Section 2.1 for the definition), it is easy to find some $0\leq y\leq 1$ such that $\dimh (K\cap (\R\times\{y\}))=\frac{\log N}{\log n}$, where $\dimh$ is the Hausdorff dimension. But an oblique self-embedding similitude sends this horizontal slice into an oblique one, and if one can show that any oblique slice of $K$ has Hausdorff dimension strictly less than $\frac{\log N}{\log n}$, then the existence of oblique self-embeddings is denied. A natural question is: when do we have such an upper bound for oblique slices? By Marstrand's slicing theorem~\cite{Mar54}, for any oblique line $L$ in $\R^2$, \begin{equation}\label{eq:marstrand} \dimh (K\cap (L+a)) \leq \max\{\dimh K-1,0\} \quad \text{for a.e. } a\in L^\perp. \end{equation} So if $K$ has non-uniform horizontal fibres (that is, there are two rows containing different numbers of selected rectangles), it is easy to check (see~\eqref{eq:dimhk} for the formula of $\dimh K$) that \[ \dimh K-1 < \frac{1}{\log m}\cdot\log \Big( m\cdot N^\frac{\log m}{\log n} \Big)-1 = \frac{\log N}{\log n}. \] So almost all oblique slices have Hausdorff dimension strictly less than $\frac{\log N}{\log n}$. When $\frac{\log n}{\log m}\notin\Q$, it is highly probable that the upper bound specified in~\eqref{eq:marstrand} holds for all oblique slices rather than for only typical ones, which leads to a quick proof for certain special cases of Theorem~\ref{thm:main1}. This slicing problem is formally stated in the nice survey~\cite{Fra21} and seems quite challenging. See~\cite{Alg20,AW23} for recent attempts. However, in the dependent case this upper bound may not hold, as indicated in~\cite{BR14}. On the other hand, one cannot hope to extend the isometry statement in Theorem~\ref{thm:AH19} to the dependent case. For example, when $n=m^2$, it is easy to construct a Bedford-McMullen carpet which is actually a self-similar carpet. In such instances, there exist numerous non-isometric self-embedding similitudes. Nevertheless, we are able to prove a logarithmic commensurability conclusion as follows. \begin{theorem}\label{thm:main2} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet with $\frac{\log n}{\log m}\in\Q$ and let $f$ be a similitude sending $K$ into itself. If $K$ is not contained in any line, then $\frac{\log\lambda}{\log n}\in\Q$, where $\lambda$ denotes the similarity ratio of $f$. \end{theorem} We outline here some rough ideas on how to prove the above two results. For the non-obliqueness statement, we rescale $f^k(K)$ in an appropriate manner so that, in the absence of vacant rows, a geometric observation can lead us to the desired conclusion for both independent and dependent cases. On the other hand, if there is a vacant row, the independent case can be directly addressed using results on the dimensions of projections of Bedford-McMullen carpets as formulated in~\cite{FJS10} . For the dependent case, we first rescale $f^k(K)$ in a new way so that there is another oblique self-embedding $g$ for which $g(K)\subset C\times\R$, where $C$ is a ``generalized'' Cantor-$n$ set. Utilizing a nontrivial extension of the logarithmic commensurability theorem in~\cite{FW09}, we establish the non-existence of such a $g$, thereby deriving Theorem~\ref{thm:main2}. Finally, we consider the self-similar case when $n=m$, which can be regarded as the ``most dependent'' case. While one might anticipate a non-oblique statement in this context, it is not hard to to construct a generalized Sierpi\'nski carpet which is symmetric with respect to an appropriate oblique line, thus permitting an oblique reflectional self-embedding. See Example~\ref{exa:nonobsiercat}. So the non-oblique statement fails in the self-similar setting. However, considering rotational self-embeddings in lieu of reflectional ones and under the assumption of the strong separation condition, we indeed eliminate almost all possibilities. Here a generalized Sierpi\'nski carpet $K=K(n,\Lambda)$ is said to satisfy the \emph{strong separation condition} if elements in $\{\varphi_{i,j}(K)\}_{(i,j)\in\Lambda}$ are mutually disjoint. \begin{theorem}\label{thm:main3} Let $K=K(n,\Lambda)$ be a generalized Sierpi\'nski carpet satisfying the strong separation condition. Let $\theta\in\R$ and let $R_\theta$ be the counterclockwise rotation at the origin by angle $\theta$. If $f(x)=\lambda R_\theta x+a$ is an oblique similitude sending $K$ into itself, then $\|\tan\theta\|=1$. \end{theorem} Theorem~\ref{thm:main3} is achieved as follows. Note that the convex hull of any non-degenerate Sierpi\'nski carpet is a polygon. On the one hand, we show that if there exists an oblique rotational $f$, then the rotational angle is a rational multiple of $\pi$, and simultaneously can be expressed as the sum of finitely many inner angles (modulus $2\pi$) of the above polygon. On the other hand, we prove that all of these inner angles have rational tangents. So the rotation angle has a rational tangent, and the desired conclusion follows directly from a theorem of Niven (see Lemma~\ref{lem:niven}). Here the strong separation condition plays a crucial role in deriving the logarithmic commensurability of the contraction ratio of $f$ (by a result in~\cite{EKM10}). In fact, by employing some mass comparison arguments, the separation condition in Theorem~\ref{thm:main3} could be relaxed to the requirement of total disconnectedness; however, this does not present a substantial improvement and we will not explore it further within the scope of this paper. \vspace{4pt} \paragraph{{\bf Organization}} In the subsequent section, we provide some useful observations on Bedford-McMullen carpets and deleted-digit sets. In Section 3, we present our new proof of the non-obliqueness statement in the independent case. Section 4 is devoted to the proofs of Theorems~\ref{thm:main1} and~\ref{thm:main2}. Finally, we establish Theorem~\ref{thm:main3} in Section 5. \vspace{4pt} \paragraph{{\bf Notation}} We write $\mathcal{S}(\R^d)$ to be the collection of all similitudes on $\R^d$ with positive similarity ratio. For any collection $\mathcal{A}$ of sets in $\R^d$, $\bigcup\mathcal{A}$ is the union of sets in $\mathcal{A}$. For $A\subset\R^d$ and $x\in\R^d$, we denote by $A+x$ (or $x+A$) the translation of $A$ by $x$, that is, $A+x:=\{a+x: a\in A\}$. For any oblique line $L\subset\R^2$, $\pi_L$ denotes the orthogonal projection onto $L$. On the other hand, we write $\pi_1$ and $\pi_2$ to be the orthogonal projections onto the $x$-axis and $y$-axis, respectively. Finally, the concatenation $fg$ of any two mappings $f,g$ simply means their composition $f\circ g$. \section{Preliminaries} \subsection{Bedford-McMullen carpets} A Bedford-McMullen carpet $K=K(n,m,\Lambda)$ can be obtained by a standard iteration process as follows. One first divides the unit square $[0,1]^2$ into an $n\times m$ grid, selecting a subset of rectangles formed by the grid (called the \emph{initial pattern}) and then repeatedly substituting the initial pattern on each of the selected rectangles. The limit set is just $K$. For $k\geq 1$, we call every element in \[ \{\varphi_{i_1,j_1}\circ\cdots\circ\varphi_{i_k,j_k}([0,1]^2): (i_1,j_1),\ldots,(i_k,j_k)\in\Lambda\} \] a level-$k$ rectangle. For any level-$k$ rectangle $R$, we write $\varphi_R$ to be the natural affine map sending $[0,1]^2$ to $R$. We first record two geometric observations of Bedford-McMullen carpets, which rely heavily on the self-affinity (more precisely, the fact that $n>m$) and might be of independent interest. \begin{proposition}\label{prop:projofkisinf} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet. If there is at most one selected rectangle in each row, then for any oblique line $L$, $\pi_L(K)$ is an infinite set. \end{proposition} The condition that there is at most one selected rectangle in each row is not necessary, but the above version will suffice for our purpose. \begin{proof} It suffices to consider when $L$ passes through the origin. In this case, there is an angle $\theta$ such that $\pi_L(x)$ is essentially given by $\pi_1(x)\cos\theta+\pi_2(x)\sin\theta$. Since $L$ is oblique, $\cos\theta$ and $\sin\theta$ are both nonzero. Let $p\geq 1$ and pick a large $k$ so that $\frac{m^{k+p}}{n^k}<\|\frac{\sin\theta}{\cos\theta}\|$. Let $R$ be any level-$k$ rectangle. Since there is at most one selected rectangle in each row, there are at least $(\#\Lambda)^p$ many level-$(k+p)$ rectangles in $R$ that lies in different rows in the $n^{-k-p}\times m^{-k-p}$ grid. Since every level-$(k+p)$ rectangle intersects $K$, we can select $(\#\Lambda)^p/2$ many points in $\varphi_R(K)$ with $m^{-k-p}$-separated $y$-coordinates. Then their projections under $\pi_L$ are mutually distinct: otherwise, there are two of these points, say $a,b$, such that $\pi_L(a)=\pi_L(b)$, which implies that \begin{align} \frac{\|\sin\theta\|}{\|\cos\theta\|} &= \frac{\|\pi_1(a)-\pi_1(b)\|}{\|\pi_2(a)-\pi_2(b)\|} && \text{(since $\pi_L(a)=\pi_L(b)$)} \\ &\leq \frac{n^{-k}}{\|\pi_2(a)-\pi_2(b)\|} && \text{(since $a,b\in R$)} \\ &\leq \frac{n^{-k}}{m^{-k-p}} && \text{(by the choice of that subset)} \\ &= \frac{m^{k+p}}{n^k} < \frac{\|\sin\theta\|}{\|\cos\theta\|}, && \text{(by our choice of $k$)} \end{align} a contradiction. In particular, $\pi_L(K)\supset\pi_L(\varphi_R(K))$ contains at least $(\#\Lambda)^p/2$ many points. Since $\#\Lambda\geq 2$ and $p$ is arbitrary, $\pi_L(K)$ must be an infinite set. \end{proof} \begin{proposition}\label{prop:noobliquelines} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet. Then $K$ does not contain any oblique segment of positive length. \end{proposition} When $n=m$, this proposition sometimes fails. For example, when $\Lambda=\{(i,i):0\leq i\leq n-1\}$, the carpet is nothing but a diagonal of the unit square. \begin{proof} Since $\#\Lambda<mn$, there are $0\leq i_0\leq n-1,0\leq j_0\leq m-1$ such that $(i_0,j_0)\notin\Lambda$. Denote by $Q$ the open rectangle $=(\frac{i_0}{n},\frac{i_0+1}{n})\times(\frac{j_0}{m},\frac{j_0+1}{m})$. Since $(i_0,j_0)\notin\Lambda$, $Q\cap K=\varnothing$. By the self-affinity of $K$, for any $k\geq 1$ and any level-$k$ rectangle $R$, $\varphi_R(Q)\cap K=\varnothing$. Note that if $R$ is of the form $[\frac{p}{n^k},\frac{p+1}{n^k}]\times[\frac{q}{m^k},\frac{q+1}{m^k}]$, then \begin{equation}\label{eq:thevacantrect} \varphi_R(Q) = \Big( \frac{p}{n^k}+\frac{i_0}{n^{k+1}},\frac{p}{n^k}+\frac{i_0+1}{n^{k+1}} \Big) \times \Big(\frac{q}{m^k}+\frac{j_0}{m^{k+1}},\frac{q}{m^k}+\frac{j_0+1}{m^{k+1}} \Big). \end{equation} Suppose $K$ contains an oblique line segment $\ell$ of length $\|\ell\|>0$. Let $u$ be the slope of $\ell$. Since $\ell$ is oblique, we may assume without loss of generality that $u>0$. Fix $k$ so large that $\frac{m^ku}{n^k}<\frac{1}{2}$ and that there exists a level-$k$ rectangle $R$ such that $\ell$ meets both the left and right edges of $R$. The existence of such a rectangle can be found at the end of the proof. For convenience, we also assume that $\ell\subset R$ (considering the truncation $\ell\cap R$ instead). In particular, $\ell\subset\varphi_R(K)$ and hence $\varphi_R^{-1}(\ell)\subset K$. Note that $\varphi_{R}^{-1}(\ell)$ has slope $\frac{m^ku}{n^k}$. Pick a large integer $p$ so that $\frac{1}{m^p}<\frac{m^ku}{n^k}$ and $\frac{1}{n^{p+2}}<\frac{1}{m^{p+3}}$. Let $(0,a)$ be the intersection of $\varphi_{R}^{-1}(\ell)$ and the left edge of $\varphi_{R}^{-1}(R)=[0,1]^2$. So $\varphi_R^{-1}(\ell)$ meets the right edge of $[0,1]^2$ at $(1,a+\frac{m^ku}{n^k})$. Since $\frac{m^ku}{n^k}>\frac{1}{m^p}\geq \frac{2}{m^{p+1}}$, there is some integer $j$ such that $[\frac{j}{m^{p+1}},\frac{j+1}{m^{p+1}}]\subset [a,a+\frac{m^ku}{n^k}]$. For convenience, write $g(x):=\frac{m^ku}{n^k}x+a$ to be the linear function whose graph contains $\varphi_{R}^{-1}(\ell)$. For $0\leq t\leq n^{p+2}$, set $x_t:=\frac{t}{n^{p+2}}$ and $y_t:=g(x_t)$. By our choices of $p$ and $k$, \begin{equation}\label{eq:noobliqueline2} y_{t+1}-y_t = g(x_{t+1})-g(x_t) = \frac{m^ku}{n^k}\cdot\frac{1}{n^{p+2}}<\frac{1}{2}\cdot\frac{1}{m^{p+3}}. \end{equation} Therefore, there must exist some $0\leq t_0\leq n^{p+2}-1$ such that \begin{equation}\label{eq:noobliqueline4} [y_{t_0},y_{t_0+1}] \subset \Big[ \frac{j}{m^{p+1}}+\frac{j_0}{m^{p+3}}, \frac{j}{m^{p+1}}+\frac{j_0+1}{m^{p+3}} \Big] \end{equation} because the interval on the right hand side has length $m^{-(p+3)}$. Write \[ \widetilde{R} := [x_{t_0},x_{t_0+1}] \times \Big[ \frac{j}{m^{p+1}},\frac{j}{m^{p+1}}+\frac{1}{m^{p+2}} \Big]. \] By~\eqref{eq:noobliqueline4}, $\varphi_{R}^{-1}(\ell)$ passes the interior of $\widetilde{R}$. So $\widetilde{R}$ is a level-$(p+2)$ rectangle. By~\eqref{eq:thevacantrect}, \[ \varphi_{\widetilde{R}}(Q) = \Big( x_{t_0}+\frac{i_0}{n^{p+3}}, x_{t_0}+\frac{i_0+1}{n^{p+3}} \Big) \times \Big( \frac{j}{m^{p+1}}+\frac{j_0}{m^{p+3}}, \frac{j}{m^{p+1}}+\frac{j_0+1}{m^{p+3}} \Big). \] Letting $x_:=x_{t_0}+\frac{i_0}{n^{p+3}}+\frac{1}{2n^{p+3}}$, we have by~\eqref{eq:noobliqueline4} that \[ g(x_) \in [g(x_{t_0}), g(x_{t_0+1})] = [y_{t_0},y_{t_0+1}] \subset \Big[ \frac{j}{m^{p+1}}+\frac{j_0}{m^{p+3}}, \frac{j}{m^{p+1}}+\frac{j_0+1}{m^{p+3}} \Big] \] and hence the point $(x_, g(x_))\in\varphi_{\widetilde{R}}(Q)$. But from the definition of $g$, this point is also in $\varphi_{R}^{-1}(\ell)\subset K$, which contradicts that $\varphi_{\widetilde{R}}(Q)\cap K=\varnothing$. The existence of $R$ can be deduced in a similar way we choose $\widetilde{R}$. Let $(a_1,b_1)$ be the left endpoint of the original $\ell$ (before the truncation). Pick $k$ large enough such that $n^{-k}$ is much smaller than $\|\pi_1(\ell)\|$. Let $x'_0:=\min\{\frac{i}{n^k}: \frac{i}{n_k}\geq a_1\}$ and $y'_0$ be such that $(x'_0,y'_0)\in\ell$. Then define recursively that \[ x'_t = x'_{t-1}+\frac{1}{n^k} \text{ and } y'_t := y'_{t-1}+\frac{u}{n^k}, \quad 1\leq t\leq \lfloor\|\pi_1(\ell)\|n^k\rfloor-1, \] where $\lfloor\cdot\rfloor$ denotes the integer part. So $(x'_t,y'_t)\in\ell$ because $u$ is the slope of $\ell$. Since $\frac{m^ku}{n^k}<\frac{1}{2}$, $\|y'_t-y'_{t-1}\|=\frac{u}{n^k}<\frac{1}{2m^k}$. So there must exist some $t_0$ such that $[y'_{t_0-1},y'_{t_0}]$ is entirely contained in some $m$-adic subinterval of $[0,1]$ of the form $[\frac{j'}{m^k},\frac{j'+1}{m^k}]$. In particular, $y'_{t_0-1}\geq\frac{j'}{m^k}$ and $y'_{t_0}\leq\frac{j'+1}{m^k}$. Then it suffices to take \[ R = [x'_{t_0-1},x'_{t_0}]\times \Big[ \frac{j'}{m^k},\frac{j'+1}{m^k} \Big]. \] \end{proof} In the rest of this paper, we adopt the following notations. For $K=K(n,m,\Lambda)$, \begin{itemize} \item $J:=\{0\leq j\leq m-1: \exists i \text{ s.t. } (i,j)\in\Lambda\}$, which collects the digits of non-empty rows in the initial pattern. \item $I:=\{0\leq i\leq n-1: \exists j \text{ s.t. } (i,j)\in\Lambda\}$, which collects the digits of non-empty columns in the initial pattern. \item For $0\leq j\leq m-1$, $I_j:=\{0\leq i\leq n-1: (i,j)\in\Lambda\}$, which collects the digits of selected rectangles in the $j$-th row. \item For $0\leq i\leq n-1$, $J_i:=\{0\leq j\leq m-1: (i,j)\in\Lambda\}$, which collects the digits of selected rectangles in the $i$-th column. \item $N:=\max_{0\leq j\leq m-1}\#I_j$. \item $K_x:=\{y: (x,y)\in K\}$ (vertical slice) and $K^y:=\{x: (x,y)\in K\}$ (horizontal slice). \end{itemize} Note for every $y\in\pi_2(K)$ with a unique expansion $y=\sum_{k=1}^\infty y_km^{-k}$, where $y_k\in J$, \begin{align} K^y &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: \Big( \sum_{k=1}^\infty \frac{x_k}{n^k},\sum_{k=1}^\infty\frac{y_k}{m^k} \Big) \in K \Big\} \notag \\ &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: (x_k,y_k)\in\Lambda \text{ for all } k \Big\} \notag\\ &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: x_k\in I_{y_k} \text{ for all } k \Big\}. \label{eq:expressionofky} \end{align} For future use, we write \begin{equation}\label{eq:kyp} K^y_p:= \bigcup\Big\{ [x,x+n^{-p}]: x\in \Big\{ \sum_{k=1}^p \frac{x_k}{n^k}: x_k\in I_{y_k} \text{ for } 1\leq k\leq p \Big\} \Big\} \end{equation} and call the interior of every interval in the above union a \emph{basic open interval} of $K^y_p$. It is easy to see that $\{K^y_p\}_{p=1}^\infty$ is decreasing, $\bigcap_{p=1}^\infty K^y_p=K^y$ and \begin{equation}\label{eq:dimlbofky} \dimh K^y \leq \underline{\dim}_{\textup{B}}\, K^y \leq \liminf_{M\to\infty} \frac{\log (\prod_{k=1}^M \#I_{y_k})}{-\log n^M}\leq\frac{\log N}{\log n}, \end{equation} where $\underline{\dim}_{\textup{B}}$ denotes the lower box dimension. Furthermore, it is standard to show that $\h^{\log N/\log n}(K^y)\leq 1<\infty$ for all $y$ and we omit the proof. Here and afterwards, $\h^s$ denotes the $s$-dimensional Hausdorff measure. The following lemma records some basic facts about these horizontal slices. \begin{lemma}\label{lem:bunchoffacts} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet. Then for any $y\in\pi_2(K)$ with a unique expansion $y=\sum_{k=1}^\infty y_km^{-k}$, $\{y_k\}_{k=1}^\infty\subset J$, \begin{enumerate} \item For every $p\geq 1$ and every pair of basic open intervals $U,V$ of $K^y_p$, $K^y\cap U$ is simply a translated copy of $K^y\cap V$. \item For every $p,t\geq 1$ and every basic open interval $U$ of $K^y_p$, $U\cap K^y_p$ contains $\prod_{k=p+1}^{p+t}\# I_{y_k}$ many basic open intervals of $K^y_{p+t}$. \end{enumerate} In addition, if $\#I_{y_k}\geq 2$ for all large $k$, then the following properties hold. \begin{enumerate} \setcounter{enumi}{2} \item $\inf_{p\geq 1}\min\{\diam(K^y\cap U)/\|U\|: U\text{ is a basic open interval of $K^y_p$}\}>0$, where $\diam(\cdot)$ is the diameter. \item Calling an open interval $G$ a \emph{gap} of $K^y_p$ if $G$ is a connected component of $U\setminus K^y$, where $U$ is some basic open interval of $K^y_p$, we have \[ \sup_{p\geq 1}\max\{\|G\|n^{p}: G \text{ is a gap of } K^y_p\}<1. \] \end{enumerate} \end{lemma} \begin{proof} Let $p\geq 1$ and let $U$ be any basic open interval of $K^y_p$. Writing $a$ to be the left endpoint of $U$, we immediately see from~\eqref{eq:expressionofky} and~\eqref{eq:kyp} that \begin{align} K^y \cap U &= K^y\cap (a,a+n^{-p}) \\ &= \Big\{ a+\sum_{k=p+1}^\infty \frac{z_k}{n^k}: z_k\in I_{y_k} \text{ for all } k \Big\}\setminus\{a,a+n^{-p}\} \\ &= a + \Big( \Big\{ \sum_{k=p+1}^\infty \frac{z_k}{n^k}: z_k\in I_{y_k} \text{ for all } k \Big\}\setminus\{0,n^{-p}\} \Big). \end{align} This proves (1). Also, \[ K^y_{p+t}\cap U = \bigcup\Big\{\Big[ a+\sum_{k=p+1}^{p+t} \frac{x_k}{n^k}, \Big( a+\sum_{k=p+1}^{p+t} \frac{x_k}{n^k} \Big)+n^{-p-t}\Big] : x_k\in I_{y_k} \Big\}\setminus\{a,a+n^{-p}\}, \] which implies (2). Next, let $p$ be so large that $\#I_{y_k}\geq 2$ whenever $k\geq p$. Let $G\subset U$ be any gap of $K^y_p$. By (2), $U$ clearly contains at least $4$ basic open intervals of $K^{y}_{p+2}$. Since these intervals are disjoint and each of them contains some point in $K^y$, $\diam(K^y\cap U)\geq n^{-p-2}$ (which proves (3)) and $\|G\|\leq \|U\|-2n^{-p-2}= (1-2n^{-2})n^{-p}$, which gives (4). \end{proof} It is also worthy of mentioning that the Hausdorff dimension of Bedford-McMullen carpets was obtained by Bedford~\cite{Bed84} and McMullen~\cite{Mcm84} independently as follows: \begin{equation}\label{eq:dimhk} \dimh K = \frac{1}{\log m}\cdot\log\Big( \sum_{j=0}^{m-1} (\# I_j)^{\log m/\log n} \Big). \end{equation} \subsection{Deleted-digit sets} Deleted-digit sets naturally appear in the study of horizontal and vertical slices and projections of Bedford-McMullen carpets. \begin{definition}[Deleted-digit set] Let $n\geq 2$ be an integer. For any non-empty set $\D\subset\{0,1,\ldots,n-1\}$, we write $E(n,\D)$ to be the self-similar set associated with the IFS $\{\frac{x+i}{n}:i\in\D\}$ on $\R$. In other words, \begin{equation} E(n,\D) = \Big\{ \sum_{k=1}^\infty \varepsilon_kn^{-k}: \varepsilon_k\in\D \Big\}. \end{equation} \end{definition} A common example is the middle-third Cantor set, where $n=3$ and $\D=\{0,2\}$. Note that for any deleted-digit set $E(n,\D)$, the associated IFS $\{\frac{x+i}{n}:i\in\D\}$ always satisfies the open set condition, see~\cite{Fal14}. Since $\frac{[0,1]+i}{n}\subset[0,1]$ for all $n\geq 1$ and all $0\leq i\leq n-1$, we can use the unit interval $[0,1]$ for the standard iteration process to get $E(n,\D)$. More precisely, letting $E_0(n,\D):=[0,1]$ and recursively define \begin{equation}\label{eq:eknd} E_k(n,\D) := \bigcup_{i\in\D} \frac{E_{k-1}(n,\D)+i}{n}, \quad k\geq 1, \end{equation} we get a decreasing sequence $\{E_{k}(n,\D)\}_{k=0}^\infty$ such that $\bigcap_{k=0}^\infty E_k(n,\D)=E(n,\D)$. It is clear that $E(n,\D)$ has non-empty interior if and only if $\#\D=n$. In particular, if $\dimh E(n,\D)<1$ then it must be totally disconnected. It is easy to check by induction that $E_k(n,\D)$ is a union of intervals of the form $[\frac{i}{n^k},\frac{i+1}{n^k}]$, and each pair of such intervals are either adjacent (that is, they share a common endpoint) or disjoint. Many horizontal and vertical slices of a Bedford-McMullen carpet $K=K(n,m,\Lambda)$ are deleted-digit sets or a finite union of them. For example, if $y=\sum_{k} jm^{-k}$ for some $j\in J$, then we have by~\eqref{eq:expressionofky} that $K^y = E(n,I_j)$. Similarly, if $x=\sum_{k=1}^\infty in^{-k}$ for some $i\in I$, then $K_{x}=E(m,J_i)$. On the other hand, the projections of $K$ onto the principal axes are also deleted-digit sets: \begin{align} \pi_1(K) &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: \Big( \sum_{k=1}^\infty \frac{x_k}{n^k},\sum_{k=1}^\infty \frac{y_k}{m^k} \Big) \in K \Big\} \\ &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: \exists \{y_k\} \text{ such that } (x_k,y_k)\in\Lambda \Big\}\\ &= \Big\{ \sum_{k=1}^\infty \frac{x_k}{n^k}: x_k\in I \text{ for all $k$} \Big\} = E(n,I) \end{align} and similarly, $\pi_2(K)=E(m,J)$. \begin{lemma}\label{lem:fw09} Let $E=E(n,\D)$ be a deleted-digit set with $1<\#\D<n$. If $g$ is a similitude sending $E$ into itself, then the similarity ratio of $g$ is a rational power of $n$. \end{lemma} \begin{proof} This is a special case of~\cite[Theorem 1.1]{FW09}. \end{proof} A generalization of this logarithmic commensurability theorem as follows is one of the main ingredients in our proof of the dependent and self-similar cases. Recall that $N:=\max_{0\leq j\leq m-1}\#I_j$. \begin{proposition}\label{prop:allnthenrational} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet with $2\leq N\leq n-1$ and let $y\in\pi_2(K)$. Write $\alpha:=\frac{\log N}{\log n}$. If $0<\h^\alpha(K^y)<\infty$ and there is a contraction $h\in\S(\R)$ sending $K^y$ into itself, then the contraction ratio of $h$ is a rational power of $n$. \end{proposition} Some idea of the proof stems from an ongoing work of the author and Rao~\cite{RXtodedone}. \begin{proof} Considering $h^2$ if necessary, we may assume that $h$ is orientation-preserving. If $y\in\pi_2(K)$ has more than one expansion with coefficients in $J$, $y\in\bigcup_{k\geq 1}\frac{\Z}{m^k}\cap[0,1]$. In this case, it is easy to see that $K^y$ is a finite union of scaled copies of $K^0$ and $K^1$, say $K^y=\bigcup_{s\in\Omega} K^{0,s}\cup \bigcup_{t\in\Gamma}K^{1,t}$, where $\Omega,\Gamma\subset\Z$ and \[ K^{0,s}:=\frac{K^0+s}{n^k} \quad\text{and}\quad K^{1,s}:=\frac{K^1+t}{n^k} \] for some large integer $k$. Since $h(K^y)\subset K^y$, by Baire's theorem, some $K^{0,s}$ or $K^{1,s}$, say $K^{0,s}$, must contain an (relative) interior part of $h(K^y)$ and hence of $h(K^{0,s})$. So we can find a scaled copy $E$ of $K^0$ contracting by a factor $n^{-q}$ for some large integer $q$ such that $h(E)\subset K^{0,s}$. Since $K^0=E(n,I_0)$ is a deleted-digit set with positive Hausdorff dimension, by Lemma~\ref{lem:fw09}, the contraction ratio of $h$ is a rational power of $n$. Now suppose $y$ has a unique expansion $y=\sum_{k=1}^\infty y_km^{-k}$, $\{y_k\}\subset J$. We first prove that $\#I_{y_k}=N$ for all large $k$. Suppose on the contrary that $\#I_{y_k}<N$ for infinitely many $k$. Then for any $p\geq 1$, when $\widetilde{M}$ is large enough, we have \begin{equation}\label{eq:newproofofalln} \prod_{k=1}^{\widetilde{M}} \#I_{y_k} \leq (N-1)^pN^{\widetilde{M}-p} = \Big( \frac{N-1}{N} \Big)^p N^{\widetilde{M}}. \end{equation} Since $0<\h^\alpha(K^y)<\infty$, there is a constant $\delta_0$ such that $\h^\alpha_\delta(K^y)>\h^\alpha(K^y)/2>0$ when $0<\delta\leq\delta_0$ (for the definition of $\h^\alpha_\delta$, see~\cite{Fal14}). However, we have by~\eqref{eq:newproofofalln} that \[ \h^\alpha_{n^{-\widetilde{M}}} (K^y) \leq \Big( \frac{N-1}{N} \Big)^p N^{\widetilde{M}} \cdot (n^{-\widetilde{M}})^\alpha = \Big( \frac{N-1}{N} \Big)^p < \frac{1}{2}\h^\alpha(K^y) \] as long as $p$ and $\widetilde{M}$ are large enough. This is a contradiction. Denote by $\rho$ the contraction ratio of $h$. Suppose on the contrary that $\frac{\log\rho}{\log n}\notin\Q$. Let $\varepsilon>0$ be a small fixed constant (will be specified later) and pick positive integers $s,t$ such that $-\varepsilon\leq s\cdot\frac{\log\rho}{-\log n}-t<0$. Equivalently, $n^{-t}<\rho^s\leq n^{\varepsilon-t}$. Since $\#I_{y_k} \leq N\leq n-1$ for all $k$, it is not hard to check that \[ M:= \sup_{p\geq 1}\max\Big\{\|B\|n^p: B \text{ is a connected component of } K^y_p \Big\} \leq 2N<\infty, \] that is, for all $p$, every connected component of $K^y_p$ contains at most $2N$ basic open intervals of $K^y_p$. Pick a large $p_0\geq 1$ such that there is a connected component $B$ of $K^y_{p_0}$ consisting of exactly $M$ basic open intervals of $K^y_{p_0}$ and $\#I_{y_k}=N$ for all $k\geq p_0$. Note that $B$ is a closed interval and $h^s(K^y)\subset K^y$. We distinguish two cases. {\bf Case 1}: $h^s(B)$ intersects exactly one connected component of $K^y_{p_0+t}$, say $B'$. In this case, $h^s(B\cap K^y)\subset B'\cap K^y$. Since $B$ consists of $M$ basic open intervals of $K^y_{p_0}$ and their intersections with $K^y$ are distinct from one another by only a translation (recall Lemma~\ref{lem:bunchoffacts}(1)), one can find a $\rho^sn^{-p_0}$-separated subset of $h^s(B\cap K^y)$ that has cardinality $M$. Since $\rho^sn^{-p_0}>n^{-p_0-t}$, every basic open interval of $K^y_{p_0+t}$ in $B'$, which has length $n^{-p_0-t}$, contains at most one point in that subset. Thus by the maximality of $M$, $B'$ contains exactly $M$ basic open intervals of $K^y_{p_0+t}$. Recall that $0<\h^\alpha(K^y)<\infty$. Writing $c$ to be the $\alpha$-Hausdorff measure of the intersection of $K^y$ and any basic open interval in $K^y_{p_0+t}$, we see from Lemma~\ref{lem:bunchoffacts}(1) that $0<c<\infty$ and $\h^\alpha(B'\cap K^y)=Mc$. On the other hand, by Lemma~\ref{lem:bunchoffacts}(2), each basic open interval in $B$ contains exactly $\prod_{k=p_0+1}^{p_0+t} \#I_{y_k} = N^{t}$ ones in $K^y_{p_0+t}$. So $\h^\alpha(B\cap K^y)=MN^tc$ and \begin{align} \h^\alpha(h^s(B\cap K^y)) = \rho^{\alpha s}\h^\alpha(B\cap K^y) &= \rho^{\alpha s}\cdot MN^tc \notag \\ &> n^{-\alpha t}MN^tc \notag \\ &= N^{-t}\cdot M\cdot N^tc = Mc = \h^\alpha(B'\cap K^y), \label{eq:rationpropcase1} \end{align} which contradicts that $h^s(B\cap K^y)\subset B'\cap K^y$. {\bf Case 2}: $h^s(B)$ intersects at least two connected components of $K^y_{p_0+t}$. In this case, $h^s(B)$ contains a hole between those components. More precisely, there exists some integer $i$ such that $(\frac{i}{n^{p_0+t}},\frac{i+1}{n^{p_0+t}})\cap K^y=\varnothing$ and $(\frac{i}{n^{p_0+t}},\frac{i+1}{n^{p_0+t}})\subset h^s(B)$. Write $V:=(\frac{i}{n^{p_0+t}},\frac{i+1}{n^{p_0+t}})$. Comparing the length, we can find either exactly one or two adjacent basic open intervals of $K^y_{p_0}$ in $B$ of which the images under $h^s$ meet $V$. {\bf Case 2.1}: $V\subset h^s(\widetilde{I})$ for only one basic open interval $\widetilde{I}$ in $B$ (see Figure~\ref{fig:vi1i2}(A)). Then, since $h^s(\widetilde{I} \cap K^y)\subset K^y$, $h^s(\widetilde{I}\cap K^y)\cap V=\varnothing$. This in turn tells us that $\widetilde{I}$ contains a gap of length $\geq\|h^{-s}(V)\|$. Therefore, \begin{equation}\label{eq:rationpropcase2} \max\{\|G\|n^{p_0}: G \text{ is a gap of } K^y_{p_0}\} \geq \|h^{-s}(V)\|n^{p_0} = \rho^{-s}n^{-p_0-t}\cdot n^{-p_0} > n^{-\varepsilon}. \end{equation} Choosing $\varepsilon$ small enough at the beginning, this contradicts Lemma~\ref{lem:bunchoffacts}(4). {\bf Case 2.2}: There are two basic open intervals $\widetilde{I}_1,\widetilde{I}_2$ of $K^y_{p_0}$ in $B$ such that $V\cap h^s(\widetilde{I}_i)\neq\varnothing$, $1\leq i\leq 2$. Without loss of generality, assume that $\widetilde{I}_1$ is to the left of $\widetilde{I}_2$. Denote by $a_i$ the left endpoint of $h^s(\widetilde{I}_i)$. See Figure~\ref{fig:vi1i2}(B) for an illustration. \begin{figure}[htbp] \centering \subfloat[One-interval case] { \begin{minipage}[t]{155pt} \centering \begin{tikzpicture}[scale=1] \draw[thick] (0.1,0) to (1.9,0); \node at(1,0.4) {$h^s(\widetilde{I})$}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(0,0) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(2,0) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(0.2,-1) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(1.8,-1) {}; \draw[thick] (0.3,-1) to (1.7,-1); \node at(1,-1.3) {$V$}; \draw[thick,dashed] (0.2,-1) to (0.2,0); \draw[thick,dashed] (1.8,-1) to (1.8,0); \end{tikzpicture} \end{minipage} } \subfloat[Two-intervals case] { \begin{minipage}[t]{155pt} \centering \begin{tikzpicture}[scale=1] \draw[thick] (0.1,0) to (1.9,0); \draw[thick] (2.1,0) to (3.9,0); \node at(1,0.4) {$h^s(\widetilde{I}_1)$}; \node at(3,0.4) {$h^s(\widetilde{I}_2)$}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(0,0) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(2,0) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(4,0) {}; \node at(0,-0.3) {$a_1$}; \node at(2,-0.3) {$a_2$}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(1,-1) {}; \node[draw,shape=circle,inner sep=1.5pt,thick] at(2.6,-1) {}; \draw[thick] (1.1,-1) to (2.5,-1); \node at(1.8,-1.3) {$V$}; \draw[thick,dashed] (1,-1) to (1,0); \draw[thick,dashed] (2,-1) to (2,0); \draw[thick,dashed] (2.6,-1) to (2.6,0); \end{tikzpicture} \end{minipage} } \caption{An illustration of the local behavior of Case 2} \label{fig:vi1i2} \end{figure} By Lemma~\ref{lem:bunchoffacts}(1), \begin{align} \big( a_1,a_1+\|h^s(\widetilde{I}_2)\cap V\| \big)\cap K^y &= \big( a_2,a_2+\|h^s(\widetilde{I}_2)\cap V\| \big)\cap K^y \\ &= h^s(\widetilde{I}_2)\cap V \cap K^y \\ &= \varnothing. \end{align} Thus \[ h^s(\widetilde{I}_1)\cap K^y \subset (h^s(\widetilde{I}_1)\setminus V) \setminus \big( a_1,a_1+\|h^s(\widetilde{I}_2)\cap V\| \big). \] But the right hand side is an interval of length \begin{equation}\label{eq:rationpropcase3} \|h^s(\widetilde{I}_1)\| -\|h^s(\widetilde{I_1})\cap V\|- \|h^s(\widetilde{I}_2)\cap V\| = \rho^s n^{-p_0}-\|V\| < n^{\varepsilon-p_0-t}-n^{-p_0-t}. \end{equation} Choosing $\varepsilon$ small enough at the beginning, this contradicts Lemma~\ref{lem:bunchoffacts}(3). \end{proof} For convenience, for any compact set $A\subset\R$ and $a\in A$, we call $a$ a \emph{left} (resp. \emph{right}) \emph{isolated point} of $A$ if there is some $r>0$ such that $(a-r,a)\cap A=\varnothing$ (resp. $(a,a+r)\cap A=\varnothing$). \begin{corollary}\label{cor:leftrightendpt} Let $K,N,y,h$ be as in Proposition~\ref{prop:allnthenrational}. Then there are left and right isolated points $a,b$ of $K^y$ such that $h(a),h(b)$ (or $h(b),h(a)$ if $h$ is orientation-reversing) are also left and right isolated points of $K^y$. Moreover, we can guarantee that $\pi_2(R_k)=\pi_2(R'_k)$ for all large $k$, where $R_k,R'_k$ are two level-$k$ rectangles with $(a,y)\in\varphi_{R_k}(K)$ and $(b,y)\in\varphi_{R'_k}(K)$, respectively. \end{corollary} Roughly speaking, the ``moreover'' part just requires that the two points $(a,y),(b,y)$ lie in the same row at all stages during the iteration process. This is needed for some technical reason. \begin{proof} Suppose first that $y$ has a unique expansion with coefficients in $J$. For the first statement, it suffices to prove it when $h$ is orientation-preserving. In fact, if $h$ is orientation-reversing, then $h^2$ is orientation-preserving. If $a,h^2(a)$ are both left (resp. right) isolated points of $K^y$, then $h(a)$ must be a right (resp. left) isolated point and we are done. Similarly as before, let $\rho$ be the contraction ratio of $h$. By Proposition~\ref{prop:allnthenrational}, $\rho=n^{-p/q}$ for some positive integers $p,q$. Applying the argument in the proof of Proposition~\ref{prop:allnthenrational} to $s=q$, $t=p$ (so $\varepsilon=s\cdot\frac{\log\rho}{\log n}-t=0$), we see that Case 2 in that proof is impossible because: (1) in Case 2.1, the right hand side of~\eqref{eq:rationpropcase2} now equals $1$, i.e., $\widetilde{I}$ contains a gap of the same length as itself, which contradicts that $\widetilde{I}\cap K^y\neq\varnothing$; (2) in Case 2.2 the right hand side of~\eqref{eq:rationpropcase3} now equals $0$, i.e., $\|h^q(\widetilde{I_1}\cap K^y)\|=0$, which contradicts that $\h^\alpha(h^q(\widetilde{I_1}\cap K^y))>0$. On the other hand, for Case 1 in that proof, we now have $\h^\alpha(h^q(B\cap K^y))=\h^\alpha(B'\cap K^y)$. We claim that \begin{equation}\label{eq:hqkyhasnonemptyinte} h^q(B\cap K^y)=B'\cap K^y. \end{equation} Otherwise, since $h^q(B\cap K^y)\subset B'\cap K^y$, there exists some $x\in B'\cap K^y\setminus h^q(B\cap K^y)$. So we can find a large $t$ and a basic open interval $W$ of $K^y_t$ such that $x\in W\cap B'$ but $W\cap h^q(B\cap K^y)=\varnothing$. However, since $0<\h^\alpha(K^y)<\infty$, we have again by Lemma~\ref{lem:bunchoffacts}(1) that $\h^\alpha(W\cap K^y)>0$. Thus \[ \h^\alpha(h^q(B\cap K^y)) \leq \h^\alpha(B'\cap K^y)-\h^\alpha(W\cap K^y) < \h^\alpha(B'\cap K^y), \] which is a contradiction. By the claim, $h^q$ maps all the left and right isolated points of $B\cap K^y$ to the corresponding isolated points of $B'\cap K^y$, respectively. Pick any pair of them, say $a'$ and $b'$. It is easy to check by contradiction that $(h^{q})^{-1}(a'),(h^{q})^{-1}(b')$ are left and right isolated points of $K^y$, respectively. This establishes the first requirement. For the second one, since $y$ has a unique expansion, $R_k$ and $R'_k$ are unique for all $k$. Moreover, $y$ belongs to the interior of both of $\pi_2(R_k)$ and $\pi_2(R'_k)$, which implies that $\pi_2(R_k)=\pi_2(R'_k)$. Now, suppose $y$ has more than one expansion. Similarly as in the proof of Proposition~\ref{prop:allnthenrational}, we can write $K^y=\bigcup_{i\in\Omega} K^{0,i} \cup \bigcup_{j\in\Gamma} K^{1,j}$, where $\Omega,\Gamma\subset\Z$ and \[ K^{0,i}:=\frac{K^0+i}{n^k} \quad\text{and}\quad K^{1,j}:=\frac{K^1+j}{n^k} \] for some large integer $k$. By Baire's theorem again, we may assume without loss of generality that some $K^{0,i}$ contains an interior part of $h(K^y)$. In particular, there is a scaled copy of $K^{0,i}$, say $E=\frac{K^0+z}{n^c}\subset K^{0,i}$, such that $h(E)\subset K^{0,i}$. Note that \[ h(E)\subset K^{0,i} \Longleftrightarrow h\Big( \frac{K^0+z}{n^c} \Big) \subset \frac{K^0+i}{n^k}. \] We can easily reformulate this as $g(K^0)\subset K^0$ for some similitude $g$ with contraction ratio $\rho n^{k-c}$. Since $0$ has a unique expansion, as in the first part of this proof, there are nondegenerate open intervals $U,V$ such that $g(U\cap K^0)=V\cap K^0\neq\varnothing$ (recall~\eqref{eq:hqkyhasnonemptyinte}). In particular, $g$ maps all the one-sided isolated points of $K^0$ in $U$ to those of $K^0$ in $V$. Since the inclusion $g(K^0)\subset K^0$ is just a reformulation of $h(E)\subset K^{0,i}$, $h$ maps all one-sided isolated points of $K^{0,i}$ in $U'$ (for some $U'$) to one-sided isolated points of $K^{0,i}$ in $V'$ (for some $V'$). It remains to prove that these points are also one-sided isolated points of $K^y$ (for all large $k$, one can pick $R_k,R'_k$ to be those that lie above adjacent to $\R\times\{y\}$ and thus $\pi_2(R_k)=\pi_2(R'_k)$). To get the desired statement, note first that we may pick $V'$ properly so that \begin{equation}\label{eq:koslocallygoodv} K^{0,i}\cap V'=K^y\cap V'. \end{equation} In fact, if $j\neq i$ for all $j\in\Gamma$, then we already have~\eqref{eq:koslocallygoodv}. If some $j$ equals $i$, then $K^{0,i}$ and $K^{1,j}=K^{1,i}$ may overlap. However, note that $\dimh K^{0,i}\geq\dimh K^y\geq\dimh K^{1,i}$ (recall that $h(K^y)\subset K^{0,i}$), we have either $K^{1,i}=K^{0,i}$ or these two deleted-digit sets have different patterns (in particular, $K^{0,i}\neq K^{1,i}$). So there is a small open interval $V''\subset V'$ such that $(K^{0,i}\cap V'') \cap K^{1,i}=\varnothing$ but $K^{0,i}\cap V''\neq\varnothing$. Replacing $V'$ with $V''$ gives us~\eqref{eq:koslocallygoodv}. In particular, all left (resp. right ) isolated points of $K^{0,i}$ in $V'$ are left (resp. right) isolated points of $K^y$. Next, if a one-sided isolated point $a$ of $K^{0,i}$ in $U'$ is not a one-sided isolated point of $K^y$, there is a sequence $\{x_n\}\subset K^y$ such that $x_n\to a$ from the left or from the right. Then $h(x_n)\to h(a)$ from one direction. Since $h(x_n)\in K^y$, this contradicts the fact that $h(a)$ is a one-sided isolated point of $K^y$ in $V'$, which completes the proof. \end{proof} \section{The independent case} One significant benefit of the independence assumption regarding $n$ and $m$ is that it elucidates the dimensions of the orthogonal projections of $K$ onto all lines. \begin{lemma}[\cite{FJS10}]\label{lem:projofcarpet} Let $K=K(n,m,\Lambda)$ be a Bedford-McMullen carpet. If $\frac{\log n}{\log m}\notin\Q$, then $\dimh \pi_L(K)=\min\{\dimh K,1\}$ for every oblique line $L$. \end{lemma} \begin{theorem}\label{thm:notoblique} Let $K,f$ be as in Theorem~\ref{thm:main1}. If $\frac{\log n}{\log m}\notin\Q$, then $f$ is not oblique. \end{theorem} We distinguish three cases. Recall that $N=\max_j\#I_j$. \paragraph{\noindent{\bf Case 1}} $\#J<m$. In other words, there is an empty row in the initial pattern of $K$. \begin{proof}[Proof of Theorem~\ref{thm:notoblique} under Case 1] If $f$ is oblique, then by Lemma~\ref{lem:projofcarpet}, \begin{equation}\label{eq:pitklowerbd} \dimh \pi_t(K) \geq \dimh \pi_t(f(K)) = \min\{\dimh K,1\}, \quad t=1,2. \end{equation} If there is some $j$ such that $\#I_j\geq 2$, then \begin{align} \min\{1,\dimh K\} &> \frac{1}{\log m}\cdot \log(\# J) &&\text{(since $\#J<m$ and~\eqref{eq:dimhk})}\\ &= \dimh E(m,J) \\ & =\dimh \pi_2(K), &&\text{(since $E(m,J)=\pi_2(K)$)} \end{align} which contradicts~\eqref{eq:pitklowerbd}. However, if $\#I_j=1$ for all $j\in J$, then $\#\Lambda=\#J$. Therefore, \[ \dimh\pi_1(K) = \dimh E(n,I) \leq \frac{\log\#\Lambda}{\log n} < \frac{\log\#J}{\log m} = \dimh\pi_2(K) \leq \min\{\dimh K,1\}, \] which again contradicts~\eqref{eq:pitklowerbd}. \end{proof} \paragraph{\noindent{\bf Case 2}} $\#J=m$ and $N=n$. In other words, there is no empty row but a full row in the initial pattern of $K$. \begin{proof}[Proof of Theorem~\ref{thm:notoblique} under Case 2] In this case, pick $j$ such that $\#I_j=n$. Writing $y:=\sum_{k=1}^\infty jm^{-k}$, we have seen that $K^{y}=E(n,I_j)$, which is a horizontal line segment. So if $f$ is oblique, then $K\supset f(K)$ contains an oblique segment, which contradicts Proposition~\ref{prop:noobliquelines}. \end{proof} \paragraph{\noindent{\bf Case 3}} $\#J=m$ but $N<n$. In other words, one can find a selected rectangle and an unselected one in each row in the initial pattern of $K$. \begin{proof}[Proof of Theorem~\ref{thm:notoblique} under Case 3] In this case, $K$ does not contain any horizontal line segment. On the other hand, $\pi_2(K)=[0,1]$ and hence every horizontal slice of $K$ is non-empty. Let $0<\lambda\leq 1$ be the similarity ratio of $f$. If $f$ is oblique, without loss of generality, assume that $f(x\text{-axis})$ is supported on a line of slope $u>0$. Fix a large integer $p>0$ so that $m^{-p}\lambda\leq\frac{1}{9m^2n}$. Let us pick a sequence $\{R_k\}_{k=1}^\infty$, where each $R_k$ is a level-$k$ rectangle and $R_{k+1}\subset R_k$ for all $k\geq 1$. Write $a_k$ to be the left bottom vertex of $R_k$ and $Q_k$ to be any $n^{-k+p}\times m^{-k+p}$ grid rectangle containing $f(a_k)$. For convenience, we also let $\ell^k,\ell_k$ be the bottom and left edges of $R_k$, respectively. So $\ell^k\cap\ell_k=\{a_k\}$. See Figure~\ref{fig:rkfrkinvfrk} for an illustration. Below we record two simple but key facts. Here we abuse notation slightly by writing $\varphi_{Q_k}$ to be the natural affine map sending $[0,1]^2$ onto $Q_k$. \begin{enumerate} \item Since $\ell^k$ is parallel to the $x$-axis and $f(x\text{-axis})$ is supported on a line of slope $u>0$, the line supporting $\varphi_{Q_k}^{-1}f(\ell^k)$ has slope $\frac{m^{k-p}u}{n^{k-p}}$, which tends to $0$ as $k\to\infty$. Moreover, $\varphi_{Q_k}^{-1}f(\ell^k)$ has length \begin{align} \|\varphi_{Q_k}^{-1}f(\ell^k)\| &= \sqrt{n^{2(k-p)}\|\pi_1(f(\ell^k))\|^2+m^{2(k-p)}\|\pi_2(f(\ell^k))\|^2} \notag \\ &= \sqrt{n^{2(k-p)}\cdot\lambda^2\|\ell^k\|^2\cdot\frac{1}{1+u^2} + m^{2(k-p)}\cdot\lambda^2\|\ell^k\|^2\cdot\frac{u^2}{1+u^2}} \notag \\ &= \sqrt{n^{2(k-p)}\cdot\frac{\lambda^2}{n^{2k}}\cdot\frac{1}{1+u^2} + m^{2(k-p)}\cdot\frac{\lambda^2}{n^{2k}}\cdot\frac{u^2}{1+u^2}} \notag \\ &\to \frac{n^{-p}\lambda}{\sqrt{1+u^2}}, \quad k\to\infty \label{eq:lengthofvflk} \end{align} \item Similarly, the line supporting $\varphi_{Q_k}^{-1}f(\ell_k)$ has slope $-\frac{m^{k-p}}{n^{k-p}u}$, which also tends to $0$ as $k\to\infty$. Moreover, $\varphi_{Q_k}^{-1}f(\ell_k)$ has length \begin{align} \|\varphi_{Q_k}^{-1}f(\ell_k)\| &= \sqrt{n^{2(k-p)}\|\pi_1(f(\ell_k))\|^2+m^{2(k-p)}\|\pi_2(f(\ell_k))\|^2} \\ &= \sqrt{n^{2(k-p)}\cdot\frac{\lambda^2}{m^{2k}}\cdot\frac{1}{1+u^{-2}}+m^{2(k-p)}\cdot\frac{\lambda^2}{m^{2k}}\cdot\frac{u^{-2}}{1+u^{-2}}} \\ &\to\infty, \quad k\to\infty. \end{align} \end{enumerate} Roughly speaking, for all large $k$, $\varphi_{Q_k}^{-1}f(R_k)$ is a very flat parallelogram with one edge short and the other one pretty long. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=0.6] \draw[thick] (0,0) to (1,0) to (1,5) to (0,5) to (0,0); \node at (0,0)[circle,fill,inner sep=1.5pt, red]{}; \node at (-0.4,-0.4)[circle,inner sep=2pt,red]{$a_k$}; \node at (0.5,2.5)[circle,inner sep=2pt]{$R_k$}; \node at (2,2.5)[circle, inner sep=2pt]{$\xrightarrow{f}$}; \draw[thick] (3,20/11+3/2) to (3+10/11,1.5) to (3+14/11,1.5+2/11) to (3+4/11,3.5) to (3,20/11+3/2); \node at (3+10/11,1.5)[circle,fill, inner sep=1.5pt, red]{}; \node at (3+10/11+0.9,1.2)[circle, inner sep=2pt, red]{$f(a_k)$}; \draw[thick,dashed] (3,0) to (4,0) to (4,5) to (3,5) to (3,0); \node at (3.5,0.5)[circle,inner sep=2pt]{$Q_k$}; \node at (5.5,2.5)[circle, inner sep=2pt]{$\xrightarrow{\varphi_{Q_k}^{-1}}$}; \draw[thick] (7,1.5+5/4) to (7+25/4,1.5) to (7+15/2,1.5+1/4) to (7+5/4,3) to (7,1.5+5/4); \node at (7+13/4,1.3)[circle,inner sep=2pt]{$\varphi_{Q_k}^{-1}f(R_k)$}; \node at (7+25/4,1.5)[circle,fill, inner sep=1.5pt,red]{}; \node at (7+25/4,0.9)[circle, inner sep=2pt,red]{$\widetilde{a}_k$}; \node at (-0.4,2) {$\ell_k$}; \node at (0.5,-0.4) {$\ell^k$}; \end{tikzpicture} \caption{The evolution from $R_k$ to $f(R_k)$ to $\varphi_{Q_k}^{-1}f(R_k)$ (color online)} \label{fig:rkfrkinvfrk} \end{figure} Write $\widetilde{a}_k :=\varphi_{Q_k}^{-1}f(a_k)$. Since $f(a_k)\in Q_k$, $\widetilde{a}_k\in [0,1]^2$. Fix a large $k$ (will be specified later) so that $\frac{m^{k-p}}{n^{k-p}}\cdot\max\{u^{-1},u\}<\lambda$ and $\|\varphi_{Q_k}^{-1}f(\ell^k)\|\leq \frac{2n^{-p}\lambda}{\sqrt{1+u^2}}$ (recall~\eqref{eq:lengthofvflk}). Let us truncate the parallelogram $\varphi_{Q_k}^{-1}f(R_k)$ as $\bigcup_{t\in\Z} \widetilde{E}_t$, where \[ \widetilde{E}_t := \varphi_{Q_k}^{-1}f(R_k) \cap \{x\in\R^2: \pi_1(x)\in[-t,-t+1]\}. \] Note that for every $t$, \begin{align} \|\pi_2(\widetilde{E}_t)\| &\leq \|\pi_2(\varphi_{Q_k}^{-1}f(R_k))\| \notag \\ &\leq \|\pi_2(\varphi_{Q_k}^{-1}f(\ell^k))\|+\|\pi_2(\varphi_{Q_k}^{-1}f(\ell_k))\| \notag \\ &= m^{k-p}\cdot\frac{\lambda}{n^k}\cdot\frac{u}{\sqrt{1+u^2}}+m^{k-p}\cdot\frac{\lambda}{m^k}\cdot\frac{u^{-1}}{\sqrt{1+u^{-2}}} < 2m^{-p}\lambda.\label{eq:lessthan13m} \end{align} Let us prove Case 3 under the following Condition A first and then show its validity. \paragraph{{\bf Condition A}} There are integers $t_0\geq -1$ and $0\leq j_0\leq m$ such that $\|\pi_1(\widetilde{E}_{t_0})\|=1$ while $\pi_2(\widetilde{E}_{t_0})\subset [\tfrac{j_0}{m},\tfrac{j_0+1}{m}]$. Assume that Condition A holds. Let $j'_0:=j_0\pmod {m}$. Recall that $\varphi_{Q_k}^{-1}f(\ell_k)$ is supported on a line of slope $-\frac{m^{k-p}}{n^{k-p}u}$, say $g(x)=-\frac{m^{k-p}}{n^{k-p}u}x+c$ (the equation of that line). Since $\#I_{j'_0}<n$, there is some $0\leq i_0\leq n-1$ such that $K$ does not intersect the open rectangle $(\frac{i_0}{n},\frac{i_0+1}{n})\times(\frac{j'_0}{m},\frac{j'_0+1}{m})=:U$. Write $e$ to be the point $(-t_0,\lfloor\frac{j_0}{m}\rfloor)$ and $x_0:=-t_0+\frac{i_0+1/2}{n}$. Then the point $\xi_0:=(x_0,g(x_0))$ lies on the segment $\varphi_{Q_k}^{-1}f(\ell_k)$, and \begin{align} U+e &= \Big( \frac{i_0}{n}-t_0,\frac{i_0+1}{n}-t_0 \Big)\times \Big( \frac{j'_0}{m}+\lfloor\frac{j_0}{m}\rfloor, \frac{j'_0+1}{m}+\lfloor\frac{j_0}{m}\rfloor \Big) \\ &= \Big( \frac{i_0}{n}-t_0,\frac{i_0+1}{n}-t_0 \Big)\times\Big( \frac{j_0}{m}, \frac{j_0+1}{m} \Big). \end{align} Since $\pi_2(\widetilde{E}_{t_0})\subset [\tfrac{j_0}{m},\tfrac{j_0+1}{m}]$, it is not hard to check that \begin{equation}\label{eq:sliceinupluse} \xi_0+\varphi_{Q_k}^{-1}f(\ell^k) \subset U+e. \end{equation} See Figure~\ref{fig:localofthehole} for an illustration. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[thick,dashed] (0,0) to (0,4); \draw[thick,dashed] (4,0) to (4,4); \draw[thick,dashed] (-1.5,1) to (5.5,1); \draw[thick,dashed] (-1.5,2.5) to (5.5,2.5); \node at (2,3.5) {$\widetilde{E}_{t_0}$}; \draw[thick] (-1, 1.75) to (5,1.45); \draw[thick] (-1,2.05) to (5.5,1.725); \draw[thick,red] (5,1.45) to (5.5,1.725); \draw[thick,red] (2,1.6) to (2.5,1.875); \node at (2,1.6)[circle,fill, inner sep=1.5pt]{}; \node at (2,1.3) {$\xi_0$}; \node at (5.8,1.3) [red] {$\varphi_{Q_k}^{-1}f(\ell^k)$}; \node at (-1.9,1) {$\tfrac{j_0}{m}$}; \node at (-1.9,2.5) {$\tfrac{j_0+1}{m}$}; \draw[thick,blue] (1.4,1) to (2.6,1) to (2.6,2.5) to (1.4,2.5) to (1.4,1); \node at (2,0.7) [blue] {$U+e$}; \end{tikzpicture} \caption{$\varphi_{Q_k}^{-1}f(\ell^k)+\xi_0$ is contained in the hole $U+e$ (color online)} \label{fig:localofthehole} \end{figure} Since $\xi_0+\varphi_{Q_k}^{-1}f(\ell^k)$ is a translated copy of $\varphi_{Q_k}^{-1}f(\ell^k)$ and is contained in $\varphi_{Q_k}^{-1}f(R_k)$, $f^{-1}\varphi_{Q_k}(\xi_0)+\ell^k$ is a translated copy of $\ell^k$ and is contained in $R_k$. Since $\ell^k$ is the bottom edge of the level-$k$ rectangle $R_k$, $\ell^k\cap \varphi_{R_k}(K)=\varphi_{R_k}(K^0)$. So there is $y_0\in[0,1]$ such that \[ (f^{-1}\varphi_{Q_k}(\xi_0)+\ell^k)\cap\varphi_{R_k}(K) = \varphi_{R_k}(K^{y_0}\times\{y_0\})\neq\varnothing, \] where the non-empty conclusion is because $K^{y}\neq\varnothing$ for all $y$ (as pointed out at the beginning of Case 3). This in turn implies that \begin{equation}\label{eq:tranisanotherhorislice} (\xi_0+\varphi_{Q_k}^{-1}f(\ell^k)) \cap \varphi_{Q_k}^{-1}f\varphi_{R_k}(K) = \varphi_{Q_k}^{-1}f\varphi_{R_k}(K^{y_0}\times\{y_0\}) \neq\varnothing. \end{equation} Thus \begin{align} \varnothing &\neq (\xi_0+\varphi_{Q_k}^{-1}f(\ell^k)) \cap \varphi_{Q_k}^{-1}f\varphi_{R_k}(K) && \text{(just~\eqref{eq:tranisanotherhorislice})} \\ &\subset (U+e) \cap \varphi_{Q_k}^{-1}f\varphi_{R_k}(K) && \text{(by~\eqref{eq:sliceinupluse})} \\ &\subset (U+e) \cap \varphi_{Q_k}^{-1}f(K) \\ &\subset (U+e) \cap \Big( \bigcup_{\substack{\text{$R$ is a level-$(k-p)$ rectangle} \\ \text{$R\cap f(K)\neq\varnothing$}}} \varphi_{Q_k}^{-1}\varphi_R(K) \Big) && \text{(since $f(K)\subset K$)} \\ &\subset (U+e) \cap \bigcup_{z\in\Z^2}(K+z) \\ &= (U+e) \cap (K+e) = \varnothing, && \text{(since $e$ is a lattice point)} \end{align} which leads to a contradiction. It remains to prove Condition A. The key is to note that for every $y$, $\varphi_{Q_k}^{-1}f(R_k)\cap(\R\times\{y\})$ (which is a horizontal slice of the parallelogram) has length at most $Cn^{-p}\lambda$. To see this, just look at Figure~\ref{fig:thetriangle} and recall that the two acute edges of that triangle has slope $\frac{m^{k-p}u}{n^{k-p}}$ and $-\frac{m^{k-p}}{n^{k-p}u}$, respectively. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[thick] (0,0) to (2,0.5); \draw[thick,dashed] (2,-0.25) to (-4,0.5); \draw[thick,dashed] (2,0.5) to (6,0); \draw[thick] (0,0) to (6,0); \draw[thick,dashed] (2,0.5) to (2,0); \node at (0,-0.3) {$\widetilde{a}_k$}; \node at (0.45,0.56) {$\varphi_{Q_k}^{-1}f(\ell^k)$}; \end{tikzpicture} \caption{Length of horizontal slices of $\varphi_{Q_k}^{-1}f(R_k)$} \label{fig:thetriangle} \end{figure} If Condition A holds for $t_0=1$ and some suitable $j_0$ then we are done. Otherwise, there exists some $j_$ such that $\widetilde{E}_{1}\cap(\R\times\{\frac{j_}{m}\})\neq\varnothing$. Since $\|\varphi_{Q_k}^{-1}f(R_k)\cap(\R\times\{\frac{j_}{m}\})\|\leq Cn^{-p}\lambda$, picking $k$ large at the beginning, it is not hard to check that Condition A holds for $t_$ and $j_$, where $t_:= \min\{t\geq 2: \widetilde{E}_t\cap(\R\times\{\tfrac{j_}{m}\})=\varnothing\}$. ll[teal,opacity=0.2] (0,0.9) to (4,0.5) to (4,0.9) to (0,1.3) to (0,0.9); ll[purple,opacity=0.2] (0,1.8) to (4,1.4) to (4,1.8) to (0,2.2) to (0,1.8); ll[purple,opacity=0.2](0,0.4) to (4,0) to (4,-0.4) to (0,0) to (0,0.4); \end{proof} \begin{remark}\label{rem:deindwork} Note that in the proof of Cases 2 and 3, we do not need the independence assumption $\frac{\log n}{\log m}\notin\Q$ but only the fact that $n>m$. This will save us much effort in the dependent self-affine case. \end{remark} \section{The dependent self-affine case} In this section, we fix a Bedford-McMullen carpet $K=K(n,m,\Lambda)$ that is not supported in any line and assume that $\frac{\log n}{\log m}\in\Q$. Without loss of generality, write $n=m^{p/q}$, where $p,q\in\Z^+$ are coprime. It follows from $n>m$ that $p>q$. Recall that $N=\max_{j} \#I_j$. Unlike the independent case, now we only know the dimension of the projections of $K$ in almost all directions (by Marstrand's projection theorem). This limitation renders the argument in Case 1 of Theorem~\ref{thm:notoblique} invalid. Instead, we adopt a rescaling approach as follows. \begin{lemma}\label{lem:dep1} Let $f\in\S(\R^2)$ be a contracting map. If $f(K)\subset K$, then we can find $g\in\mathcal{S}(\R^2)$ and a compact set $C\subset\R$ with $\dimh C \leq \frac{\log N}{\log n}$ such that $g(K)\subset C\times\R$. Moreover, if $f$ is oblique, then $g$ can be picked oblique as well. \end{lemma} \begin{proof} Let $\lambda$ be the contraction ratio of $f$. For all positive integers $t$ with $\lambda^t\leq m^{-1}$, write \begin{equation}\label{eq:ktinlemma4.1} k_t := \max\{k\geq 0: m^{-pk-1}\geq \lambda^t\}. \end{equation} Since the contraction ratio of $f^t$ is $\lambda^t$, $K\subset[0,1]^2$ and $f^t(K)\subset K$, $f^t(K)$ is contained in at most $4$ squares of side length $m^{-pk_t}=n^{-qk_t}$. More precisely, there are $x\in n^{-qk_t}\mathbb{N}, y\in m^{-pk_t}\mathbb{N}$ such that \[ f^t(K) \subset (x,x+2n^{-qk_t}) \times (y,y+2m^{-pk_t}) =: Q_t. \] Denote by $h_t$ the homothety sending $(0,1)^2$ to the square $(x,x+n^{-qk_t}) \times (y,y+m^{-pk_t})=:Q'_t$, the ``left bottom square'' of $Q_t$. Note that for all $t$, \[ h_t^{-1}f^t(K) \subset h_t^{-1}(Q_t) = (0,2)^2. \] The contraction ratio of $h_t^{-1}f^t$ equals $n^{qk_t}\lambda^t$, which is at most $n^{qk_t}\cdot m^{-pk_t-1}=m^{-1}$ and at least $n^{qk_t}\cdot m^{-p(k_t+1)-1}=m^{-p-1}$. So the sequence $\{h_t^{-1}f^t\}_{t}$ has accumulation points in $\mathcal{S}(\R^2)$. For simplicity, assume that $h_t^{-1}f^t$ converges to some $g$. Suppose $g(K)\cap(0,1)^2\neq\varnothing$. Then $h_t^{-1}f^t(K)\cap(0,1)^2\neq\varnothing$ for all large $t$. For each of these $t$, write $y=\sum_{k=1}^{pk_t}y_km^{-k}$, where $y_k\in\{0,\ldots,m-1\}$. Since $p>q$, the collection of all level-$pk_t$ rectangles in $\overline{Q'_t}$, denoted by $\mathcal{R}_t$, is \begin{equation} \Big\{ \Big[ x+\sum_{k=qk_t+1}^{pk_t} \frac{x_k}{n^k}, \Big( x+\sum_{k=qk_t+1}^{pk_t} \frac{x_k}{n^k} \Big)+n^{-pk_t} \Big] \times [y,y+m^{-pk_t}]: (x_k,y_k)\in\Lambda \Big\}. \end{equation} Then we have \begin{align} h_t^{-1}f^t(K) \cap (0,1)^2 &= h_t^{-1}f^t(K) \cap h_t^{-1}(Q'_t) \notag \\ &= h_t^{-1}(f^t(K)\cap Q'_t) \notag \\ &\subset h_t^{-1}\Big( \bigcup\mathcal{R}_t \Big) \label{eq:rtisnotempty} \\ &\subset \bigcup_{\substack{(x_k,y_k)\in\Lambda, \notag \\ qk_t+1\leq k\leq pk_t}} \Big[ \sum_{k=qk_t+1}^{pk_t} \frac{x_k}{n^{k-qk_t}}, n^{-(p-q)k_t}+ \sum_{k=qk_t+1}^{pk_t} \frac{x_k}{n^{k-qk_t}} \Big] \times\R \notag \\ &=: E_t \times\R. \label{eq:htinversesupset} \end{align} Moreover, write $\j_t:=y_{qk_t+1}\cdots y_{pk_t}$, which is a word of length $(p-q)k_t\to\infty$ as $t\to\infty$. Using Cantor's diagonal argument, it is not hard to find an infinite word $\j=j_1j_2\cdots\in J^\infty$ such that for all $M\geq 1$, there is a large $t$ such that $\|\j_t\wedge\j\|\geq M$, where $\j_t\wedge\j$ denotes their longest common prefix and $\|\cdot\|$ stands for the length. Let $z:=\sum_{k=1}^\infty j_km^{-k}$. We claim that \begin{equation}\label{eq:productstructure} g(K)\cap(0,1)^2 \subset K^{z} \times \R. \end{equation} Fix any $\delta>0$. Let $M$ be a large integer with $n^{-M}<\frac{\delta}{2}$. Pick $t$ so large that the following conditions hold: \begin{enumerate} \item $g(K)\cap (0,1)^2 \subset \mathcal{N}_{\delta/2}(h_t^{-1}f^t(K)\cap(0,1)^2)$, where $\mathcal{N}_{\delta/2}(\cdot)$ denotes the $\delta/2$-neighborhood. \item $(p-q)k_t\geq M$; \item there is some $\j_t$ such that $\|\j_t\wedge\j\|\geq M$. \end{enumerate} Since $\|\j_t\wedge\j\|\geq M$, it is not hard to see from the definition of $E_t$ and $\j_t$ that $E_t\subset \mathcal{N}_{n^{-M}}(K^z)$. Hence \begin{align} g(K)\cap(0,1)^2 &\subset \mathcal{N}_{\delta/2}(h_t^{-1}f^t(K)\cap(0,1)^2) &&\text{(condition (1))} \\ &\subset \mathcal{N}_{\delta/2}(E_t\times\R) && \text{(by~\eqref{eq:htinversesupset})} \\ &\subset \mathcal{N}_{n^{-M}+\delta/2}(K^z\times\R) \subset \mathcal{N}_{\delta}(K^z\times\R). \end{align} Since $\delta$ is arbitrary, we establish~\eqref{eq:productstructure}. Applying the same argument to the other three squares in $Q_t$, we find at most four points $z_{0,0}:=z,z_{0,1},z_{1,0},z_{1,1}\in\pi_2(K)$ such that \begin{equation}\label{eq:formofc} g(K) \subset \bigcup_{0\leq i,j\leq 1} (K^{z_{i,j}}+i)\times\R. \end{equation} Write $C:=\bigcup_{0\leq i,j\leq 1} (K^{z_{i,j}}+i)$. By~\eqref{eq:dimlbofky}, $\dimh C\leq\frac{\log N}{\log n}$. It remains to prove that if $f$ is oblique then $g$ can be picked oblique as well. Write $O$ to be the orthogonal part of $f$. If $\{O^t\}_{t=1}^\infty$ is a finite collection, then every element in it appears infinitely many times. Picking an oblique transformation $\widetilde{O}\in\{O^t\}_t$ and a subsequence $t_k$ so that $O^{t_k}=\widetilde{O}$ for all $k$, we can consider $f^{t_k}$ instead of $f^t$ at the beginning of the above arguments and get an oblique $g$ because $g$, as the limit of $h_t^{-1}f^t$, has $\widetilde{O}$ as its orthogonal part. If $\{O^t\}_{t=1}^\infty$ is an infinite collection, then since a $2\times 2$ orthogonal matrix is either a rotation or a reflection, $O$ must be an irrational rotation (i.e., the rotation angle is an irrational multiple of $\pi$) and hence every rotation matrix is an accumulation point of $\{O^t\}_{t=1}^\infty$. Then one just picks a subsequence $\{O^{t_k}\}_k$ so that $O^{t_k}$ converges to an oblique transformation, which, similarly as before, should be the orthogonal part of $g$. This completes the proof. \end{proof} \begin{remark}\label{rem:bairetofindint} In the above proof, if we apply Baire's theorem to~\eqref{eq:formofc}, some $(K^{z_{i,j}}+i)\times\R$ should contain an interior part of $g(K)$. In particular, there is an integer $k$ and a level-$k$ rectangle $R$ such that $g\varphi_R(K)\subset (K^{z_{i,j}}+i)\times\R$. As a consequence, \begin{equation}\label{eq:fvarphirkyinkz} g\varphi_R(K^y\times\{y\}) \subset (K^{z_{i,j}}+i)\times\R, \quad \forall y\in\pi_2(K). \end{equation} We also remark that the integer $k_t$ defined in~\eqref{eq:ktinlemma4.1} equals $\lfloor (-\frac{t\log\lambda}{\log m}-1)/p\rfloor$. So if $\frac{\log\lambda}{\log n}\notin\Q$, then $\frac{\log\lambda}{\log m}\notin\Q$ and hence $\{n^{qk_t}\lambda^t\}_t$ is dense in an open subinterval of $(0,1)$. Since the proof of Lemma~\ref{lem:dep1} works for all convergent subsequence of $\{h_t^{-1}f^t\}_{t=1}^\infty$, one can find such an accumulation map $\widetilde{g}$ so that $\widetilde{g}(K)\subset C_{\widetilde{g}}\times\R$, where $C_{\widetilde{g}}$ is a closed set with $\dimh C_{\widetilde{g}}\leq\frac{\log N}{\log n}$ (just as $C$ in Lemma~\ref{lem:dep1}), and the contraction ratio of $\widetilde{g}$ is an irrational power of $n$. \end{remark} Now we are able to prove the non-obliqueness theorem for the dependent case. The proof is separated into two parts, depending on whether $f$ is a contracting map or an isometry. \begin{theorem}\label{thm:dependentaffine} If $f$ is a similitude sending $K$ into itself, then $f$ is not oblique. \end{theorem} \begin{proof}[Proof of Theorem~\ref{thm:dependentaffine}: the contracting case] By Proposition~\ref{prop:noobliquelines}, it suffices to consider when $N\leq n-1$. Suppose on the contrary that $f$ is oblique. Pick $g, C$ as in Lemma~\ref{lem:dep1} with $g$ oblique and denote by $\rho$ the contraction ratio of $g$. If $N=1$, then it is not hard to see that every horizontal slice of $K$ is a finite set. From the form of $C$ (recall~\eqref{eq:formofc}), $C$ is a finite set. But by Proposition~\ref{prop:projofkisinf} and the obliqueness of $g$, $\pi_1(g(K))$ is an infinite set. This contradicts $\pi_1(g(K))\subset \pi_1(C\times\R)= C$. Now suppose $N\geq 2$. By Remark~\ref{rem:bairetofindint} (see~\eqref{eq:fvarphirkyinkz}), we can find $z\in\pi_2(K)$, $i\in\Z$, $k\geq 1$ and a level-$k$ rectangle $R$ such that \begin{equation}\label{eq:origvarphir} g\varphi_R(K^y\times\{y\}) \subset (K^z+i)\times\R, \quad \forall y\in\pi_2(K). \end{equation} Since $g$ is a similitude, there is a unit directional vector $v:=(v_1,v_2)$ such that for every $y\in\pi_2(K)$, $g\varphi_R$ maps the horizontal line $\R\times\{y\}$ to $L_y:=\{rv+c_y:r\in\R\}$ for some $c_y\in\R^2$. Together with~\eqref{eq:origvarphir}, we have \begin{equation}\label{eq:gky0slice} g\varphi_R(K^{y}\times\{y\}) \subset L_y \cap ((K^z+i)\times \R),\quad \forall y\in\pi_2(K). \end{equation} Since $g$ is oblique, $v_1,v_2$ are both nonzero. Without loss of generality, assume that $v_1,v_2>0$ (other cases can be similarly discussed). Note that for each $y\in\pi_2(K)$, $g\varphi_R(K^y\times\{y\})$ can be regarded as a similar copy of $K^y$ contracted by $n^{-k}\rho$, while $L_y\cap((K^z+i)\times\R)$ can be regarded as a translated copy of $\frac{\|v\|}{v_1}K^z$. In particular, one can obtain from~\eqref{eq:gky0slice} a similitude $h_y\in\S(\R)$ with contraction ratio $n^{-k}\rho\cdot\frac{v_1}{\|v\|}$ (which is independent of $y$) such that \begin{equation}\label{eq:hykysubsetc} h_y(K^y)\subset K^{z}, \quad \forall y\in\pi_2(K). \end{equation} Pick $j_\in J$ so that $\#I_{j_}=N$ and let $y_:=\sum_{k=1}^\infty j_m^{-k}$. Then $K^{y_}=E(n,I_{j_})$ (recall Section 2.2). By~\eqref{eq:hykysubsetc}, $h_{y_}(K^{y_})\subset K^{z}$. So $0<\h^\alpha(K^{z})<\infty$, where $\alpha:=\frac{\log N}{\log n}$ (recall the observation before Lemma~\ref{lem:bunchoffacts}). It might be suitable for us to point out an observation here (which will be of help later but not in this proof): since~\eqref{eq:hykysubsetc} also implies $h_z(K^z)\subset K^z$, it follows from Proposition~\ref{prop:allnthenrational} that \begin{equation}\label{eq:thenewlogformula} \frac{\log(n^{-k}\rho\cdot v_1/\|v\|)}{\log n}\in\Q. \end{equation} Next, since $h_z(K^z)\subset K^z$, by Corollary~\ref{cor:leftrightendpt}, $h_z$ maps a left isolated point $a$ and a right one $b$ both to one-sided isolated points of $K^z$. Since $h_z(K^z)\subset K^z$ is simply a one-dimensional reformulation of the inclusion $g\varphi_R(K^z\times\{z\})\subset L_z\cap ((K^z+i)\times\R)$, this in turn gives us two points $\widetilde{a},\widetilde{b}\in g\varphi_R(K^{z}\times\{z\})$ and some $\delta>0$ such that both of the tubes $(\pi_1(\widetilde{a})- \delta,\pi_1(\widetilde{a})) \times\R$ and $(\pi_1(\widetilde{b}),\pi_1(\widetilde{b})+\delta)\times\R$ do not intersect $(K^{z}+i)\times\R$. Moreover, there are two level-$t$ rectangles $R_t,R'_t\subset R$ such that $\widetilde{a}\in g\varphi_{R_t}(K)$ and $\widetilde{b}\in g\varphi_{R'_t}(K)$, respectively, and (again by Corollary~\ref{cor:leftrightendpt}) $\pi_2(R_t)=\pi_2(R'_t)$ for all $t\geq k_0$ for some $k_0$. See Figure~\ref{fig:leftrightendpt} for an illustration. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[thick] (0,0) to (0,4); \draw[thick] (4,0) to (4,4); ll[pattern=dots] (-1,0) rectangle (0,4); ll[pattern=dots] (4,0) rectangle (5,4); \draw[thick] (1,0) to (1,4); \draw[thick] (1.4,0) to (1.4,4); ll[pattern=dots] (1,0) rectangle (1.4,4); \draw[thick] (1.7,0) to (1.7,4); \draw[thick] (2,0) to (2,4); ll[pattern=dots] (1.7,0) rectangle (2,4); \draw[thick] (2.3,0) to (2.3,4); \draw[thick] (2.5,0) to (2.5,4); ll[pattern=dots] (2.3,0) rectangle (2.5,4); \draw[thick] (2.65,0) to (2.65,4); \draw[thick] (3,0) to (3,4); ll[pattern=dots] (2.65,0) rectangle (3,4); \node at (1,2)[circle,fill, inner sep=1.5pt]{}; \node at (3,2.4)[circle,fill, inner sep=1.5pt]{}; \draw[thick,dashed] (-1.5,1.5) to (5.5,2.9); \node at (0.8,1.7) {$\widetilde{a}$}; \node at (3.2,2.1) {$\widetilde{b}$}; \node at (5.25,2.55) {$L_z$}; \draw[thick,red] (0.95,1.75) to (1.13,1.79) to (1.01,2.4) to (0.82,2.36) to (0.95,1.75); \draw[thick,red] (2.95,2.15) to (3.13,2.19) to (3.01,2.8) to (2.82,2.76) to (2.95,2.15); \node at (0.55,2.6) [red] {$g(R_t)$}; \node at (3.5,2.9) [red] {$g(R'_t)$}; \end{tikzpicture} \caption{An illustration of $\widetilde{a},\widetilde{b},g(R_t)$ and $g(R'_t)$, where $(K^z+c)\times\R$ is supported in the shaded region. (color online)} \label{fig:leftrightendpt} \end{figure} Pick $x,x'\in K$ with $\pi_2(x)=\max\pi_2(K)$ and $\pi_2(x')=\min\pi_2(K)$. Since $\widetilde{a}\in g\varphi_{R_t}(K)$, $\widetilde{b}\in g\varphi_{R'_t}(K)$, $x$ can be expressed as \begin{equation}\label{eq:expressionofx} x = (g\varphi_{R_t})^{-1}(\widetilde{a})+ \Big(\begin{matrix} c_t \\ d_t \end{matrix}\Big) = (g\varphi_{R'_t})^{-1}(\widetilde{b})+\Big(\begin{matrix} \gamma_t \\ \eta_t \end{matrix}\Big) \end{equation} for some $-1\leq c_t,d_t,\gamma_t,\eta_t\leq 1$. Since $\pi_2(x)$ is the maximum, $d_t,\eta_t\geq 0$. Since $\widetilde{a}$, $\widetilde{b}$ are on the same slice and $\pi_2(R_t)=\pi_2(R'_t)$ for $t\geq k_0$, we have $d_t=\eta_t$ for $t\geq k_0$. Similarly, there are $-1\leq c'_t,\gamma'_t\leq 1$ and $-1\leq d'_t\leq 0$ such that for all $t\geq k_0$, \begin{equation}\label{eq:expressionofxprime} x' = (g\varphi_{R_t})^{-1}(\widetilde{a})+\Big(\begin{matrix} c'_t \\ d'_t \end{matrix}\Big) = (g\varphi_{R'_t})^{-1}(\widetilde{b})+\Big(\begin{matrix} \gamma'_t \\ \eta'_t \end{matrix}\Big). \end{equation} Fix a large $t\geq k_0$ (will be specified later). Note that \[ d_t-d'_t=\pi_2(x-x')=\max\pi_2(K)-\min\pi_2(K)=:\beta>0. \] So at least one of $d_t,-d'_t$ is no less than $\beta/2$. Writing $O=(\begin{smallmatrix} o_{1,1} & o_{1,2} \\ o_{2,1} & o_{2,2} \end{smallmatrix})$ to be the orthogonal part of $g$, we have by the obliqueness of $g$ that $o_{1,1},o_{1,2}\neq 0$. We distinguish four simple cases. \paragraph{{\bf Case 1}: $d_t\geq\beta/2$ and $o_{1,2}>0$} In this case, we have by~\eqref{eq:expressionofx} that \[ g\varphi_{R'_t}(x) - \widetilde{b} = \Big( \begin{matrix} o_{1,1} & o_{1,2} \\ o_{2,1} & o_{2,2} \end{matrix}\Big)\cdot\Big(\begin{matrix} \rho n^{-t}\gamma_t \\ \rho m^{-t}d_t \end{matrix}\Big). \] Since $d_t\geq\beta/2$, $n>m$ and $\|\gamma_t\|\leq 1$, $0<\rho n^{-t}\gamma_to_{1,1}+ \rho m^{-t}d_to_{1,2}<\delta$ when $t$ is large and hence \[ \pi_1(g\varphi_{R'_t}(x)) = \pi_1(\widetilde{b})+\rho n^{-t}\gamma_to_{1,1}+\rho m^{-t}d_to_{1,2} \in (\pi_1(\widetilde{b}), \pi_1(\widetilde{b})+\delta). \] In other words, $g\varphi_{R'_t}(x)$ lies in the tube $(\pi_1(\widetilde{b}),\pi_1(\widetilde{b})+\delta)\times\R$. But we have seen that this tube does not intersect $(K^{z}+c)\times\R\supset g\varphi_R(K)\ni g\varphi_{R'_t}(x)$, which leads to a contradiction. \paragraph{{\bf Case 2}: $d_t\geq\beta/2$ and $o_{1,2}<0$} In this case, we have by~\eqref{eq:expressionofx} that \[ g\varphi_{R_t}(x) - \widetilde{a} = \Big( \begin{matrix} o_{1,1} & o_{1,2} \\ o_{2,1} & o_{2,2} \end{matrix}\Big)\cdot\Big(\begin{matrix} \rho n^{-t}c_t \\ \rho m^{-t}d_t \end{matrix}\Big). \] Since $d_t\geq\beta/2$, $n>m$ and $\|c_t\|\leq 1$, we have $-\delta<\rho n^{-t}c_to_{1,1}+ \rho m^{-t}d_to_{1,2}<0$ when $t$ is large and hence \[ \pi_1(g\varphi_{R_t}(x)) = \pi_1(\widetilde{a})+ \rho n^{-t}c_to_{1,1}+ \rho m^{-t}d_to_{1,2} \in (\pi_1(\widetilde{a})-\delta, \pi_1(\widetilde{a})). \] In other words, $g\varphi_{R_t}(x)$ lies in the tube $(\pi_1(\widetilde{a})-\delta,\pi_1(\widetilde{a}))\times\R$. Again, we have seen that this tube does not intersect $(K^{z}+c)\times\R\supset g\varphi_R(K)\ni g\varphi_{R_t}(x)$, which leads to a contradiction. \paragraph{{\bf Case 3}: $-d'_t\geq\beta/2$ and $o_{1,2}>0$} Applying the argument in Case 2 to~\eqref{eq:expressionofxprime}, we have \[ \pi_1(g\varphi_{R_t}(x')) = \pi_1(\widetilde{a})+ \rho n^{-t}c'_to_{1,1}+\rho m^{-t}d'_to_{1,2} \in (\pi_1(\widetilde{a})-\delta, \pi_1(\widetilde{a})) \] when $t$ is large, which leads to a contradiction. \paragraph{{\bf Case 4}: $-d'_t\geq\beta/2$ and $o_{1,2}<0$} Applying the argument in Case 1 to~\eqref{eq:expressionofxprime}, we have \[ \pi_1(g\varphi_{R'_t}(x')) = \pi_1(\widetilde{b})+\rho n^{-t}\gamma'_to_{1,1}+ \rho m^{-t}d'_to_{1,2} \in (\pi_1(\widetilde{b}), \pi_1(\widetilde{b})+\delta) \] when $t$ is large, which leads to a contradiction. \end{proof} For the isometry case, we need a rescaling lemma of which the argument is analogous to the one of Lemma~\ref{lem:dep1}. \begin{lemma} Let $f\in\S(\R^2)$ be an oblique isometry. If $f(K)\subset K$, then we can find some $z\in\pi_2(K)$ and a nondegenerate scaled copy $E$ of $E(m,J)$ such that $E\subset K^z$. \end{lemma} \begin{proof} Pick any sequence $\{R_k\}_{k=1}^\infty$, where each $R_k$ is a level-$k$ rectangle and $R_{k+1}\subset R_k$ for all $k\geq 1$. Since $f$ is an isometry, we have \[ \max\{\|\pi_1(f(R_k))\|,\|\pi_2(f(R_k))\|\} < 2m^{-k}, \quad \forall k\geq 1. \] As a result, $f(R_{pk})$ is contained in at most $4$ squares of side length $m^{-pk}=n^{-qk}$. More precisely, there are $x\in n^{-qk}\mathbb{Z}, y\in m^{-pk}\mathbb{Z}$ such that \[ f\varphi_{R_{pk}}(K) \subset (x,x+2n^{-qk}) \times (y,y+2m^{-pk}) =: Q_k. \] Denote by $h_k$ the homothety sending $(0,1)^2$ to the square $(x,x+n^{-qk}) \times (y,y+m^{-pk})=:Q'_k$, the ``left bottom square'' of $Q_k$. Since $E(m,J)=\pi_2(K)$, it is easy to see that $d_H(\varphi_{R_{pk}}(K),\varphi_{R_{p_k}}(\{0\}\times E(m,J)))\leq n^{-pk}$, where $d_H$ denotes the Hausdorff distance (see~\cite{Fal14}). So \begin{equation}\label{eq:tendstotheleftproj} d_H(h_k^{-1}f\varphi_{R_{pk}}(K), h_k^{-1}f\varphi_{R_{pk}}(\{0\}\times E(m,J))) \leq n^{(q-p)k} \to 0, \quad k\to\infty. \end{equation} Moreover, writing $O$ to be the orthogonal part of $f$, each $h_k^{-1}f\varphi_{R_{pk}}(\{0\}\times E(m,J))$ is simply a translated copy of $O(\{0\}\times E(m,J))$, say $O(\{0\}\times E(m,J))+a_k$. So~\eqref{eq:tendstotheleftproj} becomes \[ d_H(h_k^{-1}f\varphi_{R_{pk}}(K), O(\{0\}\times E(m,J))+a_k) \leq n^{(q-p)k} \to 0, \quad k\to\infty. \] By the definition of $h_k$, the sequence $\{a_k\}_k$ lies in a bounded region in the plane. For notational simplicity, let us assume without loss of generality that $\{a_k\}_k$ is convergent, say $a_k\to a$. Then under the Hausdorff distance, \[ h_k^{-1}f\varphi_{R_{pk}}(K) \to O(\{0\}\times E(m,J))+a, \quad k\to\infty. \] Applying the rescaling argument in the proof of Lemma~\ref{lem:dep1} (now to $\{h_k^{-1}f\varphi_{R_{pk}}\}$ instead of $\{h_t^{-1}f^t\}_t$), we can find at most four points $z_{i,j}$, $0\leq i,j\leq 1$, such that \[ O(\{0\}\times E(m,J))+a \subset \bigcup_{0\leq i,j\leq 1} (K^{z_{i,j}}+i)\times\R. \] Since $O$ is oblique, projecting the above two sets to the $x$-axis, we see that $\bigcup_{0\leq i,j\leq 1} (K^{z_{i,j}}+i)$ contains a nondegenerate scaled copy of $E(m,J)$. Then by Baire's theorem, some $K^{z_{i,j}}+i$ must contain an interior part of that copy of $E(m,J)$. Since $E(m,J)$ is self-similar, this completes the proof. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:dependentaffine}: the isometry case] By Remark~\ref{rem:deindwork} and Proposition~\ref{prop:noobliquelines}, it suffices to consider when $N\leq n-1$ and $\#J\leq m-1$. Since $K$ is not contained in any line, $\#J>1$. Thus $E(m,J)$ is a deleted-digit set with Hausdorff dimension $\frac{\log\#J}{\log m}\in(0,1)$. Assume on the contrary that $f$ is oblique. Note that \[ f(K^y\times\{y\}) \subset f(K)\subset K \subset \R\times\pi_2(K) = \R\times E(m,J), \quad \forall y\in\pi_2(K). \] Therefore, $\pi_2(f(K^y\times\{y\})) \subset E(m,J)$ for all $y$. Since $f$ is oblique, the left hand side is a nondegenerate scaled copy of $K^y$. Picking $y_$ as in the proof of Theorem~\ref{thm:dependentaffine} (before~\eqref{eq:thenewlogformula}), we have $\dimh E(m,J)\geq\dimh K^{y_}=\frac{\log N}{\log n}=:\alpha$. Pick $z\in\pi_2(K)$ according to the above lemma. So there are also $\beta_2,c_2\in\R$ such that $E(m,J)\subset\beta_2K^z+c_2$. On the other hand, we have seen that $E(m,J)$ contains a scaled copy of $K^z$, say $\beta_1K^z+c_1$. So $\dimh E(m,J)\leq\dimh K^z\leq\alpha$ and hence $\dimh E(m,J)=\alpha$. Since $\alpha>0$, $N\geq 2$. Since $\h^\alpha(E(m,J))>0$, $\h^\alpha(K^z)\in(0,\infty)$ (again, the $<\infty$ part is an observation before Lemma~\ref{lem:bunchoffacts}). From $f(K^z\times\{z\})\subset\R\times E(m,J)\subset \R\times (\beta_2 K^z+c_2)$ and the above observation, we can apply Corollary~\ref{cor:leftrightendpt} and argue as in the second part of the proof of Theorem~\ref{thm:dependentaffine} (the one after~\eqref{eq:thenewlogformula}) and obtain a contradiction (the only difference is that in the previous proof, we embedded some oblique slice into a union of vertical lines, whereas now we embed some oblique slice into a union of horizontal lines).\end{proof} Combining with these results and Proposition~\ref{prop:allnthenrational}, we can quickly prove Theorem~\ref{thm:main2}. \begin{proof}[Proof of Theorem~\ref{thm:main2}] Let $K,n,m,f$ be as in Theorem~\ref{thm:main2}. If $\lambda=1$ then there is nothing to prove. If $\lambda<1$, then Theorem~\ref{thm:dependentaffine} tells us that $f$ is not oblique. So it is easy to find some integer $k\geq 1$ such that $f^k(x)=\lambda^kx+\eta$ for some $\eta\in\R^2$, i.e., the orthogonal part of $f^k$ is simply the identity matrix. Note that $f^k$ sends the $x$-axis to a horizontal line. If $\frac{\log\lambda}{\log n}\notin\Q$, we can apply Lemma~\ref{lem:dep1} to $f^k$ and get (recall the second paragraph in Remark~\ref{rem:bairetofindint}) some $g\in\S(\R^2)$ of which the contraction ratio $\rho$ is an irrational power of $n$. Also, since $f^k$ involves no rotations nor reflections, so does $g$. However, if $2\leq N\leq n-1$, applying the argument in the proof of the above theorem (where now $v=(1,0)$), we can still arrive at~\eqref{eq:thenewlogformula} and hence $\frac{\log\rho}{\log n}\in\Q$. This is a contradiction. If $N=1$, then every row contains at most one selected rectangle in the initial pattern of $K$. In particular, $\#\Lambda= \#J\leq m<n$. So there must be a vacant column. On the other hand, since $\#\Lambda\geq 2$ and $K$ is not contained in any vertical line, $\#I\geq 2$. Therefore, $\pi_1(K)=E(n,I)$ is a deleted-digit set with Hausdorff dimension $0<\frac{\log\#I}{\log n}\leq\frac{\log\#\Lambda}{\log n}<1$. Since $f^k(K)\subset K$, $\pi_1(f^k(K)) \subset \pi_1(K)=E(n,I)$. So \[ \lambda^kE(n,I)+\pi_1(\eta) = \lambda^k\pi_1(K)+\pi_1(\eta) = \pi_1(f^k(K)) \subset E(n,I). \] Then it follows from Lemma~\ref{lem:fw09} that $\frac{\log\lambda}{\log n}\in\Q$. Finally, assume that $N=n$. Let $J':=\{j\in J: \#I_{j}=n\}$. Note that \begin{equation}\label{eq:ycontainsintervals} \{y\in\pi_2(K): K^y \text{ contains an interval}\} \subset \bigcup_{t\geq 0}\bigcup_{z\in\Z} \frac{E(m,J')+z}{n^t}. \end{equation} The case when $\#J'=1$ has been settled in~\cite[Section 6.6]{AH19}. If $\#J'\geq 2$ (clearly, $\#J'\leq m-1$), then $0<\dimh E(m,J')<1$. Note that $K^y$ is an interval of length $1$ whenever $y\in E(m,J')$. Since $f^k(K)\subset K$, it follows from~\eqref{eq:ycontainsintervals} that \[ \lambda^k E(m,J')+\pi_2(\eta) \subset \bigcup_{t\geq 0}\bigcup_{z\in\Z} \frac{E(m,J')+z}{n^t}. \] By Baire's theorem, there are some $t,z$ such that $\frac{E(m,J')+z}{n^t}$ contains an interior part of $\lambda^k E(m,J')+\pi_2(\eta)$. By Lemma~\ref{lem:fw09}, $\frac{\log\lambda}{\log n}\in\Q$, which completes the proof. \end{proof} \section{The dependent self-similar case} Finally, we consider the case when $m=n$, that is, $K=K(n,\Lambda)$ is a generalized Sierpi\'nski carpet. However, there does exist such a self-similar carpet allowing oblique embeddings. \begin{example}\label{exa:nonobsiercat} Consider the carpet \[ K = \Big\{ \sum_{k=1}^\infty \frac{(x_k,y_k)}{4^k}: (x_k,y_k)\in\{(0,1),(1,3),(2,0),(3,2)\} \Big\}. \] See Figure~\ref{fig:specialcarpet}. Let $O=\big(\begin{smallmatrix} \frac{3}{5} & -\frac{4}{5}\\ -\frac{4}{5} & -\frac{3}{5} \end{smallmatrix}\big)$, which is the matrix for the reflection with respect to some line of slope $-1/2$. It is easy to check that \begin{align} OK &+ ( \tfrac{9}{5}, \tfrac{18}{5} ) \\ &= \Big\{ \sum_{k=1}^\infty \frac{(\frac{3}{5}x_k-\frac{4}{5}y_k+\frac{9}{5},-\frac{4}{5}x_k-\frac{3}{5}y_k+\frac{18}{5})}{4^k}: (x_k,y_k)\in\{(0,1),(1,3),(2,0),(3,2)\} \Big\} \\ &= \Big\{ \sum_{k=1}^\infty \frac{(x_k,y_k)}{4^k}: (x_k,y_k)\in\{(0,1),(1,3),(2,0),(3,2)\} \Big\} = K. \end{align} Therefore, $K$ allows an oblique self-embedding $f(x)=Ox+(\frac{9}{5},\frac{18}{5})$. \begin{figure}[htbp] \centering \includegraphics[width=3.8cm]{specialcarpet.eps} \caption{A carpet allowing oblique self-embeddings} \label{fig:specialcarpet} \end{figure} \end{example} In the rest of this section, we only consider rotational self-embeddings of $K$ and hope to obtain some information on the rotation angle. Recall the notation~\eqref{eq:varphiij}. The following result is well known (see e.g.~\cite{SW99}). \begin{lemma}\label{lem:convandfixpt} If $K$ is not contained in any line, then the convex hull of $K$ is a polygon. Moreover, every vertex of this polygon is the fixed point of $\varphi_{i,j}$ for some $(i,j)\in\Lambda$. \end{lemma} From now on, let us fix a generalized Sierpi\'nski carpet $K$ satisfying the strong separation condition and not supported in any line. Denote by $P$ the convex hull of $K$. By the above lemma, $P$ is a polygon with finitely many vertices, say $v_1,\ldots,v_p$. For $1\leq t\leq p$, we write $\alpha_t$ to be the interior angle of $P$ at the vertex $v_t$, and write $\ell_t$ to be the edge of $P$ joining $v_t$ and $v_{t+1}$. We claim that every $v_t$ is not isolated in $K$ even along its adjacent edges. \begin{corollary}\label{cor:limitptonell} For $1\leq t\leq p$, there exists $\{x_k\}_{k=1}^\infty\subset\ell_t\cap K$ such that $x_k\to v_t$ as $k\to\infty$. \end{corollary} \begin{proof} Fix any $1\leq t\leq p$. By Lemma~\ref{lem:convandfixpt}, there is some $(i,j)\in\Lambda$ such that $v_t$ is the fixed point of $\varphi_{i,j}$. Then \[ v_t -\varphi_{i,j}^k(v_{t+1}) = \varphi_{i,j}^k(v_{t})-\varphi_{i,j}^k(v_{t+1}) = n^{-k}(v_t-v_{t+1}), \quad \forall k\geq 1. \] In particular, $\varphi_{i,j}^k(v_{t+1})$ must lie on $\ell_t$ and tends to $v_t$ as $k\to\infty$. \end{proof} \begin{corollary}\label{cor:rationaltangent} For each $1\leq t\leq p$, $\tan\alpha_t\in\Q$. \end{corollary} \begin{proof} For $1\leq t\leq p$, we again find by Lemma~\ref{lem:convandfixpt} some $(i_t,j_t)\in\Lambda$ such that $v_t$ is the fixed point of $\varphi_{i_t,j_t}$. Therefore, $v_t=(\frac{i_t}{n-1},\frac{j_t}{n-1})$. This implies that the angle made by every edge $\ell_t$ and the $x$-axis has a rational tangent. It follows directly that $\tan\alpha_t\in\Q$ for all $t$. \end{proof} A ``local openness'' result as follows will be of great help. \begin{lemma}[{\cite[Theorems~4.5 and 4.9]{EKM10}}]\label{lem:ekmtworesults} Let $f\in\S(\R^2)$. If $f(K)\subset K$, then there is an open set $U\subset\R^2$ such that $U\cap f(K)=U\cap K\neq\varnothing$. Moreover, $\{O^t\}_{t=1}^\infty$ is a finite collection, where $O$ is the orthogonal part of $f$. \end{lemma} \begin{proposition}\label{prop:rationalangles} If $f\in\S(\R^2)$ sends $K$ into itself, then for every $1\leq t\leq p$, we can find $t'$ and two squares $R,R'$ (of level $k$ and $k'$, respectively) such that the following properties hold: \begin{enumerate} \item $f\varphi_{R'}(v_{t'})=\varphi_R(v_t)$, \item either $f\varphi_{R'}(\ell_{t'})\subset\varphi_R(\ell_t)$ or $f\varphi_{R'}(\ell_{t'+1})\subset\varphi_{R}(\ell_{t})$. \end{enumerate} \end{proposition} \begin{proof} Fix $1\leq t\leq p$. Without loss of generality, assume that $v_t$ lies in the ``left bottom area'' of $[0,1]^2$, that is, $v_t\in[0,1/2]^2$ (other cases can be similarly discussed). Let us pick an integer $k$ and a level-$k$ square $R=[\frac{i}{n^k},\frac{i+1}{n^k}]\times[\frac{j}{n^k},\frac{j+1}{n^k}]$ such that $\varphi_R(K)\subset f(K)$ and \[ K \cap \big( (\tfrac{i-1}{n^k},\tfrac{i}{n^k})\times(\tfrac{j-1}{n^k},\tfrac{j+1}{n^k}) \big) = \varnothing \quad\text{and}\quad K \cap \big( (\tfrac{i}{n^k},\tfrac{i+1}{n^k})\times(\tfrac{j-1}{n^k},\tfrac{j}{n^k}) \big) = \varnothing. \] The existence of $R$ will be proved at the end of this proof. As a consequence, there is a small $r>0$ such that \begin{equation}\label{eq:localareaofphirvt} K \cap B(\varphi_R(v_t),r) = \varphi_R(K) \cap B(\varphi_R(v_t),r) = K \cap \varphi_R(P) \cap B(\varphi_R(v_t),r), \end{equation} where $B(x,r)$ denotes the ball centered at $x$ and of radius $r$. Since $\varphi_R(v_t)\in\varphi_R(K)\subset f(K)$, there is a level-$k'$ square $R'$ (for some large $k'$) such that $\varphi_R(v_t)\in f\varphi_{R'}(K)$. We claim that $\varphi_R(v_t)$ is a vertex of the polygon $f\varphi_{R'}(P)$ if $k'$ is picked large. In particular, there exists $t'$ such that $\varphi_R(v_t)=f\varphi_{R'}(v_{t'})$, which establishes (1). To see the claim, note that since $P$ is the convex hull of $K$, $\varphi_R(v_t)\in f\varphi_{R'}(K)\subset f\varphi_{R'}(P)$. So if $k'$ is large enough, then \begin{equation}\label{eq:fvarphirinball} f\varphi_{R'}(P) \subset B(\varphi_{R}(v_t),r). \end{equation} Therefore, \begin{align} f\varphi_{R'}(K) &\subset f(K) \cap f\varphi_{R'}(P) && \text{(since $K\subset P$)} \\ &\subset K\cap B(\varphi_R(v_t),r) && \text{(by $f(K)\subset K$ and~\eqref{eq:fvarphirinball})} \\ &\subset \varphi_R(P) &&\text{(by~\eqref{eq:localareaofphirvt})}. \end{align} So $f\varphi_{R'}(P)\subset\varphi_R(P)$. Since $f\varphi_{R'}(P)$, $\varphi_R(P)$ are both polygons and $\varphi_R(v_t)\in f\varphi_{R'}(P)$, $\varphi_R(v_t)$ must be a vertex of $f\varphi_{R'}(P)$. If (2) is false, we see by (1) that $\varphi_{R}(\ell_t)$ meets $f\varphi_{R'}(P)$ only at the vertex $\varphi_R(v_t)$. By Corollary~\ref{cor:limitptonell}, there is a sequence $\{x_p\}_{p=1}^\infty\subset\ell_t\cap K$ such that $x_p\to v_t$ as $p\to\infty$. So $\varphi_R(x_p)\to\varphi_R(v_t) \in f\varphi_{R'}(K)$ and $\varphi_R(x_p)\in\varphi_R(\ell_t)$. Since $\varphi_{R}(\ell_t)$ meets $f\varphi_{R'}(P)$ only at the vertex $\varphi_R(v_t)$, $\varphi_R(x_p)\notin f\varphi_{R'}(K)$. But recalling that $\varphi_R(K)\subset f(K)$, we have $\varphi_R(x_p)\in f(K)$ for all $p$. This implies that \[ \dist\big( f\varphi_{R'}(K), f(K)\setminus f\varphi_{R'}(K) \big) \leq \inf_p\dist(f\varphi_{R'}(K),\{\varphi_R(x_p)\})= 0, \] which contradicts the strong separation condition. It remains to prove the existence of $R$. By Lemma~\ref{lem:ekmtworesults}, there is some $k_0$ and a level-$k_0$ square $Q$ such that $\varphi_Q(K)\subset f(K)$. Since $K$ is not contained in any line, we have $\#I\geq 2$ and $\#J\geq 2$ (here we abuse slightly these notations from Section 2.1). Iterating the initial pattern if necessary, we may assume that there are $(i_1,j_1),(i_2,j_2),(i_3,j_3)\in\Lambda$ such that \[ (i_1,j_1-1), (i_2-1,j_2),(i_3-1,j_3-1) \in \{0,1,\ldots,n-1\}^2\setminus\Lambda. \] Also, write $\underline{i}:=\min I$ and $\underline{j}:= \min J$. If there is some $1\leq i\leq n-1$ such that $(i,\underline{j})\in\Lambda$ but $(i-1,\underline{j})\notin\Lambda$, then it suffices to take $R=\varphi_Q\varphi_{i_1,j_1}\varphi_{i,\underline{j}}([0,1]^2)$. If such a digit does not exist, then $(0,\underline{j})\in\Lambda$. If $I_{\underline{j}}\supsetneq\{0\}$, then $2\leq\#I_{\underline{j}}\leq n-1$, where the upper bound holds because $K$ does not contain horizontal segments. So there are at least $4$ level-$2$ squares in the bottom row and it is not hard to see that at least one of them, say $Q'$, satisfies that the grid square of side $n^{-2}$ left adjacent to $Q'$ is contained in $[0,1]^2$ but unselected. Then it suffices to take $R=\varphi_Q\varphi_{i_1,j_1}(Q')$. So we may assume that $I_{\underline{j}}=\{0\}$. Similarly (looking at $(i_2,j_2)$ instead), it suffices to consider when $J_{\underline{i}}=\{0\}$. Since $(0,\underline{j})\in\Lambda$, $\underline{i}=0$. So $(0,0)\in\Lambda$ but $\{(0,j),(j,0):j\geq 1\}\cap\Lambda=\varnothing$. Then it is easy to see that taking $R=\varphi_Q\varphi_{i_3,j_3}\varphi_{0,0}([0,1]^2)$ would work. \end{proof} \begin{lemma}[Niven's theorem]\label{lem:niven} If $\theta\in\pi\Q$ and $\tan\theta\in\Q$, then $\tan\theta\in\{0,\pm 1\}$. \end{lemma} \begin{proof} For a proof, see~\cite{PV21}. \end{proof} \begin{corollary} Let $f(x)=\lambda R_\theta x+a$ be as in Theorem~\ref{thm:main3}. If $\frac{\theta}{\pi}\in\Q$ and $f$ is an oblique self-embedding similitude of $K$, then $\|\tan\theta\|=1$. \end{corollary} \begin{proof} By Lemma~\ref{lem:niven} and the obliqueness of $f$, it suffices to show that $\tan\theta\in\Q$. Fix any $1\leq t\leq p$ and pick $R,R',k,k',t'$ accordingly as in Proposition~\ref{prop:rationalangles}. Without loss of generality, assume that $f\varphi_{R'}(\ell_{t'})\subset\varphi_{R}(\ell_t)$, which is the first case in Proposition~\ref{prop:rationalangles}(2). \begin{figure}[htbp] \centering \begin{tikzpicture}[rotate=30, scale=0.5,>=stealth] \draw[thick] (0,0) to (-2,0); \draw[thick,red] (-2,0) to (-3,-2); \draw[thick] (-3,-2) to (-2.5,-2.8); \draw[thick] (-2.5,-2.8) to (-1.5,-4.8); \draw[thick,red] (-1.5,-4.8) to (0,-4.8); \node at (-3,-1) [red] {$\ell_{t'}$}; \node at (-0.5,-5.3) [red] {$\ell_t$}; \node at (-2,0)[circle,fill, inner sep=1.5pt]{}; \node at (-2.5,0.3) {$v_{t'}$}; \node at (-1.5,-4.8)[circle,fill, inner sep=1.5pt]{}; \node at (-1.8,-5.3) {$v_t$}; \draw[thick,dashed] (-3,-2) to (-3.8,-3.6); \draw[thick,dashed] (-1.5,-4.8) to (-1,-5.8); \coordinate (A) at (-2,0); \coordinate (B) at (-3,-2); \coordinate (C) at (-2.5,-2.8); \coordinate (D) at (-1.5,-4.8); \coordinate (E) at (0,0); \coordinate (F) at (0,-4.8); \draw pic["$\alpha_{t'}$", draw=black, -, angle eccentricity=2.3,angle radius=0.2cm]{angle = B--A--E}; \draw pic["$\alpha_{t'+1}$", draw=black, -, angle eccentricity=2.6,angle radius=0.2cm]{angle = C--B--A}; \draw pic["$\alpha_{t}$", draw=black, -, angle eccentricity=1.8,angle radius=0.2cm]{angle = F--D--C}; \end{tikzpicture} \caption{The rotation effect of $f$ (color online)} \label{fig:rotationoff} \end{figure} Since $\varphi_R(\ell_{t})$ is parallel to $\varphi_{R'}(\ell_{t})$ and $\varphi_{R'},\varphi_R$ involve no rotations nor reflections, $f$ sends a line parallel to $\ell_{t'}$ to another line parallel to $\ell_t$. Without loss of generality, assume that $t'<t$ and $v_1,\ldots,v_p$ are in counterclockwise order. Then a simple geometric observation (see Figure~\ref{fig:rotationoff}) tells us that $R_\theta$ can be realized as follows: first rotate the line containing $\ell_{t'}$ to the one containing $\ell_{t'+1}$, and then to the one containing $\ell_{t'+2}$ and so on until to $\ell_t$. Thus \[ \theta = \sum_{p=1}^{t-t'} (\pi-\alpha_{t'+p}) \pmod{2\pi}. \] By Corollary~\ref{cor:rationaltangent}, $\tan\theta\in\Q$ as desired. \end{proof} With all these observations in hand, Theorem~\ref{thm:main3} can be easily proved. \begin{proof}[Proof of Theorem~\ref{thm:main3}] Again, we may assume that $K$ is not supported in any line. By Lemma~\ref{lem:ekmtworesults}, $\frac{\theta}{\pi}\in\Q$. Then the statement follows directly from the above corollary. \end{proof} \bigskip \noindent{\bf Acknowledgements.} I thank Professor Huo-Jun Ruan for reading the manuscript carefully and pointing out several mistakes and a number of typos in an early version of it. I also thank Dingkun Hu for a nice observation regarding the setting of Condition A in the proof of Theorem~\ref{thm:notoblique} (under Case 3). \small \bibliographystyle{amsplain} \begin{thebibliography}{100} \bibitem{Alg201} A. Algom, Affine embeddings of Cantor sets in the plane, J. Anal. Math. {\bf 140} (2020), 695--757. \bibitem{Alg20} A. Algom, Slicing theorems and rigidity phenomena for self-affine carpets, Proc. Lond. Math. Soc. {\bf 121} (2020), 312--353. \bibitem{AH19} A. Algom and M. Hochman, Self embeddings of Bedford-McMullen carpets, Ergod. Th. \& Dynam. Sys. {\bf 39} (2019), 577--603. \bibitem{AW23} A. Algom and M. Wu, Improved versions of some Furstenberg type slicing theorems for self-affine carpets, Int. 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2412.02118v2	http://arxiv.org/abs/2412.02118v2	Algebraic properties of Indigenous semirings	\documentclass[psamsfonts]{amsart} \usepackage{amssymb,amsfonts} \usepackage[all,arc]{xy} \usepackage{enumerate} \usepackage{mathrsfs} \usepackage{graphicx} \usepackage{orcidlink} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{definitions}[theorem]{Definitions} \newtheorem{construction}[theorem]{Construction} \newtheorem{example}[theorem]{Example} \newtheorem{examples}[theorem]{Examples} \newtheorem{notation}[theorem]{Notation} \newtheorem{notations}[theorem]{Notations} \newtheorem{addendum}[theorem]{Addendum} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{warning}[theorem]{Warning} \newtheorem{scholium}[theorem]{Scholium} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\Nil}{Nil} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\FId}{FId} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\Max}{Max} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Sub}{Sub} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\zd}{zd} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\IG}{IG} \setcounter{section}{-1} \makeatletter \makeatother \numberwithin{equation}{section} \begin{document} \title{Algebraic properties of Indigenous semirings} \author[H. Behzadipour]{Hussein Behzadipour\orcidlink{0000-0001-7037-4110}} \address{Hussein Behzadipour\\ Department of Mathematics \\ Sharif University of Technology\\ Tehran\\ Iran} \email{[email protected]} \author[H. Koppelaar]{Henk Koppelaar\orcidlink{0000-0001-7487-6564}} \address{Henk Koppelaar\\ Faculty of Electrical Engineering, Mathematics and Computer Science\\ Delft University of Technology\\ Delft\\ The Netherlands} \email{[email protected]} \author[P. Nasehpour]{Peyman Nasehpour\orcidlink{0000-0001-6625-364X}} \address{Peyman Nasehpour\\ Education Department\\ The New York Academy of Sciences \\ New York, NY, USA} \email{[email protected]} \subjclass[2010]{16Y60, 13A15, 01A07.} \keywords{Indigenous semirings, Information algebras, Graph invariants} \begin{abstract} In this paper, we introduce Indigenous semirings and show that they are examples of information algebras. We also attribute a graph to them and discuss their diameters, girths, and clique numbers. On the other hand, we prove that the Zariski topology of any Indigenous semiring is the Sierpi\'{n}ski space. Next, we investigate their algebraic properties (including ideal theory). In the last section, we characterize units and idempotent elements of formal power series over Indigenous semirings. \end{abstract} \maketitle \section{Introduction} Inspired by the Indigenous number systems (cf. \cite{BenderBeller2021}, \cite{Everett2005}, \cite{Gordon2004}, \S1 in \cite{Ifrah2000}, and \cite{Vandendriessche2022}) and the Indigenous presemiring $\mathcal{M} = \{1,2,3,m\}$ proposed by Sibley in Example 5 on p. 419 of his educational book on abstract algebra \cite{Sibley2021}, we give a general definition for Indigenous semirings and investigate their algebraic properties. It turns out that the Indigenous semirings belong to a particular family of semirings called information algebras which have applications in various fields of science and engineering and have attracted the interests of some authors since 1972 (see Remark \ref{Informationalgebrasrem}). From an algebraic perspective, one could argue that a number is not its physical appearance or digit representation. A number is an abstract mathematical object, whereas its appearance is a sequence of symbols on a paper (or a sequence of bits in computer memory, or a sequence of sounds if read aloud). We never see a number itself, but always its representation. So, we become accustomed to identifying a number by its representation. In Ethnomathematics (one of the subjects of this paper) it is similar. This confusing identification exposes one of the difficulties of the field. Other difficulties are exemplified in Examples \ref{historicalexamplesIndigenous} of this paper. A plea to alleviate the study problems of indigenous number systems is to develop cultural tools for numerical cognition \cite{BenderBeller2018}. For instance, the Yapese indigenous money counting is based on stone disks called ``rai stones''. A typical ``rai stone'' is carved out of crystalline limestone and shaped like a disk with a hole in the center. The smallest may be 3.5 centimeters in diameter while the largest extant stone is 3.6 meters in diameter and 50 centimeters thick, and weighs 4,000 kilograms \cite{Gillilland1975}. This paper develops the underlying algebra (presemiring) of indigenous number systems to an unprecedented degree. Since the language of semiring-like algebraic structures is not standardized yet, we first introduce some terminology. Recall that a bimagma $(R,+,\cdot)$ is a ringoid \cite[p. 206]{Rosenfeld1968} if the binary operation ``$\cdot$'' (multiplication) distributes on the binary operation ``$+$'' (addition) from both sides. A ringoid $(R,+,\cdot)$ is a presemiring if $(R,+)$ is a commutative semigroup and $(R,\cdot)$ a semigroup \cite[Definition 4.2.1]{GondranMinoux2008}. A presemiring is commutative if $(R,\cdot)$ is a commutative semigroup. In this paper, all presemirings are supposed to be commutative. A presemiring $S$ is a semiring if it has a neutral element $0$ for its addition which is also an absorbing element for its multiplication and it has also a neutral element $1 \neq 0$ for its multiplication \cite[p. 1]{Golan1999(b)}. A semiring $S$ is entire if it is zero-divisor free, i.e., $ab = 0$ implies $a = 0$ or $b = 0$, for all $a$ and $b$ in $S$. A semiring $S$ is zerosumfree if $a+b = 0$ implies $a = b = 0$, for all $a$ and $b$ in $S$. A semiring $S$ is an information algebra if it is both zero-divisor free and zerosumfree \cite[p. 4]{Golan1999(b)}. A nonempty set $I$ of a semiring $S$ is an ideal of $S$ if $(I,+)$ is a submonoid of $(S,+)$ and $SI \subseteq I$ \cite{Bourne1951}. We collect all ideals of a semiring $S$ in $\Id(S)$. An ideal $M$ of a semiring $S$ is maximal if there is no ideal properly between $M$ and $S$. A semiring $S$ is local if it has a unique maximal ideal. An ideal $P$ of a semiring $S$ is prime if $P \neq S$ and $IJ \subseteq S$ implies $I \subseteq S$ or $J \subseteq S$, for all ideals $I$ and $J$ of $S$. All prime ideals of a semiring $S$ are collected in $\Spec(S)$. In a (commutative) semiring $S$, an ideal $P \neq S$ is prime if and only if $ab \in P$ implies either $a \in P$ or $b \in P$, for all elements $a$ and $b$ in $S$ \cite[Corollary 7.6]{Golan1999(b)}. The first section of the paper is devoted to some results for entire semirings. Recall that for each ideal $I$ of a semiring $S$ \[V(I) = \{P \in \Spec(S) : P \supseteq I\}.\] It is, then, easy to verify that $\mathcal{C} = \{V(I) : I \in \Id(S)\}$ is the family of closed sets for a topology on $X = \Spec(S)$, called the Zariski topology \cite[p. 89]{Golan1999(b)}. A topological space with exactly two points and three closed subsets is called the Sierpi\'{n}ski space (see \cite[p. 17]{ArensDugundji1951} and \cite[Exercise 1.7]{Rotman1988}). In Theorem \ref{Zariskitopologyentiresemiringwithtwoprimes}, we show that the Zariski topology of an entire semiring with exactly one nonzero prime ideal is the Sierpi\'{n}ski space. Then, we proceed to show that the localization of an information algebra is an information algebra (see Proposition \ref{Localizationofinformationalgebras}). Let us recall that if $S$ is a semiring, then $\Id(S)$ equipped with addition and multiplication of ideals is a semiring \cite[Proposition 6.29]{Golan1999(b)}. In Theorem \ref{Semiringidealsinformationalgebra}, we prove that $\Id(S)$ is an information algebra if and only if $S$ is an entire semiring. Inspired by Example 5 on p. 419 in \cite{Sibley2021}, in \S\ref{sec:indigenousgraphs} of our paper, we define Indigenous addition and multiplication on $I_k = \mathbb{N}_k \cup \{m\}$, where $\mathbb{N}_k$ is the set of positive integer numbers less than $k+1$ and $m$ is just a symbol standing for ``many'' (see Definition \ref{Indigenouspresemiringdef}), and next in Proposition \ref{Indigenoussemiringpro}, we show that the bimagma $(I_k, \oplus, \odot)$ is a unital presemiring. For some historical examples of Indigenous presemirings, check Examples \ref{historicalexamplesIndigenous}. In Definition \ref{Indigenousgraphs}, we attribute a graph $\IG_k$ to any Indigenous presemiring $I_k$ where its vertices' set is $I_k$ and $\{a,b\}$ is an edge of $\IG_k$ if $a \neq b$ are elements of $I_k$ with $ab = m$. In the rest of \S\ref{sec:indigenousgraphs}, we discuss some graph invariants of Indigenous graphs. For example in Theorem \ref{diamIG}, we prove that the Indigenous graph $\IG_k$ is a connected graph with $\diam(\IG_1) = 1$ and $\diam(\IG_k) = 2$ if $k > 1$. In Theorem \ref{girthIG}, we show that the girth of the Indigenous graph $\IG_k$ is $3$ if $k \geq 3$, and is infinity, otherwise. Finally in Theorem \ref{cliqueIG}, we prove that the clique number of the Indigenous graph $\IG_k$ is at least $\left\lfloor \frac{k}{2} \right\rfloor + 1$, for any positive integer $k$. By annexing $0$ to the Indigenous presemiring $I_k$, we define the Indigenous semiring $S_k$ and in Theorem \ref{GeneralpropertiesofIndigenoussemirings}, we discuss algebraic properties of the Indigenous semirings. In this theorem with 12 items, we show that, for instance, any Indigenous semiring $S_k$ is a local information algebra and $\{0\}$ and $S_k \setminus \{1\}$ are the only prime ideals. A corollary to this statement is that the Zariski topology of any Indigenous semiring $S_k$ is the Sierpi\'{n}ski space. Next, we verify that $S_k$ is not a semidomain (recall that a semiring $S$ is a semidomain \cite{Nasehpour2019} if $S\setminus\{0\}$ is a multiplicatively cancellative monoid). In the same result, we show that any Indigenous semiring is austere and discuss the algebra of the ideals of the Indigenous semirings. In fact, we show that $(\Id(S_k)\setminus\{\mathbf{0}\}, \cdot)$ is a nilpotent monoid with the absorbing element $\mathfrak{s}_k = \{0,m\}$, where by $\mathbf{0}$, we mean the zero ideal $\{0\}$ of $S_k$. In \S\ref{sec:distinguishedelements}, we characterize units and idempotent elements of polynomials and formal power series over the Indigenous semirings (see Proposition \ref{UnitsIndigenoussemirings}, Proposition \ref{IdempotentsIndigenoussemirings1}, and Theorem \ref{IdempotentsIndigenoussemirings2}). Golan's book \cite{Golan1999(b)} is a general reference for semiring theory, and our terminology closely follows it. \section{Some results in entire semirings}\label{sec:entiresemirings} \begin{theorem}\label{Zariskitopologyentiresemiringwithtwoprimes} The Zariski topology of an entire semiring with exactly one nonzero prime ideal is the Sierpi\'{n}ski space. \end{theorem} \begin{proof} Let $S$ be an entire semiring with exactly one nonzero prime ideal. This means that $\Spec(S) = \{\{0\},\mathfrak{m}\}$, with $\mathfrak{m} \neq \{0\}$. It follows that the only closed subsets of the Zariski topology of $S$ are \begin{itemize} \item $V(S) = \emptyset$, \item $V(I) = V(\mathfrak{m}) = \{\mathfrak{m}\}$, for any nonzero proper ideal $I$ of $S$. \item $V(\{0\}) = \{\{0\}, \mathfrak{m}\}$ \end{itemize} Therefore, the Zariski topology of $S$ has two points with three closed subsets which is the Sierpi\'{n}ski space. This completes the proof. \end{proof} \begin{remark}\label{Informationalgebrasrem} Due to the applications of information algebras, they configure an important family of semirings. Perhaps the oldest example for information algebras is the semiring of non-negative integer numbers $\mathbb{N}_0$ equipped with usual addition and multiplication of numbers. The term ``information algebra" was introduced by Jean Kuntzmann \cite{Kuntzmann1972}. Traditionally, information algebras had some applications in graph theory \cite{Kuntzmann1972} and the theory of discrete-event dynamical systems \cite[p. 7]{Golan2003}. The other example of information algebras is the min-plus semiring $(\mathbb{R} \cup \{+\infty\}, \oplus, \otimes)$ in which its operations are defined as \[a \oplus b = \min \{a,b\} \text{~and~} a \otimes b = a+b.\] The min-plus semiring has essential applications in the shortest path problem in optimization \cite[Example 1.22]{Golan1999(b)} and is used extensively in tropical geometry \cite[\S1.1]{MaclaganSturmfels2015}. Information algebras have attracted the interests of some authors working recently on factorization problems \cite{BaethGotti2020,BaethSampson2020,ChenZhaoLiu2015}. \end{remark} A nonempty subset $U$ of a semiring $S$ is multiplicatively closed if $(U,\cdot)$ is a submonoid of $(S,\cdot)$. The localization of a semiring $S$ at a multiplicatively closed set $U$ of $S$, denoted by $U^{-1}S$, is defined similar to its counterpart in commutative ring theory (for the details see \S5 in \cite{Nasehpour2018S}). \begin{proposition}\label{Localizationofinformationalgebras} Let $U \subseteq S \setminus \{0\}$ be a multiplicatively closed subset of a semiring $S$. Then, the following statements hold: \begin{enumerate} \item If $S$ is entire, then so is $U^{-1}S$. \item If $S$ is an information algebra, then so is $U^{-1}S$. \end{enumerate} \end{proposition} \begin{proof} (1): This is straightforward because $a/u = 0/1$ if and only if $a = 0$, for all $a \in S$ and $u \in U$. (2): Let $(a/u) + (b/v) = 0/1$. It follows that $va + ub = 0$. Since $S$ is zerosumfree, $va = 0$ and $ub = 0$. Now, since $u$ and $v$ are nonzero and $S$ is entire, $a$ and $b$ are both zero, and so, $a/u = 0/1$ and $b/v = 0/1$. Thus $U^{-1}S$ is an information algebra and the proof is complete. \end{proof} \begin{theorem}\label{Semiringidealsinformationalgebra} Let $S$ be a semiring. Then, the following statements hold: \begin{enumerate} \item\label{Semiringidealszerosumfree} $\Id(S)$ is a partially ordered zerosumfree and additively idempotent semiring. \item $\Id(S)$ is an information algebra if and only if $S$ is entire. \end{enumerate} \end{theorem} \begin{proof} (\ref{Semiringidealszerosumfree}): By Proposition 6.29 in \cite{Golan1999(b)}, $\Id(S)$ is zerosumfree and additively idempotent. By Proposition 2.3 in \cite{Nasehpour2018P}, $(\Id(S), \subseteq)$ is a partially ordered semiring. (2): Let $S$ be entire. If $I$ and $J$ are nonzero ideals, then there are nonzero elements $a \in I$ and $b \in J$. So, $ab \in IJ$ is nonzero showing that $\Id(S)$ is entire. Now, by the statement (\ref{Semiringidealszerosumfree}), $\Id(S)$ is an information algebra. Conversely, if $\Id(S)$ is an information algebra and $a$ and $b$ are nonzero elements of $S$, then the principal ideals $(a)$ and $(b)$ are nonzero. It follows that $(a)(b)$ is also a nonzero ideal of $S$. However, $(a)(b) = (ab)$ because $S$ is commutative. Therefore, $ab$ is nonzero, i.e., $S$ is entire and the proof is complete. \end{proof} \section{Indigenous presemirings and their graphs}\label{sec:indigenousgraphs} \begin{definition} A bimagma $(R,+,\cdot)$ is a (commutative) presemiring if $(R,+)$ is a commutative semigroup, $(R,\cdot)$ is a (commutative) semigroup, and $\cdot$ distributes on $+$ from both sides (see Definition 4.2.1 in \cite{GondranMinoux2008}). The presemiring $R$ is unital if there is an element $1$ in $R$ such that $r \cdot 1 = 1 \cdot r = r$, for all $r \in R$. \end{definition} \begin{example} The bimagma $(\mathbb{N},+,\cdot)$ is a unital presemiring. \end{example} Inspired by Example 5 on p. 419 in \cite{Sibley2021}, we give the following definition: \begin{definition}\label{Indigenouspresemiringdef} Let $k$ be a positive integer. Set $I_k = \mathbb{N}_k \cup \{m\}$, where $m$ is just a symbol standing for ``many'' and is not in $\mathbb{N}_k$. Define Indigenous addition and multiplication on $I_k$ as follows: If $a, b \in \mathbb{N}_k$, then \[ a \oplus b = \begin{cases} a+b & \text{if $ a+b \leq k$} \\ m & \text{if $a+b > k$} \end{cases} \text{, and~} a \odot b = \begin{cases} ab & \text{if $ab\leq k$} \\ m & \text{if $ab > k$} \end{cases} \] and if either $a= m$ or $b = m$, then \[a \oplus b = a \odot b = m.\] \end{definition} \begin{proposition}\label{Indigenoussemiringpro} The bimagma $(I_k, \oplus, \odot)$ defined in Definition \ref{Indigenouspresemiringdef} is a unital presemiring and an epimorphic image of the presemiring $\mathbb{N}$. \end{proposition} \begin{proof} It is easy to see that $(I_k, \oplus)$ is a commutative semigroup and $(I_k,\odot,1)$ is a commutative monoid. Now, let $a$, $b$, and $c$ be elements of $I_k$. If $ab + ac$ is less than $k+1$, then $ab$ and $ac$ are also less than $k+1$, and so, the distributive laws hold because the computation is done in natural numbers. If at least one of the elements $ab$ and $ac$ are greater than $k$, then their addition is $m$ and we have \[a \odot (b \oplus c) = m = (a \odot b) \oplus (a \odot c).\] This means that $(I_k, \oplus, \odot)$ is a unital presemiring. It is easy to check that the function $f: \mathbb{N} \rightarrow I_k$ defined by \[f(x) = \begin{cases} x & \text{if $x \leq k$} \\ m & \text{if $x>k$} \end{cases}.\] is a presemiring epimorphism and the proof is complete. \end{proof} \begin{examples}\label{historicalexamplesIndigenous} Let $k$ be a positive integer number. We call the presemiring $I_k$ an Indigenous presemiring of order $k$. In the following, we give some examples mainly discussed in the literature. \begin{enumerate} \item Gordon illustrated that the Pirah\~{a} applied a numerical vocabulary corresponding to the terms ``h\'{o}i'' (for ``one''), ``ho\'{i}'' (for ``two''), and ``baagiso'' (for ``many'') \cite{Gordon2004}. One may correspond this to the Indigenous presemiring of order $2$. \item (Sibley's Indigenous presemiring) As we have already explained, we were inspired by an example given on p. 419 in Sibley's book \cite{Sibley2021}. Sibley's Indigenous presemiring is the Indigenous presemiring of order $3$. \item On p. 5 in his book \cite{Ifrah2000} on the universal history of numbers, Ifrah explains that the Botocudos had only two real terms for numbers: one for ``one'', and the other for ``a pair''. With these lexical items they could manage to express three and four by saying something like ``one and two'' and ``two and two''. However, these people had as much difficulty conceptualizing a number above four. In fact, for larger numbers, some of the Botocudos just pointed to their hair as if they were trying to say there are as ``many'' as there are hairs on their head. This may correspond to the Indigenous presemiring of order $4$. \item In his academic book \cite{Sommerfelt1938}, Sommerfelt reports that the Aranda had only two number terms, ``ninta'' for one, and ``tara'' for two. Three and four were expressed as ``tara-mi-ninta'' (i.e., two and one) and ``tara-ma-tara'' (i.e., two and two), respectively, and the number series of Aranda stopped there. For larger quantities, imprecise terms resembling ``a lot'', ``several'', and so on were used. This may also correspond to the Indigenous presemiring of order $4$. \end{enumerate} \end{examples} \begin{definition}\label{Indigenousgraphs} We define the Indigenous graph $\IG_k$ as follows: \begin{enumerate} \item The set of the vertices of $\IG_k$ is $I_k = \mathbb{N}_k \cup \{m\}$. \item The doubleton $\{a,b\}$ is an edge of $\IG_k$ if $a \neq b$ and their multiplication $a \odot b$ equals $m$. \end{enumerate} \end{definition} Let us recall that the diameter of a graph $G$, denoted by $\diam(G)$, is the greatest distance between two vertices of $G$ \cite[\S3.1.7]{BondyMurty2008}. \begin{theorem}\label{diamIG} For any positive integer $k$, the Indigenous graph $\IG_k$ is a connected graph with $\diam(\IG_1) = 1$ and $\diam(\IG_k) = 2$ if $k > 1$. \end{theorem} \begin{proof} The Indigenous graph $\IG_1$ has only two vertices $1$ and $m$ and they are connected because $1 \odot m = m$. Therefore, $\IG_1$ is the complete graph $K_2$ which is a connected graph with $\diam(\IG_1) = 1$. Now, let $k \geq 2$ and $a \neq b$ be arbitrary elements in $\mathbb{N}_k$. Since $a \odot m = b \odot m = m$, $a$ is connected to $m$ and $m$ is connected to $b$. Therefore, $\IG_k$ is a connected graph with $\diam(\IG_k) \leq 2$. However, $1$ and $k$ are not connected. So, $\diam(\IG_k) > 1$. This completes the proof. \end{proof} Let us recall that if a graph $G$ has at least one cycle, the length of a shortest cycle is its girth \cite[p. 42]{BondyMurty2008}. If a graph has no cycle, its girth is defined to be infinity. The girth of a graph is usually denoted by $g(G)$. \begin{theorem}\label{girthIG} The girth of the Indigenous graph $\IG_k$ is $3$ if $k \geq 3$, and is infinity, otherwise. \end{theorem} \begin{proof} Let $k \geq 3$. Then, the vertices $k$ and $k-1$ are connected because $k(k-1) > k$ in $\mathbb{N}$, and so, $k \odot (k-1) = m$ in $I_k$. Therefore, the triangle with the vertices $k-1$, $k$, and $m$ is a subgraph of $\IG_k$, and so, $g(\IG_k) = 3$. It is evident that the graph $\IG_1$ has no cycle. Also, in the graph $\IG_2$, the vertices $1$ and $2$ are not connected and again $\IG_2$ has no cycle. This completes the proof. \end{proof} Recall that a clique of a graph is a set of mutually adjacent vertices, and that the maximum size of a clique of a graph $G$, the clique number of $G$, is denoted by $\omega(G)$ \cite[p. 296]{BondyMurty2008}. \begin{proposition}\label{cliqueIGleq4} The clique number of $\IG_1$, $\IG_2$, $\IG_3$, and $\IG_4$ is $2$, $2$, $3$, and $4$, respectively. \end{proposition} \begin{proof} We compute the clique number of $\IG_k$ for $k \leq 4$ as follows: \begin{enumerate} \item In $\IG_1$, the vertices $1$ and $m$ are adjacent and its clique number of $\IG_1$ is $2$. \item In $\IG_2$, the vertex $m$ is adjacent to the vertices $1$ and $2$ but the vertices $1$ and $2$ are not adjacent. So, the clique number of $\IG_2$ is again $2$. \item In $\IG_3$, the vertices $m$, $2$, and $3$ are mutually adjacent while $1$ is not connected to $2$ and $3$. It follows that the clique number of $\IG_3$ is $3$. \item In $\IG_4$, the vertices $m$, $2$, $3$, and $4$ are mutually adjacent while $1$ is not connected to $2$, $3$, and $4$. This means that the clique number of $\IG_4$ is $4$. \end{enumerate} This completes the proof. \end{proof} \begin{theorem}\label{cliqueIG} The clique number of the Indigenous graph $\IG_k$ is at least $\left\lfloor \frac{k}{2} \right\rfloor + 1$, for any positive integer $k$. \end{theorem} \begin{proof} In view of Proposition \ref{cliqueIGleq4}, the inequality $\omega(\IG_k) \geq \left\lfloor \frac{k}{2} \right\rfloor + 1$ holds for all $k \leq 4$. Now, let $k \geq 5$. We distinguish two cases. Case I. If $k = 2 \alpha$ is an even number with $\alpha \geq 3$, then \[\left(k - \left\lfloor \frac{k}{2} \right\rfloor \right)^2 - k = \alpha^2 - 2\alpha > 0.\] Case II. If $k = 2\alpha + 1$ is an odd number with $\alpha \geq 2$, then \[\left(k - \left\lfloor \frac{k}{2} \right\rfloor \right)^2 - k = (\alpha+1)^2 - (2\alpha + 1) = \alpha^2 > 0.\] Therefore, all vertices \[\left(k - \left\lfloor \frac{k}{2} \right\rfloor\right), \left(k - \left\lfloor \frac{k}{2} \right\rfloor\right) + 1, \dots, k, m\] are mutually adjacent to each other in $\IG_k$. Thus $\omega(\IG_k) \geq \left\lfloor \frac{k}{2} \right\rfloor + 1$ for any positive integer number $k$ and the proof is complete. \end{proof} Recall that the smallest number of colors needed to color the vertices of a graph $G$ such that no two adjacent vertices share the same color is called the chromatic number of $G$, denoted by $\chi(G)$ \cite[p. 358]{BondyMurty2008}. It is clear that $\chi(G) \geq \omega(G)$ \cite[p. 359]{BondyMurty2008}. So, we have the following corollary: \begin{corollary}\label{chromaticIG} The chromatic number of the Indigenous graph $\IG_k$ is at least $\left\lfloor \frac{k}{2} \right\rfloor + 1$, for any positive integer $k$. \end{corollary} \section{Indigenous semirings and their ideals}\label{sec:indigenousideals} \begin{proposition}\label{presemiringembedhemiring} Any presemiring can be embedded into a hemiring. \end{proposition} \begin{proof} Let $E$ be a presemiring and suppose that $0 \notin E$. Set $E^{\prime} = E \cup \{0\}$ and define $+^{\prime}$ and ${\cdot}^{\prime}$ on $E^{\prime}$ as follows: \begin{itemize} \item $a +^{\prime} b = a + b$ for all $a,b\in E$ and $a +^{\prime} 0 = 0 +^{\prime} a = a$ for all $a\in E^{\prime}$. \item $a {\cdot}^{\prime} b = a \cdot b$ for all $a,b\in E$ and $a {\cdot}^{\prime} 0 = 0 {\cdot}^{\prime} a = 0$ for all $a\in E^{\prime}$. \end{itemize} One can easily check that $(E^{\prime}, +^{\prime}, {\cdot}^{\prime})$ is also a presemiring and the element $0$ is an identity element for addition and an absorbing element for multiplication. Now, define $\varphi : E \rightarrow E^{\prime}$ by $\varphi(x) = x$. It is clear that $\varphi$ is a presemiring monomorphism. This completes the proof. \end{proof} \begin{corollary}\label{Indigenoussemiringcor} The Indigenous presemiring $I_k$ can be embedded into the semiring $I_k \cup \{0\}$, for each $k \in \mathbb{N}$. \end{corollary} \begin{definition}\label{Indigenoussemiringdef} For any positive integer $k$, we call the semiring $I_k \cup \{0\}$ given in Corollary \ref{Indigenoussemiringcor}, the Indigenous semiring and denote it by $S_k$. \end{definition} \begin{theorem}\label{GeneralpropertiesofIndigenoussemirings} Let $k$ be a positive integer and $S_k$ the Indigenous semiring. Then, the following statements hold: \begin{enumerate} \item\label{IndigenoustotallyorderedIA} The semiring $S_k$ is a totally ordered information algebra with the smallest element $0$ and the largest element $m$. However, $S_k$ is not a semidomain. \item\label{LocalizationIndigenous} Let $k > 1$ and $U$ be a multiplicatively closed set in $S_k$ having a positive integer $a > 1$ and $0 \notin U$. Then, the localization $U^{-1} S_k$ of $S_k$ at $U$ is isomorphic to the Boolean semiring $\mathbb{B} = \{0,1\}$. \item\label{Indigenouslocal} $(S_k, S_k \setminus\{1\})$ is a local semiring. \item\label{misinanynonzeroideal} The set $\mathfrak{s}_k = \{0,m\}$ is the smallest nonzero ideal of the Indigenous semiring $S_k$. In particular, any nonzero ideal of $S_k$ possesses $m$. \item\label{Subtractiveidealsindigenoussemirings} The semiring $S_k$ is austere, i.e., the only subtractive ideals of $S_k$ are $\{0\}$ and $S_k$ (see p. 71 in \cite{Golan1999(b)}). \item\label{PrimeidealsIndigenoussemiring} The only prime ideals of the Indigenous semiring $S_k$ are $\{0\}$ and $S_k \setminus \{1\}$. \item\label{Nonzeroprincipalprime} $S_k$ has a nonzero principal prime ideal if and only if $k\leq 2$. \item\label{ZariskitopologyIndigenoussemiring} The Zariski topology of the Indigenous semiring $S_k$ is the Sierpi\'{n}ski space. \item\label{RadicalidealsIndigenoussemirings} If $I$ is a nonzero proper ideal of $S_k$, then $\sqrt{I} = S_k \setminus \{1\}$. In particular, the only radical ideals of $S_k$ are $\{0\}$, $S_k \setminus \{1\}$, and $S_k$. \item\label{SemiringidealsIndigenousinformationalgebra} $\Id(S_k)$ is a partially ordered information algebra and if $I$ is a proper nonzero ideal of $S_k$, then \[\{0\} \subseteq \mathfrak{s}_k \subseteq I \subseteq \mathfrak{m}_k \subseteq S_k.\] \item\label{Absorbingelementnonzeroideals} $(\Id(S_k)\setminus\{\mathbf{0}\}, \cdot)$ is a monoid with the absorbing element $\mathfrak{s}_k$, where by $\mathbf{0}$, we mean the zero ideal $\{0\}$ of $S_k$. \item\label{Monoidnonzeroidealsnilpotent} If $n$ is a positive integer number such that $2^n > k$, then for any nonzero proper ideals $\{I_i\}_{i=1}^{n}$ of $S_k$, we have $\prod_{i=1}^{n} I_i= \mathfrak{s}_k$; in other words, the multiplicative monoid $M = \Id(S_k)\setminus\{\mathbf{0}\}$ is nilpotent (i.e., there is a positive integer number $n$ with $M^n = \{\mathfrak{s}_k\}$). \end{enumerate} \end{theorem} \begin{proof} (\ref{IndigenoustotallyorderedIA}): If $a$ and $b$ are nonzero elements of $S_k$, then their multiplication (addition) is either a positive integer number less than $k+1$ or $m$. So, $S_k$ is entire and zerosumfree. If we set \[0 < 1 < \cdots < k < m,\] then it is easy to see that $a \leq b$ implies $a+c \leq b+c$ and $ac \leq bc$, for all $a$, $b$, and $c$ in $S_k$. This means that $S_k$ is a totally ordered information algebra with the smallest element $0$ and the largest element $m$. Note that while $m \neq 1$, we always have \[m \cdot 1 = m = m \cdot m\] which means that $S_k$ is not a semidomain. (\ref{LocalizationIndigenous}): Since $a \in U$ and $U$ is multiplicatively closed, $a^n \in U$ for each natural number $n$. It is clear that for sufficiently large enough $n$, $a^n > k$ in $\mathbb{N}$, and so, $a^n$ which is $m$ in $S_k$ is an element of $U$. Now, let $a / u$ be a nonzero element in $U^{-1} S_k$. Note that since $m \in U$ and $m / m = 1/1$ is the multiplicative identity of the semiring $U^{-1} S_k$, we have \[a/u = (a/u)(m/m) = (am)/(mm) = m/m\] showing that $U^{-1} S_k$ has only two elements $0/1$ and $1/1$. On the other hand, \[(1/1) + (1/1) = (m/m) + (m/m) = (m+m)/m = m/m = 1/1.\] This shows that $U^{-1} S_k$ is isomorphic to the Boolean semiring $\mathbb{B}$. (\ref{Indigenouslocal}): Let $a \neq 1$ and $b \neq 1$. It is easy to see that $a+b \neq 1$. Also, since $xy = 1$ if and only if $x = 1$ and $y = 1$ in $S_k$, we see that if $s \in S_k$ is arbitrary and $a \neq 1$, then $sa \neq 1$. This means that $\mathfrak{m}_k = S_k \setminus \{1\}$ is an ideal of $S_k$. However, there is no other ideals strictly between $\mathfrak{m}_k$ and $S_k$. So, $\mathfrak{m}_k$ is a maximal ideal of $S_k$. On the other hand, an ideal $I$ of a semiring is proper if and only if $1 \notin I$. Therefore, any proper ideal of $S_k$ is a subset of $\mathfrak{m}_k$. Thus $\mathfrak{m}_k$ is the only maximal ideal of $S_k$. (\ref{misinanynonzeroideal}): It is easy to see that $\mathfrak{s}_k = \{0,m\}$ is an ideal of $S_k$. Now, let $I$ be a nonzero ideal of $S_k$. If $s$ is a nonzero element of $I$, then $m = ms$ must be in $I$. (\ref{Subtractiveidealsindigenoussemirings}): Let $I$ be a nonzero proper ideal of $S_k$. Then by the statement (\ref{misinanynonzeroideal}), $m \in I$ while $1 \notin I$. However, $m+1 = m \in I$. This means that $I$ is not subtractive. It is evident that $\{0\}$ and $S_k$ are subtractive. (\ref{PrimeidealsIndigenoussemiring}): By the statement (\ref{IndigenoustotallyorderedIA}), $\{0\}$ in prime. In view of the statement (\ref{Indigenouslocal}) and Corollary 7.13 in \cite{Golan1999(b)}, $S_k \setminus \{1\}$ is also prime. Now, let $P$ be a nonzero prime ideal of $S_k$. If $a \in S_k \setminus \{0,1\}$, then either $2 \leq a \leq k$ or $a = m$. In any case, there is a positive integer $n$ such that $a^n = m$. On the other hand, by the statement (\ref{misinanynonzeroideal}), $m$ is an element of each nonzero ideal of $S_k$. Consequently, $a^n \in P$. Since $P$ is prime, we have $a \in P$. Therefore, $P = S_k \setminus \{1\}$. (\ref{Nonzeroprincipalprime}): In view of the statement (\ref{PrimeidealsIndigenoussemiring}), in $S_1$, the principal ideal $(m) = S_1 \setminus \{0\}$ is prime, and in $S_2$, $(2) = S_2 \setminus \{1\}$ is also prime. Now, let $k \geq 3$. The principal ideal $(2)$ is not prime because a suitable power of $3$ is $m \in (2)$ but $3$ is not an element of $(2)$. Now, let $p > 2$ be a prime number in $\mathbb{N}_k$. The principal ideal $(p)$ is not prime because a suitable power of $2$ is $m \in (p)$ while $2$ is not in $(p)$. If $c$ is a composite number in $\mathbb{N}_k$, then the principal ideal $(c)$ is clearly not prime. Also, $(m) = \{0,m\}$ is not prime because a suitable power of $2$ is $m \in (m)$ while $2$ is not in $(m)$. (\ref{ZariskitopologyIndigenoussemiring}): By the statement (\ref{PrimeidealsIndigenoussemiring}), the only prime ideals of $S_k$ are $\{0\}$ and $\mathfrak{m}_k = S_k \setminus \{1\}$. Therefore, by Theorem \ref{Zariskitopologyentiresemiringwithtwoprimes}, the Zariski topology of $S_k$ is the Sierpi\'{n}ski space. (\ref{RadicalidealsIndigenoussemirings}): For any ideal $I$ of $S_k$, the radical $\sqrt{I}$ of $I$ is the intersection of prime ideals of $S_k$ containing $I$ (cf. Theorem 3.2 in \cite{Nasehpour2018P}). Now, if $I$ is nonzero, then $\sqrt{I} = S_k \setminus \{1\}$ because by the statement (\ref{PrimeidealsIndigenoussemiring}), $S_k \setminus \{1\}$ is the only prime ideal of $S_k$ containing $I$. Therefore, a nonzero proper ideal of $S_k$ is radical if and only if $I = S_k \setminus \{1\}$. Now, since by the statement (\ref{IndigenoustotallyorderedIA}), $S_k$ is entire, $\{0\}$ is a radical ideal. It is evident that $S_k$ is a radical ideal. (\ref{SemiringidealsIndigenousinformationalgebra}): By Theorem \ref{Semiringidealsinformationalgebra}, $\Id(S_k)$ is a partially ordered information algebra. By the statement (\ref{misinanynonzeroideal}), $\mathfrak{s}_k$ is the smallest nonzero ideal. By the statement (\ref{Indigenouslocal}), $\mathfrak{m}_k = S_k \setminus \{1\}$ is the largest proper ideal. (\ref{Absorbingelementnonzeroideals}): Since $\Id(S_k)$ is an entire semiring, $(\Id(S_k)\setminus\{\mathbf{0}\}, \cdot)$ is a monoid. Now, let $I$ be a nonzero ideal. We need to prove that $I \cdot \mathfrak{s}_k = \mathfrak{s}_k$. By the statement (\ref{misinanynonzeroideal}), $\mathfrak{s}_k$ is the smallest nonzero ideal of $S_k$. On the other hand, \[I \cdot \mathfrak{s}_k \subseteq I \cap \mathfrak{s}_k = \mathfrak{s}_k.\] (\ref{Monoidnonzeroidealsnilpotent}): Let $n$ be a positive integer number with $2^n > k$. Let $\{I_i\}_{i=1}^{n}$ be arbitrary nonzero proper ideals of $S_k$. By the statement (\ref{misinanynonzeroideal}), $m \in I_i$ while $1 \notin I_i$. Consider $a_i \in I_i$. Observe that if at least one of the $a_i \in I_i$ is zero, then $\prod_{i=1}^{n} a_i = 0$. Also, if all of the $a_i$s are nonzero and at least one of them is $m$, then $\prod_{i=1}^{n} a_i = m$. Now, let $a_i \notin \{0,m\}$. This means that $2 \leq a_i$, for each $i$, and so, in $\mathbb{N}$, we have \[\prod_{i=1}^{n} a_i \geq 2^n > k.\] This means that $\prod_{i=1}^{n} a_i = m$ in $S_k$ and the proof is complete. \end{proof} \begin{theorem} Let $S_k$ be the Indigenous semiring and $M$ a commutative monoid. Then, the following statements hold: \begin{enumerate} \item The monoid semiring $S_k[M]$ is an information algebra. \item If $M$ is a totally ordered commutative monoid, then the function \[\deg: (S_k[M],+,\cdot,0,1) \rightarrow (M_{\infty},\max,+,-\infty,0)\] is a semiring morphism. \end{enumerate} \end{theorem} \begin{proof} (1): By Theorem 4.10 in \cite{Nasehpour2025}, if $M$ is a commutative monoid and $S$ an information algebra, then the monoid semiring $S[M]$ is also an information algebra. Thus in view of Theorem \ref{GeneralpropertiesofIndigenoussemirings}, $S_k[M]$ is an information algebra. (2): In view of Theorem \ref{GeneralpropertiesofIndigenoussemirings}, this is a special case of Corollary 4.18 in \cite{Nasehpour2025}. This completes the proof. \end{proof} \section{Distinguished elements of polynomials and formal power series over the Indigenous semirings}\label{sec:distinguishedelements} In this section, $S_k$ denotes the Indigenous semiring defined in Definition \ref{Indigenoussemiringdef}. Since $S_k$ is an entire semiring, $S_k[X]$ \cite[Corollary 2.4]{Nasehpour2021} and $S_k[[X]]$ \cite[Lemma 43]{Nasehpour2016} are also entire semirings. Therefore, $S_k$, $S_k[X]$, and $S_k[[X]]$ have no nontrivial zero-divisors (and nilpotent elements). Now, we proceed to discuss their units and idempotent elements. \begin{proposition}\label{UnitsIndigenoussemirings} The only unit element of $S_k$, $S_k[X]$, and $S_k[[X]]$ is $1$. \end{proposition} \begin{proof} Obviously in $S_k$, $ab = 1$, implies that $a = b = 1$. So, the only unit element of $S_k$ is $1$. Let $f,g \in S_k[X]$ with $fg = 1$. Since $S_k$ is an entire semiring, we obtain that $\deg(f) = \deg(g) = 0$. Therefore, $f = g = 1$. Now, let $f, g \in S_k[[X]]$ with $fg = 1$. Suppose that $f = \sum_{i=0}^{+\infty} a_i X^i$ and $g = \sum_{j=0}^{+\infty} b_j X^j$. Clearly, $a_0 = b_0 = 1$. Since $S_k$ is zerosumfree, from $fg = 1$, we obtain that $a_i = b_i = 0$, for all $i \geq 1$. This completes the proof. \end{proof} \begin{proposition}\label{IdempotentsIndigenoussemirings1} The only idempotent elements of $S_k$ and $S_k[X]$ are $0$, $1$, and $m$. \end{proposition} \begin{proof} In $S_k$, if $a$ is different from $0$, $1$, and $m$, then $2 \leq a \leq k$. If $a^2 \leq k$, then $a^2 \neq a$. If $a^2 > k$, then $a^2 = m$ which means that again $a^2 \neq a$. Thus the only idempotent elements of $S_k$ are $0$, $1$, and $m$. Now, let $f \in S_k[X]$. Since $S_k$ is an entire semiring, then \[\deg(f^2) > \deg(f)\] except the case that $f$ is a constant polynomial. Therefore, $f^2 = f$ if and only if $f = a$, where $a \in S_k$. This means that $f$ is idempotent in $S_k[X]$ if and only if $f$ is either $0$ or $1$ or $m$. This completes the proof. \end{proof} \begin{theorem}\label{IdempotentsIndigenoussemirings2} An element $f$ in $S_k[[X]]$ is idempotent if and only if either $f = 0$ or $f = 1$ or $f = m$ or \[f = a_0 + \sum_{i=1}^{+\infty} m X^{s_i},\] where $a_0 = 1,m$ and the set $\{s_i\}_{i=1}^{+\infty}$ is a subsemigroup of $(\mathbb{N},+)$. \end{theorem} \begin{proof} Let $f = \sum_{i=0}^{+\infty} a_i X^i$ be idempotent. It follows that $a^2_0 = a_0$, and so, by using Proposition \ref{IdempotentsIndigenoussemirings1}, we can distinguish three cases: \begin{enumerate} \item The case $a_0 = 0$. If $a_0 = 0$, then $f = \sum_{i=1}^{+\infty} a_i X^i$. Now, if $a_i \neq 0$, for some $i > 0$, then obviously $f^2 \neq f$. Therefore, $f = 0$. \item The case $a_0 = 1$. Our claim is that $a_i \in \{0,m\}$, for any $i > 0$. This is because if $a_i \neq 0,m$, for some $i > 0$, then the coefficient of $X^i$ in $f^2$ is at least $2a_i$ in $\mathbb{N}$ which is never $a_i$ in $S_k$. Now, consider \[f = 1 + mX^{s_1} + mX^{s_2} + \cdots.\] Our claim is that $f$ is idempotent if and only if the set $E = \{s_i\}_{i=1}^{+\infty}$ is a subsemigroup of $(\mathbb{N},+)$. For the direct implication, we need to show that $s_i + s_j \in E$, for all $s_i \in E$ and $s_j \in E$. Consider the monomials $mX^{s_i}$ and $mX^{s_j}$ in $f$. Therefore, $mX^{s_i + s_j}$ is a monomial in $f^2$. Since $f^2 = f$, $s_i + s_j$ needs to be one of the exponents of the monomials of $f$ which means that $E$ is a subsemigroup of $\mathbb{N}$. For the converse implication, observe that an easy calculation shows that if $\{s_i\}_{i=1}^{+\infty}$ is a subsemigroup of $\mathbb{N}$, then $f^2 = f$. \item The case $a_0 = m$. Our claim is that in this case also $a_i \in \{0,m\}$, for any $i > 0$. This is because if $a_i \neq 0,m$, for some $i > 0$, then the coefficient of $X^i$ in $f^2$ is \[a_0 a_i + \dots + a_ia_0 = m a_i + \dots + a_i m = m\] which is never the same as $a_i$, i.e., the coefficient $X^i$ in $f$. Now, consider \[f = m + mX^{s_1} + mX^{s_2} + \cdots.\] Similar to the second case, one can easily check that $f$ is idempotent if and only if the set $\{s_i\}_{i=1}^{+\infty}$ is a subsemigroup of $(\mathbb{N},+)$. \end{enumerate} This completes the proof. \end{proof} \begin{theorem} Let $\alpha \neq 0$ and $\beta$ be elements of the Indigenous semiring $S_k$ and $X$ an indeterminate over $S_k$. Then, $f=\alpha X^2+\beta$ is irreducible if and only if one of the following cases happens: \begin{enumerate} \item $\alpha$ and $\beta$ are in $\mathbb{N}_k$ and $\gcd(\alpha, \beta) = 1$. \item $\alpha = m$ and $\beta = 1$. \item $\alpha = 1$ and $\beta = m$. \end{enumerate} \end{theorem} \begin{proof} In view of Proposition 6.2 in \cite{Nasehpour2025} and Theorem \ref{GeneralpropertiesofIndigenoussemirings} in the current paper, $f$ cannot be factored into $g = aX+b$ and $h = cX+d$ in $S_k[X]$, where $a$ and $c$ are nonzero in $S_k$. Therefore, $f$ is reducible if and only if there is a nonzero nonunit $\gamma$ in $S_k$, i.e., $\gamma \neq 0,1$ such that $f = \gamma g$, for some $g = \alpha'X^2 + \beta' \in S_k[X]$ which means that \[\alpha = \gamma \alpha' \wedge \beta = \gamma \beta'.\] Note that if $\alpha = \beta = m$, then $f$ is reducible. Therefore, if $f$ is irreducible, then at least one of the coefficients of $f$ must be a natural number. Now, observe the following: \begin{enumerate} \item If $\alpha$ and $\beta$ are natural numbers, then $\gamma \neq m$, and $f$ is reducible if and only if $ \gamma \mid \gcd(\alpha,\beta)$, i.e., $\gcd(\alpha,\beta) > 1$. \item If $\alpha = m$, then $f$ is reducible if and only if $\beta > 1$ because $mX^2 + 1$ is irreducible and \[f = mX^2 + \beta = \beta (mX^2 + 1).\] \item If $\beta = m$, then $f$ is reducible if and only if $\alpha > 1$ because $X^2 + m$ is irreducible and \[f = \alpha X^2 + m = \alpha (X^2 + m).\] \end{enumerate} This gives the characterization of all irreducible polynomials of the form $\alpha X^2+\beta$ and the proof is complete. \end{proof} \subsection*{Acknowledgments} The authors wish to thank the anonymous referees for their valuable comments and suggestions, which improved the presentation of this paper. \bibliographystyle{plain} \begin{thebibliography}{15.} \bibitem{ArensDugundji1951} Arens, R., Dugundji, J.: Topologies for function spaces. Pac. J. 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2412.02093v1	http://arxiv.org/abs/2412.02093v1	Twist Coefficients of Periodic Orbits of Minkowski Billiards	\documentclass[reqno]{amsart} \usepackage[left=1in, top=1in, right=1in, bottom=1in]{geometry} \usepackage{amsmath, amsthm, amssymb} \usepackage{mathtools} \usepackage{mathrsfs} \usepackage{graphicx} \usepackage{subcaption} \usepackage[all]{xy} \usepackage{enumerate} \usepackage{tikz} \usetikzlibrary{math,angles,quotes,intersections,calc,through} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \DeclareMathOperator{\Tr}{Tr} \def\bC{\mathbb{C}} \def\bQ{\mathbb{Q}} \def\bR{\mathbb{R}} \def\bT{\mathbb{T}} \def\bZ{\mathbb{Z}} \def\cM{\mathcal{M}} \def\cT{\mathcal{T}} \def\dsds{\frac{\partial s_1}{\partial s}} \def\dsdu{\frac{\partial s_1}{\partial u}} \def\duds{\frac{\partial u_1}{\partial s}} \def\dudu{\frac{\partial u_1}{\partial u}} \def\gt{\gamma(t)} \def\dgs{\dot{\gamma}(s)} \def\dgso{\dot{\gamma}(s_1)} \def\dgis{\gamma'_j(s)} \def\dgjs{\gamma'_k(s)} \def\dgiss{\gamma'_j(s_1)} \def\dgjss{\gamma'_k(s_1)} \def\ddgis{\gamma''_j(s)} \def\ddgiss{\gamma''_j(s_1)} \def\dgos{\gamma'_1(s)} \def\dgts{\gamma'_2(s)} \def\dgoss{\gamma'_1(s_1)} \def\dgtss{\gamma'_2(s_1)} \def\ddgos{\gamma''_1(s)} \def\ddgs{\gamma''(s)} \def\ddgss{\gamma''(s_1)} \def\ddgts{\gamma''_2(s)} \def\ddgoss{\gamma''_1(s_1)} \def\ddgtss{\gamma''_2(s_1)} \def\Fi{F_{v_j}(v)} \def\Fij{F_{v_jv_k}(v)} \def\ta{\tilde{a}} \def\tb{\tilde{b}} \def\pa{\partial} \def\ds{\displaystyle} \begin{document} \title{Twist Coefficients of Periodic Orbits of Minkowski Billiards} \author[C. Villanueva]{Carlos Villanueva$^$} \address{Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA} \email{\tt [email protected]} \thanks{C.V. is supported in part by the Department of Defense SMART Scholarship OMB NO. 0704-0466.} \thanks{$^$ Author to whom any correspondence should be addressed.} \author[P. Zhang]{Pengfei Zhang} \address{Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA} \email{\tt [email protected]} \subjclass[2020]{37G05 37C83 37E40 53B40} \keywords{Minkowski norm, Finsler norm, billiards, normal form, twist coefficient, nonlinear stability} \dedicatory{To Leonid Bunimovich on his 75th birthday} \begin{abstract} We investigate the fundamental properties of Minkowski billiards and introduce a new coordinate system $(s,u)$ on the phase space $\mathcal{M}$. In this coordinate system, the Minkowski billiard map $\mathcal{T}$ preserves the standard area form $\omega = ds \wedge du$. We then classify the periodic orbits of Minkowski billiards with period $2$ and derive formulas for the twist coefficient $\tau_1$ for elliptic periodic orbits, expressed in terms of the geometric characteristics of the billiard table. Additionally, we analyze the stability properties of these elliptic periodic orbits. \end{abstract} \maketitle \tableofcontents \section{Introduction}\label{sec.introduction} Dynamical billiards describe a system in which a particle moves within a container and reflects off its walls according to specific reflection rules. In \cite{Bi27, Bi27b} Birkhoff studied the dynamical billiards on convex domains with smooth boundaries and proved the existence of periodic orbits of period $n$ for any $n\ge 2$. In \cite{La73} Lazutkin proved the existence of caustics of such convex billiards near the boundaries of the phase space. Sinai pioneered the study of chaotic billiards. In his seminal paper \cite{Sin70}, he investigated dynamical billiards with dispersing boundaries and proved both the hyperbolicity and ergodicity of such systems. In \cite{Bu74a, Bu74}, Bunimovich observed that convex circular arcs could be used to construct chaotic billiards and introduced the stadium billiard, which became a favorite model among physicists. The construction of chaotic billiards was significantly extended by Wojtkowski \cite{Woj86}, Markarian \cite{Mar88}, Donnay \cite{Don91}, and Bunimovich \cite{Bu92}. Since then, several classes of chaotic billiards have been discovered. Notable examples include the elliptic stadium \cite{MaKaPdC}, track billiards \cite{BuDM}, asymmetric lemon billiards \cite{BZZ, JZ21}. An overview of chaotic billiards can be found in \cite{ChMa}. Birkhoff Normal Form plays an important role in the study of stability properties of elliptic periodic orbits, see \cite{Bi27, SiMo}. Under certain nonresonance assumptions, the dynamical system around an elliptic fixed point can be viewed as a perturbation of an integrable model, which is called the normal form of the map. If the integrable model has nonzero twist, then the elliptic fixed point must be nonlinearly stable \cite{Mos56, Mos73}. That is, the fixed point is contained in a nesting sequence of invariant disks whose boundaries are smooth invariant curves. Generally speaking, the twist coefficients can be represented as rational functions of the coefficients of the Taylor polynomials at the elliptic fixed point, see \cite{Moe90, CGM}. These rational functions, albeit explicit, are generally very involved. For dynamical systems with geometric background, it is expected that these twist coefficients can be represented as much simpler rational functions of the geometric quantities such as distances and curvatures. In \cite{KP05}, Kamphorst and Pinto-de-Carvalho showed that the first twist coefficient of a elliptic periodic orbit of period 2 of a planar billiard can be expressed a simple rational function in terms of the geometric quantities such as the orbit length and the radii of curvature of the boundary of the billiard table. In \cite{JZ22}, Jin and Zhang obtained an explicit formula of the second twist coefficient of periodic orbits of period 2 in terms of the same geometric quantities of the billiards. The authors also provided applications of their formula in the study of stability properties to various billiard systems. Dynamical billiards can also be defined on compact domains of Riemannian manifolds, on which a particle moves along geodesics in the interior of the domain and makes elastic reflections upon impact on the boundaries, see \cite{Vet, Zha17}. Therefore, billiard flows can be viewed as extensions of geodesic flows to Riemannian manifolds with boundaries. Interestingly, dynamical billiards on curved surfaces are related to the study of quantum magnetic confinement of non-planar 2D electron gases (2DEG) in semiconductors \cite{FLBP}, where the effect of varying the curvature of the surface corresponds to a change in the potential energy of the system. In \cite{GT02}, Gutkin and Tabachnikov expanded the theory by introducing dynamical billiards on Minkowski spaces, and more generally, on Finsler manifolds. Unlike Euclidean and Riemannian settings, these spaces are not equipped with an inner product but rather a norm. As a result, the concept of angles, which plays a crucial role in defining reflections in Euclidean and Riemannian billiards, no longer exists. Instead, in \cite{GT02}, they revisited the fundamental concept of reflections through the critical points of the trajectory length function. Since then, several notable results have emerged regarding Minkowski and Finsler billiards (see \cite{AO14, AFOR}), along with some unexpected applications in symplectic and contact geometry (see \cite{AKO14}). In this paper, we investigate the properties of periodic orbits in dynamical billiards on Minkowski planes. Let $F:\mathbb{R}^2 \to \mathbb{R}$ be a Minkowski norm and $Q \subset \mathbb{R}^2$ be a connected, bounded domain with piecewise smooth boundaries. We consider a point mass moving freely inside $Q$, with reflections occurring at the boundary according to Finsler reflection laws (see Proposition~\ref{pro.Finsler.reflection.law}), thereby defining a Minkowski dynamical billiard system (or {\it Minkowski billiards} for short). The phase space $\mathcal{M}$ comprises all unit vectors at the boundary $\partial Q$, pointing inward toward $Q$. Notably, this definition aligns with that of Euclidean billiards. However, the conventional arclength-angle coordinate system $(s, \theta)$ is no longer applicable, as the angle is undefined in a normed space. We introduce a new coordinate system $(s,u)$ for the phase space $\mathcal{M}$ and demonstrate that the Minkowski billiard map $\mathcal{T}$ is symplectic using the generating function method. Subsequently, we express the tangent map $D\mathcal{T}$ in terms of geometric quantities of the billiard table, see Proposition~\ref{pro.DT.Minkowski}. This enables classification of period-$2$ orbits as elliptic, parabolic, or hyperbolic. Under mild nonresonance conditions for elliptic orbits, we derive a concise formula for the first twist coefficient $\tau_1$ in terms of the geometric quantities of the billiard table, see Theorem~\ref{thm.t1.sym} and \ref{thm.t1.asym}. Combining with Moser's twist mapping theorem, we investigate specific Minkowski billiard families and determine conditions for nonlinear stability of periodic orbits, see Section~\ref{sec.applications}. \section{Preliminaries} \subsection{Minkowski spaces and Finsler manifolds} Finsler geometry has its origin in the study of the calculus of variations. It allows to study more expansive cases such as objects traveling in anisotropic mediums where speed depends on the direction of travel. For a thorough introduction to Minkowski Spaces and Finsler Geometry, see \cite{BCS, ChSh, Ru59}. \begin{definition}\label{def.Minkowski} A \textit{Minkowski norm} on the space $\bR^n$ is a function $F: \bR^n \rightarrow [0, \infty)$ satisfying the following conditions: \begin{enumerate}[(i)] \item Regularity: $F$ is $C^\infty$ on $\bR^n \backslash \{0\}$. \item Positive homogeneity: $F(c \, v) = c\, F(v)$ for any number $c >0$ and any vector $v \in \bR^n$. \item Strong convexity: The $n \times n$ Hessian matrix $ (g_{jk}(v)) := \bigg( \frac{\pa^2}{\pa v^j \pa v^k} \big( \frac{1}{2} F^2\big)(v) \bigg)$ is positive definite at every point in $\bR^n \backslash \{0\}$. \end{enumerate} Given a Minkowski norm $F$ on $\bR^n$, the pair $(\bR^n, F)$ is called a \textit{Minkowski space}. The subset $I=\{v\in \bR^n: F(v) =1\}$ of all $F$-unit vectors is called the \textit{indicatrix} of the Minkowski space. \end{definition} It is clear that the norm $F$ can be recovered from its indicatrix $I$. \begin{definition}\label{def.Finsler} Let $M$ be an $n$-dimensional closed manifold. A \textit{Finsler structure} on the manifold $M$ is a function $F: TM \rightarrow [0, \infty)$ satisfying the following conditions: \begin{enumerate}[(i)] \item Regularity: $F$ is $C^\infty$ on $TM \backslash \{0\}$. \item Positive homogeneity: $F(x, c \, v) = c\, F(x, v)$ for any $c >0$ and any $v \in T_xM$. \item Strong convexity: The $n \times n$ Hessian matrix $ (g_{jk}(x, v)) := \bigg(\frac{\pa^2}{\pa v^j \pa v^k} \big( \frac{1}{2} F^2\big)(x, v) \bigg)$ is positive definite at every point in $TM \backslash \{0\}$. \end{enumerate} Given a manifold $M$ and a Finsler structure $F$ on $TM$, the pair $(M,F)$ is called a \textit{Finsler manifold}. \end{definition} It is clear that restricting a Finsler structure $F$ to the tangent space $T_xM$ gives rise to a Minkowski space $(T_xM, F(x,\cdot))$ for each $x\in M$, and a Minkowski space can be viewed as a homogeneous Finsler manifold, where the Finsler structure is independent of the base point. \begin{definition} The \textit{Legendre transform} on the Minkowski space $(\bR^n, F)$ is defined by \begin{align} \mathcal{L}: I &\to (\bR^n)^{\ast} \\ u &\mapsto p=\mathcal{L}(u) \end{align} such that $\ker (p) = T_uI$ and $p(u) = 1$. \end{definition} \subsection{Mixed norm spaces} Given two numbers $a>0$ and $b\ge 0$, and a positive integer $k\ge 1$, we consider the Minkowski space $(\bR^2, F_{a,b, 2k})$, where the norm $F_{a,b, 2k}$ is given by \begin{equation} F_{a,b,2k}(v) = a\lVert v \rVert_{2} + b\lVert v \lVert_{2k}, \end{equation} where $\lVert v \rVert_2 = \left( v_1^2 + v_2^2 \right)^{\frac{1}{2}}$ and $\lVert v \rVert_{2k} = \left(v_1^{2k} + v_2^{2k}\right)^{\frac{1}{2k}}$. Clearly, $F_{1,0,k}(v) = \lVert v \rVert_2$ is just the usual Euclidean norm on $\bR^2$. \begin{proposition} $(\bR^2, F_{a,b,2k})$ is a Minkowski space. \label{prop: Mixed norm is Minkowski} \end{proposition} \begin{proof} We only need to check that condition $(iii)$ in Definition \ref{def.Finsler} is satisfied as the first two are easy to see. Let $F = F_{a,b,2k}$ and $(g_{jk}) = \left(\left[\frac{1}{2}F^2\right]_{v_jv_k}\right)$ be the Hessian matrix of $\frac{1}{2}F^2$ whose entries are given by \begin{align} g_{11} &= \left(F_{v_1}\right)^2 + FF_{v_1v_1}, \\ g_{12} &= g_{21} = F_{v_1}F_{v_2} + FF_{v_1v_2}, \\ g_{22} &= \left(F_{v_2}\right)^2 + FF_{v_2v_2}. \end{align} It suffices to show that $g_{11} > 0$ and $\det(g_{jk})>0$. The first and second order partials of $F$ are given by \begin{align} F_{v_1} &= av_1(v_1^2 + v_2^2)^{-\frac{1}{2}} + bv_1^{2k-1}(v_1^{2k} + v_2^{2k})^{\frac{1}{2k}-1}, \\ F_{v_2} &= av_2(v_1^2 + v_2^2)^{-\frac{1}{2}} + bv_2^{2k-1}(v_1^{2k} + v_2^{2k})^{\frac{1}{2k}-1}, \\ F_{v_1v_1} &= \alpha v_2^2 + (2k-1)\beta v_1^{2k-2}v_2^{2k}, \\ F_{v_2v_2} &= \alpha v_1^2 + (2k-1)\beta v_1^{2k}v_2^{2k-2}, \\ F_{v_1v_2} &= -v_1v_2\left[ \alpha + \beta(2k-1)(v_1v_2)^{2k-2} \right], \end{align} where $\alpha = a(v_1^2 + v_2^2)^{-\frac{3}{2}}$ and $\beta = b(v_1^{2k} + v_2^{2k})^{\frac{1}{2k}-2}$. Note $F_{v_1v_1},F_{v_2v_2} > 0$ implies $g_{11} > 0$. Further, $\left( F_{v_1v_2} \right)^2 = F_{v_1v_1}F_{v_2v_2}$, which we can use to show \begin{align} \det(g_{jk}) &= F\left[(F_{v_1})^2F_{v_2v_2} - 2F_{v_1}F_{v_2}F_{v_1v_2} + F_{v_1v_1}(F_{v_2})^2 \right] \\[5pt] &= F\left(F_{v_1}\sqrt{F_{v_2v_2}}-F_{v_2}\sqrt{F_{v_1v_1}}\right)^2 > 0. \end{align} This completes the proof. \end{proof} Starting from Section~\ref{sec.billiards.symmetry}, we will restrict to the case $k=2$, and will denote $F_{a,b} = F_{a,b, 2}$ for short. \subsection{Curvatures in Minkowski planes} As mentioned in the introduction, the twist coefficients of an elliptic periodic billiard orbit depend on the geometry of the billiard table, namely the curvatures at the points of reflection and the distance between them. To analyze the stability properties of billiard orbits on Minkowski planes, it is helpful to extend the concept of curvature from Euclidean spaces to Minkowski spaces. In the following we will explain how curvatures extend to Minkowski planes with norms following the exposition \cite{BMS19}. On the Euclidean plane, given a curve $\gamma$ with Euclidean length $\ell$, we can intuitively think of its curvature as how fast the tangent field varies with respect to the distance traveled on $\gamma$. Formally we define curvature as follows. Let $\gamma:[0, \ell] \rightarrow \bR^2$ be a smooth curve with arc-length parameter $s$, $A_s$ be the area of the sector in the unit circle between $\gamma'(0)$ and $\gamma'(s)$, and $\nu(s) = 2A_s$. The (signed) curvature of the curve $\gamma$ at $\gamma(s)$ is defined as $\kappa(s) = \nu'(s)$. Equivalently, since $\gamma''$ is normal to $\gamma'$, we can define $\kappa(s)$ in terms of the positively oriented normal vector field $n(s)$ of unit length: $\gamma''(s) = \kappa(s)n(s)$. Geometrically, this means that the area of the unit circle determined by the normal field $n(s)$ is the same as the area determined by the tangent field $\gamma'$. Lastly one can show that $\kappa(s) = \frac{1}{R(s)}$, where $R(s)$ is the radius of the circle attached to $\gamma(s)$ such that their tangent lines agree. While these three definitions of curvature yield the same quantity in the Euclidean plane, they diverge and produce distinct quantities in the more general Minkowski planes, as demonstrated in \cite{BMS19}. The difference arises from using a new definition of orthogonality that is not necessarily symmetric due to the lack of an inner product structure. For the rest of this section, let $(\bR^2, F)$ be a normed plane with $\lVert \cdot \rVert = F(\cdot)$ and $I$ denote the indicatrix at an arbitrary point. \begin{definition} Given two vectors $v,w \in V$, $v$ is said to be \textit{Birkhoff orthogonal} to $w$, denoted $v \dashv_B w$, if $\lVert v \rVert \leq \lVert v + tw \rVert$ for all $t \in \bR$. \end{definition} Geometrically, $v \dashv_B w$ indicates that $T_{v/\lVert v \rVert}I$ is parallel to $w$. Birkhoff orthogonality is not symmetric, and reversing it involves the use of a different norm. In fact, closed curves that do reverse Birkhoff orthogonality are called \textit{Radon Curves}. \begin{definition} Fix a determinant form $[\cdot, \cdot]$ on a normed space $V$ i.e. a non-degenerate bi-linear form such that $[v,v] = 0$ for all $v$. The \textit{anti-norm} on $V$ is defined as \begin{equation} \lVert v \rVert_a := \sup\{ \|[w,v]\| : \lVert w \rVert = 1 \} \label{eq: anti norm} \end{equation} \end{definition} In $\bR^2$, a determinant form is unique up to a constant multiple since it amounts to fixing an orientation with unit area. Further, although $\dim V = \dim V^$, there is no canonical isomorphism between $V$ and $V^$. However, any isomorphism is given by an identification $v \mapsto \left[\cdot, v \right]$ for any $v \in V$, where $[\cdot, \cdot]$ is a non-degenerate bilinear form. It follows from Eq.~\eqref{eq: anti norm} that $\|[w,v]\| \leq \lVert w \rVert \lVert v \rVert_a$, and it is shown in \cite{MS06} that $w \dashv_B v$ if and only if $v \dashv_B^a w$, where $\dashv_B^a$ denotes Birkhoff orthogonality in the antinorm. Note that whenever $w \dashv_B v$ and $\Vert w \rVert = 1 $, Eq.~\eqref{eq: anti norm} is equal to the dual norm of the functional $[\cdot,v] \in V^$. So for example, whenever $\frac{1}{p} + \frac{1}{q} = 1$, the norms on $\ell_p$ and $\ell_q$ reverse Birkhoff orthogonality. Having redefined orthogonality we can now generalize the normal field and the first curvature concept, where curvature was given in terms of the area swept within the unit circle by the tangent field $\gamma'$. Let $n_\gamma : [0,\ell_\gamma] \rightarrow V$ be the vector field such that each $s \in [0,\ell_\gamma]$ gets mapped to the unique vector such that $\gamma'(s) \dashv_B n_\gamma(s)$ and $[\gamma'(s),n(s)] = 1$. We call $n_\gamma$ the \textit{right normal field to $\gamma$}, and in \cite{BMS19} it is shown how $n_\gamma(s)$ is unit in the anti-norm for all $s$. \begin{definition} Let $\gamma : [0,\ell_\gamma] \rightarrow \bR^2$ be a smooth curve of normed length $\ell_\gamma$ with arc-length parameter $s$ (in $\lVert \cdot \rVert$), and $\varphi: [0,2A(I)] \rightarrow \bR^2$ be a parameterization of the indicatrix $I$ by twice the area of its sectors. Identify the tangent field $\gamma'$ within the indicatrix so that \begin{equation} \gamma'(s) = \varphi(\nu(s)), \end{equation} where $\nu: [0, \ell_\gamma] \rightarrow \bR$ maps the arc-length of $T$ to twice the area of the sectors spanned by $\gamma'(0)$ and $\gamma'(s)$. The \textit{Minkowski curvature} of $\gamma$ at $\gamma(s)$ is defined as \begin{equation} \kappa_m := \nu'(s). \label{eq: Minkowski curvature} \end{equation} \end{definition} Differentiating equation (\ref{eq: Minkowski curvature}) yields \begin{equation} \gamma''(s) = \nu'(s) \frac{d \varphi}{d \nu}(\nu(s)) = \kappa_m(s)n_\gamma(s). \label{eq: d2 gamma} \end{equation} Geometrically, this means that as $\gamma'$ rotates about the indicatrix $I$, $n_\gamma$ rotates about the indicatrix of the anti-norm $I_a$. The analogous definition to equation (\ref{eq: Minkowski curvature}), but where the area is spanned by $n_\gamma$ within $I_a$ is called the \textit{normal curvature} and is denoted $\kappa_n$. However, since the indicatrix in the norm and anti-norm are generally not the same, $\gamma'$ and $n_\gamma$ sweep different areas and thus yield different values for $\kappa_m(s)$ and $\kappa_n(s)$. For later convenience, it will be useful to express equation \eqref{eq: d2 gamma} in terms of a Euclidean structure to facilitate the calculation of the higher order derivatives of $\gamma''(s)$ needed in the Taylor expansion of the billiard map $\mathcal{T}$. Given a parameterization $r(\theta(s))$ of the indicatrix $I$ such that $\gamma'(s) = r(\theta(s))\cdot (\cos\theta(s),\sin\theta(s))$ for $\theta(s) \in [0,2\pi]$, \cite{BMS19} shows how $\kappa_m$ and $n_\gamma$ may be expressed as \begin{align} \kappa_m(s) &= k_e(s)r(\theta(s))^3 \label{eq: km in Euclid}\\[10pt] n_\gamma(s) &= \frac{1}{r(\theta(s))^2}\frac{dr}{d\theta}(\theta(s)) \cdot (\cos\theta(s),\sin\theta(s)) \nonumber \\ &\qquad + \frac{1}{r(\theta(s))} \cdot (-\sin\theta(s),\cos\theta(s)). \label{eq: n in Euclid} \end{align} \subsection{Periodic orbits}\label{sec.dysy} Let $X$ be a topological space and $f: X \rightarrow X$ a continuous map. Then the pair $(X, f)$ is called a dynamical system. The iterates of the map $f$ can be defined recursively via $f^{n+1} = f\circ f^{n}$ for any $n\ge 0$. If $f$ is a homeomorphism, then we can define the backward iterates $f^{-n}$, $n\ge 1$. A point $x \in X$ is {periodic} if $f^n(x) = x$ for some $n \ge 1$, and the {period} is the smallest $n$ satisfying this condition. Given a periodic point $x$, its orbit $\mathcal{O}(x)$ is a {periodic orbit}. If $f(x) = x$, then $x$ is a {fixed point} of $f$. In the following we will consider a smooth surface $S$ (possibly with boundaries) with smooth area form $\omega$. A diffeomorphism $f: S\to S$ is said to be symplectic if it preserves the area form $\omega$. Let $p$ be a fixed point of the symplectic map $f$, $(U, \phi)$ be a local coordinate chart around $p$. Then the tangent map $D_pf : T_p S \to T_p S$ can be viewed as a $2\times 2$ matrix with determinant $1$. Let $\lambda(p,f)$ be an eigenvalue of the matrix $D_pf$ (the other one will be $\frac{1}{\lambda(p,f)}$, if exists). Even though the entries of $D_p f$ depend on the choice of the local coordinate system $(U, \phi)$, its eigenvalues and hence the trace $\Tr(D_p f)$, do not depend on such choices. The fixed point $p$ is said to be \begin{enumerate} \item parabolic if $\lambda(p,f) =\pm 1$ (or equally, $\|\Tr(D_p f)\| =2$); \item hyperbolic if $\|\lambda(p,f)\| \neq 1$ (or equally, $\|\Tr(D_p f)\| > 2$); \item elliptic if $\|\lambda(p,f)\|= 1$ and $\lambda(p,f)\neq \pm 1$ (or equally, $\|\Tr(D_p f)\| < 2$). \end{enumerate} \subsection{Euclidean billiards} Let $Q \subset \bR^2$ be a bounded and connected domain with (piecewise) smooth boundary $\gamma=\pa Q$. Consider a point mass that moves within $Q$ freely and makes elastic reflects off the boundary. The induced system is called a Euclidean dynamical billiard, and $Q$ is called the billiard table. For convenience, we will assume that $Q$ is strictly convex and $\gamma$ is smooth. The term Euclidean billiards is not standard. However we will use it repeatedly to distinguish it from a Minkowski billiard system that will be introduced later. All billiard systems are Euclidean for the remainder of this subsection. The phase space $\cM$ of the dynamical billiard on the table $Q$ consists of tangent vectors $(q,v)$, where $q\in \gamma$, and $v\in T_q Q$ is a unit vector pointing to the interior of $Q$. Let $s$ be the arc-length parameter of $\gamma$ oriented counterclockwise, and $\theta \in [0,\pi]$ be the angle between $v$ and $\dgs$. Then each point $(p,v)$ can be expressed in terms of a pair of new coordinates $(s,\theta)$. Let $\ell_\gamma$ be the length of $\gamma$. Under the assumption that $Q$ is strictly convex, this gives rise to a new coordinate system on the phase space: $\mathcal{M} = \{(s,\theta): 0 \leq s \leq \ell_\gamma,\ 0 \leq \theta \leq \pi \}/\sim$ with the identification of the endpoints $s=0$ and $s=\ell_{\gamma}$. That is, the phase space $\mathcal{M}$ can be viewed as a cylinder $\bR/\ell_{\gamma} \times [0,\pi]$. Let $(q_0,v_0) \in \mathcal{M}$ be the initial position and direction of the billiard ball. Let $q_1 \in \gamma$ be the first point of reflection, so that at $q_1$, the billiard changes direction from $v_0$ to $v_1$. Using their corresponding coordinates $(s_i, \theta_i)$, $i=0, 1$, we can thus define the billiard map $T:\mathcal{M}\rightarrow \mathcal{M}$, $(s_0, \theta_0) \mapsto (s_1, \theta_1)$. For strictly convex billiard tables (even in $\bR^n$) whose boundary is $C^k$ for $k \geq 2$, then the billiard map $T$ is of class $C^{k-1}$. In particular, if the boundary is smooth, then $T$ is as well. By defining the billiard map, we can convert our billiard model to a discrete dynamical system on the cylinder $\bR/\ell_{\gamma} \times [0,\pi]$. Let $L = L(s,s_1)$ denote the distance between two points $\gamma(s)$ and $\gamma(s_1)$, and let $\kappa(s)$ and $\kappa(s_1)$ be the signed curvatures of $\gamma$ at $s$ and $s_1$ respectively. \begin{proposition} \label{pro:DT.euclid} The billiard map $T$ preserves the area form $\omega = \sin\theta ds \wedge d\theta$ on the phase space $\cM$. Moreover, the differential of the billiard map $T: (s, \theta) \mapsto (s_1, \theta_1)$ is given by \begin{equation}\label{eq: DpT} D_{(s,\theta)}T = \frac{1}{\sin\theta_1} \begin{bmatrix} L\kappa(s)\kappa(s_1) & L \\ L\kappa(s)\kappa(s_1) - \kappa(s_1)\sin\theta - \kappa(s)\sin\theta_1 & \qquad L\kappa(s_1)-\sin\theta_1 \end{bmatrix}. \end{equation} \end{proposition} See \cite[Section 2.11]{ChMa} for more details. Our proof below is slightly different, which provides the setup that will be used again to derive the tangent map of Minkowski billiards, see Section~\ref{sec.su} and Proposition~\ref{pro.DT.Minkowski} for more details. \begin{proof} First note that $\sin\theta > 0$ so that $\omega$ is in fact an area form on $\cM$. Let $\gamma(s)$ and $\gamma(s_1)$ be distinct points on the boundary of the billiard table, and $L=L(s,s_1)$ be as above. Then $\frac{\partial L}{\partial s} = -\cos\theta$ and $\frac{\partial L}{\partial s_1} = \cos\theta_1$. It follows that \begin{align} dL &= \frac{\partial L}{\partial s}ds + \frac{\partial L}{\partial s_1}ds_1 = -\cos\theta ds + \cos\theta_1 ds_1, \label{eq.dLa}\\ d^2L &= \sin\theta \ d\theta \wedge ds - \sin\theta_1 \ d\theta_1 \wedge ds_1 = 0. \nonumber \end{align} This proves the first part of the theorem. To compute the differential $DT$, first note that since $\gamma$ is parameterized by arc-length, $\dgs \cdot \left(\gamma(s_1)-\gamma(s) \right) = L(s,s_1)\cos\theta$ and $\dgso \cdot \left(\gamma(s_1)-\gamma(s) \right) = L(s,s_1)\cos\theta_1$. Hence \begin{align} \cos\theta &= \frac{1}{L(s,s_1)}\left(\gamma(s_1)-\gamma(s)\right)\cdot\dgs, \label{eq: costheta} \\ \cos\theta_1 &= \frac{1}{L(s,s_1)}\left(\gamma(s_1)-\gamma(s)\right)\cdot\dgso. \label{eq: costheta1} \end{align} Taking the differential of equation (\ref{eq: costheta}) yields \begin{align} -\sin\theta d\theta &= \bigg[-\frac{1}{L^2}\frac{\partial L}{\partial s} \left(\gamma(s_1)-\gamma(s)\right) \cdot \dgs + \frac{1}{L} \bigg( \left( \gamma(s_1)-\gamma(s) \right)\cdot \ddgs -\dgs \cdot \dgs \bigg) \bigg]ds \\[5pt] &\qquad + \bigg[ -\frac{1}{L^2}\frac{\partial L}{\partial s_1} \left(\gamma(s_1)-\gamma(s)\right) \cdot \dgs + \bigg]ds_1 \\[5pt] &= \left[\frac{\cos^2\theta-1}{L} + \kappa(s)\sin\theta \right]ds + \left[\frac{-\cos\theta\cos\theta_1}{L} + \frac{\cos(\theta + \theta_1)}{L}\right]ds_1 \\[5pt] &= \left[\frac{-\sin^2\theta}{L} + \kappa(s)\sin\theta \right]ds - \frac{\sin\theta\theta_1}{L}ds_1. \end{align} Solving for $\sin\theta_1ds_1$ gives \begin{equation} \sin\theta_1ds_1 = \left[L \kappa(s)-\sin\theta\right]ds + L d\theta. \label{eq: sintheta1} \end{equation} It follows that \begin{align} \frac{\partial s_1}{\partial s} &= \frac{L\kappa(s)-\sin\theta}{\sin\theta_1},\\[5pt] \frac{\partial s_1}{\partial \theta} &= \frac{L}{\sin\theta_1}. \end{align} Similarly, taking the differential of equation (\ref{eq: costheta1}) yields \begin{align} -\sin\theta_1 d\theta_1 &= \left[\frac{\cos\theta\cos\theta_1}{L} - \frac{\cos(\theta+\theta_1)}{L}\right]ds + \left[\frac{1-\cos^2\theta_1}{L}-\kappa(s_1)\sin\theta_1 \right]ds_1. \end{align} Simplifying the equation, we have \begin{align} \sin\theta_1d\theta_1 = -\frac{\sin\theta\sin\theta_1}{L}ds + \left[\kappa(s_1)-\frac{\sin\theta_1}{L}\right]\sin\theta_1ds_1. \end{align} Plugging in equation (\ref{eq: sintheta1}) into the above equation, we get \begin{equation} \sin\theta_1d\theta_1 = \bigg[L \kappa(s)\kappa(s_1)-\kappa(s_1)\sin\theta - \sin\theta_1\kappa(s)\bigg]ds + \bigg[\kappa(s_1)L-\sin\theta_1\bigg]d\theta. \end{equation} It follows that \begin{align} \frac{\partial \theta_1}{\partial s} &= \frac{1}{\sin\theta_1}\bigg[L\kappa(s)\kappa(s_1) - \kappa(s_1)\sin\theta - \kappa(s)\sin\theta_1 \bigg], \\[5pt] \frac{\partial \theta_1}{\partial \theta} &= \frac{1}{\sin\theta_1}\bigg[L\kappa(s_1)-\sin\theta_1\bigg]. \end{align} Collecting terms, we complete the proof of the proposition. \end{proof} Here we consider another variable $u= - \cos\theta$, which gives rise to an equivalent coordinate system of the phase space $\cM$ via $\bR/\ell_{\gamma} \times [-1, 1]$. In particular, the corresponding area form $\omega$ on $\cM$ becomes the standard area: \begin{align} \omega = \sin\theta ds \wedge d\theta = ds \wedge du, \label{eq.Tsymp} \end{align} and Eq.~\eqref{eq.dLa} becomes \begin{align} dL &=\ -\cos\theta ds + \cos\theta_1 ds_1 = u ds - u_1 ds_1. \label{eq.dLu} \end{align} In this way, the function $L=L(s,s_1)$ is a generating function of the billiard map $T: (s, u)\mapsto (s_1, u_1)$ in the sense that \begin{align} u = \frac{\pa L}{\pa s}(s, s_1), \quad u_1 = -\frac{\pa L}{\pa s_1}(s, s_1). \label{eq.def.u} \end{align} This change is necessary when studying Minkowski billiards since the angles are no long defined. \subsection{Stability of periodic billiard orbits} Now that we have introduced some basic properties of dynamical billiards, it is only natural to ask whether periodic orbits exist, and if so study their properties. Despite its simplicity, this is a surprisingly deep question. For example, in 1775 Fagnano proved that when the billiard table is an acute triangle, there always exists a periodic orbit - namely the \textit{orthic triangle}. However, the case of an obtuse triangle has proven to be much more elusive. It wasn't until the last fifteen years that \cite{Sch09} proved the existence of periodic orbits when the billiard table is a triangle whose angles are all at most one hundred degrees. In contrast, Birkhoff \cite{Bi27} showed that every convex billiard table with a sufficiently smooth boundary has many periodic orbits. Given a periodic orbit a natural question to ask is: What if we vary the initial conditions (maybe ever so slightly)? Do we obtain a trajectory that closely resembles the periodic orbit, or do we get something vastly different? And if the result is vastly different, is it at least in some sense 'predictable', or does it appear to be totally random? So the first step is to classify the periodic orbit according to the eigenvalues of the tangent map iterated up to period, see Section \ref{sec.dysy}. It follows from the Hartman-Grobman theorem that hyperbolic fixed points have the stable and unstable manifolds on the phase space and are locally unstable. On the other hand, the geometric characterizations for general elliptic fixed points can be very different. Given an elliptic fixed point $x^$, the linearization $D_{x^}f$ acts as a rigid rotation around the fixed point. However, the local dynamics around an elliptic fixed point can be very rich for the underlying map $f$. This is in contrast to the hyperbolic case where $f$ and $D_pf$ always posses the similar dynamic behaviors. In this section we give sufficient conditions for when the elliptic islands are inherited by $f$. For more details, see the classic text \cite{SiMo}. \begin{definition} A fixed point $x^$ of $f: U \subseteq \bR^2 \rightarrow \bR^2$ is said to be \textit{Moser stable} or \textit{nonlinearly stable} if there exists a nested sequence of $f$-invariant neighborhoods $U_n$, $n\ge 1$, containing $x^$, whose boundaries $\partial U_n$ are invariant circles and $\bigcap\limits_{n\geq1} U_n = \{x^\}$. \end{definition} The definition of nonlinearly stable elliptic periodic points of period $n$ can be done using the iterate $f^n$. Note that elliptic periodic points are not always nonlinearly stable and there are cases for which elliptic periodic points are not surrounded by any invariant circle. As an example we consider lemon billiards first introduced in \cite{He93}. Let $Q(L)$ denote the class lemon billiard tables formed by intersecting two unit circles where $\ell$ represents the distance between the two centers. Denote by $\mathcal{O}_2(L) = \{P, \cT(P)\}$ the periodic 2-orbit reflecting between $\gamma_0(0)$ and $\gamma_1(0)$ (See Figure \ref{fig.lemon}). It is shown in \cite{JZ22} that $\mathcal{O}_2(L)$ is elliptic for $L \in (0,2)$, and nonlinearly stable for $L \in (0,2) \backslash \{1\}$. However, numerical calculations in \cite{Ch13} appear to demonstrate that the billiard map is ergodic on $Q(1)$ so that in particular, $\mathcal{O}_2(1)$ is not nonlinearly stable. \begin{figure}[htbp] \tikzmath{ \r=2; \a=sqrt(3); } \begin{tikzpicture} \draw[domain=-60:60, thick , samples=100] plot ({\rcos(\x)}, {\rsin(\x)}); \draw[domain=120:240, thick, samples=100] plot ({\r+\rcos(\x)}, {\rsin(\x)}); ll (0,0) node[left]{$\gamma_{0}$} circle[radius=0.05]; ll (2,0) node[right]{$\gamma_{1}$} circle[radius=0.05]; \draw[thick] (0,0) -- (2, 0); \end{tikzpicture} \caption{The lemon table $Q(1)$.}\label{fig.lemon} \end{figure} \subsection{Birkhoff normal form and Moser's twist mapping theorem}\label{sec.BNF} Let $f: U\to \bR^2$ be a symplectic embedding and $P=(0,0)$ be an elliptic fixed point of $f$. That is, the eigenvalues of the tangent matrix $D_P f$ satisfies $\|\lambda\| =1$ and $\lambda \neq \pm 1$. Then the point $P$ is said to be non-resonant if $\lambda^n \neq 1$ for any $n\ge 3$. For any $N\ge 1$, there exists an area preserving change of coordinates of the form \begin{align} h_N: U \rightarrow \bR^2, \quad \begin{bmatrix} x \\ y \end{bmatrix} \mapsto \begin{bmatrix} x + p_2(x,y) + \cdots + p_{2N+1}(x,y) \\ y + q_2(x,y) + \cdots + q_{2N+1}(x,y) \end{bmatrix} + \text{h.o.t} \label{eq: Birkhoff transformation} \end{align} where $p_j$ and $q_j$ are $j^\text{th}$ degree polynomials for every $j=2,3,...,2N+1$, under which one can express \begin{equation} h_N^{-1}\circ f \circ h_N \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} \cos\Theta(r^2) & -\sin\Theta(r^2) \\ \sin\Theta(r^2) & \cos\Theta(r^2) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \text{h.o.t.} \label{eq: BNF} \end{equation} where $r^2 = x^2 + y^2$, \begin{equation} \Theta(r^2) = \theta + \tau_1 r^2 + \tau_2 r^4 + \cdots + \tau_N r^N, \label{eq: Theta} \end{equation} and h.o.t stands for "higher order terms". \label{prop: BNF} The symplectic transformation (\ref{eq: Birkhoff transformation}) is called the $N$-th order \textit{Birkhoff transformation}, (\ref{eq: BNF}) is called the $N$-th order \textit{Birkhoff normal form} of the map $f$ around the elliptic fixed point $P$, and the coefficient $\tau_j$ is called the \textit{$j$-{th} twist coefficient} (or Birkhoff coefficient) of $f$ at $P$. The cases such that $\lambda^n = 1$ for some $n\ge 3$ are called \textit{resonances}. The name "twist coefficient" is given because geometrically, $\Theta(r^2)$ defined in (\ref{eq: Theta}) measures the amount of rotation about the fixed point. The same results hold for elliptic periodic orbits. See \cite{SiMo, Mos73} for more details. Moreover, Moeckel \cite{Moe90} studied the behavior of the first twist coefficient near an elliptic fixed point for a one-parameter family of area-preserving diffeomorphisms and gave a method for calculating $\tau_1$. These twist coefficients play an important role when determining the nonlinear stability of the elliptic fixed points. \begin{theorem}[Moser's twist mapping theorem]\label{thm: Moser twist} If for an area-preserving map of the form (\ref{eq: BNF}), the polynomial $\Theta(r^2)$ defined in (\ref{eq: Theta}) is not identically zero, then the elliptic fixed point $P$ is nonlinearly stable. \end{theorem} There have been many applications of Moser's Twist Mapping Theorem in the study of dynamical billiards, see \cite{KP01, DKP03, BuGr, XiZh14} and the references therein. In \cite{KP05} the authors obtained an explicit formula of the first twist coefficients of elliptic $2$-periodic orbits in terms of the geometric characterizations of the billiard table. In \cite{JZ22} the authors studied the Birkhoff normal form around elliptic periodic points for a large class of billiards and used it to give explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. The authors were then able to apply the formulas of the twist coefficients to obtain characterizations of the nonlinear stability and local analytic integrability of billiards around elliptic periodic points. \section{Minkowski Billiards} Minkowski billiards and Finsler billiards were introduced by Gutkin and Tabachnikov in \cite{GT02} as a natural generalization of Euclidean billiards that use a Finsler structure to represent a billiard traveling on an anisotropic and inhomogeneous medium. On Euclidean billiard systems, a billiard travels with uniform motion at every point on the interior of the billiard table -- a consequence of the indicatrix of the Euclidean metric being a unit circle at each point. However, the indicatrix of a Finsler structure is not necessarily round or symmetric, but only a strictly convex curve that may vary from point to point. Hence the motion of the billiard depends on its location and direction. In the remainder of this section we will restate the reflection law for Minkowski billiard systems given the lack of angles in Minkowski geometry, and prove that the Minkowski billiard map is symplectic under an appropriate set of coordinates. \subsection{Reflection laws in Minkowski spaces}\label{ss.reflectionM} In Euclidean billiard systems, the reflection law is that the angle of incidence equals the angle of reflection. However, because Minkowski structures are not induced by an inner product, Minkowski geometry lacks the notion of angles. Hence to work with billiards in the Minkowski setting, a reflection law must be specified. Let $(\bR^2, F)$ be a Minkowski space, $Q\subset \bR^2$ be a bounded and strictly convex domain with (piecewise) smooth boundary $\partial Q$. As usual, $Q$ will be called the billiard table and $\pa Q$ will be parameterized by a piece-wise smooth curve $\gamma(s)$ (counterclockwise) with $F(\dot\gamma(s)) =1$ when it exists. Further, let $y_0$ and $y_1$ be two points in the interior of $Q$ and $x$ be a point on the boundary of $Q$. Then the Minkowski distances from $y_0$ to $x$ and from $x$ to $y_1$ are given as $F(x-y_0)$ and $F(y_1-x)$ respectively. Then $xy_1$ is the reflected ray of $y_0x$ if and only if $x$ is a critical point of the distance function $F(x-y_0) + F(y_1-x)$. In \cite{GT02} Gutkin and Tabachnikov established the reflection law for Finsler billiards using the geometry of the indicatrix as follows: \begin{proposition}[Finsler reflection law]\label{pro.Finsler.reflection.law} Let $y_0,y_1 \in Q$, and $x \in \pa Q$ be as above so that $xy_1$ is the billiard reflection of $y_0x$, and let $u,v \in I_x$ be the $F$-unit vectors traveling along $y_0x$ and $xy_1$ respectively. Then $\mathcal{L}(v)-\mathcal{L}(u)$ is conormal to $T_x \partial Q$ i.e. $\left(\mathcal{L}(v)-\mathcal{L}(u)\right)(w) = 0 $ for $w \in T_x \partial Q$. \end{proposition} Given an incident billiard trajectory, we can use Proposition \ref{pro.Finsler.reflection.law} to find the reflected trajectory in the following manner: \begin{enumerate} \item $T_uI_x$ is not parallel to $T_x\partial Q$: the line $T_uI_x$ intersects $T_x\partial Q$ at a unique point, say $w \in T_x\partial Q$. The Finsler reflection law states that the reflected vector $v \in T_xI$ satisfies $\mathcal{L}(u)(w) = \mathcal{L}(v)(w) = 1$. Therefore, so $w \in T_vI_x$. This determines $v$, see Fig.~\ref{fig.F.reflection}. \item $T_uI_x$ is parallel to $T_x\partial Q$: $\mathcal{L}(u)$ is proportional to a covector that is conormal to $T_x\partial Q$. The Finsler reflection law states that $\mathcal{L}(v)$, where $v$ is the reflected ray, is also proportional to $\mathcal{L}(u)$. Hence $T_vI_x$ is parallel to $T_uI_x$ thus determining the reflected vector $v$. \end{enumerate} It is important to note that Finsler billiards are generally irreversible, i.e. given an incident and reflected billiard trajectory, we cannot reverse their roles unless the norm $F$ reversible: $F(x,-\tilde{v}) = F(x,\tilde{v})$ for $\tilde{v} \in T_xM$. It is clear that the reflection is that of equal angles when the indicatrix is a circle centered at the origin. \begin{figure}[htbp] \tikzmath{ \a=18; \b=155; } \begin{tikzpicture}[scale=1.5] \draw [domain=0:360, samples=100] plot ({2cos(\x) + 1}, {0.5sin(\x)}) node[yshift=-0.2in]{$I_x$}; \draw [thick, domain=-0.6:0.7, samples=100] plot ({\x}, {\x-\x\x}) node[xshift=-0.6in,yshift=-0.6in]{$\pa Q$}; \draw (-1,-1) -- (2,2); \draw ({2cos(\a) + 1}, {0.5sin(\a)}) -- (1.5,1.5) node[pos=0.5, above right]{$T_v I_x$} node(w){} -- ({2cos(\b) + 1}, {0.5sin(\b)}) node[pos=0.5, above left]{$T_u I_x$}; \draw ({2cos(\a) + 1}, {0.5sin(\a)}) node(v){} -- (0,0) node[pos=0.03, sloped]{$>$} node[pos=0.5](A){}; \draw[dashed] ({2cos(\b) + 1}, {0.5sin(\b)}) node(u){} -- (0,0) node[pos=0.1, sloped]{$<$} node[pos=3](B){}; \draw[thick] (0,0) -- (A) node[pos=0.5, sloped]{$>$}; \draw[thick] (0,0) -- (B) node[pos=0.5, sloped]{$<$}; ll (0,0) circle(0.06) node[below]{$x$}; ll (u) circle(0.05) node[left]{$u$}; ll (v) circle(0.05) node[right]{$v$}; ll (w) circle(0.05) node[left]{$w$}; \end{tikzpicture} \caption{An illustration of the Finsler reflection law.}\label{fig.F.reflection} \end{figure} \subsection{New coordinate system on $\cM$ for Minkowski billiards}\label{sec.su} Let $(\bR^2, F)$ be a Minkowski plane, $Q\subset \bR^2$ be a strictly convex domain with (piecewise) smooth boundary. The phase space $\mathcal{M}$ of the Minkowski billiard on $Q$ is given by all points $(x, v)$ where $x \in \partial Q$, and $v \in I_x$ points to the interior of $Q$, as we have had for Euclidean billiards. The Minkowski billiard map $\mathcal{T}$ is defined using the new reflection law in Proposition \ref{pro.Finsler.reflection.law}. That is, let $x_0 = \gamma(s_0)$ and $x_1 = \gamma(s_1)$ be points on $\partial Q$, $v_0 \in I_{x_0}$ be the $F$-unit vector at $x_0$ pointing to $x_1$, and $v_1 \in I_{x_1}$ the $F$-unit vector pointing in the direction of the reflected billiard trajectory. Then the Minkowski billiard map is $\mathcal{T}: \mathcal{M} \rightarrow \mathcal{M}$, $(s_0,v_0) \mapsto (s_1,v_1)$. Next we introduce a new variable $u=u(s, s_1)$ and a new coordinate system $(s,u)$ on $\cM$ for the billiard map $\cT$ using the generating function method mimicking what has been done in Eq.~\eqref{eq.def.u}. \begin{definition}\label{def.new.variable} Let $Q \subset (\bR^2, F)$ be a compact and convex domain with piecewise smooth boundary $\pa Q$, $\gamma(s)$ be a parametrization of $\pa Q$ with the $F$-arc-length parameter $s$ in the sense that $F(\dot \gamma(s)) =1$, $L(s,s_1) = F(\gamma(s_1)-\gamma(s))$ the Minkowski length from $\gamma(s)$ to $\gamma(s_1)$. Then the following defines a new variable \begin{align} u = u(s, s_1) := \frac{\partial L}{\partial s}(s,s_1). \label{def.u} \end{align} \end{definition} It is easy to see that this introduces a new coordinate system $(s,u)$ on $\cM$. Note that the lower and upper bounds of $u$ depend on the first coordinate $s$, and hence the phase space $\cM$ becomes a cylinder with varying lower and upper boundaries. Applying the above definition to the next iterate of the billiard orbit, we get that $u_1 =u(s_1,s_2)$. Then the Minkowski billiard map can be rewritten using the new coordinate as $\mathcal{T}: (s,u) \mapsto (s_1,u_1)$. \begin{remark} It is worth pointing out that the second identity in Eq.~\eqref{eq.def.u} should not be interpreted as the definition of the coordinate $u_1$. Rather, it represents a property of the Euclidean billiard map. We will verify that the same identity does hold for Minkowski billiard maps, see \eqref{eq.verify.u1} in the proof of Proposition \ref{pro.DT.Minkowski}. \end{remark} \subsection{The tangent map $D\cT$} With these new coordinates we can derive the tangent maps for Minkowski billiards as we have done in Proposition \ref{pro:DT.euclid} for Euclidean billiards. \begin{proposition} \label{pro.DT.Minkowski} The Minkowski billiard map $\mathcal{T}$ preserves the area form $\omega = ds \wedge du$. Moreover, the tangent map $D_{(s,u)}\mathcal{T} = \begin{bmatrix} \frac{\partial s_1}{\partial s} & \frac{\partial s_1}{\partial u} \\ \frac{\partial u_1}{\partial s} & \frac{\partial u_1}{\partial u}, \end{bmatrix}$ with entires given by \begin{align} \dsds &= \frac{\sum\limits_{i,j=1}^2 \Fij\dgis\dgjs - \sum\limits_{i=1}^2\Fi\ddgis}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss}, \label{eq: dsds} \\[10pt] \dsdu &= \frac{-1}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss}, \label{eq: dsdu} \\[10pt] \duds &= \sum\limits_{i,j}^2\Fij \dgiss \dgjs - \bigg(\sum\limits_{i,j=1}^2\Fij \dgiss \dgjss \nonumber \\[10pt] & \qquad + \sum\limits_{i=1}^2 \Fi \ddgiss \bigg) \left( \frac{\sum\limits_{i,j=1}^2 \Fij\dgis\dgjs - \sum\limits_{i=1}^2\Fi\ddgis}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss} \right), \label{eq: duds} \\[10pt] \dudu &= \frac{\sum\limits_{i,j=1}^2\Fij \dgiss \dgjss + \sum\limits_{i=1}^2\Fi \ddgiss}{\sum\limits_{i,j=1}^{2}\Fij \dgis \dgjss}. \label{eq: dudu} \end{align} \end{proposition} \begin{proof} We proceed in much the same as in Proposition \ref{pro:DT.euclid}. Let $\gamma(s),\gamma(s_1)$, and $\gamma(s_2)$ be points on the boundary $\pa Q$ such the the segment $\gamma(s_1)\gamma(s_2)$ is the reflected billiard trajectory of the billiard trajectory $\gamma(s)\gamma(s_1)$. Recall the distance function $L(s,s_1) = F(\gamma(s_1)-\gamma(s))$ and the variable $u =u(s,s_1) = \frac{\partial L}{\partial s}(s,s_1)$ as above. Moreover, $u_1 = u(s_1,s_2)$. By the billiard reflection law, $s_1$ is a critical point of the combined distance $L(s,s_1) + L(s_1,s_2)$. Differentiating $s_1$, we get \begin{align} \frac{\partial}{\partial s_1}\left[ L(s,s_1) + L(s_1,s_2) \right] =\frac{\partial L}{\partial s_1}(s,s_1)+ u(s_1,s_2) =\frac{\partial L}{\partial s_1}(s,s_1)+ u_1= 0. \end{align} Therefore, \begin{align} u_1 = -\frac{\partial L}{\partial s_1}(s,s_1). \label{eq.verify.u1} \end{align} This establishes the second identity in \eqref{eq.def.u} for Minkowski billiards. Applying the newly obtained identity \eqref{eq.verify.u1}, we can rewrite the total differential of $L(s, s_1)$ as \begin{equation} dL(s,s_1) = \frac{\partial L}{\partial s}(s,s_1) ds + \frac{\partial L}{\partial s_1}(s,s_1) ds_1 = u \, ds - u_1 \, ds_1. \label{eq.dL.Minkowski} \end{equation} Taking exterior differential of \eqref{eq.dL.Minkowski} again, we get $ddL = du \wedge ds - du_1 \wedge ds_1 = 0$, thus proving the first part of the proposition. Next we will derive the differential of the Minkowski billiard map $\cT: (s,u) \mapsto (s_1, u_1)$. To shorten our expressions, we will use $v = \gamma(s_1)-\gamma(s)$. Then $L(s,s_1) =F(v)$ and \begin{align} u &= \frac{\partial L}{\partial s}(s,s_1) = - \sum\limits_{j=1}^2 F_{v_j}(v)\dgis, \\ u_1 &= -\frac{\partial L}{\partial s_1}(s,s_1) = -\sum\limits_{j=1}^2 F_{v_j}(v)\dgiss, \end{align} where we have applied \eqref{eq.verify.u1} in the second equation. Taking the differentials of $u$ and $u_1$ yield \begin{align} du &= \left[ \sum\limits_{j,k =1}^2 F_{v_jv_k}(v)\dgis\dgjs - \sum\limits_{j=1}^2 F_{v_j}(v)\ddgis \right]ds \nonumber \\[10pt] &\qquad - \sum\limits_{j,k=1}^2 F_{v_jv_k}(v)\dgis\dgjss \ ds_1 \label{eq: du} \\[10pt] du_1 &= \sum\limits_{j=1}^2 F_{v_jv_k}(v) \dgiss \dgjs \ ds - \bigg[ \sum\limits_{j,k=1}^2 F_{v_jv_k}(v) \dgiss \dgjss \nonumber \\[10pt] &\qquad + \sum\limits_{j=1}^2 F_{v_j}(v) \ddgiss \bigg]ds_1. \label{eq: du1} \end{align} Solving for $ds_1$ in Eq.~(\ref{eq: du}) gives \begin{align} ds_1 &= \frac{\sum\limits_{i,j=1}^2 \Fij\dgis\dgjs - \sum\limits_{i=1}^2\Fi\ddgis}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss} \ ds - \frac{du}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss} \label{eq: ds1}. \end{align} Substituting Eq.~\eqref{eq: ds1} into Eq.~\eqref{eq: du1} and collecting $ds$ and $du$ terms yield \begin{align} du_1 &= \left[ \sum\limits_{i,j}^2\Fij \dgiss \dgjs - \bigg(\sum\limits_{i,j=1}^2\Fij \dgiss \dgjss \right. \nonumber \\[10pt] & \qquad \left. + \sum\limits_{i=1}^2 \Fi \ddgiss \bigg) \left( \frac{\sum\limits_{i,j=1}^2 \Fij\dgis\dgjs - \sum\limits_{i=1}^2\Fi\ddgis}{\sum\limits_{i,j=1}^2\Fij\dgis\dgjss} \right) \right] \ ds \nonumber \\[10pt] & \qquad + \frac{\sum\limits_{i,j=1}^2\Fij \dgiss \dgjss + \sum\limits_{i=1}^2\Fi \ddgiss}{\sum\limits_{i,j=1}^{2}\Fij \dgis \dgjss}du \label{eq: du1 second}. \end{align} Then the entries of the tangent map $D\cT$ follows from Eq.~\eqref{eq: ds1} and Eq.~\eqref{eq: du1 second}. \end{proof} \section{Billiards in Mixed Norm Spaces}\label{sec.billiards.symmetry} In this section we will study the properties of periodic 2-orbits of Minkowski billiards on the Minkowski plane $(\bR^2,F)$, where the Minkowski norm $F(v) = F_{a,b}(v) = a\lVert v \rVert_2 + b\lVert v \rVert_4$ with $a > 0$, $b \geq 0$, and $a+b=1$. We have showed that $(\bR^2,F)$ is a Minkowski space in Proposition \ref{prop: Mixed norm is Minkowski}, and in fact is a normed space so that the indicatrix $I$ given by the level curve $F(v) = 1$ is symmetric about the origin. The assumption $a+b=1$ not only makes it so that the calculations for the twist coefficient $\tau_1$ are easier to manage, but make it so that the billiard has identical motion to a Euclidean billiard system along the horizontal and vertical axes (see Figure \ref{fig: Mixed norm indicatrix}). \begin{figure}[htbp] \centering \subcaptionbox{$a = 0.7$ and $b = 0.3$}{\includegraphics[width=0.32\textwidth]{Images/mixed_norm_0.7}} \hfill \subcaptionbox{$a = 0.4$ and $b = 0.6$}{\includegraphics[width=0.32\textwidth]{Images/mixed_norm_0.4}} \hfill \subcaptionbox{$a = 0.1$ and $b = 0.9$}{\includegraphics[width=0.32\textwidth]{Images/mixed_norm_0.1}} \caption{Indicatricies of Minkowski spaces with the mixed norm $F_{a,b}$.} \label{fig: Mixed norm indicatrix} \end{figure} Let $\alpha(t) = \sum\limits_{n \geq 1} a_{2n}t^{2n}$ and $\beta(t) = \sum\limits_{n\geq1}b_{2n}t^{2n}$, $t \in (-\epsilon, \epsilon)$, be two even functions for some $\epsilon > 0$, and $\gamma_0: t\mapsto (\alpha(t),t)$ and $\gamma_1: t\mapsto (L-\beta(t),-t)$, $t \in (-\epsilon, \epsilon)$, be two curves on $\bR^2$. The class of billiard tables $Q(L,\alpha,\beta)$ we will consider are those whose boundary $\partial Q$ is given by a piece-wise smooth curve $\gamma$ consisting of two horizontal line segments connecting the curves $\gamma_0$ and $\gamma_1$ at their endpoints (see Figure \ref{fig.btable}). Rather than parameterizing $\gamma$ with a global Minkowski arc-length parameter, we will use a local arc-length parameter $s$ on each of the curves $\gamma_j$ such that $F(\gamma_j'(s)) \equiv 1$, $s(\gamma_j(0)) = 0$ for $j = 0,1$. It is clear from the Finsler reflection law that there exists a periodic 2-orbit $\mathcal{O}_2$, bouncing back and forth between $(0,0)$ and $(L,0)$ where $L$ is the Euclidean length between the two reflection points. \begin{figure}[htbp] \begin{tikzpicture} \draw[->] (-1,0) -- (5.5,0); \draw[->] (0,-1.5) -- (0,1.5); \draw [domain=-1:1, samples=100, thick] plot ({4-\x\x +0.3\x\x\x\x }, {\x}); \node at (3.9, 0.7) {$\gamma_1$}; \draw [domain=-1:1, samples=100, thick] plot ({\x\x +0.5\x\x\x\x }, {\x}); \node at (0.3,0.8) {$\gamma_0$}; \draw[thick, ->] (0,0) node[below left]{$0$} -- (1,0) node[below]{$p$}; \draw[thick, ->] (4,0) node[below right]{$L$} -- (3,0) node[below]{$\cT(p)$}; \draw[thick] (3.3, 1) -- (1.5, 1) (3.3, -1) -- (1.5, -1); ll (0,0) circle[radius=0.06]; ll (4,0) circle[radius=0.06]; \end{tikzpicture} \caption{An illustration of the billiard table.} \label{fig.btable} \end{figure} The phase space $\mathcal{M}$ is the set of unit vectors with foot on $\partial Q$ that point inside $Q$. Recall that the new coordinates $(s,u)$ on $\mathcal{M}$ introduced in Section~\ref{sec.su}, where $u = u(s, s_1) = \frac{\partial L}{\partial s}(s,s_1)$, see Definition~\ref{def.new.variable}. Therefore the billiard map $\mathcal{T}: \mathcal{M} \rightarrow \mathcal{M}$ is given by $\mathcal{T}: (s,u) \mapsto (s_1,u_1)$, where $u_1 = u(s_1,s_2)$. For example, the periodic 2-orbit in Figure \ref{fig.btable} is given by $\mathcal{O}_2 = \{p,\mathcal{T}(p)\}$ where $p = (0,0)$ and $\mathcal{T}(p) = (0,0)$ with respect to the corresponding local coordinate systems of the phase space on the two curved parts. \subsection{Symmetric Minkowski billiards}\label{ss.sym.billiards} In Section~\ref{sec.BNF} we have stated how Moser's Twist Mapping Theorem (Theorem \ref{thm: Moser twist}) utilizes the twist coefficients $\tau_j$ in the Birkhoff normal form to obtain the nonlinear stability of elliptic periodic billiard orbits. We now construct Birkhoff transformations and give an explicit formula for the first twist coefficient in terms of the geometric properties of the Minkowski billiards. We will then use the first twist coefficient to investigate the nonlinear stability property of periodic orbits of Minkowski billiards. We will start with the case $\alpha(t) = \beta(t)$, so that the billiard table $Q(L, \alpha, \alpha)$ is symmetric about the line $x = L/2$. Combining with the symmetry of the Minkowski norm $F_{a,b}$, we get that the radius of curvature functions $R_j(s)$ on the two parts $\gamma_j$ are identical, so we set $R(s) = R_0(s) = R_1(s)$. Moreover, let $\text{Rot}_\pi$ be the rotation of the table $Q(L,\alpha,\alpha)$ about the point $(\frac{L}{2},0)$ by $\pi$. Since $F(v) = F(-v)$ then $\text{Rot}_\pi$ identifies the phase space $\mathcal{M}_1$ of $\gamma_1$ with the phase space $\mathcal{M}_0$ of $\gamma_0$.\footnote{This is false for general Minkowski norms that don't satisfy $F(-v) = F(v)$.} \subsection{Reducing the two-step map using the symmetry}\label{ss.reducing.one.step} Pick a small neighborhood $U_0 \subset \mathcal{M}_0$ of $P\in \cM_0$. Then $U_1 = \text{Rot}_\pi(U_0) \subset \mathcal{M}_1$ is a small neighborhood of $\mathcal{T}(P) \in \cM_1$. Let $\cT_i = \cT\|_{U_i}: U_i \to \cM_{1-i}$ be the restriction of the billiard map $\cT$ to $U_i$, $i=0,1$. Observe that $ \text{Rot}_\pi\circ\cT_0 = \cT_1\circ\text{Rot}_\pi$. It follows that, restricting on $U_0$, \begin{align} \cT^2 = \cT_1\circ \cT_0 = \text{Rot}_\pi\circ\cT_0 \circ \text{Rot}_\pi\circ\cT_0 =(\text{Rot}_\pi\circ\cT_0)^2. \end{align} If follows that the twist coefficient $\tau_1(\cT^2, P)= 2\tau_1(\text{Rot}_\pi\circ\cT_0, P)$. Identifying $\cM_1$ with $\cM_0$ using $\text{Rot}_\pi$, we can identify the one-step billiard map $\cT_0$ with the abstract self-map $\text{Rot}_\pi\circ\cT_0: U_0\to \cM_0$. In the following we will abuse our notation and use $\cT$ for the abstract map $\text{Rot}_\pi\circ\cT_0$. In particular, $\tau_1(\cT^2, P)= 2\tau_1(\cT, P)$. \subsection{Tangent matrix of the billiard map}\label{ss.tangent.map} We start with computing the tangent matrix of the billiard map $\mathcal{T}$. The first and second order partial derivatives of the Minkowski norm $F(v)=a \\|v\\|_2 + b\\|v\\|_4$ are given by \begin{equation} \begin{aligned}[c] F_{v^1} &= \frac{av_1}{\sqrt{v_1^2 + v_2^2}} + \frac{bv_1}{\left( v_1^4 + v_2^4 \right)^{\frac{3}{4}}}\\[10pt] F_{v^2} &= \frac{av_2}{\sqrt{v_1^2 + v_2^2}} + \frac{bv_2}{\left( v_1^4 + v_2^4 \right)^{\frac{3}{4}}} \end{aligned} \qquad\qquad \begin{aligned}[c] F_{v^1v^1} &= \frac{av_2^2}{\left(v_1^2 + v_2^2 \right)^{\frac{3}{2}}} + \frac{3bv_1^2v_2^4}{\left( v_1^4 + v_2^4 \right)^{\frac{7}{4}}} \\[10pt] F_{v^1v^2} &= -\frac{av_1v_2}{\left(v_1^2 + v_2^2\right)^{\frac{3}{2}}} - \frac{3bv_1^3v_2^3}{\left(v_1^4 + v_2^4 \right)} \\[10pt] F_{v^2v^2} &= \frac{av_1^2}{\left(v_1^2 + v_2^2 \right)^{\frac{3}{2}}} + \frac{3bv_1^4v_2^2}{\left( v_1^4 + v_2^4 \right)^{\frac{7}{4}}}. \end{aligned} \end{equation} Note that the tangent vector at $(0,0)$ and $(L,0)$ are given by $\gamma'_0(0) = \langle 0, -1 \rangle$ and $\gamma'_1(0) = \langle 0, 1 \rangle$ respectively. Recall $\gamma''(s) = \kappa_m(s)n_\gamma(s)$ where $\kappa_m$ and $n_\gamma$ can be written in terms of a parameterization $r(\theta(s))$ of the indicatrix as shown in equations (\ref{eq: km in Euclid}) and (\ref{eq: n in Euclid}). Given the norm $F(v) = a\lVert v \rVert_2 + b\lVert v \rVert_4$ for $a>0$ and $b \geq 0$, the parameterization of $\gamma_0$ is given by \begin{equation} r(\theta(s)) = \frac{1}{a + b \left(\cos^4\theta(s) + \sin^4\theta(s) \right)^{\frac{1}{4}}}. \end{equation} Moreover, from equation (\ref{eq: n in Euclid}) we see that $\kappa_m$ may be written in terms of the Euclidean radius of curvature since $\kappa_e(s) = \frac{1}{R(s)}$. Letting $R(0) = R$ and $a+b = 1$, Eq.~\eqref{eq: dsds}--\eqref{eq: dudu} at $P = (0,0)$ simplify to: \begin{align} \dsds(0,0) &= \frac{L}{aR}-1, \label{eq: dsds mixed norm} \\[10pt] \dsdu(0,0) &= \frac{L}{a}, \label{eq: dsdu mixed norm} \\[10pt] \duds(0,0) &= \frac{L-2aR}{aR^2}, \label{eq: duds mixed norm} \\[10pt] \dudu(0,0) &= \frac{L}{aR}-1. \label{eq: dudu mixed norm} \end{align} The tangent matrix of the one-step billiard map $\mathcal{T}$ at $P$ is thus given by \begin{equation} D_p\mathcal{T} = \begin{bmatrix} \ta_{10} & \ta_{01} \\ \tb_{10} & \ta_{01} \end{bmatrix} := \begin{bmatrix} \frac{L}{aR}-1 & \frac{L}{a} \\[10pt] \frac{L-2aR}{aR^2} & \frac{L}{aR}-1 \end{bmatrix}. \label{eq: DpT mixed norm} \end{equation} Note that in the Euclidean case where the indicatrix is a unit circle ($a = 1$, $b = 0$), Eq.~\eqref{eq: DpT mixed norm} reduces to the more familiar tangent matrix $D_P\mathcal{T} = \begin{bmatrix} \frac{L}{R} -1 & L \\ \frac{L}{R^2} - \frac{2}{R} & \frac{L}{R} -1 \end{bmatrix}$, see \cite[Eq.~(1.3)]{JZ22}. Because $\mathcal{T}$ is symplectic, the eigenvalues $\lambda$ of $D_p\mathcal{T}$ satisfy $\lambda^2 - \Tr(D_P\mathcal{T})\lambda + 1 = 0$, where $\Tr(D_P\mathcal{T}) = \frac{2L}{aR} - 2$. It follows that $D_P\cT$ is parabolic if $L = 2aR$, hyperbolic if $L > 2aR$, and elliptic if $ 0 < L < 2aR$. Going forward, we will restrict our discussion to the elliptic case. Then the eigenvalues of the tangent mattrix $D_P\cT$ are given by $\lambda = \ta_{10} - i\sqrt{-\ta_{01}\tb_{10}}$ and $\overline{\lambda} = \frac{1}{\lambda}$, see Eq.~\eqref{eq: DpT mixed norm}. The following non-resonance condition is needed to find the Birkhoff normal form and the first twist coefficient: \begin{itemize} \item [(A)] $\lambda^4 \neq 1 \text{ or equivalently, } L \in \left(0, 2aR \right) \backslash \bigl \{aR \bigl \}$. \end{itemize} \begin{theorem}\label{thm.t1.sym} Let $0< a \le 1$, $a + b=1$, $(\bR^2, F)$ be the Minkowski plane with the mixed norm $F=F_{a,b}$, $Q(L,\alpha, \beta)$ be the symmetric Minkowski billiard table, $R_0 = R_1 = R$ be the curvature at $t=0$ (the vertices). Suppose the nonresonance condition (A) is satisfied. Then the first twist coefficient of the one-step billiard map $\mathcal{T}:(s,u) \mapsto (s_1, u_1)$ at the elliptic periodic point $P$ is given by \begin{equation} \tau_1(\cT, P) = \frac{a^2 L R R''+(4-3 a) a L+2 (2 a-9) a R+12 R}{8 a^2 R (L-2 a R)} \label{eq: symmetric twist} \end{equation} \end{theorem} \begin{remark} As done in \cite{JZ22}, the functions $\alpha(t)$ and $\beta(t)$ are assumed to be even polynomials in order to simplify the computation of $\tau_1$. Without these assumptions, the expression for $\tau_1$ would be significantly more complex as it would also depend on $R'$. The proof of Theorem \ref{thm.t1.sym} is given over the subsequent sections and follows closely the approaches given in \cite{JZ22, Moe90}. Plugging in $a=1$ and $b=0$, we recover the first twist coefficient for the periodic orbit of period $2$ for Euclidean billiards: $\ds \tau_1 = \frac{1}{8}\left(\frac{1}{R}-\frac{LR''}{2R-L}\right)$, see also \cite{KP05, JZ22}. \end{remark} \subsection{Taylor polynomials of the billiard map}\label{ss.Taylor.coefficients} To begin, we first compute the Taylor expansion of the billiard map $\mathcal{T}(s,u) = (s_1,u_1)$ at $P = (0,0)$. We have computed the first order partial derivatives of $s_1$ and $u_1$ at $P$ which are given in equations \eqref{eq: dsds mixed norm}--\eqref{eq: dudu mixed norm}. To compute the higher order derivatives, we can apply the chain rule to equations \eqref{eq: dsds}--\eqref{eq: dudu}. More details for this calculation can be found in \cite{Vil}. Note that $R(s)$ has critical points at $\gamma_0(0)$ and $\gamma_1(0)$. Thus $R'(0) = 0$, and because of this, further calculations show that $\frac{\partial^{j+k} s_1}{\partial s^j \partial u^k}(0,0) = \frac{\partial^{j+k} u_1}{\partial s^j \partial u^k}(0,0) = 0$ for $j+k = 2$. Denote $R^{(j)}(0) = R^{(j)}$, we list the third order partial derivatives of $s_1$ and $u_1$ needed for the calculation of $\tau_1$: \begin{flalign} \frac{\partial^3 s_1}{\partial s^3}(0,0) &= \frac{1}{a^4R^5} \bigg[2a^4LR^2-3a^3R \left(L^2-LR+2R^2\right)-a^3LR^3R'' \nonumber \\ &\qquad + a^2 \left(L^3+3 L R^2\right)-3 a L R (L-3 R)-6 L R^2 \bigg], \end{flalign} \begin{flalign} \frac{\partial^3 s_1}{\partial s^2 \partial u} (0,0) &= \frac{1}{a^4R^4} \bigg[ 2a^4LR^2-2a^3R \left(L^2+R^2\right)+a^2 \left(L^3+2LR^2\right) \nonumber \\ &\qquad -3aLR(L-3R)-6LR^2 \bigg], \end{flalign} \begin{flalign} \frac{\partial^3 s_1}{\partial s \partial u^2}(0,0) &= \frac{L \left(-a^3 L R+a^2 \left(L^2+R^2\right)-3 a R (L-3 R)-6 R^2\right)}{a^4 R^3}, \end{flalign} \begin{flalign} \frac{\partial^3 s_1}{\partial u^3}(0,0) &= \frac{L \left(a^2 L^2-3 a R (L-3 R)-6 R^2\right)}{a^4 R^2}, \end{flalign} \begin{flalign} \frac{\partial^3 u_1}{\partial s^3}(0,0) &= \frac{1}{a^4R^6} \bigg[ 6LR^2-8a^5R^3-12a^3LR(L+R)+3a^2L^2(L + 3R) + 2a^4R^2(8L + 3R)\nonumber \\ &\qquad - 3aL(L^2 - LR + 3R^2)+ aR(L^3-3aL^2R+4a^2LR^2-2a^3R^3) R^{''} \bigg], \end{flalign} \begin{flalign} \frac{\partial^3 u_1}{\partial s^2 \partial u}(0,0) &= \frac{-1}{a^4R^5} \bigg[ 6a^4 LR^2-a^3R \left(9L^2+3 LR + 2R^2\right) +a^2 L \left(3 L^2+6 L R+R^2\right) \nonumber \\ &\qquad- 3aL\left(L^2- LR + 3R^2\right) + aLRR''(L-a R)^2 + 6LR^2 \bigg], \end{flalign} \begin{flalign} \frac{\partial^3 u_1}{\partial s \partial u^2}(0,0) &= \frac{-1}{a^4R^4} \bigg[ 6LR^2 + 2a^4 LR^2 + a^2L(3L^2 + 3LR + 2R^2) \nonumber \\ &\qquad - 3aL(L^2 - LR + 3R^2) - 2a^3(3L^2R + R^3) + aL^2R(L - aR)R^{''} \bigg], \end{flalign} \begin{flalign} \frac{\partial^3 u_1}{\partial u^3}(0,0) &= \frac{3 (a-1) L \left(a^2 L R-a \left(L^2-L R+R^2\right)+2 R^2\right)-a L^3 R R''}{a^4 R^3}. \end{flalign} The Taylor polynomial of $\mathcal{T}(s,u) = (s_1, u_1)$ at $P = (0,0)$ is thus given by \begin{align} s_1(s,u) &= \ta_{10}s + \ta_{01}u + \sum_{j+k = 3} \ta_{jk}s^ku^k + \text{h.o.t} \label{eq: s1 mixed norm}\\ u_1(s,u) &= \tb_{10}s + \tb_{01}u + \sum_{j+k = 3} \tb_{jk}s^ku^k + \text{h.o.t}, \label{eq: u1 mixed norm} \end{align} where $\ta_{jk} = \frac{1}{j!k!} \frac{\partial^{j+k} s_1}{\partial s^j \partial u^k}(0,0)$, $\tb_{jk} = \frac{1}{j!k!} \frac{\partial^{j+k} u_1}{\partial s^j \partial u^k}(0,0)$, and h.o.t stands for higher order terms. \subsection{The first transformation: a rigid rotation}\label{ss.rigid} Having computed the third order Taylor polynomial of the billiard map $\mathcal{T}$, we may now initiate the construction of a symplectic coordinate transformation for $\mathcal{T}$ to be in Birkhoff normal form. The first step consists of finding a coordinate transformation for which $D_p\mathcal{T}$ acts as a rigid rotation. Given $D_P\mathcal{T}$ as in equation (\ref{eq: DpT mixed norm}), $\lambda = \ta_{10} - i\sqrt{-\ta_{01}\tb_{10}}$ is an eigenvalue of $D_P\mathcal{T}$ with corresponding eigenvector $v_\lambda = \left[ \begin{smallmatrix} i\eta \\[7pt] \eta^{-1} \end{smallmatrix} \right]$, where $\eta = \left(-\ta_{01}\tb_{10}\right)^{1/4} > 0$. It follows that $D_P\mathcal{T}$ is conjugate to a rotation matrix $R_\theta$ where $\theta \in (0, 2\pi)$ is the argument of $\lambda$. That is, we can write $D_PT = BR_\theta B^{-1}$ where \begin{equation} R_\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \qquad B = \begin{bmatrix} \eta & \ 0 \\ 0 & \ \eta^{-1} \end{bmatrix}. \end{equation} Define the coordinate transformation $(s,u) \mapsto (x,y)$ given by \begin{equation} \begin{bmatrix} x \\ y \end{bmatrix} = B^{-1}\begin{bmatrix} s \\ u \end{bmatrix} = \begin{bmatrix} s/\eta \\ \eta u \end{bmatrix}. \end{equation} \noindent We abuse notation by adopting the convention that $\mathcal{T}$ will always denote the billiard map regardless of the coordinate system. This will hold true for later transformations as well. Therefore we denote $\mathcal{T}(x,y) = (x_1, y_1) = (s_1/\eta, \eta u_1)$, and it follows form equations (\ref{eq: s1 mixed norm}) and (\ref{eq: u1 mixed norm}) that \begin{align} \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} &= B^{-1} \begin{bmatrix} \ta_{10}s + \ta_{01}u + \sum\limits_{j+k =3} \ta_{jk}s^ju^k \\ \tb_{10}s + \tb_{01}u + \sum\limits_{j+k =3} \tb_{jk}s^ju^k \end{bmatrix} + \ \text{h.o.t} \label{eq: x1y1 B-1 mixed norm} \\[5pt] &= R_\theta \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} \sum\limits_{j+k = 3} a_{jk}x^jy^k \\ \sum\limits_{j+k = 3} b_{jk}x^jy^k \end{bmatrix} + \ \text{h.o.t} \label{eq: x1y1 R mixed norm} \end{align} \noindent where we obtained (\ref{eq: x1y1 B-1 mixed norm}) from (\ref{eq: x1y1 R mixed norm}) by using the relation $\begin{bmatrix} s \\ u \end{bmatrix} = B \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \eta x \\ y/\eta \end{bmatrix}$. \noindent The coefficients $a_{jk}$ and $b_{jk}$ of the billiard map $\mathcal{T}$ in the new coordinates $(x,y)$ are given by \begin{equation} a_{jk} = \eta^{j-k-1}\ta_{jk} \qquad \text{and} \qquad b_{jk} = \eta^{j-k+1}\tb_{jk} \end{equation} \noindent for $j+k = 1,3$. Finally, note that $\mathcal{T}:(x,y) \mapsto (x_1,y_1)$ is symplectic since both $B$ and $T:(s,u) \mapsto (s_1,u_1)$ are symplectic. \begin{figure}[htbp] \begin{align} \xymatrixcolsep{3pc} \xymatrixrowsep{3pc} \xymatrix{(s,u) \ar[d]_{\mathcal{T}} \ar[r]^{B^{-1}} & (x,y) \ar[d]_{\mathcal{T}}\\ (s_1,u_1) \ar[r]_{B^{-1}} & (x_1,y_1)} \end{align} \caption{The commutative diagram of the first coordinate transformation} \label{fig: first tower of transformations} \end{figure} \subsection{The second transformation: diagonalizing the linear terms}\label{ss.diagonal} Next, we now need a second coordinate transformation in which $R_\theta$ becomes the diagonal matrix $D = \begin{bmatrix} \lambda & \hspace{10pt} 0 \\ 0 & \hspace{10pt} \overline{\lambda} \end{bmatrix}$. To do so, we embed $\bR^2 \subset \bC^2$ and consider $\mathcal{T}: (x,y) \rightarrow (x_1, y_1)$ to be a complex symplectic function on a small neighborhood $(0,0)\in \bC^2$. Define new complex coordinates $z$ and $w$ as \begin{equation} z = x + iy \qquad w = x - iy \label{z,w def} \end{equation} \noindent and let $\mathcal{T}(z,w) = (z_1, w_1) = (x_1+iy_1, x_1-iy_1)$. Note that under this coordinate transformation, the subspace $\bR^2$ of $\bC^2$ is mapped to $\{(z,w) \in \bC^2 : w = \overline{z} \}$ which may also be regarded as a real subspace. Observe further that $\det D_P\mathcal{T} \neq 0$. Thus by Inverse Function Theorem, there exists a neighborhood about $P$ on which $\mathcal{T}:(z,w) \mapsto (z_1,w_1)$ is a diffeomorphism so that $w_1 = \overline{z}_1$ if and only if $w = \overline{z}$. To compute the coefficients of the billiard map $\mathcal{T}:(z,w) \mapsto (z_1,w_1)$, we plug the Taylor expansions of $x_1$ and $y_1$ given in equation \eqref{eq: x1y1 R mixed norm} into the definitions for $z_1$ and $w_1$ to get \begin{align} z_1 &= x_1 + iy_1 \nonumber \\ &= x\cos\theta - y\sin\theta + \sum_{j+k =3}a_{jk}x^jy^k + i\left( x\sin\theta + y\cos\theta + \sum_{j+k = 3} b_{jk}x^jy^k \right) + \text{h.o.t.} \nonumber \\[10pt] &= \lambda \left( z + \sum_{j+k=3} c_{jk}z^jw^k \right) + \text{h.o.t.}, \label{eq: z1} \\ \intertext{and similarly,} w_1 &= \overline{\lambda}\left( w + \sum_{j+k = 3} \overline{c}_{jk}w^jz^k \right) + \text{h.o.t.}. \label{eq: w1} \end{align} \noindent The $c_{jk}$ in equations (\ref{eq: z1}) and (\ref{eq: w1}) are given by \begin{align} c_{30} &= 2^{-3}\overline{\lambda}\left( a_{30} + ib_{30} - ia_{21} + b_{21} - a_{12} - ib_{12} + ia_{03} - b_{03} \right), \\ c_{21} &= 2^{-3}\overline{\lambda}\left( 3a_{30} + 3ib_{30} - ia_{21} + b_{21} + a_{12} + ib_{12} - 3ia_{03} + 3b_{03} \right), \\ c_{12} &= 2^{-3}\overline{\lambda}\left( 3a_{30} + 3ib_{30} + ia_{21} - b_{21} + a_{12} + ib_{12} + 3ia_{03} - 3b_{03} \right), \\ c_{03} &= 2^{-3}\overline{\lambda}\left( a_{30} + ib_{30} + ia_{21} - b_{21} - a_{12} - ib_{12} -ia_{03} + b_{03}\right). \end{align} \begin{figure}[htbp] \begin{align} \xymatrixcolsep{3pc} \xymatrixrowsep{3pc} \xymatrix{(s,u) \ar[d]_{\mathcal{T}} \ar[r]^{B^{-1}} & (x,y) \ar[d]_{\mathcal{T}} \ar[r] & (z,w) \ar[d]_{\mathcal{T}} \\ (s_1,u_1) \ar[r]_{B^{-1}} & (x_1,y_1) \ar[r] & (z_1, w_1) } \end{align} \caption{The diagram after the second coordinate transformation} \label{fig: second tower of transformations} \end{figure} Since $\mathcal{T}:(x,y) \mapsto (x_1,y_1)$ is symplectic, then $D_P\mathcal{T} = 1$ in the $(x,y)$ coordinates. Further, the transformation $(x,y) \mapsto (z,w)$ has constant Jacobian. It follows that $ \frac{\partial(z_1, w_1)}{(z,w)} = 1 + (3c_{30} + \overline{c}_{12})z^2 + (2c_{21}+2\overline{c}_{21})zw + (c_{12}+3\overline{c}_{30})w^2 + \text{h.o.t.}$ is identically equal to one so that all the non-constant terms vanish. We thus get the following relations: \begin{equation} c_{12} = -3\overline{c}_{30} \qquad c_{21} = \overline{c}_{21} \end{equation} \noindent In particular, $c_{21}$ is purely imaginary, and we can thus write $c_{21} = i \tau_1$ where $\tau_1 \in \bR$ is given by: \begin{equation} \tau_1 = \frac{a\left(a + 4b\right)L-2\left(a^2 -3ab-6b^2\right)R + a^2\left(a+b\right)^2LRR''}{8a^2R\left[\left(a+b\right)L-2aR\right]} \label{eq: twist mixed norm symmetric}. \end{equation} Shortly we will show that $\tau_1$ is in fact, the first twist coefficient of $\mathcal{T}$ at $P$. \subsection{The third transformation: killing most 3rd-order terms}\label{ss.killing} Now we need a transformation $(z,w) \mapsto (z',w')$ such that most third order terms cancel. Let this transformation be given by \begin{align} z' &:= z + p_3(z,w) = z + \sum_{j+k = 3}d_{jk}z^jw'^k, \label{eq: z' mixed norm}\\ w' &:= w + \overline{p}_3(w,z) = w + \sum_{j+k = 3}\overline{d}_{jk}w^jz^k. \end{align} Then $z'_1$ satisfies \begin{align} z'_1 &= z_1 + \sum_{j+k = 3}d_{jk}z^j_1w^k_1= \lambda \left(z + \sum_{j+k = 3}c_{jk}z^jw^k \right) + \sum_{j+k = 3}d_{jk}\lambda^{j-k}z^jw^k + \ \text{h.o.t} \\[5pt] &= \lambda \left[ z' + \sum_{j+k = 3} \left( -d_{jk} + c_{jk} + d_{jk}\lambda^{j-k-1} \right)(z')^j(w')^k \right] + \ \text{h.o.t.} \label{eq: z1' mixed norm}. \end{align} \noindent Observe how in equation (\ref{eq: z1' mixed norm}) the $d_{21}$ terms always cancel. Therefore without loss of generality, we can set $d_{21} = 0$. By setting $-d_{jk} + c_{jk} + d_{jk}\lambda^{j-k-1} = 0$ for $(j,k) \in \{ (3,0), (1,2), (2,1) \}$ we obtain \begin{equation} d_{30} = \frac{c_{30}}{1-\lambda^2} , \qquad d_{12} = \frac{c_{12}}{1-\overline{\lambda}^2} = -3\overline{d}_{30}, \qquad d_{03} = \frac{c_{03}}{1-\overline{\lambda}^4}, \end{equation} \noindent so that equation (\ref{eq: z1' mixed norm}) can be rewritten as $z'_1 = \lambda\big(z' + c_{21}(z')^2(w')\big) + \text{h.o.t.}$. The ellipticity of $D_P\mathcal{T}$ implies $\lambda \neq \pm 1$, ensuring that the expressions for $d_{30}$ and $d_{12}$ do not involve division by zero. Similarly, assumption (A1) does the same for the expression of $d_{03}$. Having $c_{21} = i\tau$ and $\lambda = e^{i\theta}$, we can further rewrite $z'_1$ if we restrict to the subspace $\{w = \overline{z}\}$: \begin{equation} z'_1 = e^{i\theta}\left(z' + i\tau \|z'\|^2z'\right) + \text{h.o.t.} = e^{i\left(\theta + \tau_1 \|z'\|^2 \right)}z' + h.o.t.. \end{equation} This shows how the linear portion of billiard map $\mathcal{T}:(z',w') \mapsto (z'_1,w'_1)$ acts as a rotation about the elliptic fixed point. However, in the $(z',w')$ coordinates, $\mathcal{T}$ is not necessarily symplectic. \begin{figure}[htbp] \begin{align} \xymatrixcolsep{3pc} \xymatrixrowsep{3pc} \xymatrix{(s,u) \ar[d]_{\mathcal{T}} \ar[r]^{B^{-1}} & (x,y) \ar[d]_{\mathcal{T}} \ar[r] & (z,w) \ar[d]_{\mathcal{T}} \ar[r]^{} & (z',w') \ar[d]_{\mathcal{T}} \ar[r] & \cdots \\ (s_1,u_1) \ar[r]_{B^{-1}} & (x_1,y_1) \ar[r] & (z_1, w_1) \ar[r] & (z'_1,w'_1) \ar[r] & \cdots} \end{align} \caption{The diagram after the third coordinate transformation} \label{fig: tower of transformations} \end{figure} \subsection{The fourth transformation: a symplectic transformation}\label{ss.symplectic} As already mentioned, the billiard map $\mathcal{T}:(z',w') \mapsto (z'_1,w'_1)$ given in the previous section is not necessarily symplectic. This is an obstacle to using Moser's twist mapping theorem to prove the nonlinear stability of periodic billiard orbits. In this section, we will use the previous transformation $(z,w) \mapsto (z',w')$ to compute a real symplectic transformation $h_N:(x,y) \mapsto (X,Y)$ such that \begin{equation} h_N^{-1} \circ T \circ h_N \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} \cos\Theta(r) & -\sin\Theta(r) \\ \sin\Theta(r) & \cos\Theta(r) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \text{h.o.t.}, \end{equation} where $r^2 = x^2 + y^2$ and $\Theta(r) = \theta + \tau_1r^2 + \tau_2r^4 + \cdots + \tau_{N-1}r^{2(N-1)}$. First, notice that $p_3(z,w)$ satisfies $\frac{\partial p_3}{\partial z} = - \frac{\partial \overline{p}_3}{\partial w}$. Hence there exists a polynomial $s_4(z,w)$ satisfying $\frac{\partial s_4}{\partial w}(z,w) = p_3(z,w)$ and $\frac{\partial s_4}{\partial z}(z,w) = -\overline{p}_3(w,z)$, given by \begin{equation} s_4(s,w) = \frac{1}{4}\frac{\overline{c}_{03}}{\lambda^4-1}z^4 - \frac{1}{4}\frac{c_{03}}{\overline{\lambda}^4-1}w^4 + \frac{\overline{c}_{30}}{\overline{\lambda}^2-1}zw^3 - \frac{c_{30}}{\lambda^2-1}z^3w, \end{equation} that is purely imaginary whenever $w = \overline{z}$. Hence we can thus define a real valued function $g_4(x+iy,x-iy) = \frac{i}{2}s_4(x+iy,x-iy)$. Next, consider the generating function $G(x,Y) := xY + g_4(x+iY, x-iY)$ of the symplectic transformation $(x,y) \mapsto (X,Y)$ where $X = X(x,y)$ and $Y = Y(x,y)$. Then \begin{alignat}{3} X &:= \frac{\partial G}{\partial Y} & &= x + \frac{i}{2} \big[ -\overline{p}_3(x-iY,x+iY)(i) + p_3(x+iY,x-iY)(-i) \big] \nonumber \\ & & &= x + \Re\left(p_3(x+iY, x-iY)\right) \nonumber \\ & & &= x + \Re\left( \sum\limits_{j+k=3}d_{jk}(x+iY)^j(x-iY)^k \right) \nonumber \\ & & &=: x + \sum_{j+k =3}p_{jk}x^jY^k ,\label{eq: X}\\ y &:= \frac{\partial G}{\partial x} & &= Y + \frac{i}{2} \bigg[ -\overline{p}_3(x-iY,x+iY) + p_3(x+iY,x-iY) \bigg] \nonumber \\ & & &= Y - \Im\left(p_3(x+iY,x-iY)\right) \nonumber \\ & & &= Y - \Im\left( \sum\limits_{j+k=3}d_{jk}(x+iY)^j(x-iY)^k \right) \nonumber\\ & & &=: Y - \sum_{j+k = 3}q_{jk}x^jY^k. \label{eq: y} \end{alignat} It follows from equations (\ref{eq: X}) and (\ref{eq: y}), that on a small neighborhood about $(x,y) = (0,0)$, we have \begin{align} X &= x + \sum_{j+k = 3}p_{jk}x^jy^k + \text{h.o.t.},\\ Y &= y + \sum_{j+k = 3}q_{jk}x^jy^k + \text{h.o.t.}. \end{align} Letting, $Z = X + iY$ and $W = X - iY$ gives \begin{align} Z &= x + iy + \Re\left(p_3(x+iy, x-iy)\right) + i\Im\left(p_3(x+iy, x-iy)\right) + \text{h.o.t.} \nonumber \\ &= z + p_3(z,w) + \text{h.o.t.} \intertext{Similarly,} W &= w + \overline{p}_3(w,z) + \text{h.o.t.}, \end{align} so that the billiard map $\mathcal{T}: (Z,W) \mapsto (Z_1,W_1)$ is given by \begin{align} Z_1 &= e^{i\left(\theta + \tau_1\|Z\| \right)}Z + \text{h.o.t.}, \label{eq: Z1}\\ W_1 &= e^{-i\left(\theta + \tau_1\|Z\|\right)}W + \text{h.o.t.}. \label{eq: W1} \end{align} Letting $\Theta = \theta + \tau_1\|Z\|^2$ and converting back to the variables $X$ and $Y$ yields the desired Birkhoff normal form, showing $\tau_1$ is the first twist coefficient of $\mathcal{T}$: \begin{equation} \begin{bmatrix} X_1 \\ Y_1 \end{bmatrix} = \begin{bmatrix} \cos\Theta & -\sin\Theta \\ \sin\Theta & \cos\Theta \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} + \text{h.o.t.} \end{equation} \begin{figure}[htbp] \begin{align} \xymatrixcolsep{3pc} \xymatrixrowsep{3pc} \xymatrix{(s,u) \ar[d]_{\mathcal{T}} \ar[r]^{B^{-1}} & (x,y) \ar[d]_{\mathcal{T}} \ar[r] & (X,Y) \ar[d]_{\mathcal{T}} \ar[r]^{} & (Z,W) \ar[d]_{\mathcal{T}}\\ (s_1,u_1) \ar[r]_{B^{-1}} & (x_1,y_1) \ar[r] & (X_1, Y_1) \ar[r] & (Z_1,W_1)} \end{align*} \caption{The diagram after the fourth coordinate transformation.} \label{fig: fourth transformation} \end{figure} \subsection{The first twist coefficient for asymmetric Minkowski billiards}\label{ss.asymmetry} In this subsection we study the general case where the (Euclidean) radius of curvature functions $R_j(s)$, have different values around $s=0$. In particular, $R_0(0) = R_0$ and $R_1(0) = R_1$ may be different. For the class of billiard tables $Q(L,\alpha, \beta)$, this will occur whenever the even polynomials $\alpha(t) = \sum_{n\geq1}\alpha_{2n}t^{2n}$ and $\beta(t) = \sum_{n\geq1}\beta_{2n}t^{2n}$ have different values for $\alpha_2$ and $\beta_2$. Going forward we will assume that $\alpha_2 > \beta_2$ so that $R_0 \leq R_1$. As before, we will consider the periodic 2-orbit $\mathcal{O}_2 = \{P, T(P)\}$. The billiard map ${\mathcal{T}}^2$ is given by the composition $(s,u) \xrightarrow{{\mathcal{T}}} (s_1,u_1) \xrightarrow{{\mathcal{T}}} (s_2, u_2)$ where as before $s_1 = s_1(s,u)$, $u_1 = u_1(s,u)$, $ s_2=s_2(s_1,u_1)$ and $u_2 =u_2(s_1,u_1)$. The partial derivatives $\frac{\partial^{j+k}s_2}{\partial s^j \partial u^k}$ and $\frac{\partial^{j+k}u_2}{\partial s^j \partial u^k}$ are straightforward albeit tedious applications of the chain rule. To illustrate it, we give expressions for $\frac{\partial s_2}{\partial s}$ and $\frac{\partial^2 s_2}{\partial s^2}$: \begin{align} \frac{\partial s_2}{\partial s} &= \frac{\partial s_2}{\partial s_1}\dsds + \frac{\partial s_2}{\partial u_1}\duds, \\[10pt] \frac{\partial^2 s_2}{\partial s^2} &= \frac{\partial^2 s_2}{\partial s_1^2} \left(\dsds \right)^2 + 2 \frac{\partial^2 s_2}{\partial s_1 \partial u_1}\duds\dsds + \frac{\partial s_2}{\partial s_1}\frac{\partial^2 s_1}{\partial s^2} + \frac{\partial^2 s_2}{\partial u_1^2} \left(\duds \right)^2 + \frac{\partial s_2}{\partial u_1}\frac{\partial^2 u_1}{\partial s^2}. \end{align} Denote $\tilde{a}_{jk} =\frac{1}{j!k!}\frac{\partial^{j+k}s_2}{\partial s^j \pa u^k}(0,0)$ and $\tilde{b}_{jk} =\frac{1}{j!k!}\frac{\partial ^{j+k}u_2}{\partial s^j \pa u^k}(0,0)$. Calculating the remaining first order partial derivatives and evaluating at $\mathcal{O}_2$, show that $D_P{\mathcal{T}}^2$ is given by \begin{equation} D_P{\mathcal{T}}^2 = \begin{bmatrix} \ta_{10} & \ta_{01} \\ \tb_{10} & \tb_{01} \end{bmatrix} = \begin{bmatrix} \frac{L}{aR_1}-1 & \frac{L}{a} \\ \frac{L-a(R_0 + R_1)}{aR_0R_1} & \frac{L}{aR_0}-1 \end{bmatrix} \begin{bmatrix} \frac{L}{aR_0}-1 & \frac{L}{a} \\ \frac{L-a(R_0+R_1)}{aR_0R_1} & \frac{L}{aR_1}-1 \end{bmatrix}, \end{equation} where \begin{align} \ta_{10} &= \tb_{01} = \frac{a R_0 \left(a R_1-2 L\right)+2 L \left(L-a R_1\right)}{a^2 R_0 R_1}, \\[10pt] \ta_{01} &= \frac{2L}{a}\left( \frac{L}{aR_1}-1 \right), \\[10pt] \tb_{10 }&=\frac{2 \left(L-a R_0\right) \left(L-a \left(R_0+R_1\right)\right)}{a^2 R_0^2 R_1}. \end{align} Then the orbit $\mathcal{O}_2$ is parabolic for $L \in \left\{ aR_0, aR_1, a(R_0 + R_1) \right\}$, hyperbolic for $L \in \left(aR_0, aR_1\right) \cup \left(a(R_0 + R_1), \infty \right)$, and elliptic for $L \in \left(0, a(R_0 +R_1)\right) \cup \left(aR_1, a(R_0 +R_1)\right)$. We assume the following non-resonance condition: \begin{itemize} \item[(B)] $\lambda^4 \neq 1$, or equivalently, $\left(L-aR_0\right)\left(L-aR_1 \right) \neq 0$. \end{itemize} \begin{theorem}\label{thm.t1.asym} Let $F(v) = a(v_1^2 + v_2^2)^{1/2} + b(v_1^4 + b_2^4)^{1/4}$ with $a+b=1$, and $Q(\alpha(t), \beta(t),L)$ be given. Additionally, assume that the resonance condition (B) is satisfied. Then the first twist coefficient of the two-step billiard map $T^2$ is given by \begin{equation} \tau_1 = \frac{\Delta}{8aR_0R_1(L-aR_0)(L-aR_1)\left(L-a(R_0+R_1)\right)}, \end{equation} \noindent where \begin{align} \Delta &= 2 a R_0^2 \left(-L R_1 \left(a^2 L R_1''+4 a^2-18 a+12\right)+a \left(2 a^2-9 a+6\right) R_1^2+a (3 a-4) L^2\right) \nonumber \\ &+ 2 a R_0^2 \left(-L R_1 \left(a^2 L R_1''+4 a^2-18 a+12\right)+a \left(2 a^2-9 a+6\right) R_1^2+a (3 a-4) L^2\right) \nonumber \\ &+ R_0 \bigg(L^2 R_1 \left(a^2 L R_0''+a^2 L R_1''+8 a^2-36 a+24\right)-2 a L R_1^2 \left(a^2 L R_0''+4 a^2-18 a+12\right) \\ &+a^2 R_1^3 \left(a^2 L R_0''+2 a^2-9 a+6\right)+a (4-3 a) L^3 \bigg) +a^2 R_0^3 \big(R_1 \left(a^2 L R_1''+2 a^2-9 a+6\right) \nonumber \\ &+a (4-3 a) L\big)-a (3 a-4) L R_1 \left(L-a R_1\right){}^2. \nonumber \end{align} \end{theorem} \begin{proof} We only give a sketch of the computation. The detailed computation be found in \cite{Vil}. The Taylor coefficients $\ta_{jk}$ and $\tb_{jk}$ can be done by taking derivatives of the two functions $s_2$ and $u_2$ repeatedly. Combining with the assumption on two functions $\alpha$ and $\beta$ being even, we have that $\ta_{jk}$'s and $\tb_{jk}$'s for $j+k = 2$ are identically zero. The remaining of the computation for $\tau_1$ is exactly the same as in Section~\ref{ss.rigid}, \ref{ss.killing} and \ref{ss.symplectic}. \end{proof} \begin{remark} The first twist coefficient of elliptic periodic orbits of period $2$ in Euclidean billiards \cite{JZ22, KP05} is recovered by setting $a=1$ and $b=0$: \begin{equation} \tau_1 = \frac{1}{8}\bigg(\frac{R_0+R_1}{R_0R_1}-\frac{L}{R_0+R_1-L}\left(\frac{R_1-L}{R_0-L}R_0'' + \frac{R_0-L}{R_1-L}R_1''\right)\bigg). \end{equation} \end{remark} \section{Applications}\label{sec.applications} In this section we use the first twist coefficient given in Theorem \ref{eq: symmetric twist} to prove the nonlinear stability of the periodic orbits of period $2$ on various symmetric billiard tables. Using the assumption $a+b =1$, we will shorten the notation for Minkowski norm as $F_{a}= F_{a, 1-a}$ with $ 0< a \le 1$, and will denote the Minkowski billiard map as $\cT_a$ to emphasize its dependence on the parameter $a$. We have showed in Section~\ref{ss.tangent.map} that the periodic point $P$ of period $2$ is elliptic for $0<L<2aR$, parabolic for $L=2aR$, and hyperbolic for $L>2aR$. \subsection{Euclidean circular tables} Consider the Euclidean circular billiard table $Q_r$ on the Minkowski plane $(\bR^2, F_{a})$ bounded by the circle $x^2 + y^2 = r^2$. Then the orbit bouncing back and forth along the $y$-axis is a periodic orbit of period $2$, say $\mathcal{O}_2(a, r) =\{P_r, \cT_a(P_r)\}$. In this case, we have $R = r$ and $L=2r$. From the stability conditions above, it follows that the periodic orbit $\mathcal{O}_2(a, r)$ is hyperbolic for $0< a < 1$ (see Figures \ref{fig:circleorbits}(a) and \ref{fig:circlephase}) and parabolic only for $a=1$, which corresponds to the Euclidean norm $F_{1}(v) = \\|v\\|_2$. In particular, such an orbit cannot be elliptic. Additionally, numerical simulations demonstrate the nonlinear stability of the periodic 2-orbits $\{(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}),(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\}$ and $\{(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}),(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\}$ (see Figures \ref{fig:circleorbits}(b) and \ref{fig:circlephase}). Although the twist coefficient in equation (\ref{eq: symmetric twist}) does not pertain to these periodic 2-orbits, we can obtain their twist coefficient in a similar way as presented above. However, equations (\ref{eq: dsds})-(\ref{eq: dudu}) would not simplify as nicely as they do in equations (\ref{eq: dsds mixed norm})-(\ref{eq: dudu mixed norm}). \begin{figure}[htbp] \centering \subcaptionbox{100 reflections of the billiard orbit with initial point $(0,-1)$ and direction $\pi/2+0.01$}{\includegraphics[width=0.48\textwidth]{Images/B1a08_hyperbolic}} \hfill \subcaptionbox{100 reflections of the billiard orbit wit initial point $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ and initial direction $\frac{5\pi}{4}+0.1$}{\includegraphics[width=0.48\textwidth]{Images/B1a08_stable2up}} \caption{Dynamics billiards on the table bounded by $x^2 + y^2 = 1$ with Minkowski norm $F_{0.8}$.} \label{fig:circleorbits} \end{figure} \begin{figure}[htbp] \centering \includegraphics[width=0.7\textwidth]{Images/phase_a08B1} \caption{Phase space of Minkowski billiards with norm $F_{0.8}$ on the table bounded by $x^2 + y^2 =1$} \label{fig:circlephase} \end{figure} \subsection{Symmetric lemon billiards} Next, consider the symmetric billiard table $Q(L)$ discussed in Section~\ref{ss.sym.billiards}, where $\gamma_0$ and $\gamma_1$ are Euclidean circular arcs of radius $1$. In this case, $R=1, R'' = 0$, and $L\in (0,2)$. From the stability conditions, the orbit $\mathcal{O}_2(a,L) =\{P_L, \cT_a(P_L)\}$ is elliptic whenever $L<2a$ with nonresonance condition $L\neq a$. The first twist coefficient is given by \begin{equation} \tau_1(\mathcal{T}_a, P_L) = \frac{a^2 (3 L-4)+a (18-4 L)-12}{8 a^2 (2 a-L)}. \end{equation} Note that $\tau_1(\mathcal{T}_a, P_L)=0$ when $L = \frac{2\left(2a^2-9a+6\right)}{a\left(3a-4\right)}$. \begin{figure}[htbp] \centering \includegraphics[width=0.8\linewidth]{Images/lemon_symmetric_graph} \caption{Stability regions for the orbit $\mathcal{O}_2(a, L)$ on a symmetric lemon table $Q(L)$ on the Minkowski plane with norm $F_{a}$.} \label{fig:lemonsymmetricgraph} \end{figure} \begin{corollary} Let $0<a <1$ and $\mathcal{O}_2(a, L)$ be the periodic orbit on the symmetric lemon table $Q(L)$ running along the $x$-axis. Then the orbit $\mathcal{O}_2(a, L)$ is nonlinearly stable for $L \in (0,2a) \backslash \left\{ a, \frac{4a^2-18a+12}{a(3a-4)} \right\}$. \end{corollary} \begin{remark} It is not clear if the periodic orbit $\mathcal{O}_2(a, L)$ is nonlinearly stable when $L= \frac{4a^2-18a+12}{a(3a-4)}$. In this case, $\tau_1(\mathcal{T}_a, P_L) =0$, and finding the next coefficient $\tau_2(\mathcal{T}_a, P_L)$ may be helpful. \end{remark} \subsection{Euclidean elliptical tables} Let $\delta \in (0,1)$, and $E(\delta)$ be the billiard table $E(\delta)$ bounded by the Euclidean ellipse given $x^2 + \frac{y^2}{\delta^2}= 1$. It is easy to see that there is a periodic orbit $\mathcal{O}_2(a, \delta)=\{P_{\delta}, \cT_a(P_{\delta}\}$ of period $1$ running along the minor axis of the ellipse $E(\delta)$. Then $L = 2\delta$ and $R = 1/\delta$, so that $\frac{L}{R}=2\delta^2$. It follows that $\mathcal{O}_2(a, \delta)$ is elliptic for $\delta < \sqrt{a}$, parabolic for $\delta = \sqrt{a}$, and hyperbolic for $\delta > \sqrt{a}$. Consequently, $\mathcal{O}_2(a, \delta)$ is always elliptic in the Euclidean case, and the non-resonance condition is satisfied for $\delta \neq \frac{\sqrt{a}}{2}$. A short calculation shows $R''=\frac{3\left(\delta^2-1\right)}{\delta}$ so that the first twist coefficient is given by \begin{equation} \tau_1(P_{\delta}, \cT_a) = \frac{\delta\left(-6+a(9+a-4\delta^2)\right)}{8a^2(a-\delta^2)}. \end{equation} \begin{figure}[htbp] \centering \includegraphics[width=0.8\linewidth]{Images/elliptic_billiards_graph} \caption{Stability regions for the orbit $\mathcal{O}_2(a, \delta)$ on an elliptic table $E(\delta)$ on the Minkowski plane with norm $F_{a}$.} \label{fig: elliptic billiard graph} \end{figure} Note that $\tau_1(P_{\delta}, \cT_a)=0$ when $\delta = \frac{\sqrt{a^2+9a-6}}{2\sqrt{a}}$. This value of $\delta$ only makes sense when it is real and less than $\sqrt{a}$ which occurs whenever $a \in \left(\frac{\sqrt{105}-9}{2},1\right)$ \begin{corollary} On the elliptical billiard table $E(\delta)$, the periodic orbit $\mathcal{O}_2(a, \delta)$ running along the minor axis is nonlinearly stable whenever $\delta < \sqrt{a}$ and $\delta \neq \frac{\sqrt{a^2 + 9a -6}}{2\sqrt{a}}$ and $\delta \neq \frac{\sqrt{a}}{2}$. \end{corollary} \begin{figure}[htbp] \centering \includegraphics[width=0.7\textwidth]{Images/phase_a08B05} \caption{Phase space of Minkowski billiards with norm $F_{0.8}$ on the table $E(0.5)$} \end{figure} \subsection{General symmetric billiards} For the class of billiard tables $Q(\alpha, \beta, L)$ where $\alpha(t) = \beta(t)$, $R(s)$ satisfies $R = \frac{1}{2 \alpha_2}$ and $R'' = 6\alpha_2 - \frac{6\alpha_4}{\alpha_2^2}$ at both $\gamma_0(0)$ and $\gamma_1(0)$. 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2412.03601v3	http://arxiv.org/abs/2412.03601v3	Relations between average shortest path length and another centralities in graphs	\documentclass[12pt, a4paper]{extarticle} \usepackage[margin=2cm]{geometry} \usepackage[affil-it]{authblk} \usepackage[english]{babel} \usepackage{graphicx} \usepackage{wrapfig} \usepackage{amsmath,amsthm,amssymb,scrextend} \usepackage{booktabs} \usepackage{multirow} \usepackage{amsfonts} \usepackage{multicol} \usepackage[shortlabels]{enumitem} \usepackage{cutwin} \setlength{\columnsep}{1cm} \usepackage{float} \usepackage{csquotes} \usepackage{enumitem} \usepackage{tikz} \usepackage{hyperref} \def \rR {\mathbb{R}} \def \bf#1 {\textbf{#1 }} \def \sumt {\sum\limits} \providecommand{\keywords}[1] { \small \textbf{\textit{Keywords:}} #1 } \let\newproof\proof \renewenvironment{proof}{\begin{addmargin}[1em]{0em}\begin{newproof}}{\end{newproof}\end{addmargin}\qed} \newtheorem{thm}{Theorem} \newtheorem{lm}{Lemma} \newtheorem{cor}{Corollary} \newtheorem{lm}{Lemma} \newtheorem{defin}{Definition} \newtheorem{defin}{Definition} \def \sumt {\sum\limits} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\BC}{BC} \DeclareMathOperator{\Clo}{Clo} \DeclareMathOperator{\Rad}{Rad} \DeclareMathOperator{\Str}{Str} \begin{document} \title{Relations between average shortest path length and another centralities in graphs} \author{Mikhail Tuzhilin \thanks{Affiliation: Moscow State University, Electronic address: \texttt{[email protected]};} } \date{} \maketitle \begin{abstract} Relations between average shortest path length and radiality, closeness, stress centralities and average clustering coefficient were obtained for simple connected graphs. \end{abstract} \keywords{Networks, centralities, local and global properties of graphs, average shortest path length, Watts-Strogatz clustering coefficient.} \section{Introduction.} Centrality measures are important characteristics in network science which contain intrinsic hidden information about networks. Centrality is a local (with relation to a vertex) or global (with relation to a whole graph) real functions on graphs. There exist many centralities~\cite{Borgatti}-~\cite{Lee}: local efficiency, radiality, closeness, betweenness, stress centralities, etc. One of the most important centralities for ``real'' networks (or small-world networks) are average clustering coefficient and average shortest path length~\cite{Watts}. The network is called small-world if it has small average shortest path length with relation to the size and big average clustering coefficient. This measures are separately calculated to prove that current network is small-world. In this article we proved that for geodetic graphs (or graphs with odd length cycles) with additional condition there is relation between average shortest path length and average clustering coefficient. \section{Main definitions.} All subsequent definitions are given for a simple, connected, undirected graph $G$. Let's denote by \begin{itemize} \item $V(G)$ the set of vertices,\;\; $E(G)$ the set of edges and\;\; $A = \{a_{ij}\}$ adjacency matrix of graph $G$. \item Neighborhood $N(v)$ --- the induced subgraph in $G$ on vertices which adjacent to the vertex $v$, \item $N'(v)$ induced subgraph in $G$ on vertices $ = V\big(N(v)\big)\bigcup \{v\}$, \item $\bar{f}(x_1, x_2, ... , x_k)$, where $f$ is any function $V\times V \times ... \times V\rightarrow\rR$, the restriction of this function on $N'(v)$ (for example $\bar L(x,y)$ will be the average shortest path between $x$ and $y$ with restriction to subgraph $N'(v)$), \item $d_i = deg(v_i)$, \item $n = \|V(G)\|,\;\; m = \|E(G)\|$, \item $X(i) = X(v_i)$ for any $X$ --- set or function corresponding to vertex $v_i$, \end{itemize} Let's give definitions of centralities: \begin{enumerate} \item \bf{Diameter} $diam(G) = \max_{s,t\in V(G)} dist(s,t)$. \item \bf{Average shortest path length} $L(G) = \frac 1 {n(n-1)} \sumt_{s\neq t} dist(s,t)$. \item \bf{Local cluster coefficient} $c_i = c(i) = \frac {\text{number of edges in }N(i)} {\text{maximum possible number of edges in }N(i)}= \frac{2 \|E(N(i))\|} {d_i(d_i-1)}$. \item \bf{Average clustering coefficient} $C_{WS}(G) = \frac 1 {n} \sumt_{i\in V(G)} c_i = \frac 1 {n} \sumt_{i\in V(G)} \frac{2 \|E(N(i)))\|} {d_i(d_i-1)} = \frac 1 n \sumt_{i\in V(G)} \frac {\sumt_{j, k\in V(G)} a_{ij} a_{jk} a_{ki}} {d_i (d_i-1)}$. \item \bf{Global clustering coefficient} $C(G) = \frac {\text{number of closed triplets in $G$}} {\text{number of all triplets in $G$}} = \frac {\sumt_{i, j, k\in V(G)} a_{ij} a_{jk} a_{ki}} {\sumt_{i\in V(G)} d_i (d_i-1)}$. \item \bf{Closeness centrality} $Clo(v) = \frac {n-1} {\sumt_{t\in V(G)} dist(v, t)}$. \item \bf{Radiality} $Rad(v) = \frac {\sumt_{t\in V(G), t\neq v} (diam(G)+1-dist(v, t))} {n-1}$. \item \bf{Stress} $Str(i) = \sumt_{s,t\in V(G),\; s\neq t\neq i} \sigma_{st}(i)$, where $\sigma_{st}(i)$ is the total number of shortest paths from $s$ to $t$ which contains vertex $i$. \end{enumerate} Note that all centralities are non-negative and $c_i, C_{WS}, \Clo(v)$ are less or equal 1. Also let's give the definition of geodetic graph: \begin{defin} If there exists the unique shortest path between any two vertices in $G$ then the graph $G$ is called \bf{geodetic}. \end{defin} This definition is equivalent to the condition then there is no even cycles in a graph. \section{Main results.} Let's consider a induced subgraph $G'\subset G$. In general $G'$ can be not connected graph. In this case let's define the average shortest path length for vertices of $G'$ with relation to the distance $\dist$ in the ambient graph $G$. Let's call $L\big(N(i)\big)$ the local average shortest path length for the vertex $i$. Let's start with simple relations: let's proof the relation between local shortest path length and local clustering coefficient: \begin{lm}\label{lm1}\label{lm1} $$L(N(i)) = 2-c_i.$$ \end{lm} \begin{proof} $$ L(N(i)) = \frac 1 {d_i(d_i-1)} \sumt_{s,t\in N(i), s\neq t} dist(s,t) = \frac 1 {d_i(d_i-1)} \sumt_{(s,t)\in E(N(i))} dist(s,t)+ \sumt_{s,t\in N(i), (s,t)\notin E(N(i))} dist(s,t) = $$ $$ = \frac 1 {d_i(d_i-1)} (2 \|E(N(i))\| + \sumt_{(s,i), (i, t)\in E(G), (s,t) \notin E(G)} dist(s,t)) = $$ $$ = \frac 1 {d_i(d_i-1)} (2 \|E(N(i))\|+2 (d_i(d_i-1) - 2 \| E(N(i))\|)) = 2 - c_i. $$ Note that shortest paths for vertices in $N(i)$ are defined corresponding to whole graph $G$. \end{proof} Averaging by $i$ we obtain simple corollary about the relation between local shortest path length and average clustering coefficient. \begin{cor} $$C_{WS}(G) = 2- \frac 1 n \sumt_{i\in V(G)}L\big(N(i)\big).$$ \end{cor} Let's proof the relation between shortest path length in subgraph $N'(i)$ and shortest path length in $N(i)$. \begin{lm} $$L\big(N'(i)\big) = \frac {(d_i-1)L\big(N(i)\big)+2} {d_i+1}.$$ \end{lm} \begin{proof} By definition $$ L\big(N'(i)\big) = \frac 1 {(d_i+1)d_i} \sumt_{s,t\in V(N'(i)),\;s\neq t} \dist(s,t) = \frac {d_i-1} {d_i+1} L\big(N(i)\big) + \frac 2 {d_i+1}. $$ \end{proof} Let's proof the relation between shortest path length in a induced subgraph and shortest path length in ambient graph if induced subgraph is obtained from ambient graph by deleting one vertex. \begin{thm} Let a graph $G$ is obtained from a connected simple graph $G'$ by deleting one vertex and $\|V(G)\| = n$. Then $$L(G')\geq \frac n {n+1} L(G),$$ where the average shortest path length $L(G)$ is defined in the ambient graph $G'$, if $G$ is not connected. \end{thm} \begin{proof} Let's define the deleted vertex by $v$. By the triangle inequality $\forall s, t\in V(G): \dist(s,v)+\dist(v,t)\geq \dist(s,t)$, where the equality holds if, there are no paths from $s$ to $t$ in $G$. Therefore, $$\sumt_{s,t\in V(G), s\neq t}\big(\dist(s,v)+\dist(v,t)\big)\geq \sumt_{s,t\in V(G), s\neq t}\dist(s,t)$$ $$\frac {2(n-1)} {n(n-1)}\sumt_{t\in V(G)}\dist(v,t)\geq\frac 1 {n(n-1)} \sumt_{s,t\in V(G), s\neq t}\dist(s,t)$$ $$\frac {2} {n}\sumt_{t\in V(G)}\dist(v,t)\geq L(G)$$ Then, $$L(G') = \frac 1 {(n+1)n} \sumt_{s,t\in V(G'),s\neq t}\dist(s,t) = \frac 1 {(n+1)n} \Big(2 \sumt_{t\in V(G)}\dist(v,t)+\sumt_{s,t\in V(G),s\neq t}\dist(s,t)\Big) =$$ $$=\frac 1 {n+1} \frac 2 {n} \sumt_{t\in V(G)}\dist(v,t) + \frac {n-1} {n+1} L(G)\geq \frac n {n+1} L(G)$$ Note that if $G$ consists of $n$ isolated vertices then the equality holds. \end{proof} We obtain corollaries \begin{cor} Let a graph $G$ is obtained from a connected simple graph $G' \subset H$ by deleting one vertex, $\|V(G)\| = n$ and $H$ is connected and simple. Then $$L(G')\geq \frac n {n+1} L(G),$$ where the average shortest path lengths $L(G)$ and $L(G')$ are defined in the ambient graph $H$, if corresponded graphs are not connected. \end{cor} \begin{proof} The proof is the same as for the previous theorem and also if $G$ consists of $n$ isolated vertices then the equality holds. \end{proof} \begin{cor}\label{cor1} $$L\big(N'(i)\big) \geq \frac {d_i} {d_i+1}L\big(N(i)\big).$$ \end{cor} Let's proof the relation between shortest path length in a induced subgraph and shortest path length in ambient graph. \begin{thm} Let's $G'\subset G$ be induced subgraph and $\|V(G)\| = n, \|V(G')\| = n+k$. Then $$L(G')\geq \frac n {n+k} L(G),$$ where the average shortest path length $L(G)$ is defined in the ambient graph $G'$, if $G$ is not connected. \end{thm} \begin{proof} Let's construct the graph $G'$ from $G$ by adding sequentially $k$ vertices and corresponded edges. Let's first sequentially add vertices adjacent to vertices of the graph $G$. Adding these vertices one by one we obtain a sequence of graphs $G_1, G_2, ... , G_p, $ where $\|V(G_i)\| = n+i$. Further, let's add is the same way vertices adjacent to vertices of the graph $G_p$ and so on. In the end we obtain the graph $G'$. By previous corollary $$\hspace{-15pt}L(G')\geq \frac {n+k-1} {n+k} L(G_{k-1})\geq \frac {n+k-1} {n+k} \frac {n-k-2} {n-k-1} L(G_{k-2}) = \frac {n-k-2} {n+k} L(G_{k-2})\geq \cdots \geq \frac n {n+k} L(G).$$ \end{proof} Let's proof the relation between average closeness centrality and average shortest path length: \begin{lm} $$ L(G)\geq \frac n {\sumt_{v\in V(G)} \Clo(v)}. $$ \end{lm} \begin{proof} By the inequality of harmonic mean and arithmetic mean $$ \frac 1 {n} \sumt_{v\in V(G)} Clo(v) = \frac 1 n \sumt_{v\in V(G)} \frac {n-1} {\sumt_{t\in V(G)} dist(v, t)}\geq \frac {n(n-1)} {\sumt_{v, t\in V(G)} dist(v, t)} = \frac 1 {L(G)}. $$ Note that an equality holds when all average shortest path lengths from any vertex to all remaining vertices are equal. \end{proof} Let's proof the relation between average shortest path length and average radiality: \begin{lm}\label{lm3} $$ L(G) = \diam(G)+1 -\frac 1 {n} \sumt_{v\in V(G)} {\Rad}(v). $$ \end{lm} \begin{proof} The proof holds from definition $$ \frac 1 {n} \sumt_{v\in V(G)} {Rad}(v) = \frac 1 {n} \sumt_{v\in V(G)} \frac {(n-1) (diam(G)+1)- \sumt_{t\in V(G),\; t\neq v} dist(v,t))} {n-1} = diam(G)+1 -L(G). $$ \end{proof} Now let's prove theorem about a relation between average clustering coefficient and radiality using previous theorem. \begin{thm} $$ \frac {C_{WS}(G)} 2 \geq \frac 1 n \sumt_{i\in V(G)} \frac {\sumt_{v\in N(i)} \overline{\Rad}(v)} {d_i} -2 + \frac {\#\{N(i) \text{ which are complete graphs}\}} {n}. $$ \end{thm} \begin{proof} By lemmas~\ref{cor1}: $L(N'(i))\geq \frac {d_i} {d_i+1}L(N(i))\geq\frac 1 2 L(N(i)) = 1-\frac 1 2 c_i$. Therefore, $$ \frac 1 {d_i} \sumt_{v\in N(i)} \overline{\Rad}(v) = \diam(N'(i))+1-L(N'(i)) \leq \diam (N'(i))+\frac 1 2 c_i = \frac 1 2 c_i+2-\chi_{K_{d_i}}(N(i)), $$ where $ \chi_{K_{d_i}}(N(i)) = \begin{cases} 1 & \text{if $N(i) = K_{d_i}$} \\ 0 & \text{otherwise} \end{cases} $. Averaging by $i$ ends the proof. \end{proof} Let's prove a theorem about a connection between the average stress centrality and average shortest path length for geodetic graphs. \begin{thm}\label{thm1} If $G$ is geodetic, then $$ L(G) = 1+\frac {1} {n(n-1)} \sumt_{i\in V(G)}\Str(i). $$ \end{thm} \begin{proof} Let's define $$\chi_{st}(i) = \begin{cases} 1 & \text{if $i\neq s\neq t$ is the vertex of the shortest path between $s$ and $t$,}\\ 0 & \text{otherwise.} \end{cases}$$ Thus, $\Str(i) = \sumt_{s,t\in V(G)} \chi_{st}(i)$. If there exists the unique shortest path between any two vertices in $G$, then $\dist(s,t) = \sumt_{i\in V(G)} \chi_{st}(i)+1$ (otherwise, $\dist(s,t) \leq \sumt_{i\in V(G)} \chi_{st}(i)+1$). Therefore, for any $i$ $$ \sumt_{s,t\in V(G)} \dist(s,t) = 2\|E\|+\sumt_{s,t\in V(G),\,\dist(s,t)\geq 2} \dist(s,t) = 2\|E\|+\sumt_{s,t\in V(G),\,\dist(s,t)\geq 2} \Big(\sumt_{i\in V(G)} \chi_{st}(i)+1\Big) = $$ $$ = 2\|E\|+\sumt_{i\in V(G)}\Str(i)+n(n-1)-2\|E\|=\sumt_{i\in V(G)}\Str(i)+n(n-1). $$ \end{proof} \begin{cor} For any simple connected $G$ $$ L(G) \leq 1+\frac {1} {n(n-1)} \sumt_{i\in V(G)}\Str(i). $$ \end{cor} First let's proof theorem about a connection between average clustering coefficient and stress centrality and we will use it for a connection between the average shortest path length and the average local clustering coefficient further. \begin{thm}\label{thm2} If there is no pendant vertices in graph, then $$ C_{WS}(G)\geq 1- \frac 1 n \sumt_{i\in V(G)} \frac {Str(i)} {d_i(d_i-1)}. $$ \end{thm} \begin{proof} Note that $\forall j,k\in N(i): (j,k)\notin E(N(i))$ the shortest path between $j$ and $k$ is $j\rightarrow i\rightarrow k$. Therefore, $$ Str(i)\geq 2( \frac {d_i(d_i-1)} 2-\|E(N(i))\|), $$ $$ \frac 1 {d_i(d_i-1)} Str(i)\geq 1-c_i, $$ Averaging by $i$ $$ C_{WS}(G) \geq \frac 1 n \sumt_{i\in V(G)} (1- \frac {Str(i)} {d_i(d_i-1)}). $$ Note that for $diam(G) = 2$ holds an equality. \end{proof} \begin{cor} In the general case $$ C_{WS}(G)\geq 1- \frac 1 n \sumt_{i\in V(G)} \frac {Str(i)} {d_i(d_i-1)}-\frac {\text{number of pendant vertices}} n. $$ \end{cor} \begin{proof} Let's for pendant vertex define $\frac {Str(i)} {d_i(d_i-1)}$ as 0, then in the inequality $\frac 1 {d_i(d_i-1)} Str(i)\geq 1-c_i$ in the right side will be 1 and in the left side 0, thus if we add in the left side 1 for every pendant vertex, will be right equality. \end{proof} Now let's prove a theorem about a connection between the average shortest path length and the average local clustering coefficient for geodetic graphs. \begin{thm}\label{thm6} If $G$ is geodetic and $\forall i,j\in V(G)$ hold, if $d_i\leq d_j $ then $ \Str(i)\leq \Str(j)$, then $$ 1-C_{WS}(G) \leq\frac 1 n \sumt_{i\in V(G)} \frac {\big(L(G)-1\big)(n-1)} {d_i(d_i-1)}+\frac {\text{number of pendant vertices}} n. $$ \end{thm} \begin{proof} Let's re-numerate vertices such that $\forall i\leq j:d_i\leq d_j$. Then for $i\leq j$ hold $\Str(i)\leq \Str(j)$ and $d_i(d_i-1)\leq d_j(d_j-1)$. By theorems~\ref{thm1} and~\ref{thm2} for the case if there is no pendant vertices in graph $$\hspace{-150pt} 1-C_{WS}(G)\leq \frac 1 n \sumt_{i\in V(G)} \frac {Str(i)} {d_i(d_i-1)}\stackrel{\text{Chebyshev's sum inequality}}{\leq} $$ $$\hspace{70pt} \leq\Big(\frac 1 n \sumt_{i\in V(G)} {Str(i)}\Big) \Big(\frac 1 n \sumt_{i\in V(G)} \frac {1} {d_i(d_i-1)} \Big)= \frac 1 n\sumt_{i\in V(G)} \frac {\big(L(G)-1\big)(n-1)} {d_i(d_i-1)}. $$ For pendant vertices we should add $\frac {\text{number of pendant vertices}} n$ in the right side. Note that if there exist two vertices $i,j\in V(G)$ such that $d_i < d_j $ and $ \Str(i)< \Str(j)$ then the inequality in this theorem will be strict. \end{proof} Let's consider a star $G$ with $V(G) = n+1$ vertices. The star is geodetic graph. The central vertex has degree $n$, local clustering coefficient $c_i = 0$ and stress centrality $\Str(i) = n(n-1)$. Other vertices are pendant ($d_i = 1,\ c_i = 0,\ \Str(i) = 0$). Thus for this graph holds theorem~\ref{thm6}. $$ L(G) = \frac {n(2n-1)+n} {(n+1)n} = \frac {2n} {n+1},\quad C_{WS} = 0. $$ $$ 1-C_{WS}(G) = 1 = \frac {\frac {n-1} {n+1}\, n} {n(n-1)}+\frac n {n+1} = \frac 1 {n+1} \sumt_{i\in V(G)} \frac {\big(L(G)-1\big)n} {d_i(d_i-1)}+\frac {\text{number of pendant vertices}} {n+1}. $$ Thus, for this example holds equality. \begin{thebibliography}{99} \bibitem{Borgatti} Borgatti S. P., Everett M. G. A graph-theoretic perspective on centrality //Social networks. 2006. \bf{28}. №~4. 466--484. \bibitem{Kiss} Kiss C., Bichler M. Identification of influencers—measuring influence in customer networks //Decision Support Systems. 2008. \bf{46}. №~1. 233--253. \bibitem{Lee} Lee S. H. M., Cotte J., Noseworthy T. J. The role of network centrality in the flow of consumer influence //Journal of Consumer Psychology. 2010. \bf{20}. №~1. 66--77. \bibitem{Watts} Watts D. J., Strogatz S. H. Collective dynamics of ‘small-world’networks //nature. 1998. \bf{393}. №~6684. 440--442. \end{thebibliography} \end{document}
2412.02420v1	http://arxiv.org/abs/2412.02420v1	Fitting parameters of a Fokker-Planck-like equation with constraint	\documentclass[12pt]{article} \usepackage{amsmath,amsfonts,amssymb} \usepackage{mathrsfs} \usepackage{dsfont} \usepackage[utf8x]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{ucs} \usepackage{fullpage} \usepackage{float} \usepackage{graphicx} \usepackage{caption} \usepackage{array} \usepackage{multicol} \usepackage{multirow} \usepackage[format=hang, justification=raggedright]{caption} \usepackage{subfig} \usepackage{color} \usepackage{authblk} \usepackage{esint} \usepackage{enumitem,hyperref} \usepackage[normalem]{ulem} \usepackage{tikz} \usetikzlibrary{decorations} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{decorations.pathmorphing} \usepackage{tkz-tab} \usetikzlibrary{shapes} \tikzset{t style/.style={style=solid}} \newtheorem{lemma}{Lemma}[section] \newtheorem{theo}{Theorem} \newtheorem{rmk}[lemma]{Remark} \newtheorem{prop}[lemma]{Proposition} \newtheorem{defin}[lemma]{Definition} \newtheorem{coro}[lemma]{Corollary} \newtheorem{example}[lemma]{Example} \makeatletter \def\namedlabel#1#2{\begingroup #2 \def\@currentlabel{#2} \phantomsection\label{#1}\endgroup } \makeatother \makeatletter \renewcommand{\eqref}[1]{ \hyperref[{#1}]{\textup{\tagform@{\ref{#1}}}}} \makeatother \newenvironment{Proof}{\noindent \abovedisplayskip = 0.5\abovedisplayskip \belowdisplayskip=\abovedisplayskip{\bfseries Proof. }}{\QED\medskip} \newenvironment{ProofOf}[1]{\noindent \abovedisplayskip = 0.5\abovedisplayskip \belowdisplayskip=\abovedisplayskip{\bfseries Proof of #1. }}{\QED\medskip} \newcommand{\rouge}[2]{{\textcolor{blue}{#1} \textcolor{red}{#2}}} \def\dbar{{\mathchar'26\mkern-12mu \mathrm d}} \newcommand{\hslashslash}{ \raisebox{.9ex}{ \scalebox{.7}{ \rotatebox[origin=c]{18}{$-$} } }} \newcommand{\QED}{\mbox{}\hfill \raisebox{-0.2pt}{\rule{5.6pt}{6pt}\rule{0pt}{0pt}} \medskip\par} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\dd}{\mathrm{d}} \newcommand{\J}{\mathbf{J}} \newcommand{\JJ}{\mathbb{J}} \newcommand{\JJJ}{\mathcal{J}} \renewcommand{\L}{\mathbf{L}} \newcommand{\D}{\mathrm{D}} \newcommand{\ds}{\displaystyle} \newcommand{\ud}{\, {\mathrm{d}}} \newcommand{\avex}[1]{\big\langle{#1} \big\rangle_{\mathbb X^d}} \newcommand{\avev}[1]{\big\langle{#1} \big\rangle_{\mathbb R^d}} \newcommand{\Td}{\mathbb T^d} \newcommand{\TumCellunit}{[cell_{\TumVar}]} \newcommand{\ImCellunit}{[cell_{\ImVar}]} \newcommand{\TumCellunitNoBracket}{cell_{\TumVar}} \newcommand{\ImCellunitNoBracket}{cell_{\ImVar}} \newcommand{\Lunit}{[L]} \newcommand{\lunit}{[l]} \newcommand{\tunit}{[t]} \newcommand{\micrometer}{\mu m} \newcommand{\Vunit}{\micrometer^3 \cdot day^{-1}} \newcommand{\Aunit}{day^{-1}} \newcommand{\Deltaunit}{\mu m^{-3} \cdot day^{-1}} \newcommand{\Chiunit}{mm^2 \cdot mmol^{-1} \cdot day^{-1}} \newcommand{\Dunit}{mm^2 \cdot day^{-1}} \newcommand{\Punit}{day^{-1}} \newcommand{\Sunit}{\ImCellunitNoBracket \cdot mm^{-3}} \newcommand{\Gammaunit}{day^{-1}} \newcommand{\Kunit}{mm^2 \cdot day^{day^{-1}}} \newcommand{\Sigmaunit}{mmol \cdot \TumCellunitNoBracket^{-1} \cdot \micrometer^{-3} \cdot day^{-1}} \newcommand{\Alphaunit}{\TumCellunitNoBracket \cdot \micrometer^{-3}} \newcommand{\Iunit}{mol} \newcommand{\TumVar}{n} \newcommand{\DivRateVar}{a} \newcommand{\DivProdVar}{k} \newcommand{\muzero}{\mu_0} \newcommand{\muone}{\mu} \newcommand{\ImVar}{c} \newcommand{\snormal}{\nu} \newcommand{\Snormal}{\nu} \newcommand{\ChemPoVar}{\phi} \newcommand{\mukc}{\mu_k^{\ImVar}} \newcommand{\mukonec}{\mu_{k-1}^{\ImVar}} \newcommand{\muktwoc}{\mu_{k-2}^{\ImVar}} \newcommand{\muc}{\mu_0^{\ImVar}} \newcommand{\TregVar}{r} \newcommand{\com}[1]{\textcolor{red}{#1}} \newcommand{\comb}[1]{\textcolor{blue}{#1}} \title {Fitting parameters of a Fokker-Planck-like equation with constraint} \markboth{Constrainted Fokker-Planck equation }{K. Atsou, T. Goudon, P.-E. Jabin} \author[1]{Kevin Atsou\thanks{{\tt [email protected]}} } \author[2]{Thierry~Goudon\thanks{{\tt [email protected]} }} \author[3]{Pierre-Emmanuel Jabin\thanks{{\tt [email protected]}} } \affil[1]{\small Pfizer } \affil[2]{\small Universit\'e C\^ote d'Azur, CNRS, LJAD } \affil[3]{\small Penn State University, Math. Dept.} \date{} \begin{document} \maketitle \abstract{We analyse a Fokker-Planck like equation, driven by a scalar parameter in order to reach an integral constraint. We exhibit criteria guaranteeing existence-uniqueness of a solution. We also provide counter-examples. This problem is motivated by an application to the immune control of tumor growth. } \vspace{.5cm} {\small \noindent{\bf Keywords.} Fokker-Planck equation. Constrained elliptic problems. Tumor growth. Equilibrium phase. \vspace{.5cm} \noindent{\bf Math.~Subject Classification.} 35B99. 92C50, 92C17, } \section{Introduction} We are interested in the following problem: given a certain ``confining'' (the meaning of which will be made precise later on) potential $\Phi:\mathbb R^N\to \mathbb R$, $\gamma, \lambda>0$ and two non negative functions $\delta, S:\mathbb R^N\to \mathbb R$, we consider the PDE \begin{equation}\label{def_FP} \gamma u -\mu\nabla\cdot(\nabla \Phi u)-\Delta u=\mu S\end{equation} where the parameter $\mu>0$ is selected so that the associated solution $u_\mu$ satisfies the constraint \begin{equation}\label{constraint} \ds\int_{\mathbb R^N} \delta u\ud x=\lambda>0.\end{equation} This question has been introduced in \cite{aabg_pub}, motivated by the modeling of the immune response to tumor growth, in order to explain equilibrium phases where the tumor is kept under control by the action of the immune cells. Numerical simulations show that the formation of equilibria, and thus the existence and stability of solutions of \eqref{def_FP}-\eqref{constraint}, is a quite robust phenomenom, see also \cite{aabg_fr}. However, the analysis provided in \cite{aabg_pub}, by means of the implicit function theorem, is restricted to small values of the constraint parameter $\lambda$. We wish to extend the existence-uniqueness of the pair $(\mu, u_\mu)$ satisfying \eqref{def_FP}-\eqref{constraint}, associated to any $\lambda\geq 0$. In fact, our analysis shows that this requires further ``compatibility'' conditions between the potential $\Phi$, the source $S$ and the contraint function $\delta$. We provide counter-examples explaining the role of these conditions. Our arguments, which are likely of interest beyond the original application to tumor-immune system interactions, rely on properties of the underlying Fokker-Planck operator, moment propagation and duality reasonings. The paper is organized as follows. First, Section~\ref{Sec:Motiv} motivates the problem \eqref{def_FP}-\eqref{constraint} by rapidly coming back to the modeling introduced in \cite{aabg_pub}. Next, Section~\ref{Sec:FP} discusses some properties of the Fokker-Planck operator $\nabla\cdot(\nabla\Phi u)+\Delta u$ which arises in \eqref{def_FP} and are crucial for the analysis. In Section~\ref{Sec:Asymp}, we study in details the behavior of the constraint functional $$\mathscr F:\mu\mapsto \int_{\mathbb R^N} \delta u_\mu\ud x$$ for small and large $\mu$'s. Finally, in Section~\ref{Sec:Monot} we analyze the monotonicity of the mapping $\mathscr F$, depending on assumptions on the data $ \Phi, S, \delta$. The analytical results are further illustrated by a few numerical examples. \section{Motivation} \label{Sec:Motiv} We remind the reader the modeling principles that lead to the problem \eqref{def_FP}-\eqref{constraint}. The earliest stages of tumor growth can be described through the evolution of the density of tumor cells $(t,z) \mapsto \TumVar(t,z)$: the integral $\int_a^b z \TumVar(t,z)\ud z$ gives the volume of the tumor occupied at time $t$ by tumor cells having their size $z$ in the interval $(a,b)$. It is governed by two phenomena: a natural growth, embodied in the rate $z\mapsto V(z)\geq 0$ and cell division mechanisms, where a cell with size $z'$ divides into cells with respective sizes $ z$ and $z'-z$. The latter depend on the frequency of division $z\mapsto a(z)$ and the size-distribution $k(z\vert z')$ from the division of a tumor cell with size $z'$. Therefore, without any further interaction, the evolution of the tumor cells obey the initial-boundary value problem: \begin{equation}\label{evol_n} \partial_t n+\partial_z (Vn)=Q(n), \qquad n(0,z)=n_0(z),\qquad n(t,0)=0,\end{equation} with \[ Q(n)(t,z) = -a(z)n(t,z) + \int_z^{\infty}a(z')k(z\vert z')n(t,z')\ud z '. \] A basic example of such cell- division operator is given by the binary division operator \[ Q(n)(t,z) = 4a(2z)n(t,2z) -a(z)n(t,z). \] In any cases, the assumption on the kernel $k$ are such that \begin{itemize} \item The total number of tumor cells is non decreasing \[ \dfrac{\ud}{\ud t}\ds\int_0^{\infty} n(t,z)\ud z \geq 0, \] \item The total mass of the tumor is non decreasing \[ \dfrac{\ud}{\ud t} \ds\int_0^{\infty} z n(t,z)\ud z = \ds\int_0^\infty V(z) n(t,z)\ud z \geq 0. \] \end{itemize} Note that the former is due to cell division, the later to the natural growth. A remarkable fact about this growth-division equation is the existence of an eigenpair $(\lambda, N)$, with $\lambda >0$ and $z\geq 0 \mapsto N(z)$ taking non negative values, that satisfy \begin{equation} \label{tumorgrowthEigenProblem} \left\{ \begin{array}{l} \partial_z (V N) -Q( N) + \lambda N = 0 \text{ for } z \geq 0 \\ \nonumber N(0) = 0, \qquad N(z)>0 \ \text{ for $z>0$},\qquad \ds\int_{0}^{+\infty} N(z)\ud z = 1. \end{array}\right.\end{equation} We refer the reader to \cite{DoGa, Mich1,PerBk} for precise assumptions and statements with proofs relying on suitable applications of the Krein-Rutman theorem. Note that, in the specific case where $a, V$ are constant and $Q$ is the binary division operator, we have $\lambda=a$ and the profile $N$ is explicitly known, \cite{Bac, PerBk,PeRy}. Dedicated numerical methods to compute the eigenpair are presented in \cite{aabg_fr}. Furthermore, it can be shown that this eigenstate drives the large time behavior of the Cauchy problem for \eqref{evol_n}: we have $ n(t,z) \sim_{t\to \infty} \nu_0 e^{\lambda t} \overline N(z) $ where $\nu_0$ is a constant determined by the initial condition, see \cite{DGL,Mich1,MMP}. The modeling of immune response adopted in \cite{aabg_pub} assumes that the displacement of the immune cells holds at a larger scale, described by a space variable $x\in \mathbb R^N$, while the tumor is attached at a given location $x_0$. The immune cells are activated from a reservoir of resting cells, are their motion is driven by diffusion and chemotaxis directed towards the tumor. The strength of both the activation and the directed drift depends on the total tumor mass \[\mu(t)=\ds\int_0^\infty z n(t,z)\ud z.\] Let $\Phi:\mathbb R^N\rightarrow \mathbb R$ be a potential, intended to create an attractive force towards the tumor location $x_0$ (and from now on, wlog, we suppose $x_0=0$). The time evolution of the concentration of immune cells $C:(0,\infty)\times \mathbb R^N\rightarrow [0,\infty)$ is governed by \[\partial_ t C-\Delta C-\mu\nabla\cdot(\nabla\Phi C) =\mu S-\gamma C, \] where $S: \mathbb R^N\rightarrow[0,\infty)$ describes the source of resting immune cells, and $\gamma>0$ is the natural death rate of the immune cells. The action of the immune cells on the tumor cells is taken into account through a death term \[-\ds\int_{\mathbb R^N} \delta(x)C(t,x)\ud x,\] in the right hand side of \eqref{evol_n} where $\delta: \mathbb R^N\rightarrow[0,\infty)$ is intended to describe the killing effects on the tumor cells; it acts as a mollified delta-Dirac at $x_0=0$. Performing simulations of the coupled problem, one observes the formation of an equilibrium phase, with a residual tumor, having a positive mass, controlled by the action of the immune cells. Such an equilibrium can be explained by coming back to the eigenproblem \eqref{tumorgrowthEigenProblem}: the death term induced by the immune cells is expected to counterbalance the natural growth rate of the cell-division equation. The other way around, we expect that $C(t,x)$ tends to $u(x) $, solution of \eqref{def_FP}-\eqref{constraint} as $t$ goes to $\infty$, with $\ell=\lambda$, the eigenvalue determined by \eqref{tumorgrowthEigenProblem}. We refer the reader to \cite{aabg_pub, aabg_fr} for numerical illustration of such a behavior, which seems very robust. Moreover, this interpretation of the equilibrium phase by means of an eigenvalue problem permits to compute a priori the final mass of the tumor, given the biological parameters \cite{aabg_fr}. Unfortunately, a direct reasoning justifies this interpretation only for small values of $\lambda$. \begin{theo}\label{theo} If $\ell>0$ is small enough, there exists a unique $\mu(\ell)>0$ such that $u_{\mu(\ell)} $, solution of the stationary equation \eqref{def_FP}, satisfies \eqref{constraint}. \end{theo} \noindent {\bf Proof.} The framework slightly differs from \cite{aabg_pub} which deals with the problem set in a bounded domain, endowed with appropriate boundary conditions. The argument uses the results in Proposition~\ref{prop:LxM}, detailed below. We are searching for the zeroes of the mapping \[\mathscr X:(\ell,\mu)\in [0,\infty)\times [0,\infty)\longmapsto \ds\int _\Omega \delta u_{\mu} \ud x-\ell\] where $u_{\mu} $ is the solution of \eqref{def_FP} associated to $\mu$ and knowing that $\mathscr X(0,0)=0$, since $u_{0}=0$. We have $\partial_{\mu} \mathscr X(\ell,\mu)=\int _\Omega \delta u'_{\mu} \ud x$, with $u'_{\mu}$, solution of \[ \gamma u'-\Delta u'-\mu\nabla_x\cdot (u'\nabla_x\Phi)= S +\nabla_x\cdot (u_{\mu}\nabla_x\Phi). \] Since $u_0=0$ and $S\geq 0$, we get $u'_{0}>0$ (see Proposition~\ref{prop:LxM} below). It follows that $\partial_{\mu} \mathscr X(0,0)=\int _{\mathbb R^N} \delta u'_{0} \ud x>0$. The implicit function theorem tells us that there exists $\ell _\star>0$ and a mapping $\mu:\ell\in [0,\ell_\star)\mapsto\mu(\ell)$ such that $\mathscr X(\ell,\mu(\ell))=0$ holds for any $\ell\in [0,\ell_\star)$, which means that $u_\ell$ satisfies \eqref{def_FP}-\eqref{constraint}. Observe that $$ \begin{array}{l} \partial_{\ell} \mathscr X(\ell,\mu(\ell))+ \mu'(\ell)\partial_{\mu} \mathscr X(\ell,\mu(\ell)) =-1 +\mu'(\ell)\partial_{\mu} \mathscr X(\ell,\mu(\ell))=0\end{array}$$ holds with $\partial_{\mu} \mathscr X(0,0)>0$. Hence, $\ell\mapsto \mu(\ell)$ is increasing on the neighborood of $\ell=0$, and it thus takes positive values. Note that the argument cannot be extended for any $\ell$, since we do not have a direct knowledge on the sign of $\nabla_x\cdot (u_{\mu}\nabla_x\Phi)$ for $\mu\neq 0$. However, the proof do not use the confining feature of the potential $\Phi$. Hence, we are going to develop a viewpoint that further exploits these properties. \QED \section{Fundamental properties of the operator $L_\mu$} \label{Sec:FP} Let us make the following assumption on the potential $\Phi$: \begin{equation} \label{hPhi1} \begin{array}{l} \textrm{for any $x\in \mathbb R^N$, we have $\Phi(x)\geq 0$}, \\ \textrm{ and, for any $\mu>0$, $x\mapsto M_\mu(x)=e^{-\mu \Phi(x)}\in L^1(\mathbb R^N)$}. \end{array} \end{equation} As a matter of fact, the latter integrability property is guaranteed by the following strengthened convexity condition: assuming $\Phi\in C^2$, \begin{equation} \label{hPhi2} \begin{array}{l} \textrm{$\nabla \Phi(0)=0$ and there exists a constant $\Lambda>0$ such that } \\ \textrm{the hessian matrix of $\Phi$ satisfies, for any $x\in \mathbb R^N$, $D^2_{ij}\Phi(x)\geq \Lambda\mathbb I$.} \end{array} \end{equation} These conditions describe the confining feature of the potential, having an attractive effect towards $x=0$, which is a strict global minimizor of the potential. Then, we introduce the Fokker-Planck operator \begin{equation} \label{def_fp} L_\mu u =\mu\nabla\cdot(\nabla\Phi u)+\Delta u \end{equation} and its adjoint operator (defined with the standard $L^2$ inner product) \begin{equation} \label{def_fpst} L_\mu^\star \psi =-\mu\nabla\Phi \cdot\nabla \psi+\Delta \psi. \end{equation} It is convenient to recast these operators by making the function $M_\mu$ appear \[ L_\mu u=\nabla\cdot\left(M_\mu \nabla\left(\ds\frac u{M_\mu}\right)\right), \qquad L_\mu^\star \psi=\ds\frac{1}{M_\mu}\nabla\cdot\left( M_\mu \nabla\psi \right). \] Accordingly, we observe that \begin{equation}\label{vari_L} -\ds\int_{\mathrm R^N} \ds\frac u{M_\mu}L_\mu u\ud x =\ds\int_{\mathrm R^N} M_\mu \left\|\nabla\left(\ds\frac u{M_\mu}\right) \right\|^2\ud x \geq 0. \end{equation} Note that, owing to \eqref{hPhi2}, the following Sobolev inequality \begin{equation}\label{coer_L}\begin{array}{lll} \ds\int_{\mathrm R^N} M_\mu \left\|\nabla\left(\ds\frac u{M_\mu}\right) \right\|^2 \langle M_\mu\rangle\ud x&\geq& 2\Lambda\mu \ds\int_{\mathrm R^N} \left\| u-\langle u\rangle \ds\frac{M_\mu}{\langle M_\mu\rangle} \right\|^2\ds\frac{\langle M_\mu\rangle\ud x}{M_\mu} \\&\geq& 2\Lambda\mu\left( \ds\int_{\mathrm R^N} \left\| u-\langle u\rangle \ds\frac{M_\mu}{\langle M_\mu\rangle} \right\|\ud x\right)^2 , \end{array}\end{equation} holds, where $\langle u\rangle=\int_{\mathbb R^N} u\ud x$, see \cite[condition (A2), Corollary 2.18]{AMTU}. Similarly, we have \[ -\ds\int_{\mathrm R^N} M_\mu \psi L^\star_\mu \psi\ud x =\ds\int_{\mathrm R^N} M_\mu \left\| \nabla\psi \right\|^2\ud x \geq 0. \] \begin{prop} \label{prop:LxM} The following assertions hold: \begin{itemize} \item[i)] $\mathrm{Ker}(L_\mu)=\mathrm{Span}(M_\mu)$ and $\mathrm{Ker}(L^\star_\mu)=\mathrm{Span}(\mathbf 1)$, \item[ii)] Let $\gamma>0$. For any $S\in L^2(\mathbb R^N,\frac{\ud x}{M_\mu})$, there exists a unique solution $u\in L^2(\mathbb R^N,\frac{\ud x}{M_\mu})$, with $\nabla\frac u{M_\mu}\in L^2(\mathbb R^N,M_\mu \ud x)$ of $(\gamma\mathbb I-L_\mu)u=S$. Moreover, if $S\geq 0$, then $u\geq 0$ too. \item[iii)] Let $\gamma>0$. For any $\delta \in L^2(\mathbb R^N,M_\mu \ud x)$, there exists a unique solution $\psi\in L^2(\mathbb R^N,M_\mu \ud x)$, with $\nabla\psi\in L^2(\mathbb R^N,M_\mu \ud x)$ of $(\gamma\mathbb I-L^\star_\mu)\psi=\delta$. Moreover, if $\delta \geq 0$, then $\psi\geq 0$ too. \end{itemize} \end{prop} \noindent {\bf Proof.} The first item is a direct consequence of \eqref{vari_L} and \eqref{coer_L}. Next, we simply apply the Lax-Milgram theorem (or, in the present context the Riesz theorem) in the Hilbert space \[ H=\Big\{u\in L^2(\mathbb R^N,\frac{\ud x}{M_\mu}),\ \nabla \ds\frac u{M_\mu}\in L^2(\mathbb R^N,M_\mu \ud x)\Big\}\] to solve the variational problem: to find $u\in H$, such that, for any $v\in H$, we have \[ \gamma \ds\int_{\mathbb R^N}uv \ds\frac{\ud x}{M_\mu}+\ds\int_{\mathrm R^N} M_\mu \left\|\nabla\left(\ds\frac u{M_\mu}\right) \right\|^2\ud x=\ds\int_{\mathbb R^N} Sv\ds\frac{\ud x}{M_\mu}.\] We obtain the sign property by using $v=u_-$ as trial function in the variational formulation: it yields \[ \gamma \ds\int_{\mathbb R^N} u^2_- \ds\frac{\ud x}{M_\mu}+\ds\int_{\mathrm R^N} M_\mu \left\|\nabla\left(\ds\frac {u_-}{M_\mu}\right) \right\|^2\ud x=\ds\int_{\mathbb R^N} Su_-\ds\frac{\ud x}{M_\mu}\leq 0\] when $S$ takes non negative values. It implies $u_-=0$ a. e. A similar argument applies readily to the adjoint problem. \QED \section{Asymptotic behavior of $ \mu\mapsto \mathscr F(\mu)$} \label{Sec:Asymp} \begin{lemma} Suppose \eqref{hPhi2}. Let $(1+x^2)S\in L^1(\mathbb R^N)$ with $S\geq 0$ and $S\in L^2(\mathbb R^N, \frac{\ud x}{M_\mu})$ for any $\mu>0$. Let $\delta:\mathbb R^N\to [0,\infty)$ be a non negative function such that $\delta\in L^2(\mathbb R^N, M_\mu)$ for any $\mu>0$. We suppose that there exists $\eta, r>0$ such that $\delta (x)\geq \eta$ on $B(0,r)$. Let $u_\mu$ given by Proposition~\ref{prop:LxM}-ii) for right hand side $\mu S$. Then, $\lim_{\mu\to \infty}\mathscr F(\mu)=+\infty$. \end{lemma} \noindent {\bf Proof.} By integrating the equation $(\gamma -L_\mu)u_\mu=\mu S$, we get \[ \gamma\ds\int_{\mathbb R^N} u_\mu\ud x=\mu \ds\int_{\mathbb R^N} S\ud x.\] Similarly, considering the second moment and using integration by parts, we are led to \[ \begin{array}{lll} \gamma \ds\int_{\mathbb R^N} x^2u_\mu\ud x&=&\mu \ds\int_{\mathbb R^N} x^2S\ud x +\ds\int_{\mathbb R^N} x^2\nabla\cdot(\nabla u_\mu+\mu u_\mu\nabla \Phi )\ud x \\ &=& \mu \ds\int_{\mathbb R^N} x^2S\ud x +2N \ds\int_{\mathbb R^N}u_\mu\ud x-2\mu \ds\int_{\mathbb R^N} x\cdot \nabla \Phi u_\mu \ud x \\ &\leq & \mu \ds\int_{\mathbb R^N} (2N+x^2) S\ud x-2\Lambda \mu \ds\int_{\mathbb R^N} x^2u_\mu\ud x, \end{array}\] since \eqref{hPhi2} implies $x\cdot \nabla\Phi(x)=x\cdot (\nabla\Phi(x)-\nabla\Phi(0))\geq \Lambda x^2$. It follows that the second moment is bounded uniformly wrt $\mu>0$ since \[ \ds\int_{\mathbb R^N} x^2u_\mu\ud x\leq \ds\frac{\mu}{\gamma + 2\Lambda\mu} \ds\int_{\mathbb R^N} (2N+x^2) S\ud x \leq \ds\frac{1}{\Lambda} \ds\int_{\mathbb R^N} (2N+x^2) S\ud x.\] We now split, for $r>0$, \[\begin{array}{lll} \ds\int_{\mathbb R^N} \delta u_\mu\ud x&=&\ds\int_{\|x\|\leq r} \delta u_\mu\ud x+\ds\int_{\|x\|>r} \delta u_\mu\ud x \\ & \geq & \eta \ds\int_{\|x\|\leq r} u_\mu\ud x=\eta \left(\ds\int_{\mathbb R^N} u_\mu\ud x- \ds\int_{\|x\|> r} u_\mu\ud x\right) \\ & \geq & \eta \mu \ds\int_{\mathbb R^N} S\ud x- \ds\frac{\eta}{r^2}\ds\int_{\mathbb R^N} x^2 u_\mu\ud x \\&\geq& \eta \mu \ds\int_{\mathbb R^N} S\ud x- \ds\frac{\eta}{\Lambda r^2}\ds\int_{\mathbb R^N} (2N+x^2) S\ud x \end{array}\] where the RHS tends to $+\infty$ as $\mu\to \infty$. \QED In fact, we can make the behavior for large $\mu$'s more precise, by appealing to the Laplace method, which can be summarized in the following claim \cite[Theorem~15.2.2]{BaSi}. \begin{lemma} Let $f:\mathbb R^N\to \mathbb R$ be a continuous function such that $f(0)\neq 0$. Then, as $\mu$ goes to $+\infty$, $\int_{\mathbb R^N} f M_{\mu}\ud x$ is equivalent to \[\ds\frac{f(0)}{\mu^{N/2}}\sqrt{ \ds\frac{(2\pi)^N}{\mathrm{det}(\mathrm D^2\Phi(0))}} .\] \end{lemma} Let us set $$ m(\mu)=\ds\int_{\mathbb R^N} M_\mu\ud x, \qquad \varsigma(\mu)=\ds\int_{\mathbb R^N} \ds\frac{S^2}{M_\mu}\ud x$$ and introduce the following rescaling $$\tilde u_\mu(x)=\ds\frac{u_\mu(x)}{\mu\sqrt{\varsigma(\mu)}}.$$ The latter satisfies \[\gamma \tilde u_\mu-\nabla\cdot\left(M_\mu\nabla \ds\frac{\tilde u_\mu}{M_\mu}\right)=\ds\frac{S}{\sqrt{\varsigma(\mu)}}.\] In turn, we obtain the following estimate \[\gamma \ds\int_{\mathbb R^N} \ds\frac{\|\tilde u_\mu\|^2}{M_\mu}\ud x +2\ds\int_{\mathbb R^N} M_\mu \left\|\nabla \ds\frac{\tilde u_\mu}{M_\mu}\right\|^2 \ud x \leq 1,\] together with \[\langle \tilde u_\mu \rangle = \ds\frac{\langle S\rangle}{\sqrt{\varsigma(\mu)}}.\] Owing to \eqref{coer_L}, it leads to \[ \ds\int_{\mathbb R^N} \left \|\tilde u_\mu-\ds\frac{\langle S\rangle}{\sqrt{\varsigma(\mu)}} \ds\frac{M_\mu}{m(\mu)}\right\|\ud x \leq \ds\frac1{2\sqrt{\Lambda\mu}}\xrightarrow [\mu\to \infty]{}0. \] Therefore, assuming $\delta\in L^\infty(\mathbb R^N)$, $\delta $ continuous with $\delta (0)\neq 0$, we get \[\begin{array}{lll} \mathscr F(\mu)&=&\mu \sqrt{\varsigma(\mu)} \ds\int_{\mathbb R^N} \tilde u_\mu \delta \ud x \underset{\mu \to \infty}{ \sim} \mu \sqrt{\varsigma(\mu)} \ds\int_{\mathbb R^N} \ds\frac{\langle S\rangle}{\sqrt{\varsigma(\mu)}} \ds\frac{M_\mu}{m(\mu)} \delta \ud x \\&\underset{\mu \to \infty}{ \sim}& \langle S\rangle \ds\frac{\mu}{m(\mu)} \ds\frac{\delta (0)}{\mu^{N/2}}\sqrt{\ds\frac{(2\pi)^N}{\mathrm{det}(\mathrm D^2\Phi(0))}} \underset{\mu \to \infty}{ \sim} \mu \delta (0)\langle S\rangle.\end{array} \] Since the function $\mu\mapsto \mathscr F(\mu)$ is continuous, with $\mathscr F(0)=0$, we deduce the following existence result. \begin{coro} Suppose $\delta\in L^\infty(\mathbb R^N)$, $\delta $ continuous with $\delta (0)\neq 0$. For any $\ell\geq 0$, there exists at least a $\mu\in [0,\infty)$ such that \eqref{def_FP}-\eqref{constraint} holds. \end{coro} \section{Monotonicity} \label{Sec:Monot} We are going to show that, under certain compatibility conditions, the function $\mathscr F$ is increasing; to this end we use a duality argument. We introduce the solution $\psi_\mu$ of \[ (\gamma-L^\star_\mu) \psi_\mu= \delta\] so that $\mathscr F(\mu) $ recasts as \[ \ds\int_{\mathbb R^N} u_\mu\delta\ud x = \ds\int_{\mathbb R^N} u_\mu(\gamma-L^\star_\mu) \psi_\mu\ud x =\ds\int_{\mathbb R^N} (\gamma-L^\star_\mu) u_\mu\psi_\mu\ud x =\mu\ds\int_{\mathbb R^N} S\psi_\mu\ud x. \] Accordingly, showing the monotonicity of $\mathscr F$ reduces to investigating the sign of \begin{equation}\label{deriv} \ds\frac{\ud}{\ud \mu} \mathscr F(\mu) =\ds\int_{\mathbb R^N} S\psi_\mu\ud x+\mu\ds\int_{\mathbb R^N} S\psi'_\mu\ud x\end{equation} where $\psi'_\mu$ satisfies \[(\gamma-L^\star_\mu) \psi'_\mu=-\nabla\Phi\cdot \nabla\psi_\mu.\] The data $S$ and $\delta$ being non negative, the first integral in the right hand side of \eqref{deriv} is non negative. We are going to show that the second term is equally non negative, under appropriate assumptions on the data. \subsection{Problem in radial symmetry} From now on, we assume that all data $S,\delta, \Phi$ are \emph{radially symmetric}. In turn, $M_\mu$ and the solutions of the associated PDE are also radially symmetric. We write the equation in radial coordinates: we get \[ \gamma u_\mu -\mu \partial_r (u_\mu\partial_ r \Phi)-\mu\ds\frac{N-1}{r}\partial_r \Phi u_\mu-\ds\frac{1}{r^{N-1}}\partial_r(r^{N-1}\partial_r u_\mu)=\mu S.\] It casts as \[ \gamma u_\mu -\ds\frac{1}{r^{N-1}}\partial_r \left(M_\mu r^{N-1} \partial_r\Big(\ds\frac{u_\mu}{M_\mu}\Big)\right) =\mu S.\] For the adjoint equation, we obtain \begin{equation}\label{adjrad}\begin{array}{l} \gamma \psi_\mu +\mu \partial_r \Phi\partial_r \psi_\mu -\ds\frac{1}{r^{N-1}}\partial_r(r^{N-1}\partial_ r \psi_\mu)=\delta \\ \qquad\qquad = \gamma \psi_\mu - \ds\frac{1}{r^{N-1} M_\mu}\partial_r(r^{N-1}M_\mu\partial_r \psi_\mu). \end{array}\end{equation} Let us set $\chi_\mu=\partial_r\psi_\mu$. It satisfies \[\begin{array}{l} \left(\gamma + \ds\frac{N-1}{r^2}+\mu \partial^2_r \Phi\right)\chi_\mu +\mu \partial_r \Phi\partial_r \chi_\mu -\ds\frac{1}{r^{N-1}}\partial_r(r^{N-1}\partial_ r \chi_\mu)=\partial_r\delta \\\qquad\qquad =\left(\gamma + \ds\frac{N-1}{r^2}+\mu \partial^2_r \Phi\right)\chi_\mu -\ds\frac{1}{r^{N-1}M_\mu }\partial_r(r^{N-1}M_\mu \partial_ r \chi_\mu).\end{array}\] We assume the convexity/monotonicity properties \begin{equation}\label{cx}\partial_r \Phi\geq 0,\qquad \partial^2_r \Phi\geq 0,\qquad \partial_r\delta\leq 0.\end{equation} Under these assumptions, we obtain $$\chi_\mu=\partial_r\psi_\mu \leq 0.$$ Now, we go back to \eqref{adjrad}, which yields \[ \begin{array}{l} \gamma\psi'_\mu +\mu \partial_r \Phi\partial_r \psi'_\mu -\ds\frac{1}{r^{N-1}}\partial_r(r^{N-1}\partial_ r \psi'_\mu)=- \underbrace{\partial_r \Phi}_{\geq 0} \underbrace{\partial_r \psi_\mu }_{\leq 0} \\ \qquad\qquad = \gamma \psi'_\mu - \ds\frac{1}{r^{N-1} M_\mu}\partial_r(r^{N-1}M_\mu\partial_r \psi'_\mu).\end{array}\] The right hand side thus satisfies $-\partial_r \Phi\partial_r \psi_\mu\geq 0$ so that $ \psi'_\mu\geq 0$. Coming back to \eqref{deriv}, we conclude that $\mathscr F$ is non decreasing. We need to slightly improve the result, requiring further regularity on $\delta, \Phi$, say $\delta\in C^{1}$, $\Phi\in C^2$, with $\delta, \Phi$ not identically 0. We can thus apply the strong maximum principle \cite[Section~3.2]{GT} which tells us that $\psi'_\mu>0$ on $(0,\infty)$. Our findings recap as follows. \begin{theo}\label{mainth} In the radially symmetric framework, we suppose that $S, \delta\in C^1$ and $\Phi\in C^2$ take non negative values, but are not identically 0, and that \eqref{cx} is fulfilled. Then, $\mathscr F$ is increasing and, for any $\ell\geq 0$, the problem \eqref{def_FP}-\eqref{constraint} admits a unique solution $0<\mu <\infty$. \end{theo} The assumptions of radial symmetry together with \eqref{cx} are quite natural and relevant for the presented modeling: the tumor being located at $x=0$, the action of the immune cells, embodied into $\delta$, is centred on this position and the chemotactic potential $\Phi$ drives the immune cells towards the tumoral centre. Let us detail a simple example showing that these assumptions are also technically important. We consider the simplest confining potential $\Phi(x)=x^2$ and we compute the first even moments of the solutions of \eqref{def_FP}: we have already seen that $\int_{\mathbb R^N} u_\mu\ud x=\mu \int_{\mathbb R^N} S\ud x$; next we have \[\begin{array}{lll} \gamma\ds\int_{\mathbb R^N} \|x\|^2 u_\mu\ud x&=& \mu \ds\int_{\mathbb R^N} \|x\|^2 S\ud x +2N \ds\int_{\mathbb R^N} u_\mu \ud x- 2\mu \ds\int_{\mathbb R^N} u_\mu x\cdot\nabla\Phi\ud x\\ &=& \mu \ds\int_{\mathbb R^N} \|x\|^2S\ud x +2N\mu \ds\int_{\mathbb R^N} S \ud x-4 \mu \ds\int_{\mathbb R^N} \|x\|^2 u_\mu\ud x \end{array}\] so that \[ \ds\int_{\mathbb R^N} \|x\|^2 u_\mu\ud x=\ds\frac{\mu}{\gamma + 4\mu} \ds\int_{\mathbb R^N} (\|x\|^2+2N) S\ud x. \] It follows that $\mu\mapsto \int_{\mathbb R^N} x^2 u_\mu\ud x$ is an increasing function. We turn to \[ \gamma\ds\int_{\mathbb R^N} \|x\|^4 u_\mu\ud x= \mu \ds\int_{\mathbb R^N} \|x\|^4 S\ud x +(4N+8) \ds\int_{\mathbb R^N} \|x\|^2 u_\mu \ud x- 8\mu \ds\int_{\mathbb R^N} \|x\|^4 u_\mu \ud x. \] It leads to the expression \[ \ds\int_{\mathbb R^N} \|x\|^4 u_\mu\ud x=\underbrace{A\frac{\mu}{\gamma + 8\mu}+ B\frac{1}{\gamma + 8\mu}\frac{\mu}{\gamma + 4\mu}}_{:=f(\mu)}\] with \[A=\ds\int_{\mathbb R^N} \|x\|^4 S\ud x,\qquad B=(4N+8)\ds\int_{\mathbb R^N} (\|x\|^2+2N) S\ud x .\] The forth momentum is not necessarily a monotone function of $\mu$ since \[ f'(\mu)=\ds\frac{1}{(\gamma + 8\mu)^2}\left(\gamma A+B\ds\frac{\gamma}{\gamma+4\mu} -4B\ds\frac{\mu(\gamma+8\mu)}{(\gamma+4\mu)^2}\right)\] might change sign. But, in this example, $\delta(x)=\|x\|^4$ vanishes at $x=0$, contradicting the modelling assumptions. \subsection{Numerical illustrations} Dealing with the radially symmetric problem, the finite elements framework is a reliable way to get rid of the singularity at $r=0$. For realizing simulations, we consider the problem set on the slab $[0,1]$: since the phenomena are naturally quite concentrated next to the origin, we expect that this does not influence too much the final results (by the way, we indeed do not observe significant differences when imposing Dirichlet or Neumann conditions at $r=1$ or extending the domain for larger $r$'s). We introduce a discretization of $ [0,1]$ with $N+1$ points \[r_0=0<r_1=h<...<r_N=Nh=1,\qquad h=1/N.\] We introduce the associated $\mathbb P_1$ basis functions, $\chi_1,...,\chi_{N-1}$: \[\chi_j(r)=\ds\frac{r-(j-1)h}h\mathbf 1_{(j-1)h<r< jh}+ \ds\frac{(j+1)h-r}h\mathbf 1_{jh\leq r<(j+1)h},\quad \chi_0(r)=- \ds\frac{r-h}h\mathbf 1_{0\leq r<h}.\] Then, we define the matrices with coefficients \[ M_{ij}=\ds\int_0^1\chi'_i(r)\chi'_j(r) r^{n-1}\ud r,\quad A_{ij}=\ds\int_0^1\chi_i(r)\chi_j(r) r^{n-1}\ud r.\] Given the potential $\Phi$, we also define the centered difference matrix $C$, which is skew-symmetric with \[ C_{j,j+1}=\ds\frac12((j+1)h)^{n-1}\Phi'((j+1)h). \] Then, for a given source term $S$, we define the vector with components \[S(j)=\ds\int_{ih}^{(i+1)h} S(r)r^{n-1}\ud r.\] Eventually, we solve the linear system \[(\gamma A -\mu C+M)U=\mu S,\] and we compute the associated discrete version of $F(\mu)=\int_0^1 \delta u_\mu(r)r^{n-1}\ud r$. We perform the simulation with a source term given by \[S(r)=\mathbf 1_{.3\leq r\leq .5}\] which, for the application to tumor-immune system interactions, corresponds to a located reservoir of resting immune cells, for instance a blood vessel or a lymph node. We set $n=3$ and $\gamma=1$. We start with simulations of the expected situation, with a confining potential pointing towards the origin $\Phi(r)=2r^2$, and a constraint kernel peaked at the origin $$\delta(r)=\frac{e^{-r^2/\epsilon}}{(4\pi\epsilon)^{n/2}},\qquad \epsilon=10^{-3}.$$ Fig.~\ref{F_OK1} represents the profile of the solutions for relatively small values of $\mu$: as $\mu$ increases, the value at $r=0$ increases and the solution concentrates near $r=0$. We numerically check that $\mu\mapsto F(\mu)$ is non decreasing, see Fig.~\ref{F_OK2}; the solution $u_\mu$ becomes highly concentrated to $r=0$, which thus requires a very fine mesh to resolve the solution when $\mu$ becomes large. \begin{figure}[!hbtp] \begin{center} \includegraphics[height=6cm]{./Fig/CasFavo_SolMuPt.eps} \caption{Solutions $r\mapsto u_\mu(r)$ for 10 equidistant values of $\mu$ in $[0,10]$. As $\mu$ increases the solution takes larger value at the origin and presents a stiffer profile for transient radius \label{F_OK1}} \end{center} \end{figure} \begin{figure}[!hbtp] \begin{center} \includegraphics[height=6cm]{./Fig/FMu_OK.eps} \caption{ $\mu\mapsto F(\mu)$ for $\mu$ up to $10^7$. \label{F_OK2}} \end{center} \end{figure} Simulations of the counter-example detailed in the previous section are displayed in Fig.~\ref{F_delta4}: we just modify $\delta$ into \[ \delta(r)=\epsilon r^4,\qquad \epsilon=10^{-3}.\] Then the function $\mu\mapsto F(\mu)$ looses its monotony, but it is still increasing at $\mu=0$ and for large values of $\mu$. \begin{figure}[!hbtp] \begin{center} \includegraphics[height=4cm]{./Fig/Delta_r4.eps}~\includegraphics[height=4cm]{./Fig/Delta_r4Bis.eps}~\includegraphics[height=4cm]{./Fig/UMu_Delta_r4.eps} \caption{Profile of $\mu\mapsto F(\mu)$ for $\mu$ up to $100$ (left), up to $500$ (middle), and snapshots of the corresponding solution profiles $r\mapsto u_\mu(r)$ (right) \label{F_delta4}} \end{center} \end{figure} Next, we consider a situation which can find some physical motivation: the potential is given by \[ \Phi(r)=2r^2\times (r-r_0)^2,\qquad r_0=.2.\] It has a quite flat profile between the two minima $r=0$ and $r=.2$. Note that \eqref{hPhi2} is not satisfied. It describes a defect of the attractivity of the immune cells towards the tumor, due either to the geometry of the tissues around the tumor, or to pro-tumoral effects that reduce the efficacy of the immune response. Another pro-tumoral effect can result in a reduction of the capacity of the immune cells to eliminate tumor cells, that we traduce by shifting the kernel $\delta$ \[\delta(r) = \frac{e^{-(r-r_1)^2/\epsilon}}{(4\pi\epsilon)^{n/2}},\qquad \epsilon=10^{-3},\qquad r_1=.05.\] Note that it still keeps a significantly positive value at $r=0$, see Fig.~\ref{F_Bad}-Top Left. We indeed observe that $\mu\mapsto F(\mu)$ does not tend to infinity as $\mu \to \infty$, and the monotonicity is compromised, see Fig.~\ref{F_Bad}-Bottom. The solution $u_\mu$ tends to form a high peak in the interior of the domain, thus far from the tumor, Fig.~\ref{F_Bad}-Top Right. \begin{figure}[!hbtp] \begin{center} \includegraphics[height=4cm]{./Fig/Delta_Bad.eps}~\includegraphics[height=4cm]{./Fig/Bad_UMu.eps} \\ \includegraphics[height=4cm]{./Fig/F_Bad.eps}~\includegraphics[height=4cm]{./Fig/FMu_Bad2.eps} \caption{Profile of $r\mapsto \delta(r)$ (top-left), profile of $\mu\mapsto F(\mu)$ for $\mu$ up to $10^4$ (bottom-left) and $15\cdot 10^4$ (bottom-right), the solution profiles $r\mapsto u_\mu(r)$ for several $\mu$ up to $10^4$ (top-right) \label{F_Bad}} \end{center} \end{figure} Eventually, we challenge the condition that $\delta$ takes positive value near the origine: we come back to the quadratic potential, but now we work with \[\delta(r) = \frac{e^{-(r-r_1)^2/\epsilon}}{(4\pi\epsilon)^{n/2}},\qquad \epsilon=10^{-3},\qquad r_1=.1\] which (almost) vanishes at $r=0$. Results are reported in Fig.~\ref{F_Bad2}. Again, we observe that $\mu\mapsto F(\mu)$ is not monotone and does not tend to $\infty$ (it seems to be decaying for large $\mu$'s), at least as far as it can be numerically checked. \begin{figure}[!hbtp] \begin{center} \includegraphics[height=4cm]{./Fig/Delta_Bad2.eps}~\includegraphics[height=4cm]{./Fig/Bad_UMu2.eps}~ \includegraphics[height=4cm]{./Fig/F_Bad2.eps} \caption{Profile of $r\mapsto \delta(r)$ (left), profile of $\mu\mapsto F(\mu)$ for $\mu$ up to $2500$ (bottom-left) (bottom-right), snapshot on the corresponding solution profiles $r\mapsto u_\mu(r)$ (top-right) \label{F_Bad2}} \end{center} \end{figure} These numerical experiments highlight the role of the assumptions on both the potential, and the constraint kernel. Coming back to the motivation from the modeling of tumor-immune system interactions, these findings shed light on the role of the pro-tumor mechanisms which not only may promote tumor proliferation, but can also reduce the efficacy of the immune response, and eventually allow the tumor to escape to the control of the immune system \cite{aabg_pone}. \section*{Ackowledgements} This research is supported by the CNRS program ``International Emerging Actions''. We acknowledge the support of the Math.~Dept. at Penn State University. \bibliography{AGJ} \bibliographystyle{plain} \end{document}
2412.02523v1	http://arxiv.org/abs/2412.02523v1	Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadratic irrational parameters	\documentclass[12pt]{amsart} \usepackage{amssymb,latexsym,amsmath,amsthm,amscd} \usepackage{setspace} \usepackage[active]{srcltx} \usepackage[all]{xy} \xyoption{arc} \usepackage[left=3cm,top=2cm,right=3cm,bottom = 2cm]{geometry} \usepackage{graphicx} \usepackage[usenames,dvipsnames]{color} \usepackage{charter} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{conj}[thm]{Conjecture} \newtheorem{exer}[thm]{Exercise} \theoremstyle{definition} \newtheorem{dfn}[thm]{Definition} \newtheorem{ex}[thm]{Example} \newtheorem{claim}[thm]{Claim} \newtheorem{ques}[thm]{Question} \theoremstyle{remark} \newtheorem{rmk}[thm]{Remark} \newtheorem{recall}[thm]{Recall} \newcommand{\lcm}{\operatorname{lcm}} \input{preamble} \DeclareMathOperator{\denom}{denom} \pagenumbering{arabic} \pagestyle{headings} \setcounter{secnumdepth}{4} \setcounter{tocdepth}{2} \setlength{\parindent}{1cm} \begin{document} \title[Density formulas for bounded primes in hypergeometric series]{Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadratic irrational parameters} \author{Cameron Franc} \address{Department of Mathematics, McMaster University} \curraddr{} \email{[email protected]} \thanks{} \author{Nathan Heisz} \address{Department of Mathematics, McMaster University} \curraddr{} \email{[email protected]} \author{Hannah Nardone} \address{Department of Mathematics, McMaster University} \curraddr{} \email{[email protected]} \thanks{We thank NSERC for support via a Discovery grant.} \date{} \begin{abstract} We study densities of $p$-adically bounded primes for hypergeometric series in two cases: the case of generalized hypergeometric series with rational parameters, and the case of $_2F_1$ with parameters in a quadratic extension of the rational numbers. In the rational case we extend work from $_2F_1$ to $_nF_{n-1}$ for an exact formula giving the density of bounded primes for the series. The density is shown to be one exactly in accordance with the case of finite monodromy as classified by Beukers-Heckmann. In the quadratic irrational case, we obtain an unconditional lower bound on the density of bounded primes. Assuming the normality of the $p$-adic digits of quadratic irrationalities, this lower bound is shown to be an exact formula for the density of bounded primes. In the quadratic irrational case, there is a trivial upper bound of $1/2$ on the density of bounded primes. In the final section of the paper we discuss some results and computations on series that attain this bound. In particular, all such examples we have found are associated to imaginary quadratic fields, though we do not prove this is always the case. \end{abstract} \maketitle \tableofcontents \setcounter{tocdepth}{1} \section{Introduction} A classical problem in the study of hypergeometric series is to determine which series are integral. This problem is closely related to integrality of ratios of factorials, algebraicity of generalized hypergeometric series, and even the Riemann hypothesis --- see \cite{Bober} for a review of some of this work, and the recent paper \cite{AdolphsonSperber} for an extension of some of these questions to the case of GKZ-hypergeometric series. The related problem of determining when a generalized hypergeometric series has finite monodromy was solved by Beukers-Heckmann \cite{beukers}, preceded in the case of $_2F_1$ by classical work of Schwarz. In these cases the algebraic series have rational hypergeometric parameters, and it is known by a classical result of Eisenstein that while the series need not have integral coefficients, there are at worst a finite number of primes that arise in the denominators of the series. Thus, while algebraic generalized hypergeometric series need not be integral, the density of primes where such a series is $p$-adically bounded equals one. In this paper we study densities of $p$-adically bounded primes for various hypergeometric series, typically non-algebraic. The study of \(p\)-adic boundedness of a hypergeometric series, spearheaded by Dwork \cite{dwork} and Christol \cite{christol}, provided necessary and sufficient conditions for a generalized hypergeometric series \(_nF_{n-1}\) to be \(p\)-adically bounded. Franc-Gannon-Mason \cite{FGM} showed that for all series $_2F_1$ with rational hypergeometric parameters, there is a corresponding density of $p$-adically bounded primes, and this density is one exactly when the series is algebraic, that is, it comes from Schwarz's classical list. Franc et.al. in \cite{fetal} built on this work to establish an exact formula for the Dirichlet densities of bounded primes in the denominators of series \(_2F_1\) with rational parameters. Our first main result Theorem \ref{t:rational} extends this density formula to generalized hypergeometric series \(_mF_n\) and furthermore shows that the only case of interest in terms of boundedness of primes is the case where \(m = n+1\). These results can be summarized as follows: \begin{thm} \label{t:main1} Let $F(z) = {} _mF_n(\alpha_1,\ldots, \alpha_m; \beta_1,\ldots, \beta_n;z)$ denote a generalized hypergeometric series with rational hypergeometric parameters, $\alpha_i,\beta_j \in \QQ$. Then the set $B$ of primes where $F(z)$ is $p$-adically bounded has a Dirichlet density. This density is one if $m > n+1$, and it is zero of $m < n+1$. When $m=n+1$, the density can be computed exactly using a simple formula given in Corollary \ref{c:rationaldensityformula}. In this case, the density is one if and only if $F(z)$ is an algebraic function. \end{thm} The next case we discuss, starting in Section \ref{s:quadratic}, is that of hypergeometric series of the form \[ f(z) = {}_2F_1(a+b\sqrt{D},a-b\sqrt{D};c;z) \] where $a,b,c,D \in \QQ$. Such series have rational expansions yet, if $D$ is not square, then they are transcendental functions by \cite{beukers}. The \(N\)-integrality of hypergeometric series \(_mF_n\) with parameters from quadratic fields has been studied by Hong-Wang \cite{hong}, and specific cases of \(_2F_1\) with quadratic irrational parameters have appeared as solutions to a differential equation concerning the arithmetic of modular forms on \(\Gamma_0(2)\) in work by Gottesman \cite{gottesman}. Some of our results in Section \ref{s:quadratic} can be summarized as follows: \begin{thm} \label{t:main2} Let $f(z) \in \pseries{\QQ}{z}$ be a hypergeometric series as above, where $D$ is not a perfect square, and let $B$ be the set of primes where $f(z)$ is $p$-adically bounded. Then $B$ has a Dirichlet density that is at most $1/2$, and which is bounded below by explicit formulas described in Propositions \ref{prop1} and \ref{prop2}. If one assumes that the $p$-adic digits of quadratic irrationalities are normal, then the density bounds in Propositions \ref{prop1} and \ref{prop2} are exact formulas. \end{thm} In the rational case of Theorem \ref{t:main1}, our arguments rely on the periodic $p$-adic expansions of rational numbers. In the quadratic irrational case of Theorem \ref{t:main2}, the expansions are no longer periodic, and so normality serves as a convenient replacement for this property. Actually, to establish our conjectural density formulas in Section \ref{s:quadratic}, we only need to assume a result about $p$-adic expansions of quadratic irrationalities, Conjecture \ref{conj} below, that is implied by $p$-adic normality of digits. We do not know if Conjecture \ref{conj} is in fact equivalent to $p$-adic normality. The density in Theorem \ref{t:main2} is bounded above by $1/2$ for the simple reason that a series such as $f(z)$ can only be $p$-adically bounded for primes $p$ that split in $\QQ(\sqrt{D}$), cf. Proposition \ref{p:split}. The reason why there are two density formulas, cf. Propositions \ref{prop1} and \ref{prop2}, is because the formulas are different in the cases where $2a$ is integral or not. Since the trace of $a+b\sqrt{D}$ plays an important role in the proofs, whether the trace $2a$ has a truncating or periodic $p$-adic expansion affects the argument and the formulas. Section \ref{s:schwarz} is inspired by the Schwarz list for ${}_2F_1$, which essentially classifies series with density of bounded primes equal to one. For the case of series such as $f(z)$ above, a natural quadratic irrational analogue is to classify series where the density of bounded primes is $1/2$. We do not resolve this question, but we discuss some computations towards it, and also establish some upper-bounds that indicate that it should be reasonably rare for a series such as $f(z)$ to have a density of bounded primes equal to $1/2$. For example, in all of our examples, we only ever observed a density of $1/2$ when the field $K=\QQ(\sqrt{D})$ is imaginary quadratic. See Table \ref{tab:dvaryk} for some examples with $a$ and $c$ of small height. This rarity of large densities is in-line with the classical Schwarz list being rather sparse among all rational hypergeometric series. An interesting open question is how the irrational case extends to higher values of \(n\) for a general series \(_nF_{n-1}\) with irrational but algebraic hypergeometric parameters. For example, one could consider series \(_3F_2\) with a pair of quadratic conjugates in the numerator, along with a third rational numerator parameter, and a separate pair of conjugate quadratic irrationalities in the denominator. This will still have rational coefficients, and presumably also a density of bounded primes. Or, one could consider $_3F_2$ with three cyclic Galois conjugates of degree three as the numerator parameters --- again, is there a density of bounded primes for such series? While we find these questions interesting, the array of conceivable patterns only grows more vast as \(n\) increases, so it is difficult to imagine what sort of general result might exist. Nevertheless, the density results in the quadratic irrational case of ${}_2F_1$ are surprisingly clean, and they require a nice application of $p$-adic normality of algebraic numbers --- or its a priori weaker form of Conjecture \ref{conj} --- that could provide an impetus for work in the area of $p$-adic metric number theory. \section{Background} The rising factorials are defined for integers $k\geq 0$ as $(x)_0 = 1$ and otherwise \[ (x)_k \df x(x+1)\cdots (x+k-1). \] Let $\alpha = (\alpha_1,\ldots ,\alpha_{m})$ and $\beta = (\beta_1,\ldots, \beta_m)$ denote complex numbers, and assume that $\beta_j \not \in \ZZ_{\leq 0}$ for all $j=1,\ldots, n$. The corresponding \emph{generalized hypergeometric series}, or just hypergeometric series, ${}_mF_n(\alpha;\beta;z)$ is defined by the infinite series \[ {}_mF_n(\alpha;\beta;z) = \sum_{k\geq 0} \frac{(\alpha_1)_k\cdots (\alpha_m)_k}{(\beta_1)_k\cdots (\beta_n)_k} \frac{z^k}{k!}. \] These series satisfy hypergeometric differential equations, and the equations are regular singular exactly when $m=n+1$. The case of ${}_2F_1$ has a long history dating back at least to work of Gauss and Riemann. The finite monodromy groups of these equations were classified by Schwarz, and the case of ${}_mF_n$ was treated more recently by Beukers-Heckmann\cite{beukers}. Let $p$ be a rational prime, and let $\nu_p$ denote the corresponding $p$-adic valuation normalized so that $\nu_p(p)=1$. \begin{dfn} A hypergeometric series for parameters $(\alpha,\beta)$ is said to be \emph{rational} if ${}_mF_n(\alpha;\beta;z) \in \pseries{\QQ}{z}$. \end{dfn} \begin{rmk} Obviously, hypergeometric parameters such that $\alpha_i\in \QQ$ and $\beta_j \in \QQ$ for every $i$ and $j$ yield rational series, but there exist other examples such as $\alpha = (a+b\sqrt{2},a-b\sqrt{2})$ and $\beta = (c)$ for any rational numbers $a,b,c$ with $c \not \in \ZZ_{\leq 0}$. \end{rmk} \begin{dfn} A prime $p$ is said to be \emph{bounded} for a rational hypergeometric series if the $p$-adic valuations of the coefficients of ${}_mF_n(\alpha;\beta;z)$ are bounded below. Equivalently, there exists $N \in \ZZ$ such that \[ p^N{}_mF_n(\alpha;\beta;z) \in \pseries{\ZZ_p}{z}. \] Otherwise $p$ is said to be \emph{unbounded} for the given series. \end{dfn} In \cite{FGM} it was shown that if $\alpha,\beta,\gamma \in \QQ$ with $\gamma \not \in \ZZ_{\leq 0}$, then the set of primes such that the classical hypergeometric series ${}_2F_1(\alpha,\beta;\gamma;z)$ has bounded denominators has a natural density. In Theorem 4 of \cite{fetal}, precision was added to this result in the form of an explicitly computable formula for this natural density: \begin{thm}\label{t:fetal} Let $a,b$ and $c$ denote rational numbers with $0 < a,b,c, < 1$, where $c \neq a,b$. Let $N$ denote the least common multiple of the denominators of $a-1$, $b-1$ and $c-1$, and define \[ B(a,b;c) \df \{u \in (\ZZ/N\ZZ)^\times \mid \textrm{ for all } j \in \ZZ,~ \{-u^jc\} \leq \max(\{-u^ja\},\{-u^jb\})\}. \] Then for all primes $p > N$, the series ${}_2F_1(a,b;c;z)$ is $p$-adically bounded if and only if $p$ is congruent to an element of $B(a,b;c)$ mod $N$. Therefore, the set of bounded primes for ${}_2F_1(a,b;c;z)$ has a density equal to $\tfrac{\abs{B(a,b;c)}}{\phi(N)}$. \end{thm} \begin{proof} This is Theorem 4 of \cite{fetal}. \end{proof} Our goals below are twofold: \begin{enumerate} \item generalize Theorem \ref{t:fetal} to the case of general $m$ and $n$; \item generalize Theorem \ref{t:fetal} when $m=2$, $n=1$, to the case of certain rational parameters living in quadratic number fields. \end{enumerate} In the case of quadratic fields, our density result is conditional on a seemingly plausible conjecture about the normality of the $p$-adic expansions of irrational algebraic numbers at split primes. Such a conjecture is a substitute for the lack of periodicity of the $p$-adic expansions of the hypergeometric parameters in these cases. To study the $p$-adic boundedness of hypergeometric series, following \cite{FGM} we introduce: \begin{dfn} \label{d:carries} Let $c_p$ denote the function \[c_p \colon \ZZ_p\times \ZZ_{\geq 0} \to \ZZ_{\geq 0} \cup \{\infty\}\] where $c_p(x,k)$ computes the number of carries needed to evaluate the $p$-adic sum $x+k$ using the usual add-and-carry method. \end{dfn} In \cite{FGM} it was observed that this function is locally constant in $x$, and using a classical result of Kummer, one in fact has the following: \begin{thm}[Kummer] \label{t:kummer} One has $\nu_p\binom{x+k}{k} = c_p(x,k)$. \end{thm} \begin{proof} See Theorem 2.5 of \cite{FGM}. \end{proof} The following result is a slight generalization of Theorem 3.4 of \cite{FGM}, to the case of general $m$ and $n$. \begin{cor} \label{c:FGM} Let $\alpha$ and $\beta$ denote hypergeometric parameters contained in $\ZZ_p$, such that no $\beta$ parameter is a negative integer. Then if we write ${}_m F_n(\alpha;\beta;z) = \sum_{k\geq 0}A_kz^k$ for $A_k \in \QQ_p$, we have \[ \nu_p(A_k) = \sum_{i=1}^mc_p(\alpha_i-1,k)-\sum_{j=1}^n c_p(\beta_j-1,k) + (m-n-1)\nu_p(k!). \] \end{cor} \begin{proof} By a direct computation one sees that \[ A_k=\frac{\prod_{i=1}^m\binom{\alpha_i-1+k}{k}}{\prod_{j=1}^n\binom{\beta_j-1+k}{k}}\cdot (k!)^{m-n-1} \] Therefore, this Corollary follows immediately from Theorem \ref{t:kummer}. \end{proof} \begin{rmk} \label{r:nontrivial} Since $\nu_p(k!) = O(k)$, while $c_p(x,k) = O(\log k)$ away from the poles $x$ of $c_p(x,k)$, one can use Corollary \ref{c:FGM} to prove that for rational hypergeometric parameters, all primes are bounded if $m > n+1$, whereas no primes are bounded if $m < n+1$. Thus, as far as densities go, for rational parameters the only interesting case is when $m=n+1$ and the corresponding hypergeometric differential equation is regular singular. \end{rmk} \section{Rational parameters} \label{s:rational} This section considers the case of rational parameters $\alpha,\beta$. In light of Remark \ref{r:nontrivial}, we restrict also to the case of ${}_nF_{n-1}$. We begin by recalling some facts on the $p$-adic expansions of rational numbers, which are always eventually periodic. In fact, if $a$ is a rational number satisfying $-1 < a < 0$ and $a \in \ZZ_p^\times$, then the $p$-adic expansion of $a$ is purely periodic, of period equal to the multiplicative order of $p$ modulo the denominator of $a$. Since the density of bounded primes for a given set of hypergeometric parameters only depends on the rational parameters mod $\ZZ$, we will normalize our parameters to lie in the interval $(0,1)$. We state the following lemma on the digits of our hypergeometric parameters in a way which will make the application of Corollary \ref{c:FGM} more straightforward. \begin{lem} \label{l:digitformula} Let $a = \tfrac uv$ denote a rational number with $\gcd(u,v) =1$ satisfying $0 < a < 1$, and let $p$ denote a prime such that $a-1$ is a $p$-adic unit. Let $M$ denote the multiplicative order of $p \pmod{v}$, and let the periodic expansion of $a-1$ be denoted as: \[ a-1 = \overline{\alpha_0\alpha_1\cdots \alpha_{M-1}}. \] Then for each $j$ with $0\leq j \leq M-1$ we have \[ \alpha_j = \floor{\{-p^{M-1-j}a\}p}. \] In particular, if $p > v$, then each $\alpha_j$ is nonzero. \end{lem} \begin{proof} Most of this is Lemma 2.4 of \cite{FGM}. All that remains to observe is the final claim that each $\alpha_j$ is nonzero. For this, begin with the observation that \[ \{-p^{M-1-j}a\} -\tfrac 1p < \tfrac{\alpha_j}{p} \leq \{-p^{M-1-j}a\}. \] Since $p$ is coprime to $a$, then $\{-p^{M-1-j}a\}$ is a rational number of the form $k/v$ for some integer $1\leq k \leq v-1$. Hence, \[ 0<\frac{1}{v}-\frac 1p < \frac{\alpha_j}{p} \] as claimed. \end{proof} \begin{dfn} A prime $p$ is \emph{good} for hypergeometric parameters $\alpha$, $\beta$ provided that all of $\alpha_i-1$ and $\beta_i-1$ are $p$-adic units, and such that $p$ does not divide any of the differences $\alpha_i-\beta_j$. \end{dfn} Note that as long as no $\alpha_i$ is equal to a $\beta_j$, a harmless hypothesis that we enforce below, then the set of good primes for a set of hypergeometric parameters contains all primes that are large enough. Hence, as far as densities of bounded primes go, it is harmless to restrict to considering good primes only. Let $p$ denote a good prime for some rational hypergeometric parameters, and let $M$ denote the order of $p$ modulo the least common multiple of the denominators of the hypergeometric parameters. Then by Lemma \ref{l:digitformula}, the $p$-adic periods of each parameter all divide $M$. \begin{dfn} \label{d:numeratormajorized} Let $\alpha = (\alpha_i)_{i=1}^n$ and $\beta = (\beta_i)_{i=1}^{n-1}$ denote rational hypergeometric parameters, and let $N$ denote the least common multiple of the denominators of the parameters. Fix an invertible congruence class $u\pmod{N}$. Then a set of hypergeometric parameters is said to be \emph{numerator majorized} for this congruence class provided that for all integers $j\geq 0$, there exists a permutation $\sigma_j \in S_n$ such that \[ \{-u^j\beta_i\} \leq \{-u^j\alpha_{\sigma_j(i)}\} \] for all $i=1,\ldots, n-1$. \end{dfn} Definition \ref{d:numeratormajorized} is reminiscient of the interlacing of roots of unity condition that appears in the study of finite monodromy in \cite{Landau} and \cite{beukers}. For a given set of parameters, it is a finite computation to determine which congruence classes are numerator majorized or not. \begin{ex} \label{ex1} Consider the hypergeometric series \[ {}_3F_2(\tfrac 15, \tfrac 25, \tfrac 35; \tfrac 45, \tfrac 12;z). \] In this case $N=10$, and so we only need to test the numerator majorization condition for $j=0,1,2,3$, since $\phi(10) = 4$, and thus $p^4 \equiv 1 \pmod{10}$. \begin{table} \renewcommand{\arraystretch}{1.5} \begin{tabular}{c\|\|c\|c\|c\|\|c\|c} $p\pmod{10}$ & $\tfrac 15$ & $\tfrac 25$ & $\tfrac 35$ & $\tfrac 45$ & $\tfrac 12$\\ \hline $1$ &$\left[\frac{4}{5}, \frac{4}{5}, \frac{4}{5}, \frac{4}{5}\right]$&$\left[\frac{3}{5}, \frac{3}{5}, \frac{3}{5}, \frac{3}{5}\right]$&$\left[\frac{2}{5}, \frac{2}{5}, \frac{2}{5}, \frac{2}{5}\right]$&$\left[\frac{1}{5}, \frac{1}{5}, \frac{1}{5}, \frac{1}{5}\right]$&$\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right]$\\ $3$ &$\left[\frac{4}{5}, \frac{2}{5}, \frac{1}{5}, \frac{3}{5}\right]$&$\left[\frac{3}{5}, \frac{4}{5}, \frac{2}{5}, \frac{1}{5}\right]$&$\left[\frac{2}{5}, \frac{1}{5}, \frac{3}{5}, \frac{4}{5}\right]$&$\left[\frac{1}{5}, \frac{3}{5}, \frac{4}{5}, \frac{2}{5}\right]$&$\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right]$\\ $7$ &$\left[\frac{4}{5}, \frac{3}{5}, \frac{1}{5}, \frac{2}{5}\right]$&$\left[\frac{3}{5}, \frac{1}{5}, \frac{2}{5}, \frac{4}{5}\right]$&$\left[\frac{2}{5}, \frac{4}{5}, \frac{3}{5}, \frac{1}{5}\right]$&$\left[\frac{1}{5}, \frac{2}{5}, \frac{4}{5}, \frac{3}{5}\right]$&$\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right]$\\ $9$ &$\left[\frac{4}{5}, \frac{1}{5}, \frac{4}{5}, \frac{1}{5}\right]$&$\left[\frac{3}{5}, \frac{2}{5}, \frac{3}{5}, \frac{2}{5}\right]$&$\left[\frac{2}{5}, \frac{3}{5}, \frac{2}{5}, \frac{3}{5}\right]$&$\left[\frac{1}{5}, \frac{4}{5}, \frac{1}{5}, \frac{4}{5}\right]$&$\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right]$ \end{tabular} \caption{The values $\{-p^j\alpha_i\}$ and $\{-p^j\beta_i\}$ for the hypergeometric parameters $\alpha = (\tfrac 15,\tfrac 25, \tfrac 35)$ and $\beta = (\tfrac 45, \tfrac 12)$.} \label{t1} \end{table} Table \ref{t1} shows that the class $1 \pmod{10}$ is numerator majorized for these parameters. The class $3\pmod{10}$ fails the numerator majorization condition for $j=2$, as the third entry of the list for the denominator parameter $\tfrac 45$ in that case is $\tfrac 45$, which is maximal and thus can't be numerator majorized. Similarly, $7 \pmod{10}$ and $9\pmod{10}$ fail the numerator majorization condition in this example. \end{ex} \begin{lem} \label{l:equivalent} Let $\alpha = (\alpha_i)_{i=1}^n$ and $\beta = (\beta_i)_{i=1}^{n-1}$ denote rational hypergeometric parameters taken from the interval $(0,1)$, and let $N$ denote the least common multiple of the denominators of the parameters. Then these parameters are numerator majorized for a a congruence class $u\pmod{N}$ if and only if for all large enough primes $p\equiv u \pmod{N}$, if we write the $p$-adic digits of $\alpha_i-1$ as $\alpha_{i,j}(p)$ and similarly for $\beta_i-1$ and $\beta_{i,j}(p)$, then for all integers $j\geq 0$ there exists a permutation $\sigma_j \in S_n$ such that \[ \beta_{i,j}(p) \leq \alpha_{\sigma_j(i),j}(p) \] for all $i=1,\ldots, n-1$. \end{lem} \begin{proof} For all primes $p\equiv u \pmod{N}$, being numerator majorized is equivalent to the condition \[ \{-p^j\beta_i\}p \leq \{-p^j\alpha_{\sigma_j(i)}\}p. \] If $p > 1/(\alpha_u-\beta_v)$ for all $u,v$, then this inequality is \emph{equivalent} to the inequality obtained by taking floors above: \[ \floor{\{-p^j\beta_i\}p} \leq \floor{\{-p^j\alpha_{\sigma_j(i)}\}p}. \] Therefore, the equivalence of these two formulations of numerator majorization follows from Lemma \ref{l:digitformula}. \end{proof} \begin{thm} \label{t:rational} Let $\alpha = (\alpha_i)_{i=1}^n$ and $\beta = (\beta_i)_{i=1}^{n-1}$ denote hypergeometric parameters, and let $N$ denote the least common multiple of the denominators of the parameters, and fix an invertible congruence class $u\pmod{N}$. Then the corresponding hypergeometric series ${}_nF_{n-1}(\alpha,\beta,z)$ is $p$-adically bounded for all sufficiently large primes $p\equiv u \pmod{N}$ if and only if the hypergeometric parameters are numerator majorized with respect to the congruence class $p \equiv u\pmod{N}$. \end{thm} \begin{proof} Assume that $p > \max_{i,j}(\alpha_i-\beta_j)^{-1}$ so that Lemma \ref{l:equivalent} holds for such primes. Assume that the series is numerator majorized with respect to $u \pmod{N}$, so that the $p$-adic digits are numerator majorized by Lemma \ref{l:equivalent}. Now Corollary \ref{c:FGM} shows that the negative contributions for carries from terms $c_p(\beta_j-1,k)$ are compensated for by carries $c_p(\alpha_i-1,k)$. Hence $\nu_p(a_k) \geq 0$ for such primes $p$, so that in fact ${}_nF_{n-1}(\alpha,\beta,z) \in \pseries{\ZZ_p}{z}$. This proves one direction of the Theorem. Assume conversely that the parameters are not numerator majorized with respect to a congruence class $u\pmod{N}$. This means there exists a digit index $j$ such that for every $\sigma \in S_n$, there exists a hypergeometric parameter index $i$ such that \[ \beta_{i,j}(p) > \alpha_{\sigma(i),j}(p). \] First suppose that there is an index $i$ such that $\beta_{i,j}(p) > \alpha_{k,j}(p)$ for all $k$. For each integer $A \geq 0$, consider the term of the hypergeometric series indexed by \[k_A \df (p-\beta_{i,j}(p))\sum_{u=0}^Ap^{j+u\phi(N)}.\] Since $\phi(N)$ is a common period for the $p$-adic expansions of all of the hypergeometric parameters, we see that $c_p(\beta_i-1,k_A) = A+1$, but otherwise by construction $c_p(\beta_u-1,k_A) = 0$ for $u\neq i$ and $c_p(\alpha_v-1,k_A) = 0$ for all $v$. Therefore Corollary \ref{c:FGM} shows that in this case $v_p(a_{k_A}) \geq A+1$, where the $a_{k}$ denote the coefficients of ${}_nF_{n-1}(\alpha,\beta,z)$. This treats the case where one of the $\beta_{i,j}(p)$ digits is maximal among all of the $j$-th coefficients of the given set of hypergeometric parameters. Otherwise, we can without loss of generality assume that $\alpha_{1,j}(p) \geq \cdots \geq \alpha_{n,j}(p)$ and similarly $\beta_{1,j}(p) \geq \cdots \geq \beta_{n-1,j}(p)$. We have treated in the previous paragraph the case when $\alpha_{1,j}(p) < \beta_{1,j}(p)$, and since numerator majorization fails, there must exist a smallest index $i$ such that $\alpha_{u,j}(p) \geq \beta_{u,j}(p)$ for $1\leq u \leq i-1$ but $\beta_{i,j}(p) > \alpha_{i,j}(p)$. Now defining $k_A$ as above, this time we see that carries coming from the $\beta_{u,j}(p)$ are compensated for by carries coming from $\alpha_{u,j}(p)$ for $1 \leq u \leq i-1$, but we have $c_p(\alpha_{i,j}(p),k_A) \geq A+1$ and $c_p(\beta_{i,j}(p),k_A) =0$. Therefore we again find that $\nu_p(a_{k_A}) \leq -A-1$, and so in all cases, if the parameters are not numerator majorized for $u \pmod{N}$, then the corresponding series is not $p$-adically bounded for all primes $p > \max_{i,j}(\alpha_i-\beta_j)^{-1}$ satisfying $p\equiv u\pmod{N}$. This concludes the proof of the theorem. \end{proof} \begin{rmk} \label{r:explicit} We emphasize that this proof shows that if the parameters are numerator majorized for $u\pmod{N}$, then for all primes $p > \max_{i,j}(\alpha_i-\beta_j)^{-1}$ satisfying $p\equiv u \pmod{N}$, we have ${}_nF_{n-1}(\alpha,\beta,z) \in \pseries{\ZZ_p}{z}$. \end{rmk} \begin{cor} \label{c:rationaldensityformula} Let $\alpha = (\alpha_i)_{i=1}^n$ and $\beta = (\beta_i)_{i=1}^{n-1}$ denote hypergeometric parameters, and let $N$ denote the least common multiple of the denominators of the parameters. Then for every prime $p > \max_{i,j}(\alpha_i-\beta_j)^{-1}$, the series ${}_nF_{n-1}(\alpha,\beta,z)$ is $p$-adically bounded if and only if $p$ is congruent to an element of the following set: \[ B(\alpha;\beta) =\{u \in (\ZZ/N\ZZ)^\times \mid \alpha \textrm{ and } \beta \textrm{ are numerator majorized for } u\}. \] In fact, for such primes $p$, the series ${}_nF_{n-1}(\alpha,\beta,z)$ is $p$-adically integral. In particular, the set of bounded primes for $\alpha,\beta$ has a Dirichlet density equal to $\tfrac{\abs{B(\alpha;\beta)}}{\phi(N)}$. \end{cor} \begin{proof} This follows directly from Theorem \ref{t:rational} and Remark \ref{r:explicit}. \end{proof} \begin{rmk} \label{r:cyclic} Notice that the set $B(\alpha;\beta)$, if nonempty, is a union of cyclic subgroups of $(\ZZ/N\ZZ)^\times$. \end{rmk} \begin{ex} \label{ex2} In Example \ref{ex1} we saw that for \[ {}_3F_2(\tfrac 15, \tfrac 25, \tfrac 35; \tfrac 45, \tfrac 12;z) \] we have $N = 10$ and $B(\tfrac 15, \tfrac 25, \tfrac 35; \tfrac 45, \tfrac 12) = \{1\} \subseteq (\ZZ/10\ZZ)^\times$. Therefore, the bounded primes $p > 10$ for this series are precisely those satisfying $p\equiv 1 \pmod{10}$, so that the density of bounded primes in this case is $\tfrac 14$. In fact, this series is $p$-integral for such primes, so that the only primes appearing in the denominators of this series satisfy $p \leq 7$ or $p\equiv 3,7,9 \pmod{10}$. \end{ex} The following result generalizes Theorem 4.14 from \cite{FGM}. \begin{cor} Let $\alpha = (\alpha_i)_{i=1}^n$ and $\beta = (\beta_i)_{i=1}^{n-1}$ denote hypergeometric parameters, and assume that they are ordered in increasing order of their fractional parts. Then the density of bounded primes for ${}_nF_{n-1}(\alpha,\beta,z)$ is zero if and only if there exists an index $i$ such that \[ \{\beta_i\} < \{\alpha_i\}. \] \end{cor} \begin{proof} By Corollary \ref{c:rationaldensityformula} and Remark \ref{r:cyclic}, the density of bounded primes is zero if and only if $1 \not \in B(\alpha,\beta)$. The numerator majorization failure for $u=1$ is equivalent to the existence of an index $i$ with $\{-\beta_i\} > \{-\alpha_i\}$, or equivalently, $\{\beta_i\} < \{\alpha_i\}$ as claimed. \end{proof} \section{Quadratic irrational parameters} \label{s:quadratic} Let now $D \in \QQ$ be a nonsquare rational, and let $K = \QQ(\sqrt{D})$. Notice that for $a,b,c\in\QQ$ with $c \not \in \ZZ_{\leq 0}$, the series \[ F_D(a,b,c,z) \df {}_2F_1(a+b\sqrt{D},a-b\sqrt{D};c;z) \] has rational coefficients. We will assume $b \neq 0$, so that $F_D$ does not have rational parameters. \begin{prop} \label{p:split} For all but finitely many primes $p$, a necessary condition for $F_D(a,b,c,z)$ to be $p$-adically bounded is that $p$ splits in $K$. \end{prop} \begin{proof} Let $A_n$ denote the $n$th coefficient of $F_D(a,b,c,z)$, and let $P(x)$ denote the minimal polynomial of $a+b\sqrt{D}$: \[P(x) = x^2 - 2ax + a^2-b^2D.\] Then we find that \[ A_n = \frac{(a+b\sqrt{D})_n(a-b\sqrt{D})_n}{(c)_nn!} =\frac{\prod_{j=0}^{n-1}P(-j)}{(c)_nn!}. \] Since $a+b\sqrt{D}$ generates $K/\QQ$, and $P(x)$ is its minimal polynomial, with at most finitely many exceptions only rational primes that split in $K$ divide the values $P(j)$ in the numerator of $A_n$. On the other hand, the denominator is $(c)_nn!$, and all but finitely many exceptions depending only on $c$ divide this denominator to larger and larger powers as $n$ grows. Therefore, a necessary condition to have cancellation in $A_n$ for large enough primes is that $p$ be split in $K$. \end{proof} \begin{rmk} We say \(u\) splits in \(K\) if primes equivalent to \(u\mod M\) split in \(K\). This is well defined as the splitting condition of a prime in a number field \(K\) relies on the congruence class of the prime \(\mod \Delta_K\), where \(\Delta_K\) denotes the discriminant of the number field \(K\). Since \(M = \lcm(\denom(a),\denom(c),\Delta_K)\), all primes equivalent to \(u\mod M\) split in \(K\) if any prime equivalent to \(u\mod M\) splits in \(K\). \end{rmk} Let $S$ denote the set of rational primes that split in $K$. Notice that Corollary \ref{c:FGM} still applies to $F_D(a,b,c,z)$ for all but finitely many primes $p \in S$. In the rational case we are able to leverage the periodicity of $p$-adic expansions in conjunction with Corollary \ref{c:FGM} to produce unbounded denominators. But the $p$-adic expansions of the irrational hypergeometric parameters $\alpha = (a+b\sqrt{D},a-b\sqrt{D})$ are no longer periodic. However, it turns out that we can still study the density of bounded primes in a similar way if these $p$-adic irrational numbers have sufficiently randomly distributed digits. To begin, we recall the following definition. \begin{dfn} \label{d:normal} A $p$-adic integer $\alpha \in \ZZ_p$ is said to be \emph{normal} provided that every sequence of $p$-adic digits occurs equally often in the following asymptotic sense: let $\alpha = \sum_{n\geq 0}\alpha_np^n$ be the $p$-adic expansion of $\alpha$ for $\alpha_n \in \{0,1,\ldots, p-1\}$ for all $n$, and let $B = B_1B_2\cdots B_{m}$ be a bit string, where $B_j \in \{0,1,\ldots, p-1\}$ for all $j$. Then \[ \lim_{N\to \infty}\frac{\abs{\{j\leq N \mid \alpha_j\alpha_{j+1}\cdots \alpha_{j+m-1} = B\}}}{N} = \frac{1}{p^m}. \] This should hold for any choice of $p$-adic bit string $B$. \end{dfn} Normality is an established property over the real numbers. For more information see a text on metric number theory, such as \cite{MNT}. There exist $p$-adic transcendental numbers that are not normal. Computations suggest that $p$-adic numbers that are algebraic over $\QQ$ of degree $d\geq 2$ may always be normal, though this is currently an open question. The following conjecture may be regarded as a weakened form of normality. \begin{conj} \label{conj} Let $\alpha \in \overline \QQ\setminus \QQ$, and let $S$ be the set of rational primes that are totally split in $\QQ(\alpha)$. Let $r$ and $s$ be integers, and let $u,v$ be real numbers with $0\leq u < v \leq 1$. Let $\alpha_n(p,\sigma)$ denote the $n$th $p$-adic digit of $\alpha$ for $p \in S$ with respect to the field embedding $\sigma \colon \QQ(\alpha) \to \QQ_p$. Then for all but finitely many primes $p \in S$ (where the finite exceptional set depends on $\alpha$, $r$, $s$, $u$ and $v$), the digit $\alpha_n(p,\sigma)$ is contained in the interval $(u(p-1),v(p-1))$ for infinitely many integers $n\geq 0$ in the arithmetic progression $n = rm+s$. \end{conj} \begin{prop} \label{p:normalimpliesconj} Suppose that $\alpha \in \overline\QQ \setminus \QQ$ has the property that $\sigma(\alpha)$ is $p$-adically normal for any embedding $\sigma \colon \QQ(\alpha) \to \QQ_p$. Then Conjecture \ref{conj} holds for $\alpha$. \end{prop} \begin{proof} We will show that given an arithmetic progression $n = rm+s$, then \emph{any} digit occurs infinitely often in the sequence $(\alpha_{rm+s}(p,\sigma))_{m\geq 0}$ of $p$-adic digits, which clearly implies Conjecture \ref{conj}. Suppose that a digit $b$ only occurs finitely many times along this progression, and let $B$ be the string which is $m$ copies of $b$. Then the quantity \[ \abs{\{j\leq N \mid \alpha_j(p,\sigma)\alpha_{j+1}(p,\sigma)\cdots \alpha_{j+m-1}(p,\sigma) = B\}} \] is bounded absolutely, independently of $N$, since at least one of the terms $\alpha_{j+i}(p,\sigma)$ for $0\leq i \leq m-1$ must meet the sequence $(\alpha_{rm+s}(p,\sigma))_{m\geq 0}$. As this contradicts the normality of $\sigma(\alpha)$, it thus verifies our claim that every digit appears infinitely often in $(\alpha_{rm+s}(p,\sigma))_{m\geq 0}$. This concludes the proof. \end{proof} Before establishing our density results, we reformulate the condition of $p$-adic unboundedness for $F_D$ slightly: \begin{thm} Let $a,b,c$ denote rational numbers as above, and write the $p$-adic digits of $a+b\sqrt{D}-1$, $a-b\sqrt{D}-1$, and $c-1$ as $\alpha_j(p)$, $\alpha_j'(p)$ and $\gamma_j(p)$, respectively. Then the following are equivalent for all but finitely many primes that split in $K$: \begin{enumerate} \item the inequality $\gamma_j(p) > \max(\alpha_j(p),\alpha'_j(p))$ holds for infinitely many $j$; \item $F_D(a,b;c;z)$ has $p$-adically unbounded coefficients. \end{enumerate} \end{thm} \begin{proof} First suppose that (1) holds and let $j_1,j_2,\ldots$ denote the sequence of indices where $\gamma_j(p) > \max(\alpha_j(p),\alpha'_j(p))$. If we set \[ m_r=\sum_{k=1}^r (p-\gamma_{j_k}(p))p^{j_k} \] and let $A_n$ denote the $n$th coefficient of $F_D(a,b;c;z)$, then we find by Corollary \ref{c:FGM} that $\nu_p(A_{m_r}) \leq -r$. Thus (1) implies (2). Conversely, if (2) holds, then by Corollary \ref{c:FGM} there are infinitely many terms where the $j$th digit of $c-1$ is larger than the $j$th digits of both $a\pm b\sqrt{D}-1$. \end{proof} Before establishing our density results we prove a simple lemma: \begin{lem} \label{l:digitsum} Let $x,y\in\ZZ_p$ and set $z = x+y$. Then \[ x[j] + y[j] \geq z[j]-1. \] \end{lem} \begin{proof} Since $p$-adic addition involves carries, we deduce that \[ x[j]+y[j]= \begin{cases} z[j]& \textrm{no carries at digits $j-1$ and $j$,}\\ z[j]-1 & \textrm{carry at $j-1$, not at $j$,}\\ z[j] + \alpha p& \textrm{carry at $j$, not at $j-1$,}\\ z[j]+\alpha p-1& \textrm{carries at $j-1$ and $j$,} \end{cases} \] where $\alpha \in \{1,2\}$. Therefore the lemma holds for elementary reasons. \end{proof} The density results differ depending on whether $2a$ is an integer or not. We first treat the case where $2a$ is an integer: \begin{prop} \label{prop1} Let $a$ be rational number with $2a \in \ZZ$, let $b$ be rational and nonzero, and let \( c\) be rational and different from a negative integer. Let $K = \QQ(\sqrt{D})$ with $D \in \ZZ$ not a perfect square. Let $M$ be the least common multiple of the denominator of \(c\), and twice the discriminant of \(K\). Consider the hypergeometric series defined by: \[ F_D= {}_2F_1\left(a+b\sqrt D, a-b\sqrt D; c ; z\right) \] Then there exists a density $\delta$ of $p$-adically bounded primes for $F_D$. If we set \[ B_K(a;c) \df \{u\in \left(\ZZ/M\ZZ\right)^\times {\vert} \{-u^jc\} \leq \{-u^j\tfrac 12\} \text{ for all j and \(u\) splits in K}\}, \] then $\delta$ satisfies \[ \frac{\abs{B_K(a;c)}}{\phi(M)} \leq \delta \leq \frac 12. \] Furthermore, if Conjecture \ref{conj} holds, then $\delta = \frac{\abs{B_K(a;c)}}{\phi(M)}$. In particular, the density is conjecturally independent of $b$. \end{prop} \begin{proof} In order to apply Lemma \ref{l:digitformula} below, we need to adjust our rational numbers $a$ and $c$ so that they lie in the range $0 < a,c < 1$. This can be done using classical transformation formulas for ${}_2F_1$ relating series whose parameters differ by integers. These transformation formulas will change denominators of the series at finitely many primes, so that it it harmless to assume that $0 < a,c <1$. Henceforth we assume this throughout the proof. In particular, we now take $a = 1/2$. Since $M$ is even, any integer \(u\) coprime to \(M\) must be odd, the inequality in the definition of $B_K$ simplifies to \[ \{-u^j c\} \le \frac 12. \] By Lemma \ref{l:digitformula}, this implies that if $p$ is congruent to an element of $B_K(a,c)$ mod $M$, then we have \begin{equation} \label{eq:cdigitsbound} (c-1)[j] \le \tfrac{p-1}2 \end{equation} for all $j$, where now $x[j]$, for a $p$-adic integer $x$, denotes the $j$th $p$-adic digit of $x$. Apply Lemma \ref{l:digitsum} with $x = \tfrac 12 + b\sqrt{D}-1$ and $y=\tfrac 12 - b\sqrt{D}-1$. We deduce that \[ (\tfrac 12 + b\sqrt{D}-1)[j]+(\tfrac 12 - b\sqrt{D}-1)[j] \geq (-1)[j]-1 = p-2. \] In particular, at least one of $(\tfrac 12 \pm b\sqrt{D}-1)[j]$ is $\geq (p-1)/2$. But then it follows by Corollary \ref{c:FGM} and equation \eqref{eq:cdigitsbound}, as in the arguments of \cite{FGM}, that carries arising from the denominator parameter $c$ are balanced by the carries arising from the numerator parameters $a\pm b\sqrt{D}$. Therefore, $F_D$ is $p$-adically bounded for all primes congruent to elements of $B_D(a;c)$. Now assume Conjecture \ref{conj}. For every prime $p$ large enough and such that $p$ is not congruent to an element of $B_K(a;c)$ we will show that $F_D$ is $p$-adically unbounded. If $p$ is such a prime, where the size bound will be identified below, then by definition of $B_K(a;c)$, there exists an integer $j$ such that \[ \{-p^jc\} > \frac 12. \] If $A$ is the multiplicative order of $p \pmod{M}$, we then obtain \[ \{-p^{j+nA}c\} > \frac 12 \] for all integers $n$, since the left side is independent of $n$. Moreover, the left hand side of this inequality is an element of $(1/M)\ZZ$, and so we deduce that: \[ \{-p^{j+nA}c\} \geq \frac 12 + \frac{1}{M} \] for all integers $n$. Now, notice by Lemma \ref{l:digitformula} that $c-1$ has a periodic $p$-adic expansion of period dividing $A$, since the denominator of $c$ divides $M$. In particular, the terms \[(c-1)[j+nA]\] are independent of $n$. Also, if $p$ is large enough, Lemma \ref{l:digitformula} and the inequality above imply that \[ (c-1)[j+nA] = \floor{\{-p^{j+nA}c\}p} \geq \floor{\frac p2+\frac pM} \geq \frac{p-1}{2}+\floor{\frac pM} \] That is, for all $n\geq 0$ we have shown that \begin{equation} \label{eq:cboundprop1} \floor{\frac pM} \leq (c-1)[j+nA] - \frac{p-1}{2}. \end{equation} Now, Conjecture \ref{conj} yields an infinite subsequence $(x_n)$ of the arithmetic progression $j+nA$ such that \begin{equation} \label{eq:cboundprop1,2} \frac{p-1}{2}-\frac 12 \floor{\frac pM} < (\tfrac 12 +b\sqrt{D}-1)[x_n] < \frac{p-1}{2}+\frac 12 \floor{\frac pM} \end{equation} for all $n\geq 0$ as long as $p$ is large enough. Notice that \[ (\tfrac 12 -b\sqrt{D}-1)[m] = p-1-(\tfrac 12 +b\sqrt{D}-1)[m] \] for all $m\geq 0$, since \[ (\tfrac 12 -b\sqrt{D}-1) + (\tfrac 12 +b\sqrt{D}-1) = -1 = \sum_{n\geq 0} (p-1)p^n. \] If we combine this observation with equation \eqref{eq:cboundprop1,2} we deduce likewise: \begin{equation} \label{eq:negativebound} \frac{p-1}{2}-\frac 12 \floor{\frac pM} < (\tfrac 12 -b\sqrt{D}-1)[x_n] < \frac{p-1}{2}+\frac 12 \floor{\frac pM} \end{equation} Now equations \eqref{eq:cboundprop1}, \eqref{eq:cboundprop1,2} and \eqref{eq:negativebound} yield \[ \max\left((\tfrac12+b\sqrt D-1)[x_n],(\tfrac12-b\sqrt D-1)[x_n] \right) < (c-1)[x_n] \] for all $n\geq 0$. Using this, it is now straightforward to construct a sequence of indices along which the coefficients of $F_D$ go to infinity in the $p$-adic absolute value, similarly to the argument in the proof of Theorem \ref{t:rational}. This concludes the proof. \end{proof} \begin{cor} Assume Conjecture \ref{conj}. Then for a set of parameters with \(2a\in\ZZ \), the series $F_D$ has a Dirichlet density of bounded primes equal to $0$ if and only if \(c<\tfrac12\). \end{cor} \begin{proof} As the set of bounded primes is a union of cyclic subgroups of $(\ZZ/M\ZZ)^\times$, the set of bounded primes is empty if and only if \(1\) is not in the bounded prime set. Thus, the Corollary follows from Proposition \ref{prop1}. \end{proof} Now we treat the case where $2a \not \in \ZZ$. \begin{prop} \label{prop2} Given two rational parameters $a$ and $c$ where $ a \neq \frac{1}{2}$, and a quadratic number field \(K\) defined by $K = \QQ(\sqrt{D})$, taking M to be the least common multiple of the denominators of \(a\), \(c\) and the discriminant of \(K\), consider the hypergeometric series defined by: \[ _2F_1(a+\sqrt D, a-\sqrt D,c ; z) := \sum_{n=0}^\infty \frac{(a+\sqrt D)_n(a-\sqrt D)_n}{(c)_n n!}z^n. \] The following set provides a lower bound on the density of bounded primes in the denominators of the series: \[ B_K(a;c) \df \{u\in \left(\ZZ/M\ZZ\right)^\times {\vert} \{-u^jc\}\leq\tfrac 12\{-2u^ja\} \text{ for all j, and \(u\) splits in K}\}. \] This set is \emph{equal} to the set of all equivalence classes such that $F_D$ is $p$-adically bounded for primes in the corresponding congruence if Conjecture \ref{conj} holds. \end{prop} \begin{proof} Let $p$ be a prime that is large enough, so that $p$ is congruent mod $M$ to an element of $B_K(a;c)$. We will show that our hypergeometric series is $p$-adically bounded for all such primes. Thus, we are assuming that $p$ satisfies \[ \{-p^jc\}\leq \tfrac 12\{-2p^ja\} \] for all $j\in \ZZ$. Multiply by $p$, take floors, and use both $\lfloor \tfrac 12 x\rfloor \leq \tfrac 12 \lfloor x\rfloor$ and Lemma \ref{l:digitformula}: \[ (c-1)[j] \leq \lfloor \tfrac 12\{-2p^ja\}p\rfloor \leq \tfrac 12 \lfloor\{-2p^ja\}p\rfloor \leq\tfrac 12((2a-1)[j]). \] That is, we are assuming that for all $j$ we have \begin{equation} \label{eq:firstdirection} (c-1)[j] \leq \tfrac 12 ((2a-1)[j]). \end{equation} Now, by Lemma \ref{l:digitsum} with $x=a+b\sqrt{D}-1$ and $y=a-b\sqrt{D}-1$, so that $z = x+y = 2a-2$, we deduce that \[ (a+b\sqrt{D}-1)[j]+(a-b\sqrt{D}-1)[j] \geq (2a-2)[j]-1 \] for all $j$. It follows that for all $j$, \[ \max ((a+b\sqrt{D}-1)[j],(a-b\sqrt{D}-1)[j]) \geq \tfrac 12((2a-2)[j]-1) \] For all but finitely many $j$ we have $(2a-2)[j] = (2a-1)[j]$, and so for all but finitely many $j$ we have \[ \max ((a+b\sqrt{D}-1)[j],(a-b\sqrt{D}-1)[j]) \geq \tfrac 12((2a-1)[j]-1). \] Combining this with inequality \eqref{eq:firstdirection}, we obtain that for all but finitely many $j$ we have \[ \max ((a+b\sqrt{D}-1)[j],(a-b\sqrt{D}-1)[j]) \geq (c-1)[j]-\tfrac 12. \] But since the left hand side is an integer, as is $(c-1)[j]$, it follows that there exists an $N$ such that for all $j > N$ we have \begin{equation} \label{eq:maxwithc} \max ((a+b\sqrt{D}-1)[j],(a-b\sqrt{D}-1)[j]) \geq (c-1)[j]. \end{equation} Now again, the proof can be completed in this case using Corollary \ref{c:FGM} and the argument of \cite{FGM}: equation \eqref{eq:maxwithc} assures that carries arising in the formula of Corollary \ref{c:FGM} from $c-1$ are balanced by carries arising from at least one of the numerator parameters $a\pm b\sqrt{D}-1$. Hence, $F_D$ is $p$-adically integral for all such primes congruence to elements of $B_K(a;c)$. Now, assume Conjecture \ref{conj}, and suppose that $p$ is a prime that is not congruent to an element of $B_K(a;c)$. This means that for some $j$ we have \[ \{-p^jc\} > \tfrac12\{-2p^ja\}. \] In the inequality above, both sides are rational numbers such that multiplying by $M$ yields an integer. Thus, the inequality above implies that \(\{-p^jc\} \ge \tfrac12\{-2p^ja\} + \frac1M\). Then we deduce \[ \floor{\{-p^jc\}p} \ge \floor{\tfrac12\{-2p^ja\}p + \frac{p}{M}} \geq \floor{\tfrac12\{-2p^ja\}p} + \floor{\frac{p}{M}}. \] Therefore, by Lemma \ref{l:digitformula} and the above mentioned inequality \(\floor{\tfrac12x}\geq\tfrac12\floor{x}\), we arrive at \begin{equation} \label{e:otherway} (c-1)[j] \geq \tfrac12((2a-1)[j])+\floor{\tfrac pM}. \end{equation} And as above, if $A$ is the order of $p \pmod{M}$, then the expansion of $c-1$ is periodic of period $A$, and the expansion of $2a-1$ is eventually periodic of period $A$. So, after possibly replacing $j$ by some large term $j+nA$ to avoid the part of the expansion of $2a-1$ that is not periodic, we deduce that \begin{equation} \label{eq:cboundprop2} \floor{\tfrac pM}+\tfrac12((2a-1)[j+nA]) \leq (c-1)[j+nA] \end{equation} for all integers $n\geq 0$, since both sequences of digits in inequality \eqref{eq:cboundprop2} are independent of $n$. Up until this point, the proof has been essentially the same as the proof of Proposition \ref{prop1}. At this point the proofs diverge, as now the relationship between the digits of $a+b\sqrt{D}-1$ and $a-b\sqrt{D}-1$ is not as simple as in the case when $a=\tfrac 12$. Let us first observe that if $p$ is large enough, where this restriction on the size only depends on $a$, then all of the $p$-adic digits of $2a-1$ in its periodic part are nonzero by the last claim in Lemma \ref{l:digitformula}. Therefore, we may suppose that $(2a-1)[j] \neq 0$. Choose a constant $K > 1$ such that \[ \frac{1}{KM} < \frac{1}{2p}((2a-1)[x_n]). \] It may look as though this constant depends on $p$, but by Lemma \ref{l:digitformula}, when $p$ is large, the digits of $(2a-1)$ are all approximately equal to rational numbers of the form $\alpha p/M$ for $\alpha =1,2,\ldots, M-1$. Hence when $p$ is large enough, the constant $K$ can be chosen independently of $p$. Conjecture \ref{conj} yields an infinite subsequence $(x_n)$ of the arithmetic progression \((j+nA)\) such that \begin{equation} \label{eq:bsqrtDprop2} \tfrac12((2a-1)[x_n])-\tfrac 1K\floor{\tfrac pM} < (a+b\sqrt D-1)[x_n] < \tfrac12((2a-1)[x_n])+\tfrac 1K\floor{\tfrac pM} \end{equation} for all $n\geq 0$, assuming $p$ is large enough. As in the proof of Lemma 4.7, we have one of the following four possibilities depending on how certain carries take place when adding $a\pm b\sqrt{D}-1$: \[ (a+b\sqrt{D}-1)[x_n] + (a-b\sqrt{D}-1)[x_n] =\begin{cases} (2a-1)[x_n] + \alpha p,\\ (2a-1)[x_n]-1 + \alpha p,\\ \end{cases} \] where $\alpha \in \{0,1,2\}$, for $j$ large enough so that $(2a-1)[x_n]=(2a-2)[x_n]$. However, from equation \eqref{eq:bsqrtDprop2} and the trivial bound $(a-b\sqrt{D}-1)[x_n] \leq p-1$ we obtain \[ (a+b\sqrt{D}-1)[x_n] + (a-b\sqrt{D}-1)[x_n]\leq \tfrac12((2a-1)[x_n])+\tfrac 1K\floor{\tfrac pM} + p-1. \] It follows that \[ \alpha \leq \tfrac {1}{pK}\floor{\tfrac pM} + 1-\tfrac{1}{2p}((2a-1)[x_n]) \leq 1+\tfrac {1}{KM}-\tfrac{1}{2p}((2a-1)[x_n]) < 1, \] where the last inequality follows by our choice of $K$. Therefore, $\alpha=0$ and we have \[ (a+b\sqrt{D}-1)[x_n] + (a-b\sqrt{D}-1)[x_n]\leq (2a-1)[x_n], \] so that if we apply the left side of inequality \eqref{eq:bsqrtDprop2} then \[ (a-b\sqrt{D}-1)[x_n] \leq \tfrac 12((2a-1)[x_n] + \tfrac{1}{K}\floor{\tfrac pM} \] Now, the line above and inequalities \eqref{eq:bsqrtDprop2} and \eqref{eq:cboundprop2} give \[ \max((a+b\sqrt{D}-1)[x_n], (a-b\sqrt{D}-1)[x_n]) < (c-1)[x_n] \] for all $n$ large enough, and thus $F_D$ is unbounded at $p$ using the same argument as in Proposition \ref{prop1} \end{proof} \begin{cor} If Conjecture \ref{conj} holds, then a set of parameters $(a,c)$ as above with \(a\not\in \tfrac12 \ZZ\) has series $F_D$ with a density of bounded primes equal to zero if and only if \(2\{-c\}>\{-2a\}\). \end{cor} \begin{proof} Notice that the set of bounded primes is a union of cyclic subgroups, if nonempty. Therefore, the set of bounded primes is empty if and only if $1$ is not in the bounded prime set. This is equivalent to the statement of the Corollary by the preceding proposition. \end{proof} \section{Towards a quadratic irrational Schwarz list} \label{s:schwarz} Obviously, due to the splitting condition on $p$ in $K = \QQ(\sqrt{D})$, the density \[ D_K(a;c) = \frac{\abs{B_K(a;c)}}{\phi(M)} \] is at most equal to $1/2$. In analogy with the Schwarz list, it is natural to ask how often this maximal density is reached. Computations suggest that frequently we have $D_K(a;c) \leq 1/4$, but this is not always the case. To understand why densities frequently do not exceed $1/4$, let us introduce some more notation: let $M_1$ denote the lcm of the denominators of $a$ and $c$, and let $M$ denote the lcm of $M_1$ and $\Delta_K$, where $\Delta_K$ is the discriminant of our field $K=\QQ(\sqrt{D})$. Define $B(a;c)$ in the same way as $B_K(a;c)$ except we use $M_1$ in place of $M$ and we ignore the splitting condition. Then define \begin{equation} \label{e:dac} D(a;c) = \frac{\abs{B(a;c)}}{\phi(M_1)}. \end{equation} Suppose that $M_1$ and $\Delta_K$ are coprime. Then by the Chinese remainder theorem we have \[ (\ZZ/M\ZZ)^\times \cong (\ZZ/M_1\ZZ)^\times \times (\ZZ/\Delta_K\ZZ)^\times \] and this isomorphism induces a bijection of sets: \begin{equation} \label{eq:coprimeDbound} B_K(a;c) \cong B(a;c) \times \{u\in (\ZZ/\Delta_K\ZZ)^\times \mid u \textrm{ splits in }K\}. \end{equation} \begin{prop} \label{p:Dbound} Suppose that $a,c$ are rationals such that \(2c\) and \(2a\) are not both integers. Then \[ D(a;c) \leq 1/2. \] \end{prop} \begin{proof} We first consider the case where \(2a\not\in\ZZ\). Notice that \[ B(a;c) \subseteq A(a;c) = \{u \in (\ZZ/M_1\ZZ)^\times \mid \{-uc\} \leq \tfrac 12 \{-2ua\}\}. \] It suffices to show that at most one of \(u\) and \(-u\) are in the set \(A(a;c)\). Assume towards a contradiction that both $u$ and $-u$ are in A. Considering $u \in A(a;c)$, we have: \begin{align} \{-uc\} &\leq \tfrac{1}{2}\{-2ua\}\\ \intertext{Noting $ \{-x\} = 1 - \{x\}$ holds when $x \in \QQ\setminus\ZZ$,} 1 - \{uc\} & \leq \tfrac{1}{2}(1 - \{2ua\})\\ 1 - \{uc\} & \leq \tfrac{1}{2} - \tfrac{1}{2}\{2ua\}\\ - \{uc\} & \leq -\tfrac{1}{2} - \tfrac{1}{2}\{2ua\}\\ \tfrac{1}{2} + \tfrac{1}{2}\{2ua\} & \leq \{uc\}. \end{align} As $-u \in A(a;c)$, we also have \[\{uc\} \leq \tfrac{1}{2}\{2ua\}.\] Therefore, \[\tfrac{1}{2} + \tfrac{1}{2}\{2ua\} \leq \{uc\} \leq \tfrac{1}{2}\{2ua\},\] a contradiction! Thus only one of \(u\) and \(-u\) are in \(A(a;c)\) in this case. In the case where \(2a\in\ZZ\), we must instead consider \[ B(a;c)\subseteq A(a;c) = \{u\in(\ZZ/M\ZZ)^\times \mid \{-uc\}\le\tfrac12\}. \] Once again, it suffices to show that at most half of all \(u\) can appear in this set. Assume towards a contradiction that \(u\) and \(-u\) are both in this set. Taking \(u\in A(a;c)\), we have: \begin{align} \{-uc\} &\le \tfrac12\\ 1-\{uc\} &\le \tfrac12\\ -\{uc\} &\le -\tfrac12\\ \tfrac12 &\le \{uc\} \end{align} By our assumptions, \(-u\in A(a;c)\), so \[ \{uc\} \le \tfrac12. \] Thus, since we have assumed that in this case \(2c\not\in\ZZ\), this is a contradiction. So \(B(a;c)\) contains at most half of the elements of \((\ZZ/M_1\ZZ)^\times\), and so \(D(a;c)\) is at most \(\tfrac12\). \end{proof} Now we can prove that the field $K$ must be ramified at some of the primes dividing our hypergeometric parameters in order to find examples $(a,c,K)$ where $D_K(a;c)=\tfrac 12$: \begin{cor} \label{cor:Dbound} Suppose that $\gcd(\Delta_K,M_1) = 1$ and that $2a$ and $2c$ are not both in $\ZZ$. Then \[ D_K(a;c) \leq 1/4. \] \end{cor} \begin{proof} Combining equation \eqref{eq:coprimeDbound} and Proposition \ref{p:Dbound} we find that \[ D_K(a;c) \leq D(a;c)\times \tfrac 12 \leq \tfrac 14. \] \end{proof} Table \ref{tab:largedensitiesnoK} lists pairs of rational parameters $a,c \in (0,1)$ with $a$ and $c$ of height at most $48$, and where $2a \not \in \ZZ$, such that $D(a,c) > 1/4$. We do not know if there exists an infinite number of such examples as the height of $a$ and $c$ grow. Since we have $D_K(a,c) \leq D(a,c)$ for all fields $K = \QQ(\sqrt{D})$, in each case from Table \ref{tab:largedensitiesnoK}, it is natural to ask whether we can find a field where equality holds. By Corollary \ref{cor:Dbound}, we know that in such examples, $D$ can't be coprime to the least common multiple of the denominators of $a$ and $c$. We thus examined all choices of $K=\QQ(\sqrt{D})$ for $D$ dividing this least common multiple. In all cases of pairs $(a,c)$ from Table \ref{tab:largedensitiesnoK} with $D(a,c)=1/2$, we found exactly one field $K$ such that $D_K(a,c)=1/2$. Further, it was always imaginary quadratic. \nocite{MNT} \nocite{kummer} \begin{table} \resizebox{\textwidth}{!}{ \renewcommand{\arraystretch}{1} \centering \begin{tabular}{p{2.8cm}p{2.8cm}p{2.8cm}p{2.8cm}p{2.8cm}p{2.8cm}} $D(a;c)=\tfrac12$ &$D(a;c)=\tfrac12$ & $D(a;c) = \tfrac38$& $D(a;c) = \tfrac38$&$D(a;c)=\tfrac5{16}$&$D(a;c)=\tfrac13$\\ \begin{tabular}{cc} $a$&$c$ \\ \hline 1/3&5/6\\ 2/3&2/3\\ 2/3&5/6\\ 1/4&3/4\\ 1/4&5/6\\ 3/4&3/4\\ 3/4&5/6\\ 1/6&2/3\\ 1/6&5/6\\ 5/6&5/6\\ 1/8&3/4\\ 1/8&5/8\\ 3/8&7/8\\ 5/8&3/4\\ 5/8&5/8\\ 7/8&7/8\\ 1/12&2/3\\ 1/12&5/6\\ 1/12&7/12\\ 5/12&11/12\\ 7/12&2/3\\ 7/12&5/6\\ 7/12&7/12\\ 11/12&11/12\\ 1/16&5/8\\ 3/16&7/8\\ 9/16&5/8\\ 11/16&7/8\\ 1/20&11/20\\ 3/20&13/20\\ 7/20&17/20\\ 9/20&19/20\\ 11/20&11/20\\ 13/20&13/20\\ 17/20&17/20\\ 19/20&19/20 \end{tabular} & \begin{tabular}{cc} $a$&$c$\\ \hline 1/24&3/4\\ 1/24& 5/6\\ 1/24&7/12\\ 1/24&13/24\\ 5/24&3/4\\ 5/24&11/12\\ 5/24&17/24\\ 7/24&5/6\\ 7/24&19/24\\ 11/24&23/24\\ 13/24&3/4\\ 13/24&5/6\\ 13/24&7/12\\ 13/24&13/24\\ 17/24&3/4\\ 17/24&11/12\\ 17/24&17/24\\ 19/24&5/6\\ 19/24&19/24\\ 23/24&23/24\\ 1/40&11/20\\ 3/40&13/20\\ 7/40&17/20\\ 9/40&19/20\\ 21/40&11/20\\ 23/40&13/20\\ 27/40&17/20\\ 29/40&19/20\\ 1/48&13/24\\ 5/48&17/24\\ 7/48&19/24\\ 11/48&23/24\\ 25/48&13/24\\ 29/48&17/24\\ 31/48&19/24\\ 35/48&23/24 \end{tabular} & \vspace{-9.215cm} \begin{tabular}{cc} $a$&$c$ \\ \hline 1/16&5/6\\ 1/16&7/12\\ 1/16&11/12\\ 1/16&17/24\\ 1/16&19/24\\ 3/16&5/6\\ 3/16&17/24\\ 3/16&19/24\\ 5/16&5/6\\ 5/16&11/12\\ 5/16&23/24\\ 7/16&23/24\\ 9/16&5/6\\ 9/16&7/12\\ 9/16&11/12\\ 9/16&17/24\\ 9/16&19/24\\ 11/16&5/6\\ 11/16&17/24\\ 11/16&19/24\\ 13/16&5/6\\ 13/16&11/12\\ 13/16&23/24\\ 15/16&23/24 \end{tabular} & \vspace{-9.215cm} \begin{tabular}{cc} $a$&$c$ \\ \hline 1/48&5/6\\ 1/48&7/12\\ 1/48&17/24\\ 1/48&23/24\\ 5/48&11/12\\ 5/48&19/24\\ 7/48&5/6\\ 7/48&17/24\\ 7/48&23/24\\ 11/48&19/24\\ 13/48&5/6\\ 17/48&11/12\\ 25/48&7/12\\ 25/48&17/24\\ 25/48&5/6\\ 25/48&23/24\\ 29/48&11/12\\ 29/48&19/24\\ 31/48&5/6\\ 31/48&17/24\\ 31/48&23/24\\ 35/48&19/24\\ 37/48&5/6\\ 41/48&11/12 \end{tabular}& \vspace{-9.215cm} \begin{tabular}{cc} $a$&$c$\\ \hline 1/32&11/12\\ 5/32&23/24\\ 9/32&11/12\\ 13/32&23/24\\ 17/32&11/12\\ 21/32&23/24\\ 25/32&11/12\\ 29/32&23/24\\ 1/40&5/6\\ 3/40&5/6\\ 7/40&5/6\\ 9/40&5/6\\ 21/40&5/6\\ 23/40&5/6\\ 27/40&5/6\\ 29/40&5/6 \end{tabular} & \vspace{-9.215cm} \begin{tabular}{cc} $a$&$c$ \\ \hline 2/7&5/6\\ 11/14&5/6\\ 1/21&5/6\\ 2/39&5/6\\ 11/39&5/6\\ 23/42&5/6 \end{tabular} \end{tabular} } \caption{All examples where $D(a;c) >1/4$ for $0<a,c<1$ with $2a\not \in \ZZ$, and the height of $a$ and $c$ are at most $48$.} \label{tab:largedensitiesnoK} \end{table} \begin{table} \renewcommand{\arraystretch}{1.3} \centering \begin{tabular}{ccc} $a$&$c$&$D$\\ \hline 1/3&5/6&-3\\ 2/3&2/3&-3\\ 2/3&5/6&-3\\ 1/4&3/4&-1\\ 1/4&5/6&-3\\ 3/4&3/4&-1\\ 3/4&5/6&-3\\ 1/6&2/3&-3\\ 1/6&5/6&-3\\ 5/6&5/6&-3\\ 1/8&3/4&-1\\ 1/8&5/8&-2\\ 3/8&7/8&-2\\ 5/8&3/4&-1\\ 5/8&5/8&-2\\ 7/8&7/8&-2\\ 1/12&2/3&-3\\ 1/12&5/6&-3\\ 1/12&7/12&-1\\ 5/12&11/12&-1\\ 7/12&2/3&-3\\ 7/12&5/6&-3\\ 7/12&7/12&-1\\ 11/12&11/12&-1\\ 1/16&5/8&-2\\ 3/16&7/8&-2\\ 9/16&5/8&-2\\ 11/16&7/8&-2\\ 1/20&11/20&-5\\ 3/20&13/20&-5\\ 7/20&17/20&-5\\ 9/20&19/20&-5\\ 11/20&11/20&-5\\ 13/20&13/20&-5\\ 17/20&17/20&-5\\ 19/20&19/20&-5 \end{tabular} \hspace{1cm} \begin{tabular}{ccc} $a$&$c$&$D$\\ \hline 1/24&3/4&-1\\ 1/24& 5/6&-3\\ 1/24&7/12&-1\\ 1/24&13/24&-6\\ 5/24&3/4&-1\\ 5/24&11/12&-1\\ 5/24&17/24&-6\\ 7/24&5/6&-3\\ 7/24&19/24&-6\\ 11/24&23/24&-6\\ 13/24&3/4&-1\\ 13/24&5/6&-3\\ 13/24&7/12&-1\\ 13/24&13/24&-6\\ 17/24&3/4&-1\\ 17/24&11/12&-1\\ 17/24&17/24&-6\\ 19/24&5/6&-3\\ 19/24&19/24&-6\\ 23/24&23/24&-6\\ 1/40&11/20&-5\\ 3/40&13/20&-5\\ 7/40&17/20&-5\\ 9/40&19/20&-5\\ 21/40&11/20&-5\\ 23/40&13/20&-5\\ 27/40&17/20&-5\\ 29/40&19/20&-5\\ 1/48&13/24&-6\\ 5/48&17/24&-6\\ 7/48&19/24&-6\\ 11/48&23/24&-6\\ 25/48&13/24&-6\\ 29/48&17/24&-6\\ 31/48&19/24&-6\\ 35/48&23/24&-6 \end{tabular} \caption{Examples of \(a,c,D\) such that 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2412.02494v2	http://arxiv.org/abs/2412.02494v2	On the hit problem for the polynomial algebra and the algebraic transfer	\PassOptionsToPackage{top=1cm, bottom=1cm, left=2cm, right=2cm}{geometry} \documentclass[final,3p,times]{elsarticle} \makeatletter \def\ps@pprintTitle{ \let\@oddhead\@empty \let\@evenhead\@empty \def\@oddfoot{\reset@font\hfil\thepage\hfil} \let\@evenfoot\@oddfoot } \makeatother \usepackage{amsthm, amssymb, amsfonts, mathrsfs, amsmath} \everymath{\displaystyle} \usepackage{palatino} \usepackage[T5]{fontenc} \usepackage{amscd} \usepackage{graphicx} \usepackage{pb-diagram} \usepackage[x11names]{xcolor} \usepackage[ colorlinks, ]{hyperref} \newcommand\rurl[1]{ \href{http://#1}{\nolinkurl{#1}}} \AtBeginDocument{\hypersetup{citecolor=blue, urlcolor=blue, linkcolor=blue}} \theoremstyle{plain} \newtheorem{thm}{Theorem}[subsection] \newtheorem{dly}[thm]{Theorem} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{corl}[thm]{Corollary} \theoremstyle{definition} \newtheorem{rema}[thm]{Remark} \newtheorem{nota}[thm]{Notation} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newtheorem{acknow}{Acknowledgments} \theoremstyle{plain} \newtheorem{therm}{Theorem}[section] \newtheorem{conj}[therm]{Conjecture} \newtheorem{propo}[therm]{Proposition} \newtheorem{lema}[therm]{Lemma} \newtheorem{corls}[therm]{Corollary} \theoremstyle{definition} \newtheorem{rems}[therm]{Remark} \newtheorem{cy}[therm]{Note} \newtheorem{notas}[therm]{Notation} \theoremstyle{definition} \newtheorem{dn}{Definition}[subsection] \newtheorem{kh}[dn]{Notation} \newtheorem{note}[thm]{Note} \newtheorem{nx}[thm]{Remark} \everymath{\displaystyle} \def\leq{\leqslant} \def\geq{\geqslant} \def\DD{D\kern-.7em\raise0.4ex\hbox{\char '55}\kern.33em} \renewcommand{\baselinestretch}{1.2} \usepackage{sectsty} \subsectionfont{\fontsize{11.5}{11.5}\selectfont} \makeatletter \def\blfootnote{\xdef\@thefnmark{}\@footnotetext} \makeatother \begin{document} \fontsize{11.5pt}{11.5}\selectfont \begin{frontmatter} \title{{\bf On the hit problem for the polynomial algebra and the algebraic transfer}} \author{\DD\d{\u A}NG V\~O PH\'UC} \address{{\fontsize{10pt}{10}\selectfont Department of Information Technology, FPT University, Quy Nhon A.I Campus,\\ An Phu Thinh New Urban Area, Quy Nhon City, Binh Dinh, Vietnam\\[2.5mm] \textbf{\textit{(Dedicated to my beloved wife and cherished son)}}}\\[2.5mm] \textit{\fontsize{11pt}{11}\selectfont \textbf{ORCID: \url{https://orcid.org/0000-0002-6885-3996}}} } \cortext[]{\href{mailto:[email protected], [email protected]}{\texttt{Email address: [email protected], [email protected]}}} \begin{abstract} Let $\mathcal A$ be the classical, singly-graded Steenrod algebra over the prime order field $\mathbb F_2$ and let $P^{\otimes h}: = \mathbb F_2[t_1, \ldots, t_h]$ denote the polynomial algebra on $h$ generators, each of degree $1.$ Write $GL_h$ for the usual general linear group of rank $h$ over $\mathbb F_2.$ Then, $P^{\otimes h}$ is an $\mathcal A[GL_h]$-module. As is well known, for all homological degrees $h \geq 6$, the cohomology groups ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb F_2, \mathbb F_2)$ of the algebra $\mathcal A$ are still shrouded in mystery. The algebraic transfer $Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_{\bullet}\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb F_2, \mathbb F_2)$ of rank $h,$ constructed by W. Singer [Math. Z. \textbf{202} (1989), 493-523], is a beneficial technique for describing the Ext groups. Singer's conjecture about this transfer states that \textit{it is always a one-to-one map}. Despite significant effort, neither a complete proof nor a counterexample has been found to date. The unresolved nature of the conjecture makes it an interesting topic of research in Algebraic topology in general and in homotopy theory in particular. \medskip The objective of this paper is to investigate Singer's conjecture, with a focus on all $h\geq 1$ in degrees $n\leq 10 = 6(2^{0}-1) + 10\cdot 2^{0}$ and for $h=6$ in the general degree $n:=n_s=6(2^{s}-1) + 10\cdot 2^{s},\, s\geq 0.$ Our methodology relies on the hit problem techniques for the polynomial algebra $P^{\otimes h}$, which allows us to investigate the Singer conjecture in the specified degrees. Our work is a continuation of the work presented by Mothebe et al. [J. Math. Res. \textbf{8} (2016), 92-100] with regard to the hit problem for $P^{\otimes 6}$ in degree $n_s$, expanding upon their results and providing novel contributions to this subject. More generally, for $h\geq 6,$ we show that the dimension of the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ in degrees $2^{s+4}-h$ is equal to the order of the factor group of $GL_{h-1}$ by the Borel subgroup $B_{h-1}$ for every $s\geq h-5.$ Especially, for the Galois field $\mathbb F_{q}$ ($q$ denoting the power of a prime number), based on Hai's recent work [C. R. Math. Acad. Sci. Paris \textbf{360} (2022), 1009-1026], we claim that the dimension of the space of the indecomposable elements of $\mathbb F_q[t_1, \ldots t_h]$ in general degree $q^{h-1}-h$ is equal to the order of the factor group of $GL_{h-1}(\mathbb F_q)$ by a subgroup of the Borel group $B_{h-1}(\mathbb F_q).$ As applications, we establish the dimension result for the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes 7}$ in degrees $n_{s+5},\, s > 0.$ Simultaneously, we demonstrate that the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$ and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2)$ do not belong to the image of the sixth Singer algebraic transfer, $Tr_6^{\mathcal A}.$ This discovery holds significant implications for Singer's conjecture concerning algebraic transfers. We further deliberate on the correlation between these conjectures and antecedent studies, thus furnishing a comprehensive analysis of their implications. \end{abstract} \begin{keyword} Hit problem; Steenrod algebra; Primary cohomology operations; Algebraic transfer \MSC[2010] 55R12, 55S10, 55S05, 55T15 \end{keyword} \end{frontmatter} \tableofcontents \section{Introduction}\label{s1} \setcounter{equation}{0} Throughout this text, we adopt the convention of working over the prime field $\mathbb F_2$ and denote the Steenrod algebra over this field by $\mathcal A$, unless otherwise stated. Let $V_h$ be the $h$-dimensional $\mathbb F_2$-vector space. It is well-known that the mod 2 cohomology of the classifying space $BV_h$ is given by $P^{\otimes h} := H^{}(BV_h) = H^{}((\mathbb RP^{\infty})^{\times h}) = \mathbb F_2[t_1, \ldots, t_h],$ where $(t_1, \ldots, t_h)$ is a basis of $H^{1}(BV_h) = {\rm Hom}(V_h, \mathbb F_2).$ The polynomial algebra $P^{\otimes h}$ is a connected $\mathbb Z$-graded algebra, that is, $P^{\otimes h} = \{P_n^{\otimes h}\}_{n\geq 0},$ in which $P_n^{\otimes h}:= (P^{\otimes h})_n$ denotes the vector subspace of homogeneous polynomials of degree $n$ with $P_0^{\otimes h}\equiv \mathbb F_2$ and $P_n^{\otimes h} = \{0\}$ for all $n < 0.$ We know that the algebra $\mathcal A$ is generated by the Steenrod squares $Sq^{k}$ ($k\geq 0$) and subject to the Adem relations (see e.g., \cite{Walker-Wood}). The Steenrod squares are the cohomology operations satisfying the naturality property. Moreover, they commute with the suspension maps, and therefore, they are stable. In particular, these squares $Sq^k$ were applied to the vector fields on spheres and the Hopf invariant one problem, which asks for which $k$ there exist maps of Hopf invariant $\pm 1$. The action of $\mathcal A$ over $P^{\otimes h} $ is determined by instability axioms. By Cartan's formula, it suffices to determine $Sq^{i}(t_j)$ and the instability axioms give $Sq^{1}(t_j)= t_j^{2}$ while $Sq^{k}(t_j) = 0$ if $k > 1.$ The investigation of the Steenrod operations and related problems has been undertaken by numerous authors (for instance \cite{Brunetti1, Singer1, Silverman, Monks, Wood}). An illustration of the importance of the Steenrod operators is their stability property, which, when used in conjunction with the Freudenthal suspension theorem \cite{Freudenthal}, enables us to make the claim that the homotopy groups $\pi_{k+1}(\mathbb S^{k})$ are non-trivial for $k\geq 2$ (see also \cite{Phuc13} for further details). The Steenrod algebra naturally acts on the cohomology ring $H^{}(X)$ of a CW-complex $X.$ In several cases, the resulting $\mathcal A$-module structure on $H^{}(X)$ provides additional information about $X$ (for instance the CW-complexes $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ have cohomology rings that agree as graded commutative $\mathbb F_2$-algebras, but are different as modules over $\mathcal A.$ We also refer the readers to \cite{Phuc10-0} for an explicit proof.) Hence the Steenrod algebra is one of the important tools in Algebraic topology. Especially, its cohomology ${\rm Ext}_{\mathcal A}^{, }(\mathbb F_2, \mathbb F_2)$ is an algebraic object that serves as the input to the Adams (bigraded) spectral sequence \cite{J.A} and therefore, computing this cohomology is of fundamental importance to the study of the stable homotopy groups of spheres. \medskip The identification of a minimal generating set for the $\mathcal A$-module $P^{\otimes h}$ has been a significant and challenging open problem in Algebraic topology in general and in homotopy theory in particular for several decades. This problem, famously known as the "hit" problem, was first proposed by Frank Peterson \cite{Peterson, Peterson2} through computations for cases where $h<2$ and has since captured the attention of numerous researchers in the field, as evidenced by works such as Kameko \cite{Kameko}, Repka and Selick \cite{Repka-Selick}, Singer \cite{Singer1}, Wood \cite{Wood}, Mothebe et al. \cite{Mothebe0, Mothebe, MKR}, Walker and Wood \cite{Walker-Wood, Walker-Wood2}, Sum \cite{Sum00, Sum1, Sum2, Sum2-0, Sum3, Sum4, Sum5}, Ph\'uc and Sum \cite{P.S1, P.S2}, Ph\'uc \cite{Phuc4, Phuc6, Phuc10, Phuc10-0, Phuc11, Phuc13, Phuc15, Phuc16}, Hai \cite{Hai}), and others. Peterson himself, as well as several works such as \cite{Priddy, Singer, Wood}, have shown that the hit problem is closely connected to some classical problems in homotopy theory. To gain a deeper understanding of this problem and its numerous applications, readers are cordially invited to refer to the excellent volumes written by Walker and Wood \cite{Walker-Wood, Walker-Wood2}. An interesting fact is that, if $\mathbb F_2$ is a trivial $\mathcal A$-module, then the hit problem is essentially the problem of finding a monomial basis for the graded vector space $$QP^{\otimes h}=\big\{QP_n^{\otimes h} := (QP^{\otimes h})_n =P_n^{\otimes h}/\overline{\mathcal A}P_n^{\otimes h} = (\mathbb F_2\otimes_{\mathcal A}P^{\otimes h})_n = {\rm Tor}_{0, n}^{\mathcal A}(\mathbb F_2, P^{\otimes h})\big\}_{n\geq 0}.$$ Here $\overline{\mathcal A}P_n^{\otimes h}:=P_n^{\otimes h}\cap \overline{\mathcal A}P^{\otimes h}$ and $\overline{\mathcal A}$ denotes the set of positive degree elements in $\mathcal A.$ The investigation into the structure of $QP^{\otimes h}$ has seen significant progress in recent years. Notably, it has been able to explicitly describe $QP^{\otimes h}$ for $h\leq 4$ and all $n > 0$ through the works of Peterson \cite{Peterson} for $h = 1, 2$, Kameko \cite{Kameko} for $h = 3$, and Sum \cite{Sum1, Sum2} for $h = 4$. Even so, current techniques have yet to full address the problem. While the information that follows may not be integral to the chief content of this paper, it will be beneficial for readers who desire a more in-depth comprehension of the hit problem. When considering the field $\mathbb F_p$, where $p$ is an odd prime, one must address the challenge of the "hit problem," which arises in the polynomial algebra $\mathbb F_p[t_1, \ldots, t_h] = H^{}((\mathbb CP^{\infty})^{\times h}; \mathbb F_p)$ on generators of degree $2$. This algebra is viewed as a module over the mod $p$ Steenrod algebra $\mathcal A_p$. Here $\mathbb CP^{\infty}$ denotes the infinite complex projective space. The action of $\mathcal A_p$ on $\mathbb F_p[t_1, \ldots, t_h]$ can be succinctly expressed by $\mathscr P^{p^{j}}(t_i^{r}) = \binom{r}{p^{j}}t_i^{r+p^{j+1}-p^{j}},\, \beta(t_i) = 0$ ($\beta\in \mathcal A_p$ being the Bockstein operator) and the usual Cartan formula. In particular, if we write $r = \sum_{j\geq 0}\alpha_j(r)p^{j}$ for the $p$-adic expansion of $r,$ then $\mathscr P^{p^{j}}(t_i^{r}) \neq 0$ if and only if $\binom{r}{p^{j}}\equiv \alpha_j(r)\, ({\rm mod}\, p)\, \neq 0.$ Since each Steenrod reduced power $\mathscr P^{j}$ is decomposable unless $j$ is a power of $p,$ a homogeneous polynomial $f$ is hit if and only if it can be represented as $\sum_{j \geq 0}\mathscr P^{p^{j}}(f_j)$ for some homogeneous polynomials $f_j\in \mathbb F_p[t_1, \ldots, t_h].$ In other words, $f$ belongs to $\overline{\mathcal A_p}\mathbb F_p[t_1, \ldots, t_h].$ (This is analogous to the widely recognized case when $p = 2.$) To illustrate, let us consider the monomial $t^{p(p+1)-1}\in \mathbb F_p[t].$ Since $\binom{2p-1}{p}\equiv \binom{p-1}{0}\binom{1}{1} \equiv 1\, ({\rm mod}\, p),$ $t^{p(p+1)-1} = \mathscr P^{p}(t^{2p-1}),$ i.e., $t^{p(p+1)-1}$ is hit. Actually, the hit problem for the algebra $ \mathbb F_p[t_1, \ldots, t_h]$ is an intermediate problem of identifying a minimal set of generators for the ring $H^{}(V; \mathbb F_p) = H^{}(BV; \mathbb F_p) = \Lambda(V^{\sharp})\otimes_{\mathbb F_p}S(V^{\sharp})$ as a module over $\mathcal A_p.$ Here $\Lambda(V^{\sharp})$ is an exterior algebra on generators of degree $1$ while $S(V^{\sharp})$ is a symmetric algebra on generators of degree $2.$ In both situations, the generators may be identified as a basis for $V^{\sharp}$, the linear dual of an elementary abelian $p$-group $V$ of rank $h$, which can be regarded as an $h$-dimensional vector space over the field $\mathbb F_p.$ Viewed as an algebra over the Steenrod algebra, $S(V^{\sharp})$ can be identified with $H^{}((\mathbb CP^{\infty})^{\times h}; \mathbb F_p)$. Consequently, the cohomology of $V$ over the field $\mathbb F_p$ can be expressed as $\Lambda(V^{\sharp})\otimes_{\mathbb F_p} \mathbb F_p[t_1, \ldots, t_h].$ Thus, the information about the hit problem for $H^{}(V; \mathbb F_p)$ as an $\mathcal A_p$-module can usuallly be obtained from the similar information about the hit problem for the $\mathcal A_p$-module $\mathbb F_p[t_1, \ldots, t_h]$ without much difficulty. With a monomial $f = t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}\in \mathbb F_p[t_1, \ldots, t_h],$ we denote its degree by $\deg(f) = \sum_{1\leq i\leq h}a_i.$ This coincides with the usual grading of $P^{\otimes h}$ for $p = 2.$ Notwithstanding, it is one half of the usual grading of $\mathbb F_p[t_1, \ldots, t_h]$ for $p$ odd. With respect to this grading, Peterson's conjecture \cite{Peterson2} is no longer true for $p$ odd in general. As a case in point, our work \cite{Phuc14} provides a detailed proof that $\alpha((i+1)p^{r}-1+1) = \alpha_r((i+1)p^{r}-1+1) = i+1 > 1,$ but the monomials $t^{(i+1)p^{r}-1}\in \mathbb F_p[t],$ for $1\leq i< p-1,\, r\geq 0,$ are not hit. Returning to the topic of the indecomposables $QP^{\otimes h}_n,$ let $\mu: \mathbb{N} \longrightarrow \mathbb{N}$ be defined by $\mu(n) = \min\bigg\{k \in \mathbb{N}: \alpha(n+k) \leq k\bigg\}$, where $\alpha(n)$ denotes the number of ones in the binary expansion of $n$. In the work of Sum \cite{Sum4}, it has been demonstrated that $\mu(n) = h$ if and only if there exists uniquely a sequence of integers $d_1 > d_2 > \cdots > d_{h-1}\geq d_h > 0$ such that $n = \sum_{1\leq j\leq h}(2^{d_j} - 1).$ On the other side, according to Wood \cite{Wood}, if $\mu(n) > h,$ then $\dim QP^{\otimes h}_{n} = 0.$ This validates also Peterson's conjecture \cite{Peterson2} in general. Singer \cite{Singer1} later proved a generalization of Wood's result, identifying a larger class of hit monomials. In \cite{Silverman}, Silverman makes progress toward proving a conjecture of Singer which would identify yet another class of hit monomials. In \cite{Monks}, Monks extended Wood's result to determine a new family of hit polynomials in $P^{\otimes h}.$ Notably, Kameko \cite{Kameko} showed that if $\mu(2n+h) = h,$ then $QP^{\otimes h}_{2n+h}\cong QP^{\otimes h}_{n}.$ This occurrence elucidates that the surjective map $(\widetilde {Sq^0_})_{2n+h}: QP^{\otimes h}_{2n+h} \longrightarrow QP^{\otimes h}_{n},\ \mbox{[}u\mbox{]}\longmapsto \left\{\begin{array}{ll} \mbox{[}y\mbox{]}& \text{if $u = \prod_{1\leq j\leq h}t_jy^{2}$},\\ 0& \text{otherwise}, \end{array}\right.$ defined by Kameko himself, transforms into a bijective map when $\mu(2n+h) = h.$ Thus it is only necessary to calculate $\dim QP^{\otimes h}_n$ for degrees $n$ in the "generic" form: \begin{equation}\label{pt} n= k(2^{s} - 1) + r\cdot 2^{s} \end{equation} whenever $k,\, s,\, r$ are non-negative integers satisfying $\mu(r) < k \leq h.$ (For more comprehensive information regarding this matter, kindly refer to Remark \ref{nxpt} in Sect.\ref{s2}.) The dual problem to the hit problem for the algebra $P^{\otimes h}$ is to ascertain a subring consisting of elements of the Pontrjagin ring $H_(BV_h) = [P^{\otimes h}]^{}$ that are mapped to zero by all Steenrod squares of positive degrees. This subring is commonly denoted by ${\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{}.$ Let $GL_h = GL(V_h)$ be the general linear group. This $GL_h$ acts on $V_h$ and then on $QP_n^{\otimes h}.$ For each positive integer $n,$ denote by $[QP_n^{\otimes h}]^{GL_h}$ the subspace of elements that are invariant under the action of $GL_h.$ It is known that there exists an isomorphism between $(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_n$ and $[QP_n^{\otimes h}]^{GL_h}$, which establishes a close relationship between the hit problem and the $h$-th algebraic transfer \cite{Singer}, $$Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_n\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2).$$ The homomorphism $Tr_h^{\mathcal A}$ was constructed by William Singer while studying the Ext groups, employing the modular invariant theory. One notable aspect is that the Singer transfer can be regarded as an algebraic formulation of the stable transfer $B(V_h)_+^{S}\longrightarrow \mathbb S^{0}.$ It is a well-established fact, as demonstrated by Liulevicius \cite{Liulevicius}, that there exist squaring operations $Sq^{i}$ for $i\geq 0$ that act on the $\mathbb F_2$-cohomology of the Steenrod algebra $\mathcal A$. These operations share many of the same properties as the Steenrod operations $Sq^{i}$ that act on the $\mathbb F_2$-cohomology of spaces. Nonetheless, $Sq^{0}$ is not the identity. On the other side, there exists an analogous squaring operation $Sq^{0}$, called the Kameko operation, which acts on the domain of the algebraic transfer and commutes with the classical $Sq^{0}$ on ${\rm Ext}_{\mathcal A}^{,}(\mathbb F_2, \mathbb F_2)$ thourgh Singer's transfer (see Sect.\ref{s2} for its precise meaning). Hence, the highly non-trivial character of the algebraic transfer establishes it as a tool of potential in the study of the inscrutable Ext groups. Moreover, the hit problem and the Singer transfer have been shown in the papers \cite{Minami0, Minami} to be significant tools for investigating the Kervaire invariant one problem. It is noteworthy that Singer made the following prediction. \begin{conj}[see \cite{Singer}]\label{gtSinger} The transfer $Tr_h^{\mathcal A}$ is a one-to-one map for any $h.$ \end{conj} Despite not necessarily resulting in a one-to-one correspondence, Singer's transfer is a valuable tool for analyzing the structure of the Ext groups. It is established, based on the works of Singer \cite{Singer} and Boardman \cite{Boardman}, that the Singer conjecture is true for homological degrees up to $3$. In these degrees, the transfer is known to be an isomorphism. We are thrilled to announce that our recent works \cite{Phuc12, Phuc10-3, Phuc10-2} has finally brought closure to the complex and long-standing problem of verifying Singer's conjecture in the case of rank four. Our study, detailed in \cite{Phuc12, Phuc10-3, Phuc10-2}, specifically establishes the truth of Conjecture \ref{gtSinger} in the case where $h=4.$ For some information on the interesting case of rank five, we recommend consulting works such as \cite{Phuc4, Phuc6, Phuc10, Phuc10-0, Phuc17, Sum4}. It is essential to underscore that the isomorphism between the domain of the homomorphism $Tr_h^{\mathcal A}$ and $(QP_n^{\otimes h})^{GL_h}$ (the subspace of $GL_h$-invariants of $QP_n^{\otimes h}$) implies that it is sufficient to explore Singer's transfer in internal degrees $n$ of the form \eqref{pt}. \medskip Despite extensive research, no all-encompassing methodology exists for the investigation of the hit problem and Singer's algebraic transfer in every positive degree. Therefore, each computation holds considerable importance and serves as an independent contribution to these subjects. By this reason, our primary objective in this work is to extend the findings of Mothebe et al. \cite{MKR} regarding the hit problem of six variables, while simultaneously verifying Singer's conjecture for all ranks $h\geq 1$ in certain internal degrees. Our methodology is based on utilizing the techniques developed for the hit problem, which have proven to be quite effective in determining the Singer transfer. More precisely, using the calculations in \cite{MKR}, we embark on an investigation of Singer's conjecture for bidegrees $(h, h+n)$, where $h\geq 1$ and $1\leq n\leq 10 = 6(2^{0}-1) + 10\cdot 2^{0}$. Subsequently, we proceed to solve the hit problem for $P^{\otimes 6} = \mathbb F_2[t_1, \ldots, t_6]$ in degrees of the form \eqref{pt}, with $k = 6$ and $r = 1$ (i.e., degree $n:=n_s= 6(2^{s}-1) + 10\cdot 2^{s}$, $s\geq 0$). Furthermore, for $h\geq 6$ and degree $2^{s+4}-h$, we establish that for each $s\geq h-5$. the dimension of the cohit module $QP^{\otimes h}_{2^{s+4}-h}$ is equal to the order of the factor group of $GL_{h-1}$ by the Borel subgroup $B_{h-1}.$ Additionally, utilizing the algebra $\pmb{A}_q$ of Steenrod reduced powers over the Galois field $\mathbb F_{q}$ of $q = p^{m}$ elements, and based on Hai's recent work \cite{Hai}, we assert that for any $h\geq 2,$ the dimension of the cohit module $\mathbb F_q[t_1, \ldots t_h]/\overline{\pmb{A}}_q\mathbb F_q[t_1, \ldots t_h]$ in degree $q^{h-1}-h$ is equal to the order of the factor group of $GL_{h-1}(\mathbb F_q)$ by a subgroup of the Borel group $B_{h-1}(\mathbb F_q).$ As a result, we establish the dimension result for the space $QP^{\otimes 7}$ in degrees $n_{s+5}$, where $s > 0$, and explicitly determine the dimension of the domain of $Tr_6^{\mathcal A}$ in degrees $n_s$. Our findings reveal that $Tr_h^{\mathcal A}$ is an isomorphism in some degrees $\leq 10$ and that $Tr_6^{\mathcal A}$ does not detect the non-zero elements $h_2^{2}g_1 = Sq^{0}(h_1Ph_1) = h_4Ph_2$ and $D_2$ in the sixth cohomology groups ${\rm Ext}_{\mathcal A}^{6, 6+n_s}(\mathbb F_2, \mathbb F_2)$. This finding carries significant implications for Singer's conjecture on algebraic transfers. Specifically, we affirm the validity of Conjecture \ref{gtSinger} for the bidegrees $(h, h+n)$ where $h\geq 1$, $1\leq n\leq n_0$, as well as for any bidegree $(6, 6+n_s).$ It is worth noting that the results have been rigorously verified using \textbf{MAGMA} \cite{Bosma}. \medskip {\bf Organization of the rest of this work.}\ In Sect.\ref{s2}, we provide a brief overview of the necessary background material. The main findings are then presented in Sect.\ref{s3}, with the proofs being thoroughly explained in Sect.\ref{s4}. As an insightful consolidation, Sect.\ref{s5} will encapsulate the core essence of the paper by distilling its key discoveries and notable contributions. Finally, in the appendix (referred to as Sect.\ref{s6}), we provide an extensive list of admissible monomials of degree $n_1 =6(2^{1}-1) + 10\cdot 2^{1}$ in the $\mathcal A$-module $P^{\otimes 6}.$ \begin{acknow} I would like to extend my heartfelt gratitude and appreciation to the editor and the anonymous referees for their meticulous review of my work and insightful recommendations, which greatly contributed to the enhancement of this manuscript. Furthermore, their constructive feedback has served as a source of inspiration for me to elevate the outcomes of my research beyond the initial expectations. I am greatly appreciative of Bill Singer for his keen interest, constructive criticism, and enlightening e-mail correspondence. My sincere thanks are also due to Bob Bruner, John Rognes and Weinan Lin for helpful discussions on the Ext groups. \textbf{A condensed version of this article was published in the Journal of Algebra, 613 (2023), 1--31.} \end{acknow} \section{Several fundamental facts}\label{s2} In order to provide the reader with necessary background information for later use, we present some related knowledge before presenting the main results. Readers may also refer to \cite{Kameko, Phuc4, Sum2} for further information. We will now provide an overview of some fundamental concepts related to the hit problem. \medskip {\bf Weight and exponent vectors of a monomial.}\ For a natural number $k,$ writing $\alpha_j(k)$ and $\alpha(k)$ for the $j$-th coefficients and the number of $1$'s in dyadic expansion of $k$, respectively. Thence, $\alpha(k) = \sum_{j\geq 0}\alpha_j(k),$ and $k$ can be written in the form $\sum_{j\geq 0}\alpha_j(k)2^j.$ For a monomial $t = t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}$ in $P^{\otimes h},$ we consider a sequence associated with $t$ by $\omega(t) :=(\omega_1(t), \omega_2(t), \ldots, \omega_i(t), \ldots)$ where $\omega_i(t)=\sum_{1\leq j\leq h}\alpha_{i-1}(a_j)\leq h,$ for all $i\geq 1.$ This sequence is called the {\it weight vector} of $t.$ One defines $\deg(\omega(t)) = \sum_{j\geq 1}2^{j-1}\omega_j(t).$ We use the notation $a(t) = (a_1, \ldots, a_h)$ to denote the exponent vector of $t.$ Both the sets of weight vectors and exponent vectors are assigned the left lexicographical order. \medskip {\bf The linear order on \mbox{\boldmath $P^{\otimes h}$.}}\ Consider the monomials $t = t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}$ and $t' = t_1^{a'_1}t_2^{a'_2}\ldots t_h^{a'_h}$ in the $\mathcal A$-module $P^{\otimes h}$. We define the relation "$<$" between these monomials as follows: $t < t'$ if and only if either $\omega(t) < \omega(t')$, or $\omega(t) = \omega(t')$ and $a(t) < a'(t').$ \medskip {\bf The equivalence relations on \mbox{\boldmath $P^{\otimes h}$.}}\ Let $\omega$ be a weight vector of degree $n$. We define two subspaces of $P_n^{\otimes h}$ associated with $\omega$ as follows: $P_n^{\otimes h}(\omega) = {\rm span}\{ t\in P_n^{\otimes h}\|\, \deg(t) = \deg(\omega) = n,\ \omega(t)\leq \omega\},$ and $P_n^{\otimes h}(< \omega) = {\rm span}\{t\in P_n^{\otimes h}\|\, \deg(t) = \deg(\omega) = n,\ \omega(t) < \omega\}.$ Let $u$ and $v$ be two homogeneous polynomials in $P_n^{\otimes h}.$ We define the equivalence relations "$\equiv$" and "$\equiv_{\omega}$" on $P_n^{\otimes h}$ by setting: $u\equiv v$ if and only if $(u+v)\in \overline{\mathcal {A}}P_n^{\otimes h}$, while $u \equiv_{\omega} v$ if and only if $u,\, v\in P_n^{\otimes h}(\omega) $ and $$(u +v)\in \overline{\mathcal {A}}P_n^{\otimes h} \cap P_n^{\otimes h}(\omega) + P_n^{\otimes h}(< \omega).$$ (In particular, if $u\equiv 0,$ then $u$ is a hit monomial. If $u \equiv_{\omega} 0,$ then $u$ is called {\it $\omega$-hit}.) We will denote the factor space of $P_n^{\otimes h}(\omega)$ by the equivalence relation $\equiv_{\omega}$ as $QP_n^{\otimes h}(\omega)$. According to \cite{Sum4, Walker-Wood}, this $QP_n^{\otimes h}(\omega)$ admits a natural $\mathbb F_2[GL_h]$-module structure, and the reader is recommended to \cite{Sum4} for a detailed proof. It is noteworthy that if we define $\widetilde{(QP^{\otimes h}_{n})^{\omega}}:= \langle \{[t]\in QP^{\otimes h}_{n}:\ \omega(t) = \omega,\, \mbox{ and $t$ is admissible} \}\rangle,$ then this $\widetilde{(QP^{\otimes h}_{n})^{\omega}}$ is an $\mathbb F_2$-subspace of $QP^{\otimes h}_n.$ Furthermore, the mapping $QP^{\otimes h}_{n}(\omega)\longrightarrow \widetilde{(QP^{\otimes h}_{n})^{\omega}}$ determined by $[t]_{\omega}\longmapsto [t]$ is an isomorphism. This implies that $ \dim QP^{\otimes h}_{n} = \sum_{\deg(\omega) = n}\dim\widetilde{(QP^{\otimes h}_{n})^{\omega}} = \sum_{\deg(\omega) = n}\dim QP^{\otimes h}_{n}(\omega).$ \medskip \begin{cy}\label{cyP} (i) The conjecture proposed by Kameko in the thesis \cite{Kameko} asserts that $\dim QP^{\otimes h}_{n}\leq \prod_{1\leq j\leq h}(2^{j}-1)$ for all values of $h$ and $n$. While this inequality has been proven for $h \leq 4$ and all $n$, counterexamples provided by Sum in \cite{Sum00, Sum2} demonstrate that it is wrong when $h > 4$. It is worth noting, however, that the local version of Kameko's conjecture, which concerns the inequality $\dim QP^{\otimes h}_{n}(\omega)\leq \prod_{1\leq j\leq h}(2^{j}-1)$, remains an open question. (ii) As it is known, the algebra of divided powers $[P^{\otimes h}]^{} = H_{}(BV_h)= \Gamma(a_1, a_2, \ldots, a_h)$ is generated by $a_1, \ldots, a_h,$ each of degree $1.$ Here $a_i = a_i^{(1)}$ is dual to $t_i\in P^{\otimes h}_1,$ with duality taken with respect to the basis of $P^{\otimes h}$ consisting of all monomials in $t_1, \ldots, t_h.$ Kameko defined in \cite{Kameko} a homomorphism of $\mathbb F_2[GL_h]$-modules $\widetilde{Sq}^{0}: [P^{\otimes h}]^{}=H_{}(BV_h)\longrightarrow [P^{\otimes h}]^{} = H_{}(BV_h),$ which is determined by $\widetilde{Sq}^{0}(a_1^{(i_1)}\ldots a_h^{(i_h)}) = a_1^{(2i_1+1)}\ldots a_h^{(2i_h+1)}.$ The dual of this $\widetilde{Sq}^{0}$ induced the homomorphism $(\widetilde {Sq^0_})_{2n+h}: QP^{\otimes h}_{2n+h} \longrightarrow QP^{\otimes h}_{n}$ (see Sect.\ref{s1}). Further, as $Sq^{2k+1}_\widetilde{Sq}^{0} = 0$ and $Sq_{}^{2k}\widetilde{Sq}^{0} = \widetilde{Sq}^{0}Sq_{}^{k},$ $\widetilde{Sq}^{0}$ maps ${\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{}$ to itself. Here we write $Sq^u_: H_{}(BV_h)\longrightarrow H_{-u}(BV_h)$ for the operation on homology which by duality of vector spaces is induced by the square $Sq^{u}: H_{}(BV_h)\longrightarrow H_{+u}(BV_h).$ The Kameko $Sq^{0}$ is defined by $Sq^{0}: \mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{}\longrightarrow \mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{},$ which commutes with the classical $Sq^{0}$ on the $\mathbb F_2$-cohomology of $\mathcal A$ through the Singer algebraic transfer. Thus, for any integer $n\geq 1,$ the following diagram commutes: $$ \begin{diagram} \node{(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_n} \arrow{e,t}{Tr_h^{\mathcal A}}\arrow{s,r}{Sq^0} \node{{\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)} \arrow{s,r}{Sq^0}\\ \node{(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_{2n+h}} \arrow{e,t}{Tr_h^{\mathcal A}} \node{{\rm Ext}_{\mathcal A}^{h, 2h+2n}(\mathbb F_2, \mathbb F_2)} \end{diagram}$$ Thus, Kameko's $Sq^0$ is known to be compatible via the Singer transfer with $Sq^0$ on ${\rm Ext}_{\mathcal A}^{,}(\mathbb F_2, \mathbb F_2)$. Moreover, the $GL_h$-coinvariants $(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_n$ form a bigraded algebra and the Singer algebraic transfers $Tr_^{\mathcal A}$ yield a morphism of bigraded algebras with values in ${\rm Ext}_{\mathcal A}^{,}(\mathbb F_2, \mathbb F_2).$ These compatibilities are suggestive of a far closer relationship between these structures. In addition, the operations $Sq^0$ and the algebra structure on ${\rm Ext}_{\mathcal A}^{,}(\mathbb F_2, \mathbb F_2)$ are key ingredients in understanding the image of the algebraic transfer. Unfortunately, detecting the image of the Singer transfer by mapping \textit{out} of ${\rm Ext}_{\mathcal A}^{,}(\mathbb F_2, \mathbb F_2)$ is not easy. For example, Lannes and Zarati \cite{LZ} constructed an algebraic approximation to the Hurewicz map: for an unstable $\mathcal A$-module $M$ this is of the form ${\rm Ext}_{\mathcal A}^{h, h+n}(\Sigma^{-h}M, \mathbb F_2)\longrightarrow [(\mathbb F_2\otimes_{\mathcal A}\mathscr R_hM)_n]^,$ where $\mathscr R_h$ is the $h$-th Singer functor (as defined by Lannes and Zarati). However, it is conjectured by H\uhorn ng \cite{Hung0} that this vanishes for $h>2$ in positive stem, an algebraic version of the long-standing and difficult \textit{generalized spherical class conjecture} in Algebraic topology, due to Curtis \cite{Curtis}. In \cite{Hung2}, H\uhorn ng and Powell proved the weaker result that this holds on the image of the transfer homomorphism. This illustrates the difficulty of studying the Singer transfer. An analogous diagram has also been established for the case of odd primes $p$ \cite{Minami}: $$ \begin{diagram} \node{(\mathbb F_p\otimes_{GL_h(\mathbb F_p)}{\rm Ann}_{\overline{\mathcal A_p}}H_(V; \mathbb F_p))_n} \arrow{e,t}{Tr_h^{\mathcal A_p}}\arrow{s,r}{Sq^{0}} \node{{\rm Ext}_{\mathcal A_p}^{h, h+n}(\mathbb F_p, \mathbb F_p)} \arrow{s,r}{Sq^{0}}\\ \node{(\mathbb F_p\otimes_{GL_h(\mathbb F_p)}{\rm Ann}_{\overline{\mathcal A_p}}H_(V; \mathbb F_p))_{p(n+h)-h}} \arrow{e,t}{Tr_h^{\mathcal A_p}} \node{{\rm Ext}_{\mathcal A_p}^{h, p(h+n)}(\mathbb F_p, \mathbb F_p)} \end{diagram}$$ Here, the left vertical arrow represents the Kameko $Sq^0,$ and the right vertical one represents the classical squaring operation. Our recent work \cite{Phuc14} proposes a conjecture that \textit{the transfer $Tr_h^{\mathcal A_p}$ is an injective map for all $1\leq h\leq 4$ and odd primes $p$.} We have also established the validity of this conjecture in certain generic degrees. \end{cy} {\bf (Strictly) inadmissible monomial.}\ We say that a monomial $t\in P_n^{\otimes h}$ is {\it inadmissible} if there exist monomials $z_1, z_2,\ldots, z_k\in P_n^{\otimes h}$ such that $z_j < t$ for $1\leq j\leq k$ and $t = \sum_{1\leq j\leq k}z_j + \sum_{m > 0}Sq^{m}(z_m),$ for some $m\in \mathbb N$ and $z_m\in P_{n-m}^{\otimes h},\, m<n.$ Then, $t$ is said to be {\it admissible} if it is not inadmissible. A monomial $t\in P_n^{\otimes h}$ is said to be {\it strictly inadmissible} if and only if there exist monomials $z_1, z_2,\ldots, z_k$ in $P_n^{\otimes h}$ such that $z_j < t$ for $1\leq j \leq k$ and $t = \sum_{1\leq j\leq k}z_j + \sum_{0\leq m \leq s-1}Sq^{2^{m}}(z_m),$ where $s = {\rm max}\{i\in\mathbb N: \omega_i(t) > 0\}$ and suitable polynomials $z_m\in P^{\otimes h}_{n-2^{m}}.$ \medskip Note that every strictly inadmissible monomial is inadmissible but the converse is not generally true. For example, consider the monomial $t = t_1t_2^{2}t_3^{2}t_4^{2}t_5^{6}t_6\in P^{\otimes 6}_{14},$ we see that this monomial is not strictly inadmissible, despite its inadmissibility. This can be demonstrated through the application of the Cartan formula, which yields $$t = Sq^{1}(t_1^{4}t_2t_3t_4t_5^{5}t_6) + Sq^{3}(t_1^{2}t_2t_3t_4t_5^{5}t_6) + Sq^{6}(t_1t_2t_3t_4t_5^{3}t_6) + \sum_{X<t}X.$$ \begin{therm}[{\bf Criteria on inadmissible monomials}]\label{dlKS} The following claims are each true: \begin{itemize} \item[(i)] Let $t$ and $z$ be monomials in $P^{\otimes h}.$ For an integer $r >0,$ assume that there exists an index $i>r$ such that $\omega_i(t) = 0.$ If $z$ is inadmissible, then $tz^{2^r}$ is, too {\rm (see Kameko \cite{Kameko})}; \medskip \item[(ii)] Let $z, w$ be monomials in $P^{\otimes h}$ and let $r$ be a positive integer. Suppose that there is an index $j > r$ such that $\omega_j(z) = 0$ and $\omega_r(z)\neq 0.$ If $z$ is strictly inadmissible, then so is, $zw^{2^{r}}$ {\rm (see Sum \cite{Sum2})}. \end{itemize} \end{therm} \medskip We shall heavily rely on the arithmetic function $\mu: \mathbb{N}\longrightarrow \mathbb{N},$ as well as Kameko's map $(\widetilde{Sq^0_})_{2n+h}: QP^{\otimes h}_{2n+h}\longrightarrow QP^{\otimes h}_{n}$, both of which are elucidated in Sect. \ref{s1}. The technical theorem below related to the $\mu$-function holds crucial significance. \begin{therm}\label{dlWS} The following statements are each true: \begin{itemize} \item[(i)] {\rm (cf. Sum \cite{Sum4})}. $\mu(n) = r\leq h$ if and only if there exists uniquely a sequence of positive integers $d_1 > d_2 > \cdots > d_{r-1} \geq d_r$ such that $n = \sum_{1\leq i\leq r}(2^{d_i} - 1).$ \medskip \item[(ii)] {\rm (cf. Wood \cite{Wood})}. For each positive integer $n,$ the space $QP^{\otimes h}_n$ is trivial if and only if $\mu(n) > h.$ \medskip \item[(iii)] {\rm (cf. Kameko \cite{Kameko})}. The homomorphism $(\widetilde {Sq^0_})_{2n+h}$ is an isomorphism of $\mathbb F_2$-vector spaces if and only if $\mu(2n+h) = h.$ \end{itemize} \end{therm} \begin{rems}\label{nxpt} In Sect.\ref{s1}, it was noted that the hit problem needs to be solved only in degrees of the form \eqref{pt}. Furthermore, in \cite[Introduction]{Sum2}, Sum made the remark that for every positive integer $n$, the condition $3\leq \mu(n)\leq h$ holds if and only if there exist uniquely positive integers $s$ and $r$ satisfying $1\leq \mu(n)-2\leq \mu(r) = \alpha(r+\mu(r))\leq \mu(n)-1$ and $n = \mu(n)(2^{s}-1) + r\cdot2^{s}.$ This can be demonstrated straightforwardly by utilizing Theorem \ref{dlWS}(i). Suppose that $\mu(n) = k\geq 3.$ Then, the "only if" part has been shown in \cite{Phuc4}. The "if" part is established as follows: if $n = k(2^s-1) + r\cdot 2^s$ and $1\leq k-2\leq \mu(r) = \alpha(r+\mu(r)) \leq k-1.$ Then, either $\mu(r) = k-2$ or $\mu(r) = k-1.$ We set $\mu(r) = \ell$ and see that by Theorem \ref{dlWS}(i), there exist uniquely a sequence of integers $c_1 > c_2>\ldots>c_{\ell-1} \geq c_{\ell} > 0$ such that $r = 2^{c_1} + 2^{c_2} + \cdots + 2^{c_{\ell-1}} + 2^{c_{\ell}} - \ell.$ Obviously, $\alpha(r + \ell) = \ell,$ and so, $n =k(2^s-1) + r\cdot2^{s} = 2^{c_1 + s} + 2^{c_2 + s} + \cdots + 2^{c_{\ell} + s} + 2^{s}(s-\ell) - s.$ Now, if $\ell = k-2,$ then $n = 2^{c_1 + s} + 2^{c_2 + s} + \cdots + 2^{c_{\ell} + s} + 2^{s}(s-\ell) - s = 2^{c_1 + s} + 2^{c_2 + s} + \cdots + 2^{c_{k-2} + s} + 2^{s} + 2^{s} - k.$ Let $u_i = c_i + s$ with $1\leq i\leq k-2$ and let $u_{k-1} = u_k = s.$ Since $u_1 > u_2 > \cdots > u_{k-2} > u_{k-1} = u_k,$ by Theorem \ref{dlWS}(i), $\mu(n) = k.$ Finally, if $\ell = k-1$ then $ n = 2^{c_1 + s} + 2^{c_2 + s} + \cdots + 2^{c_{k-1} + s} + 2^{s} - k.$ We put $v_i = c_i + s$ where $1\leq i\leq k-1$ and $v_k = s.$ Since $v_1 > v_2 > \cdots > v_{k-1} > v_k,$ according to Theorem \ref{dlWS}(i), one derives $\mu(n) = k.$ \end{rems} {\bf Spike monomial.}\ A monomial $t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}$ in $P^{\otimes h}$ is called a {\it spike} if every exponent $a_j$ is of the form $2^{\beta_j} - 1.$ In particular, if the exponents $\beta_j$ can be arranged to satisfy $\beta_1 > \beta_2 > \ldots > \beta_{r-1}\geq \beta_r \geq 1,$ where only the last two smallest exponents can be equal, and $\beta_j = 0$ for $ r < j \leq h,$ then the monomial $t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}$ is called a {\it minimal spike}. \medskip \begin{therm}[see Ph\'uc and Sum \cite{P.S1}]\label{dlPS} All the spikes in $P^{\otimes h}$ are admissible and their weight vectors are weakly decreasing. Furthermore, if a weight vector $\omega = (\omega_1, \omega_2, \ldots)$ is weakly decreasing and $\omega_1\leq h,$ then there is a spike $z\in P^{\otimes h}$ such that $\omega(z) = \omega.$ \end{therm} The subsequent information demonstrates the correlation between minimal spike and hit monomials. \begin{therm}[{\bf Singer's criterion on hit monomials} {\rm \cite{Singer1}}]\label{dlSin} Suppose that $t\in P^{\otimes h}$ and $\mu(\deg(t))\leq h.$ Consequently, if $z$ is a minimal spike in $P^{\otimes h}$ such that $\omega(t) < \omega(z),$ then $t\equiv 0$ (or equivalently, $t$ is hit). \end{therm} It is of importance to observe that the converse of Theorem \ref{dlSin} is generally not valid. As a case in point, let us consider $z = t_1^{31}t_2^{3}t_3^{3}t_4^{0}t_5^{0}\in P^{\otimes 5}_{37}$ and $t = t_1(t_2t_3t_4t_5)^{9}\in P^{\otimes 5}_{37}.$ One has $\mu(37) = 3 < 5,$ and $t = fg^{2^3},$ where $f = t_1t_2t_3t_4t_5$ and $g =t_2t_3t_4t_5.$ Then $\deg(f) = 5 < (2^{3}-1)\mu(\deg(g)),$ and so, due to Silverman \cite[Theorem 1.2]{Silverman2}, we must have $t\equiv 0.$ It can be observed that despite $z$ being the minimal spike of degree $37$ in the $\mathcal A$-module $P^{\otimes 5},$ the weight $\omega(t) = (5,0,0,4,0)$ exceeds the weight of $z,$ which is $\omega(z) = (3,3,1,1,1).$ The reader may also refer to \cite{Phuc16} for further information regarding the cohit module $QP^{\otimes 5}_{37}.$ \begin{notas} We will adopt the following notations for convenience and consistency: \begin{itemize} \item [$\bullet$] Let us denote by $(P^{\otimes h})^{0}:= {\rm span}\bigg\{\prod_{1\leq j\leq h}t_j^{\alpha_j} \in P^{\otimes h}\bigg\|\, \prod_{1\leq j\leq h}\alpha_j = 0\bigg\}$ and $(P^{\otimes h})^{> 0}:= {\rm span}\bigg \{\prod_{1\leq j\leq h}t_j^{\alpha_j} \in P^{\otimes h}\bigg\|\, \prod_{1\leq j\leq h}\alpha_j > 0\bigg\}.$ It can be readily observed that these spaces are $\mathcal A$-submodules of $P^{\otimes h}.$ Moreover, for each positive degree $n,$ we have $QP_n^{\otimes h} \cong (QP_n^{\otimes h})^0\,\bigoplus\, (QP_n^{\otimes h})^{>0},$ where $(QP_n^{\otimes h})^0:= (Q(P^{\otimes h})^{0})_n = (\mathbb F_2\otimes_{\mathcal A} (P^{\otimes h})^{0})_n$ and $(QP_n^{\otimes h})^{>0}:= (Q(P^{\otimes h})^{>0})_n = (\mathbb F_2\otimes_{\mathcal A} (P^{\otimes h})^{>0})_n$ are the $\mathbb F_2$-subspaces of $QP_n^{\otimes h}.$ \medskip \item [$\bullet$] Given a monomial $t\in P^{\otimes h}_n,$ we write $[t]$ as the equivalence class of $t$ in $QP^{\otimes h}_n.$ If $\omega$ is a weight vector of degree $n$ and $t\in P_n^{\otimes h}(\omega),$ we denote by $[t]_\omega$ the equivalence class of $t$ in $QP^{\otimes h}_n(\omega).$ Noting that if $\omega$ is a weight vector of a minimal spike, then $[t]_{\omega} = [t].$ For a subset $C\subset P^{\otimes h}_n,$ we will often write $\|C\|$ to denote the cardinality of $C$ and use notation $[C] = \{[t]\, :\, t\in C\}.$ If $C\subset P_n^{\otimes h}(\omega),$ then we denote $[C]_{\omega} = \{[t]_{\omega}\, :\, t\in C\}.$ \medskip \item [$\bullet$] Write $\mathscr {C}^{\otimes h}_{n},\, (\mathscr {C}^{\otimes h}_{n})^{0}$ and $(\mathscr {C}^{\otimes h}_{n})^{>0}$ as the sets of all the admissible monomials of degree $n$ in the $\mathcal A$-modules $P^{\otimes h},$\ $(P^{\otimes h})^{0}$ and $(P^{\otimes h})^{>0},$ respectively. If $\omega$ is a weight vector of degree $n.$ then we put $\mathscr {C}^{\otimes h}_{n}(\omega) := \mathscr {C}^{\otimes h}_{n}\cap P_n(\omega),$\ $(\mathscr {C}^{\otimes h}_{n})^{0}(\omega) := (\mathscr {C}^{\otimes h}_{n})^{0}\cap P_n^{\otimes h}(\omega),$ and $(\mathscr {C}^{\otimes h}_{n})^{>0}(\omega) := (\mathscr {C}^{\otimes h}_{n})^{>0}\cap P_n^{\otimes h}(\omega).$ \end{itemize} \end{notas} \section{Statement of main results}\label{s3} We are now able to present the principal findings of this paper. The demonstration of these results will be exhaustively expounded in subsequent section. As previously alluded to, our examination commences with a critical analysis of the hit problem for the polynomial algebra $P^{\otimes 6}$ in degree $n_s:= 6(2^{s}-1) + 10\cdot 2^{s}$, wherein $s$ is an arbitrary non-negative integer. \medskip {\bf Case $\pmb{s = 0.}$}\ Mothebe et al. demonstrated in \cite{MKR} the following outcome. \begin{therm}[see \cite{MKR}]\label{dlMKR} For each integer $h\geq 2,$ $\dim QP^{\otimes h}_{n_0} = \sum_{2\leq j\leq n_0}C_j\binom{h}{j},$ where $\binom{h}{j} = 0$ if $h < j$ and $C_2 = 2,$ $C_3 = 8,$ $C_4 = 26,$ $C_5 = 50,$ $C_6 = 65,$ $C_7 = 55,$ $C_8 = 28,$ $C_9 = 8,$ $C_{n_0} = 1.$ This means that there exist exactly $945$ admissible monomials of degree $n_0$ in the $\mathcal A$-module $P^{\otimes 6}.$ \end{therm} The following corollary is readily apparent. \begin{corls}\label{hq10-1} \begin{itemize} \item[(i)] One has an isomorphism of $\mathbb F_2$-vector spaces: $ QP^{\otimes 6}_{n_0}\cong \bigoplus_{1\leq j\leq 5} QP^{\otimes 6}_{n_0}(\overline{\omega}^{(j)}),$ where $\overline{\omega}^{(1)}:= (2,2,1),$\ $\overline{\omega}^{(2)}:=(2,4),$\ $\overline{\omega}^{(3)}:=(4,1,1),$\ $\overline{\omega}^{(4)}:=(4,3),$ and $\overline{\omega}^{(5)}:=(6,2).$ \item[(ii)] $(QP^{\otimes 6}_{n_0})^{0}\cong \bigoplus_{1\leq j\leq 4}(QP^{\otimes 6}_{n_0})^{0}(\overline{\omega}^{(j)})$ and $(QP^{\otimes 6}_{n_0})^{>0}\cong \bigoplus_{2\leq j\leq 5}(QP^{\otimes 6}_{n_0})^{>0}(\overline{\omega}^{(j)}),$ and \centerline{\begin{tabular}{c\|\|cccccc} $j$ &$1$ & $2$ & $3$ & $4$& $5$\cr \hline \hline \ $\dim (QP^{\otimes 6}_{n_0})^{0}(\overline{\omega}^{(j)})$ & $400$ & $30$ & $270$ &$180$ &$0$ \cr \hline \hline \ $\dim (QP^{\otimes 6}_{n_0})^{>0}(\overline{\omega}^{(j)})$ & $0$ & $4$ & $10$ &$36$ & $15$ \cr \end{tabular}} \end{itemize} \end{corls} It is worth noting that the epimorphism of $\mathbb{F}_2$-vector spaces, Kameko's squaring operation $(\widetilde {Sq^0_})_{n_0}: QP^{\otimes 6}_{n_0} \longrightarrow (QP^{\otimes 6}_{2})^{0}$, implies that $QP^{\otimes 6}_{n_0}$ is isomorphic to ${\rm Ker}((\widetilde {Sq^0_})_{n_0})\bigoplus \psi((QP_{2}^{\otimes 6})^{0})$. Here, $\psi: (QP^{\otimes 6}_{2})^{0}\longrightarrow QP^{\otimes 6}_{n_0}$ is induced by the up Kameko map $\psi: (P^{\otimes 6}_2)^{0}\longrightarrow P^{\otimes 6}_{n_0},\, t\longmapsto t_1t_2\ldots t_6t^{2}$. Hence, by virtue of Corollary \ref{hq10-1}, one has the isomorphisms: ${\rm Ker}((\widetilde {Sq^0_})_{n_0})\cong \bigoplus_{1\leq j\leq 4}QP^{\otimes 6}_{n_0}(\overline{\omega}^{(j)}),$ and $\psi((QP_{2}^{\otimes 6})^{0}) \cong QP^{\otimes 6}_{n_0}(\overline{\omega}^{(5)}).$ \begin{rems}\label{nxp10} Let us consider the set $\mathcal L_{h, k} = \{J = (j_1, \ldots, j_k):\, 1\leq j_1 < j_2 < \cdots < j_k\leq h \},\ 1\leq k < h.$ Obviously, $\|\mathcal L_{h, k}\| = \binom{h}{k}.$ For each $J\in \mathcal L_{h, k},$ we define the the homomorphism $\varphi_J: P^{\otimes k}\longrightarrow P^{\otimes h}$ of algebras by setting $\varphi_J(t_{u}) = t_{j_{u}},\ 1\leq u\leq h.$ It is straightforward to see that this homomorphism is also a homomorphism of $\mathcal A$-modules. For each $1\leq k<h,$ we have the isomorphism of $\mathbb F_2$-vector spaces $Q(\varphi_J((P^{\otimes k})^{>0}))_{n}(\omega) = (\mathbb F_2\otimes_{\mathcal A} \varphi_J((P^{\otimes k})^{>0}))_{n}\cong (QP_{n}^{\otimes k})^{>0}(\omega),$ wherein $\omega$ is a weight vector of degree $n.$ As a consequence of this, and based on the work of \cite{Walker-Wood}, we get $$ (QP_{n}^{\otimes h})^{0}(\omega)\cong \bigoplus_{1\leq k\leq h-1}\bigoplus_{J\in\mathcal L_{h, k}}Q(\varphi_J((P^{\otimes k})^{>0}))_{n}(\omega) \cong\bigoplus_{1\leq k\leq h-1}\bigoplus_{1\leq d\leq \binom{h}{k}}(QP_{n}^{\otimes k})^{>0}(\omega), $$ which implies $ \dim (QP_{n}^{\otimes h})^{0}(\omega) = \sum_{1\leq k\leq h-1}\binom{h}{k}\dim (QP_{n}^{\otimes k})^{> 0}(\omega).$ By utilizing Theorem \ref{dlWS}(ii) in combination, we obtain $$ \dim (QP_{n}^{\otimes h})^{0}(\omega) = \sum_{\mu(n)\leq k\leq h-1}\binom{h}{k}\dim (QP_{n}^{\otimes k})^{> 0}(\omega). $$ \end{rems} Through a straightforward calculation utilizing Theorem \ref{dlMKR}, we can claim the following. \begin{corls}\label{hq10-1-0} Let $\overline{\omega}^{(j)}$ be the weight vectors as in Corollary \ref{hq10-1} with $1\leq j\leq 5.$ Then, for each $h\geq 7,$ the dimension of $(QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{(j)})$ is determined by the following table: \centerline{\begin{tabular}{c\|\|cccccccccccccccc} $j$ &&$1$ && $2$ && $3$ && $4$&& $5$\cr \hline \hline \ $\dim (QP^{\otimes 7}_{n_0})^{>0}(\overline{\omega}^{(j)})$ && $0$ && $0$ && $0$ &&$20$ &&$35$ \cr \hline \hline \ $\dim (QP^{\otimes 8}_{n_0})^{>0}(\overline{\omega}^{(j)})$ && $0$ && $0$ && $0$ &&$0$ && $20$ \cr \hline \hline \ $\dim (QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{(j)}),\, h \geq 9$ && $0$ && $0$ && $0$ &&$0$ && $0$ \cr \end{tabular}} \end{corls} Through a basic computation, in conjunction with Remark \ref{nxp10}, Corollaries \ref{hq10-1}, \ref{hq10-1-0}, as well as the preceding outcomes established by \cite{Peterson}, \cite{Kameko}, and \cite{Sum2}, we are able to deduce the subsequent corollary. \begin{corls}\label{hq10-1-1} Let $\overline{\omega}^{(j)}$ be the weight vectors as in Corollary \ref{hq10-1} with $1\leq j\leq 5.$ Then, for each $h\geq 7,$ the dimension of $(QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{(j)})$ is given as follows: $$ \dim (QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{(j)}) = \left\{\begin{array}{ll} 2\bigg[\binom{h}{2} + 4\binom{h}{3} + 6\binom{h}{4}\bigg] + 5\binom{h}{5}&\mbox{if $j = 1$},\\[1mm] 5\binom{h}{5}+ 4\binom{h}{6} &\mbox{if $j = 2$},\\[1mm] 10\bigg[\binom{h}{4}+2\binom{h}{5} + \binom{h}{6}\bigg]&\mbox{if $j = 3$},\\[1mm] 812&\mbox{if $j = 4,\, h =7$},\\[1mm] 4\bigg[\binom{h}{4}+ 5\binom{h}{5}+9\binom{h}{6} + 5\binom{h}{7}\bigg] &\mbox{if $j = 4,\, h \geq 8$},\\[1mm] 105 &\mbox{if $j = 5,\, h =7$},\\[1mm] 700&\mbox{if $j = 5,\, h = 8$},\\[1mm] 5\bigg[3\binom{h}{6}+ 7\binom{h}{7}+ 4\binom{h}{8}\bigg]&\mbox{if $j = 5,\, h \geq 9$}. \end{array}\right.$$ \end{corls} As a direct implication of the findings presented in \cite{MKR}, we derive \begin{corls}\label{hq10-2} For each integer $h\geq 6,$ consider the following weight vectors of degree $n = h+4$: $$\overline{\omega}^{(1,\, h)}:= (h-4, 4),\ \ \overline{\omega}^{(2,\, h)}:= (h-2, 1,1),\ \ \overline{\omega}^{(3,\, h)}:= (h-2, 3),\ \ \overline{\omega}^{(4,\, h)}:= (h, 2).$$ Then, for each rank $h\geq 6,$ the dimension of $(QP^{\otimes h}_{h+4})^{>0}(\overline{\omega}^{(j,\, h)})$ is determined by the following table: \centerline{\begin{tabular}{c\|\|cccccccccccc} $j$ &&$1$ && $2$ && $3$ && $4$\cr \hline \hline \ $\dim (QP^{\otimes h}_{h+4})^{>0}(\overline{\omega}^{(j,\, h)})$ && $\binom{h-1}{4}-1$ && $\binom{h-1}{2}$ && $h\binom{h-2}{2}$ &&$\binom{h}{2}.$ \cr \end{tabular}} \end{corls} Owing to Corollary \ref{hq10-1}, one has $\overline{\omega}^{(j,\, h)} = \overline{\omega}^{(j+1)}$ for $h = 6$ and $1\leq j\leq 4.$ So we can infer that the dimension of $(QP^{\otimes 6}_{n_0})^{>0}(\overline{\omega}^{(j)}),\, 2\leq j\leq 5$ in Corollary \ref{hq10-1} can be derived from Corollary \ref{hq10-2}. Furthermore, in light of Corollaries \ref{hq10-1-0}, \ref{hq10-1-1}, \ref{hq10-2}, as well as the previous results established by Peterson \cite{Peterson}, Kameko \cite{Kameko}, Sum \cite{Sum2}, and Mothebe et al. \cite{MKR}, we are also able to confirm that a local version of Kameko's conjecture (as articulated in Note \ref{cyP}) holds true for certain weight vectors of degrees $h+4$, where $h\geq 1.$ \medskip As is well-known, Mothebe et al. \cite{MKR} computed the dimension of $QP_n^{\otimes h}$ for $h\geq 1$ and degrees $n$ satisfying $1\leq n\leq 9$. The following theorem provides more details. \begin{therm}[see \cite{MKR}]\label{dlMM} Given any $h\geq 1,$ the dimension of $QP_n^{\otimes h}$ is determined as follows: $$ \begin{array}{ll} \medskip \dim QP_1^{\otimes h} &=h ,\ \ \dim QP_2^{\otimes h} = \binom{h}{2},\ \ \dim QP_3^{\otimes h} = \sum_{1\leq j\leq 3}\binom{h}{j},\\ \medskip \dim QP_4^{\otimes h} &= 2\binom{h}{2} + 2\binom{h}{3}+\binom{h}{4},\ \ \dim QP_5^{\otimes h} = 3\binom{h}{3} + 3\binom{h}{4}+\binom{h}{5},\\ \medskip \dim QP_6^{\otimes h} &=\binom{h}{2} + 3\binom{h}{3}+6\binom{h}{4} + 4\binom{h}{5} + \binom{h}{6},\\ \medskip \dim QP_7^{\otimes h} &= \binom{h}{1} + \binom{h}{2}+4\binom{h}{3} + 9\binom{h}{4} + 10\binom{h}{5}+ 5\binom{h}{6} + \binom{h}{7},\\ \medskip \dim QP_8^{\otimes h} &= 3\binom{h}{2}+6\binom{h}{3} + 13\binom{h}{4} + 19\binom{h}{5}+ 15\binom{h}{6} + 6\binom{h}{7} + \binom{h}{8},\\ \medskip \dim QP_9^{\otimes h} &= 7\binom{h}{3} + 18\binom{h}{4} + 31\binom{h}{5}+ 34\binom{h}{6} + 21\binom{h}{7} + 7\binom{h}{8}+\binom{h}{9}, \end{array}$$ wherein the binomial coefficients $\binom{h}{k}$ are to be interpreted modulo 2 with the usual convention $\binom{h}{k} = 0$ if $k \geq h+1.$ \end{therm} The theorem has also been demonstrated by Peterson \cite{Peterson} for $h\leq 2$, by Kameko's thesis \cite{Kameko} for $h = 3$ and by Sum \cite{Sum2} for $h =4.$ Using Theorems \ref{dlMKR}, \ref{dlMM}, and Corollaries \ref{hq10-1-0}, \ref{hq10-1-1}, we aim to analyze the behavior of the Singer transfer in bidegree $(h, h+n)$ for $1\leq n\leq n_0$ and any $h\geq 1.$ As a result of our investigation, we establish the following first main result. \begin{therm}\label{dlc0} For any integer $n$ satisfying $1\leq n\leq n_0,$ the algebraic transfer $$Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_{n}\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)$$ is a trivial isomorphism for all $h\geq 1$, except for the cases of rank 5 in degree 9 and rank 6 in degree $n_0$. In these exceptional cases, $Tr_h^{\mathcal A}$ is a monomorphism. Consequently, Singer's Conjecture \ref{gtSinger} holds true in bidegrees $(h, h+n)$ for $h\geq 1$ and $1\leq n\leq n_0.$ \end{therm} The theorem has been proven by Singer \cite{Singer} for $1\leq h\leq 2$, by Boardman \cite{Boardman} for $h = 3$, by Sum \cite{Sum2-1} and the author \cite{Phuc12} for $h = 4,$ by Sum \cite{Sum0, Sum2-0, Sum3} for $h =5$ and $n = 4,\, 5,\, n_0,$ by Sum and T\'in \cite{Sum-Tin0, Sum-Tin} for $h = 5$ and $n =1,\, 2,\, 3,\, 7,\, 9.$ The present writer has established the theorem for the case $h=5$ and degree $n=6, 8,$ as well as for the cases $6\leq h\leq 8$ and any degree $n$, as shown in \cite{Phuc0, Phuc6, Phuc11}. It should be brought to the attention of the readers that, $Tr_h^{\mathcal A}$ is not an epimorphism for rank 5 in degree 9 \cite{Singer}, and also for rank 6 in degree $n_0$ \cite{CHa, CHa0}. These imply that $Ph_1\not\in {\rm Im}(Tr_5^{\mathcal A})$ and $h_1Ph_1\not\in {\rm Im}(Tr_6^{\mathcal A}),$ wherein $\{Ph_1\}\subset {\rm Ext}_{\mathcal A}^{5, 5+9}(\mathbb F_2, \mathbb F_2)$ and $\{h_1Ph_1\}\subset {\rm Ext}_{\mathcal A}^{6, 6+n_0}(\mathbb F_2, \mathbb F_2)$ are sets that generate ${\rm Ext}_{\mathcal A}^{5, 5+9}(\mathbb F_2, \mathbb F_2)$ and ${\rm Ext}_{\mathcal A}^{6, 6+n_0}(\mathbb F_2, \mathbb F_2),$ respectively. \medskip {\bf Case $\pmb{s = 1}.$}\ We notice that $n_1 = 6(2^{1}-1) + 10\cdot 2^{1} = 26$ and make the following observation. \begin{rems}\label{nxp0} Let us consider the Kameko map $(\widetilde {Sq^0_})_{n_1}: QP^{\otimes 6}_{n_1} \longrightarrow QP^{\otimes 6}_{n_0}$ which is an epimorphism of the $\mathbb F_2$-vector spaces and is determined by $(\widetilde {Sq^0_})_{n_1}([u]) = [t]$ if $u = t_1t_2\ldots t_6t^{2}$ and $(\widetilde {Sq^0_})_{n_1}([u]) = 0$ otherwise. Then, since the homomorphism $q: {\rm Ker}((\widetilde {Sq^0_})_{n_1})\longrightarrow QP^{\otimes 6}_{n_1}$ is an embedding, we have a short exact sequence of $\mathbb F_2$-vector spaces.: $0\longrightarrow {\rm Ker}((\widetilde {Sq^0_})_{n_1})\longrightarrow QP^{\otimes 6}_{n_1}\longrightarrow QP^{\otimes 6}_{n_0}\longrightarrow 0.$ Let us consider the up Kameko map $\psi: P_{n_0}^{\otimes 6}\longrightarrow P_{n_1}^{\otimes 6},$ which is determined by $\psi(t) = t_1t_2\ldots t_6t^{2}$ for any $t\in P_{n_0}^{\otimes 6}.$ This $\psi$ induces a homomorphism $\psi: QP_{n_0}^{\otimes 6}\longrightarrow QP_{n_1}^{\otimes 6}.$ These data imply that the above exact sequence is split and so, $QP^{\otimes 6}_{n_1} \cong {\rm Ker}((\widetilde {Sq^0_})_{n_1})\bigoplus QP^{\otimes 6}_{n_0}.$ Furthermore, as well known, $(QP^{\otimes 6}_{n_1})^{0}$ and ${\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}$ are the $\mathbb F_2$-vector subspaces of ${\rm Ker}((\widetilde {Sq^0_})_{n_1}$ and $QP^{\otimes 6}_{n_1}\cong (QP^{\otimes 6}_{n_1})^{0}\bigoplus (QP^{\otimes 6}_{n_1})^{>0},$ one gets $$ QP^{\otimes 6}_{n_1} \cong (QP^{\otimes 6}_{n_1})^{0}\bigoplus \big({\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}\big)\bigoplus QP^{\otimes 6}_{n_0}. $$ Through the combination of the previously mentioned remark along with the utilization of Corollary \ref{hq10-1}, we arrive at the following conclusion. \begin{corls}\label{hqMKR} We have an isomorphism of $\mathbb F_2$-vector spaces: $$ QP_{n_0}^{\otimes 6}\cong \langle \{[t_1t_2\ldots t_6t^{2}]:\, t\in \mathscr C^{\otimes 6}_{n_0}\} \rangle \cong \bigoplus_{1\leq j\leq 5}(QP^{\otimes 6}_{n_1})^{>0}(\overline{\omega}^{(j)}),$$ where $\overline{\omega}^{(1)}:= (6,2,2,1),$\ $\overline{\omega}^{(2)}:=(6,2,4),$\ $\overline{\omega}^{(3)}:= (6,4,1,1),$\ $\overline{\omega}^{(4)}:=(6,4,3)$ and $\overline{\omega}^{(5)}:=(6,6,2),$ and the dimension of $(QP^{\otimes 6}_{n_1})^{>0}(\overline{\omega}^{(j)}) $ is determined by $$\dim (QP^{\otimes 6}_{n_1})^{>0}(\overline{\omega}^{(j)}) = \dim (QP^{\otimes 6}_{n_0})^{0}(\overline{\omega}^{(j)})+\dim (QP^{\otimes 6}_{n_0})^{>0}(\overline{\omega}^{(j)}),\ \mbox{for $1\leq j\leq 5.$}$$ Here the dimensions of $(QP^{\otimes 6}_{n_0})^{0}(\overline{\omega}^{(j)})$ and $(QP^{\otimes 6}_{n_0})^{>0}(\overline{\omega}^{(j)})$ are given as in Corollary \ref{hq10-1}. \end{corls} \end{rems} We must now determine the dimensions of $(QP^{\otimes 6}_{n_1})^{0}$ and ${\rm Ker}((\widetilde {Sq^0})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}.$ To accomplish this, we invoke a well-known outcome concerning the dimension of $QP^{\otimes 5}$ at degree $n_1$. \begin{therm}[see Walker and Wood \cite{Walker-Wood2}]\label{dlWW} In any minimal generating set for the $\mathcal A$-module $P^{\otimes h},$ there are $2^{\binom{h}{2}}$ elements in degree $2^{h}-(h+1).$ Consequently, $QP^{\otimes 5}_{n_1}$ is an $\mathbb F_2$-vector of dimension $1024.$ \end{therm} Walker and Wood proved this by considering the special case of the Steinberg representation ${\rm St}_h,$ using the hook formula to count the number of semistandard Young tableaux. More precisely, they claim that by the hook formula, the dimension of the cohit module $QP^{\otimes h}_{2^{h}-h-1}$ is upper bounded by $2^{\binom{h}{2}}.$ The equality then follows from the first occurrence of the Steinberg representation in this degree. Thus, $QP^{\otimes h}_{2^{h}-h-1}\cong {\rm St}_h$ for the first occurrence degree $2^{h}-h-1.$ It would also be interesting to see that the dimension of this cohit module is equal to the order of the Borel subgroup $B_h$ of $GL_h.$ \medskip We also employ the following homomorphisms: Let $h$ be a fixed integer with $5\leq h\leq 6$, and for each $l\in\mathbb Z$ such that $1\leq l\leq h$, we define a homomorphism $\mathsf{q}_{l}: P^{\otimes (h-1)}\longrightarrow P^{\otimes h}$ of algebras by setting $\mathsf{q}_{l}(t_j) = t_j$ for $1\leq j \leq l-1$ and $\mathsf{q}_{l}(t_j) = t_{j+1}$ for $l\leq j \leq h-1.$ Obviously, this $\mathsf{q}_{l}$ is also a homomorphism of $\mathcal A$-modules. The following comment is a crucial factor in computing $(QP^{\otimes 6}_{n_1})^{0}$ and ${\rm Ker}((\widetilde {Sq^0})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}.$ \begin{rems}\label{nxp1} \begin{itemize} \item[(i)] It is patently obvious that the weight vector of the minimal spike $t_1^{15}t_2^{7}t_3^{3}t_4$ of degree $n_1$ in the $\mathcal A$-module $P^{\otimes 6}$ is $(4,3,2,1),$ In \cite{MKR}, Mothebe et.al proved that the cohit $QP^{\otimes 6}$ has dimension $1205$ in degree $11.$ So, $\omega(\underline{t})\in\big\{(3,2,1),\, (3,4),\, (5,1,1),\, (5,3)\big\}.$ As an immediate consequence of these results and Theorems \ref{dlKS}(i), \ref{dlSin}, we state that if $t$ is an admissible monomial in $P^{\otimes 6}_{n_1}$ such that $[t]$ belongs to the kernel of Kameko's map $(\widetilde {Sq^0_})_{n_1},$ then $\omega(t)\in \big\{(4, 3,2,1),\ (4, 3,4),\ (4, 5,1,1),\ (4, 5,3)\big\}$ and $t$ can represented as $t_it_jt_kt_{\ell}\underline{t}^2,$ where $\underline{t}$ is an admissible monomial of degree $11$ in $P^{\otimes 6}$ and $1\leq i<j<k<\ell\leq 6.$ \item[(ii)] Since $QP^{\otimes 6}_{n_1}(\omega) \cong (QP^{\otimes 6}_{n_1})^{0}(\omega)\bigoplus (QP^{\otimes 6}_{n_1})^{> 0}(\omega),$ where $\omega$ is a weight vector of degree $n_1,$ one obtains an isomorphism: $QP^{\otimes 6}_{n_1} \cong (QP^{\otimes 6}_{n_1})^{0}\bigoplus \big(\bigoplus_{\deg(\omega)=n_1}(QP^{\otimes 6}_{n_1})^{>0}(\omega)\big).$ On the other hand, by Remark \ref{nxp0} and Corollary \ref{hqMKR}, we infer that $$QP^{\otimes 6}_{n_1} \cong (QP^{\otimes 6}_{n_1})^{0}\bigoplus \big({\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}\big)\bigoplus \big(\bigoplus_{1\leq j\leq 5}(QP^{\otimes 6}_{n_1})^{>0}(\omega^{(j)})\big),$$ where $(QP^{\otimes 6}_{n_1})^{0}\subset {\rm Ker}((\widetilde {Sq^0_})_{n_1}).$ Hence, by invoking the aforementioned argument (i), an isomorphism will be established as follows: ${\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}\cong U_1\bigoplus U_2,$ wherein $$U_1:=(QP^{\otimes 6}_{n_1})^{>0}(4,5,1,1)\bigoplus (QP^{\otimes 6}_{n_1})^{>0}(4,5,3),\ \mbox{ and }\ U_2:= (QP^{\otimes 6}_{n_1})^{>0}(4,3,2,1)\bigoplus (QP^{\otimes 6}_{n_1})^{>0}(4,3,4).$$ \end{itemize} \end{rems} Drawing on Remark \ref{nxp1}, we establish the second main result of this paper. \begin{therm}\label{dlc1} With the above notation, the following assertions are true: \begin{itemize} \item[(i)] $ \dim (QP^{\otimes 6}_{n_1})^{0}(\omega) = \left\{\begin{array}{ll} 5184&\mbox{if $\omega = (4,3,2,1)$},\\[1mm] 0&\mbox{if $\omega\neq (4,3,2,1)$}. \end{array}\right.$\\[1mm] Consequently, $(QP^{\otimes 6}_{n_1})^{0}$ is isomorphic to $(QP^{\otimes 6}_{n_1})^{0}(4,3,2,1)$ and $(\mathscr C^{\otimes 6}_{n_1})^{0} = (\mathscr C^{\otimes 6}_{n_1})^{0}(4,3,2,1)$ has all $5184$ admissible monomials. \item[(ii)] $\dim U_1 = 546$ and $\dim U_2 = 3090.$ These imply that there exist exactly $9765$ admissible monomials of degree $n_1$ in the $\mathcal A$-module $P^{\otimes 6}.$ Consequently, the cohit $QP^{\otimes 6}_{n_1}$ is $9765$-dimensional. \end{itemize} \end{therm} We will now recall a previously established result on the Kameko squaring operation. \begin{therm}[see Kameko \cite{Kameko}]\label{dlMK} The homomorphism $(\widetilde {Sq^0_})_{2n+h}: QP^{\otimes h}_{2n+h} \longrightarrow QP^{\otimes h}_{n}$ is an isomorphism of the $\mathbb F_2$-vector spaces if and only if $\mu(2n+h) = h.$ Then, one has an inverse homomorphism $ \psi: QP^{\otimes h}_{n}\longrightarrow QP^{\otimes h}_{2n+h}$ of $(\widetilde {Sq^0_})_{2n+h},$ which is induced by the mapping $\psi: P^{\otimes h}\longrightarrow P^{\otimes h},$ $t\longmapsto \prod_{1\leq j\leq h}t_jt^{2}.$ \end{therm} Write $\mathbb F_q$ for the Galois field of size $q$ ($q$ being a power of the prime characteristic $p$ of this field), let $B_h(\mathbb F_q)$ be the Borel subgroup of the general linear group $GL_h(\mathbb F_q)$ over $\mathbb F_q.$ When $q = 2,$ we put $GL_h:= GL_h(\mathbb F_2)$ and $B_h:= B_h(\mathbb F_2).$ Note that $\mathbb F_{q = p^{m}}\cong \mathbb F_p^{\oplus m}$ as groups (in fact as $\mathbb F_p$-modules). For the sake of completeness, let us remind the readers that the algebra of Steenrod $q$-th reduced powers $\pmb {A}_q$ can be defined as an algebra over $\mathbb F_q$ by generators $\mathscr P^{j},\, j\geq 0,$ subject to the relation $\mathscr P^{0} = 1$ and the Adem relations, $\mathscr P^{a}\mathscr P^{b} = \sum_{0\leq j\leq [a/q]}(-1)^{i+j}\binom{(q-1)(b-j)-1}{a-qj}\mathscr P^{a+b-j}\mathscr P^{j},\ a<qb.$ For $q = p,$ as a subalgebra of the mod $p$ Steenrod algebra $\mathcal A_p,$ the element $\mathscr P^{k}$ is given the degree $2k(p-1),$ but for simplicity, one regrade $\pmb{A}_q$ by giving $\mathscr P^{j}$ the "reduced" degree $k.$ So, when $q = p =2,$ $\mathscr P^{k}$ will mean Steenrod squares $Sq^{k},$ and not $Sq^{2k}$ (see also \cite{Smith}). Consider an $h$-dimensional vector space $\pmb{V}_h$ over $\mathbb F_q,$ the symmetric power algebra $S(\pmb{V}^{}_h)$ on the dual $\pmb{V}_h^{} = {\rm Hom}_{\mathbb F_q}(\pmb{V}_h, \mathbb F_q)$ of $\pmb{V}_h$ is identified with the polynomial algebra $\mathbb F_q[t_1, \ldots, t_h],$ where $\deg(t_i) = 1$ for every $i$ and $\{t_1, \ldots, t_h\}$ is a basis of $\pmb{V}^{}_h.$ Applying Theorem \ref{dlMK} in conjunction with the work by Hai \cite{Hai}, we derive the following corollaries. \begin{corls} In degree $q^{h-1}-h,$ we have $$ \dim_{\mathbb F_q} (\mathbb F_q[t_1, \ldots t_h]/\overline{\pmb{A}}_q\mathbb F_q[t_1, \ldots t_h])_{q^{h-1}-h} = {\rm ord}(GL_{h-1}(\mathbb F_q)/B^{}_{h-1}(\mathbb F_q)) = \prod_{1\leq j\leq h-1}(q^{j}-1),$$ on which $B_{h-1}^{}(\mathbb F_q)\subset B_{h-1}(\mathbb F_q)\cap {\rm Ker}(\det)$ and each element of $B_{h-1}^{}(\mathbb F_q)$ has 1's in the main diagonal. Here $\det$ denotes the $\mathbb F_q$-linear map $GL_{h-1}(\mathbb F_q)\longrightarrow \mathbb F_q^{}.$ \end{corls} \begin{corls}\label{hqs22} Let $h \geq 6$ be a given fixed integer. Setting $n_{h, s}:=2^{s+4}-h,$ then, for each $s\geq h-5,$ we have $$ \dim QP^{\otimes h}_{n_{h, s}} = {\rm ord}(GL_{h-1}/B_{h-1}) = \prod_{1\leq j\leq h-1}(2^{j}-1).$$ Moreover, $QP^{\otimes h}_{n_{h, s}}\cong {\rm Ker}((\widetilde {Sq^0_})_{n_{h, h-5}})\bigoplus QP^{\otimes h}_{2^{h-2}-h}$ for any $s\geq h-5.$ \end{corls} Indeed, we have that the order of the Borel subgroup $B_h(\mathbb F_q)$ is $q^{\binom{h}{2}}\prod_{1\leq j\leq h}(q-1),$ since elements in the main diagonal are taken from $\mathbb F_q^{}$ and elements above to the main diagonal can be any element of $\mathbb F_q.$ The order of $GL_h(\mathbb F_q)$ is determined as follows: the first row $u_1$ of the matrix can be anything but the $0$-vector, so there are $q^{h}-1$ possibilities for the first row. For any one of these possibilities, the second row $u_2$ can be anything but a multiple of the first row, giving $q^{h}-q$ possibilities. For any choice $u_1,\, u_2$ of the first two rows, the third row can be anything but a linear combination of $u_1$ and $u_2.$ The number of linear combinations $\sum_{1\leq i\leq 2}\gamma_iu_i$ is just the number of choices for the pair $(\gamma_1,\gamma_2),$ and there are $q^{2}$ of these. It follows that for every $u_1$ and $u_2,$ there are $q^{h}-q^{2}$ possibilities for the third row. For any allowed choice $u_1, u_2, u_3,$ the fourth row can be anything except a linear combination $\sum_{1\leq i\leq 3}\gamma_iu_i$ of the first three rows. Thus for every allowed $u_1, u_2, u_3$ there are $q^{3}$ forbidden fourth rows, and therefore $q^{h}-q^{3}$ allowed fourth rows. In the same way, the number of non-singular matrices is $(q^{h}-1)(q^{h}-q)\ldots (q^{h}-q^{h-1}),$ and so, $${\rm ord}(GL_h(\mathbb F_q)) = \prod_{0\leq j\leq h-1}(q^{h}-q^{j}) = q^{\binom{h}{2}}\prod_{1\leq j\leq h}(q^{j}-1).$$ Given the $\mathbb F_q$-linear $\det: GL_h(\mathbb F_q)\longrightarrow \mathbb F_q^{},$ consider the subsets $B^{}_h(\mathbb F_q)$ of the groups $B_h(\mathbb F_q)\cap {\rm Ker}(\det),$ where each element of $B^{}_h(\mathbb F_q)$ has 1's in the main diagonal. Then, $B^{}_h(\mathbb F_q)$ is also a self-conjugate subgroup of $GL_h(\mathbb F_q).$ It is straightforward to see that the order of $B^{}_h(\mathbb F_q)$ is $q^{\binom{h}{2}}.$ In particular, when $q = 2,$ we have ${\rm Ker}(\det) = GL_h$ and $B^{}_h = B_h.$ Thus ${\rm ord}(B_h(\mathbb F_q)) = {\rm ord}(B^{}_h(\mathbb F_q))\prod_{1\leq j\leq h}(q-1)$ and ${\rm ord}(GL_h(\mathbb F_q)) = {\rm ord}(B^{}_h(\mathbb F_q))\prod_{1\leq j\leq h}(q^{j}-1).$ In \cite[Theorem 4]{Hai}, by considering a variant of a family of finite quotient rings of $\mathbb F_q[t_1, \ldots, t_h]$, Hai proved that the space of the indecomposable elements of $\mathbb F_q[t_1, \ldots, t_h]$ has dimension $(q-1)(q^{2}-1)\ldots (q^{h-1}-1)$ in degree $q^{h-1}-h.$ From these data, we get $$ \dim_{\mathbb F_q}(\mathbb F_q[t_1, \ldots t_h]/\overline{\pmb{A}}_q\mathbb F_q[t_1, \ldots t_h])_{q^{h-1}-h} = \prod_{1\leq j\leq h-1}(q^{j}-1) = {\rm ord}(GL_{h-1}(\mathbb F_q)/B^{}_{h-1}(\mathbb F_q)).$$ (The reader should also keep in mind that the product $\prod_{1\leq j\leq h-1}(q^{j}-1)$ is also a well known formula for the degree of a cuspidal character of $GL_h(\mathbb F_q).$ The cuspidal characters are of great importance for characters of $GL_h(\mathbb F_q)$ since each character of this linear group is build up from cuspidal characters.) Now, with the field $\mathbb F_2$ and degree $n_{h, s} = 2^{s+4}-h,$ since $$n_{h, s} = (2^{s+3}-1) + (2^{s+2}-1) + (2^{s+1}-1) + \cdots + (2^{s-(h-5)}-1)+(2^{s-(h-5)}-1),$$ $\mu(n_{h, s}) = h$ for any $s \geq h-4,$ and so, by Theorem \ref{dlMK}, the iterated Kameko squaring operation $(\widetilde {Sq^0_})^{s-h+5}_{n_{h, s}}: QP^{\otimes h}_{n_{h, s}} \longrightarrow QP^{\otimes h}_{n_{h, h-5}}$ is an isomorphism for every $s \geq h-5.$ Combining this with the facts that $\dim QP^{\otimes h}_{n_{h, h-5}} = \prod_{1\leq j\leq h-1}(2^{j}-1)$ and $B_{h-1}^{} = B_{h-1},$ we must have $$ \dim QP^{\otimes h}_{n_{h, s}} = \prod_{1\leq j\leq h-1}(2^{j}-1) = {\rm ord}(GL_{h-1}/B^{}_{h-1})= {\rm ord}(GL_{h-1}/B_{h-1}),\ \mbox{for all $s\geq h-5.$}$$ Moreover, as the Kameko homomorphism $(\widetilde {Sq^0_})_{n_{h, s}}: QP^{\otimes h}_{n_{h, s}}\longrightarrow QP^{\otimes h}_{n_{h, s-1}}$ is an epimorphism and $QP^{\otimes h}_{n_{h, s}}\cong QP^{\otimes h}_{n_{h, h-5}},$ one gets $QP^{\otimes h}_{n_{h, s}}\cong {\rm Ker}((\widetilde {Sq^0_})_{n_{h, h-5}})\bigoplus QP^{\otimes h}_{2^{h-2}-h}$ for arbitrary $s\geq h-5.$ \medskip Let us take notice that, in the case of $q=2$ and $h=6$, the dimensionality of $QP^{\otimes 6}_{n_1}$ is equal to $(2^{1}-1)\ldots (2^{6-1}-1) = 9765$, a result that can be gleaned from Theorem \ref{dlc1}. Therefore, our research stands independently of Hai's, and our approach is completely distinct. Furthermore our work offers a precise and unambiguous description of a monomial basis for the cohit module $QP^{\otimes 6}_{2^{6-1}-6= n_1}$, which serves as a representation of $GL_6$. Theoretically, our technique can be extended to any values of $h$ and $n.$ Nonetheless, the process of calculation becomes increasingly complex as the dimensions of $QP^{\otimes h}_n$ grow larger with increasing $h$ and $n$. \begin{rems} Consider general degree $n_h = 2^{h-2}-h,\, h\geq 4,$ we have $\dim QP^{\otimes 4}_{n_4} = \dim \mathbb F_2 = 1$, $\dim QP^{\otimes 5}_{n_5} = 7$ (see Theorem \ref{dlMM}) and $\dim QP^{\otimes 6}_{n_6} = 945$ (see Theorem \ref{dlMKR}). Given any $h\geq 7,$ by Corollary \ref{hqs22}, $\dim QP^{\otimes h}_{n_h} = 3.7\ldots (2^{h-1}-1)-\dim {\rm Ker}((\widetilde {Sq^0_})_{n_{h, h-5}}).$ Hence, in order to determine the dimension of $QP^{\otimes h}_{n_h}$ for all $h>6,$ it suffices to calculate the dimension of the kernel of Kameko's map $(\widetilde {Sq^0_})_{n_{h, h-5}}$. However, this aspect will be investigated in a separate study. Utilizing a result from Hai \cite[Corollary 3]{Hai}, we have $QP^{\otimes (h-2)}_{n_h}\cong {\rm St}_{h-2}\otimes_{\mathbb F_2}{\rm det}^{1},$ where ${\rm det}^{1}$ denotes the first power of the determinant representation of $GL_{h-2}$ and ${\rm St}_{h-2}$ is the Steinberg module (a.k.a the Steinberg representation). Remarkably, for $h = 8,\, 9,$ since the cohomology groups ${\rm Ext}_{\mathcal A}^{h-2, n_h + h-2}(\mathbb F_2, \mathbb F_2)$ are trivial \cite{Bruner}, the Singer conjecture is wrong if $\dim[QP^{\otimes (h-2)}_{n_h}]^{GL_{h-2}} > 0.$ For $h =4,$ one has an isomorphism $(\mathbb F_2 \otimes_{GL_2} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 2}]^{})_{0} \cong \mathbb F_2\cong {\rm Ext}_{\mathcal A}^{2, 2}(\mathbb F_2, \mathbb F_2),$ which implies that the Singer conjecture holds for bidegree $(2, 2).$ For $h = 5,\, 6,\, 7,$ Singer's conjecture for bidegree $(h-2, n_h+h-2)$ has been verified by Boardman \cite{Boardman} for $h = 5,$ by the present author \cite{Phuc12} for $h = 6$ and by Sum \cite{Sum3} for $h = 7.$ By these, it would also be of significant interest to determine explicit generators of $QP^{\otimes (h-2)}_{n_h}.$ The dimension of this cohit module was determined by Peterson \cite{Peterson} for $h = 4,$ by Kameko \cite{Kameko} for $h = 5,$ and by Sum \cite{Sum2, Sum3} for $h = 6,\, 7.$ (See also Theorems \ref{dlMKR} and \ref{dlMM} for the cases where $4\leq h\leq 6.$) \end{rems} Building upon Corollary \ref{hqs22} and the calculations in \cite{Tangora, Bruner, Bruner2, Lin2}, we can see that with degree $n_{h, s}$ as in Corollary \ref{hqs22}, $$ \begin{array}{ll} {\rm Ext}_{\mathcal A}^{7, 7+n_{7, s}}(\mathbb F_2, \mathbb F_2)=\left\{\begin{array}{ll} 0 &\mbox{if $s = 1,\, 4$},\\[1mm] \langle Q_2(0)\rangle &\mbox{if $s = 2$},\\[1mm] \langle \{Q_2(1), h_6D_2 \} \rangle &\mbox{if $s = 3$},\\[1mm] \end{array}\right.\\ \\ {\rm Ext}_{\mathcal A}^{8, 8+n_{8, s}}(\mathbb F_2, \mathbb F_2)=\left\{\begin{array}{ll} 0 &\mbox{if $s = 1,\, 2$},\\[1mm] \langle h_{6}Q_2(0)\rangle &\mbox{if $s = 3$},\\[1mm] \langle x_{n_{8, 4}, 8}\rangle &\mbox{if $s = 4$}, \end{array}\right. \end{array}$$ where $x_{n_{8, 4}, 8}$ is an indecomposable element. We believe that the following prediction would be of significant interest to investigate regarding Conjecture \ref{gtSinger} in high homological degrees. \newpage \begin{conj}\label{gtbsm} The family $\big\{Q_2(k):\, k\geq 0\big\}$ is a finite $Sq^{0}$-family. Furthermore, we have that: \begin{itemize} \item[(i)] the transfer $Tr_7^{\mathcal A}$ does not detect the non-zero elements $Q_2(0),\, Q_2(1)$ and $h_6D_2;$ \item[(ii)] the transfer $Tr_8^{\mathcal A}$ does not detect the non-zero elements $h_{6}Q_2(0)$ and $x_{n_{8, 4}, 8}.$ \end{itemize} \end{conj} Note that an $Sq^{0}$ -family is called \textit{finite} if it has only finitely many nonzero elements, \textit{infinite} if all of its elements are nonzero \cite{Hung}. Due to Corollary \ref{hqs22}, it is observed that the conjecture for items (i) and (ii) is valid under the following circumstances: $$ \begin{array}{ll} (\mathbb F_2 \otimes_{GL_7} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 7}]^{})_{n_{7, 1}}\cong (\mathbb F_2 \otimes_{GL_7} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 7}]^{})_{n_{7, 2}} \cong [{\rm Ker}((\widetilde {Sq^0_})_{n_{7, 2}})]^{GL_7} = 0,\\ (\mathbb F_2 \otimes_{GL_8} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 8}]^{})_{n_{8, 2}}\cong (\mathbb F_2 \otimes_{GL_8} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 8}]^{})_{n_{8, 3}} \cong [{\rm Ker}((\widetilde {Sq^0_})_{n_{8, 3}})]^{GL_8} = 0. \end{array}$$ Our approach for determining the domain of $Tr_7^{\mathcal A}$ with respect to degrees $n_{7, 1}$ and $n_{7, 2}$, as well as the domain of $Tr_8^{\mathcal A}$ with respect to degree $n_{8, 3}$, will involve the use of Theorems \ref{dlMM} and \ref{dlWW}, alongside Corollary \ref{hqs22}. Nevertheless, the calculation at hand seems to be rather daunting. Adopting an alternative perspective, H\uhorn ng \cite{Hung} proposed an interesting notion concerning a \textit{critical element} that exists within ${\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)$. Specifically, a non-zero element $\zeta$ in ${\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)$ is deemed \textit{critical} if it satisfies two conditions: firstly, $\mu(2n+h) = h$, and secondly, the image of $\zeta$ under the classical squaring operation $Sq^{0}$ is zero. It is well-established that $Sq^0$ is a monomorphism in positive stems of ${\rm Ext}_{\mathcal A}^{h, }(\mathbb F_2, \mathbb F_2)$ for $h < 5,$ thereby implying the absence of any critical element for $h < 5.$ Remarkably, H\uhorn ng's work \cite[Theorem 5.9]{Hung} states that Singer's Conjecture \ref{gtSinger} is not valid, if the algebraic transfer detects the critical elements. Now, given the non-zero elements $Q_2(1)$ and $h_6D_2$, we are able to deduce that $\mu(2{\rm Stem}(Q_2(1))+7) = \mu(2{\rm Stem}(h_6D_2)+7) = 7.$ Furthermore, it is worth noting that $Sq^{0}(Q_2(1)) = 0 = Sq^{0}(h_6D_2)$, an observation which can be attributed to the fact that ${\rm Ext}_{\mathcal A}^{7, 7+n_{7,4}}(\mathbb F_2, \mathbb F_2) = 0$, as previously discussed. Thus $Q_2(1)$ and $h_6D_2$ must be critical elements. By this reason, in the event that Conjecture \ref{gtbsm}(i) is proven to be false, it would entail the refutation of Singer's conjecture in general. \medskip We now turn our attention to the hit problem for $P^{\otimes 6}$ in degree $n_s$ with $s > 1.$ \medskip {\bf Cases $\pmb{s > 1}.$}\ By Theorem \ref{dlc1} and Corollary \ref{hqs22}, $\dim QP^{\otimes 6}_{n_s} = \dim QP^{\otimes 6}_{n_1} = 9765$ for any $s > 0.$ Moreover, since the iterated homomorphism $((\widetilde {Sq^0_})_{n_{s}})^{s-1}: QP^{\otimes 6}_{n_s} \longrightarrow QP^{\otimes 6}_{n_1}$ is an isomorphism, for every positive integer $s,$ we have immediately the below corollary. \begin{corls}\label{dlc2} For each integer $s\geq 2,$ the set $\big\{[t]:\ t\in \psi^{s-1}(\mathscr C^{\otimes 6}_{n_1})\big\}$ is a monomial basis of the $\mathbb F_2$-vector space $QP^{\otimes 6}_{n_s},$ on which $\psi: P^{\otimes 6}\longrightarrow P^{\otimes 6},\, t\longmapsto t_1t_2\ldots t_6t^{2}$ and $\psi^{s-1}(\mathscr C^{\otimes 6}_{n_1}) = \bigg\{\prod_{1\leq j\leq 6}t^{2^{s-1}-1}_ju^{2^{s-1}}:\ u\in \mathscr C^{\otimes 6}_{n_1}\bigg\}.$ \end{corls} The next contribution of this work is to apply the aforementioned results into the investigation of the cohit module $QP^{\otimes 7}$ in general degree $n_{s+5}$ and the behavior of the sixth algebraic transfer in internal degrees $n_s$ for any $s > 0.$ To achieve this goal, we will begin by recalling an interesting result on an inductive formula for the dimension of $QP^{\otimes h}_n.$ \begin{therm}[see Sum \cite{Sum2}]\label{dlS} Consider the degree $n$ of the form \eqref{pt} with $k = h-1,$ and $s, r$ positive integers such that $1\leq h-3\leq \mu(r)\leq h-2,$ and $\mu(r) = \alpha(r + \mu(r)).$ Then for each $s\geq h-1,$ we have $\dim QP^{\otimes h}_n = (2^{h}-1)\dim QP^{\otimes (h-1)}_r.$ \end{therm} \begin{rems}\label{nxP} With the general degrees $n_s:= (h-1)(2^{s}-1) + r\cdot 2^{s},$ assume there is a non-negative integer $\zeta$ such that $\zeta < s$ and $1\leq h-3\leq \mu(n_{\zeta}) = \alpha(n_{\zeta}+ \mu(n_{\zeta}))\leq h-2.$ Let us consider generic degrees of the form $k(2^{s-\zeta + h-1}-1) + r\cdot 2^{s-\zeta + h-1},$ where $k = h-1$, $r = n_{\zeta}$ and $s\geq \zeta\geq 0.$ Consequently, due to $\mu(r) = \alpha(r+\mu(r)),$ we have the following inductive formula, which is deduced from Theorem \ref{dlS} and the proof of this theorem on pages 445-446 of \cite{Sum2}: $$ \dim QP^{\otimes h}_{(h-1)(2^{s-\zeta + h-1}-1) + n_{\zeta}2^{s-\zeta + h-1}} = (2^{h}-1)\dim QP^{\otimes (h-1)}_{n_s},\ \mbox{ for every $s\geq \zeta.$}$$ Now, with $h = 7,$ $r = 10$, $\zeta = 1,$ and degree $n_s,$ we have $\mu(n_{1}) = 4 = \alpha(n_{1}+ \mu(n_{1})).$ Hence the following is immediate from Corollary \ref{hqs22} and Remark \ref{nxP}. \end{rems} \begin{corls}\label{dlc3} For every positive integer $s,$ the cohit module $QP^{\otimes 7}$ has dimension $1240155$ in degree $n_{s+5} = 6(2^{s+5}-1) + 10\cdot 2^{s+5}.$ \end{corls} As a consequence of Theorem \ref{dlc1} and the computations done in \cite{Tangora, Bruner, Bruner2}, we are able to establish the third main result of this paper. \begin{therm}\label{dlc4} For each integer $s > 0,$ the coinvariant $(\mathbb F_2 \otimes_{GL_6} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 6}]^{})_{n_s}$ is trivial. Consequently, the algebraic transfer $Tr_6^{\mathcal A}: (\mathbb F_2 \otimes_{GL_6} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 6}]^{})_{n_s}\longrightarrow {\rm Ext}_{\mathcal A}^{6, 6+n_s}(\mathbb F_2, \mathbb F_2)$ is a monomorphism, but it is not an epimorphism for $0< s < 3.$ This means that the transfer $Tr_6^{\mathcal A}$ does not detect the non-zero elements $h_4Ph_2 = h_2^{2}g_1\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$ and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2).$ When $s = 3,$ the transfer $Tr_6^{\mathcal A}$ is a trivial isomorphism. \end{therm} By invoking Theorems \ref{dlc0} and \ref{dlc4}, it is possible to deduce that in the bidegrees $(6, 6+n_s),$ Singer's transfer is a monomorphism, but not an epimorphism for $0\leq s\leq 2.$ In the case where $s\geq 3,$ the transfer is a trivial isomorphism if its codomain is zero, and a monomorphism otherwise. These lead to an immediate consequence \begin{corls}\label{hqbs} Conjecture \ref{gtSinger} is valid in the bidegrees of $(6, 6+n_s)$ for any non-negative integer $s$. \end{corls} {\bf Final remarks.} Drawing upon the findings in \cite{Tangora, Bruner, Bruner2, Lin, Chen, Chen2, Lin2} on the structure of ${\rm Ext}_{\mathcal A}^{s, }(\mathbb F_2, \mathbb F_2)$ for $s\leq 6,$ we end this section by presenting the following conjecture, which predicts the structure of the sixth cohomology group ${\rm Ext}_{\mathcal A}^{6, 6+n_s}(\mathbb F_2, \mathbb F_2)$ for all $s\geq 0$. \begin{conj}\label{gtP} With degree $n_s = 6(2^{s}-1) + 10\cdot 2^{s},$ we have \begin{equation} {\rm Ext}_{\mathcal A}^{6, 6+n_s}(\mathbb F_2, \mathbb F_2)=\left\{\begin{array}{ll} \langle h_1Ph_1 \rangle &\mbox{if $s = 0$},\\[1mm] \langle h_2^{2}g_1\rangle &\mbox{if $s = 1$},\\[1mm] \langle D_2 \rangle &\mbox{if $s = 2$},\\[1mm] \langle \{h_s^{2}h_{s+1}^{2}h_{s+2}^{2},\, h_{s+1}^{2}g_s= h_{s+1}h_{s+3}g_{s-1},\, h_sh_{s+2}f_{s-1},\, h_sh_{s+1}^{2}c_s\} \rangle = 0 &\mbox{if $s \geq 3$},\\[1mm] \end{array}\right. \end{equation} where $P$ denotes the Adams periodicity operator and \begin{equation} \begin{array}{ll} \medskip &h_1Ph_1=[\lambda_1\lambda_2 \lambda_0^3 \lambda_7 + \lambda_1^2 \lambda_2 \lambda_4 \lambda_1^2 + \lambda_1^3 \lambda_2 \lambda_4 \lambda_1 + \lambda_1^2 \lambda_2 \lambda_1^2 \lambda_4]\neq 0,\\ & h_2^{2}g_1= Sq^{0}(h_1Ph_1) = h_4Ph_2= [\lambda_{15}\lambda_4\lambda_0^3 \lambda_7 + \lambda_{15}\lambda_3\lambda_5 \lambda_1^3 + \lambda_{15}\lambda_3\lambda_2\lambda_4 \lambda_1^2 + \lambda_{15}\lambda_3\lambda_1\lambda_2 \lambda_4 \lambda_1 \\ \medskip &\hspace{5cm} + \lambda_{15}\lambda_3\lambda_2\lambda_1^2\lambda_4 + \lambda_{15}\lambda_2\lambda_2 \lambda_0^2 \lambda_7 + \lambda_{15}\lambda_1\lambda_1 \lambda_2\lambda_0 \lambda_7]\neq 0,\\ &D_2 = [\lambda_0^{4}\lambda_{11}\lambda_{47}]\neq 0. \end{array} \end{equation} \end{conj} Note that $h_{s+1}g_s = h_{s+3}g_{s-1}$ for any $s\geq 2.$ Given the calculations presented in \cite{Tangora, Bruner, Lin2}, it has been unequivocally established that the conjecture holds for $s\leq 3.$ Additionally, if the conjecture is confirmed to be accurate in general, then Singer's algebraic transfer would be a trivial isomorphism in the bidegrees $(6, 6+n_s)$ for $s\geq 3$. The readers will observe that Singer's Conjecture \ref{gtSinger} in these bidegrees would be disproven if the dimension of the invariant $[QP^{\otimes 6}_{n_1}]^{GL_6}$ is equal to 1. However, as demonstrated in Theorem \ref{dlc4}, this eventuality did not transpire. Inspired by the calculations in \cite{Bruner2}, we are confident that Conjecture \ref{gtP} also holds true for all $s > 3$. From another perspective, by virtue of the calculations set forth in the works \cite{Lin, Chen2, Bruner2}, and by making use of the fundamental property that $Sq^{0}$ is an algebraic homomorphism, it follows that $Sq^{0}(h_2^{2}g_1) = Sq^{0}(h_4Ph_2)= h_5h_3g_1 = 0.$ On the other hand, in \cite{Bruner}, Bruner claimed that ${\rm Ext}_{\mathcal A}^{6, 6+n_3}(\mathbb F_2, \mathbb F_2)$ is trivial. So, $Sq^{0}$ must send the indecomposable element $D_2$ to zero. Thus, since $\mu(2n_1+6) = 6= \mu(2n_2+6),$ both $h_2^{2}g_1$ and $D_2$ are critical elements. (Additionally, considering the fact that $2^{4} = {\rm Stem}(Ph_2) +5 < 4({\rm Stem}(Ph_2))^2$ and that $Ph_2\in {\rm Ext}_{\mathcal A}^{5, 16}(\mathbb F_2, \mathbb F_2)$ is a critical element, as noted in \cite{Hung}, it follows that $h_2^{2}g_1=h_4Ph_2 $ is a critical element. This explanation further strengthens the aforementioned assertion.) An interesting observation from H\uhorn ng's paper \cite[Lemma 5.3]{Hung} is that the condition $2^{m}\geq \max\big\{4d^{2}, d+h\big\}$ is insufficient to identify all critical elements of the form $h_mx$ in ${\rm Ext}_{\mathcal A}^{h, }(\mathbb F_2, \mathbb F_2)$ when $x$ is also a critical element and $d = {\rm Stem}(x).$ This can be inferred from the foregoing facts that $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2)$ and $h_6D_2\in {\rm Ext}_{\mathcal A}^{7, 7+n_{7, 3}}(\mathbb F_2, \mathbb F_2)$ are critical elements, but $2^{6} < \max\big\{4({\rm Stem}(D_2))^{2},\, {\rm Stem}(D_2)+6\big\}.$ In summary, even thourgh the elements $h_4Ph_2$ and $D_2$ are critical, they cannot be detected by the algebraic transfer. (It should be brought to the attention of the readers that the research conducted by Qu\`ynh \cite{Quynh} demonstrates that the indecomposable element $Ph_2$ does not belong to the image of the fifth transfer.) This reinforces the conclusion that Conjecture \ref{gtSinger} continues to hold for the bidegrees $(6, 6+n_1)$ and $(6, 6+n_2),$ as established in Theorem \ref{dlc4} and Corollary \ref{hqbs}. \section{Proofs of the main results}\label{s4} This section is devoted to proving Theorems \ref{dlc0}, \ref{dlc1} and \ref{dlc4}. To begin with, we need the following homomorphisms and a helpful remark below. One should note that $V_h\cong \langle \{t_1, \ldots, t_h\}\rangle \subset P^{\otimes h}.$ For $1\leq d\leq h,$ we define the $\mathbb F_2$-linear map $\sigma_d: V_h\longrightarrow V_h$ by setting $$ \left\{\begin{array}{ll} \sigma_d(t_d) = t_{d+1},\\[1mm] \sigma_d(t_{d+1}) = t_d,\\[1mm] \sigma_d(t_i) = t_i,\ \mbox{for $i\neq d, d +1,\; 1\leq d\leq h-1,$}\\[1mm] \sigma_h(t_1) = t_1 + t_2,\ \ \sigma_h(t_i) = t_i,\ \mbox{for $2\leq i \leq h.$} \end{array}\right.$$ \medskip Denote by $\Sigma_h\subset GL_h$ the symmetric group of degree $h.$ Then, $\Sigma_h$ is generated by the ones associated with $\sigma_1,\, \ldots, \sigma_{h-1}.$ For each permutation in $\Sigma_h$, consider corresponding permutation matrix; these form a group of matrices isomorphic to $\Sigma_h.$ Indeed, consider the following map $\Delta: \Sigma_h\longrightarrow \mathcal P_{h\times h},$ where the latter is the set of permutation matrices of order $h.$ This map is defined as follows: given $\sigma\in \Sigma_h,$ the $i$-th column of $\Delta(\sigma)$ is the column vector with a $1$ in the $\rho(i)$-th position, and $0$ elsewhere. It is easy to see that $\Delta(\rho)$ is indeed a permutation matrix, since a $1$ occurs in any position if and only if that position is described by $(\rho(i), i),$ for any $1\leq i\leq h.$ The map $\Delta$ is clearly multiplicative. (It is to be noted that because these are matrices, it is enough to show that each corresponding entry is equal. So let us take the entry $(i, j)$ of each matrix.) Then, $\Delta(\rho\circ\rho')_{ij} =1$ if and only if $i = \rho\circ \rho'(j).$ Note also that by ordinary matrix multiplication, one has $(\Delta(\rho)\Delta(\rho'))_{ij} = \sum_{1\leq k\leq h}\Delta(\rho)_{ik}\Delta(\rho')_{kj}.$ Now, we know that $\Delta(\rho)_{ik} = 1$ only when $i = \rho(k).$ Similarly, $\Delta(\rho')_{kj} = 1$ only when $k = \rho'(j).$ Hence, their product is one precisely when both of these happen: $i = \rho(k),$ and $k = \rho'(j).$ If both these do not happen simultaneously, then whenever one of $\Delta(\rho)_{ik},\, \Delta(\rho')_{kj}$ is one of the other will be zero, so the whole sum will be zero. However, this is the same as saying that the sum is one exactly when $i = \rho\circ\rho'(j).$ This description matches with the description for $\Delta(\rho\circ \rho')_{ij}$ given earlier. Hence, entry by entry these matrices are the same. Therefore the matrices are the same, and hence $\Delta$ is a homomorphism between the two spaces, an isomorphism as it has trivial kernel and the sets are of the same cardinality. Thus, $GL_h\cong GL(V_h),$ and $GL_h$ is generated by the matrices associated with $\sigma_1, \ldots, \sigma_h.$ \medskip Let $T = t_1^{a_1}t_2^{a_2}\ldots t_h^{a_h}$ be a monomial in $P_n^{\otimes h}$. Then, the weight vector $\omega(T)$ is invariant under the permutation of the generators $t_j,\ j = 1, 2, \ldots, h;$ hence $QP_n^{\otimes h}(\omega(T))$ also has a $\Sigma_h$-module structure. We see that the linear map $\sigma_d$ induces a homomorphism of $\mathcal A$-algebras which is also denoted by $\sigma_d: P^{\otimes h}\longrightarrow P^{\otimes h}.$ So, a class $[T]_{\omega(T)}\in QP_n^{\otimes h}(\omega)$ is an $GL_h$-invariant if and only if $\sigma_d(T) \equiv_{\omega(T)} T$ for $1\leq d\leq h.$ If $\sigma_d(T) \equiv_{\omega(T)} T$ for $1\leq d\leq h-1,$ then $[T]_{\omega(T)}$ is an $\Sigma_h$-invariant. (We must stress that the explicit calculation of the $GL_h$-invariants of $QP_n^{\otimes h}(\omega)$ in every positive degree $n$ is a non-trivial undertaking. Nonetheless, this computation becomes significantly more tractable when a monomial basis of $QP_n^{\otimes h}(\omega)$ is precisely determined.) \subsection{Proof of Theorem \ref{dlc0}}\label{s2.0} Undoubtedly if $h > n,$ then $QP^{\otimes h}_{n} \cong (QP^{\otimes h}_{n})^{0}.$ So, the coinvariant $(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_{n}$ vanishes for any $1\leq n\leq n_0$ and $h\geq n+1.$ Let us consider the following weight vectors: $$ {\omega}^{}_{(1)}:=(3,1,1), \ \ {\omega}^{}_{(2)}:=(3,3), \ \ {\omega}^{}_{(3)}:=(5,2), \ \ {\omega}^{}_{(4)}:=(7,1),\ \ {\omega}^{}_{(5)}:=(9,0).$$ Consequently $\deg({\omega}^{}_{(1)}) = \deg({\omega}^{}_{(2)}) = \deg({\omega}^{}_{(3)}) = \deg({\omega}^{}_{(4)}) = \deg({\omega}^{}_{(5)}) = 9.$ It can be seen that $(QP^{\otimes 9}_{9})^{>0}\cong (\widetilde {Sq^0_})_{9}(QP^{\otimes 9}_{9}) \cong \mathbb F_2$, and so, $(QP^{\otimes 9}_{9})^{>0}= \mathbb F_2\big[\prod_{1\leq i\leq 9}t_i\big]_{\omega^{}_{(5)}}.$ In combination with the earlier studies by Peterson \cite{Peterson}, Kameko \cite{Kameko}, Sum \cite{Sum2}, Sum and T'in \cite{Sum-Tin}, and Mothebe et al. \cite{MKR}, the following isomorphisms are obtained: $$ (QP^{\otimes h}_{9})^{>0}\cong \left\{\begin{array}{ll} (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(1)})\bigoplus (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(2)})&\mbox{if $3\leq h\leq 4$},\\[2mm] (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(1)})\bigoplus (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(2)})\bigoplus (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(3)})&\mbox{if $h = 5$},\\[2mm] (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(2)})\bigoplus (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(3)})&\mbox{if $h = 6$},\\[2mm] (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(3)})\bigoplus (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(4)})&\mbox{if $h = 7$},\\[2mm] (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(4)})&\mbox{if $h = 8$},\\[2mm] (QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(5)})&\mbox{if $h = 9$},\\[2mm] O&\mbox{if $h \geq n_0.$} \end{array}\right.$$ Hence the dimensions of the indecomposables $(QP^{\otimes h}_{9})^{>0}({\omega}^{}_{(j)})$ are determined as follows: \centerline{\begin{tabular}{c\|\|cccccccccccccccc} $j$ &&$1$ && $2$ && $3$ && $4$ && $5$ \cr \hline \hline \ $\dim (QP^{\otimes 3}_{9})^{>0}(\omega^{}_{(j)})$ && $6$ && $1$ && $0$ &&$0$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 4}_{9})^{>0}({\omega}^{}_{(j)})$ && $12$ && $6$ && $0$ &&$0$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 5}_{9})^{>0}({\omega}^{}_{(j)})$ && $6$ && $15$ && $10$ &&$0$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 6}_{9})^{>0}({\omega}^{}_{(j)})$ && $0$ && $10$ && $24$ &&$0$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 7}_{9})^{>0}({\omega}^{}_{(j)})$ && $0$ && $0$ && $14$ &&$7$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 8}_{9})^{>0}({\omega}^{}_{(j)})$ && $0$ && $0$ && $0$ &&$7$ &&$0$ \cr \hline \hline \ $\dim (QP^{\otimes 9}_{9})^{>0}({\omega}^{}_{(j)})$ && $0$ && $0$ && $0$ &&$0$ &&$1$ \cr \end{tabular}} Through a straightforward calculation utilizing the aforementioned data and Remark \ref{nxp10}, we obtain $$ \dim (QP^{\otimes h}_{9})^{0}(\omega^{}_{(j)})= \left\{\begin{array}{ll} 6\binom{h}{3} &\mbox{if $j = 1$ and $h=4$},\\[2mm] 6\binom{h}{3} + 12\binom{h}{4}&\mbox{if $j = 1$ and $h=5$},\\[2mm] 6\binom{h}{3} + 12\binom{h}{4} + 6\binom{h}{5}&\mbox{if $j = 1$ and $h\geq 6$},\\[2mm] \binom{h}{3} &\mbox{if $j = 2$ and $h=4$},\\[2mm] \binom{h}{3} + 6\binom{h}{4}&\mbox{if $j = 2$ and $h=5$},\\[2mm] \binom{h}{3} + 6\binom{h}{4} + 15\binom{h}{5}&\mbox{if $j = 2$ and $h=6$},\\[2mm] \binom{h}{3} + 6\binom{h}{4} + 15\binom{h}{5} + 10\binom{h}{6}&\mbox{if $j = 2$ and $h\geq 7$},\\[2mm] 10\binom{h}{5} &\mbox{if $j = 3$ and $h=6$},\\[2mm] 10\binom{h}{5} + 24\binom{h}{6}&\mbox{if $j = 3$ and $h=7$},\\[2mm] 10\binom{h}{5} + 24\binom{h}{6} + 14\binom{h}{7} &\mbox{if $j = 3$ and $h\geq 8$},\\[2mm] 7\binom{h}{7} &\mbox{if $j = 4$ and $h=8$},\\[2mm] 7\bigg(\binom{h}{7} + \binom{h}{8}\bigg) &\mbox{if $j = 4$ and $h\geq 9.$} \end{array}\right.$$ Then, for each $h\geq n_0$ we have an isomorphism $QP^{\otimes h}_{9} \cong \bigoplus_{1\leq j\leq 5}(QP^{\otimes h}_{9})^{0}(\omega^{}_{(j)}).$ \medskip $\bullet$ For $h = 9$ and $n = 9,$ since $(\widetilde {Sq^0_})_{9}: QP^{\otimes 9}_{9} \longrightarrow \mathbb F_2$ is an epimorphism, $$QP^{\otimes 9}_{9} \cong \mathbb F_2\bigoplus {\rm Ker}((\widetilde {Sq^0_})_{9}) \cong \mathbb F_2\big[\prod_{1\leq i\leq 9}t_i\big]_{\omega^{}_{(5)}}\bigoplus \big(\bigoplus_{1\leq j\leq 4}(QP^{\otimes 9}_{9})^{0}(\omega^{}_{(j)})\big).$$ This shows that $(QP^{\otimes 9}_{9})^{0}\cong \bigoplus_{1\leq j\leq 4}(QP^{\otimes 9}_{9})^{0}(\omega^{}_{(j)}).$ Hence, $\dim (QP^{\otimes 9}_{9})^{0} = \sum_{1\leq j\leq 4}\dim (QP^{\otimes 9}_{9})^{0}(\omega^{}_{(j)}) = 10437$ and $\dim QP^{\otimes 9}_{9} = 10438.$ Using this result and the homomorphisms $\sigma_d: P^{\otimes 9}\longrightarrow P^{\otimes 9},\, 1\leq d\leq 9,$ we claim that $[QP^{\otimes 9}_{9}]^{GL_9}$ is zero, and so is, $(\mathbb F_2\otimes_{GL_9}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 9}]^{})_{9}.$ \medskip $\bullet$ For $9\leq h\leq n_0$ and $n = n_0,$ by a simple computation using Theorem \ref{dlMKR} and Corollaries \ref{hq10-1-0}, \ref{hq10-1-1}, one has the following isomorphisms: $$ \begin{array}{ll} \medskip QP^{\otimes h}_{n_0}&\cong \bigoplus_{1\leq j\leq 6}(QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{j})\bigoplus (QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{6}),\ \mbox{for $h = 9$},\\ QP^{\otimes h}_{n_0}&\cong \bigoplus_{1\leq j\leq 6}(QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{j})\bigoplus (QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{7}),\ \mbox{for $h = n_0,$} \end{array}$$ where $\overline{\omega}^{6} := (8,1)$ and $\overline{\omega}^{7} := (n_0,0).$ It is to be noted that $\bigoplus_{1\leq j\leq 6}(QP^{\otimes n_0}_{n_0})^{0}(\overline{\omega}^{j}) \cong {\rm Ker}((\widetilde {Sq^0_})_{n_0}),$ where ${\rm Ker}((\widetilde {Sq^0_})_{n_0})$ is the kernel of the Kameko homomorphism $(\widetilde {Sq^0_})_{n_0}: QP^{\otimes n_0}_{n_0}\longrightarrow \mathbb F_2.$ The dimensions of the cohit spaces $(QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{j}),\, 1\leq j\leq 5$ are explicitly determined as in Corollary \ref{hq10-1-1}. Based on Theorem \ref{dlMKR} and direct calculations, we find that $$ \dim (QP^{\otimes h}_{n_0})^{0}(\overline{\omega}^{6}) = \left\{\begin{array}{ll} 72&\mbox{if $h = 9$},\\[1mm] 8\bigg(\binom{n_0}{8} + \binom{n_0}{9}\bigg) = 125 &\mbox{if $h = n_0$}, \end{array}\right.$$ $$ \dim (QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{j}) = \left\{\begin{array}{ll} 8&\mbox{if $h = 9$ and $j = 6$},\\[1mm] \dim \mathbb F_2 = 1 &\mbox{if $h = n_0$ and $j = 7$}. \end{array}\right.$$ Using these data and the homomorphisms $\sigma_d,\, 1\leq d\leq n_0,$ we state that $[QP^{n_0}_{n_0}]^{GL_{n_0}} = 0$ and that for each $1\leq j\leq 6,$ the invariant $[(QP^{\otimes h}_{n_0})^{>0}(\overline{\omega}^{j})]^{GL_9}$ is zero, and so is, $(\mathbb F_2\otimes_{GL_9}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 9}]^{})_{n_0}.$ We will describe explicitly $[(QP^{\otimes 9}_{n_0})^{>0}(\overline{\omega}^{j})]^{GL_9}$ for $j = 6.$ The others can be obtained by similar calculations. As shown above, $(QP^{\otimes 9}_{n_0})^{>0}(\overline{\omega}^{6})$ is an $\mathbb F_2$-vector space of dimension $8$ with a monomial basis represented by the following admissible monomials: \begin{center} \begin{tabular}{llr} ${\rm a}_{73}=t_1t_2t_3t_4t_5t_6t_7t_8t_9^{2}$, & ${\rm a}_{74}=t_1t_2t_3t_4t_5t_6t_7t_8^{2}t_9$, & \multicolumn{1}{l}{${\rm a}_{75}=t_1t_2t_3t_4t_5t_6t_7^{2}t_8t_9$,} \\ ${\rm a}_{76}=t_1t_2t_3t_4t_5t_6^{2}t_7t_8t_9$, & ${\rm a}_{77}=t_1t_2t_3t_4t_5^{2}t_6t_7t_8t_9$, & \multicolumn{1}{l}{${\rm a}_{78}=t_1t_2t_3t_4^{2}t_5t_6t_7t_8t_9$,} \\ ${\rm a}_{79}=t_1t_2t_3^{2}t_4t_5t_6t_7t_8t_9$, & ${\rm a}_{80}=t_1t_2^{2}t_3t_4t_5t_6t_7t_8t_9.$ & \end{tabular}\end{center} Suppose $[f]_{\overline{\omega}^{6}}\in [(QP^{\otimes 9}_{n_0})^{>0}(\overline{\omega}^{6})]^{\Sigma_9}.$ Then, we have $f\equiv_{\overline{\omega}^{6}} \sum_{73\leq i\leq 80} \gamma_{i}{\rm a}_{i}$ where $\gamma_i\in \mathbb F_2$ for every $i.$ Let us consider the homomorphisms $\sigma_d: P^{\otimes 9}\longrightarrow P^{\otimes 9},\, 1\leq d\leq 8.$ An easy calculation shows: $$ \begin{array}{ll} \sigma_1(f) &\equiv_{\overline{\omega}^{6}}\sum_{73\leq j\leq 79} \gamma_{j}{\rm a}_{j} + \gamma_{80}t_1^{2}t_2t_3t_4t_5t_6^{2}t_7t_8t_9\\ &\equiv_{\overline{\omega}^{6}} \sum_{73\leq j\leq 79}(\gamma_{j} + \gamma_{80}){\rm a}_{j} + \gamma_{80}{\rm a}_{80},\\ &(\mbox{since $t_1^{2}t_2t_3t_4t_5t_6^{2}t_7t_8t_9 = Sq^{1}(t_1t_2t_3t_4t_5t_6^{2}t_7t_8t_9) + \sum_{73\leq j\leq 80}{\rm a}_{j}$}),\\ \medskip \sigma_2(f) &\equiv_{\overline{\omega}^{6}}\sum_{73\leq j\leq 78} \gamma_{j}{\rm a}_{j} + \gamma_{79}{\rm a}_{80} + \gamma_{80}{\rm a}_{79},\ \ \sigma_3(f) \equiv_{\overline{\omega}^{6}}\sum_{j\neq 78,\, 79} \gamma_{j}{\rm a}_{j} + \gamma_{78}{\rm a}_{79} + \gamma_{79}{\rm a}_{78} ,\\ \medskip \sigma_4(f) &\equiv_{\overline{\omega}^{6}}\sum_{j\neq 77,\, 78} \gamma_{j}{\rm a}_{j} + \gamma_{77}{\rm a}_{78} + \gamma_{78}{\rm a}_{77},\ \ \sigma_5(f) \equiv_{\overline{\omega}^{6}}\sum_{j\neq 76,\, 77} \gamma_{j}{\rm a}_{j} + \gamma_{76}{\rm a}_{77} + \gamma_{77}{\rm a}_{76},\\ \medskip \sigma_6(f) &\equiv_{\overline{\omega}^{6}}\sum_{j\neq 75,\, 76} \gamma_{j}{\rm a}_{j} + \gamma_{75}{\rm a}_{76} + \gamma_{76}{\rm a}_{75},\ \ \sigma_7(f) \equiv_{\overline{\omega}^{6}}\sum_{j\neq 74,\, 75} \gamma_{j}{\rm a}_{j} + \gamma_{74}{\rm a}_{75} + \gamma_{75}{\rm a}_{74},\\ \sigma_8(f) &\equiv_{\overline{\omega}^{6}}\sum_{j\neq 73,\, 74} \gamma_{j}{\rm a}_{j} + \gamma_{73}{\rm a}_{74} + \gamma_{74}{\rm a}_{73}. \end{array}$$ By these equalities and the relations $\sigma_d(f) + f\equiv_{\overline{\omega}^{6}} 0,\, 1\leq d\leq 8$ we get $\gamma_{i} = 0,\, 73\leq i\leq 80.$ Thus, $[QP^{\otimes 9}_{n_0}(\overline{\omega}^{6})]^{GL_9} = [(QP^{\otimes 9}_{n_0})^{0}(\overline{\omega}^{6})]^{GL_9}.$ Note that $(QP^{\otimes 9}_{n_0})^{0}(\overline{\omega}^{6})$ is an $\mathbb F_2$-vector space of dimension $72$ with a monomial basis represented by the following admissible monomials: \begin{center} \begin{tabular}{llll} ${\rm a}_{1}=t_2t_3t_4t_5t_6t_7t_8t_9^{3}$, & ${\rm a}_{2}=t_2t_3t_4t_5t_6t_7t_8^{3}t_9$, & ${\rm a}_{3}=t_2t_3t_4t_5t_6t_7^{3}t_8t_9$, & ${\rm a}_{4}=t_2t_3t_4t_5t_6^{3}t_7t_8t_9$, \\ ${\rm a}_{5}=t_2t_3t_4t_5^{3}t_6t_7t_8t_9$, & ${\rm a}_{6}=t_2t_3t_4^{3}t_5t_6t_7t_8t_9$, & ${\rm a}_{7}=t_2t_3^{3}t_4t_5t_6t_7t_8t_9$, & ${\rm a}_{8}=t_2^{3}t_3t_4t_5t_6t_7t_8t_9$, \\ ${\rm a}_{9}=t_1t_3t_4t_5t_6t_7t_8t_9^{3}$, & ${\rm a}_{10}=t_1t_3t_4t_5t_6t_7t_8^{3}t_9$, & ${\rm a}_{11}=t_1t_3t_4t_5t_6t_7^{3}t_8t_9$, & ${\rm a}_{12}=t_1t_3t_4t_5t_6^{3}t_7t_8t_9$, \\ ${\rm a}_{13}=t_1t_3t_4t_5^{3}t_6t_7t_8t_9$, & ${\rm a}_{14}=t_1t_3t_4^{3}t_5t_6t_7t_8t_9$, & ${\rm a}_{15}=t_1t_3^{3}t_4t_5t_6t_7t_8t_9$, & ${\rm a}_{16}=t_1^{3}t_3t_4t_5t_6t_7t_8t_9$, \\ ${\rm a}_{17}=t_1t_2t_4t_5t_6t_7t_8t_9^{3}$, & ${\rm a}_{18}=t_1t_2t_4t_5t_6t_7t_8^{3}t_9$, & ${\rm a}_{19}=t_1t_2t_4t_5t_6t_7^{3}t_8t_9$, & ${\rm a}_{20}=t_1t_2t_4t_5t_6^{3}t_7t_8t_9$, \\ ${\rm a}_{21}=t_1t_2t_4t_5^{3}t_6t_7t_8t_9$, & ${\rm a}_{22}=t_1t_2t_4^{3}t_5t_6t_7t_8t_9$, & ${\rm a}_{23}=t_1t_2^{3}t_4t_5t_6t_7t_8t_9$, & ${\rm a}_{24}=t_1^{3}t_2t_4t_5t_6t_7t_8t_9$, \\ ${\rm a}_{25}=t_1t_2t_3t_5t_6t_7t_8t_9^{3}$, & ${\rm a}_{26}=t_1t_2t_3t_5t_6t_7t_8^{3}t_9$, & ${\rm a}_{27}=t_1t_2t_3t_5t_6t_7^{3}t_8t_9$, & ${\rm a}_{28}=t_1t_2t_3t_5t_6^{3}t_7t_8t_9$, \\ ${\rm a}_{29}=t_1t_2t_3t_5^{3}t_6t_7t_8t_9$, & ${\rm a}_{30}=t_1t_2t_3^{3}t_5t_6t_7t_8t_9$, & ${\rm a}_{31}=t_1t_2^{3}t_3t_5t_6t_7t_8t_9$, & ${\rm a}_{32}=t_1^{3}t_2t_3t_5t_6t_7t_8t_9$, \\ ${\rm a}_{33}=t_1t_2t_3t_4t_6t_7t_8t_9^{3}$, & ${\rm a}_{34}=t_1t_2t_3t_4t_6t_7t_8^{3}t_9$, & ${\rm a}_{35}=t_1t_2t_3t_4t_6t_7^{3}t_8t_9$, & ${\rm a}_{36}=t_1t_2t_3t_4t_6^{3}t_7t_8t_9$, \\ ${\rm a}_{37}=t_1t_2t_3t_4^{3}t_6t_7t_8t_9$, & ${\rm a}_{38}=t_1t_2t_3^{3}t_4t_6t_7t_8t_9$, & ${\rm a}_{39}=t_1t_2^{3}t_3t_4t_6t_7t_8t_9$, & ${\rm a}_{40}=t_1^{3}t_2t_3t_4t_6t_7t_8t_9$, \\ ${\rm a}_{41}=t_1t_2t_3t_4t_5t_7t_8t_9^{3}$, & ${\rm a}_{42}=t_1t_2t_3t_4t_5t_7t_8^{3}t_9$, & ${\rm a}_{43}=t_1t_2t_3t_4t_5t_7^{3}t_8t_9$, & ${\rm a}_{44}=t_1t_2t_3t_4t_5^{3}t_7t_8t_9$, \\ ${\rm a}_{45}=t_1t_2t_3t_4^{3}t_5t_7t_8t_9$, & ${\rm a}_{46}=t_1t_2t_3^{3}t_4t_5t_7t_8t_9$, & ${\rm a}_{47}=t_1t_2^{3}t_3t_4t_5t_7t_8t_9$, & ${\rm a}_{48}=t_1^{3}t_2t_3t_4t_5t_7t_8t_9$, \\ ${\rm a}_{49}=t_1t_2t_3t_4t_5t_6t_7t_9^{3}$, & ${\rm a}_{50}=t_1t_2t_3t_4t_5t_6t_7^{3}t_9$, & ${\rm a}_{51}=t_1t_2t_3t_4t_5t_6^{3}t_7t_9$, & ${\rm a}_{52}=t_1t_2t_3t_4t_5^{3}t_6t_7t_9$, \\ ${\rm a}_{53}=t_1t_2t_3t_4^{3}t_5t_6t_7t_9$, & ${\rm a}_{54}=t_1t_2t_3^{3}t_4t_5t_6t_7t_9$, & ${\rm a}_{55}=t_1t_2^{3}t_3t_4t_5t_6t_7t_9$, & ${\rm a}_{56}=t_1^{3}t_2t_3t_4t_5t_6t_7t_9$, \\ ${\rm a}_{57}=t_1t_2t_3t_4t_5t_6t_8t_9^{3}$, & ${\rm a}_{58}=t_1t_2t_3t_4t_5t_6t_8^{3}t_9$, & ${\rm a}_{59}=t_1t_2t_3t_4t_5t_6^{3}t_8t_9$, & ${\rm a}_{60}=t_1t_2t_3t_4t_5^{3}t_6t_8t_9$, \\ ${\rm a}_{61}=t_1t_2t_3t_4^{3}t_5t_6t_8t_9$, & ${\rm a}_{62}=t_1t_2t_3^{3}t_4t_5t_6t_8t_9$, & ${\rm a}_{63}=t_1t_2^{3}t_3t_4t_5t_6t_8t_9$, & ${\rm a}_{64}=t_1^{3}t_2t_3t_4t_5t_6t_8t_9$, \\ ${\rm a}_{65}=t_1t_2t_3t_4t_5t_6t_7t_8^{3}$, & ${\rm a}_{66}=t_1t_2t_3t_4t_5t_6t_7^{3}t_8$, & ${\rm a}_{67}=t_1t_2t_3t_4t_5t_6^{3}t_7t_8$, & ${\rm a}_{68}=t_1t_2t_3t_4t_5^{3}t_6t_7t_8$, \\ ${\rm a}_{69}=t_1t_2t_3t_4^{3}t_5t_6t_7t_8$, & ${\rm a}_{70}=t_1t_2t_3^{3}t_4t_5t_6t_7t_8$, & ${\rm a}_{71}=t_1t_2^{3}t_3t_4t_5t_6t_7t_8$, & ${\rm a}_{72}=t_1^{3}t_2t_3t_4t_5t_6t_7t_8.$ \end{tabular}\end{center} \medskip Suppose $[g]_{\overline{\omega}^{6}}\in [QP^{\otimes 9}_{n_0}(\overline{\omega}^{6}]^{\Sigma_9}.$ Then, one has $g\equiv_{\overline{\omega}^{6}} \sum_{1\leq j\leq 72} \beta_{i}{\rm a}_{i},$ in which $\beta_i\in \mathbb F_2,\, 1\leq i\leq 72.$ Using the homomorphisms $\sigma_d: P^{\otimes 9}\longrightarrow P^{\otimes 9},$ for $1\leq d\leq 8,$ we obtain the following equalities: $$ \begin{array}{ll} \sigma_1(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 8}\beta_{i}{\rm a}_{i+8} + \sum_{9\leq i\leq 16}\beta_{i}{\rm a}_{i-8} + \sum_{17\leq i\leq 22}\beta_{i}{\rm a}_{i} + \beta_{23}{\rm a}_{24} + \beta_{24}{\rm a}_{23}\\ &\quad + \sum_{25\leq i\leq 30}\beta_{i}{\rm a}_{i} + \beta_{31}{\rm a}_{32} + \beta_{32}{\rm a}_{31} + \sum_{33\leq i\leq 38}\beta_{i}{\rm a}_{i} + \beta_{39}{\rm a}_{40} + \beta_{40}{\rm a}_{39} \\ &\quad+ \sum_{41\leq i\leq 46}\beta_{i}{\rm a}_{i} + \beta_{47}{\rm a}_{48} + \beta_{48}{\rm a}_{47} + \sum_{49\leq i\leq 54}\beta_{i}{\rm a}_{i} + \beta_{55}{\rm a}_{56} + \beta_{56}{\rm a}_{55}\\ \medskip &\quad + \sum_{57\leq i\leq 62}\beta_{i}{\rm a}_{i} + \beta_{63}{\rm a}_{64} + \beta_{64}{\rm a}_{63}+\sum_{65\leq i\leq 70}\beta_{i}{\rm a}_{i} + \beta_{71}{\rm a}_{72} + \beta_{72}{\rm a}_{71},\\ \sigma_2(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 6}\beta_{i}{\rm a}_{i} +\beta_{7}{\rm a}_{8} + \beta_{8}{\rm a}_{7} + \sum_{9\leq i\leq 16}\beta_{i}{\rm a}_{i+8} + \sum_{17\leq i\leq 24}\beta_{i}{\rm a}_{i-8}\\ &\quad + \sum_{25\leq i\leq 29}\beta_{i}{\rm a}_{i} + \beta_{30}{\rm a}_{31} + \beta_{31}{\rm a}_{30} + \sum_{32\leq i\leq 37}\beta_{i}{\rm a}_{i} + \beta_{38}{\rm a}_{39} + \beta_{39}{\rm a}_{38}\\ &\quad + \sum_{40\leq i\leq 45}\beta_{i}{\rm a}_{i} + \beta_{46}{\rm a}_{47} + \beta_{47}{\rm a}_{46} + \sum_{48\leq i\leq 53}\beta_{i}{\rm a}_{i} + \beta_{54}{\rm a}_{55} + \beta_{55}{\rm a}_{54}\\ \medskip &\quad+ \sum_{56\leq i\leq 61}\beta_{i}{\rm a}_{i} + \beta_{62}{\rm a}_{63} + \beta_{63}{\rm a}_{62}+\sum_{64\leq i\leq 69}\beta_{i}{\rm a}_{i} + \beta_{70}{\rm a}_{71} + \beta_{71}{\rm a}_{70} + \beta_{72}{\rm a}_{72},\\ \sigma_3(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 5}\beta_{i}{\rm a}_{i} +\beta_{6}{\rm a}_{7} + \beta_{7}{\rm a}_{6} + \sum_{8\leq i\leq 13}\beta_{i}{\rm a}_{i} +\beta_{14}{\rm a}_{15} + \beta_{15}{\rm a}_{14} + \beta_{16}{\rm a}_{16} \\ &\quad+ \sum_{17\leq i\leq 24}\beta_{i}{\rm a}_{i+8} + \sum_{25\leq i\leq 32}\beta_{i}{\rm a}_{i-8} + \sum_{33\leq i\leq 36}\beta_{i}{\rm a}_{i} + \beta_{37}{\rm a}_{38} + \beta_{38}{\rm a}_{37}\\ &\quad + \sum_{39\leq i\leq 44}\beta_{i}{\rm a}_{i} + \beta_{45}{\rm a}_{46} + \beta_{46}{\rm a}_{45} + \sum_{47\leq i\leq 52}\beta_{i}{\rm a}_{i} + \beta_{53}{\rm a}_{54} + \beta_{54}{\rm a}_{53}\\ \medskip &\quad+ \sum_{55\leq i\leq 60}\beta_{i}{\rm a}_{i} + \beta_{61}{\rm a}_{62} + \beta_{62}{\rm a}_{61}+\sum_{63\leq i\leq 68}\beta_{i}{\rm a}_{i} + \beta_{69}{\rm a}_{70} + \beta_{70}{\rm a}_{69} + \sum_{71\leq i\leq 72}\beta_{i}{\rm a}_{i},\\ \sigma_4(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 4}\beta_{i}{\rm a}_{i} +\beta_{5}{\rm a}_{6} + \beta_{6}{\rm a}_{5} + \sum_{7\leq i\leq 12}\beta_{i}{\rm a}_{i} +\beta_{13}{\rm a}_{14} + \beta_{14}{\rm a}_{13}\\ &\quad + \sum_{15\leq i\leq 20}\beta_{i}{\rm a}_{i} +\beta_{21}{\rm a}_{22} + \beta_{22}{\rm a}_{21} + \sum_{23\leq i\leq 24}\beta_{i}{\rm a}_{i} + \sum_{25\leq i\leq 32}\beta_{i}{\rm a}_{i+8} \\ &\quad+ \sum_{33\leq i\leq 40}\beta_{i}{\rm a}_{i-8} + \sum_{41\leq i\leq 43}\beta_{i}{\rm a}_{i} + \beta_{44}{\rm a}_{45} + \beta_{45}{\rm a}_{44}+ \sum_{46\leq i\leq 51}\beta_{i}{\rm a}_{i} + \beta_{52}{\rm a}_{53} + \beta_{53}{\rm a}_{52}\\ \medskip &\quad+\sum_{54\leq i\leq 59}\beta_{i}{\rm a}_{i} + \beta_{60}{\rm a}_{61} + \beta_{61}{\rm a}_{60} + \sum_{62\leq i\leq 67}\beta_{i}{\rm a}_{i}+ \beta_{68}{\rm a}_{69} + \beta_{69}{\rm a}_{68} + \sum_{70\leq i\leq 72}\beta_{i}{\rm a}_{i},\\ \end{array}$$ \newpage $$ \begin{array}{ll} \sigma_5(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 3}\beta_{i}{\rm a}_{i} +\beta_{4}{\rm a}_{5} + \beta_{5}{\rm a}_{4} + \sum_{6\leq i\leq 11}\beta_{i}{\rm a}_{i} +\beta_{12}{\rm a}_{13} + \beta_{13}{\rm a}_{12}\\ &\quad + \sum_{14\leq i\leq 19}\beta_{i}{\rm a}_{i} +\beta_{20}{\rm a}_{21} + \beta_{21}{\rm a}_{20} + \sum_{22\leq i\leq 27}\beta_{i}{\rm a}_{i} +\beta_{28}{\rm a}_{29} + \beta_{29}{\rm a}_{28} + \sum_{30\leq i\leq 32}\beta_{i}{\rm a}_{i} \\ &\quad+ \sum_{33\leq i\leq 40}\beta_{i}{\rm a}_{i+8} + \sum_{41\leq i\leq 48}\beta_{i}{\rm a}_{i-8} + \beta_{49}{\rm a}_{50} + \beta_{50}{\rm a}_{49}+ \beta_{51}{\rm a}_{52} + \beta_{52}{\rm a}_{51}\\ \medskip &\quad + \sum_{53\leq i\leq 58}\beta_{i}{\rm a}_{i} + \beta_{59}{\rm a}_{60} + \beta_{60}{\rm a}_{59}+\sum_{61\leq i\leq 66}\beta_{i}{\rm a}_{i} + \beta_{67}{\rm a}_{68} + \beta_{68}{\rm a}_{67} + \sum_{69\leq i\leq 72}\beta_{i}{\rm a}_{i},\\ \sigma_6(g)&\equiv_{\overline{\omega}^{6}} \sum_{1\leq i\leq 2}\beta_{i}{\rm a}_{i} +\beta_{3}{\rm a}_{4} + \beta_{4}{\rm a}_{3} + \sum_{5\leq i\leq 10}\beta_{i}{\rm a}_{i} +\beta_{11}{\rm a}_{12} + \beta_{12}{\rm a}_{11}\\ &\quad + \sum_{13\leq i\leq 18}\beta_{i}{\rm a}_{i} +\beta_{19}{\rm a}_{20} + \beta_{20}{\rm a}_{19} + \sum_{21\leq i\leq 26}\beta_{i}{\rm a}_{i} +\beta_{27}{\rm a}_{28} + \beta_{28}{\rm a}_{27} + \sum_{29\leq i\leq 34}\beta_{i}{\rm a}_{i} \\ &\quad +\beta_{35}{\rm a}_{36} + \beta_{36}{\rm a}_{35} + \sum_{37\leq i\leq 40}\beta_{i}{\rm a}_{i} + \sum_{41\leq i\leq 48}\beta_{i}{\rm a}_{i+16} + \beta_{49}{\rm a}_{49} + \beta_{50}{\rm a}_{51}+ \beta_{51}{\rm a}_{50} \\ \medskip &\quad + \sum_{52\leq i\leq 56}\beta_{i}{\rm a}_{i}+ \sum_{57\leq i\leq 64}\beta_{i}{\rm a}_{i-16} + \beta_{65}{\rm a}_{65} + \beta_{66}{\rm a}_{67} + \beta_{67}{\rm a}_{66} +\sum_{68\leq i\leq 72}\beta_{i}{\rm a}_{i},\\ \sigma_7(g)&\equiv_{\overline{\omega}^{6}} \beta_{1}{\rm a}_{1} + \beta_{2}{\rm a}_{3}+\beta_{3}{\rm a}_{2} + \sum_{4\leq i\leq 9}\beta_{i}{\rm a}_{i} +\beta_{10}{\rm a}_{11} + \beta_{11}{\rm a}_{10} + \sum_{12\leq i\leq 17}\beta_{i}{\rm a}_{i} +\beta_{18}{\rm a}_{19} \\ &\quad + \beta_{19}{\rm a}_{18} + \sum_{20\leq i\leq 25}\beta_{i}{\rm a}_{i} +\beta_{26}{\rm a}_{27} + \beta_{27}{\rm a}_{26} + \sum_{28\leq i\leq 33}\beta_{i}{\rm a}_{i} +\beta_{34}{\rm a}_{35} + \beta_{35}{\rm a}_{34} \\ &\quad + \sum_{36\leq i\leq 41}\beta_{i}{\rm a}_{i} + \beta_{42}{\rm a}_{43}+\beta_{43}{\rm a}_{42} + \sum_{44\leq i\leq 48}\beta_{i}{\rm a}_{i}+ \sum_{49\leq i\leq 56}\beta_{i}{\rm a}_{i+8} + \sum_{57\leq i\leq 64}\beta_{i}{\rm a}_{i-8} \\ \medskip &\quad + \beta_{65}{\rm a}_{66} + \beta_{66}{\rm a}_{65} + \sum_{67\leq i\leq 72}\beta_{i}{\rm a}_{i},\\ \sigma_8(g)&\equiv_{\overline{\omega}^{6}} \beta_{1}{\rm a}_{2} + \beta_{2}{\rm a}_{1}+ \sum_{3\leq i\leq 8}\beta_{i}{\rm a}_{i} +\beta_{9}{\rm a}_{10} + \beta_{10}{\rm a}_{9} + \sum_{11\leq i\leq 16}\beta_{i}{\rm a}_{i} +\beta_{17}{\rm a}_{18} + \beta_{18}{\rm a}_{17} + \sum_{19\leq i\leq 24}\beta_{i}{\rm a}_{i} \\ &\quad +\beta_{25}{\rm a}_{26} + \beta_{26}{\rm a}_{25} + \sum_{27\leq i\leq 32}\beta_{i}{\rm a}_{i} +\beta_{33}{\rm a}_{34} + \beta_{34}{\rm a}_{33} + \sum_{35\leq i\leq 40}\beta_{i}{\rm a}_{i} + \beta_{41}{\rm a}_{42}+\beta_{42}{\rm a}_{41} \\ &\quad + \sum_{43\leq i\leq 48}\beta_{i}{\rm a}_{i}+ \sum_{49\leq i\leq 56}\beta_{i}{\rm a}_{i+16} +\beta_{57}{\rm a}_{58} + \beta_{58}{\rm a}_{57} + \sum_{59\leq i\leq 64}\beta_{i}{\rm a}_{i} + \sum_{65\leq i\leq 72}\beta_{i}{\rm a}_{i-16}. \end{array}$$ Then, from the relations $\sigma_d(g)\equiv_{\overline{\omega}^{6}} g,\, 1\leq d\leq 8,$ one gets $\beta_i = \beta_1$ for all $i,\, 2\leq i\leq 72.$ This means that $[QP^{\otimes 9}_{n_0}(\overline{\omega}^{6}]^{\Sigma_9} = \mathbb F_2\big[\sum_{1\leq i\leq 72}{\rm a}_{i}\big]_{\overline{\omega}^{6}}.$ Then, given any $[h]_{\overline{\omega}^{6}}\in [QP^{\otimes 9}_{n_0}(\overline{\omega}^{6}]^{GL_9},$ we have $h\equiv_{\overline{\omega}^{6}} \zeta\sum_{1\leq j\leq 72}{\rm a}_{i}$ with $\zeta\in \mathbb F_2.$ Since $\sigma_9(h) + h\equiv_{\overline{\omega}^{6}} 0,$ $\zeta = 0$, and therefore, $[QP^{\otimes 9}_{n_0}(\overline{\omega}^{6}]^{GL_9} = 0.$ Now, the theorem can be derived from the above results, in conjunction with the following facts: firstly, the transfer $\{Tr_h^{\mathcal A}\}_{h\geq 0}: \{\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{}\}_{h\geq 0}\longrightarrow \{{\rm Ext}_{\mathcal A}^{h, }(\mathbb F_2, \mathbb F_2)\}_{h\geq 0}$ is an algebra homomorphism (see Singer \cite{Singer}), and secondly, according to \cite{Tangora, Bruner, Lin, Chen, Lin2}, the cohomology groups ${\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2),$ where $h\geq 1$ and $1\leq n\leq n_0$, can be identified as follows: $$ {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)=\left\{\begin{array}{ll} \mathbb F_2 h_1 &\mbox{if $h = 1$ and $n = 1,$}\\[1mm] 0 &\mbox{if $h \geq 2$ and $n = 1,$}\\[1mm] \mathbb F_2 h^{2}_1 &\mbox{if $h = 2$ and $n = 2,$}\\[1mm] 0 &\mbox{if $h \geq 1,\, h\neq 2$ and $n = 2,$}\\[1mm] \mathbb F_2 h_2 &\mbox{if $h = 1$ and $n = 3,$}\\[1mm] \mathbb F_2 h_0h_2 &\mbox{if $h = 2$ and $n = 3,$}\\[1mm] \mathbb F_2 h_1^{3} &\mbox{if $h = 3$ and $n = 3,$}\\[1mm] 0 &\mbox{if $h \geq 4$ and $n = 3,$}\\[1mm] 0 &\mbox{if $h \geq 1$ and $4\leq n\leq 5,$}\\[1mm] \mathbb F_2 h^{2}_2 &\mbox{if $h = 2$ and $n = 6,$}\\[1mm] \end{array}\right.$$ \newpage $$ {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2)=\left\{\begin{array}{ll} 0 &\mbox{if $h \geq 1,\, h\neq 2$ and $n = 6,$}\\[1mm] \mathbb F_2 h_3 &\mbox{if $h = 1$ and $n = 7,$}\\[1mm] \mathbb F_2 h_0h_3 &\mbox{if $h = 2$ and $n = 7,$}\\[1mm] \mathbb F_2 h_0^{2}h_3 &\mbox{if $h = 3$ and $n = 7,$}\\[1mm] \mathbb F_2 h_0^{3}h_3 &\mbox{if $h = 4$ and $n = 7,$}\\[1mm] 0 &\mbox{if $h \geq 5$ and $n = 7,$}\\[1mm] \mathbb F_2 h_1h_3 &\mbox{if $h = 2$ and $n = 8,$}\\[1mm] \mathbb F_2 c_0 &\mbox{if $h = 3$ and $n = 8,$}\\[1mm] 0 &\mbox{if $h\geq 1,\, h \neq 2,\, 3$ and $n = 8,$}\\[1mm] \mathbb F_2 h^{3}_2 &\mbox{if $h = 3$ and $n = 9,$}\\[1mm] \mathbb F_2 h_1c_0 &\mbox{if $h = 4$ and $n = 9,$}\\[1mm] \mathbb F_2 Ph_1 &\mbox{if $h = 5$ and $n = 9,$}\\[1mm] 0 &\mbox{if $h\geq 1,\, h \neq 3,\, 4,\, 5$ and $n = 9,$}\\[1mm] \mathbb F_2 h_1Ph_1 &\mbox{if $h = 6$ and $n = n_0,$}\\[1mm] 0&\mbox{if $h\geq 1,\, h \neq 6$ and $n = n_0.$} \end{array}\right.$$ \subsection{Proof of Theorem \ref{dlc1}}\label{s2.1} Recall that by Remark \ref{nxp10}, one gets $\dim (QP_{n_1}^{\otimes h})^{0}(\omega) = \sum_{\mu(n_1) = 4\leq k\leq h-1}\binom{h}{k}\dim (QP_{n_1}^{\otimes k})^{> 0}(\omega),$ where $\omega$ is a weight vector of degree $n_1.$ Sine $\mu(n_1) = 4,$ by Theorem \ref{dlWS}(iii), the Kameko homomorphism $(\widetilde {Sq^0_})_{n_1}: QP^{\otimes 4}_{n_1}\longrightarrow QP^{\otimes 4}_{11}$ is an isomorphism. So, we have the inverse homomorphism $((\widetilde {Sq^0_})_{n_1})^{-1}: QP^{\otimes 4}_{11}\longrightarrow QP^{\otimes 4}_{n_1},$ determined by $((\widetilde {Sq^0_})_{n_1})^{-1}([y]) = [t_1t_2t_3t_4y^{2}]$ for all $[y]\in QP^{\otimes 4}_{11}.$ So, a basis of $QP^{\otimes 4}_{n_1} = (QP^{\otimes 4}_{n_1})^{>0}$ is the set of all the equivalence classes represented by the admissible monomials of the form $t_1t_2t_3t_4y^{2}$ where $y\in \mathscr C^{\otimes 4}_{11}.$ By Sum \cite{Sum2}, $\|\mathscr C^{\otimes 4}_{11}\| = 64$, which means that $\dim (QP^{\otimes 4}_{n_1})^{>0}(\omega) = \dim (QP^{\otimes 4}_{n_1})^{>0} = 64$ if $\omega = (4,3,2,1)$ and $(QP^{\otimes 4}_{n_1})^{>0}(\omega) = 0$ otherwise. So, by the above formula, $\dim (QP_{n_1}^{\otimes 5})^{0}(\omega) = \binom{5}{4}\dim (QP_{n_1}^{\otimes 4})^{> 0}(\omega) = 320$ if $\omega = (4,3,2,1)$ and $(QP^{\otimes 5}_{n_1})^{>0}(\omega) = 0$ otherwise. On the other side, since $QP_{n_1}^{\otimes 5}\cong (QP_{n_1}^{\otimes 5})^{0}\bigoplus (QP_{n_1}^{\otimes 5})^{>0},$ by Theorem \ref{dlWW}, we derive $(QP^{\otimes 5}_{n_1})^{>0}(\omega) = 0$ if $\omega\neq (4,3,2,1)$ and $\dim (QP_{n_1}^{\otimes 5})^{>0}(4,3,21) = \dim (QP_{n_1}^{\otimes 5})^{>0}(4,3,2,1) = 1024 - 320 = 704.$ Therefore, $$\dim (QP_{n_1}^{\otimes 6})^{0}(4,3,2,1) = \sum_{\mu(n_1) = 4\leq k\leq 5}\binom{6}{k}\dim (QP_{n_1}^{\otimes k})^{> 0}(4,3,2,1)= 5184, $$ and $(QP^{\otimes 6}_{n_1})^{0}(\omega) = 0$ if $\omega\neq (4,3,2,1).$ Since $(QP^{\otimes 6}_{n_1})^{0}\cong \bigoplus_{\deg(\omega) = n_1}(QP^{\otimes 6}_{n_1})^{0}(\omega),$ by the above calculations, we obtain $(QP^{\otimes 6}_{n_1})^{0}\cong (QP^{\otimes 6}_{n_1})^{0}(4,3,2,1).$ This completes the proof of Part (i). We will now proceed to prove Part (ii) of the theorem. Our first step is to compute the space $U_1$ by explicitly determining the monomial bases of $(QP_{n_1}^{\otimes 6})^{>0}(4,5,1,1)$ and $(QP_{n_1}^{\otimes 6})^{>0}(4,5,3).$ It is important to note that according to Remark \ref{nxp1}(i), if $t\in (P_{n_1}^{\otimes 6})^{>0}(4,5,1,1)$ or $t\in (P_{n_1}^{\otimes 6})^{>0}(4,5,3)$ such that $[t]\in {\rm Ker}((\widetilde {Sq^0_})_{n_1}),$ then $t$ can be represented as $t_it_jt_kt_l\underline{t}^2,$ where $1\leq i<j<k<l\leq 6$ and $\underline{t}\in \mathscr C^{\otimes 6}_{11}.$ Therefore, in order to compute a monomial basis of $U_1,$ we need to determine all admissible monomials of degree $11$ in $P^{\otimes 6}.$ In \cite{MKR}, Mothebe et al. showed that $QP^{\otimes 6}_{11}$ has dimension $1205.$ Utilizing this result, we can explicitly determine all monomials of the form $t_it_jt_kt_l\underline{t}^2\in P^{\otimes 6}_{n_1}.$ In particular, utilizing the dimension result for $QP^{\otimes h}_{11}$ in \cite{MKR} and conducting a straightforward computation with Remark \ref{nxp10}, Corollary \ref{hq10-2}, and our previous work in \cite{Phuc11}, we obtain the following corollary. \begin{corl}\label{hqT} Let us consider the weight vectors of degree $11$: $$ \begin{array}{ll} & \widehat{\omega}_{(1)}:= (3,2,1),\ \widehat{\omega}_{(2)}:= (3,4),\ \widehat{\omega}_{(3)}:= (5,1,1),\ \widehat{\omega}_{(4)}:= (5,3),\\ & \widehat{\omega}_{(5)}:= (7,2),\ \widehat{\omega}_{(6)}:= (9,1),\ \widehat{\omega}_{(7)}:= (11,0). \end{array}$$ Then, for each $h\geq 6,$ we have the isomorphisms: $$ QP^{\otimes h}_{11}\cong \left\{\begin{array}{ll} \bigoplus_{1\leq j\leq 4}QP^{\otimes h}_{11}(\widehat{\omega}_{(j)}) &\mbox{if $h = 6$},\\[1mm] \bigoplus_{1\leq j\leq 5}QP^{\otimes h}_{11}(\widehat{\omega}_{(j)}) &\mbox{if $7\leq h\leq 8$},\\[1mm] \bigoplus_{1\leq j\leq 6}QP^{\otimes h}_{11}(\widehat{\omega}_{(j)}) &\mbox{if $9\leq h\leq 10$},\\[1mm] \bigoplus_{1\leq j\leq 7}QP^{\otimes h}_{11}(\widehat{\omega}_{(j)}) &\mbox{if $h \geq 11,$} \end{array}\right.$$ where $QP^{\otimes h}_{11}(\widehat{\omega}_{(j)})\cong (QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(j)})\bigoplus (QP^{\otimes h}_{11})^{>0}(\widehat{\omega}_{(j)}).$ The dimensions of $(QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(j)})$ and $(QP^{\otimes h}_{11})^{>0}(\widehat{\omega}_{(j)})$ are determined as follows: $$ \dim (QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(j)})=\left\{\begin{array}{ll} 8\binom{h}{3} + 32\binom{h}{4} + 40\binom{h}{5} &\mbox{if $h = 6,\, j=1$ {\rm(see \cite{Phuc11})}},\\[1mm] 10\binom{h}{5} &\mbox{if $h = 6,\, j=2$ {\rm(see \cite{Phuc11})}},\\[1mm] 15\binom{h}{5} &\mbox{if $h = 6,\, j=3$ {\rm(see \cite{Phuc11})}},\\[1mm] 10\binom{h}{5} &\mbox{if $h = 6,\, j=4$ {\rm(see \cite{Phuc11})}},\\[1mm] 0 &\mbox{if $h = 6,\, 5\leq j\leq 7$},\\[1mm] 8\binom{h}{3} + 32\binom{h}{4} + 40\binom{h}{5} + 16\binom{h}{6} &\mbox{if $h\geq 7,\, j=1$ {\rm (see \cite{Phuc11})}},\\[1mm] 10\binom{h}{5} +24\binom{h}{6} &\mbox{if $h = 7,\, j=2$ {\rm(see \cite{Phuc11})}},\\[1mm] 15\binom{h}{5} + 30\binom{h}{6} &\mbox{if $h = 7,\, j=3$ {\rm(see \cite{Phuc11})}},\\[1mm] 10\binom{h}{5}+45\binom{h}{6} &\mbox{if $h = 7,\, j=4$ {\rm(see \cite{Phuc11})}},\\[1mm] 0&\mbox{if $h = 7,\, 5\leq j\leq 7$},\\[1mm] 10\binom{h}{5} +24\binom{h}{6} +14\binom{h}{7} &\mbox{if $h \geq 8,\, j=2$},\\[1mm] 15\binom{h}{5} + 30\binom{h}{6} + 15\binom{h}{7} &\mbox{if $h \geq 8,\, j=3$},\\[1mm] 10\binom{h}{5}+45\binom{h}{6} + 70\binom{h}{7} &\mbox{if $h = 8,\, j=4$},\\[1mm] 21\binom{h}{7} &\mbox{if $h = 8,\, j=5$},\\[1mm] 0 &\mbox{if $h = 8,\, 6\leq j\leq 7$},\\[1mm] 10\binom{h}{5}+45\binom{h}{6} + 70\binom{h}{7} +35\binom{h}{8} &\mbox{if $h \geq 9,\, j=4$},\\[1mm] 21\binom{h}{7}+48\binom{h}{8} &\mbox{if $h \geq 9,\, j=5$},\\[1mm] 9\binom{h}{9} &\mbox{if $h = 10,\, j=6$},\\[1mm] 9\bigg(\binom{h}{9} + \binom{h}{10}\bigg) &\mbox{if $h \geq 11,\, j=6$},\\[1mm] 0 &\mbox{if $h = 11,\, j=7$},\\[1mm] \binom{h}{11} &\mbox{if $h \geq 12,\, j=7$}, \end{array}\right.$$ \newpage $$ \dim (QP^{\otimes h}_{11})^{> 0}(\widehat{\omega}_{(j)})=\left\{\begin{array}{ll} 16 &\mbox{if $h = 6,\, j = 1$ {\rm(see \cite{Phuc11})}},\\[1mm] 24 &\mbox{if $h = 6,\, j = 2$ {\rm(see \cite{Phuc11})}},\\[1mm] 30 &\mbox{if $h = 6,\, j = 3$ {\rm(see \cite{Phuc11})}},\\[1mm] 45 &\mbox{if $h = 6,\, j = 4$ {\rm(see \cite{Phuc11})}},\\[1mm] 0 &\mbox{if $h = 6,\, 5\leq j\leq 7$},\\[1mm] 0 &\mbox{if $h = 7,\, j = 1,\, 6,\, 7$},\\[1mm] \dim (QP^{\otimes h}_{h+4})^{> 0}(\overline{\omega}^{(j-1,\, h)}) &\mbox{if $h = 7,\, 2\leq j\leq 5$ {\rm(see Corollary \ref{hq10-2})}},\\[1mm] 0 &\mbox{if $h = 8,\, j\neq 4,\, 5$},\\[1mm] 35 &\mbox{if $h = 8,\, j = 4$},\\[1mm] 48 &\mbox{if $h = 8,\, j = 5$},\\[1mm] 0 &\mbox{if $h = 9,\, j\neq 5,\, 6$},\\[1mm] 27 &\mbox{if $h = 9,\, j = 5$},\\[1mm] \dim (QP^{\otimes h}_{1})^{0} = 9 &\mbox{if $h = 9,\, j = 6$},\\[1mm] 9 &\mbox{if $h = 10,\, j = 6$},\\[1mm] 0 &\mbox{if $h \geq 10,\, j \neq 6$}. \end{array}\right.$$ \end{corl} It must be noted that for $h = 7,\, 9,\, 11,$ we only need to compute the kernels of the Kameko squaring operations $(\widetilde {Sq^0_})_{11}: QP^{\otimes h}_{11}\longrightarrow QP^{\otimes h}_{(11-h)/2}.$ Furthermore, we can readily deduce the following: $$ {\rm Ker}((\widetilde {Sq^0_})_{11})\cong \left\{ \begin{array}{ll} (QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(1)})\bigoplus \bigg(\bigoplus_{2\leq j\leq 4}QP^{\otimes h}_{11}(\widehat{\omega}_{(j)})\bigg) &\mbox{if $h = 7$},\\[1mm] \bigg(\bigoplus_{1\leq j\leq 4}(QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(j)})\bigg)\bigoplus QP^{\otimes h}_{11}(\widehat{\omega}_{(5)})\bigoplus (QP^{\otimes h}_{11})^{>0}(\widehat{\omega}_{(6)})&\mbox{if $h = 9$},\\[1mm] \bigoplus_{1\leq j\leq 6}(QP^{\otimes h}_{11})^{0}(\widehat{\omega}_{(j)}) &\mbox{if $h = 11$}. \end{array}\right.$$ \medskip The following relevant observation is indispensable in order to establish the theorem: For each positive integer $n,$ the \textit{up Kameko map} $\psi: P_{n}^{\otimes 6}\longrightarrow P_{2n+6}^{\otimes 6}$ is an injective linear map defined on monomials by $\psi(t) = t_1t_2t_3t_4t_5t_6t^2.$ Then, from the calculations in \cite{Mothebe}, we deduce that for each $1\leq d\leq 4,\, d\in\mathbb Z,$ if $t\in (\mathscr {C}^{\otimes 5}_{n+1-2^{d}})^{>0},$ then $t_l^{2^{d}-1}\mathsf{q}_{l}(t)\in (\mathscr {C}^{\otimes 6}_{n})^{>0}$ for any $1\leq l\leq 6.$ In other words, if $t$ is an admissible monomial of degree $n+1-2^{d}$ in the $\mathcal A$-module $P^{\otimes 5},$ then $t_l^{2^{d}-1}\mathsf{q}_{l}(t)$ is an admissible monomial of degree $n$ in $\mathcal A$-module $P^{\otimes 6}.$ We set $ (\mathscr C(d, n))^{>0}:= \big\{t_l^{2^{d}-1}\mathsf{q}_{l}(t)\|\, t\in (\mathscr {C}^{\otimes 5}_{n+1-2^{d}})^{>0},\, 1\leq l\leq 6\big\},\ 1\leq d\leq 4.$ Notice that when $n = n_1$ and $h = 6$, Kameko's maps can be rewritten as $ (\widetilde {Sq^0_})_{n_1}: QP^{\otimes h}_{n_1} \longrightarrow QP^{\otimes h}_{\frac{n_1-h}{2}},\ \ \psi: P_{\frac{n_1-h}{2}}^{\otimes h}\longrightarrow P_{n_1}^{\otimes h}.$ So, $\psi\big(\mathscr {C}^{\otimes 6}_{\frac{n_1-h}{2}}\big)\subset (\mathscr C(d, n_1))^{>0}$ and $\widetilde {Sq^0_}([u]) = [0]$ for all $u\in (\mathscr C(d, n_1))^{>0}\setminus \psi\big(\mathscr {C}^{\otimes 6}_{\frac{n_1-h}{2}}\big).$ These lead to $[u]\in {\rm Ker}((\widetilde {Sq^0_})_{n_1}).$ According to the works in \cite{Sum3, Tin}, we have $$ \big\|(\mathscr {C}^{\otimes 5}_{n_1+1-2^{d}})^{>0}\big\| =\left \{\begin{array}{ll} 720 &\mbox{if $d = 1$},\\[1mm] 610 &\mbox{if $d = 2$},\\[1mm] 642 &\mbox{if $d = 3$},\\[1mm] 75&\mbox{if $d = 4$}. \end{array}\right.$$ Thanks to the results, a straightforward computation yields $$ \bigcup_{1\leq d\leq 4}(\mathscr C(d, n_1))^{>0} = D_1\bigcup D_2 \bigcup E,$$ wherein $$ \begin{array}{ll} &D_1 = \big\{c_j:\ 1\leq j\leq 336\big\},\ \ D_1\subset (\mathscr {C}^{\otimes 6}_{n_1})^{>0}(4,5,1,1),\\[1mm] &D_2 = \big\{d_j:\ 1\leq j\leq 210 \big\},\ \ D_2\subset (\mathscr {C}^{\otimes 6}_{n_1})^{>0}(4,5,3),\\[1mm] &E\subset \big((\mathscr {C}^{\otimes 6}_{n_1})^{>0}(4,3,2,1)\bigcup (\mathscr {C}^{\otimes 6}_{n_1})^{>0}(4,3,4)\bigcup \psi(\mathscr {C}^{\otimes 6}_{n_0})\big),\\[1mm] \end{array}$$ and the admissible monomials $c_j$ and $d_j$ are respectively described by Appendices \ref{s51} and \ref{s52}. The results obtained are based on filtering and removing the same monomials. For each monomial $u\in D_1,$ we notice that for any $[t]_{(4,5,1,1)}\in (QP^{\otimes 6}_{n_1})^{>0}(4,5,1,1),$ if $t$ is not equal to $u,$ then $t$ must take one of the following forms: \begin{enumerate} \item[$$] $t_i^{3}t_j^{15}t_k^{2}t_{\ell}t_m^{2}t_p^{3}$,\ $t_i^{3}t_j^{15}t_k^{2}t_{\ell}^{2}t_mt_p^{3}$,\ $t_i^{3}t_j^{2}t_kt_{\ell}^{2}t_m^{7}t_p^{11}$,\ $t_i^{3}t_j^{2}t_k^{2}t_{\ell}t_m^{7}t_p^{11}$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item [$$] $t_i^{3}t_j^{2}t_kt_{\ell}^{14}t_m^{3}t_p^{3}$,\ $t_i^{3}t_j^{2}t_k^{14}t_{\ell}t_m^{3}t_p^{3}$,\ $t_i^{3}t_j^{14}t_kt_{\ell}^{2}t_m^{3}t_p^{3}$,\ $t_i^{3}t_j^{14}t_k^{2}t_{\ell}t_m^{3}t_p^{3}$,\ $t_1t_2^{2}t_3^{14}t_{4}^{3}t_5^{3}t_6^{3},$ where $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6)$ and $j < k < \ell;$ \medskip \item [$$] $t_i^{3}t_j^{2}t_kt_{\ell}^{7}t_m^{3}t_p^{10},$\ $t_i^{3}t_j^{2}t_k^{7}t_{\ell}t_m^{3}t_p^{10},$\ $t_i^{3}t_j^{7}t_kt_{\ell}^{2}t_m^{3}t_p^{10},$\ $t_i^{3}t_j^{7}t_k^{2}t_{\ell}t_m^{3}t_p^{10},$\ $t_i^{3}t_j^{2}t_k^{5}t_{\ell}^{10}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{2}t_k^{6}t_{\ell}^{9}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{6}t_k^{2}t_{\ell}^{9}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{6}t_k^{9}t_{\ell}^{2}t_m^{3}t_p^{3},$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item [$$] $t_i^{3}t_j^{3}t_k^{6}t_{\ell}t_m^{3}t_p^{10},$\ $t_1t_2^{6}t_3^{3}t_{4}^{3}t_5^{3}t_6^{10},$\ $t_1t_2^{6}t_3^{3}t_{4}^{3}t_5^{10}t_6^{3},$\ $t_1t_2^{6}t_3^{3}t_{4}^{10}t_5^{3}t_6^{3},$\ $t_1t_2^{6}t_3^{10}t_{4}^{3}t_5^{3}t_6^{3},$\ $t_i^{3}t_j^{2}t_k^{2}t_{\ell}^{5}t_m^{3}t_p^{11},$\ $t_i^{3}t_j^{2}t_k^{5}t_{\ell}^{2}t_m^{3}t_p^{11},$\\ $t_i^{3}t_j^{2}t_k^{13}t_{\ell}^{2}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{2}t_k^{2}t_{\ell}^{13}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{2}t_k^{2}t_{\ell}^{9}t_m^{7}t_p^{3},$\ $t_i^{3}t_j^{2}t_k^{9}t_{\ell}^{2}t_m^{7}t_p^{3},$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item [$$] $t_1t_j^{2}t_k^{6}t_{\ell}^{3}t_m^{3}t_p^{11},$\ $t_1t_j^{6}t_k^{2}t_{\ell}^{3}t_m^{3}t_p^{11},$ $t_1t_j^{6}t_k^{3}t_{\ell}^{2}t_m^{3}t_p^{11},$ $t_1^{3}t_j^{2}t_kt_{\ell}^{6}t_m^{3}t_p^{11},$\ $t_1^{3}t_j^{2}t_k^{6}t_{\ell}t_m^{3}t_p^{11},$ $t_1^{3}t_j^{6}t_kt_{\ell}^{2}t_m^{3}t_p^{11},$\ $t_1^{3}t_j^{6}t_k^{2}t_{\ell}t_m^{3}t_p^{11}$ with $j < k < \ell,$ and $(j, k, \ell, m, p)$ is a premutation of $(2, 3, 4, 5, 6);$ \medskip \item [$$] $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12},$\ $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{2}t_m^{4}t_p^{11},$\ $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{7}t_m^{2}t_p^{8},$\ $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{4}t_p^{10},$\ $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{6}t_p^{8},$ where $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6).$ \end{enumerate} It is straightforward to check that these monomials are strictly inadmissible, and so, they are inadmissible. To exemplify, let us consider the monomials $t_it_j^{6}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{10}$ and $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12}.$ It is clear that $\omega(t_it_j^{6}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{10}) = \omega(t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12}) = (4,5,1,1).$ As well known, the action of the Steenrod algebra $\mathcal A$ on the polynomial algebra $P^{\otimes 6}$ is given by the rule $$ Sq^k(t_j) = \left\{\begin{array}{ll} t_j &\mbox{if $k = 0,$}\\[1mm] t_j^2& \mbox{if $k = 1,$}\ \ \mbox{(\textit{the instability condition})},\\[1mm] 0 &\mbox{otherwise,} \end{array}\right.$$ and the Cartan formula $Sq^k(fg) = \sum_{a+b = k}Sq^{a}(f)Sq^{b}(g),$ for all $f,\, g\in P^{\otimes 6}.$ Note that for each $t\in P^{\otimes 1}$ and each positive integer $n,$ $Sq^{a}(t^{n}) = \binom{n}{a}t^{n+a},$ where the binomial coefficients $\binom{n}{a}$ are to be interpreted modulo 2 with the usual convention $\binom{n}{a} = 0$ if $n < a.$ Therefore, through a direct calculation, we obtain: $$ \begin{array}{ll} \medskip t_it_j^{6}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{10} &= Sq^{1}(t_i^{2}t_j^{3}t_k^{5}t_{\ell}^{3}t_m^{3}t_p^{9} + t_i^{2}t_j^{5}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{9} + t_i^{2}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{5}t_p^{9} + t_i^{2}t_j^{3}t_k^{3}t_{\ell}^{5}t_m^{3}t_p^{9} )\\ \medskip &\quad + Sq^{2}(t_it_j^{3}t_k^{5}t_{\ell}^{3}t_m^{3}t_p^{9} + t_it_j^{5}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{9} + t_it_j^{3}t_k^{3}t_{\ell}^{3}t_m^{5}t_p^{9} + t_it_j^{3}t_k^{3}t_{\ell}^{5}t_m^{3}t_p^{9})\\ \medskip &\quad + t_it_j^{3}t_k^{6}t_{\ell}^{3}t_m^{3}t_p^{10} + t_it_j^{3}t_k^{3}t_{\ell}^{3}t_m^{6}t_p^{10} + t_it_j^{3}t_k^{3}t_{\ell}^{6}t_m^{3}t_p^{10} \mod P_{26}^{\otimes 6}(< (4,5,1,1)),\\ t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12}&=Sq^{1}(t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_mt_p^{12}) \mod P_{n_1}^{\otimes 6}(< (4,5,1,1)), \end{array}$$ which imply $t_it_j^{6}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{10}\equiv_{(4,5,1,1)} (t_it_j^{3}t_k^{6}t_{\ell}^{3}t_m^{3}t_p^{10} + t_it_j^{3}t_k^{3}t_{\ell}^{3}t_m^{6}t_p^{10} + t_it_j^{3}t_k^{3}t_{\ell}^{6}t_m^{3}t_p^{10}),$ and $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12}\equiv_{(4,5,1,1)} 0.$ Hence, the monomials $ t_it_j^{6}t_k^{3}t_{\ell}^{3}t_m^{3}t_p^{10}$ and $t_i^{3}t_j^{3}t_k^{3}t_{\ell}^{3}t_m^{2}t_p^{12}$ are strictly inadmissible and $(4,5,1,1)$-hit, respectively. Since the monomials in $D_1$ are admissible, $\mathscr (C_{n_1}^{\otimes 6})^{>0}(4,5,1,1) = D_1.$ Thus, it may be claimed that $\dim (QP^{\otimes 6}_{n_1})^{>0}(4,5,1,1) = \|D_1\| = 336.$ Next, we observe that for each monomial $\widehat{u}\in D_2,$ if $[\widehat{t}]_{(4,5,3)}\in (QP^{\otimes 6}_{n_1})^{>0}(4,5,3),$ and $\widehat{t}\neq \widehat{u},$ then $\widehat{t}$ must have one of the following forms: \begin{enumerate} \item[$-$] $t_i^{3}t_j^{2}t_k^{2}t_{\ell}^{5}t_m^{7}t_p^{7}$,\ $t_i^{3}t_j^{2}t_k^{5}t_{\ell}^{2}t_m^{7}t_p^{7}$,\ $t_i^{7}t_j^{2}t_kt_{\ell}^{2}t_m^{7}t_p^{7}$,\ $t_i^{7}t_j^{2}t_k^{2}t_{\ell}t_m^{7}t_p^{7},$\ $t_i^{3}t_j^{6}t_k^{5}t_{\ell}^{6}t_m^{3}t_p^{3},$\ $t_i^{3}t_j^{6}t_k^{6}t_{\ell}^{5}t_m^{3}t_p^{3},$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item[$-$] $t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7},$\ $t_it_j^{6}t_k^{2}t_{\ell}^{3}t_m^{7}t_p^{7},$\ $t_it_j^{6}t_k^{3}t_{\ell}^{2}t_m^{7}t_p^{7},$\ $t_i^{3}t_j^{2}t_k^{6}t_{\ell}t_m^{7}t_p^{7},$\ $t_i^{3}t_j^{6}t_k^{2}t_{\ell}t_m^{7}t_p^{7},$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item[$-$] $t_i^{3}t_j^{6}t_kt_{\ell}^{6}t_m^{3}t_p^{7}$,\ $t_i^{3}t_j^{6}t_k^{6}t_{\ell}t_m^{3}t_p^{7},$\ $t_it_j^{6}t_k^{3}t_{\ell}^{6}t_m^{3}t_p^{7},$\ $t_it_j^{6}t_k^{6}t_{\ell}^{3}t_m^{3}t_p^{7},$\ $t_i^{3}t_j^{2}t_k^{5}t_{\ell}^{6}t_m^{3}t_p^{7},$\ $t_i^{3}t_j^{2}t_k^{6}t_{\ell}^{5}t_m^{3}t_p^{7},$\ $t_i^{3}t_j^{6}t_k^{2}t_{\ell}^{5}t_m^{3}t_p^{7},$\\ $t_i^{3}t_j^{6}t_k^{5}t_{\ell}^{2}t_m^{3}t_p^{7},$ where $j < k < \ell,$ and $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6);$ \medskip \item[$-$] $t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{6}$,\ $t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{7}t_m^{2}t_p^{4},$ where $(i, j, k, \ell, m, p)$ is a premutation of $(1, 2, 3, 4, 5, 6).$ \end{enumerate} It is indeed facile to simply assert that these monomials are inadmissible. As an illustration, let us consider the monomials $t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7}$ and $t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{6}.$ Then, $\omega(t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7}) = \omega(t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{6}) = (4,5,3)$ and by a simple computation, we get $$ \begin{array}{ll} \medskip t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7} &= Sq^{2}(t_it_jt_k^{3}t_{\ell}^{5}t_m^{7}t_p^{7} + t_it_jt_k^{5}t_{\ell}^{3}t_m^{7}t_p^{7} + t_it_jt_k^{3}t_{\ell}^{3}t_m^{7}t_p^{9} + t_it_jt_k^{3}t_{\ell}^{3}t_m^{9}t_p^{7})\\ \medskip &\quad + Sq^{1}( t_i^{2}t_jt_k^{3}t_{\ell}^{5}t_m^{7}t_p^{7} + t_i^{2}t_jt_k^{5}t_{\ell}^{3}t_m^{7}t_p^{7}) + t_it_j^{2}t_k^{3}t_{\ell}^{6}t_m^{7}t_p^{7} \mod P_{n_1}^{\otimes 6}(< (4,5,3)),\\ t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{6}&= Sq^{1}(t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{5}) \mod P_{n_1}^{\otimes 6}(< (4,5,3)). \end{array} $$ These equalities show that $ t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7}$ is strictly inadmissible (since $t_it_j^{2}t_k^{3}t_{\ell}^{6}t_m^{7}t_p^{7}< t_it_j^{2}t_k^{6}t_{\ell}^{3}t_m^{7}t_p^{7}$) and that $t_i^{3}t_j^{3}t_k^{7}t_{\ell}^{3}t_m^{4}t_p^{6}$ is $(4,5,3)$-hit. \medskip Since the monomials in $D_2$ are admissible, $\mathscr (C_{n_1}^{\otimes 6})^{>0}(4,5,3) = D_2.$ So, $\dim (QP^{\otimes 6}_{n_1})^{>0}(4,5,3) = \|D_2\| = 210.$ Incorporating the above-mentioned computation, we arrive at the conclusion that the dimension of $U_1$ is equal to $\|D_1\| + \|D_2\| = 336 + 210 = 546.$ \medskip The next step is to determine the dimension of $U_2$. Directly computing the dimension of this cohit module is a task of considerable complexity. Nevertheless, the methods employed to compute it are akin to those utilized in our earlier publications \cite{P.S1, Phuc4, Phuc6, Phuc10}. Therefore, to avoid redundancy, we shall present only a rough outline of the calculations for this $U_2.$ We commence by considering the set \begin{equation} \mathcal{N}_h := \bigg\{(i; I)\;\|\; I = (i_1,i_2,\ldots,i_r), 1\leqslant i < i_1<i_2 < \ldots < i_r\leqslant h, 0\leqslant r < h\bigg\},\ 5\leq h\leq 6. \end{equation} Here we also use the notation $r=\ell(I)$ to denote the length of $I.$ If $r = 0,$ then by convention we take $I$ to be the empty set. For each pair $(i; I)\in\mathcal{N}_h,$ we define an $\mathcal A$-homomorphism $p_{(i; I)}: P^{\otimes h}\longrightarrow P^{\otimes (h-1)}$ by the following substitutions: $p_{(i; I)}(t_j) = t_j$ if $1\leqslant j < i,$ $p_{(i; I)}(t_j) = \sum\limits_{k\in I}t_{k-1}$ if $j = i,$ and $p_{(i; I)}(t_j) = t_{j-1}$ if $i < j \leqslant h.$ In particular, $p_{(i; \emptyset)}(t_i) = 0$ for every $i.$ and $p_{(i; I)}(\mathsf{q}_i(t)) = t$ for all $(i; I)\in \mathcal {N}_h$ and any $t\in P^{\otimes (h-1)}.$ Moreover, as shown in our previous work \cite{P.S1}, we can observe that $p_{(i; I)}$ induces a homomorphism from $QP^{\otimes h}_{n_1}(\omega)$ to $QP^{\otimes (h-1)}_{n_1}(\omega),$ where $\deg(\omega) = n_1.$ \medskip For each the pair $(i; I)\in\mathcal{N}_h,$ and $1\leq u < r,$ let us denote by $t_{(I;\,u)} := t_{i_u}^{2^{r-1} + 2^{r-2} +\, \cdots\, + 2^{r-u}}\prod_{u < d\leq r}t_{i_d}^{2^{r-d}},$ where $t_{(\emptyset; 1)} = 1.$ In \cite{Sum2}, Sum gives an interesting $\mathbb F_2$-linear function $\phi_{(i; I)}: P^{\otimes (h-1)}\longrightarrow P^{\otimes h},$ which is determined by $\phi_{(i; I)}(\prod_{1\leq j\leq h-1}t_j^{a_j}) = \dfrac{t_i^{2^{r} - 1}\mathsf{q}_{i}(\prod_{1\leq k\leq h-1}t_k^{a_k})}{t_{(I;\,u)}}, $ if there exists $u$ such that: \begin{equation} \left\{ \begin{array}{ll} a_{i_1 - 1} +1= \cdots = a_{i_{(u-1)} - 1} +1 = 2^{r},\\[1mm] a_{i_{u} - 1} + 1 > 2^{r},\\[1mm] \alpha_{r-d}(a_{i_{u} - 1}) -1 = 0,\, 1\leq d\leq u, \\[1mm] \alpha_{r-d}(a_{i_{d}-1}) -1 = 0,\, \ u+1 \leq d \leq r, \end{array}\right. \end{equation} and $\phi_{(i; I)}(\prod_{1\leq j\leq h-1}t_j^{a_j}) = 0$ otherwise. One has the following observation. If $I = \emptyset,$ then $\phi_{(i; I)} = \mathsf{q}_{i}$ for $1\leq i\leq h.$ It is in fact not hard to show that if $\phi_{(i; I)}( \prod_{1\leq j\leq h-1}t_j^{a_j})\neq 0,$ then $\omega(\phi_{(i; I)}( \prod_{1\leq j\leq h-1}t_j^{a_j})) = \omega(\prod_{1\leq j\leq h-1}t_j^{a_j}).$ By an elementary calculation, it is not so difficult to assert the following. \begin{lem}\label{bdbs} With the notations as above, if $t$ is an admissible monomial in the $\mathcal A$-module $P^{\otimes (h-1)},$ then $\phi_{(i; I)}(t)$ is also admissible monomial in the $\mathcal A$-module $P^{\otimes h}$ for $5\leq h\leq 6.$ \end{lem} We also believe that this result can be extended to any $h.$ For the convenience of readers, let us provide an example to illustrate this point. We put $\Gamma_6 = \{1,2,\ldots, 6\},\ T_{S} = T_{\{j_1,j_2,\ldots, j_d\}} = \prod_{j\in\Gamma_6\setminus S}t_j,$ where $S = \{j_1,j_2,\ldots, j_d\}\subseteq \Gamma_6.$ Now for $Y = \prod_{1\leq j\leq 6}t_j^{a_j}\in P^{\otimes 6}$ and for each non-negative integer $\gamma,$ we denote by $S_{\gamma}(Y) = \{j\in \Gamma_6:\; \alpha_{\gamma}(a_j) = 0\}\subseteq \Gamma_6.$ Then, the monomial $Y$ can be represented as $Y = \prod_{\gamma\geq 0}T_{S_{\gamma}(Y)}^{2^{\gamma}}.$ Given the monomial $Y =T_{\{6\}} = \prod_{1\leq j\leq 5}t_j\in P^{\otimes 6}_{5}$. Let $m$ be a positive integer such that $m > r = \ell(I).$ Then, one has the following equality: $$ \begin{array}{lll} \medskip \phi_{(i; I)}(Y^{2^m-1})&=& \dfrac{t_i^{2^r-1}t_2^{2^m-1}\ldots t_{i_1}^{2^m-1}\ldots t_{i_r}^{2^m-1}\ldots t_6^{2^m-1}}{t_{i_1}^{2^{r-1}}t_{i_2}^{2^{r-2}}\ldots t_{i_r}^{2^{r-r}}}= t_i^{2^r-1}t_{i_1}^{2^m-2^{r-1}-1}\ldots t_{i_2}^{2^m- 2^{r-2}-1}\ldots t_{i_r}^{2^m- 2^{r-r}-1}T^{2^m-1}_{\{i, i_1, \ldots, i_r\}}\\ &=&t_i^{2^r-1}\prod_{1\leq k\leq r}t_{i_k}^{2^m-2^{r-k}-1}T^{2^m-1}_{\{i, i_1, \ldots, i_r\}}\neq 0. \end{array}$$ Now, for each integer $q,\, 2\leq q \leq 6,$ if $r = q - 1,$ then $m\geq q.$ Let us consider the pair $(i; I)$ where $i = 1$ and $I = (i_1, i_2, \ldots, i_r) = (2, 3, \ldots, q).$ A direct computation shows that $$ \phi_{(1; I)}(Y^{2^m-1}) = t_1^{2^{q-1}-1}\prod_{1\leq k \leq q-1}t_{j}^{2^m-2^{q-1-k}-1}T^{2^m-1}_{\{1, 2, 3,\ldots, q\}}= t_1^{2^{q-1}-1}\ldots t_j^{2^m-2^{q-j}-1}\ldots t_q^{2^m-2}t_{q+1}^{2^m-1}\ldots t_6^{2^m-1}. $$ Owing to Theorem \ref{dlPS}, we know that $Y^{2^m-1}\in P^{\otimes 5}$ is admissible. On the other side, following the work of Mothebe \cite{Mothebe0}, the monomials $t_1^{2^{q-1}-1}\ldots t_j^{2^m-2^{q-j}-1}\ldots t_q^{2^m-2}t_{q+1}^{2^m-1}\ldots t_6^{2^m-1}$ are admissible. As a result, the non-zero monomial $\phi_{(1; I)}(Y^{2^m-1})\in P^{\otimes 6}$ is admissible as well. \medskip Applying a result in Sum \cite{Sum2} and combining it with Theorem \ref{dlWW}, we obtain $$\dim (QP^{\otimes 5}_{n_1})^{>0} = \dim (QP^{\otimes 5}_{n_1})^{>0}(4,3,2,1) =2^{\binom{5}{2}}-64\binom{5}{4} = 704.$$ Furthermore, a simple computation shows $(\mathscr C^{\otimes 5}_{n_1})^{>0} = \bigcup_{(i; I)\in \mathcal N_5}\phi_{(i; I)}(\mathscr C^{\otimes 4}_{n_1}) \bigcup S_1\bigcup S_2,$ where $$\begin{array}{ll} \medskip &S_1:= \bigcup_{1\leq d\leq 4}\big\{t_l^{2^{d}-1}\mathsf{q}_{l}(t)\|\, t\in (\mathscr {C}^{\otimes 4}_{n_1+1-2^{d}})^{>0},\, 1\leq l\leq 5\big\},\ \mathsf{q}_{l}: P^{\otimes 4}\longrightarrow P^{\otimes 5}, \\ &S_2 := \bigg\{t_1^{3}t_2^{12}t_3^{3}t_4^{3}t_5^{5},\, t_1^{3}t_2^{12}t_3^{3}t_4^{5}t_5^{3},\, t_1^{3}t_2^{4}t_3^{3}t_4^{5}t_5^{11},\, t_1^{7}t_2^{9}t_3^{6}t_4t_5^{3},\, t_1^{7}t_2^{9}t_3^{6}t_4^{3}t_5,\, t_1^{7}t_2^{8}t_3^{3}t_4^{3}t_5^{5},\, t_1^{7}t_2^{8}t_3^{3}t_4^{5}t_5^{3}\bigg\}. \end{array}$$ The readers should also take note that, $QP^{\otimes 4}_{n_1} = (QP^{\otimes 4}_{n_1})^{>0}\cong QP^{\otimes 4}_{11},$ a cohit module of dimension $64$ (see \cite{Sum2}). This, in conjunction with Theorems \ref{dlKS}, \ref{dlSin}, \ref{dlWW}, and Lemma \ref{bdbs}, implies that $U_2 = \langle [\mathcal S] \rangle,$ wherein \begin{equation} \begin{array}{ll} \mathcal S &= \bigg(E\setminus \psi(\mathscr C^{\otimes 6}_{n_0})\bigg)\bigcup F \bigcup \bigg(\bigcup_{(i; I)\in \mathcal N_6}\phi_{(i; I)}((\mathscr C^{\otimes 5}_{n_1})^{>0})\setminus (\mathscr C^{\otimes 6}_{n_1})^{0}\bigg)\\ & = \bigg\{z_j\in (P^{\otimes 6}_{n_1})^{>0}:\, \mbox{$z_j$ is admissible},\, \forall j,\, 1\leq j\leq 3090\bigg\}. \end{array} \end{equation} Here $\|F\| = 3090 - \bigg\|E\setminus \psi(\mathscr C^{\otimes 6}_{n_0})\bigg\| - \bigg\|\bigg(\bigcup_{(i; I)\in \mathcal N_6}\phi_{(i; I)}((\mathscr C^{\otimes 5}_{n_1})^{>0})\setminus (\mathscr C^{\otimes 6}_{n_1})^{0}\bigg)\bigg\|,$ where the set $E$ was described earlier above. Hence, it may be asserted that $\|\mathcal S\| = 250 + 2840 = 3090.$ Suppose there exists a linear relation $\sum_{z\in \mathcal S}\gamma_zz\equiv 0,$ where $\gamma_z\in \mathbb F_2$ are coefficients. Using Theorems \ref{dlSin}, \ref{dlWW} and the homomorphisms $p_{(i; I)},$ for each pair $(i; I)\in \mathcal N_6,\, \ell(I) > 0,$ we determine explicitly $p_{(i; I)}(\sum_{z\in \mathcal S}\gamma_zz)$ in admissible terms in $(\mathscr C^{\otimes 5}_{n_1})^{>0}$ modulo $\overline{\mathcal A}P^{\otimes 5}_{n_1}.$ Then, since $p_{(i; I)}(\sum_{z\in\mathcal S}\gamma_zz)\equiv 0,$ for all $(i; I)\in \mathcal N_6,$ we obtain $\gamma_z = 0$ for all $z.$ These data lead to $U_2$ being an $\mathbb F_2$-vector space of dimension $\|\mathcal S\|.$ Thus, from the above calculations, ${\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}$ is an $\mathbb F_2$-vector of dimension $3636.$ Finally, as $QP^{\otimes 6}_{n_1} \cong (QP^{\otimes 6}_{n_1})^{0}\bigoplus \big({\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0}\big)\bigoplus QP^{\otimes 6}_{n_0},$ we conclude that $$ \begin{array}{ll} \medskip \dim QP_{n_1}^{\otimes 6} &= \dim (QP_{n_1}^{\otimes 6})^{0} + \dim {\rm Ker}((\widetilde {Sq^0_})_{n_1})\cap (QP^{\otimes 6}_{n_1})^{>0} + \dim QP_{n_0}^{\otimes 6}\\ & = 5184 + 3636+ 945 = 9765. \end{array}$$ The proof of the theorem is complete. \newpage To close this subsection, we would like to provide some insightful observations and remarks regarding the indecomposables $Q^{\otimes h}_{11}.$ \begin{nx} \begin{itemize} \item[(i)] With Corollary \ref{hqT} and a result from \cite{Tin} concerning $\dim Q^{\otimes 5}_{11}$ as our basis, we assert that the localized version of Kameko's conjecture in Note \ref{cyP}(i) holds true for all $h$ and degree $11$. \item[(ii)] Clearly, Corollary \ref{hqT} implies that the coinvariant $(\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{})_{11}$ is trivial for all $h\geq 12.$ Hence, Singer's Conjecture \ref{gtSinger} is true for bidegrees $(h, h+11)$ with $h>11.$ Moreover, in \cite{Phuc11}, we have demonstrated that the conjecture is also true for $6\leq h\leq 8.$ This is achieved by explicitly computing the dimensions of the invariants $[{\rm Ker}((\widetilde {Sq^0_})_{11})]^{GL_7},\, [QP^{\otimes 7}_2]^{GL_7},$ and $[QP^{\otimes h}_{11}(\widehat{\omega}_{(j)})]^{GL_h}$ for $h = 6,\, 8,\, 1\leq j\leq 5.$ We then prove that the transfer homomorphism $Tr_h^{\mathcal A}$ is a monomorphism if $6\leq h\leq 7$ and is a trivial isomorphism if $h = 8.$ Here the weight vectors $\widehat{\omega}_{(j)}$ are given as in Corollary \ref{hqT}. It is noteworthy to mention that, the works of Ch\ohorn n and H\`a \cite{CHa, CHa0} demonstrated the non-surjectivity of the transfer in the bidegrees $(6, 6+11)$ and $(7, 7+11).$ According to the research conducted by \cite{Tangora, Bruner, Bruner2, Lin2}, it can be concluded that for every $h\geq 6,$ $$ {\rm Ext}_{\mathcal A}^{h, h+11}(\mathbb F_2, \mathbb F_2) = \left\{\begin{array}{ll} \mathbb F_2h_0Ph_2&\mbox{if $h = 6$},\\[1mm] \mathbb F_2h_0^{2}Ph_2 = \mathbb F_2 h_1^{2}Ph_1&\mbox{if $h = 7$},\\[1mm] 0&\mbox{if $h\geq 8$}. \end{array}\right.$$ For $h = 9,$ according to Corollary \ref{hqT}, we have $QP^{\otimes 9}_{1} = (QP^{\otimes 9}_{1})^{0} = \langle \{[t_i]\}_{1\leq i\leq 9}\rangle,$ and $QP^{\otimes 9}_{11}(\widehat{\omega}_{(j)}) = (QP^{\otimes 9}_{11})^{0}(\widehat{\omega}_{(j)})$ for $1\leq j\leq 4.$ So the invariants $[QP^{\otimes 9}_{1}]^{GL_9}$ and $QP^{\otimes 9}_{11}(\widehat{\omega}_{(j)}),\, 1\leq j\leq 4$ are trivial. By combining these data with Corollary \ref{hqT} and taking into account that the mapping $(\widetilde {Sq^0_})_{11}: QP^{\otimes 9}_{11}\longrightarrow QP^{\otimes 9}_2$ is a surjective, we can derive an estimate $$\dim (\mathbb F_2\otimes_{GL_{9}}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 9}]^{})_{11}=\dim [QP^{\otimes 9}_{11}]^{GL_9}\leq \dim [{\rm Ker}((\widetilde {Sq^0_})_{11})]^{GL_9}\leq \dim [QP^{\otimes 9}_{11}(\widehat{\omega}_{(5)})]^{GL_9}.$$ Owing to Corollary \ref{hqT}, one has an isomorphism $QP^{\otimes 9}_{11}(\widehat{\omega}_{(5)})\cong (QP^{\otimes 9}_{11})^{0}(\widehat{\omega}_{(5)})\bigoplus (QP^{\otimes 9}_{11})^{>0}(\widehat{\omega}_{(5)})$ where $\dim (QP^{\otimes 9}_{11})^{0}(\widehat{\omega}_{(5)}) = 21\binom{9}{7} + 48\binom{9}{8} = 1188$ and $\dim (QP^{\otimes 9}_{11})^{>0}(\widehat{\omega}_{(5)}) = 27.$ When $h=10$, it follows from Corollary \ref{hqT} that the invariants $QP^{\otimes 10}_{11}(\widehat{\omega}_{(j)})$ are trivial for $1\leq j\leq 5.$ As a result, we can establish an inequality $$\dim (\mathbb F_2\otimes_{GL_{10}}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 10}]^{})_{11}=\dim [QP^{\otimes 10}_{11}]^{GL_{10}}\leq \dim [QP^{\otimes 10}_{11}(\widehat{\omega}_{(6)})]^{GL_{10}}.$$ According to Corollary \ref{hqT}, $QP^{\otimes 10}_{11}(\widehat{\omega}_{(6)})\cong (QP^{\otimes 10}_{11})^{0}(\widehat{\omega}_{(6)})\bigoplus (QP^{\otimes 10}_{11})^{>0}(\widehat{\omega}_{(6)})$ where $$\dim (QP^{\otimes 10}_{11})^{0}(\widehat{\omega}_{(6)}) = 9\binom{10}{9} =90\ \mbox{and}\ \dim (QP^{\otimes 10}_{11})^{>0}(\widehat{\omega}_{(6)}) = 9.$$ In the case where $h=11$, it is possible to derive from Corollary \ref{hqT} that $$QP^{\otimes 11}_{11}\cong \bigoplus_{1\leq j\leq 6}(QP^{\otimes 11}_{11})^{0}(\widehat{\omega}_{(j)})\bigoplus (QP^{\otimes 11}_{11})^{>0}(\widehat{\omega}_{(7)}),$$ where $(QP^{\otimes 11}_{11})^{>0}(\widehat{\omega}_{(7)}) \cong \mathbb F_2[t_1t_2\ldots t_{11}]_{\widehat{\omega}_{(7)}}.$ Using the $\mathcal A$-homomorphisms $\sigma_d: P^{\otimes 11}\longrightarrow P^{\otimes 11},$ for $1\leq d\leq 11,$ one gets $\dim (\mathbb F_2\otimes_{GL_{11}}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 11}]^{})_{11} =\dim [QP^{\otimes 11}_{11}]^{GL_{11}} = 0 = \dim {\rm Ext}_{\mathcal A}^{11, 22}(\mathbb F_2, \mathbb F_2).$ Hence, Singer's transfer is a trivial isomorphism in bidegree $(11, 22).$ Thus if the invariants $[QP^{\otimes 9}_{11}(\widehat{\omega}_{(5)})]^{GL_9}$ and $[QP^{\otimes 10}_{11}(\widehat{\omega}_{(6)})]^{GL_{10}}$ are trivial, then Singer's Conjecture \ref{gtSinger} also holds for bidegrees $(h, h+11)$ with $9\leq h\leq 11.$ This matter will be thoroughly investigated and discussed in another context. \end{itemize} \end{nx} \subsection{Proof of Theorem \ref{dlc4}} In what follows, suppose that $\omega$ is a weight vector of degree $n_1.$ We denote by $\mathscr C^{\otimes 6}_{n_1}(\omega)$ the set of all admissible monomials in $P^{\otimes 6}_{n_1}(\omega)$ and by $[\mathscr C^{\otimes 6}_{n_1}(\omega)]_{\omega} = \{[t]_{\omega}:\ t\in \mathscr C^{\otimes 6}_{n_1}(\omega)\}.$ For $z_1, z_2,\ldots, z_m\in P^{\otimes 6}_{n_1}(\omega)$ with $m\geq 1,$ we put $$ \begin{array}{ll} \medskip \Sigma_6(z_1, \ldots, z_m) &= \big\{\theta(z_j):\ \theta\in \Sigma_6,\, 1\leq j\leq m\big\},\\ \medskip [\mathscr C(z_1, \ldots, z_m)]_{\omega}&= [\mathscr C^{\otimes 6}_{n_1}(\omega)]_{\omega}\cap \langle [\Sigma_6(z_1, \ldots, z_m)]_{\omega} \rangle,\\ \widehat{p(z)}&= \sum_{y\in \mathscr C^{\otimes 6}_{n_1}(\omega)\cap \Sigma_6(z)}y. \end{array}$$ $\langle [\Sigma_6(z_1, \ldots, z_m)]_{\omega} \rangle$ is manifestly a $\Sigma_6$-submodule of $QP^{\otimes 6}_{n_1}(\omega).$ As we have pointed out before, $$ {\rm Ker}((\widetilde {Sq^0_})_{n_1})\cong (QP_{n_1}^{\otimes 6})(4,3,2,1)\bigoplus QP_{n_1}^{\otimes 6}(4,3,4)\bigoplus QP_{n_1}^{\otimes 6}(4,5,1,1)\bigoplus QP_{n_1}^{\otimes 6}(4,5,3), $$ where $$ \begin{array}{ll} \medskip &(QP_{n_1}^{\otimes 6})(4,3,4) = (QP_{n_1}^{\otimes 6})^{>0}(4,3,4),\ \ QP_{n_1}^{\otimes 6}(4,5,1,1) = (QP_{n_1}^{\otimes 6})^{>0}(4,5,1,1),\\ &QP_{n_1}^{\otimes 6}(4,5,3) = (QP_{n_1}^{\otimes 6})^{>0}(4,5,3). \end{array}$$ So, one gets an estimate $$\begin{array}{ll} \dim {\rm [}{\rm Ker}((\widetilde {Sq^0_})_{n_1}){\rm]}^{GL_6}&\leq \dim {\rm [}QP_{n_1}^{\otimes 6}(4,3,2,1){\rm ]}^{GL_6} +\dim {\rm [}QP_{n_1}^{\otimes 6}(4,3,4){\rm ]}^{GL_6} \\ &\quad + \dim {\rm [}QP_{n_1}^{\otimes 6}(4,5,1,1){\rm ]}^{GL_6} + \dim {\rm [}QP_{n_1}^{\otimes 6}(4,5,3){\rm ]}^{GL_6}. \end{array}$$ By using the monomial basis of the space $QP_{n_1}^{\otimes 6}(\omega)$ with $\omega\in \{(4,3,2,1), (4,3,4), (4,5,1,1), (4,5,3)\}$ (see Theorem \ref{dlc1}) and the homomorphisms $\sigma_d$ for $1\leq d\leq 6,$ we find that the invariants $[QP_{n_1}^{\otimes 6}(\omega)]^{GL_6}$ are zero. Indeed, we will prove this claim for the invariants $[QP_{n_1}^{\otimes 6}(4,5,1,1)]^{GL_6}$ and $[QP_{n_1}^{\otimes 6}(4,5,3)]^{GL_6}$ in detail. Similarly, we also obtain the results for the other spaces. \medskip We set $\widetilde{\omega}:= (4,5,1,1)$ and $\omega:= (4,5,3).$ We first describe the space $[QP_{n_1}^{\otimes 6}(\widetilde{\omega})]^{GL_6}.$ Following the proof of Theorem \ref{dlc1}, the space $QP_{n_1}^{\otimes 6}(\widetilde{\omega}) = (QP_{n_1}^{\otimes 6})^{>0}(\widetilde{\omega})$ is $336$-dimensional with the monomial basis $\{[c_j]_{\widetilde{\omega}}:\ 1\leq j\leq 336\}.$ Note that the admissible monomials $c_j$ are explicitly described as in Subsect. \ref{s51}. Let us consider the following admissible monomials: $$ \begin{array}{ll} \medskip c_1 &= t_1^{3}t_2t_3^{15}t_4^{2}t_5^{2}t_6^{3}, \ \ c_{61} = t_1^{3}t_2^{7}t_3^{11}t_4t_5^{2}t_6^{2},\ \ c_{121} = t_1t_2^{3}t_3^{14}t_4^{2}t_5^{2}t_6^{3},\\ \medskip c_{156} &= t_1^{3}t_2^{3}t_3^{13}t_4^{2}t_5^{2}t_6^{3}, \ \ c_{166} = t_1^{3}t_2t_3^{2}t_4^{3}t_5^{6}t_6^{11},\ \ c_{196} = t_1^{3}t_2^{5}t_3^{11}t_4^{2}t_5^{2}t_6^{3},\\ c_{211} &= t_1t_2^{7}t_3^{10}t_4^{2}t_5^{3}t_6^{3}, \ \ c_{286} = t_1^{3}t_2^{7}t_3^{9}t_4^{2}t_5^{2}t_6^{3},\ \ c_{301} = t_1^{3}t_2t_3^{3}t_4^{3}t_5^{6}t_6^{10}. \end{array}$$ It is easy to see that $[c_{156}]_{\widetilde{\omega}} = [\theta_1(c_{121})]_{\widetilde{\omega}},$\ $[c_{196}]_{\widetilde{\omega}} = [\theta_2(c_{166})]_{\widetilde{\omega}},$ and $[c_{286}]_{\widetilde{\omega}} = [\theta_3(c_{211})]_{\widetilde{\omega}},$ where $\theta_j$ are suitable permutations in $\Sigma_6.$ So, the following spaces are $\Sigma_6$-submodules of $QP_{n_1}^{\otimes 6}(\widetilde{\omega})$: $$ \begin{array}{ll} \medskip \langle[\Sigma_6(c_{1})]_{\widetilde{\omega}}\rangle &= \langle \{[c_j]_{\widetilde{\omega}}:\ 1\leq j\leq 60\} \rangle,\ \ \langle[\Sigma_6(c_{61})]_{\widetilde{\omega}}\rangle = \langle \{[c_j]_{\widetilde{\omega}}:\ 61\leq j\leq 120\} \rangle,\\ \medskip \langle[\Sigma_6(c_{121})]_{\widetilde{\omega}}\rangle &= \langle \{[c_j]_{\widetilde{\omega}}:\ 121\leq j\leq 165\} \rangle,\ \ \langle[\Sigma_6(c_{166})]_{\widetilde{\omega}}\rangle = \langle \{[c_j]_{\widetilde{\omega}}:\ 166\leq j\leq 210\} \rangle,\\ \langle[\Sigma_6(c_{211})]_{\widetilde{\omega}}\rangle &= \langle \{[c_j]_{\widetilde{\omega}}:\ 211\leq j\leq 300\} \rangle,\ \ \langle[\Sigma_6(c_{301})]_{\widetilde{\omega}}\rangle = \langle \{[c_j]_{\widetilde{\omega}}:\ 301\leq j\leq 336\} \rangle. \end{array}$$ So, we have an isomorphism $$ \begin{array}{ll} \medskip QP_{n_1}^{\otimes 6}(\widetilde{\omega}) &\cong \langle[\Sigma_6(c_{1})]_{\widetilde{\omega}}\rangle\bigoplus \langle[\Sigma_6(c_{61})]_{\widetilde{\omega}}\rangle \bigoplus \langle[\Sigma_6(c_{121})]_{\omega}\rangle\\ &\quad\bigoplus \langle[\Sigma_6(c_{166})]_{\widetilde{\omega}}\rangle\bigoplus \langle[\Sigma_6(c_{211})]_{\widetilde{\omega}}\rangle\bigoplus \langle[\Sigma_6(c_{301})]_{\widetilde{\omega}}\rangle. \end{array}$$ By direct calculations using the homomorphisms $\sigma_i: P^{\otimes 6}\longrightarrow P^{\otimes 6}$ for $1\leq i\leq 5,$ we obtain the following results: $$ \begin{array}{ll} \medskip \langle[\Sigma_6(c_{j})]_{\widetilde{\omega}}\rangle^{\Sigma_6} &= 0,\ \ \mbox{for $j = 121,\, 166,\, 211,$}\\ \langle[\Sigma_6(c_{j})]_{\widetilde{\omega}}\rangle^{\Sigma_6} &= \langle [\widehat{p(c_j)}]_{\widetilde{\omega}}\rangle,\ \ \mbox{for $j = 1,\, 61,\, 301$}, \end{array}$$ where $\widehat{p(c_j)} = \sum_{c_j\in \mathscr C(c_j)} c_j$ with $\mathscr C(c_1) = \{c_j:\ 1\leq j\leq 60\},$\ $\mathscr C(c_{61}) = \{c_j:\ 61\leq j\leq 120\},$ and $\mathscr C(c_{301}) = \{c_j:\ 301\leq j\leq 336\}.$ Note that the sets $[\mathscr C(c_j)]_{\widetilde{\omega}}$ are the bases of the spaces $\langle[\Sigma_6(c_{j})]_{\widetilde{\omega}}\rangle$ for $j = 1,\, 61,\, 301.$ Thus, one gets $[QP_{n_1}^{\otimes 6}(\widetilde{\omega})]^{\Sigma_6} = \langle [\widehat{p(c_1)}]_{\widetilde{\omega}},\ [\widehat{p(c_{61})}]_{\widetilde{\omega}},\ [\widehat{p(c_{301})}]_{\widetilde{\omega}} \rangle.$ Now, assume that $[g]_{\widetilde{\omega}}\in [QP_{n_1}^{\otimes 6}(\widetilde{\omega})]^{GL_6},$ then because $\Sigma_6\subset GL_6,$ we must have that $g\equiv_{\widetilde{\omega}}\gamma_1\widehat{p(c_1)} + \gamma_2\widehat{p(c_{61})} + \gamma_3\widehat{p(c_{301})},$ in which $\gamma_j\in \mathbb F_2$ for every $j.$ By a simple computation using the homomorphism $\sigma_6$ and the relation $\sigma_6(g) +g\equiv_{\widetilde{\omega}}0,$ we get $\sigma_6(g) +g\equiv_{\widetilde{\omega}} (\gamma_1c_5 + (\gamma_1 + \gamma_2)(c_{23} + c_{24} + c_{25}) + \gamma_{3}c_{308} + \ \mbox{other terms })\equiv_{\widetilde{\omega}}0,$ which implies that $\gamma_1 = \gamma_2 = \gamma_3 = 0.$ Hence, the invariant $[QP_{n_1}^{\otimes 6}(\widetilde{\omega})]^{GL_6}$ is equal to $0.$ \medskip Next, we compute the space $[QP_{n_1}^{\otimes 6}(\omega)]^{GL_6}.$ Following the proof of Theorem \ref{dlc1}, $\dim QP_{n_1}^{\otimes 6}(\omega) = 210,$ and $QP_{n_1}^{\otimes 6}(\omega) = \langle \{[d_j]_{\omega}:\ 1\leq j\leq 210\}\rangle,$ where the admissible monomials $d_j$ are explicitly described as in Subsect. \ref{s52}. By a simple computation, we have a direct summand decomposition of the $\Sigma_6$-submodules: $$ QP_{n_1}^{\otimes 6}(\omega) = \langle[\Sigma_6(d_1)]_{\omega}\rangle\bigoplus \langle[\Sigma_6(d_{21})]_{\omega}\rangle \bigoplus \langle[\Sigma_6(d_{111})]_{\omega}\rangle\bigoplus \langle[\Sigma_6(d_{201})]_{\omega}\rangle,$$ where $$ \begin{array}{ll} \medskip \langle[\Sigma_6(d_1)]_{\omega}\rangle\ &=\langle \{[d_j]_{\omega}:\ 1\leq j\leq 20\} \rangle, \ \ \ \langle[\Sigma_6(d_{21})]_{\omega}\rangle =\langle \{[d_j]_{\omega}:\ 21\leq j\leq 110\} \rangle,\\ \langle[\Sigma_6(d_{111})]_{\omega}\rangle &=\langle \{[d_j]_{\omega}:\ 111\leq j\leq 200\} \rangle, \ \ \ \langle[\Sigma_6(d_{201})]_{\omega}\rangle =\langle \{[d_j]_{\omega}:\ 201\leq j\leq 210\} \rangle. \end{array}$$ We first compute the action of the symmetric group $\Sigma_6$ on $ QP_{n_1}^{\otimes 6}(\omega).$ We find that $$[QP_{n_1}^{\otimes 6}(\omega)]^{\Sigma_6} = \big\langle \{[\sum_{1\leq j\leq 20}d_j]_{\omega},\ [\sum_{21\leq j\leq 110}d_j]_{\omega},\ [\sum_{111\leq j\leq 200}d_j]_{\omega}\}\big \rangle.$$ This is immediate from the following assertions: \begin{enumerate} \item[i)] $ \langle[\Sigma_6(d_1)]_{\omega}\rangle^{\Sigma_6} = \langle [\widehat{p(d_1)}]_{\omega}$ with $\widehat{p(d_1)}:= \sum_{1\leq j\leq 20}d_j;$ \medskip \item[ii)] $\langle[\Sigma_6(d_{21})]_{\omega}\rangle^{\Sigma_6} = \langle [\widehat{p(d_{21})}]_{\omega} \rangle$ with $\widehat{p(d_{21})}:= \sum_{21\leq j\leq 110}d_j;$ \medskip \item[iii)] $\langle[\Sigma_6(d_{111})]_{\omega}\rangle^{\Sigma_6} = \langle [\widehat{p(d_{111})}]_{\omega} \rangle$ with $\widehat{p(d_{111})}:= \sum_{111\leq j\leq 200}d_j;$ \medskip \item[iv)] $\langle[\Sigma_6(d_{201})]_{\omega}\rangle^{\Sigma_6} = 0.$ \end{enumerate} We compute the cases i) and iv) and leave the rest to the reader. It is straightforward to see that the sets $[\mathscr C(d_1)]_{\omega} = \{[d_j]_{\omega}:\ 1\leq j\leq 20\}$ and $[\mathscr C(d_{201})]_{\omega} =\{[d_j]_{\omega}:\ 201\leq j\leq 210\}$ are the bases of the spaces $\langle[\Sigma_6(d_1)]_{\omega}\rangle$ and $\langle[\Sigma_6(d_{201})]_{\omega}\rangle,$ respectively. Suppose that $[f]_{\omega}\in \langle[\Sigma_6(d_1)]_{\omega}\rangle^{\Sigma_6}$ and $[g]_{\omega}\in \langle[\Sigma_6(d_{201})]_{\omega}\rangle^{\Sigma_6}.$ Then, one has that $f\equiv_{\omega}\sum_{1\leq j\leq 20}\gamma_jd_j,$ and $g\equiv_{\omega}\sum_{201\leq j\leq 210}\beta_jd_j,$ in which the coefficients $\gamma_j$ and $\beta_j$ belong to $\mathbb F_2$ for all $j.$ By direct calculations using the homomorphisms $\sigma_i: P^{\otimes 6}\longrightarrow P^{\otimes 6}$ for $1\leq i\leq 5,$ and the relations $\sigma_i(f) + f\equiv_{\omega}0,$ and $\sigma_i(g) + g\equiv_{\omega}0,$ we obtain the following equalities: $$ \begin{array}{ll} \sigma_1(f) +f& \equiv_{\omega} (\gamma_5 + \gamma_{11})(d_5 + d_{11}) + (\gamma_6 + \gamma_{12})(d_6 + d_{12}) + (\gamma_7 + \gamma_{13})(d_7 + d_{13}) \\ \medskip &\quad + (\gamma_8 + \gamma_{14})(d_8 + d_{14})+ (\gamma_9 + \gamma_{15})(d_9 + d_{15}) + (\gamma_{10} + \gamma_{16})(d_{10} + d_{16}) \equiv_{\omega} 0,\\ \sigma_2(f) + f&\equiv_{\omega} (\gamma_2 + \gamma_{5})(d_2 + d_{5}) + (\gamma_3 + \gamma_{6})(d_3 + d_{6}) + (\gamma_4 + \gamma_{7})(d_4 + d_{7})\\ \medskip &\quad + (\gamma_{14} + \gamma_{17})(d_{14} + d_{17}) + (\gamma_{15} + \gamma_{18})(d_{15} + d_{18}) + (\gamma_{16} + \gamma_{19})(d_{16} + d_{19})\equiv_{\omega} 0,\\ \sigma_3(f) + f&\equiv_{\omega} (\gamma_1 + \gamma_{2})(d_1 + d_{2}) + (\gamma_6 + \gamma_{8})(d_6 + d_{8}) + (\gamma_7 + \gamma_{9})(d_7 + d_{9}) \\ \medskip &\quad + (\gamma_{12} + \gamma_{14})(d_{12} + d_{14}) + (\gamma_{13} + \gamma_{15})(d_{13} + d_{15}) + (\gamma_{19} + \gamma_{20})(d_{19} + d_{20})\equiv_{\omega} 0,\\ \sigma_4(f) + f&\equiv_{\omega} (\gamma_2 + \gamma_{3})(d_2 + d_{3}) + (\gamma_5 + \gamma_{6})(d_5 + d_{6}) + (\gamma_9 + \gamma_{10})(d_9 + d_{10})\\ \medskip &\quad + (\gamma_{11} + \gamma_{12})(d_{11} + d_{12}) + (\gamma_{15} + \gamma_{16})(d_{15} + d_{16}) + (\gamma_{18} + \gamma_{19})(d_{18} + d_{19})\equiv_{\omega} 0,\\ \sigma_5(f) + f&\equiv_{\omega} (\gamma_3 + \gamma_{4})(d_3 + d_{4}) + (\gamma_6 + \gamma_{7})(d_6 + d_{7}) + (\gamma_8 + \gamma_{9})(d_8 + d_{9})\\ \medskip &\quad + (\gamma_{12} + \gamma_{13})(d_{12} + d_{13}) + (\gamma_{14} + \gamma_{15})(d_{14} + d_{15}) + (\gamma_{17} + \gamma_{18})(d_{17} + d_{18}) \equiv_{\omega} 0,\\ \end{array}$$ \newpage $$ \begin{array}{ll} \medskip \sigma_1(g) + g &\equiv_{\omega} \beta_{205}d_{205} + (\beta_{205} + \beta_{208} + \beta_{209})(d_{205} + d_{208} + d_{209})\\ &\quad + (\beta_{206} + \beta_{208} + \beta_{210})(d_{206} + d_{208} + d_{210})\\ \medskip &\quad + (\beta_{207} + \beta_{209} + \beta_{210})(d_{207} + d_{209} + d_{210}) \equiv_{\omega} 0,\\ \sigma_2(g) + g &\equiv_{\omega} (\beta_{202} + \beta_{205})(d_{202} + d_{205}) + (\beta_{203} + \beta_{206})(d_{203} + d_{206})\\ \medskip &\quad + (\beta_{204} + \beta_{207})(d_{204} + d_{207}) \equiv_{\omega} 0,\\ \sigma_3(g) + g &\equiv_{\omega} (\beta_{201} + \beta_{202})(d_{201} + d_{202}) + (\beta_{206} + \beta_{208})(d_{206} + d_{208})\\ \medskip &\quad + (\beta_{207} + \beta_{209})(d_{207} + d_{209})\equiv_{\omega} 0,\\ \sigma_4(g) + g &\equiv_{\omega} (\beta_{202} + \beta_{203})(d_{202} + d_{203}) + (\beta_{205} + \beta_{206})(d_{205} + d_{206})\\ \medskip &\quad + (\beta_{209} + \beta_{210})(d_{209} + d_{210})\equiv_{\omega} 0,\\ \sigma_5(g) + g &\equiv_{\omega} (\beta_{203} + \beta_{204})(d_{203} + d_{204}) + (\beta_{206} + \beta_{207})(d_{206} + d_{207})\\ &\quad + (\beta_{208} + \beta_{209})(d_{208} + d_{209}) \equiv_{\omega} 0. \end{array}$$ The above equalities imply that $\gamma_{j} = \gamma_{1}$ for all $j,\, 2\leq j\leq 20,$ and $\beta_{201} = \beta_{202} = \cdots = \beta_{210} = 0.$ Next, we compute the action of the general linear group $GL_6$ on $QP_{n_1}^{\otimes 6}(\omega).$ Since $\Sigma_6\subset GL_6,$ if the equivalence class $[h]_{\omega}$ belongs to the invariant $[QP_{n_1}^{\otimes 6}(\omega)]^{GL_6},$ then $$h\equiv_{\omega} \xi_1\widehat{p(d_1)} + \xi_2\widehat{p(d_{21})} + \xi_3\widehat{p(d_{111})},\ \ \xi_j\in \mathbb F_2,\ j = 1,\, 2,\, 3.$$ Using the homomorphism $\sigma_6: P^{\otimes 6}\longrightarrow P^{\otimes 6}$ and the relation $\sigma_6(h) + h\equiv_{\omega} 0,$ we get $$ \sigma_6(h) + h\equiv_{\omega} \big(\xi_1\big(\sum_{5\leq j\leq 10}d_j\big) + \xi_2d_{27} + \xi_3d_{111} + \mbox{other terms }\big)\equiv_{\omega} 0.$$ The above equality indicates that $\xi_1 = \xi_2 = \xi_3 = 0,$ and therefore $[QP_{n_1}^{\otimes 6}(\omega)]^{GL_6}$ is zero. \medskip Thus, from the above calculations, one gets $[{\rm Ker}((\widetilde {Sq^0_})_{n_1})]^{GL_6} = 0$. On the other hand, because $$\dim [QP_{n_1}^{\otimes 6}]^{GL_6}\leq \dim [{\rm Ker}((\widetilde {Sq^0_})_{n_1})]^{GL_6} + \dim [QP_{n_0}^{\otimes 6}]^{GL_6}$$ and $[QP_{n_0}^{\otimes 6}]^{GL_6} \cong (\mathbb F_2 \otimes_{GL_6} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 6}]^{})_{n_0} = 0$ (see \cite{Phuc11}), one gets $[QP_{n_1}^{\otimes 6}]^{GL_6} = 0.$ By this and Corollary \ref{hqs22}), the coinvariant $(\mathbb F_2 \otimes_{GL_6} {\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 6}]^{})_{n_s}$ vanishes for every positive integer $s.$ Now, it is well-known (see, for instance, Tangora \cite{Tangora}) that the only elements $h_2^{2}g_1 = h_4Ph_2$, and $D_2$ are non-zero in ${\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$, and ${\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2),$ respectively. These data and the above calculations indicate that the sixth algebraic transfer, $Tr_6^{\mathcal A}$ is a monomorphism, but not an epimorphism in degrees $n_s,$ for $0 < s\leq 2.$ Therefore, Singer's transfer $Tr_6^{\mathcal A}$ does not detect the non-zero elements $h_4Ph_2$ and $D_2.$ One can observe that the result for $s=3$ is a consequence of the fact (as referenced in \cite{Bruner, Bruner2, Lin2}) that the sixth cohomology group ${\rm Ext}_{\mathcal A}^{6,6+n_3}(\mathbb F_2, \mathbb F_2)$ is trivial. The proof of the theorem is complete. \section{Conclusion}\label{s5} The central emphasis of our work is to further investigate the hit problem for the polynomial algebra $P^{\otimes 6} = \mathbb F_2[t_1, \ldots, t_6]$ in degree $n_s = 16\cdot 2^{s}-6$ for general $s$, building on the results from a previous work in \cite{MKR} which addressed the case $s = 0$. We have shown that the cohit $\mathbb F_2$-module $(P^{\otimes h}/\overline{\mathcal A}P^{\otimes h}){n_{h, s}}$ is equal to the order of the factor group $GL_{h-1}/B_{h-1}$ in general degrees $n_{h, s}=2^{s+4}-h$ with $h\geq 6$ and $s\geq h-5$. Also, based on a previously established result in \cite{Hai}, we have indicated that the cohit $\mathbb F_q$-module $(\mathbb F_q[t_1,\ldots, t_h]/\overline{\pmb A}_q\mathbb F_q[t_1,\ldots, t_h])_{q^{h-1}-h}$ is equal to the order of the factor group $GL_{h-1}(\mathbb F_q)/B^{}_{h-1}(\mathbb F_q).$ As applications, we have established the dimension result for the cohit module $(\mathbb F_2\otimes_{\mathcal A}P^{\otimes 7})_{n_{s+5}}$ and confirmed Singer's Conjecture \ref{gtSinger} for bidegrees $(h, h+n)$ with $h\geq 1$ and $1\leq n\leq n_0$, as well as for bidegrees $(6, 6+n_s)$ with $s\geq 1$. One of the important corollaries then states that the sixth algebraic transfer does not detect the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$ and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2).$ The research conducted in this article advances the understanding of the Peterson hit problem and Singer's algebraic transfer, representing a significant contribution to the literature on this topic. In particular, the application of the hit problem technique in this study has showcased its effectiveness as a powerful tool for investigating the algebraic transfer. We therefore can expect that this approach will lead to more significant breakthroughs in the future. \section{Appendix}\label{s6} In this section, we first enumerate all admissible monomials in the spaces $(P^{\otimes 6}_{n_1})^{> 0}(4,5,1,1) = P^{\otimes 6}_{n_1}(4,5,1,1)$ and $(P^{\otimes 6}_{n_1})^{> 0}(4,5,3) = P^{\otimes 6}_{n_1}(4,5,3).$ \subsection{Admissible monomials in \mbox{\boldmath $(P^{\otimes 6}_{n_1})^{> 0}(4,5,1,1)$}}\label{s51} According to the proof of Theorem \ref{dlc1}, the dimension of $(QP^{\otimes 6}_{n_1})^{>0}(4,5,1,1)$ is equal to the cardinality of $D_1,$ where $D_1 = \big\{c_j:\ 1\leq j\leq 336\big\}$ and the admissible monomials $c_j$ are given as follows: \begin{center} \begin{tabular}{llll} ${\rm c}_{1}=t_1^{3}t_2t_3^{15}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{2}=t_1^{3}t_2t_3^{15}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{3}=t_1^{3}t_2t_3^{15}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{4}=t_1^{3}t_2^{3}t_3^{15}t_4t_5^{2}t_6^{2}$, \\ ${\rm c}_{5}=t_1t_2^{3}t_3^{15}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{6}=t_1t_2^{3}t_3^{15}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{7}=t_1t_2^{3}t_3^{15}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{8}=t_1t_2^{2}t_3^{15}t_4^{2}t_5^{3}t_6^{3}$, \\ ${\rm c}_{9}=t_1t_2^{2}t_3^{15}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{10}=t_1t_2^{2}t_3^{15}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{11}=t_1^{3}t_2t_3^{2}t_4^{2}t_5^{3}t_6^{15}$, & ${\rm c}_{12}=t_1^{3}t_2t_3^{2}t_4^{2}t_5^{15}t_6^{3}$, \\ ${\rm c}_{13}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{2}t_6^{15}$, & ${\rm c}_{14}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{15}t_6^{2}$, & ${\rm c}_{15}=t_1^{3}t_2t_3^{2}t_4^{15}t_5^{2}t_6^{3}$, & ${\rm c}_{16}=t_1^{3}t_2t_3^{2}t_4^{15}t_5^{3}t_6^{2}$, \\ ${\rm c}_{17}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{2}t_6^{15}$, & ${\rm c}_{18}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{15}t_6^{2}$, & ${\rm c}_{19}=t_1^{3}t_2t_3^{3}t_4^{15}t_5^{2}t_6^{2}$, & ${\rm c}_{20}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{2}t_6^{15}$, \\ ${\rm c}_{21}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{15}t_6^{2}$, & ${\rm c}_{22}=t_1^{3}t_2^{3}t_3t_4^{15}t_5^{2}t_6^{2}$, & ${\rm c}_{23}=t_1^{3}t_2^{15}t_3t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{24}=t_1^{3}t_2^{15}t_3t_4^{2}t_5^{3}t_6^{2}$, \\ ${\rm c}_{25}=t_1^{3}t_2^{15}t_3t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{26}=t_1^{3}t_2^{15}t_3^{3}t_4t_5^{2}t_6^{2}$, & ${\rm c}_{27}=t_1^{15}t_2t_3^{2}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{28}=t_1^{15}t_2t_3^{2}t_4^{3}t_5^{2}t_6^{3}$, \\ ${\rm c}_{29}=t_1^{15}t_2t_3^{2}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{30}=t_1^{15}t_2t_3^{3}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{31}=t_1^{15}t_2t_3^{3}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{32}=t_1^{15}t_2t_3^{3}t_4^{3}t_5^{2}t_6^{2}$, \\ ${\rm c}_{33}=t_1^{15}t_2^{3}t_3t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{34}=t_1^{15}t_2^{3}t_3t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{35}=t_1^{15}t_2^{3}t_3t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{36}=t_1^{15}t_2^{3}t_3^{3}t_4t_5^{2}t_6^{2}$, \\ ${\rm c}_{37}=t_1t_2^{3}t_3^{2}t_4^{2}t_5^{3}t_6^{15}$, & ${\rm c}_{38}=t_1t_2^{3}t_3^{2}t_4^{2}t_5^{15}t_6^{3}$, & ${\rm c}_{39}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{2}t_6^{15}$, & ${\rm c}_{40}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{15}t_6^{2}$, \\ ${\rm c}_{41}=t_1t_2^{3}t_3^{2}t_4^{15}t_5^{2}t_6^{3}$, & ${\rm c}_{42}=t_1t_2^{3}t_3^{2}t_4^{15}t_5^{3}t_6^{2}$, & ${\rm c}_{43}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{2}t_6^{15}$, & ${\rm c}_{44}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{15}t_6^{2}$, \\ ${\rm c}_{45}=t_1t_2^{3}t_3^{3}t_4^{15}t_5^{2}t_6^{2}$, & ${\rm c}_{46}=t_1t_2^{15}t_3^{2}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{47}=t_1t_2^{15}t_3^{2}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{48}=t_1t_2^{15}t_3^{2}t_4^{3}t_5^{3}t_6^{2}$, \\ ${\rm c}_{49}=t_1t_2^{15}t_3^{3}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{50}=t_1t_2^{15}t_3^{3}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{51}=t_1t_2^{15}t_3^{3}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{52}=t_1t_2^{2}t_3^{3}t_4^{2}t_5^{3}t_6^{15}$, \\ ${\rm c}_{53}=t_1t_2^{2}t_3^{3}t_4^{2}t_5^{15}t_6^{3}$, & ${\rm c}_{54}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{2}t_6^{15}$, & ${\rm c}_{55}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{15}t_6^{2}$, & ${\rm c}_{56}=t_1t_2^{2}t_3^{3}t_4^{15}t_5^{2}t_6^{3}$, \\ ${\rm c}_{57}=t_1t_2^{2}t_3^{3}t_4^{15}t_5^{3}t_6^{2}$, & ${\rm c}_{58}=t_1t_2^{2}t_3^{2}t_4^{3}t_5^{15}t_6^{3}$, & ${\rm c}_{59}=t_1t_2^{2}t_3^{2}t_4^{15}t_5^{3}t_6^{3}$, & ${\rm c}_{60}=t_1t_2^{2}t_3^{2}t_4^{3}t_5^{3}t_6^{15}$, \\ ${\rm c}_{61}=t_1^{3}t_2^{7}t_3^{11}t_4t_5^{2}t_6^{2}$, & ${\rm c}_{62}=t_1^{7}t_2^{3}t_3^{11}t_4t_5^{2}t_6^{2}$, & ${\rm c}_{63}=t_1t_2^{7}t_3^{11}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{64}=t_1^{7}t_2t_3^{11}t_4^{2}t_5^{2}t_6^{3}$, \\ ${\rm c}_{65}=t_1t_2^{7}t_3^{11}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{66}=t_1^{7}t_2t_3^{11}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{67}=t_1t_2^{7}t_3^{11}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{68}=t_1^{7}t_2t_3^{11}t_4^{3}t_5^{2}t_6^{2}$, \\ ${\rm c}_{69}=t_1^{3}t_2t_3^{2}t_4^{2}t_5^{7}t_6^{11}$, & ${\rm c}_{70}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{2}t_6^{11}$, & ${\rm c}_{71}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{11}t_6^{2}$, & ${\rm c}_{72}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{2}t_6^{11}$, \\ ${\rm c}_{73}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{74}=t_1^{3}t_2t_3^{7}t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{75}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{76}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{11}t_6^{2}$, \\ ${\rm c}_{77}=t_1^{3}t_2^{7}t_3t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{78}=t_1t_2^{3}t_3^{2}t_4^{2}t_5^{7}t_6^{11}$, & ${\rm c}_{79}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{2}t_6^{11}$, & ${\rm c}_{80}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{11}t_6^{2}$, \\ ${\rm c}_{81}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{82}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{83}=t_1t_2^{3}t_3^{7}t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{84}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{2}t_6^{11}$, \\ ${\rm c}_{85}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{86}=t_1^{7}t_2^{3}t_3t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{87}=t_1t_2^{2}t_3^{3}t_4^{2}t_5^{7}t_6^{11}$, & ${\rm c}_{88}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{2}t_6^{11}$, \\ ${\rm c}_{89}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{11}t_6^{2}$, & ${\rm c}_{90}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{91}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{92}=t_1t_2^{7}t_3^{3}t_4^{11}t_5^{2}t_6^{2}$, \\ ${\rm c}_{93}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{94}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{95}=t_1^{7}t_2t_3^{3}t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{96}=t_1^{7}t_2^{11}t_3^{3}t_4t_5^{2}t_6^{2}$, \\ ${\rm c}_{97}=t_1t_2^{2}t_3^{2}t_4^{7}t_5^{11}t_6^{3}$, & ${\rm c}_{98}=t_1t_2^{2}t_3^{7}t_4^{2}t_5^{11}t_6^{3}$, & ${\rm c}_{99}=t_1t_2^{2}t_3^{7}t_4^{11}t_5^{2}t_6^{3}$, & ${\rm c}_{100}=t_1t_2^{7}t_3^{2}t_4^{2}t_5^{11}t_6^{3}$, \\ ${\rm c}_{101}=t_1t_2^{7}t_3^{2}t_4^{11}t_5^{2}t_6^{3}$, & ${\rm c}_{102}=t_1^{7}t_2t_3^{2}t_4^{2}t_5^{11}t_6^{3}$, & ${\rm c}_{103}=t_1^{7}t_2t_3^{2}t_4^{11}t_5^{2}t_6^{3}$, & ${\rm c}_{104}=t_1^{7}t_2^{11}t_3t_4^{2}t_5^{2}t_6^{3}$, \\ ${\rm c}_{105}=t_1t_2^{2}t_3^{2}t_4^{7}t_5^{3}t_6^{11}$, & ${\rm c}_{106}=t_1t_2^{2}t_3^{7}t_4^{2}t_5^{3}t_6^{11}$, & ${\rm c}_{107}=t_1t_2^{2}t_3^{7}t_4^{11}t_5^{3}t_6^{2}$, & ${\rm c}_{108}=t_1t_2^{7}t_3^{2}t_4^{2}t_5^{3}t_6^{11}$, \\ ${\rm c}_{109}=t_1t_2^{7}t_3^{2}t_4^{11}t_5^{3}t_6^{2}$, & ${\rm c}_{110}=t_1^{7}t_2t_3^{2}t_4^{2}t_5^{3}t_6^{11}$, & ${\rm c}_{111}=t_1^{7}t_2t_3^{2}t_4^{11}t_5^{3}t_6^{2}$, & ${\rm c}_{112}=t_1^{7}t_2^{11}t_3t_4^{2}t_5^{3}t_6^{2}$, \\ ${\rm c}_{113}=t_1t_2^{2}t_3^{2}t_4^{3}t_5^{7}t_6^{11}$, & ${\rm c}_{114}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{2}t_6^{11}$, & ${\rm c}_{115}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{116}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{2}t_6^{11}$, \\ ${\rm c}_{117}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{118}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{2}t_6^{11}$, & ${\rm c}_{119}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{120}=t_1^{7}t_2^{11}t_3t_4^{3}t_5^{2}t_6^{2}$, \\ ${\rm c}_{121}=t_1t_2^{3}t_3^{14}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{122}=t_1t_2^{3}t_3^{14}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{123}=t_1^{3}t_2t_3^{14}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{124}=t_1^{3}t_2t_3^{14}t_4^{3}t_5^{2}t_6^{3}$, \\ ${\rm c}_{125}=t_1t_2^{3}t_3^{14}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{126}=t_1^{3}t_2t_3^{14}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{127}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{3}t_6^{14}$, & ${\rm c}_{128}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{14}t_6^{3}$, \\ ${\rm c}_{129}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{3}t_6^{14}$, & ${\rm c}_{130}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{14}t_6^{3}$, & ${\rm c}_{131}=t_1^{3}t_2t_3^{3}t_4^{3}t_5^{2}t_6^{14}$, & ${\rm c}_{132}=t_1^{3}t_2t_3^{3}t_4^{3}t_5^{14}t_6^{2}$, \\ ${\rm c}_{133}=t_1^{3}t_2t_3^{3}t_4^{14}t_5^{2}t_6^{3}$, & ${\rm c}_{134}=t_1^{3}t_2t_3^{3}t_4^{14}t_5^{3}t_6^{2}$, & ${\rm c}_{135}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{3}t_6^{14}$, & ${\rm c}_{136}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{14}t_6^{3}$, \\ ${\rm c}_{137}=t_1^{3}t_2^{3}t_3t_4^{3}t_5^{2}t_6^{14}$, & ${\rm c}_{138}=t_1^{3}t_2^{3}t_3t_4^{3}t_5^{14}t_6^{2}$, & ${\rm c}_{139}=t_1^{3}t_2^{3}t_3t_4^{14}t_5^{2}t_6^{3}$, & ${\rm c}_{140}=t_1^{3}t_2^{3}t_3t_4^{14}t_5^{3}t_6^{2}$, \\ ${\rm c}_{141}=t_1^{3}t_2^{3}t_3^{3}t_4t_5^{2}t_6^{14}$, & ${\rm c}_{142}=t_1^{3}t_2^{3}t_3^{3}t_4t_5^{14}t_6^{2}$, & ${\rm c}_{143}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{3}t_6^{14}$, & ${\rm c}_{144}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{14}t_6^{3}$, \\ \end{tabular} \end{center} \newpage \begin{center} \begin{tabular}{llll} ${\rm c}_{145}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{3}t_6^{14}$, & ${\rm c}_{146}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{14}t_6^{3}$, & ${\rm c}_{147}=t_1t_2^{3}t_3^{3}t_4^{3}t_5^{2}t_6^{14}$, & ${\rm c}_{148}=t_1t_2^{3}t_3^{3}t_4^{3}t_5^{14}t_6^{2}$, \\ ${\rm c}_{149}=t_1t_2^{3}t_3^{3}t_4^{14}t_5^{2}t_6^{3}$, & ${\rm c}_{150}=t_1t_2^{3}t_3^{3}t_4^{14}t_5^{3}t_6^{2}$, & ${\rm c}_{151}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{3}t_6^{14}$, & ${\rm c}_{152}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{14}t_6^{3}$, \\ ${\rm c}_{153}=t_1t_2^{2}t_3^{3}t_4^{14}t_5^{3}t_6^{3}$, & ${\rm c}_{154}=t_1t_2^{3}t_3^{2}t_4^{14}t_5^{3}t_6^{3}$, & ${\rm c}_{155}=t_1^{3}t_2t_3^{2}t_4^{14}t_5^{3}t_6^{3}$, & ${\rm c}_{156}=t_1^{3}t_2^{3}t_3^{13}t_4^{2}t_5^{2}t_6^{3}$, \\ ${\rm c}_{157}=t_1^{3}t_2^{3}t_3^{13}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{158}=t_1^{3}t_2^{3}t_3^{13}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{159}=t_1^{3}t_2^{3}t_3^{3}t_4^{13}t_5^{2}t_6^{2}$, & ${\rm c}_{160}=t_1^{3}t_2^{13}t_3^{3}t_4^{2}t_5^{2}t_6^{3}$, \\ ${\rm c}_{161}=t_1^{3}t_2^{13}t_3^{3}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{162}=t_1^{3}t_2^{13}t_3^{3}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{163}=t_1^{3}t_2^{13}t_3^{2}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{164}=t_1^{3}t_2^{13}t_3^{2}t_4^{3}t_5^{2}t_6^{3}$, \\ ${\rm c}_{165}=t_1^{3}t_2^{13}t_3^{2}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{166}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{6}t_6^{11}$, & ${\rm c}_{167}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{6}t_6^{11}$, & ${\rm c}_{168}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{2}t_6^{11}$, \\ ${\rm c}_{169}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{11}t_6^{2}$, & ${\rm c}_{170}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{6}t_6^{11}$, & ${\rm c}_{171}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{2}t_6^{11}$, & ${\rm c}_{172}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{11}t_6^{2}$, \\ ${\rm c}_{173}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{6}t_6^{11}$, & ${\rm c}_{174}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{6}t_6^{11}$, & ${\rm c}_{175}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{2}t_6^{11}$, & ${\rm c}_{176}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{11}t_6^{2}$, \\ ${\rm c}_{177}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{6}t_6^{11}$, & ${\rm c}_{178}=t_1t_2^{2}t_3^{3}t_4^{6}t_5^{11}t_6^{3}$, & ${\rm c}_{179}=t_1t_2^{3}t_3^{2}t_4^{6}t_5^{11}t_6^{3}$, & ${\rm c}_{180}=t_1t_2^{3}t_3^{6}t_4^{2}t_5^{11}t_6^{3}$, \\ ${\rm c}_{181}=t_1t_2^{3}t_3^{6}t_4^{11}t_5^{2}t_6^{3}$, & ${\rm c}_{182}=t_1^{3}t_2t_3^{2}t_4^{6}t_5^{11}t_6^{3}$, & ${\rm c}_{183}=t_1^{3}t_2t_3^{6}t_4^{2}t_5^{11}t_6^{3}$, & ${\rm c}_{184}=t_1^{3}t_2t_3^{6}t_4^{11}t_5^{2}t_6^{3}$, \\ ${\rm c}_{185}=t_1t_2^{2}t_3^{3}t_4^{6}t_5^{3}t_6^{11}$, & ${\rm c}_{186}=t_1t_2^{3}t_3^{2}t_4^{6}t_5^{3}t_6^{11}$, & ${\rm c}_{187}=t_1t_2^{3}t_3^{6}t_4^{2}t_5^{3}t_6^{11}$, & ${\rm c}_{188}=t_1t_2^{3}t_3^{6}t_4^{11}t_5^{3}t_6^{2}$, \\ ${\rm c}_{189}=t_1^{3}t_2t_3^{2}t_4^{6}t_5^{3}t_6^{11}$, & ${\rm c}_{190}=t_1^{3}t_2t_3^{6}t_4^{2}t_5^{3}t_6^{11}$, & ${\rm c}_{191}=t_1^{3}t_2t_3^{6}t_4^{11}t_5^{3}t_6^{2}$, & ${\rm c}_{192}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{2}t_6^{11}$, \\ ${\rm c}_{193}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{194}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{2}t_6^{11}$, & ${\rm c}_{195}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{196}=t_1^{3}t_2^{5}t_3^{11}t_4^{2}t_5^{2}t_6^{3}$, \\ ${\rm c}_{197}=t_1^{3}t_2^{5}t_3^{11}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{198}=t_1^{3}t_2^{5}t_3^{11}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{199}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{200}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{11}t_6^{2}$, \\ ${\rm c}_{201}=t_1^{3}t_2^{3}t_3^{5}t_4^{11}t_5^{2}t_6^{2}$, & ${\rm c}_{202}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{2}t_6^{11}$, & ${\rm c}_{203}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{11}t_6^{2}$, & ${\rm c}_{204}=t_1^{3}t_2^{5}t_3^{3}t_4^{11}t_5^{2}t_6^{2}$, \\ ${\rm c}_{205}=t_1^{3}t_2^{5}t_3^{2}t_4^{2}t_5^{11}t_6^{3}$, & ${\rm c}_{206}=t_1^{3}t_2^{5}t_3^{2}t_4^{11}t_5^{2}t_6^{3}$, & ${\rm c}_{207}=t_1^{3}t_2^{5}t_3^{2}t_4^{2}t_5^{3}t_6^{11}$, & ${\rm c}_{208}=t_1^{3}t_2^{5}t_3^{2}t_4^{11}t_5^{3}t_6^{2}$, \\ ${\rm c}_{209}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{2}t_6^{11}$, & ${\rm c}_{210}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{11}t_6^{2}$, & ${\rm c}_{211}=t_1t_2^{7}t_3^{10}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{212}=t_1t_2^{7}t_3^{10}t_4^{3}t_5^{2}t_6^{3}$, \\ ${\rm c}_{213}=t_1^{7}t_2t_3^{10}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{214}=t_1^{7}t_2t_3^{10}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{215}=t_1t_2^{7}t_3^{10}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{216}=t_1^{7}t_2t_3^{10}t_4^{3}t_5^{3}t_6^{2}$, \\ ${\rm c}_{217}=t_1^{3}t_2t_3^{2}t_4^{3}t_5^{7}t_6^{10}$, & ${\rm c}_{218}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{3}t_6^{10}$, & ${\rm c}_{219}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{10}t_6^{3}$, & ${\rm c}_{220}=t_1^{3}t_2t_3^{3}t_4^{2}t_5^{7}t_6^{10}$, \\ ${\rm c}_{221}=t_1^{3}t_2t_3^{3}t_4^{7}t_5^{2}t_6^{10}$, & ${\rm c}_{222}=t_1^{3}t_2t_3^{3}t_4^{7}t_5^{10}t_6^{2}$, & ${\rm c}_{223}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{224}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{10}t_6^{3}$, \\ ${\rm c}_{225}=t_1^{3}t_2t_3^{7}t_4^{3}t_5^{2}t_6^{10}$, & ${\rm c}_{226}=t_1^{3}t_2t_3^{7}t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{227}=t_1^{3}t_2t_3^{7}t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{228}=t_1^{3}t_2t_3^{7}t_4^{10}t_5^{3}t_6^{2}$, \\ ${\rm c}_{229}=t_1^{3}t_2^{3}t_3t_4^{2}t_5^{7}t_6^{10}$, & ${\rm c}_{230}=t_1^{3}t_2^{3}t_3t_4^{7}t_5^{2}t_6^{10}$, & ${\rm c}_{231}=t_1^{3}t_2^{3}t_3t_4^{7}t_5^{10}t_6^{2}$, & ${\rm c}_{232}=t_1^{3}t_2^{3}t_3^{7}t_4t_5^{2}t_6^{10}$, \\ ${\rm c}_{233}=t_1^{3}t_2^{3}t_3^{7}t_4t_5^{10}t_6^{2}$, & ${\rm c}_{234}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{235}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{236}=t_1^{3}t_2^{7}t_3t_4^{3}t_5^{2}t_6^{10}$, \\ ${\rm c}_{237}=t_1^{3}t_2^{7}t_3t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{238}=t_1^{3}t_2^{7}t_3t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{239}=t_1^{3}t_2^{7}t_3t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{240}=t_1^{3}t_2^{7}t_3^{3}t_4t_5^{2}t_6^{10}$, \\ ${\rm c}_{241}=t_1^{3}t_2^{7}t_3^{3}t_4t_5^{10}t_6^{2}$, & ${\rm c}_{242}=t_1t_2^{3}t_3^{2}t_4^{3}t_5^{7}t_6^{10}$, & ${\rm c}_{243}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{3}t_6^{10}$, & ${\rm c}_{244}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{10}t_6^{3}$, \\ ${\rm c}_{245}=t_1t_2^{3}t_3^{3}t_4^{2}t_5^{7}t_6^{10}$, & ${\rm c}_{246}=t_1t_2^{3}t_3^{3}t_4^{7}t_5^{2}t_6^{10}$, & ${\rm c}_{247}=t_1t_2^{3}t_3^{3}t_4^{7}t_5^{10}t_6^{2}$, & ${\rm c}_{248}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{3}t_6^{10}$, \\ ${\rm c}_{249}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{250}=t_1t_2^{3}t_3^{7}t_4^{3}t_5^{2}t_6^{10}$, & ${\rm c}_{251}=t_1t_2^{3}t_3^{7}t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{252}=t_1t_2^{3}t_3^{7}t_4^{10}t_5^{2}t_6^{3}$, \\ ${\rm c}_{253}=t_1t_2^{3}t_3^{7}t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{254}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{255}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{256}=t_1^{7}t_2^{3}t_3t_4^{3}t_5^{2}t_6^{10}$, \\ ${\rm c}_{257}=t_1^{7}t_2^{3}t_3t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{258}=t_1^{7}t_2^{3}t_3t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{259}=t_1^{7}t_2^{3}t_3t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{260}=t_1^{7}t_2^{3}t_3^{3}t_4t_5^{2}t_6^{10}$, \\ ${\rm c}_{261}=t_1^{7}t_2^{3}t_3^{3}t_4t_5^{10}t_6^{2}$, & ${\rm c}_{262}=t_1t_2^{2}t_3^{3}t_4^{3}t_5^{7}t_6^{10}$, & ${\rm c}_{263}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{3}t_6^{10}$, & ${\rm c}_{264}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{10}t_6^{3}$, \\ ${\rm c}_{265}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{266}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{267}=t_1t_2^{7}t_3^{3}t_4^{3}t_5^{2}t_6^{10}$, & ${\rm c}_{268}=t_1t_2^{7}t_3^{3}t_4^{3}t_5^{10}t_6^{2}$, \\ ${\rm c}_{269}=t_1t_2^{7}t_3^{3}t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{270}=t_1t_2^{7}t_3^{3}t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{271}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{272}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{10}t_6^{3}$, \\ ${\rm c}_{273}=t_1^{7}t_2t_3^{3}t_4^{3}t_5^{2}t_6^{10}$, & ${\rm c}_{274}=t_1^{7}t_2t_3^{3}t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{275}=t_1^{7}t_2t_3^{3}t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{276}=t_1^{7}t_2t_3^{3}t_4^{10}t_5^{3}t_6^{2}$, \\ ${\rm c}_{277}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{278}=t_1t_2^{2}t_3^{7}t_4^{10}t_5^{3}t_6^{3}$, & ${\rm c}_{279}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{280}=t_1t_2^{7}t_3^{2}t_4^{10}t_5^{3}t_6^{3}$, \\ ${\rm c}_{281}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{282}=t_1^{7}t_2t_3^{2}t_4^{10}t_5^{3}t_6^{3}$, & ${\rm c}_{283}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{3}t_6^{10}$, & ${\rm c}_{284}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{3}t_6^{10}$, \\ ${\rm c}_{285}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{3}t_6^{10}$, & ${\rm c}_{286}=t_1^{3}t_2^{7}t_3^{9}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{287}=t_1^{3}t_2^{7}t_3^{9}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{288}=t_1^{3}t_2^{7}t_3^{9}t_4^{3}t_5^{2}t_6^{2}$, \\ ${\rm c}_{289}=t_1^{7}t_2^{3}t_3^{9}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{290}=t_1^{7}t_2^{3}t_3^{9}t_4^{2}t_5^{3}t_6^{2}$, & ${\rm c}_{291}=t_1^{7}t_2^{3}t_3^{9}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{292}=t_1^{3}t_2^{3}t_3^{7}t_4^{9}t_5^{2}t_6^{2}$, \\ ${\rm c}_{293}=t_1^{3}t_2^{7}t_3^{3}t_4^{9}t_5^{2}t_6^{2}$, & ${\rm c}_{294}=t_1^{7}t_2^{3}t_3^{3}t_4^{9}t_5^{2}t_6^{2}$, & ${\rm c}_{295}=t_1^{7}t_2^{9}t_3^{3}t_4^{2}t_5^{2}t_6^{3}$, & ${\rm c}_{296}=t_1^{7}t_2^{9}t_3^{3}t_4^{2}t_5^{3}t_6^{2}$, \\ ${\rm c}_{297}=t_1^{7}t_2^{9}t_3^{3}t_4^{3}t_5^{2}t_6^{2}$, & ${\rm c}_{298}=t_1^{7}t_2^{9}t_3^{2}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{299}=t_1^{7}t_2^{9}t_3^{2}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{300}=t_1^{7}t_2^{9}t_3^{2}t_4^{3}t_5^{3}t_6^{2}$, \\ ${\rm c}_{301}=t_1^{3}t_2t_3^{3}t_4^{3}t_5^{6}t_6^{10}$, & ${\rm c}_{302}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{3}t_6^{10}$, & ${\rm c}_{303}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{10}t_6^{3}$, & ${\rm c}_{304}=t_1^{3}t_2^{3}t_3t_4^{3}t_5^{6}t_6^{10}$, \\ ${\rm c}_{305}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{3}t_6^{10}$, & ${\rm c}_{306}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{10}t_6^{3}$, & ${\rm c}_{307}=t_1^{3}t_2^{3}t_3^{3}t_4t_5^{6}t_6^{10}$, & ${\rm c}_{308}=t_1t_2^{3}t_3^{3}t_4^{3}t_5^{6}t_6^{10}$, \\ ${\rm c}_{309}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{3}t_6^{10}$, & ${\rm c}_{310}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{10}t_6^{3}$, & ${\rm c}_{311}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{312}=t_1t_2^{3}t_3^{6}t_4^{10}t_5^{3}t_6^{3}$, \\ ${\rm c}_{313}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{314}=t_1^{3}t_2t_3^{6}t_4^{10}t_5^{3}t_6^{3}$, & ${\rm c}_{315}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{3}t_6^{10}$, & ${\rm c}_{316}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{3}t_6^{10}$, \\ ${\rm c}_{317}=t_1^{3}t_2^{5}t_3^{10}t_4^{2}t_5^{3}t_6^{3}$, & ${\rm c}_{318}=t_1^{3}t_2^{5}t_3^{10}t_4^{3}t_5^{2}t_6^{3}$, & ${\rm c}_{319}=t_1^{3}t_2^{5}t_3^{10}t_4^{3}t_5^{3}t_6^{2}$, & ${\rm c}_{320}=t_1^{3}t_2^{3}t_3^{3}t_4^{5}t_5^{2}t_6^{10}$, \\ ${\rm c}_{321}=t_1^{3}t_2^{3}t_3^{3}t_4^{5}t_5^{10}t_6^{2}$, & ${\rm c}_{322}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{3}t_6^{10}$, & ${\rm c}_{323}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{324}=t_1^{3}t_2^{3}t_3^{5}t_4^{3}t_5^{2}t_6^{10}$, \\ ${\rm c}_{325}=t_1^{3}t_2^{3}t_3^{5}t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{326}=t_1^{3}t_2^{3}t_3^{5}t_4^{10}t_5^{2}t_6^{3}$, & ${\rm c}_{327}=t_1^{3}t_2^{3}t_3^{5}t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{328}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{3}t_6^{10}$, \\ ${\rm c}_{329}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{10}t_6^{3}$, & ${\rm c}_{330}=t_1^{3}t_2^{5}t_3^{3}t_4^{3}t_5^{2}t_6^{10}$, & ${\rm c}_{331}=t_1^{3}t_2^{5}t_3^{3}t_4^{3}t_5^{10}t_6^{2}$, & ${\rm c}_{332}=t_1^{3}t_2^{5}t_3^{3}t_4^{10}t_5^{2}t_6^{3}$, \\ ${\rm c}_{333}=t_1^{3}t_2^{5}t_3^{3}t_4^{10}t_5^{3}t_6^{2}$, & ${\rm c}_{334}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{10}t_6^{3}$, & ${\rm c}_{335}=t_1^{3}t_2^{5}t_3^{2}t_4^{10}t_5^{3}t_6^{3}$, & ${\rm c}_{336}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{3}t_6^{10}$. \end{tabular}\end{center} \newpage \subsection{Admissible monomials in \mbox{\boldmath $(P^{\otimes 6}_{n_1})^{> 0}(4,5,3)$}}\label{s52} By the proof of Theorem \ref{dlc1}, $\dim (QP^{\otimes 6}_{n_1})^{>0}(4,5,3) = \|D_2\| = 210,$ where $D_2 = \big\{d_j:\ 1\leq j\leq 210\big\}$ and the admissible monomials $d_j$ are determined as follows: \begin{center} \begin{tabular}{llrr} ${\rm d}_{1}=t_1t_2^{2}t_3^{2}t_4^{7}t_5^{7}t_6^{7}$, & ${\rm d}_{2}=t_1t_2^{2}t_3^{7}t_4^{2}t_5^{7}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{3}=t_1t_2^{2}t_3^{7}t_4^{7}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{4}=t_1t_2^{2}t_3^{7}t_4^{7}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{5}=t_1t_2^{7}t_3^{2}t_4^{2}t_5^{7}t_6^{7}$, & ${\rm d}_{6}=t_1t_2^{7}t_3^{2}t_4^{7}t_5^{2}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{7}=t_1t_2^{7}t_3^{2}t_4^{7}t_5^{7}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{8}=t_1t_2^{7}t_3^{7}t_4^{2}t_5^{2}t_6^{7}$,} \\ ${\rm d}_{9}=t_1t_2^{7}t_3^{7}t_4^{2}t_5^{7}t_6^{2}$, & ${\rm d}_{10}=t_1t_2^{7}t_3^{7}t_4^{7}t_5^{2}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{11}=t_1^{7}t_2t_3^{2}t_4^{2}t_5^{7}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{12}=t_1^{7}t_2t_3^{2}t_4^{7}t_5^{2}t_6^{7}$,} \\ ${\rm d}_{13}=t_1^{7}t_2t_3^{2}t_4^{7}t_5^{7}t_6^{2}$, & ${\rm d}_{14}=t_1^{7}t_2t_3^{7}t_4^{2}t_5^{2}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{15}=t_1^{7}t_2t_3^{7}t_4^{2}t_5^{7}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{16}=t_1^{7}t_2t_3^{7}t_4^{7}t_5^{2}t_6^{2}$,} \\ ${\rm d}_{17}=t_1^{7}t_2^{7}t_3t_4^{2}t_5^{2}t_6^{7}$, & ${\rm d}_{18}=t_1^{7}t_2^{7}t_3t_4^{2}t_5^{7}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{19}=t_1^{7}t_2^{7}t_3t_4^{7}t_5^{2}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{20}=t_1^{7}t_2^{7}t_3^{7}t_4t_5^{2}t_6^{2}$,} \\ ${\rm d}_{21}=t_1t_2^{2}t_3^{3}t_4^{6}t_5^{7}t_6^{7}$, & ${\rm d}_{22}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{6}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{23}=t_1t_2^{2}t_3^{3}t_4^{7}t_5^{7}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{24}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{6}t_6^{7}$,} \\ ${\rm d}_{25}=t_1t_2^{2}t_3^{7}t_4^{3}t_5^{7}t_6^{6}$, & ${\rm d}_{26}=t_1t_2^{2}t_3^{7}t_4^{7}t_5^{3}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{27}=t_1t_2^{3}t_3^{2}t_4^{6}t_5^{7}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{28}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{6}t_6^{7}$,} \\ ${\rm d}_{29}=t_1t_2^{3}t_3^{2}t_4^{7}t_5^{7}t_6^{6}$, & ${\rm d}_{30}=t_1t_2^{3}t_3^{6}t_4^{2}t_5^{7}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{31}=t_1t_2^{3}t_3^{6}t_4^{7}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{32}=t_1t_2^{3}t_3^{6}t_4^{7}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{33}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{6}t_6^{7}$, & ${\rm d}_{34}=t_1t_2^{3}t_3^{7}t_4^{2}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{35}=t_1t_2^{3}t_3^{7}t_4^{6}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{36}=t_1t_2^{3}t_3^{7}t_4^{6}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{37}=t_1t_2^{3}t_3^{7}t_4^{7}t_5^{2}t_6^{6}$, & ${\rm d}_{38}=t_1t_2^{3}t_3^{7}t_4^{7}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{39}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{40}=t_1t_2^{7}t_3^{2}t_4^{3}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{41}=t_1t_2^{7}t_3^{2}t_4^{7}t_5^{3}t_6^{6}$, & ${\rm d}_{42}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{6}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{43}=t_1t_2^{7}t_3^{3}t_4^{2}t_5^{7}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{44}=t_1t_2^{7}t_3^{3}t_4^{6}t_5^{2}t_6^{7}$,} \\ ${\rm d}_{45}=t_1t_2^{7}t_3^{3}t_4^{6}t_5^{7}t_6^{2}$, & ${\rm d}_{46}=t_1t_2^{7}t_3^{3}t_4^{7}t_5^{2}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{47}=t_1t_2^{7}t_3^{3}t_4^{7}t_5^{6}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{48}=t_1t_2^{7}t_3^{7}t_4^{2}t_5^{3}t_6^{6}$,} \\ ${\rm d}_{49}=t_1t_2^{7}t_3^{7}t_4^{3}t_5^{2}t_6^{6}$, & ${\rm d}_{50}=t_1t_2^{7}t_3^{7}t_4^{3}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{51}=t_1^{3}t_2t_3^{2}t_4^{6}t_5^{7}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{52}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{6}t_6^{7}$,} \\ ${\rm d}_{53}=t_1^{3}t_2t_3^{2}t_4^{7}t_5^{7}t_6^{6}$, & ${\rm d}_{54}=t_1^{3}t_2t_3^{6}t_4^{2}t_5^{7}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{55}=t_1^{3}t_2t_3^{6}t_4^{7}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{56}=t_1^{3}t_2t_3^{6}t_4^{7}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{57}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{6}t_6^{7}$, & ${\rm d}_{58}=t_1^{3}t_2t_3^{7}t_4^{2}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{59}=t_1^{3}t_2t_3^{7}t_4^{6}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{60}=t_1^{3}t_2t_3^{7}t_4^{6}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{61}=t_1^{3}t_2t_3^{7}t_4^{7}t_5^{2}t_6^{6}$, & ${\rm d}_{62}=t_1^{3}t_2t_3^{7}t_4^{7}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{63}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{64}=t_1^{3}t_2^{7}t_3t_4^{2}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{65}=t_1^{3}t_2^{7}t_3t_4^{6}t_5^{2}t_6^{7}$, & ${\rm d}_{66}=t_1^{3}t_2^{7}t_3t_4^{6}t_5^{7}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{67}=t_1^{3}t_2^{7}t_3t_4^{7}t_5^{2}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{68}=t_1^{3}t_2^{7}t_3t_4^{7}t_5^{6}t_6^{2}$,} \\ ${\rm d}_{69}=t_1^{3}t_2^{7}t_3^{7}t_4t_5^{2}t_6^{6}$, & ${\rm d}_{70}=t_1^{3}t_2^{7}t_3^{7}t_4t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{71}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{72}=t_1^{7}t_2t_3^{2}t_4^{3}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{73}=t_1^{7}t_2t_3^{2}t_4^{7}t_5^{3}t_6^{6}$, & ${\rm d}_{74}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{6}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{75}=t_1^{7}t_2t_3^{3}t_4^{2}t_5^{7}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{76}=t_1^{7}t_2t_3^{3}t_4^{6}t_5^{2}t_6^{7}$,} \\ ${\rm d}_{77}=t_1^{7}t_2t_3^{3}t_4^{6}t_5^{7}t_6^{2}$, & ${\rm d}_{78}=t_1^{7}t_2t_3^{3}t_4^{7}t_5^{2}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{79}=t_1^{7}t_2t_3^{3}t_4^{7}t_5^{6}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{80}=t_1^{7}t_2t_3^{7}t_4^{2}t_5^{3}t_6^{6}$,} \\ ${\rm d}_{81}=t_1^{7}t_2t_3^{7}t_4^{3}t_5^{2}t_6^{6}$, & ${\rm d}_{82}=t_1^{7}t_2t_3^{7}t_4^{3}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{83}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{84}=t_1^{7}t_2^{3}t_3t_4^{2}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{85}=t_1^{7}t_2^{3}t_3t_4^{6}t_5^{2}t_6^{7}$, & ${\rm d}_{86}=t_1^{7}t_2^{3}t_3t_4^{6}t_5^{7}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{87}=t_1^{7}t_2^{3}t_3t_4^{7}t_5^{2}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{88}=t_1^{7}t_2^{3}t_3t_4^{7}t_5^{6}t_6^{2}$,} \\ ${\rm d}_{89}=t_1^{7}t_2^{3}t_3^{7}t_4t_5^{2}t_6^{6}$, & ${\rm d}_{90}=t_1^{7}t_2^{3}t_3^{7}t_4t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{91}=t_1^{7}t_2^{7}t_3t_4^{2}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{92}=t_1^{7}t_2^{7}t_3t_4^{3}t_5^{2}t_6^{6}$,} \\ ${\rm d}_{93}=t_1^{7}t_2^{7}t_3t_4^{3}t_5^{6}t_6^{2}$, & ${\rm d}_{94}=t_1^{7}t_2^{7}t_3^{3}t_4t_5^{2}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{95}=t_1^{7}t_2^{7}t_3^{3}t_4t_5^{6}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{96}=t_1^{3}t_2^{5}t_3^{2}t_4^{2}t_5^{7}t_6^{7}$,} \\ ${\rm d}_{97}=t_1^{3}t_2^{5}t_3^{2}t_4^{7}t_5^{2}t_6^{7}$, & ${\rm d}_{98}=t_1^{3}t_2^{5}t_3^{2}t_4^{7}t_5^{7}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{99}=t_1^{3}t_2^{5}t_3^{7}t_4^{2}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{100}=t_1^{3}t_2^{5}t_3^{7}t_4^{2}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{101}=t_1^{3}t_2^{5}t_3^{7}t_4^{7}t_5^{2}t_6^{2}$, & ${\rm d}_{102}=t_1^{3}t_2^{7}t_3^{5}t_4^{2}t_5^{2}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{103}=t_1^{3}t_2^{7}t_3^{5}t_4^{2}t_5^{7}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{104}=t_1^{3}t_2^{7}t_3^{5}t_4^{7}t_5^{2}t_6^{2}$,} \\ ${\rm d}_{105}=t_1^{3}t_2^{7}t_3^{7}t_4^{5}t_5^{2}t_6^{2}$, & ${\rm d}_{106}=t_1^{7}t_2^{3}t_3^{5}t_4^{2}t_5^{2}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{107}=t_1^{7}t_2^{3}t_3^{5}t_4^{2}t_5^{7}t_6^{2}$,} & \multicolumn{1}{l}{${\rm d}_{108}=t_1^{7}t_2^{3}t_3^{5}t_4^{7}t_5^{2}t_6^{2}$,} \\ ${\rm d}_{109}=t_1^{7}t_2^{3}t_3^{7}t_4^{5}t_5^{2}t_6^{2}$, & ${\rm d}_{110}=t_1^{7}t_2^{7}t_3^{3}t_4^{5}t_5^{2}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{111}=t_1t_2^{3}t_3^{7}t_4^{3}t_5^{6}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{112}=t_1t_2^{3}t_3^{7}t_4^{6}t_5^{3}t_6^{6}$,} \\ ${\rm d}_{113}=t_1t_2^{3}t_3^{7}t_4^{6}t_5^{6}t_6^{3}$, & ${\rm d}_{114}=t_1^{3}t_2t_3^{7}t_4^{3}t_5^{6}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{115}=t_1^{3}t_2t_3^{7}t_4^{6}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{116}=t_1^{3}t_2t_3^{7}t_4^{6}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{117}=t_1^{3}t_2^{3}t_3^{7}t_4t_5^{6}t_6^{6}$, & ${\rm d}_{118}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{6}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{119}=t_1t_2^{3}t_3^{3}t_4^{6}t_5^{7}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{120}=t_1t_2^{3}t_3^{3}t_4^{7}t_5^{6}t_6^{6}$,} \\ ${\rm d}_{121}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{6}t_6^{7}$, & ${\rm d}_{122}=t_1t_2^{3}t_3^{6}t_4^{3}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{123}=t_1t_2^{3}t_3^{6}t_4^{6}t_5^{3}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{124}=t_1t_2^{3}t_3^{6}t_4^{6}t_5^{7}t_6^{3}$,} \\ ${\rm d}_{125}=t_1t_2^{3}t_3^{6}t_4^{7}t_5^{3}t_6^{6}$, & ${\rm d}_{126}=t_1t_2^{3}t_3^{6}t_4^{7}t_5^{6}t_6^{3}$, & \multicolumn{1}{l}{${\rm d}_{127}=t_1t_2^{7}t_3^{3}t_4^{3}t_5^{6}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{128}=t_1t_2^{7}t_3^{3}t_4^{6}t_5^{3}t_6^{6}$,} \\ ${\rm d}_{129}=t_1t_2^{7}t_3^{3}t_4^{6}t_5^{6}t_6^{3}$, & ${\rm d}_{130}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{6}t_6^{7}$, & \multicolumn{1}{l}{${\rm d}_{131}=t_1^{3}t_2t_3^{3}t_4^{6}t_5^{7}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{132}=t_1^{3}t_2t_3^{3}t_4^{7}t_5^{6}t_6^{6}$,} \\ ${\rm d}_{133}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{6}t_6^{7}$, & ${\rm d}_{134}=t_1^{3}t_2t_3^{6}t_4^{3}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{135}=t_1^{3}t_2t_3^{6}t_4^{6}t_5^{3}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{136}=t_1^{3}t_2t_3^{6}t_4^{6}t_5^{7}t_6^{3}$,} \\ ${\rm d}_{137}=t_1^{3}t_2t_3^{6}t_4^{7}t_5^{3}t_6^{6}$, & ${\rm d}_{138}=t_1^{3}t_2t_3^{6}t_4^{7}t_5^{6}t_6^{3}$, & \multicolumn{1}{l}{${\rm d}_{139}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{140}=t_1^{3}t_2^{3}t_3t_4^{6}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{141}=t_1^{3}t_2^{3}t_3t_4^{7}t_5^{6}t_6^{6}$, & ${\rm d}_{142}=t_1^{3}t_2^{7}t_3t_4^{3}t_5^{6}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{143}=t_1^{3}t_2^{7}t_3t_4^{6}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{144}=t_1^{3}t_2^{7}t_3t_4^{6}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{145}=t_1^{3}t_2^{7}t_3^{3}t_4t_5^{6}t_6^{6}$, & ${\rm d}_{146}=t_1^{7}t_2t_3^{3}t_4^{3}t_5^{6}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{147}=t_1^{7}t_2t_3^{3}t_4^{6}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{148}=t_1^{7}t_2t_3^{3}t_4^{6}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{149}=t_1^{7}t_2^{3}t_3t_4^{3}t_5^{6}t_6^{6}$, & ${\rm d}_{150}=t_1^{7}t_2^{3}t_3t_4^{6}t_5^{3}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{151}=t_1^{7}t_2^{3}t_3t_4^{6}t_5^{6}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{152}=t_1^{7}t_2^{3}t_3^{3}t_4t_5^{6}t_6^{6}$,} \\ ${\rm d}_{153}=t_1^{3}t_2^{3}t_3^{7}t_4^{5}t_5^{2}t_6^{6}$, & ${\rm d}_{154}=t_1^{3}t_2^{3}t_3^{7}t_4^{5}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{155}=t_1^{3}t_2^{5}t_3^{7}t_4^{2}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{156}=t_1^{3}t_2^{5}t_3^{7}t_4^{2}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{157}=t_1^{3}t_2^{5}t_3^{7}t_4^{3}t_5^{2}t_6^{6}$, & ${\rm d}_{158}=t_1^{3}t_2^{5}t_3^{7}t_4^{3}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{159}=t_1^{3}t_2^{5}t_3^{7}t_4^{6}t_5^{2}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{160}=t_1^{3}t_2^{5}t_3^{7}t_4^{6}t_5^{3}t_6^{2}$,} \\ ${\rm d}_{161}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{6}t_6^{7}$, & ${\rm d}_{162}=t_1^{3}t_2^{3}t_3^{5}t_4^{2}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{163}=t_1^{3}t_2^{3}t_3^{5}t_4^{6}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{164}=t_1^{3}t_2^{3}t_3^{5}t_4^{6}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{165}=t_1^{3}t_2^{3}t_3^{5}t_4^{7}t_5^{2}t_6^{6}$, & ${\rm d}_{166}=t_1^{3}t_2^{3}t_3^{5}t_4^{7}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{167}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{6}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{168}=t_1^{3}t_2^{5}t_3^{2}t_4^{3}t_5^{7}t_6^{6}$,} \\ ${\rm d}_{169}=t_1^{3}t_2^{5}t_3^{2}t_4^{6}t_5^{3}t_6^{7}$, & ${\rm d}_{170}=t_1^{3}t_2^{5}t_3^{2}t_4^{6}t_5^{7}t_6^{3}$, & \multicolumn{1}{l}{${\rm d}_{171}=t_1^{3}t_2^{5}t_3^{2}t_4^{7}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{172}=t_1^{3}t_2^{5}t_3^{2}t_4^{7}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{173}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{6}t_6^{7}$, & ${\rm d}_{174}=t_1^{3}t_2^{5}t_3^{3}t_4^{2}t_5^{7}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{175}=t_1^{3}t_2^{5}t_3^{3}t_4^{6}t_5^{2}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{176}=t_1^{3}t_2^{5}t_3^{3}t_4^{6}t_5^{7}t_6^{2}$,} \\ ${\rm d}_{177}=t_1^{3}t_2^{5}t_3^{3}t_4^{7}t_5^{2}t_6^{6}$, & ${\rm d}_{178}=t_1^{3}t_2^{5}t_3^{3}t_4^{7}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{179}=t_1^{3}t_2^{5}t_3^{6}t_4^{2}t_5^{3}t_6^{7}$,} & \multicolumn{1}{l}{${\rm d}_{180}=t_1^{3}t_2^{5}t_3^{6}t_4^{2}t_5^{7}t_6^{3}$,} \\ ${\rm d}_{181}=t_1^{3}t_2^{5}t_3^{6}t_4^{3}t_5^{2}t_6^{7}$, & ${\rm d}_{182}=t_1^{3}t_2^{5}t_3^{6}t_4^{3}t_5^{7}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{183}=t_1^{3}t_2^{5}t_3^{6}t_4^{7}t_5^{2}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{184}=t_1^{3}t_2^{5}t_3^{6}t_4^{7}t_5^{3}t_6^{2}$,} \\ ${\rm d}_{185}=t_1^{3}t_2^{7}t_3^{3}t_4^{5}t_5^{2}t_6^{6}$, & ${\rm d}_{186}=t_1^{3}t_2^{7}t_3^{3}t_4^{5}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{187}=t_1^{3}t_2^{7}t_3^{5}t_4^{2}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{188}=t_1^{3}t_2^{7}t_3^{5}t_4^{2}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{189}=t_1^{3}t_2^{7}t_3^{5}t_4^{3}t_5^{2}t_6^{6}$, & ${\rm d}_{190}=t_1^{3}t_2^{7}t_3^{5}t_4^{3}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{191}=t_1^{3}t_2^{7}t_3^{5}t_4^{6}t_5^{2}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{192}=t_1^{3}t_2^{7}t_3^{5}t_4^{6}t_5^{3}t_6^{2}$,} \\ \end{tabular} \end{center} \newpage \begin{center} \begin{tabular}{llrr} ${\rm d}_{193}=t_1^{7}t_2^{3}t_3^{3}t_4^{5}t_5^{2}t_6^{6}$, & ${\rm d}_{194}=t_1^{7}t_2^{3}t_3^{3}t_4^{5}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{195}=t_1^{7}t_2^{3}t_3^{5}t_4^{2}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{196}=t_1^{7}t_2^{3}t_3^{5}t_4^{2}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{197}=t_1^{7}t_2^{3}t_3^{5}t_4^{3}t_5^{2}t_6^{6}$, & ${\rm d}_{198}=t_1^{7}t_2^{3}t_3^{5}t_4^{3}t_5^{6}t_6^{2}$, & \multicolumn{1}{l}{${\rm d}_{199}=t_1^{7}t_2^{3}t_3^{5}t_4^{6}t_5^{2}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{200}=t_1^{7}t_2^{3}t_3^{5}t_4^{6}t_5^{3}t_6^{2}$,} \\ ${\rm d}_{201}=t_1^{3}t_2^{3}t_3^{3}t_4^{5}t_5^{6}t_6^{6}$, & ${\rm d}_{202}=t_1^{3}t_2^{3}t_3^{5}t_4^{3}t_5^{6}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{203}=t_1^{3}t_2^{3}t_3^{5}t_4^{6}t_5^{3}t_6^{6}$,} & \multicolumn{1}{l}{${\rm d}_{204}=t_1^{3}t_2^{3}t_3^{5}t_4^{6}t_5^{6}t_6^{3}$,} \\ ${\rm d}_{205}=t_1^{3}t_2^{5}t_3^{3}t_4^{3}t_5^{6}t_6^{6}$, & ${\rm d}_{206}=t_1^{3}t_2^{5}t_3^{3}t_4^{6}t_5^{3}t_6^{6}$, & \multicolumn{1}{l}{${\rm d}_{207}=t_1^{3}t_2^{5}t_3^{3}t_4^{6}t_5^{6}t_6^{3}$,} & \multicolumn{1}{l}{${\rm d}_{208}=t_1^{3}t_2^{5}t_3^{6}t_4^{3}t_5^{3}t_6^{6}$,} \\ ${\rm d}_{209}=t_1^{3}t_2^{5}t_3^{6}t_4^{3}t_5^{6}t_6^{3}$, & ${\rm d}_{210}=t_1^{3}t_2^{5}t_3^{6}t_4^{6}t_5^{3}t_6^{3}.$ & & \end{tabular}\end{center} \begin{thebibliography}{99} \bibitem[Ada60]{J.A} J.F. 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Here $a_i = a_i^{(1)}$ denotes the dual of $t_i\in H^{1}(BV_4) = P^{\otimes 4}_1,$ where the duality is taken with respect to the basis of $H^{}(BV_4) = P^{\otimes 4}$ consisting of all monomials in $t_1, \ldots, t_4.$ The (right) action of the algebra $\mathcal A$ on $H_(BV_4)$ is determined by the rule $(a_i^{(\ell)})Sq^k = \binom{\ell-k}{k}a_i^{(\ell-k)}$ and the Cartan formula. In accordance with the paper \cite{Sum2-1}, it is adequate to investigate the behavior of the fourth Singer algebraic transfer in the internal degrees of the following forms: $$\begin{array}{lll} \medskip {\rm (i)} & n &= 2^{k+1} - m, \ \mbox{for $1\leq m\leq 3,$} \\ \medskip {\rm (ii)} & n &= 2^{k+s+1} +2^{k + 1}-3,\\ \medskip {\rm (iii)} & n &= 2^{k+s} +2^{k}-2,\\ \medskip {\rm (iv)} & n &= 2^{k+s+u} + 2^{k+s} + 2^{k} - 3, \end{array}$$ whenever $k,\, s,$ and $u$ are positive integers. Verifying that the aforementioned degrees can be expressed in the form of \eqref{pt} is a straightforward task. The explicit description of the rank four Singer transfer has been provided by Sum \cite{Sum2-1} for item (i), and by the author \cite{Phuc10-3, Phuc12} for item (ii), as well as for item (iii) with $s\geq 1,\, s\neq 2,\, 4$. We will now conduct an investigation into $Tr_4^{\mathcal A}$ for item (iii) when $s$ is equal to 2 and 4, and also for item (iv). \begin{thm} The following assertions are true: \begin{itemize} \item[(I)] Singer's transfer is an isomorphism in bidegrees $(4, 2+5\cdot 2^k)$ and $(4, 2+17\cdot 2^k)$ for any $k > 0.$ \item[(II)] Given the generic degree $n:= n_{k,\,s,\, u} = 2^{k+s+u} + 2^{k+s} + 2^{k} - 3,$ whenever $k,\, s,$ and $u$ are positive integers. Then, the transfer homomorphism of rank four $$Tr_4^{\mathcal A}: (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,s,\, u}}\longrightarrow {\rm Ext}_{\mathcal A}^{4, 4+n_{k,\,s,\, u}}(\mathbb F_2, \mathbb F_2)$$ is also an isomorphism for all $k,\,s,\, u.$ \end{itemize} \end{thm} \begin{proof} The proof of part $(I)$ can be found in the preprint \cite{Phuc10-1}. We will now proceed to prove part $(II)$. The dimensions of the indecomposables $QP^{\otimes 4}_{n_{k,\,s,\, u}}$ are determined as follows, as a result of the work of Sum \cite{Sum2-1}: $$ \dim QP^{\otimes 4}_{n_{k,\,s,\, u}} =\left\{\begin{array}{ll} 64,&\mbox{if $s = 1$, $u = 1$ and $k=1$},\\[1mm] 120,&\mbox{if $s = 1$, $u = 1$ and $k\geq 2$},\\[1mm] 155, &\mbox{if $s = 2$, $u = 1$ and $k = 1$},\\[1mm] 210, &\mbox{if $s = 2$, $u = 1$ and $k\geq 2$},\\[1mm] 140,&\mbox{if $s\geq 3$, $u=1$ and $k= 1$},\\[1mm] 210,&\mbox{if $s\geq 3$, $u=1$ and $k\geq 2$},\\[1mm] 140,&\mbox{if $s = 1$, $u = 2$ and $k= 1$},\\[1mm] 225,&\mbox{if $s = 1$, $u = 2$ and $k\geq 2$},\\[1mm] 120,&\mbox{if $s = 1$, $u\geq 3$ and $k= 1$},\\[1mm] 210,&\mbox{if $s = 1$, $u\geq 3$ and $k\geq 2$},\\[1mm] 225,&\mbox{if $s = 2$, $u\geq 2$ and $k = 1$},\\[1mm] 210,&\mbox{if $s \geq 3$, $u\geq 2$ and $k = 1$},\\[1mm] 315,&\mbox{if $s \geq 2$, $u\geq 2$ and $k \geq 2$}. \end{array}\right.$$ Thanks to the above results, we are able to compute explicitly the action of the group $GL_4$ on the spaces $[QP^{\otimes 4}_{n_{k,\,s,\, u}}]^{GL_4}.$ By utilizing the homomorphisms $\sigma_d : P^{\otimes 4}\longrightarrow P^{\otimes 4} ,\, 1\leq d\leq 4,$ and performing similar calculations to those in the preprints \cite{Phuc10-1, Phuc10-3}, we obtain that $$ \dim (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,s,\, u}} = \dim [QP^{\otimes 4}_{n_{k,\,s,\, u}}]^{GL_4} =\left\{\begin{array}{ll} 1&\mbox{if $s = 1$, $u = 2$ and $k\geq 2$},\\[1mm] 1&\mbox{if $s = 2$, $u\geq 1$ and $k = 1$},\\[1mm] 1&\mbox{if $s \geq 2$, $u\geq 2$ and $k\geq 2$},\\[1mm] 0, &\mbox{otherwise}. \end{array}\right. $$ Moreover, the generators of $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,s,\, u}}$ can be determined as follows: $$ (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,s,\, u}} =\left\{\begin{array}{ll} \langle [\zeta_{k,\, 1,\, 2}] \rangle,&\mbox{if $s = 1$, $u = 2$ and $k\geq 2$},\\[1mm] \langle [\zeta_{1,\, 2,\, u}] \rangle,&\mbox{if $s = 2$, $u\geq 1$ and $k = 1$},\\[1mm] \langle [\zeta_{k,\, s,\, u}] \rangle,&\mbox{if $s \geq 2$, $u\geq 2$ and $k\geq 2$},\\[1mm] 0, &\mbox{otherwise}. \end{array}\right.$$ wherein the elements $$\begin{array}{ll} \medskip \zeta_{k,\, 1,\, 2}&= a_2^{(2^{k+2}-1)}a_3^{(2^{k+2}-1)}a_4^{(3.2^{k}-1)} + a_2^{(2^{k+2}-1)}a_3^{(5.2^{k}-1)}a_4^{(2^{k+1}-1)} \\ \medskip &\quad + a_2^{(6.2^{k}-1)}a_3^{(3.2^{k}-1)}a_4^{(2^{k+1}-1)} + a_2^{(7.2^{k}-1)}a_3^{(2^{k+1}-1)}a_4^{(2^{k+1}-1)},\\ \medskip \zeta_{1,\, 2,\, u} &= a_1^{(2^{u+3}-1)}a_2^{(3)}a_3^{(3)}a_4^{(2)} + a_1^{(2^{u+3}-1)}a_2^{(3)}a_3^{(4)}a_4^{(1)}+ a_1^{(2^{u+3}-1)}a_2^{(5)}a_3^{(2)}a_4^{(1)} + a_1^{(2^{u+3}-1)}a_2^{(6)}a_3^{(1)}a_4^{(1)},\\ \medskip \zeta_{k,\, s,\, u} &= a_2^{(2^{k}-1)}a_3^{(2^{k+s}-1)}a_4^{(2^{k+s+u}-1)} \end{array}$$ are $\overline{\mathcal A}$-annihilated. Thus, one has isomorphisms: $$\begin{array}{ll} \medskip (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,1,\, 2}} \cong \mathbb F_2,\ (k\geq 2),\\ \medskip (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,2,\, u}} \cong \mathbb F_2,\ (u\geq 1),\\ \medskip (\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{k,\,s,\, u}} \cong \mathbb F_2,\, (k\geq 2, s\geq 2, u\geq 2). \end{array}$$ To simplify matters, we shall describe explicitly $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,1,\, 1}}$ and $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,2,\, 1}}.$ The remaining coinvariants can be determined through analogous techniques. \medskip $\bullet$ We note that the indecomposables $QP^{\otimes 4}_{n_{1,\,1,\, 1}}$ has dimension $64$ and that $$QP^{\otimes 4}_{n_{1,\,1,\, 1}}\cong (QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{0}\bigoplus (QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{>0}.$$ Following \cite{Sum2-1}, $(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{0}$ has a monomial basis consisting of all the equivalence classes represented by the following admissible monomial: \begin{center} \begin{tabular}{llll} ${\rm adm}_{1}=t_2t_3^{3}t_4^{7}$, & ${\rm adm}_{2}=t_2t_3^{7}t_4^{3}$, & ${\rm adm}_{3}=t_2^{3}t_3t_4^{7}$, & ${\rm adm}_{4}=t_2^{3}t_3^{7}t_4$, \\ ${\rm adm}_{5}=t_2^{7}t_3t_4^{3}$, & ${\rm adm}_{6}=t_2^{7}t_3^{3}t_4$, & ${\rm adm}_{7}=t_1t_3^{3}t_4^{7}$, & ${\rm adm}_{8}=t_1t_3^{7}t_4^{3}$, \\ ${\rm adm}_{9}=t_1t_2^{3}t_4^{7}$, & ${\rm adm}_{10}=t_1t_2^{3}t_3^{7}$, & ${\rm adm}_{11}=t_1t_2^{7}t_4^{3}$, & ${\rm adm}_{12}=t_1t_2^{7}t_3^{3}$, \\ ${\rm adm}_{13}=t_1^{3}t_3t_4^{7}$, & ${\rm adm}_{14}=t_1^{3}t_3^{7}t_4$, & ${\rm adm}_{15}=t_1^{3}t_2t_4^{7}$, & ${\rm adm}_{16}=t_1^{3}t_2t_3^{7}$, \\ ${\rm adm}_{17}=t_1^{3}t_2^{7}t_4$, & ${\rm adm}_{18}=t_1^{3}t_2^{7}t_3$, & ${\rm adm}_{19}=t_1^{7}t_3t_4^{3}$, & ${\rm adm}_{20}=t_1^{7}t_3^{3}t_4$, \\ ${\rm adm}_{21}=t_1^{7}t_2t_4^{3}$, & ${\rm adm}_{22}=t_1^{7}t_2t_3^{3}$, & ${\rm adm}_{23}=t_1^{7}t_2^{3}t_4$, & ${\rm adm}_{24}=t_1^{7}t_2^{3}t_3$, \\ ${\rm adm}_{25}=t_2^{3}t_3^{3}t_4^{5}$, & ${\rm adm}_{26}=t_1^{3}t_3^{3}t_4^{5}$, & ${\rm adm}_{27}=t_1^{3}t_2^{3}t_4^{5}$, & ${\rm adm}_{28}=t_1^{3}t_2^{3}t_3^{5}$, \\ ${\rm adm}_{29}=t_2^{3}t_3^{5}t_4^{3}$, & ${\rm adm}_{30}=t_1^{3}t_3^{5}t_4^{3}$, & ${\rm adm}_{31}=t_1^{3}t_2^{5}t_4^{3}$, & ${\rm adm}_{32}=t_1^{3}t_2^{5}t_3^{3}$. \end{tabular}\end{center} It is easy to see that $[{\rm adm}_j]_{\omega({\rm adm}_j)} = [{\rm adm}_j]$ for every $j.$ By a simple computation, we have a direct summand decomposition of the $\Sigma_4$-submodules: $ (QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{0} = \Sigma_4({\rm adm}_1)\bigoplus \Sigma_4({\rm adm}_{25}),$ where $\Sigma_4({\rm adm}_1) = \langle \{[{\rm adm}_j]:\ 1\leq j\leq 24\}\rangle$ and $\Sigma_4({\rm adm}_{25}) = \langle \{[{\rm adm}_j]:\ 25\leq j\leq 32\}\rangle.$ We compute the action of $\Sigma_4$ on the $(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{0}$ and obtain $ [(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{0}]^{\Sigma_4} = \langle [\sum_{1\leq j\leq 24}{\rm adm}_j] \rangle.$ To do this, we shall show that $[\Sigma_4({\rm adm}_1)]^{\Sigma_4} = \langle [\sum_{1\leq j\leq 24}{\rm adm}_j] \rangle$ and $[\Sigma_4({\rm adm}_{25})]^{\Sigma_4} = 0.$ For simplicity, we compute the $\Sigma_4$-invariant subspace $[\Sigma_4({\rm adm}_{25})]^{\Sigma_4}$ in detail. The remaining space can be obtained in the same way. We can see that the set $\{[{\rm adm}_j]:\ 25\leq j\leq 32\}$ is a monomial basis of $\Sigma_4({\rm adm}_{25}).$ Suppose that $[t]\in [\Sigma_4({\rm adm}_{25})]^{\Sigma_4},$ then one has $t\equiv \sum_{25\leq j\leq 32}\gamma_j{\rm adm}_j,$ where $\gamma_j\in \mathbb F_2$ for all $j.$ For each $i,\, 1\leq i\leq 3,$ applying the homomorphism $\sigma_d: P^{\otimes 4}\longrightarrow P^{\otimes 4},\, 1\leq d\leq 3$ to the sum $\sum_{25\leq j\leq 32}\gamma_j{\rm adm}_j,$ one gets the following equalities: $$ \begin{array}{ll} \sigma_1\big(\sum_{25\leq j\leq 32}\gamma_j{\rm adm}_j\big) &\equiv \gamma_{25}{\rm adm}_{26} + \gamma_{26}{\rm adm}_{25} + \gamma_{27}{\rm adm}_{27} + \gamma_{28}{\rm adm}_{28} \\ \medskip &\quad + \gamma_{29}{\rm adm}_{30} + \gamma_{30}{\rm adm}_{29} +\gamma_{31}t_1^{5}t_2^{3}t_4^{3} + \gamma_{32}t_1^{5}t_2^{3}t_3^{3},\\ \sigma_2\big(\sum_{25\leq j\leq 32}\gamma_j{\rm adm}_j\big) &\equiv \gamma_{25}{\rm adm}_{25} + \gamma_{26}{\rm adm}_{27} + \gamma_{27}{\rm adm}_{26} + \gamma_{28}{\rm adm}_{32} \\ \medskip &\quad + \gamma_{29}t_2^{5}t_3^{3}t_4^{3} + \gamma_{30}{\rm adm}_{31} +\gamma_{31}{\rm adm}_{30} + \gamma_{32}{\rm adm}_{28},\\ \sigma_3\big(\sum_{25\leq j\leq 32}\gamma_j{\rm adm}_j\big) &\equiv \gamma_{25}{\rm adm}_{29} + \gamma_{26}{\rm adm}_{30} + \gamma_{27}{\rm adm}_{28} + \gamma_{28}{\rm adm}_{27} \\ \medskip &\quad + \gamma_{29}{\rm adm}_{25}+ \gamma_{30}{\rm adm}_{26} +\gamma_{31}{\rm adm}_{32} + \gamma_{32}{\rm adm}_{31}. \end{array}$$ Applying the Cartan formula, the monomials $t_1^{5}t_2^{3}t_4^{3},$ $t_1^{5}t_2^{3}t_3^{3}$ and $t_2^{5}t_3^{3}t_4^{3} $ can be written as a linear combination of terms ${\rm adm}_j,\, 1\leq j\leq 32$ as follows: $t_1^{5}t_2^{3}t_4^{3} \equiv {\rm adm}_{27} + {\rm adm}_{31},\ t_1^{5}t_2^{3}t_3^{3} \equiv {\rm adm}_{28} + {\rm adm}_{32},\ t_2^{5}t_3^{3}t_4^{3} \equiv {\rm adm}_{25} + {\rm adm}_{29}.$ Combining these calculations and the relations $\sigma_d(t)\equiv t$ for all $d,\, 1\leq d\leq 3,$ we can easily deduce that $$ \begin{array}{ll} \sigma_1(t)+t&\equiv \bigg((\gamma_{25} + \gamma_{26})({\rm adm}_{25} + {\rm adm}_{26}) + \gamma_{31}{\rm adm}_{27} + \gamma_{32}{\rm adm}_{28}\\ \medskip &\quad + (\gamma_{29} + \gamma_{30})({\rm adm}_{29} + {\rm adm}_{30})\bigg)\equiv 0,\\ \sigma_2(t)+t&\equiv \bigg((\gamma_{26} + \gamma_{27})({\rm adm}_{26} + {\rm adm}_{27}) + \gamma_{29}{\rm adm}_{25} + (\gamma_{30} + \gamma_{31})({\rm adm}_{30} + {\rm adm}_{31}) \\ \medskip &\quad+ (\gamma_{28} + \gamma_{32})({\rm adm}_{28} + {\rm adm}_{32})\bigg)\equiv 0,\\ \sigma_3(t)+t&\equiv \bigg((\gamma_{25} + \gamma_{29})({\rm adm}_{25} + {\rm adm}_{29}) + (\gamma_{26} + \gamma_{30})({\rm adm}_{26} + {\rm adm}_{30})\\ &\quad + (\gamma_{31} + \gamma_{32})({\rm adm}_{31} + {\rm adm}_{32})\bigg)\equiv 0. \end{array}$$ These equalities imply that $\gamma_j = 0$ for all $j,\, 25\leq j\leq 32.$ \medskip We now compute the $\Sigma_4$-invariants space $[(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{>0}]^{\Sigma_4}.$ Under the action of the group $\Sigma_4,$ we have $[(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{>0}]^{\Sigma_4} = \langle [\sum_{53\leq j\leq 64}{\rm adm}_j] \rangle,$ where the admissible monomials ${\rm adm}_j,\, 53\leq j\leq 64$ are described as below. Indeed, let us recall that from a result in Sum \cite{Sum2-1}, $(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{>0}$ has a monomial basis consisting of all the equivalence classes represented by the following admissible monomial: \begin{center} \begin{tabular}{llll} ${\rm adm}_{33}=t_1t_2t_3^{2}t_4^{7}$, & ${\rm adm}_{34}=t_1t_2t_3^{7}t_4^{2}$, & ${\rm adm}_{35}=t_1t_2^{2}t_3t_4^{7}$, & ${\rm adm}_{36}=t_1t_2^{2}t_3^{7}t_4$, \\ ${\rm adm}_{37}=t_1t_2^{7}t_3t_4^{2}$, & ${\rm adm}_{38}=t_1t_2^{7}t_3^{2}t_4$, & ${\rm adm}_{39}=t_1^{7}t_2t_3t_4^{2}$, & ${\rm adm}_{40}=t_1^{7}t_2t_3^{2}t_4$, \\ ${\rm adm}_{41}=t_1^{3}t_2t_3t_4^{6}$, & ${\rm adm}_{42}=t_1^{3}t_2t_3^{6}t_4$, & ${\rm adm}_{43}=t_1t_2^{2}t_3^{3}t_4^{5}$, & ${\rm adm}_{44}=t_1t_2^{2}t_3^{5}t_4^{3}$, \\ ${\rm adm}_{45}=t_1t_2^{3}t_3^{2}t_4^{5}$, & ${\rm adm}_{46}=t_1t_2^{3}t_3^{5}t_4^{2}$, & ${\rm adm}_{47}=t_1^{3}t_2t_3^{2}t_4^{5}$, & ${\rm adm}_{48}=t_1^{3}t_2t_3^{5}t_4^{2}$, \\ ${\rm adm}_{49}=t_1^{3}t_2^{5}t_3t_4^{2}$, & ${\rm adm}_{50}=t_1^{3}t_2^{5}t_3^{2}t_4$, & ${\rm adm}_{51}=t_1t_2^{3}t_3^{3}t_4^{4}$, & ${\rm adm}_{52}=t_1t_2^{3}t_3^{4}t_4^{3}$, \\ ${\rm adm}_{53}=t_1t_2t_3^{3}t_4^{6}$, & ${\rm adm}_{54}=t_1t_2t_3^{6}t_4^{3}$, & ${\rm adm}_{55}=t_1t_2^{3}t_3t_4^{6}$, & ${\rm adm}_{56}=t_1t_2^{3}t_3^{6}t_4$, \\ ${\rm adm}_{57}=t_1t_2^{6}t_3t_4^{3}$, & ${\rm adm}_{58}=t_1t_2^{6}t_3^{3}t_4$, & ${\rm adm}_{59}=t_1^{3}t_2t_3^{3}t_4^{4}$, & ${\rm adm}_{60}=t_1^{3}t_2t_3^{4}t_4^{3}$, \\ ${\rm adm}_{61}=t_1^{3}t_2^{3}t_3t_4^{4}$, & ${\rm adm}_{62}=t_1^{3}t_2^{3}t_3^{4}t_4$, & ${\rm adm}_{63}=t_1^{3}t_2^{4}t_3t_4^{3}$, & ${\rm adm}_{64}=t_1^{3}t_2^{4}t_3^{3}t_4$. \end{tabular}\end{center} Through a straightforward calculation, it becomes evident that $$ \begin{array}{ll} \medskip &\Sigma_4({\rm adm}_{33}) = \langle \{[{\rm adm}_j]:\ 33\leq j\leq 40\}\rangle, \\ &\Sigma_4({\rm adm}_{41}, {\rm adm}_{43}, {\rm adm}_{51}) = \langle \{[{\rm adm}_j]:\ 41\leq j\leq 64\}\rangle, \end{array}$$ and so, one has an isomorphism $$(QP^{\otimes 4}_{n_{1,\,1,\, 1}})^{>0} \cong \Sigma_4({\rm adm}_{33})\bigoplus \Sigma_4({\rm adm}_{41}, {\rm adm}_{43}, {\rm adm}_{51}).$$ By similar calculations as above, we obtain $$ [\Sigma_4({\rm adm}_{33})]^{\Sigma_4} = 0, \ \ [\Sigma_4({\rm adm}_{41}, {\rm adm}_{43}, {\rm adm}_{51})]^{\Sigma_4} = \langle [\sum_{53\leq j\leq 64}{\rm adm}_j] \rangle.$$ With the data in hand and under the action of the group $\Sigma_4,$ we get $$ [QP^{\otimes 4}_{n_{1,\,1,\, 1}}]^{\Sigma_4} = \big\langle\big \{[\sum_{1\leq j\leq 24}{\rm adm}_j],\ [\sum_{53\leq j\leq 64}{\rm adm}_j]\big\} \big\rangle.$$ Now, for any $[h]\in [QP^{\otimes 4}_{n_{1,\,1,\, 1}}]^{GL_4},$ since $\Sigma_4\subset GL_4,$ we have $$h\equiv \bigg(\beta_1\sum_{1\leq j\leq 24}{\rm adm}_j + \beta_2\sum_{53\leq j\leq 64}{\rm adm}_j\bigg),$$ wherein $\beta_1$ and $\beta_2$ belong to $\mathbb F_2.$ Applying the homomorphism $\sigma_4: P^{\otimes 4}\longrightarrow P^{\otimes 4}$ to the sum $S:=\beta_1\sum_{1\leq j\leq 24}{\rm adm}_j + \beta_2\sum_{53\leq j\leq 64}{\rm adm}_j$ and we compute explicitly $\sigma_4(S)$ in admissible terms ${\rm adm}_{j}$ modulo ($\overline{\mathcal A}P^{\otimes 4}_{n_{1,\,1,\, 1}}$). Then, by the relation $\sigma_4(h) + h\equiv 0,$ we can easily obtain $$ \begin{array}{ll} \sigma_4(h) + h&\equiv \big(\beta_1(\sum_{1\leq j\leq 6}{\rm adm}_j + \sum_{9\leq j\leq 12}{\rm adm}_j + {\rm adm}_{17} + {\rm adm}_{18}+{\rm adm}_{27} + {\rm adm}_{28}\\ &\quad + {\rm adm}_{33}+{\rm adm}_{34} + {\rm adm}_{47} + {\rm adm}_{48}+ \sum_{55\leq j\leq 60}{\rm adm}_j+ {\rm adm}_{63} + {\rm adm}_{64}\\ &\quad + (\beta_1 +\beta_2)({\rm adm}_{45}+{\rm adm}_{46})\big)\equiv 0. \end{array}$$ The above equality implies that $\beta_1 = \beta_2 =0.$ Thus, the coinvariant $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,1,\, 1}}$ vanishes. $\bullet$ In the case of the degree $n_{1,\,2,\,1}$, which is widely acknowledged (as stated in Sum \cite{Sum2-1}), $\dim (QP^{\otimes 4}_{n_{1,\,2,\, 1}})^0 = \dim (QP^{\otimes 4}_{n_{1,\,2,\, 1}})^0(3,2,2,1) = 56$ and $\dim(QP^{\otimes 4}_{n_{1,\,2,\, 1}})^{>0} = \dim (QP^{\otimes 4}_{n_{1,\,2,\, 1}})^{>0}(3,2,2,1) = 99.$ A direct computation then yields: $$ \begin{array}{ll} {\rm [}(QP^{\otimes 4}_{n_{1,\,2,\, 1}})^0{\rm ]}^{\Sigma_4}&= \langle [\widehat{p_1}],\, [\widehat{p_2}],\, [\widehat{p_3}] \rangle,\\ {\rm [}(QP^{\otimes 4}_{n_{1,\,2,\, 1}})^{>0}{\rm ]}^{\Sigma_4}&= \big\langle [\widehat{p_4}],\, [\widehat{p_5}:= \sum_{1\leq j\leq 3}q_j] ,\, [\widehat{p_6}:= \sum_{2\leq j\leq 6}q_j],\, [\widehat{p_7}:= \sum_{6\leq j\leq 9}q_j],\, [\widehat{p_8}:=q_5 + \sum_{9\leq j\leq 11}q_j] \big\rangle,\\ \end{array}$$ where $$ \begin{array}{ll} \widehat{p_1}&= t_2t_3^{7}t_4^{15}+ t_1t_3^{7}t_4^{15}+ t_1t_2^{7}t_4^{15}+ t_1t_2^{7}t_3^{15}+ t_2t_3^{15}t_4^{7}+ t_1t_3^{15}t_4^{7}+ t_1t_2^{15}t_4^{7}+ t_1t_2^{15}t_3^{7}+ \medskip t_2^{7}t_3t_4^{15}\\ &\quad+ t_1^{7}t_3t_4^{15}+ t_1^{7}t_2t_4^{15}+ t_1^{7}t_2t_3^{15}+ t_2^{7}t_3^{15}t_4+ t_1^{7}t_3^{15}t_4+ t_1^{7}t_2^{15}t_4+ t_1^{7}t_2^{15}t_3+ t_2^{15}t_3t_4^{7}+ \medskip t_1^{15}t_3t_4^{7}\\ &\quad+ t_1^{15}t_2t_4^{7}+ t_1^{15}t_2t_3^{7}+ t_2^{15}t_3^{7}t_4+ t_1^{15}t_3^{7}t_4+ t_1^{15}t_2^{7}t_4+ \medskip t_1^{15}t_2^{7}t_3,\\ \widehat{p_2}&= t_2^{3}t_3^{5}t_4^{15}+ t_1^{3}t_3^{5}t_4^{15}+ t_1^{3}t_2^{5}t_4^{15}+ t_1^{3}t_2^{5}t_3^{15}+ t_2^{3}t_3^{15}t_4^{5}+ t_1^{3}t_3^{15}t_4^{5}+ \medskip t_1^{3}t_2^{15}t_4^{5}\\ &\quad+ t_1^{3}t_2^{15}t_3^{5} t_2^{15}t_3^{3}t_4^{5}+ t_1^{15}t_3^{3}t_4^{5}+ t_1^{15}t_2^{3}t_4^{5}+ \medskip t_1^{15}t_2^{3}t_3^{5},\\ \widehat{p_3}&=t_2^{3}t_3^{13}t_4^{7}+ t_1^{3}t_3^{13}t_4^{7}+ t_1^{3}t_2^{13}t_4^{7}+ t_1^{3}t_2^{13}t_3^{7}+ t_2^{7}t_3^{3}t_4^{13}+ t_1^{7}t_3^{3}t_4^{13}+ \medskip t_1^{7}t_2^{3}t_4^{13}\\ &\quad+ t_1^{7}t_2^{3}t_3^{13}+ t_2^{7}t_3^{11}t_4^{5}+ t_1^{7}t_3^{11}t_4^{5}+ t_1^{7}t_2^{11}t_4^{5}+ t_1^{7}t_2^{11}t_3^{5}+ t_2^{7}t_3^{7}t_4^{9}+ \medskip t_1^{7}t_3^{7}t_4^{9}\\ &\quad+ \medskip t_1^{7}t_2^{7}t_4^{9}+ t_1^{7}t_2^{7}t_3^{9},\\ \widehat{p_4}&= t_1t_2t_3^{6}t_4^{15}+ t_1t_2t_3^{15}t_4^{6}+ t_1t_2^{6}t_3t_4^{15}+ t_1t_2^{6}t_3^{15}t_4+ t_1t_2^{15}t_3t_4^{6}+ \medskip t_1t_2^{15}t_3^{6}t_4\\ &\quad+ t_1^{15}t_2t_3t_4^{6}+ t_1^{15}t_2t_3^{6}t_4+ t_1^{3}t_2t_3^{4}t_4^{15}+ t_1^{3}t_2t_3^{15}t_4^{4}+ t_1^{3}t_2^{15}t_3t_4^{4}+ \medskip t_1^{15}t_2^{3}t_3t_4^{4}\\ &\quad+ t_1^{3}t_2^{4}t_3t_4^{15}+ t_1^{3}t_2^{4}t_3^{15}t_4+ t_1^{3}t_2^{15}t_3^{4}t_4+ \medskip t_1^{15}t_2^{3}t_3^{4}t_4,\\ \end{array}$$ \newpage $$ \begin{array}{ll} \medskip q_1&= t_1t_2^{3}t_3^{14}t_4^{5} + t_1^{3}t_2^{3}t_3^{12}t_4^{5},\\ \medskip q_2&= t_1t_2^{3}t_3^{7}t_4^{12} + t_1^{3}t_2^{4}t_3^{11}t_4^{5} + t_1t_2^{6}t_3^{11}t_4^{5} + t_1t_2^{6}t_3^{7}t_4^{9},\\ \medskip q_3&= t_1^3t_2^{5}t_3^{6}t_4^{9} + t_1^{3}t_2^{5}t_3^{7}t_4^{8},\\ q_4&= t_1t_2^{3}t_3^{5}t_4^{14}+ t_1^{3}t_2t_3^{5}t_4^{14}+ t_1^{3}t_2^{5}t_3t_4^{14}+ t_1^{3}t_2^{5}t_3^{14}t_4+ t_1^{3}t_2^{7}t_3t_4^{12}+ t_1^{3}t_2^{3}t_3^{5}t_4^{12}+ \medskip t_1^{3}t_2^{7}t_3^{12}t_4,\\ q_5&=t_1^{3}t_2t_3^{14}t_4^{5}+ t_1t_2^{7}t_3^{3}t_4^{12}+ t_1^{7}t_2t_3^{3}t_4^{12}+ t_1^{3}t_2^{13}t_3^{2}t_4^{5}+ t_1^{3}t_2^{13}t_3^{3}t_4^{4}+ t_1t_2^{7}t_3^{10}t_4^{5}+ \medskip t_1^{7}t_2t_3^{10}t_4^{5}\\ &\quad+ t_1^{3}t_2^{7}t_3^{8}t_4^{5}+ t_1^{7}t_2^{9}t_3^{2}t_4^{5}+ \medskip t_1^{7}t_2^{9}t_3^{3}t_4^{4},\\ q_6&= t_1^{7}t_2^{3}t_3t_4^{12}+ t_1^{3}t_2^{7}t_3^{9}t_4^{4}+ t_1^{7}t_2^{7}t_3t_4^{8}+ t_1^{7}t_2^{3}t_3^{12}t_4+ \medskip t_1^{7}t_2^{7}t_3^{8}t_4,\\ q_7&= t_1t_2t_3^{7}t_4^{14}+ t_1t_2^{7}t_3t_4^{14}+ t_1t_2^{7}t_3^{14}t_4+ t_1^{7}t_2t_3t_4^{14}+ t_1^{7}t_2t_3^{14}t_4+ t_1t_2^{3}t_3^{13}t_4^{6}+ \medskip t_1^{3}t_2t_3^{13}t_4^{6}\\ &\quad+ t_1^{3}t_2^{13}t_3t_4^{6}+ t_1^{3}t_2^{13}t_3^{6}t_4+ t_1^{7}t_2^{11}t_3t_4^{4}+ \medskip t_1^{7}t_2^{11}t_3^{4}t_4,\\ q_8&= t_1t_2^{3}t_3^7t_4^{12}+ t_1t_2^{6}t_3^{7}t_4^9+ \medskip t_1^{3}t_2^{4}t_3^{7}t_4^{9},\\ q_9&= t_1^{7}t_2^{3}t_3^{9}t_4^{4}+ t_1t_2^{7}t_3^{7}t_4^{8}+ t_1^{7}t_2t_3^{7}t_4^{8}+ t_1t_2^{7}t_3^{6}t_4^{9}+ t_1^{7}t_2t_3^{6}t_4^{9}+ t_1^{3}t_2^{7}t_3^{4}t_4^{9}+ \medskip t_1^{7}t_2^{3}t_3^{4}t_4^{9},\\ q_{10}&= t_1t_2^{3}t_3^{7}t_4^{12}+ t_1^{3}t_2^{4}t_3^{11}t_4^{5}+ \medskip t_1t_2^{6}t_3^{11}t_4^{5},\\ q_{11}&= t_1t_2^{3}t_3^{6}t_4^{13}+ t_1t_2^{6}t_3^{3}t_4^{13}+ t_1^{3}t_2t_3^{7}t_4^{12}+ t_1^{3}t_2^{3}t_3^{4}t_4^{13}+ t_1^{3}t_2^{3}t_3^{13}t_4^{4}+ t_1^{3}t_2^{4}t_3^{3}t_4^{13}. \end{array}$$ Let $[w]\in [QP^{\otimes 4}_{n_{1,\,2,\, 1}}]^{GL_4}$ be arbitrary. Then, we can express $w$ as $w\equiv \sum_{1\leq i\leq 8}\gamma_i\widehat{p_i},$ where $\gamma_i\in\mathbb F_2$ for every $i.$ Utilizing the homomorphism $\sigma_4: P^{\otimes 4}\longrightarrow P^{\otimes 4}$ and the relation $\sigma_4(w) +w \equiv 0,$ we obtain $$ \begin{array}{ll} \medskip \sigma_4(w) +w &\equiv \gamma_1t_2t_3^{7}t_4^{15} + (\gamma_1+\gamma_2)t_1t_2^{7}t_4^{15} + (\gamma_1+\gamma_3)t_1t_2^{15}t_4^{7} + (\gamma_3+\gamma_5+\gamma_6)t_2^7t_3^{7}t_4^{9}\\ \medskip &\quad + (\gamma_4+\gamma_7)t_1t_2^{15}t_3t_4^{6} + (\gamma_1+\gamma_6+\gamma_7)t_1t_2^{7}t_3t_4^{14}\\ &\quad + (\gamma_1+\gamma_3+\gamma_6 + \gamma_8)t_1t_2^{3}t_3^5t_4^{14} + \ \mbox{other terms} \equiv 0, \end{array}$$ which leads to $\gamma_1 = \gamma_2 = \gamma_3 = 0$ and $\gamma_4 = \gamma_5 = \gamma_6 = \gamma_7 = \gamma_8.$ Let us consider the following element in $H_(BV_4)$: $$\zeta_{1,\,2,\, 1} = a_1^{(15)}a_2^{(3)}a_3^{(3)}a_4^{(2)} + a_1^{(15)}a_2^{(3)}a_3^{(4)}a_4^{(1)}+ a_1^{(15)}a_2^{(5)}a_3^{(2)}a_4^{(1)} + a_1^{(15)}a_2^{(6)}a_3^{(1)}a_4^{(1)}.$$ Upon performing a straightforward computation, it can be shown that $ (\zeta_{1,\,2,\, 1})Sq^{2^k}= 0$ for all $k\geq 2$ and $$\begin{array}{ll} \medskip (a_1^{(15)}a_2^{(3)}a_3^{(3)}a_4^{(2)})Sq^{1}&=(a_1^{(15)}a_2^{(3)}a_3^{(4)}a_4^{(1)})Sq^{1}= a_1^{(15)}a_2^{(3)}a_3^{(3)}a_4^{(1)},\\ \medskip (a_1^{(15)}a_2^{(5)}a_3^{(2)}a_4^{(1)} )Sq^{1}&=(a_1^{(15)}a_2^{(6)}a_3^{(1)}a_4^{(1)})Sq^{1}=a_1^{(15)}a_2^{(5)}a_3^{(1)}a_4^{(1)},\\ \medskip (a_1^{(15)}a_2^{(3)}a_3^{(3)}a_4^{(2)})Sq^{2}&=(a_1^{(15)}a_2^{(6)}a_3^{(1)}a_4^{(1)})Sq^2 = 0,\\ (a_1^{(15)}a_2^{(3)}a_3^{(4)}a_4^{(1)})Sq^{2}&=(a_1^{(15)}a_2^{(5)}a_3^{(2)}a_4^{(1)})Sq^2 = a_1^{(15)}a_2^{(3)}a_3^{(2)}a_4^{(1)}. \end{array}$$ So, $\zeta_{1,\,2,\, 1}$ is an $\overline{\mathcal A}$-annihilated element. (In reality, due to the unstable condition, we only need to consider the effects of the Steenrod operations $Sq^{2^{k}}$ for $k = 0,\, 1.$) Furthermore, it is straightforward to check that $<\zeta_{1,\,2,\, 1},\, \sum_{4\leq i\leq 8}\widehat{p_i}> = 1,$ where $<- ,- >$ denotes the canonical non-singular pairing $H_(BV_4)\times H^(BV_4)\longrightarrow \mathbb F_2.$ Therefore, it may be concluded that $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,2,\, 1}}$ is 1-dimensional and $(\mathbb F_2\otimes_{GL_4}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes 4}]^{})_{n_{1,\,2,\, 1}} = \mathbb F_2[\zeta_{1,\,2,\, 1}].$ \medskip Now, according to Lin \cite{Lin}, the $\mathbb F_2$-cohomology ${\rm Ext}_{\mathcal A}^{4, }(\mathbb F_2, \mathbb F_2)$ is generated by $$h_ih_jh_{\ell}h_m,\, h_uc_v,\, d_t,\, e_t,\, f_t,\, g_{t+1},\, p_t,\, D_3(t),\, p'_t$$ for $m\geq \ell\geq j\geq i\geq 0$, $u,\ v,\ t\geq 0$, and subject to the relations $h_ih_{i+1} = 0,\ h_ih_{i+2}^{2} = 0,\, h_i^3 = h^2_{i-1}h_{i+1},\, h^2_ih^2_{i+3} = 0,\, h_{v-1}c_v = 0,\, h_{v}c_v = 0,\, h_{v+2}c_v = 0$ and $h_{v+3}c_v = 0.$ Then, one gets $${\rm Ext}_{\mathcal A}^{4, 4+n_{k,\, s,\, u}}(\mathbb F_2, \mathbb F_2) = \left\{\begin{array}{ll} \langle h_0c_k \rangle&\mbox{if $s = 1$, $u = 2$, and $k\geq 2$},\\ \langle h_{u+3}c_0 \rangle&\mbox{if $s = 2$, $u\geq 1$, and $k = 1$},\\ \langle h_0h_kh_{k+s}h_{k+s+u} \rangle&\mbox{if $s \geq 2$, $u\geq 2$, and $k\geq 2$},\\ 0&\mbox{otherwise}. \end{array}\right. $$ Therefrom Part $(II)$ can be obtained from the aforementioned calculations, in combination with the following known facts: the indecomposable elements $h_j$ and $c_j$ belong respectively to the image of $Tr_1^{\mathcal A}$ and $Tr_3^{\mathcal A},$ and the total transfer $\{Tr_h^{\mathcal A}\}_{h\geq 0}: \{\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{}\}_{h\geq 0}\longrightarrow \{{\rm Ext}_{\mathcal A}^{h, }(\mathbb F_2, \mathbb F_2)\}_{h\geq 0}$ of algebras detects the subalgebra generated by the $Sq^{0}$-family $\{h_j\in {\rm Ext}_{\mathcal A}^{1, 2^{j}}(\mathbb F_2, \mathbb F_2) \}_{j\geq 0}.$ The theorem is proved. \end{proof}
2412.02620v1	http://arxiv.org/abs/2412.02620v1	The Dimension of the Disguised Toric Locus of a Reaction Network	\documentclass[11pt]{article} \usepackage{amsmath,amsfonts,amssymb,amsthm} \usepackage{enumerate} \usepackage{xcolor} \usepackage{url} \usepackage{tcolorbox} \usepackage{hyperref} \usepackage{multicol, latexsym} \usepackage{latexsym} \usepackage{psfrag,import} \usepackage{verbatim} \usepackage{color} \usepackage{epsfig} \usepackage[outdir=./]{epstopdf} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan } \usepackage[title]{appendix} \usepackage{geometry} \usepackage{mathtools} \usepackage{enumerate} \usepackage{enumitem} \usepackage{multicol} \usepackage{booktabs} \usepackage{enumitem} \usepackage{parcolumns} \usepackage{thmtools} \usepackage{xr} \usepackage{epstopdf} \usepackage{mathrsfs} \usepackage{subcaption} \usepackage{soul} \usepackage{float} \parindent 1ex \parskip1ex \usepackage{comment} \usepackage{authblk} \usepackage{setspace} \usepackage{cleveref} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{notation}[theorem]{Notation} \newtheorem{remark}[theorem]{Remark} \theoremstyle{remark} \newtheorem{claim}{Claim} \numberwithin{equation}{section} \parskip=0pt plus 1pt \setlength{\parindent}{20pt} \newcommand\RR{\mathbb{R}} \newcommand\GG{\mathcal{G}} \newcommand\bla{\boldsymbol{\lambda}} \newcommand\by{\boldsymbol{y}} \newcommand\bypi{\boldsymbol{y'_i}} \newcommand\byi{\boldsymbol{y_i}} \newcommand\bypj{\boldsymbol{y'_j}} \newcommand\byj{\boldsymbol{y_j}} \newcommand\be{\boldsymbol{e}} \newcommand\bep{\boldsymbol{\varepsilon}} \newcommand\bc{\boldsymbol{c}} \renewcommand\bf{\boldsymbol{f}} \newcommand\bh{\boldsymbol{h}} \newcommand\bk{\boldsymbol{k}} \newcommand\bw{\boldsymbol{w}} \newcommand\bb{\boldsymbol{b}} \newcommand\bW{\boldsymbol{W}} \newcommand\bu{\boldsymbol{u}} \newcommand\bg{\boldsymbol{g}} \newcommand\bx{\boldsymbol{x}} \newcommand\bv{\boldsymbol{v}} \newcommand\bz{\boldsymbol{z}} \newcommand\bY{\boldsymbol{Y}} \newcommand\bA{\boldsymbol{A}} \newcommand\bB{\boldsymbol{B}} \newcommand\bC{\boldsymbol{C}} \newcommand\bF{\boldsymbol{F}} \newcommand\bG{\boldsymbol{G}} \newcommand\bH{\boldsymbol{H}} \newcommand\bI{\boldsymbol{I}} \newcommand\bq{\boldsymbol{q}} \newcommand\bp{\boldsymbol{p}} \newcommand\br{\boldsymbol{r}} \newcommand\bJ{\boldsymbol{J}} \newcommand\bj{\boldsymbol{j}} \newcommand\hbJ{\hat{\boldsymbol{J}}} \newcommand{\mK}{\mathcal{K}} \newcommand{\dK}{\mathcal{K}_{\RR\text{-disg}}} \newcommand{\pK}{\mathcal{K}_{\text{disg}}} \newcommand{\mJ}{\mathcal{J}_{\RR}} \newcommand{\eJ}{\mathcal{J}_{\textbf{0}}} \newcommand{\mD}{\mathcal{D}_{\textbf{0}}} \newcommand{\mS}{\mathcal{S}} \newcommand{\mSG}{\mathcal{S}_G} \newcommand{\hPsi}{\hat{\Psi}} \newcommand{\hbx}{\hat{\bx}} \newcommand{\hbk}{\hat{\bk}} \newcommand{\hbp}{\hat{\bp}} \newcommand{\hbq}{\hat{\bq}} \newcommand{\hmJ}{\hat{\mJ}} \newcommand\bd{\boldsymbol{d}} \newcommand{\defi}{\textbf} \DeclareMathOperator{\spn}{span} \begin{document} \title{ The Dimension of the Disguised Toric Locus of a Reaction Network } \author[1]{ Gheorghe Craciun } \author[2]{ Abhishek Deshpande } \author[3]{ Jiaxin Jin } \affil[1]{\small Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison} \affil[2]{Center for Computational Natural Sciences and Bioinformatics, \protect \\ International Institute of Information Technology Hyderabad} \affil[3]{\small Department of Mathematics, University of Louisiana at Lafayette} \date{} \maketitle \begin{abstract} Under mass-action kinetics, complex-balanced systems emerge from biochemical reaction networks and exhibit stable and predictable dynamics. For a reaction network $G$, the associated dynamical system is called \emph{disguised toric} if it can yield a complex-balanced realization on a possibly different network $G_1$. This concept extends the robust properties of toric systems to those that are not inherently toric. In this work, we study the \emph{disguised toric locus} of a reaction network — i.e., the set of positive rate constants that make the corresponding mass-action system disguised toric. Our primary focus is to compute the exact dimension of this locus. We subsequently apply our results to Thomas-type and circadian clock models. \end{abstract} \begin{NoHyper} \tableofcontents \end{NoHyper} \section{Introduction} Mathematical models of biochemical interaction networks can generally be described by {\em polynomial dynamical systems}. These dynamical systems are ubiquitous in models of biochemical reaction networks, infectious diseases, and population dynamics~\cite{craciun2022homeostasis,deshpande2014autocatalysis}. However, analyzing these systems is a challenging problem in general. Classical nonlinear dynamical properties like multistability, oscillations, or chaotic dynamics are difficult to examine~\cite{Ilyashenko2002, yu2018mathematical}. Studying the dynamical properties of reaction networks is crucial for understanding the behavior of chemical and biological systems. In this paper, we will focus on a class of dynamical systems generated by reaction networks called {\em complex-balanced systems} (also known as {\em toric dynamical systems}~\cite{CraciunDickensteinShiuSturmfels2009} owing to their connection with toric varieties~\cite{dickenstein2020algebraic}). Complex-balanced systems are known to exhibit remarkably robust dynamics, which {\em rules out} multistability, oscillations, and even chaotic dynamics~\cite{horn1972general}. More specifically, there exists a strictly convex Lyapunov function, which implies that all positive steady states are locally asymptotically stable~\cite{horn1972general, yu2018mathematical}. In addition, a unique positive steady state exists within each affine invariant polyhedron. They are also related to the \emph{Global Attractor Conjecture}~\cite{CraciunDickensteinShiuSturmfels2009} which states that complex-balanced dynamical systems have a globally attracting steady state within each stoichiometric compatibility class. Several special cases of this conjecture have been proved~\cite{anderson2011proof,gopalkrishnan2014geometric, pantea2012persistence, craciun2013persistence, boros2020permanence}, and a proof in full generality has been proposed in~\cite{craciun2015toric}. An important limitation of the classical theory of complex-balanced systems is that to be applicable for a large set of parameter values (i.e., choices of reaction rate constants) the reaction network under consideration must satisfy two special properties: {\em weak reversibility} and {\em low deficiency} (see \cite{yu2018mathematical} for definitions). Our focus here will be on understanding how one can take advantage of the notion of {\em dynamical equivalence} in order to greatly relax both of these restrictions. Dynamical equivalence relies on the fact that two different reaction networks can generate the same dynamics for well-chosen parameter values. This phenomenon has also been called \emph{macro-equivalence}~\cite{horn1972general} or {\em confoundability}~\cite{craciun2008identifiability}. Recently, this phenomenon has found applications in the design of efficient algorithms for finding weakly reversible single linkage class and weakly reversible deficiency one realizations~\cite{WR_df_1, WR_DEF_THM}. Moreover, it has also been used to show the existence of infinitely many positive states for weakly reversible and endotactic dynamical systems~\cite{boros2020weakly,kothari2024endotactic}. More recently, it has been used to generate the necessary and sufficient conditions for the existence of realizations using weakly reversible dynamical systems~\cite{kothari2024realizations}. In this paper, we consider the notion of a disguised toric locus for a given reaction network $G$. The \emph{disguised toric locus} is the set of positive reaction rate vectors in $G$ for which the corresponding dynamical system can be realized as a complex-balanced system by a network $G_1$. In other words, this locus consists of positive reaction rate vectors $\bk$ such that the mass-action system $(G, \bk)$ is dynamically equivalent to a complex-balanced system $(G_1, \bk_1)$. Additionally, if the rate constants are allowed to take any real values, we refer to the set of reaction rate vectors in $G$ that satisfy this property as the \emph{$\mathbb{R}$-disguised toric locus} of $G$. The concept of a disguised toric locus was first introduced in \cite{2022disguised}. Since then, several general properties of both the disguised toric locus and the $\mathbb{R}$-disguised toric locus have been established. For example, it was demonstrated in \cite{haque2022disguised} that the disguised toric locus is invariant under invertible affine transformations of the network. Furthermore, \cite{disg_1} showed that both loci are path-connected, and \cite{disg_2} provided a lower bound on the dimension of the $\mathbb{R}$-disguised toric locus. Consider for example the Thomas-type model (E-graph $G$) shown in Figure \ref{fig:thomas_model_intro}. \begin{figure}[!ht] \centering \includegraphics[scale=0.7]{thomas_model.eps} \caption{ (a) The E-graph $G$ represents a Thomas-type model, with all edges labeled by the reaction rate constants $\bk$. (b) The E-graph $G_1$ is weakly reversible, with all edges labeled by the reaction rate constants $\bk_1$. The mass-action system $(G_1, \bk_1)$ is complex-balanced. } \label{fig:thomas_model_intro} \end{figure} Since $G$ is not weakly reversible, the system $(G, \bk)$ is not complex-balanced, so classical complex-balanced theory offers limited insight into the dynamics of $(G, \bk)$. However, by direct computation, $(G, \bk)$ is dynamically equivalent to the complex-balanced system $(G_1, \bk_1)$, which enables us to deduce its dynamical properties. Thus, $\bk$ can be viewed as a “good” reaction rate vector for $G$. The disguised toric locus of $G$ consists of such reaction rate vectors $\bk$. In this paper, we develop a general framework to compute the exact dimensions of both the disguised toric locus and the $\mathbb{R}$-disguised toric locus of a reaction network. Building on \cite{disg_2}, we construct a mapping on the $\mathbb{R}$-disguised toric locus of $G$ and show that this mapping is a homeomorphism, allowing us to determine the dimensions of both the disguised toric locus and the $\mathbb{R}$-disguised toric locus. When applied to Figure \ref{fig:thomas_model_intro}, the disguised toric locus of $G$ is shown to be full-dimensional, significantly larger than its toric locus, which is empty (see details in Example \ref{ex:thomas}). \bigskip \textbf{Structure of the paper.} In Section~\ref{sec:reaction_networks}, we introduce the basic terminology of reaction networks. Section~\ref{sec:flux_systems} presents flux systems and analyzes their properties. In Section~\ref{sec:disguised_locus}, we recall the key concepts of the toric locus, the $\RR$-disguised toric locus, and the disguised toric locus. Section~\ref{sec:map} constructs a continuous bijective map $\hPsi$ connecting the $\RR$-disguised toric locus to a specific flux system. In Section~\ref{sec:continuity}, we first establish key lemmas \ref{lem:key_1} - \ref{lem:key_4} and then use them to prove that $\hPsi$ is a homeomorphism in Theorem \ref{thm:hpsi_homeo}. Section~\ref{sec:dimension} leverages this homeomorphism to establish precise bounds on the dimension of the disguised toric locus and the $\RR$-disguised toric locus, as shown in Theorem~\ref{thm:dim_kisg_main}. In Section~\ref{sec:applications}, we apply our results to Thomas-type models and circadian clock models, showing both disguised toric loci are full-dimensional. Finally, Section~\ref{sec:discussion} summarizes our findings and outlines potential directions for future research. \bigskip \textbf{Notation.} We let $\mathbb{R}_{\geq 0}^n$ and $\mathbb{R}_{>0}^n$ denote the set of vectors in $\mathbb{R}^n$ with non-negative entries and positive entries respectively. For vectors $\bx = (\bx_1, \ldots, \bx_n)^{\intercal}\in \RR^n_{>0}$ and $\by = (\by_1, \ldots, \by_n)^{\intercal} \in \RR^n$, we define: \begin{equation} \notag \bx^{\by} = \bx_1^{y_{1}} \ldots \bx_n^{y_{n}}. \end{equation} For any two vectors $\bx, \by \in \RR^n$, we define $\langle \bx, \by \rangle = \sum\limits^{n}_{i=1} x_i y_i$. For E-graphs (see Definition \ref{def:e-graph}), we always let $G, G'$ denote arbitrary E-graphs, and let $G_1$ denote a weakly reversible E-graph. \section{Reaction networks} \label{sec:reaction_networks} We start with the introduction of the concept of a {\em reaction network} as a directed graph in Euclidean space called {\em E-graph}, and describe some of its properties. \begin{definition}[\cite{craciun2015toric, craciun2019polynomial,craciun2020endotactic}] \label{def:e-graph} \begin{enumerate}[label=(\alph)] \item A \textbf{reaction network} $G=(V,E)$ is a directed graph, also called a \textbf{Euclidean embedded graph} (or \textbf{E-graph}), such that $V \subset \mathbb{R}^n$ is a finite set of \textbf{vertices} and the set $E\subseteq V\times V$ represents a finite set of \textbf{edges}. We assume that there are neither self-loops nor isolated vertices in $G=(V, E)$. \item A directed edge $(\by,\by')\in E$ connecting two vertices $\by, \by' \in V$ is denoted by $\by \rightarrow \by' \in E$ and represents a reaction in the network. Here $\by$ is called the \textbf{source vertex}, and $\by'$ is called the \textbf{target vertex}. Further, the difference vector $\by' - \by \in\mathbb{R}^n$ is called the \textbf{reaction vector}. \end{enumerate} \end{definition} \begin{definition} Consider an E-graph $G=(V,E)$. Then \begin{enumerate}[label=(\alph)] \item $G$ is \textbf{weakly reversible}, if every reaction in $G$ is part of an oriented cycle. \item $G$ is a \textbf{(directed) complete} graph, if $\by\rightarrow \by'\in E$ for every two distinct vertices $\by, \by'\in V$. \item An E -graph $G' = (V', E')$ is a \textbf{subgraph} of $G$ (denoted by $G' \subseteq G$), if $V' \subseteq V$ and $E' \subseteq E$. In addition, we let $G' \sqsubseteq G$ denote that $G'$ is a weakly reversible subgraph of $G$. \item We denote the \defi{complete graph on $G$} by $G_c$, which is obtained by connecting every pair of source vertices in $V$. One can check that $G_c$ is weakly reversible and $G \subseteq G_c$. \end{enumerate} \end{definition} \begin{figure}[!ht] \centering \includegraphics[scale=0.4]{euclidean_embedded_graph.eps} \caption{\small (a) An E-graph with two reactions. The stoichiometric subspace corresponding to this graph is $\RR^2$. (b) A weakly reversible E-graph. (c) A directed complete E-graph with three vertices. Note that the E-graph in (b) is a weakly reversible subgraph of the E-graph in (c).} \label{fig:e-graph} \end{figure} \begin{definition}[\cite{adleman2014mathematics,guldberg1864studies,voit2015150,gunawardena2003chemical,yu2018mathematical,feinberg1979lectures}] Consider an E-graph $G=(V,E)$. Let $k_{\by\to \by'}$ denote the \textbf{reaction rate constant} corresponding to the reaction $\by\to \by'\in E$. Further, we let ${\bk} :=(k_{\by\to \by'})_{\by\to \by' \in E} \in \mathbb{R}_{>0}^{E}$ denote the \textbf{vector of reaction rate constants} (\textbf{reaction rate vector}). The \textbf{associated mass-action system} generated by $(G, \bk)$ on $\RR^n_{>0}$ is given by \begin{equation} \label{def:mas_ds} \frac{d\bx}{dt} = \displaystyle\sum_{\by \rightarrow \by' \in E}k_{\by\rightarrow\by'}{\bx}^{\by}(\by'-\by). \end{equation} We denote the \defi{stoichiometric subspace} of $G$ by $\mathcal{S}_G$, which is \begin{equation} \notag \mathcal{S}_G = \spn \{ \by' - \by: \by \rightarrow \by' \in E \}. \end{equation} \cite{sontag2001structure} shows that if $V \subset \mathbb{Z}_{\geq 0}^n$, the positive orthant $\mathbb{R}_{>0}^n$ is forward-invariant under system \eqref{def:mas_ds}. Any solution to \eqref{def:mas_ds} with initial condition $\bx_0 \in \mathbb{R}_{>0}^n$ and $V \subset \mathbb{Z}_{\geq 0}^n$, is confined to $(\bx_0 + \mathcal{S}_G) \cap \mathbb{R}_{>0}^n$. Thus, the set $(\bx_0 + \mathcal{S}_G) \cap \mathbb{R}_{>0}^n$ is called the \textbf{invariant polyhedron} of $\bx_0$. \end{definition} \begin{definition} Let $(G, \bk)$ be a mass-action system. \begin{enumerate}[label=(\alph)] \item A point $\bx^ \in \mathbb{R}^n_{>0}$ is called a \defi{positive steady state} of the system if \begin{equation} \label{eq:steady_statez} \displaystyle\sum_{\by\rightarrow \by' \in E } k_{\by\rightarrow\by'}{(\bx^)}^{\by}(\by'-\by)=0. \end{equation} \item A point $\bx^ \in \mathbb{R}^n_{>0}$ is called a \defi{complex-balanced steady state} of the system if for every vertex $\by_0 \in V$, \begin{eqnarray} \notag \sum_{\by_0 \rightarrow \by \in E} k_{\by_0 \rightarrow \by} {(\bx^)}^{\by_0} = \sum_{\by' \rightarrow \by_0 \in E} k_{\by' \rightarrow \by_0} {(\bx^)}^{\by'}. \end{eqnarray} Further, if the mass-action system $(G, \bk)$ admits a complex-balanced steady state, then it is called a \defi{complex-balanced (dynamical) system} or \defi{toric dynamical system}. \end{enumerate} \end{definition} \begin{remark} \label{rmk:complex_balance_property} Complex-balanced systems are known to exhibit robust dynamical properties. As mentioned in the introduction, they are connected to the \emph{Global Attractor Conjecture}, which proposes that complex-balanced systems possess a globally attracting steady state within each stoichiometric compatibility class. Several important special cases of this conjecture and related open problems have been proven. In particular, it has been shown that complex-balanced systems consisting of a single linkage class admit a globally attracting steady state \cite{anderson2011proof}. Additionally, two- and three-dimensional endotactic networks are known to be permanent \cite{craciun2013persistence}. Strongly endotactic networks have also been proven to be permanent \cite{gopalkrishnan2014geometric}. Furthermore, complex-balanced systems that are permanent always admit a globally attracting steady state \cite{yu2018mathematical}. \end{remark} \begin{theorem}[\cite{horn1972general}] \label{thm:cb} Consider a complex-balanced system $(G, \bk)$. Then \begin{enumerate} \item[(a)] The E-graph $G = (V, E)$ is weakly reversible. \item[(b)] Every positive steady state is a complex-balanced steady state. Given any $\bx_0 \in \mathbb{R}_{>0}^n$, there is exactly one steady state within the invariant polyhedron $(\bx_0 + \mathcal{S}_G) \cap \mathbb{R}_{>0}^n$. \end{enumerate} \end{theorem} \begin{theorem}[\cite{johnston2012topics}] \label{thm:jacobian} Consider a weakly reversible E-graph $G = (V, E)$ with the stoichiometric subspace $\mS_G$. Suppose $(G, \bk)$ is a complex-balanced system given by \begin{equation} \label{eq:jacobian} \frac{\mathrm{d} \bx}{\mathrm{d} t} = \bf (\bx) = \displaystyle\sum_{\by\rightarrow \by' \in E} k_{\by\rightarrow\by'}{\bx}^{\by}(\by'-\by). \end{equation} For any steady state $\bx^* \in \RR^n_{>0}$ of the system \eqref{eq:jacobian}, then \begin{equation} \label{eq:jacobian_ker} \Big( \ker \big( \mathbf{J}_{\bf} \|_{\bx = \bx^} \big) \Big)^{\perp} = \mS_G, \end{equation} where $\mathbf{J}_{\bf}$ represents the Jacobian matrix of $\bf (\bx)$. \end{theorem} \begin{definition} \label{def:de} Consider two mass-action systems $(G,\bk)$ and $(G',\bk')$. Then $(G,\bk)$ and $(G',\bk')$ are said to be \defi{dynamically equivalent} if for every vertex\footnote{ Note that when $\by_0 \not\in V$ or $\by_0 \not\in V'$, the corresponding side is considered as an empty sum} $\by_0 \in V \cup V'$, \begin{eqnarray} \notag \displaystyle\sum_{\by_0 \rightarrow \by\in E} k_{\by_0 \rightarrow \by} (\by - \by_0) = \displaystyle\sum_{\by_0 \rightarrow \by'\in E'} k'_{\by_0 \rightarrow\by'} (\by' - \by_0). \end{eqnarray} We let $(G,\bk)\sim (G', \bk')$ denote that two mass-action systems $(G,\bk)$ and $(G',\bk')$ are dynamically equivalent. \end{definition} \begin{remark}[\cite{horn1972general,craciun2008identifiability,deshpande2022source}] \label{rmk:de_ss} Following Definition \ref{def:de}, two mass-action systems $(G, \bk)$ and $(G', \bk')$ are dynamically equivalent if and only if for all $\bx \in \RR_{>0}^{n}$, \begin{equation} \label{eq:eqDE} \sum_{\by_i \to \by_j \in E} k_{\by_i \to \by_j} \bx^{\by_i} (\by_j - \by_i) = \sum_{\by'_i \to \by'_j \in E'} k'_{\by'_i \to \by'_j} \bx^{\by'_i} (\by'_j - \by'_i), \end{equation} and thus two dynamically equivalent systems share the same set of steady states. \end{remark} \begin{definition} \label{def:d0} Consider an E-graph $G=(V, E)$. Let $\bla = (\lambda_{\by \to \by'})_{\by \to \by' \in E} \in \RR^{\|E\|}$. The set $\mD(G)$ is defined as \begin{equation} \notag \mD (G):= \{\bla \in \RR^{\|E\|} \, \Big\| \, \sum_{\by_0 \to \by \in E} \lambda_{\by_0 \to \by} (\by - \by_0) = \mathbf{0} \ \text{for every vertex } \by_0 \in V \}. \end{equation} We can check that $\mD (G)$ is a linear subspace of $\RR^E$. \end{definition} \begin{lemma}[\cite{disg_2}] \label{lem:d0} Consider two mass-action systems $(G, \bk)$ and $(G, \bk')$. Then $\bk' - \bk \in \mD (G)$ if and only if $(G, \bk) \sim (G, \bk')$. \end{lemma} \section{Flux systems} \label{sec:flux_systems} Due to the non-linearity of the dynamical systems, we now introduce linear systems arising from the network structure: the flux systems, and the complex-balanced flux systems, and study their properties. \begin{definition} Consider an E-graph $G=(V, E)$. Then \begin{enumerate}[label=(\alph)] \item Let $J_{\by \to \by'} > 0$ denote the \textbf{flux} corresponding to the edge $\by \to \by'\in E$. Further, we let $\bJ = (J_{\by \to \by'})_{\by \to \by' \in E} \in \RR_{>0}^E$ denote the \textbf{flux vector} corresponding to the E-graph $G$. The \defi{associated flux system} generated by $(G, \bJ)$ is given by \begin{equation} \label{eq:flux} \frac{\mathrm{d} \bx}{\mathrm{d} t} = \sum_{\byi \to \byj \in E} J_{\byi \to \byj} (\byj - \byi). \end{equation} \item Consider two flux systems $(G,\bJ)$ and $(G', \bJ')$. Then $(G,\bJ)$ and $(G', \bJ')$ are said to be \defi{flux equivalent} if for every vertex\footnote{Note that when $\by_0 \not\in V$ or $\by_0 \not\in V'$, the corresponding side is considered as an empty sum} $\by_0 \in V \cup V'$, \begin{equation} \notag \sum_{\by_0 \to \by \in E} J_{\by_0 \to \by} (\by - \by_0) = \sum_{\by_0 \to \by' \in E'} J'_{\by_0 \to \by'} (\by' - \by_0). \end{equation} We let $(G, \bJ) \sim (G', \bJ')$ denote that two flux systems $(G, \bJ)$ and $(G', \bJ')$ are flux equivalent. \end{enumerate} \end{definition} \begin{definition} Let $(G,\bJ)$ be a flux system. A flux vector $\bJ \in \RR_{>0}^E$ is called a \defi{steady flux vector} to $G$ if \begin{equation} \notag \frac{\mathrm{d} \bx}{\mathrm{d} t} = \sum_{\byi \to \byj \in E} J_{\byi \to \byj} (\byj - \byi) = \mathbf{0}. \end{equation} A steady flux vector $\bJ\in \RR^{E}_{>0}$ is called a \defi{complex-balanced flux vector} to $G$ if for every vertex $\by_0 \in V$, \begin{eqnarray} \notag \sum_{ \by_0 \to \by \in E} J_{\by_0 \to \by} = \sum_{\by' \to \by_0 \in E} J_{\by' \to \by_0}, \end{eqnarray} and then $(G, \bJ)$ is called a \defi{complex-balanced flux system}. Further, let $\mathcal{J}(G)$ denote the set of all complex-balanced flux vectors to $G$ as follows: \begin{equation} \notag \mathcal{J}(G):= \{\bJ \in \RR_{>0}^{E} \mid \bJ \text{ is a complex-balanced flux vector to $G$} \}. \end{equation} \end{definition} \begin{definition} \label{def:j0} Consider an E-graph $G=(V, E)$. Let $\bJ = ({J}_{\byi \to \byj})_{\byi \to \byj \in E} \in \RR^E$. The set $\eJ (G)$ is defined as \begin{equation} \label{eq:J_0} \eJ (G): = \{{\bJ} \in \mD (G) \, \bigg\| \, \sum_{\by \to \by_0 \in E} {J}_{\by \to \by_0} = \sum_{\by_0 \to \by' \in E} {J}_{\by_0 \to \by'} \ \text{for every vertex } \by_0 \in V \}. \end{equation} Note that $\eJ(G) \subset \mD (G)$ is a linear subspace of $\RR^E$. \end{definition} \begin{lemma}[\cite{disg_2}] \label{lem:j0} Let $(G, \bJ)$ and $(G, \bJ')$ be two flux systems. Then \begin{enumerate} \item[(a)] $(G, \bJ) \sim (G, \bJ')$ if and only if $\bJ' - \bJ \in \mD (G)$. \item[(b)] If $(G, \bJ)$ and $(G, \bJ')$ are both complex-balanced flux systems, then $(G, \bJ) \sim (G, \bJ')$ if and only if $\bJ' - \bJ \in \eJ(G)$. \end{enumerate} \end{lemma} \begin{proposition}[\cite{craciun2020efficient}] \label{prop:craciun2020efficient} Consider two mass-action systems $(G, \bk)$ and $(G', \bk')$. Let $\bx \in \RR_{>0}^n$. Define the flux vector $\bJ (\bx) = (J_{\by \to \by'})_{\by \to \by' \in E}$ on $G$, such that for every $\by \to \by' \in E$, \begin{equation} \notag J_{\by \to \by'} = k_{\by \to \by'} \bx^{\by}. \end{equation} Further, define the flux vector $\bJ' (\bx) = (J'_{\by \to \by'})_{\by \to \by' \in E'}$ on $G'$, such that for every $\by \to \by' \in E$, \begin{equation} \notag J'_{\by \to \by'} = k'_{\by \to \by'} \bx^{\by}. \end{equation} Then the following are equivalent: \begin{enumerate} \item[(a)] The mass-action systems $(G, \bk)$ and $(G', \bk')$ are dynamically equivalent. \item[(b)] The flux systems $(G, \bJ(\bx))$ and $(G', \bJ')$ are flux equivalent for all $\bx \in \RR_{>0}^n$. \item[(c)] The flux systems $(G, \bJ(\bx))$ and $(G', \bJ'(\bx))$ are flux equivalent for some $\bx \in \RR_{>0}^n$ \end{enumerate} \end{proposition} \section{Toric locus, disguised toric locus and \texorpdfstring{$\RR$}{R}-disguised toric locus} \label{sec:disguised_locus} In this section, we introduce the key concepts in this paper: the Toric locus, the Disguised toric locus, and the $\RR$-disguised toric locus. \begin{definition}[\cite{disg_2}] \label{def:mas_realizable} Let $G=(V, E)$ be an E-graph. Consider a dynamical system \begin{equation} \label{eq:realization_ode} \frac{\mathrm{d} \bx}{\mathrm{d} t} = \bf (\bx). \end{equation} It is said to be \defi{$\RR$-realizable} (or has a \defi{$\RR$-realization}) on $G$, if there exists some $\bk \in \mathbb{R}^{E}$ such that \begin{equation} \label{eq:realization} \bf (\bx) = \sum_{\by_i \rightarrow \by_j \in E}k_{\by_i \rightarrow \by_j} \bx^{\by_i}(\by_j - \by_i). \end{equation} Further, if $\bk \in \mathbb{R}^{E}_{>0}$ in \eqref{eq:realization}, the system \eqref{eq:realization_ode} is said to be \defi{realizable} (or has a \defi{realization}) on $G$. \end{definition} \begin{definition} Consider an E-graph $G=(V, E)$. \begin{enumerate} \item[(a)] Define the \defi{toric locus} of $G$ as \begin{equation} \notag \mK (G) := \{ \bk \in \mathbb{R}_{>0}^{E} \ \big\| \ \text{the mass-action system generated by } (G, \bk) \ \text{is toric} \}. \end{equation} \item[(b)] Consider a dynamical system \begin{equation} \label{eq:def_cb_realization} \frac{\mathrm{d} \bx}{\mathrm{d} t} = \bf (\bx). \end{equation} It is said to be \defi{disguised toric} on $G$ if it is realizable on $G$ for some $\bk \in \mK (G)$. Further, we say the system \eqref{eq:def_cb_realization} has a \defi{complex-balanced realization} on $G$. \end{enumerate} \end{definition} \begin{definition} \label{def:de_realizable} Consider two E-graphs $G =(V,E)$ and $G' =(V', E')$. \begin{enumerate} \item[(a)] Define the set $\mK_{\RR}(G', G)$ as \begin{equation} \notag \mK_{\RR}(G', G) := \{ \bk' \in \mK (G') \ \big\| \ \text{the mass-action system } (G', \bk' ) \ \text{is $\RR$-realizable on } G \}. \end{equation} \item[(b)] Define the set $\dK(G, G')$ as \begin{equation} \notag \dK(G, G') := \{ \bk \in \mathbb{R}^{E} \ \big\| \ \text{the dynamical system} \ (G, \bk) \ \text{is disguised toric on } G' \}. \end{equation} Note that $\bk$ may have negative or zero components. \item[(c)] Define the \defi{$\RR$-disguised toric locus} of $G$ as \begin{equation} \notag \dK(G) := \displaystyle\bigcup_{G' \sqsubseteq G_{c}} \ \dK(G, G'). \end{equation} Note that in the above definition of $\RR$-disguised toric locus of $G$, we take a union over only those E-graphs which are weakly reversible subgraphs of $G_c$. This follows from a result in~\cite{craciun2020efficient} which asserts that if a dynamical system generated by $G$ has a complex-balanced realization using some graph $G_1$, then it also has a complex-balanced realization using $G'\sqsubseteq G_{c}$. \item[(d)] Define the set $\pK (G, G')$ as \begin{equation} \notag \pK (G, G') := \dK(G, G') \cap \mathbb{R}^{E}_{>0}. \end{equation} Further, define the \defi{disguised toric locus} of $G$ as \begin{equation} \notag \pK (G) := \displaystyle\bigcup_{G' \sqsubseteq G_{c}} \ \pK(G, G'). \end{equation} Similar to the $\RR$-disguised toric locus, it is sufficient for us to include those E-graphs which are weakly reversible subgraphs of $G_c$~\cite{craciun2020efficient}. \end{enumerate} \end{definition} \begin{lemma}[\cite{disg_2}] \label{lem:semi_algebaic} Let $G = (V, E)$ be an E-graph. \begin{enumerate} \item[(a)] Suppose that $G_1 = (V_1, E_1)$ is a weakly reversible E-graph, then $\dK(G,G_1)$ and $\pK(G,G_1)$ are semialgebraic sets. \item[(b)] Both $\dK(G)$ and $\pK(G)$ are semialgebraic sets. \end{enumerate} \end{lemma} \begin{proof} For part $(a)$, following Lemma 3.6 in \cite{disg_2}, we obtain that $\dK(G, G_1)$ is a semialgebraic set. The positive orthant is also a semialgebraic set since it can be defined by polynomial inequalities on all components. Since finite intersections of semialgebraic sets are semialgebraic sets, together with Definition \ref{def:de_realizable}, we conclude that $\pK(G, G_1)$ is a semialgebraic set. \smallskip For part $(b)$, since finite unions of semialgebraic sets are semialgebraic sets~\cite{coste2000introduction}, together with Definition \ref{def:de_realizable} and part $(a)$, we conclude that $\dK(G)$ and $\pK(G)$ are semialgebraic sets. \end{proof} \begin{remark}[\cite{lee2010introduction}] \label{rmk:semi_algebaic} From Lemma \ref{lem:semi_algebaic} and \cite{lee2010introduction}, on a dense open subset of any semialgebraic set $\dK(G, G_1)$ or $\pK(G, G_1)$, it is locally a \textbf{submanifold}. The dimension of $\dK(G, G_1)$ or $\pK(G, G_1)$ can be defined to be the largest dimension at points at which it is a submanifold. \end{remark} \begin{remark} \label{rmk:mJ_dK} Let $G_1 = (V_1, E_1)$ be a weakly reversible E-graph and let $G = (V, E)$ be an E-graph. From Definition \ref{def:de_realizable}, it follows that $\dK (G, G_1)$ is empty if and only if $\mK_{\RR} (G_1, G)$ is empty. \end{remark} Analogous to the $\RR$-disguised toric locus, we also introduce the $\RR$-realizable complex-balanced flux system, which plays a crucial role in the rest of the paper. \begin{definition} \label{def:flux_realizable} Consider a flux system $(G', \bJ')$. It is said to be \defi{$\RR$-realizable} on $G$ if there exists some $\bJ \in \mathbb{R}^{E}$, such that for every vertex\footnote{Note that when $\by_0 \not\in V$ or $\by_0 \not\in V'$, the corresponding side is considered as an empty sum} $\by_0 \in V \cup V'$, \begin{equation} \notag \sum_{\by_0 \to \by \in E} J_{\by_0 \to \by} (\by - \by_0) = \sum_{\by_0 \to \by' \in E'} J'_{\by_0 \to \by'} (\by' - \by_0). \end{equation} Further, define the set $\mJ (G', G)$ as \begin{equation} \notag \mJ (G', G) := \{ \bJ' \in \mathcal{J} (G') \ \big\| \ \text{the flux system } (G', \bJ') \ \text{is $\RR$-realizable on } G \}. \end{equation} Proposition \ref{prop:craciun2020efficient} implies that $\dK (G, G')$ is empty if and only if $\mJ(G', G)$ is empty. \end{definition} \begin{lemma}[{\cite[Lemma 2.33]{disg_2}}] \label{lem:j_g1_g_cone} Consider a weakly reversible E-graph $G_1 = (V_1, E_1)$ and let $G = (V, E)$ be an E-graph. Then we have the following: \begin{enumerate} \item[(a)] There exists a vectors $\{ \bv_1, \bv_2, \ldots, \bv_k \} \subset \RR^{\|E_1\|}$, such that \begin{equation} \label{j_g1_g_generator} \mJ (G_1, G) = \{ a_1 \bv_1 + \cdots a_k \bv_k \ \| \ a_i \in \RR_{>0}, \bv_i \in \RR^{\|E_1\|} \}. \end{equation} \item[(b)] $\dim (\mJ (G_1, G)) = \dim ( \spn \{ \bv_1, \bv_2, \ldots, \bv_k \} )$. \item[(c)] If $\mJ (G_1, G) \neq \emptyset$, then \[ \eJ(G_1) \subseteq \spn \{ \bv_1, \bv_2, \ldots, \bv_k \}. \] \end{enumerate} \end{lemma} \section{The map \texorpdfstring{$\hPsi$}{hPsi}} \label{sec:map} The goal of this section is to study the properties of a map $\hat{\Psi}$ (see Definition \ref{def:hpsi}) that relates the sets $\dK(G, G_1)$ and $\hat{\mJ} (G_1, G)$ (see Equation \eqref{def:hat_j_g1_g}). In particular, we show the map $\hat{\Psi}$ is bijective and continuous. \paragraph{Notation.} We introduce the following notation that will be used for the entire section. Let $G = (V, E)$ be an E-graph. Let $b$ denote the dimension of the linear subspace $\mD(G)$, and denote an orthonormal basis of $\mD(G)$ by \[ \{\bB_1, \bB_2, \ldots, \bB_b\}. \] Further, we consider $G_1 = (V_1, E_1)$ to be a weakly reversible E-graph. Let $a$ denote the dimension of the linear subspace $\eJ(G_1)$, and denote an orthonormal basis of $\eJ(G_1)$ by \[ \{\bA_1, \bA_2, \ldots, \bA_a \}. \] \qed \medskip Recall the set $\mJ (G_1,G)$. Now we define the set $\hat{\mJ} (G_1,G) \subset \RR^{\|E_1\|}$ as \begin{equation} \label{def:hat_j_g1_g} \hat{\mJ} (G_1,G) = \{ \bJ + \sum\limits^a_{i=1} w_i \bA_i \ \| \ \bJ \in \mJ (G_1,G), \text{ and } w_i \in \RR \text{ for } 1 \leq i \leq a \}. \end{equation} Further, we define the set $\hat{\mathcal{J}} (G_1) \subset \RR^{\|E_1\|}$ as \begin{equation} \label{def:hat_j_g1} \hat{\mathcal{J}} (G_1) = \{\bJ \in \RR^{E} \mid \sum_{\by \to \by_0 \in E} J_{\by \to \by_0} = \sum_{\by_0 \to \by' \in E} J_{\by_0 \to \by'} \text{ for every vertex $\by_0 \in V_1$}\}. \end{equation} \begin{remark} \label{rmk:hat_j_g1_g} Following~\eqref{def:hat_j_g1_g}, it is clear that $\mJ (G_1,G) \subset \hat{\mJ} (G_1,G)$. Further, from $\{\bA_i \}^{a}_{i=1} \in \eJ(G)$ and Lemma \ref{lem:j0}, we conclude that \[\hat{\mJ} (G_1,G) \cap \RR^{\|E_1\|}_{>0} = \mJ (G_1,G). \] Similarly, we have $\hat{\mathcal{J}} (G_1) \cap \RR^{\|E_1\|}_{>0} = \mathcal{J} (G_1)$. \end{remark} \begin{remark} Note that $\hat{\mathcal{J}} (G_1)$ is a linear subspace of $\RR^{\|E_1\|}$, while the sets $\hat{\mJ} (G_1,G)$, $\mJ (G_1,G)$ and $\mathcal{J} (G_1)$ are not linear subspaces. \end{remark} \begin{definition} \label{def:hpsi} Given a weakly reversible E-graph $G_1 = (V_1, E_1)$ with its stoichiometric subspace $\mS_{G_1}$. Consider an E-graph $G = (V, E)$ and $\bx_0\in\mathbb{R}^n_{>0}$, define the map \begin{equation} \label{eq:hpsi} \hPsi: \hat{\mJ} (G_1,G) \times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \RR^b \rightarrow \dK(G,G_1) \times \RR^a, \end{equation} such that for $(\hat{\bJ}, \bx, \bp) \in \hat{\mJ} (G_1,G) \times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$, \begin{equation} \notag \hat{\Psi} (\hat{\bJ},\bx, \bp) : = (\bk, \bq), \end{equation} where \begin{equation} \label{def:hpsi_k} (G, \bk) \sim (G_1, \hat{\bk}_1) \ \text{ with } \ \hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}_{\by\rightarrow \by'}}{{\bx}^{\by}}, \end{equation} and \begin{equation} \label{def:hpsi_kq} \bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle), \ \ \bq = ( \langle \hat{\bJ}, \bA_1 \rangle, \langle \hat{\bJ}, \bA_2 \rangle, \ldots, \langle \hat{\bJ}, \bA_a \rangle ). \end{equation} \end{definition} Recall Remark \ref{rmk:mJ_dK}, $\dK (G, G_1)$ is empty if and only if $\mJ(G_1, G)$ is empty. If $\mJ(G_1, G) = \dK (G, G_1) = \emptyset$, then the map $\hPsi$ is trivial. However, we are interested in the case when $\dK (G, G_1) \neq \emptyset$, therefore we assume both $\mJ(G_1, G)$ and $\dK (G, G_1)$ are non-empty sets in the rest of the paper. \begin{lemma} \label{lem:hpsi_well_def} The map $\hPsi$ in Definition \ref{def:hpsi} is well-defined. \end{lemma} \begin{proof} Consider any point $(\hbJ^, \bx^, \bp^) \in \hat{\mJ} (G_1,G)\times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$. From Equation\eqref{def:hat_j_g1_g}, there exist a $\bJ^ = (J^_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \in \mJ (G_1,G)$ and $w^_i \in \RR$ for $1 \leq i \leq a$, such that \[ \hbJ^* = \bJ^* + \sum\limits^a_{i=1} w^_i \bA_i. \] Since $\{ \bA_i \}^a_{i=1}$ is an orthonormal basis of the subspace $\eJ(G_1)$, we obtain \begin{equation} \label{eq:psi_wd_1} (G_1, \hbJ^) \sim (G_1, \bJ^). \end{equation} From $\bJ^ \in \mJ (G_1,G) \subset \bJ (G_1)$, set $\bk_1 = (k_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with $k_{1, \by\rightarrow \by'} = \frac{J^_{\by \rightarrow \by'} }{ (\bx^)^{\by} }$. Then \begin{equation} \label{eq:psi_wd_2} \bk_1 \in \mK_{\RR} (G_1,G) \subset \mK(G_1). \end{equation} Moreover, $\bx^$ is the complex-balanced steady state of $(G_1, \bk_1)$. Set $\hbk_1 = (\hat{k}_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with $\hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}^_{\by \rightarrow \by'} }{ (\bx^)^{\by} }$. From Equation\eqref{eq:psi_wd_1} and Proposition \ref{prop:craciun2020efficient}, we have \begin{equation} \label{eq:psi_wd_3} (G_1, \bk_1) \sim (G_1, \hat{\bk}_1). \end{equation} From Equation\eqref{eq:psi_wd_2}, there exists a $\bk \in \dK(G,G_1) \subset \RR^{\|E\|}$, such that $(G, \bk) \sim (G_1, \bk_1)$. Now suppose $\bp^ = (p^_1, p^_2, \ldots, p^_b) \in \RR^b$, we construct the vector $\bk^ \in \RR^{\|E\|}$ as \[ \bk^* = \bk + \sum\limits^{b}_{i=1} (p^_i - \langle \bk, \bB_i \rangle ) \bB_i. \] Since $\{ \bB_i \}^b_{i=1}$ is an orthonormal basis of the subspace $\mD(G)$, then for $1 \leq j \leq b$, \begin{equation} \label{eq:kp} \langle \bk^, \bB_j \rangle = \langle \bk + \sum\limits^{b}_{i=1} (p^_i - \langle \bk, \bB_i \rangle ) \bB_i, \bB_j \rangle = \langle \bk, \bB_j \rangle + (p^_j - \langle \bk, \bB_j \rangle ) = p^_j. \end{equation} Using Lemma \ref{lem:d0}, together with $\sum\limits^{b}_{i=1} (p^_i - \bk \bB_i ) \bB_i \in \mD(G)$ and \eqref{eq:psi_wd_3}, we obtain \begin{equation} \label{eq:psi_wd_4} (G, \bk^) \sim (G, \bk) \sim (G_1, \hat{\bk}_1). \end{equation} Therefore, $\bk^$ satisfies Equations\eqref{def:hpsi_k} and \eqref{def:hpsi_kq}. \smallskip \noindent Let us assume that there exists $\bk^{} \in \dK(G,G_1)$ satisfying Equations\eqref{def:hpsi_k} and \eqref{def:hpsi_kq}, i.e., \[(G, \bk^{}) \sim (G_1, \hat{\bk}_1) \ \text{ and } \ \bp^* = ( \langle \bk^{}, \bB_1 \rangle, \langle \bk^{}, \bB_2 \rangle, \ldots, \langle \bk^{}, \bB_b \rangle). \] This implies that $(G, \bk^{}) \sim (G, \bk^)$. From Lemma \ref{lem:d0}, we obtain \[ \bk^{} - \bk^{} \in \mD(G). \] Using \eqref{eq:kp}, we get \[ \langle \bk^, \bB_j \rangle = \langle \bk^{}, \bB_j \rangle = p^_j \ \text{ for any } \ 1 \leq j \leq b. \] Recall that $\{ \bB_i \}^b_{i=1}$ is an orthonormal basis of $\mD(G)$. Therefore, we get \[ \bk^{*} = \bk^{}. \] This implies that $\bk^* \in \dK(G,G_1)$ is well-defined. Moreover, from \eqref{def:hpsi_kq} we obtain \[ \bq^* = ( \langle \hbJ^, \bA_1 \rangle, \langle \hbJ^, \bA_2 \rangle, \ldots, \langle \hbJ^, \bA_a \rangle ) \ \text{ is well-defined}. \] This implies that we get \[ \hPsi (\hbJ^, \bx^, \bp^) = (\bk^, \bq^), \] and thus the map $\hPsi$ is well-defined. \end{proof} The following is a direct consequence of Lemma \ref{lem:hpsi_well_def}. \begin{corollary} \label{cor:hpsi_ss} Consider the map $\hPsi$ in Definition \ref{def:hpsi}. Suppose that $\hat{\Psi} (\hat{\bJ},\bx, \bp) = (\bk, \bq)$, then $\bx$ is a steady state of the system $(G, \bk)$. \end{corollary} \begin{proof} It is clear that $\hat{\bJ} \in \hat{\mJ} (G_1,G)$ and $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$. From Equation\eqref{def:hat_j_g1_g}, there exist some $\bJ^* = (J^_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \in \mJ (G_1,G)$, such that \[ \hbJ - \bJ^ \in \spn \{\bA_i \}^{a}_{i=1}. \] Using \eqref{eq:psi_wd_2} and setting $\bk_1 = (k_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with $k_{1, \by\rightarrow \by'} = \frac{J^_{\by \rightarrow \by'} }{ (\bx^)^{\by} }$, we derive \[ \bk_1 \in \mK_{\RR} (G_1,G), \] and $\bx^$ is the complex-balanced steady state of $(G_1, \bk_1)$. Finally, using Equations\eqref{eq:psi_wd_3} and \eqref{eq:psi_wd_4}, together with Remark \ref{rmk:de_ss}, we obtain $(G, \bk) \sim (G_1, \bk_1)$ and conclude that $\bx$ is a steady state of the system $(G, \bk)$. \end{proof} \begin{lemma} \label{lem:hpsi_bijective} The map $\hPsi$ in Definition \ref{def:hpsi} is bijective. \end{lemma} \begin{proof} First, we show the map $\hPsi$ is injective. Suppose two elements $(\hbJ^, \bx^, \bp^)$ and $(\hbJ^{}, \bx^{}, \bp^{*})$ of $\hat{\mJ} (G_1,G) \times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$ satisfy \[ \hPsi (\hbJ^, \bx^, \bp^) = \hPsi (\hbJ^{}, \bx^{}, \bp^{*}) = (\bk, \bq) \in \dK(G,G_1)\times \RR^a. \] From \eqref{def:hat_j_g1_g}, there exist $\bJ^ = (J^_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \in \mJ (G_1,G)$ and $\bJ^{} = (J^{}_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \in \mJ (G_1,G)$, such that \begin{equation} \label{eq:hpsi_bijective_1} \hbJ^ - \bJ^* \in \spn \{ \bA_i \}^{a}_{i=1} \ \text{ and } \ \hbJ^{} - \bJ^{} \in \spn \{ \bA_i \}^{a}_{i=1}. \end{equation} Then we set $\bk^* = (k^_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ and $\bk^{} = (k^{}_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with \[ k^_{\by\rightarrow \by'} = \frac{J^_{\by\rightarrow \by'}}{{(\bx^)}^{\by}} \ \text{ and } \ k^{}_{\by\rightarrow \by'} = \frac{J^{}_{\by\rightarrow \by'}}{{(\bx^)}^{\by}}. \] Using Propositions\ref{prop:craciun2020efficient} and \eqref{def:hpsi_k}, we get \[\bk^, \bk^{*} \in \mK_{\RR} (G_1,G) \ \text{ and } \ (G, \bk) \sim (G_1, \bk^) \sim (G_1, \bk^{*}). \] Moreover, two complex-balanced system $(G_1, \bk^)$ and $(G_1, \bk^{*})$ admit steady states \[ \bx^ \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0} \ \text{ and } \ \bx^{*} \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}, \ \text{respectively}. \] Since every complex-balanced system has a unique steady state within each invariant polyhedron and $(G_1, \bk^) \sim (G_1, \bk^{*})$, then \[ \bx^ = \bx^{*}. \] Now applying Proposition \ref{prop:craciun2020efficient} and Lemma \ref{lem:j0}, we get \begin{equation} \label{eq:hpsi_bijective_2} (G_1, \bJ^) \sim (G_1, \bJ^{}) \ \text{ and } \ \bJ^{} - \bJ^* \in \eJ(G_1). \end{equation} Since $\eJ(G_1) = \spn \{ \bA_i \}^{a}_{i=1}$, using \eqref{eq:hpsi_bijective_1} and \eqref{eq:hpsi_bijective_2}, we have \begin{equation} \label{eq:hpsi_bijective_3} \hbJ^{*} - \hbJ^ \in \spn \{ \bA_i \}^{a}_{i=1}. \end{equation} On the other hand, Equation\eqref{def:hpsi_kq} shows that \[ \bq = ( \langle \hbJ^, \bA_1 \rangle, \langle \hbJ^, \bA_2 \rangle, \ldots, \langle \hbJ^, \bA_a \rangle ) = ( \langle \hbJ^{}, \bA_1 \rangle, \langle \hbJ^{}, \bA_2 \rangle, \ldots, \langle \hbJ^{}, \bA_a \rangle ). \] Since $\{\bA_i \}^{a}_{i=1}$ is an orthonormal basis of the subspace $\eJ(G)$, together with \eqref{eq:hpsi_bijective_3}, then \[ \hbJ^ = \hbJ^{*}. \] Furthermore, from \eqref{def:hpsi_kq} we obtain \[ \bp^ = \bp^{*} = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle). \] Therefore, we show $(\bJ^, \bx^, \bp^) = (\bJ^{}, \bx^{}, \bp^{*})$ and conclude the injectivity. \medskip We now show that the map $\hPsi$ is surjective. Assume any point $(\bk, \bq) \in \dK(G,G_1)\times \RR^a$. Since $\bk \in \dK (G, G_1)$, there exists some $\bk_1 \in \mK (G_1, G)$, such that \begin{equation} \label{eq:gk_g1k1} (G, \bk) \sim (G_1, \bk_1) \ \text{ with } \ \bk_1 = (k_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}. \end{equation} From Theorem \ref{thm:cb}, the complex-balanced system $(G_1, \bk_1)$ has a unique steady state $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$. We set the flux vector $\bJ_1$ as \[ \bJ_1 = (J_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \ \text{ with } \ J_{1, \by\rightarrow \by'} = k_{1, \by\rightarrow \by'} {\bx}^{\by}. \] It is clear that $\bJ_1 \in \mJ (G_1,G)$ and the flux system $(G_1, \bJ_1)$ gives rise to the complex-balanced system $(G_1, \bk_1)$ with a steady state $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$. Now suppose $\bq = (q_1, q_2, \ldots, q_a)$, we construct a new flux vector $\hbJ$ as follows: \[ \hbJ = \bJ_1 + \sum\limits^{a}_{i=1} (q_i - \langle \bJ_1, \bA_i \rangle ) \bA_i. \] Using the fact that $\{ \bA_i \}^a_{i=1}$ is an orthonormal basis of the subspace $\eJ(G_1)$, we can compute \begin{equation} \notag \langle \hbJ, \bA_i \rangle = \hat{q}_i \ \text{ for any } \ 1 \leq i \leq a. \end{equation} From Lemma \ref{lem:j0} and $\sum\limits^{a}_{i=1} (q_i - \langle\bJ_1 \bA_i\rangle ) \bA_i \in \eJ(G_1)$, we obtain \[ (G, \hbJ) \sim (G_1, \bJ_1). \] Let $\hbk_1 = (k_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with $\hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}_{\by\rightarrow \by'}}{{\bx}^{\by}}$. From Proposition \ref{prop:craciun2020efficient} and \eqref{eq:gk_g1k1}, we have \[ (G, \bk) \sim (G_1, \bk_1) \sim (G, \hbk_1). \] Finally, let $\bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle)$ and derive that \[ \hat{\Psi} (\hat{\bJ},\bx, \bp) = (\bk, \bq). \] Therefore, we prove the map $\hat{\Psi}$ is surjective. \end{proof} \begin{lemma} \label{lem:hpsi_cts} The map $\hPsi$ in Definition \ref{def:hpsi} is continuous. \end{lemma} \begin{proof} Consider any fixed point $(\hbJ, \bx, \bp) \in \hmJ (G_1,G)\times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$, such that \[ \hPsi (\hbJ, \bx, \bp) = (\bk, \bq). \] From \eqref{def:hpsi_kq} in Definition \ref{def:hpsi}, $\bq$ is defined as \[ \bq = ( \langle \hat{\bJ}, \bA_1 \rangle, \langle \hat{\bJ}, \bA_2 \rangle, \ldots, \langle \hat{\bJ}, \bA_a \rangle ). \] It follows that $\bq$ is a continuous function of $\hbJ$. \smallskip Now it remains to show that $\bk$ is also a continuous function of $(\hbJ,\bx,\bq)$. Recall \eqref{def:hpsi_k} in Definition \ref{def:hpsi}, $\bk$ is defined as \[ (G, \bk) \sim (G_1, \hat{\bk}_1) \ \text{ with } \ \hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}_{\by\rightarrow \by'}}{{\bx}^{\by}}. \] Together with \eqref{def:hpsi_kq}, we get \begin{equation} \label{eq:k_ct_2} \bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle), \end{equation} and for every vertex $\by_0 \in V \cup V_1$, \begin{equation} \label{eq:k_ct_1} \sum_{\by_0 \to \by \in E} k_{\by_0 \to \by} (\by - \by_0) = \sum_{\by_0 \to \by' \in E_1} \frac{\hat{J}_{\by_0 \rightarrow \by'}}{{\bx}^{\by_0}} (\by' - \by_0). \end{equation} Note that $\hbJ$ and $\bx$ are fixed, then \eqref{eq:k_ct_1} can be rewritten as \begin{equation} \label{eq:k_ct_1_1} \sum_{\by_0 \to \by \in E} k_{\by_0 \to \by} (\by - \by_0) = \text{constant}. \end{equation} Assume $\bk'$ is another solution to \eqref{eq:k_ct_1_1}, then \[ (G, \bk) \sim (G, \bk'). \] Using Lemma \ref{lem:d0}, we obtain that \[ \bk' - \bk \in \mD (G). \] Together with the linearity of $\mD (G)$, the solutions to \eqref{eq:k_ct_1_1} form an affine linear subspace. Hence, the tangent space of the solution to \eqref{eq:k_ct_1_1} at $(\bJ, \bx, \bp)$ is $\mD(G)$. Analogously, given fixed $\bp$, the solutions to \eqref{eq:k_ct_2} also form an affine linear subspace, whose tangent space at $(\bJ, \bx, \bp)$ is tangential to \begin{equation} \notag \spn \{\bB_1, \bB_2, \ldots, \bB_b\} = \mD(G). \end{equation} This indicates that two tangent spaces at $(\bJ, \bx, \bp)$ are complementary, and thus intersect transversally~\cite{guillemin2010differential}. From Lemma \ref{lem:hpsi_well_def}, $\bk$ is the unique solution to \eqref{eq:k_ct_2} and \eqref{eq:k_ct_1}. Therefore, we conclude that $\bk$ as the unique intersection point (solution) of two equations \eqref{eq:k_ct_2} and \eqref{eq:k_ct_1} must vary continuously with respect to parameters $(\hbJ, \bx, \bp)$. \end{proof} \section{Continuity of \texorpdfstring{$\hPsi^{-1}$}{hPsi-1}} \label{sec:continuity} In this section, we first introduce the map $\Phi$ (see Definition \ref{def:phi}) and prove $\Phi = \hPsi^{-1}$ is well-defined. Then we show the map $\Phi$ is continuous, i.e. $\hPsi^{-1}$ is also continuous. \begin{definition} \label{def:phi} Given a weakly reversible E-graph $G_1 = (V_1, E_1)$ with its stoichiometric subspace $\mS_{G_1}$. Consider an E-graph $G = (V, E)$ and $\bx_0\in\mathbb{R}^n_{>0}$, define the map \begin{equation} \label{eq:phi} \Phi: \dK(G,G_1)\times \RR^a \rightarrow \hat{\mJ} (G_1,G) \times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \RR^b, \end{equation} such that for $(\bk, \bq) \in \dK(G,G_1)\times \RR^a$, \begin{equation} \notag \Phi (\bk, \bq) := (\hat{\bJ},\bx, \bp), \end{equation} where $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G, \bk)$, and \begin{equation} \label{def:phi_k} (G, \bk) \sim (G_1, \hat{\bk}_1) \ \text{ with } \ \hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}_{\by\rightarrow \by'}}{{\bx}^{\by}}, \end{equation} and \begin{equation} \label{def:phi_kq} \bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle), \ \ \bq = ( \langle \hat{\bJ}, \bA_1 \rangle, \langle \hat{\bJ}, \bA_2 \rangle, \ldots, \langle \hat{\bJ}, \bA_a \rangle ). \end{equation} \end{definition} \medskip \begin{lemma} \label{lem:phi_wd} The map $\Phi$ in Definition \ref{def:phi} is well-defined, and $\Phi = \hPsi^{-1}$ is bijective. \end{lemma} \begin{proof} Assume any point $(\bk^, \bq^) \in \dK(G,G_1)\times \RR^a$. There exists $\bk_1 \in \mK_{\RR} (G_1,G)$ satisfying \begin{equation} \label{eq:phi_wd_1} (G, \bk^) \sim (G_1, \bk_1). \end{equation} From Theorem \ref{thm:cb}, $(G_1, \bk_1)$ has a unique steady state $\bx^* \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$. Further, Remark \ref{rmk:de_ss} shows that $(G, \bk^)$ and $(G_1, \bk_1)$ share the same steady states, thus $\bx^ \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is also the unique steady state of $(G, \bk^)$, i.e. $\bx^$ is well-defined. Moreover, from \eqref{def:phi_kq} we obtain \begin{equation} \label{eq:phi_wd_2} \bp^* = ( \langle \bk^, \bB_1 \rangle, \langle \bk^, \bB_2 \rangle, \ldots, \langle \bk^, \bB_b \rangle), \end{equation} which is well-defined. Since $\bk_1 \in \mK_{\RR} (G_1,G)$, then $(G_1, \bk_1)$ and its steady state $\bx^$ give rise to the complex-balanced flux system $(G_1, \bJ^)$, such that \[ \bJ^ = (J^_{\by\rightarrow \by'})_{\by\rightarrow \by' \in E_1} \in \mJ (G_1,G) \ \text{ with } \ J^_{\by\rightarrow \by'} = k_{1, \by\rightarrow \by'} (\bx^)^{\by}. \] Suppose $\bq^ = (q^_1, q^_2, \ldots, q^_a) \in \RR^a$, we construct the vector $\hbJ^ \in \RR^{\|E\|}$ as \[ \hbJ^* = \bJ^* + \sum\limits^a_{i=1} (q^_i - \langle \bJ^, \bA_i \rangle ) \bA_i \in \hat{\mJ} (G_1,G). \] Note that $\{ \bA_i \}^a_{i=1}$ is an orthonormal basis of $\eJ(G_1)$, together with Lemma \ref{lem:j0}, we obtain \begin{equation} \notag \bq^* = ( \langle \hbJ^, \bA_1 \rangle, \langle \hbJ^, \bA_2 \rangle, \ldots, \langle \hbJ^, \bA_a \rangle ) \ \text{ and } \ (G_1, \hbJ^) \sim (G_1, \bJ^). \end{equation} Using Proposition \ref{prop:craciun2020efficient} and \eqref{eq:phi_wd_1}, we set $\hbk_1 = (\hat{k}_{1, \by\rightarrow \by'})_{\by\rightarrow \by' \in E_1}$ with $\hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}^_{\by\rightarrow \by'}}{{(\bx^)}^{\by}}$ and derive \begin{equation} \notag (G_1, \hat{\bk}_1) \sim (G_1, \bk_1) \sim (G, \bk^). \end{equation} Together with \eqref{eq:phi_wd_2}, we conclude that $(\hbJ^, \bx^, \bp^)$ satisfies \eqref{def:phi_k} and \eqref{def:phi_kq}. Now suppose there exists another $(\hbJ^{}, \bx^{}, \bp^{}) \in \hat{\mJ} (G_1,G)\times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$, which also satisfies \eqref{def:phi_k} and \eqref{def:phi_kq}. From Definition \ref{def:hpsi}, we deduce \begin{equation} \notag \hPsi (\hbJ^, \bx^, \bp^) = \hPsi (\hbJ^{}, \bx^{}, \bp^{*}) = (\bk^, \bq^). \end{equation} Since $\hPsi$ is proved to be bijective in Lemma \ref{lem:hpsi_bijective}, then \begin{equation} \notag (\hbJ^, \bx^, \bp^) = (\hbJ^{}, \bx^{}, \bp^{*}). \end{equation} Thus, we conclude that $\Phi$ is well-defined. \smallskip Next, for any $(\hbJ, \bx, \bp) \in \hat{\mJ} (G_1,G)\times [(\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}] \times \mathbb{R}^b$, suppose that \begin{equation} \label{eq:phi_wd_3} \hPsi (\hbJ, \bx, \bp) = (\bk, \bq) \in \dK(G,G_1)\times \RR^a. \end{equation} From Definition \ref{def:hpsi} and Corollary \ref{cor:hpsi_ss}, together with \eqref{def:phi_k} and \eqref{def:phi_kq}, we have \begin{equation} \label{eq:phi_wd_4} \Phi (\bk, \bq) = (\hbJ, \bx, \bp). \end{equation} This implies $\Phi = \hPsi^{-1}$. Recall that $\hPsi$ is bijective, thus its inverse $\hPsi^{-1}$ is well-defined and bijective. Therefore, we prove the lemma. \end{proof} \begin{lemma} \label{lem:inverse_cts_q} Consider the map $\Phi$ in Definition \ref{def:phi}, suppose any fixed $\bk \in \dK(G,G_1)$ and $\bq_1, \bq_2 \in \RR^a$, then \begin{equation} \label{eq:inverse_cts_q_1} \Phi (\bk, \bq_1) - \Phi (\bk, \bq_2) = \left(\sum\limits^{a}_{i=1} \varepsilon_i \bA_i, \mathbf{0}, \mathbf{0}\right), \end{equation} where $\bq_1 - \bq_2 := (\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_a) \in \RR^a$. \end{lemma} \begin{proof} Given fixed $\bk \in \dK(G,G_1)$, consider any $\bq \in \RR^a$, such that \begin{equation} \notag \Phi (\bk, \bq) = (\hat{\bJ},\bx, \bp). \end{equation} From Definition \ref{def:phi}, $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G, \bk)$. Further, we have \begin{equation} \label{eq:inverse_cts_q_3} (G, \bk) \sim (G_1, \hat{\bk}_1) \ \text{ with } \ \hat{k}_{1, \by\rightarrow \by'} = \frac{\hat{J}_{\by\rightarrow \by'}}{{\bx}^{\by}}, \end{equation} and \begin{equation} \label{eq:inverse_cts_q_4} \bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle), \ \ \bq = ( \langle \hat{\bJ}, \bA_1 \rangle, \langle \hat{\bJ}, \bA_2 \rangle, \ldots, \langle \hat{\bJ}, \bA_a \rangle ). \end{equation} \smallskip Now consider any vector $\bep = (\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_a) \in \RR^a$, it follows that \eqref{eq:inverse_cts_q_1} is equivalent to show the following: \begin{equation} \label{eq:inverse_cts_q_2} \Phi (\bk, \bq + \bep) = (\hat{\bJ} + \sum\limits^{a}_{i=1} \varepsilon_i \bA_i,\bx, \bp). \end{equation} Suppose $\Phi (\bk, \bq + \bep) = (\hbJ^{\bep}, \bx^{\bep}, \bp^{\bep})$. From Definition \ref{def:phi} and Lemma \ref{lem:phi_wd}, $\bx^{\bep}$ is the unique steady state of $(G, \bk)$ in the invariant polyhedron $ (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$. Recall that $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is also the steady state of $(G, \bk)$, thus we have \begin{equation} \label{eq:inverse_cts_q_6} \bx = \bx^{\bep}. \end{equation} Since $\hat{\bJ} \in \hmJ (G_1,G)$ and $\{ \bA_i \}^a_{i=1}$ is an orthonormal basis of $\eJ(G_1)$, we get \[ (G_1, \hat{\bJ}) \sim (G_1, \hat{\bJ} + \sum\limits^{a}_{i=1} \varepsilon_i \bA_i). \] Using Proposition \ref{prop:craciun2020efficient} and \eqref{eq:inverse_cts_q_3}, by setting $\hat{J}_{\by\rightarrow \by'} + \sum\limits^{a}_{i=1} \varepsilon_i \bA_{i, \by\rightarrow \by'} = \hat{k}^{\bep}_{1, \by\rightarrow \by'} \bx^{\by}$, we obtain \begin{equation} \label{eq:inverse_cts_q_5} (G_1, \hat{\bk}^{\bep}_1) \sim (G_1, \hat{\bk}_1) \sim (G, \bk). \end{equation} Under direct computation, for $1 \leq i \leq a$, \begin{equation} \notag \langle \hat{\bJ} + \sum\limits^{a}_{i=1} \varepsilon_i \bA_i, \bA_i \rangle = \langle \hat{\bJ}, \bA_i \rangle + \langle \sum\limits^{a}_{i=1} \varepsilon_i \bA_i, \bA_i \rangle = \langle \hat{\bJ}, \bA_i \rangle + \varepsilon_i. \end{equation} From Lemma \ref{lem:phi_wd} and \eqref{eq:inverse_cts_q_5}, we get \begin{equation} \label{eq:inverse_cts_q_7} \hbJ^{\bep} = \hat{\bJ} + \sum\limits^{a}_{i=1} \varepsilon_i \bA_i. \end{equation} Finally, from Definition \ref{def:phi} and \eqref{eq:inverse_cts_q_4}, it is clear that \begin{equation} \label{eq:inverse_cts_q_8} \bp^{\bep} = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle ) = \bp. \end{equation} Combining Equations~\eqref{eq:inverse_cts_q_6}, \eqref{eq:inverse_cts_q_7} and \eqref{eq:inverse_cts_q_8}, we prove \eqref{eq:inverse_cts_q_2}. \end{proof} Here we present Proposition \ref{prop:inverse_cts_k}, which is the key for the continuity of $\hPsi^{-1}$. \begin{proposition} \label{prop:inverse_cts_k} Consider the map $\Phi$ in Definition \ref{def:phi} and any fixed $\bq \in \RR^a$, then $\Phi (\cdot, \bq)$ is continuous with respect to $\bk$. \end{proposition} To prove Proposition~\ref{prop:inverse_cts_k}, we need to show Lemmas \ref{lem:key_1} - \ref{lem:key_3} and Proposition \ref{lem:key_4}. The following is the overview of the process. First, Lemma \ref{lem:key_1} shows that if two reaction rate vectors in $\dK (G, G_1)$ are close enough, then there exist two reaction rate vectors (dynamically equivalent respectively) in $\mK (G_1, G_1)$ such that their distance can be controlled. Second, in Lemma \ref{lem:key_2} we show that given a complex-balanced rate vector $\bk_1 \in \mK (G_1)$, there exists a neighborhood around $\bk_1$ of $\RR^{E_1}_{>0}$, in which the steady states of the system associated with the rate constants vary continuously. Combining Lemma \ref{lem:key_1} with \ref{lem:key_2}, we prove in Lemma \ref{lem:key_3} that given a reaction rate vector $\bk \in \dK (G, G_1)$, there exists an open neighborhood $\bk \in U \subset \RR^{E}$, such that the steady states of the system associated with the rate vectors in $U$ vary continuously. Finally, in Proposition \ref{lem:key_4} we prove that given a complex-balanced rate vector $\bk^ \in \mK (G_1, G_1)$, for any sequence $\bk_i \to \bk^$ in $\mK (G_1, G_1)$, there exists another sequence of reaction rate vectors (dynamically equivalent respectively) $\hbk_i \to \bk^$ in $\RR^{E_1}$, and all associated fluxes from reaction rate vectors have the same projections on $\eJ (G_1)$. \medskip \begin{lemma} \label{lem:key_1} Let $\bk \in \dK (G,G_1)$. Then we have the following: \begin{enumerate}[label=(\alph)] \item There exists $\bk_1 \in \mK (G_1)$ satisfying $(G, \bk) \sim (G_1, \bk_1)$. \item There exist constants $\varepsilon = \varepsilon (\bk) > 0$ and $C = C (\bk) > 0$, such that for any $\hbk \in \dK (G,G_1)$ with $\\| \hbk - \bk \\| \leq \varepsilon$, there exists $\hbk_1 \in \mK (G_1,G_1)$ that satisfies \begin{enumerate}[label=(\roman)] \item $\\|\hbk_1 - \bk_1 \\| \leq C \varepsilon $. \item $(G,\hbk) \sim (G_1, \hbk_1)$. \end{enumerate} \end{enumerate} \end{lemma} \begin{proof} For part $(a)$, from Definitions \ref{def:mas_realizable} and \ref{def:de_realizable}, given $\bk \in \dK (G,G_1)$, the system $(G, \bk)$ is disguised toric on $G_1$, that is, there exists $\bk_1 \in \mK_{\RR} (G_1, G) \subset \mK (G_1)$ with $(G, \bk) \sim (G_1, \bk_1)$. \smallskip Now we prove part $(b)$.\\ \textbf{Step 1: } Let $\by \in G \cup G_1$ be a fixed vertex and consider the following vector space: \begin{equation} \notag W_{\by} = \spn \{ \by' - \by: \by \rightarrow \by' \in G_1 \}. \end{equation} Let $d(\by) = \dim (W_{\by})$. Then there exists an orthogonal basis of $W_{\by}$ denoted by: \begin{equation} \label{eq:key_1_1} \{ \bw_1, \bw_2, \ldots, \bw_{d (\by)} \}. \end{equation} For each $\bw_i$ in \eqref{eq:key_1_1}, there exist positive $\{ c_{i, \by \rightarrow \by'} \}_{\by \rightarrow \by' \in G_1}$, that satisfy \begin{equation} \label{eq:key_1_2} \bw_i = \sum\limits_{\by \rightarrow \by' \in G_1} c_{i, \by \rightarrow \by'} (\by' - \by). \end{equation} Let $\hbk \in \dK (G,G_1)$. From Definition \ref{def:de_realizable}, $\sum\limits_{\by \rightarrow \tilde{\by} \in G} \hbk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by)$ is realizable on $G_1$ at the vertex $\by \in G \cup G_1$. This implies that \begin{equation} \label{eq:key_1_3} \sum\limits_{\by \rightarrow \tilde{\by} \in G} \hbk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by) \in W_{\by}. \end{equation} Since $\bk \in \dK (G,G_1)$, together with Equation~\eqref{eq:key_1_3}, we obtain \begin{equation} \label{eq:key_1_Delta} \Delta_{\by} (\hbk, \bk) := \sum\limits_{\by \rightarrow \tilde{\by} \in G} ( \hbk_{\by \rightarrow \tilde{\by}} - \bk_{\by \rightarrow \tilde{\by}}) (\tilde{\by} - \by) \in W_{\by}. \end{equation} Assume that $\\| \hbk - \bk \\| \leq \varepsilon$. Consider all reaction vectors in $G$ and let $m = \max\limits_{\by \rightarrow \tilde{\by} \in G} \\| \tilde{\by} - \by \\|$, then there exists a constant $C_1 = m \|E\|$, such that \[ \\| \Delta_{\by} (\hbk, \bk) \\| \leq \sum\limits_{\by \rightarrow \tilde{\by} \in G} m \varepsilon = C_1 \varepsilon. \] On the other side, from \eqref{eq:key_1_1}, $\Delta_{\by} (\hbk, \bk)$ can be expressed as \begin{equation} \label{eq:key_1_4} \Delta_{\by} (\hbk, \bk) = \sum\limits^{d(\by)}_{i=1} \delta_i \bw_i \ \text{ with } \ \delta_i \in \RR. \end{equation} Using \eqref{eq:key_1_4} and the orthogonal basis in \eqref{eq:key_1_1}, for any $1 \leq i \leq d (\by)$, \begin{equation} \label{eq:key_1_5} \| \delta_i \| \leq \\| \Delta_{\by} (\hbk, \bk) \\| \leq C_1 \varepsilon. \end{equation} Inputting \eqref{eq:key_1_2} into \eqref{eq:key_1_4}, we get \begin{equation} \label{eq:key_1_6} \Delta_{\by} (\hbk, \bk) = \sum\limits^{d(\by)}_{i=1} \delta_i \big( \sum\limits_{\by \rightarrow \by' \in G_1} c_{i, \by \rightarrow \by'} (\by' - \by) \big) = \sum\limits_{\by \rightarrow \by' \in G_1} \big( \sum\limits^{d(\by)}_{i=1} \delta_i c_{i, \by \rightarrow \by'} \big) (\by' - \by). \end{equation} From \eqref{eq:key_1_5} and \eqref{eq:key_1_6}, there exists a constant $C_2$, such that for any $\by \rightarrow \by' \in G_1$, \begin{equation} \label{eq:key_1_7} \big\| \hat{c}_{\by \rightarrow \by'} := \sum\limits^{d(\by)}_{i=1} \delta_i c_{i, \by \rightarrow \by'} \big\| \leq C_2 \varepsilon. \end{equation} Then we construct $\hbk_1$ as follows: \begin{equation} \label{eq:key_1_8} \hbk_{1, \by \rightarrow \by'} := \bk_{1, \by \rightarrow \by'} + \hat{c}_{\by \rightarrow \by'} \ \text{ for any } \ \by \rightarrow \by' \in G_1. \end{equation} Consider all reaction vectors in $G_1$, together with \eqref{eq:key_1_7}, we derive \begin{equation} \label{eq:key_1_estimate} \\| \hbk_1 - \bk_1 \\| \leq \sum\limits_{\by \rightarrow \by' \in G_1} \|\hat{c}_{\by \rightarrow \by'}\| \leq \sum\limits_{\by \rightarrow \by' \in G_1} C_2 \varepsilon \leq C_2 \|E_1\| \varepsilon. \end{equation} Similarly, we can go through all vertices in $G \cup G_1$, and take the above steps to update $\hbk_1$. For every vertex, we can derive an estimate similar to \eqref{eq:key_1_estimate}. Collecting the estimates on all vertices, we can find a constant $C$, such that \[ \\| \hbk_1 - \bk_1 \\| \leq C \varepsilon \ \text{ for any } \ \\| \hbk - \bk \\| \leq \varepsilon. \] \textbf{Step 2: } We claim that there exists a sufficiently small constant $\varepsilon = \varepsilon (\bk) > 0$, such that for any $\hbk$ with $\\| \hbk - \bk \\| \leq \varepsilon$, then $\hbk_1$ defined in \eqref{eq:key_1_8} satisfies \begin{equation} \label{eq:key_1_claim} (G, \hbk) \sim (G_1, \hbk_1) \ \text{ and } \ \hbk_1 \in \mK (G_1,G_1). \end{equation} Recall \eqref{eq:key_1_3} and \eqref{eq:key_1_Delta}, at vertex $\by \in G \cup G_1$, \begin{equation} \label{eq:key_1_9} \Delta_{\by} (\hbk, \bk) = \sum\limits_{\by \rightarrow \tilde{\by} \in G} \hbk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by) - \sum\limits_{\by \rightarrow \tilde{\by} \in G} \bk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by). \end{equation} On the other hand, from \eqref{eq:key_1_6}-\eqref{eq:key_1_8}, at vertex $\by \in G \cup G_1$, \begin{equation} \label{eq:key_1_10} \Delta_{\by} (\hbk, \bk) = \sum\limits_{\by \rightarrow \by' \in G_1} \hbk_{1, \by \rightarrow \by'} (\by' - \by) - \sum\limits_{\by \rightarrow \by' \in G_1} \bk_{1, \by \rightarrow \by'} (\by' - \by). \end{equation} Note that $(G, \bk) \sim (G_1, \bk_1)$ implies that, at vertex $\by \in G \cup G_1$, \[ \sum\limits_{\by \rightarrow \tilde{\by} \in G} \bk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by) = \sum\limits_{\by \rightarrow \by' \in G_1} \bk_{1, \by \rightarrow \by'} (\by' - \by). \] Together with \eqref{eq:key_1_9} and \eqref{eq:key_1_10}, we have, at vertex $\by \in G \cup G_1$, \begin{equation} \sum\limits_{\by \rightarrow \tilde{\by} \in G} \hbk_{\by \rightarrow \tilde{\by}} (\tilde{\by} - \by) = \sum\limits_{\by \rightarrow \by' \in G_1} \hbk_{1, \by \rightarrow \by'} (\by' - \by). \end{equation} Hence, we derive $(G, \hbk) \sim (G_1, \hbk_1)$. Moreover, since $\hbk \in \dK (G,G_1)$, there exists $\hbk^* \in \mK (G_1)$ with $(G, \hbk) \sim (G_1, \hbk^)$, and thus \[ (G_1, \hbk_1) \sim (G_1, \hbk^). \] Recall that $\bk_1 \in \mK (G_1) \subset \RR^{E_1}_{>0}$, together with \eqref{eq:key_1_estimate}, there must exist a constant $\varepsilon = \varepsilon (\bk) > 0$, such that for any $\hbk$ with $\\| \hbk - \bk \\| \leq \varepsilon$, we have $\hbk_1 \in \RR^{E_1}_{>0}$. Therefore, we obtain $\hbk_1 \in \mK (G_1,G_1)$ and prove the claim. \end{proof} \begin{lemma} \label{lem:key_2} Suppose $\bx_0 \in \mathbb{R}^n_{>0}$ and $\bk_1 \in \mK (G_1)$, then there exists an open set $U \subset \RR^{E_1}_{>0}$ containing $\bk_1$, such that there exists a unique continuously differentiable function \begin{equation} \label{lem:key_2_1} T : U \rightarrow (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}. \end{equation} such that for any $\hbk \in U$, \begin{equation} \label{lem:key_2_2} T (\hbk) = \hbx, \end{equation} where $\hbx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G_1, \hbk)$. \end{lemma} \begin{proof} Given $\bx_0 \in \mathbb{R}^n_{>0}$ and $\bk_1 \in \mK (G_1)$, Theorem \ref{thm:cb} shows the system $(G_1, \bk_1)$ has a unique steady state $\bx^* \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. Consider the system $(G_1, \bk_1)$ as follows: \begin{equation} \label{eq:key_2_0} \frac{d\bx}{dt} = \bf (\bk_1, \bx) := (\bf_1, \bf_2, \ldots, \bf_n)^{\intercal} = \sum_{\by_i \rightarrow \by_j \in E_1} k_{1, \by_i \rightarrow \by_j} \bx^{\by_i}(\by_j - \by_i). \end{equation} Suppose $\dim (\mS_{G_1}) = s \leq n$. This implies that there exist exactly $s$ linearly independent components among $\bf (\bk_1, \bx)$. Without loss of generality, we assume that $\{\bf_1, \ldots, \bf_s \}$ are linearly independent components, and every $\bf_i$ with $s+1 \leq i \leq n$ can be represented as a linear combination of $\{\bf_i \}^{s}_{i=1}$. Using Theorem~\ref{thm:jacobian}, we obtain that \begin{equation} \notag \ker \Big( \big[ \frac{\partial \bf_i}{ \partial \bx_j} \big]_{1 \leq i, j \leq n} \big\|_{\bx = \bx^} \Big) = \mS^{\perp}_{G_1}. \end{equation} Together with the linear dependence among $\{ \bf_i (\bx) \}^{n}_{i=1}$, we derive \begin{equation} \label{eq:key_2_1} \ker \Big( \big[ \frac{\partial \bf_i}{ \partial \bx_j} \big]_{1 \leq i \leq s, 1 \leq j \leq n} \big\|_{\bx = \bx^} \Big) = \mS^{\perp}_{G_1}. \end{equation} Consider the orthogonal complement $\mS^{\perp}_{G_1}$ to the stoichiometric subspace in $\mathbb{R}^n$, which admits an orthonormal basis given by \[ \{\bv_1, \bv_2, \ldots, \bv_{n-s} \}. \] Now we construct a system of $n$ equations $\bg (\bk, \bx) = (\bg_1, \bg_2, \ldots, \bg_n )^{\intercal}$ as follows: \begin{equation} \label{eq:key_2_2} \bg_i (\bk, \bx) = \begin{cases} \bf_i (\bk, \bx), & \text{ for } 1 \leq i \leq s, \\[5pt] \bx \cdot \bv_{i-s} - \bx_0 \cdot \bv_{i-s}, & \text{ for } s+1 \leq i \leq n. \end{cases} \end{equation} From \eqref{eq:key_2_0}, we can check that $\bg (\bk, \bx) = \mathbf{0}$ if and only if $\bx \in \bx_0 + \mS_{G_1}$ is the steady state of the system $(G_1, \bk)$. Thus, $(\bk_1, \bx^)$ can be considered as a solution to $\bg (\bk, \bx) = \mathbf{0}$, that is, $\bg (\bk_1, \bx^) = \mathbf{0}$. Computing the Jacobian matrix of $\bg (\bk, \bx)$ as in Equation~\eqref{eq:key_2_2}, we get \begin{equation} \notag \mathbf{J}_{\bg, \bx} = \begin{pmatrix} \big[ \frac{\partial \bf_i}{ \partial \bx_j} \big]_{1 \leq i \leq s, 1 \leq j \leq n} \\[5pt] \bv_1 \\ \ldots \\ \bv_{n-s} \end{pmatrix}. \end{equation} From~\eqref{eq:key_2_1}, we have \[ \ker \big( \mathbf{J}_{\bg, \bx} \|_{\bk = \bk_1, \bx = \bx^} \big) \subseteq \mS^{\perp}_{G_1}. \] Since the last $n-s$ rows of $\mathbf{J}_{\bg} (\bx)$, $\{\bv_1, \bv_2, \ldots, \bv_{n-s} \}$, is a orthonormal basis of $\mS^{\perp}_{G_1}$, we derive \begin{equation} \label{eq:key_2_3} \det \big( \mathbf{J}_{\bg, \bx} \|_{\bk = \bk_1, \bx = \bx^} \big) \neq 0. \end{equation} Hence, the Jacobian matrix $\mathbf{J}_{\bg, \bx}$ is invertible at $(\bk, \bx) = (\bk_1, \bx^)$. Further, note that $\bg (\bk, \bx)$ is continuously differentiable. Using the implicit function theorem, for any $\hbk \in U$, we have \begin{equation} \notag T (\hbk) = \hbx, \end{equation} where $\hbx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G_1, \hbk)$. \end{proof} \begin{lemma} \label{lem:key_3} Suppose $\bx_0\in\mathbb{R}^n_{>0}$ and $\bk \in \dK (G,G_1)$, then there exists an open set $U \subset \dK (G,G_1)$ containing $\bk$, such that there exists a unique continuous function \begin{equation} \label{eq:key_3_1} h : U \rightarrow (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}. \end{equation} such that for any $\hbk \in U$, \begin{equation} \label{eq:key_3_2} h (\hbk) = \hbx, \end{equation} where $\hbx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G, \hbk)$. \end{lemma} \begin{proof} Given $\bk \in \dK (G, G_1)$ and $\bx_0 \in \mathbb{R}^n_{>0}$, there exists $\bk_1 \in \mK (G_1)$ such that \[ (G, \bk) \sim (G_1, \bk_1). \] Theorem \ref{thm:cb} shows the system $(G_1, \bk_1)$ has a unique steady state $\bx^ \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. Since $(G, \bk) \sim (G_1, \bk_1)$, $\bx^* \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$ is also the unique steady state of the system $(G, \bk)$. Analogously, for any $\hbk \in \dK (G,G_1)$, it has a unique steady state of the system $(G, \hbk)$ in $(\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. Thus, the function $h$ in \eqref{eq:key_3_1}-\eqref{eq:key_3_2} is well-defined. It remains to prove that there exists an open set $U \subset \dK (G, G_1)$ containing $\bk$ and $h$ is continuous with respect to the domain $U$. From Lemma~\ref{lem:key_2}, there exists an open set $U_1 \subset \RR^{E_1}_{>0}$ containing $\bk_1$, such that there exists a unique continuously differentiable function \begin{equation} \label{eq:key_3_4} T : U_1 \rightarrow (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}. \end{equation} such that for any $\hbk \in U_1$, \begin{equation} \notag T (\hbk) = \hbx, \end{equation} where $\hbx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G_1, \hbk)$. Using \eqref{eq:key_3_4}, we can find a constant $\varepsilon_1 = \varepsilon_1 (\bk)$ such that \begin{equation} \label{eq:key_3_B} B = \{ \bk^* \in \RR^{E_1}_{>0}: \\|\bk^* - \bk_1 \\| \leq \varepsilon_1 \} \subseteq U_1. \end{equation} Hence, it is clear that $T$ is continuous with respect to the domain $B$. On the other hand, from Lemma \ref{lem:key_1}, there exist $\varepsilon = \varepsilon (\bk) > 0$ and $C = C (\bk) > 0$, such that for any $\hbk \in \dK (G,G_1)$ with $\\| \hbk - \bk \\| \leq \varepsilon$, there exists $\hbk_1 \in \mK (G_1,G_1)$ satisfying \begin{equation} \label{eq:key_3_3} \\|\hbk_1 - \bk_1 \\| \leq C \varepsilon \ \text{ and } \ (G,\hbk) \sim (G_1, \hbk_1). \end{equation} Now pick $\varepsilon_2 = \min ( \varepsilon, \varepsilon_1 / C)$, and consider the following set: \begin{equation} \notag U := \{ \bk^* \in \RR^{E}_{>0}: \\|\bk^* - \bk \\| < \varepsilon_2 \} \ \cap \ \dK (G,G_1). \end{equation} Using~\eqref{eq:key_3_3}, we have that for any $\bk^* \in U$, there exists $\bk^_1 \in \mK (G_1,G_1)$ such that \begin{equation} \label{eq:key_3_5} \\| \bk^_1 - \bk_1 \\| \leq C \varepsilon_2 = \varepsilon_1 \ \text{ and } \ (G, \bk^) \sim (G_1, \bk^_1). \end{equation} From \eqref{eq:key_3_B}, this shows that $\bk^_1 \in B$. Further, from \eqref{eq:key_3_4} and \eqref{eq:key_3_3}, we obtain \[ h (\bk^) = T (\bk^_1) \] Since $T$ is continuous with respect to the domain $B$, together with \eqref{eq:key_3_5} and $\bk^_1 \in B$, we conclude that $h$ is continuous on $U$. \end{proof} \begin{proposition} \label{lem:key_4} Suppose $\bx_0 \in \RR^n_{>0}$ and $\bk^* \in \mK (G_1) \subset \mK (G_1,G_1)$. For any $\bk \in \mK (G_1,G_1)$, then we have the following: \begin{enumerate}[label=(\alph)] \item The system $(G_1, \bk^)$ has a unique steady state $\bx^* \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. \item The system $(G_1, \bk)$ has a unique steady state $\bx \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. \item Consider the steady state $\bx^$ in part $(a)$ and $\bx$ obtained in part $(b)$. Then there exists a unique $\hbk \in \RR^{E_1}$, such that \begin{enumerate}[label=(\roman)] \item \label{lem:key_4_a} $(G_1, \bk) \sim (G_1, \hbk)$. \item\label{lem:key_4_b} $\hbJ := (\hat{k}_{\by \to \by'} \bx^{\by})_{\by \to \by' \in E_1} \in \hat{\mathcal{J}} (G_1)$. \item \label{lem:key_4_c} $\langle \hbJ, \bA_i \rangle = \langle \bJ^, \bA_i \rangle$ for any $1 \leq i \leq a$, where $\bJ^ := (k^_{\by \to \by'} (\bx^)^{\by})_{\by \to \by' \in E_1}$. \end{enumerate} \item For any sequence $\{ \bk_i \}^{\infty}_{i = 1}$ in $\mK (G_1,G_1)$ converging to $\bk^$, there exist a unique corresponding sequence $\{ \hbk_i \}^{\infty}_{i = 1}$ obtained from part $(c)$. Moreover, the sequence $\{ \hbk_i \}^{\infty}_{i = 1}$ satisfies \begin{equation} \notag \hbk_i \to \bk^ \ \text{ as } \ i \to \infty. \end{equation} \end{enumerate} \end{proposition} \begin{proof} For part (a), since $\bk^* \in \mK (G_1)$, Theorem \ref{thm:cb} shows that the system $(G_1, \bk^)$ has a unique steady state $\bx^ \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. \smallskip For part (b), given $\bk \in \mK (G_1,G_1)$, there exists some $\bk' \in \mK (G_1)$, such that \begin{equation} \label{eq:key_4_3} (G_1, \bk) \sim (G_1, \bk'). \end{equation} Thus, by Theorem \ref{thm:cb}, the systems $(G_1, \bk)$ and $(G_1, \bk')$ share a unique steady state in $(\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$, denoted by $\bx$. \smallskip For part (c), define $\bJ' := (k'_{\by \to \by'} \bx^{\by})_{\by \to \by' \in E_1}$, then we construct a flux vector on $G_1$ as follows: \begin{equation} \label{eq:key_4_4} \hbJ := \bJ' + \sum\limits^{a}_{i=1} (\langle \bJ^, \bA_i \rangle - \langle \bJ', \bA_i \rangle) \bA_i. \end{equation} Under direct computation, we have \begin{equation} \label{eq:key_4_5} \langle \hbJ, \bA_i \rangle = \langle \bJ^, \bA_i \rangle \ \text{ for any } \ 1 \leq i \leq a. \end{equation} Note that $\bk' \in \mK (G_1)$ and $\{\bA_i \}^{a}_{i=1} \in \eJ(G) \subset \hat{\mathcal{J}} (G_1)$, then \eqref{eq:key_4_4} show that \begin{equation} \label{eq:key_4_5.5} \bJ' \in \mathcal{J} (G_1) \ \text{ and } \ \hbJ \in \hat{\mathcal{J}} (G_1). \end{equation} Consider the flux vector $\bJ := (k_{\by \to \by'} \bx^{\by})_{\by \to \by' \in E_1}$. Using Proposition \ref{prop:craciun2020efficient} and \eqref{eq:key_4_3}, we deduce \begin{equation} \notag (G_1, \bJ) \sim (G_1, \bJ'). \end{equation} From Lemma \ref{lem:j0}, this shows $\bJ' - \bJ \in \mD (G_1)$. Together with \eqref{eq:key_4_4}, we get \begin{equation} \notag \hbJ - \bJ \in \mD (G_1). \end{equation} Hence, we rewrite $\hbJ$ as \begin{equation} \label{eq:key_4_6} \hbJ = \bJ + \bv \ \text{ with } \ \bv \in \mD (G_1). \end{equation} Now we set the reaction rate vector as \begin{equation} \label{eq:key_4_6.5} \hbk := ( \frac{\hbJ}{\bx^{\by}} )_{\by \to \by' \in E_1} \in \RR^{E_1}. \end{equation} Using Proposition \ref{prop:craciun2020efficient} and \eqref{eq:key_4_6}, we obtain $(G_1, \bk) \sim (G_1, \hbk)$. Together with \eqref{eq:key_4_5} and \eqref{eq:key_4_5.5}, we derive that the reaction rate vector $\hbk$ satisfies conditions \ref{lem:key_4_a}, \ref{lem:key_4_b} and \ref{lem:key_4_c}. We now show the uniqueness of the vector $\hbk$. Suppose there exists another reaction rate vector $\hbk_1$ satisfying conditions \ref{lem:key_4_a}-\ref{lem:key_4_c}. From the condition \ref{lem:key_4_a}, we have \[ (G_1, \hbk) \sim (G_1, \hbk_1). \] From the condition \ref{lem:key_4_b}, we get \[ \hbJ_1 := (\hat{k}_{1, \by \to \by'} \bx^{\by})_{\by \to \by' \in E_1} \in \hat{\mathcal{J}} (G_1). \] Then Proposition \ref{prop:craciun2020efficient} and Lemma \ref{lem:j0} show \[ (G_1, \hbJ) \sim (G_1, \hbJ_1) \ \text{ and } \ \hbJ_1 - \hbJ \in \eJ (G_1). \] Using the condition \ref{lem:key_4_c}, we obtain \[ \langle \hbJ, \bA_i \rangle = \langle \hbJ_1, \bA_i \rangle \ \text{ for any } \ 1 \leq i \leq a. \] Since $\{\bA_i \}^{a}_{i=1}$ is an orthonormal basis of the subspace $\eJ(G)$, this implies that \[ \hbJ_1 - \hbJ \in \big( \eJ (G_1) \big)^{\perp}. \] Hence, $\hbJ_1 - \hbJ = \mathbf{0}$ and $\hbk_1 = \hbk$. Therefore, we conclude the uniqueness. \smallskip For part (d), we will prove it in a sequence of three steps. \smallskip \textbf{Step 1: } Assume a sequence of reaction rate vectors $\bk_i \in \mK (G_1,G_1)$ with $i \in \mathbb{N}$, such that \[ \bk_i \to \bk^* \ \text{ as } \ i \to \infty. \] Analogously, there exists some $\bk'_i \in \mK (G_1)$, such that $(G_1, \bk_i) \sim (G_1, \bk'_i)$. Moreover, two systems $(G_1, \bk_i)$ and $(G_1, \bk'_i)$ share a unique steady state $\bx^i \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$. Follow the steps in \eqref{eq:key_4_3}-\eqref{eq:key_4_5}, we obtain the corresponding sequences of flux vector as follows: \begin{equation} \begin{split} \label{eq:key_4_7} & \bJ_i := (k_{i, \by \to \by'} (\bx^i)^{\by})_{\by \to \by' \in E_1} \ \text{ with } \ i \in \mathbb{N}, \\& \bJ'_i := (k'_{i, \by \to \by'} (\bx^i)^{\by})_{\by \to \by' \in E_1} \ \text{ with } \ i \in \mathbb{N}. \end{split} \end{equation} and \begin{equation} \label{eq:key_4_8} \hbJ_i := \bJ'_i + \sum\limits^{a}_{j=1} (\langle \bJ^, \bA_j \rangle - \langle \bJ'_i, \bA_j \rangle) \bA_j \ \text{ with } \ i \in \mathbb{N}. \end{equation} Under direct computation, for any $i \in \mathbb{N}$, \begin{equation} \label{eq:key_4_8.5} \langle \hbJ_i, \bA_j \rangle = \langle \bJ^, \bA_j \rangle \ \text{ for any } \ 1 \leq j \leq a, \end{equation} and similar from \eqref{eq:key_4_5.5}, we have \begin{equation} \label{eq:key_4_12} \hbJ_i \in \hat{\mathcal{J}} (G_1) \ \text{ for any } \ i \in \mathbb{N}. \end{equation} Using Proposition \ref{prop:craciun2020efficient} and $(G_1, \bk_i) \sim (G_1, \bk'_i)$, we deduce \begin{equation} \notag (G_1, \bJ_i) \sim (G_1, \bJ'_i) \ \text{ for any } \ i \in \mathbb{N}. \end{equation} From Lemma \ref{lem:j0}, together with \eqref{eq:key_4_8}, we get \begin{equation} \notag \hbJ_i - \bJ_i \in \mD (G_1) \ \text{ for any } \ i \in \mathbb{N}. \end{equation} Thus, for any $i \in \mathbb{N}$, $\hbJ_i$ can be expressed as \begin{equation} \label{eq:key_4_9} \hbJ_i = \bJ_i + \bv^i \ \text{ with } \ \bv^i \in \mD (G_1). \end{equation} On the other hand, using Lemma \ref{lem:key_2}, together with $\bk_i \to \bk^$ as $i \to \infty$, we have \begin{equation} \notag \bx^i \to \bx^ \ \text{ as } \ i \to \infty. \end{equation} Combining with \eqref{eq:key_4_7}, we derive that \begin{equation} \label{eq:key_4_10} \bJ_i \to \bJ^* \ \text{ as } \ i \to \infty. \end{equation} \smallskip \textbf{Step 2: } Now we claim that \begin{equation} \label{eq:key_4_13} \\| \bv^i \\|_{\infty} \to 0 \ \text{ as } \ i \to \infty. \end{equation} We prove this by contradiction. Suppose not, w.l.o.g. there exists a subsequence $\{\bv^{i_l} \}^{\infty}_{l=1}$, such that for any $l \in \mathbb{N}$, \begin{equation} \notag \\| \bv^{i_l} \\|_{\infty} \geq 1. \end{equation} Then we consider the sequence $\{ \bw^l \}^{\infty}_{l=1}$ as follows: \begin{equation} \label{eq:key_4_14} \bw^{l} = \frac{\bv^{i_l}}{\\| \bv^{i_l} \\|_{\infty}} \ \text{ with } \ l \in \mathbb{N}. \end{equation} It is clear that $\\| \bw^{l} \\|_{\infty} = 1$ for any $l \in \mathbb{N}$. From the Bolzano–Weierstrass theorem, there exists a subsequence $\{ \bw^{l_j} \}^{\infty}_{j=1}$, such that \begin{equation} \notag \bw^{l_j} \to \bw^* \ \text{ as } \ j \to \infty. \end{equation} Recall from \eqref{eq:key_4_9} and \eqref{eq:key_4_14}, we have for any $j \in \mathbb{N}$, \begin{equation} \label{eq:key_4_15} \bw^{l_j} = \frac{\bv^{i_{l_j}}}{\\| \bv^{i_{l_j}} \\|_{\infty}} = \frac{1}{\\| \bv^{i_{l_j}} \\|_{\infty}} \big( \hbJ_{i_{l_j}} - \bJ_{i_{l_j}} \big). \end{equation} Since $\bv^i \in \mD (G_1)$, together with $\\| \bv^{i_l} \\|_{\infty} \geq 1$, we obtain that \[ \bw^{l_j} \in \mD (G_1). \] Note that $\mD (G_1)$ is a linear subspace of finite dimension. Therefore, $\bw^{l_j} \to \bw^$ implies \begin{equation} \label{eq:key_4_16} \bw^ \in \mD (G_1). \end{equation} Let $\bz \in \big( \hat{\mathcal{J}} (G_1) \big)^{\perp}$. From \eqref{eq:key_4_12}, we have for any $j \in \mathbb{N}$, \begin{equation} \label{eq:key_4_17} \langle \hbJ_{i_{l_j}}, \bz \rangle = 0. \end{equation} From \eqref{eq:key_4_10} and $\bJ \in \mathcal{J} (G_1)$, we obtain \begin{equation} \label{eq:key_4_18} \langle \bJ_{i_{l_j}}, \bz \rangle \to \langle \bJ, \bz \rangle = 0 \ \text{ as } \ j \to \infty. \end{equation} Using \eqref{eq:key_4_15}, \eqref{eq:key_4_17} and \eqref{eq:key_4_18}, together with $\\| \bv^{i_l} \\|_{\infty} \geq 1$ and $\bw^{l_j} \to \bw^$, we derive \begin{equation} \notag \langle \bw^{l_j}, \bz \rangle \to \langle \bw^, \bz \rangle = 0. \end{equation} Since $\bz$ is arbitrary in $\big( \hat{\mathcal{J}} (G_1) \big)^{\perp}$, this shows $\bw^* \in \hat{\mathcal{J}} (G_1)$. Together with \eqref{eq:key_4_16}, we get \begin{equation} \label{eq:key_4_19} \bw^* \in \eJ (G_1). \end{equation} Recall that $\{\bA_i \}^{a}_{i=1}$ is an orthonormal basis of the subspace $\eJ(G)$. Without loss of generality, we pick $\bA_1 \in \eJ(G)$. From \eqref{eq:key_4_8.5} and \eqref{eq:key_4_10}, we get \begin{equation} \notag \langle \hbJ_{i_{l_j}} - \bJ_{i_{l_j}}, \bA_1 \rangle = \langle \bJ^, \bA_1 \rangle - \langle \bJ_{i_{l_j}}, \bA_1 \rangle \to 0 \ \text{ as } \ j \to \infty. \end{equation} Together with $\\| \bv^{i_l} \\|_{\infty} \geq 1$ and $\bw^{l_j} \to \bw^$, we derive \begin{equation} \notag \langle \bw^{l_j}, \bA_1 \rangle \to \langle \bw^, \bA_1 \rangle = 0. \end{equation} Analogously, we can get $\langle \bw^, \bA_j \rangle = 0$ for any $1 \leq j \leq a$. This shows that \begin{equation} \label{eq:key_4_20} \bw^* \in \big( \eJ (G_1) \big)^{\perp}. \end{equation} Combining \eqref{eq:key_4_19} with \eqref{eq:key_4_20}, we conclude that $\bw^* = \mathbf{0}$. Since $\\| \bw^{l} \\|_{\infty} = 1$ for any $l \in \mathbb{N}$, this contradicts with $\bw^{l_j} \to \bw^$ as $j \to \infty$. Therefore, we prove the claim. \smallskip \textbf{Step 3: } Using \eqref{eq:key_4_9}, \eqref{eq:key_4_10} and \eqref{eq:key_4_13}, we derive that \begin{equation} \label{eq:key_4_21} \hbJ_i = \bJ_i + \bv^i \to \bJ^ \ \text{ as } \ i \to \infty. \end{equation} Since $\bJ \in \mathcal{J} (G_1) \subset \RR^{E_1}_{>0}$, there exists sufficiently large $N$, such that \begin{equation} \notag \hbJ_i \in \RR^{E_1}_{>0} \ \text{ for any } \ i > N. \end{equation} Together with \eqref{eq:key_4_12} and Remark \ref{rmk:hat_j_g1_g}, we obtain that \[ \hbJ_i \in \hat{\mathcal{J}} (G_1) \cap \RR^{\|E_1\|}_{>0} = \mathcal{J} (G_1) \ \text{ for any } \ i > N. \] Following \eqref{eq:key_4_6.5}, we set $\{ \hbk_i\}^{\infty}_{i=1}$ as follows: \begin{equation} \label{eq:key_4_22} \hbk_i := \big( \frac{\hat{J}_{i, \by \to \by'} }{(\bx^i)^{\by}} \big)_{\by \to \by' \in E_1} \ \text{ with } \ i \in \mathbb{N}. \end{equation} Note that $\bx^i \in (\bx_0 + \mS_{G_1}) \cap \mathbb{R}^n_{>0}$ and $\hbJ_i \in \mathcal{J} (G_1)$ for any $i > N$, we get \begin{equation} \notag \hbk_i \in \mK (G_1) \ \text{ for any } \ i > N. \end{equation} Using \eqref{eq:key_4_9} and Proposition \ref{prop:craciun2020efficient}, we derive \begin{equation} \notag (G_1, \bk_i) \sim (G_1, \hbk_i). \end{equation} Finally, using $\hbJ_i \to \bJ^$ and $\bx^i \to \bx^$, together with $\bJ^* = (k^_{\by \to \by'} (\bx^)^{\by})_{\by \to \by' \in E_1}$, we have \begin{equation} \hbk_i \to \bk^* \ \text{ as } \ i \to \infty. \end{equation} Therefore, we conclude the proof of this Proposition. \end{proof} Now we are ready to prove Proposition~\ref{prop:inverse_cts_k}. \begin{proof}[Proof of Proposition \ref{prop:inverse_cts_k}] Given fixed $\bq = (q_1, q_2, \ldots, q_a) \in \RR^a$, consider $\bk \in \dK(G,G_1)$ such that \begin{equation} \notag \Phi (\bk, \bq) = (\hat{\bJ},\bx, \bp). \end{equation} Follow definition, there exists $\bk_1 \in \mK (G_1) \subset \mK_{\RR} (G_1,G)$ satisfying \[ (G, \bk) \sim (G_1, \bk_1). \] Remark \ref{rmk:de_ss} shows $\bx \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G_1, \bk_1)$ and $(G, \bk)$. From Lemma \ref{lem:phi_wd}, by setting \begin{equation} \label{eq:cts_k_1} \bJ = \big( k_{1, \by\rightarrow \by'} \bx^{\by} \big)_{\by\rightarrow \by' \in E_1}, \end{equation} then we obtain \begin{equation} \label{eq:cts_k_2} \hbJ = \bJ + \sum\limits^a_{j=1} (q_j - \langle \bJ, \bA_j \rangle ) \bA_j \in \hat{\mJ} (G_1,G). \end{equation} Moreover, from \eqref{def:phi_kq} we obtain \begin{equation} \notag \bp = ( \langle \bk, \bB_1 \rangle, \langle \bk, \bB_2 \rangle, \ldots, \langle \bk, \bB_b \rangle), \end{equation} which is continuous with respect to $\bk$. \smallskip Now assume any sequence $\{ \bk^i \}^{\infty}_{i = 1}$ in $\dK(G,G_1)$, such that \begin{equation} \label{eq:cts_k_3} \bk^i \to \bk \ \text{ as } \ i \to \infty. \end{equation} Suppose $\Phi (\bk^i, \bq) = (\hbJ^i, \bx^i, \bp^i)$ with $i \in \mathbb{N}$, then $\bx^i \in (\bx_0 + \mS_{G_1} )\cap\mathbb{R}^n_{>0}$ is the steady state of $(G_1, \bk^i)$. Using Lemma \ref{lem:key_3}, together with $\bk^i \to \bk$ in \eqref{eq:cts_k_3}, we derive \begin{equation} \label{eq:cts_k_4} \bx^i \to \bx \ \text{ as } \ i \to \infty. \end{equation} From Lemma \ref{lem:key_1}, there exists a sequence $\{ \bk^i_1 \}^{\infty}_{i = 1}$ in $\mK (G_1,G_1)$, such that \begin{equation} \notag (G, \bk^i) \sim (G_1, \bk^i_1) \ \text{ for any } \ i \in \mathbb{N}, \end{equation} and \begin{equation} \label{eq:cts_k_5} \bk^i_1 \to \bk_1 \ \text{ as } \ i \to \infty. \end{equation} Then apply Proposition \ref{lem:key_4}, there exists a corresponding sequence $\{ \hbk_i \}^{\infty}_{i = 1}$, such that \begin{equation} \notag (G_1, \hbk_i) \sim (G_1, \bk^i_1) \ \text{ for any } \ i \in \mathbb{N}, \end{equation} Set $\hbJ_i = (\hat{k}_{i, \by \to \by'} (\bx^i)^{\by})_{\by \to \by' \in E_1}$, then for any $i \in \mathbb{N}$, \begin{equation} \label{eq:cts_k_6} \hbJ_i \in \hat{\mathcal{J}} (G_1) \ \text{ and } \ \langle \hbJ_i, \bA_j \rangle = \langle \bJ, \bA_j \rangle \ \text{ for any } \ 1 \leq j \leq a. \end{equation} Moreover, from $\bk^i_1 \to \bk_1$ in \eqref{eq:cts_k_5}, we have \begin{equation} \notag \hbk_i \to \bk_1 \ \text{ as } \ i \to \infty. \end{equation} Together with $\bx^i \to \bx$ in \eqref{eq:cts_k_4} and $\bJ$ in \eqref{eq:cts_k_1}, we derive that \begin{equation} \label{eq:cts_k_7} \hbJ_i \to \bJ \ \text{ as } \ i \to \infty. \end{equation} Since $\bJ \in \mathcal{J} (G_1)$ and $\hbJ_i \in \hat{\mathcal{J}} (G_1)$, this shows there exists a sufficiently large $N$, such that \begin{equation} \label{eq:cts_k_8} \hbJ_i \in \mathcal{J} (G_1) \ \text{ for any } \ i > N. \end{equation} Note that $(G_1, \hbk_i) \sim (G_1, \bk^i_1) \sim (G_1, \bk^i)$, thus $\bx^i$ is also the steady state of $(G_1, \hbk_i)$. Since $\hbJ_i = (\hat{k}_{i, \by \to \by'} (\bx^i)^{\by})_{\by \to \by' \in E_1}$, together with \eqref{eq:cts_k_8}, we deduce \begin{equation} \notag \hbk_i \in \mK (G_1) \ \text{ for any } \ i > N. \end{equation} Note that $\Phi (\bk^i, \bq) = (\hbJ^i, \bx^i, \bp^i)$. From \eqref{eq:cts_k_2}, we obtain \begin{equation} \notag \hbJ^i = \hbJ_i + \sum\limits^a_{j=1} (q_j - \langle \hbJ_i, \bA_j \rangle ) \bA_j \ \text{ for any } \ i > N. \end{equation} Using \eqref{eq:cts_k_6} and \eqref{eq:cts_k_7}, we have \begin{equation} \notag \hbJ^i \to \bJ \ \text{ as } \ i \to \infty. \end{equation} Recall that $\Phi (\bk, \bq) = (\bJ, \bx, \bp)$. Suppose any sequence $\bk^i \to \bk$ with $\Phi (\bk^i, \bq) = (\hbJ^i, \bx^i, \bp^i)$, we show the continuity on $\bp$, $\bx^i \to \bx$ and $\hbJ^i \to \bJ$. Therefore, we conclude that $\Phi (\cdot, \bq)$ is continuous with respect to $\bk$. \end{proof} Here we state the first main theorem in this paper. \begin{theorem} \label{thm:inverse_cts} Consider the map $\hPsi$ in Definition \ref{def:hpsi}, then the map $\hPsi^{-1}$ is continuous. \end{theorem} \begin{proof} From Lemma \ref{lem:phi_wd}, consider the map $\Phi$ in Definition \ref{def:phi}, then $\Phi = \hPsi^{-1}$ is well-defined and bijective. Thus, it suffices to show the map $\Phi$ is continuous. Suppose any $(\bk, \bq) \in \dK(G,G_1) \times \RR^a$. Consider any positive real number $\varepsilon > 0$. From Proposition \ref{prop:inverse_cts_k}, $\Phi (\cdot, \bq)$ is continuous with respect to $\bk$. Thus, there exists some positive real number $\delta_1 > 0$, such that for any $\tilde{\bk} \in \dK(G,G_1)$ with $\\| \tilde{\bk} - \bk \\| < \delta_1$, then \begin{equation} \label{eq:inverse_cts_1} \big\\| \Phi (\tilde{\bk}, \bq) - \Phi (\bk, \bq) \big\\| < \frac{\varepsilon}{2}. \end{equation} Note that $\{\bA_1, \bA_2, \ldots, \bA_a \}$ is an orthonormal basis of $\eJ(G_1) \subset \RR^a$, there exists some positive real number $\delta_2 > 0$, such that for any $\bv = (v_1, v_2, \ldots, v_a) \in \RR^a$ with $\\| \bv \\| < \delta_2$, then \begin{equation} \label{eq:inverse_cts_2} \big\\| \sum\limits^{a}_{i=1} v_i \bA_i \big\\| < \frac{\varepsilon}{2}. \end{equation} Let $\delta = \min \{ \delta_1, \delta_2 \}$, consider any $(\hbk, \hbq) \in \dK(G,G_1) \times \RR^a$ with $\| (\hbk, \hbq) - (\bk, \bq) \| < \delta$. This implies $\\| \hbk - \bk \\| < \delta$ and $\\| \hbq - \bq \\| < \delta$. Then we compute that \begin{equation} \label{eq:inverse_cts_3} \Phi (\hbk, \hbq) - \Phi (\bk, \bq) = \big( \Phi (\hbk, \hbq) - \Phi (\bk, \hbq) \big) + \big( \Phi (\bk, \hbq) - \Phi (\bk, \bq) \big). \end{equation} From \eqref{eq:inverse_cts_1} and $\\| \hbk - \bk \\| < \delta \leq \delta_1$, we have \begin{equation} \label{eq:inverse_cts_4} \big\\| \Phi (\hbk, \hbq) - \Phi (\bk, \hbq) \big\\| < \frac{\varepsilon}{2}. \end{equation} Using Lemma \ref{lem:inverse_cts_q} and setting $\hbq - \bq := (v_1, v_2, \ldots, v_a) \in \RR^a$, we have \begin{equation} \notag \Phi (\bk, \hbq) - \Phi (\bk, \bq) = \sum\limits^{a}_{i=1} v_i \bA_i, \end{equation} Together with \eqref{eq:inverse_cts_2} and $\\| \hbq - \bq \\| < \delta \leq \delta_2$, we obtain \begin{equation} \label{eq:inverse_cts_5} \big\\| \Phi (\bk, \hbq) - \Phi (\bk, \bq) \big\\| = \big\\| \sum\limits^{a}_{i=1} v_i \bA_i \big \\| < \frac{\varepsilon}{2}. \end{equation} Inputting \eqref{eq:inverse_cts_4} and \eqref{eq:inverse_cts_5} into \eqref{eq:inverse_cts_3}, we derive \begin{equation} \notag \big\\| \Phi (\hbk, \hbq) - \Phi (\bk, \bq) \big\\| \leq \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon. \end{equation} Therefore, $\Phi$ is continuous and we conclude this theorem. \end{proof} The following result is a direct consequence of Theorem \ref{thm:inverse_cts}. \begin{theorem} \label{thm:hpsi_homeo} The map $\hPsi$ in Definition \ref{def:hpsi} is a homeomorphism. \end{theorem} \begin{proof} From Lemma \ref{lem:hpsi_bijective} and \ref{lem:hpsi_cts}, we derive that $\hPsi$ is bijective and continuous. On the other hand, Proposition \ref{thm:inverse_cts} shows the inverse map $\hPsi^{-1}$ is also continuous. Therefore, we conclude that the map $\hPsi$ is a homomorphism. \end{proof} \section{Dimension of \texorpdfstring{$\dK(G,G_1)$}{KGG1} and \texorpdfstring{$\pK(G,G_1)$}{pKGG1} } \label{sec:dimension} In this section, we give a precise bound on the dimension of $\dK(G, G_1)$, where $G_1 \sqsubseteq G_c$. Further, we show the dimension of $\pK(G, G_1)$ when $\pK(G, G_1) \neq \emptyset$. Finally, we remark on the dimension of {\em $\RR$-disguised toric locus} $\dK(G)$ and {\em disguised toric locus} $\pK(G)$. \begin{lemma} \label{lem:hat_j_g1_g_cone} Let $G_1 = (V_1, E_1)$ be a weakly reversible E-graph and let $G = (V, E)$ be an E-graph. If $\mJ (G_1, G) \neq \emptyset$, then $\hat{\mJ} (G_1, G)$ is a convex cone, which satisfies \begin{equation} \label{hat_j_g1_g_generator_dim} \dim (\hat{\mJ} (G_1, G)) = \dim (\mJ (G_1, G)). \end{equation} \end{lemma} \begin{proof} From Lemma \ref{lem:j_g1_g_cone}, suppose there exists a set of vectors $\{ \bv_1, \bv_2, \ldots, \bv_k \} \subset \RR^{\|E_1\|}$, such that \begin{equation} \notag \mJ (G_1, G) = \{ a_1 \bv_1 + \cdots a_k \bv_k \ \| \ a_i \in \RR_{>0} \}. \end{equation} Using \eqref{def:hat_j_g1_g}, $\hat{\mJ} (G_1, G)$ can be represented as the positive combination of the following vectors: \begin{equation} \label{hj_g1g_basis} \{ \bv_1, \bv_2, \ldots, \bv_k, \pm \bA_1, \pm \bA_2, \ldots, \pm \bA_a \}. \end{equation} This shows $\hat{\mJ} (G_1, G)$ is a convex cone. Moreover, we have \begin{equation} \notag \dim (\hat{\mJ} (G_1, G)) =\dim ( \spn \{ \bv_1, \bv_2, \ldots, \bv_k, \bA_1, \bA_2, \ldots, \bA_a \} ). \end{equation} Since $\mJ (G_1, G) \neq \emptyset$, Lemma \ref{lem:j_g1_g_cone} shows that \begin{equation} \notag \spn \{ \bA_i \}^a_{i=1} = \eJ(G_1) \subseteq \spn \{ \bv_1, \bv_2, \ldots, \bv_k \}. \end{equation} Therefore, we conclude that \begin{equation} \notag \dim (\hat{\mJ} (G_1, G)) = \dim ( \spn \{ \bv_1, \bv_2, \ldots, \bv_k \} ) = \dim (\mJ (G_1, G)). \end{equation} \end{proof} \begin{theorem} \label{thm:dim_kisg} Let $G_1 = (V_1, E_1)$ be a weakly reversible E-graph with its stoichiometric subspace $\mS_{G_1}$. Suppose an E-graph $G = (V, E)$, recall $\mJ (G_1,G)$, $\mD(G)$ and $\eJ(G_1)$ defined in Definitions~\ref{def:flux_realizable}, \ref{def:d0} and \ref{def:j0} respectively. \begin{enumerate}[label=(\alph)] \item\label{part_a} Consider $\dK(G,G_1)$ from Definition~\ref{def:de_realizable}, then \begin{equation} \label{eq:dim_kisg} \begin{split} & \dim(\dK(G,G_1)) = \dim (\mJ(G_1,G)) + \dim (\mS_{G_1}) + \dim(\eJ(G_1)) - \dim(\mD(G)). \end{split} \end{equation} \item\label{part_b} Further, consider $\pK (G, G_1)$ from Definition~\ref{def:de_realizable} and assume that $\pK (G, G_1) \neq \emptyset$. Then \begin{equation} \label{eq:dim_kdisg} \dim(\pK (G,G_1)) = \dim(\dK(G,G_1)). \end{equation} \end{enumerate} \end{theorem} \begin{proof} For part $(a)$, recall we prove that $\hat{\Psi}$ is a homeomorphism in Theorem \ref{thm:hpsi_homeo}. Using the invariance of dimension theorem \cite{hatcher2005algebraic,munkres2018elements}, together with Remark \ref{rmk:semi_algebaic} and \eqref{hat_j_g1_g_generator_dim} in Lemma \ref{lem:hat_j_g1_g_cone}, we obtain \begin{equation} \notag \dim (\dK(G, G_1)) + \dim(\mD(G)) = \dim (\mJ (G_1, G)) + \dim (\mS_{G_1}) + \dim(\eJ(G_1)), \end{equation} and conclude \eqref{eq:dim_kisg}. Further, we emphasize that on a dense open subset of $\dK(G, G_1)$, it is locally a submanifold. The homomorphism indicates that all such submanifolds have the same dimension. \smallskip For part $(b)$, since $\pK (G, G_1) \neq \emptyset$, together with Lemma \ref{lem:semi_algebaic} and Remark \ref{rmk:semi_algebaic}, there exists a $\bk \in \pK(G, G_1)$ and a neighborhood of $\bk$ in $\pK(G, G_1)$, denoted by $U$, such that \[ \bk \in U \subset \pK(G, G_1), \] where $U$ is a submanifold with $\dim (U) = \dim (\pK(G, G_1))$. Moreover, $\pK (G, G_1) = \dK(G, G_1) \cap \mathbb{R}^{E}_{>0}$ implies that $U$ is also a neighborhood of $\bk$ in $\dK(G, G_1)$. From part $(a)$, we obtain that on a dense open subset of $\dK(G, G_1)$, all local submanifolds have the same dimension. Therefore, we conclude \eqref{eq:dim_kdisg}. \end{proof} \begin{theorem} \label{thm:dim_kisg_main} Consider an E-graph $G = (V, E)$. \begin{enumerate}[label=(\alph)] \item Consider $\dK(G)$ from Definition~\ref{def:de_realizable}, then \begin{equation} \notag \dim (\dK(G) ) = \max_{G'\sqsubseteq G_c} \Big\{ \dim (\mJ(G',G)) + \dim (\mS_{G'}) + \dim(\eJ(G')) - \dim(\mD(G)) \Big\}, \end{equation} where $\mJ (G',G)$, $\mD(G)$ and $\eJ(G')$ are defined in Definitions~\ref{def:flux_realizable}, \ref{def:d0} and \ref{def:j0} respectively. \item Further, consider $\pK (G)$ from Definition~\ref{def:de_realizable} and assume that $\pK (G) \neq \emptyset$. Then \begin{equation} \notag \begin{split} & \dim (\pK(G) ) \\& = \max_{ \substack{ G'\sqsubseteq G_c, \\ \pK(G, G') \neq \emptyset } } \Big\{ \dim (\mJ(G',G)) + \dim (\mS_{G'}) + \dim(\eJ(G')) - \dim(\mD(G)) \Big\}. \end{split} \end{equation} \end{enumerate} \end{theorem} \begin{proof} Note that from Lemma~\ref{lem:semi_algebaic}, both $\pK (G, G_1)$ and $\dK(G, G_1)$ are semialgebraic sets. Further, the dimension of the union of finitely many semialgebraic sets is the maximum of the dimensions of these semialgebraic sets \cite{coste2000introduction,basu201738,lairez2021computing}. The result then follows from Definition~\ref{def:de_realizable}, Lemma \ref{lem:semi_algebaic} and Theorem~\ref{thm:dim_kisg}. \end{proof} \section{Examples} \label{sec:applications} \begin{example}[Thomas type models {\cite[Chapter 6]{murray2007mathematical}}] \label{ex:thomas} This model illustrates the substrate inhibition mechanism, capturing the interaction between uric acid and oxygen catalyzed by the enzyme uricase. The E-graph corresponding to this model, shown in Figure~\ref{fig:thomas_model} (a), is denoted by $G$. The system is governed by the following differential equations: \begin{equation} \begin{split} \frac{du}{dt} & = c - u - uv, \\ \frac{dv}{dt} & = \beta(d-v) - uv, \end{split} \end{equation} where $u$ denotes the concentration of uric acid, $v$ represents the concentration of oxygen, and the parameters $c$, $d$, and $\beta$ are positive constants. As demonstrated in \cite{CraciunDickensteinShiuSturmfels2009}, the toric locus associated with $G$ forms a toric variety of codimension one, implying that it has measure zero. \begin{figure}[!ht] \centering \includegraphics[scale=0.7]{thomas_model_2.eps} \caption{ (a) The E-graph $G$ represents a Thomas-type model, where $U$ denotes uric acid and $V$ denotes oxygen. (b) The E-graph $G_1$ is a weakly reversible subgraph of the complete graph formed by the source vertices of $G$. } \label{fig:thomas_model} \end{figure} The graph $G$ contains 6 reactions, hence $\pK(G) \subseteq \mathbb{R}^6_{>0}$. We claim that \[ \rm{dim}(\pK(G)) = 6. \] This implies that the disguised toric locus corresponding to $G$ is of positive measure. To prove this claim, we consider the E-graph $G_1$ and compute $\dim (\pK(G, G_1))$. First, consider a flux vector $\bJ \in \mJ(G_1, G) \subset \mathbb{R}^7$. Since $\bJ$ is a complex-balanced flux vector in $G_1$, this imposes $4 - 1 = 3$ constraints on $\bJ$, as shown in \cite{craciun2020efficient}. Additionally, being $\mathbb{R}$-realizable in $G$ imposes no further constraints, since every flux vector in $G_1$ can be transformed into $G$. Therefore, we have \[ \dim (\mJ(G_1, G)) = 7 - 0 - 3 = 4. \] Second, upon inspecting the graphs $G$ and $G_1$, we obtain the following: \[ \dim (\mathcal{S}_{G_1}) = 2, \ \ \rm{dim}(\mD (G)) = 0 \ \text{ and } \ \dim (\eJ(G_1)) = 0. \] Using Theorem~\ref{thm:dim_kisg} (a), we get that \begin{equation} \begin{split} \notag \rm{dim}(\dK(G, G_1)) & = \dim (\mJ(G_1, G)) + \dim (\mathcal{S}_{G_1}) + \dim(\mD (G)) - \dim(\eJ (G_1)) \\& = 4 + 2 + 0 - 0 = 6. \end{split} \end{equation} Since $G_1$ is weakly reversible, $\mK ({G_1}) \neq \emptyset$. From Figure~\ref{fig:thomas_model}, given any $\bk_1 \in \mK ({G_1})$ there exists $\bk$ such that $(G,\bk)\sim (G_1, \bk_1)$. This implies that $\pK (G, G_1) \neq \emptyset$, and thus Theorem~\ref{thm:dim_kisg} (b) further shows \[ \rm{dim}(\pK(G,G_1)) = \rm{dim}(\dK(G,G_1)) = 6. \] This, together with Theorem \ref{thm:dim_kisg_main}, implies that $\rm{dim}(\pK(G)) = 6$. A lengthy computation (based on the matrix-tree theorem, see \cite{CraciunDickensteinShiuSturmfels2009}) can be used to derive that \[ \pK(G) \ \supseteq \ \{ \bk \in \mathbb{R}^{7}_{>0} \mid \frac{k_{V \to \emptyset} k_{U \to \emptyset}}{k_{U+V \to U}} > k_{\emptyset \to V} \geq k_{\emptyset \to U} \ \text{ or } \ \frac{k_{V \to \emptyset} k_{U \to \emptyset}}{k_{U+V \to U}} > k_{\emptyset \to U} \geq k_{\emptyset \to V} \}, \] which, informally speaking, implies that at least 50\% of all positive parameter choices belong to $\pK(G)$. Finally, we remark that the Thomas-type model $G$ and the corresponding weakly reversible E-graph $G_1$ in Figure~\ref{fig:thomas_model} are both two-dimensional. From Remark \ref{rmk:complex_balance_property}, for any $\bk \in \pK(G, G_1)$, the system $(G, \bk)$ has a globally attracting steady state within each stoichiometric compatibility class. \qed \end{example} \begin{example}[Circadian clock models \cite{leloup1999chaos}] \label{ex:circadian} The circadian clock forms an integral part of the behavioral, physical, and biological changes in the body over a 24-hour cycle. The dynamics of this network are known to exhibit exotic behaviors, such as oscillations and limit cycles. For our analysis, we utilize the model of the circadian clock introduced in \cite{leloup1999chaos}, described by the following reactions: \begin{equation} \notag P + T \rightleftharpoons C \rightarrow \emptyset, \ \ P \rightleftharpoons \emptyset, \ \ T \rightleftharpoons \emptyset, \end{equation} where $P$ denotes period, $T$ denotes time, and $C$ denotes the period-time complex. The E-graph corresponding to this model, shown in Figure~\ref{fig:circadian_clock} (a), is denoted by $G$. Since $G$ is not weakly reversible, \cite{CraciunDickensteinShiuSturmfels2009} demonstrates that the toric locus associated with $G$ is empty. \begin{figure}[!ht] \centering \includegraphics[scale=0.43]{circadian_clock.eps} \caption{ (a) The E-graph $G$ represents a circadian clock model. (b) The E-graph $G_1$ is a weakly reversible subgraph of the complete graph formed by the source vertices of $G$. } \label{fig:circadian_clock} \end{figure} Note that the graph $G$ contains 7 reactions, so $\pK(G) \subseteq \mathbb{R}^7_{>0}$. We claim that \[ \rm{dim}(\pK(G)) = 7. \] This implies that the disguised toric locus corresponding to $G$ is of positive measure. To prove this claim, we consider the E-graph $G_1$ and compute $\dim (\pK(G, G_1))$. First, consider a flux vector $\bJ \in \mJ(G_1, G) \subset \mathbb{R}^8$. Since $\bJ$ is a complex-balanced flux vector in $G_1$, this imposes $5 - 1 = 4$ constraints on $\bJ$, as shown in \cite{craciun2020efficient}. Additionally, being $\mathbb{R}$-realizable in $G$ imposes no further constraints, since every flux vector in $G_1$ can be transformed into $G$. Therefore, we have \[ \dim (\mJ(G_1, G)) = 8 - 0 - 4 = 4. \] Second, upon inspecting the graphs $G$ and $G_1$, we obtain the following: \[ \dim (\mathcal{S}_{G_1}) = 3, \ \ \rm{dim}(\mD (G)) = 0 \ \text{ and } \ \dim (\eJ(G_1)) = 0. \] Using Theorem~\ref{thm:dim_kisg} (a), we get that \begin{equation} \begin{split} \notag \rm{dim}(\dK(G, G_1)) & = \dim (\mJ(G_1, G)) + \dim (\mathcal{S}_{G_1}) + \dim(\mD (G)) - \dim(\eJ (G_1)) \\& = 4 + 3 + 0 - 0 = 7. \end{split} \end{equation} Since $G_1$ is weakly reversible, $\mK ({G_1}) \neq \emptyset$. From Figure~\ref{fig:circadian_clock}, given any $\bk_1 \in \mK ({G_1})$ there exists $\bk$ such that $(G,\bk)\sim (G_1, \bk_1)$. This implies that $\pK (G, G_1) \neq \emptyset$, and thus Theorem~\ref{thm:dim_kisg}~(b) further shows \[ \rm{dim}(\pK(G,G_1) = \rm{dim}(\dK(G,G_1)) = 7. \] This, together with Theorem \ref{thm:dim_kisg_main}, implies that $\rm{dim}(\pK(G)) = 7$. Actually, a computation (based on the matrix-tree theorem~\cite{CraciunDickensteinShiuSturmfels2009}) can be used to derive that $\pK(G) = \mathbb{R}^{7}_{>0}. $ Finally, note that for the circadian clock model $G$, the corresponding weakly reversible E-graph $G_1$ in Figure~\ref{fig:circadian_clock} contains a single linkage class. From Remark \ref{rmk:complex_balance_property} it follows that for any $\bk \in \pK(G, G_1)$, the system $(G, \bk)$ has a globally attracting steady state within each stoichiometric compatibility class. \qed \end{example} \section{Discussion} \label{sec:discussion} Due to their remarkably robust dynamics, complex-balanced dynamical systems form an important class of dynamical systems. In particular, they are notable for their connection to the \emph{Global Attractor Conjecture}~\cite{horn1972general, CraciunDickensteinShiuSturmfels2009}, which asserts that these systems possess a globally attracting steady state within each invariant polyhedron. The set of rate constants of a network that generate complex-balanced systems form a variety called the \emph{toric locus}~\cite{craciun2020structure}, and its codimension is determined by the deficiency of the reaction network~\cite{CraciunDickensteinShiuSturmfels2009}. A generalization of the toric locus concerns the set of rate constants that yield the same dynamics as complex-balanced systems. When the rate constants can take both positive and negative values, this set is referred to as the \emph{$\mathbb{R}$-disguised toric locus}; if only positive values are allowed, it is called the \emph{disguised toric locus} \cite{2022disguised}. In~\cite{disg_1}, it was demonstrated that both the disguised toric locus and the $\mathbb{R}$-disguised toric locus are path-connected. A recent study~\cite{disg_2} established a lower bound for the dimensions of both loci. The key contribution of this paper is to estimate the exact dimensions of both the disguised and $\mathbb{R}$-disguised toric loci through the construction of an explicit homomorphism. Specifically, Theorem~\ref{thm:dim_kisg_main} provides these dimensions as follows: {\small \begin{equation} \notag \begin{split} &\dim (\dK(G) ) = \max_{G'\sqsubseteq G_c} \Big\{ \dim (\mJ(G',G)) + \dim (\mS_{G'}) + \dim(\eJ(G')) - \dim(\mD(G)) \Big\}, \\& \dim (\pK(G) ) = \max_{ \substack{ G'\sqsubseteq G_c, \\ \pK(G, G') \neq \emptyset } } \Big\{ \dim (\mJ(G',G)) + \dim (\mS_{G'}) + \dim(\eJ(G')) - \dim(\mD(G)) \Big\}. \end{split} \end{equation} } This characterization allows us to identify regions of parameter space that produce robust dynamics, thus refining the modeling and prediction of system responses. The results outlined above generate several new directions for future research. One direction involves the efficient computation of the dimensions of both the $\mathbb{R}$-disguised toric locus and the disguised toric locus for a given network. Among the various quantities on the right-hand side of the equations, the most challenging to characterize is $\dim (\mJ(G', G))$, which we specifically aim to address in our upcoming work~\cite{disg_4}. Specifically, computing $\dim (\mJ(G', G))$ is equivalent to solving a linear feasibility problem that incorporates both dynamical equivalence and complex-balancing constraints. Another promising direction is to identify the conditions under which the homeomorphisms constructed in this paper can be shown to be diffeomorphisms. A recent work~\cite{smoothness} has shown that the toric locus of a weakly reversible network is a smooth manifold. Consequently, we aim to establish the conditions on the network that ensure the disguised toric locus is also a smooth manifold, with the corresponding steady states varying continuously along this locus. \section*{Acknowledgements} This work was supported in part by the National Science Foundation grant DMS-2051568. \bibliographystyle{unsrt} \bibliography{Bibliography} \end{document}
2412.02681v1	http://arxiv.org/abs/2412.02681v1	On Rank of Multivectors in Geometric Algebras	\documentclass[AMA,STIX1COL]{WileyNJD-v2} \usepackage{moreverb} \def\cl{{C}\!\ell} \def\R{{\mathbb R}} \def\F{{\mathbb F}} \def\CC{{\mathbb C}} \def\C{\mathcal {G}} \def\P{{\rm P}} \def\A{{\rm A}} \def\B{{\rm B}} \def\Q{{\rm Q}} \def\Z{{\rm Z}} \def\H{{\rm H}} \def\Aut{{\rm Aut}} \def\ker{{\rm ker}} \def\OO{{\rm O}} \def\SO{{\rm SO}} \def\Pin{{\rm Pin}} \def\Spin{{\rm Spin}} \def\ad{{\rm ad}} \def\mod{{\rm \;mod\; }} \newcommand{\BR}{\mathbb{R}} \newcommand{\BC}{\mathbb{C}} \newcommand{\Mat}{{\rm Mat}} \newcommand{\Det}{{\rm Det}} \newcommand{\tr}{{\rm tr}} \newcommand{\rank}{{\rm rank}} \newcommand{\spn}{{\rm span}} \newcommand{\diag}{{\rm diag}} \newcommand{\Adj}{{\rm Adj}} \def\cl{\mathcal {G}} \newcommand{\U}{{\rm U}} \newcommand{\G}{{\rm G}} \newcommand{\T}{{\rm T}} \newtheorem{example}{Example} \newcommand\BibTeX{{\rmfamily B\kern-.05em \textsc{i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} \articletype{Research article} \received{<day> <Month>, <year>} \revised{<day> <Month>, <year>} \accepted{<day> <Month>, <year>} \begin{document} \title{On Rank of Multivectors in Geometric Algebras\protect\thanks{The article was prepared within the framework of the project “Mirror Laboratories” HSE University “Quaternions, geometric algebras and applications”.}} \author[1,2]{Dmitry Shirokov} \authormark{DMITRY SHIROKOV} \address[1]{ \orgname{HSE University}, \orgaddress{\state{Moscow}, \country{Russia}}} \address[2]{ \orgname{Institute for Information Transmission Problems of Russian Academy of Sciences}, \orgaddress{\state{Moscow}, \country{Russia}}} \corres{Dmitry Shirokov. \email{[email protected]}} \presentaddress{HSE University, 101000, Moscow, Russia} \abstract[Abstract]{We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.} \keywords{characteristic polynomial; Clifford algebra; geometric algebra; rank; singular value decomposition; unitary group} \jnlcitation{\cname{\author{D. Shirokov}} (\cyear{2024}), \ctitle{On Rank of Multivectors in Geometric Algebras}} \maketitle \section{Introduction} The notion of rank of matrix is one of the most important concepts of the matrix theory, which is used in different applications -- data analysis, physics, engineering, control theory, computer sciences, etc. The Clifford geometric algebras can be regarded as unified language of mathematics \cite{ABS, Porteous, Helm}, physics \cite{Hestenes, Doran, BT, Snygg}, engineering \cite{Bayro2}, and computer science \cite{Dorst, Bayro1}. The Clifford geometric algebras are isomorphic to the classical matrix algebras. In particular, the complexified Clifford geometric algebras $\cl^\BC_{p,q}:=\BC\otimes \cl_{p,q}$ are isomorphic to the following complex matrix algebras: \begin{eqnarray} \cl^\BC_{p,q}\simeq \begin{cases} \Mat(2^{\frac{n}{2}}, \BC), &\mbox{if $n$ is even,}\\ \Mat(2^{\frac{n-1}{2}}, \BC)\oplus\Mat(2^{\frac{n-1}{2}}, \BC), &\mbox{if $n$ is odd.} \end{cases} \end{eqnarray} An arbitrary element $M\in\cl^\BC_{p,q}$ (a multivector) can be represented as a complex matrix of the corresponding size $$N:=2^{[\frac{n+1}{2}]},$$ where square brackets mean taking the integer part. In the case of odd $n$, we deal with block-diagonal matrices with two nonzero blocks of the same size $2^{\frac{n-1}{2}}$. In this regard, the problem arises of determining the rank of multivectors $M\in\cl^\BC_{p,q}$ without using the matrix representation and using only the operations in Clifford geometric algebras. In this paper, we solve this problem in the case of any dimension. To do this, we use our previous results on SVD and characteristic polynomial in Clifford geometric algebras. Theorems \ref{thrankpr}, \ref{thrankpr2}, \ref{thrank}, \ref{thrankherm} are new. New explicit formulas (\ref{exp1}), (\ref{exp2}) for the cases of dimensions $3$ and $4$ can be used in various applications of geometric algebras in physics, engineering, and computer science. The paper is organized as follows. In Section \ref{secGA}, we discuss real and complexified geometric algebras (GA) and introduce the necessary notation. In Section \ref{secbeta}, we discuss an operation of Hermitian conjugation in GA, introduce a positive scalar product, a norm, unitary space and unitary groups in GA. Also we discuss faithful representations of GA and present an explicit form on one of them. In Section \ref{secSVD}, we discuss singular value decomposition of multivectors in GA. In Section \ref{secDet}, we discuss a realization of the determinant and other characteristic polynomial coefficients in GA. In Section \ref{secRank}, we introduce a notion of rank of multivector in GA and prove a number of properties of this notion. We prove that this notion does not depend on the choosing of matrix representation and present another equivalent definition of this notion using only GA operations. Examples for cases of small dimensions are presented. In Section \ref{secRankherm}, we consider the special case of normal multivectors, for which rank can be determined more simply. The conclusions follow in Section \ref{secConcl}. \section{Real and Complexified Geometric Algebras}\label{secGA} Let us consider the real Clifford geometric algebra $\cl_{p,q}$ \cite{Hestenes,Lounesto,Doran,Bulg} with the identity element $e\equiv 1$ and the generators $e_a$, $a=1, 2, \ldots, n$, where $n=p+q\geq 1$. The generators satisfy the conditions $$ e_a e_b+e_b e_a=2\eta_{ab}e,\qquad \eta=(\eta_{ab})=\diag(\underbrace{1, \ldots , 1}_p, \underbrace{-1, \ldots, -1}_{q}) $$ Consider the subspaces $\cl^k_{p,q}$ of grades $k=0, 1, \ldots, n$, which elements are linear combinations of the basis elements $e_A=e_{a_1 a_2 \ldots a_k}=e_{a_1}e_{a_2}\cdots e_{a_k}$, $1 \leq a_1<a_2<\cdots< a_k \leq n$, with ordered multi-indices of length $k$. An arbitrary element (multivector) $M\in\cl_{p,q}$ has the form $$ M=\sum_A m_A e_A\in\cl_{p,q},\qquad m_A\in\BR, $$ where we have a sum over arbitrary multi-index $A$ of length from $0$ to $n$. The projection of $M$ onto the subspace $\cl^k_{p,q}$ is denoted by $\langle M \rangle_k$. The grade involution and reversion of a multivector $M\in\cl_{p,q}$ are denoted by \begin{eqnarray} \widehat{M}=\sum_{k=0}^n(-1)^{k}\langle M \rangle_k,\qquad \widetilde{M}=\sum_{k=0}^n (-1)^{\frac{k(k-1)}{2}} \langle M \rangle_k. \end{eqnarray} We have \begin{eqnarray} \widehat{M_1 M_2}=\widehat{M_1} \widehat{M_2},\qquad \widetilde{M_1 M_2}=\widetilde{M_2} \widetilde{M_1},\qquad \forall M_1, M_2\in\cl_{p,q}.\label{invol} \end{eqnarray} Let us consider the complexified Clifford geometric algebra $\cl_{p,q}^\BC:=\BC\otimes\cl_{p,q}$ \cite{Bulg}. An arbitrary element of $M\in\cl^\BC_{p,q}$ has the form $$ M=\sum_A m_A e_A\in\cl^\BC_{p,q},\qquad m_A\in\BC. $$ Note that $\cl^\BC_{p,q}$ has the following basis of $2^{n+1}$ elements: \begin{eqnarray} e, ie, e_1, ie_1, e_2, i e_2, \ldots, e_{1\ldots n}, i e_{1\ldots n}.\label{basisC} \end{eqnarray} In addition to the grade involution and reversion, we use the operation of complex conjugation, which takes complex conjugation only from the coordinates $m_A$ and does not change the basis elements $e_A$: $$ \overline{M}=\sum_A \overline{m}_A e_A\in\cl^\BC_{p,q},\qquad m_A\in\BC,\qquad M\in\cl^\BC_{p,q}. $$ We have $$ \overline{M_1 M_2}=\overline{M_1}\,\, \overline{M_2},\qquad \forall M_1, M_2\in\cl^\BC_{p,q}. $$ \section{Hermitian conjugation and unitary groups in Geometric Algebras}\label{secbeta} Let us consider an operation of Hermitian conjugation $\dagger$ in $\cl^\BC_{p,q}$ (see \cite{unitary,Bulg}): \begin{eqnarray} M^\dagger:=M\|_{e_A \to (e_A)^{-1},\,\, m_A \to \overline{m}_A}=\sum_A \overline{m}_A (e_A)^{-1}.\label{herm} \end{eqnarray} We have the following two equivalent definitions of this operation: \begin{eqnarray} &&M^\dagger=\begin{cases} e_{1\ldots p} \overline{\widetilde{M}}e_{1\ldots p}^{-1}, & \mbox{if $p$ is odd,}\\ e_{1\ldots p} \overline{\widetilde{\widehat{M}}}e_{1\ldots p}^{-1}, & \mbox{if $p$ is even,}\\ \end{cases}\\ &&M^\dagger= \begin{cases} e_{p+1\ldots n} \overline{\widetilde{M}}e_{p+1\ldots n}^{-1}, & \mbox{if $q$ is even,}\\ e_{p+1\ldots n} \overline{\widetilde{\widehat{M}}}e_{p+1\ldots n}^{-1}, & \mbox{if $q$ is odd.}\\ \end{cases} \end{eqnarray} The operation\footnote{Compare with the well-known operation $M_1 M_2:=\langle \widetilde{M_1} M_2 \rangle_0$ in the real geometric algebra $\cl_{p,q}$, which is positive definite only in the case of signature $(p,q)=(n,0)$.} $$(M_1, M_2):=\langle M_1^\dagger M_2 \rangle_0$$ is a (positive definite) scalar product with the properties \begin{eqnarray} &&(M_1, M_2)=\overline{(M_2, M_1)},\\ &&(M_1+M_2, M_3)=(M_1, M_3)+(M_2, M_3),\quad (M_1, \lambda M_2)=\lambda (M_1, M_2),\\ &&(M, M)\geq 0,\quad (M, M)=0 \Leftrightarrow M=0.\label{\|\|M\|\|} \end{eqnarray} Using this scalar product we introduce inner product space over the field of complex numbers (unitary space) in $\cl^\BC_{p,q}$. We have a norm \begin{eqnarray} \|\|M\|\|:=\sqrt{(M,M)}=\sqrt{\langle M^\dagger M \rangle_0}.\label{norm} \end{eqnarray} Let us consider the following faithful representation (isomorphism) of the complexified geometric algebra \begin{eqnarray} \beta:\cl^\BC_{p,q}\quad \to\quad \begin{cases} \Mat(2^{\frac{n}{2}}, \BC), &\mbox{if $n$ is even,}\\ \Mat(2^{\frac{n-1}{2}}, \BC)\oplus\Mat(2^{\frac{n-1}{2}}, \BC), &\mbox{if $n$ is odd.} \end{cases}\label{isom} \end{eqnarray} Let us denote the size of the corresponding matrices by $$N:=2^{[\frac{n+1}{2}]},$$ where square brackets mean taking the integer part. Let us present an explicit form of one of these representations of $\cl^\BC_{p,q}$ (we use it also for $\cl_{p,q}$ in \cite{det} and for $\cl^\BC_{p,q}$ in \cite{LMA}). We denote this fixed representation by $\beta'$. Let us consider the case $p = n$, $q = 0$. To obtain the matrix representation for another signature with $q\neq 0$, we should multiply matrices $\beta'(e_a)$, $a = p + 1, \ldots, n$ by imaginary unit $i$. For the identity element, we always use the identity matrix $\beta'(e)=I_N$ of the corresponding dimension $N$. We always take $\beta'(e_{a_1 a_2 \ldots a_k}) = \beta' (e_{a_1}) \beta' (e_{a_2}) \cdots \beta'(e_{a_k})$. In the case $n=1$, we take $\beta'(e_1)=\diag(1, -1)$. Suppose we know $\beta'_a:=\beta'(e_a)$, $a = 1, \ldots, n$ for some fixed odd $n = 2k + 1$. Then for $n = 2k + 2$, we take the same $\beta'(e_a)$, $a = 1, \ldots , 2k + 1$, and $$\beta'(e_{2k+2})=\left( \begin{array}{cc} 0 & I_{\frac{N}{2}} \\ I_{\frac{N}{2}} & 0 \end{array} \right).$$ For $n = 2k + 3$, we take $$\beta'(e_{a})= \left(\begin{array}{cc} \beta'_a & 0 \\ 0 & -\beta'_a \end{array} \right),\qquad a=1, \ldots, 2k+2,$$ and $$\beta'(e_{2k+3})=\left(\begin{array}{cc} i^{k+1}\beta'_1\cdots \beta'_{2k+2} & 0 \\ 0 & -i^{k+1}\beta'_1\cdots \beta'_{2k+2} \end{array} \right).$$ This recursive method gives us an explicit form of the matrix representation $\beta'$ for all $n$. Note that for this matrix representation we have $$ (\beta'(e_a))^\dagger=\eta_{aa} \beta'(e_a),\qquad a=1, \ldots, n, $$ where $\dagger$ is the Hermitian transpose of a matrix. Using the linearity, we get that Hermitian conjugation of matrix is consistent with Hermitian conjugation of corresponding multivector: \begin{eqnarray} \beta'(M^\dagger)=(\beta'(M))^\dagger,\qquad M\in\cl^\BC_{p,q}.\label{sogl} \end{eqnarray} Note that the same is not true for an arbitrary matrix representations $\beta$ of the form (\ref{isom}). It is true the matrix representations $\gamma=T^{-1}\beta' T$ obtained from $\beta'$ using the matrix $T$ such that $T^\dagger T= I$. Let us consider the group \begin{eqnarray} \U\cl^\BC_{p,q}=\{M\in \cl^\BC_{p,q}: M^\dagger M=e\}, \end{eqnarray} which we call a unitary group in $\cl^\BC_{p,q}$. Note that all the basis elements $e_A$ of $\cl_{p,q}$ belong to this group by the definition. Using (\ref{isom}) and (\ref{sogl}), we get the following isomorphisms to the classical matrix unitary groups: \begin{eqnarray} \U\cl^\BC_{p,q}\simeq\begin{cases} \U(2^{\frac{n}{2}}), &\mbox{if $n$ is even,}\\ \U(2^{\frac{n-1}{2}})\times\U(2^{\frac{n-1}{2}}), &\mbox{if $n$ is odd,} \end{cases}\label{isgr} \end{eqnarray} where \begin{eqnarray} \U(k)=\{A\in\Mat(k, \BC),\quad A^\dagger A=I\}. \end{eqnarray} \section{Singular Value Decomposition in Geometric Algebras}\label{secSVD} The method of singular value decomposition was discovered independently by E. Beltrami in 1873 \cite{Beltrami} and C. Jordan in 1874 \cite{Jordan1,Jordan2}. We have the following well-known theorem on singular value decomposition of an arbitrary complex matrix \cite{For,Van}. For an arbitrary $A\in\BC^{n\times m}$, there exist matrices $U\in \U(n)$ and $V\in\U(m)$ such that \begin{eqnarray} A=U\Sigma V^\dagger,\label{SVD} \end{eqnarray} where $$ \Sigma=\diag(\lambda_1, \lambda_2, \ldots, \lambda_k),\qquad k=\min(n, m),\qquad \BR\ni\lambda_1, \lambda_2, \ldots, \lambda_k\geq 0. $$ Note that choosing matrices $U\in \U(n)$ and $V\in\U(m)$, we can always arrange diagonal elements of the matrix $\Sigma$ in decreasing order $\lambda_1\geq \lambda_2 \geq \cdots \geq \lambda_k\geq 0$. Diagonal elements of the matrix $\Sigma$ are called singular values, they are square roots of eigenvalues of the matrices $A A^\dagger$ or $A^\dagger A$. Columns of the matrices $U$ and $V$ are eigenvectors of the matrices $A A^\dagger$ and $A^\dagger A$ respectively. \begin{theorem}[SVD in GA]\cite{SVDAACA}\label{th1} For an arbitrary multivector $M\in\cl^\BC_{p,q}$, there exist multivectors $U, V\in \U\cl^\BC_{p,q}$, where $$ \U\cl^\BC_{p,q}=\{U\in \cl^\BC_{p,q}: U^\dagger U=e\},\qquad U^\dagger:=\sum_A \overline{u}_A (e_A)^{-1}, $$ such that \begin{eqnarray} M=U\Sigma V^\dagger,\label{SVDMC} \end{eqnarray} where multivector $\Sigma$ belongs to the subspace $K\in\cl^\BC_{p,q}$, which is a real span of a set of $N=2^{[\frac{n+1}{2}]}$ fixed basis elements (\ref{basisC}) of $\cl^\BC_{p,q}$ including the identity element~$e$. \end{theorem} \section{Determinant and other characteristic polynomial coefficients in Geometric Algebras}\label{secDet} Let us consider the concept of determinant \cite{rudn,acus} and characteristic polynomial \cite{det} in geometric algebra. Explicit formulas for characteristic polynomial coefficients are discussed in \cite{Abd,Abd2}, applications to Sylvester equation are discussed in \cite{Sylv,Sylv2}, the relation with noncommutative Vieta theorem is discussed in \cite{Vieta1,Vieta2}, applications to calculation of elementary functions in geometric algebras are discussed in \cite{Acus}. We can introduce the notion of determinant $$\Det(M):=\det(\beta(M))\in\BR,\qquad M\in\cl^\BC_{p,q},$$ where $\beta$ is (\ref{isom}), and the notion of characteristic polynomial \begin{eqnarray} &&\varphi_M(\lambda):=\Det(\lambda e-M)=\lambda^N-C_{(1)}\lambda^{N-1}-\cdots-C_{(N-1)}\lambda-C_{(N)}\in\cl^0_{p,q}\equiv\BR,\nonumber\\ &&M\in\cl^\BC_{p,q},\quad N=2^{[\frac{n+1}{2}]},\quad C_{(k)}=C_{(k)}(M)\in\cl^0_{p,q}\equiv\BR,\quad k=1, \ldots, N.\label{char} \end{eqnarray} The following method based on the Faddeev--LeVerrier algorithm allows us to recursively obtain basis-free formulas for all the characteristic coefficients $C_{(k)}$, $k=1, \ldots, N$ (\ref{char}): \begin{eqnarray} &&M_{(1)}:=M,\qquad M_{(k+1)}=M(M_{(k)}-C_{(k)}),\label{FL0}\\ &&C_{(k)}:=\frac{N}{k}\langle M_{(k)} \rangle_0,\qquad k=1, \ldots, N. \label{FL}\end{eqnarray} In particular, we have \begin{eqnarray} C_{(1)}=N \langle M \rangle_0=\tr(\beta(M)). \end{eqnarray} In this method, we obtain high coefficients from the lowest ones. The determinant is minus the last coefficient \begin{eqnarray} \Det(M)=-C_{(N)}=-M_{(N)}=U(C_{(N-1)}-M_{(N-1)})\label{laststep} \end{eqnarray} and has the properties (see \cite{rudn,det}) \begin{eqnarray} &&\Det(M_1 M_2)=\Det(M_1) \Det (M_2),\qquad M_1, M_2\in\cl^\BC_{p,q},\label{detpr}\\ &&\Det(M)=\Det(\widehat{M})=\Det(\widetilde{M})=\Det(\overline{M})=\Det(M^\dagger),\qquad \forall M\in\cl^\BC_{p,q}.\label{detpr2} \end{eqnarray} The inverse of a multivector $M\in\cl^\BC_{p,q}$ can be computed as \begin{eqnarray} M^{-1}=\frac{\Adj(M)}{\Det(M)}=\frac{C_{(N-1)}-M_{(N-1)}}{\Det(M)},\qquad \Det(M)\neq 0.\label{inv} \end{eqnarray} The presented algorithm and formulas (\ref{FL0}), (\ref{FL}), (\ref{inv}) are actively used to calculate inverse in GA \cite{inv1,inv2,inv3}. \section{Rank in Geometric Algebras}\label{secRank} Let us introduce the notion of rank of a multivector $M\in\cl^\BC_{p,q}$: \begin{eqnarray} \rank(M):=\rank(\beta(M))\in\{0, 1, \ldots, N\},\label{rank} \end{eqnarray} where $\beta$ is (\ref{isom}). Below we present another equivalent definition, which does not depend on the matrix representation $\beta$ (Theorem \ref{thrank}). We use the fact that rank is the number of nonzero singular values in the SVD and Vieta formulas. \begin{lemma}\label{lemmawell} The rank of multivector $\rank(M)$ (\ref{rank}) is well-defined, i.e. it does not depend on the representation $\beta$ (\ref{isom}). \end{lemma} \begin{proof} In the case of even $n$, for an arbitrary representation $\beta$ of type (\ref{isom}), by the Pauli theorem \cite{Pauli}, there exists $T$ such that $\beta(e_a)=T^{-1}\beta'(e_a) T$, where $\beta'$ is fixed matrix representation from Section \ref{secbeta}. We get $\beta(M)=T^{-1}\beta'(M) T$ and $\rank(\beta(M))=\rank(\beta'(M))$. In the case of odd $n$, for an arbitrary representation $\beta$ of type (\ref{isom}), by the Pauli theorem \cite{Pauli}, there exists $T$ such that $\beta(e_a)=T^{-1}\beta'(e_a) T$ or $\beta(e_a)=-T^{-1}\beta'(e_a) T$. In the first case, we get $\rank(\beta(M))=\rank(\beta'(M))$ similarly to the case of even $n$. In the second case, we get $\beta(M)=T^{-1}\beta'(\widehat{M}) T$ and $\rank(\beta(M))=\rank(\beta'(\widehat{M}))$. The equality $\rank(\beta'(\widehat{M}))=\rank(\beta'(M))$ is verified using the explicit form of representation $\beta'$ from Section \ref{secbeta}. Namely, the matrices $\beta'(e_a)=\diag(\beta'_a, -\beta'_a)$, $a=1, \ldots, n$, are block-diagonal matrices with two blocks differing in sign on the main diagonal by construction. Thus the matrix $\beta'(e_{ab})=\beta'(e_a)\beta'(e_b)=\diag(\beta'_a \beta'_b, \beta'_a \beta'_b)$ has two identical blocks. We conclude that the even part of multivector $M$ has the matrix representation $\diag(A, A)$ with two identical blocks, and the odd part of multivector $M$ has the matrix representation $\diag(B, -B)$ with two blocks differing in sign. Finally, we obtain $\rank(\beta'(\widehat{M})=\rank(\diag(A-B, A+B))=\rank(\diag(A+B, A-B))=\rank(\beta'(M))$. \end{proof} \begin{theorem}\label{thrankpr} We have the following properties of the rank of arbitrary multivectors $M_1, M_2, M_3\in\cl^\BC_{p,q}$: \begin{eqnarray} &&\rank(M_1 U)=\rank(U M_1)=\rank (M_1),\qquad \forall \,\,\mbox{invertible}\,\,U\in\cl^\BC_{p,q},\\ &&\rank(M_1 M_2)\leq \min(\rank(M_1), \rank(M_2)),\\ &&\rank(M_1 M_2)+\rank(M_2 M_3)\leq \rank(M_1 M_2 M_3)+\rank(M_2),\\ &&\rank(M_1 )+\rank(M_3)\leq \rank(M_1 M_3)+N. \end{eqnarray} \end{theorem} \begin{proof} These properties are the corollary of the corresponding properties of rank of matrices. \end{proof} \begin{theorem}\label{thrankpr2} We have \begin{eqnarray} &&\rank(M)=\rank(\widehat{M})=\rank(\widetilde{M})=\rank(\overline{M})\\ &&\qquad=\rank(M^\dagger)=\rank(M^\dagger M)=\rank(M M^\dagger),\qquad \forall M\in\cl^\BC_{p,q}. \end{eqnarray} \end{theorem} \begin{proof} Let us prove $\rank(M)=\rank(\widehat{M})$. In the case of even $n$, we have $\rank(\widehat{M})=\rank(e_{1\ldots n}M e_{1\ldots n}^{-1})=\rank (M)$. In the case of odd $n$, we have already proved the statement in the proof of Lemma \ref{lemmawell}. Let us prove $\rank(M)=\rank(\widetilde{M})$. We have the following relation between the reversion (or the superposition of reversion and grade involution) and the transpose (see \cite{nspinors,LMA}): \begin{eqnarray} (\beta'(M))^\T=\begin{cases} \beta'(e_{b_1 \ldots b_k}\widetilde{M}e_{b_1\ldots b_k}^{-1}), & \mbox{if $k$ is odd,}\\ \beta'(e_{b_1 \ldots b_k}\widehat{\widetilde{M}}e_{b_1\ldots b_k}^{-1}), & \mbox{if $k$ is even,} \end{cases} \end{eqnarray} for some fixed basis element $e_{b_1\ldots b_k}$, where $k$ is the number of symmetric matrices among $\beta'(e_a)$, $a=1, \ldots, n$. We get $\rank(M)=\rank(\beta'(M))=\rank((\beta'(M))^\T)=\rank(\widetilde{M})$. Using (\ref{sogl}), we obtain the other formulas for the Hermitian conjugation and complex conjugation, which is a superposition of Hermitian conjugation and transpose. \end{proof} \begin{lemma}\label{lemmaB} Suppose that a square matrix $A\in\BC^{N\times N}$ is diagonalizable. Then \begin{eqnarray} &&\rank(A)=N \quad \Leftrightarrow \quad C_{(N)}\neq 0;\\ && \rank(A)=k\in\{1, \ldots, N-1\} \, \Leftrightarrow \, C_{(k)}\neq 0,\,\, C_{(j)}=0,\, j=k+1, \ldots, N;\\ &&\rank(A)=0 \quad \Leftrightarrow \quad A=0. \end{eqnarray} \end{lemma} \begin{proof} We use Vieta formulas for the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_N$: \begin{eqnarray} C_{(1)}&=&\lambda_1+\cdots+\lambda_N,\\ C_{(2)}&=&-(\lambda_1 \lambda_2+\lambda_1 \lambda_3+\cdots+\lambda_{N-1}\lambda_N),\\ && \cdots\\ C_{(N)}&=&-\lambda_1 \cdots \lambda_N. \end{eqnarray} To the right, all statements are obvious. To the left, they are proved by contradiction. \end{proof} \begin{lemma}\label{lemmaC} For an arbitrary multivector $M\in\cl^\BC_{p,q}$, we have \begin{eqnarray} C_{(N)}(M^\dagger M)=0 &\Longleftrightarrow& C_{(N)}(M)=0,\\ C_{(1)}(M^\dagger M)=0 &\Longleftrightarrow& M=0. \end{eqnarray} \end{lemma} \begin{proof} We have \begin{eqnarray} C_{(N)}(M^\dagger M)&=&-\Det(M^\dagger M)=-\Det(M^\dagger) \Det(M)\\ &=&-(\Det M)^2=(C_{(N)}(M))^2,\\ C_{(1)}(M^\dagger M)&=&N \langle M^\dagger M \rangle_0=N \|\|M\|\|^2, \end{eqnarray} where we use (\ref{detpr}), (\ref{detpr2}), (\ref{norm}), and (\ref{\|\|M\|\|}). \end{proof} \begin{theorem}[Rank in GA]\label{thrank} Let us consider an arbitrary multivector $M\in\cl^\BC_{p,q}$ and $T:=M^\dagger M$. We have \begin{eqnarray} \rank(M)=\begin{cases} N,\quad &\mbox{if $C_{(N)}(M)\neq 0$,}\\ N-1,\quad &\mbox{if $C_{(N)}(M)=0$ and $C_{(N-1)}(T)\neq 0$,}\\ N-2\qquad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(T)=0$ and}\\ &\mbox{$C_{(N-2)}(T)\neq 0$,}\\ \cdots &\\ 2,\quad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(T)=\cdots=C_{(3)}(T)=0$ and}\\ &\mbox{$C_{(2)}(T)\neq 0$,}\\ 1,\quad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(T)=\cdots=C_{(2)}(T)=0$ and}\\ &\mbox{$M\neq 0$,}\\ 0,\quad &\mbox{if $M=0$.}\label{rank22} \end{cases} \end{eqnarray} \end{theorem} \begin{proof} We use the fact that the rank of a matrix equals the number of non-zero singular values, which is the same as the number of non-zero diagonal elements of the matrix $\Sigma$ in the singular value decomposition $A=U\Sigma V^\dagger$ (\ref{SVD}): $\rank(A)=\rank(U\Sigma V^\dagger)=\rank(\Sigma)$. The number of non-zero diagonal elements of the matrix $\Sigma$ can be written in terms of zero and non-zero characteristic polynomial coefficients of the matrix $A^\dagger A$ (see Lemma \ref{lemmaB}). Then we use Lemma~\ref{lemmaC}. \end{proof} \begin{example} For an arbitrary $M\in\cl^\BC_{p,q}$, $n=p+q=1$, we have \begin{eqnarray} \rank(M)=\begin{cases} $2,\quad$ &\mbox{if $M\widehat{M}\neq 0$,}\\ $1,\quad$ &\mbox{if $M\widehat{M}=0$ and $M\neq 0$,}\\ $0,\quad$ &\mbox{if $M=0$.} \end{cases}\label{exp-1} \end{eqnarray} \end{example} \begin{example} For an arbitrary $M\in\cl^\BC_{p,q}$, $n=p+q=2$, we have \begin{eqnarray} \rank(M)=\begin{cases} $2,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\neq 0$,}\\ $1,\quad$ &\mbox{if $M\widetilde{\widehat{M}}=0$ and $M\neq 0$,}\\ $0,\quad$ &\mbox{if $M=0$.} \end{cases}\label{exp0} \end{eqnarray} \end{example} \begin{example} For an arbitrary $M\in\cl^\BC_{p,q}$, $n=p+q=3$, we have \begin{eqnarray} \rank(M)=\begin{cases} $4,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}\neq 0$,}\\ $3,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=0$ and $T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T\widehat{T}\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}\widetilde{T}\neq 0$,}\\ $2,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T\widehat{T}\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}\widetilde{T}=0$ and} \\ &\mbox{$T\widetilde{\widehat{T}}+T\widehat{T}+T\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}+\widetilde{\widehat{T}}\widetilde{T}+\widehat{T}\widetilde{T}\neq 0$,}\\ $1,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T\widehat{T}\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}\widetilde{T}=$}\\ &\mbox{$=T\widetilde{\widehat{T}}+T\widehat{T}+T\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}+\widetilde{\widehat{T}}\widetilde{T}+\widehat{T}\widetilde{T}=0$ and $M\neq 0$,}\\ $0,\quad$ &\mbox{if $M=0$,} \end{cases}\label{exp1} \end{eqnarray} where $T:=M^\dagger M$. \end{example} \begin{example} Let us consider the $\bigtriangleup$-operation \cite{det} \begin{eqnarray} M^{\bigtriangleup}:=\sum_{k=0}^n (-1)^{\frac{k(k-1)(k-2)(k-3)}{24}}\langle M \rangle_k=\!\!\!\!\!\sum_{k=0, 1, 2, 3\!\!\!\!\mod 8}\!\!\!\!\!\langle M \rangle_k-\!\!\!\!\!\sum_{k=4, 5, 6, 7\!\!\!\!\mod 8}\!\!\!\!\!\langle M \rangle_k.\label{opconj} \end{eqnarray} Note that we have $(M_1 M_2)^\bigtriangleup\neq M_1^\bigtriangleup M_2^\bigtriangleup$ and $(M_1 M_2)^\bigtriangleup\neq M_2^\bigtriangleup M_1^\bigtriangleup $ in the general case. For an arbitrary $M\in\cl^\BC_{p,q}$, $n=p+q=4$, we have \begin{eqnarray} \rank(M)=\begin{cases} $4,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup\neq 0$,}\\ $3,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=0$ and} \\ &\mbox{$T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T(\widehat{T}\widetilde{T})^\bigtriangleup+\widetilde{\widehat{T}}(\widehat{T}\widetilde{T})^\bigtriangleup\neq 0$,}\\ $2,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T(\widehat{T}\widetilde{T})^\bigtriangleup+\widetilde{\widehat{T}}(\widehat{T}\widetilde{T})^\bigtriangleup=0$} \\ &\mbox{and $T\widetilde{\widehat{T}}+T\widehat{T}+T\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}+\widetilde{\widehat{T}}\widetilde{T}+(\widehat{T}\widetilde{T})^\bigtriangleup\neq 0$,}\\ $1,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=T\widetilde{\widehat{T}}\widehat{T}+T\widetilde{\widehat{T}}\widetilde{T}+T(\widehat{T}\widetilde{T})^\bigtriangleup+\widetilde{\widehat{T}}(\widehat{T}\widetilde{T})^\bigtriangleup=$}\\ &\mbox{$=T\widetilde{\widehat{T}}+T\widehat{T}+T\widetilde{T}+\widetilde{\widehat{T}}\widehat{T}+\widetilde{\widehat{T}}\widetilde{T}+(\widehat{T}\widetilde{T})^\bigtriangleup=0$ and $M\neq 0$,}\\ $0,$ &\mbox{if $M=0$,} \end{cases}\label{exp2} \end{eqnarray} where $T:=M^\dagger M$. \end{example} You can get an explicit form of formulas from Theorem \ref{thrank} for the cases $n=5$ and $n=6$ using the explicit formulas for the characteristic polynomial coefficients $C_{(k)}$, $k=1, 2, \dots, N$, from \cite{Abd2}. We do not present them here because they are quite cumbersome. These formulas involve only the operations of summation, multiplication, $\widehat{\quad}$, $\widetilde{\quad}$, and $\,\,^\bigtriangleup$. \section{The Case of Normal Multivectors}\label{secRankherm} We call \textit{a normal multivector} $M\in\cl^\BC_{p,q}$ a multivector with the property $M^\dagger M=M M^\dagger$, where $\dagger$ is (\ref{herm}). \textit{Hermitian multivectors} $M^\dagger=M$, \textit{anti-Hermitian multivectors} $M^\dagger=-M$, \textit{unitary multivectors }$M^\dagger M=e$ are the particular cases of normal multivectors. For example, the basis elements $e_A$ are unitary by the definition. Note that all unitary multivectors have rank equal to $N$. \begin{theorem}\label{thrankherm} Let us consider a normal ($M^\dagger M=M M^\dagger$) multivector $M\in\cl^\BC_{p,q}$. We have \begin{eqnarray} \rank(M)=\begin{cases} N,\quad &\mbox{if $C_{(N)}(M)\neq 0$,}\\ N-1,\quad &\mbox{if $C_{(N)}(M)=0$ and $C_{(N-1)}(M)\neq 0$,}\\ N-2\qquad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(M)=0$ and}\\ &\mbox{$C_{(N-2)}(M)\neq 0$,}\\ \cdots &\\ 2,\quad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(M)=\cdots=C_{(3)}(M)=0$ and}\\ &\mbox{$C_{(2)}(M)\neq 0$,}\\ 1,\quad &\mbox{if $C_{(N)}(M)=C_{(N-1)}(M)=\cdots=C_{(2)}(M)=0$ and}\\ &\mbox{$M\neq 0$,}\\ 0,\quad &\mbox{if $M=0$.}\label{rankherm} \end{cases} \end{eqnarray} \end{theorem} \begin{proof} We use that normal matrix is always diagonalizable. \end{proof} \begin{example} For an arbitrary normal ($M^\dagger M=M M^\dagger$) multivector $M\in\cl^\BC_{p,q}$, $n=p+q=3$, we have \begin{eqnarray} \rank(M)=\begin{cases} $4,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}\neq 0$,}\\ $3,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=0$ and $M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M\widehat{M}\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}\widetilde{M}\neq 0$,}\\ $2,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M\widehat{M}\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=0$ and} \\ &\mbox{$M\widetilde{\widehat{M}}+M\widehat{M}+M\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}+\widetilde{\widehat{M}}\widetilde{M}+\widehat{M}\widetilde{M}\neq 0$,}\\ $1,\quad$ &\mbox{if $M\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M\widehat{M}\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}\widetilde{M}=$}\\ &\mbox{$=M\widetilde{\widehat{M}}+M\widehat{M}+M\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}+\widetilde{\widehat{M}}\widetilde{M}+\widehat{M}\widetilde{M}=0$ and $M\neq 0$,}\\ $0,\quad$ &\mbox{if $M=0$.} \end{cases}\label{exp12} \end{eqnarray} \end{example} \begin{example} For an arbitrary normal ($M^\dagger M=M M^\dagger$) multivector $M\in\cl^\BC_{p,q}$, $n=p+q=4$, we have \begin{eqnarray} \rank(M)=\begin{cases} $4,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup\neq 0$,}\\ $3,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=0$ and} \\ &\mbox{$M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M(\widehat{M}\widetilde{M})^\bigtriangleup+\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup\neq 0$,}\\ $2,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M(\widehat{M}\widetilde{M})^\bigtriangleup+\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=0$} \\ &\mbox{and $M\widetilde{\widehat{M}}+M\widehat{M}+M\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}+\widetilde{\widehat{M}}\widetilde{M}+(\widehat{M}\widetilde{M})^\bigtriangleup\neq 0$,}\\ $1,$ &\mbox{if $M\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=M\widetilde{\widehat{M}}\widehat{M}+M\widetilde{\widehat{M}}\widetilde{M}+M(\widehat{M}\widetilde{M})^\bigtriangleup+\widetilde{\widehat{M}}(\widehat{M}\widetilde{M})^\bigtriangleup=$}\\ &\mbox{$=M\widetilde{\widehat{M}}+M\widehat{M}+M\widetilde{M}+\widetilde{\widehat{M}}\widehat{M}+\widetilde{\widehat{M}}\widetilde{M}+(\widehat{M}\widetilde{M})^\bigtriangleup=0$ and $M\neq 0$,}\\ $0,$ &\mbox{if $M=0$.} \end{cases}\label{exp22} \end{eqnarray} \end{example} Note that formulas (\ref{rankherm}), (\ref{exp12}), (\ref{exp22}) differ from (\ref{rank22}), (\ref{exp1}), (\ref{exp2}), respectively, by replacing the multivector $T=M^\dagger M$ with the multivector $M$. \section{Conclusions}\label{secConcl} In this paper, we implement the notion of rank of multivector in complexified Clifford geometric algebras without using the corresponding matrix representations. Theorem \ref{thrank} involves only operations in geometric algebras. To obtain the results, we use the fact that rank of the matrix $A\in\Mat(N, \BC)$ is the number of nonzero (positive) singular values of $A$, that is, the number of nonzero (positive) eigenvalues of the matrix $A^\dagger A$. Theorems \ref{thrankpr}, \ref{thrankpr2}, \ref{thrank}, \ref{thrankherm} are new. New explicit formulas (\ref{exp1}), (\ref{exp2}) for the cases of dimensions $3$ and $4$ can be used in various applications of geometric algebras in physics, engineering, and computer science. Note that the results of this work are valid not only for complexified Clifford geometric algebras, but also for real Clifford geometric algebras, since we can use the same matrix representations in the real case (but these matrix representations will have non-minimal dimension in this case). 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2412.02679v1	http://arxiv.org/abs/2412.02679v1	On z-Superstable and Critical Configurations of Chip Firing Pairs	\documentclass[11pt,twoside]{amsart} \title{On z-superstable and critical configurations of chip-firing pairs} \author{Zach Benton} \address{Stanford University} \email{[email protected]} \author{Jane Kwak} \address{UCLA} \email{[email protected]} \author{Suho Oh} \address{Texas State University} \email{[email protected]} \author{Mateo Torres} \address{University of Delaware} \email{[email protected]} \author{Mckinley Xie} \address{Texas A\&M University} \email{[email protected]} \usepackage{prefix} \date{} \begin{document} \begin{abstract} It is well known that there is a duality map between the superstable configurations and the critical configurations of a graph. This was extended to all M-matrices in (Guzm\`an-Klivans 2015). We show a natural way to extend this to all $(L,M)$-chip firing pairs introduced in (Guzm\`an-Klivans 2016). In addition, we study various properties of this map. \end{abstract} \maketitle \input{introandprelim} \section{The Duality map for chip-firing pairs} In this section we establish the duality between the superstable configurations and critical configurations of $(L,M)$-pairs, that extends the canonical duality between superstable and critical configurations of $M$-matrices. We are mainly going to be dealing with the pre-images of the configurations in $R^{+}$. The idea is to group up the preimages that have the same floor and then map such groupings into different groupings. \begin{remark} Recall that for chip-firing pairs, the coordinate-wise maximal critical configuration does not necessarily exist, as was the case in \cref{ex:nocmax}. However, for $M$-matrices it does. Throughout the paper, given any $(L,M)$-pair and we write $\cm$, it stands for $\cm$ of $M$. \end{remark} Consider the map $\sst(M) \rightarrow \sst(M)$ that sends $\vec s \in \sst(M)$ to $\sstab(\cmax-\vec s)$, where $\sstab(\vec{v})$ means we are taking the unique superstable configuration in the equivalence class that contains $\vec{v}$ (we use $\crit(\vec{v}$ for the unique critical configuration in the same class). This map only relies on $M$ and is completely independent of $L$. This map sends a superstable configuration to the unique superstable configuration in the same equivalence class (under $M$) as the image of $\vec s$ under the usual duality map. Now we are going to use the information of $L$, to put a mask on this map: \begin{definition} \label{def:ivomap} For any chip-firing pair $(L,M)$, we define the map $\ivo : \sst(M) \rightarrow \sst(M)$: \[ \ivo(\vec s) = \begin{cases} \vec s &\text{if $\{\LM2\vec s\:\} = \{\LM\cm\}$, } \\ \sstab(\cmax-\vec s) &\text{otherwise.} \end{cases} \] \end{definition} \begin{proposition} The map $\ivo$ is an involution. \end{proposition} \begin{proof} If $\{\LM2\vec s\:\} = \{\LM\cm\}$, we have the identity map. If that condition does not hold, then we have \begin{align} \ivo(\ivo(\vec s)) &= \sstab(\cm-\sstab(\cm-\vec s))\\ &= \sstab(\cm-(\cm-\vec s))\\ &= \sstab(\vec s)\\ &= \vec s \end{align} Therefore, $\ivo(\ivo(\vec s)) = \vec s$. \end{proof} In the special case where $L=M$ (this recovers the usual chip-firing on $M$-matrices, and in particular when $M$ is the Laplacian of a graph, the classical chip-firing) the above involution is simply the identity map: since $\LM2\vec s$ is an integer vector for any integer vector $\vec s$. \begin{lemma} \label{lem:equivsameblock} Let $\vec{a},\vec{b}$ be integer vectors such that they are equivalent under $M$. Let $\vec{f}$ be a vector such that every entry $f_i$ satisfies $0 \leq f_i < 1$. Then $\vec{a}+\vec{f} \in R^{+}$ if and only if $\vec{b}+\vec{f} \in R^{+}$. \end{lemma} \begin{proof} Recall that we have a nonnegative rational vector $\vec{v} \in R^{+}$ if and only if $LM^{-1}\vec{v}$ is a integer vector. If $\vec{a} \equiv_M \vec{b}$ then we may write $\vec{b} = \vec{a} + M\vec{z}$ where $z$ is an integer vector. This gives us $LM^{-1}\vec{b} = LM^{-1}\vec{a} + L\vec{z}$ and since $L$ and $\vec{z}$ are integral, we get the desired claim. \end{proof} \begin{example} Consider the signed graph from \cref{ex:usualbad}. We start by focusing on the chip-firing of the underlying graph (ignoring $L$ for now). From the configuration $\vec a = (1, 2, 0)$, we can fire vertex 2 to obtain the configuration $\vec b = (2, 0, 1)$, so these configurations are firing-equivalent (equivalent under $M$). Take some non-negative rational vectors where the entries are bounded above by $1$, say $\vec f_1 = (\nicefrac 1 3, \nicefrac 2 3, 0)$ and $\vec f_2 = (\nicefrac 2 3, \nicefrac 1 3, 0)$. Looking at the images under $LM^{-1}$ we get the following table. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \end{center} \begin{center} \begin{tabular}{c \| c} Preimage & Image under $LM^{-1}$ \\ \hline $\vec a + \vec f_1 = (\nicefrac 4 3, \nicefrac 8 3, 0)$ & $(8, 7, 0)$ \\ $\vec b + \vec f_1 = (\nicefrac 7 3, \nicefrac 2 3, 1)$ & $(7, 5, 1)$ \\ \hline $\vec a + \vec f_2 = (\nicefrac 5 3, \nicefrac 7 3, 0)$ & $(8, \nicefrac{27}{4}, 0)$\\ $\vec b + \vec f_2 = (\nicefrac 8 3, \nicefrac 1 3, 1)$ & $(7, \nicefrac{19}{4}, 1)$ \end{tabular} \end{center} Notice from the table that we have $\vec{a} + \vec f_1$ and $\vec{b} + \vec f_1$ are both in $R^{+}$ whereas $\vec{a}+ \vec f_2$ and $\vec{b} + \vec f_2$ are both not in $R^{+}$, which is consistent with \cref{lem:equivsameblock}. \end{example} \begin{theorem} Let $(L,M)$ be any chip-firing pair. The map $\dual : \sst(L,M) \rightarrow \crit(L,M)$ given by $\ss \mapsto \cm-\ivo(\floor{\ss}) + \frp{\ss}$ is a bijection. \end{theorem} \begin{proof} We approach this proof in two cases. When $\ivo(\floor{\vec s})=\sstab(\cm-\floor{\vec s})$, we have \begin{align} \cm - \ivo(\floor{\vec s}) &= \cm - \sstab(\cm - \floor{\vec s}).\\ \intertext{Since $\cm - \vec s$ is critical for any superstable configuration $\vec s$ of $M$,} &= \crit(\cm - (\cm - \floor{\vec s}))\\ &= \crit(\floor{\vec s}). \end{align} From \cref{lem:equivsameblock}, since $\vec s = \floor{\vec s} + \{\vec s\} \in R^+$, we have that $\crit(\floor{\vec s}) + \{\vec s\}$ is in $R^+$, and is a critical configuration of $(L,M)$ thanks to \cref{thm:22floor}. In the second case when $\ivo(\floor{\vec s})=\floor{\vec s}$, recall that \begin{align} \ivo(\floor{\vec s})=\floor{\vec s} &\iff \{\LM(2\floor{\vec s})\} = \{\LM(\cm)\} \\&\iff \{\LM(\floor{\vec s})\} = \{\LM(\cm - \floor{\vec s})\}. \end{align} Together with the fact that $LM^{-1}(\floor{\vec s} + \{\vec s\})$ is integral, the above tells us that $LM^{-1}(\cm-\floor{\vec s} + \{\vec s\})$ is also integral. Hence $\cm-\floor{\vec s} + \{\vec s\} \in R^{+}$ and is a critical configuration of $(L,M)$ thanks to \cref{thm:22floor}. \end{proof} \begin{example} Again take a look at the signed graph from \cref{ex:usualbad}. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \end{center} Start from the superstable configuration $(5,4,0)$ in $S^{+}$. As can be seen in the table in \cref{ex:usualbad}, its corresponding preimage is $\vec{s} = (\nicefrac{4}{3}, \nicefrac{7}{6},0)$. We can check that $\{LM^{-1}2\floor{\vec s}\} = (0, \nicefrac{1}{2}, 0) \neq (0,0,0) = \{LM^{-1}\cmax\}$, so $\ivo (\floor{\vec s}) = \sstab(\cmax - \floor{\vec s})$. For this graph, we have $\cmax = (2,1,2)$, so $\ivo (\floor{\vec s}) = (0,0,1)$ as $M^{-1}(\cmax - \floor{\vec s} - (0,0,1)) \in \ZZ$. Then \[\dual(\vec s) = \cmax - \ivo(\floor{\vec s}) + \{\vec s\} = (2,1,2) - (0,0,1) + \paren{\nicefrac{1}{3}, \nicefrac{1}{6}, 0} = \paren{\nicefrac{7}{3}, \nicefrac{7}{6}, 1}\] gives a critical preimage in $R^+$, which corresponds to the critical configuration $(8,6,1)$ of $S^+$. The table of superstables and criticals in \cref{ex:usualbad} is aligned in a way so that the superstable configuration and the critical configuration obtained from this duality map are in the same row. \end{example} When we are looking at chip-firing pairs $(M,M)$ the above duality map is exactly same as the previously known duality map for $M$-matrices: sending $\vec{c}$ to $c_{max} - \vec{c}$. \begin{example} Consider the graph from \cref{ex:unsignedkyle}. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-unsigned} \end{tikzpicture} \end{center} We have that $(0,0,1)$ and $(1,1,0)$ are superstable configurations (and also superstable preimages, since $LM^{-1}$ is the identity matrix). Since they are integral, they are fixed points in our involution $\ivo$. Then \[\dual(0,0,1) = \cm-\ivo(\floor{(0,0,1)}) + \frp{(0,0,1)} = (2,1,2) - (0,0,1) + (0,0,0) = (2,1,1),\] \[\dual(1,1,0) = \cm-\ivo(\floor{(1,1,0)}) + \frp{(1,1,0)} = (2,1,2) - (1,1,0) + (0,0,0) = (1,0,2),\] We can see that this aligns with the classical duality map between superstable and critical configurations for graphs. \end{example} The inverse of the above duality map looks like the following. \begin{corollary} Let $(L,M)$ be any chip-firing pair. The map $\dual^{-1} : \crit(L,M) \rightarrow \sst(L,M)$ given by $\vec{c} \mapsto \ivo(\cm - \floor{\vec{c}}) + \frp{\vec{c}}$ is a bijection. \end{corollary} \section{Frackets} In this section, we focus on \newword{frackets}, subgroups of the critical group constructed by looking at the fractional parts of preimages. We provide an elegant formula for calculating the cardinality of such groups, and then use that result to enumerate the number of fixed points of the involution map studied in the previous section. \subsection{The definition of frackets} For any integral, invertible matrix $L$, we let $\critgroup(L)$ to denote the group we get by looking at the equivalence classes given by $\equiv_L$. Let $L$ and $M$ be any integral, invertible $n$-by-$n$ matrices. We call such pair $(L,M)$ as an \newword{ii-pair}. When a vector $\vec{f}$ satisfies $0 \leq f_i < 1$ for every coordinate $i$, we call it a \newword{fractional vector}. Recall that for any vector $\vec{f}$, we use $\{\vec f\}$ to denote its fractional part. \begin{definition} Given any ii-pair $(L,M)$ and any fractional vector $\vec f$, we define the $L$-fracket $F^L_{\vec f}$ as the subset of $\mathcal K(L)$ consisting of every equivalence class that has a vector representation $\vv \in \ZZ^n$ such that $\{ML^{-1}\vv\} = \vec{f}$. \end{definition} In other words, an $L$-fracket consists of all equivalence classes of configurations whose images under $ML^{-1}$ have the same fractional part. It is clear from the definition that for any chip-firing pair $(L,M)$ it is also an ii-pair. Moreover $(M,L)$ is also an ii-pair as well. \begin{example} Let us again consider the chip-firing pair $(L,M)$ studied in \cref{table:signed_toy_ss}. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \begin{tabular}{c \| c} Element of $\critgroup(L)$ & Preimages in $R^{+}$ \\ \hline $(7, 6, 8)$ & $(\nicefrac 1 2, 0, \nicefrac 5 2)$ \\ $(8, 6, 7)$ & $(\nicefrac 5 2, 0, \nicefrac 1 2)$ \end{tabular} \end{center} Their preimages both have fractional part $(\nicefrac 1 2, 0, \nicefrac 1 2)$, so we have $[(7,6,8)]_L, [(8,6,7)]_L \in F^L_{(\nicefrac 1 2, 0, \nicefrac 1 2)}$. \begin{center} \begin{tabular}{c \| c} Element of $\critgroup(M)$ & Image under $LM^{-1}$ \\ \hline $(0, 0, 0)$ & $(0, 0, 0)$ \\ $(0, 1, 0)$ & $(3, 3, 3)$ \end{tabular} \end{center} The image under $LM^{-1}$ of these configurations both have fractional part $\vec 0$, so we have $[(0,0,0)]_M,[(0,1,0)]_M \in F^M_0$. \end{example} Let us now establish several basic properties on frackets. In general, $F^L_{\vec{f}}$ is not necessarily a group, but the zero fracket is closed under addition: \begin{lemma} \label{lem:zerofrackgroup} The zero fracket $F^L_0$ is a subgroup of $\mathcal{K} (L)$. \end{lemma} Since all cosets have the same size, we have the following result. \begin{lemma} \label{lem:fracketsizesame} For any fractional vectors $\vec{f}$ and $\vec{g}$ such that $F^L_{\vec{f}}$ and $F^L_{\vec{g}}$ are both non-empty, we have that $\|F^L_{\vec{f}}\| = \|F^L_{\vec{g}}\|$. \end{lemma} We can go between the elements of $F^L_0$ and $F^M_0$ by the invertible linear transformations $ML^{-1}$ and $LM^{-1}$. This map preserves vector addition and integrality, so we get the following: \begin{proposition}\label{prop:frack_equals_s_frack} For an arbitrary ii-pair $(L,M)$, we have $F^L_0 \cong F^M_0$. \end{proposition} Recall that the cardinality of the critical group $\critgroup(L)$ is given by the determinant of $L$, denoted by $\|L\|$. Since $F^L_0$ is a subgroup of $\mathcal{K}(L)$ and $F^M_0$ is a subgroup of $\mathcal{K}(M)$ and from \cref{prop:frack_equals_s_frack}, we immediately see that $\|F^L_0\|$ divides $\gcd(\|L\|, \|M\|)$. Moreover, from \cref{lem:fracketsizesame} and \cref{prop:frack_equals_s_frack} we get the following result. \begin{corollary} \label{cor:fracketallsamesize} For any ii-pair $(L,M)$, all $L$-frackets and all $M$-frackets have the same size. \end{corollary} \begin{lemma} \label{lem:equivalencefrackets} Suppose $\vv, \vec u$ are integral vectors with $\vv \equiv_L \vec u$. Then, $\vv$ and $\vec u$ belong to the same $L$-fracket. \end{lemma} \begin{proof} From $\vv \equiv_L \vec u$, we get $\vv = \vec u + L \vec z$ for some integer vector $\vec z$. Then, $ML^{-1}\vv - ML^{-1}\vec u = M \vec z$. Since $M$ is integral, $M \vec z$ is integral, so $ML^{-1} \vv - ML^{-1} \vec u$ is integral. Therefore, we get $\{ML^{-1} \vv\} = \{ML^{-1} \vec u\}$, so $\vv$ and $\vec u$ belong to the same fracket. \end{proof} Since $F^L_0$ is a subgroup of the group $\mathcal{K}(L)$, we can consider the quotient group $\mathcal{K}(L) / F^L_0$. The elements of this group are the images of the fractional vectors indexing all the frackets of $L$. \begin{example} \label{ex:Lfracketquotient} Consider the $ii$-pair coming from the $(L,M)$ pair studied in \cref{ex:usualbad}. We can look at the 2nd column (or the 5th column) of \cref{table:toy_stats} to extract the frackets. There is an $L$-fracket corresponding to each of the following fractional vectors: \[(0, 0, 0), (\nicefrac 1 3, \nicefrac 1 6, 0), (\nicefrac 2 3, \nicefrac 1 3, 0), (0, \nicefrac 1 2, 0) (\nicefrac 1 3, \nicefrac 2 3, 0), (\nicefrac 2 3, \nicefrac 5 6, 0).\] Beware that the image of these fractional vectors under the map $LM^{-1}$ is not necessarily an integral vector. However, there exists a vector in each of the corresponding frackets. By taking a vector from each fracket, we get the elements of $\mathcal{K}(L) / F^L_0$: $$(0,0,0),(3,2,0),(4,3,2),(1,1,0),(4,3,0),(5,4,2).$$ \end{example} \subsection{Computing the size of the frackets} Recall that from \cref{cor:fracketallsamesize}, for any ii-pair $(L,M)$, all $L$-frackets and all $M$-frackets have the same size. So we only need to be able to compute the size of $F^L_0$ in order to obtain the size of all frackets. In this subsection, we show an elegant formula for computing $\|F^L_0\|$. We begin with a lemma that describes the largest invariant factor of each of $\mathcal{K}(L) / F^L_0$ and $\mathcal{K}(M) / F^M_0$. Given a matrix $M$ with rational entries, we use $\flcm(M)$ to denote the least common multiple of all the denominators of the entries (written in irreducible fractions). Given a matrix $M$ with integer entries, we use $\gcd(M)$ to denote the greatest common divisor of all entries. Given matrices $L,M$ with integer entries, we use $\gcd(L,M)$ to denote the greatest common divisor of all entries of $L$ and $M$. \begin{lemma}\label{lem:frackets_largest_inv_factors} The largest invariant factors for $\mathcal{K}(M) / F^M_0$ and $\mathcal{K}(L) / F^L_0$ are $\flcm(\LM)$ and $\flcm(\ML)$, respectively. \end{lemma} \begin{proof} Let $k = \flcm(\LM)$. Then for any integral vector $\vec{v}$, since $k\LM$ is an integral matrix, we have $k \vec{v} \in F^M_0$. And $k$ is the smallest integer that have this property, since we can use the unit vectors for our $\vec{v}$ as well. So $k$ is the smallest value such that $k\vec{v} = 0$ for all $\vec{v} \in \critgroup(M) / F^M_0$. Thus, $k$ must be the size of the largest invariant factor of $\critgroup(M) / F^M_0$. The other argument comes from applying the claim we just proved on the ii-pair $(M,L)$. \end{proof} \begin{example} Let us revisit the signed graph from our running example. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \end{center} Recall that from \cref{ex:Lfracketquotient}, the nonempty frackets are indexed by the following fractional vectors: \[(0, 0, 0), (0, \nicefrac 1 2, 0), (\nicefrac 2 3, \nicefrac 1 3, 0), (\nicefrac 2 3, \nicefrac 5 6, 0), (\nicefrac 1 3, \nicefrac 1 6, 0), (\nicefrac 1 3, \nicefrac 2 3, 0).\] The collection of $L$-frackets has group structure $\mathbb Z_6$. One way to see this is by considering $(\nicefrac 1 3, \nicefrac 1 6, 0)$ as a generator. Therefore, the largest invariant factor of $\critgroup(L) / \fracket^L_0$ has size 6. If we look at \begin{gather} ML^{-1} = \begin{pmatrix} \nicefrac 4 3 & - \nicefrac 4 3 & - \nicefrac 1 3 \\ - \nicefrac 5 6 & \nicefrac 4 3 & - \nicefrac 1 6 \\ 0 & 0 & 1 \end{pmatrix}, \end{gather} we can see that $\flcm(ML^{-1}) = 6$, which does indeed match the size of the largest invariant factor of $\critgroup(L) / \fracket^L_0$. \end{example} From \cref{lem:frackets_largest_inv_factors}, we can make a general statement about the size of the zero fracket. \begin{theorem}\label{theorem:zero_fracket_size} Let $(L,M)$ be any $ii$-pair. Let $p_M$ be the product of the invariant factors of $\SK(M)/F_0^M$ excluding the largest invariant factor, and let $p_L$ be the product of the invariant factors of $\SK(M)/F_0^M$ excluding the largest invariant factor. Then, $\|\fracket^L_0\| = \frac{\gcd(\|L\|\ML,\|M\|\LM)}{\gcd(p_M, p_L)}$. \end{theorem} \begin{proof} Let $p_M$ denote the product of the invariant factors of $\critgroup(M) / F^M_0$, excluding the largest invariant factor. Then, we see that from \cref{lem:frackets_largest_inv_factors}, together with $\flcm(LM^{-1}) = \|M\| / \gcd(\|M\|\LM)$, we get: \begin{align} \|F^M_0\| p_M &= \|F^M_0\| \paren{ \dfrac {\| \critgroup(M) / F^M_0\|} { \|M\| / \gcd(\|M\|\LM)} } \\ &= \|F^M_0\| \frac{\|M\| / \|F^M_0\|}{\|M\| / \gcd(\|M\|\LM)} \\ &= \gcd(\|M\|\LM). \end{align} Similarly, we have $\|\fracket^L_0\|p_L = \gcd(\|L\|\ML)$. Using \cref{prop:frack_equals_s_frack}, we have $\|\fracket^L_0\|p_M = \|F^M_0\| p_M = \gcd(\|M\|\LM)$. Combining these results together, we get \[\|\fracket^L_0\| = \frac{\gcd(\|L\|\ML, \|M\|\LM)}{ \gcd(p_M, p_L)}.\] \end{proof} \begin{example}\label{example:naive_zero_fracket} Let us revisit the graph from our running example. \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \end{center} \begin{multicols}{2} \begin{center} \begin{tabular}{c \| c} $L$-Frackets & $M$-Frackets \\ \hline $(0, 0, 0)$ & $(0, 0, 0)$ \\ $(\nicefrac 1 3, \nicefrac 1 6, 0)$ & $(0, \nicefrac 1 4, 0)$ \\ $(\nicefrac 2 3, \nicefrac 1 3, 0)$ & $(0, \nicefrac 1 2, 0)$ \\ $(0, \nicefrac 1 2, 0)$ & $(0, \nicefrac 3 4, 0)$\\ $(\nicefrac 1 3, \nicefrac 2 3, 0)$ & \\ $(\nicefrac 2 3, \nicefrac 5 6, 0)$ & \end{tabular} \end{center}\columnbreak \[\|L\| ML^{-1} = \begin{pmatrix} 16 & - 16 & - 4 \\ - 10 & 16 & -2 \\ 0 & 0 & 2 \end{pmatrix}\]\[ \|M\| LM^{-1} = \begin{pmatrix} 16 & 16 & 8 \\ 10 & 16 & 6 \\ 0 & 0 & 8 \end{pmatrix}\] \end{multicols} From the table, we see that $\mathcal{K}(M) / F^M_0 \cong \ZZ_4$, and $\mathcal{K}(L) / \fracket^L_0 \cong \ZZ_6$. Then $p_M = 1$ and $p_L = 1$ since there is only one invariant factor. We also see that the greatest common divisor of $\|M\|LM^{-1}$ and $\|L\|ML^{-1}$ is $2$, so we expect the zero fracket to have size $2$. Indeed, $F^L_0$ consists only of the equivalence classes of $(0, 0, 0)$ and $(3, 3, 3)$. \end{example} In the above example, we needed to compute the structure of $\mathcal{K}(M) / F^M_0$ and $\mathcal{K}(L) / \fracket_0$ in order to obtain the size of the frackets. However, in many cases (conjecturally most cases for chip-firing pairs coming from graphs \cite[Conjecture 2]{clancy2015}), we can bypass this computation. If either $\mathcal{K}(M) / F^M_0$ or $\mathcal{K}(L) / \fracket^L_0$ is cyclic, we get a much simpler expression for $\|\fracket^L_0\|$. \begin{corollary} \label{cor:cyclic_fracket_size} Choose an arbitrary ii-pair $(L, M)$. Then, $\critgroup(M) / F^M_0$ is cyclic if and only if $\|\fracket^L_0\| = \gcd(\|M\| \LM)$. Similarly, $\critgroup(L) / \fracket_0^L$ is cyclic if and only if $\|\fracket^L_0\| = \gcd(\|L\|\ML)$. \end{corollary} \begin{proof} If $\SK(M)/F_0^M$ is cyclic, then $\|\SK(M)/F_0^M\| = \flcm(LM^{-1})$ by \cref{lem:frackets_largest_inv_factors}. Thus, \begin{gather} \frac{\|M\|}{\|F_0^M\|} = \|\SK(M)/F_0^M\| = \flcm(LM^{-1}) = \frac{\|M\|}{\gcd(\|M\|LM^{-1})}, \end{gather} establishing that $\|F_0^M\| = \gcd(\|M\|LM^{-1})$. For the other direction of the proof, suppose $\|\fracket^M_0\| = \gcd(\|M\| \LM)$. Then, $\|\critgroup(M) / F^M_0\| = \|M\| / \gcd(\|M\| \LM)$. By \cref{lem:frackets_largest_inv_factors}, the largest invariant factor of $\critgroup(M) / F^M_0$ also has size $\flcm (LM^{-1}) = \|M\| / \gcd(\|M\| \LM)$. Therefore, $\critgroup(M) / F^M_0$ has exactly one invariant factor, which means it is cyclic. \end{proof} Therefore, if there somehow is a guarantee that either one of $\mathcal{K}(M) / F^M_0$ or $\mathcal{K}(L) / \fracket^L_0$ is cyclic, \cref{cor:cyclic_fracket_size} gives us a way to enumerate the size of the frackets in a very simple manner. For example, consider the ii-pair $(L,M)$ coming from a signed graph. If we know that the underlying graph has a cyclic critical group $\mathcal{K}(M)$, then $\mathcal{K}(M) / F^M_0$ must also be cyclic. \begin{example}\label{example:soph_zero_fracket} Let us revisit \cref{example:naive_zero_fracket} using \cref{cor:cyclic_fracket_size}, given that the underlying graph's critical group is cyclic. All we need is to compute $\|M\|LM^{-1}$ (which we have already done in \cref{example:naive_zero_fracket}). We see that $\gcd(\|M\|LM^{-1}) = 2$, which matches $\|\fracket_0^L\|$. \end{example} \subsection{Analyzing the number of fixed points in the involution map} In \cref{def:ivomap} we defined an involution $\ivo$ on the set of preimages of superstable configurations of an $M$-matrix, given a chip-firing pair $(L,M)$. In this section, we use the techniques developed in the previous subsections to count the number of fixed points of $\ivo$. Recall that the fixed points of $\ivo$ are $\vec s \in \sstab(M)$ such that $\{\LM(\cm - 2\ss)\} = \vec{0}.$ \begin{theorem}\label{thm:numsols} The number of $\ss \in \sstab(M)$, satisfying $\cm - 2\ss \in F^M_0$ is equal to either $0$ or $\|F^M_0\|d$, where $d$ is the number of elements of $\mathcal{K}(M) / F^M_0$ with order at most 2. \end{theorem} \begin{proof} Suppose that there exists some $s \in \sst(M)$ such that $\cm - 2s \in F^M_0$. Note that every element $g \in \SK(M)$ is uniquely expressible as $s + h + f$ for some $h \in \SK(M)/F^M_0$, $f \in F^M_0$. Then, $\cm - 2g = (\cm - 2s) - 2f - 2h$. Since $\cm - 2s \in F^M_0$ and $2f \in F^M_0$, we have that $\cm - 2g \in F^M_0$ if and only if $2h \in F^M_0$. There are $\|F_0^M\|$ possible choices for $f$, and $d$ possible choices for $h$, so there are $\|F_0^M\|d$ such options for $g$. \end{proof} \begin{example} Let's revisit the running example of a $(L,M)$-pair coming from a signed graph. \begin{center} \begin{tabular}{c \| c} $\sstab(M)$ & Its image (under $LM^{-1}$) \\ \hline $(0, 0, 0)$ & $(0, 0, 0)$ \\ $(0, 0, 1)$ & $(1, \nicefrac 3 4, 1)$ \\ $(0, 0, 2)$ & $(2, \nicefrac 3 2, 2)$ \\ $(0, 1, 0)$ & $(2, 2, 0)$ \\ $(0, 1, 1)$ & $(3, \nicefrac{11}{4}, 1)$ \\ $(1, 0, 0)$ & $(2, \nicefrac 5 4, 1)$ \\ $(1, 1, 0)$ & $(4, \nicefrac{13}{4}, 1)$ \\ $(2, 0, 0)$ & $(4, \nicefrac 5 2, 2)$ \\ \hline $\cm = (2, 1, 2)$ & $(8, 6, 4)$ \end{tabular} \end{center} From the above table, we see that $\cm \in F^M_0$. Therefore, the $4$ superstable preimages that $2\ss$ is in the zero fracket of $M$ will be the fixed point of $\mu$. Recall that $\mathcal{K}(M) = \ZZ_8$ and $\|F^M_0\| = 2$ (from \cref{example:soph_zero_fracket}). Therefore, $\mathcal{K}(M) / F^M_0 \cong \ZZ_4$, which has $2$ elements of order at most $2$. Using $\cref{thm:numsols}$ with $d = 2$ and $\|F^M_0\| = 2$, we can verify that we get $4$ fixed points. \end{example} The remainder of this subsection will be on other more specific observations and corollaries of \cref{thm:numsols} and \cref{cor:cyclic_fracket_size}. \begin{proposition}\label{oddorder} If $\cm$ has odd order in $\mathcal{K}(M) / F^M_0$, then the number of fixed points of $\ivo$ is nonzero. \end{proposition} \begin{proof} Let $H$ be the subgroup of $\mathcal{K}(M) / F^M_0$ generated by $\cm$. Since $\cm$ has odd order, we know that $\|H\|$ is odd. There is no element of $H$ with order 2, so it follows that there is some element $g \in H$ such that $2g = \cm$. Then, $g$ is a fixed point. \end{proof} If $\mathcal{K}(M) / F^M_0$ is cyclic and $\cm$ has even order, we have an exact criterion to check whether the number of fixed points is zero or not. \begin{corollary} Suppose $\mathcal{K}(M) / F^M_0$ is cyclic and $\cm$ has even order in $\mathcal{K}(M) / F^M_0$. Then, the number of fixed points of $\ivo$ is nonzero if and only if $\frac{\|\mathcal{K}(M) / F^M_0\|}{\ord(\cm)}$ is even. \end{corollary} \begin{proof} Suppose $g$ generates $\mathcal{K}(M) / F^M_0$, so $\cm = kg$ for some positive integer $k$. Since $\ord(\cm) k g = 0$ and $\mathcal{K}(M) / F^M_0$ is cyclic, we know that $\|\mathcal{K}(M) / F^M_0\|$ divides $\ord(\cm)k$. Therefore, $\|\mathcal{K}(M) / F^M_0\| / \ord(\cm)$ must divide $k$. Since $\frac{\|\mathcal{K}(M) / F^M_0\|}{\ord(\cm)}$ is even, we see that $k$ must be even as well. Therefore, $(k/2)g$ is a fixed point of $\ivo$. For the other direction of the proof, assume there is some fixed point $g$ of $\ivo$. then we have $2g = \cm$ in $\mathcal{K}(M) / F^M_0$. We know that $2 \ord(\cm)g = 0$, so $\ord(g)$ must divide $2 \ord(\cm)$. Let $H$ be the subgroup of $\mathcal{K}(M) / F^M_0$ generated by $g$. Then, $\cm \in H$ and $\|H\|$ cyclic, so $\ord(\cm)$ must divide $\|H\| = \ord(g)$. So we have $\ord(g) \| 2\ord(\cm)$ and $\ord(\cm) \| \ord(g)$. Thus, one of the two happens: $\ord(g) = \ord(\cm)$ or $\ord(g) = 2 \ord(\cm)$. Suppose for the sake of contradiction that $\ord(g) = \ord(\cm)$. In other words, $\ord(g) = \ord(2g)$. This means that $\ord(g)$ is odd. By our assumption that $\ord(g) = \ord(\cm)$, it follows that $\ord(\cm)$ is also odd, contradicting our assumption that $\cm$ is of even order. Therefore, $\ord(g) = 2\ord(\cm)$. Then $\frac{\|\mathcal{K}(M) / F^M_0\|}{\ord(\cm)} = 2 \frac{\|\mathcal{K}(M) / F^M_0\|}{\ord(g)}$, so $\frac{\|\mathcal{K}(M) / F^M_0\|}{\ord(\cm)}$ must be even. \end{proof} \subsection{Chip-firing pairs coming from complete graphs} In this subsection, we will be focusing on chip-firing pairs coming from signed graphs, where the underlying graph is the complete graph. Same as the chip-firing pairs coming from signed graphs we have seen before, $L$ is going to denote the (reduced) Laplacian of the signed graph and $M$ is going to denote the (reduced) Laplacian of the underlying complete graph $K_n$, where $n$ is even. \begin{lemma}\label{KLMinv.den} For such $(L,M)$ pairs coming from a complete graph $K_n$ where $n$ is even, \[\frac{n}{2}\LM \text{ is an integer matrix.}\] \end{lemma} \begin{proof} We will show that every term in $\LM$ has a denominator dividing $n/2$. The entries of the inverse of $M$ can be described as the following: \[\Mi_{i,j} = \begin{cases} \frac{2}{n} & \text{ if } i = j \\ \frac{1}{n} & \text{ if } i \neq j. \end{cases}\] From this we analyze the entry $(LM^{-1})_{i,j} = \sum_{k = 1}^{n-1} L_{i,k}\Mi_{k,j}$. Let $\alpha_i$ to denote the number of $-1$'s that appear in row $i$. Then there will be exactly $n-2-\alpha_i$ many $+1$'s in that row. We do a case-by-case analysis. In the case $i=j$, we have that $$(\LM)_{i,j} =\sum_{k = 1}^{n-1} L_{i,k}\Mi_{k,j}= \frac{2(n-1)}{n}-\frac{\alpha_i}{n}+\frac{n-2-\alpha_i}{n} = \frac{3n-4-2\alpha_i}{n}.$$ If $L_{i,j} = -1$, then we have $$(\LM)_{i,j} = \sum_{k = 1}^{n-1} L_{i,k}\Mi_{k,j}= \frac{n-1}{n}-\frac{2}{n} - \frac{\alpha_i-1}{n} +\frac{n-2-\alpha_i}{n} = \frac{2n-4-2\alpha_i}{n}.$$ Finally when $L_{i,j} = 1$ we have \[(\LM)_{i,j} = \sum_{k = 1}^{n-1} L_{i,k}\Mi_{k,j}= \frac{n-1}{n} + \frac{2}{n} - \frac{\alpha_i}{n} +\frac{n-3-\alpha_i}{n} = \frac{2n-2-2\alpha_i}{n}.\] Since $n$ is even, the numerators in all cases are also even. So $\frac{n}{2}\LM$ is an integer matrix. \end{proof} Using the above, we now show that any signed graph of a complete graph $K_n$ where $n$ is a fixed even number, contain a nontrivial common subgroup. Actually we show something stronger, that the zero frackets (a subgroup of the critical group) of any signed graph with underlying graph being the complete graph $K_n$ where $n$ is even, contain a nontrivial common subgroup. \begin{theorem} \label{thm:z0common} The zero fracket $F_0^L$ of any $(L,M)$-pair coming from some signed graph where the underlying graph is the complete graph $K_n$ where $n$ is even, has a subgroup isomorphic to $\ZZ_2^{n-2}$. \end{theorem} \begin{proof} Consider the configuration $\vec s_i := \frac{n}{2} \vec{e_i}$. Then by \cref{KLMinv.den}, we have that $\LM\paren{\frac{n}{2}\vec{e_i}}$ is an integer vector, so this is a valid configuration in $R^+$. From \cref{thm:22floor}, we have that $\vec s_i$ is a superstable perimage. Additionally, $n \vec{e_i} = M(\vec{1} + \vec{e_i})$, so $\vec s_i$ has order 2 in $F_0^L$. For each $I \subseteq \{1\dots n\}$ such that $0 \leq \|I\| \leq \frac{n-2}{2}$, we know that $\sum_{i \in I} s_i$ is a superstable preimage: when we fire some set $S$, for any $v \in S$ we start with either zero or $\frac{n}{2}$ chips, will lose $n-1$ chips, and gain back at most $\|I\|-1 \leq \frac{n-4}{2}$ chips, resulting in that vertex having at most $-1$ chips after the firing. And since $\sum_{i \in I} s_i$ is a sum of elements of order $2$, it has order $2$ as well in $F_0^L$. There are $\frac{2^{n-1}}{2}$ possible choices for $I$ such that $0 \leq \|I\| \leq \frac{n-2}{2}$, so the set of $\sum_{i \in I} s_i$'s form a subgroup of $F_0^L$ isomorphic to $\ZZ_2^{n-2}$. \end{proof} \begin{example} Consider the graph $K_6$. Over all possible signed graphs coming from $K_6$, there are seven possible critical groups up to isomorphism: \begin{gather} \mathbb Z_6 \oplus \ZZ_6 \oplus \ZZ_6 \oplus \ZZ_6, \\ \ZZ_{36} \oplus \ZZ_{12} \oplus \ZZ_2 \oplus \ZZ_2, \\ \ZZ_{78} \oplus \ZZ_6 \oplus \ZZ_2 \oplus \ZZ_2, \\ \ZZ_{50} \oplus \ZZ_{10} \oplus \ZZ_{2} \oplus \ZZ_{2}, \\ \ZZ_{64} \oplus \ZZ_8 \oplus \ZZ_2 \oplus \ZZ_{2}, \\ \ZZ_{132} \oplus \ZZ_4 \oplus \ZZ_2 \oplus \ZZ_{2}, \\ \ZZ_{36} \oplus \ZZ_4 \oplus \ZZ_4 \oplus \ZZ_4. \end{gather} We can check that $\mathbb Z_2^4$ is a subgroup of all cases above. \end{example} \section{Further questions} In this section, we discuss the remaining questions that naturally follow this study. The duality between $z$-superstable and critical configurations of $(L,M)$-pairs constructed in Section $3$, relied on the involution $\ivo$ on the set of superstable configurations of $M$. When we constructed $\ivo$, we used the equivalence relation given by $M$. It would be interesting if we can skip that process as well. \begin{question} Can one construct a duality between the set of $z$-superstable configurations and the set of critical configuration of chip-firing pairs without ever relying on the equivalence class computation (of $L$ or $M$)? \end{question} A lot of examples we used came from signed graphs and their Laplacian. It would be interesting if we can generalize \cref{thm:z0common} to more general graphs: \begin{question} Let $(L,M)$ be a chip-firing pair coming from a signed graph with an underlying graph $G$. Is there a way to obtain the biggest group that all critical groups of the chip-firing pairs coming from $G$ contain as a subgroup up to isomorphism? \end{question} For signed graphs, there is an operation called \newword{vertex-switching} \cite{Zaslavsky}. It is known that there is an isomorphism between the critical groups of signed graphs that are switching equivalent to one another. So it is natural to ask the following question: \begin{question} Can one construct a natural isomorphism between the critical groups of switching equivalent $(L, M)$ pairs (without any reliance on the obvious bijection coming from the equivalence classes)? \end{question} Recall that in \cref{thm:numsols} we could only describe the number of fixed points of $\ivo$ when we were guaranteed that it was nonzero. It would be interesting if we have an elegant way to describe exactly when the number of fixed points of $\ivo$ is nonzero in terms of $L$ and $M$: \begin{question} Is there a way to describe exactly when the map $\ivo$ has zero fixed points? \end{question} \section{Acknowledgements} The work presented here was conducted as part of an REU at Texas State University in the summer of 2024, sponsored by NSF. We thank NSF and Texas State for the support and the stimulating work environment. \printbibliography \end{document} \section{Introduction} Chip-firing is a game that takes place on a connected graph $G$, where chips are distributed across the vertices of $G$ and moved to adjacent vertices based on a straightforward rule. This dynamical system has a profound theory that links to various fields in mathematics and physics \cite{BakTangWiesenfeld88}, \cite{Dhar90}, \cite{Gabrielov93}. For further details, please see the recent textbooks \cite{Klivans} and \cite{CorryPerk}. Consider a graph $G$ where one specific vertex is designated as the \emph{sink}, and chips are placed on each non-sink vertex. The allocation of chips is described by an integer vector $\vec{c} \in \mathbb{Z}^{n}$, known as a \textit{(chip) configuration}. A non-sink vertex with chips equal to or greater than its degree can fire, distributing one chip to each adjacent vertex. A configuration $\vec{c}$ is termed \emph{stable} if no non-sink vertex is able to fire. For a connected graph $G$, any starting configuration will eventually reach stability via sequence of firings, with some chips moving to the sink vertex. Letting $v_1, ..., v_{n}$ represent the non-sink vertices of $G$, the chip-firing rules can be described using $L_G$, the $n \times n$ \textit{reduced Laplacian} matrix of $G$. The outcome of firing a vertex $v_i$ on a chip configuration $\vec{c}$ is represented by $\vec{c} - L\vec{e_i}$, where $\vec{e_i}$ denotes the $i$-th standard basis vector. The matrix $L_G$ establishes an equivalence relation among the vectors in ${\mathbb Z}^{n}$, with $\vec{c}$ and $\vec{d}$ being \emph{firing equivalent} if $\vec{c} - \vec{d}$ lies within the image of $L_G$. This determines the \emph{critical group} of $G$, as described by $\mathcal{K}(G) := {\mathbb Z}^n/\Img L_G$ \cite{Biggs99}. A configuration $\vec{c}$ is considered \emph{valid} (or \emph{effective}) if for all $i = 1, \dots, n$, the condition $c_i \geq 0$ holds. The goal is to identify notable valid configurations within each equivalence class $[\vec{c}] \in {\mathcal K}(G)$. It turns out that each $[\vec{c}]$ contains a distinct valid configuration that is \emph{critical}, which means it is stable and can be derived from a sufficiently large configuration $\vec{b}$. Stabilizing the sum of two critical configurations results in a critical configuration. One can show that each $[\vec{c}]$ contains a unique valid configuration that is \emph{superstable}, meaning that it is stable under \emph{set-firings}. A superstable configuration represents a solution to an energy minimization problem and aligns with the concept of a \emph{$G$-parking function}. For a connected graph $G$, a straightforward bijection exists linking the set of critical configurations and the set of superstable configurations, which are both correspondingly in bijection with the set of spanning trees of $G$. This simple map (take a configuration, negate it coordinate-wise from a certain maximal configuration) is what is called the \newword{duality map} between superstable configurations and critical configurations, and our focus is to extend this map to more general models. Recently, chip-firing has been extended to more general settings, where the reduced Laplacian of a graph is replaced by other matrices (see for instance \cite{gabrielov} and \cite{GuzKliMmatrices}). We use a matrix $M$ to define a firing rule that mimics the graphical setting: firing $v_i$ now takes a configuration $\vec{c}$ to $\vec{c} - M \vec{e_i}$, where $\vec{e_i}$ stands for the unit vector with $1$ at the $i$-th coordinate. For a well-defined notion of chip-firing we require that $M$ satisfies an \emph{avalanche finite} property, so that repeated firings of any initial configuration eventually stabilize in an appropriate sense. The class of matrices with this property are known as \emph{$M$-matrices}, and can be characterized in a number of ways (see \cref{def:Mmatrix} below). In \cite{GuzKliMmatrices}, Guzm\'an and Klivans have shown that the chip-firing theory defined by an $M$-matrix leads to good notions of critical and superstable configurations. They further generalized this model by introducing an invertible matrix $L$, called chip-firing pairs, and extended the definition of critical and superstable configurations to that model \cite{GuzKlivans}. Our goal is to extend the duality between critical and superstable configurations to $(L,M)$-chip firing pairs. In Section~$2$ we summarize the Guzm\'an-Klivans theory of chip-firing pairs and also review the previously known duality between superstable and critical configurations of $M$-matrices. In Section~$3 $ we provide our main result on extending the duality map to chip-firing pairs. In Section~$4$ we study some properties of the map discussed in the main result. \section{Prerequisites} In this section we review the definition of chip-firing pairs. After that we review the duality map between superstable configurations and critical configurations for $M$-matrices. Then we will go over the tools developed in \cite{cho2024} that we will use to deal with the $z$-superstable and critical configurations of chip-firing pairs. \subsection{Chip firing pairs} In \cite{GuzKliMmatrices}, Guzm\'an and Klivans generalized the chip-firing on graphs to \newword{$M$-matrices}. $M$-matrices are used in various fields such as economics or scientific computing \cite{BurmanErn05}, \cite{CiarletRaviart73}, \cite{Leontief41}, \cite{Plemmons77}. Guzm\'an and Klivans further generalized this by introducing an invertible integer matrix $L$ in \newword{chip-firing pairs} $(L,M)$ introduced in \cite{GuzKlivans}. \begin{definition} \label{def:Mmatrix} Suppose $M$ is an $n \times n$ matrix such that $(M)_{ii} > 0$ for all $i$ and $(M)_{ij} \leq 0$ for all $i \neq j$. Then $M$ is called an (invertible) \emph{$M$-matrix} if any of the following equivalent conditions hold: \begin{enumerate} \item $M$ is avalanche finite; \item The real part of the eigenvalues of $M$ are positive; \item The entries of $M^{-1}$ are non-negative; \item There exists a vector $\vec{x} \in {\mathbb R}^{n}$ with ${\vec x} \geq \vec{0}$ such that $M \vec{x}$ has all positive entries. \end{enumerate} \end{definition} The pair $(L,M)$, an $M$-matrix $M$ together with an invertible integer matrix $L$, is called a \emph{chip-firing pair}. The relevant (chip) \emph{configurations} $\vec{c} \in \mathbb{Z}^{n}$ are simply integer vectors with $n$ entries, and chip-firing is dictated by the matrix $L$. In particular, $(M,M)$ recovers the chip-firing on $M$-matrices and $(L_G,L_G)$ when $L_G$ is the (reduced) Laplacian of a graph recovers the classical chip-firing model on graphs. \begin{definition}\label{defn:valid} Suppose $(L,M)$ is a chip-firing pair. A configuration $\vec{c}$ is \emph{valid} if $\vec{c} \in S^+$, where \[S^+ = \{LM^{-1} \vec{x} : LM^{-1}\vec{x} \in \mathbb{Z}^{n}, \vec{x} \in \mathbb{R}^n_{\geq 0}\}. \] Equivalently, a configuration $\vec{c}$ is valid if $ML^{-1}\vec{c} \in R^+$, where \[R^+ = \{ \vec{x} \in \mathbb{R}^n_{\geq 0} : LM^{-1}\vec{x} \in \mathbb{Z}^{n} \}.\] \end{definition} In particular, for $(M,M)$, being valid is exactly the same as being a nonnegative integer vector. \begin{definition} Suppose $(L,M)$ is a chip-firing pair, and suppose that $\vec{c} \in S^+$ is a valid configuration. A site $i \in \{1,\dots, n\}$ is \emph{ready to fire} if \[\vec{c} - L\vec{e}_i \in S^+,\] so that the vector obtained by subtracting the $i$th row of $L$ from $\vec{c}$ is also valid. Similarly, suppose $\vec{x} \in R^+$. Then a site $i \in \{1,\dots, n\}$ is \emph{ready to fire} if \[\vec{x} - M\vec{e}_i \in R^+.\] A configuration $\vec{c}$ (in $S^+$ or $R^+$) is \emph{stable} if no site is ready to fire. \end{definition} If $i$ is ready to fire, we declare that $\vec{b} = \vec{c} - L \vec{e}_i \in S^+$ is derived from $\vec{c}$ through a \emph{legal firing}. Repeating this process, a vector $\vec{a} \in S^+$ is said to be derived from $\vec{c}$ through a \emph{sequence of legal firings}. For a configuration $\vec{c} \in S^+$ (or conversely, $\vec{d} \in R^+$), we define $\text{stab}_{S^+}(\vec{c})$ (and $\text{stab}_{R^+}(\vec{d})$) as the resulting configuration after executing a series of legal firings until no site remains eligible to fire. Adapting the proof presented in \cite[Theorem 2.2.2]{Klivans}, it can be established that both $\text{stab}_{S^+}(\vec{c})$ and $\text{stab}_{R^+}(\vec{d})$ are uniquely determined. When it is clear whether we are dealing with $S^{+}$ or $R^{+}$, we use $\text{stab}(\vec{x})$ to refer to the stabilization of the configuration $\vec{x}$. The definition of critical and superstable configurations in this model are as follows. \begin{definition} Given an $(L,M)$ pair, a configuration $\vec c \in S^+$ is reachable if there exists some configuration $\vec d \in S^+$ satisfying: \begin{itemize} \item $\vec d - L \vec e_i \in S^+$ for all $1 \leq i \leq n$ \item $\vec c = \vec d - \sum_{j = 1}^k L \vec e_j$ and $\vec d - \sum_{j = 1}^\ell L \vec e_j \in S^+$ for all $\ell < k$. \end{itemize} Given an $(L,M)$ pair, a configuration $\vec c \in S^+$ is critical if $\vec c$ is both stable and reachable. \end{definition} \begin{definition}[{\cite[Definition 4.3]{GuzKliMmatrices}}] A vector $f \in \ZZ^n$ with $f \geq 0$ is \newword{$z$-superstable} if for every $z \in \ZZ^n$ with $z \geq 0$ and $z \neq 0$ there exists $1 \leq i \leq n$ such that $f_i - (Lz)_i < 0$. \end{definition} It turns out that in the equivalence class given by the matrix $L$ in $S^{+}$, we can always find a unique representative that is critical and a unique representative that is $z$-superstable. \begin{theorem}[{\cite[Theorems 3.5, 4.3, 5.5]{GuzKlivans}\label{thm:class}}] Suppose $(L,M)$ is a chip-firing pair. Then there exists exactly one $z$-superstable configuration and one critical configuration in each equivalence class $[\vec{c}]_L$. \end{theorem} \begin{remark} For chip-firing pairs, there is the notion of $\chi$-superstable configurations and $z$-superstable configurations. From \cref{thm:class}, it is the $z$-superstable configurations that have the same size as the critical configurations. We will only focus on the $z$-superstable configurations, and from now on throughout the paper, we will just call them the superstable configurations of the chip-firing pair, omitting the letter $z$. \end{remark} In the next subsection, we go over the duality that is known to exist when $L=M$. \subsection{Duality for \texorpdfstring{$M$}{M}-matrices} If we take a chip-firing pair $(M,M)$, it recovers the chip-firing on $M$-matrices studied in \cite{GuzKliMmatrices}. Chip-firing on $M$-matrices generalizes many properties and results of the classical chip-firing on graphs, and one of them is the duality between superstable and critical configurations. Given an $M$-matrix, it turns out that there is a critical configuration that is a coordinate-wise greater or equal to every other critical configuration. We call this configuration $\cm$, given by taking all diagonal entries of $M$ minus one and forming a vector (in the classical case, this corresponds to having $\deg(v)-1$ chips for each vertex $v$). \begin{theorem}[\cite{GuzKliMmatrices}] \label{thm:usualMduality} Let $M$ be an $M$-matrix. Let $\cm$ denote the vector where each entry is coming from the corresponding diagonal entry $M_{ii}$ minus one. Then we have a bijection between superstable and critical configurations by the map $\vec{c} \rightarrow \cm - \vec{c}$. \end{theorem} \begin{example} \label{ex:unsignedkyle} Consider the following graph that has the reduced Laplacian to be $L_G = \begin{pmatrix} 3 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 3 \end{pmatrix}.$ \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-unsigned} \end{tikzpicture} \end{center} The superstable configurations and critical configurations are given in the following table: \begin{center}\label{table:unsigned_toy_ss} \begin{tabular}{ c \| c } Superstables & Criticals \\ \hline $(0, 0, 0)$ & $(2, 1, 2)$\\ $(0, 0, 1)$ & $(2, 1, 1)$\\ $(0, 0, 2)$ & $(2, 1, 0)$\\ $(0, 1, 0)$ & $(2, 0, 2)$\\ \end{tabular} \begin{tabular}{ c \| c } Superstables & Criticals \\ \hline $(0, 1, 1)$ & $(2, 0, 1)$\\ $(1, 0, 0)$ & $(1, 1, 2)$\\ $(1, 1, 0)$ & $(1, 0, 2)$\\ $(2, 0, 0)$ & $(0, 1, 2)$ \end{tabular} \end{center} \end{example} Notice that in the above example, we have a bijection between superstable configurations and critical configurations via the map $\vec{c} \rightarrow (2,1,2) - \vec{c}$. Recall that our goal is to extend this to $(L,M)$ chip-firing pairs. Next subsection will show that the map $\vec{c} \rightarrow \cmax - \vec{c}$ does not work for chip-firing pairs. \subsection{The usual duality map does not work for chip-firing pairs} Recall that the usual duality map between the superstable and critical configurations for graphs (and also $M$-matrices) is given by the map $\vec{c} \rightarrow \cm - \vec{c}$ for some fixed $\cm$. As can be seen in the example below, this does not work for $(L,M)$-pairs in general. The examples from this point throughout will be using $(L,M)$-pairs coming from a signed graph. The systematic study of signed graphs and their Laplacian was initiated by Zaslavsky in \cite{Zaslavsky} and also studied in \cite{MR666857}, \cite{MR4641710}, \cite{FundBapat}. \begin{example} \label{table:signed_toy_ss} We take $L$ to be the (reduced) Laplacian of the following signed graph and $M$ to be the (reduced) Laplacian of the underlying unsigned graph. The Laplacian of the signed graph is simply obtained from the Laplacian of the underlying graph, by changing the signs of entries corresponding to negative edges. \begin{figure}[ht] \centering \begin{minipage}{0.3\linewidth} \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed.tex} \end{tikzpicture} \end{center} \end{minipage} \begin{minipage}{0.6\linewidth} $$M = \begin{pmatrix} 3 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 3 \end{pmatrix} \hspace{5mm} L = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 2 & -1 \\ -1 & -1 & 3 \end{pmatrix}$$ \end{minipage} \label{fig:Signed-Kyle-S+} \end{figure} \begin{center} Configurations in $S^+$ \renewcommand{\arraystretch}{1.2} \begin{tabular}{ c \| c } \label{table:toy_ss} Superstables & Criticals \\ \hline $(0, 0, 0) $ & $(6, 4, 2) $ \\ $(1, 1, 0) $ & $(7, 5, 2) $ \\ $(4, 3, 2) $ & $(8, 6, 0) $ \\ $(5, 4, 2) $ & $(9, 7, 0) $ \\ \end{tabular} \begin{tabular}{c \| c} Superstables & Criticals \\\hline $(2, 2, 0) $ & $(8, 6, 2) $ \\ $(3, 3, 0) $ & $(9, 7, 2) $ \\ $(3, 2, 0) $ & $(6, 4, 1) $ \\ $(4, 3, 0) $ & $(7, 5, 1) $ \\ \end{tabular} \begin{tabular}{c \| c} Superstables & Criticals \\\hline $(5, 4, 0) $ & $(8, 6, 1) $ \\ $(6, 5, 0) $ & $(9, 7, 1) $ \\ $(6, 4, 0) $ & $(6, 5, 2) $ \\ $(7, 5, 0) $ & $(7, 6, 2)$ \end{tabular} \end{center} \end{example} Notice that in the table of superstable and critical configurations of the chip-firing pair, the coordinate-wise maximal critical configuration is $(9,7,2)$. However if we take the superstable configuration $(1,1,0)$, the vector we get by applying the traditional duality map $(9,7,2) - (1,1,0) = (8,6,2)$ is not a critical configuration. Even worse, there are many cases where $c_{max}$, the critical configuration that has coordinate-wise maximal entries does not even exist. \begin{example} \label{ex:nocmax} Consider the $(L,M)$-pair coming from the signed graph below. The underlying graph is $C_6$, the cycle on six vertices. \begin{figure}[ht] \centering \renewcommand{\arraystretch}{1.2} \begin{minipage}{.45\linewidth} \centering \begin{tikzpicture}[scale=.85] \input{tikz/c6-signed} \end{tikzpicture} \end{minipage} \begin{minipage}{.45\linewidth} \centering \begin{tabular}{ c } Critical configurations \\ \hline (9, 15, 17, 15, 9) \\ (12, 20, 23, 21, 13) \\ (13, 21, 23, 20, 12) \\ (7, 11, 12, 11, 7) \\ (10, 16, 18, 17, 11) \\ (11, 17, 18, 16, 10) \\ \end{tabular} \end{minipage} \label{fig:Cycle-5} \end{figure} As can be checked from the table of critical configurations above, there is no critical configuration that is the maximal in all coordinates. \end{example} \subsection{Finding the superstable/critical configurations of chip firing pairs.} In this subsection, we go over an alternative way to find the ($z$)-superstable and critical configurations of $(L,M)$ chip-firing pairs, developed in \cite{cho2024}. Let $\sstab(M)$ denote the set of superstable configurations of an $M$-matrix $M$ and let $\crit(M)$ denote the set of critical configurations. However, beware we are not going to be using $\sstab(L,M)$ to denote the set of superstable configurations of $(L,M)$ and same for $\crit(L,M)$. It turns out that for configurations in $S^{+}$, it is important to look at their preimages in $R^{+}$. Given any vector $\vec{f}$, we use $\floor{\vec{f}}$ to denote the vector obtained from $f$ by taking the floor at every coordinate. \begin{theorem}[{\cite[Theorem 3.2]{cho2024}}] \label{thm:22floor} Given an $(L, M)$ pair, a configuration $\vec c \in S^+$ is superstable/critical if and only if $\floor{ML^{-1} \vec c}$ is a superstable/critical configuration of $M$. \end{theorem} For example, we look at our running example coming from a signed graph. \begin{example} \label{ex:usualbad} Consider the signed graph studied in \cref{table:signed_toy_ss}. The table lists all superstable and critical configurations in $S^{+}$, their preimage in $R^{+}$, and the floor of the preimage. We can notice that the floor of the preimages are the superstable and critical configurations of the underlying graph we saw in \cref{ex:unsignedkyle} (however, not all superstable/critical configurations of the underlying graph are used). \begin{center} \begin{tikzpicture}[scale=2] \input{tikz/kyle-signed} \end{tikzpicture} \end{center} \begin{center}\label{table:toy_stats} \renewcommand{\arraystretch}{1.2} \begin{tabular}{ c \| c \| c \| c \| c \| c} $\LM(\sstab)$ & $\sstab$ & $\floor{\sstab}$ & $\LM(\crit)$& $\crit$ & $\floor{\crit}$\\\hline $ (0, 0, 0) $ & $(0, 0, 0) $ & $ (0, 0, 0) $ & $(6,4,2) $& $ (2, 0, 2) $ & $ (2, 0, 2) $ \\ $ (1, 1, 0) $ & $(0, \nicefrac{1}{2}, 0) $ & $ (0, 0, 0) $ & $(7,5,2) $& $ (2, \nicefrac{1}{2}, 2) $ & $ (2, 0, 2) $ \\ $ (4, 3, 2) $ & $(\nicefrac{2}{3}, \nicefrac{1}{3}, 2) $ & $ (0, 0, 2) $ & $(8,6,0) $& $ (\nicefrac{8}{3}, \nicefrac{4}{3}, 0) $ & $ (2, 1, 0) $ \\ $ (5, 4, 2) $ & $(\nicefrac{2}{3}, \nicefrac{5}{6}, 2) $ & $ (0, 0, 2) $ & $(9,7,0) $& $ (\nicefrac{8}{3}, \nicefrac{11}{6}, 0) $ & $ (2, 1, 0) $ \\ $ (2, 2, 0) $ & $(0, 1, 0) $ & $ (0, 1, 0) $ & $(8,6,2) $& $ (2, 1, 2) $ & $ (2, 1, 2) $ \\ $ (3, 3, 0) $ & $(0, \nicefrac{3}{2}, 0) $ & $ (0, 1, 0) $ & $(9,7,2) $& $ (2, \nicefrac{3}{2}, 2) $ & $ (2, 1, 2) $ \\ $ (3, 2, 0) $ & $(\nicefrac{4}{3}, \nicefrac{1}{6}, 0) $ & $ (1, 0, 0) $ & $(6,4,1) $& $ (\nicefrac{7}{3}, \nicefrac{1}{6}, 1) $ & $ (2, 0, 1) $ \\ $ (4, 3, 0) $ & $(\nicefrac{4}{3}, \nicefrac{2}{3}, 0) $ & $ (1, 0, 0) $ & $(7,5,1) $& $ (\nicefrac{7}{3}, \nicefrac{2}{3}, 1) $ & $ (2, 0, 1) $ \\ $ (5, 4, 0) $ & $(\nicefrac{4}{3}, \nicefrac{7}{6}, 0) $ & $ (1, 1, 0) $ & $(8,6,1) $& $ (\nicefrac{7}{3}, \nicefrac{7}{6}, 1) $ & $ (2, 1, 1) $ \\ $ (6, 5, 0) $ & $(\nicefrac{4}{3}, \nicefrac{5}{3}, 0) $ & $ (1, 1, 0) $ & $(9,7,1) $& $ (\nicefrac{7}{3}, \nicefrac{5}{3}, 1) $ & $ (2, 1, 1) $ \\ $ (6, 4, 0) $ & $(\nicefrac{8}{3}, \nicefrac{1}{3}, 0) $ & $ (2, 0, 0) $ & $(6,5,2) $& $ (\nicefrac{2}{3}, \nicefrac{4}{3}, 2) $ & $ (0, 1, 2) $ \\ $ (7, 5, 0) $ & $(\nicefrac{8}{3}, \nicefrac{5}{6}, 0) $ & $ (2, 0, 0) $ & $(7,6,2) $& $ (\nicefrac{2}{3}, \nicefrac{11}{6}, 2) $ & $ (0, 1, 2) $ \end{tabular} \end{center} \end{example} Thanks to \cref{thm:22floor}, it is much more convenient to deal with the preimages of configurations, especially when trying to check if it is superstable or critical. Given a superstable configuration in $S^{+}$, we are going to denote its preimage in $R^{+}$ as \newword{superstable preimage} and for a critical configuration in $S^{+}$, we are going to denote its preimage in $R^{+}$ as \newword{critical preimage}. We are also going to use $\sstab(L,M)$ to denote the set of superstable preimages of a $(L,M)$ chip-firing pair, and use $\crit(L,M)$ to denote the set of critical preimages. \begin{scope}[every node/.style={circle,draw}] \node (1) at (0,1) {$1$}; \node (2) at (1,1) {$2$}; \node (3) at (1,0) {$3$}; \node (q) at (0,0) {$q$}; \end{scope} \begin{scope}[every node/.style={draw=black}, every edge/.style={draw=black}] \path (1) edge (3); \path (1) edge (q); \path (3) edge (2); \path (3) edge (q); \path (1) edge (2); \end{scope} \begin{scope}[every node/.style={circle,draw}] \node (1) at (0,1) {$1$}; \node (2) at (1,1) {$2$}; \node (3) at (1,0) {$3$}; \node (q) at (0,0) {$q$}; \end{scope} \begin{scope}[every node/.style={draw=black}, every edge/.style={draw=black}] \path (1) edge (3); \path (1) edge (q); \path (3) edge (2); \path (3) edge (q); \end{scope} \begin{scope}[every node/.style={text=red,font=\bfseries}, every edge/.style={draw=red}] \path (1) edge node[above] {$-$} (2); \end{scope} \def\r{2cm} \def\a{360/6} \begin{scope}[every node/.style={circle,draw}] \node (1) at (0:\r) {$1$}; 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\path (3) edge node[right] {$-$} (2); \path (1) edge node[above] {$-$} (2); \end{scope} \begin{scope}[every node/.style={circle,draw}] \node (1) at (0,1) {$1$}; \node (2) at (1,1) {$2$}; \node (3) at (1,0) {$3$}; \node (q) at (0,0) {$q$}; \end{scope} \begin{scope}[every node/.style={draw=black}, every edge/.style={draw=black}] \path (2) edge (q); \path (1) edge (q); \path (3) edge (2); \path (3) edge (q); \path (1) edge (2); \end{scope} \documentclass[11pt,reqno]{amsart} \usepackage{graphicx} \usepackage{xcolor} \usepackage{stackengine} \begin{document} :D This is what abuse of notation actually is \Huge \scalebox{3}[1]{$\mathcal{o}$} \scalebox{1}[3]{$\mathcal{o}$} \scalebox{5}[1]{$\mathcal{o}$} \scalebox{1}[5]{$\mathcal{o}$} \scalebox{5}[1]{$\mathcal{o}$} \scalebox{1}[5]{$\mathcal{o}$} \scalebox{10}[1]{$\mathcal{o}$} \scalebox{1}[10]{$\mathcal{o}$} \scalebox{10}[0.5]{$\mathcal{o}$} \scalebox{1}[5]{$\mathcal{o}$} \scalebox{10}[0.5]{$\mathcal{o}$} \scalebox{1}[5]{$\mathcal{o}$} \scalebox{10}[0.5]{$\mathcal{o}$} \stackinset{c}{}{c}{}{\rotatebox{45}{\scalebox{25}[0.5]{$\mathcal{o}$}}}{\rotatebox{-45}{\scalebox{25}[0.5]{$\mathcal{o}$}}} \color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!15}$\mathcal{o}$\color{black!30}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!90}$\mathcal{o}$\color{black!75}$\mathcal{o}$\color{black!60}$\mathcal{o}$\color{black!45}$\mathcal{o}$\color{black!30}$\mathcal{o}$ \end{document}
2412.02787v1	http://arxiv.org/abs/2412.02787v1	The Ehrhart series of alcoved polytopes	\documentclass[11pt,reqno]{amsart} \usepackage{amsmath, amssymb, amsthm, wasysym,todonotes, graphicx,manfnt} \usepackage{stmaryrd} \usepackage{mathrsfs,mathtools} \usepackage[margin=3cm]{geometry} \usepackage{color} \usepackage{hyperref} \hypersetup{ colorlinks, citecolor=black, filecolor=black, linkcolor=black, urlcolor=black } \usepackage[capitalize,nameinlink,noabbrev,nosort]{cleveref} \usepackage[maxbibnames=99,sorting=nyt,giveninits,maxnames=10,backend=bibtex,style=alphabetic,block=space,url=false]{biblatex} \addbibresource{main.bib} \usepackage{enumerate} \usepackage{caption} \usepackage{subcaption} \usepackage{url} \usepackage{floatrow} \usepackage{chngcntr} \usepackage{manfnt} \usepackage{esvect} \usepackage{verbatim} \usepackage{comment} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usepackage{tikz} \usetikzlibrary{shapes.geometric} \usetikzlibrary{math} \usetikzlibrary{calc,arrows} \usepackage{tikz-3dplot} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{observation}[theorem]{Observations} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{theorem}{Theorem} \newcommand \acknowledgements{\paragraph{\textbf{Acknowledgements}}} \newcommand\inner[2]{\langle #1, #2 \rangle} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathcal{F}} \newcommand{\D}{\mathcal{D}} \newcommand{\precdot}{\prec\cdot ~} \newcommand{\cube}{\text{\mancube}} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\vertex}{vert} \DeclareMathOperator{\des}{des} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\EE}{EE} \DeclareMathOperator{\volu}{vol} \DeclareMathOperator{\cover}{cover} \newcommand{\ts}{\textsuperscript} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\trace}{trace} \newcommand{\hv}{\mathcal{H}_{\mathcal{V}}} \newcommand{\shift}{\overset{\mathrm{shift}}{=\joinrel=}} \renewcommand{\P}{\mathcal{P}} \newcommand{\YJ}[1]{{$\spadesuit$ \textcolor{blue}{YJ: #1}}} \newcommand{\id}{\Delta_0} \newcommand{\fan}{\text{Fan}} \newcommand{\dist}{\text{dist}} \DeclareMathOperator{\wt}{wt} \DeclareMathOperator{\Ehr}{Ehr} \newcommand{\aff}{\mathrm{aff}} \DeclarePairedDelimiter{\bbracket}{\llbracket}{\rrbracket} \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \title{The Ehrhart series of alcoved polytopes} \author{Elisabeth Bullock} \address{Massachusetts Institute of Technology} \email{[email protected]} \author{Yuhan Jiang} \address{Harvard University} \email{[email protected]} \date{\today} \begin{document} \begin{abstract} Alcoved polytopes are convex polytopes which are the closure of a union of alcoves in an affine Coxeter arrangement. They are rational polytopes, and therefore have Ehrhart quasipolynomials. We describe a method for computing the generating function of the Ehrhart quasipolynomial, or Ehrhart series, of any alcoved polytope via a particular shelling order of its alcoves. We also conjecture a connection between Early's decorated ordered set partitions and this shelling order for the hypersimplex $\Delta_{2,n}$. \end{abstract} \maketitle \setcounter{tocdepth}{1} \section{Introduction} Let $\Phi \subset V$ be an irreducible \emph{crystallographic root system} and $W$ be the corresponding Weyl group. Associated to $\Phi$ is an infinite hyperplane arrangement known as the affine Coxeter arrangement. The connected components of this hyperplane arrangement are simplices called \emph{alcoves}. Lam and Postnikov defined a \emph{proper alcoved polytope} to be the closure of a union of alcoves. In the special situation of the root system $\Phi = A_n$, examples of alcoved polytopes include \emph{hypersimplices} and \emph{positroid polytopes}. For any root system, Fomin and Zelevinsky's \emph{generalized associahedra} are examples of polytopes which can be realized as alcoved polytopes \cite{fz03,chapoton}. Alcoved polytopes are rational polytopes, i.e., their vertices have rational coordinates. Let $P \subset \R^n$ be a rational convex polytope. Ehrhart \cite{ehrhart} showed that the number of lattice points in the $t$-dilate of $P$ is a \emph{quasipolynomial} in $t$. A quasipolynomial with period $d$ is a function $p: \Z \to \Z$ such that there exist periodic functions $p_i: \Z \to \Z$ with period $d$ so $p(z) = \sum_{i=0}^n p_i(z) z^i$. We call $E(P,t) = \#(t\cdot P) \cap \Z^n$ the \emph{Ehrhart quasipolynomial} of $P$. The generating series $\sum_{t=0}^\infty E(P,t) z^t$ is a rational function in $z$. While the Ehrhart theory of lattice polytopes has been widely studied, the Ehrhart theory of rational polytopes is a younger subject \cite{beck22,BBKV13,linke11}. Ehrhart theory extends naturally to polytopes with some facets removed. The Ehrhart series are \emph{additive}, in the sense that if two half-open polytopes are disjoint, then the Ehrhart series of their union is equal to the sum of their Ehrhart series. Our main result stated imprecisely is the following: \begin{theorem}\label{thm:main-imprecise} Fix an irreducible crystallographic root system $\Phi\subset V$, where $\dim(V)=n$. Let $P$ be an alcoved polytope and let $\Gamma_P=(V,E)$ be the dual graph to the alcove triangulation of $P$. Pick some $v_0\in V$ and orient the edges of $\Gamma_P$ so that for all $\{u,w\}\in E$, $u\to w$ if and only if $u$ appears before $v$ in the breadth first search algorithm starting at $v_0$. There exists a weighting of the edges $E$ and parameters $\ell_1,\cdots,\ell_n$ depending only on $\Phi$ such that the Ehrhart series of $P$ is equal to $$\Ehr(P,z) = \frac{\sum_{w \in V} z^{\wt(w)}}{\prod_{i=0}^n (1-z^{\ell_i})}$$ where $\wt(w) = \sum_{u \to w} \wt((u,w))$ is the sum of the weights of the ingoing edges to $w$. \end{theorem} We will prove our main result using the additivity of Ehrhart series. We will decompose each alcoved polytopes into disjoint union of half-open alcoves, and then add up all their Ehrhart series. \begin{figure}[H] \centering \begin{tikzpicture}[scale=1.5] \coordinate (A) at (0,0); \coordinate (B) at (2,0); \coordinate (C) at (1,1); \draw[thick] (A) -- (B) -- (C) -- cycle; \draw node [left, below] at (A) {(0,0)}; \draw node [right, below] at (B) {(1,0)}; \draw node [above] at (C) {($\frac{1}{2}, \frac{1}{2}$)}; \end{tikzpicture} \hspace{3cm} \begin{tikzpicture}[scale=1.5] \coordinate (A) at (0,0); \coordinate (B) at (2,0); \coordinate (C) at (2,2); \coordinate (D) at (3,1); \coordinate (E) at (1,1); \draw[thick] (A) -- (B) -- (D) -- (C) -- cycle; \draw[thick] (E) -- (B); \draw[thick] (B) -- (C); \coordinate (1) at (1,1/2); \coordinate (2) at (3/2,1); \coordinate (3) at (5/2,1); \draw node at (1) {$\bullet$}; \draw node at (2) {$\bullet$}; \draw node at (3) {$\bullet$}; \draw[thick] (1) -- (2) node [midway, above] {1}; \draw[thick] (2) -- (3) node [midway, above] {2}; \end{tikzpicture} \caption{On the left, we have the fundamental alcove of type $B_2$. On the right, we have an alcoved polytope of type $B_2$ and the graph of its alcoved triangulation. The Ehrhart series of the square is $\frac{1+z+z^2}{(1-z)^2 (1-z^2)}$.} \label{fig:B22} \end{figure} \begin{figure}[H] \centering \begin{tikzpicture} \coordinate (A) at (0,0); \coordinate (B) at ($(0,{-2tan(60)})$); \coordinate (C) at ($(B) - (2,0)$); \draw[thick] (A) -- (B) -- (C) --cycle; \draw node [above] at (A) {$(0,0,0)$}; \draw node [right] at (B) {$(\frac{1}{2},0,-\frac{1}{2})$}; \draw node [left] at (C) {$(\frac{2}{3}, -\frac{1}{3}, -\frac{1}{3})$}; \end{tikzpicture} \hspace{3cm} \begin{tikzpicture} \coordinate (A) at (0,0); \coordinate (B) at (2,0); \coordinate (C) at ($(A) - (0,{2tan(60)})$); \coordinate (D) at (-2,0); \coordinate (E) at ($(C) - (2,0)$); \draw[thick] (A) -- (B) -- (C) -- (E) -- (D) --cycle; \draw[thick] (E) -- (A) -- (C); \coordinate (F) at ($ (A)!.333!(B)!.333!(C) $); \coordinate (G) at ($ (A)!.333!(E)!.333!(C) $); \coordinate (H) at ($ (A)!.333!(D)!.333!(E) $); \draw[thick] (F) -- (G) node [midway, above] {3}; \draw[thick] (G) -- (H) node [midway, above] {2}; \draw node at (F) {$\bullet$}; \draw node at (G) {$\bullet$}; \draw node at (H) {$\bullet$}; \end{tikzpicture} \caption{On the left, we have the fundamental alcove of type $G_2$. On the right, we have an alcoved polytope of type $G_2$. The Ehrhart series of the trapezoid is $\frac{1+z^2+z^3}{(1-z)(1-z^2)(1-z^3)}$.} \label{fig:G22} \end{figure} After proving this result, we conjecture a relationship between this formula and another formula for the $h^$-polynomial of the second hypersimplex $\Delta_{2,n}=\{(x_1,\dots,x_n)\in[0,1]^n~\|~\sum_{i=1}^nx_i=2\}$ which is given in terms of combinatorial objects called decorated ordered set partitions (DOSPs). The only known proofs of the latter formula rely on algebraic manipulation and enumerative combinatorics; our conjecture is aimed towards providing a more geometric reason for why DOSPs appear in the $h^$-polynomial of the hypersimplex. \subsection{Organization} In \cref{sec:prelim}, we cover relevant background materials, including root systems, alcoved polytopes, and Coxeter groups. In \cref{sec:ehr}, we review some general Ehrhart theory of rational polytopes and prove a few lemmas that will be used to show our main result. In \cref{sec:shell}, we review Coxeter complexes and the notion of a convex subset of a Coxeter group. In this section, we also describe a shelling order of the Coxeter complex given by \cite{bjorner} and prove a corollary about shelling subcomplexes of the Coxeter complex (\cref{cor:cox}) which will be used to show our main result. We precisely state and prove our main result (\cref{thm:main}) in \cref{sec:proof}. In \cref{sec:2n}, we recall a formula for the $h^$-polynomial of the hypersimplex $\Delta_{k,n}$ in terms of hypersimplicial decorated ordered set partitions \cite{kim2020combinatorial} and conjecture a connection with the formula yielded by our main theorem in the case $k=2$. \vspace{.5cm} \acknowledgements{We thank Alex Postnikov for suggesting this topic. We thank Lauren Williams for the helpful discussions and comments on the manuscript. We thank Nick Early for helpful discussions.} \section{Preliminaries}\label{sec:prelim} In this section, we recall the relevant background for \emph{root systems} which will be used to define \emph{alcoved polytopes}. We recall notations from \emph{Coxeter groups} which will be used to describe our shellings of the Coxeter complexes related to the alcove triangulation of an alcoved polytope. We follow the conventions in \cite{humphreys} and \cite{alcove2}. \subsection{Root systems}\label{sec:rootsys} Let $V$ be a real Euclidean space of rank $n$ with nondegenerate symmetric inner product $(\cdot,\cdot)$. Let $\Phi \subset V$ be an irreducible \emph{crystallographic root system} with a choise of basis of \emph{simple roots} $\alpha_1, \dots, \alpha_n$. Let $\Phi^+ \subset \Phi$ be the corresponding set of \emph{positive roots}. The \emph{coweight lattice} $\Lambda^\vee$ is the integer lattice defined by $\Lambda^\vee = \Lambda^\vee(\Phi) = \{ \lambda \in V \mid (\lambda, \alpha) \in \Z, \text{ for all } \alpha \in \Phi\}$. Let $\omega_1, \dots, \omega_n \subset V$ be the basis dual to the basis of simple roots, i.e., $(\omega_i, \alpha_j) = \delta_{ij}$. The $\omega_i$ are called the \emph{fundamental coweights}. They generate the coweight lattice $\Lambda^\vee$. Let $\rho = \omega_1 + \cdots + \omega_n$. The \emph{height} of a root $\alpha$ is the number $(\rho, \alpha)$ of simple roots that add up to $\alpha$. Since we assumed that $\Phi$ is irreducible, there exists a unique \emph{highest root} $\theta \in \Phi^+$ of maximal possible height. For convenience we set $\alpha_0 = -\theta$. Let $a_0 = 1$ and $a_1, \dots, a_n$ be the positivie integers given by $a_i = (\omega_i, \theta)$. The \emph{dual Coxeter number} is defined as $h^\vee = (\rho, \theta) + 1 = a_0 + a_1 + \cdots + a_n$. \subsection{Alcoved polytopes}\label{sec:alcove} Let $\Phi\subset V$ be a crystallographic root system. The set of affine hyperplanes of the form $H_{\alpha,k} = \{\lambda \in V \mid (\lambda, \alpha) = k \}$, where $\alpha\in\Phi^+,k\in\mathbb{Z}$, divides $V$ into \emph{open alcoves}: \begin{definition} An (open) \emph{alcove} is the set $$ A = \{ \lambda \in V \mid m_\alpha < (\lambda, \alpha) < m_\alpha+1, \text{ for } \alpha \in \Phi^+\} $$ where $m_\alpha$ is a collection of integers associated with the alcove $A$. A \emph{closed alcove} is the closure of an alcove. \end{definition} \begin{definition}\label{def:fundamental} The \emph{fundamental alcove} is the simplex given by \begin{align} A_\circ &= \{ \lambda \in V \mid 0 < (\lambda, \alpha) < 1, \text{ for } \alpha \in \Phi^+\} \\ &= \text{Convex Hull of the points } 0, \omega_1/a_1, \dots, \omega_r/a_r. \end{align} \end{definition} The \emph{affine Weyl group} $W_{\aff}$ associated with the root system $\Phi$ is generated by the reflections $s_{\alpha,k}: V \to V, \alpha \in \Phi, k \in \Z$, with respect to the affine hyperplanes $H_{\alpha,k}$, and acts simply transitively on the collection of all alcoves \cite{humphreys}. In particular, the closure of each alcove has the same Ehrhart series, and we can use alcoves as the building blocks for a family of polytopes with natural triangulations in which all top-dimensional simplices have equal volume: \begin{definition} An \textit{alcoved polytope} is a polytope which is the union of some collection of faces of alcoves. A \textit{proper alcoved polytope} is a polytope which is a union of alcoves, equivalently a top-dimensional alcoved polytope. \end{definition} In other words, every alcoved polytope is of the form $$ P = \{ \lambda \in V \mid k_\alpha \leq (\lambda, \alpha) \leq K_\alpha, \text{ for } \alpha \in \Phi^+ \}, $$ where $k_\alpha, K_\alpha$ are two collections of integers indexd by the positive roots $\alpha \in \Phi^+$. \begin{definition}\label{def:graphlabel} Let $P$ be an alcoved polytope. We associate a graph $\Gamma_P = (V,E)$ with labeled edges to the alcoved triangulation of $P$. We will abuse notation and also use $\Gamma_P$ to denote the simplicial complex of the alcove triangulation of $P$. The vertex set $V$ consists of closed alcoves in $P$, and the edge set $E$ consists of $(A, A')$ if $A$ and $A'$ share a common facet. Let $(A, A')$ be an edge of $G$, and let $F = A \cap A'$ be the facet it represents. Then $F$ can be transformed to a facet $F_\circ$ of the fundamental alcove $A_\circ$ under the action of the affine Weyl group. Let $\omega_i/a_i$ be the vertex of $A_\circ$ that does not belong to $F_\circ$. Then $(A,A')$ has \emph{weight} $\ell_i$, denoted $\wt((A,A')) = \ell_i$, in which $\ell_i$ is the least common multiple of the denominators in $\omega_i/a_i\in\mathbb{Q}^n$ for $i=1,\dots,n$ and $\ell_0 = 1$. \end{definition} \subsection{Coxeter Groups} For a root system $\Phi$, the affine Weyl group $W_\aff$ is an example of a Coxeter group: \begin{definition} Let $S$ be a set. A matrix $m: S \times S \to \{1,2,\dots,\infty\}$ is called a \emph{Coxeter matrix} if it satisfies \begin{align} m(s,s') &= m(s',s) \\ m(s,s') = 1 & \iff s = s'. \end{align} A Coxeter matrix $m$ determines a group $W$ with the presentation $$ \langle s \in S \mid (ss')^{m(s,s')} = e \text{ for all } m(s,s') \neq \infty \rangle, $$ where $e$ is the identity element. The pair $(W,S)$ is called a \emph{Coxeter system} and the group $W$ is the \emph{Coxeter group}. \end{definition} \begin{definition} Let $(W, S)$ be a Coxeter system. Each element $w \in W$ can be written as a product of generators $w = s_1 s_2 \cdots s_k$ for $s_i \in S$. If $k$ is minimal among all such expressions for $w$, then $k$ is called the \emph{length} of $w$ (written $\ell(w) = k$) and the word $s_1 s_2 \cdots s_k$ is called a \emph{reduced word} for $w$. \end{definition} \begin{definition} Let $(W, S)$ be a Coxeter system, and let $u, v \in W$. Define the \emph{right weak order} as the partial order $\leq$ on $W$ such that $u \leq w$ if and only if $w = u s_1 s_2 \cdots s_k$ for some $s_i \in S$ such that $\ell(u s_1 s_2 \cdots s_i) = \ell(u) + i$ for all $0 \leq i \leq k$. \end{definition} \begin{definition} Let $W$ be a Coxeter group. For $w \in W$, define the \emph{descent set} $\D(w) = \{s \in S \mid w s < w\}$. For $J \subseteq S$, let $W_J$ be the subgroup of $W$ generated by the set $J$, and let $W^J = \{ w \in W \mid w s > w \text{ for all } s \in J\}$ be a system of distinct coset representatives modulo $W_J$. \end{definition} \section{Ehrhart series of half-open rational simplices}\label{sec:ehr} A key ingredient of the proof of our main theorem is to write an alcoved polytope $P$ as a disjoint union of half-open simplices, or simplices with some facets removed. Since the number of lattice points in $P_1\sqcup P_2$ is the sum of lattice points in $P_1$ and $P_2$, it suffices to look at the Ehrhart series of these \emph{half-open} simplices. If we remove no facets from an alcove, we may use the following theorem to write its Ehrhart series: \begin{theorem}[Theorem 1.3, \cite{stanley80}] \label{rat-simp} Suppose $S$ is a $k$-simplex in $\R^m$ with rational vertices $\beta_0, \dots, \beta_k$. Let $\ell_i$ be the least positive integer $t$ for which $t\beta_i \in \Z^m$. Consider the $(k+1) \times (m+1)$-matrix whose rows are the vectors $(\ell_i \beta_i, \ell_i)$. If the greatest common divisor of all $(k+1)\times(k+1)$ minors of the matrix is equal to 1, then the Ehrhart series of the simplex $S$ is equal to $$ \Ehr(S, z) = \frac{1}{\prod_{i=0}^k (1-z^{\ell_i})}. $$ \end{theorem} The above theorem serves as a base case in the proof of the following more general formula for half-open simplices: \begin{lemma}\label{lem:half-opensimplex} Suppose $S$ is a $k$-simplex in $\R^m$ with rational vertices $\beta_0, \dots, \beta_k$. Let $\ell_i$ be the least positive integer $t$ for which $t\beta_i \in \Z^m$. Let $F_i$ be the facet of $S$ that does not contain the vertex $\beta_i$. Let $\mathcal{F} \subseteq \{0, \dots, k\}$ label a subset of facets of $S$. Then the Ehrhart series of $S$ with facets $\{F_i \mid i \in \mathcal{F}\}$ removed is $$ \Ehr(S \setminus \bigcup_{i \in \mathcal{F}} F_i, z) = \frac{ \prod_{i \in \mathcal{F}} z^{\ell_i}}{\prod_{i=0}^k (1-z^{\ell_i})}. $$ \end{lemma} \begin{proof} Without loss of generality assume $\mathcal{F}=\{0,\ldots,m\}$. We induct on $m$. First, if we remove no facets, we are done by Theorem \ref{rat-simp}. By inclusion-exclusion and our inductive hypothesis, \begin{align} \Ehr\left(S\setminus\bigcup_{i \in \mathcal{F}} F_i, z\right) &=\Ehr(S,z)+\sum_{\mathcal{G}\subseteq\mathcal{F}}(-1)^{\|\mathcal{G}\|}\Ehr\left(\bigcap_{F\in\mathcal{G}}F,z\right)\\ &=\frac{1}{\prod_{i=0}^k(1-z^{\ell_i})}+\sum_{\mathcal{G}\subseteq\mathcal{F}}\frac{(-1)^{\|\mathcal{G}\|}}{\prod_{i\in V(\mathcal{G})}(1-z^{\ell_i})} \end{align} where $V(\mathcal{G})$ is the set of indices of vertices $\bigcap_{F\in\mathcal{G}}F$. We have $V(\mathcal{G})=\{0,\ldots,k\}\setminus\mathcal{G}$. Clearing denominators, \begin{align} \Ehr\left(S\setminus\bigcup_{i \in \mathcal{F}} F_i, z\right) &=\frac{1}{\prod_{i=0}^k(1-z^{\ell_i})}\left[\sum_{\mathcal{G}\subseteq\mathcal{F}}(-1)^{\|\mathcal{G}\|}\prod_{i\in\mathcal{G}}(1-z^{\ell_i})\right]. \end{align} Using inclusion-exclusion again, the last sum is equal to $\prod_{i\in\mathcal{F}}z^{\ell_i}$. \end{proof} \section{A shelling order}\label{sec:shell} The goal of this section is to build up to a concrete decomposition of any alcoved polytope into half-open alcoves. This is done via a \emph{shelling order}. In particular, we relate the alcove structure to a simplicial complex called the coxeter complex and use existing results about the latter to define a shelling order. We begin by recalling the notion of Coxeter complexes, following \cite{bjornerbrenti}. \begin{definition} An \emph{abstract simplicial complex} $\Delta$ on the vertex set $V$ is a collection $\Delta$ of finite subsets of $V$, called \emph{faces}, such that $x \in V$ implies $\{x\} \in \Delta$ and $F \subseteq F' \in \Delta$ implies $F \in \Delta$. The dimension of a face is $\dim F = \|F\| - 1$, and $\dim \Delta = \sup_{F \in \Delta} \dim F$. A complex is \emph{pure $d$-dimensional} if every face is contained in some $d$-dimensional face. In this case, we denote the set of $d$-dimensional faces (called facets) by $\mathscr{F}(\Delta)$. Two facets $A, A'$ are \emph{adjacent} if $\dim(A \cap A') = d-1$. If $F \in \Delta$, let $\bar{F}$ be the \emph{simplex} $\{E \mid E \subseteq F\}$. \end{definition} \begin{definition} Let $(W,S)$ be a Coxeter system with $\|S\|<\infty$. We write $(s) := S \setminus \{s\}$, for $s \in S$, and $V:= \bigcup_{s \in S} W/W_{(s)}$ for the collection of all left cosets of all maximal parabolic subgroups. The \emph{Coxeter complex} $\Delta(W, S)$ is by definition the pure $(\|S\|-1)$-dimensional simplicial complex on the vertex set $V$ with facets $C_w := \{w W_{(s)} \mid s \in S\}$, for $w \in W$. For a face $F \in \Delta(W, S)$, define its \emph{type} $\tau(F) = \{s \in S \mid F \cap W/W_{(s)} \neq \emptyset\}$. \end{definition} Let $(W, S)$ be a Coxeter system and $u,v \in W$. A path from $u$ to $v$ is a sequence $u = w_0 \to w_2 \to \cdots \to w_s = v$ such that $w_{i+1} = w_i s$ for some simple reflection $s \in S$. A subset $K \subseteq W$ is called \emph{convex} if for every $u, v \in K$ we have that any shortest path from $u$ to $v$ lies in $K$. This mirrors the notion of convexity for unions of closed alcoves: \begin{proposition}[Prop. 3.5, \cite{alcove2}]\label{prop:convex} Let $P$ be a bounded subset which is a union of closed alcoves. Then $P$ is a convex polytope if and only if one of the following conditions hold: \begin{enumerate}[(1)] \item For any two alcoves $A, B \subset P$, any shortest path from $A= A_0 \to A_1 \to A_2 \to \cdots \to A_s = B$ lies in $P$. Here $A_i$ are alcoves and $A' \to A''$ means that the closure of the two alcoves $A'$ and $A''$ share a facet. \item The subset $W_P = \{w \in W_\aff \mid w(A_\circ) \subset P\}$ of the affine Weyl group is a convex subset. \end{enumerate} \end{proposition} Let $(W, S)$ be a Coxeter system and let $\mathscr{F}(\Delta(W, S))$ denote the set of all facets of $\Delta(W, S)$. By \cite[pp. 40-44]{bourbaki} and \cite[Chap. 2]{tits}, there is a bijection between $W$ and $\mathscr{F}(\Delta(W, S))$ given by $w \mapsto C_w$. Two facets $C_w$ and $C_{w'}$ are adjacent if and only if $w' = ws$ for some $s \in S$. By \cite{humphreys}, there is a bijection between $\mathscr{F}(\Delta(W, S))$ and the set of alcoves in the affine Coxeter arrangement. This bijection maps $C_e$ to the fundamental alcove $A_\circ$ (see \cref{def:fundamental}), and maps $C_w$ to the alcove $A$ such that there is a shortest path $A_\circ \overset{s_1}{\to} A_1 \overset{s_2}{\to} \cdots \overset{s_k}{\to} A$ where $s_1 s_2 \cdots s_k$ is a reduced word for $w$. The group $W$ acts on $\Delta(W, S)$ by left translation $w: v W_{(s)} \mapsto wv W_{(s)}$, and this action is \emph{type-preserving}, i.e., $\tau(w(F)) = \tau(F)$ for all faces $F \in \Delta(W, S)$. The key property of Coxeter complexes of interest in this paper is that they have a \emph{shelling order}: \begin{definition}\label{def:shelling} Let $\Delta$ be a pure $d$-dimensional complex of at most countable cardinality. A \emph{shelling} of $\Delta$ is a linear order $A_1, A_2, A_3, \dots$ on the set of facets of $\Delta$ such that $\bar{A_k} \cap \Delta_{k-1}$ is pure $(d-1)$-dimensional for $k = 2,3,\dots$, where $\Delta_{k-1} = \bar{A}_1 \cup \cdots \cup \bar{A}_{k-1}$. \end{definition} Given a shelling, define the \emph{restriction} of a facet $A_k$ by $$\mathscr{R}(A_k) = \{x \in A_k \mid A_k - \{x\} \in \Delta_{k-1}\}.$$ \begin{theorem}[Theorem 2.1, \cite{bjorner}] Let $(W, S)$ be a Coxeter system, $\|S\|<\infty$. Then any linear extension of the weak ordering of $W$ assigns a shelling order to the facets of $\Delta(W, S)$. \end{theorem} In particular, Bj\"orner showed that the restriction of this shelling is $$ \mathscr{R}(C_w) = \{w W_{(s)} \mid s \in \D(w)\} .$$ \begin{corollary}\label{cor:cox} Let $(W, S)$ be a Coxeter system, $\|S\|<\infty$. Let $\Gamma_P$ be the subcomplex of $\Delta(W, S)$ induced by a convex subset $P$ of the Coxeter group $W$ that contains the identity element $e$. Any linear extension of the weak order is a shelling of $\Gamma_P$. \end{corollary} \begin{proof} Since $P$ contains the identity and is convex, if $w \in P$, then all the shortest paths from $e$ to $w$ are contained in $P$. That is, all the reduced words of $w$ are contained in $P$. Let $C_1, C_2, \dots$ denote a shelling of $\Delta(W, S)$, and let $C_{a_1}, C_{a_2}, \dots$ denote the subsequence of $C_1, C_2, \dots$ consisting of facets that are in $\Gamma_P$. For $k \geq 2$, let $\Delta_{k-1} = \bar{C}_1 \cup \cdots \cup \bar{C}_{k-1}$ and let $\Delta'_{k-1} = \bar{C}_{a_1} \cup \cdots \cup \bar{C}_{a_{k-1}}$. Then by definition, $\Delta'_{k-1} \subseteq \Delta_{k-1}$, so $C_{a_k} \cap \Delta'_{k-1} \subseteq C_{a_k} \cap \Delta_{{a_k}-1}$. Suppose $C_{a_k} = C_w$ for some $w \in W$. Since all the reduced words of $w$ are contained in $P$, for each $s \in \D(w)$, we have $C_w - \{ w W_{(s)} \} \in \Delta'_{k-1}$, so $C_{a_k} \cap \Delta'_{k-1} = C_{a_k} \cap \Delta_{{a_k}-1}$ is pure $(\|S\|-1)$-dimensional, concluding the proof. \end{proof} \begin{figure} \centering \tdplotsetmaincoords{60}{70} \begin{tikzpicture}[tdplot_main_coords, scale=5] \coordinate (A) at (1, 0, 0); \coordinate (B) at (1, 1, 0); \coordinate (C) at (0.5, 0.5, 0); \coordinate (D) at (1, 1, 1); \coordinate (E) at (0.5, 0.5, 0.5); \coordinate (F) at (1.5, 0.5, 0); \coordinate (G) at (1.5, 0.5, 0.5); \coordinate (H) at (1, 0.5, 0.5); \node[left,below] at (A) {$(1, 0, 0)$}; \node[right] at (B) {$(1, 1, 0)$}; \node[left] at (C) {$(\frac{1}{2}, \frac{1}{2}, 0)$}; \node[right,above] at (D) {$(1, 1, 1)$}; \node[above] at (E) {$(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$}; \node[right,below] at (F) {$(\frac{3}{2}, \frac{1}{2}, 0)$}; \node[right] at (G) {$(\frac{3}{2}, \frac{1}{2}, \frac{1}{2})$}; \draw[thick] (A) -- (G) -- (D) -- (E) -- cycle; \draw[thick] (B) -- (D); \draw[thick] (A) -- (F) -- (B); \draw[thick,dotted] (B) -- (C); \draw[thick,dotted] (A) -- (C); \draw[thick,dotted] (E) -- (C); \draw[thick] (G) -- (F); \draw[thick] (A) -- (D); \draw[thick] (E) -- (G); \draw[thick] (G) -- (B); \draw[thick,dotted] (H) -- (B); \draw[thick,dotted] (A) -- (B); \draw[thick,dotted] (E) -- (B); \end{tikzpicture} \begin{tikzpicture} \coordinate (A) at (0,-1); \coordinate (B) at (1,0); \coordinate (C) at (1,1); \coordinate (D) at (3,0); \coordinate (E) at (3,1); \coordinate (F) at (4,-1); \node at (A) {$\bullet$}; \node at (B) {$\bullet$}; \node at (C) {$\bullet$}; \node at (D) {$\bullet$}; \node at (E) {$\bullet$}; \node at (F) {$\bullet$}; \draw[thick] (A) -- (B) node[midway,above] {2}; \draw[thick] (B) -- (C) node[midway,left] {1}; \draw[thick] (D) -- (E) node[midway,right] {1}; \draw[thick] (C) -- (E) node[midway,above] {2}; \draw[thick] (B) -- (D) node[midway,below] {2}; \draw[thick] (D) -- (F) node[midway,above] {2}; \end{tikzpicture} \caption{The generalized hypersimplex for $\Phi = B_3$ and $k = 2$. The Ehrhart series of $\Delta^{B_3}_2$ is $\Ehr(\Delta^{B_3}_2,z) = \frac{1+z+3z^2+z^3}{(1-z)^2 (1-z^2)^2}$} \label{fig:B32} \end{figure} \section{Proof of the main theorem}\label{sec:proof} We can now translate the shelling order of subcomplexes of the Coxeter complex in Corollary \cref{cor:cox} into a shelling order of the alcoves of an alcoved polytope $P$, and use this shelling order to compute the Ehrhart series of $P$. \begin{definition}\label{def:partialorder} Let $\Gamma = (V,E)$ be an undirected graph, and let $v_0 \in V$ be an arbitrary vertex of $\Gamma$. Define the \emph{breadth-first search order of $\Gamma$ with root $v_0$} as the partial order $(\P_{v_0,\Gamma},\prec)$ on $V$ such that for two distinct vertices $u, v \in V$, $u \prec v$ if and only if there is a shortest path from $v_0$ to $v$ passing through $u$. \end{definition} The following is a corollary of \cref{cor:cox}. \begin{corollary}\label{cor:shelling} Let $P$ be an alcoved polytope and let $\Gamma_P = (V, E)$ be the graph of the alcoved triangulation of $P$. For any $v_0 \in V$, any linear extension of the partial order $(\P_{v_0,\Gamma_P},\prec)$ is a shelling order of the alcove triangulation of $P$. \end{corollary} \begin{proof} Let $W = W_\aff$ be the affine Weyl group that acts on the alcoves in $P$. Let $W_P = \{w \in W \mid w(A_\circ) \subset \}$, which is convex by \cref{prop:convex}. Let $v_0$ be the alcove $C_w$ for some $w \in W$. Then $w^{-1}(W_P)$ is a convex subset of $W$ that contains the identity, and $w^{-1}(\mathcal{P}_{v_0, \Gamma_P},\prec)$ is the weak order on the facets of $w^{-1}(\Gamma_P)$. By \cref{cor:shelling}, any linear extension of $w^{-1}(\mathcal{P}_{v_0, \Gamma_P},\prec)$ is a a shelling order of $w^{-1}(\Gamma_P)$. Since $W$ acts transitively on the set of all alcoves, we conclude the proof. \end{proof} \begin{lemma}\label{lem:disjointunion} Let $P$ be an alcoved polytope. Let $\Gamma_P = (V,E)$ be the graph of the alcoved triangulation of $P$. Fix some $v_0 \in V$ and let $(\P_{v_0,\Gamma_P}, \prec)$ be the breadth-first search order on $\Gamma_P$ with root $v_0$. For each alcove $A$ in $P$, let $\mathscr{I}_A=\{\bar{A} \cap \bar{A'} \mid A' \precdot A \}$ be a subset of facets of the closure $A$. Then the set of half-open alcoves $A^\circ = \bar{A} \setminus (\cup \mathscr{I}_A)$, are mutually disjoint, and their union is equal to $P$, i.e., $$ P = \bigsqcup_{A \in V} A^\circ. $$ \end{lemma} \begin{proof} We prove this by induction. Let $A \in V$ be an alcove in $P$. Let $\mathcal{A} = \{A' \in V \mid A' \prec A \}$ be the set of alcoves that comes before $A$ in the topological sort of $\Gamma_P$. We show that \begin{enumerate}[(1)] \item $A^\circ \cap \bigcup_{A' \in \mathcal{A}} A'^{\circ} = \emptyset$; \label{disjoint} \item $A^\circ \cup \bigcup_{A' \in \mathcal{A}} A'^{\circ} = \bar{A} \cup (\bigcup_{A' \in \mathcal{A}} \bar{A'})$.\label{union} \end{enumerate} To show (\ref{disjoint}), assume there exists $A' \in \mathcal{A}$ such that $A^\circ \cap A'^{\circ} \neq \emptyset$. Then $\bar{A} \cap \bar{A}' \neq \emptyset$ and it must be contained in a facet of $\bar{A}$ in $\bar{A} \cap \bigcup_{A' \in \mathcal{A}} \bar{A}'$ because $\prec$ is a shelling. This facet is equal to $\bar{A} \cap \bar{A}'$ for $A' \precdot A$, and therefore will be excluded in $A^\circ$. To show (\ref{union}), we first use the induction hypothesis that $\bigcup_{A' \in \mathcal{A}} A'^{\circ} = \bigcup_{A' \in \mathcal{A}} \bar{A}'$. Then, since $\cup \mathscr{I}_A \subset \bigcup_{A' \in \mathcal{A}} \bar{A}'$, we have $A^\circ \cup (\bigcup_{A' \in \mathcal{A}} \bar{A}' ) = \bar{A} \cup (\bigcup_{A' \in \mathcal{A}} \bar{A}' ) $. \end{proof} We now have all the ingredients we need to compute the Ehrhart polynomial of an arbitrary alcoved polytope: \begin{theorem}\label{thm:main} Let $P$ be an alcoved polytope and let $\Gamma_P = (V, E)$ be the edge-weighted graph of its alcoved triangulation (see \cref{def:graphlabel} for details). Given any $v_0 \in V$, let $\mathcal{P}_{v_0,\Gamma_P}$ be the breadth-first search order of $\Gamma_P$ with root $v_0$ (see \cref{def:partialorder}). The Ehrhart series of $P$ is equal to $$\Ehr(P,z) = \frac{\sum_{v \in V} z^{\wt(v)}}{\prod_{i=0}^n (1-z^{\ell_i})}$$ where $\wt(v) = \sum_{u \precdot v} \wt((u,v))$ is the sum of the weights of the edges between $v$ and the elements it covers. \end{theorem} \begin{proof}[Proof of \cref{thm:main}] By \cref{lem:disjointunion} and \cref{lem:half-opensimplex} and the \emph{additivity} of Ehrhart series, we conclude the proof. \end{proof} \begin{example} Consider the $n$-dimensional hypercube $\cube_n=[0,1]^n$. The graph of alcoved triangulation of a hypercube is the weak Bruhat graph of the symmetric group $S_n$. Therefore, $h^(\cube_n,z) = \sum_{w \in S_n} z^{\des(w)}$, which is the Eulerian polynomial. This is a well known result from \cite{stanley80}. \end{example} \section{The hypersimplex $\Delta_{2,n}$}\label{sec:2n} We now speculate on a relationship between our shelling order formula and a combinatorial formula for a well-studied polytope. The hypersimplex $\Delta_{k,n}$ is the subset of $[0,1]^n\subset\mathbb{R}^n$ consisting of points $\{(x_1,\dots,x_n) \mid x_1 + \cdots + x_n = k\}$. Under the linear transformation $$ y_i = x_1+ \cdots + x_i, $$ the hypersimplex $\Delta_{k,n}$ can be realized as an alcoved polytope defined by $0 \leq y_i - y_{i-1} \leq 1$ and $y_n = k$ for all $i =1,\dots,n$ with the convention $y_0 = 0$. The coefficients of the $h^$-polynomial form the \emph{$h^$-vector}. The $h^$-vector of the hypersimplex $\Delta_{k,n}$ was proved to be enumerated by \emph{hypersimplicial decorated ordered set partitions}. \begin{definition}[\cite{early2017conjectures}] A \emph{decorated ordered set partition} $((S_1)_{s_1},\dots,(S_p)_{s_p})$ of type $(k,n)$ consists of an ordered set partition $(S_1, \dots, S_p)$ of $[n]$ and an $p$-tuple of integers $(s_1, \dots, s_p) \in \Z^{p}$ such that $\sum_{i=1}^p s_i = k$ and $s_i \geq 1$. We call each $S_i$ a block and we place them on a circle in the clockwise fashion then think of $s_i$ as the clockwise distance between adjacent block $S_i$ and $S_{i+1}$. We regard them up to cyclic rotation of blocks, so $((S_1)_{s_1},(S_2)_{s_2},\dots,(S_p)_{s_p})$ is the same as $((S_2)_{s_2},\dots,(S_p)_{s_p},(S_1)_{s_1})$. A decorated ordered set partition is \emph{hypersimplicial} if $1 \leq s_i \leq \|S_i\|-1$ for all $i$. The \emph{winding vector} of a decorated ordered set partition is an $n$-tuple of integers $(l_1,\dots,l_n)$ such that $l_i$ is the distance of the path starting from the block containing $i$ to the block containing $(i+1)$ moving clockwise. If $i$ and $(i+1)$ are in the same block then $l_i = 0$. If $l_1 + \cdots + l_n = kd$, then we define the \emph{winding number} to be $d$. \end{definition} \begin{theorem}[\cite{Kimh}] Let $h^(\Delta_{k,n},z) = h_0^(\Delta_{k,n}) + h_1^(\Delta_{k,n}) z + \cdots + h_{n-1}^(\Delta_{k,n}) z^{n-1}$ be the $h^$-polynomial of the hypersimplex $\Delta_{k,n}$. The number of hypersimplicial decorated ordered set partitions of type $(k,n)$ and winding number $d$ is $h_d^(\Delta_{k,n})$. \end{theorem} In this section, we conjecture the relation between our formula in \cref{thm:main} for the hypersimplex $\Delta_{2,n}$ to the formula of Early and Kim given by the hypersimplicial ordered set partitions. \begin{definition} If a hypersimplicial decorated ordered set partition of type $(2,n)$ with nonzero winding number is of the form $((A)_1, (B)_1)$ for nonempty disjoint subsets $A, B \subseteq [n]$ with $A \cup B = [n]$, then we simply denote it by the set of sets $\{A,B\}$. This notation also emphasizes the fact that we regard each hypersimplicial decorated ordered set partition up to cyclic rotation of blocks. \end{definition} Let $\Gamma_{2,n}$ be the graph of alcoved triangulation of the hypersimplex $\Delta_{2,n}$, see \cref{def:graphlabel}. We associate a hypersimplicial decorated ordered set partition of type $(2,n)$ with winding number 1 to each edge in $\Gamma_{2,n}$. \begin{definition}\label{def:edgelabels} Let $A,A' \subseteq \Delta_{2,n}$ be two alcoves such that $A$ and $A'$ share a common facet. Then $A \cap A'$ is of the form $y_j - y_i = 1$ for some $i \not\equiv j \pm 1 \bmod n$. We associate the hypersimplicial decorated ordered set partition $\{\{i,\dots,j-1\}, \{j,\dots,i-1\}\}$ to the edge $(A,A')$ in the graph $\Gamma_{2,n}$. \end{definition} \begin{definition} Two hypersimplicial decorated ordered set partitions of type $(2,n)$ and winding number 1 are \emph{adjacent} if they can be associated to facets of the same alcove. \end{definition} \begin{example} The hypersimplicial decorated ordered set partitions $\{\{2,3\},\{4,5,1\}\}$ is adjacent to $\{\{1,2\},\{3,4,5\}$ and $\{\{3,4\},\{5,1,2\}\}$, but $\{\{2,3\},\{4,5,1\}\}$ is not adjacent to $\{\{4,5\},\{1,2,3\}\}$. \end{example} To relate the formula in \cref{thm:main} to hypersimplicial decorated ordered set partitions, we define a map $\psi$ from a set of $d$ adjacent hypersimplicial decorated ordered set partitions of winding number 1 to a single hypersimplicial decorated ordered set partition of winding number $d$. \begin{definition}\label{def:newdosp} For two hypersimplicial decorated ordered set partitions $\{A, B\}, \{A', B'\}$ of type $(2,n)$ with nonzero winding numbers, we define $\psi(\{A, B\}, \{A', B'\})$ to be a new hypersimplicial decorated ordered set partition $\{A'', B''\}$ of type $(2,n)$ such that $A'' = A \triangle A'$ is the symmetric difference between $A$ and $A'$ and $B'' = [n] \setminus A''$ is the complement of $A''$. For a collection of $d$ adjacent hypersimplicial decorated ordered set partitions $\{A_1,B_1\},\dots, \{A_d, B_d\}$ of type $(2,n)$ and winding number 1, we define $\psi(\{A_1,B_1\},\dots, \{A_d, B_d\})$ recursively to be $$\psi(\psi(\{A_1,B_1\},\dots, \{A_{d-1}, B_{d-1}\}), \{A_d, B_d\}).$$ \end{definition} \begin{example}\label{ex:1223} Consider $\{\{2,3\},\{4,5,1\}\}$ and $\{\{1,2\},\{3,4,5\}$ of type $(2,5)$ and winding number 1, then $\psi(\{\{2,3\},\{4,5,1\}\}, \{\{1,2\},\{3,4,5\}) = \{\{1,3\},\{2,4,5\}\}$. \end{example} \begin{example} Consider $\{\{1,2,3\},\{4,5,6\}\}$ and $\{\{2,3,4\},\{5,6,1\}\}$ of type $(2,6)$ and winding number 1, then $\psi(\{\{1,2,3\},\{4,5,6\}\}, \{\{2,3,4\},\{5,6,1\}\}) = \{\{1,4\},\{2,3,5,6\}\}$. \end{example} \begin{example} Consider $\{\{1,2,3\},\{4,5,6\}\}$, $\{\{2,3,4\},\{5,6,1\}\}$, and $\{\{3,4,5\},\{6,1,2\}\}$ of type $(2,6)$ and winding number 1, we have \begin{align} &\psi(\{\{1,2,3\},\{4,5,6\}\}, \{\{2,3,4\},\{5,6,1\}\}, \{\{3,4,5\},\{6,1,2\}\}) \\ &= \psi(\{\{1,4\},\{2,3,5,6\}\}, \{\{3,4,5\},\{6,1,2\}\}) \\ &= \{\{1,3,5\},\{2,4,6\}\}. \end{align} \end{example} \begin{conjecture}\label{conj:hypersimplicial decorated ordered set partition2n} For any $n$ and for any alcove $A_0$ in $\Delta_{2,n}$, let $\mathcal{P}_{A_0} = (V, E)$ be the breadth-first search order of $\Gamma_{2,n}$ with root $A_0$ (see \cref{def:partialorder}). For $v \in V$, let $\cover(v) = \# \{u \in V \mid u \precdot v \text{ in } \mathcal{P}_{A_0 }\}$ be the number of elements $v$ covers. The map $\psi$ is a bijection from the set $\{v \in V \mid \cover(v) = d\}$ and the set of hypersimplicial decorated ordered set partitions of type $(2,n)$ with winding number $d$. \end{conjecture} \begin{figure} \centering \begin{tikzpicture}[scale=5] \node[inner sep=2pt, scale=.8] (4123) at (0:1) {$\{3,5\}_1, \{1,2,4\}_1$}; \node[inner sep=2pt, scale=.8] (1243) at (72:1) {$\{2,4\}_1,\{1,3,5\}_1$}; \node[inner sep=2pt, scale=.8] (1324) at (144:1) {$\{1,3\}_1,\{2,4,5\}_1$}; \node[inner sep=2pt, scale=.8] (2134) at (216:1) {$\{2,5\}_1,\{1,3,4\}_1$}; \node[inner sep=2pt, scale=.8] (2341) at (288:1) {$\{1,4\}_1,\{2,3,5\}_1$}; \node[inner sep=2pt, scale=.8] (1423) at (36:0.6) {$\{3,4\}_1,\{1,2,5\}_1$}; \node[inner sep=2pt, scale=.8] (3124) at (108:0.6) {$\{2,3\}_1,\{1,4,5\}_1$}; \node[inner sep=2pt, scale=.8] (1342) at (180:0.6) {$\{1,2\}_1,\{3,4,5\}_1$}; \node[inner sep=2pt, scale=.8] (2314) at (252:0.6) {$\{1,5\}_1,\{2,3,4\}_1$}; \node[inner sep=2pt, scale=.8] (3412) at (324:0.6) {$\{4,5\}_1,\{1,2,3\}_1$}; \node[inner sep=2pt, scale=.8] (3142) at (0,0) {$\{1,2,3,4,5\}_2$}; \draw[thick,->] (3142) -- (1423); \draw[thick,->] (3142) -- (3124); \draw[thick,->] (3142) -- (1342); \draw[thick,->] (3142) -- (2314); \draw[thick,->] (3142) -- (3412); \draw[thick,->,orange] (3124) -- (1324); \draw[thick,->] (3124) -- (1243); \draw[thick,->] (1423) -- (4123); \draw[thick,->] (1423) -- (1243); \draw[thick,->] (3412) -- (4123); \draw[thick,->] (3412) -- (2341); \draw[thick,->] (2314) -- (2134); \draw[thick,->] (2314) -- (2341); \draw[thick,->] (1342) -- (2134); \draw[thick,->,orange] (1342) -- (1324); \end{tikzpicture} \caption{This is the graph of alcoved triangulation of the hypersimplex $\Delta_{2,5}$. Our root alcove $A_0$ is at the center. The arrows indicate cover relations in $\mathcal{P}_{A_0}$, pointing in increasing directions. The orange arrows are facets representing the cover relations of the alcove labeled by $\{1,3\}_1,\{2,4,5\}_1$.} \label{fig:hypersimplex25} \end{figure} \begin{example} In \cref{fig:hypersimplex25}, we choose $A_0$ to be the simplex in the center of the graph $\Gamma_{2,5}$ and we label it by the unique hypersimplicial decorated ordered set partition of type $(2,5)$ with winding number 0, which is $\{1,2,3,4,5\}_2$. The arrows are cover relations, and they point in increasing directions in $\mathcal{P}_{A_0}$. The alcove labeled by $\{1,3\}_1,\{2,4,5\}_1$ covers two alcoves in the poset $\mathcal{P}_{A_0}$, through facets colored by orange labeled by $\{1,2\}_1,\{3,4,5\}_1$ and $\{2,3\}_1,\{1,4,5\}_1$. This agrees with previous \cref{ex:1223}. \end{example} \printbibliography \end{document}
2412.02777v1	http://arxiv.org/abs/2412.02777v1	How to quantify the coherence of a set of beliefs	\documentclass[11pt,reqno]{amsart} \usepackage{amsmath,amsthm,verbatim,enumerate} \usepackage{bbm} \pdfoutput=1 \addtolength{\oddsidemargin}{-.2in} \addtolength{\evensidemargin}{-.2in} \addtolength{\textwidth}{.4in} \def\topfraction{.9} \def\floatpagefraction{.8} \newcommand{\arxiv}[1]{{\tt \href{http://arxiv.org/abs/#1}{arXiv:#1}}} \usepackage[utf8]{inputenc}\usepackage{mathtools} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{algpseudocode} \usepackage{xcolor} \usepackage{graphicx} \usepackage{subcaption} \usepackage{float} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\rsp}[1]{\langle#1\rangle} \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\deriv}[2]{\frac{d #1}{d #2}} \newcommand{\F}{\mathbb{F}} \newcommand{\R}{\mathbb{R}} \newcommand{\prob}{\mathbb{P}} \newcommand{\bec}{\boldsymbol} \usepackage{listings} \usepackage{array} \usepackage[backend=biber,style=numeric,sorting=none]{biblatex} \addbibresource{main.bib} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary} \theoremstyle{definition} \newtheorem{definition}{Definition} \DeclareMathOperator{\argmax}{arg\,max} \DeclareMathOperator{\argmin}{arg\,min} \newenvironment{proofS}{ \renewcommand{\proofname}{Proof Sketch}\proof}{\endproof} \usepackage{enumitem} \newenvironment{enro}{\begin{enumerate}[label=(\roman)]}{\end{enumerate}} \usepackage{ bbold } \usepackage{hyperref} \hypersetup{colorlinks} \author[R. Hess]{Rowan Hess} \address{Rowan Hess. Cornell University, Ithaca, NY 14853.} \email{[email protected]} \thanks{LL was partially supported by a grant from Open Philanthropy.} \author[L.\ Levine]{Lionel Levine} \address{Lionel Levine. Department of Mathematics, Cornell University, Ithaca, NY 14853. } \email{[email protected]} \begin{document} \title{How to quantify the coherence of a set of beliefs} \date{\today} \begin{abstract} Given conflicting probability estimates for a set of events, how can we quantify how much they conflict? How can we find a single probability distribution that best encapsulates the given estimates? One approach is to minimize a loss function such as binary KL-divergence that quantifies the dissimilarity between the given estimates and the candidate probability distribution. Given a set of events, we characterize the facets of the polytope of coherent probability estimates about those events. We explore two applications of these ideas: eliciting the beliefs of large language models, and merging expert forecasts into a single coherent forecast. \end{abstract} \maketitle \section{Introduction} \noindent When two or more predictions conflict, what is a principled way to combine them into a single meta-prediction? \\\\ This kind of question comes up in several different domains: \begin{itemize} \item Multiple experts are independently asked to make predictions, and we would like to integrate their predictions into a single meta-prediction that expresses the ``wisdom of the crowd'' \cite{galton_vox_1907, Ungar2012TheGJ, sempere2023alignment, mcandrew_aggregating_2021}. \item A single individual may hold incoherent beliefs without being aware of the incoherence \cite{herzog_harnessing_2014}. However, when someone points out that their beliefs are incoherent they may want to update to the ``closest'' coherent set of beliefs. \item A market maker that has beliefs about underlying probabilities of events may want to give a coherent set of quotes so as to avoid giving a ``Dutch book" that is exploitable via risk-free arbitrage \cite{dutch_book}. \item The expressed opinions of a large language model can be very sensitive to small changes in the prompt. To investigate whether the model holds an internally coherent belief and extract that belief, one approach is to ensemble many weak predictors derived from the model's internal state into a single meta-prediction \cite{burns2022discovering}. \end{itemize} \noindent Given a ground set $\Omega$, a list of events $\mathtt E = E_1, \ldots, E_n \subset \Omega$, and a list of \textit{credences} $q_1,\ldots,q_n$, the vector $\bec q = (q_1,\ldots,q_n)$ is called \textit{coherent} with respect to $\mathtt E$ if there is a probability measure $P$ (on the Boolean algebra generated by $E_1,\ldots,E_n$) such that $q_i = P(E_i)$ for all $i$. \\\\ For example, when predicting the next day's weather, suppose three of our events are ``rainy'', ``warm and rainy'', ``cold and rainy''. Then we have the coherence condition \[ P(\text{warm and rainy}) + P(\text{cold and rainy}) = P(\text{rainy}). \] If ``warm or rainy'' is also one of our given events $E_i$, then we have the additional coherence condition \[ P(\text{warm}) + P(\text{rainy}) - P(\text{warm and rainy}) = P(\text{warm or rainy}). \] These two examples of coherence conditions are both the result of probabilities being overdetermined: fixing the probabilities of some events determines the probabilities of others. There is also a second kind of coherence condition that results from probabilities being between by 0 and 1. For example, even if ``warm or rainy'' is not one of our events, we still have the coherence condition \[ P(\text{warm}) + P(\text{rainy}) - P(\text{warm and rainy}) \leq 1. \] \noindent Given a list of events $\mathtt E$ and credences $\bec q$ we would like a systematic way of measuring incoherence: a loss function $L^:[0, 1]^n\rightarrow \R\cup \set\infty$ satisfying $L^(\bec q) \geq 0$, with equality if and only if $\bec q$ is coherent. One natural method of constructing such a loss is via some notion of ``distance'' from the vector $\bec q$ to the set $C(\mathtt E)$ of coherent vectors: \begin{equation} \label{eq:theloss} L^(\bec q) = \inf \{L(\bec p,\bec q) \,:\, \bec p \in C(\mathtt E) \}. \end{equation} We will discuss several choices of ``distance" function $L(\bec p,\bec q)$. We write distance in quotation marks because $L$ need not be a metric (e.g. it might not be symmetric in $\bec p$ and $\bec q$). \\\\ The forecaster seeking to combine the beliefs of others (or individual seeking to correct their incoherent beliefs) can adopt the minimizer $\bec{p^}$ of \eqref{eq:theloss}. In Theorem \ref{punique} below we give conditions for existence and uniqueness of $\bec{p^}$. \subsection{Plan of the paper} We start Section \ref{sec:seperateBeliefs} defining the process of loss function minimization more precisely and giving conditions for when there is a unique minimizer $\bec{p^}$. We proceed by summarizing a result from Predd et al.\ \cite{predd_probabilistic_2009} that can motivate one's choice of loss function and then by looking at two specific examples of loss functions (binary KL divergence and its transpose). We discuss natural justifications for these loss functions in Theorems \ref{thm:coherentBetter} and \ref{thm:fo}.\\\\ Section \ref{sec:nn} explores a possible application of our method to eliciting beliefs from large language models. We use loss functions to generalize the method suggested in \cite{burns2022discovering} and consider possible modifications that this generalization allows. \\\\ Section \ref{sec:inequalities} explores the geometry of $C(\mathtt{E})$, the set of coherent beliefs, which is a convex polytope. It is easy to describe the vertices of this polytope (Lemma~\ref{lem:convex}), but somewhat more challenging to describe its facets (Theorem~\ref{thm:restatement}). \\\\ In Section \ref{sec:indCohExp}, we consider how predictions should be aggregated in a setting where multiple experts each make internally coherent predictions. We seek a method to aggregate these predictions that leverages the internal coherence of each expert. \\\\ Finally, in Section \ref{sec:ex}, we illustrate these methods in the scenario of masked letter prediction and discuss the results. \subsection{Related work} This problem of the reconciliation has been examined in the past \cite{capotorti_correction_2010, thimm_inconsistency_2013}. In particular, \cite{capotorti_correction_2010} proposes the use of binary KL divergence as a loss function and discusses several of its properties. \subsection{Preliminaries} We will assume that there are $n$ events $E_i$ for $i\in \set{1, \dots, n}$ and that for each event $E_i$, we receive some estimate $q_i$ of its probability. These estimates may come from a single expert or multiple experts. We discuss the implications of one expert giving multiple beliefs in Section \ref{sec:indCohExp}. \begin{definition} A \textit{credence base} is the ordered pair $(\mathtt E, \textbf q)$ where $\mathtt E = (E_i)_{i=1}^{n}$ is a sequence containing the (potentially repeated) events whose probabilities are being estimated and $\bec q\in [0, 1]^n$ is the vector whose $i$th entry is $q_i$. \end{definition} \noindent Recalling that every finite Boolean algebra is the power set of a set, let $2^\Omega$ be the Boolean algebra generated by $\mathtt E$, where $\Omega=\set{\omega_1, \dots, \omega_N}$ is finite as $\mathtt E$ is finite. For example, if $\mathtt E = (\set{\text{rock}}, \set{\text{paper}}, \set{\text{scissors},\text{rock}})$ then $\Omega = \set{\text{rock}, \text{paper}, \text{scissors}}$. \begin{definition} A credence base $(\mathtt E, \bec q)$ is \textit{coherent} if there exists a probability distribution $\pi:2^\Omega\rightarrow [0, 1]$ where $\pi(E_i)=q_i$ for all $i\in\set{1,\dots, n}$. We also call $\bec q$ \textit{coherent} with respect to $\mathtt E$ if $(\mathtt E, \bec q)$ is coherent. \end{definition} \begin{definition} The \textit{set of coherent beliefs} over events $\mathtt E$ is \[ C(\mathtt E)=\set{\boldsymbol p\in [0, 1]^n \,:\, (\mathtt E, \boldsymbol{p}) \text{ is coherent}}. \] \end{definition} \noindent \section{Loss Functions that Quantify Incoherence} \label{sec:seperateBeliefs} \noindent In this section, we examine a class of loss functions $L(\bec p,\bec q)$ that are additive over events. We give conditions that guarantee that for any credence base $(\mathtt E,\bec q)$ there is a unique coherent credence base $(\mathtt E,\bec{p^})$ minimizing the loss $L(\bec p,\bec q)$. Then we state a theorem of Predd et al.\ \cite{predd_probabilistic_2009} showing that in a certain forecasting setting it is never advantageous to submit an incoherent forecast. We proceed by giving natural justifications for two examples of loss functions: binary KL divergence and its transpose. \\\\ Recall that a function $f$ is \textit{lower semi-continuous} at $x$ if $$\liminf_{x_n\rightarrow x} f(x_n) \ge f(x).$$ Also, we say that a function $f$ is \textit{strictly convex when finite} if for all $t\in(0, 1)$ and $x, x'$ in its domain, then $$tf(x) + (1-t)f(x') \ge f(tx+(1-t)x')$$ and the inequality is strict unless both sides are infinite. \begin{definition} A \textit{dissimilarity function} $\ell(p, q)$ is a function $\ell:[0, 1]\times [0, 1]\rightarrow [0, \infty]$ satisfying \begin{enro} \item $\ell$ is lower semi-continuous with respect to $p$ \item $\ell$ is strictly convex when finite with respect to $p$ \item $\ell(p, q)=0$ if and only if $p=q$. \end{enro} \end{definition} Some examples of dissimilarity functions are: \begin{itemize} \item $f(p, q) = p\ln \frac pq + (1-p) \ln \frac{1-p}{1-q}$ \item $f^o(p, q) = f(q, p)$ \item $\ell(p, q) = (p-q)^2$ \item $\ell(p, q) = \left\{\begin{array}{cc} 0 & p=q \\ \infty & p\ne q \end{array}\right..$ \end{itemize} \noindent Given dissimilarity functions $\ell_1, \dots, \ell_n$, we define an associated \textit{loss function} $L:[0, 1]^n\times [0, 1]^n\rightarrow [0, \infty]$ given by \begin{equation} L(\textbf p, \textbf q)=\sum_{i=1}^n \ell_i(p_i, q_i).\label{eqn:lossfunction} \end{equation} \noindent We typically consider all dissimilarity functions $\ell_i$ to be equal to some common $\ell$. We denote the corresponding loss function $L_\ell$ and call it the loss function derived from dissimilarity function $\ell$. We omit the subscript when it is clear from context. \\\\ Then, given a loss function $L$ with $n$ terms and a credence base $\mathcal{Q}=(\mathtt E, \bec q)$ with $n$ events, we can define the \emph{incoherence} of $\mathcal Q$ to be \begin{equation} L^(\boldsymbol q):=\min_{\boldsymbol p\in C(\mathtt E)} L(\textbf p, \textbf q).\label{eqn:lstar} \end{equation} We denote the minimizing coherent belief by $\bec p^(\bec q)$. The next theorem, which extends \cite[Cor.~7]{capotorti_correction_2010} to a more general family of loss functions, gives conditions under which this minimizer is unique. \begin{theorem} \label{punique} Given a credence base $(\mathtt E, \bec q)$ and a loss function $L$ of the form in (\ref{eqn:lossfunction}), if there exists a coherent $\bec p\in C(\mathtt E)$ such that $L(\bec p, \bec q)$ is finite, then there exists a unique $\bec {p^}$ such that $$L(\bec{p^}, \bec q) = \min_{\bec p\in C(\mathtt E)} L(\bec p, \bec q).$$ Moreover, $\bec{p^} = \bec q$ if and only if $\bec q\in C(\mathtt E)$. \end{theorem} \begin{proof} As the sum of lower semi-continuous and strictly convex when finite functions, $L(\cdot, \bec q)$ is also lower semi-continuous and strictly convex when finite. Choose coherent $\bec p\in C(\mathtt E)$ such that $L(\bec p, \bec q)$ is finite. As the set $C(\mathtt E)$ is compact, these is a sequence $\bec p_1, \bec p_2, \dots C(\mathtt E)$ that converges to some $\bec p^$ so that $$\lim_{k\rightarrow \infty} L(\bec p_k, \bec q) = \inf_{\bec p\in C(\mathtt E)} L(\bec p, \bec q).$$ Then, by lower semi-continuity, $$\inf_{\bec p\in C(\mathtt E)} L(\bec p, \bec q) \le L(\bec p^, \bec q) \le \lim_{k\rightarrow \infty} L(\bec p_k, \bec q) =\inf_{\bec p\in C(\mathtt E)} L(\bec p, \bec q) $$ so $\bec p^$ is a minimizer of $L(\cdot, \bec q)$ in $C(\mathtt E)$. \\\\ As for uniqueness, if $L(\bec p, \bec q)=L(\bec p', \bec q)$ for $\bec p \ne \bec p'$, $$L\left(\frac 12 \bec p+\frac 12 \bec p', \bec q\right)<\frac 12 L(\bec p, \bec q)+\frac 12 L(\bec p', \bec q)=L(\bec p, \bec q)$$ because $L(\bec p, \bec q)$ and $L(\bec p', \bec q)$ must be finite. Therefore, $\bec p$ is not a minimizer of $L(\cdot, \bec q)$. \\\\ If $\bec q\in C(\mathtt E)$, then $L(\bec q, \bec q)=0$ by property (iii). Therefore, as $L$ can only take non-negative values and the minimizer is unique, $\bec{p^} = \bec q$. \end{proof} \noindent We discuss how to compute $\bec p^$ in Appendix \ref{sec:findpstar} and the continuity of this $\bec p^$ and $L^$ in Appendix \ref{sec:continuity}. \subsection{The coherence theorem of Predd et al.}\label{section:predd} The number $L^(\bec q)$ is a way of quantifying the incoherence of the credence base $(\mathtt E, \bec q)$. But which dissimilarity function $\ell$ should we use? One way to motivate the choice of $\ell$ arises from forecasting competitions: A forecaster submits a list of predicted probabilities $\bec q$ for events $\mathtt E$, and these are scored after the outcomes of all events are known. If event $E_i$ occurred, then the forecaster receives penalty $s(1, q_i)$ for that event; if $E_i$ did not occur, then the forecaster receives penalty $s(0, q_i)$ for that event. The forecaster aims to minimize the expectation of the random variable \[S(\mathtt E, \bec q)=\sum_{i=1}^n s(1_{E_i}, q_i).\] \noindent How should $s$ be chosen in order to elicit the true beliefs of each forecaster? If a forecaster believes that event $E_i$ will happen with probability $q_i$, then the scoring rule should incentivize them to submit prediction $q_i$ in order to minimize their expected score. A function $s$ with this property is called a \textit{proper scoring rule}. More formally, a proper scoring rule is a function $s:\set{0, 1}\times [0, 1] \rightarrow [0, \infty]$ satisfying \cite{predd_probabilistic_2009, winkler_good_1968}: \begin{enro} \item For each $p\in[0, 1]$, the quantity \[ ps(1, q)+(1-p)s(0, q)\] is uniquely minimized at $q=p$. \item $s$ is continuous in $q$. Note that we allow $s$ to take the value $+\infty$, but $(i)$ ensures that all values of $s$ are finite except possibly $s(0, 1)$ and $s(1, 0)$. \end{enro} \noindent Predd et al.\ \cite{predd_probabilistic_2009} prove that for any proper scoring rule $s$, there is a dissimilarity function $\ell$ so that for any incoherent $\bec q$, the prediction $\bec{p^}(\bec q)$ using $\ell$ scores better than $\bec q$ no matter which events actually happen! Namely, let \begin{equation}\ell(p, q)=p[s(1, q)-s(1, p)]+(1-p)[s(0, q)-s(0, p)]\label{eqn:ell}\end{equation} be the expected excess penalty for predicting $q$ rather than $p$ when the true probability of an event is $p$. Note that for loss functions $L^$ derived from such dissimilarity functions, we have $L^(\bec q)\ge 0$ with equality if and only if $(\mathtt E, \bec q)$ is coherent, because such functions $\ell$ are uniquely minimized when $p=q$ as $s$ is a proper scoring rule. \begin{theorem}[\cite{predd_probabilistic_2009}] Let $s$ be any proper scoring rule, and let $\ell$ be given by (\ref{eqn:ell}). For any credence base $(\mathtt E, \bec q)$, let $\bec{p^}(\bec q)\in C(\mathtt E)$ be the coherent minimizer of Theorem~\ref{punique}. Then, for all $\omega \in \Omega$, $${S(\mathtt E, \bec q)(\omega)-S(\mathtt{E}, \bec{p^}(\bec q))(\omega)\ge L^(\bec q)}.$$ \label{thm:coherentBetter} \end{theorem} \noindent Equation (\ref{eqn:ell}) still leaves a lot of possible choices of dissimilarity function $\ell$, since there are many choices of proper scoring rule $s$. Next we examine the case of the widely-used logarithmic scoring rule. The corresponding dissimilarity function $\ell$ turns out to be a variant of the Kulback-Leibler (KL) divergence \cite{kullback_information_1951}. \subsection{Binary KL divergence} Capotorti, Regoli, and Vattari \cite{capotorti_correction_2010} proposed correcting incoherent probability assessments using the binary KL-divergence \begin{equation} f(p, q)=p \ln \frac pq + (1-p)\ln \frac{1-p}{1-q}.\label{eqn:kldivergence} \end{equation} The resulting loss $L(\bec p,\bec q) = \sum_{i=1}^n f(p_i,q_i)$ can be interpreted as the \emph{total expected excess surprise of all experts}, when expert $i$ predicts probability $q_i$ for event $E_i$ and the ground truth probability of $E_i$ is $p_i$. \\\\This loss function can also be motivated by the logarithmic scoring rule $$s(i, q)=i\ln\frac 1 q + (1-i)\ln\frac{1}{1-q}$$ with the goal to minimize total score $S(\mathtt E, \bec q)=\sum_{i=1}^n s(1_{E_i}, q_i)$. By Theorem \ref{thm:coherentBetter}, if $(\mathtt E, \bec{q})$ is incoherent then the prediction $\bec{p^}(\bec q)$ using $\ell=f$ scores strictly better than the prediction $\bec q$ no matter which events actually occur. \subsection{Transposed binary KL divergence} Next we consider the same loss with the roles of $\bec p$ and $\bec q$ interchanged: $$f^{o}(p, q)=f(q, p).$$ In Table~\ref{fig:Lminimizer} we describe the coherent minimizer $\bec p^$ determined by $f$ and $f^o$ in each of two basic scenarios: \begin{enumerate} \item Partition: Fix $n$ and let $\Omega = \set{\omega_1, \omega_2, \dots, \omega_n}$. For all $1\le i \le n$, we ;et $E_i = \set{\omega_i}$. In this scenario all experts give estimates for different events, exactly one of which occurs. \item Repetition: Let $\Omega = \set{\omega_1,\omega_2}$ and for all $1\le i \le n$, let $ E_i=\set{\omega_1}$. In this scenario all experts give estimates for the same event. \end{enumerate} \begin{table}[h] \begin{center} \begin{tabular}{\|m{4.5em} \| m{16.5em}\| m{13em} \|} \hline \cline{2-3}& Binary KL loss & Transposed binary KL loss \\ \hline Partition & $\ln\frac{p^_i}{1-p^_i}-\ln\frac{q_i}{1-q_i}$ does not depend on $i$& $\frac{q_i-p^_i}{p^_i(1-p^_i)}$ does not depend on $i$\\ \hline Repitition & $\frac{p^}{1-p^}=\left(\prod_{i} \frac{q_i}{1-q_i}\right)^{1/n}$& $p^=\frac{1}{n}\sum_{i} q_i$ \\\hline \end{tabular} \caption{Description of the coherent minimizer $\bec{p^}$ in each of two scenarios, with two different loss functions: Binary KL loss ($\ell=f$ as in (\ref{eqn:kldivergence})) and its transpose $(\ell=f^{o})$.} \label{fig:Lminimizer} \end{center} \end{table} \noindent The motivation for switching the order of the variables in $f$ comes from the following theorem, showing that the minimizer of loss functions determined by $f^{o}$ is now, in a certain scenario, the maximum likelihood estimation of $\boldsymbol{p}$. \\\\ Suppose the $i$th expert independently observes some number $w_i$ of independent trials of event $E_i$, and submits the estimate $q_i = \frac{k_i}{w_i}$ where $k_i$ is the number of trials in which $E_i$ occurred. Note that if $P(E_i)=p_i$, then $k_i$ has the binomial distribution $\text{Bin}(w_i,p_i)$. \\\\ To represent that each expert has a different amount of information, we can multiply each summand in (\ref{eqn:lossfunction}) by some weight $w_i$. \begin{theorem}\label{thm:fo} The maximum likelihood estimator $\bec{\hat p}$ given $\boldsymbol q$ is \[\argmin_{\boldsymbol{p}\in C(\mathtt E)}\sum_{i=1}^nw_if^{o}(p_i, q_i).\] \end{theorem} \begin{proof} The probability of receiving the estimations $\boldsymbol q$ is $$P(\boldsymbol q \| \boldsymbol p)=\prod_{i=1}^n\binom{w_i}{q_iw_i}p_i^{q_iw_i}(1-p_i)^{(1-q_i)w_i}.$$ Next, if $\boldsymbol{\hat p}$ is the maximum likelihood estimation, then $$\boldsymbol{\hat p}= \argmax_{\boldsymbol p\in C(\mathtt E)}\prod_{i=1}^n\binom{w_i}{q_iw_i}p_i^{q_iw_i}(1-p_i)^{(1-q_i)w_i}$$ $$=\argmax_{\boldsymbol p\in C(\mathtt E)}\sum_{i=1}^nq_iw_i\ln p_i+(1-q_i)w_i\ln (1-p_i)$$ $$=\argmax_{\boldsymbol p\in C(\mathtt E)}\sum_{i=1}^nw_i(q_i\ln p_i + (1-q_i)\ln(1-p_i))$$ $$=\argmax_{\boldsymbol p\in C(\mathtt E)}\sum_{i=1}^nw_i\left(q_i\ln \frac{p_i}{q_i} + (1-q_i)\ln \frac{1-p_i}{1-q_i}\right)$$ $$=\argmin_{\boldsymbol p\in C(\mathtt E)}\sum_{i=1}^nw_i\left(q_i\ln \frac{q_i}{p_i} + (1-q_i)\ln \frac{1-q_i}{1-p_i}\right)$$ $$=\argmin_{\boldsymbol p\in C(\mathtt E)}\sum_{i=1}^nw_if^o(p_i, q_i).\qedhere$$ \end{proof} \noindent In general, $f$ produces more extreme probabilities than $f^o$ and gives more weight to extreme probability estimates, as demonstrated numerically in Appendices \ref{sec:compare} and \ref{sec:ex}. \section{Application: Eliciting Latent Beliefs from Language Models} \label{sec:nn} \noindent Large language models sometimes express beliefs that they ``know'' to be false, for example because the prompt includes false statements. Burns et al \cite{burns2022discovering} propose a method to elicit the model's actual belief about a natural language claim corresponding to an event $E$ from its hidden state $\phi(\text{des}_E)$ where $\text{des}_E$ is an assertion in natural language that the event $E$ holds. This method learns a linear probe $b(\phi)$ minimizing the loss \begin{equation} (q_{E} + q_{E^c} - 1)^2 + \min(q_{E},q_{E^c})^2\label{eqn:burns} \end{equation} where $q_{E} =\sigma(b(\phi(\text{des}_E)))= \frac{1}{1+e^{-b(\phi(\text{des}_E))}}$ and $q_{E^c} = \sigma(b(\phi(\text{des}_{E^c})))$ are intended to estimate the model's credence in the claims $\text{des}_E$ and $\text{des}_{E^c}$. The idea is that the term $(q_{E} + q_{E^c} - 1)^2$ is a measure of the incoherence of the linear probe, while the $\min(q_{E},q_{E^c})^2$ term encourages the linear probe to be decisive. We propose three possible elaborations of this approach, all of which train probes to minimize an expression of the form $$\mathcal I(\bec q) + \mathcal{J}(\bec q)$$ where $\mathcal I$ is a measure of incoherence and $\mathcal J$ is a measure of indecisiveness. \subsection{The Case of a Single Event and its Complement}\label{subsec:singleEvent} The following proposition shows that the measure of incoherence $(q_E+q_{E^c}-1)^2$ can be derived from the dissimilarity function $\ell(p, q)=2(p-q)^2$. \begin{lemma} When using dissimilarity function $\ell(p, q)=2(p-q)^2$, the incoherence function becomes $$L_\ell^(q_E, q_{E^c}) = (q_{E} + q_{E^c} - 1)^2.$$ \end{lemma} \begin{proof} See Lemma \ref{lem:burns_loss} in the appendix. \end{proof} \noindent This approach could be generalized to choosing $\ell$ to be either the functions $f$ or $f^o$ discussed previously. The next two lemmas give approximations for $L^$ when $q_E+q_{E^c}-1$ is small \begin{lemma} When using dissimilarity function $$\ell(p, q)=f(p, q)=p\ln \frac pq + (1-p)\ln \frac{1-p}{1-q}$$ the loss function becomes $$L_f^(q_E, q_{E^c}) = \frac{(q_E + q_{E^c} - 1)^2}{(1-q_E+q_{E^c})(q_E+1-q_{E^c})}+O((q_E + q_{E^c} - 1)^3).$$\label{lem:burns_f2} \end{lemma} \begin{proof} See Lemma \ref{lem:burns_f} in the appendix. \end{proof} \begin{lemma} When using dissimilarity function $$\ell(p, q)=f^o(p, q)=q\ln \frac qp + (1-q)\ln \frac{1-q}{1-p}$$ the loss function becomes $$L_{f^o}^(q_E, q_{E^c}) = \frac{(q_E + q_{E^c} - 1)^2}{(1-q_E+q_{E^c})(q_E+1-q_{E^c})}+O((q_E + q_{E^c} - 1)^3).$$\label{lem:burns_fo2} \end{lemma} \begin{proof} See Lemma \ref{lem:burns_fo} in the appendix. \end{proof} \noindent The reason that the $(1-q_E+q_{E^c})(q_E+1-q_{E^c})$ term might be desirable to have in the denominator is that it increases the loss for the same absolute error when the probabilities are more extreme, which mirrors the intuition behind common scoring functions such as log-loss. \subsection{Multiple Probes and Multiple Events} A natural extension of \cite{burns2022discovering} is to train $k$ probes $b_1, \dots, b_k$ where $b_i$ is trained to elicit the model's credence from its $i$th layer hidden state $\phi_i$ using $2m$ rephrasings $\text{des}_{E, j}$ and $\text{des}_{E^c, j}$ of a natural language assertion about an event $E$ and its complement $E^c$. Then, letting $q_i(\text{des}) = \sigma(b_i(\phi_i(\text{des})))$, one could replace the $(q_E + q_{E^c}-1)^2$ term of (\ref{eqn:burns}) with $$\mathcal I(\bec q) = L^\left(q_i(\text{des}_{E, j}), q_i(\text{des}_{E^c, j}) \right)_{i=1}^{k}\left._{j=1}^m\right..$$ This could be further generalized to a set of $n$ interrelated events $\set{E_1, \dots, E_n}$ each of which has $m$ natural language assertions $\text{des}_{E_h, 1}, \dots, \text{des}_{E_h, m}$ that it holds, using the coherence term $$\mathcal I(\bec q) = L^\left(q_i(\text{des}_{E_h, j})\right)_{i=1}^{k}\left._{h=1}^n\right.\left._{j=1}^{m}\right..$$ Theorem \ref{thm:holderImpliesDiff} in Appendix \ref{sec:continuity} gives a formula for the gradient of $L^$ assuming $\bec p^$ is sufficiently smooth. \\\\ Why would multiple probes ($k> 1$) help elicit beliefs? We hypothesize that the language model's beliefs -- when they exist -- arise from a bundle of internal heuristics that sometimes conflict. Each probe $b_i$ represents one such heuristic. In the case that $L^$ is close to zero, the model's internal heuristics are mostly self-consistent (the credences are close to an actual probability distribution) and in this case we might say that the model ``has beliefs" about the claims $E_1,...,E_n$. These beliefs could be certain or uncertain: if all the probes think a coin flip is 50\% likely to land heads, then that is a coherent belief. On the other hand, if $L^$ is large, then it is possible that the model ``does not have beliefs" about the claims $E_i$, or else that the probes simply did not discover the right heuristics. \\\\ Why would multiple events ($n> 2$) help elicit beliefs? By minimizing $L^$ applied to more events, we are encouraging not just $q_E+q_{E^c}=1$ but also $q_{E\cup F} = q_E + q_F - q_{E\cap F}$ and so on. The philosophy is that to find ``beliefs", one should look for functions of the hidden state that obey the laws of probability. The logical dependencies between events are incorporated in the definition of $L^$ as a minimum over $C(\mathtt E)$, as defined in equation (\ref{eqn:lstar}). We further study the geometry of $C(\mathtt E)$ in Section \ref{sec:inequalities}. \subsection{Generalizing the Decisiveness Term} Recall that the $\min(q_{E},q_{E^c})^2$ term in Equation (\ref{eqn:burns}) is present to encourage the linear probe to be decisive. However, in the scenarios above with more than 2 estimates, it is unclear what should replace it. We discuss some options below: \begin{itemize} \item We want the probes to be decisive so as to maximize information they give about the language model's internal state. One way to quantify the information content of $\bec p^$ is by the maximum entropy of a probability distribution that gives $\bec p^$. Hence, one could use the decisiveness term $$\mathcal J(\bec q) = \max \set{H(\pi) \, : \pi(\bec{ \mathtt E}) = \bec p^(\bec q)}$$ where $$H(\pi) = \sum_{\omega\in \Omega} -\pi(\omega) \ln \pi(\omega).$$ \item The above notion can be generalized. Recall that ${s(i, q)=i\ln q + (1-i) \ln (1-q)}$ is the log-loss scoring rule. Then, we can write entropy as $$H(\pi) = E_\pi\left[ \sum_{\omega\in \Omega}1_\omega s(1, \pi(\omega))\right].$$ Recall that in Section \ref{section:predd}, we discussed how a proper scoring rule $s$ can be transformed into a loss function $\ell$. If using such a loss function, it might make sense to replace the log-loss function above with the proper scoring rule that $\ell$ is based on, making the decisiveness term $$\mathcal J(\bec q) = \max \set{E_\pi\left[ \sum_{\omega\in \Omega}1_\omega s(1, \pi(\omega))\right]: \pi(\bec{ \mathtt E}) = \bec p^(\bec q)}$$ for a general proper scoring rule $s$. \item Using the idea that the least decisive distribution $\pi$ is the distribution that maximizes $$E_\pi\left[ \sum_{\omega\in \Omega}1_\omega s(1, \pi(\omega))\right],$$ let $\bec u = \pi(\mathtt E)$ be this least decisive set of coherent beliefs and use either $\mathcal J(\bec q) = -L(\bec u, \bec p^(\bec q))$ or $\mathcal J(\bec q) = -L(\bec p^(\bec q), \bec u)$ to reward the probe for being more decisive.\\ \item When using $f$ or $f^o$ as a measure of dissimilarity, the incoherence term by itself might be enough to incentivize decisiveness due to the increased sensitivity of the sigmoid function to noise when its input is close to $0$. To illustrate this, we revisit the scenario where we use one linear probe to find $q_E$ and $q_{E_c}$ from one natural language description of the event $E$ and one of its complement. We suppose that that the value of the linear probe $b(\phi(\text{des}_E))$ has normal distribution $N(\sigma^{-1}(p), S^2)$ and $b(\phi(\text{des}_{E^c}))\sim N(\sigma^{-1}(1-p), S^2)$ independently where $p$ is the probe's goal credence in event $E$. Regardless of which of $f$ or $f^o$ we choose as a dissimilarity function the expected value of the incoherence becomes approximately, using the quadratic approximation given by Lemmas \ref{lem:burns_f2} and \ref{lem:burns_fo2}: $$\mathbb E[L^((q_{E}, q_{E^c}))]\approx \frac{\mathbb E[(q_{E} + q_{E^c} - 1)^2]}{4p(1-p)}.$$ Noting $q_{E} + q_{E^c} - 1$ has mean $0$ and is the sum of two independent random variables with approximate variances $\sigma'(\sigma^{-1}(p))^2S^2=\sigma'(\sigma^{-1}(1-p))^2S^2$ by the $\delta$-method, this becomes $$\mathbb E[L^((q_{E}, q_{E^c}))]\approx \frac{2 S^2}{4p(1-p)}\sigma'(\sigma^{-1}(p))^2=\frac{S^2p(1-p)}{2}.$$ As this is minimized when $p$ is close to $0$ or $1$, the incoherence term by itself incentivizes the probe to be decisive. \end{itemize} \section{The Polytope of Coherent Beliefs} \label{sec:inequalities} \noindent Here, we explore the shape of the space of coherent beliefs. We first describe it as the convex hull of a finite set of $0,1$-vectors in Lemma \ref{lem:convex} and end by describing the inequalities that bound this polytope in Theorem \ref{thm:restatement}. \\\\ Given a set of events $\mathtt E$ with beliefs $\bec q$, let $\textbf E_i\in \set{0, 1}^N$ be the vector whose $j$th entry is $1$ if $\omega_j\in E_i$ and $0$ otherwise. Then, we can encode the structure of events $\mathtt E$ in a matrix $V$ whose $i$th row is the vector $\bec E_i$. Because $V$ contains the same information as $\mathtt E$, it will be convenient to use the pair $(V, \bec q)$ to represent the credence base $(\mathtt E, \bec q)$. We also encode a probability distribution $\pi$ on $\Omega$ with the vector $\bec \pi \in [0, 1]^N$ whose $j$th entry $\bec \pi \cdot \bec e_j = \pi(\set{\omega_j})$. \begin{definition} A vector $\bec\pi\in [0, 1]^N$ is a probability vector if $\sum_{i=1}^N \pi_i=1$. \end{definition} \begin{lemma} \cite{capotorti_correction_2010} A credence base $\mathcal{Q}=(V, \bec q)$ is coherent if and only if $\boldsymbol q$ is in the convex hull of the columns of $V$. \label{lem:convex} \end{lemma} \begin{proof} By definition $\mathcal Q$ is coherent if and only if there is a probability vector $\bec\pi \in [0, 1]^{N}$ where $V \bec \pi = \boldsymbol{ q}$. If $V_i$ is the $i$th column of $V$, that is true if and only if $$\boldsymbol q = \sum_{i=1}^{N} \pi_i \boldsymbol V_i.\qedhere$$ \end{proof} \noindent For example, if we receive three beliefs: one about the probability of it being warm, one about the probability of it raining, and one of the probability of it being warm and raining, then the matrix $V$ becomes $$\begin{pmatrix} 1 & 1 & 0 & 0\\ 1 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 \end{pmatrix}.$$ So the polytope of coherent beliefs is the tetrahedron depicted in Figure \ref{fig:poytopeVisual}. \begin{figure}[h] \begin{subfigure}[b]{.4\textwidth} \centering \includegraphics[width=\linewidth]{polytope_view2.png} \end{subfigure} \begin{subfigure}[b]{.5\textwidth} \begin{itemize}[leftmargin=] \item The facet not containing vertex $(1, 1, 1)$ corresponds to the inequality $P(\text{warm and rainy})\ge 0$. \item The facets not containing either $(1, 0, 0)$ or $(0, 1, 0)$ correspond to the inequalities $P(\text{warm})\ge P(\text{warm and rainy})$ and $P(\text{rainy})\ge P(\text{warm and rainy})$. \item The facet not containing vertex $(0, 0, 0)$ corresponds to the inequality ${1 + P(\text{warm and rainy}) \ge P(\text{warm}) + P(\text{rainy})}$. \end{itemize} \end{subfigure} \caption{The polytope of coherent beliefs about $P(\text{warm}), P(\text{rainy})$, and $P(\text{warm and rainy})$ and the inequalities corresponding to its facets as described in Theorem \ref{thm:restatement}.} \label{fig:poytopeVisual} \end{figure} \noindent \\Let $\bar V$ be the $(n+1)\times N$ matrix obtained by appending a final row of $1$s to $V$. Similarly, take $\bar{\textbf q}\in [0, 1]^{n+1}$ to have the same first $n$ entries as $\textbf q$ and $n+1$th entry $1$ and $E_{n+1}=\bec 1$, the vector whose entries are all $1$s. These represent the fact that $\pi(\Omega)=1$. For two vectors $\bec v, \bec w$, write $\textbf u \geq \bec v$ if $u_i\ge v_i$ for all $i$. Finally, let $O_+$ be $\R_{\ge 0}^N$ and for a matrix $M$, let $\rsp{M}$ denote the row span of $M$. \\\\ The following is a statement of the Dutch Book Theorem \cite{dutch_book}. \begin{lemma} The following are equivalent: \begin{enro} \item $\mathcal{Q}$ is coherent. \item For all $(a_1, ..., a_{n+1})\in \R^{n+1}$, if $\sum_{i=1}^{n+1}a_i\boldsymbol E_i\geq \boldsymbol 0$, then $\sum_{i=1}^{n+1}a_i \bar q_i\ge 0$. \end{enro} \label{lem:qcoherent} \end{lemma} \begin{proof} Suppose $(ii)$. By Farkas' Lemma \cite{Farkas}, as there is no $\bec a \in \R^{n+1}$ with all entries in $\bar V^T \bec a$ non-negative and $\bec a \cdot \bec{ \bar q} < 0$, it must be that there is a solution $\bec \pi \in O_+$ to $\bar V \bec \pi = \bec{\bar q}$, with the row of $1$s in $\bar V$ ensuring that $\bec \pi$ is a probability vector. \\\\ For the converse, consider $\mathcal{Q}$ coherent. Then, there exists $\boldsymbol \pi \in [0, 1]^N$ where $\bec E_i\cdot \boldsymbol \pi = \bar q_i$ for all $1\le i \le n+1$. But, if $\bar V^T \boldsymbol a \ge \bec 0$, then $$\boldsymbol a \cdot \boldsymbol{\bar q}=\boldsymbol a \cdot (\bar V \boldsymbol \pi)=(\bar V^T \boldsymbol a) \cdot \boldsymbol \pi\ge0$$ as the dot product of two non-negative vectors. \end{proof} \begin{lemma} Suppose that $\bar{V}^T\boldsymbol a=\boldsymbol 0$ for some non-zero vector $\boldsymbol a \in \R^{n+1}$ where $\boldsymbol{\bar q} \cdot \boldsymbol a \neq 0$. Then, $\mathcal{Q}$ is incoherent. \label{lem:ConsistencyOnRemoval1} \end{lemma} \begin{proof} If $\boldsymbol{\bar q} \cdot \boldsymbol a<0$, then we have $\bar V^T\boldsymbol a =\bec 0 \ge \bec 0$ but $\bec{\bar q} \cdot \boldsymbol a<0$. By Lemma \ref{lem:qcoherent}, $\mathcal{Q}$ is incoherent. \\\\ If $\boldsymbol {\bar q} \cdot \boldsymbol a>0$, then $\bec{\bar q}\cdot -\bec a<0$, so $\mathcal{Q}$ is incoherent. \end{proof} \begin{lemma} Suppose that $\bar{V}^T\boldsymbol a=\boldsymbol 0$ for some non-zero vector $\boldsymbol a \in \R^{n+1}$ where $\boldsymbol{\bar q} \cdot \boldsymbol a = 0$. For any $1\le i \le n$ where $a_i\ne 0$, we can remove the $i$th entry of $\boldsymbol{ q}$ and the $i$th row from $ V$ to obtain $\boldsymbol{ r}$ and $ W$ so that the credence base $(\boldsymbol r, W)$ is coherent if and only if $\mathcal{Q}$ is coherent. \label{lem:ConsistencyOnRemoval2} \end{lemma} \begin{proof} Without loss of generality, $\boldsymbol E_1 = \sum_{i=2}^{n+1} c_i \boldsymbol E_i$ and $q_1=\sum_{i=2}^{n+1} c_i q_i$. Call this vector $\boldsymbol c\in R^n$. Consider removing the first entry of $\boldsymbol q$ and the first row of $ V$ to yield $\boldsymbol r$ and $W$. \\\\ For any $\boldsymbol a$ where $\bar V^T \boldsymbol a \ge \bec 0$, if $\boldsymbol a_p$ is is the vector $\boldsymbol a$ with the first entry removed, then $\bar V^T \boldsymbol a = \bar W^T(\boldsymbol a_p + \boldsymbol c)$. Then, also $\boldsymbol{\bar q} \cdot \boldsymbol a = \boldsymbol{\bar r} \cdot (\boldsymbol a_p+\boldsymbol c)$. Therefore, if $(\boldsymbol r, W)$ is coherent, then $\mathcal{Q}$ is coherent. \\\\ If $(\boldsymbol r, W)$ were incoherent, then $\sum_{i=2}^{n+1}c_i \boldsymbol E_i\geq 0$ but $\sum_{i=2}^{n+1}c_i q_i< 0$, so $\mathcal{Q}$ is also incoherent. \end{proof} \noindent Based on Lemmas \ref{lem:ConsistencyOnRemoval1} and \ref{lem:ConsistencyOnRemoval2}, from here on, we will only consider $\bar V$ of rank $n+1$. Then, as the rows of $\bar V$ are linearly independent, the function $Q:\rsp{\bar V}\rightarrow \R$ where $Q(\boldsymbol o)=\boldsymbol a \cdot \boldsymbol{\bar q}$ if $\bar V ^T \boldsymbol a = \boldsymbol o$ is well-defined. By Lemma \ref{lem:qcoherent}, $\mathcal{Q}$ is coherent if and only if $Q(\boldsymbol o)\geq 0$ for all $\boldsymbol o \in \rsp{\bar V}\cap O_+$. \\\\ Vectors $\bec a\in \R^{n+1}$ can be thought of as ``bets", where the payout of the bet in world $\omega$ is equal to $$\sum_{i=1}^{n+1} a_i 1_{E_i}(\omega)$$ where $E_{n+1}=\Omega$. We consider the bets to be against a bookmaker with credence base $\mathcal Q$ who expects no edge. \\\\ For such a bet $\bec a$, the vector $\bec b = \bar V^T \bec a\in \R^N$ represents the payout of $\bec a$ atomwise: if atom $\omega_j$ happens, then the payout of bet $\bec a$ will be $b_j=\bec e_j \cdot (\bar V^T \bec a)$. Because we assume that $\bar V$ has full rank, there is a bijection between bets $\bec a\in \R^n$ and atomwise payouts $\bec b\in \rsp{\bar V}$. Note that it does not make sense to talk about atomwise payouts outside of $\rsp{\bar V}$ as it is impossible to make bets with such payouts. \\\\ For finding the coherence of a credence base, it is somewhat more helpful to think in terms of atomwise payouts rather than bets. For instance, if all entries of an atomwise payout $\bec b$ are non-negative, then, if $\mathcal Q$ is coherent, the cost $Q(\bec b)$ of the bet should be non-negative. The Dutch Book Theorem \ref{lem:qcoherent} tells us that having this condition for all atomwise payouts $\bec b\in \rsp{\bar V}$ is equivalent to coherence. Recalling that the set of coherent beliefs is a polytope, each bet whose atomwise payout is non-negative corresponds to an inequality that all elements of the polytope satisfy, and if a belief is outside the polytope, it violates an inequality corresponding to some facet, which is a bet that can be made. The following lemma allows us to just consider a positive spanning set $B$ of atomwise payouts that need to be checked to ensure coherence. \begin{definition} A \textit{positive spanning set} of a set $S\subseteq \R^N$ is a subset $B\subseteq S$ with the property that for any $\boldsymbol s \in S$, there exist coefficients $c_{\boldsymbol b}\in \R_{\ge 0}$ for each $\boldsymbol b \in B$ where only a finite number of $c_{\bec b}$ are non-zero and $$\boldsymbol s = \sum_{\boldsymbol b \in B} c_{\boldsymbol b}\boldsymbol b.$$ \end{definition} \begin{lemma} Let $B$ be a positive spanning set of $\rsp{\bar V}\cap O_+$. Then, $\mathcal{Q}$ is coherent if and only if $Q(\boldsymbol b)\geq 0$ for all $\boldsymbol b \in B$. \end{lemma} \begin{proof} If $Q(\boldsymbol b)<0$, for some $\boldsymbol b \in B$, then $\mathcal{Q}$ is incoherent by Lemma \ref{lem:qcoherent}. \\\\ Suppose that $Q(\boldsymbol b)\ge 0$ for all $\boldsymbol b \in B$. Then, consider any $\boldsymbol o\in \rsp{\bar V}\cap O_+$. As $B$ is a positive spanning set, we can represent $$\boldsymbol o =\sum_{\boldsymbol b \in \mathcal{B}} c_{\boldsymbol b}\boldsymbol b.$$ As $Q$ is linear, $$Q(\boldsymbol o)=\sum_{\boldsymbol b \in \mathcal{B}} c_{\boldsymbol b}Q(\boldsymbol b)\ge 0$$ as the sum of the product of non-negative reals. Therefore, $\mathcal{Q}$ is coherent. \end{proof} \begin{definition} The set of \textit{extremal vectors} of $\rsp{\bar V}\cap O_+$ is $$M:=\set{\boldsymbol b\in \rsp{\bar V}\cap O_+:\forall \boldsymbol o \in \rsp{\bar V}\cap O_+, \bec o\le \bec b \implies \boldsymbol o = \lambda \boldsymbol b \text{ for some $\lambda \in \R$}}.$$ \end{definition} \noindent Let $O_1=\set{\boldsymbol o \in \rsp{\bar V}\cap O_+:\text{the first non-zero entry of }\boldsymbol o \text{ is 1}}$ where we fix an arbitrary ordering of the finite number of entries of $\bec o$. We will show that $M\cap O_1$ is a minimal positive spanning set of $\rsp{\bar V} \cap O_+$. \\\\ \noindent If we think of each positive spanning set as a set of atomwise payouts that the bookmaker has to check is positive to ensure coherence, the following gives some atomwise payouts that the bookmaker definitely has to check. \begin{lemma} Let $B$ be a positive spanning set of $\rsp{\bar V} \cap O_+$ where $B\subseteq O_1$. Then, $$O_1\cap M\subseteq B.$$ \end{lemma} \begin{proof} If $B$ did not include such a $\boldsymbol b\in O_1\cap M$, as $B$ is a positive spanning set, then there is a finite set $S$ where $\boldsymbol b = \sum_{s\in S} c_s\boldsymbol b_s$ for some $c_s\geq 0$. But as not all $c_s = 0$, without loss of generality, suppose $c_1\neq 0$. Then, $\boldsymbol b - c_1\boldsymbol b_1\geq 0$, so $\boldsymbol b = \lambda \boldsymbol b_1$. As the first non-zero entry of each is $1$, $\boldsymbol b = \boldsymbol b_1$. \end{proof} \begin{definition} Let $A$ be a subset of $\R^n$. Consider $Z:A\rightarrow 2^{\set{1, \dots, n}}$ where $Z(\boldsymbol o)=\set{i:o_i=0}$. A vector $\boldsymbol v$ is \textit{maximally-$0$} in $A$ if there there is no $\boldsymbol w\in A\symbol{92}\set{\boldsymbol 0}$ where $Z(\boldsymbol v)\subsetneq Z(\boldsymbol w)$. \end{definition} \begin{lemma} For any vector $\boldsymbol o$ that is maximally-$0$ in $\rsp{\bar V}$, if $Z(\boldsymbol v)=Z(\boldsymbol o)$ for some vector $\boldsymbol v\in \rsp{\bar V}$, then $\boldsymbol v \in \rsp{\boldsymbol o}$. \label{lem:z=maximally0} \end{lemma} \begin{proof} Suppose that $\boldsymbol v \notin \rsp{\boldsymbol o}$. Let the first non-zero entry of $\boldsymbol o$ be $\lambda$ and the first non-zero entry of $\boldsymbol v$ be $\mu$. Then, $\boldsymbol 0\neq\boldsymbol o - \frac \lambda \mu \boldsymbol v$ and $Z(\boldsymbol o) \subsetneq Z(\boldsymbol o-\frac \lambda \mu\boldsymbol v)$. \end{proof} \noindent It turns out that it suffices to just check the maximally-0 elements of $\rsp{\bar V}$, as shown by the following to lemmas. \begin{lemma} A vector $\boldsymbol o \in \rsp{\bar V}\cap O_+$ is maximally-$0$ in $\rsp{\bar V}$ if and only if $\boldsymbol o \in M$\label{lem:MequivMax0}. \end{lemma} \begin{proof} Suppose that $\boldsymbol o \not \in M$. Then, there exists $\boldsymbol v \in \rsp{\bar V}\cap O_+\symbol{92}\rsp{\boldsymbol o}$ where $\boldsymbol o - \boldsymbol v\geq 0$. Then, $Z(\boldsymbol o) \subseteq Z(\boldsymbol v)$, so by Lemma \ref{lem:z=maximally0}, $\boldsymbol o$ is not maximally-0 in $\rsp{\bar V}$. \\\\ Suppose that $\boldsymbol o\in M$. If $\boldsymbol o$ were not maximally-$0$ in $\rsp{\bar V}\cap O_+$, then we would have $\boldsymbol v\in \rsp{\bar V}\cap O_+\setminus\set{\bec 0}$ with with $Z(\bec o) \subsetneq Z(\bec v)$. Let $v_{\uparrow}$ be the maximum entry of $\boldsymbol v$ and $o_{\downarrow}$ be the minimum entry of $\boldsymbol o$. Then, $\boldsymbol o - \frac{o_{\downarrow}}{v_{\uparrow}}\boldsymbol v \geq \bec 0$, which is impossible as $\boldsymbol v$ cannot be in the span of $\boldsymbol o$ by the definition of $M$. Therefore, $\bec o$ is maximally-$0$ in $\rsp{\bar V}\cap O_+$ by contradiction. \\\\ Next, it suffices to show that if $\bec o$ is maximally-$0$ in $\rsp{\bar V}\cap O_+$, then $\bec o$ is maximally-$0$ in $\rsp{\bar V}$. To see this, suppose not and that there exists some $\boldsymbol v\in \rsp{\bar V}\symbol{92}\set {\boldsymbol {0}}$, where $Z(\boldsymbol o)\subsetneq Z(\boldsymbol v)$. Then, $\boldsymbol h =\boldsymbol o - \frac{o_{\downarrow}}{v_{\uparrow}}\boldsymbol v \geq \bec 0$ and $Z(\boldsymbol v)\subseteq Z(\boldsymbol h)$. But $\bec h\in \rsp{\bar V}\cap O_+$ and we have $Z(\boldsymbol o)\subsetneq Z(\boldsymbol v)\subseteq Z(\boldsymbol h)$, which contradicts the fact that $\bec o$ is maximally-$0$ in $\rsp{\bar V}\cap O_+$. \end{proof} \begin{lemma} $O_1\cap M$ is a positive spanning set of $\rsp{\bar V} \cap O_+$. \end{lemma} \begin{proof} For the sake of contradiction, suppose that $O_1\cap M$ is not a positive spanning set and consider $\boldsymbol o\in\rsp{\bar V} \cap O_+$ to be a maximally-$0$ vector not its positive span. As $\boldsymbol o$ is not in the span of any element of $O_1\cap M$, $\boldsymbol o$ is not maximally-$0$ in $\rsp{\bar V}$. Therefore, there exists $0\neq \boldsymbol v \in \rsp{\bar V}$ where $Z(\boldsymbol o)\subsetneq Z(\boldsymbol v)$. For all non-zero elements of $\boldsymbol v$, consider $r_i=\frac{o_i}{v_i}$ and let $r=\min\set{r_i:r_i>0}$. Then, $\boldsymbol w = \boldsymbol o - r \boldsymbol v\in \rsp{\bar V} \cap O_+$, so as $Z(\boldsymbol o) \subsetneq Z(\boldsymbol w)$, $\boldsymbol w$ is in the positive span of $B$. For $w_i\neq 0$, let $s_i=\frac{o_i}{w_i}$ and $s = \min \set{s_i}$. Then, $\boldsymbol o - s \boldsymbol w \in \rsp{\bar V} \cap O_+$ and $Z(\boldsymbol o)\subsetneq Z(\boldsymbol o - s \boldsymbol w)$. Therefore, $\boldsymbol o - s \boldsymbol w$ and $\boldsymbol w$ are both elements of the positive span of $O_1\cap M$, so $\boldsymbol o$ must also be an element of the positive span of $O_1\cap M$. \end{proof} \begin{theorem} The hyperplane $\set{\bec x\in \R^n : \bec a \cdot \bec x = c}$ is a facet of the polytope whose vertices are the columns of $V$ if and only if $$\bar V^T \begin{pmatrix} \bec a\\ -c \end{pmatrix}\in M.$$ \label{thm:restatement} \end{theorem} \begin{proof} By \cite{Ziegler2000}, the hyperplane $\set{\bec x : \bec a \cdot \bec x = c}$ is a facet of a polytope $P$ if and only if for all vertices $\bec v$ of $P$, $\bec a \cdot \bec v \ge c$ with equality for a maximal subset of vertices. This happens if and only if the vector $$V^T \bec a \ge c\bec 1$$ with equality for a maximal subset of the entries, which is true if and only if $$\bec b :=\bar V^T \begin{pmatrix} \bec a\\ -c \end{pmatrix}\ge \bec 0$$ and is maximally-$0$ among vectors in $\rsp{\bar V}$. By Lemma \ref{lem:MequivMax0}, this is equivalent to $\bec b\in M$. \end{proof} \begin{corollary} There is a one to one correspondence between elements of $O_1 \cap M$ and facets of the polytope whose vertices are the columns of $V$. \end{corollary} \noindent By \cite{Ziegler2000}, every non-redundant set of inequalities bounding a polytope $P$ has exactly one inequality for every face. This means that if a bookmaker verifies that they are not giving a Dutch book by checking a fixed set of inequalities, it suffices to check the inequalities $Q(\bec b)\ge 0$ for all $\bec b\in M\cap O_1$, and every non-redundant set of inequalities they must check consists of rescaled versions of these inequalities. \section{Merging Individually Coherent Experts} \label{sec:indCohExp} \noindent In this section we consider a setting in which each of several experts submits a set of internally coherent credences, but the union of all expert credences is possibly incoherent. To aggregate these credences into a single coherent set of beliefs, we could use a loss function of the form (\ref{eqn:lossfunction}), but that ignores the additional information that beliefs from the same expert are coherent. Since logical inferences can be made from an expert's stated credences, the expert can express the same information in multiple ways. As long as the set of possible inferences about an expert's beliefs is the same, the loss function should be invariant under the specific information the expert shares, a property that the method for separate beliefs previously discussed does not have if naively applied here. We call this property \textit{content invariance} and discuss it further below. \\\\ For example, if a meteorologist says that there is a $40\%$ chance of rain tomorrow and a $70\%$ chance of clouds, it is reasonable to infer that there is a $30\%$ chance of sun and a $30\%$ chance of clouds without rain. In such a situation, the meteorologist might also explicitly say the chance of it being sunny, however, because that statement does not provide new information, the loss function should be invariant under whether or not they say it. \\\\ Recall that $\bar V$ obtained by appending a row of $1$s to $V$ and that $\rsp{\bar V}$ is the row span of $\bar V$. \begin{definition} Two coherent credence bases $(V, \bec q)$ and $(V', \bec q')$ are \textit{content equivalent} if $\rsp{\bar V}=\rsp{\bar V'}$ and the credence base $((V, V'), (\bec q, \bec q'))$ is coherent. \end{definition} \begin{lemma} If $(V, \bec q)$ and $(V', \bec q')$ are content equivalent and $\bec \pi$ is a probability vector, then $$V\bec \pi = \bec q \iff V'\bec \pi = \bec q'.$$ \label{lem:piSameContent} \end{lemma} \begin{proof} Since $\ker(\bar V) = \rsp{\bar V}^\perp$, the orthogonal space to the row span of $\bar V$, if $\rsp{\bar V}=\rsp{\bar V'}$, then $\ker(\bar V)=\ker(\bar V')$. By the coherence of $((V, V'), (\bec q, \bec q'))$, there is some $\bec{\tilde\pi}$ so that $V\bec{\tilde\pi}=\bec q$ and $V'\bec{\tilde\pi}=\bec q'$. Then, for any probability vector $\bec \pi$, as $\bec\pi -\bec{\tilde\pi}\in \ker(\bec 1^T)$, $$V\bec \pi = \bec q \iff \bec \pi - \tilde{\bec \pi}\in \ker(\bar V)\iff \bec \pi - \tilde{\bec \pi}\in \ker(\bar V')\iff V'\bec \pi = \bec q'.\qedhere$$ \end{proof} \noindent Lemma \ref{lem:piSameContent} implies that content equivalence is transitive, so is an equivalence relation. \begin{definition} Let $\phi$ be a function of a credence base. $\phi$ is \textit{content invariant} if for content equivalent credence bases $\mathcal{Q}$ and $\mathcal{Q'}$ we have $\phi(\mathcal{Q})=\phi(\mathcal{Q}')$. \end{definition} \subsection{Content Invariance}\label{subsec:contentInvariance} Suppose that $(V, \bec q)$ is a coherent credence base where the $n+1$ rows of $\bar V$ are linearly independent. Let $I=\rsp{\bar V}\cap \set{0, 1}^N$, which the next lemma shows is the set of events whose probabilities are linearly inferable. \begin{lemma} Consider the statements \begin{enro} \item $\bec E\in I$. \item There is a unique belief $q$ so that the credence base $((\bec E, V), (q, \bec q))$ is coherent. \end{enro} (i) implies (ii). Moreover, if there is a probability vector $\bec \pi$ so that $V \bec \pi = \bec q$ where none of the entries of $\bec \pi$ are $0$ or $1$, then (ii) implies (i). \end{lemma} \begin{proof} Suppose (i). The belief $q=\bec E \cdot \bec \pi$ makes $((\bec E, V), (q, \bec q))$ coherent as $(V, \bec q)$ is coherent. For uniqueness, suppose we there are two coherent beliefs $q, q'$ about some event for $\bec E\in I$. Then as there is some $\bec a\in \R^n$ where $V^T \bec a = \bec E$, there are two probability vectors $\bec \pi$ and $\bec \pi'$ where $q = (V^t \bec a) \cdot \bec \pi$ and $q' = (V^t \bec a) \cdot \bec \pi'$ with $\bec q = V \bec \pi = V \bec \pi'$. This means that $$q=(V^t \bec a) \cdot \bec \pi = \bec a^T V \bec\pi = \bec a^T \bec q = \bec a^T V \bec\pi' = q'.$$ Suppose not (i) and let $\bar V'$ be the matrix $\bar V$ with the vector $\bec E$ appended as the last row. Then, there is a vector $\bec v\in \ker(\bar V)\symbol{92}\ker(\bar V')$. By the assumption that no entries of $\bec \pi$ are $0$ or $1$, there is an $\epsilon>0$ so that for all $\|a\|<\epsilon$, $\bec \pi + a\bec v$ is also a probability vector. Then, for all such $a$, taking $q=\bec E \cdot (\bec \pi + a\bec v)$ makes $((\bec E, V), (q, \bec q))$ coherent. As $\bec v \in \ker(\bar V)\symbol{92}\ker(\bar V')$, we have $\bec E\cdot \bec v\neq 0$, so each distinct value of $a$ yields a distinct $q$. \end{proof} \begin{lemma} Let $R$ be the reduced row-echelon form of $\bar V$. Then, $$I=\set{R^T \boldsymbol v:\boldsymbol v \in \set{0, 1}^{n+1}, R^T \boldsymbol v \in \set{0, 1}^N}.$$ \end{lemma} \begin{proof} Firstly, $$\set{R^T \boldsymbol v:\boldsymbol v \in \set{0, 1}^{n+1}, R^T \boldsymbol v \in \set{0, 1}^N}\subseteq \rsp{\bar V}\cap\set{0, 1}^N=I.$$ Note that all of $R^T$'s columns have a leading $1$ which is the only entry in its row, so each of $\boldsymbol v$'s entries appear somewhere in $R^T\boldsymbol v$. Therefore, if $R^T\boldsymbol v \in \set{0, 1}^N$, then $\boldsymbol v \in \set{0, 1}^{n+1}$. \end{proof} \begin{corollary} The maximum size of $I$ is $2^{n+1}$. \end{corollary} \noindent Because $I$ is defined only in terms of the row span $\rsp{\bar V}$, it is content invariant. However, $I$ tends to be quite large even for well-tamed events, so a smaller, content invariant set might be desirable. \\\\ Let $B\subseteq I$ be a minimal positive spanning set of $I$. \begin{lemma} Any maximally-$0$ element of $I$ must be part of $B$. \end{lemma} \begin{proof} Suppose that $\bec v\in I$ is maximally-$0$ in $I$. As $B$ is a positive spanning set, we can write $$\bec v = \sum_{\bec b\in B} c_{\bec b} \bec b.$$ As $\bec v\ne \bec 0$, there is some $\bec b\in B$ where $c_{\bec b}\ne \bec 0$, so $\bec b \le \bec v$ and $Z(\bec b)\subseteq Z(\bec v)$. As $\bec v$ is maximally-$0$, $Z(\bec b)= Z(\bec v)$ which means that $\bec v = \bec b$ as both are $0,1$-vectors. \end{proof} \begin{theorem} \label{thm:Bmax0}$B$ is the set of maximally-$0$ elements of $I$. \end{theorem} \begin{proof} It suffices to show that the maximally-$0$ elements of $I$ are a positive spanning set. Suppose that this were not the case. Let $\boldsymbol v$ be a maximally-$0$ vector in $I\symbol{92}\rsp{B}_+$, the maximal set positively spanned by $B$. Note that $\boldsymbol v$ is not maximally-$0$ in $I$ or it would be a member of $B$. Then, there exists $\boldsymbol w\in I\symbol{92}\set{0}$ where $Z(\boldsymbol v)\subsetneq Z(\boldsymbol w)$. Note that $Z(\boldsymbol v)\subsetneq Z(\boldsymbol v-\bec w)$, meaning $\boldsymbol w$ and $\boldsymbol v-\boldsymbol w$ are both in the positive span of $B$, so $\boldsymbol v$ is also in the positive span of $B$. \end{proof} \begin{lemma} Let $Q:I \rightarrow [0, 1]$ be the implied belief function, taking an event $\bec E$ to the unique belief $q$ that makes the credence base $((\bec E, V), (q, \bec q))$ coherent. The function $Q$ as well as the sets $I$ and $B$ are content invariant. \label{lem:I_content_invariant} \end{lemma} \begin{proof} Let $(V, \bec q)$ and $(V', \bec q')$ be two content equivalent coherent credence bases with set of inferable events $I=\rsp{\bar V} \cap \set{0, 1}^N$ and $I' = \rsp{\bar V'} \cap \set{0, 1}^N$, positive bases $B$ and $B'$ of $I$ and $I'$, and implied belief functions $Q$ and $Q'$. Then, by definition, $\rsp{\bar V} = \rsp{\bar V'}$, so $I=\rsp{\bar V} \cap \set{0, 1}^N=\rsp{\bar V'} \cap \set{0, 1}^N=I'$ is content invariant. Also, because $B$ and $B'$ are defined solely in terms of $I$ and $I'$ respectively, $B=B'$. \\\\ To show that $Q$ is content invariant, if $Q(\bec E) = q$, then by coherence, there is a probability vector $\bec \pi$ so that $V\bec \pi = \bec q$ and $\bec E \cdot \bec \pi = q$. By Lemma \ref{lem:piSameContent}, we have $V' \bec \pi = \bec q'$, meaning that $\bec \pi$ certifies that $((\bec E, V'), (q, \bec q'))$ is coherent, so by uniqueness, $Q'(\bec E) = q$. \end{proof} \subsection{Loss Functions with Content Invariance} Suppose that there are $k$ experts, the $i$th of which tells us the coherent credence base of their beliefs $\mathcal E_i = (V_i, \bec q_i)$, with event matrix $V_i$ and belief vector $\bec q_i$ defined previously. Let $\mathcal Q$ be the combined credence base of all experts, formed by concatenating $V_1, \dots, V_k$ and $\bec q_1, \dots, \bec q_k$. We will consider loss functions of the form $$L(\bec p, \mathcal{Q})=\sum_{i=1}^{k} \mathcal D_i(\bec p, \mathcal{E}_i)$$ for some content invariant measure of disagreement $\mathcal D_i$. In contrast to Section \ref{sec:seperateBeliefs}, the sum is over experts rather than over individual events. As before, $$L^(\mathcal{Q})=\min_{\bec p \in C(\mathtt E)}L( \mathcal{Q})=L(\bec{p^}, \mathcal Q)$$ is a measure of the incoherence of the set of experts as a group. \\\\ For the $i$th expert, let $I_i=\rsp{\bar V_i} \cap \set{0, 1}^N$ be the set of events for which we can infer a probability from expert $i$'s stated beliefs, let $Q_i:I_i\rightarrow [0, 1]$ be this inferred probability, and let $B_i$ be the minimal positive spanning set of $I_i$, as defined in Section \ref{subsec:contentInvariance}. Also, write $P(\bec E)$ be the entry of $\bec p$ corresponding to event $E$. \\\\ As $Q_i$ is content invariant one approach is to let \begin{equation} \mathcal D_i = D_{i, S_i} := \frac{1}{\#S_i} \sum_{\bec b\in S_i} \ell(P(\bec b), Q_i(\bec b))\label{eqn:contentinvariantloss} \end{equation} where $S_i\subseteq I_i$ is a content invariant set of events and $\ell$ is a dissimilarity function as in Section \ref{sec:seperateBeliefs}. As $I_i$ and $B_i$ are both content invariant, they are both choices for $S_i$. \\\\ One possible justification for summing over $B_i$ instead of $I_i$ is that $\#I_i$ can be much larger than $\#B_i$. $B_i$ contains the maximally-$0$ elements of $I_i$, or the elements whose probabilities are smallest. As the dissimilarity functions $f$ and $f^o$ punish more for the same error when $q$ is smaller ($f(0.101, 0.001)>f(0.6, 0.5)$), summing over $B_i$ is summing over the elements of $I_i$ whose relative errors will be greatest while not considering the other terms for ease of computation. \\\\ The normalizing coefficient $\frac{1}{\#S}$ is included in the loss function in order to give each expert equal weight, regardless of how many beliefs they express. If the measures of disagreement were not normalized, then an expert might become twice as influential on the overall loss function by expressing one additional belief because $\#I_i$ and $\#B_i$ can grow exponentially with the number of beliefs expressed. \subsection{Removing the Symmetry between an Event and its Complement} \noindent Looking at the dissimilarity function $f$, there are two terms: $p\ln \frac pq$ representing the expected surprise from the event occurring and $(1-p)\ln \frac{1-p}{1-q}$, representing the expected surprise from the event not occurring. However, if we sum the dissimilarity function over a minimal positive basis, then being surprised by the complement of an event happening is redundant; when one event does not happen, we are surprised instead by other events happening. Alternatively, an agent observing the world might only keep track of which events do happen, so cannot be surprised by an event not happening. \\\\ This motivates the removal the term representing the surprise of an event not happening from the loss function. However, one needs to be careful in doing this: so far, we have required that loss functions $L(\bec p, \bec q)$ be non-negative and equal to $0$ if and only if $\bec p = \bec q$. For example, $p\ln \frac pq$ by itself is not a dissimilarity function, so the corresponding ``loss function" $L$ does not have this property in general. The following lemmas motivates two possible content invariant disagreement functions that have this property: \begin{lemma} Let $(V, \bec p)$ and $(V, \bec q)$ be coherent credence bases and suppose that $V^T \bec 1=k\bec 1$ for some $k\in \mathbb N$. Then, $$\sum_{i=1}^n p_i \ln \frac{p_i}{q_i}\ge 0$$ with equality if and only if $\bec p = \bec q$. \label{lem:log-sum} \end{lemma} \begin{proof} By Lemma \ref{lem:ConsistencyOnRemoval1}, since $V^T \bec 1 - k\bec 1 = \bec 0$, we have $$\sum_{i=1}^n p_i - k = \sum_{i=1}^n q_i - k = 0.$$ Therefore, $\sum_{i=1}^n p_i = k = \sum_{i=1}^n q_i$, so applying the log-sum inequality, Theorem 2.7.1 of \cite{infoTheory}, implies that $$\sum_{i=1}^n p_i \ln \frac{p_i}{q_i}\ge \left(\sum_{i=1}^n p_i\right) \ln \frac{\sum_{i=1}^n p_i}{\sum_{i=1}^n q_i} = 0$$ with equality if and only if $\frac{p_i}{q_i}$ is constant. By coherence, this happens if and only if $\bec p = \bec q$. \end{proof} \noindent Recall from Section \ref{section:predd} that for a proper scoring rule $s(i, q)$, there is an associated dissimilarity function $\ell(p, q) = p(s(1, p) - s(1, q)) + (1-p)(s(0, p) -s(0, q))$. When using the log-loss scoring rule, this becomes $\ell = f$. Can we generalize Lemma \ref{lem:log-sum} to apply to half-dissimilarity functions \begin{equation}\label{eqn:tilde} \tilde{\ell}(p, q):=ps(1, p) - ps(1, q) \end{equation} for a proper scoring rule $s$? No. The following lemma shows that log-loss is the only differentiable proper scoring rule for which Lemma \ref{lem:log-sum} holds. \begin{lemma} Let $s(i, q)$ be a proper scoring rule that is differentiable in $q$ for $q\in (0, 1)$ with $\tilde \ell$ as defined in equation (\ref{eqn:tilde}). Suppose that for all coherent credence bases $(V, \bec q)$ where $V^T \bec 1=k\bec 1$ for some $k\in \mathbb N$, $$\bec q = \argmin_{\bec p \in C(V)} \sum_{i=1}^n \tilde \ell(p_i, q_i)$$ Then for some $\lambda < 0$ and $c\in \R$, $${s(i, q) = \lambda(i\ln q + (1-i)\ln (1-q)) + c}.$$\label{lem:loglossunique} \end{lemma} \begin{proof} Note that by continuity, the value of $s$ when $q\in \set{0, 1}$ are determined by its values on the interval $(0, 1)$. Write $$L(\bec p, \bec q) := \sum_{i=1}^n \tilde \ell(p_i, q_i)$$ Let $V$ be the $3\times 3$ identity matrix and let $\lambda = \frac 12 s'(1, \frac 12)$ where $s'$ is the derivative of $s$ in $q$. For any $0<q < \frac 12$, consider the coherent belief vector $\bec q = (q, \frac 12, \frac 12 - q)^T$. By the assumption that $L(\bec p, \bec q)$ is minimized among coherent $\bec p$ when $\bec p = \bec q$, Lagrange multipliers yield that $$c\bec 1 = \pderiv{L}{\bec p}(\bec q, \bec q) = \begin{pmatrix} \lambda \\ q s'(1, q) \\ (\frac{1}{2}-q) s'(1, \frac{1}{2}-q) \end{pmatrix}$$ so $q s'(1, q)=\lambda$ for all $0<q < \frac 12$. For $\frac 12 < q <1$, take $V$ to be the $2\times 2$ identity matrix and $\bec q = (q, 1 - q)$. The same argument as above shows that $q s'(1, q) = (1-q)s'(1, 1-q) = \lambda$. Therefore, for all $q\in (0, 1)$, we have $q s'(1, q) = \lambda$, so $s'(1, q) = \frac{\lambda}{q}$ and $s(1, q) = \lambda \ln q + c_1$ for some $c_1\in \R$. \\\\ As $s$ is a proper scoring rule, the quantity $p s(1, q) + (1-p)s(0, q)$ is uniquely minimized in $q$ when $p=q$, so $-(1-p) s'(0, p) = ps'(1, p) =\lambda$, meaning that $s'(0, p) = \frac{-1}{1-p}$ and $s(0, p) = \lambda \ln (1-q)+c_2$ for some $c_2\in \R$. Therefore, we have $$s(i, q) = \lambda(i\ln q + (1-i)\ln (1-q)) + c_1+c_2.$$ In order for $s$ to be a proper scoring rule, we must have $\lambda < 0$. \end{proof} \noindent Letting $O_i$ be the set of all subsets of $B_i$ whose sum is $\boldsymbol 1$ and $M_i=\sum_{S\in O_i} \#S$, we can use the dissimilarity function \begin{equation} \mathcal D_i = \tilde D_{i, B_i} = \frac{1}{M_i}\sum_{S \in O_i} \sum_{\boldsymbol b \in S}\tilde \ell(P(\bec b), Q(\bec b)).\label{eqn:assymetricvariant} \end{equation} Lemma \ref{lem:log-sum} implies that when using $\tilde \ell(p, q) = \tilde f(p, q) := p\ln \frac pq$ or $\tilde \ell(p, q) = \tilde f^o(p, q) := q\ln \frac qp$, the measure of disagreement $\tilde D_{i, B_i}$ is minimized when $P(\bec b) = Q_i(\bec b)$ for all $\bec b \in B_i$. \\\\ We compare the methods presented here in Appendix \ref{sec:specialcompare} as well as in Appendix \ref{sec:ex}. \color{black} \section{Acknowledgements} We thank Kenny Easwaran, Jonathan Gabor, Oliver Hopcroft, David Krueger, Yuval Peres, Suvadip Sana, Luchen Shi, and Ariel Yadin for inspiring discussions. \pagestyle{plain} \printbibliography \pagestyle{headings} \appendix \section{Numerical Comparison between $f$ and $f^o$\label{sec:compare}} \noindent To contrast the two settings in Section \ref{sec:seperateBeliefs}: One should use the dissimilarity function $f$ when submitting multiple forecasts based on their possibility incoherent internal beliefs (and expects these forecasts to be scored with the logarithmic proper scoring rule); its transpose $f^o$ should be used in aggregating independent sources of information, each of which gives information about one event, in order to find the most likely true probability distribution. \\\\ Recall that $\textbf E_i\in \set{0, 1}^N$ is the vector whose $j$th entry is $1$ if $\omega_j\in E_i$ and $0$ otherwise. Additionally, $\textbf e_i$ denotes the $i$th standard basis vector and $\textbf 1$ the vector of all $1$s. For any distribution $\pi$, we associate a vector $\boldsymbol \pi$ with it, whose $j$th entry is $\pi(\omega_j)$, the probability of the $j$th atom in the Boolean algebra. \\\\ Table~\ref{fig:Lnumeric} contains some examples of the numerical differences between using $f$ and $f^o$ as dissimilarity functions. \begin{table}[h] \begin{center} \begin{tabular}{\|m{12em} \| m{10em}\| m{9em} \|} \hline Situation & \multicolumn{2}{c\|}{$\bec{p^}$} \\ \cline{2-3}& $\ell=f$ & $\ell=f^o$\\ \hline Exactly one event occurs: & & \\$\boldsymbol E_1=\boldsymbol e_1, q_1 = 0.1$ & $p_1=0.01$&$p_1=0.04$ \\$\boldsymbol E_2=\boldsymbol e_2, q_2 = 0.6$ & $p_2=0.11$&$p_2=0.30$ \\$\boldsymbol E_3=\boldsymbol e_3, q_3 = 0.99$ & $p_3=0.89$&$p_3=0.66$ \\\hline All estimates are for the same event: & & \\${\boldsymbol E_1 = \boldsymbol E_2=\boldsymbol E_3=(1 \, 0)^T}$ & & \\ $q_1 = 0.1, q_2=0.3, q_3=0.5$ &$p_1=p_2=p_3=0.27$ & $p_1=p_2=p_3=0.3$ \\\hline $\boldsymbol E_1 = (1 \,1 \,0 \,0)^T$, $q_1=0.99$ &$p_1=0.87$ & $p_1=0.73$ \\ $\boldsymbol E_2 = (1 \,0 \,1 \,0)^T$, $q_2=0.5$ &$p_2=0.40$ & $p_2=0.47$ \\ $\boldsymbol E_3 = (1 \, 0 \,0 \,0)^T$, $q_3=0.1$ &$p_3=0.27$ & $p_3=0.20$ \\ $\boldsymbol E_4 = (0 \,1 \,0 \,0)^T$, $q_4=0.4$ &$p_4=0.60$ & $p_4=0.53$ \\ $\boldsymbol E_5 = (0 \,0 \,1 \,0)^T$, $q_5=0.4$ &$p_5=0.13$ & $p_5=0.27$ \\\hline \end{tabular} \caption{Three examples of correcting incoherent probabilities with $f$ and $f^o$. By Theorem \ref{punique}, $\bec{p^}$ is unique.} \label{fig:Lnumeric} \end{center} \end{table} \begin{figure}[h] \centering \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{pf} \caption{Using $\ell=f$} \label{fig:p:h=f} \end{subfigure} \hfill \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{pfo} \caption{Using $\ell=f^o$} \label{fig:p:h=fo} \end{subfigure} \hfill \caption{Contour plots of $p^(\bec q)$ when $E_1=E_2$ are the events being estimated. Contour distance is 0.02. \label{fig:pcomparisons} } \end{figure} \\\\ The differences in $f$ and $f^o$ also leads the function $L^$ to behave differently when using the two functions, as illustrated in Figure \ref{fig:Lcomparisons}, where $E_1$ and $E_2=E_1^c$ are the events being estimated. \begin{figure}[h] \centering \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{Lf} \caption{Using $\ell=f$} \label{fig:L:h=f} \end{subfigure} \hfill \begin{subfigure}[b]{0.4\textwidth} \centering \includegraphics[width=\textwidth]{Lfo} \caption{Using $\ell=f^o$} \label{fig:L:h=fo} \end{subfigure} \hfill \caption{Contour plots of $L^(\bec q)$ when $E_1$ and $ E_2=E_1^c$ are the events being estimated. Near the line $q_1+q_2=1$, both functions are approximately quadratic and can be approximated by $\frac{(q_1+q_2-1)^2}{(1-q_1+q_2)(1+q_1-q_2)}$. Also, when using $\ell=f$, $L^$ is not bounded, while it is when using $\ell=f^o$. Contour distance is 0.1.} \label{fig:Lcomparisons} \end{figure} \section{Computing \texorpdfstring{$\bec{p^}$}{p} and \texorpdfstring{$L^$}{L}\label{sec:findpstar}} \noindent One way to find $L^$ and $\boldsymbol{p^}$ numerically is to define a function that, given a vector $\boldsymbol \pi$ and the $n\times N$ matrix $V$ whose $i$th row is $\textbf E_i$, and the credences $\boldsymbol q$, returns $L(V\boldsymbol\pi, \boldsymbol q)$. Then, one can use a gradient descent algorithm to minimize $L(V\boldsymbol\pi, \boldsymbol q)$ under the conditions that the entries of $\boldsymbol \pi$ are nonnegative and sum to $1$. This algorithm is illustrated below in psuedocode.\\ \begin{algorithmic} \State $\bec {\pi_0} \gets \frac{1}{N} \bec{1}$ \Comment{Initial guess is that all atoms have equal probability} \State \Return gradientDescent($\bec \pi \mapsto L(V\boldsymbol\pi, \boldsymbol q)$,\Comment{The loss function in terms of $\bec \pi$}\\ \hspace{1.47in} $\bec{\pi_0}$, \Comment{Initial guess of $\bec \pi$}\\ \hspace{1.47in} $\bec \pi \cdot \bec 1 = 1$, $\bec 0 \le \bec \pi \le \bec 1$) \Comment{Conditions $\bec \pi$ must satisfy}\\ \end{algorithmic} Computing $\bec{p^}(\bec q)$ to within an accuracy of $\epsilon$ takes $O\left(nN\log\left(\frac{N}{\epsilon}\right)\right)$ time \cite{bubeck_convex_2015}. Python code to compute $\bec{p^}$ and $L^$ can be found on \url{https://github.com/scim142/quantifying_coherence}. \section{Holder Continuity of $\bec{p^}$ and $L^$\label{sec:continuity}} \noindent Let $\|\|\cdot \|\|$ denote the Euclidean norm. \begin{lemma} If $\ell$ is (strictly) convex/$n$-times differentiable/Lipschitz in $p$/$q$/both, then $L$ will also be. \label{lem:2.1} \end{lemma} \begin{proof} (convexity in $p$) Let $\ell_i=w_i\ell(\textbf p \cdot \textbf e_i, \textbf q \cdot \textbf e_i)$. Then, for $t\in[0, 1]$, $$\ell_i(t\textbf p + (1-t)\textbf p', \textbf q)=w_i \ell(tp_i+(1-t)p_i', q_i)$$ $$\le w_i t\ell(p_i, q_i)+w_i(1-t)\ell(p_i', q_i)=t\ell_i(\textbf p, \textbf q)+(1-t)\ell_i(\textbf p', \textbf q)$$ so $\ell_i$ is convex. As $L(\textbf p, \textbf q) = \sum_{i=1}^{n} \ell_i(\textbf p, \textbf q)$ is the sum of convex functions, $L$ is convex. \end{proof} \begin{lemma} \label{Lqsim} If $\ell$ is Lipchitz in $q$ in some region $D\subseteq [0, 1]^2$, then $L^$ will be Lipchitz on the set $\bar{D}:=\set{\textbf q\in [0, 1]^n:(\boldsymbol{p^}(\textbf q), \textbf q)\in D}$ with Lipschitz constant equal to that of $L$ in $\boldsymbol q$. \end{lemma} \begin{proof} Let $\boldsymbol q, \boldsymbol{q'}\in \bar D$ and consider the case when $L^(\textbf q)\le L^(\boldsymbol{q'})$. Let $\boldsymbol \delta = \boldsymbol{q'} - \boldsymbol q$ and $k$ be the Lipschitz constant of $L$. Then, $$L^(\textbf q)\le L^(\textbf q + \boldsymbol \delta)\le L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q + \boldsymbol \delta)\le L^(\boldsymbol q) + k \|\|\boldsymbol \delta\|\|$$ If $L^(\boldsymbol q)> L^(\boldsymbol{q'})$, then the above shows that $L^(\boldsymbol q)-k\|\|\boldsymbol \delta\|\|\le L^(\boldsymbol{q'})$. Therefore, regardless of the relative sizes of $L^(\boldsymbol q)$ and $L^(\boldsymbol{q'})$, we have $$L^(\boldsymbol q)-k\|\|\boldsymbol \delta\|\|\le L^(\boldsymbol{q'})\le L^(\boldsymbol q)+k\|\|\boldsymbol \delta\|\|.\qedhere$$ \end{proof} \begin{lemma} If $\ell$ is strictly convex and twice differentiable in $(0, 1)^n$, then $\boldsymbol{p^}(\boldsymbol q)$ is continuous. \label{lem:pcontinuous} \end{lemma} \begin{proof} Fix any $\bec q \in (0, 1)^n$ and $k_q$, $k_p$ greater than the Lipschitz constants of $L$ with respect to $\bec q$ and $\bec p$ respectively in the neighborhood of $(\bec{p^}(\bec q), \bec q)$. Let $D$ be a closed region in which $L$ is Lipschitz in $\boldsymbol{p}$ with Lipschitz constant $k_p$ and Lipshcitz in $\boldsymbol{q}$ with Lipschitz constant $k_q$ and $\bar D$ be a closed region contained in $\set{\boldsymbol q':(\boldsymbol{p^}(\boldsymbol{q'}), \boldsymbol{q'})\in D}$ that contains $(\bec{p^}(\bec q), \bec q)$.\\\\ Let $\boldsymbol{q_1}, \boldsymbol{q_2}, \dots$ be a sequence in $\bar D$ with limit $\boldsymbol q$. As $(\boldsymbol{p^}(\boldsymbol a), \boldsymbol{a})\in D$ for all $\boldsymbol a\in \bar D$, and $D$ is sequentially compact, the sequence $\boldsymbol{p^}(\boldsymbol{q_1}), \boldsymbol{p^}(\boldsymbol{q_2}), \dots$ has a subsequence with a limit. Let $\boldsymbol{p'}$ be one of those limit and $\boldsymbol{a_1}, \boldsymbol{a_2}, \dots$ be a subsequence of $\boldsymbol{q_1}, \boldsymbol{q_2}, \dots$ where the limit of $\boldsymbol{p^}(\boldsymbol{a_1}), \boldsymbol{p^}(\boldsymbol{a_2}), \dots$ is $\boldsymbol{p'}$. Then, for any $\epsilon >0$, there exists $N$ where for all $n>N$, $\|\|\boldsymbol{a_n}-\boldsymbol q\|\|<\epsilon$ and $\|\|\boldsymbol{p^}(\boldsymbol{a_n})-\boldsymbol{p'}\|\|<\epsilon$. Then, by Lemma~\ref{Lqsim}, $$\|\|L(\boldsymbol{p^}(\boldsymbol{a_n}), \boldsymbol{a_n})-L(\boldsymbol{p^}(\boldsymbol{q}), \boldsymbol{a})\|\|\le k_q\epsilon.$$ Using the fact that $L$ is Lipschitz in $\boldsymbol p$ and $\boldsymbol q$, by the triangle inequality $$\|\|L(\boldsymbol{p'}, \boldsymbol{q})-L(\boldsymbol{p^}(\boldsymbol{q}), \boldsymbol{a})\|\|-k_p\epsilon-k_q\epsilon\le \|\|L(\boldsymbol{p^}(\boldsymbol{a_n}), \boldsymbol{a_n})-L(\boldsymbol{p^}(\boldsymbol{q}), \boldsymbol{a})\|\|$$ $$\|\|L(\boldsymbol{p'}, \boldsymbol{q})-L(\boldsymbol{p^}(\boldsymbol{q}), \boldsymbol{a})\|\|\le (2k_q+k_p)\epsilon.$$ As this is true for all $\epsilon>0$, we must have $\|\|L(\boldsymbol{p'}, \boldsymbol{q})-L(\boldsymbol{p^}(\boldsymbol{q}), \boldsymbol{a})\|\|=0$, so by Theorem \ref{punique}, $\boldsymbol{p'}=\boldsymbol{p^}(\boldsymbol{q})$ and $\boldsymbol{p^}$ is continuous at $\boldsymbol q$. \end{proof} \noindent Let $\nabla L(\bec p, \bec q)$ be the gradient of $L$ with respect to $\bec p$ at $(\bec p, \bec q)$. \begin{theorem} \label{holder} If $\ell$ is strictly convex and twice differentiable in $(0, 1)^n$, then $\boldsymbol{p^}(\boldsymbol q)$ is Holder-$\frac 12$. \label{thm:holder} \end{theorem} \begin{proof} Fix any $\bec q \in (0, 1)^n)$ and let $\lambda$ be the least eigenvalue of the Hessian $H$ of $L$ with respect to $\boldsymbol p$ at $(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$, let $D$ be some open region over which $L$ is Lipschitz that contains $(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$, and let $\bar{D}=\set{\textbf q\in [0, 1]^n:(\boldsymbol{p^}(\textbf q), \textbf q)\in D}$. Additionally, let $k$ be the Lipschitz constant of $L$ with respect to $\boldsymbol q$ on $\bar D$. Then, first of all, by Lemma~\ref{Lqsim}, for all $\boldsymbol \delta$ where $\boldsymbol q + \boldsymbol \delta \in \bar D$, $$\|L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q+\boldsymbol \delta)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)\|\le k \|\|\boldsymbol \delta\|\|.$$ Then, by the triangle inequality, as $L$ is Lipschitz in $\boldsymbol q$ $$\|L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)\|$$ $$\le \|L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol\delta), \boldsymbol q)-L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q+\boldsymbol \delta)\|+\|L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q+\boldsymbol \delta)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)\|\le2 k \|\|\boldsymbol \delta\|\|.$$ So, $$L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)\le 2k\|\|\boldsymbol \delta\|\|.$$ Also, as $L$ is twice differentiable in $\boldsymbol p$ and $ \boldsymbol{p^}$ is continuous at $\boldsymbol q$ by Lemma \ref{lem:pcontinuous}, $$0=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)-\nabla L(\bec{p^}(\bec q), \bec q)\cdot \Delta \bec{p^}-\Delta \boldsymbol{p^{}}^T H \Delta \boldsymbol{p^}}{\|\|\Delta \boldsymbol{p^}\|\|^2}$$ where $\Delta \boldsymbol{p^}=\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta)-\boldsymbol{p^}(\boldsymbol q)$ and $\nabla L(\bec{p^}(\bec q), \bec q)\cdot \Delta \bec{p^}\ge 0$ as in Theorem \ref{punique}. Then, for any $0<\epsilon <\lambda$, for there exists $\Delta > 0$ where for $0<\|\|\boldsymbol \delta\|\|<\Delta$ $$-\epsilon<\frac{L(\boldsymbol{p^}(\boldsymbol q+\boldsymbol \delta), \boldsymbol q)-L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)-\nabla L(\bec{p^}(\bec q), \bec q)\cdot \Delta \bec{p^}-\Delta \boldsymbol p^{T} H \Delta \boldsymbol{p^}}{\|\|\Delta \boldsymbol{p^}\|\|^2}$$ $$\le\frac{2k\|\|\boldsymbol \delta\|\|-\lambda \|\|\Delta \boldsymbol{p^}\|\|^2}{\|\|\Delta \boldsymbol{p^}\|\|^2}.$$ So $$\frac{\|\|\Delta \boldsymbol p^\|\|}{\sqrt{\|\|\boldsymbol\delta\|\|}}\le\sqrt{\frac{2k}{\lambda-\epsilon}}.$$ And $\boldsymbol{p^}$ is Holder-$\frac{1}{2}$ in the open ball of radius $\Delta$ at $\boldsymbol q$. \end{proof} \noindent Let $\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$ be the gradient of $L(\boldsymbol p, \boldsymbol q)$ as $\boldsymbol q$ is changed at the point $(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$. \begin{theorem} \label{thm:holderImpliesDiff} Suppose there is some open region $\bar D\subseteq(0, 1)^n$ where $\boldsymbol{p^}$ is Holder-$\frac{1+\alpha}{2}$ for some $\alpha>0$, and $\ell$ is twice differentiable at $(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$ for $\boldsymbol q \in \bar D$. Then, $L^$ is differentiable in $\bar D$ with derivative $\nabla L^(\boldsymbol q)=\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$. \end{theorem} \begin{proof} Let $k$ be the Holder-$\frac{1+\alpha}{2}$ coefficient of $p^$ and $\lambda$ the least eigenvalue of the Hessian $H$ of $L$ with respect to $\boldsymbol p$ at $(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$. Then, consider $$\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q + \boldsymbol \delta) -L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q + \boldsymbol \delta) +L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}$$ $$=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) - \nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)=0$$ as $\boldsymbol{p^}$ and $\nabla_{\boldsymbol q} L$ are both continuous. Then, $$\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L^(\boldsymbol q + \boldsymbol\delta)-L^(\boldsymbol q)-\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}$$ $$=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)+L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q + \boldsymbol \delta) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)-\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}$$ $$=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}$$ $$=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol{p^}(\boldsymbol v + \boldsymbol \delta)-\boldsymbol{p^}(\boldsymbol v)\|\|^2}\frac{\|\|\boldsymbol{p^}(\boldsymbol v + \boldsymbol \delta)-\boldsymbol{p^}(\boldsymbol v)\|\|^2}{\|\|\boldsymbol \delta\|\|}$$ $$\le \lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\lambda \frac{\|\|\boldsymbol{p^}(\boldsymbol v + \boldsymbol \delta)-\boldsymbol{p^}(\boldsymbol v)\|\|^2}{\|\|\boldsymbol \delta\|\|}=\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\lambda \|\|\boldsymbol \delta\|\|^{2\alpha}\frac{\|\|\boldsymbol{p^}(\boldsymbol v + \boldsymbol \delta)-\boldsymbol{p^}(\boldsymbol v)\|\|^2}{\|\|\boldsymbol \delta\|\|^{1+2\alpha}}\le\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\lambda k \|\|\boldsymbol\delta\|\|^{2\alpha}=0$$ Note that by the minimum property of $L^$ $$0\le\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L(\boldsymbol{p^}(\boldsymbol q + \boldsymbol \delta), \boldsymbol q) -L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}$$ So, $$\lim_{\boldsymbol \delta \rightarrow \boldsymbol 0}\frac{L^(\boldsymbol q + \boldsymbol\delta)-L^(\boldsymbol q)-\nabla_q L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)}{\|\|\boldsymbol \delta\|\|}=0$$ and $\nabla L^(\boldsymbol q)=\nabla_{\boldsymbol q} L(\boldsymbol{p^}(\boldsymbol q), \boldsymbol q)$. \end{proof} \section{Proofs of Section \ref{subsec:singleEvent}} \begin{lemma} When using dissimilarity function $\ell(p, q)=2(p-q)^2$, the loss function becomes $$L^(q_E, q_{E^c}) = (q_{E} + q_{E^c} - 1)^2.\label{lem:burns_loss}$$ \end{lemma} \begin{proof} Lagrange multipliers yield $p^_E=\frac{q_E + 1 - q_{E^c}}{2}=1-p^_{E^c}$. Therefore \begin{align} L^(q_{E}, q_{E^c})&=L((p_{E}, p_{E^c}), (q_{E}, q_{E^c}))\\ &=2(p_E - q_{E})^2+2(p_{E^c} - q_{E^c})^2\\ &=(q_{E} + q_{E^c} - 1)^2.\qedhere \end{align} \end{proof} \begin{lemma} When using dissimilarity function $$\ell(p, q)=f(p, q)=p\ln \frac pq + (1-p)\ln \frac{1-p}{1-q}$$ the loss function becomes $$L^(q_E, q_{E^c}) = \frac{(q_E + q_{E^c} - 1)^2}{(1-q_E+q_{E^c})(q_E+1-q_{E^c})}+O((q_E + q_{E^c} - 1)^3).\label{lem:burns_f}$$ \end{lemma} \begin{proof} Lagrange multipliers yield that the closest coherent probability $p^_{E}$ satisfies $\frac{p^_{E}}{1-p^_{E}}=\left(\frac{q_E(1-q_{E^c})}{(1-q_{E})q_{E^c}}\right)^{1/2}$, giving the loss \begin{align} L^(q_E, q_{E^c})&=2p^_{E} \ln \frac{p^_E}{1-p^_E} - p^_E\ln\frac{q_E}{1-q_E} - p^_E\ln\frac{q_{E_c}}{1-q_{E^c}}+\ln \frac{(1-p^_E)^2}{(1-q_E)q_{E^c}}\\ &=\ln \frac{(1-p^_E)^2}{(1-q_E)q_{E^c}}. \end{align} Letting $o_{E}= \frac{q_E}{1-q_E}$ and $o'_E=\frac{1-q_{E^c}}{q_{E^c}}$ be the odds of $E$ derived from the two probes with $q=q_E$ and $q'=1-q_{E^c}$ the probabilities derived from the probes and Taylor expanding in $o$ about $o=o'$ yields: \begin{align} L^(q_E, q_{E^c})&=\ln \frac{(1+o)(1+o')}{(1+\sqrt{oo'})^2}\\ &\approx \frac{1}{2}(o-o')^2 \deriv{^2}{o^2}\ln \frac{(1+o)(1+o')}{(1+\sqrt{oo'})^2}+O((o-o')^3)\\ &=\frac{1}{2}(o-o')^2\left(-\frac{1}{(1+o')^2}+\frac{1+2o'}{2o'(1+o')^2}\right)+O((o-o')^3)\\ &=\frac{1}{4o'}(o-o')^2+O((o-o')^3)\\ &=\frac{(q-q')^2}{4q'(1-q')}+O((q-q')^3)\\ &\approx \frac{(q - q')^2}{(q+q')(2-q-q')}+(q-q')^2O(q-q')+O((q-q')^3)\\ &=\frac{(q - q')^2}{(q+q')(2-q-q')}+O((q - q')^3). \qedhere \end{align} \end{proof} \begin{lemma} When using dissimilarity function $$\ell(p, q)=f^o(p, q)=q\ln \frac qp + (1-q)\ln \frac{1-q}{1-p}$$ the loss function becomes $$L^(q_E, q_{E^c}) = \frac{(q_E + q_{E^c} - 1)^2}{(1-q_E+q_{E^c})(q_E+1-q_{E^c})}+O((q_E + q_{E^c} - 1)^3).$$\label{lem:burns_fo} \end{lemma} \begin{proof} Again letting $q=q_E$ and $q'=1-q_{E^c}$, Lagrange multipliers show that $p^_E = \frac{q+q'}{2}$, which we will denote by $p$ for ease of notation. Taking the derivative of $L^$ in $q'$ gives $$\pderiv{}{q'}L^(q, q') = \ln\frac{q'}{1-q'}-\ln \frac{p}{1-p}$$ and $$\pderiv{^2}{q'^2}L^(q, q') = \frac{1}{q'(1-q')}-\frac{1}{2p(1-p)}.$$ Then, Taylor expanding around $q=q'$ in $q'$ gives \begin{align} L^(q_E, q_{E^c}) &\approx \frac{1}{2}(q-q')^2\pderiv{^2}{q'^2}L^(q, q') + O((q-q')^3)\\ &=\frac{(q-q')^2}{2q'(1-q')}-\frac{q-q')^2}{4p(1-p)}+ O((q-q')^3\\ &\approx \frac{(q-q')^2}{4p(1-p)}+(q-q')^2O((q-q'))+ O((q-q')^3\\ &=\frac{(q-q')^2}{(q+q')(2-q-q')} + O((q-q')^3).\qedhere \end{align} \end{proof} \section{Comparison of Expert Aggregation Methods\label{sec:specialcompare}} \noindent Here, we will look at the following scenario and consider what various methods from Section \ref{sec:indCohExp} give us as estimates of the probabilities of various events: \\\\ We take $N=4$ and consider two experts. The first expert submits probability estimates for two events $$V_1=\begin{pmatrix} 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 0 \end{pmatrix}, \boldsymbol{q}_1=\begin{pmatrix} 0.5\\0.9 \end{pmatrix}$$ and the second expert submits estimates for three events $$V_2=\begin{pmatrix} 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}, \boldsymbol{q}_2=\begin{pmatrix} 0.3\\0.2\\0.6 \end{pmatrix}.$$ Table \ref{table:assymetriccomparison} summarizes the aggregated beliefs $\bec p^$ using four different methods and two different dissimilarity functions.\color{black} \begin{table}[H] \begin{center} \begin{tabular}{\|p{15em} \|m{2em}\|m{2em}\| m{6em} \| m{6em} \|} \hline Aggregation Method & $\bec q_1$& $\bec q_2$ &$\bec p^$ with $\ell=f$ & $\bec p^$ with $\ell=f^o$ \\\hline Sum over stated beliefs only (not content invariant): \newline $\mathcal D_i = D_{i, \mathtt E_i}$&$\begin{matrix} 0.5 \\ 0.9 \\ - \\ - \end{matrix}$ & $\begin{matrix} 0.3 \\ - \\ 0.2 \\ 0.6 \end{matrix}$ &$\begin{matrix} 0.43 \\ 0.68 \\ 0.25 \\ 0.32 \end{matrix}$ &$\begin{matrix} 0.41 \\ 0.64 \\ 0.22 \\ 0.36 \end{matrix}$\\ \hline Sum over all inferable beliefs: \newline $\mathcal D_i=D_{i, I_i}$ &$\begin{matrix} 0.5 \\ 0.9 \\ - \\ - \end{matrix}$ & $\begin{matrix} 0.3 \\ - \\ 0.2 \\ 0.6 \end{matrix}$& $\begin{matrix} 0.46 \\ 0.73 \\ 0.46 \\ 0.27 \end{matrix}$ &$\begin{matrix} 0.43 \\ 0.71 \\ 0.47 \\ 0.29 \end{matrix}$\\ \hline Sum over the minimal positive basis: \newline $\mathcal D_i= D_{i, B_i}$ &$\begin{matrix} 0.5 \\ 0.9 \\ - \\ - \end{matrix}$ & $\begin{matrix} 0.3 \\ - \\ 0.2 \\ 0.6 \end{matrix}$& $\begin{matrix} 0.46 \\ 0.72 \\ 0.42 \\ 0.28 \end{matrix}$&$\begin{matrix} 0.42 \\ 0.69 \\ 0.41 \\ 0.31 \end{matrix}$\\ \hline Sum over the minimal positive basis, asymmetric variant: \newline $\mathcal D_i=\tilde D_{i, B_i}$ &$\begin{matrix} 0.5 \\ 0.9 \\ - \\ - \end{matrix}$ & $\begin{matrix} 0.3 \\ - \\ 0.2 \\ 0.6 \end{matrix}$& $\begin{matrix} 0.48 \\ 0.74 \\ 0.42 \\ 0.26 \end{matrix}$&$\begin{matrix} 0.41 \\ 0.69 \\ 0.41 \\ 0.31 \end{matrix}$\\ \hline \end{tabular} \color{black} \caption{Minimizer of $L$ for various loss functions, of four possible forms defined in equations (\ref{eqn:contentinvariantloss}) and (\ref{eqn:assymetricvariant}), three of which are content invariant, each using one of two possible dissimilarity functions. $\bec p^$ is the nearest coherent vector of beliefs. Observe that all three content invariant aggregation methods give similar estimates $\bec p^$ provided they use the same loss function.}\label{table:assymetriccomparison} \end{center} \end{table} \noindent It should be noted that the first method assigns a probability of $0$ to atom $\omega_2$, but that it is given non-zero probability by all the content invariant methods. This is because the event $\bec e_2$ is not present in the sum over stated beliefs only, but is present in all the other sums. \section{Extended Example: Masked Letters}\label{sec:ex} \noindent We illustrate the methods of Section \ref{sec:indCohExp} in the following scenario: given a word with one masked letter, we seek to predict the masked letter by combining two heuristics, which will play the role of experts in Section \ref{sec:indCohExp}. One heuristic uses the preceding letters to make predictions and the other heuristic uses the succeeding letters to make its predictions. \\\\For example, given the word EMIL, with the third letter masked, the first expert would use the 3-gram EM* to make its predictions $Q_1(\text{A})=0.16, Q_1(\text{B})=0.08, \dots$. The second heuristic would use the 3-gram IL to make the predictions $Q_2(\text{A})=0.32, Q_2(\text{B})=0.27, \dots$. \\\\ If each heuristic is derived from counting 3-gram frequencies in a corpus, then it will be internally coherent, so we use a content invariant loss function as outlined in Section \ref{sec:indCohExp}. As $\#I_1=\#I_2=2^{26}$, the size of the set of events whose probabilities can be inferred, is much too large, one should either sum over the minimal spanning sets $B_i$ or sum over subsets of the minimal spanning sets that add to $\bec 1$. We will outline the calculations using dissimilarity function $\ell=f$ and compare the results with $\ell = f^o$. \subsection{\texorpdfstring{Method 1: Summing over a Positive Basis}{Method 1: Summing over a Positive Basis}} Here we consider the loss function determined by $$\mathcal D_i=D_{i, B_i} = \frac{1}{B_i}\sum_{\boldsymbol b \in B_i}\ell(P(\boldsymbol b), Q_i(\boldsymbol b))$$ Recall that $M_i$ is the normalizing constant, equal to the number of terms in the summand. Additionally, in this situation, the heuristics' predictions are exactly the minimal spanning set. So $$L(\bec p, (\bec q_1, \bec q_2))=L(V\bec \pi, (\bec q_1,\bec q_2))=\frac{1}{26}\sum_{\alpha\in \set{\text{A}, \text{B}, \dots}} \ell(\pi_\alpha, q_{1, \alpha})+\ell(\pi_\alpha, q_{2, \alpha}).$$ Then, if all entries in $\bec \pi$ are strictly positive (which they will be for $f$ and $f^o$ as long as all entries in $\bec q_1$ and $\bec q_2$ are non-zero), by Lagrange multipliers as $\sum_{\alpha\in \set{\text{A}, \text{B}, \dots}}\pi_\alpha=1$, there is some constant $k$ where for all $\alpha$ $$\frac k{26}=\pderiv{L}{\pi_\alpha}=\pderiv{\ell(\pi_\alpha, q_{1, \alpha})}{\pi_\alpha}+\pderiv{\ell(\pi_\alpha, q_{2, \alpha})}{\pi_\alpha}.$$ Taking $\ell=f$ and calculating shows $$k=26\pderiv{L}{\pi_\alpha}=\ln\frac{\pi_\alpha}{1-\pi_\alpha}-\ln\frac{q_{1, \alpha}}{1-q_{1, \alpha}}+\ln\frac{\pi_\alpha}{1-q_\alpha}-\ln\frac{q_{2, \alpha}}{1-q_{2, \alpha}}$$ so $$\ln\frac{\pi_\alpha}{1-\pi_\alpha}-\frac12\left(\ln\frac{q_{1, \alpha}}{1-q_{1, \alpha}}+\ln\frac{q_{2, \alpha}}{1-q_{2, \alpha}}\right)$$ does not depend on $\alpha$. Computing $\bec \pi$ analytically from here is difficult, so some numerical algorithm should be used. \subsection{\texorpdfstring{Method 2: Using an Asymmetric Loss Function}{Method 2: Using an Asymmetric Loss Function}} Here we consider the loss function determined by $$\mathcal D_i=\tilde D_{i, B_i} = \frac{1}{\#M_i}\sum_{S \in O_i} \sum_{\boldsymbol b \in S}\tilde{\ell}(P(\bec b), Q_i(\bec b))$$ Here $O_i\subset 2^{B_i}$ is the set of subsets $S$ of $B_i$ whose sum is $\bec 1$. In this case, $O_i$ is the singleton set $\set{B_i}$ as the heuristic's predictions are the maximal spanning set and their sum is $1$, so we sum over the same events as previously. However, instead of using the function $\ell$, we use $\tilde \ell$ and ignore the terms related to the surprise of a letter not being chosen. As $M_1=M_2=26$ is the number of terms in the summand, the loss function is $$L(V\bec \pi, (\bec q_1,\bec q_2))=\frac{1}{26}\sum_{\alpha\in \set{\text{A}, \text{B}, \dots}} \tilde \ell(\pi_\alpha, q_{1, \alpha})+\tilde \ell(\pi_\alpha, q_{2, \alpha}).$$ As we are using $\ell=f$, so $\tilde \ell(p, q)=p\ln \frac pq$, this becomes $$=\frac{1}{26}\sum_{\alpha\in \set{\text{A}, \text{B}, \dots}} \pi_\alpha \ln \frac{\pi_\alpha}{q_{1, \alpha}}+\pi_\alpha\ln \frac{\pi_\alpha}{q_{2, \alpha}}.$$ And again using Lagrange multipliers to take the derivative, the quantity $$26\pderiv{L}{\pi_\alpha} - 2=\ln \frac{\pi_\alpha}{q_{1, \alpha}} + \ln \frac{\pi_\alpha}{q_{2, \alpha}}$$ does not depend on $\alpha$. Therefore, for some $k$ that does not depend on $\alpha$, $$\pi_\alpha = k(q_{1, \alpha}q_{2, \alpha})^\frac{1}{2}.$$ Using the fact that $\sum_\alpha \pi_\alpha = 1$, $$\pi_\alpha=(q_{1, \alpha}q_{2, \alpha})^\frac{1}{2}\left(\sum_\beta (q_{1, \beta}q_{2, \beta})^\frac12\right)^{-1}.$$ \subsection{Comparison of Methods} Using a list of 10,000 common English words \cite{price_wordlist}, there are 7 letters that appear both after EM and before IL, not necessarily in the same word (A, B, E, M, O, P, S). Applying the methods described above yields Table \ref{fig:extended_ex}: \begin{table}[h] \begin{center} \begin{tabular}{\|m{3em}\|m{2em}\|m{2em}\| m{9em} \|m{9em}\| m{5em} \|} \hline Letter &$\bec q_1$ & $\bec q_2$ & \multicolumn{2}{c\|}{$\ell = f$} & $\ell = f^o$ \\\cline{4-5} & & & $\bec p^$ with $\mathcal D_i=D_{i, B_i}$ & $\bec p^$ with $\mathcal D_i = \tilde{{\scriptstyle D}}_{i, B_i}$ & $\bec p^$ \\\hline $\begin{matrix} A \\ B \\ E \\ M \\ O \\ P \\ S \end{matrix}$ & $\begin{matrix} 0.16\\0.08\\ 0.39\\ 0.01\\ 0.15\\ 0.17\\ 0.04 \end{matrix}$ & $\begin{matrix} 0.32\\ 0.27\\ 0.02\\ 0.22\\ 0.03\\ 0.07\\ 0.07 \end{matrix}$ &$\begin{matrix} 0.29 \\ 0.20 \\ 0.14 \\ 0.07 \\ 0.09 \\ 0.14 \\ 0.07 \end{matrix}$ &$\begin{matrix} 0.31 \\ 0.20 \\ 0.13 \\ 0.05 \\ 0.09\\0.15\\0.07 \end{matrix}$ & $\begin{matrix} 0.24\\0.18\\0.20\\0.12\\0.09\\0.12\\0.05 \end{matrix}$ \\\hline \end{tabular} \end{center} \caption{The resulting $\bec p^$s when reconciling the heuristic $\bec q_1$ based on the 3-gram EM and the heuristic $\bec q_2$ based on the 3-gram *IL with various methods.} \label{fig:extended_ex} \end{table} \noindent \\\\Firstly, in this setup, when using $f^o$, it does not matter whether one includes the $\tilde f^o(1-p, 1-q)$ terms. In either case, the method amounts to setting $p_\alpha = \frac{q_{1, \alpha}+q_{2, \alpha}}{2}$, which is coherent as the set of coherent beliefs is convex. This will not always be the case, and was so here because of the simplicity of the events included in the sum. Looking at the results of using $f$, there is little difference, although low probabilities ($q_{1, m}=0.01$) seem to be given more weight when excluding $\tilde f(1-p, 1-q)$ terms. As usual, $f$ gives more extreme estimates than $f^o$ with either method. \\\\ Here, both methods with either loss function were correct in assigning the letter A the highest probability in EMAIL. However, this is not usually the case. Over all the 5 letter words in \cite{price_wordlist}, both methods using $f$ gave the highest probability to the actual letter about $34\%$ of the time and both methods using $f^o$ were correct about $33\%$ of the time. The code to reproduce this experiment can be found at \url{https://github.com/scim142/quantifying_coherence}. \end{document}
2412.02902v1	http://arxiv.org/abs/2412.02902v1	$\left(p,q\right)$-adic Analysis and the Collatz Conjecture	\documentclass[11pt,oneside,english,refpage,intoc,openany]{book} \usepackage{lmodern} \renewcommand{\sfdefault}{lmss} \renewcommand{\ttdefault}{lmtt} \renewcommand{\familydefault}{\rmdefault} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \usepackage{geometry} \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} \setcounter{tocdepth}{3} \usepackage{color} \usepackage{babel} \usepackage{varioref} \usepackage{textcomp} \usepackage{mathrsfs} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} \usepackage{cancel} \usepackage{makeidx} \makeindex \usepackage{graphicx} \PassOptionsToPackage{normalem}{ulem} \usepackage{ulem} \usepackage{nomencl} \providecommand{\printnomenclature}{\printglossary} \providecommand{\makenomenclature}{\makeglossary} \makenomenclature \usepackage[unicode=true, bookmarks=true,bookmarksnumbered=true,bookmarksopen=false, 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\providecommand{\factname}{Fact} \providecommand{\lemmaname}{Lemma} \providecommand{\notationname}{Notation} \providecommand{\problemname}{Problem} \providecommand{\propositionname}{Proposition} \providecommand{\questionname}{Question} \providecommand{\remarkname}{Remark} \providecommand{\theoremname}{Theorem} \begin{document} \fancypagestyle{plain}{ \fancyhf{} \fancyhead[R]{\thepage} \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt} }\frontmatter \global\long\def\headrulewidth{0pt} \thispagestyle{fancy}\fancyfoot[L]{May 2022}\fancyfoot[R]{Maxwell Charles Siegel}\pagenumbering{gobble} \begin{center} $\left(p,q\right)$-ADIC ANALYSIS AND THE COLLATZ CONJECTURE \par\end{center} \vphantom{} \begin{center} by \par\end{center} \vphantom{} \begin{center} Maxwell Charles Siegel \par\end{center} \vphantom{} \vphantom{} \vphantom{} \begin{center} A Dissertation Presented to the \par\end{center} \begin{center} FACULTY OF THE USC DORNSIFE COLLEGE OF LETTERS AND SCIENCES \par\end{center} \begin{center} UNIVERSITY OF SOUTHERN CALIFORNIA \par\end{center} \begin{center} In Partial Fulfillment of the \par\end{center} \begin{center} Requirements for the Degree \par\end{center} \begin{center} DOCTOR OF PHILOSOPHY \par\end{center} \begin{center} (MATHEMATICS) \par\end{center} \vphantom{} \vphantom{} \vphantom{} \begin{center} May 2022 \par\end{center} \newpage\pagenumbering{roman}\setcounter{page}{2} \addcontentsline{toc}{chapter}{Dedication} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{center} {\large{}{}I dedicate this dissertation to those who listened, and to those who cared. Also, to my characters; I'm sorry it took so long.}{\large\par} \par\end{center} \begin{center} {\large{}{}I'll try to do better next time.}{\large\par} \par\end{center} \tableofcontents\pagebreak \listoftables \newpage{} \section{Abstract} \pagestyle{fancy}\fancyfoot{}\fancyhead[L]{\sl ABSTRACT}\fancyhead[R]{\thepage}\addcontentsline{toc}{chapter}{Abstract} What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer\textemdash courtesy of the half-century old theory of non-archimedean functional analysis: \emph{not much}. It turns out this judgment was premature. ``$\left(p,q\right)$-adic analysis'' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$, one can construct a function $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ for an appropriate choice of distinct primes $p,q$ with the property that $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ if and only if there is a $p$-adic integer $\mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} $ so that $\chi_{H}\left(\mathfrak{z}\right)=x$. By generalizing Monna-Springer integration theory and establishing a $\left(p,q\right)$-adic analogue of the Wiener Tauberian Theorem, one can show that the question ``is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$'' is essentially equivalent to ``is the span of the translates of the Fourier transform of $\chi_{H}\left(\mathfrak{z}\right)-x$ dense in an appropriate non-archimedean function space?'' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice $\mathbb{Z}^{d}$ for any $d\geq1$. \newpage{} \section{Preface} \fancyhead[L]{\sl PREFACE}\addcontentsline{toc}{chapter}{Preface} Like with so many other of the world's wonders, my first encounter with the Collatz Conjecture was on Wikipedia. I might have previously stumbled across it as an undergraduate, but it wasn't until around the time I started graduate school (Autumn 2015) that I gave it any serious consideration. I remember trying to understand what $p$-adic numbers were solely through what Wikipedia said they were, hoping to apply it to the Conjecture, but then gave up after a day or two. I spent a while fooling around with trying to prove the holomorphy of tetration, and then fiddled with fractional differentiation and the iterated Laplace transform, mostly mindlessly. But then, in March of 2017, the Collatz ``bug'' bit\textemdash and bit \emph{hard}. I realized I could encode the action of the Collatz map as a transformation of holomorphic functions on the disk, and\textemdash with complex analysis being near and dear to my heart\textemdash I was hooked. Obsession was \emph{instantaneous}. As of the writing of this preface some five years later, I suppose the ardor of my obsession has dimmed, somewhat, primarily under the weight of external pressures\textemdash angst, doubt, apprehension and a healthy dose of circumspection, among them. Although all PhD dissertations are uphill battles\textemdash frequently Sisyphean\textemdash the lack of a real guiding figure for my studies made my scholarly journey particularly grueling. The mathematics department of my graduate school\textemdash the University of Southern California\textemdash has many strengths. Unfortunately, these did not include anything in or adjacent to the wide variety of sub-disciplines I drew from in my exploration of Collatz: harmonic analysis, boundary behavior of power series, analytic number theory, $p$-adic analysis, and\textemdash most recently\textemdash non-archimedean functional analysis. I was up the proverbial creek without a paddle: I did not get to benefit from the guidance of a faculty advisor who could give substantive feedback on my mathematical work. The resulting intellectual isolation was vaster than anything I'd previously experienced\textemdash and that was \emph{before }the COVID-19 pandemic hit. Of course, I alone am to blame for this. I \emph{chose }to submit to my stubborn obsession, rather than fight it. The consequences of this decision will most likely haunt me for many years to come. as much trouble as my doltish tenacity has caused me, I still have to thank it. I wouldn't have managed to find a light at the end of the tunnel without it. As I write these words, I remain apprehensive\textemdash even fearful\textemdash of what the future holds for me, mathematical or otherwise. No course of action comes with a guarantee of success, not even when you follow your heart. But, come what may, I can say with confidence that I am proud of what I have accomplished in my dissertation, and prouder to still be able to share it with others. Connectedness is success all its own. But enough about \emph{me}. In the past five years, my research into the Collatz Conjecture can be grouped into two-and-a-half different categories. The first of these was the spark that started it all: my independent discovery that the Collatz Conjecture could be reformulated in terms of the solutions of a \emph{functional equation} defined over the open unit disk $\mathbb{D}$ in $\mathbb{C}$. Specifically, the equation: \begin{equation} f\left(z\right)=f\left(z^{2}\right)+\frac{z^{-1/3}}{3}\sum_{k=0}^{2}e^{-\frac{4k\pi i}{3}}f\left(e^{\frac{2k\pi i}{3}}z^{2/3}\right)\label{eq:Collatz functional equation} \end{equation} where the unknown $f\left(z\right)$ is represented by a power series of the form: \begin{equation} f\left(z\right)=\sum_{n=0}^{\infty}c_{n}z^{n}\label{eq:Power series} \end{equation} In this context, the Collatz Conjecture becomes equivalent to the assertion that the set of holomorphic functions $\mathbb{D}\rightarrow\mathbb{C}$ solving equation (\ref{eq:Collatz functional equation}) is precisely: \begin{equation} \left\{ \alpha+\frac{\beta z}{1-z}:\alpha,\beta\in\mathbb{C}\right\} \end{equation} This discovery was not new. The observation was first made by Meinardus and Berg \cite{Berg =00003D000026 Meinardus,Meinardus and Berg} (see also \cite{Meinardus,Opfer}). What made my take different was its thrust and the broadness of its scope. I arrived at equation (\ref{eq:Collatz functional equation}) in the more general context of considering a linear operator (what I call a \textbf{permutation operator}) induced by a map $H:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$; here $\mathbb{N}_{0}$ is the set of integers $\geq0$). The permutation operator $\mathcal{Q}_{H}$ induced by $H$ acts upon the space of holomorphic functions $\mathbb{D}\rightarrow\mathbb{C}$ by way of the formula: \begin{equation} \mathcal{Q}_{H}\left\{ \sum_{n=0}^{\infty}c_{n}z^{n}\right\} \left(z\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{H\left(n\right)}z^{n}\label{eq:Definition of a permutation operator} \end{equation} It is not difficult to see that the space of functions fixed by $\mathcal{Q}_{H}$ is precisely: \begin{equation} \left\{ \sum_{v\in V}z^{v}:V\textrm{ is an orbit class of }H\textrm{ in }\mathbb{N}_{0}\right\} \label{eq:orbit class set-series} \end{equation} A simple computation shows that the functional equation (\ref{eq:Collatz functional equation}) is precisely the equation $\mathcal{Q}_{T_{3}}\left\{ f\right\} \left(z\right)=f\left(z\right)$, where $T_{3}$ is the \textbf{Shortened Collatz map}, with the general \textbf{Shortened}\index{$qx+1$ map} \textbf{$qx+1$ map} being defined by \index{$3x+1$ map} \begin{equation} T_{q}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{n}{2} & \textrm{if }n=0\mod2\\ \frac{qn+1}{2} & \textrm{if }n=1\mod2 \end{cases}\label{eq:qx plus 1 shortened} \end{equation} where $q$ is any odd integer $\geq3$. Maps like these\textemdash I call them \textbf{Hydra maps}\textemdash have been at the heart of my research in the past five years. Indeed, they are the central topic of this dissertation. My investigations of functional equations of the form $\mathcal{Q}_{H}\left\{ f\right\} \left(z\right)=f\left(z\right)$ eventually led me to a kind of partial version of a famous lemma\footnote{\textbf{Theorem 3.1 }in \cite{On Wiener's Lemma}.} due to Norbert Wiener. I used this to devise a novel kind of Tauberian theory, where the boundary behavior of a holomorphic function on $\mathbb{D}$ was represented by a singular measure on the unit circle\footnote{This viewpoint is intimately connected with the larger topic of representing holomorphic functions on $\mathbb{D}$ in terms of their boundary values, such as by the Poisson Integral Formula \cite{Bounded analytic functions,function classes on the unit disc} or the Cauchy transform and its fractional analogues \cite{Ross et al,Fractional Cauchy transforms}.}. Of particular import was that these measures could also be interpreted as \emph{functions} in the space $L^{2}\left(\mathbb{Q}/\mathbb{Z},\mathbb{C}\right)$\textemdash all complex-valued functions on the additive group $\mathbb{Q}/\mathbb{Z}=\left[0,1\right)\cap\mathbb{Q}$ with finite $L^{2}$-norm with respect to that group's Haar measure (the counting measure): \begin{equation} L^{2}\left(\mathbb{Q}/\mathbb{Z},\mathbb{C}\right)=\left\{ f:\mathbb{Q}/\mathbb{Z}\rightarrow\mathbb{C}\mid\sum_{t\in\mathbb{Q}/\mathbb{Z}}\left\|f\left(t\right)\right\|^{2}<\infty\right\} \label{eq:L2 of Q/Z} \end{equation} Because the Pontryagin dual of $\mathbb{Q}/\mathbb{Z}$ is the additive group of profinite integers\footnote{The direct product $\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$ of the rings of $p$-adic integers, over all primes $p$.}, I decided to interpret these measures/functions as the Fourier transforms of complex-valued functions on the profinite integers. Detailed explanations of these can be found in my paper \cite{Dreancatchers for Hydra Maps} on arXiv. The notation in that paper is a mess, and I will likely have to re-write it at some point in the future. During these investigations, my \emph{ide fixe }was to use Fourier analysis and exploit the Hilbert space structure of $L^{2}\left(\mathbb{Q}/\mathbb{Z},\mathbb{C}\right)$ to prove what I called \emph{rationality theorems}. These were generalizations of the notion of a \emph{sufficient set} \index{sufficient set}(see \cite{Andaloro - first paper on sufficient sets,Monks' sufficiency paper for 3x+1}); a set $S\subseteq\mathbb{N}_{0}$ is said to be ``sufficient'' for the Collatz Conjecture whenever the statement ``the Collatz map iterates every element of $S$ to $1$'' is sufficient to prove the Collatz Conjecture in full. \cite{Remmert} showed that, for example, the set $\left\{ 16n+1:n\geq0\right\} $ was sufficient for the Collatz map; Monks et. al. \cite{The many Monks paper} extended this, showing that any infinite arithmetic\index{arithmetic progression} progression (``IAP'') (a set of the form $\left\{ an+b:n\geq0\right\} $ for integers $a,b$, with $a\neq0$) was sufficient for the Collatz map. Unfortunately, all I could manage to prove was a weaker form of this conclusion: namely, that for any Hydra map satisfying certain simple conditions, if an orbit class $V\subseteq\mathbb{N}_{0}$ of the Hydra map could be written as the union of a finite set and finitely many IAPs, then $V$ contained all but at most finitely many elements of $\mathbb{N}_{0}$. The one advantage of my approach was that it \emph{appeared} to generalize to ``multi-dimensional'' Hydra maps\textemdash a Collatz-type map defined on the ring of integers a finite-degree field extension of $\mathbb{Q}$, or, equivalently, on the lattice $\mathbb{Z}^{d}$. I say ``appeared'' because, while the profinite integer Hilbert space part of the generalization worked out without any difficulty, it is not yet clear if (or how) the crux of the original argument could be extended to power series of several complex variables. For the curious, this crux was a beautiful theorem\footnote{See Section 4 in Chapter 11 of \cite{Remmert}.} due to Gbor Szeg\H{o}\index{SzegH{o}, Gbor@Szeg\H{o}, Gbor} regarding the analytic continuability of (\ref{eq:Power series}) in the case where the set: \begin{equation} \left\{ c_{n}:n\in\mathbb{N}_{0}\right\} \end{equation} is finite. Taking the time to investigate this issue would allow for the results of \cite{Dreancatchers for Hydra Maps} to be extended to the multi-dimensional case, the details of which\textemdash modulo the Szeg\H{o} gap\textemdash I have written up but not yet published or uploaded in any form. I will most likely re-write \cite{Dreancatchers for Hydra Maps} at a future date\textemdash assuming I have not already done so by the time you are reading this. The \emph{first-and-a-halfth} category of my work branched off from my quest for rationality theorems. I turned my focus to using $L^{2}\left(\mathbb{Q}/\mathbb{Z},\mathbb{C}\right)$ to establish the possible value(s) of the growth constant $r>0$ \index{growth exponent} for orbit classes of Hydra maps. The essentials are as follows. Let $H$ be a Hydra map. Then, letting $\omega$ be a positive real number and letting $f\left(z\right)=\left(1-z\right)^{-\omega}$, an elementary computation with limits establishes that the limit \begin{equation} \lim_{x\uparrow1}\left(1-x\right)^{\omega}\mathcal{Q}_{H}\left\{ f\right\} \left(x\right)=1\label{eq:Eigenrate Equation} \end{equation} occurs if and only if $\omega$ satisfies a certain transcendental equation. For the case of the Shortened $qx+1$ map, the values of $\omega$ satisfying (\ref{eq:Eigenrate Equation}) are precisely the solutions of the equation: \begin{equation} 2^{\omega}-q^{\omega-1}=1 \end{equation} My hope was to prove that for: \begin{equation} \omega_{0}\overset{\textrm{def}}{=}\min\left\{ \omega\in\left(0,1\right]:2^{\omega}-q^{\omega-1}=1\right\} \end{equation} given any orbit class $V\subseteq\mathbb{N}_{0}$ of $T_{q}$ such that the iterates $T_{q}\left(v\right),T_{q}\left(T_{q}\left(v\right)\right),\ldots$ were bounded for each $v\in V$, we would have: \begin{equation} \left\|\left\{ v\in V:v\leq N\right\} \right\|=o\left(N^{\omega_{0}+\epsilon}\right)\textrm{ as }N\rightarrow\infty \end{equation} for any $\epsilon>0$. While initially quite promising, obtaining this asymptotic result hinged on justifying an interchange of limits, a justification which appears to be at least as difficult as establishing this asymptotic directly, \emph{without appealing} to $\mathcal{Q}_{H}$ and complex analysis as a middleman. Although still conjecture as of the writing of this dissertation, the $2^{\omega}-q^{\omega-1}=1$ formula would provide an elegant resolution to a conjecture of Kontorovich and Lagarias regarding the growth exponent for the orbit class of all integers that $T_{5}$ iterates to $1$ \cite{Lagarias-Kontorovich Paper}. I have also studied this issue from profinite integer. This turns out to be equivalent to studying the iterations of $H$ on the profinite integer and subgroups thereof, such as $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$, the ring of ``$\left(2,3\right)$-adic integers''. I have obtained minor results in that regard by exploiting the known ergodicity of the $qx+1$ maps on $\mathbb{Z}_{2}$ (see \textbf{Theorem \ref{thm:shift map}}\vpageref{thm:shift map}). I will likely write this work up for publication at a later date. Whereas the above approach was grounded on the unit disk and harmonic analysis on subgroups of the circle, the second-and-a-halfth branch of my research\textemdash the one considered in this dissertation\textemdash concerns a completely different line of reasoning, from a completely different subspecialty of analysis, no less: $p$-adic and non-archimedean analysis. By considering the different branches of a given, suitably well-behaved Hydra map, I discovered a function $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ (where $p$ and $q$ are distinct primes which depend on $H$) with the remarkable property\footnote{I call this the \textbf{Correspondence Principle}; see \textbf{Theorem \ref{thm:CP v1} }and its other three variants, \textbf{Corollaries \ref{cor:CP v2}}, \textbf{\ref{cor:CP v3}}, and \textbf{\ref{cor:CP v4}}, starting from page \pageref{thm:CP v1}.} that the set of periodic points of $H$ in $\mathbb{Z}$ was \emph{completely determined} by the elements of the image of $\chi_{H}$ which lay in $\mathbb{Z}_{q}\cap\mathbb{Z}$. While I developed this approach in late 2019 and early 2020, and it was only in mid-2020 that I realized that the $\chi_{H}$ corresponding to the Shortened Collatz map played a pivotal role in a paper on the Collatz Conjecture published by Terence Tao at the end of 2019 \cite{Tao Probability paper}. However, due to Tao's \emph{deliberate choice} not\emph{ }to use a fully $p$-adic formalism\textemdash one which would have invoked $\chi_{H}$ in full\textemdash it appears he failed to notice this remarkable property. The relation between Tao's work an my own is discussed in detail starting on page \pageref{subsec:2.2.4 Other-Avenues}. It would be interesting to see my approach investigated from Tao's perspective. As of the writing of this dissertation, I have yet to grok Tao's probabilistic arguments, and have therefore refrained from attempting to employ or emulate them here. To the extent there even is such a thing as ``Collatz studies'', the subject is bazaar filled with odd and ends\textemdash \emph{ad hoc}s\emph{ }of the moment formed to deal with the latest line of thought about the Collatz map \emph{itself}. On the other hand, there have been few, if any, noteworthy attempts to center the study of Collatz-type arithmetical dynamical systems on a sound \emph{theoretical} underpinning\textemdash and by ``sound \emph{theoretical }underpinning'', I mean ``non-probabilistic investigations of multiple Collatz-type maps which \emph{do not }''. While there \emph{are }some works that address larger families of Collatz-type maps (\cite{dgh paper,Matthews' slides,Matthews and Watts,Moller's paper (german)}, for instance), they tend to be probabilistic, and are usually replete with conjectures, heuristic results, or specific findings that do not inspire much confidence in the possibility of a broader underlying non-probabilistic theory. My greatest aspiration for this dissertation\textemdash my labor of love, madness, obsession, and wonder\textemdash is that it will help pull Collatz studies out of the shadows to to emerge as a commendable, \emph{recommendable }field of study, complete with a theoretical foundation worth admiring. If I can do even half of that, I will consider my efforts successful, and all the angst I had to wade through along the way will, at last, be vindicated. \vphantom{}Los Angeles, California, USA March 14, 2022 \vphantom{} \vphantom{} \vphantom{} P.S. If you have questions, or have found typos or grammatical errors, please e-mail me at [email protected] (yes, I went to UCLA as an undergraduate). \newpage{} \pagebreak\pagestyle{headings} \mainmatter \chapter{Introduction \label{chap:Introduction}} \thispagestyle{headings} \includegraphics[scale=0.45]{./PhDDissertationEroica1.png} \vphantom{} Just like the title says, this is the introductory chapter of my PhD dissertation. Section \ref{sec:1.1 A-Guide-for-the-Perplexed} begins with an introduction the general notational conventions and idiosyncrasies used throughout this tome (Subsection \ref{subsec:1.1.1 Notational-Idiosyncrasies}). Subsection \ref{subsec:1.1.2 Some-Much-Needed-Explanations} provides the eponymous much needed explanations regarding my main results, and the agenda informing my (and our) pursuit of them\textemdash not to mention a \emph{mea culpa }regarding my dissertation's prodigious length. Speaking of length, Subsection \ref{subsec:1.1.3 An-Outline-of} contains a chapter-by-chapter outline of the dissertation as a whole, one I hope the reader will find useful. Section \ref{sec:1.2 Dynamical-Systems-Terminology} gives a brief exposition of basic notions from the theory of discrete dynamical systems\textemdash orbit classes and the like. The slightly longer Section \ref{sec:1.3 Crash course in ultrametric analysis} is a crash course in $p$-adic numbers and ultrametric analysis, written under the assumption that the reader has never heard of either topic. That being said, even an expert should take a gander at Subsections \ref{subsec:1.3.3 Field-Extensions-=00003D000026}, \ref{subsec:1.3.4 Pontryagin-Duality-and}, and \ref{subsec:1.3.5 Hensel's-Infamous-Blunder}, because my approach will frequently bring us into dangerous territory where we mix-and-match $p$-adic topologies, either with the real or complex topology, or for different primes $p$. \section{\emph{\label{sec:1.1 A-Guide-for-the-Perplexed}A Guide for the Perplexed}} \subsection{\label{subsec:1.1.1 Notational-Idiosyncrasies}Notational Idiosyncrasies} In order to resolve the endless debate over whether or not $0$ is a natural number, given any real number $x$, I write $\mathbb{N}_{x}$\nomenclature{$\mathbb{N}_{k}$}{the set of integers greater than or equal to $k$ \nopageref} to denote the set of all elements of $\mathbb{Z}$ which are $\geq x$. An expression like $\mathbb{N}_{x}^{r}$, where $r$ is an integer $\geq1$, then denotes the cartesian product of $r$ copies of $\mathbb{N}_{x}$. Likewise, $\mathbb{Z}^{d}$ denotes the cartesian product of $d$ copies of $\mathbb{Z}$. As a matter of personal preference, I use $\mathfrak{fraktur}$ (fraktur, \textbackslash mathfrak) font ($\mathfrak{z},\mathfrak{y}$, etc.) to denote $p$-adic (or $q$-adic) variables, and use the likes of $\mathfrak{a},\mathfrak{b},\mathfrak{c}$ to denote $p$-adic (or $q$-adic) constants. I prefer my notation to be as independent of context as possible, and therefore find it useful and psychologically comforting to distinguish between archimedean and non-archimedean quantities. Also, to be frank, I think the fraktur letters look \emph{cool}. I write $\left[\cdot\right]_{p^{n}}$ to denote the ``output the residue of the enclosed object modulo $p^{n}$'' operator. The output is \emph{always }an element of the set of integers $\left\{ 0,\ldots,p^{n}-1\right\} $. $\left[\cdot\right]_{1}$ denotes ``output the residue of the enclosed object modulo $1$''; hence, any integer (rational or $p$-adic) is always sent by $\left[\cdot\right]_{1}$ to $0$. $\left\|\cdot\right\|_{p}$ and $\left\|\cdot\right\|_{q}$ denote the standard $p$-adic and $q$-adic absolute values, respectively, while $v_{p}$ and $v_{q}$ denote the standard $p$-adic and $q$-adic valuations, respectively, with $\left\|\cdot\right\|_{p}=p^{-v_{p}\left(\cdot\right)}$, and likewise for $\left\|\cdot\right\|_{q}$ and $v_{q}$. Similarly, I write $\mathbb{Z}/p\mathbb{Z}$ as a short-hand for the set $\left\{ 0,\ldots,p-1\right\} $, in addition to denoting the group thereof formed by addition modulo $1$. I write $\hat{\mathbb{Z}}_{p}$ to denote the Pontryagin dual of the additive group of $p$-adic integers. I identify $\hat{\mathbb{Z}}_{p}$ with the set $\mathbb{Z}\left[1/p\right]/\mathbb{Z}$ of rational numbers in $\left[0,1\right)$ of the form $k/p^{n}$ where $n$ is an integer $\geq0$ and where $k$ is a non-negative integer which is either $0$ or co-prime to $p$. $\hat{\mathbb{Z}}_{p}=\mathbb{Z}\left[1/p\right]/\mathbb{Z}$ is made into an additive group with the operation of addition modulo $1$. All functions defined on $\hat{\mathbb{Z}}_{p}$ are assumed to be $1$-periodic; that is, for such a function $\hat{\phi}$, $\hat{\phi}\left(t+1\right)=\hat{\phi}\left(t\right)$ for all $t\in\hat{\mathbb{Z}}_{p}$. $\left\{ \cdot\right\} _{p}$ denotes the $p$-adic fractional part, viewed here as a homomorphism from the additive group $\mathbb{Q}_{p}$ of $p$-adic rational numbers to the additive group $\mathbb{Q}/\mathbb{Z}$ of rational numbers in $\left[0,1\right)$, equipped with the operation of addition modulo $1$. Because it will be frequently needed, I write $\mathbb{Z}_{p}^{\prime}$ to denote $\mathbb{Z}_{p}\backslash\mathbb{N}_{0}$ (the set of all $p$-adic integers which are not in $\left\{ 0,1,2,3,\ldots\right\} $). I use the standard convention of writing $\mathcal{O}_{\mathbb{F}}$ to denote the ring of integers of a number field $\mathbb{F}$\nomenclature{$\mathcal{O}_{\mathbb{F}}$}{the ring of $\mathbb{F}$-integers \nopageref}. With regard to embeddings (and this discussion will be repeated in Subsections \ref{subsec:1.3.3 Field-Extensions-=00003D000026} and \ref{subsec:1.3.4 Pontryagin-Duality-and}), while an algebraist might be comfortable with the idea that for a primitive third root of unity $\xi$ in $\mathbb{Q}_{7}$, the expression $\left\|2-3\xi\right\|_{7}$ is not technically defined, for my analytic purposes, this simply \emph{will not do}. As such, throughout this dissertation, given an odd prime $q$, in addition to writing $e^{2\pi i/\left(q-1\right)}$ to denote the complex number $\cos\left(2\pi/\left(q-1\right)\right)+i\sin\left(2\pi/\left(q-1\right)\right)$, I also write $e^{2\pi i/\left(q-1\right)}$ to denote the \emph{unique} primitive $\left(q-1\right)$th root of unity $\xi$ in $\mathbb{Z}_{q}^{\times}$ so that the value of $\xi$ modulo $q$ (that is, the first digit in the $q$-adic representation of $\xi$) is the smallest integer in $\left\{ 2,\ldots,q-2\right\} $ which is a primitive root modulo $q-1$. By far, the most common convention for expressing numerical congruences is of the form: \begin{equation} x=k\mod p \end{equation} However, I have neither any intention nor any inclination of using this cumbersome notation. Instead, I write $\overset{p}{\equiv}$ to denote congruence modulo $p$. This is a compact and extremely malleable notation. For example, given two elements $s,t\in\hat{\mathbb{Z}}_{p}$, I write $s\overset{1}{\equiv}t$ \nomenclature{$\overset{1}{\equiv}$}{congruence modulo $1$ \nopageref}to indicate that $s$ is congruent to $t$ modulo $1$; i.e., $s-t\in\mathbb{Z}$, which\textemdash of course\textemdash makes $s$ and $t$ different representatives of the same element of $\hat{\mathbb{Z}}_{p}$ (example: $s=1/p$ and $t=\left(p+1\right)/p$). This congruence notation is also compatible with $p$-adic numbers (integer or rational): $\mathfrak{z}\overset{p^{n}}{\equiv}k$\nomenclature{$\overset{p^{n}}{\equiv}$}{congruence modulo $p^{n}$ \nopageref} holds true if and only if $\mathfrak{z}$ is an element of $k+p^{n}\mathbb{Z}_{p}$. My congruence notation is particularly useful in conjunction with the indispensable Iverson bracket notation. Given a statement or expression $S$, the Iverson bracket is the convention of writing $\left[S\right]$ to denote a quantity which is equal to $1$ when $S$ is true, and which is $0$ otherwise. For example, letting $k$ be an integer constant and letting $\mathfrak{z}$ be a $p$-adic integer variable, $\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]$ is then the indicator function for the set $k+p^{n}\mathbb{Z}_{p}$, being equal to $1$ if $\mathfrak{z}$ is congruent to $k$ mod $p^{n}$ (that is, $\left[\mathfrak{z}\right]_{p^{n}}=k$), and being $0$ otherwise. Given a point $s\in\hat{\mathbb{Z}}_{p}$, I write $\mathbf{1}_{s}\left(t\right)$ to denote the function $\left[t\overset{1}{\equiv}s\right]$ of the variable $t\in\hat{\mathbb{Z}}_{p}$. That is, $\mathbf{1}_{s}\left(t\right)$ is $1$ if and only if $t-s$ is in $\mathbb{Z}$, and is $0$ for all other values of $t$. I write $\mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)$ to denote the function $\left[\mathfrak{z}=\mathfrak{a}\right]$ of the variable $\mathfrak{z}\in\mathbb{Z}_{p}$, which is $1$ if and only if $\mathfrak{z}=\mathfrak{a}$ and is $0$ otherwise. Because the absolute-value-induced-metric-topology with which we will make sense of limits of sequences or sums of infinite series will often frequently change, I have cultivated a habit of writing the space in which the convergence occurs over the associated equal sign. Thus, $\lim_{n\rightarrow\infty}f\left(x_{n}\right)\overset{\mathbb{Z}_{p}}{=}c$ means that the sequence $f\left(x_{n}\right)$ converges to $c$ in $p$-adic absolute value. Note that this necessarily forces $c$ and the $f\left(x_{n}\right)$s to be quantities whose $p$-adic absolute values are meaningfully defined. Writing $\overset{\overline{\mathbb{Q}}}{=}$ or $\overset{\mathbb{Q}}{=}$ indicates that no limits are actually involved; the sums in question are not infinite, but consist of finitely many terms, all of which are contained in the field indicated above the equals sign. I write $\overset{\textrm{def}}{=}$\nomenclature{$\overset{\textrm{def}}{=}$}{"by definition" \nopageref} to mean ``by definition''. The generalizations of these notations to the multi-dimensional case are explained on pages \pageref{nota:First MD notation batch}, \pageref{nota:Second batch}, and \pageref{nota:third batch}. These also include extensions of other notations introduced elsewhere to the multi-dimensional case, as well as the basic algebraic conventions we will use in the multi-dimensional case (see page \pageref{nota:First MD notation batch}). Also, though it will be used infrequently, I write \nomenclature{$\ll$}{Vinogradov notation (Big O)}$\ll$ and $\gg$ to denote the standard Vinogradov notation (an alternative to Big $O$), indicating a bound on a real quantity which depends on an implicit constant ($f\left(x\right)\ll g\left(x\right)$ iff $\exists$ $K>0$ so that $f\left(x\right)\leq Kg\left(x\right)$). Given a map $T$, I write $T^{\circ n}$ to denote the composition of $n$ copies of $T$. Finally, there is the matter of self-reference. Numbers enclosed in (parenthesis) refer to the numeric designation assigned to one of the many, many equations and expressions that occur in this document. Whenever a Proposition, Lemmata, Theorem, Corollary or the like is invoked, I do so by writing it in bold along with the number assigned to it; ex: \textbf{Theorem 1.1}. Numbers in parenthesis, like (4.265), refer to the equation with that particular number as its labels. References are cited by their assigned number, enclosed in {[}brackets{]}. I also sometimes refer to vector spaces as linear spaces; ergo, a $\overline{\mathbb{Q}}$-linear space is a vector space over $\overline{\mathbb{Q}}$. \subsection{Some Much-Needed Explanations \label{subsec:1.1.2 Some-Much-Needed-Explanations}} Given that this document\textemdash my PhD Dissertation\textemdash is over four-hundred fifty pages long, I certainly owe my audience an explanation. I have explanations in spades, and those given in this incipit are but the first of many. I have spent the past five years studying the Collatz Conjecture from a variety of different analytical approaches. My journey eventually led me into the \emph{ultima Thule }of non-Archimedean analysis, which\textemdash to my surprise\textemdash turned out to be far less barren and far more fruitful than I could have ever suspected. With this monograph, I chronicle my findings. To set the mood\textemdash this \emph{is }a study of the Collatz Conjecture, after all\textemdash let me give my explanations in threes. Besides the obvious goal of securing for myself a hard-earned doctoral degree, the purpose of this dissertation is three-fold: \begin{itemize} \item To present a analytical programme for studying a wide range of Collatz-type arithmetical dynamical systems in a unified way; \item To detail new, unforeseen aspects of the subspecialty of non-archimedean analysis involving the study of functions from the $p$-adic integers to the $q$-adic (complex) numbers (I call this ``\emph{$\left(p,q\right)$-adic analysis}''); \item To use $\left(p,q\right)$-adic analysis to study the dynamics of Collatz-type maps\textemdash primarily the question of whether or not a given integer is a periodic point of such a map. \end{itemize} In this respect\textemdash as indicated by its title\textemdash this dissertation is as much about the novel methods of non-archimedean analysis as it is about using those methods to describe and better understand Collatz-type dynamical systems. The reader has a winding road ahead of them, monotonically increasing in complexity over the course of their six-chapter journey. To me, among the many ideas we will come across, three innovations stand out among the rest: \begin{itemize} \item Given a kind of a Collatz-type map $H:\mathbb{Z}\rightarrow\mathbb{Z}$ which I call a \textbf{Hydra map} (see page \pageref{def:p-Hydra map}), provided $H$ satisfies certain simple qualitative properties, there are distinct primes $p,q$ and a function $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$\textemdash the \textbf{Numen} of $H$ (see page \pageref{def:Chi_H on N_0 in strings})\textemdash with the property that an integer $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ \emph{if and only if }there is a $p$-adic integer $\mathfrak{z}_{0}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,3,\ldots\right\} $ so that $\chi_{H}\left(\mathfrak{z}_{0}\right)=x$. I call this phenomenon \textbf{the} \textbf{Correspondence Principle}\footnote{Actually, we can say more: modulo certain simple qualitative conditions on $H$, if there is an ``irrational'' $p$-adic integer $\mathfrak{z}_{0}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$ so that $\chi_{H}\left(\mathfrak{z}_{0}\right)\in\mathbb{Z}$, then the sequence of iterates $\chi_{H}\left(\mathfrak{z}_{0}\right),H\left(\chi_{H}\left(\mathfrak{z}_{0}\right)\right),H\left(H\left(\chi_{H}\left(\mathfrak{z}_{0}\right)\right)\right),\ldots$ is unbounded with respect to the standard archimedean absolute value on $\mathbb{R}$. I have not been able to establish the converse of this result, however.} (see page \pageref{cor:CP v4}). \item My notion of \textbf{frames} (presented in Subsection \ref{subsec:3.3.3 Frames}), which provide a formalism for dealing with functions $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ (where $p$ and $q$ are distinct primes) defined by infinite series such that the \emph{topology of convergence} used to sum said series \emph{varies from point to point}. The archetypical example is that of a series in a $p$-adic integer variable $\mathfrak{z}$ which converges in the $q$-adic topology whenever $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0,1,2,3,\ldots\right\} $ and converges in the topology of the reals whenever $\mathfrak{z}\in\left\{ 0,1,2,3,\ldots\right\} $. Using frames, we can significantly enlarge the class of $\left(p,q\right)$-adic functions which can be meaningfully ``integrated''\textemdash I call such functions \textbf{quasi-integrable }functions. With these tools, I then establish corresponding $\left(p,q\right)$-adic generalizations of the venerable \textbf{Wiener Tauberian Theorem }(\textbf{WTT}) from harmonic analysis (Subsection \ref{subsec:3.3.7 -adic-Wiener-Tauberian}). \item A detailed $\left(p,q\right)$-adic Fourier-analytical study of $\chi_{H}$. I show that, for many Hydra maps $H$, the numen $\chi_{H}$ is in fact quasi-integrable. Not only does this come with explicit non-trivial $\left(p,q\right)$-adic series representations of $\chi_{H}$, and formulae for its Fourier transforms ($\hat{\chi}_{H}$), but\textemdash in conjunction with the Correspondence Principle and the $\left(p,q\right)$-adic Wiener Tauberian Theorem\textemdash I show that the question ``is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$?'' is essentially equivalent to ``is the span of the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ dense in the appropriate function space?'' In particular, the answer to the former is ``no'' whenever the density of the span of the translates occurs (\textbf{Theorem \ref{thm:Periodic Points using WTT}}\vpageref{thm:Periodic Points using WTT}). I call this the \textbf{Tauberian Spectral Theorem }for $H$.\footnote{If the spans of the translates are \emph{not }dense, then, at present, my methods then guarantee that $x$ is either a periodic point of $H$ or the sequence of iterates $x,H\left(x\right),H\left(H\left(x\right)\right),\ldots$ is unbounded with respect to the standard archimedean absolute value on $\mathbb{R}$.} \end{itemize} I consider the above my ``main results''. It is worth mentioning that these results hold not just for Collatz-type maps $H:\mathbb{Z}\rightarrow\mathbb{Z}$, but for Collatz-type maps $H:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ for any integer $d\geq1$\textemdash the $d\geq2$ case being the ``\textbf{multi-dimensional case}'' (Chapters 5 \& 6). These arise from studying generalizations of Collatz-type maps from $\mathbb{Z}$ to $\mathcal{O}_{\mathbb{F}}$, the ring of integers of $\mathbb{F}$, a finite-dimensional field extension of $\mathbb{Q}$; such maps were first considered by Leigh (1981) and subsequently explored by his colleague K. R. Matthews \cite{Leigh's article,Matthews' Leigh Article,Matthews' slides}. I briefly corresponded with Matthews in 2017, and again in 2019\textemdash the man is a veteran Collatz scholar\textemdash and he informed me that other than the works just cited, he knew of no others that have pursued Collatz-type maps on rings of algebraic integers. My final explanatory triplet addresses the forty-foot dragon in the corner of the room with page after page of \emph{pages }clutched between his claws: \emph{why is this dissertation over four-hundred-fifty ($450$) pages long?} \begin{itemize} \item Unusually for ``hard'' analysis\textemdash generally a flexible, well-mixed, free-ranged discipline\textemdash the \emph{sub}-subspecialty of $\left(p,q\right)$-adic analysis is incredibly arcane, even within the already esoteric setting of non-archimedean analysis as a whole. While it is not anywhere near the level of, say, algebraic geometry, non-archimedean analysis behooves the reader a good deal of legwork. Many readers will find in these pages at least one (and perhaps many) ideas they never would have thought of considering; common courtesy and basic decency demands I give a thorough accounting of it. That my work is left field even by the standards of non-archimedean analysis only strengths my explanatory resolve. \item It is a general conviction of mine\textemdash and one strongly held, at that\textemdash that thought or knowledge without warmth, patience, openness, charity, and accessibility is indistinguishable from Brownian motion or Gaussian noise. Mathematics is difficult and frustrating enough as it is. I refuse to allow my writing\textemdash or, well, any potential \emph{lack} thereof\textemdash to cause or perpetuate pain, suffering, or discomfort which could otherwise have been avoided. Also\textemdash as befits a budding new theory\textemdash many of the most important methods and insights in this document arise directly from demanding computational considerations. The formulas provide the patterns, so it would make little sense (nor kindness, for that matter) not to explore and discuss them at length. \item I am a \emph{wordy} person, and\textemdash in addition to my passion for mathematics\textemdash I have a passion for writing. The Collatz obsession infected me just as I was finishing up my second novel (woefully stillborn), and persisted throughout the writing of my \emph{third} novel, coeval with the research and writing that led to my dissertation. \end{itemize} All I can say is, I hope you enjoy the ride. \subsection{\label{subsec:1.1.3 An-Outline-of}An Outline of this Dissertation} \subsubsection{Chapter 1} If this dissertation were a video game, Chapter 1 would be the Tutorial level. Aside from my verbal excess, Chapter 1 provides the first significant portion of necessary background material. Section \ref{sec:1.2 Dynamical-Systems-Terminology} reviews elementary facts and terminology from the theory of dynamical systems (periodic points, divergent trajectories, orbit classes, etc.). The largest section of Chapter 1 is \ref{sec:1.3 Crash course in ultrametric analysis}, which is basically a crash course in ultrametric analysis, with a special focus on the $p$-adic numbers in Subsection \ref{subsec:1.3.1. The-p-adics-in-a-nutshell}. Most of the material from Subsection \ref{subsec:1.3.2. Ultrametrics-and-Absolute} is taken from Schikhof's \emph{Ultrametric Calculus }\cite{Ultrametric Calculus}, which I highly recommend for anyone interested in getting a first taste of the subject as a whole. Subsection \ref{subsec:1.3.3 Field-Extensions-=00003D000026} briefly mentions $p$-adic field extensions, Galois actions, and spherical completeness. Subsection \ref{subsec:1.3.4 Pontryagin-Duality-and} introduces Pontryagin duality and the Fourier transform of a \emph{complex-valued }function on a locally compact abelian group, again laser-focused on the $p$-adic case. While Subsections \ref{subsec:1.3.1. The-p-adics-in-a-nutshell} through \ref{subsec:1.3.4 Pontryagin-Duality-and} can be skipped by a reader already familiar with their contents, I must recommend that \emph{everyone }read \ref{subsec:1.3.5 Hensel's-Infamous-Blunder}. Throughout this dissertation, we will skirt grave danger by intermixing and interchanging archimedean and non-archimedean topologies on $\overline{\mathbb{Q}}$. Subsection \ref{subsec:1.3.5 Hensel's-Infamous-Blunder} explains this issue in detail, using Hensel's infamous $p$-adic ``proof'' of the irrationality of $e$ as a springboard. The reason we will not suffer a similar, ignominious fate is due to the so-called ``universality'' of the geometric series. \subsubsection{Chapter 2} Chapter 2 presents the titular Hydra maps, my name for the class of Collatz-type maps studied in this dissertation, as well as the other Collatz-related work discussed in the Preface. Similar generalizations have been considered by \cite{Matthews and Watts,Moller's paper (german),dgh paper}; the exposition given here is an attempt to consolidate these efforts into a single package. In order to avoid excessive generality, the maps to be considered will be grounded over the ring of integers ($\mathbb{Z}$) and finite-degree extensions thereof. In his slides, Matthews considers Collatz-type maps defined over rings of polynomials with coefficients in finite fields \cite{Matthews' slides}. While, in theory, it may be possible to replicate the $\chi_{H}$ construction by passing from rings of polynomials with coefficients in a finite field $\mathbb{F}$ to the local field of formal Laurent series with coefficients in $\mathbb{F}$, in order to avoid the hassle of dealing with fields of positive characteristic, this dissertation will defer exploration of Hydra maps on fields of positive characteristic to a later date. My presentation of Hydra maps deals with two types: a ``one-dimensional'' variant\textemdash Hydra maps defined on $\mathbb{Z}$\textemdash and a ``multi-dimensional'' variant; given a number field $\mathbb{F}$, a Hydra map defined on the ring of integers $\mathcal{O}_{\mathbb{F}}$ is said to be of dimension $\left[\mathbb{F}:\mathbb{Q}\right]=\left[\mathcal{O}_{\mathbb{F}}:\mathbb{Z}\right]$. Equivalently, the multi-dimensional variant can be viewed as maps on $\mathbb{Z}^{d}$. The study of the multi-dimensional variant is more or less the same as the study of the one-dimensional case, albeit with a sizable dollop of linear algebra. After presenting Hydra maps, Chapter 2 proceeds to demonstrate how, to construct the function $\chi_{H}$\textemdash the ``numen''\footnote{My original instinct was to call $\chi_{H}$ the ``characteristic function'' of $H$ (hence the use of the letter $\chi$). However, in light of Tao's work\textemdash where, in essence, one works with the characteristic function (in the Probability-theoretic sense) of $\chi_{H}$, this name proved awkward. I instead chose to call it the ``numen'' of $H$ (plural: ``numina''), from the Latin, meaning ``the spirit or divine power presiding over a thing or place''.} of the Hydra map $H$\textemdash and establishes the conditions required of $H$ in order for $\chi_{H}$ to exist. Provided that $H\left(0\right)=0$, we will be able to construct $\chi_{H}$ as a function from $\mathbb{N}_{0}$ to $\mathbb{Q}$. Additional conditions (see page \pageref{def:Qualitative conditions on a p-Hydra map}) are then described which will guarantee the existence of a unique extension/continuation of $\chi_{H}$ to a $q$-adic valued function of a $p$-adic integer variable for appropriate $p$ and $q$ determined by $H$ and the corresponding generalization for multi-dimensional Hydra maps. This then leads us to the cornerstone of this dissertation: the \textbf{Correspondence Principle}, which I give in four variants, over pages \pageref{thm:CP v1}-\pageref{cor:CP v4}. This principle is the fact that\textemdash provided $H$ satisfies an extra condition beyond the one required for $\chi_{H}$ to have a non-archimedean extension\textemdash there will be a one-to-one correspondence between the periodic points of $H$ and the rational integer values attained by the non-archimedean extension of $\chi_{H}$. In certain cases, this principle even extends to cover the points in divergent orbits of $H$! Understandably, the Correspondence Principle motivates all the work that follows. Chapter 2 concludes with a brief discussion of the relationship between $\chi_{H}$ and other work, principally Tao's paper \cite{Tao Probability paper} (discussed in this dissertation starting \vpageref{subsec:Connections-to-Tao}), transcendental number theory, the resolution of \textbf{Catalan's Conjecture }by P. Mih\u{a}ilescu in 2002 \cite{Cohen Number Theory}, and the implications thereof for Collatz studies\textemdash in particular, the fundamental inadequacy of the current state of transcendental number theory that precludes it from being able to make meaningful conclusions on the Collatz Conjecture. \subsubsection{Chapter 3} This is the core theory-building portion of this dissertation, and is dedicated to presenting the facets of non-archimedean analysis which I believe will finally provide Hydra maps and the contents of Chapter 2 with a proper theoretical context. After a historical essay on the diverse realizations of ``non-archimedean analysis'' (Subsection \ref{subsec:3.1.1 Some-Historical-and}), the first section of Chapter 3 is devoted to explaining the essentials of what I call ``$\left(p,q\right)$-adic analysis'', the specific sub-discipline of non-archimedean analysis which concerns functions of a $p$-adic variable taking values in $\mathbb{Q}_{q}$ (or a field extension thereof), where $p$ and $q$ are distinct primes. Since much of the literature in non-archimedean analysis tends to be forbiddingly generalized\textemdash much more so than the ``hard'' analyst is usually comfortable with\textemdash I have taken pains to minimize unnecessary generality as much as possible. As always, the goal is to make the exposition of the material as accessible as possible without compromising rigor. The other subsections of \ref{sec:3.1 A-Survey-of} alternate between the general and the specific. \ref{subsec:3.1.2 Banach-Spaces-over} addresses the essential aspects of the theory of non-archimedean Banach spaces, with an emphasis on which of the major results of the archimedean theory succeed or fail. Subsection \ref{subsec:3.1.3 The-van-der} presents the extremely useful \textbf{van der Put basis }and \textbf{van der Put series representation }for the space of continuous $\mathbb{F}$-valued functions defined on the $p$-adic integers, where $\mathbb{F}$ is a metrically complete valued field (either archimedean or non-archimedean). The existence of this basis makes computations in $\left(p,q\right)$-adic analysis seem much more like its classical, archimedean counterpart than is usually the case in more general non-archimedean analysis. Subsections \ref{subsec:3.1.4. The--adic-Fourier} and \ref{subsec:3.1.5-adic-Integration-=00003D000026} are devoted to the $\left(p,q\right)$-adic Fourier(-Stieltjes) transform. The contents of these subsections are a mix of W. M. Schikhof's 1967 PhD dissertation,\emph{ Non-Archimedean Harmonic Analysis} \cite{Schikhof's Thesis}, and my own independent re-discovery of that information prior to learning that \emph{Non-Archimedean Harmonic Analysis} already existed. For the classical analyst, the most astonishing feature of this subject is the near-total absence of \emph{nuance}: $\left(p,q\right)$-adic functions are integrable (with respect to the $\left(p,q\right)$-adic Haar probability measure, whose construction is detailed in Subsection \ref{subsec:3.1.5-adic-Integration-=00003D000026}) if and only if they are continuous, and are continuous if and only if they have a $\left(p,q\right)$-adic Fourier series representation which converges uniformly \emph{everywhere}. Whereas Subsections \ref{subsec:3.1.4. The--adic-Fourier} and \ref{subsec:3.1.5-adic-Integration-=00003D000026} are primarily oriented toward practical issues of computation, \ref{subsec:3.1.6 Monna-Springer-Integration} gives a thorough exposition of the chief theory of integration used in non-archimedean analysis as a whole, that of\textbf{ }the\textbf{ Monna-Springer Integral}. My primary motivation for introducing Monna-Springer theory is to make it better known, with an eye toward equipping other newcomers to the subject with a guide for decoding the literature. The novel content of Chapter 3 occurs in Sections \ref{sec:3.2 Rising-Continuous-Functions} and \ref{sec:3.3 quasi-integrability}, particularly the latter. Section \ref{sec:3.2 Rising-Continuous-Functions} presents and expands upon material which is given but a couple exercise's\footnote{Exercise 62B, page 192, and its surroundings\cite{Ultrametric Calculus}.} worth of attention in Schikhof's \emph{Ultrametric Calculus}. Instead, in Subsection \ref{subsec:3.2.1 -adic-Interpolation-of}, we will develop into a theory of what I call \textbf{rising-continuous functions}. In brief, letting $\mathbb{F}$ be a metrically complete valued field, these are functions $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{F}$ satisfying the point-wise limit condition: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{\mathbb{F}}{=}\lim_{n\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} where, as indicated by the $\overset{\mathbb{F}}{=}$, the convergence occurs in the topology of $\mathbb{F}$. These functions arise naturally when one considers interpolating functions on $\mathbb{N}_{0}$ to ones on $\mathbb{Z}_{p}$. After this expos, Subsection \ref{subsec:3.2.2 Truncations-=00003D000026-The} demonstrates that the set of rising-continuous functions forms a non-archimedean Banach algebra which extends the Banach algebra of continuous $\left(p,q\right)$-adic functions. The conditions required for a rising-continuous function to be a unit\footnote{I.E., the reciprocal of the function is also rising-continuous.} of this algebra are investigated, with the \textbf{Square Root Lemma} (see page \pageref{lem:square root lemma}) of Subsection \ref{subsec:3.2.2 Truncations-=00003D000026-The} providing a necessary and sufficient condition. In the \emph{sub}-subsection starting \vpageref{subsec:3.2.2 Truncations-=00003D000026-The}, the theory of Berkovitch spaces from $p$-adic (algebraic) geometry is briefly invoked to prove some topological properties of rising-continuous functions. Subsection \ref{subsec:3.2.2 Truncations-=00003D000026-The} also introduces the crucial construction I call \textbf{truncation}. The $N$th truncation of a $\left(p,q\right)$-adic function $\chi$, denoted $\chi_{N}$, is the function defined by: \begin{equation} \chi_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{equation} Even if $\chi$ is merely rising-continuous, its $N$th truncation is continuous\textemdash in fact, locally constant\textemdash for all $N$, and converges point-wise to $\chi$ everywhere as $N\rightarrow\infty$. We also explore the interplay between truncation, van der Put series, and the Fourier transform as it regards continuous $\left(p,q\right)$-adic functions. The heart of section \ref{sec:3.3 quasi-integrability}, however, is the exposition of my chief innovations non-archimedean analysis: \textbf{frames }and \textbf{quasi-integrability}. Because an arbitrary rising-continuous function is not going to be continuous, it will fail to be integrable in the sense of Monna-Springer theory. This non-integrability precludes a general rising-continuous function from possessing a well-defined Fourier transform. Quasi-integrability is the observation that this difficulty can, in certain cases, be overcome, allowing us to significantly enlarge the class of $\left(p,q\right)$-adic functions we can meaningfully integrate. Aside from facilitating a more detailed analysis of $\chi_{H}$, these innovations demonstrate that $\left(p,q\right)$-adic analysis is far more flexible\textemdash and, one would hope, \emph{useful}\textemdash than anyone had previously thought. Instead of diving right into abstract axiomatic definitions, in Subsection \ref{subsec:3.3.1 Heuristics-and-Motivations}, I take the Arnoldian approach (otherwise known as the \emph{sensible} approach) of beginning with an in-depth examination of the specific computational examples that led to and, eventually, necessitated the development of frames and quasi-integrability. To give a minor spoiler, the idea of quasi-integrability, in short, is that even if $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ is not continuous, \emph{if we can find }a function $\hat{\chi}$ (defined on $\hat{\mathbb{Z}}_{p}$) so that the partial sums: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} converge to $\chi$ as $N\rightarrow\infty$ in an appropriate sense for certain $\mathfrak{z}$, we can then use that $\hat{\chi}$ to define \emph{a}\footnote{Alas, it is not \emph{the} Fourier transform; it is only unique modulo the Fourier-Stieltjes transform of what we will call a \textbf{degenerate measure}.}\textbf{ Fourier transform }of $\chi$. In this case, we say that $\chi$ is \textbf{quasi-integrable}. The name is due to the fact that the existence of $\hat{\chi}$ allows us to define the integral $\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\chi\left(\mathfrak{z}\right)d\mathfrak{z}$ for any continuous $\left(p,q\right)$-adic function $f$. We view $\chi$ as the ``derivative'' of the $\left(p,q\right)$-adic measure $\chi\left(\mathfrak{z}\right)d\mathfrak{z}$; this makes $\hat{\chi}$ the Fourier-Stieltjes transform of this measure. The success of this construction is of particular interest, given that, in general \cite{van Rooij - Non-Archmedean Functional Analysis}, the Radon-Nikodym theorem fails to hold in non-archimedean analysis. As a brief\textemdash but fascinating\textemdash tangent, starting from page \pageref{eq:Definition of (p,q)-adic Mellin transform}, I showcase how quasi-integrability can be used to meaningfully define the \textbf{Mellin Transform }of a continuous $\left(p,q\right)$-adic function. Subsection \ref{subsec:3.3.2 The--adic-Dirichlet} shows how the limiting process described above can be viewed as a $\left(p,q\right)$-adic analogue of summability kernels, as per classical Fourier analysis. All the most important classical questions in Fourier Analysis of functions on the unit interval (or circle, or torus) come about as a result of the failure of the \textbf{Dirichlet Kernel }to be a well-behaved approximate identity. On the other hand, $\left(p,q\right)$-adic analysis has no such troubles: the simple, natural $\left(p,q\right)$-adic incarnation of the Dirichlet kernel is, in that setting, an approximate identity\textemdash no questions asked! In subsection \pageref{subsec:3.3.3 Frames}, the reader will be introduced to the notion of \textbf{frames}.\textbf{ }A frame is, at heart, an organizing tool that expedites and facilitates clear, concise discussion of the ``spooky'' convergence behaviors of the partial sums of Fourier series generated by $q$-adically bounded functions $\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$. This subsection is primarily conceptual, as is its counterpart \ref{subsec:3.3.5 Quasi-Integrability}, where quasi-integrability with respect to a given frame is formally defined. Subsection \ref{subsec:3.3.4 Toward-a-Taxonomy} makes the first steps toward what will hopefully one day be an able-bodied theory of $\left(p,q\right)$-adic measures, and introduces several (re-)summation formulae for $\left(p,q\right)$-adic Fourier series that will take central stage in our analysis of $\chi_{H}$. Subsection \ref{subsec:3.3.6 L^1 Convergence} also shows that the hitherto neglected topic of absolute convergence of $\left(p,q\right)$-adic integrals\textemdash that is, $L^{1}$ convergence\textemdash in non-archimedean analysis seems to provide the logical setting for studying quasi-integrability at the theoretical level. Lastly, Subsection \ref{subsec:3.3.7 -adic-Wiener-Tauberian} introduces \textbf{Wiener's Tauberian Theorem }and states and proves two new $\left(p,q\right)$-adic analogues of this chameleonic result, one for continuous functions, and one for measures. Sub-subsection \ref{subsec:A-Matter-of} then explores how the question of the existence and continuity of the reciprocal of a $\left(p,q\right)$-adic function can be stated and explored in terms of the spectral theory of \textbf{circulant matrices}. \subsubsection{Chapter 4} In Chapter 4, the techniques developed in Chapter 3 are mustered in a comprehensive $\left(p,q\right)$-adic Fourier analysis of $\chi_{H}$. After a bit of preparatory work (Subsection \ref{sec:4.1 Preparatory-Work--}), Subsection \ref{sec:4.2 Fourier-Transforms-=00003D000026} launches into the detailed, edifying computation needed to establish the quasi-integrability of $\chi_{H}$ for a wide range of Hydra maps. The method of proof is a kind of asymptotic analysis: by computing the the Fourier transforms of the \emph{truncations} of $\chi_{H}$, we can identify fine structure which exists independently of the choice of truncation. From this, we derive an explicit formula for Fourier transform of $\chi_{H}$ and demonstrate its convergence with respect to the appropriate frame. A recurring motif of this analysis is the use of functional equation characterizations to confirm that our formulae do, in fact, represent the desired functions. As a capstone, my $\left(p,q\right)$-adic generalizations of Wiener's Tauberian Theorem are then invoked to establish a near-equivalence of the problem of characterizing the periodic points and divergent points of $H$ to determining those values of $x$ for which the span of the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$; this is the content of the \textbf{Tauberian Spectral Theorem }for $\chi_{H}$ (\textbf{Theorem \ref{thm:Periodic Points using WTT}}, on page \pageref{thm:Periodic Points using WTT}). Section \ref{sec:4.3 Salmagundi} is a potpourri of miscellaneous results about $\chi_{H}$. Some of these build upon the work of Section \ref{sec:4.2 Fourier-Transforms-=00003D000026}; others which do not. I suspect the lines of inquiry covered in Section \ref{sec:4.3 Salmagundi} will be useful for future explorations of this subject. Of particular interest is what I call the ``$L^{1}$ method'' (\textbf{Theorem \ref{thm:The L^1 method}} on \pageref{thm:The L^1 method}), a means for obtaining upper bounds on the \emph{archimedean }absolute value of $\chi_{H}\left(\mathfrak{z}\right)$ over certain appropriately chosen subsets of $\mathbb{Z}_{p}$. \subsubsection{Chapter 5} Although my methods are applicable to Collatz-type maps on rings of algebraic integers\textemdash equivalently $\mathbb{Z}^{d}$\textemdash I prefer to cover the multi-dimensional case only after the one-dimensional case, so as to let the underlying ideas shine through, unobscured by technical considerations of multi-dimensional linear algebra. Chapter 5 contains the multi-dimensional case of the content of the one-dimensional case as given in Chapters 2 and 3. Section \ref{sec:5.1 Hydra-Maps-on} deals with the issue of converting a Collatz-type on some ring of algebraic integers $\mathcal{O}_{\mathbb{F}}$ to an analogous map acting on a lattice $\mathbb{Z}^{d}$ of appropriate dimension, with Subsection \ref{subsec:5.1.2 Co=00003D0000F6rdinates,-Half-Lattices,-and} providing the working definition for a Hydra map on $\mathbb{Z}^{d}$. Section \ref{sec:5.2 The-Numen-of} is the multi-dimensional counterpart of Chapter 2, and as such constructs $\chi_{H}$ and establishes the multi-dimensional version of the \textbf{Correspondence Principle}. Sections \ref{sec:5.3 Rising-Continuity-in-Multiple} and \ref{sec:5.4. Quasi-Integrability-in-Multiple} treat rising-continuity and $\left(p,q\right)$-adic Fourier theory, respectively, as they occur in the multi-dimensional context, with the tensor product being introduced in Section \ref{sec:5.3 Rising-Continuity-in-Multiple}. \ref{sec:5.4. Quasi-Integrability-in-Multiple} presents multi-dimensional frames and quasi-integrability \subsubsection{Chapter 6} Chapter 6 is a near-verbatim copy of Chapter 5, extended to the multi-dimensional case, though without a corresponding extension of the salmagundi of Section \ref{sec:4.3 Salmagundi}\textemdash I simply did not have enough time. \newpage{} \section{\label{sec:1.2 Dynamical-Systems-Terminology}Elementary Theory of Discrete Dynamical Systems} In this brief section, we give an overview of the essential terminology from the theory of discrete dynamical systems\index{discrete dynamical systems}. A good reference for a beginner is \cite{Dynamical Systems}; it also covers continuous systems. For our purposes, it suffices to consider dynamical systems on $\mathbb{Z}^{d}$, where $d$ is an integer $\geq1$, which is to say, there is some map $T:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ whose iterates we wish to study. Recall that we write $T^{\circ n}$ to denote the $n$-fold iteration of $T$: \begin{equation} T^{\circ n}=\underbrace{T\circ\cdots\circ T}_{n\textrm{ times}} \end{equation} \begin{defn} Given an $\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{Z}^{d}$, the \textbf{forward orbit }of $\mathbf{x}$ under $T$ is the sequence of iterates $\mathbf{x},T\left(\mathbf{x}\right),T\left(T\left(\mathbf{x}\right)\right),\ldots,T^{\circ n}\left(\mathbf{x}\right),\ldots$. This is also called the \textbf{trajectory }of $\mathbf{x}$ under $T$. More generally, a \textbf{trajectory }of $T$ refers to the forward orbit of some $\mathbf{x}$ under $T$. \end{defn} \vphantom{} For us, the most important classical construction of the theory of dynamical systems is that of the \textbf{orbit class}\index{orbit class}: \begin{defn} \label{def:orbit class}Define a relation $\sim$ on $\mathbb{Z}^{d}$ by $\mathbf{x}\sim\mathbf{y}$ if and only if there are integers $m,n\geq0$ so that $T^{\circ m}\left(\mathbf{x}\right)=T^{\circ n}\left(\mathbf{y}\right)$, where, recall, $T^{\circ0}$ is the identity map. It can be verified that $\sim$ is an equivalence relation on $\mathbb{Z}^{d}$. The distinct equivalence classes of $\mathbb{Z}^{d}$ under $\sim$ are called the \textbf{irreducible orbit classes }of $T$ in $\mathbb{Z}^{d}$. More generally, an \textbf{orbit class }is the union of irreducible orbit classes of $T$; we say an orbit class is \textbf{reducible }precisely when it can be written as a union of two or more non-empty orbit classes, and say it is \textbf{irreducible }when no such decomposition exists. \end{defn} \vphantom{} For those who have not seen this concept before, it is equivalent to the idea of a watershed in earth sciences and hydrology: all elements of a single irreducible orbit class of $T$ have the same long-term behavior under iteration by $T$. \begin{prop} A set $V\subseteq\mathbb{Z}^{d}$ is an orbit class of $T$ (possibly reducible) if and only if $T^{-1}\left(V\right)=V$. \end{prop} \vphantom{} Using orbit classes, we can distinguish between sets according to how $T$ behaves on them. \begin{defn} We say $\mathbf{x}\in\mathbb{Z}^{d}$ is a \textbf{periodic point} of $T$ whenever there is an integer $n\geq1$ so that $T^{\circ n}\left(\mathbf{x}\right)=\mathbf{x}$. The \textbf{period }of $\mathbf{x}$ is the smallest integer $n\geq1$ for which $T^{\circ n}\left(\mathbf{x}\right)=\mathbf{x}$ holds true. The set $\left\{ T^{\circ m}\left(\mathbf{x}\right):m\geq0\right\} $ (the forward orbit of $\mathbf{x}$) is then called the \textbf{cycle generated by $\mathbf{x}$}. More generally, we say a set $\Omega\subseteq\mathbb{Z}^{d}$ is a \textbf{cycle }of $T$ whenever there is an $\mathbf{x}\in\Omega$ which is a periodic point of $T$ such that $\Omega=\left\{ T^{\circ m}\left(\mathbf{x}\right):m\geq0\right\} $. \end{defn} \vphantom{} The following facts are easily verified directly from the above definitions: \begin{prop} Let $\Omega\subseteq\mathbb{Z}^{d}$ be a cycle of $\Omega$. Then: \vphantom{} I. $\Omega$ is a finite set. \vphantom{} II. Every element of $\Omega$ is a periodic point of $T$ of period $\left\|\Omega\right\|$. \vphantom{} III. $T\mid_{\Omega}$ (the restriction of $T$ to $\Omega$) is a bijection of $\Omega$. \vphantom{} IV. $\Omega\subseteq T^{-1}\left(\Omega\right)$ \vphantom{} V. Either there exists an $\mathbf{x}\in\mathbb{Z}^{d}\backslash\Omega$ so that $T\left(\mathbf{x}\right)\in\Omega$ or $T^{-1}\left(\Omega\right)=\Omega$. \end{prop} (IV) justifies the next bit of terminology: \begin{defn} Given a cycle $\Omega$ of $T$, we say $\Omega$ is \textbf{attracting }if $T^{-1}\left(\Omega\right)\backslash\Omega$ is non-empty\footnote{That is, if there are points not in $\Omega$ which $T$ sends into $\Omega$.}; we say $\Omega$ is \textbf{isolated }(or, in analogy with continuous dynamics, \textbf{repelling}) whenever $T^{-1}\left(\Omega\right)=\Omega$. Additionally, we say $\mathbf{x}\in\mathbb{Z}^{d}$ is a \textbf{pre-periodic point }of $T$ whenever there is an $n\geq0$ so that $T^{\circ n}\left(\mathbf{x}\right)$ is a periodic point of $T$ (where, recall, $T^{\circ0}$ is defined to be the identity map). Given a cycle $\Omega$, we say $\mathbf{x}$ is \textbf{pre-periodic into $\Omega$ }if $T^{\circ n}\left(\mathbf{x}\right)\in\Omega$ occurs for some $n\geq0$. \end{defn} \begin{prop} Let $\mathbf{x}\in\mathbb{Z}^{d}$ be arbitrary. Then, either $\mathbf{x}$ is a pre-periodic point of $T$, or the forward orbit of $\mathbf{x}$ is unbounded. In particular, if $\mathbf{x}$ is not a pre-periodic point, then: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert T^{\circ n}\left(\mathbf{x}\right)\right\Vert _{\infty}=+\infty\label{eq:Divergent Trajectory Definition} \end{equation} where for any $\mathbf{y}=\left(y_{1},\ldots,y_{d}\right)\in\mathbb{Z}^{d}$: \begin{equation} \left\Vert \mathbf{y}\right\Vert _{\infty}\overset{\textrm{def}}{=}\max\left\{ \left\|y_{1}\right\|,\ldots,\left\|y_{d}\right\|\right\} \label{eq:Definition of ell infinity norm on Z^d} \end{equation} \end{prop} Proof: If $\mathbf{x}$ is not pre-periodic, then each element in the forward orbit of $\mathbf{x}$ is unique; that is: for any integers $m,n\geq0$, we have that $T^{\circ n}\left(\mathbf{x}\right)=T^{\circ m}\left(\mathbf{x}\right)$ occurs if and only if $m=n$. So, letting $N>0$ be arbitrary, note that there are only finitely many $\mathbf{y}\in\mathbb{Z}^{d}$ with $\left\Vert \mathbf{y}\right\Vert _{\infty}\leq N$. As such, if $\mathbf{x}$ is not pre-periodic, there can exist at most finitely many $n\geq0$ for which $\left\Vert T^{\circ n}\left(\mathbf{x}\right)\right\Vert _{\infty}\leq N$. So, for any $N$, $\left\Vert T^{\circ n}\left(\mathbf{x}\right)\right\Vert _{\infty}$ will be strictly greater than $N$ for all sufficiently large $n$. This establishes the limit (\ref{eq:Divergent Trajectory Definition}). Q.E.D. \begin{defn} We say $\mathbf{x}$ is a \textbf{divergent point }of $T$, or is \textbf{divergent under $T$ }whenever the limit (\ref{eq:Divergent Trajectory Definition})\textbf{ }occurs. We use the terms \textbf{divergent trajectory}\index{divergent!trajectory}\textbf{ }and\textbf{ divergent orbit }to refer to a set of the form $\left\{ T^{\circ n}\left(\mathbf{x}\right):n\geq0\right\} $ where $\mathbf{x}$ is divergent under $T$. \end{defn} \begin{defn}[\textbf{Types of orbit classes}] Let $V\subseteq\mathbb{Z}^{d}$ be an orbit class of $T$. We say $V$ is: \vphantom{} I. \textbf{attracting},\textbf{ }whenever it contains an attracting cycle of $T$. \vphantom{} II. \textbf{isolated}, whenever it contains an isolated cycle of $T$. \vphantom{} III. \index{divergent!orbit class}\textbf{divergent}, whenever it contains a divergent trajectory of $T$. \end{defn} \vphantom{} It is a straight-forward and useful exercise to prove the following: \begin{lem} If $V\subseteq\mathbb{Z}^{d}$ is an irreducible orbit class of $T$, then one and only one of the following occurs: \vphantom{} I. $V$ is attracting. Additionally, there exists a unique attracting cycle $\Omega\subset V$ so that every $\mathbf{x}\in V$ is pre-periodic into $\Omega$. \vphantom{} II. $V$ is isolated. Additionally, $V$ itself is then an isolated cycle of $T$. \vphantom{} III. $V$ is divergent. Additionally, $\lim_{n\rightarrow\infty}\left\Vert T^{\circ n}\left(\mathbf{x}\right)\right\Vert _{\infty}=+\infty$ for all $\mathbf{x}\in V$. \end{lem} \vphantom{} Since the irreducible orbit classes of $T$ partition $\mathbb{Z}^{d}$, they furnish a complete characterization of $T$'s dynamics on $\mathbb{Z}^{d}$. \begin{thm} \label{thm:orbit classes partition domain}Let $T:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ be a map. Then, the irreducible orbit classes constitute a partition of $\mathbb{Z}^{d}$ into at most countably infinitely many pair-wise disjoint sets. In particular, for any $\mathbf{x}\in\mathbb{Z}^{d}$, exactly one of the following occurs: \vphantom{} I. $\mathbf{x}$ is a periodic point of $T$ in an isolated cycle. \vphantom{} II. $\mathbf{x}$ is a pre-periodic point of $T$ which is either in or eventually iterated \emph{into }an attracting cycle of $T$. \vphantom{} III. $\mathbf{x}$ has a divergent trajectory under $T$. \end{thm} \newpage{} \section{\label{sec:1.3 Crash course in ultrametric analysis}$p$-adic Numbers and Ultrametric Analysis} THROUGHOUT THIS SECTION, $p$ DENOTES A PRIME NUMBER. \subsection{\label{subsec:1.3.1. The-p-adics-in-a-nutshell}The $p$-adics in a Nutshell} We owe the conception of the $p$-adic numbers to the mathematical work of\emph{ }Kurt Hensel\index{Hensel, Kurt} in 1897. Although there are many ways of defining these numbers, I find Hensel's own approach to be the most enlightening: the $p$-adics as \emph{power series in the ``variable'' $p$} \cite{Gouvea's introudction to p-adic numbers book}. In fact, many of the fundamental concepts associated with power series turn out to be the inspiration for much of the paradigm-shifting developments in number theory and algebraic geometry during the twentieth century. Likely the most important of these is the simple observation that, in general, a power series for an analytic function $f:\mathbb{C}\rightarrow\mathbb{C}$ about a given point $z_{0}\in\mathbb{C}$ gives us a formula for the function which is valid only in a \emph{neighborhood} of $z_{0}$. Quite often, we need to compute power series expansion about different points (a process known as \emph{analytic continuation}) in order to get formulae for the function on all the points in its domain. That is to say, the function itself is a \emph{global }object, which we study \emph{locally }(near the point $z_{0}$) by expanding it in a power series about a point. \begin{defn} The set of $p$-\textbf{adic integers}\index{$p$-adic!integers}, denoted \nomenclature{$\mathbb{Z}_{p}$}{the set of $p$-adic integers \nopageref}$\mathbb{Z}_{p}$, is the set of all (formal) sums $\mathfrak{z}$ of the form: \begin{equation} \mathfrak{z}=c_{0}+c_{1}p+c_{2}p^{2}+\ldots\label{eq:Definition of a p-adic integer} \end{equation} where the $c_{n}$s are elements of the set $\left\{ 0,1,\ldots,p-1\right\} $. \end{defn} \vphantom{} We can think of $p$-adic integers as ``power series in $p$''. Note that every non-negative integer is automatically a $p$-adic integer, seeing as every non-negative integer $x$ can be uniquely written as a \emph{finite }sum $\mathfrak{z}=\sum_{n=0}^{N}c_{n}p^{n}$, where the $c_{n}$s are the\index{$p$-adic!digits} $p$\textbf{-ary/adic digits }of $\mathfrak{z}$. The $p$-adic representation of $\mathfrak{z}\in\mathbb{Z}_{p}$ is the expression: \begin{equation} \mathfrak{z}=\centerdot_{p}c_{0}c_{1}c_{2}\ldots\label{eq:Definition of the p-adic digit representation} \end{equation} where the subscript $p$ is there to remind us that we are in base $p$; this subscript can (and will) be dropped when there is no confusion as to the value of $p$. $\mathbb{Z}_{p}$ becomes a commutative, unital ring when equipped with the usual addition and multiplication operations, albeit with the caveat that \begin{equation} cp^{n}=\left[c\right]_{p}p^{n}+\left(\frac{c-\left[c\right]_{p}}{p}\right)p^{n+1} \end{equation} for all $n,c\in\mathbb{N}_{0}$. \begin{example} As an example, for $a\in\left\{ 0,p-1\right\} $: \begin{equation} \centerdot_{p}00\left(p+a\right)=\left(p+a\right)p^{2}=ap^{2}+1p^{3}=\centerdot_{p}00a1 \end{equation} where, in $\centerdot_{p}00\left(p+a\right)$, $p+a$ is the value of the third $p$-adic digit, and where $\left[c\right]_{p}\in\left\{ 0,\ldots,p-1\right\} $ denotes the residue class of $c$ mod $p$. That is to say, like an odometer, \emph{we carry over to the next $p$-adic digit's place whenever a digit reaches $p$}. Thus, in the $2$-adics: \begin{eqnarray} 3 & = & 1\times2^{0}+1\times2^{1}=\centerdot_{2}11\\ 5 & = & 1\times2^{0}+0\times2^{1}+1\times2^{2}=\centerdot_{2}101 \end{eqnarray} When we add these numbers, we ``carry over the $2$'': \[ 3+5=\centerdot_{2}11+\centerdot_{2}101=\centerdot_{2}211=\centerdot_{2}021=\centerdot_{2}002=\centerdot_{2}0001=1\times2^{3}=8 \] Multiplication is done similarly. \end{example} \vphantom{} Using this arithmetic operation, we can write negative integers $p$-adically. \begin{example} The $p$-adic number $\mathfrak{y}$ whose every $p$-adic digit is equal to $p-1$: \[ \mathfrak{y}=\centerdot_{p}\left(p-1\right)\left(p-1\right)\left(p-1\right)\left(p-1\right)\ldots=\sum_{n=0}^{\infty}\left(p-1\right)p^{n} \] satisfies: \begin{eqnarray} 1+\mathfrak{y} & = & \centerdot_{p}\left(p\right)\left(p-1\right)\left(p-1\right)\left(p-1\right)\ldots\\ & = & \centerdot_{p}0\left(p-1+1\right)\left(p-1\right)\left(p-1\right)\ldots\\ & = & \centerdot_{p}0\left(p\right)\left(p-1\right)\left(p-1\right)\ldots\\ & = & \centerdot_{p}00\left(p-1+1\right)\left(p-1\right)\ldots\\ & = & \centerdot_{p}00\left(p-1+1\right)\left(p-1\right)\ldots\\ & = & \centerdot_{p}000\left(p-1+1\right)\ldots\\ & \vdots\\ & = & \centerdot_{p}000000\ldots\\ & = & 0 \end{eqnarray} and thus, $\mathfrak{y}=-1$ in $\mathbb{Z}_{p}$. \end{example} \vphantom{} The beauty of the power series conception of these numbers is that it makes such formulae explicit: \begin{eqnarray} 1+y & = & 1+\sum_{n=0}^{\infty}\left(p-1\right)p^{n}\\ & = & 1+\sum_{n=0}^{\infty}p^{n+1}-\sum_{n=0}^{\infty}p^{n}\\ & = & p^{0}+\sum_{n=1}^{\infty}p^{n}-\sum_{n=0}^{\infty}p^{n}\\ & = & \sum_{n=0}^{\infty}p^{n}-\sum_{n=0}^{\infty}p^{n}\\ & = & 0 \end{eqnarray} Consequently, just as when given a power series $\sum_{n=0}^{\infty}a_{n}z^{n}$, we can compute its multiplicative inverse $\sum_{n=0}^{\infty}b_{n}z^{n}$ recursively by way of the equations: \begin{eqnarray} 1 & = & \left(\sum_{m=0}^{\infty}a_{m}z^{m}\right)\left(\sum_{n=0}^{\infty}b_{n}z^{n}\right)=\sum_{k=0}^{\infty}\left(\sum_{n=0}^{k}a_{n}b_{k-n}\right)z^{n}\\ & \Updownarrow\\ 1 & = & a_{0}b_{0}\\ 0 & = & a_{0}b_{1}+a_{1}b_{0}\\ 0 & = & a_{0}b_{2}+a_{1}b_{1}+a_{2}b_{0}\\ & \vdots \end{eqnarray} we can use the same formula to compute the multiplicative inverse of a given $p$-adic integer\textemdash \emph{assuming} it exists. Working through the above equations, it can be seen that the $p$-adic integer $\mathfrak{z}=\sum_{n=0}^{\infty}c_{n}p^{n}$ has a multiplicative inverse if and only if $c_{0}\in\left\{ 0,\ldots,p-1\right\} $ is multiplicatively invertible mod $p$ (i.e., $c_{0}$ is co-prime to $p$). This is one of several reasons why we prefer to study $p$-adic integers for prime $p$: every non-zero residue class mod $p$ is multiplicatively invertible modulo $p$. \begin{defn} Just as we can go from the ring of power series to the field of Laurent series, we can pass from the ring of $p$-adic integers $\mathbb{Z}_{p}$ to the field of \textbf{$p$-adic (rational) numbers}\index{$p$-adic!rational numbers} \nomenclature{$\mathbb{Q}_{p}$}{the set of $p$-adic rational numbers \nopageref}$\mathbb{Q}_{p}$\textbf{ }by considering ``Laurent series'' in $p$. Every $\mathfrak{z}\in\mathbb{Q}_{p}$ has a unique representation as: \begin{equation} \mathfrak{z}=\sum_{n=n_{0}}^{\infty}c_{n}p^{n}\label{eq:Laurent series representation of a p-adic rational number} \end{equation} for some $n_{0}\in\mathbb{Z}$. We write the ``fractional part'' of $\mathfrak{z}$ (the terms with negative power of $p$) to the right of the $p$-adic point $\centerdot_{p}$. \end{defn} \begin{example} Thus, the $3$-adic number: \begin{equation} \frac{1}{3^{2}}+\frac{2}{3^{1}}+0\times3^{0}+1\times3^{1}+1\times3^{2}+1\times3^{3}+\ldots \end{equation} would be written as: \begin{equation} 12\centerdot_{3}0\overline{1}=12\centerdot_{3}0111\ldots \end{equation} where, as usual, the over-bar indicates that we keep writing $1$ over and over again forever. \end{example} \begin{defn} We can equip $\mathbb{Q}_{p}$ with the structure of a metric space by defining the \index{$p$-adic!valuation}\textbf{$p$-adic valuation} \textbf{$v_{p}$ }and $p$\textbf{-absolute value}\footnote{Although this text and most texts on non-archimedean analysis by native-English-speaking authors make a clear distinction between the $p$-adic absolute value and the $p$-adic valuation, this distinction is not universal adhered to. A non-negligible proportion of the literature (particularly when authors of Russian extraction are involved) use the word ``valuation'' interchangeably, to refer to either $v_{p}$ or $\left\|\cdot\right\|_{p}$, depending on the context.}\textbf{ }$\left\|\cdot\right\|_{p}$: \begin{equation} v_{p}\left(\sum_{n=n_{0}}^{\infty}c_{n}p^{n}\right)\overset{\textrm{def}}{=}n_{0}\label{eq:Definition of the p-adic valuation} \end{equation} \begin{equation} \left\|\sum_{n=n_{0}}^{\infty}c_{n}p^{n}\right\|_{p}\overset{\textrm{def}}{=}p^{-n_{0}}\label{eq:Definition of the p-adic absolute value} \end{equation} that is, $\left\|\mathfrak{z}\right\|_{p}=p^{-v_{p}\left(\mathfrak{z}\right)}$. We also adopt the convention that $v_{p}\left(0\right)\overset{\textrm{def}}{=}\infty$. The \textbf{$p$-adic metric}\index{p-adic@\textbf{$p$}-adic!metric}\textbf{ }is then the distance formula defined by the map: \begin{equation} \left(\mathfrak{z},\mathfrak{y}\right)\in\mathbb{Z}_{p}\times\mathbb{Z}_{p}\mapsto\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}\label{eq:Definition of the p-adic metric} \end{equation} This metric (also known as an \textbf{ultrametric})\textbf{ }is said to \textbf{non-archimedean} because, in addition to the triangle inequality we all know and love, it also satisfies the \textbf{strong triangle inequality} (also called the \textbf{ultrametric inequality}): \begin{equation} \left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}\leq\max\left\{ \left\|\mathfrak{z}\right\|_{p},\left\|\mathfrak{y}\right\|_{p}\right\} \label{eq:the Strong Triangle Inequality} \end{equation} \emph{Crucially}\textemdash and I \emph{cannot} emphasize this enough\textemdash (\ref{eq:the Strong Triangle Inequality}) holds \emph{with equality} whenever $\left\|\mathfrak{z}\right\|_{p}\neq\left\|\mathfrak{y}\right\|_{p}$. The equality of (\ref{eq:the Strong Triangle Inequality}) when $\left\|\mathfrak{z}\right\|_{p}\neq\left\|\mathfrak{y}\right\|_{p}$ is one of the most subtly powerful tricks in ultrametric analysis, especially when we are trying to contradict an assumed upper bound. Indeed, this very method is at the heart of my proof of th $\left(p,q\right)$-adic Wiener Tauberian Theorem. \end{defn} \begin{rem} Although it might seem unintuitive that a \emph{large} power of $p$ should have a very small $p$-adic absolute value, this viewpoint becomes extremely natural when viewed through Hensel's original conception of $p$-adic integers as number theoretic analogues of power series \cite{Hensel-s original article,Gouvea's introudction to p-adic numbers book,Journey throughout the history of p-adic numbers}. It is a well-known and fundamental fact of complex analysis that a function $f:U\rightarrow\mathbb{C}$ holomorphic on an open, connected, non-empty set $U\subseteq\mathbb{C}$ which possesses a zero of infinite degree in $U$ must be identically zero on $U$. In algebraic terms, if $f$ has a zero at $z_{0}$ and $f$ is not identically zero, then $f\left(z\right)/\left(z-z_{0}\right)^{n}$ can only be divided by the binomial $z-z_{0}$ finitely many times before we obtain a function which is non-zero at $z_{0}$. As such, a function holomorphic on an open neighborhood of $z_{0}$ which remains holomorphic after being divided by $z-z_{0}$ arbitrarily many times is necessarily the constant function $0$. Hensel's insight was to apply this same reasoning to numbers. Indeed, the uniform convergence to $0$ of the function sequence $\left\{ z^{n}\right\} _{n\geq0}$ on any compact subset of the open unit disk $\mathbb{D}\subset\mathbb{C}$ is spiritually equivalent to the convergence of the sequence $\left\{ p^{n}\right\} _{n\geq0}$ to $0$ in the $p$-adics: just as the only holomorphic function on $\mathbb{D}$ divisible by arbitrarily large powers of $z$ is the zero function, the only $p$-adic integer divisible by arbitrarily high powers of $p$ is the integer $0$. Indeed, in the context of power series and Laurent series\textemdash say, a series $\sum_{n=n_{0}}^{\infty}c_{n}\left(z-z_{0}\right)^{n}$ about some $z_{0}\in\mathbb{C}$\textemdash the $p$-adic valuation corresponds to the \textbf{zero degree }of the series at $z_{0}$. If $n_{0}$ is negative, then the function represented by that power series has a pole of order $-n_{0}$ at $z_{0}$; if $n_{0}$ is positive, then the function represented by that power series has a \emph{zero }of order $n_{0}$ at $z_{0}$. Thus, for all $n\in\mathbb{Z}$: \begin{equation} \left\|p^{n}\right\|_{p}=p^{-n} \end{equation} This is especially important, seeing as the $p$-adic absolute value is a multiplicative group homomorphism from $\mathbb{Z}_{p}$ to $\mathbb{R}^{+}$: \begin{equation} \left\|xy\right\|_{p}=\left\|x\right\|_{p}\left\|y\right\|_{p}\label{eq:Multiplicativity of p-adic absolute value} \end{equation} as such, the level sets of $\mathbb{Z}_{p}$ are: \begin{equation} \left\{ x\in\mathbb{Z}_{p}:\left\|x\right\|_{p}=p^{-n}\right\} =p^{n}\mathbb{Z}_{p}\overset{\textrm{def}}{=}\left\{ p^{n}y:y\in\mathbb{Z}_{p}\right\} \label{eq:Definition of p^n times Z_p} \end{equation} for all $n\in\mathbb{N}_{0}$. \end{rem} \vphantom{} Returning to the matter at hand, by using absolute values, we see that the $p$-adic integers are precisely those elements of $\mathbb{Q}_{p}$ with $p$-adic absolute value $\leq1$. Consequentially, the metric space obtained by equipping $\mathbb{Z}_{p}$ with the $p$-adic metric is \emph{compact}. $\mathbb{Q}_{p}$, meanwhile, is locally compact, just like $\mathbb{R}$ and $\mathbb{C}$. \begin{defn} We write \nomenclature{$\mathbb{Z}_{p}^{\times}$}{the group of multiplicatively invertible $p$-adic integers \nopageref}$\mathbb{Z}_{p}^{\times}$ to denote the set of all units of $\mathbb{Z}_{p}$\textemdash that is, elements of $\mathbb{Z}_{p}$ whose reciprocals are contained in $\mathbb{Z}_{p}$. This is an abelian group under multiplication, with $1$ as its identity element. Note, also, that: \begin{equation} \mathbb{Z}_{p}^{\times}=\left\{ \mathfrak{z}\in\mathbb{Z}_{p}:\left\|\mathfrak{z}\right\|_{p}=1\right\} \end{equation} \end{defn} \begin{defn} \textbf{Congruences }are extremely important when working with $p$-adic integers; this is an intrinsic feature of the ``projective limit'' more algebraic texts frequently use to define the $p$-adic integers. As we saw, every $p$-adic integer $\mathfrak{z}$ can be written uniquely as: \begin{equation} \mathfrak{z}=\sum_{n=0}^{\infty}c_{n}p^{n}\label{eq:p-adic series representation of a p-adic integer} \end{equation} for constants $\left\{ c_{n}\right\} _{n\geq0}\subseteq\left\{ 0,\ldots,p-1\right\} $. The series representation (\ref{eq:p-adic series representation of a p-adic integer}) of a $p$-adic integer $\mathfrak{z}$is sometimes called the \textbf{Hensel series}\index{Hensel series}\index{series!Hensel}series\textbf{ }or \textbf{Henselian series }of $\mathfrak{z}$; Amice uses this terminology, for example \cite{Amice}. With this representation, given any integer $m\geq0$, we can then define \nomenclature{$\left[\mathfrak{z}\right]_{p^{n}}$}{Projection of a $p$-adic integer modulo $p^n$ nomnorefpage} $\left[\mathfrak{z}\right]_{p^{m}}$, the \textbf{projection of $\mathfrak{z}$ mod $p^{m}$} like so: \begin{equation} \left[\mathfrak{z}\right]_{p^{m}}\overset{\textrm{def}}{=}\sum_{n=0}^{m-1}c_{n}p^{n}\label{eq:Definition of the projection of z mod p to the m} \end{equation} where the right-hand side is defined to be $0$ whenever $m=0$. Since $\left[\mathfrak{z}\right]_{p^{m}}$ is a finite sum of integers, it itself is an integer. Given $\mathfrak{z},\mathfrak{y}\in\mathbb{Z}_{p}$, we write $\mathfrak{z}\overset{p^{m}}{\equiv}\mathfrak{y}$ if and only if $\left[\mathfrak{z}\right]_{p^{m}}=\left[\mathfrak{y}\right]_{p^{m}}$. Moreover, there is an equivalence between congruences and absolute values: \begin{equation} \mathfrak{z}\overset{p^{m}}{\equiv}\mathfrak{y}\Leftrightarrow\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}\leq p^{-m} \end{equation} There is a very useful notation for denoting subsets (really, ``open neighborhoods'') of $p$-adic integers. Given $\mathfrak{z}\in\mathbb{Z}_{p}$ and $n\in\mathbb{N}_{0}$, we write: \begin{equation} \mathfrak{z}+p^{n}\mathbb{Z}_{p}\overset{\textrm{def}}{=}\left\{ \mathfrak{y}\in\mathbb{Z}_{p}:\mathfrak{y}\overset{p^{n}}{\equiv}\mathfrak{z}\right\} +\left\{ \mathfrak{y}\in\mathbb{Z}_{p}:\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}\leq p^{-n}\right\} \label{eq:Definition of co-set notation for p-adic neighborhoods} \end{equation} In particular, note that: \begin{align} \mathfrak{z}+p^{n}\mathbb{Z}_{p} & =\mathfrak{y}+p^{n}\mathbb{Z}_{p}\\ & \Updownarrow\\ \mathfrak{z} & \overset{p^{n}}{\equiv}\mathfrak{y} \end{align} Additionally\textemdash as is crucial for performing integration over the $p$-adics\textemdash for any $n\geq0$, we can partition $\mathbb{Z}_{p}$ like so: \begin{equation} \mathbb{Z}_{p}=\bigcup_{k=0}^{p^{n}-1}\left(k+p^{n}\mathbb{Z}_{p}\right) \end{equation} where the $k+p^{n}\mathbb{Z}_{p}$s are pair-wise disjoint with respect to $k$. \end{defn} \vphantom{} Finally, it is worth mentioning that $\mathbb{N}_{0}$ is dense in $\mathbb{Z}_{p}$, and that $\mathbb{Q}$ is dense in $\mathbb{Q}_{p}$. This makes both $\mathbb{Z}_{p}$ and $\mathbb{Q}_{p}$ into separable topological spaces. Moreover, as topological spaces, they\textemdash and any field extension thereof\textemdash are \emph{totally disconnected}. This has profound implications for $p$-adic analysis, and for ultrametric analysis in general. \subsection{\label{subsec:1.3.2. Ultrametrics-and-Absolute}An Introduction to Ultrametric Analysis} While the $p$-adic numbers are surely the most well-known non-archimedean spaces, they are far from the only ones. The study of function theory, calculus, and the like on generic non-archimedean spaces is sometimes called \textbf{Ultrametric analysis}\index{ultrametric!analysis}. As will be addressed at length in the historical essay of Subsection \ref{subsec:3.1.1 Some-Historical-and}, the sub-disciplines that go by the names of non-archimedean analysis, $p$-adic analysis, ultrametric analysis, and the like are sufficiently divers that it is worth being aware of the undercurrent of common terminology. \begin{defn} Let $X$ be a set, and let $d:X\times X\rightarrow\left[0,\infty\right)$ be a metric; that is, a function satisfying: \vphantom{} I. $d\left(x,y\right)\geq0$ $\forall x,y\in X$, with equality if and only if $x=y$; \vphantom{} II. $d\left(x,y\right)=d\left(y,x\right)$ $\forall x,y\in X$; \vphantom{} III. $d\left(x,y\right)\leq d\left(x,z\right)+d\left(y,z\right)$ $\forall x,y,z\in X$. \vphantom{} We say $d$ is a \textbf{non-archimedean} \textbf{metric }or \textbf{ultrametric} whenever it satisfies the \textbf{Strong Triangle Inequality }(a.k.a. \textbf{Ultrametric Inequality}): \begin{equation} d\left(x,y\right)\leq\max\left\{ d\left(x,z\right),d\left(y,z\right)\right\} ,\forall x,y,z\in X\label{eq:Generic Strong Triangle Inequality} \end{equation} Like with the specific case of the $p$-adic ultrametric inequality, the general ultrametric inequality holds \emph{with equality} whenever $d\left(x,z\right)\neq d\left(y,z\right)$. An \index{ultrametric!space}\textbf{ultrametric space }is a pair $\left(X,d\right)$, where $X$ is a set and $d$ is an ultrametric on $X$. \end{defn} \vphantom{} The most important ultrametrics are those that arise from \textbf{absolute values} on abelian groups, particularly fields. \begin{defn} Let $K$ be an abelian group, written additively, and with $0$ as its identity element. An \textbf{absolute value}\index{absolute value (on a field)}\textbf{ }on $K$ is a function $\left\|\cdot\right\|_{K}:K\rightarrow\left[0,\infty\right)$ satisfying the following properties for all $x,y\in K$: \vphantom{} I. $\left\|0\right\|_{K}=0$; \vphantom{} II. $\left\|x+y\right\|_{K}\leq\left\|x\right\|_{K}+\left\|y\right\|_{K}$; \vphantom{} III. $\left\|x\right\|_{K}=0$ if and only if $x=0$; \vphantom{} IV. If $K$ is a ring, we also require $\left\|x\cdot y\right\|_{K}=\left\|x\right\|_{K}\cdot\left\|y\right\|_{K}$. \vphantom{} Finally, we say that $\left\|\cdot\right\|_{K}$ is a \textbf{non-archimedean absolute value} if, in addition to the above, $\left\|\cdot\right\|_{K}$ satisfies the \textbf{Ultrametric Inequality}\index{ultrametric!inequality}\index{triangle inequality!strong}: \vphantom{} V. $\left\|x+y\right\|_{K}\leq\max\left\{ \left\|x\right\|_{K},\left\|y\right\|_{K}\right\} $ (with equality whenever $\left\|x\right\|_{K}\neq\left\|y\right\|_{K}$),. \vphantom{} If (V) is not satisfied, we call $\left\|\cdot\right\|_{K}$ an \textbf{archimedean absolute value}. Finally, note that if $K$ is a field, any absolute value $\left\|\cdot\right\|_{K}$ on a field induces a metric $d$ on $K$ by way of the formula $d\left(x,y\right)=\left\|x-y\right\|_{K}$. We call the pair $\left(K,\left\|\cdot\right\|_{K}\right)$ a \textbf{valued group }(resp. \textbf{valued ring}; resp. \index{valued field}\textbf{valued field}) whenever $K$ is an abelian group (resp. ring\footnote{The ring need not be commutative.}; resp., field), $\left\|\cdot\right\|_{K}$ is an absolute value on and say it is \textbf{archimedean }whenever $\left\|\cdot\right\|_{K}$ is archimedean, and say that it is \textbf{non-archimedean }whenever $\left\|\cdot\right\|_{K}$. Let $K$ be a non-archimedean valued ring. \vphantom{} I. Following Schikhof \cite{Schikhof Banach Space Paper}, let $B_{K}\overset{\textrm{def}}{=}\left\{ x\in K:\left\|x\right\|_{K}\leq1\right\} $ and let $B_{K}^{-}\overset{\textrm{def}}{=}\left\{ x\in K:\left\|x\right\|_{K}<1\right\} $. Both $B_{K}$ and $B_{K}^{-}$ are rings under the addition and multiplication operations of $K$, with $B_{K}^{-}$ being an ideal\footnote{In fact, $B_{K}^{-}$ is a maximal ideal in $B_{K}$, and is the unique non-zero prime ideal of $B_{K}$. If $K$ is a ring, $K$ is then called a \textbf{local ring}; if $K$ is a field, it is then called a \textbf{local field}, in which case $B_{K}$ is the ring of $K$-integers, and $K$ is the field of fractions of $B_{K}$.} in $B_{K}$. The ring $B_{K}/B_{K}^{-}$ obtained by quotienting $B_{K}$ out by $B_{K}^{-}$ is called the \textbf{residue field }/ \textbf{residue class field }of $K$. We say $K$ is\textbf{ $p$-adic} (where $p$ is a prime) when the residue field of $K$ has characteristic $p$. In the case $K$ is $p$-adic, in an abuse of notation, we will write $\left\|\cdot\right\|_{p}$ to denote the absolute value on $K$. Also, note that if $K$ is a field, $B_{K}$ is then equal to $\mathcal{O}_{K}$, the \textbf{ring of integers }of $K$. \vphantom{} II. The set: \begin{equation} \left\|K\backslash\left\{ 0\right\} \right\|_{K}\overset{\textrm{def}}{=}\left\{ \left\|x\right\|_{K}:x\in K\backslash\left\{ 0\right\} \right\} \label{eq:Definition of the value group of a field} \end{equation} is called the \index{value group}\textbf{value group}\footnote{Not to be confused with a \emph{valued }group.}\textbf{ }of $K$. This group is said to be \textbf{dense }if it is dense in the interval $\left(0,\infty\right)\subset\mathbb{R}$ in the standard topology of $\mathbb{R}$, and is said to be \textbf{discrete }if it is not dense. \end{defn} \vphantom{} Both the residue field and the value group of $K$ play a crucial role in some of the fundamental properties of analysis on $K$ (or on normed vector spaces over $K$). For example, they completely determine whether or not $K$ is locally compact. \begin{thm} A non-archimedean valued field $K$ is locally compact if and only if its residue field is finite and its value group is discrete\footnote{\textbf{Theorem 12.2} from \cite{Ultrametric Calculus}.}. \end{thm} \vphantom{} The term ``non-archimedean'' comes from the failure of the \textbf{archimedean property} of classical analysis, that being the intuitive notion that \emph{lengths add up}. Non-archimedean spaces are precisely those metric spaces where lengths \emph{need not} add up. In particular, we have the following delightful result: \begin{thm} Let $\left(K,\left\|\cdot\right\|_{K}\right)$ be a valued field, and let $1$ denote the multiplicative identity element of $K$. Then, $\left(K,\left\|\cdot\right\|_{K}\right)$ is non-archimedean if and only if: \begin{equation} \left\|1+1\right\|_{K}\leq\left\|1\right\|_{K} \end{equation} \end{thm} Proof: Exercise. Q.E.D. \vphantom{} The following list, adapted from Robert's book \cite{Robert's Book}, gives an excellent summary of the most important features of ultrametric analysis. \begin{fact}[\textbf{The Basic Principles of Ultrametric Analysis}] \label{fact:Principles of Ultrametric Analysis}Let $\left(K,\left\|\cdot\right\|_{K}\right)$ be a non-archimedean valued group with additive identity element $0$. Then: \vphantom{} I. \textbf{\emph{The Strongest Wins}}\emph{:} \vphantom{} \begin{equation} \left\|x\right\|_{K}>\left\|y\right\|_{K}\Rightarrow\left\|x+y\right\|_{K}=\left\|x\right\|_{K}\label{eq:The Strongest Wins} \end{equation} II. \textbf{\emph{Equilibrium}}\emph{:} All triangles are isosceles (or equilateral): \begin{equation} a+b+c=0\textrm{ \& }\left\|c\right\|_{K}<\left\|b\right\|_{K}\Rightarrow\left\|a\right\|_{K}=\left\|b\right\|_{K}\label{eq:Equilibrium} \end{equation} \vphantom{} III. \textbf{\emph{Competition}}\emph{:} If: \[ a_{1}+\cdots+a_{n}=0 \] then there are distinct $i,j$ so that $\left\|a_{i}\right\|_{K}=\left\|a_{j}\right\|_{K}=\max_{k}\left\|a_{k}\right\|_{K}$. \vphantom{} IV. \textbf{\emph{The Freshman's Dream}}\emph{:} If the metric space $\left(K,\left\|\cdot\right\|_{K}\right)$ is complete, a series $\sum_{n=0}^{\infty}a_{n}$ converges in $K$ if and only if $\left\|a_{n}\right\|_{K}\rightarrow0$ as $n\rightarrow\infty$. Consequently: \vphantom{} i. (The infinite version of (III) holds) $\sum_{n=0}^{\infty}a_{n}=0$ implies there are distinct $i,j$ so that $\left\|a_{i}\right\|_{K}=\left\|a_{j}\right\|_{K}=\max_{k}\left\|a_{k}\right\|_{K}$. \vphantom{} ii. The convergence of $\sum_{n=0}^{\infty}\left\|a_{n}\right\|_{K}$ in $\mathbb{R}$ implies the convergence of $\sum_{n=0}^{\infty}a_{n}$ in $K$, but the converse is not true: $\sum_{n=0}^{\infty}a_{n}$ converging in $K$ need not imply $\sum_{n=0}^{\infty}\left\|a_{n}\right\|_{K}$ converges in $\mathbb{R}$. \vphantom{} iii. For any $K$-convergent series $\sum_{n=0}^{\infty}a_{n}$: \begin{equation} \left\|\sum_{n=0}^{\infty}a_{n}\right\|_{K}\leq\sup_{n\geq0}\left\|a_{n}\right\|_{K}=\max_{n\geq0}\left\|a_{n}\right\|_{K}\label{eq:Series Estimate} \end{equation} The right-most equality indicates that there is going to be an $n$ for which the absolute value is maximized. \vphantom{} V. \textbf{\emph{The Sophomore's Dream}}\emph{:} A sequence $\left\{ a_{n}\right\} _{n\geq0}$ is Cauchy if and only if $\left\|a_{n+1}-a_{n}\right\|_{K}\rightarrow0$ as $n\rightarrow\infty$. \vphantom{} \vphantom{} VI. \textbf{\emph{Stationarity of the absolute value}}\emph{:} If $a_{n}$ converges to $a$ in $K$, and if $a\neq0$, then there is an $N$ so that $\left\|a_{n}\right\|_{K}=\left\|a\right\|_{K}$ for all $n\geq N$. \end{fact} \vphantom{} We also have the following results regarding infinite series: \begin{prop}[\index{series!re-arrangement}\textbf{Series re-arrangement}\footnote{Given on page 74 of \cite{Robert's Book}.}] \label{prop:series re-arrangement}Let $\left(K,\left\|\cdot\right\|_{K}\right)$ be a complete non-archimedean valued group, and let $\left\{ a_{n}\right\} _{n\geq0}$ be a sequence in $K$ which tends to $0$ in $K$, so that $\sum_{n=0}^{\infty}a_{n}$ converges in $K$ to $s$. Then, no matter how the terms of the sum are grouped or rearranged, the resultant series will still converge in $K$ to $s$. Specifically: \vphantom{} I. For any bijection $\sigma:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$, $\sum_{n=0}^{\infty}a_{\sigma\left(n\right)}$ converges in $K$ to $s$. \vphantom{} II. For any partition of $\mathbb{N}_{0}$ into sets $I_{1},I_{2},\ldots$, the series: \begin{equation} \sum_{k}\left(\sum_{n\in I_{k}}a_{n}\right) \end{equation} converges in $K$ to $s$. \end{prop} \begin{prop}[\index{series!interchange}\textbf{Series interchange}\footnote{Given on page 76 of \cite{Robert's Book}.}] \label{prop:series interchange} Let $\left(K,\left\|\cdot\right\|_{K}\right)$ be a complete non-archimedean valued group, and let $\left\{ a_{m,n}\right\} _{m,n\geq0}$ be a double-indexed sequence in $K$. If, for any $\epsilon>0$, there are only finitely many pairs $\left(m,n\right)$ so that $\left\|a_{m,n}\right\|_{K}>\epsilon$, then the double sum: \begin{equation} \sum_{\left(m,n\right)\in\mathbb{N}_{0}^{2}}a_{m,n} \end{equation} converges in $K$, and, moreover: \begin{equation} \sum_{\left(m,n\right)\in\mathbb{N}_{0}^{2}}a_{m,n}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{\infty}a_{m,n}\right)=\sum_{n=0}^{\infty}\left(\sum_{m=0}^{\infty}a_{m,n}\right) \end{equation} where all equalities are in $K$. \end{prop} \vphantom{} The topological properties of ultrametric spaces are drastically different from Euclidean spaces (ex: $\mathbb{R}^{n}$). \begin{defn} Let $\left(X,d\right)$ be an ultrametric space. \vphantom{} I. A closed ball\index{ball} in $X$ of radius $r$ (where $r$ is a positive real number) centered at $x\in X$, written $B\left(x,r\right)$, is the set: \begin{equation} B\left(x,r\right)\overset{\textrm{def}}{=}\left\{ y\in X:d\left(x,y\right)\leq r\right\} \label{eq:Definition of a closed ball} \end{equation} Open balls are obtained by making the inequality $\leq r$ strict ($<r$). However, as we will see momentarily, this doesn't actually amount to much of a distinction. \vphantom{} II. Given any non-empty subset $Y\subseteq X$, the \textbf{diameter }of $Y$, denoted $d\left(Y\right)$, is defined by: \begin{equation} d\left(Y\right)\overset{\textrm{def}}{=}\sup\left\{ d\left(a,b\right):a,b\in Y\right\} \label{eq:Definition of the diameter of an ultrametric set} \end{equation} \end{defn} \begin{rem} For any ball $B\left(x,r\right)\subseteq X$: \begin{equation} d\left(B\right)=\inf\left\{ r^{\prime}>0:B\left(x,r^{\prime}\right)=B\left(x,r\right)\right\} \label{eq:Diameter of an ultrametric ball in terms of its radius} \end{equation} \end{rem} \begin{prop} In\footnote{\textbf{Propositions 18.4} and \textbf{18.5} from \cite{Ultrametric Calculus}.} an ultrametric space\emph{ \index{ultrametric!balls}}$\left(X,d\right)$: \vphantom{} I. All open balls are closed, and all closed balls are open\footnote{However, it is not necessarily the case that $\left\{ y\in X:d\left(x,y\right)\leq r\right\} $ and $\left\{ y\in X:d\left(x,y\right)<r\right\} $ will be the same set. For example, if $X=\mathbb{Z}_{3}$, then $\left\{ \mathfrak{y}\in\mathbb{Z}_{3}:\left\|\mathfrak{z}-\mathfrak{y}\right\|_{3}\leq\frac{1}{2}\right\} =\left\{ \mathfrak{y}\in\mathbb{Z}_{3}:\left\|\mathfrak{z}-\mathfrak{y}\right\|_{3}\leq\frac{1}{3}\right\} $, because the value group of $\mathbb{Z}_{3}$ is $\left\{ 3^{-n}:n\in\mathbb{N}_{0}\right\} $. On the other hand, for the same reason, $\left\{ \mathfrak{y}\in\mathbb{Z}_{3}:\left\|\mathfrak{z}-\mathfrak{y}\right\|_{3}<\frac{1}{3}\right\} =\left\{ \mathfrak{y}\in\mathbb{Z}_{3}:\left\|\mathfrak{z}-\mathfrak{y}\right\|_{3}\leq\frac{1}{9}\right\} $.}. \vphantom{} II. All points inside a ball are at the center of the ball; that is, given $x,y\in K$ and $r>0$ such that $d\left(x,y\right)\leq r$, we have that $B\left(x,r\right)=B\left(y,r\right)$. \vphantom{} III. Given two balls $B_{1},B_{2}\subseteq X$, either $B_{1}\cap B_{2}=\varnothing$ or either $B_{1}\subseteq B_{2}$ or $B_{2}\subseteq B_{1}$. \vphantom{} IV. Given any ball $B\left(x,r\right)$, there are infinitely many real numbers $r^{\prime}$ for which $B\left(x,r^{\prime}\right)=B\left(x,r\right)$. \end{prop} \begin{fact} The \textbf{Heine-Borel Property}\footnote{A set is compact if and only if it is both closed and bounded.}\index{Heine-Borel property} also holds in $\mathbb{Q}_{p}$, as well as in any finite-dimensional, locally compact field extension thereof. \end{fact} \vphantom{} We also have the following result regarding open covers in an arbitrary ultrametric space: \begin{thm}[\textbf{Open Set Decomposition Theorem}\footnote{\textbf{Theorem 18.6} on page 48 of \cite{Ultrametric Calculus}.}] Let $X$ be an ultrametric space. Then, every non-empty open set $U\subseteq X$ can be written as the union of countably many pair-wise disjoint balls. \end{thm} \begin{cor} Let $X$ be a compact ultrametric space (such as $\mathbb{Z}_{p}$) can be written as the union of finitely many pair-wise disjoint balls. \end{cor} Proof: Let $U\subseteq\mathbb{Z}_{p}$ be non-empty and clopen. Since $U$ is clopen, it is closed, and since $U$ is in $\mathbb{Z}_{p}$, it is bounded in $\mathbb{Q}_{p}$. Since $\mathbb{Q}_{p}$ possess the Heine-Borel property, the closedness and boundedness of $U$ then force $U$ to be compact. Now, by the theorem, since $U$ is clopen and non-empty, it is open and non-empty, and as such, $U$ can be written as a union of countably many disjoint balls. Since the balls are clopen sets, this collection of balls forms an open cover for $U$. By the compactness of $U$, this open cover must contain a finite sub-cover, which shows that $U$ can, in fact, be written as the union of finitely many clopen balls. Q.E.D. \vphantom{} By this point, the reader might have noticed that this exposition of basic ultrametric analysis has yet to say a word about functions. This is intentional. Non-archimedean function theory is one of the central overarching concerns of this dissertation. Nevertheless, there are two important pieces of non-archimedean function theory I can introduce here and now. \begin{defn}[\textbf{Locally constant functions}\footnote{Taken from \cite{Ultrametric Calculus}.}] Let $X$ be an ultrametric space, and let $K$ be any valued field. We say a function $f:X\rightarrow K$ is \textbf{locally constant }if, for each $x\in X$ there is an open neighborhood $U\subseteq X$ containing $x$ so that $f$ is constant on $U\cap X$. \end{defn} \begin{example} The most important example we will be working with are functions $\mathbb{Z}_{p}\rightarrow K$ of the form: \begin{equation} \sum_{n=0}^{p^{N}-1}a_{n}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{equation} where the $a_{n}$s are scalars in $K$. Such a function's value at any given $\mathfrak{z}$ is entirely determined by the value of $\mathfrak{z}$ modulo $p^{N}$. \end{example} \vphantom{} Our second appetizer in non-archimedean function theory is, in all honesty, a matter of non-archimedean \emph{functional }theory. It is also one of several reasons why integration of $p$-adic-valued functions of one or more $p$-adic variables is ``cursed'' as the kids these days like to say. \begin{defn} Letting $C\left(\mathbb{Z}_{p},K\right)$ denote the space of continuous functions from $\mathbb{Z}_{p}$ to $K$, where $K$ is a metrically complete field extension of $\mathbb{Q}_{p}$. We say a linear functional $\varphi:C\left(\mathbb{Z}_{p},K\right)\rightarrow K$ is\index{translation invariance} \textbf{translation invariant }whenever $\varphi\left(\tau_{1}\left\{ f\right\} \right)=\varphi\left(f\right)$ for all $f\in C\left(\mathbb{Z}_{p},K\right)$ where: \begin{equation} \tau_{1}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}f\left(\mathfrak{z}+1\right) \end{equation} \end{defn} \begin{thm} Let $K$ be a metrically complete field extension of $\mathbb{Q}_{p}$. Then, the only translation invariant linear functional $\varphi:C\left(\mathbb{Z}_{p},K\right)\rightarrow K$ is the zero functional ($\varphi\left(f\right)=0$ for all $f$)\footnote{\textbf{Corollary 3 }on page 177 of Robert's book \cite{Robert's Book}.}. \end{thm} \subsection{\label{subsec:1.3.3 Field-Extensions-=00003D000026}Field Extensions and Spherical Completeness} THROUGHOUT THIS SUBSECTION, WE WORK WITH A FIXED PRIME $p$. \vphantom{} It is often said that students of mathematics will not truly appreciate the beauty of analytic functions until after they have endured the pathological menagerie of functions encountered in a typical introductory course in real analysis. In my experience, $p$-adic analysis does much the same, only for fields instead than functions. The fact that adjoining a square root of $-1$ to the real numbers results in a metrically complete algebraically closed field is one of the greatest miracles in all mathematics. Working with the $p$-adic numbers, on the other hand, gives one an appreciation of just how \emph{awful} field extensions can be. The problem begins before we even take our first step up the tower of $p$-adic field extensions. Anyone who has ever struggled with the concept of negative numbers should be thankful that we do not exist in a macroscopically $p$-adic universe. In real life, the existence of negative integers is equivalent to the fact that $\mathbb{Z}^{\times}$\textemdash the group of multiplicative;y invertible elements of $\mathbb{Z}$, a.k.a. $\left\{ 1,-1\right\} $\textemdash is isomorphic to the cyclic group $\mathbb{Z}/2\mathbb{Z}$. The isomorphism $\mathbb{Z}^{\times}\cong\mathbb{Z}/2\mathbb{Z}$ is also responsible for the existence of a meaningful ordering on $\mathbb{Z}$ given by $<$ and friends. Unfortunately, for any grade-school students who happen to live in a $p$-adic universe, things are much less simple, and this has grave implications for the study of field extensions of $\mathbb{Q}_{p}$. To see why, for a moment, let us proceed navely. Let $K$ be a finite-degree Galois extension\footnote{Recall, this means that the set: \[ \left\{ \mathfrak{y}\in K:\sigma\left(\mathfrak{y}\right)=\mathfrak{y},\textrm{ }\forall\sigma\in\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)\right\} \] is equal to $\mathbb{Q}_{p}$.} of $\mathbb{Q}_{p}$. Because we are interested in doing analysis, our first order of business is to understand how we might extend the $p$-adic absolute value on $\mathbb{Q}_{p}$ to include elements of $K$. To do this, observe that since any $\sigma\in\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)$ is a field automorphism of $K$, any candidate for a \emph{useful }absolute value $\left\|\cdot\right\|_{K}$ on $K$ ought to satisfy: \begin{equation} \left\|\sigma\left(\mathfrak{y}\right)\right\|_{K}=\left\|\mathfrak{y}\right\|_{K},\textrm{ }\forall\mathfrak{y}\in K \end{equation} Moreover, because $\mathbb{Q}_{p}$ is a subset of $K$, it is not unreasonable to require $\left\|\mathfrak{y}\right\|_{K}$ to equal $\left\|\mathfrak{y}\right\|_{p}$ for all $\mathfrak{y}\in K$ that happen to be elements of $\mathbb{Q}_{p}$. To that end, let us get out the ol' determinant: \begin{defn} Define $N_{K}:K\rightarrow\mathbb{Q}_{p}$ by: \begin{equation} N_{K}\left(\mathfrak{y}\right)\overset{\textrm{def}}{=}\prod_{\sigma\in\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)}\sigma\left(\mathfrak{y}\right)\label{eq:Definition of N_K} \end{equation} \end{defn} \begin{rem} $N_{K}$ is $\mathbb{Q}_{p}$-valued here precisely because $K$ is a Galois extension. \end{rem} \vphantom{} With this, we have that for all $\mathfrak{y}\in K$: \[ \left\|N_{K}\left(\mathfrak{y}\right)\right\|_{p}=\left\|N_{K}\left(\mathfrak{y}\right)\right\|_{K}=\left\|\prod_{\sigma\in\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)}\sigma\left(\mathfrak{y}\right)\right\|_{K}=\prod_{\sigma\in\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)}\left\|\sigma\left(\mathfrak{y}\right)\right\|_{K}=\left\|\mathfrak{y}\right\|_{K}^{d} \] where $d$ is the order of $\textrm{Gal}\left(K/\mathbb{Q}_{p}\right)$\textemdash or, equivalently (since $K$ is a Galois extension), the degree of $K$ over $\mathbb{Q}_{p}$. In this way, we can compute $\left\|\mathfrak{y}\right\|_{K}$ by taking the $d$th root of $\left\|N_{K}\left(\mathfrak{y}\right)\right\|_{p}$, which is already defined, seeing as $N_{K}\left(\mathfrak{y}\right)$ is an element of $\mathbb{Q}_{p}$. With a little work, making this argument rigorous leads to an extremely useful theorem: \begin{thm} \label{thm:Galois conjugates}Let $K$ be a Galois extension of $\mathbb{Q}_{p}$ of degree $d$, and let $\left\|\cdot\right\|_{p}$ be the $p$-adic absolute value. Then, the map: \begin{equation} \mathfrak{y}\mapsto\left\|N_{K}\left(\mathfrak{y}\right)\right\|_{p}^{1/d} \end{equation} defines the unique absolute value $\left\|\cdot\right\|_{K}$ on $K$ which extends the $p$-adic absolute value of $\mathbb{Q}_{p}$. \end{thm} \begin{rem} In fact, with a slight modification of $N_{K}$, one can show that this method gives the absolute value on $K$ for \emph{any }finite-degree extension $K$ of $\mathbb{Q}_{p}$, not only Galois extensions. Robert does this on page 95 of \cite{Robert's Book}. \end{rem} \begin{example} \label{exa:incorrect galois}As a sample application, consider $\mathbb{Q}_{5}$. Let $\xi$ denote a primitive fourth root of unity, which we adjoin to $\mathbb{Q}_{5}$. Then, since: \begin{align} \left(1+2\xi\right)\left(1+2\xi^{3}\right) & =1+2\xi^{3}+2\xi+4\xi^{4}\\ \left(\xi+\xi^{3}=-1-\xi^{2}\right); & =5+2\left(-1-\xi^{2}\right)\\ \left(\xi^{2}=-1\right); & =5 \end{align} and so, by \textbf{Theorem \ref{thm:Galois conjugates}}: \begin{equation} \left\|1+2\xi\right\|_{5}=\left\|1+2\xi^{3}\right\|_{5}=\sqrt{\left\|5\right\|_{5}}=\frac{1}{\sqrt{5}} \end{equation} Alas, this\emph{ }is \emph{wrong!} While this argument would be perfectly valid if our base field was $\mathbb{Q}$ or $\mathbb{R}$, it fails for $\mathbb{Q}_{5}$, due to the fact that $\xi$ was \emph{already an element of $\mathbb{Q}_{5}$.} \end{example} \vphantom{} With the sole exception of $\mathbb{Z}_{2}$, whose only roots of unity are $1$ and $-1$, given any odd prime $p$, $\mathbb{Z}_{p}$ contains more roots of unity than just $\left\{ 1,-1\right\} $, and this ends up complicating the study of field extensions. Fortunately, the situation isn't \emph{too }bad; it is not difficult to determine the roots of unity naturally contained in $\mathbb{Z}_{p}^{\times}$ (and hence, in $\mathbb{Z}_{p}$ and $\mathbb{Q}_{p}$): \begin{thm} When $p$ is an odd prime, $\mathbb{Z}_{p}^{\times}$ contains all $\left(p-1\right)$th roots of unity. $\mathbb{Z}_{2}^{\times}$, meanwhile, contains only second roots of unity ($-1$ and $1$). \end{thm} Proof: (Sketch) If $\mathfrak{z}\in\mathbb{Z}_{p}^{\times}$ satisfies $\mathfrak{z}^{n}=1$ for some $n\geq1$, it must also satisfy $\mathfrak{z}^{n}\overset{p}{\equiv}1$, so, the first $p$-adic digit of $\mathfrak{z}$ must be a primitive root of unity modulo $p$. Conversely, for each primitive root $u\in\left\{ 1,2,\ldots,p-1\right\} $ of unity modulo $p$, with a little work (specifically, \textbf{Hensel's Lemma }(see Section 6 of Chapter 1 of \cite{Robert's Book} for details)), we can construct a unique $\mathfrak{z}\in\mathbb{Z}_{p}^{\times}$ with $u$ as its first digit. Because the group of multiplicative units of $\mathbb{Z}/p\mathbb{Z}$ is isomorphic to $\mathbb{Z}/\left(p-1\right)\mathbb{Z}$, this then yields the theorem. Q.E.D. \vphantom{} Note that in order to compute the $p$-adic absolute value of an expression involving a native root of unity of $\mathbb{Z}_{p}$, we need to use said root's $p$-adic digit expansion. \begin{example} The roots of unity in $\mathbb{Z}_{7}$ are the $6$th roots of unity. Of these, note that only two are primitive. By the argument used to prove the previous theorem, note that these two roots of unity will have $3$ and $5$, respectively, as their first $7$-adic digits, seeing as those are the only two primitive roots of unity in $\mathbb{Z}/7\mathbb{Z}$. If we let $\xi$ and $\zeta$ denote the the $3$-as-first-digit and $5$-as-first-digit roots of unity, we then have that: \begin{equation} 2\xi+1\overset{7}{\equiv}2\cdot3+1\overset{7}{\equiv}0 \end{equation} and so, $\left\|2\xi+1\right\|_{7}\leq1/7$. By using \textbf{Hensel's Lemma} to compute more and more digits of $\xi$, one can determine the number of $0$s that occur at the start of $2\xi+1$'s $7$-adic expansion and thereby determine its $7$-adic absolute value. On the other hand: \[ 2\zeta+1\overset{7}{\equiv}2\cdot5+1\overset{7}{\equiv}4 \] so, $\left\|2\zeta+1\right\|_{7}=1$, because $2\zeta+1$ is a $7$-adic integer which is not congruent to $0$ modulo $7$. In addition to Hensel's Lemma, there is a more algebraic approach to computing the $7$-adic absolute values of these quantities. Since $\xi$ is a primitive $6$th root of unity, the other primitive root of unity (the one we denote by $\zeta$) must be $\xi^{5}$; $\xi^{2}$ and $\xi^{4}$ will be primitive $3$rd roots of unity, while $\xi^{3}$ will be the unique primitive square root of unity otherwise known as $-1$. As such, writing $\zeta$ as $\xi^{5}$, observe that: \begin{align} \left(2\xi+1\right)\left(2\xi^{5}+1\right) & =4\xi^{6}+2\xi+2\xi^{5}+1\\ & =4+2\left(\xi+\xi^{5}\right)+1\\ & =5+2\xi\left(1+\xi^{4}\right)\\ \left(\textrm{let }\omega=\xi^{2}\right); & =5+2\xi\left(1+\omega^{2}\right) \end{align} Because $\omega$ is a primitive $3$rd root of unity, $1+\omega^{2}=-\omega=-\xi^{2}$, and so: \begin{align} \left(2\xi+1\right)\left(2\xi^{5}+1\right) & =5+2\xi\left(1+\omega^{2}\right)\\ & =5+2\xi\left(-\xi^{2}\right)\\ & =5-2\xi^{3}\\ \left(\xi^{3}=-1\right); & =5+2\\ & =7 \end{align} Hence: \[ \left\|2\xi+1\right\|_{7}\underbrace{\left\|2\xi^{5}+1\right\|_{7}}_{=\left\|2\zeta+1\right\|_{7}=1}=\left\|7\right\|_{7}=\frac{1}{7} \] So, $\left\|2\xi+1\right\|_{7}$ is precisely $1/7$. What this shows in order to of said root in order to use complex exponential notation to write roots of unity for any given odd prime $p$, we need to specify the $p$-adic digits of $\xi_{p-1}\overset{\textrm{def}}{=}e^{2\pi i/\left(p-1\right)}$ so that we can then have a uniquely defined $p$-adic absolute value for any polynomial in $\xi_{p-1}$ with coefficients in $\mathbb{Q}_{p}$. \end{example} \vphantom{} In general, field extensions of $\mathbb{Q}_{p}$ are obtained in the usual way, by adjoining to $\mathbb{Q}_{p}$ the root of a polynomial with coefficients in $\mathbb{Q}_{p}$. Note that since every algebraic number $\alpha$ is, by definition, the root of a polynomial with coefficients in $\mathbb{Q}$, given any algebraic number $\alpha$, we can create a $p$-adic field containing $\alpha$ by adjoining $\alpha$ to $\mathbb{Q}_{p}$, seeing as $\mathbb{Q}_{p}$ contains $\mathbb{Q}$ as a subfield. As described above, the absolute value on $p$ extends with these extensions in the natural way, maintaining its multiplicativity and ultrametric properties in the extension. Thus, for example, in any extension $K$ of $\mathbb{Q}_{p}$, all roots of unity will have a $p$-adic absolute value of $1$. More generally, in any finite-degree extension $K$ of $\mathbb{Q}_{p}$, the $p$-adic absolute value of an arbitrary element of $K$ will be a number of the form $p^{r}$, where $r\in\mathbb{Z}\cup V$, where $V$ is a finite set whose elements are non-integer rational numbers. Far more problematic, however, is the issue of algebraic closure. \begin{defn} We write \nomenclature{$\overline{\mathbb{Q}}_{p}$}{$p$-adic algebraic numbers} $\overline{\mathbb{Q}}_{p}$, the \textbf{algebraic closure of $\mathbb{Q}_{p}$}, the field obtained by adjoining to $\mathbb{Q}_{p}$ the roots of every polynomial with coefficients in $\mathbb{Q}_{p}$.\index{algebraic closure} \end{defn} \vphantom{} It is not an overgeneralization to say that algebraic number theory is the study of the Galois group $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$, and that the continued interest of the subject hinges upon the fact that\textemdash unlike with $\mathbb{R}$\textemdash the algebraic closure of $\mathbb{Q}$ is not a finite dimensional extension of $\mathbb{Q}$. The same is true in the $p$-adic context: $\overline{\mathbb{Q}}_{p}$\textemdash the algebraic closure of $\mathbb{Q}_{p}$\textemdash is an infinite-dimensional extension of $\mathbb{Q}_{p}$. However, the parallels between the two algebraic closures run deeper still. Just as $\overline{\mathbb{Q}}$ is incomplete as a metric space with respect to the complex absolute value, so too is $\overline{\mathbb{Q}}_{p}$ incomplete as a metric spaces with respect to the extension of $\left\|\cdot\right\|_{p}$. Moreover, as infinite-dimensional extensions of their base fields, neither $\overline{\mathbb{Q}}$ nor $\overline{\mathbb{Q}}_{p}$ is locally compact. Just as taking the metric closure of $\overline{\mathbb{Q}}$ with respect to the usual absolute value yields $\mathbb{C}$, the field of complex numbers, taking the metric closure of $\overline{\mathbb{Q}}_{p}$ with respect to the $p$-adic absolute value yields the $p$-adic complex numbers: \begin{defn} We write \nomenclature{$\mathbb{C}_{p}$}{the set of $p$-adic complex numbers \nomnorefpage} $\mathbb{C}_{p}$ to denote the field of \index{$p$-adic!complex numbers}\textbf{$p$-adic complex numbers}, defined here as the metric closure of $\overline{\mathbb{Q}}_{p}$ \end{defn} \vphantom{}. One of the many reasons complex analysis is the best form of analysis is because $\mathbb{R}$ is locally compact and has $\mathbb{C}$ as a one-dimensional extension of $\mathbb{R}$; this then guarantees that $\mathbb{C}$ is also locally compact. Unfortunately, the same does not hold for $\mathbb{C}_{p}$; $\mathbb{C}_{p}$ is \emph{not }locally compact. Worse still, $\mathbb{C}_{p}$ lacks a property most analysts take for granted: \textbf{spherical completeness}. Ordinarily, an (ultra)metric space is complete if and only if any nested sequence of balls $B_{1}\supseteq B_{2}\supseteq\cdots$ whose diameters/radii tend to $0$, the intersection $\bigcap_{n=1}^{\infty}B_{n}$ is non-empty. In the real/complex world, removing the requirement that the diameters of the $B_{n}$s tend to $0$ would not affect the non-emptiness of the intersection. Bizarrely, this is \emph{not }always the case in an ultrametric space. \begin{defn} An ultrametric space $X$ is said to be \index{spherically complete}\textbf{spherically complete }whenever every nested sequence of balls in $X$ has a non-empty intersection. We say $X$ is \textbf{spherically incomplete }whenever it is not spherically complete. \end{defn} \begin{prop} Any locally compact field is spherically complete. Consequently, for any prime number $p$, $\mathbb{Q}_{p}$ is spherically complete, as is any \emph{finite dimensional }field extension of $\mathbb{Q}_{p}$. \end{prop} Proof: Compactness. Q.E.D. \vphantom{} Despite this, there exist non-locally compact fields which are spherically complete (see \cite{Robert's Book} for more details; Robert denotes the spherical completion of $\mathbb{C}_{p}$ by $\Omega_{p}$). \begin{thm} Let $K$ be a non-archimedean valued field. Then, $K$ is spherically complete whenever it is metrically complete, and has a discrete value group. In particular, all locally compact metrically complete valued fields (archimedean or not) are spherically complete\footnote{\textbf{Corollary 20.3} from \cite{Ultrametric Calculus}.}. \end{thm} As it regards $\mathbb{C}_{p}$, the key properties are as follows: \begin{thm} $\mathbb{C}_{p}$ is \textbf{not}\emph{ }spherically complete\emph{ \cite{Robert's Book}}. \end{thm} \begin{thm} $\mathbb{C}_{p}$ is not\emph{ }separable\footnote{That is, it contains no dense \emph{countable} subset.}\emph{ \cite{Robert's Book}}. \end{thm} \vphantom{} As will be mentioned in Subsection \ref{subsec:3.1.2 Banach-Spaces-over}, whether or not a non-archimedean field is spherically complete has significant impact on the nature of the Banach spaces which can be built over it. At present, it suffices to say that in most ``classical'' applications of $p$-adic analysis (especially to number theory and algebraic geometry), spherical incompleteness is considered an undesirable property, particularly because of its destructive interaction with non-archimedean analogues of the \textbf{Hahn-Banach Theorem}\footnote{The thrust of this theorem, recall, is that we can take a continuous linear functional $\phi$ defined on a subspace $\mathcal{Y}$ of a given Banach space $\mathcal{X}$ and extend $\phi$ to a continuous linear functional on $\mathcal{X}$ itself.}\textbf{ }of functional analysis \cite{Robert's Book}. Despite this, the spherical \emph{incompleteness} of the fields in which our $\left(p,q\right)$-adic functions will take their values plays a pivotal role in our work, because of the effect this spherical incompleteness has upon the Fourier analytic properties of our functions. See, for instance, \textbf{Theorem \ref{thm:FST is an iso from measures to ell infinity}} on page \pageref{thm:FST is an iso from measures to ell infinity} about the $\left(p,q\right)$-adic Fourier-Stieltjes transform. Finally, we must give a brief mention of ``complex exponentials'' in the $p$-adic context. Although one can proceed to define a \textbf{$p$-adic exponential function}\index{$p$-adic!exponential function} by way of the usual power series $\exp_{p}:\mathbb{Q}_{p}\rightarrow\mathbb{Q}_{p}$: \begin{equation} \exp_{p}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}\frac{\mathfrak{z}^{n}}{n!}\in\mathbb{Q}_{p}\label{eq:Definition of the p-adic exponential function} \end{equation} unlike its classical counterpart, this construction is quite unruly, converging only for those $\mathfrak{z}$ with $\left\|\mathfrak{z}\right\|_{p}<p^{-\frac{1}{p-1}}$. Even worse, thanks to the properties of $p$-adic power series (specifically, due to \textbf{Strassman's Theorem} \cite{Robert's Book}), $\exp_{p}\left(\mathfrak{z}\right)$ is not a periodic function of $\mathfrak{z}$. As such, instead of working natively, when $p$-adic analysis wants to make use of complex exponentials in a manner comparable to their role in complex analysis, we need to take a different approach, as we discuss in the next subsection. \subsection{\label{subsec:1.3.4 Pontryagin-Duality-and}Pontryagin Duality and Embeddings of Fields} For those who haven't had the pleasure of meeting it, \textbf{Pontryagin duality}\index{Pontryagin duality}\textbf{ }is that lovely corner of mathematics where the concept of representing functions as Fourier series is generalized to study complex-valued functions on spaces\textemdash specifically, \textbf{locally compact abelian groups} (\textbf{LCAG}s)\footnote{Such a space is an abelian group that has a topology in which the group's operations are continuous; moreover, as a topological space, the group is locally compact.}\textemdash other than the circle ($\mathbb{R}/\mathbb{Z}$), the line ($\mathbb{R}$), and their cartesian products. Essentially, given a LCAG $G$, you can decompose functions $G\rightarrow\mathbb{C}$ as series or integrals of ``elementary functions'' (\textbf{characters}\index{character}) which transform in a simple, predictable way when they interact with the group's operation. In classical Fourier analysis, the characters are exponential functions $e^{2\pi ixt}$, $e^{-i\theta}$, and the like, and their ``well-behavedness'' is their periodicity\textemdash their invariance with respect to translations/shifts of the circle / line. In terms of the literature, Folland's book \cite{Folland - harmonic analysis} gives an excellent introduction and is very much worth reading in general. For the reader on the go, Tao's course notes on his blog \cite{Tao Fourier Transform Blog Post} are also quite nice. However, because this dissertation will approach Haar measures and integration theory primarily from the perspective of functional analysis, we will not need to pull out the full workhorse of Pontryagin duality and abstract harmonic analysis. Still, we need to spend at least \emph{some }amount of time discussing characters and our conventions for using them. A character\textbf{ }is, in essence, a generalization of the complex exponential. Classically, given an abelian group $G$, a \textbf{(complex-valued) character }on $G$ is a group homomorphism $\chi:G\rightarrow\mathbb{T}$, where $\left(\mathbb{T},\times\right)$ is the unit circle in $\mathbb{C}$, realized as an abelian group with the usual complex-number multiplication operation. Every character can be written as $\chi\left(g\right)=e^{2\pi i\xi\left(g\right)}$, where $\xi:G\rightarrow\mathbb{R}/\mathbb{Z}$ is a \textbf{frequency}, a group homomorphism from $G$ to $\mathbb{R}/\mathbb{Z}$\textemdash the group of real numbers in $\left[0,1\right)$\textemdash equipped with the operation of addition modulo $1$. The set of all characters of $G$ forms an abelian group under point-wise multiplication, and the set of all frequencies of $G$ forms an abelian group under point-wise addition. These two groups isomorphic to one another, and are both realizations of the \textbf{(Pontryagin) dual }of $G$, denoted $\hat{G}$. Depending on $G$, it can take some work to classify the characters and frequencies and figure out explicit formulae to use for them. Thankfully, since all of our work will be done over the additive group $\mathbb{Z}_{p}$ (where $p$ is a prime), we only need to familiarize ourselves with one set of formulae. \begin{defn}[$\hat{\mathbb{Z}}_{p}$] \ \vphantom{} I. We write \nomenclature{$\hat{\mathbb{Z}}_{p}$}{$\overset{\textrm{def}}{=}\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}$}$\hat{\mathbb{Z}}_{p}$ to denote the group $\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}$. This is the set of all $p$-ary rational numbers in $\left[0,1\right)$: \begin{equation} \hat{\mathbb{Z}}_{p}\overset{\textrm{def}}{=}\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}=\left\{ \frac{k}{p^{n}}:n\in\mathbb{N}_{0},k\in\left\{ 0,\ldots,p^{n}-1\right\} \right\} \label{eq:Definition of Z_p hat} \end{equation} and is a group under the operation of addition modulo $1$. This is the \textbf{additive}\footnote{The \emph{multiplicative} $p$-Prfer group, generally denoted $\mathbb{Z}\left(p^{\infty}\right)$, is the set of all $p$-power roots of unity in $\mathbb{T}$\textemdash that is, all $\xi\in\mathbb{T}$ so that $\xi^{p^{n}}=1$ for some $n\in\mathbb{N}_{0}$\textemdash made into a group by multiplication.}\textbf{ $p$-Prfer group}\index{$p$-Prfer group}. As the notation $\hat{\mathbb{Z}}_{p}$ suggests, this group is one realization of the \textbf{Pontryagin dual} of the additive group of $p$-adic integers: $\left(\mathbb{Z}_{p},+\right)$. \vphantom{} II. Recall that every $p$-adic rational number $\mathfrak{z}\in\mathbb{Q}_{p}$ can be written as: \begin{equation} \mathfrak{z}=\sum_{n=n_{0}}^{\infty}c_{n}p^{n} \end{equation} for constants $n_{0}\in\mathbb{Z}$ and $\left\{ c_{n}\right\} _{n\geq-n_{0}}\subseteq\left\{ 0,\ldots,p-1\right\} $. The\textbf{ $p$-adic fractional part}\index{$p$-adic!fractional part}, denoted \nomenclature{$\left\{ \cdot\right\} _{p}$}{$p$-adic fractional part} $\left\{ \cdot\right\} _{p}$, is the map from $\mathbb{Q}_{p}$ to $\hat{\mathbb{Z}}_{p}$ defined by: \begin{equation} \left\{ \mathfrak{z}\right\} _{p}=\left\{ \sum_{n=-n_{0}}^{\infty}c_{n}p^{n}\right\} _{p}\overset{\textrm{def}}{=}\sum_{n=n_{0}}^{-1}c_{n}p^{n}\label{eq:Definition of the p-adic fractional part} \end{equation} where the sum on the right is defined to be $0$ whenever $n_{0}\geq0$. The \textbf{$p$-adic integer part}\index{$p$-adic!integer part}, denoted $\left[\cdot\right]_{1}$ \nomenclature{$ \left[\cdot\right]_{1}$}{$p$-adic integer part}, is the map from $\mathbb{Q}_{p}$ to $\mathbb{Z}_{p}$ defined by: \begin{equation} \left[\mathfrak{z}\right]_{1}=\left[\sum_{n=-n_{0}}^{\infty}c_{n}p^{n}\right]_{1}\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{n}p^{n}\label{eq:Definition of the p-adic integer part} \end{equation} \end{defn} \vphantom{} Note that $\left[\mathfrak{z}\right]_{1}=\mathfrak{z}-\left\{ \mathfrak{z}\right\} _{p}$. Indeed, the $p$-adic fractional and integer parts are projections from $\mathbb{Q}_{p}$ onto $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ and $\mathbb{Z}_{p}$, respectively. In fact, $\hat{\mathbb{Z}}_{p}$ and $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ are isomorphic as additive groups. Moreover, we can express the group $\left(\mathbb{Q}_{p},+\right)$ as the sum: \begin{equation} \mathbb{Q}_{p}=\mathbb{Z}\left[\frac{1}{p}\right]+\mathbb{Z}_{p}\label{eq:Sum decomposition of Q_p} \end{equation} where the right-hand side is group whose elements are of the form $t+\mathfrak{z}$, where $t\in\mathbb{Z}\left[1/p\right]$ and $\mathfrak{z}\in\mathbb{Z}_{p}$. Note, this sum is \emph{not }direct\footnote{For people who enjoy staring at diagrams of (short) exact sequences, Subsection 5.4 (starting at page 40) of Robert's book \cite{Robert's Book} gives a comprehensive discussion of (\ref{eq:Sum decomposition of Q_p})'s failure to be a direct sum. Even though the \emph{set-theoretic }bijection $\iota:\mathbb{Q}_{p}\rightarrow\hat{\mathbb{Z}}_{p}\times\mathbb{Z}_{p}$ given by $\iota\left(\mathfrak{y}\right)\overset{\textrm{def}}{=}\left(\left\{ \mathfrak{y}\right\} _{p},\left[\mathfrak{y}\right]_{1}\right)$ does more than your ordinary set-theoretic bijection (for example, its restriction to $\mathbb{Z}_{p}$ \emph{is }an isomorphism of groups), $\iota$ itself is not an isomorphism of groups. For example: \[ \iota\left(\frac{1}{p}+\frac{p-1}{p}\right)=\iota\left(1\right)=\left(0,1\right) \] \[ \iota\left(\frac{1}{p}\right)+\iota\left(\frac{p-1}{p}\right)=\left(\frac{1}{p},0\right)+\left(\frac{p-1}{p},0\right)=\left(1,0\right)=\left(0,0\right) \] }. For example: \begin{equation} \frac{1}{p}+2=\left(\frac{1}{p}+1\right)+1 \end{equation} Despite this, every $p$-adic number is uniquely determined by its fractional and integer parts, and we can view $\hat{\mathbb{Z}}_{p}$ as a subset of $\mathbb{Q}_{p}$ by identifying it with the subset of $\mathbb{Q}_{p}$ whose elements all have $0$ as the value of their $p$-adic integer part. In this way we can add and multiply $t$s in $\hat{\mathbb{Z}}_{p}$ with $\mathfrak{z}$s in $\mathbb{Z}_{p}$ or $\mathbb{Q}_{p}$. \begin{example} For $t=\frac{1}{p}$ and $\mathfrak{z}=-3$, $t\mathfrak{z}=-\frac{3}{p}\in\mathbb{Q}_{p}$. \end{example} \vphantom{} Since every $p$-ary rational number $t\in\hat{\mathbb{Z}}_{p}$ is an element of $\mathbb{Q}_{p}$, we can consider $t$'s $p$-adic valuation and its $p$-adic absolute value. Specifically: \begin{fact}[\textbf{Working with $v_{p}$ and $\left\|\cdot\right\|_{p}$ on $\hat{\mathbb{Z}}_{p}$}] \ \vphantom{} I. $v_{p}\left(t\right)$ will be $\infty$ when $t=0$, and, otherwise, will be $-n$, where $n$ is the unique integer $\geq1$ so that $t$ can be written as $t=k/p^{n}$, where $k\in\left\{ 1,\ldots,p^{n}-1\right\} $ is co-prime to $p$. \vphantom{} II. For non-zero $t$, $\left\|t\right\|_{p}=p^{-v_{p}\left(t\right)}$ will be the value in the denominator of $t$. Specifically, $\left\|t\right\|_{p}=p^{n}$ if and only if the irreducible fraction representing $t$ is $t=k/p^{n}$, where $k\in\left\{ 1,\ldots,p^{n}-1\right\} $ is co-prime to $p$. \vphantom{} III. For non-zero $t$, $t\left\|t\right\|_{p}$ is the numerator $k$ of the irreducible fraction $k/p^{n}$ representing $t$. \vphantom{} IV. The $p$-adic fractional part also encodes information about congruences. In particular, we have the following identity: \begin{equation} \left\{ t\mathfrak{z}\right\} _{p}\overset{1}{\equiv}t\left[\mathfrak{z}\right]_{\left\|t\right\|_{p}},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:p-adic fractional part identity modulo 1} \end{equation} where, $\left[\mathfrak{z}\right]_{\left\|t\right\|_{p}}$ is the projection of $\mathfrak{z}$ modulo the power of $p$ in the denominator of $t$, and where, recall, $\overset{1}{\equiv}$ means that the left and the right hand side are equal to one another as real numbers modulo $1$\textemdash that is, their difference is an integer. \end{fact} \begin{example} Let $\mathfrak{z}\in\mathbb{Z}_{2}$. Then, we have the following equality in $\mathbb{C}$ \begin{equation} e^{2\pi i\left\{ \frac{\mathfrak{z}}{4}\right\} _{2}}\overset{\mathbb{C}}{=}e^{2\pi i\frac{\left[\mathfrak{z}\right]_{4}}{4}} \end{equation} More generally, for any $\mathfrak{y}\in\mathbb{Q}_{p}$, we can define: \begin{equation} e^{2\pi i\left\{ \mathfrak{y}\right\} _{p}}\overset{\mathbb{C}}{=}\sum_{n=0}^{\infty}\frac{\left(2\pi i\left\{ \mathfrak{y}\right\} _{p}\right)^{n}}{n!} \end{equation} because $\left\{ \mathfrak{y}\right\} _{p}\in\hat{\mathbb{Z}}_{p}=\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}\subset\left[0,1\right)\subset\mathbb{R}\subset\mathbb{C}$. \end{example} \vphantom{} There is also the issue of what happens when we ``mix our $p$s and $q$s''. \begin{example} Let $\mathfrak{z}\in\mathbb{Z}_{2}$. Then, since $\left\|\frac{1}{3}\right\|_{2}=1$, we have that $\frac{1}{3}\in\mathbb{Z}_{2}$, and thus, that $\frac{\mathfrak{z}}{3}\in\mathbb{Z}_{2}$. However: \begin{equation} \left\{ \frac{\mathfrak{z}}{24}\right\} _{2}=\left\{ \frac{1}{8}\frac{\mathfrak{z}}{3}\right\} _{2}\neq\frac{1}{24}\left[\mathfrak{z}\right]_{8} \end{equation} Rather, we have that: \begin{equation} \left\{ \frac{\mathfrak{z}}{24}\right\} _{2}=\left\{ \frac{1}{8}\frac{\mathfrak{z}}{3}\right\} _{2}\overset{1}{\equiv}\frac{1}{8}\left[\frac{\mathfrak{z}}{3}\right]_{8}\overset{1}{\equiv}\frac{1}{8}\left[\mathfrak{z}\right]_{8}\left[\frac{1}{3}\right]_{8}\overset{1}{\equiv}\frac{\left[3\right]_{8}^{-1}}{8}\left[\mathfrak{z}\right]_{8}\overset{1}{\equiv}\frac{3}{8}\left[\mathfrak{z}\right]_{8} \end{equation} where $\left[3\right]_{8}^{-1}$ denotes the unique integer in $\left\{ 1,\ldots,7\right\} $ which is the multiplicative inverse of $3$ modulo $8$\textemdash in this case, $3$, since $3$ is a square root of unity modulo $8$. Lastly, to return to our discussion of characters, it turns out that every complex-valued character on $\mathbb{Z}_{p}$ is a function of the form $\mathfrak{z}\in\mathbb{Z}_{p}\mapsto e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\in\mathbb{C}$ for some unique $t\in\hat{\mathbb{Z}}_{p}$; this is an instance of the isomorphism between the character group of $\mathbb{Z}_{p}$ and $\hat{\mathbb{Z}}_{p}$. \end{example} \vphantom{} Note that all of the above discussion was predicated on the assumption that our characters were \emph{complex-valued}. Importantly, because $\left\{ \cdot\right\} _{p}$ outputs elements of $\hat{\mathbb{Z}}_{p}$, each of which is a rational number, note also that the output of any complex-valued character $\mathfrak{z}\in\mathbb{Z}_{p}\mapsto e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\in\mathbb{C}$ of $\mathbb{Z}_{p}$ is a $p$-power root of unity. Since every $p$-power root of unity is the root of a polynomial of the form $X^{p^{n}}-1$ for some sufficiently large $n\geq1$, every $p$-power root of unity is an \emph{algebraic }number. As such, our $p$-adic characters are not \emph{true} $\mathbb{C}$-valued functions, but $\overline{\mathbb{Q}}$-valued functions. When working with functions taking values in a non-archimedean field like $\mathbb{Q}_{q}$, where $q$ is a prime, it is generally more useful to study them not by using Fourier series involving complex-valued ($\mathbb{C}$-valued) characters, but \textbf{$q$-adic (complex) valued characters}\textemdash those taking values in $\mathbb{C}_{q}$. However, regardless of whether we speak of the complex-valued or $q$-adic-valued characters of $\mathbb{Z}_{p}$, our characters will still end up taking values in $\overline{\mathbb{Q}}$. The reason there even exists a verbal distinction between complex-valued and $q$-adic-valued characters on $\mathbb{Z}_{p}$ is due to the needs of (algebraic) number theorists. Ordinarily, in algebraic number theory, the question of how a given field $\mathbb{F}$ is embedded in a larger field $K$ is very important, and it is not uncommon to allow the embeddings to vary, or to even do work by considering all possible embeddings \emph{simultaneously}. However, in this dissertation, we will do exactly the opposite. Our goal is to use the easy, comforting complex-exponential $e$-to-the-power-of-$2\pi i$-something notation for roots of unity. Because the $\mathbb{C}$-valued characters of $\mathbb{Z}_{p}$ are $\overline{\mathbb{Q}}$-valued, all of the algebraic rules for manipulating expressions like $\left\{ t\mathfrak{z}\right\} _{p}$ and $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ will remain valid when we view $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ as an element of $\mathbb{C}_{q}$, because we will be identifying $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ with its counterpart in the copy of $\overline{\mathbb{Q}}$ that lies in $\mathbb{C}_{q}$\textemdash a copy which is isomorphic to the version of $\overline{\mathbb{Q}}$ that lies in $\mathbb{C}$. If the reader is nervous about this, they can simply think of $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ as our notation of writing an abstract unitary character on $\mathbb{Z}_{p}$, without reference to an embedding. \begin{assumption}[Embedding Convention] For any primes $p$ and $q$, and any integer $n\geq1$, we will write $e^{2\pi i/p^{n}}$ to denote a specific choice of a primitive $p^{n}$th root of unity in $\mathbb{C}_{q}$, and shall then identify $e^{2\pi ik/p^{n}}$ as the $k$th power of that chosen root of unity. We also make these choices so that the are compatible in the natural way, with \begin{equation} \left(e^{2\pi i/p^{n+1}}\right)^{p}=e^{2\pi i/p^{n}},\textrm{ }\forall n\geq1 \end{equation} More generally, when $q\geq5$, since the set of roots of unity in $\mathbb{Q}_{q}$ is generated by a primitive $\left(q-1\right)$th root of unity, we define $e^{2\pi i/\left(q-1\right)}$ to be the primitive $\left(q-1\right)$th root of unity in $\mathbb{Z}_{q}^{\times}$ whose value mod $q$ is $r_{q}$, the smallest element of $\left(\mathbb{Z}/q\mathbb{Z}\right)^{\times}$ which is a primitive $\left(q-1\right)$th root of unity mod $q$. Then, for every other root of unity $\xi$ in $\mathbb{Q}_{q}$, there is a unique integer $k\in\left\{ 0,\ldots,q-2\right\} $ so that $\xi=\left(e^{2\pi i/\left(q-1\right)}\right)^{k}$, and as such, we define $\xi$'s value mod $q$ to be the value of $r_{q}^{k}$ mod $q$. \end{assumption} In this convention, any \emph{finite }$\overline{\mathbb{Q}}$-linear combination of roots of unity will then be an element of $\overline{\mathbb{Q}}$, and, as such, we can and will view the linear combination as existing in $\mathbb{C}$ and $\mathbb{C}_{q}$ \emph{simultaneously}. When working with infinite sums, the field to which the sum belongs will be indicated, when necessary, by an explicit mention of the field in which the convergence happens to be occurring. \emph{The practical implications of all this is that the reader will not need to worry about considerations of local compactness, spherical completeness, or field embeddings when doing or reading through computations with complex exponentials in this dissertation. }Except for when limits and infinite series come into play (and even then, only rarely), it will be as if everything is happening in $\mathbb{C}$. The reason everything works out is thanks to the fundamental identity upon which all of our Fourier analytic investigations will be based: \begin{equation} \left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{a}\right]\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p^{n}}\sum_{\left\|t\right\|_{p}\leq p^{n}}e^{-2\pi i\left\{ t\mathfrak{a}\right\} _{p}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z},\mathfrak{a}\in\mathbb{Z}_{p} \end{equation} where the sum is taken over all $t\in\hat{\mathbb{Z}}_{p}$ whose denominators are at most $p^{n}$. This is the Fourier series for the indicator function of the co-set $\mathfrak{a}+p^{n}\mathbb{Z}_{p}$. As indicated, the equality holds in $\overline{\mathbb{Q}}$. Moreover, observe that this identity is invariant under the action of $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$, seeing as the left-hand side is always rational, evaluating to $1$ when $\mathfrak{z}$ is in $\mathfrak{a}+p^{n}\mathbb{Z}_{p}$ and evaluating to $0$ for all other $\mathfrak{z}\in\mathbb{Z}_{p}$. Because of this, nothing is lost in choosing a particular embedding, and the theories we develop will be equally valid regardless of our choice of an embedding. So, nervous number theorists can set aside their worries and breathe easily. For staying power, we will repeat this discussion in Subsection \ref{subsec:1.3.3 Field-Extensions-=00003D000026}. One of the quirks of our approach is that when we actually go about computing Fourier transforms and integrals for $q$-adic valued functions of one or more $p$-adic variables, the computational formalisms will be identical to their counterparts for real- or complex-valued functions of one or more $p$-adic variables. As such, the overview of non-archimedean Fourier analysis given in Subsections \ref{subsec:3.1.4. The--adic-Fourier} and \ref{subsec:3.1.5-adic-Integration-=00003D000026} will include all the necessary computational tools which will be drawn upon in this dissertation. That being said, I highly recommend the reader consult a source like \cite{Bell - Harmonic Analysis on the p-adics} in case they are not familiar with the nitty-gritty details of doing computations with (real)-valued Haar measures on the $p$-adics and with the Fourier transform of real- or complex-valued functions of a $p$-adic variable. A more comprehensive\textemdash and rigorous\textemdash expositions of this material can be found in the likes of \cite{Taibleson - Fourier Analysis on Local Fields,Automorphic Representations}; \cite{Automorphic Representations} is a book on representation theory which deals with the methods of $p$-adic integration early on because of its use in that subject, whereas \cite{Taibleson - Fourier Analysis on Local Fields} is dedicated entirely to the matter of the Fourier analysis of complex-valued functions on $\mathbb{Q}_{p}$ and other local fields. \subsection{\label{subsec:1.3.5 Hensel's-Infamous-Blunder}Hensel's Infamous Blunder and Geometric Series Universality} In discussions with strangers over the internet, it became apparent that some of the number-theoretic liberties I take in my work are a source of controversy\textemdash and for good reason. Despite the extraordinary importance that $p$-adic numbers would eventually take in number theory, the $p$-adics' immediate reception by the mathematical community of the \emph{fin de sicle }was one of reservation and concerned apprehension. There are a variety of reasons for this: the cultural attitudes of his contemporaries; Hensel\index{Hensel, Kurt}'s unabashed zeal for his discovery; the nascent state of the theory of topology\footnote{Poincar published the installments of \emph{Analysis Situs }from 1895 through to 1904, contemporaneous with Hensel's introduction of the $p$-adics in 1897; \emph{Analysis Situs }is to topology what Newton's \emph{Principia }was to physics.} at that particular day and age; and, especially, Hensel's own fatal blunder: a foundational flawed 1905 $p$-adic ``proof'' that $e$ was a transcendental number \cite{Gouvea's introudction to p-adic numbers book,Gouvea's p-adic number history slides,Koblitz's book,Conrad on p-adic series}. Gouvea presents this blunder like so: \begin{quotation} Hensel's incorrect proof goes like this. Start from the equation: \[ e^{p}=\sum_{n=0}^{\infty}\frac{p^{n}}{n!} \] Hensel checks that this series converges in $\mathbb{Q}_{p}$ and concludes that it satisfies an equation of the form $y^{p}=1+p\varepsilon$ with $\varepsilon$ a $p$-adic unit. If $e$ is algebraic, it follows that $\left[\mathbb{Q}\left(e\right):\mathbb{Q}\right]\geq p$. But $p$ was arbitrary, so we have a contradiction, and $e$ is transcendental \cite{Gouvea's p-adic number history slides}. \end{quotation} Hensel's error is as much \emph{topological} it is number theoretic. Specifically, he \textbf{\emph{incorrectly}} assumes the following statement is true: \begin{assumption} \emph{Let $\mathbb{F}$ and $\mathbb{K}$ be any field extensions of $\mathbb{Q}$, equipped with absolute values $\left\|\cdot\right\|_{\mathbb{F}}$ and $\left\|\cdot\right\|$$_{\mathbb{K}}$, respectively, which make $\left(\mathbb{F},\left\|\cdot\right\|_{\mathbb{F}}\right)$ and $\left(\mathbb{K},\left\|\cdot\right\|_{\mathbb{K}}\right)$ into complete metric spaces. Let $\left\{ x_{n}\right\} _{n\geq1}$ be a sequence in $\mathbb{Q}$ such that there are elements $\mathfrak{a}\in\mathbb{F}$ and $\mathfrak{b}\in\mathbb{K}$ so that $\lim_{n\rightarrow\infty}\left\|\mathfrak{a}-x_{n}\right\|_{\mathbb{F}}=0$ and $\lim_{n\rightarrow\infty}\left\|\mathfrak{b}-x_{n}\right\|_{\mathbb{F}}=0$. Then $\mathfrak{a}=\mathfrak{b}$.} \end{assumption} \vphantom{} So, Hensel presumed that if $\mathbb{F}$ is a field extension of $\mathbb{Q}$ in which the series $\sum_{n=0}^{\infty}\frac{p^{n}}{n!}$ converges, the sum of the series was necessarily $e^{p}$. This, of course, is wrong. One might object and say ``but this is obvious: not every $p$-adic integer can be realized as an ordinary real or complex number!'', however, the situation is even more subtle than that. For example, the statement in the previous paragraph remains false even if we required $\mathfrak{a}\in\mathbb{F}\cap\mathbb{Q}$ and $\mathfrak{b}\in\mathbb{K}\cap\mathbb{Q}$. Consider the following: \begin{example}[\textbf{A Counterexample for Kurt Hensel}\footnote{Taken from \cite{Conrad on p-adic series}.}] Let $a_{n}=\frac{1}{1+p^{n}}$, and let $r_{n}=a_{n}-a_{n-1}$, so that $\sum_{n}r_{n}=\lim_{n\rightarrow\infty}a_{n}$. Then: \begin{align} \sum_{n=0}^{\infty}r_{n} & \overset{\mathbb{R}}{=}0\\ \sum_{n=0}^{\infty}r_{n} & \overset{\mathbb{Q}_{p}}{=}1 \end{align} because $a_{n}$ tends to $0$ in $\mathbb{R}$ and tends to $1$ in $\mathbb{Q}_{p}$. \end{example} \vphantom{} This is, quite literally, the oldest mistake in the subject, and still worth pointing out to neophytes over a century later; I, myself, fell for it in an earlier iteration of this dissertation. Nevertheless, in this work, we are going to do what Hensel \emph{thought }he could do, but in our case, \emph{it will be justified}. Such shenanigans will occur multiple times in this dissertation, so it is worth taking a moment to explain in detail what we are going to do, and why it actually \emph{works}. And the exemplary exception that saves us is none other than the tried and true geometric series formula. \begin{fact}[\textbf{\emph{Geometric Series Universality}}] \textbf{\label{fact:Geometric series universality}} Let\index{geometric series universality} $r\in\mathbb{Q}$, and suppose there is a valued field $\left(\mathbb{F},\left\|\cdot\right\|_{\mathbb{F}}\right)$ extending $\mathbb{Q}$ so that the series $\sum_{n=0}^{\infty}r^{n}$ converges in $\mathbb{F}$ (i.e., for which $\left\|r\right\|_{\mathbb{F}}<1$), with the sum converging in $\mathbb{F}$ to $R\in\mathbb{F}$. Then, $R\in\mathbb{Q}$, and, for any valued field $\left(\mathbb{K},\left\|\cdot\right\|_{\mathbb{K}}\right)$ extending $\mathbb{Q}$ for which the series $\sum_{n=0}^{\infty}r^{n}$ converges in $\mathbb{K}$ (i.e., for which $\left\|r\right\|_{\mathbb{K}}<1$), the sum converges in $\mathbb{K}$ to $R$. \emph{\cite{Conrad on p-adic series}} \end{fact} \vphantom{} In practice, in our use of this universality, we will have a formal sum $\sum_{n=0}^{\infty}a_{n}$ of rational numbers $\left\{ a_{n}\right\} _{n\geq0}\subseteq\mathbb{Q}$ such that $\sum_{n=0}^{\infty}a_{n}$ can be written as the sum of finitely many geometric series: \begin{equation} \sum_{n=0}^{\infty}b_{1}r_{1}^{n}+\sum_{n=0}^{\infty}b_{2}r_{2}^{n}+\sum_{n=0}^{\infty}b_{M}r_{M}^{n} \end{equation} where the $b_{m}$s and $r_{m}$s are rational numbers. In particular, there will exist an integer $c\geq1$ so that: \begin{equation} \sum_{n=0}^{cN-1}a_{n}\overset{\mathbb{Q}}{=}\sum_{n=0}^{N-1}b_{1}r_{1}^{n}+\sum_{n=0}^{N-1}b_{2}r_{2}^{n}+\sum_{n=0}^{N-1}b_{M}r_{M}^{n},\textrm{ }\forall N\geq1\label{eq:Partial sum decomposed as sum of partial sums of geometric series} \end{equation} Then, if there is a prime $p$ so that $\left\|r_{m}\right\|<1$ and $\left\|r_{m}\right\|_{p}<1$ for all $m\in\left\{ 1,\ldots,M\right\} $, the series: \begin{equation} \sum_{n=0}^{N-1}b_{m}r_{m}^{n} \end{equation} will converge to $\frac{b_{m}}{1-r_{m}}$ in both $\mathbb{R}$ \emph{and }$\mathbb{Q}_{p}$ as $N\rightarrow\infty$. \begin{example} The geometric series identity: \[ \sum_{n=0}^{\infty}\left(\frac{3}{4}\right)^{n}=\frac{1}{1-\frac{3}{4}}=4 \] holds in both $\mathbb{R}$ and $\mathbb{Q}_{3}$, in that the series converges in both fields' topologies, and the limit of its partial sum is $4$ in both of those topologies. Indeed: \begin{equation} \sum_{n=0}^{N-1}\left(\frac{3}{4}\right)^{n}=\frac{1-\left(\frac{3}{4}\right)^{N}}{1-\frac{3}{4}}=4-4\left(\frac{3}{4}\right)^{N} \end{equation} and so: \[ \lim_{N\rightarrow\infty}\left\|4-\sum_{n=0}^{N-1}\left(\frac{3}{4}\right)^{n}\right\|=\lim_{N\rightarrow\infty}4\left(\frac{3}{4}\right)^{N}\overset{\mathbb{R}}{=}0 \] and: \begin{equation} \lim_{N\rightarrow\infty}\left\|4-\sum_{n=0}^{N-1}\left(\frac{3}{4}\right)^{n}\right\|_{3}=\lim_{N\rightarrow\infty}\left\|4\left(\frac{3}{4}\right)^{N}\right\|_{3}=\lim_{N\rightarrow\infty}3^{-N}\overset{\mathbb{R}}{=}0 \end{equation} \end{example} \vphantom{} Our second technique for applying universality is the observation that, \emph{for any prime $p$, if a geometric series $\sum_{n=0}^{\infty}r^{n}$ (where $r\in\mathbb{Q}\cap\mathbb{Q}_{p}$) converges in $\mathbb{Q}_{p}$, then its sum, $\frac{1}{1-r}$, is }\textbf{\emph{also}}\emph{ a rational number. }In light of these two tricks, we will consider situations like the following. Here, let $p$ and $q$ be primes (possibly with $p=q$), and consider a function $f:\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{q}$. Then, if there is a subset $U\subseteq\mathbb{Z}_{p}$ such that, for each $\mathfrak{z}\in U$, $f\left(\mathfrak{z}\right)$ is expressible a finite linear combination of geometric series: \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{Q}_{q}}{=}\sum_{m=1}^{M}\sum_{n=0}^{\infty}b_{m}r_{m}^{n} \end{equation} where the $b_{m}$s and the $r_{m}$s are elements of $\mathbb{Q}$, we can view the restriction $f\mid_{U}$ as a rational-valued function $f\mid_{U}:U\rightarrow\mathbb{Q}$. More generally, we can view $f\mid_{U}$ as a real-, or even complex-valued function on $U$. Depending on the values of $\left\|r\right\|$ and $\left\|r\right\|_{q}$, we may be able to compute the sum $\sum_{m=1}^{M}\sum_{n=0}^{\infty}b_{m}r_{m}^{n}$ solely in the topology of $\mathbb{R}$ or $\mathbb{C}$, solely in the topology of $\mathbb{Q}_{q}$, or in both, simultaneously. \chapter{\label{chap:2 Hydra-Maps-and}Hydra Maps and their Numina} \includegraphics[scale=0.45]{./PhDDissertationEroica2.png} \vphantom{} Chapter 2 splits neatly (though lop-sidedly) into two halves. Section \ref{sec:2.1 Hydra-Maps-=00003D000026 history} introduces Hydra maps\textemdash my attempt to provide a taxonomy for some of the most important Collatz-type maps, accompanied with numerous examples. Subsection \ref{subsec:2.1.2 It's-Probably-True} is a historical essay on the Collatz Conjecture and most of the most notable efforts to lay bare its mysteries. Section \ref{sec:2.2-the-Numen}, meanwhile, is where we begin with new results. The focus of \ref{sec:2.2-the-Numen} is the construction of the titular \textbf{numen}, a $\left(p,q\right)$-adic function we can associate to any sufficiently well-behaved Hydra map, and the elucidation of its most important properties, these being its characterization as the unique solution of a system of $\left(p,q\right)$-adic functional equations\textemdash subject to a continuity-like condition\textemdash as well as the all-important \textbf{Correspondence Principle}, the first major result of my dissertation. Presented in Subsection \ref{subsec:2.2.3 The-Correspondence-Principle}\textemdash and in four different variants, no less\textemdash the Correspondence Principle establishes a direct correspondence between the values taken by a numen and the periodic points and divergent points of the Hydra map to which it is associated. \ref{sec:2.2-the-Numen} concludes with Subsection \ref{subsec:2.2.4 Other-Avenues}, in which further avenues of exploration are discussed, along with connections to other recent work, principally Mih\u{a}ilescu's resolution of \textbf{Catalan's Conjecture }and the shiny new state-of-the-art result on Collatz due to Tao in 2019-2020 \cite{Tao Probability paper}. \section{\label{sec:2.1 Hydra-Maps-=00003D000026 history}History and Hydra Maps} Much has been written about the Collatz Conjecture, be it efforts to solve it, or efforts to dissuade others from trying. Bibliographically speaking, I would place Lagarias' many writings on the subject\footnote{Such as his papers \cite{Lagarias-Kontorovich Paper,Lagarias' Survey}, and particularly his book \cite{Ultimate Challenge}.}, Wirsching's Book \cite{Wirsching's book on 3n+1}, and\textemdash for this dissertation's purposes, Tao's 2019 paper \cite{Tao Probability paper}\textemdash near the top of the list, along with K.R. Matthews maintains a dedicated $3x+1$ page on his website, \href{http://www.numbertheory.org/3x\%2B1/}{page of Matthews' website}, filled with interesting links, pages, and, of course, his fascinating powerpoint slides \cite{Matthews' slides,Matthews' Leigh Article,Matthews and Watts}. The interested reader can survey these and other sources referenced by this dissertation to get the ``lay of the land''\textemdash to the extent that ``the land'' exists in this particular context. The purpose of this section is two-fold: to introduce this dissertation's fundamental objects of study\textemdash Hydra maps\textemdash and to provide a short, likely inadequate history of the work that has been done the Collatz Conjecture. \subsection{\emph{\label{subsec:2.1.1 Release-the-Hydras!}Release the Hydras! }- An Introduction to Hydra Maps} We begin with the definition: \begin{defn} \label{def:p-Hydra map}\nomenclature{$H$}{a Hydra map}\index{$p$-Hydra map} \index{Hydra map!one-dimensional}\index{Hydra map}Fix an integer $p\geq2$. A \textbf{$p$-Hydra map} is a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$ of the form: \begin{equation} H\left(n\right)=\begin{cases} \frac{a_{0}n+b_{0}}{d_{0}} & \textrm{if }n\overset{p}{\equiv}0\\ \frac{a_{1}n+b_{1}}{d_{1}} & \textrm{if }n\overset{p}{\equiv}1\\ \vdots & \vdots\\ \frac{a_{p-1}n+b_{p-1}}{d_{p-1}} & \textrm{if }n\overset{p}{\equiv}p-1 \end{cases},\textrm{ }\forall n\in\mathbb{Z}\label{eq:Def of a Hydra Map on Z} \end{equation} and where $a_{j}$, $b_{j}$, and $d_{j}$ are integer constants (with $a_{j},d_{j}\geq0$ for all $j$) so that the following two properties hold: \vphantom{} I. (``co-primality'') $a_{j},d_{j}>0$ and $\gcd\left(a_{j},d_{j}\right)=1$ for all $j\in\left\{ 0,\ldots,p-1\right\} $. \vphantom{} II. (``one-sidedness'') $H\left(\mathbb{N}_{0}\right)\subseteq\mathbb{N}_{0}$ and $H\left(-\mathbb{N}_{1}\right)\subseteq-\mathbb{N}_{1}$, where $-\mathbb{N}_{1}=\left\{ -1,-2,-3,\ldots\right\} $. \vphantom{} I call $H$ an \textbf{integral }Hydra map\index{Hydra map!integral}, if, in addition, it satisfies: \vphantom{} III. (``integrality''\footnote{For future or broader study, this condition might be weakened, or even abandoned. It is not needed for the construction of the numen, for example.}) For each $j\in\left\{ 0,\ldots,p-1\right\} $, $\frac{a_{j}n+b_{j}}{d_{j}}$ is an integer if and only if $n\overset{p}{\equiv}j$. \vphantom{} Hydra maps not satisfying (III) are said to be \textbf{non-integral}.\index{Hydra map!non-integral} Finally, $H$ is said to be \textbf{prime }whenever $p$ is prime. \end{defn} \vphantom{} When speaking of or working with these maps, the following notions are quite useful: \begin{defn}[\textbf{Branches of $H$}] For each $j\in\mathbb{Z}/p\mathbb{Z}$, we write $H_{j}:\mathbb{R}\rightarrow\mathbb{R}$ to denote the \textbf{$j$th branch}\index{Hydra map!$j$th branch} of $H$, defined as the function: \begin{equation} H_{j}\left(x\right)\overset{\textrm{def}}{=}\frac{a_{j}x+b_{j}}{d_{j}}\label{eq:Definition of H_j} \end{equation} \end{defn} Next, as a consequence of (\ref{eq:Def of a Hydra Map on Z}), observe that for all $m\in\mathbb{N}_{0}$ and all $j\in\left\{ 0,\ldots,p-1\right\} $, we can write: \[ \underbrace{H\left(pm+j\right)}_{\in\mathbb{N}_{0}}=\frac{a_{j}\left(pm+j\right)+b_{j}}{d_{j}}=\frac{pa_{j}m}{d_{j}}+\frac{ja_{j}+b_{j}}{d_{j}}=\frac{pa_{j}}{d_{j}}m+\underbrace{H\left(j\right)}_{\in\mathbb{N}_{0}} \] Setting $m=1$, we see that the quantity $\frac{pa_{j}}{d_{j}}=H\left(pm+j\right)-H\left(j\right)$ must be an integer for each value of $j$. This quantity occurs often enough in computations to deserve getting a symbol of its own. \begin{defn}[$\mu_{j}$] We write \nomenclature{$\mu_{j}$}{$\overset{\textrm{def}}{=}\frac{pa_{j}}{d_{j}}$}$\mu_{j}$ to denote: \begin{equation} \mu_{j}\overset{\textrm{def}}{=}\frac{pa_{j}}{d_{j}}\in\mathbb{N}_{1},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Def of mu_j} \end{equation} In particular, since $a_{j}$ and $d_{j}$ are always co-prime, \textbf{note that $d_{j}$ must be a divisor of $p$}. In particular, this then forces $\mu_{0}/p$ (the derivative of $H_{0}$) to be a positive rational number which is not equal to $1$. \end{defn} \begin{rem} With this notation, we then have that: \begin{equation} H\left(pm+j\right)=\mu_{j}m+H\left(j\right),\textrm{ }\forall m\in\mathbb{Z},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Value of rho m + j under H} \end{equation} \end{rem} \vphantom{} For lack of a better or more descriptive name, I have decided to call maps of this form $p$-Hydra maps, in reference to the many-headed monster of Greek myth. Much like their legendary namesake, Hydra maps are not easily conquered. Questions and conjectures regarding these maps' dynamical properties\textemdash such as the number of map's periodic points, the existence of divergent trajectories for a given map, or the total number of the map's orbit classes\textemdash rank among the most difficult problems in all mathematics \cite{Lagarias' Survey}. Case in point, the quintessential example of a Hydra map\textemdash for the handful of people who aren't already aware of it\textemdash is the infamous Collatz map: \begin{defn} \label{def:The-Collatz-map}The\index{Collatz!map}\index{$3x+1$ map} \textbf{Collatz map} $C:\mathbb{Z}\rightarrow\mathbb{Z}$ is the function defined by: \begin{equation} C\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{n}{2} & \textrm{if }n\overset{2}{\equiv}0\\ 3n+1 & \textrm{if }n\overset{2}{\equiv}1 \end{cases}\label{eq:Collatz Map} \end{equation} \end{defn} First proposed by Lothar Collatz\footnote{Actually, Collatz came up with the map in 1929-1930; Matthews' website has a translation of a letter by Collatz himself in which the man states as such. It arose, apparently, as a diversion during time Collatz spent attending classes by Perron and Schur, among others \cite{Collatz Letter}.} in 1937 \cite{Collatz Biography}, the eponymous map's well-deserved mathematical infamy stems from the egregiously simple, yet notoriously difficult conjecture of the same name \cite{Lagarias' Survey,Collatz Biography}: \begin{conjecture}[\textbf{Collatz Conjecture}\footnote{Also known as the $3n+1$ problem, the $3x+1$ problem, the Syracuse Problem, the Hailstone problem, to give but a few of its many monickers.} \index{Collatz!Conjecture}] \label{conj:Collatz}For every $n\in\mathbb{N}_{1}$, there exists a $k\geq0$ so that $C^{\circ k}\left(n\right)=1$. \end{conjecture} \begin{rem} An equivalent reformulation of this result is the statement: \emph{$\mathbb{N}_{1}$ is an irreducible orbit class of $C$.} \end{rem} \vphantom{} A moment's thought reveals that the Collatz Conjecture actually consists of \emph{two} separate statements: \begin{conjecture}[\textbf{Weak Collatz Conjecture}\footnote{Also known as the ``(No) Finite Cycles'' Conjecture.}] \label{conj:Weak Collatz}The only periodic points of $C$ in $\mathbb{N}_{1}$ are $1$, $2$, and $4$. \end{conjecture} \begin{conjecture}[\textbf{Divergent Trajectories Conjecture}] $C$ has no divergent trajectories in $\mathbb{N}_{1}$. That is to say, for each $n\in\mathbb{N}_{1}$: \begin{equation} \sup_{k\geq0}C^{\circ k}\left(n\right)<\infty\label{eq:No divergent trajectories} \end{equation} \end{conjecture} \vphantom{} Since $3n+1$ will be even whenever $n$ is odd, it is generally more expedient to alter $C$ by inserting an extra division by $2$ to obtain the so-called Shortened Collatz map: \begin{defn} \label{def:The-Shortened-Collatz Map}\nomenclature{$T_{3}$}{shortened Collatz map}The\index{$3x+1$ map} \index{Collatz!map}\textbf{Shortened Collatz map }$T_{3}:\mathbb{Z}\rightarrow\mathbb{Z}$ is defined by: \begin{equation} T_{3}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{n}{2} & \textrm{if }n\overset{2}{\equiv}0\\ \frac{3n+1}{2} & \textrm{if }n\overset{2}{\equiv}1 \end{cases}\label{eq:Definition of T_3} \end{equation} \end{defn} \vphantom{} The general dynamics of which are identical to that of $C$, with $T_{3}\left(n\right)=C\left(n\right)$ for all even $n$ and $T_{3}\left(n\right)=C\left(C\left(n\right)\right)$ for all odd $n$. The use of $T$ or $T_{3}$ to denote this map is standard in the literature (ex. \cite{Lagarias-Kontorovich Paper}, \cite{Wirsching's book on 3n+1}). After the Collatz map $C$ and its shortened counterpart, $T_{3}$, the next most common level of generalization comes from considering the larger family of ``shortened $qx+1$ maps'': \begin{defn} \label{def:Shortened qx plus 1 map}\nomenclature{$T_{q}$}{shortened $qx+1$ map}Let $q$ be an odd integer $\geq1$. Then, the \textbf{Shortened $qx+1$ map}\index{$qx+1$ map} $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ is the function defined by: \begin{equation} T_{q}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{n}{2} & \textrm{if }n\overset{2}{\equiv}0\\ \frac{qn+1}{2} & \textrm{if }n\overset{2}{\equiv}1 \end{cases}\label{eq:Definition of T_q} \end{equation} \end{defn} \vphantom{} Of the (shortened) $qx+1$ maps, the most well-known are $T_{3}$ and $T_{5}$, with $T_{5}$ being of interest because its behavior seems to be diametrically opposed to that of $T_{3}$: one the one hand, the set of positive integers iterated by $T_{3}$ to $\infty$ has zero density \cite{Lagarias' Survey,Terras 76,Terras 79}; on the other hand, $T_{5}$ iterates almost every positive integer to $\infty$ \cite{Lagarias-Kontorovich Paper}. Nevertheless, despite the probabilistic guarantee that almost every positive integer should belong to a divergent trajectory, it has yet to be proven that any \emph{specific} positive integer belongs to a divergent trajectory \cite{Lagarias-Kontorovich Paper}! A notable feature of Collatz Studies is its overriding single-mindedness. Discussions of $C$ and its various shortened versions\footnote{Tao, for example, considers a variant which sends an $n$ odd integer to the next odd integer $n^{\prime}$ in the forward orbit of $n$ under $C$; that is $n\mapsto\left(3n+1\right)\left\|3n+1\right\|_{2}$, where $\left\|\cdot\right\|_{2}$ is the $2$-adic absolute value \cite{Tao Blog}.} predominate, with most in-depth number theoretic studies of Hydra maps being focused on $C$. While other Hydra maps do receive a certain degree of attention, it is generally from a broader, more holistic viewpoint. Some of the best examples of these broader studies can be found works of \index{Matthews, K. R.}K. R. Matthews and his colleagues and collaborators, going all the way back to the 1980s. The \emph{leitmotif} of this body of work is the use of Markov chains to model the maps' dynamics. These models provide a particularly fruitful heuristic basis for conjectures regarding the maps' ``typical'' trajectories, and the expected preponderance of cycles relative to divergent trajectories and vice-versa \cite{dgh paper,Matthews' Conjecture,Matthews' Leigh Article,Matthews' slides,Matthews and Watts}. This line of investigation dates back to the work of H. Mller in the 1970s \cite{Moller's paper (german)}, who formulated a specific type of generalized multiple-branched Collatz map and made conjectures about the existence of cycles and divergent trajectories. Although probabilistic techniques of this sort will not on our agenda here, Matthews and his colleagues' investigations provide ample ``non-classical'' examples of what I call Hydra maps. \begin{example} Our first examples are due to Leigh (\cite{Leigh's article,Matthews' Leigh Article}); we denote these as $L_{1}$ and $L_{2}$, respectively: \begin{equation} L_{1}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{n}{4} & \textrm{if }n\overset{8}{\equiv}0\\ \frac{n+1}{2} & \textrm{if }n\overset{8}{\equiv}1\\ 20n-40 & \textrm{if }n\overset{8}{\equiv}2\\ \frac{n-3}{8} & \textrm{if }n\overset{8}{\equiv}3\\ 20n+48 & \textrm{if }n\overset{8}{\equiv}4\\ \frac{3n-13}{2} & \textrm{if }n\overset{8}{\equiv}5\\ \frac{11n-2}{4} & \textrm{if }n\overset{8}{\equiv}6\\ \frac{n+1}{8} & \textrm{if }n\overset{8}{\equiv}7 \end{cases} \end{equation} \begin{equation} L_{2}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} 12n-1 & \textrm{if }n\overset{4}{\equiv}0\\ 20n & \textrm{if }n\overset{4}{\equiv}1\\ \frac{3n-6}{4} & \textrm{if }n\overset{4}{\equiv}2\\ \frac{n-3}{4} & \textrm{if }n\overset{4}{\equiv}3 \end{cases} \end{equation} Note that neither $L_{1}$ nor $L_{2}$ satisfy the integrality condition, seeing as they have branches which are integer-valued for all integer inputs. Also, while $L_{1}$ fixes $0$, $L_{2}$ sends $0$ to $-1$, and as such, $L_{2}$ is not a $4$-Hydra map \emph{as written}; this can be rectified by conjugating $L_{2}$ by an appropriately chosen affine linear map\textemdash that is, one of the form $an+b$, where $a$ and $b$ are integers; here, $a$ needs to be co-prime to $p=4$. Setting $a=1$ gives: \begin{equation} L_{2}\left(n-b\right)+b=\begin{cases} 12n-11b-1 & \textrm{if }n\overset{4}{\equiv}b\\ 20n-19b & \textrm{if }n\overset{4}{\equiv}b+1\\ \frac{3n+b-6}{4} & \textrm{if }n\overset{4}{\equiv}b+2\\ \frac{n+3b-3}{4} & \textrm{if }n\overset{4}{\equiv}b+3 \end{cases} \end{equation} Setting $b=1$ then yields the map $\tilde{L}_{2}:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$: \begin{equation} \tilde{L}_{2}\left(n\right)\overset{\textrm{def}}{=}L_{2}\left(n-1\right)+1=\begin{cases} \frac{n}{4} & \textrm{if }n\overset{4}{\equiv}0\\ 12n-12 & \textrm{if }n\overset{4}{\equiv}1\\ 20n-19 & \textrm{if }n\overset{4}{\equiv}2\\ \frac{3n-5}{4} & \textrm{if }n\overset{4}{\equiv}3 \end{cases}\label{eq:Conjugated Hydra map} \end{equation} which, unlike $L_{2}$, satisfies conditions (I) and (II) for in (\ref{eq:Def of a Hydra Map on Z}). Conjugating maps in this way is a useful tool to put them in a ``standard form'' of sorts, one in which the conjugated map sends the non-negative integers $\mathbb{N}_{0}$ to $\mathbb{N}_{0}$ and, preferably, also fixes $0$. \end{example} \vphantom{} Our next example of a map is also non-integral, yet, it is significant because it possess \emph{provably} divergent trajectories\index{divergent!trajectory}. Moreover, it may be useful to broaden the notion of Hydra maps to include maps like these, seeing as, despite its integrality failure, we can still construct its numen using the methods of Section \ref{sec:2.2-the-Numen}. \begin{example}[\textbf{Matthews' Map \& Divergent Trajectories}] \label{exa:Matthews' map}The following map is due to Matthews \cite{Matthews' Conjecture}: \begin{equation} M\left(n\right)\overset{\textrm{def}}{=}\begin{cases} 7n+3 & \textrm{if }n\overset{3}{\equiv}0\\ \frac{7n+2}{3} & \textrm{if }n\overset{3}{\equiv}1\\ \frac{n-2}{3} & \textrm{if }n\overset{3}{\equiv}2 \end{cases}\label{eq:Matthews' Conjecture Map} \end{equation} That this map fails to satisfy the integrality requirement can be seen in the fact that its $0$th branch $7n+3$ outputs integers even when $n$ is \emph{not }congruent to $0$ mod $3$. That being said, as defined, note that $-1$ is the only fixed point of $M$ in $\mathbb{Z}$. For our purposes, we will need to work with maps that fix $0$. To do so, we can conjugate $M$ by $f\left(n\right)=n-1$ ($f^{-1}\left(n\right)=n+1$) to obtain: \begin{equation} \tilde{M}\left(n\right)\overset{\textrm{def}}{=}\left(f^{-1}\circ M\circ f\right)\left(n\right)=M\left(n-1\right)+1=\begin{cases} \frac{n}{3} & \textrm{if }n\overset{3}{\equiv}0\\ 7n-3 & \textrm{if }n\overset{3}{\equiv}1\\ \frac{7n-2}{3} & \textrm{if }n\overset{3}{\equiv}2 \end{cases}\label{eq:Matthews' Conjecture Map, conjugated} \end{equation} Conjugating this map by $g\left(n\right)=-n$ ($g^{-1}=g$) then allows us to study $\tilde{T}$'s behavior on the negative integers: \begin{equation} \tilde{T}_{-}\left(n\right)\overset{\textrm{def}}{=}-\tilde{T}\left(-n\right)=\begin{cases} \frac{n}{3} & \textrm{if }n\overset{3}{\equiv}0\\ \frac{7n+2}{3} & \textrm{if }n\overset{3}{\equiv}1\\ 7n+3 & \textrm{if }n\overset{3}{\equiv}2 \end{cases}\label{eq:Matthews' conjecture map, conjugated to the negatives} \end{equation} In general, after conjugating a map (Hydra or not) so that it fixes $0$, conjugating by $g\left(n\right)=-n$ is how one studies the behavior of the maps on the non-positive integers. \end{example} \begin{rem} It may be of interest to note that the $1$st and $2$nd branches $\tilde{M}_{1}\left(x\right)=7x-3$ and $\tilde{M}_{2}\left(x\right)=\frac{7x-2}{3}$ commute with one another: \begin{equation} \tilde{M}_{1}\left(\tilde{M}_{2}\left(x\right)\right)=7\left(\frac{7x-2}{3}\right)-3=\frac{49x-14}{3}-3=\frac{49x-23}{3} \end{equation} \begin{equation} \tilde{M}_{2}\left(\tilde{M}_{1}\left(x\right)\right)=\frac{7\left(7x-3\right)-2}{3}=\frac{49x-21-2}{3}=\frac{49x-23}{3} \end{equation} \end{rem} \vphantom{} As was mentioned above, the import of $M$ lies in the fact that it has provably divergent trajectories: \begin{prop} \label{prop:Matthews' map}$M$ has at least two divergent trajectories over $\mathbb{Z}$: \vphantom{} I. The set of all non-negative integer multiples of $3$ (which are iterated to $+\infty$); \vphantom{} II. The set of all negative integer multiples of three (which are iterated to $-\infty$). \end{prop} Proof: Since $7n+3$ is congruent to $0$ mod $3$ whenever $n$ is congruent to $0$ mod $3$, any non-negative (resp. negative) integer multiple of $3$ is iterated by $M$ to positive (resp. negative) infinity. Q.E.D. \vphantom{} For extra incentive, it is worth mentioning that there is money on the table. \index{Matthews, K. R.}Matthews has put a bounty of $100$ Australian dollars, to be given to anyone who can bring him the severed, fully-proven head of the following conjecture: \begin{conjecture}[\textbf{Matthews' Conjecture}\index{Matthews' Conjecture} \cite{Matthews' Conjecture}] \label{conj:Matthews conjecture}The irreducible orbit classes of $M$ in $\mathbb{Z}$ are the divergent orbit classes associated to the trajectories (I) and (II) described in \textbf{\emph{Proposition \ref{prop:Matthews' map}}}, and the orbit classes corresponding to the attracting cycles $\left\{ -1\right\} $ and $\left\{ -2,-4\right\} $. \end{conjecture} \vphantom{} Matthews' powerpoint slides \cite{Matthews' slides} (freely accessible from his website) contain many more examples, both along these lines, and in further, more generalized forms. The slides are quite comprehensive, detailing generalizations and conjectures due to Hasse and Herbert Mller\index{Mller, Herbert}, as well as delving into the details of the aforementioned Markov-chain-based analysis of the maps and their dynamics. The slides also detail how the Markov methods can be extended along with the maps from $\mathbb{Z}$ to spaces of $p$-adic integers, as well as to the ring of profinite integers\index{profinite integers} (a.k.a,\emph{ polyadic integers}) given by the direct product $\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$. However, for our purposes, the most significant offering in Matthews' slides are the generalizations given in which simple Collatz-type maps are defined over discrete spaces other than $\mathbb{Z}$. \begin{example}[\textbf{A ``Multi-Dimensional'' Hydra Map}] The first example of such an extension appears to from Leigh in 1983\cite{Matthews' slides}. Leigh considered the map\footnote{The notation $T_{3,\sqrt{2}}$ is my own.} $T_{3,\sqrt{2}}:\mathbb{Z}\left[\sqrt{2}\right]\rightarrow\mathbb{Z}\left[\sqrt{2}\right]$ defined by: \begin{equation} T_{3,\sqrt{2}}\left(z\right)\overset{\textrm{def}}{=}\begin{cases} \frac{z}{\sqrt{2}} & \textrm{if }z\overset{\sqrt{2}}{\equiv}0\\ \frac{3z+1}{\sqrt{2}} & \textrm{if }z\overset{\sqrt{2}}{\equiv}1 \end{cases},\textrm{ }\forall z\in\mathbb{Z}\left[\sqrt{2}\right]\label{eq:Definition of T_3,root 2} \end{equation} which acts on the ring of quadratic integers of the form $a+b\sqrt{2}$, where $a,b\in\mathbb{Z}$. Here, as indicated, a congruence $z\overset{\sqrt{2}}{\equiv}w$ for $z,w\in\mathbb{Z}\left[\sqrt{2}\right]$ means that $z-w$ is of the form $v\sqrt{2}$, where $v\in\mathbb{Z}\left[\sqrt{2}\right]$. Thus, for example, $3+2\sqrt{2}\overset{\sqrt{2}}{\equiv}1$, since: \begin{equation} 3+2\sqrt{2}-1=2+2\sqrt{2}=\left(\sqrt{2}+2\right)\sqrt{2} \end{equation} Leigh's \index{multi-dimensional!Hydra map}map is the prototypical example of what I call a \index{Hydra map!multi-dimensional}\textbf{Multi-Dimensional Hydra Map}. The ``multi-dimensionality'' becomes apparent after applying a bit of linear algebra. We can establish a ring isomorphism of $\mathbb{Z}\left[\sqrt{2}\right]$ and $\mathbb{Z}^{2}$ by associating $z=a+b\sqrt{2}\in\mathbb{Z}\left[\sqrt{2}\right]$ with the column vector $\left(a,b\right)$. A bit of algebra produces: \begin{equation} T_{3,\sqrt{2}}\left(a+b\sqrt{2}\right)=\begin{cases} b+\frac{a}{2}\sqrt{2} & \textrm{if }a=0\mod2\\ 3b+\frac{3a+1}{2}\sqrt{2} & \textrm{if }a=1\mod2 \end{cases},\textrm{ }\forall\left(a,b\right)\in\mathbb{Z}^{2}\label{eq:T_3,root 2 of a plus b root 2} \end{equation} Expressing the effect of $T_{3,\sqrt{2}}$ on $\left(a,b\right)$ in terms of these coordinates then gives us a map $\tilde{T}_{3,\sqrt{2}}:\mathbb{Z}^{2}\rightarrow\mathbb{Z}^{2}$ defined by: \begin{eqnarray} \tilde{T}_{3,\sqrt{2}}\left(\left[\begin{array}{c} a\\ b \end{array}\right]\right) & \overset{\textrm{def}}{=} & \begin{cases} \left[\begin{array}{cc} 1 & 0\\ 0 & 2 \end{array}\right]^{-1}\left(\left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]\left[\begin{array}{c} a\\ b \end{array}\right]\right) & \textrm{if }a=0\mod2\\ \left[\begin{array}{cc} 1 & 0\\ 0 & 2 \end{array}\right]^{-1}\left(\left[\begin{array}{cc} 0 & 3\\ 3 & 0 \end{array}\right]\left[\begin{array}{c} a\\ b \end{array}\right]+\left[\begin{array}{c} 0\\ 1 \end{array}\right]\right) & \textrm{if }a=1\mod2 \end{cases}\label{eq:Definition of the Lattice analogue of T_3,root 2} \end{eqnarray} which is a ``multi-dimensional'' analogue of a Hydra map; the branch: \begin{equation} n\mapsto\frac{an+b}{d} \end{equation} \textemdash an affine linear map on $\mathbb{Q}$\textemdash has been replaced by an affine linear map: \begin{equation} \mathbf{n}\mapsto\mathbf{D}^{-1}\left(\mathbf{A}\mathbf{n}+\mathbf{b}\right) \end{equation} on $\mathbb{Q}^{2}$, for $2\times2$ invertible matrices $\mathbf{A}$ and $\mathbf{D}$ and a $2$-tuple $\mathbf{b}$, all of which have integer entries. In this way, notion of Hydra map defined at the beginning of this subsection is really but a ``one-dimensional'' incarnation of the more general notion of a map on a finitely generated module over a principal ideal domain. Note in (\ref{eq:Definition of the Lattice analogue of T_3,root 2}) the presence of the permutation matrix: \begin{equation} \left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right] \end{equation} which swaps the coordinate entries of a given $2$-tuple. As will be seen in Section \ref{sec:5.1 Hydra-Maps-on}, this intertwining of an affine linear map involving diagonal matrices with permutation matrices is characteristic of the matrix representations of multi-dimensional Hydra maps, illustrating the much greater degrees of freedom and flexibility engendered by stepping out of $\mathbb{Z}$ and into spaces of higher dimensionality. \end{example} \vphantom{} Other than one additional example in Leigh's vein\textemdash presented at the very end of Subsection \ref{subsec:5.1.2 Co=00003D0000F6rdinates,-Half-Lattices,-and}\textemdash Matthews told me in our 2019 correspondence that he knew of no other examples of Hydra maps of this type in the literature. For all intents and purposes, the subject is simply unstudied. All the more reason, then, for someone to try to whip it into shape. \cite{Matthews' slides} goes even further, presenting near the end examples of Hydra-like maps defined on a ring of polynomials with coefficients in a finite field. While it may be possible to extend my techniques to study such maps, I have yet to give consideration to how my methods will apply when working with Hydra-like maps on spaces of positive characteristic. Finally, it should be mentioned that, for simplicity's sake, this dissertation only concerns prime Hydra maps\textemdash those for which $p$ is prime. While it remains to be seen whether or note my construction of $\chi_{H}$ holds water in the case where $p$ is an arbitrary composite integer, it seems pretty straightforward to modify the theory presented here to deal with the case where $p$ is the power of a prime\textemdash so, a $p^{n}$-Hydra map, where $p$ is prime and $n\geq1$. \newpage{} \subsection{\emph{\label{subsec:2.1.2 It's-Probably-True}}A Short History of the Collatz Conjecture} \begin{quote} \begin{flushright} \emph{As you know, the $3x+1$ problem is a notorious problem, with perhaps my favorite quote calling it a Soviet conspiracy to slow down American mathematics.} \par\end{flushright} \begin{flushright} \textemdash Stephen J. Miller\footnote{Professor Miller made this remark to me as part of an e-mail correspondence with me back in June of 2021.} \par\end{flushright} \end{quote} \index{Collatz!Conjecture} The Collatz Conjecture is infamous for playing hard to get. With other difficult problems, lack of progress toward a solution often comes with new ideas and insights worth studying in their own right. On the other hand, for Collatz and its ilk, the exact opposite appears to hold: a tool or insight of use for Collatz seems to have little if any applicability to the outside mathematical world. For people\textemdash like myself\textemdash with a vested interest in mainstreaming Collatz studies this is especially problematic. It isn't much help with conquering Collatz, either, given how much mathematical progress owes to the discovery of intriguing new structures, relations, and techniques, even if those novelties might not be immediately applicable to the ``big'' questions we would like to solve. The purpose of this essay is to give an overview of the primary ways in which the mathematical community at large has attempted to frame and understand the Collatz Conjecture and related arithmetical dynamical systems. That my approach is essentially orthogonal to all of the ones we are about to encounter is actually of the utmost relevance to this discussion; I hope it will inspire readers to find other ways of looking beyond the ``standard'' methods, so as to improve our chances of finding exactly the sort of unexpected connections which might be needed to better understand problems like Collatz. To the extent that Collatz studies exists as a coherent subject, however, the most preponderant trends are statistical in nature, where studies center around defining a handful of interesting quantities\textemdash hoping to bottle a sliver of the lightning animating Collatz's behavior\textemdash and then demonstrating that interesting conclusions can be drawn. Arguably the most impactful techniques of the statistical school have their roots in the groundbreaking work done by Riho Terras\index{Terras, Riho} in the 1970s (\cite{Terras 76,Terras 79}; see also Lagarias' survey \cite{Lagarias' Survey}). Terras' paradigm rests upon two statistics: the \textbf{stopping time} and the \textbf{parity sequence} (or \textbf{parity vector}). \begin{defn}[\textbf{Stopping times}\footnote{Adapted from \cite{Lagarias-Kontorovich Paper}.}] Let $\lambda$ be any positive real number, and let $T:\mathbb{Z}\rightarrow\mathbb{Z}$ be a map. \vphantom{} I. For any $n\in\mathbb{Z}$, the \textbf{$\lambda$-decay stopping time}\index{stopping time}\textbf{ }of $n$ under $T$, denoted $\sigma_{T,\lambda}^{-}\left(n\right)$, is the smallest integer $k\geq0$ so that $T^{\circ k}\left(n\right)<\lambda n$: \begin{equation} \sigma_{T,\lambda}^{-}\left(n\right)\overset{\textrm{def}}{=}\inf\left\{ k\geq0:\frac{T^{\circ k}\left(n\right)}{n}<\lambda\right\} \label{eq:Definition of the lambda decay stopping time} \end{equation} \vphantom{} II. For any $n\in\mathbb{Z}$, the \textbf{$\lambda$-growth stopping time }of $n$ under $T$, denoted $\sigma_{T,\lambda}^{+}\left(n\right)$, is the smallest integer $k\geq0$ so that $T^{\circ k}\left(n\right)>\lambda n$: \begin{equation} \sigma_{T,\lambda}^{+}\left(n\right)\overset{\textrm{def}}{=}\inf\left\{ k\geq0:\frac{T^{\circ k}\left(n\right)}{n}>\lambda\right\} \label{eq:Definition of the lambda growth stopping time} \end{equation} \end{defn} \vphantom{} Terras' claim to fame rests on his ingenious implementation of these ideas to prove: \begin{thm}[\textbf{Terras' Theorem} \cite{Terras 76,Terras 79}] Let $S$ be set of positive integers with finite \index{Terras' Theorem}$1$-decay\footnote{i.e., $\lambda$-decay, with $\lambda=1$.} stopping time under the Collatz map $C$. Then, $S$ has a natural density of $1$; that is to say: \begin{equation} \lim_{N\rightarrow\infty}\frac{\left\|S\cap\left\{ 1,2,3,\ldots,N\right\} \right\|}{N}=1\label{eq:Terras' Theorem} \end{equation} \end{thm} In words, Terras' Theorem is effectively the statement \emph{almost every positive integer does not belong to a divergent trajectory of $C$}; or, equivalently, \emph{the set of divergent trajectories of $C$ in $\mathbb{N}_{1}$ has density $0$}.\emph{ }This result is of double importance: not only does it prove that $S$ \emph{possesses }a density, it also shows that density to be $1$. One of the frustrations of analytic number theory is that the limit (\ref{eq:Terras' Theorem}) defining the natural density of a set $S\subseteq\mathbb{N}_{1}$ does not exist for every\footnote{As an example, the set $S=\bigcup_{n=0}^{\infty}\left\{ 2^{2n},\ldots,2^{2n+1}-1\right\} $ does not have a well-defined natural density. For it, the $\limsup$ of (\ref{eq:Terras' Theorem}) is $2/3$, while the $\liminf$ is $1/3$.} $S\subseteq\mathbb{N}_{1}$. Tao's breakthrough was to improve ``almost every positive integer does not go to infinity'' to ``almost all orbits of the Collatz map attain almost bounded values''\textemdash indeed, that is the very title of his paper \cite{Tao Probability paper}. It is worth noting that by doing little more than replacing every instance of a $3$ in Terras' argument with a $5$\textemdash and adjusting all inequalities accordingly\textemdash an analogous result can be obtained for the Shortened $5x+1$ map\index{$5x+1$ map} (or $T_{q}$, for any odd $q\geq5$) \cite{Lagarias-Kontorovich Paper}: \begin{thm}[\textbf{Terras' Theorem for $qx+1$}] Let $S$ be set of positive integers with finite $1$-growth\footnote{i.e., $\lambda$-growth, with $\lambda=1$.} stopping time under the Shortened $qx+1$ map $T_{q}$. Then, $S$ has a natural density of $1$. \end{thm} \vphantom{} That is to say, while the set of positive integers which belong to divergent trajectories of $T_{3}$ has density $0$, the set of divergent trajectories of $T_{q}$ for $q\geq5$ has density $1$. Surreally, for $q\geq5$, even though we know that almost every positive integer should be iterated to $\infty$ by $T_{q}$, mathematics has yet to prove that any specific positive integer \emph{suspected}\footnote{Such as $7$, for the case of the $5x+1$ map.} of membership in a divergent trajectory actually belongs to one! Next, we have Terras' notion of the \index{parity sequence}\textbf{parity sequence} of a given input $n$. The idea is straightforward: since the behavior of an integer $n$ under iterations of one of the $T_{q}$ maps is determined by the parity of those iterates\textemdash that is, the value mod $2$\textemdash it makes sense to keep track of those values. As such, the parity sequence of $n$ under $T_{q}$ is the sequence $\left[n\right]_{2},\left[T_{q}\left(n\right)\right]_{2},\left[T_{q}^{\circ2}\left(n\right)\right]_{2},\ldots$ consisting of the values (mod $2$) of the iterates of $n$ under the map $T_{q}$ (\ref{eq:Definition of T_q}) \cite{Lagarias' Survey}. More generally, since the iterates of $n$ under an arbitrary $p$-Hydra map $H$ are determined not by their iterates' parities, but by their iterates' values mod $p$, it will be natural to generalize Terras' parity sequence into the \textbf{$p$-ity sequence}\index{$p$-ity sequence}\textbf{ }$\left[n\right]_{p},\left[H\left(n\right)\right]_{p},\left[H^{\circ2}\left(n\right)\right]_{p},\ldots$ consisting of the values mod $p$ of the iterates of $n$ under $H$; doing so will play a crucial role early on in our construction of the numen of $H$. It should not come as much of a surprise that the parity sequence of a given $n$ under $T_{q}$ ends up being intimately connected to the behavior of $T_{q}$'s extension to the $2$-adic integers (see for instance \cite{Parity Sequences}), where we apply the even branch of $T_{q}$ to those $\mathfrak{z}\in\mathbb{Z}_{2}$ with $\mathfrak{z}\overset{2}{\equiv}0$ and apply the odd branch of $T_{q}$ to those $\mathfrak{z}\in\mathbb{Z}_{2}$ with $\mathfrak{z}\overset{2}{\equiv}1$ Extending the environment of study from $\mathbb{Z}$ to $\mathbb{Z}_{2}$ in this manner is another one of Terras' innovations, one which is relevant to our approach for all the obvious reasons. Despite these promising connections, trying to conquer Collatz simply by extending it to the $2$-adics is a quixotic undertaking, thanks to the following astonishing result\footnote{Adapted from \cite{Lagarias-Kontorovich Paper}.}: \begin{thm} \label{thm:shift map}For any odd integer $q\geq1$, the $2$-adic extension of $T_{q}$ is a homeomorphism of $\mathbb{Z}_{2}$, and is topologically and measurably conjugate to the $2$-adic shift map $\theta:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{2}$ defined by: \begin{equation} \theta\left(c_{0}+c_{1}2^{1}+c_{2}2^{2}+\cdots\right)\overset{\textrm{def}}{=}c_{1}+c_{2}2^{1}+c_{3}2^{2}+\cdots\label{eq:Definition of the 2-adic shift map} \end{equation} meaning that for every odd integer $q\geq1$, there exists a homeomorphism $\Phi_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{2}$ which preserves the (real-valued) $2$-adic Haar probability measure\footnote{$\mu\left(\Phi_{q}^{-1}\left(U\right)\right)=\mu\left(U\right)$, for all measurable $U\subseteq\mathbb{Z}_{2}$.} so that: \begin{equation} \left(\Phi_{q}\circ T_{q}\circ\Phi_{q}^{-1}\right)\left(\mathfrak{z}\right)=\theta\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Conjugacy of T_q and the shift map} \end{equation} \end{thm} \vphantom{} As this theorem shows, in both a topological and measure theoretic perspective, for any odd integers $p,q\geq1$, the dynamics $T_{p}$ and $T_{q}$ on $\mathbb{Z}_{2}$ are utterly indistinguishable from one another, and from the shift map, which is ergodic on $\mathbb{Z}_{2}$ in a very strong sense \cite{Lagarias-Kontorovich Paper}. Because $\mathbb{Z}$ has zero real-valued Haar measure in $\mathbb{Z}_{2}$, it was already a given that employing tools of \index{ergodic theory}ergodic theory and the measure-theoretic branches of dynamical systems would be something of a wash; those methods cannot detect phenomena occurring in sets of measure zero. However, the conjugate equivalence of the different $T_{q}$ maps would appear to suggest that, at least over the $2$-adics, probabilistic methods will useless for understanding the dynamics of the $qx+1$ maps, let alone the Collatz map. Nevertheless, as shown in \cite{Parity Sequences} and related works, there may yet be fruitful work to be done in the $2$-adic setting, not by directly studying the dynamics of $T_{3}$, but by how certain functions associated to $T_{3}$ affect certain subsets of $\mathbb{Z}_{2}$. Prior to Tao's innovations \cite{Tao Probability paper}, among the motley collection of techniques used to study the Collatz Conjecture, the least \emph{in}effective arguments in support of the Conjecture's truth were based on the so-called ``difference inequalities'' of I. Krasikov\index{Krasikov, Ilia} \cite{Krasikov}. In the 1990s, Applegate and Lagarias\index{Lagarias, Jeffery} used Krasikov's inequalities (along with a computer to assist them with a linear program in $\left(3^{9}-1\right)/2$ variables! \cite{Applegate and Lagarias - Difference inequalities}) to establish the asymptotic: \begin{thm}[\textbf{Applegate-Krasikov-Lagarias} \cite{Applegate and Lagarias - Difference inequalities}] For any integer $a$, define $\pi_{a}:\mathbb{R}\rightarrow\mathbb{N}_{0}$ by: \begin{equation} \pi_{a}\left(x\right)\overset{\textrm{def}}{=}\left\|\left\{ n\in\mathbb{Z}:\left\|n\right\|\leq x\textrm{ and }T^{\circ k}\left(n\right)=a\textrm{ for some }k\in\mathbb{N}_{0}\right\} \right\|\label{eq:Definition of Pi_a} \end{equation} Then, for any $a\in\mathbb{Z}$ which is not a multiple of $3$, there is a positive real constant $c_{a}$ so that: \begin{equation} \pi_{a}\left(x\right)\geq c_{a}x^{0.81},\textrm{ }\forall x\geq a\label{eq:Apple-Lag-Kra asymptotic theorem} \end{equation} \end{thm} \vphantom{} This school of work \cite{Applegate and Lagarias - Trees,Applegate and Lagarias - Difference inequalities,Krasikov,Wirsching's book on 3n+1} intersects with graph-theoretic considerations (such as \cite{Applegate and Lagarias - Trees}), a well-established player in probabilistic number theory and additive combinatorics in the twentieth century and beyond. With the exception of Rozier's work in \cite{Parity Sequences}, all of the avenues of inquiry discussed above\textemdash even Tao's!\textemdash suffer the same weakness, inherent to any \emph{probabilistic }approach: they can only establish truths modulo ``small'' sets (zero density, zero measure, etc.). Although results in this tradition provide often deeply insightful, psychologically satisfying substitutes for resolutions of the Collatz Conjecture, in part or in whole, they nevertheless fail to capture or address whatever number-theoretic mechanisms are ultimately responsible for the Collatz's dynamics, and the dynamics of Hydra maps in general. In this respect, the Markov-chain\index{Markov chains}-based work of Mller, Matthews, and their school offers an intriguing contrast, if only because their methods show\textemdash or, at least, \emph{suggest}\textemdash how the dynamical properties of Hydra maps might be related to the constants used to define them. Mller's generalization of Collatz began by fixing co-prime integers $m,d>1$, with $m>d$ and considering the map $T_{\textrm{Mller}}:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by \cite{Moller's paper (german),Matthews' slides}: \begin{equation} T_{\textrm{Mller}}\left(x\right)\overset{\textrm{def}}{=}\begin{cases} \frac{x}{d} & \textrm{if }x\overset{d}{\equiv}0\\ \frac{mx-1}{d} & \textrm{if }x\overset{d}{\equiv}\left[m\right]_{d}^{-1}\\ \frac{mx-2}{d} & \textrm{if }x\overset{d}{\equiv}2\left[m\right]_{d}^{-1}\\ \vdots & \vdots\\ \frac{mx-\left(d-1\right)}{d} & \textrm{if }x\overset{d}{\equiv}\left(d-1\right)\left[m\right]_{d}^{-1} \end{cases},\textrm{ }\forall x\in\mathbb{Z}\label{eq:Moller's map} \end{equation} Mller's analysis led him to conjecture: \begin{conjecture} $T_{\textrm{Mller}}$ eventually iterates every $x\in\mathbb{Z}$ if and only if $m<d^{d/\left(d-1\right)}$. Also, regardless of whether or not $m<d^{d/\left(d-1\right)}$, $T_{\textrm{Mller}}$ has finitely many cycles. \end{conjecture} \vphantom{} Matthews \cite{Matthews' slides} generalizes Mller's work like so. Let $d\geq2$, and let $m_{0},\ldots,m_{d-1}$ be non-zero integers (not necessarily positive). Also, for $j\in\left\{ 0,\ldots,d-1\right\} $, let $r_{j}\in\mathbb{Z}$ satisfy $r_{j}\overset{d}{\equiv}jm_{j}$. Then, define $T_{\textrm{Matthews}}:\mathbb{Z}\rightarrow\mathbb{Z}$ by: \begin{equation} T_{\textrm{Matthews}}\left(x\right)\overset{\textrm{def}}{=}\begin{cases} \frac{m_{0}x-r_{0}}{d} & \textrm{if }x\overset{d}{\equiv}0\\ \vdots & \vdots\\ \frac{m_{d-1}x-r_{d-1}}{d} & \textrm{if }x\overset{d}{\equiv}d-1 \end{cases},\textrm{ }\forall x\in\mathbb{Z}\label{eq:Matthews' Moller-map} \end{equation} He calls this map \textbf{relatively prime }whenever $\gcd\left(m_{j},d\right)=1$ for all $j\in\mathbb{Z}/d\mathbb{Z}$, and then makes the following conjectures (given in \cite{Matthews' slides}): \begin{conjecture} For the relatively prime case: \vphantom{} I. Every trajectory of $T_{\textrm{Matthews}}$ on $\mathbb{Z}$ is eventually periodic whenever: \begin{equation} \prod_{j=0}^{d-1}\left\|m_{j}\right\|<d^{d} \end{equation} \vphantom{} II. If: \begin{equation} \prod_{j=0}^{d-1}\left\|m_{j}\right\|>d^{d} \end{equation} then the union of all divergent orbit classes of $T_{\textrm{Matthews}}$ has density $1$ in $\mathbb{Z}$. \vphantom{} III. Regardless of the inequalities in (I) and (II), $T_{\textrm{Matthews}}$ always has a finite, non-zero number of cycles in $\mathbb{Z}$. \vphantom{} IV. Regardless of the inequalities in (I) and (II), for any $x\in\mathbb{Z}$ which belongs to a divergent orbit class of $T_{\textrm{Matthews}}$, the iterates of $x$ under $T$ are uniformly distributed modulo $d^{n}$ for every $n\geq1$; that is: \begin{equation} \lim_{N\rightarrow\infty}\frac{1}{N}\left\|\left\{ k\in\left\{ 0,\ldots,N-1\right\} :T^{\circ k}\left(x\right)\overset{d^{n}}{\equiv}j\right\} \right\|=\frac{1}{d^{n}}\label{eq:Matthews' Conjecture on uniform distribution modulo powers of rho of iterates of elements of divergent trajectories} \end{equation} \end{conjecture} \vphantom{} Even though these are conjectures, rather than results, they have the appeal of asserting a connection between the maps' parameters and the maps' dynamics. Even though probabilistic methods are not my strong suit, I am fond of approaches like these because of their open-mindedness, as well as the similarities between them and some of my currently unpublished investigations (principally ergodic theoretic) into the actions of Hydra maps on $\check{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$, the ring of profinite integers (the product is taken over all primes $p$) and its subgroups (ex: for Collatz, $\mathbb{Z}_{2}$ and $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$). As Lagarias has frequently told me (or intimated) in my on-and-off correspondences with him over the past five years, any significant progress on the Collatz Conjecture should either shed light on the dynamics of close relatives like the $5x+1$ map, or give insight into why the arguments used depend on the $3$ in $3x+1$. As such, if we're going to try to surmount an insurmountable obstacle like the Collatz Conjecture, we might as well cast wide nets, in search of broader structures, avoiding any unnecessary dotage over $3x+1$ until we think there might be a way to tackle it. That, of course, is exactly what I intend to do here. \newpage{} \section{\label{sec:2.2-the-Numen}$\chi_{H}$, the Numen of a Hydra Map} THROUGHOUT THIS SECTION, WE ASSUME $p$ IS A PRIME AND THAT $H$ IS A $p$-HYDRA MAP WITH $H\left(0\right)=0$. \vphantom{} In 1978, Bhm and Sontacchi \cite{Bohm and Sontacchi} (see also \cite{Wirsching's book on 3n+1}) established what has since become a minor (and oft-rediscovered) cornerstone of Collatz literature: \begin{thm}[\textbf{Bhm-Sontacchi Criterion}\footnote{A monicker of my own invention.}] \label{thm:Bohm-Sontacchi}Let\index{Bhm-Sontacchi criterion}\index{diophantine equation} $x\in\mathbb{Z}$ be a periodic point of the Collatz map $C$ (\ref{eq:Collatz Map}). Then, there are integers $m,n\geq1$ and a strictly increasing sequence of positive integers $b_{1}<b_{2}<\cdots<b_{n}$ so that: \begin{equation} x=\frac{\sum_{k=1}^{n}2^{m-b_{k}-1}3^{k-1}}{2^{m}-3^{n}}\label{eq:The Bohm-Sontacchi Criterion} \end{equation} \end{thm} \vphantom{} (\ref{eq:The Bohm-Sontacchi Criterion}) is arguably the clearest expression of how the Collatz Conjecture reaches out of the gyre of recreational mathematics and up toward more refined spheres of study. Not only is (\ref{eq:The Bohm-Sontacchi Criterion}) an exponential diophantine equation\textemdash an area of number theory as classical as it is difficult\textemdash but is, worse yet, a \emph{family }of exponential diophantine equations. Especially noteworthy is the denominator term, $2^{m}-3^{n}$, which evidences (\ref{eq:The Bohm-Sontacchi Criterion}) as falling under the jurisdiction of of transcendental number theory\footnote{A slightly more in-depth explanation of this connection is given beginning from page \pageref{subsec:Baker,-Catalan,-and}.}, in which the study of lower bounds for quantities such as $\left\|2^{m}-3^{n}\right\|$ is of the utmost importance. The most significant result of the present section\textemdash and, one of this dissertation's three principal achievements\textemdash is the \textbf{Correspondence Principle }(proved in Subsection \ref{subsec:2.2.3 The-Correspondence-Principle}), which significantly generalizes the \textbf{Bhm-Sontacchi Criterion}, and in two respects. First, it shows that the integers $m$, $n$, and the $b_{k}$s can all be parameterized as functions of a single variable in $\mathbb{Z}$. Secondly, it generalizes the criterion to apply to a much larger family of Hydra maps. Not only that, but, we will be able to do the same for Hydra maps on $\mathbb{Z}^{d}$ in Chapter 5! It should be noted that a generalized form of the Criterion is given in Matthews' slides \cite{Matthews' slides}, but the version there is not as broadly minded as mine, and lacks the single-variable parameterization of the integer parameters in (\ref{eq:The Bohm-Sontacchi Criterion}) produced my $\chi_{H}$ formalism. The ideas needed to make all this work emerged from considering the $p$-ity vector associated to a given integer $n$ under the particular $p$-Hydra map $H$ we are studying. To illustrate this, let us consider the Shortened Collatz map. We will denote the two branches of $T_{3}$ ($x/2$ and $\left(3x+1\right)/2$) by $H_{0}\left(x\right)$ and $H_{1}\left(x\right)$, respectively. Letting $n\geq1$, note that as we successively apply $T_{3}$ to $n$, the iterates $T_{3}\left(n\right),T_{3}\left(T_{3}\left(n\right)\right),\ldots$ can be expressed in terms of compositions of $H_{0}$ and $H_{1}$: \begin{example} \ \begin{align} T_{3}\left(1\right) & =2=\frac{3\left(1\right)+1}{2}=H_{1}\left(1\right)\\ T_{3}\left(T_{3}\left(1\right)\right) & =1=\frac{T_{3}\left(1\right)}{2}=H_{0}\left(T_{3}\left(1\right)\right)=H_{0}\left(H_{1}\left(1\right)\right)\\ & \vdots \end{align} \end{example} \vphantom{} Given integers $m,n\geq1$, when we express the $m$th iterate of $n$ under $T_{3}$: \begin{equation} T_{3}^{\circ m}\left(n\right)=\underbrace{\left(T_{3}\circ T_{3}\circ\cdots\circ T_{3}\right)}_{m\textrm{ times}}\left(n\right) \end{equation} as a composition of $H_{0}$ and $H_{1}$, the sequence of the subscripts $0$ and $1$ that occur in this sequence of maps is clearly equivalent to the parity vector generated by $T_{3}$ over $n$. In this ``classical'' view of a parity vector, the parity vector is \emph{subordinated }to $n$, being obtained \emph{from }$n$. My approach is to turn this relationship on its head: what happens if we reverse it? That is, what happens if we let the \emph{parity vector} (or, more generally, $p$-ity vector) determine $n$? For $T_{3}$, we can see this reversal in action by considering the kinds of composition sequences of $H_{0}$ and $H_{1}$ that occur when we are given an integer $x$ which is a periodic point $x$ of $T_{3}$. For such an $x$, there is an integer $m\geq1$ so that $T_{3}^{\circ m}\left(x\right)=x$, which is to say, there is some length-$m$ composition sequence of $H_{0}$ and $H_{1}$ that has $x$ as a fixed point. Instead of starting with $x$ as the given, however, we treat it as an unknown to be solved for. \begin{example} Suppose $x$ is fixed by the sequence: \begin{equation} x=H_{1}\left(H_{1}\left(H_{0}\left(H_{1}\left(H_{0}\left(x\right)\right)\right)\right)\right) \end{equation} Expanding the right-hand side, we obtain: \begin{equation} x=\frac{1}{2}\left(3\left[\frac{1}{2}\left(3\left[\frac{1}{2}\left(\frac{3\left(\frac{x}{2}\right)+1}{2}\right)\right]+1\right)\right]+1\right)=\frac{3^{3}}{2^{5}}x+\frac{3^{2}}{2^{4}}+\frac{3}{2^{2}}+\frac{1}{2}\label{eq:Bohm-Sontacci Example} \end{equation} and hence: \[ x=\frac{3^{2}\cdot2+3\cdot2^{3}+2^{4}}{2^{5}-3^{3}}=\frac{58}{5} \] Since $58/5$ is not an integer, we can conclude that there is no integer fixed point of $T_{3}$ whose iterates have the parity sequence generated by the string $0,1,0,1,1$. \end{example} \vphantom{} In general, to any \emph{finite }sequence $j_{1},j_{2},j_{3},\ldots,j_{N}$ consisting of $0$s and $1$s, we can solve the equation: \begin{equation} x=\left(H_{j_{1}}\circ H_{j_{2}}\circ\cdots\circ H_{j_{N}}\right)\left(x\right)\label{eq:Periodic point set up for T3} \end{equation} for $x$, and, in doing so, obtain a map\footnote{When we generalize to consider an arbitrary $p$-Hydra map $H$ instead of $T_{3}$, we will obtain a particular $x$ for each $H$.} from the space of all such sequences to the set of rational numbers. That this works is because $H_{0}$ and $H_{1}$ are bijections of $\mathbb{Q}$. Thus, any finite composition sequence $S$ of these two branches is also a bijection, and as such, there is a unique $x\in\mathbb{Q}$ so that $x=S\left(x\right)$, which can be determined in the manner shown above. To put it another way, we can \textbf{\emph{parameterize}} the quantities $m$, $n$, $x$, and $b_{k}$ which appear in the Bhm-Sontacchi Criterion (\ref{eq:Bohm-Sontacci Example}) in terms of the sequence $j_{1},\ldots,j_{N}$ of branches of $T_{3}$ which we applied to map $x$ to itself. Viewing (\ref{eq:Bohm-Sontacci Example}) as a particular case of this formula, note that in the above example, the values of $m$, $n$, the $b_{k}$s in (\ref{eq:Bohm-Sontacci Example}), and\textemdash crucially\textemdash $x$ itself were determined by our choice of the composition sequence: \begin{equation} H_{1}\circ H_{1}\circ H_{0}\circ H_{1}\circ H_{0} \end{equation} and our assumption that this sequence mapped $x$ to $x$. In this way, we will be able to deal with all the possible variants of (\ref{eq:Bohm-Sontacci Example}) in the context of a single object\textemdash to be called $\chi_{3}$. Just as we used sequences $j_{1},\ldots,j_{N}$ in $\left\{ 0,1\right\} $ to deal with the $2$-Hydra map $T_{3}$, for the case of a general $p$-Hydra map, our sequences $j_{1},\ldots,j_{N}$ will consist of integers in the set $\left\{ 0,\ldots,p-1\right\} $, with each sequence corresponding to a particular composition sequences of the $p$ distinct branches of the $p$-Hydra map $H$ under consideration. In doing so, we will realize these expressions as specific cases of a more general functional equation satisfied the function I denote by $\chi_{H}$. We will initially define $\chi_{H}$ as a rational-valued function of finite sequences $j_{1},j_{2},\ldots$. However, by a standard bit of identifications, we will re-contextualize $\chi_{H}$ as a rational function on the non-negative integers $\mathbb{N}_{0}$. By placing certain qualitative conditions on $H$, we can ensure that $\chi_{H}$ will remain well-defined when we extend its inputs from $\mathbb{N}_{0}$ to $\mathbb{Z}_{p}$, thereby allowing us to interpolate $\chi_{H}$ to a $q$-adic valued function over $\mathbb{Z}_{p}$ for an appropriate choice of primes $p$ and $q$. \subsection{\label{subsec:2.2.1 Notation-and-Preliminary}Notation and Preliminary Definitions} Fix a $p$-Hydra map $H$, and \textbf{suppose that $b_{0}=0$} so as to guarantee that $H\left(0\right)=0$; note also that this then makes $H_{0}\left(0\right)=0$. In our study, \emph{it is imperative that }\textbf{\emph{$b_{0}=0$}}. We begin by introducing formalism\textemdash \textbf{strings}\textemdash to used to denote sequences of the numbers $\left\{ 0,\ldots,p-1\right\} $. \begin{defn}[\textbf{Strings}] We write \nomenclature{$\textrm{String}\left(p\right)$}{ }$\textrm{String}\left(p\right)$ to denote the set of all finite sequences whose entries belong to the set $\left\{ 0,\ldots,p-1\right\} $; we call such sequences \textbf{strings}. We also include the empty set ($\varnothing$) as an element of $\textrm{String}\left(p\right)$, and refer to it as the \textbf{empty string}. Arbitrary strings are usually denoted by $\mathbf{j}=\left(j_{1},\ldots,j_{n}\right)$, where $n$ is the \textbf{length }of the string, denoted by \nomenclature{$\left\|\mathbf{j}\right\|$}{ }$\left\|\mathbf{j}\right\|$. In this manner, any $\mathbf{j}$ can be written as $\mathbf{j}=\left(j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}\right)$. We define the empty string as having length $0$, making it the unique string of length $0$. We say the string $\mathbf{j}$ is \textbf{non-zero }if $\mathbf{j}$ contains at least one non-zero entry. \vphantom{} We write \nomenclature{$\textrm{String}_{\infty}\left(p\right)$}{ }$\textrm{String}_{\infty}\left(p\right)$ to denote the set of all all sequences (finite \emph{or }infinite) whose entries belong to the set $\left\{ 0,\ldots,p-1\right\} $. A \textbf{finite }string is one with finite length; on the other hand, any string in $\textrm{String}_{\infty}\left(p\right)$ which is not finite is said to be \textbf{infinite}, and its length is defined to be $+\infty$. We also include $\varnothing$ in $\textrm{String}_{\infty}\left(p\right)$, again referred to as the empty string. \end{defn} \begin{defn} Given any $\mathbf{j}\in\textrm{String}\left(p\right)$, we define the \index{composition sequence}\textbf{composition sequence }$H_{\mathbf{j}}:\mathbb{Q}\rightarrow\mathbb{Q}$ as the affine linear map: \nomenclature{$H_{\mathbf{j}}\left(x\right)$}{$\overset{\textrm{def}}{=}\left(H_{j_{1}}\circ\cdots\circ H_{j_{\left\|\mathbf{j}\right\|}}\right)\left(x\right)$} \begin{equation} H_{\mathbf{j}}\left(x\right)\overset{\textrm{def}}{=}\left(H_{j_{1}}\circ\cdots\circ H_{j_{\left\|\mathbf{j}\right\|}}\right)\left(x\right),\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:Def of composition sequence} \end{equation} When writing by hand, one can use vector notation: $\vec{j}$ instead of $\mathbf{j}$. \end{defn} \begin{rem} Note the near-equivalence of $\mathbf{j}$ with $p$-ity vectors. Indeed, given any $m,n\in\mathbb{N}_{1}$, there exists a unique $\mathbf{j}\in\textrm{String}\left(p\right)$ of length $m$ so that $H^{\circ m}\left(n\right)=H_{\mathbf{j}}\left(n\right)$. Said $\mathbf{j}$ is the $p$-ity vector for the first $m$ iterates of $n$ under $H$, albeit written in reverse order. While, admittedly, this reverse-order convention is somewhat unnatural, it does have one noteworthy advantage. In a moment, we will identify $\mathbf{j}$ with a $p$-adic integer $\mathfrak{z}$ by viewing the entries of $\mathbf{j}$ as the $p$-adic digits of $\mathfrak{z}$, written left-to-right in order of increasing powers of $p$. Our ``reverse'' indexing convention guarantees that the left-to-right order in which we write the $H_{j_{k}}$s, the left-to-right order in which we write the entries of $\mathbf{j}$, and the left-to-write order in which we write the $p$-adic digits of the aforementioned $\mathfrak{z}$ are all the same. \end{rem} \vphantom{} Strings will play a vital role in this chapter. Not only do they simplify many arguments, they also serve as a go-between for the dynamical-system motivations (considering arbitrary composition sequences of $H_{j}$s) and the equivalent reformulations in terms of non-negative and/or $p$-adic integers. As was mentioned above and is elaborated upon below, we will identify elements of $\textrm{String}\left(p\right)$ with the sequences of base $p$ digits of non-negative integers. Likewise, we will identify elements of $\textrm{String}_{\infty}\left(p\right)$ with the sequences of $p$-adic digits of $p$-adic integers. We do this by defining the obvious map for converting strings into ($p$-adic) integers: \begin{defn} We write \nomenclature{$\textrm{Dig}\textrm{Sum}_{p}$}{ }$\textrm{Dig}\textrm{Sum}_{p}:\textrm{String}_{\infty}\left(p\right)\rightarrow\mathbb{Z}_{p}$ by: \begin{equation} \textrm{DigSum}_{p}\left(\mathbf{j}\right)\overset{\textrm{def}}{=}\sum_{k=1}^{\left\|\mathbf{j}\right\|}j_{k}p^{k-1}\label{eq:Definition of DigSum_p of bold j} \end{equation} \end{defn} \begin{defn}[\textbf{Identifying Strings with Numbers}] Given $\mathbf{j}\in\textrm{String}_{\infty}\left(p\right)$ and $\mathfrak{z}\in\mathbb{Z}_{p}$, we say $\mathbf{j}$ \textbf{represents }(or \textbf{is} \textbf{associated to})\textbf{ }$\mathfrak{z}$ (and vice-versa), written $\mathbf{j}\sim\mathfrak{z}$ or $\mathfrak{z}\sim\mathbf{j}$ whenever $\mathbf{j}$ is the sequence of the $p$-adic digits of $n$; that is: \begin{equation} \mathfrak{z}\sim\mathbf{j}\Leftrightarrow\mathbf{j}\sim\mathfrak{z}\Leftrightarrow\mathfrak{z}=j_{1}+j_{2}p+j_{3}p^{2}+\cdots\label{eq:Definition of n-bold-j correspondence.} \end{equation} equivalently: \begin{equation} \mathbf{j}\sim\mathfrak{z}\Leftrightarrow\textrm{DigSum}_{p}\left(\mathbf{j}\right)=\mathfrak{z} \end{equation} As defined, $\sim$ is then an equivalence relation on $\textrm{String}\left(p\right)$ and $\textrm{String}_{\infty}\left(p\right)$. We write: \begin{equation} \mathbf{i}\sim\mathbf{j}\Leftrightarrow D_{p}\left(\mathbf{i}\right)=D_{p}\left(\mathbf{j}\right)\label{eq:Definition of string rho equivalence relation, rational integer version} \end{equation} and we have that $\mathbf{i}\sim\mathbf{j}$ if and only if both $\mathbf{i}$ and $\mathbf{j}$ represent the same $p$-adic integer. Note that in both $\textrm{String}\left(p\right)$ and $\textrm{String}_{\infty}\left(p\right)$, the shortest string representing the number $0$ is then the empty string. Finally, we write \nomenclature{$\textrm{String}\left(p\right)/\sim$}{ }$\textrm{String}\left(p\right)/\sim$ and \nomenclature{$\textrm{String}_{\infty}\left(p\right)/\sim$}{ }$\textrm{String}_{\infty}\left(p\right)/\sim$ to denote the set of equivalence classes of $\textrm{String}\left(p\right)$ and $\textrm{String}_{\infty}\left(p\right)$ under this equivalence relation. \end{defn} \begin{prop} \label{prop:string number equivalence}\ \vphantom{} I. $\textrm{String}\left(p\right)/\sim$ and $\textrm{String}_{\infty}\left(p\right)/\sim$ are in bijective correspondences with $\mathbb{N}_{0}$ and $\mathbb{Z}_{p}$, respectively, by way of the map $\textrm{Dig}\textrm{Sum}_{p}$. \vphantom{} II. Two finite strings $\mathbf{i}$ and $\mathbf{j}$ satisfy $\mathbf{i}\sim\mathbf{j}$ if and only if either: \vphantom{} a) $\mathbf{i}$ can be obtained from $\mathbf{j}$ by adding finitely many $0$s to the right of $\mathbf{j}$. \vphantom{} b) $\mathbf{j}$ can be obtained from $\mathbf{i}$ by adding finitely many $0$s to the right of $\mathbf{i}$. \vphantom{} III. A finite string $\mathbf{i}$ and an infinite string $\mathbf{j}$ satisfy $\mathbf{i}\sim\mathbf{j}$ if and only if $\mathbf{j}$ contains finitely many non-zero entries. \end{prop} Proof: Immediate from the definitions. Q.E.D. \vphantom{} The principal utility of this formalism is the way in which composition sequences $H_{\mathbf{j}}$ interact with concatenations of strings. \begin{defn}[\textbf{Concatenation}] We introduce the \textbf{concatenation operation}\index{concatenation!operation} $\wedge:\textrm{String}\left(p\right)\times\textrm{String}_{\infty}\left(p\right)\rightarrow\textrm{String}_{\infty}\left(p\right)$, defined by: \begin{equation} \mathbf{i}\wedge\mathbf{j}=\left(i_{1},\ldots,i_{\left\|\mathbf{i}\right\|}\right)\wedge\left(j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}\right)\overset{\textrm{def}}{=}\left(i_{1},\ldots,i_{\left\|\mathbf{i}\right\|},j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}\right)\label{eq:Definition of Concatenation} \end{equation} with the definition being modified in the obvious way to when $\mathbf{j}$ is of infinite length. \vphantom{} Additionally, for any integer $m\geq1$ and any finite string $\mathbf{j}$, we write $\mathbf{j}^{\wedge m}$ to denote the concatenation of $m$ copies of $\mathbf{j}$: \begin{equation} \mathbf{j}^{\wedge m}\overset{\textrm{def}}{=}\left(\underbrace{j_{1},\ldots,j_{\left\|\mathbf{j}\right\|},j_{1},\ldots,j_{\left\|\mathbf{j}\right\|},\ldots,j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}}_{m\textrm{ times}}\right)\label{eq:Definition of concatenation exponentiation} \end{equation} \end{defn} \vphantom{} In the next subsection, the proposition below will be the foundation for various useful functional equations: \begin{prop} \label{prop:H string formalism}\ \begin{equation} H_{\mathbf{i}\wedge\mathbf{j}}\left(x\right)=H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(x\right)\right),\textrm{ }\forall x\in\mathbb{R},\textrm{ }\forall\mathbf{i},\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:H string formalism} \end{equation} \end{prop} Proof: \begin{equation} H_{\mathbf{i}\wedge\mathbf{j}}\left(x\right)=H_{i_{1},\ldots,i_{\left\|\mathbf{i}\right\|},j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}}\left(x\right)=\left(H_{i_{1},\ldots,i_{\left\|\mathbf{i}\right\|}}\circ H_{j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}}\right)\left(x\right)=H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(x\right)\right) \end{equation} Q.E.D. \vphantom{} In working with strings and the ($p$-adic) integers they represent, the following functions will be useful to help bridge string-world and number-world. \begin{defn}[$\lambda_{p}$ \textbf{and} $\#_{p:0},\#_{p:1},\ldots,\#_{p:p-1}$] We write $\lambda_{p}:\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$ to denote the function:\nomenclature{$\lambda_{p}\left(n\right)$}{$\overset{\textrm{def}}{=}\left\lceil \log_{p}\left(n+1\right)\right\rceil$ \nopageref } \begin{equation} \lambda_{p}\left(n\right)\overset{\textrm{def}}{=}\left\lceil \log_{p}\left(n+1\right)\right\rceil ,\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:definition of lambda rho} \end{equation} which gives the number of $p$-adic digits of $n$. Every $n\in\mathbb{N}_{0}$ can be uniquely written as: \begin{equation} n=c_{0}+c_{1}p+\cdots+c_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1} \end{equation} where the $c_{j}$s are integers in $\left\{ 0,\ldots,p-1\right\} $. Additionally, note that: \begin{equation} \left\lceil \log_{p}\left(n+1\right)\right\rceil =\left\lfloor \log_{p}n\right\rfloor +1,\textrm{ }\forall n\in\mathbb{N}_{1}\label{eq:Floor and Ceiling expressions for lambda_rho} \end{equation} and that $\lambda_{p}\left(n\right)\leq\left\|\mathbf{j}\right\|$ is satisfied for any string $\mathbf{j}$ representing $n$. Next, for each $n\geq1$ and each $k\in\left\{ 0,\ldots,p-1\right\} $, we write $\#_{p:k}\left(n\right)$\nomenclature{$\#_{p:k}\left(n\right)$}{the number of $k$s in the $p$-adic digits of $n$ } to denote the number of $k$s present in the $p$-adic expansion of $n$. We define $\#_{p:k}\left(0\right)$ to be $0$ for all $k$. In a minor abuse of notation, we also use $\#_{p:k}$ to denote the number of $k$s which occur in a given string: \begin{equation} \#_{p:k}\left(\mathbf{j}\right)\overset{\textrm{def}}{=}\textrm{number of }k\textrm{s in }\mathbf{j},\textrm{ }\forall k\in\mathbb{Z}/p\mathbb{Z},\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:Definition of number of ks in rho adic digits of bold j} \end{equation} \end{defn} \begin{rem} $\lambda_{2}\left(n\right)$ is what computer scientists called the number of \textbf{bits }in $n$. \end{rem} \begin{example} For: \begin{align} n & =1\cdot6^{0}+2\cdot6^{1}+0\cdot6^{2}+5\cdot6^{3}+0\cdot6^{4}+2\cdot6^{5}\\ & =1+2\cdot6+5\cdot6^{3}+2\cdot6^{5}\\ & =16645 \end{align} we have: \begin{align} \#_{6:0}\left(16645\right) & =2\\ \#_{6:1}\left(16645\right) & =1\\ \#_{6:2}\left(16645\right) & =2\\ \#_{6:3}\left(16645\right) & =0\\ \#_{6:4}\left(16645\right) & =0\\ \#_{6:5}\left(16645\right) & =1 \end{align} \end{example} \vphantom{} All important functions in this dissertation satisfy equally important functional equations. Here are the functional equations for $\lambda_{p}$ and the $\#_{p:k}$s. \begin{prop}[\textbf{Functional Equations for $\lambda_{p}$ and the $\#_{p:k}$s}] \ \vphantom{} I. For $k\in\left\{ 1,\ldots,p-1\right\} $: \begin{align} \#_{p:k}\left(p^{n}a+b\right) & =\#_{p:k}\left(a\right)+\#_{p:k}\left(b\right),\textrm{ }\forall a\in\mathbb{N}_{0},\textrm{ }\forall n\in\mathbb{N}_{1},\textrm{ }\forall b\in\left\{ 0,\ldots,p^{n}-1\right\} \label{eq:number-symbol functional equations} \end{align} For $k=0$, we have: \begin{equation} \#_{p:0}\left(p^{n}a+b\right)=\#_{p:0}\left(a\right)+\#_{p:0}\left(b\right)+n-\lambda_{p}\left(b\right),\textrm{ }\forall a\in\mathbb{N}_{1},\textrm{ }\forall n\in\mathbb{N}_{1},\textrm{ }\forall b\in\left\{ 0,\ldots,p^{n}-1\right\} \label{eq:number-symbol functional equations, k is 0} \end{equation} \vphantom{} II. \begin{align} \lambda_{p}\left(p^{n}a+b\right) & =\lambda_{p}\left(a\right)+n,\textrm{ }\forall a\in\mathbb{N}_{1},\textrm{ }\forall n\in\mathbb{N}_{1},\textrm{ }\forall b\in\left\{ 0,\ldots,p^{n}-1\right\} \label{eq:lambda functional equations} \end{align} \vphantom{} III. \begin{equation} \sum_{k=0}^{p-1}\#_{p:k}\left(n\right)=\lambda_{p}\left(n\right)\label{eq:Relation between lambda_rho and the number of rho adic digits} \end{equation} \end{prop} Proof: A straightforward computation. (\ref{eq:number-symbol functional equations, k is 0}) follows from using (\ref{eq:Relation between lambda_rho and the number of rho adic digits}) to write: \[ \#_{p:0}\left(n\right)=\lambda_{p}\left(n\right)-\sum_{k=1}^{p-1}\#_{p:k}\left(n\right) \] and then applying (\ref{eq:number-symbol functional equations}) and (\ref{eq:lambda functional equations}) to compute $\#_{p:0}\left(p^{n}a+b\right)$. Q.E.D. \subsection{\label{subsec:2.2.2 Construction-of}Construction of $\chi_{H}$} Since the $H_{j}$s are affine linear maps, so is any composition sequence $H_{\mathbf{j}}$. As such, for any $\mathbf{j}\in\textrm{String}\left(p\right)$ we can write: \begin{equation} H_{\mathbf{j}}\left(x\right)=H_{\mathbf{j}}^{\prime}\left(0\right)x+H_{\mathbf{j}}\left(0\right),\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:ax+b formula for h_bold_j} \end{equation} If $\mathbf{j}$ is a string for which $H_{\mathbf{j}}\left(x\right)=x$, this becomes: \begin{equation} x=H_{\mathbf{j}}^{\prime}\left(0\right)x+H_{\mathbf{j}}\left(0\right)\label{eq:affine formula for little x} \end{equation} from which we obtain: \begin{equation} x=\frac{H_{\mathbf{j}}\left(0\right)}{1-H_{\mathbf{j}}^{\prime}\left(0\right)}\label{eq:Formula for little x in terms of bold j} \end{equation} Note that the map which sends a string $\mathbf{j}$ to the rational number $x$ satisfying $H_{\mathbf{j}}\left(x\right)=x$ can then be written as: \begin{equation} \mathbf{j}\mapsto\frac{H_{\mathbf{j}}\left(0\right)}{1-H_{\mathbf{j}}^{\prime}\left(0\right)}\label{eq:Formula for X} \end{equation} While we could analyze this map directly, the resultant function is ill-behaved because of the denominator term. Consequently, instead of studying (\ref{eq:Formula for X}) directly, we will focus on the map $\mathbf{j}\mapsto H_{\mathbf{j}}\left(0\right)$, which we denote by $\chi_{H}$. \begin{defn}[$\chi_{H}$] \label{def:Chi_H on N_0 in strings}Let $H$ be a $p$-Hydra map that fixes $0$. Then, for any $\mathbf{j}\in\textrm{String}\left(p\right)$, we write: \begin{equation} \chi_{H}\left(\mathbf{j}\right)\overset{\textrm{def}}{=}H_{\mathbf{j}}\left(0\right)\label{eq:Definition of Chi_H of bold j} \end{equation} Identifying $H_{\varnothing}$ (the composition sequence associated to the empty string) with the identity map, we have that $\chi_{H}\left(\varnothing\right)=0$. \end{defn} \begin{prop} \label{prop:Chi_H is well defined mod twiddle}$\chi_{H}$ is well-defined on $\textrm{String}\left(p\right)/\sim$. That is, $\chi_{H}\left(\mathbf{i}\right)=\chi_{H}\left(\mathbf{j}\right)$ for all $\mathbf{i},\mathbf{j}\in\textrm{String}\left(p\right)$ for which $\mathbf{i}\sim\mathbf{j}$. \end{prop} Proof: Let $\mathbf{j}\in\textrm{String}\left(p\right)$ be any non-empty string. Then, by \textbf{Proposition \ref{prop:H string formalism}}: \begin{equation} \chi_{H}\left(\mathbf{j}\wedge0\right)=H_{\mathbf{j}\wedge0}\left(0\right)=H_{\mathbf{j}}\left(H_{0}\left(0\right)\right)=H_{\mathbf{j}}\left(0\right)=\chi_{H}\left(\mathbf{j}\right) \end{equation} since $H_{0}\left(0\right)=0$. \textbf{Proposition \ref{prop:string number equivalence}}, we know that given any two equivalent finite strings $\mathbf{i}\sim\mathbf{j}$, we can obtain one of the two by concatenating finitely many $0$s to the right of the other. Without loss of generality, suppose that $\mathbf{i}=\mathbf{j}\wedge0^{\wedge n}$, where $0^{\wedge n}$ is a string of $n$ consecutive $0$s. Then: \begin{equation} \chi_{H}\left(\mathbf{i}\right)=\chi_{H}\left(\mathbf{j}\wedge0^{\wedge n}\right)=\chi_{H}\left(\mathbf{j}\wedge0^{\wedge\left(n-1\right)}\right)=\cdots=\chi_{H}\left(\mathbf{j}\right) \end{equation} Hence, $\chi_{H}$ is well defined on $\textrm{String}\left(p\right)/\sim$. Q.E.D. \begin{lem}[$\chi_{H}$ \textbf{on} $\mathbb{N}_{0}$] \index{chi_{H}@$\chi_{H}$}\label{lem:Chi_H on N_0}We can realize $\chi_{H}$ as a function $\mathbb{N}_{0}\rightarrow\mathbb{Q}$ by defining: \begin{equation} \chi_{H}\left(n\right)\overset{\textrm{def}}{=}\chi_{H}\left(\mathbf{j}\right)=H_{\mathbf{j}}\left(0\right)\label{eq:Definition of Chi_H of n} \end{equation} where $\mathbf{j}\in\textrm{String}\left(p\right)$ is any string representing $n$; we define $\chi_{H}\left(0\right)$ to be $\chi_{H}\left(\varnothing\right)$ (the empty string), which is just $0$. \end{lem} Proof: By \textbf{Proposition \ref{prop:string number equivalence}}, $\textrm{String}\left(p\right)/\sim$ is in a bijection with $\mathbb{N}_{0}$. \textbf{Proposition \ref{prop:Chi_H is well defined mod twiddle} }tells us that $\chi_{H}$ is well-defined on $\textrm{String}\left(p\right)/\sim$, so, by using the aforementioned bijection to identify $\mathbb{N}_{0}$ with $\textrm{String}\left(p\right)/\sim$, the rule $\chi_{H}\left(n\right)\overset{\textrm{def}}{=}\chi_{H}\left(\mathbf{j}\right)$ is well-defined, since it is independent of which $\mathbf{j}\in\textrm{String}\left(p\right)$ we choose to represent $n$. Q.E.D. \begin{defn}[$M_{H}$] \index{$M_{H}$}\label{def:M_H on N_0 in strings}Let $H$ be a $p$-Hydra map. Then, we define $M_{H}:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ by: \begin{equation} M_{H}\left(n\right)\overset{\textrm{def}}{=}M_{H}\left(\mathbf{j}\right)\overset{\textrm{def}}{=}H_{\mathbf{j}}^{\prime}\left(0\right),\textrm{ }\forall n\geq1\label{eq:Definition of M_H} \end{equation} where $\mathbf{j}\in\textrm{String}\left(p\right)$ is the shortest element of $\textrm{String}\left(p\right)$ representing $n$. We define $M_{H}\left(0\right)$ to be $1$. \end{defn} \begin{example} \emph{Unlike} $\chi_{H}$, observe that $M_{H}$ is \emph{not well-defined over} $\textrm{String}\left(p\right)/\sim$ there can be $H$ and distinct strings $\mathbf{i},\mathbf{j}\in\textrm{String}\left(p\right)$ so that $\mathbf{i}\sim\mathbf{j}$, yet $M_{H}\left(\mathbf{i}\right)\neq M_{H}\left(\mathbf{j}\right)$. For example, $H=T_{3}$, the strings $\left(0,1\right)$ and $\left(0,1,0\right)$ both represent the integer $2$. Nevertheless: \begin{align} H_{0,1}\left(x\right) & =H_{0}\left(H_{1}\left(x\right)\right)=\frac{3}{4}x+\frac{1}{4}\\ H_{0,1,0}\left(x\right) & =H_{0}\left(H_{1}\left(H_{0}\left(x\right)\right)\right)=\frac{3}{8}x+\frac{1}{4} \end{align} Thus, $\left(0,1\right)\sim\left(0,1,0\right)$, but $M_{T_{3}}\left(0,1\right)=3/4$, while $M_{T_{3}}\left(0,1,0\right)=3/8$. With these definitions in place, we can now use the string formalism to establish vital string-based functional equation identities for $\chi_{H}$ and $M_{H}$. I will refer to these identities as \textbf{concatenation identities}\index{concatenation!identities}. \end{example} \begin{prop}[\textbf{An expression for $H_{\mathbf{j}}$}] \label{prop:H_boldj in terms of M_H and Chi_H}Let $H$ be a $p$-Hydra map. Then: \begin{equation} H_{\mathbf{j}}\left(x\right)=M_{H}\left(\mathbf{j}\right)x+\chi_{H}\left(\mathbf{j}\right),\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right),\textrm{ }\forall x\in\mathbb{Q}\label{eq:Formula for composition sequences of H} \end{equation} \end{prop} Proof: Since the $H_{j}$s are maps of the form $ax+b$, any composition sequence of the $H_{j}$s will also be of that form. Consequently, the venerable slope-intercept formula yields: \[ H_{\mathbf{j}}\left(x\right)=H_{\mathbf{j}}^{\prime}\left(0\right)x+H_{\mathbf{j}}\left(0\right)\overset{\textrm{def}}{=}M_{H}\left(\mathbf{j}\right)x+\chi_{H}\left(\mathbf{j}\right) \] Q.E.D. \vphantom{} The lemmata below give the concatenation identities for $\chi_{H}$ and $M_{H}$. \begin{lem}[$\chi_{H}$ \textbf{Concatenation Identity}] \label{lem:Chi_H concatenation identity}\ \begin{equation} \chi_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)=H_{\mathbf{i}}\left(\chi_{H}\left(\mathbf{j}\right)\right),\textrm{ }\forall\mathbf{i},\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:Chi_H concatenation identity} \end{equation} That is to say, for $\mathbf{i}=\left(i_{1},i_{2},\ldots,i_{m}\right)\in\textrm{String}\left(p\right)$, where $m\geq1$, we have: \begin{equation} \chi_{H}\left(i_{1}+i_{2}p+\cdots+i_{m}p^{m-1}+p^{m}n\right)=H_{\mathbf{i}}\left(\chi_{H}\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Chi_H concatenation identity, alternate version} \end{equation} \end{lem} Proof: Letting $\mathbf{i}$, $\mathbf{j}$, and $x$ be arbitrary, we have that: \begin{align} H_{\mathbf{i}\wedge\mathbf{j}}\left(x\right) & =H_{\mathbf{i}\wedge\mathbf{j}}^{\prime}\left(0\right)x+H_{\mathbf{i}\wedge\mathbf{j}}\left(0\right)\\ & =H_{\mathbf{i}\wedge\mathbf{j}}^{\prime}\left(0\right)x+H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(0\right)\right) \end{align} Setting $x=0$ yields: \[ \underbrace{H_{\mathbf{i}\wedge\mathbf{j}}\left(0\right)}_{\chi_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)}=\underbrace{H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(0\right)\right)}_{H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{j}\right)\right)} \] Q.E.D. \begin{prop}[$M_{H}$ \textbf{Concatenation Identity}] \label{prop:M_H concatenation identity}\ \begin{equation} M_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)=M_{H}\left(\mathbf{i}\right)M_{H}\left(\mathbf{j}\right),\textrm{ }\forall\mathbf{i},\mathbf{j}\in\textrm{String}\left(p\right),\textrm{ }\left\|\mathbf{i}\right\|,\left\|\mathbf{j}\right\|\geq1\label{eq:M_H concatenation identity} \end{equation} That is to say, for $\mathbf{i}=\left(i_{1},i_{2},\ldots,i_{m}\right)\in\textrm{String}\left(p\right)$, where $m\geq1$, we have: \begin{equation} M_{H}\left(i_{1}+i_{2}p+\cdots+i_{m}p^{m-1}+p^{m}n\right)=M_{H}\left(\mathbf{i}\right)M_{H}\left(n\right),\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Inductive identity for M_H} \end{equation} \end{prop} Proof: We have: \begin{equation} H_{\mathbf{i}\wedge\mathbf{j}}\left(x\right)=H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(x\right)\right) \end{equation} Differentiating both sides with respect to $x$ and applying the Chain rule gives: \[ H_{\mathbf{i}\wedge\mathbf{j}}^{\prime}\left(x\right)=H_{\mathbf{i}}^{\prime}\left(H_{\mathbf{j}}\left(x\right)\right)H_{\mathbf{j}}^{\prime}\left(x\right) \] Now, set $x=0$: \begin{equation} \underbrace{H_{\mathbf{i}\wedge\mathbf{j}}^{\prime}\left(0\right)}_{M_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)}=H_{\mathbf{i}}^{\prime}\left(H_{\mathbf{j}}\left(0\right)\right)\underbrace{H_{\mathbf{j}}^{\prime}\left(0\right)}_{M_{H}\left(\mathbf{j}\right)} \end{equation} Since $H_{\mathbf{i}}\left(x\right)$ is an affine linear map of the form $H_{\mathbf{i}}^{\prime}\left(0\right)x+H_{\mathbf{i}}\left(0\right)$, its derivative is the constant function $H_{\mathbf{i}}^{\prime}\left(y\right)=H_{\mathbf{i}}^{\prime}\left(0\right)$. Thus: \begin{equation} M_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)=H_{\mathbf{i}}^{\prime}\left(H_{\mathbf{j}}\left(0\right)\right)M_{H}\left(\mathbf{j}\right)=H_{\mathbf{i}}^{\prime}\left(0\right)M_{H}\left(\mathbf{j}\right)=M_{H}\left(\mathbf{i}\right)M_{H}\left(\mathbf{j}\right) \end{equation} Q.E.D. \vphantom{} We build a bridge between string-world and number-world by restating the concatenation identities in terms of systems of functional equations involving integer inputs. \begin{prop}[\textbf{Functional Equations for} $M_{H}$] \label{prop:M_H functional equation}\index{functional equation!$M_{H}$}\ \end{prop} \begin{equation} M_{H}\left(pn+j\right)=\frac{\mu_{j}}{p}M_{H}\left(n\right),\textrm{ }\forall n\geq0\textrm{ \& }\forall j\in\mathbb{Z}/p\mathbb{Z}:pn+j\neq0\label{eq:M_H functional equations} \end{equation} \begin{rem} To be clear, this functional equation \emph{does not hold} when $pn+j=0$. \end{rem} Proof: Let $n\geq1$, and let $\mathbf{j}=\left(j_{1},\ldots,j_{m}\right)$ be the shortest string in $\textrm{String}\left(p\right)$ which represents $n$. Then, observe that for $\mathbf{k}=\left(j,j_{1},j_{2},\ldots,j_{m}\right)$, where $j\in\mathbb{Z}/p\mathbb{Z}$ is arbitrary, we have that: \[ \mathbf{k}=j\wedge\mathbf{j}\sim\left(j+pn\right) \] \textbf{Proposition \ref{prop:M_H concatenation identity}} lets us write: \[ M_{H}\left(pn+j\right)=M_{H}\left(j\wedge\mathbf{j}\right)=M_{H}\left(j\right)M_{H}\left(\mathbf{j}\right)=\frac{\mu_{j}}{p}M_{H}\left(n\right) \] When $n=0$, these equalities hold for all $j\in\left\{ 1,\ldots,p-1\right\} $, seeing as: \[ \left(j+p\cdot0\right)\sim j\wedge\varnothing \] As for the one exceptional case\textemdash $n=j=0$\textemdash note that we obtain: \[ M_{H}\left(p\cdot0+0\right)=M_{H}\left(0\right)\overset{\textrm{def}}{=}1 \] but: \[ \frac{\mu_{0}}{p}M_{H}\left(0\right)\overset{\textrm{def}}{=}\frac{\mu_{0}}{p}\times1=\frac{\mu_{0}}{p} \] and, as we saw, $\mu_{0}/p\neq1$. Q.E.D. \vphantom{} Using the concatenation identity for $\chi_{H}$, we can establish the first of two characterizations for it in terms of functional equations. \begin{lem}[\textbf{Functional Equations for} $\chi_{H}$] \label{lem:Chi_H functional equation on N_0 and uniqueness}$\chi_{H}$ is the unique rational-valued function on $\mathbb{N}_{0}$ satisfying the system of functional equations:\index{chi{H}@$\chi_{H}$!functional equation}\index{functional equation!chi_{H}@$\chi_{H}$} \begin{equation} \chi_{H}\left(pn+j\right)=\frac{a_{j}\chi_{H}\left(n\right)+b_{j}}{d_{j}},\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall\mathbb{Z}/p\mathbb{Z}\label{eq:Chi_H functional equations} \end{equation} \end{lem} \begin{rem} These functional equations can be written more compactly as: \begin{equation} \chi_{H}\left(pn+j\right)=H_{j}\left(\chi_{H}\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Alternate statement of Chi_H functional equations} \end{equation} \end{rem} Proof: I. Let $\mathbf{i}\sim n$ and let $j\in\mathbb{Z}/p\mathbb{Z}$ be arbitrary. Then $pn+j\sim j\wedge\mathbf{i}$, and hence, by $\chi_{H}$'s concatenation identity (\textbf{Lemma \ref{lem:Chi_H concatenation identity}}): \begin{equation} \chi_{H}\left(pn+j\right)=\chi_{H}\left(j\wedge\mathbf{i}\right)=H_{j}\left(\chi_{H}\left(\mathbf{i}\right)\right)=H_{j}\left(\chi_{H}\left(n\right)\right) \end{equation} Thus, $\chi_{H}$ is a solution of the given system of functional equations. The reason why we need not exclude the case where $n=j=0$ (as we had to do with $M_{H}$) is because $H_{0}\left(\chi_{H}\left(0\right)\right)=H_{0}\left(0\right)=0$. \vphantom{} II. On the other hand, let $f:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ be any function so that: \begin{equation} f\left(pn+j\right)=H_{j}\left(f\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z} \end{equation} Setting $n=j=0$ gives: \begin{align} f\left(0\right) & =H_{0}\left(f\left(0\right)\right)\\ \left(H_{0}\left(x\right)=H_{0}^{\prime}\left(0\right)x+\underbrace{H_{0}\left(0\right)}_{0}\right); & =H_{0}^{\prime}\left(0\right)f\left(0\right) \end{align} Since $H$ is a $p$-Hydra map: \[ H_{0}^{\prime}\left(0\right)=\frac{\mu_{0}}{p}\notin\left\{ 0,1\right\} \] Thus, $f\left(0\right)=H_{0}^{\prime}\left(0\right)f\left(0\right)$ forces $f\left(0\right)=0$. Plugging in $n=0$ then leaves us with: \[ f\left(j\right)=H_{j}\left(f\left(0\right)\right)=H_{j}\left(0\right),\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z} \] Writing $n$ $p$-adically as: \begin{equation} n=j_{1}+j_{2}p+\cdots+j_{L}p^{L-1} \end{equation} the identity $f\left(pn+j\right)=H_{j}\left(f\left(n\right)\right)$ is then equivalent to: \begin{align} f\left(j+j_{1}p+j_{2}p^{2}+\cdots+j_{L}p^{L}\right) & =H_{j}\left(f\left(j_{1}+j_{2}p+\cdots+j_{L}p^{L-1}\right)\right)\\ & =H_{j}\left(H_{j_{1}}\left(f\left(j_{2}+j_{3}p+\cdots+j_{L}p^{L-2}\right)\right)\right)\\ & \vdots\\ & =\left(H_{j}\circ H_{j_{1}}\circ\cdots\circ H_{j_{L}}\right)\left(f\left(0\right)\right)\\ & =\left(H_{j}\circ H_{j_{1}}\circ\cdots\circ H_{j_{L}}\right)\left(0\right)\\ & =H_{j,j_{1},\ldots,j_{L}}\left(0\right) \end{align} So, for any string $\mathbf{j}^{\prime}=\left(j,j_{1},\ldots,j_{L}\right)$, we have: \[ f\left(\mathbf{j}^{\prime}\right)=H_{\mathbf{j}^{\prime}}\left(0\right)\overset{\textrm{def}}{=}\chi_{H}\left(\mathbf{j}^{\prime}\right) \] where, note: $\mathbf{j}^{\prime}=j\wedge\mathbf{j}$ and $\mathbf{j}^{\prime}\sim n$. In other words, if $f$ solves the given system of functional equations, it is in fact equal to $\chi_{H}$ at every finite string, and hence, at every non-negative integer. This shows the system's solutions are unique. Q.E.D. \vphantom{} Next we compute explicit formulae for $M_{H}$ and $\chi_{H}$ in terms of the numbers present in $H$. These will be needed in to make the $q$-adic estimates needed to establish the extension/interpolation of $\chi_{H}$ from a function $\mathbb{N}_{0}\rightarrow\mathbb{Q}$ to a function $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$, for an appropriate choice of a prime $q$. \begin{prop} \label{prop:Explicit Formulas for M_H}$M_{H}\left(n\right)$ and $M_{H}\left(\mathbf{j}\right)$ can be explicitly given in terms of the constants associated to $H$ by the formulae: \begin{equation} M_{H}\left(n\right)=\frac{1}{p^{\lambda_{p}\left(n\right)}}\prod_{k=0}^{p-1}\mu_{k}^{\#_{p:k}\left(n\right)}=\prod_{k=0}^{p-1}\frac{a_{k}^{\#_{p:k}\left(n\right)}}{d_{k}^{\#_{p:k}\left(n\right)}}\label{eq:Formula for M_H of n} \end{equation} and: \begin{equation} M_{H}\left(\mathbf{j}\right)=\prod_{k=1}^{\left\|\mathbf{j}\right\|}\frac{a_{j_{k}}}{d_{j_{k}}}=\frac{1}{p^{\left\|\mathbf{j}\right\|}}\prod_{k=1}^{\left\|\mathbf{j}\right\|}\mu_{j_{k}}=\frac{1}{p^{\left\|\mathbf{j}\right\|}}\prod_{k=0}^{p-1}\mu_{k}^{\#_{p:k}\left(\mathbf{j}\right)}=\prod_{k=0}^{p-1}\frac{a_{k}^{\#_{p:k}\left(\mathbf{j}\right)}}{d_{k}^{\#_{p:k}\left(\mathbf{j}\right)}}\label{eq:formula for M_H of bold-j} \end{equation} respectively. We adopt the convention that the $k$-product in \emph{(\ref{eq:formula for M_H of bold-j})} is defined to be $1$ when $\left\|\mathbf{j}\right\|=0$. Finally, for any $\mathfrak{z}\in\mathbb{Z}_{p}$ and any $N\geq1$, we have the formula: \begin{equation} M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\frac{\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}}{p^{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}}\label{eq:M_H of z mod rho to the N} \end{equation} obtained by setting $n=\left[\mathfrak{z}\right]_{p^{N}}$ in \emph{(\ref{eq:Formula for M_H of n})}. \end{prop} Proof: The proof is purely computational. I omit it, seeing as it occurs in the proof of \textbf{Proposition \ref{prop:Explicit formula for Chi_H of bold j}}, given below. Q.E.D. \begin{prop} \label{prop:Explicit formula for Chi_H of bold j} \begin{equation} \chi_{H}\left(\mathbf{j}\right)=\sum_{m=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}}=\sum_{m=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m},\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:Formula for Chi_H in terms of bold-j} \end{equation} where the $k$-product is defined to be $1$ when $m=1$. \end{prop} Proof: Let $\mathbf{j}=\left(j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}\right)\in\textrm{String}\left(p\right)$ be arbitrary. Since $\chi_{H}\left(\mathbf{j}\right)=\chi_{H}\left(\mathbf{i}\right)$ for any $\mathbf{i}\in\textrm{String}\left(p\right)$ for which $\mathbf{i}\sim\mathbf{j}$, we can assume without loss of generality that $\mathbf{j}$'s right-most entry is non-zero; that is: $j_{\left\|\mathbf{j}\right\|}\neq0$. The proof follows by examining: \begin{equation} H_{\mathbf{j}}\left(x\right)=M_{H}\left(\mathbf{j}\right)x+\chi_{H}\left(\mathbf{j}\right),\textrm{ }\forall x\in\mathbb{R} \end{equation} and carefully writing out the composition sequence in full: \begin{align} H_{\mathbf{j}}\left(x\right) & =H_{\left(j_{1},\ldots,j_{\left\|\mathbf{j}\right\|}\right)}\left(x\right)\\ & =\frac{a_{j_{1}}\frac{a_{j_{2}}\left(\cdots\right)+b_{j_{2}}}{d_{j_{2}}}+b_{j_{1}}}{d_{j_{1}}}\\ & =\overbrace{\left(\prod_{k=1}^{\left\|\mathbf{j}\right\|}\frac{a_{j_{k}}}{d_{j_{k}}}\right)}^{M_{H}\left(\mathbf{j}\right)}x+\overbrace{\frac{b_{j_{1}}}{d_{j_{1}}}+\frac{a_{j_{1}}}{d_{j_{1}}}\frac{b_{j_{2}}}{d_{j_{2}}}+\frac{a_{j_{2}}}{d_{j_{2}}}\frac{a_{j_{1}}}{d_{j_{1}}}\frac{b_{j_{3}}}{d_{j_{3}}}+\cdots+\left(\prod_{k=1}^{\left\|\mathbf{j}\right\|-1}\frac{a_{j_{k}}}{d_{j_{k}}}\right)\frac{b_{j_{\left\|\mathbf{j}\right\|}}}{d_{j_{\left\|\mathbf{j}\right\|}}}}^{\chi_{H}\left(\mathbf{j}\right)}\\ & =M_{H}\left(\mathbf{j}\right)x+\underbrace{\sum_{m=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}}}_{\chi_{H}\left(\mathbf{j}\right)} \end{align} where we use the convention that $\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}}=1$ when $m=1$. This proves: \begin{equation} \chi_{H}\left(\mathbf{j}\right)=\sum_{m=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}} \end{equation} Next, using $\mu_{j}=\frac{pa_{j}}{d_{j}}$, for $m>1$, the $m$th term $\frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}}$ can be written as: \begin{align} \frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}} & =\frac{1}{p}\cdot\frac{b_{j_{m}}}{a_{j_{m}}}\cdot\frac{pa_{j_{m}}}{d_{j_{m}}}\cdot\prod_{k=1}^{m-1}\left(\frac{pa_{j_{k}}}{d_{j_{k}}}\cdot\frac{1}{p}\right)\\ & =\frac{1}{p}\frac{b_{j_{m}}}{a_{j_{m}}}\frac{\mu_{j_{m}}}{p^{m-1}}\prod_{k=1}^{m-1}\mu_{j_{k}}\\ & =\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m} \end{align} On the other hand, when $m=1$, by our $k$-product convention, we have: \begin{equation} \frac{b_{j_{1}}}{d_{j_{1}}}\underbrace{\prod_{k=1}^{1-1}\frac{a_{j_{k}}}{d_{j_{k}}}}_{1}=\frac{b_{j_{1}}}{d_{j_{1}}} \end{equation} Thus: \begin{align} \chi_{H}\left(\mathbf{j}\right) & =\frac{b_{j_{1}}}{d_{j_{1}}}+\sum_{m=2}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m}\\ & =\frac{1}{p}\frac{b_{j_{1}}}{a_{j_{1}}}\frac{pa_{j_{1}}}{d_{j_{1}}}+\sum_{m=2}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m}\\ & =\frac{b_{j_{1}}}{a_{j_{1}}}\cdot\mu_{j_{1}}\cdot p^{-1}+\sum_{m=2}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m}\\ & =\sum_{m=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m} \end{align} as desired. Q.E.D. \vphantom{} To interpolate $\chi_{H}$ from a rational-valued function on $\mathbb{N}_{0}$ to a $q$-adic-valued function on $\mathbb{Z}_{p}$, we need to define some qualitative properties of Hydra maps so as to distinguish those cases where this interpolation will actually exist. \begin{defn}[\textbf{Qualitative Terminology for $p$-Hydra maps}] \label{def:Qualitative conditions on a p-Hydra map}Let $H$ be a $p$-Hydra map. For the definitions below, we DO NOT require $p$ to be prime. \vphantom{} I. We say $H$ is \textbf{simple }if $d_{j}=p$ for all $j$. \vphantom{} II. We say $H$ is \textbf{semi-simple }if $\gcd\left(a_{j},d_{k}\right)=1$ for all $j,k\in\left\{ 0,\ldots,p-1\right\} $. \vphantom{} III. We say $H$ is \textbf{degenerate }if $a_{j}=1$ for some non-zero $j$. If $a_{j}\neq1$ for all non-zero $j$, we say $H$ is \textbf{non-degenerate}. \vphantom{} IV\footnote{This definition will not be relevant until Chapter 3.}. We say $H$ is \textbf{contracting }if $a_{0}/d_{0}$ (a.k.a. $\mu_{0}/p$) is $<1$, and say $H$ is \textbf{expanding }if $a_{0}/d_{0}$ (a.k.a. $\mu_{0}/p$) is $>1$. \vphantom{} V. We say $H$ is \textbf{monogenic}\footnote{This term is based on the Greek for ``one kind/species''.} whenever: \begin{equation} \gcd\left(\left\{ a_{j}:j\in\left\{ 0,\ldots,p-1\right\} \textrm{ \& }a_{j}\neq1\right\} \right)\geq2\label{eq:Monogenic gcd} \end{equation} When this $\textrm{gcd}$ is $1$, we say $H$ is \textbf{polygenic}. If $H$ is monogenic, we then write $q_{H}$\nomenclature{$q_{H}$}{ } to denote the smallest prime divisor of (\ref{eq:Monogenic gcd}). For brevity, sometimes we will drop the $H$ and just write this as $q$ instead of $q_{H}$. \vphantom{} VI. We say $H$ is \textbf{basic }if $H$ is simple, non-degenerate, and monogenic. \vphantom{} VII. We say $H$ is \textbf{semi-basic }if $H$ is semi-simple, non-degenerate, and monogenic. \end{defn} \begin{rem} If $H$ is simple, note that $\mu_{j}=a_{j}$ for all $j$. \end{rem} \begin{rem} It is worth noting that all of the results in this section hold when $q_{H}$ is \emph{any }prime divisor of (\ref{eq:Monogenic gcd}). \end{rem} \vphantom{} Next, a minor technical result which shows that our qualitative definitions are actually useful. \begin{prop}[\textbf{Co-primality of $d_{j}$ and $q_{H}$}] \label{prop:co-primality of d_j and q_H}Let $H$ be a semi-basic $p$-Hydra map. Then, $\gcd\left(d_{j},q_{H}\right)=1$ for all $j\in\left\{ 0,\ldots,p-1\right\} $. \end{prop} Proof: By way of contradiction, suppose there is a $k\in\left\{ 0,\ldots,p-1\right\} $ so that $\gcd\left(d_{k},q_{H}\right)>1$. Since $H$ is semi-basic, $H$ is monogenic and non-degenerate, and so $q_{H}$ is $\geq2$ and, moreover, $a_{j}\neq1$ for any $j\in\left\{ 1,\ldots,p-1\right\} $. Thus, there is a $k$ so that $d_{k}$ and $q_{H}$ have a non-trivial common factor, and there is a $j$ so that $q_{H}$ and $a_{j}$ have a non-trivial common factor (namely, $q_{H}$). Since every factor of $q_{H}$ divides $a_{j}$, the non-trivial common factor of $d_{k}$ and $q_{H}$ must also divide $a_{j}$. This forces $\gcd\left(a_{j},d_{k}\right)>1$. However, as a semi-basic $p$-Hydra map, $H$ is semi-simple, and so $\gcd\left(a_{j},d_{k}\right)=1$ for all $j,k\in\left\{ 0,\ldots,p-1\right\} $\textemdash and that is a contradiction! Thus, it must be that $\gcd\left(d_{k},q_{H}\right)=1$ for all $k\in\left\{ 0,\ldots,p-1\right\} $. Q.E.D. \vphantom{} While our next result is also technical, it is not minor in the least. It demonstrates that $M_{H}\left(\mathbf{j}\right)$ will be $q_{H}$-adically small whenever $\mathbf{j}$ has many non-zero entries. This is the essential ingredient for the proof of the existence of $\chi_{H}$'s $p$-adic interpolation. \begin{prop}[\textbf{$q_{H}$-adic behavior of $M_{H}\left(\mathbf{j}\right)$ as $\left\|\mathbf{j}\right\|\rightarrow\infty$}] \label{prop:q-adic behavior of M_H of j as the number of non-zero digits tends to infinity}Let $H$ be a semi-basic $p$-Hydra map which fixes $0$. Then: \begin{equation} \left\|M_{H}\left(\mathbf{j}\right)\right\|_{q_{H}}\leq q_{H}^{-\sum_{k=1}^{p-1}\#_{p:k}\left(\mathbf{j}\right)}\label{eq:M_H of bold j has q-adic valuation at least as large as the number of non-zero digits of bold-j} \end{equation} In particular, for any sequence of strings $\left\{ \mathbf{j}_{n}\right\} _{n\geq1}$ in $\textrm{String}\left(p\right)$ so that $\lim_{n\rightarrow\infty}\left\|\mathbf{j}_{n}\right\|=\infty$, we have that $\lim_{n\rightarrow\infty}\left\|M_{H}\left(\mathbf{j}_{n}\right)\right\|_{q_{H}}=0$ whenever the number of non-zero entries in $\mathbf{j}_{n}$ tends to $\infty$ as $n\rightarrow\infty$. \end{prop} Proof: Let $\mathbf{j}$ be a non-zero element of $\textrm{String}\left(p\right)$. Then, by \textbf{Proposition \ref{prop:Explicit Formulas for M_H}}: \[ M_{H}\left(\mathbf{j}\right)=\prod_{\ell=1}^{\left\|\mathbf{j}\right\|}\frac{a_{j_{\ell}}}{d_{j_{\ell}}} \] Since $H$ is semi-basic, \textbf{Proposition \ref{prop:co-primality of d_j and q_H}} tells us that every $d_{j}$ is co-prime to $q_{H}$, and so $\left\|d_{j}\right\|_{q_{H}}=1$ for any $j$. On the other hand, the non-degeneracy and monogenicity of $H$ tells us that $a_{j}$ is a non-zero integer multiple of $q_{H}$ for all $j\in\left\{ 1,\ldots,p-1\right\} $; thus, $\left\|a_{j}\right\|_{q_{H}}\leq1/q_{H}$ for every $\ell$ for which $j_{\ell}\neq0$. Taking $q_{H}$-adic absolute values of $M_{H}\left(\mathbf{j}\right)$ then gives: \begin{equation} \left\|M_{H}\left(\mathbf{j}\right)\right\|_{q_{H}}=\prod_{\ell=1}^{\left\|\mathbf{j}\right\|}\left\|\frac{a_{j_{\ell}}}{d_{j_{\ell}}}\right\|_{q_{H}}=\prod_{\ell:a_{j_{\ell}}\neq0}\left\|a_{j_{\ell}}\right\|_{q_{H}}\leq q_{H}^{-\left\|\left\{ \ell\in\left\{ 1,\ldots,\left\|\mathbf{j}\right\|\right\} :j_{\ell}\neq0\right\} \right\|} \end{equation} Here, $\left\|\left\{ \ell\in\left\{ 1,\ldots,\left\|\mathbf{j}\right\|\right\} :j_{\ell}\neq0\right\} \right\|$ is precisely the number of non-zero entries of $\mathbf{j}$, which is: \begin{equation} \left\|\left\{ \ell\in\left\{ 1,\ldots,\left\|\mathbf{j}\right\|\right\} :j_{\ell}\neq0\right\} \right\|=\sum_{k=1}^{p-1}\#_{p:k}\left(\mathbf{j}\right) \end{equation} For a sequence of $\mathbf{j}_{n}$s tending to $\infty$ in length, this also shows that $\left\|M_{H}\left(\mathbf{j}_{n}\right)\right\|_{q_{H}}\rightarrow0$ as $n\rightarrow\infty$, provided that the number of non-zero entries in $\mathbf{j}_{n}$ also tends to $\infty$ as $n\rightarrow\infty$. Q.E.D. \vphantom{} Now the main result of this subsection: the $\left(p,q\right)$-adic characterization of $\chi_{H}$. \begin{lem}[\textbf{$\left(p,q\right)$-adic Characterization of }$\chi_{H}$] \label{lem:Unique rising continuation and p-adic functional equation of Chi_H}Let $H$ be a semi-basic $p$-Hydra map which fixes $0$. Then, the limit: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{n\rightarrow\infty}\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Rising Continuity Formula for Chi_H} \end{equation} exists for all $\mathfrak{z}\in\mathbb{Z}_{p}$, and thereby defines an interpolation of $\chi_{H}$ to a function $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$. Moreover: \vphantom{} I. \begin{equation} \chi_{H}\left(p\mathfrak{z}+j\right)=\frac{a_{j}\chi_{H}\left(\mathfrak{z}\right)+b_{j}}{d_{j}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Functional Equations for Chi_H over the rho-adics} \end{equation} \vphantom{} II. The interpolation $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$ defined by the limit \emph{(\ref{eq:Rising Continuity Formula for Chi_H})} is the \textbf{unique} function $f:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$ satisfying the functional equations: \emph{ \begin{equation} f\left(p\mathfrak{z}+j\right)=\frac{a_{j}f\left(\mathfrak{z}\right)+b_{j}}{d_{j}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:unique p,q-adic functional equation of Chi_H} \end{equation} }along with the \textbf{rising-continuity}\footnote{As discussed in Section \ref{sec:3.2 Rising-Continuous-Functions}, I say a function is \textbf{rising-continuous }whenever it satisfies the limit condition (\ref{eq:unique p,q-adic rising continuity of Chi_H}). (\ref{eq:Rising Continuity Formula for Chi_H}) shows that $\chi_{H}$ is rising-continuous.} \textbf{condition}:\textbf{ } \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{n\rightarrow\infty}f\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:unique p,q-adic rising continuity of Chi_H} \end{equation} \end{lem} Proof: First, we show the existence of the limit (\ref{eq:Rising Continuity Formula for Chi_H}). If $\mathfrak{z}\in\mathbb{N}_{0}\cap\mathbb{Z}_{p}$, then $\left[\mathfrak{z}\right]_{p^{N}}=\mathfrak{z}$ for all sufficiently large $N$, showing that the limit (\ref{eq:Rising Continuity Formula for Chi_H}) exists in that case. So, suppose $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. Next, let $\mathbf{j}_{n}=\left(j_{1},\ldots,j_{n}\right)$ be the shortest string representing $\left[\mathfrak{z}\right]_{p^{n}}$; that is: \begin{equation} \left[\mathfrak{z}\right]_{p^{n}}=\sum_{k=1}^{n}j_{k}p^{k-1} \end{equation} By \textbf{Lemma \ref{eq:Definition of Chi_H of n}}, (\ref{eq:Rising Continuity Formula for Chi_H}) can be written as: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)=\lim_{n\rightarrow\infty}\chi_{H}\left(\mathbf{j}_{n}\right) \end{equation} Using \textbf{Proposition \ref{prop:Explicit formula for Chi_H of bold j}}, we can write: \begin{equation} \chi_{H}\left(\mathbf{j}_{n}\right)=H_{\mathbf{j}_{n}}\left(0\right)=\sum_{m=1}^{\left\|\mathbf{j}_{n}\right\|}\frac{b_{j_{m}}}{d_{j_{m}}}\prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}} \end{equation} where the product is defined to be $1$ when $m=1$. Then, using \textbf{Proposition \ref{prop:Explicit Formulas for M_H}}, we get: \begin{equation} \prod_{k=1}^{m-1}\frac{a_{j_{k}}}{d_{j_{k}}}=M_{H}\left(\mathbf{j}_{m-1}\right) \end{equation} where, again, the product is defined to be $1$ when $m=1$. So, our previous equation for $\chi_{H}\left(\mathbf{j}_{n}\right)$ can be written as: \begin{equation} \chi_{H}\left(\mathbf{j}_{n}\right)=\sum_{m=1}^{\left\|\mathbf{j}_{n}\right\|}\frac{b_{j_{m}}}{d_{j_{m}}}M_{H}\left(\mathbf{j}_{m-1}\right) \end{equation} Taking limits yields: \begin{equation} \lim_{n\rightarrow\infty}\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=\lim_{n\rightarrow\infty}\chi_{H}\left(\mathbf{j}_{n}\right)=\lim_{n\rightarrow\infty}\sum_{m=1}^{\infty}\frac{b_{j_{m}}}{d_{j_{m}}}M_{H}\left(\mathbf{j}_{m-1}\right)\label{eq:Formal rising limit of Chi_H} \end{equation} Thanks to the ultrametric topology of $\mathbb{Z}_{q_{H}}$, the series on the right will converge in $\mathbb{Z}_{q_{H}}$ if and only if is $m$th term tends to $0$ $q_{H}$-adically: \begin{equation} \lim_{m\rightarrow\infty}\left\|\frac{b_{j_{m}}}{d_{j_{m}}}M_{H}\left(\mathbf{j}_{m-1}\right)\right\|_{q_{H}}=0 \end{equation} Because the $b_{j}$s and $d_{j}$s belong to a finite set, with the $d_{j}$s being all non-zero: \begin{equation} \sup_{j\in\left\{ 0,\ldots,p-1\right\} }\left\|\frac{b_{j}}{d_{j}}\right\|_{q_{H}}<\infty \end{equation} and so: \begin{equation} \left\|\frac{b_{j_{m}}}{d_{j_{m}}}M_{H}\left(\mathbf{j}_{m-1}\right)\right\|_{q_{H}}\ll\left\|M_{H}\left(\mathbf{j}_{m-1}\right)\right\|_{q_{H}} \end{equation} Because $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, $\mathfrak{z}$ has infinitely many non-zero digits. As such, both the length and number of non-zero entries of $\mathbf{j}_{m-1}$ tend to infinity as $m\rightarrow\infty$. By\textbf{ Proposition \ref{prop:q-adic behavior of M_H of j as the number of non-zero digits tends to infinity}}, this forces $\left\|M_{H}\left(\mathbf{j}_{m-1}\right)\right\|_{q_{H}}\rightarrow0$ as $m\rightarrow\infty$. So, the $m$th term of our series decays to zero $q_{H}$-adically, which then guarantees the convergence of $\lim_{n\rightarrow\infty}\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ in $\mathbb{Z}_{q_{H}}$. \vphantom{} To prove (I), by \textbf{Lemma \ref{lem:Chi_H functional equation on N_0 and uniqueness}}, we know that $\chi_{H}$ satisfies the functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}) for $\mathfrak{z}\in\mathbb{N}_{0}$. Taking limits of the functional equations using (\ref{eq:Rising Continuity Formula for Chi_H}) then proves the equations hold for all $\mathfrak{z}\in\mathbb{Z}_{p}$. \vphantom{} For (II), we have just shown that $\chi_{H}$ satisfies the interpolation condition and the functional equations. So, conversely, suppose $f:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$ satisfies (\ref{eq:unique p,q-adic functional equation of Chi_H}) and (\ref{eq:unique p,q-adic rising continuity of Chi_H}). Then, by \textbf{Lemma \ref{lem:Chi_H functional equation on N_0 and uniqueness}}, the restriction of $f$ to $\mathbb{N}_{0}$ must be equal to $\chi_{H}$. (\ref{eq:Rising Continuity Formula for Chi_H}) shows that the values $f\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{n\rightarrow\infty}f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ obtained by taking limits necessarily forces $f\left(\mathfrak{z}\right)=\chi_{H}\left(\mathfrak{z}\right)$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. Q.E.D. \subsection{\label{subsec:2.2.3 The-Correspondence-Principle}The Correspondence Principle} THROUGHOUT THIS SUBSECTION, WE ASSUME $H$ IS INTEGRAL. \vphantom{} Our goal\textemdash the \textbf{Correspondence Principle}\textemdash relates the periodic points of $H$ in $\mathbb{Z}$ to the set $\mathbb{Z}\cap\chi_{H}\left(\mathbb{Z}_{p}\right)$, where we view both $\mathbb{Z}$ and $\chi_{H}\left(\mathbb{Z}_{p}\right)$ as being embedded in $\mathbb{Z}_{q_{H}}$. The distinction between basic and semi-basic $p$-Hydra maps will play a key role here, as there is an additional property (\textbf{propriety}, to be defined below) satisfied by semi-basic integral $p$-Hydra maps which is needed to ensure that the set $\mathbb{Z}\cap\chi_{H}\left(\mathbb{Z}_{p}\right)$ fully characterizes the periodic points of $H$. Before proceeding, we will also need a certain self-map of $\mathbb{Z}_{p}$ which I denote by $B_{p}$, so as to formulate one more functional equation satisfied by $\chi_{H}$, one directly implicated in the Correspondence Principle. First, however, a useful\textemdash and motivating\textemdash observation. \begin{prop} \label{prop:Concatenation exponentiation}Let $p$ be an integer $\geq2$, let $n\in\mathbb{N}_{1}$, and let $\mathbf{j}\in\textrm{String}\left(p\right)$ be the shortest string representing $n$. Then, for all $m\in\mathbb{N}_{1}$: \begin{equation} \mathbf{j}^{\wedge m}\sim n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}}\label{eq:Proposition 1.2.10} \end{equation} That is to say, the quantity on the right is the integer whose sequence of $p$-adic digits consists of $m$ concatenated copies of the $p$-adic digits of $n$ (or, equivalently, the entries of $\mathbf{j}$). \end{prop} Proof: Since: \[ n=j_{1}+j_{2}p+\cdots+j_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1} \] we have that: \begin{align} \mathbf{j}^{\wedge m} & \sim j_{1}+j_{2}p+\cdots+j_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}\\ & +p^{\lambda_{p}\left(n\right)}\left(j_{1}+j_{2}p+\cdots+j_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}\right)\\ & +\cdots\\ & +p^{\left(m-1\right)\lambda_{p}\left(n\right)}\left(j_{1}+j_{2}p+\cdots+j_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}\right)\\ & =n+np^{\lambda_{p}\left(n\right)}+np^{2\lambda_{p}\left(n\right)}+\cdots+np^{\left(m-1\right)\lambda_{p}\left(n\right)}\\ & =n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}} \end{align} as desired. Q.E.D. \begin{defn}[$B_{p}$] \nomenclature{$B_{p}$}{ }Let $p$ be an integer $\geq2$. Then, we define $B_{p}:\mathbb{N}_{0}\rightarrow\mathbb{Q}\cap\mathbb{Z}_{p}$ by: \begin{equation} B_{p}\left(n\right)\overset{\textrm{def}}{=}\begin{cases} 0 & \textrm{if }n=0\\ \frac{n}{1-p^{\lambda_{p}\left(n\right)}} & \textrm{if }n\geq1 \end{cases}\label{eq:Definition of B projection function} \end{equation} \end{defn} \begin{rem} $B_{p}$ has a simple interpretation in terms of $p$-adic expansions: it sends $n$ to the $p$-adic integer whose sequence of $p$-adic digits consists of infinitely many concatenated copies of the sequence of $p$-adic digits of $n$: \begin{equation} B_{p}\left(n\right)\overset{\mathbb{Z}_{p}}{=}\lim_{m\rightarrow\infty}n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}}\overset{\mathbb{Z}_{p}}{=}\frac{n}{1-p^{\lambda_{p}\left(n\right)}} \end{equation} In particular, since the sequence of $p$-adic digits of $B_{p}\left(n\right)$ is, by construction, periodic, the geometric series formula in $\mathbb{Z}_{p}$ guarantees the $p$-adic integer $B_{p}\left(n\right)$ is in fact an element of $\mathbb{Q}$. \end{rem} \begin{rem} Note that $B_{p}$ extends to a function on $\mathbb{Z}_{p}$, one whose restriction to $\mathbb{Z}_{p}^{\prime}$ is identity map. \end{rem} \vphantom{} Using $\chi_{H}$ and $B_{p}$, we can compactly and elegantly express the diophantine equation (\ref{eq:The Bohm-Sontacchi Criterion}) of the Bhm-Sontacchi Criterion in terms of a functional equation. The \textbf{Correspondence Principle }emerges almost immediately from this functional equation. \begin{lem}[\textbf{Functional Equation for }$\chi_{H}\circ B_{p}$] \label{lem:Chi_H o B_p functional equation}Let\index{functional equation!chi_{H}circ B_{p}@$\chi_{H}\circ B_{p}$} $H$ be semi-basic. Then: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)\overset{\mathbb{Z}_{qH}}{=}\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)},\textrm{ }\forall n\in\mathbb{N}_{1}\label{eq:Chi_H B functional equation} \end{equation} \end{lem} Proof: Let $n\in\mathbb{N}_{1}$, and let $\mathbf{j}\in\textrm{String}\left(p\right)$ be the shortest string representing $n$. Now, for all $m\geq1$ and all $k\in\mathbb{N}_{1}$: \begin{equation} H_{\mathbf{j}^{\wedge m}}\left(k\right)=H_{\mathbf{j}}^{\circ m}\left(k\right) \end{equation} Since: \[ H_{\mathbf{j}}\left(k\right)=M_{H}\left(\mathbf{j}\right)k+\chi_{H}\left(\mathbf{j}\right) \] the geometric series formula gives us: \begin{equation} H_{\mathbf{j}^{\wedge m}}\left(k\right)=H_{\mathbf{j}}^{\circ m}\left(k\right)=\left(M_{H}\left(\mathbf{j}\right)\right)^{m}k+\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(\mathbf{j}\right)}\chi_{H}\left(\mathbf{j}\right) \end{equation} Since we also have: \begin{equation} H_{\mathbf{j}^{\wedge m}}\left(k\right)=M_{H}\left(\mathbf{j}^{\wedge m}\right)k+\chi_{H}\left(\mathbf{j}^{\wedge m}\right) \end{equation} this then yields: \begin{equation} M_{H}\left(\mathbf{j}^{\wedge m}\right)k+\chi_{H}\left(\mathbf{j}^{\wedge m}\right)=\left(M_{H}\left(\mathbf{j}\right)\right)^{m}k+\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(\mathbf{j}\right)}\chi_{H}\left(\mathbf{j}\right) \end{equation} By \textbf{Proposition \ref{prop:M_H concatenation identity}}, we see that $M_{H}\left(\mathbf{j}^{\wedge m}\right)k=\left(M_{H}\left(\mathbf{j}\right)\right)^{m}k$. Cancelling these terms from both sides leaves us with: \begin{equation} \chi_{H}\left(\mathbf{j}^{\wedge m}\right)=\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(\mathbf{j}\right)}\chi_{H}\left(\mathbf{j}\right)=\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(n\right)}\chi_{H}\left(n\right)\label{eq:Chi_H B functional equation, ready to take limits} \end{equation} Now, by \textbf{Proposition \ref{prop:Concatenation exponentiation}}, we have that: \begin{equation} \chi_{H}\left(\mathbf{j}^{\wedge m}\right)=\chi_{H}\left(n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}}\right) \end{equation} where the equality is of rational numbers in $\mathbb{R}$. Since $n\in\mathbb{N}_{1}$, we have that: \begin{equation} B_{p}\left(n\right)=\frac{n}{1-p^{\lambda_{p}\left(n\right)}} \end{equation} is a $p$-adic integer. Moreover, as is immediate from the proof of \textbf{Proposition \ref{prop:Concatenation exponentiation}}, the projection of this $p$-adic integer modulo $p^{m}$ is: \begin{equation} \left[B_{p}\left(n\right)\right]_{p^{m}}=n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}} \end{equation} which is, of course, exactly the rational integer represented by the string $\mathbf{j}^{\wedge m}$. In other words: \begin{equation} \frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(n\right)}\chi_{H}\left(n\right)=\chi_{H}\left(\mathbf{j}^{\wedge m}\right)=\chi_{H}\left(n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}}\right)=\chi_{H}\left(\left[B_{p}\left(n\right)\right]_{p^{m}}\right) \end{equation} By \textbf{Lemma \ref{lem:Unique rising continuation and p-adic functional equation of Chi_H}}, we have that: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)\overset{\mathbb{Z}_{q}}{=}\lim_{m\rightarrow\infty}\chi_{H}\left(\left[B_{p}\left(n\right)\right]_{p^{m}}\right)=\lim_{m\rightarrow\infty}\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(n\right)}\chi_{H}\left(n\right) \end{equation} Finally, because $H$ is semi-basic, it follows by \textbf{Proposition \ref{prop:q-adic behavior of M_H of j as the number of non-zero digits tends to infinity} }that $\left\|M_{H}\left(\mathbf{j}\right)\right\|_{q}<1$ (since $\mathbf{j}$ has a non-zero entry), and hence, that $\left(M_{H}\left(\mathbf{j}\right)\right)^{m}$ tends to $0$ in $\mathbb{Z}_{q}$ as $m\rightarrow\infty$. Thus: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)\overset{\mathbb{Z}_{q}}{=}\lim_{m\rightarrow\infty}\frac{1-\left(M_{H}\left(\mathbf{j}\right)\right)^{m}}{1-M_{H}\left(n\right)}\chi_{H}\left(n\right)\overset{\mathbb{Z}_{q}}{=}\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)} \end{equation} Q.E.D. \begin{rem} As seen in the above proof, (\ref{eq:Chi_H B functional equation}) is really just the geometric series summation formula, convergent in $\mathbb{Z}_{q}$ whenever $\left\|M_{H}\left(n\right)\right\|_{q_{H}}<1$. In computing $\chi_{H}\left(B_{p}\left(n\right)\right)$ by using (\ref{eq:Rising Continuity Formula for Chi_H}) with $\mathfrak{z}=B_{p}\left(n\right)$ and $n\geq1$, the series in (\ref{eq:Formal rising limit of Chi_H}) will reduce to a geometric series, one which converges in $\mathbb{Z}_{q}$ to $\chi_{H}\left(n\right)/\left(1-M_{H}\left(n\right)\right)$. Additionally, for any $n\geq1$ for which the \emph{archimedean }absolute value $\left\|M_{H}\left(n\right)\right\|$ is less than $1$, the \emph{universality} of the geometric series formula: \begin{equation} \sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x} \end{equation} guarantees that the limit to which the series in (\ref{eq:Formal rising limit of Chi_H}) converges to in $\mathbb{R}$ in this case is the same as the limit to which it converges in $\mathbb{Z}_{q_{H}}$ \cite{Conrad on p-adic series}. This is worth mentioning because, as we shall see, the values of $n\geq1$ for which $\chi_{H}\left(n\right)/\left(1-M_{H}\left(n\right)\right)$ is a periodic point of $H$ in $\mathbb{N}_{1}$ are exactly those values of $n\geq1$ for which the non-archimedean absolute value $\left\|M_{H}\left(n\right)\right\|_{q_{H}}$ is $<1$. \end{rem} \vphantom{} Now we introduce the final bit of qualitative controls on $H$'s behavior which we will needed to prove the \textbf{Correspondence Principle}. First, however, a proposition about $H$'s extendibility to the $p$-adics. \begin{prop} \label{prop:p-adic extension of H}Let $H:\mathbb{Z}\rightarrow\mathbb{Z}$ be a $p$-Hydra map. Then, $H$ admits an extension to a continuous map $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ defined by: \begin{equation} H\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{j=0}^{p-1}\left[\mathfrak{z}\overset{p}{\equiv}j\right]\frac{a_{j}\mathfrak{z}+b_{j}}{d_{j}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:p-adic extension of H} \end{equation} meaning that, $H\left(\mathfrak{z}\right)=H_{j}\left(\mathfrak{z}\right)$ for all $j\in\left\{ 0,\ldots,p-1\right\} $ and all $\mathfrak{z}\in j+p\mathbb{Z}_{p}$. Moreover, the function $f:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ defined by: \begin{equation} f\left(\mathfrak{z}\right)=\sum_{j=0}^{p-1}\left[\mathfrak{z}\overset{p}{\equiv}j\right]\frac{a_{j}\mathfrak{z}+b_{j}}{d_{j}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} is the unique continuous function $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ whose restriction to $\mathbb{Z}$ is equal to $H$. \end{prop} Proof: Let $f\left(\mathfrak{z}\right)=\sum_{j=0}^{p-1}\left[\mathfrak{z}\overset{p}{\equiv}j\right]\frac{a_{j}\mathfrak{z}+b_{j}}{d_{j}}$; continuity is immediate, as is the fact that the restriction of $f$ to $\mathbb{Z}$ is equal to $H$. As for uniqueness, if $f:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is any continuous function whose restriction on $\mathbb{Z}$ is equal to $H$, the density of $\mathbb{Z}$ in $\mathbb{Z}_{p}$ coupled with the continuity of $f$ then forces $f\left(\mathfrak{z}\right)=\sum_{j=0}^{p-1}\left[\mathfrak{z}\overset{p}{\equiv}j\right]\frac{a_{j}\mathfrak{z}+b_{j}}{d_{j}}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. Hence, the extension of $H$ to the $p$-adics is unique. Q.E.D. \vphantom{} As mentioned in \textbf{Subsection \ref{subsec:2.1.2 It's-Probably-True}}, the $p$-adic extension of a Hydra map given by \textbf{Proposition \ref{prop:p-adic extension of H}} has been studied in conjunction with investigations of the Collatz Conjecture and the $qx+1$ maps, however\textemdash paradoxically\textemdash doing so seems to lose most information about the map, viz. \textbf{Theorem \ref{thm:shift map}}, which shows that the $2$-adic extension of the Shortened $qx+1$ map is equivalent to the shift map. For us, however, the utility of the $p$-adic extension of $H$ is in keeping track of what happens if we apply a ``wrong'' choice of branch to a given rational number, as described below. \begin{defn}[\textbf{Wrong Values and Propriety}] \label{def:1D wrong values and properiety}\index{Hydra map!wrong value}\index{Hydra map!proper} Let $H$ be a $p$-Hydra map. \vphantom{} I. We say a $p$-adic number $\mathfrak{y}\in\mathbb{Q}_{p}$ is a \textbf{wrong value for $H$ }whenever there is a $\mathbf{j}\in\textrm{String}\left(p\right)$ and a $\mathfrak{z}\in\mathbb{Z}_{p}$ so that $\mathfrak{y}=H_{\mathbf{j}}\left(\mathfrak{z}\right)$ and $H_{\mathbf{j}}\left(\mathfrak{z}\right)\neq H^{\circ\left\|\mathbf{j}\right\|}\left(\mathfrak{z}\right)$. We then call $\mathfrak{z}$ a \textbf{seed }of $\mathfrak{y}$. \vphantom{} II. We say $H$ is \textbf{proper }if $\left\|H_{j}\left(\mathfrak{z}\right)\right\|_{p}>1$ holds for any $\mathfrak{z}\in\mathbb{Z}_{p}$ and any $j\in\mathbb{Z}/p\mathbb{Z}$ so that $\left[\mathfrak{z}\right]_{p}\neq j$. \end{defn} \begin{rem} If $\mathfrak{y}$ is a wrong value of $H$ with seed $\mathfrak{z}$ and string $\mathbf{j}$ so that $\mathfrak{y}=H_{\mathbf{j}}\left(\mathfrak{z}\right)$, note that for any $\mathbf{i}\in\textrm{String}\left(p\right)$, $H_{\mathbf{i}}\left(\mathfrak{y}\right)=H_{\mathbf{i}}\left(H_{\mathbf{j}}\left(\mathfrak{z}\right)\right)=H_{\mathbf{i}\wedge\mathbf{j}}\left(\mathfrak{z}\right)$ will \emph{also }be a wrong value of $H$ with seed $\mathfrak{z}$. Thus, branches of $H$ will always map wrong values to wrong values. \end{rem} \begin{rem} Propriety is a stronger version of the integrality condition. If $H$ is an integral Hydra map, then $H_{j}\left(n\right)\notin\mathbb{Z}$ for any $n\in\mathbb{Z}$ and $j\in\mathbb{Z}/p\mathbb{Z}$ with $\left[n\right]_{p}\neq j$. On the other hand, if $H$ is proper, then $H_{j}\left(\mathfrak{z}\right)\notin\mathbb{Z}_{p}$ for any $\mathfrak{z}\in\mathbb{Z}_{p}$ and $j\in\mathbb{Z}/p\mathbb{Z}$ with $\left[\mathfrak{z}\right]_{p}\neq j$. \end{rem} \vphantom{} Like with the qualitative definitions from the previous subsection, we some technical results to show that our new definitions are useful. \begin{prop} \label{prop:Q_p / Z_p prop}Let $H$ be any $p$-Hydra map. Then, $H_{j}\left(\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}\right)\subseteq\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$ for all $j\in\mathbb{Z}/p\mathbb{Z}$. \end{prop} \begin{rem} Here, we are viewing the $H_{j}$s as functions $\mathbb{Q}_{p}\rightarrow\mathbb{Q}_{p}$. \end{rem} Proof: Let $\mathfrak{y}\in\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$ and $j\in\mathbb{Z}/p\mathbb{Z}$ be arbitrary. Note that as a $p$-Hydra map, we have $p\mid d_{j}$ and $\gcd\left(a_{j},d_{j}\right)=1$. Consequently, $p$ does not divide $a_{j}$. and so, $\left\|a_{j}\right\|_{p}=1$. Thus: \begin{equation} \left\|a_{j}\mathfrak{y}\right\|_{p}=\left\|\mathfrak{y}\right\|_{p}>1 \end{equation} So, $a_{j}\mathfrak{y}\in\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. Since $b_{j}$ is a rational integer, the $p$-adic ultrametric inequality guarantees that $a_{j}\mathfrak{y}+b_{j}\in\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. Finally, $p\mid d_{j}$ implies $\left\|d_{j}\right\|_{p}<1$, and so: \begin{equation} \left\|H_{j}\left(\mathfrak{y}\right)\right\|_{p}=\left\|\frac{a_{j}\mathfrak{y}+b_{j}}{d_{j}}\right\|_{p}\geq\left\|a_{j}\mathfrak{y}+b_{j}\right\|_{p}>1 \end{equation} which shows that $H_{j}\left(\mathfrak{y}\right)\in\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. Q.E.D. \vphantom{} The lemmata given below are the key ingredients in our proofs of the \textbf{Correspondence Principle}. These will utilize \textbf{Proposition \ref{prop:Q_p / Z_p prop}}. \begin{lem} \label{lem:integrality lemma}Let $H$ be a semi-basic $p$-Hydra map, where $p$ is prime. Then, $H$ is proper if and only if $H$ is integral. \end{lem} Proof: Let $H$ be semi-basic. I. (Proper implies integral) Suppose $H$ is proper, and let $n\in\mathbb{Z}$ be arbitrary. Then, clearly, when $j=\left[n\right]_{p}$, we have that $H_{j}\left(n\right)\in\mathbb{Z}$. So, suppose $j\in\mathbb{Z}/p\mathbb{Z}$ satisfies $j\neq\left[n\right]_{p}$. Since $H$ is proper, the fact that $n\in\mathbb{Z}\subseteq\mathbb{Z}_{p}$ and $j\neq\left[n\right]_{p}$ tells us that $\left\|H_{j}\left(n\right)\right\|_{p}>1$. Hence, $H_{j}\left(n\right)$ is not a $p$-adic integer, and thus, cannot be a rational integer either. This proves $H$ is integral. \vphantom{} II. (Integral implies proper) Suppose $H$ is integral, and\textemdash by way of contradiction\textemdash suppose $H$ is \emph{not}\textbf{ }proper. Then, there is a $\mathfrak{z}\in\mathbb{Z}_{p}$ and a $j\in\mathbb{Z}/p\mathbb{Z}$ with $j\neq\left[\mathfrak{z}\right]_{p}$ so that $\left\|H_{j}\left(\mathfrak{z}\right)\right\|_{p}\leq1$. Now, writing $\mathfrak{z}=\sum_{n=0}^{\infty}c_{n}p^{n}$: \begin{align} H_{j}\left(\mathfrak{z}\right) & =\frac{a_{j}\sum_{n=0}^{\infty}c_{n}p^{n}+b_{j}}{d_{j}}\\ & =\frac{a_{j}c_{0}+b_{j}}{d_{j}}+\frac{1}{d_{j}}\sum_{n=1}^{\infty}c_{n}p^{n}\\ \left(c_{0}=\left[\mathfrak{z}\right]_{p}\right); & =H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)+\frac{p}{d_{j}}\underbrace{\sum_{n=1}^{\infty}c_{n}p^{n-1}}_{\textrm{call this }\mathfrak{y}} \end{align} Because $H$ is a $p$-Hydra map, $d_{j}$ must be a divisor of $p$. The primality of $p$ then forces $d_{j}$ to be either $1$ or $p$. In either case, we have that $p\mathfrak{y}/d_{j}$ is an element of $\mathbb{Z}_{p}$. Our contradictory assumption $\left\|H_{j}\left(\mathfrak{z}\right)\right\|_{p}\leq1$ tells us that $H_{j}\left(\mathfrak{z}\right)$ is also in $\mathbb{Z}_{p}$, and so: \begin{equation} H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)=H_{j}\left(\mathfrak{z}\right)-\frac{p\mathfrak{y}}{d_{j}}\in\mathbb{Z}_{p} \end{equation} Since $H$ was given to be integral, $j\neq\left[\mathfrak{z}\right]_{p}$ implies $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)\notin\mathbb{Z}$. As such, $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)$ is a $p$-adic integer which is \emph{not }a rational integer. Since $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)$ is a non-integer rational number which is a $p$-adic integer, the denominator of $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)=\frac{a_{j}\left[\mathfrak{z}\right]_{p}+b_{j}}{d_{j}}$ must be divisible by some prime $q\neq p$, and hence, $q\mid d_{j}$. However, we saw that $p$ being prime forced $d_{j}\in\left\{ 1,p\right\} $\textemdash this is impossible! Thus, it must be that $H$ is proper. Q.E.D. \begin{lem} \label{lem:wrong values lemma}Let $H$ be a proper $p$-Hydra map. All wrong values of $H$ are elements of $\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. \end{lem} Proof: Let $H$ be a proper $p$-Hydra map, let $\mathfrak{z}\in\mathbb{Z}_{p}$, and let $i\in\mathbb{Z}/p\mathbb{Z}$ be such that $\left[\mathfrak{z}\right]_{p}\neq i$. Then, by definition of properness, the quantity: \begin{equation} H_{i}\left(\mathfrak{z}\right)=\frac{a_{i}\mathfrak{z}+b_{i}}{d_{i}} \end{equation} has $p$-adic absolute value $>1$. By \textbf{Proposition \ref{prop:Q_p / Z_p prop}}, this then forces $H_{\mathbf{j}}\left(H_{i}\left(\mathfrak{z}\right)\right)$ to be an element of $\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$ for all $\mathbf{j}\in\textrm{String}\left(p\right)$. Since every wrong value with seed $\mathfrak{z}$ is of the form $H_{\mathbf{j}}\left(H_{i}\left(\mathfrak{z}\right)\right)$ for some $\mathbf{j}\in\textrm{String}\left(p\right)$, some $\mathfrak{z}\in\mathbb{Z}_{p}$, and some $i\in\mathbb{Z}/p\mathbb{Z}$ for which $\left[\mathfrak{z}\right]_{p}\neq i$, this shows that every wrong value of $H$ is in $\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. So, $H$ is proper. Q.E.D. \begin{lem} \label{lem:properness lemma}Let $H$ be a proper $p$-Hydra map, let $\mathfrak{z}\in\mathbb{Z}_{p}$, and let $\mathbf{j}\in\textrm{String}\left(p\right)$. If $H_{\mathbf{j}}\left(\mathfrak{z}\right)=\mathfrak{z}$, then $H^{\circ\left\|\mathbf{j}\right\|}\left(\mathfrak{z}\right)=\mathfrak{z}$. \end{lem} Proof: Let $H$, $\mathfrak{z}$, and $\mathbf{j}$ be as given. By way of contradiction, suppose $H^{\circ\left\|\mathbf{j}\right\|}\left(\mathfrak{z}\right)\neq\mathfrak{z}$. But then $\mathfrak{z}=H_{\mathbf{j}}\left(\mathfrak{z}\right)$ implies $H^{\circ\left\|\mathbf{j}\right\|}\left(\mathfrak{z}\right)\neq H_{\mathbf{j}}\left(\mathfrak{z}\right)$. Hence, $H_{\mathbf{j}}\left(\mathfrak{z}\right)$ is a wrong value of $H$ with seed $\mathfrak{z}$. \textbf{Lemma} \ref{lem:wrong values lemma} then forces $H_{\mathbf{j}}\left(\mathfrak{z}\right)\in\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. However, $H_{\mathbf{j}}\left(\mathfrak{z}\right)=\mathfrak{z}$, and $\mathfrak{z}$ was given to be in $\mathbb{Z}_{p}$. This is impossible! Consequently, $H_{\mathbf{j}}\left(\mathfrak{z}\right)=\mathfrak{z}$ implies $H^{\circ\left\|\mathbf{j}\right\|}\left(\mathfrak{z}\right)=\mathfrak{z}$. Q.E.D. \vphantom{} Next, we need an essentially trivial observation: if $H_{\mathbf{j}}\left(n\right)=n$, $H\left(0\right)=0$, and $\left\|\mathbf{j}\right\|\geq2$, then the $p$-ity vector $\mathbf{j}$ must contain \emph{at least one} non-zero entry. \begin{prop} \label{prop:essentially trivial observation}Let $H$ be a non-degenerate $p$-Hydra map which fixes $0$, let $n\in\mathbb{Z}$ be a periodic point of $H$, and let $\Omega$ be the cycle of $H$ which contains $n$. Letting $\mathbf{j}=\left(j_{1},\ldots,j_{\left\|\Omega\right\|}\right)\in\left(\mathbb{Z}/p\mathbb{Z}\right)^{\left\|\Omega\right\|}$ be the unique string so that: \begin{equation} H_{\mathbf{j}}\left(n\right)=H^{\circ\left\|\Omega\right\|}\left(n\right)=n \end{equation} we have that either $\left\|\Omega\right\|=1$ (i.e., $n$ is a fixed point of $H$), or, there is an $\ell\in\left\{ 1,\ldots,\left\|\Omega\right\|\right\} $ so that $j_{\ell}\neq0$. \end{prop} Proof: Let $H$, $n$, and $\mathbf{j}$ be as given, and suppose $\left\|\Omega\right\|\geq2$. Since $H$ is non-degenerate, $a_{j}\neq1$ for any $j\in\left\{ 1,\ldots,p-1\right\} $. Now, \emph{by way of contradiction}, suppose that $j_{\ell}=0$ for every $\ell$. Since $\mathbf{j}$ contains only $0$s, we can write: \begin{equation} n=H_{\mathbf{j}}\left(n\right)=H_{0}^{\circ\left\|\Omega\right\|}\left(n\right) \end{equation} Because $H_{0}$ is an affine linear map of the form $H_{0}\left(x\right)=\frac{a_{0}x+b_{0}}{d_{0}}$ where $b_{0}=0$, $x=0$ is the one and only fixed point of $H_{0}$ in $\mathbb{R}$. In fact, the same is true of $H_{0}^{\circ m}\left(x\right)$ for all $m\geq1$. Thus, $n=H_{0}^{\circ\left\|\Omega\right\|}\left(n\right)$ forces $n=0$. So, $H\left(n\right)=H_{0}\left(n\right)=n$, which is to say, the cycle $\Omega$ to which $n$ belongs contains only one element: $n$ itself. This contradicts the assumption that $\left\|\Omega\right\|\geq2$. Q.E.D. \vphantom{} Now, at last, we can prove the \textbf{Correspondence Principle}. This is done in four different ways, the last of which (\textbf{Corollary \ref{cor:CP v4}}) is the simplest and most elegant. The first version\textemdash \textbf{Theorem \ref{thm:CP v1}}, proved below\textemdash is more general than \textbf{Corollary \ref{cor:CP v4}}, which is proved by using \textbf{Theorem \ref{thm:CP v1} }as a stepping stone. Whereas \textbf{Corollary \ref{cor:CP v4}} establishes a correspondence between the values attained by $\chi_{H}$ and the periodic \emph{points }of $H$, \textbf{Theorem \ref{thm:CP v1} }establishes a slightly weaker correspondence between the values attained by $\chi_{H}$ and the \emph{cycles }of $H$. \textbf{Corollary \ref{cor:CP v4}} refines this correspondence to one with individual periodic points of $\chi_{H}$. The other two versions of the Correspondence Principle are closer to the Bhm-Sontacchi Criterion, with \textbf{Corollary \ref{cor:CP v3}} providing Bhm-Sontacchi-style diophantine equations for the non-zero periodic points of $H$. In that respect, \textbf{Corollary \ref{cor:CP v3} }is not entirely new; a similar diophantine equation characterization of periodic points for certain general Collatz-type maps on $\mathbb{Z}$ can be found in Matthews' slides \cite{Matthews' slides}. The main innovations of my approach is the reformulation of these diophantine equations in terms of the function $\chi_{H}$, along with the tantalizing half-complete characterization $\chi_{H}$ provides for the divergent trajectories of $H$ (\textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}}). \begin{thm}[\textbf{Correspondence Principle, Ver. 1}] \label{thm:CP v1}Let\index{Correspondence Principle}$H$ be semi-basic. Then: \vphantom{} I. Let $\Omega$ be any cycle of $H$ in $\mathbb{Z}$, with $\left\|\Omega\right\|\geq2$. Then, there exist $x\in\Omega$ and $n\in\mathbb{N}_{1}$ so that: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)\overset{\mathbb{Z}_{q_{H}}}{=}x \end{equation} That is, there is an $n$ so that the infinite series defining $\chi_{H}\left(B_{p}\left(n\right)\right)$ converges\footnote{This series is obtained by evaluating $\chi_{H}\left(\left[B_{p}\left(n\right)\right]_{p^{m}}\right)$ using \textbf{Proposition \ref{prop:Explicit formula for Chi_H of bold j}}, and then taking the limit in $\mathbb{Z}_{q}$ as $m\rightarrow\infty$.} in $\mathbb{Z}_{q_{H}}$ to $x$. \vphantom{} II. Suppose also that $H$ is integral, and let $n\in\mathbb{N}_{1}$. If the rational number $\chi_{H}\left(B_{p}\left(n\right)\right)$ is in $\mathbb{Z}_{p}$, then $\chi_{H}\left(B_{p}\left(n\right)\right)$ is a periodic point of $H$ in $\mathbb{Z}_{p}$. In particular, if $\chi_{H}\left(B_{p}\left(n\right)\right)$ is in $\mathbb{Z}$, then $\chi_{H}\left(B_{p}\left(n\right)\right)$ is a periodic point of $H$ in $\mathbb{Z}$. \end{thm} Proof: I. Let $\Omega$ be a cycle of $H$ in $\mathbb{Z}$, with $\left\|\Omega\right\|\geq2$. Given any periodic point $x\in\Omega$ of $H$, there is going to be a string $\mathbf{j}\in\textrm{String}\left(p\right)$ of length $\left\|\Omega\right\|$ so that $H_{\mathbf{j}}\left(x\right)=x$. In particular, since $\left\|\Omega\right\|\geq2$,\textbf{ Proposition \ref{prop:essentially trivial observation}} tells us that $\mathbf{j}$ contains \emph{at least one} non-zero entry. A moment's thought shows that for any $x^{\prime}\in\Omega$ \emph{other }than $x$, the entries of the string $\mathbf{i}$ for which $H_{\mathbf{i}}\left(x^{\prime}\right)=x^{\prime}$ must be a cyclic permutation of the entries of $\mathbf{j}$. For example, for the Shortened Collatz Map and the cycle $\left\{ 1,2\right\} $, our branches are: \begin{align} H_{0}\left(x\right) & =\frac{x}{2}\\ H_{1}\left(x\right) & =\frac{3x+1}{2} \end{align} The composition sequence which sends $1$ back to itself first applies $H_{1}$ to $1$, followed by $H_{0}$: \begin{equation} H_{0}\left(H_{1}\left(1\right)\right)=H_{0}\left(2\right)=1 \end{equation} On the other hand, the composition sequence which sends $2$ back to itself first applies $H_{0}$ to $2$, followed by $H_{1}$: \begin{equation} H_{1}\left(H_{0}\left(2\right)\right)=H_{1}\left(1\right)=2 \end{equation} and the strings $\left(0,1\right)$ and $\left(1,0\right)$ are cyclic permutations of one another. In this way, note that there must then exist an $x\in\Omega$ and a $\mathbf{j}$ (for which $H_{\mathbf{j}}\left(x\right)=x$) so that \emph{the right-most entry of $\mathbf{j}$ is non-zero}. From this point forward, we will fix $x$ and $\mathbf{j}$ so that this condition is satisfied. That being done, the next observation to make is that because $H_{\mathbf{j}}$ sends $x$ to $x$, further applications of $H_{\mathbf{j}}$ will keep sending $x$ back to $x$. Expressing this in terms of concatenation yields: \begin{equation} x=H_{\mathbf{j}}^{\circ m}\left(x\right)=\underbrace{\left(H_{\mathbf{j}}\circ\cdots\circ H_{\mathbf{j}}\right)}_{m\textrm{ times}}\left(x\right)=H_{\mathbf{j}^{\wedge m}}\left(x\right) \end{equation} Writing the affine map $H_{\mathbf{j}^{\wedge m}}$ in affine form gives us: \begin{equation} x=H_{\mathbf{j}^{\wedge m}}\left(x\right)=M_{H}\left(\mathbf{j}^{\wedge m}\right)x+\chi_{H}\left(\mathbf{j}^{\wedge m}\right)=\left(M_{H}\left(\mathbf{j}\right)\right)^{m}x+\chi_{H}\left(\mathbf{j}^{\wedge m}\right) \end{equation} where the right-most equality follows from $M_{H}$'s concatenation identity (\textbf{Proposition \ref{prop:M_H concatenation identity}}). Since $\mathbf{j}$ the right-most entry of $\mathbf{j}$ is non-zero, \textbf{Proposition \ref{prop:q-adic behavior of M_H of j as the number of non-zero digits tends to infinity}} tells us that $\left\|M_{H}\left(\mathbf{j}\right)\right\|_{q_{H}}<1$. As such, $\left\|\left(M_{H}\left(\mathbf{j}\right)\right)^{m}\right\|_{q}\rightarrow0$ as $m\rightarrow\infty$. So, taking limits as $m\rightarrow\infty$, we get: \begin{equation} x\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{m\rightarrow\infty}\left(\left(M_{H}\left(\mathbf{j}\right)\right)^{m}x+\chi_{H}\left(\mathbf{j}^{\wedge m}\right)\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{m\rightarrow\infty}\chi_{H}\left(\mathbf{j}^{\wedge m}\right) \end{equation} Now, let $n$ be the integer represented by $\mathbf{j}$. Since $\mathbf{j}$'s right-most entry is non-zero, $\mathbf{j}$ is then the \emph{shortest} string representing $n$; moreover, $n$ is non-zero. In particular, we have that $\lambda_{p}\left(n\right)=\left\|\mathbf{j}\right\|>0$, and so, using \textbf{Proposition \ref{prop:Concatenation exponentiation}},\textbf{ }we can write: \begin{equation} \mathbf{j}^{\wedge m}\sim n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}},\textrm{ }\forall m\geq1 \end{equation} So, like in the proof of \textbf{Lemma \ref{lem:Chi_H o B_p functional equation}}, we then have: \begin{align} x & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{m\rightarrow\infty}\chi_{H}\left(\mathbf{j}^{\wedge m}\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{m\rightarrow\infty}\chi_{H}\left(n\frac{1-p^{m\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}}\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\chi_{H}\left(\frac{n}{1-p^{\lambda_{p}\left(n\right)}}\right)\\ & =\chi_{H}\left(B_{p}\left(n\right)\right) \end{align} This proves the existence of the desired $x$ and $n$. \vphantom{} II. Suppose $H$ is integral. First, before we assume the hypotheses of (II), let us do a brief computation. \begin{claim} \label{claim:2.2}Let $n$ be any integer $\geq1$, and let $\mathbf{j}\in\textrm{String}\left(p\right)$ be the shortest string representing $n$. Then, $H_{\mathbf{j}}$ is a continuous map $\mathbb{Z}_{q_{H}}\rightarrow\mathbb{Z}_{q_{H}}$, and, moreover, the $q_{H}$-adic integer $\chi_{H}\left(B_{p}\left(n\right)\right)$ is fixed by $H_{\mathbf{j}}$. Proof of claim: Let $n$ and $\mathbf{j}$ be as given. Using $\chi_{H}$'s concatenation identity (\textbf{Lemma \ref{lem:Chi_H concatenation identity}}), we can write: \begin{equation} \chi_{H}\left(\mathbf{j}^{\wedge k}\right)=H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{j}^{\wedge\left(k-1\right)}\right)\right)=\cdots=H_{\mathbf{j}^{\wedge\left(k-1\right)}}\left(\chi_{H}\left(\mathbf{j}\right)\right)\label{eq:Concatenation identity for Chi_H} \end{equation} By \textbf{Proposition \ref{prop:Concatenation exponentiation}}, we know that $B_{p}\left(n\right)$ represents $\lim_{k\rightarrow\infty}\mathbf{j}^{\wedge k}$ (i.e. $\textrm{DigSum}_{p}\left(\lim_{k\rightarrow\infty}\mathbf{j}^{\wedge k}\right)=B_{p}\left(n\right)$). In particular, we have that: \begin{equation} \left[B_{p}\left(n\right)\right]_{p^{k}}=\mathbf{j}^{\wedge k}=n\frac{1-p^{k\lambda_{p}\left(n\right)}}{1-p^{\lambda_{p}\left(n\right)}} \end{equation} Letting $k\rightarrow\infty$, the limit (\ref{eq:Rising Continuity Formula for Chi_H}) tells us that: \begin{equation} \lim_{k\rightarrow\infty}H_{\mathbf{j}^{\wedge\left(k-1\right)}}\left(\chi_{H}\left(n\right)\right)\overset{\mathbb{Z}_{q_{H}}}{=}\chi_{H}\left(\frac{n}{1-p^{\lambda_{p}\left(n\right)}}\right)=\chi_{H}\left(B_{p}\left(n\right)\right)\label{eq:Iterating H_bold-j on Chi_H} \end{equation} The heuristic here is that: \begin{equation} H_{\mathbf{j}}\left(\lim_{k\rightarrow\infty}H_{\mathbf{j}^{\wedge\left(k-1\right)}}\right)=\lim_{k\rightarrow\infty}H_{\mathbf{j}^{\wedge\left(k-1\right)}} \end{equation} and, hence, that $\chi_{H}\left(B_{p}\left(n\right)\right)$ will be fixed by $H_{\mathbf{j}}$. To make this rigorous, note that $\gcd\left(a_{j},p\right)=1$ for all $j$, because $H$ is semi-basic. As such, for any element of $\textrm{String}\left(p\right)$\textemdash such as our $\mathbf{j}$\textemdash the quantities $H_{\mathbf{j}}^{\prime}\left(0\right)$ and $H_{\mathbf{j}}\left(0\right)$ are then rational numbers which lie in $\mathbb{Z}_{q_{H}}$. This guarantees that the function $\mathfrak{z}\mapsto H_{\mathbf{j}}^{\prime}\left(0\right)\mathfrak{z}+H_{\mathbf{j}}\left(0\right)$ (which is, of course $H_{\mathbf{j}}\left(\mathfrak{z}\right)$) is then a well-defined \emph{continuous} map $\mathbb{Z}_{q_{H}}\rightarrow\mathbb{Z}_{q_{H}}$. Applying $H_{\mathbf{j}}$ to $\chi_{H}\left(B_{p}\left(n\right)\right)$, we obtain: \begin{align} H_{\mathbf{j}}\left(\chi_{H}\left(B_{p}\left(n\right)\right)\right) & \overset{\mathbb{Z}_{q_{H}}}{=}H_{\mathbf{j}}\left(\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{j}^{\wedge\left(k-1\right)}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{j}^{\wedge\left(k-1\right)}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{j}}\left(H_{\mathbf{j}^{\wedge\left(k-1\right)}}\left(0\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{j}^{\wedge k}}\left(0\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{j}^{\wedge k}\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\chi_{H}\left(B_{p}\left(n\right)\right) \end{align} The interchange of $H_{\mathbf{j}}$ and $\lim_{k\rightarrow\infty}$ on the second line is justified by the continuity of $H_{\mathbf{j}}$ on $\mathbb{Z}_{q_{H}}$. This proves the claim. \end{claim} \vphantom{} Now, let us actually \emph{assume} that $n$ satisfies the hypotheses of (II): suppose\emph{ }$\chi_{H}\left(B_{p}\left(n\right)\right)\in\mathbb{Z}_{p}$. By \textbf{Claim \ref{claim:2.2}}, we know that $H_{\mathbf{j}}\left(\chi_{H}\left(B_{p}\left(n\right)\right)\right)=\chi_{H}\left(B_{p}\left(n\right)\right)$. Because $H$ was given to be integral, the primality of $p$ and $H$'s semi-basicness guarantee that $H$ is proper (\textbf{Lemma \ref{lem:integrality lemma}}), and so, we can apply \textbf{Lemma \ref{lem:properness lemma}} to conclude that: \begin{equation} H^{\circ\left\|\mathbf{j}\right\|}\left(\chi_{H}\left(B_{p}\left(n\right)\right)\right)=\chi_{H}\left(B_{p}\left(n\right)\right) \end{equation} where $\left\|\mathbf{j}\right\|=\lambda_{p}\left(n\right)\geq1$. So, $\chi_{H}\left(B_{p}\left(n\right)\right)$ is a periodic point of $H$ in $\mathbb{Z}_{p}$. In particular, if $\chi_{H}\left(B_{p}\left(n\right)\right)$ is in $\mathbb{Z}$, then every element of the cycle generated by applying the $p$-adic extension of $H$ to $\chi_{H}\left(B_{p}\left(n\right)\right)$ is an element of $\mathbb{Z}$, and $\chi_{H}\left(B_{p}\left(n\right)\right)$ is then a periodic point of $H$ in $\mathbb{Z}$. Q.E.D. \vphantom{} Using the functional equation in \textbf{Lemma \ref{lem:Chi_H o B_p functional equation}}, we can restate the Correspondence Principle in terms of $\chi_{H}$ and $M_{H}$. \begin{cor}[\textbf{Correspondence Principle, Ver. 2}] \label{cor:CP v2}\index{Correspondence Principle} Suppose that $H$ is semi-basic. \vphantom{} I. Let $\Omega\subseteq\mathbb{Z}$ be any cycle of $H$. Then, viewing $\Omega\subseteq\mathbb{Z}$ as a subset of $\mathbb{Z}_{q_{H}}$, the intersection $\chi_{H}\left(\mathbb{Z}_{p}\right)\cap\Omega$ is non-empty. Moreover, for every $x\in\chi_{H}\left(\mathbb{Z}_{p}\right)\cap\Omega$, there is an $n\in\mathbb{N}_{1}$ so that: \begin{equation} x=\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)} \end{equation} \vphantom{} II. Suppose in addition that $H$ is integral, and let $n\in\mathbb{N}_{1}$. If the quantity $x$ given by: \begin{equation} x=\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)} \end{equation} is a $p$-adic integer, then $x$ is a periodic point of $H$ in $\mathbb{Z}_{p}$; if $x$ is in $\mathbb{Z}$, then $x$ is a periodic point of $H$ in $\mathbb{Z}$. Moreover, if $x\in\mathbb{Z}$, then $x$ is positive if and only if $M_{H}\left(n\right)<1$, and $x$ is negative if and only if $M_{H}\left(n\right)>1$. \end{cor} Proof: Re-write the results of \textbf{Theorem \ref{thm:CP v1}} using \textbf{Lemma \ref{lem:Chi_H o B_p functional equation}}. The positivity/negativity of $x$ stipulated in (II) follows by noting that $\chi_{H}\left(n\right)$ and $M_{H}\left(n\right)$ are positive rational numbers for all $n\in\mathbb{N}_{1}$. Q.E.D. \vphantom{} Next, we have Bhm-Sontacchi-style diophantine equations characterizing of $H$'s periodic points. \begin{cor}[\textbf{Correspondence Principle, Ver. 3}] \label{cor:CP v3}Let $H$ be a semi-basic $p$-Hydra map. \index{Bhm-Sontacchi criterion}\index{diophantine equation}Then: \vphantom{} I. Let $\Omega$ be a cycle of $H$ in $\mathbb{Z}$ containing at least two elements. Then, there is an $x\in\Omega$, and a $\mathbf{j}\in\textrm{String}\left(p\right)$ of length $\geq2$ so that: \begin{equation} x=\frac{\sum_{n=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{n}}}{a_{j_{n}}}\left(\prod_{k=1}^{n}\mu_{j_{k}}\right)p^{\left\|\mathbf{j}\right\|-n}}{p^{\left\|\mathbf{j}\right\|}-\prod_{k=1}^{\left\|\mathbf{j}\right\|}\mu_{j_{k}}}\label{eq:Generalized Bohm-Sontacchi criterion-1} \end{equation} \vphantom{} II. Suppose $H$ is integral, and let $x\left(\mathbf{j}\right)=x\left(j_{1},\ldots,j_{N}\right)$ denote the quantity: \begin{equation} x\left(\mathbf{j}\right)\overset{\textrm{def}}{=}\frac{\sum_{n=1}^{\left\|\mathbf{j}\right\|}\frac{b_{j_{n}}}{a_{j_{n}}}\left(\prod_{k=1}^{n}\mu_{j_{k}}\right)p^{\left\|\mathbf{j}\right\|-n}}{p^{\left\|\mathbf{j}\right\|}-\prod_{k=1}^{\left\|\mathbf{j}\right\|}\mu_{j_{k}}},\textrm{ }\forall\mathbf{j}\in\textrm{String}\left(p\right)\label{eq:definition of x of the js (bold version)} \end{equation} If $\mathbf{j}\in\textrm{String}\left(p\right)$ has $\left\|\mathbf{j}\right\|\geq2$ and makes $x\left(\mathbf{j}\right)\in\mathbb{Z}$, then $x\left(\mathbf{j}\right)$ is a periodic point of $H$. \end{cor} Proof: Use \textbf{Propositions \ref{prop:Explicit Formulas for M_H}} and \textbf{\ref{prop:Explicit formula for Chi_H of bold j}} on (I) and (II) of \textbf{Corollary \ref{cor:CP v2}}. Q.E.D. \vphantom{} Finally, we can state and prove the most elegant version of the Correspondence Principle for integral semi-basic Hydra maps. This is also the most \emph{powerful }version of the Correspondence Principle, because it will allow us to establish a connection between $\chi_{H}$ and $H$'s \textbf{\emph{divergent points}}. \begin{cor}[\textbf{Correspondence Principle, Ver. 4}] \label{cor:CP v4} \index{Correspondence Principle}Let $H$ be an integral semi-basic $p$-Hydra map. Then, the set of all non-zero periodic points of $H$ in $\mathbb{Z}$ is equal to $\mathbb{Z}\cap\chi_{H}\left(\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}\right)$, where $\chi_{H}\left(\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}\right)$ is viewed as a subset of $\mathbb{Z}_{q_{H}}$. \end{cor} \begin{rem} The implication ``If $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point, then $x\in\chi_{H}\left(\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}\right)$'' \emph{does not }require $H$ to be integral. \end{rem} Proof: First, note that $p$ is prime and since $H$ is integral and semi-basic, \textbf{Lemma \ref{lem:integrality lemma}} tell us that $H$ is proper. I. Let $x$ be a non-zero periodic point of $H$, and let $\Omega$ be the unique cycle of $H$ in $\mathbb{Z}$ which contains $x$. By Version 1 of the Correspondence Principle (\textbf{Theorem \ref{thm:CP v1}}), there exists a $y\in\Omega$ and a $\mathfrak{z}=B_{p}\left(n\right)\subset\mathbb{Z}_{p}$ (for some $n\in\mathbb{N}_{1}$) so that $\chi_{H}\left(\mathfrak{z}\right)=y$. Since $y$ is in $\Omega$, there is an $k\geq1$ so that $x=H^{\circ k}\left(y\right)$. In particular, there is a \emph{unique} length $k$ string $\mathbf{i}\in\textrm{String}\left(p\right)$ so that $H_{\mathbf{i}}\left(y\right)=H^{\circ k}\left(y\right)=x$. Now, let $\mathbf{j}\in\textrm{String}_{\infty}\left(p\right)$ represent $\mathfrak{z}$; note that $\mathbf{j}$ is infinite and that its entries are periodic. Using $\chi_{H}$'s concatenation identity (\textbf{Lemma \ref{lem:Chi_H concatenation identity}}), we can write: \begin{equation} x=H_{\mathbf{i}}\left(y\right)=H_{\mathbf{i}}\left(\chi_{H}\left(\mathfrak{z}\right)\right)=\chi_{H}\left(\mathbf{i}\wedge\mathbf{j}\right) \end{equation} Next, let $\mathfrak{x}$ denote the $p$-adic integer $\mathfrak{x}$ whose sequence of $p$-adic digits is $\mathbf{i}\wedge\mathbf{j}$; note that $\mathfrak{x}$ is then \emph{not} an element of $\mathbb{N}_{0}$. By the above, we have that $\chi_{H}\left(\mathfrak{x}\right)=x$, and hence that $x\in\mathbb{Z}\cap\chi_{H}\left(\mathbb{Z}_{p}\right)$. Finally, since $\mathfrak{z}=B_{p}\left(n\right)$, its $p$-adic digits are periodic, which forces $\mathfrak{z}$ to be an element of $\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$. Indeed, $\mathfrak{z}$ is not in $\mathbb{N}_{0}$ because $n\geq1$, and $B_{p}\left(n\right)\in\mathbb{N}_{0}$ if and only if $n=0$. So, letting $m$ be the rational integer represented by length-$k$ string $\mathbf{i}$, we have: \begin{equation} \mathfrak{x}\sim\mathbf{i}\wedge\mathbf{j}\sim m+p^{\lambda_{p}\left(m\right)}\mathfrak{z} \end{equation} This shows that $\mathfrak{x}\in\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$, and hence, that $x=\chi_{H}\left(\mathfrak{x}\right)\in\mathbb{Z}\cap\chi_{H}\left(\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}\right)$. \vphantom{} II. Suppose $x\in\mathbb{Z}\cap\chi_{H}\left(\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}\right)$, with $x=\chi_{H}\left(\mathfrak{z}\right)$ for some $\mathfrak{z}\in\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$. As a rational number which is both a $p$-adic integer \emph{and} not an element of $\mathbb{N}_{0}$, the $p$-adic digits of $\mathfrak{z}$ are \emph{eventually} periodic. As such, there are integers $m$ and $n$ (with $n\neq0$) so that: \begin{equation} \mathfrak{z}=m+p^{\lambda_{p}\left(m\right)}B_{p}\left(n\right) \end{equation} Here, $n$'s $p$-adic digits generate the periodic part of $\mathfrak{z}$'s digits, while $m$'s $p$-adic digits are the finite-length sequence in $\mathfrak{z}$'s digits that occurs before the periodicity sets in. Now, let $\mathbf{i}$ be the finite string representing $m$, and let $\mathbf{j}$ be the infinite string representing $B_{p}\left(n\right)$. Then, $\mathfrak{z}=\mathbf{i}\wedge\mathbf{j}$. So, by \textbf{Lemmata \ref{lem:Chi_H concatenation identity}} and \textbf{\ref{lem:Chi_H o B_p functional equation}} (the concatenation identity and $\chi_{H}\circ B_{p}$ functional equation, respectively): \begin{equation} x=\chi_{H}\left(\mathbf{i}\wedge\mathbf{j}\right)=H_{\mathbf{i}}\left(\chi_{H}\left(\mathbf{j}\right)\right)=H_{\mathbf{i}}\left(\chi_{H}\left(B_{p}\left(n\right)\right)\right)=H_{\mathbf{i}}\left(\underbrace{\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)}}_{\textrm{call this }y}\right) \end{equation} where $y\overset{\textrm{def}}{=}\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)}$ is a rational number. \begin{claim} $\left\|y\right\|_{p}\leq1$. Proof of claim: First, since $y$ is a rational number, it lies in $\mathbb{Q}_{p}$. So, by way of contradiction, suppose $\left\|y\right\|_{p}>1$. By \textbf{Lemma \ref{lem:wrong values lemma}}, because $H$ is proper, every branch specified by $\mathbf{i}$ maps $\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$ into $\mathbb{Q}_{p}\backslash\mathbb{Z}_{p}$. Consequently, $\left\|H_{\mathbf{i}}\left(y\right)\right\|_{p}>1$. However, $H_{\mathbf{i}}\left(y\right)=x$, and $x$ is a rational integer; hence, $1<\left\|H_{\mathbf{i}}\left(y\right)\right\|_{p}=\left\|x\right\|_{p}\leq1$. This is impossible! So, it must be that $\left\|y\right\|_{p}\leq1$. This proves the claim. \end{claim} \begin{claim} \label{claim:2.3}$x=H^{\circ\left\|\mathbf{i}\right\|}\left(y\right)$ Proof of claim: Suppose the equality failed. Then $H_{\mathbf{i}}\left(y\right)=x\neq H^{\circ\left\|\mathbf{i}\right\|}\left(y\right)$, and so $x=H_{\mathbf{i}}\left(y\right)$ is a wrong value of $H$ with seed $y$. Because $H$ is proper,\textbf{ Lemma \ref{lem:wrong values lemma}} forces $\left\|x\right\|_{p}=\left\|H_{\mathbf{i}}\left(y\right)\right\|_{p}>1$. However, $\left\|x\right\|_{p}\leq1$. This is just as impossible as it was in the previous paragraph. This proves the claim. \end{claim} \vphantom{} Finally, let $\mathbf{v}$ be the shortest string representing $n$, so that $\mathbf{j}$ (the string representing $B_{p}\left(n\right)$) is obtained by concatenating infinitely many copies of $\mathbf{v}$. Because $q_{H}$ is co-prime to all the $d_{j}$s, \emph{note that $H_{\mathbf{v}}$ is continuous on $\mathbb{Z}_{q_{H}}$}. As such: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{v}^{\wedge k}\right) \end{equation} implies: \begin{align} H_{\mathbf{v}}\left(\chi_{H}\left(B_{p}\left(n\right)\right)\right) & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{v}}\left(\chi_{H}\left(\mathbf{v}^{\wedge k}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{v}^{\wedge\left(k+1\right)}\right)\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\chi_{H}\left(B_{p}\left(n\right)\right) \end{align} Hence, $H_{\mathbf{v}}\left(y\right)=y$. Since $\left\|y\right\|_{p}\leq1$, the propriety of $H$ allows us to apply \textbf{Lemma \ref{lem:properness lemma}} to conclude $H_{\mathbf{v}}\left(y\right)=H^{\circ\left\|\mathbf{v}\right\|}\left(y\right)$. Thus, $y$ is a periodic point of $H:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$. By \textbf{Claim \ref{claim:2.3}}, $H$ iterates $y$ to $x$. Since $y$ is a periodic point of $H$ in $\mathbb{Z}_{p}$, this forces $x$ and $y$ to belong to the same cycle of $H$ in $\mathbb{Z}_{p}$, with $x$ being a periodic point of $H$. As such, just as $H$ iterates $y$ to $x$, so too does $H$ iterate $x$ to $y$. Likewise, since $x$ is a rational integer, so too is $y$. Thus, $x$ belongs to a cycle of $H$ in $\mathbb{Z}$, as desired. Q.E.D. \begin{example} To illustrate the Correspondence Principle in action, observe that the cycle $1\rightarrow2\rightarrow1$ of the $2$-Hydra map $H=T_{3}$ applies the even branch ($H_{0}$) second and the odd branch ($H_{1}$) first. Thus, the string $\mathbf{j}$ such that $H_{\mathbf{j}}\left(1\right)=1$ is $\mathbf{j}=\left(0,1\right)$: \begin{equation} 1=H_{0}\left(H_{1}\left(1\right)\right)=H_{0}\left(2\right)=1 \end{equation} The integer $n$ represented by $\mathbf{j}$ is $n=0\cdot2^{0}+1\cdot2^{1}=2$. Thus: \begin{equation} \centerdot01010101\ldots\overset{\mathbb{Z}_{2}}{=}B_{2}\left(2\right)=\frac{2}{1-2^{\lambda_{2}\left(2\right)}}=\frac{2}{1-4}=-\frac{2}{3} \end{equation} and so: \begin{equation} \chi_{3}\left(-\frac{2}{3}\right)=\chi_{3}\left(B_{2}\left(2\right)\right)=\frac{\chi_{3}\left(2\right)}{1-M_{3}\left(2\right)}=\frac{\frac{1}{4}}{1-\frac{3}{4}}=\frac{1}{4-3}=1 \end{equation} where: \begin{equation} \chi_{3}\left(2\right)=\frac{1}{2}\chi_{3}\left(1\right)=\frac{1}{2}\left(\frac{3\chi_{3}\left(0\right)+1}{2}\right)=\frac{1}{2}\cdot\frac{0+1}{2}=\frac{1}{4} \end{equation} and: \begin{equation} M_{3}\left(2\right)=\frac{3^{\#_{1}\left(2\right)}}{2^{\lambda_{2}\left(2\right)}}=\frac{3^{1}}{2^{2}}=\frac{3}{4} \end{equation} \end{example} \vphantom{} We end our exploration of the Correspondence Principle with a question mark. As infamous as Collatz-type problems' difficulty might be, a combination of familiarization with the available literature and personal exploration of the problems themselves suggests that, as difficult as the study of these maps' periodic points might be, the question of their divergent trajectories could very well be an order of magnitude more difficult. The diophantine equation characterizations of periodic points given in \textbf{Corollary \ref{cor:CP v3} }show that, at the bare minimum, the question of periodic points can has an \emph{expressible }non-trivial reformulation in terms of an equation of finitely many variables. Yes, there might be infinitely many possible equations to consider, and solving any single one of them\textemdash let alone all of them\textemdash is difficult\textemdash maybe even extraordinarily difficulty. But, at least there is something we can \emph{attempt }to confront. The finitude of any cycle of $H$ guarantees at least this much. For divergent trajectories, on the other hand, the prospects are bleak indeed. The only finite aspect of their existence that come to mind are the fact that they are bounded from below (if divergent to $+\infty$) or above (if divergent to $-\infty$). I am not certain if it has been shown, given an $n\in\mathbb{Z}$ belonging to a divergent trajectory of $T_{3}$, that there are iterates $T_{3}^{\circ k}\left(n\right)$ which are divisible by arbitrarily large powers of $2$; that is: \begin{equation} \sup_{k\geq0}v_{2}\left(T_{3}^{\circ k}\left(n\right)\right)=\infty \end{equation} where $v_{2}$ is the $2$-adic valuation. Even with a positive or negative answer to this question, it is not at all clear how it might be useful for tackling the question of divergent trajectories, assuming it would be useful at all. The extra difficulty of divergent trajectories is all the more startling considering the likes of the shortened $5x+1$ map, $T_{5}$. We know that the set of divergent trajectories of $T_{5}$ in $\mathbb{N}_{1}$ has density $1$, yet not a single positive integer has been proven to belong to a divergent trajectory! \cite{Lagarias-Kontorovich Paper} Late in the writing of this dissertation (March 2022), I was surprised to realize that, for an almost trivial reason, the Correspondence Principle implies a connection between $\chi_{H}$ and $H$'s divergent trajectories, and a particularly beautiful one at that. \textbf{Corollary \ref{cor:CP v4} }shows that the periodic points are controlled by values in $\mathbb{Z}$ attained by $\chi_{H}$ for \emph{rational }$p$-adic inputs; that is, those $p$-adic integers whose $p$-adic digit sequences are eventually periodic. But... what about \emph{irrational }$p$-adic integers?\textemdash that is, $\mathbb{Z}_{p}\backslash\mathbb{Q}$, the set of $\mathfrak{z}\in\mathbb{Z}_{p}$ whose sequence of $p$-adic digits are not eventually periodic. Modulo some simple conditions on $H$, we can prove that irrational $\mathfrak{z}$s which make $\chi_{H}\left(\mathfrak{z}\right)$ an integer then make $\chi_{H}\left(\mathfrak{z}\right)$ an element of a divergent trajectory; this is \textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}}. \begin{prop} \label{prop:Preparing for application to divergent trajectories}Let $H$ be a semi-basic $p$-Hydra map, and suppose that $\left\|H_{j}\left(0\right)\right\|_{q_{H}}=1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. Then $\chi_{H}\left(\mathfrak{z}\right)\neq0$ for any $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$. \end{prop} Proof: Let $H$ as given. By way of contradiction, suppose that $\chi_{H}\left(\mathfrak{z}\right)=0$ for some $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$. Now, let $j\in\left\{ 0,\ldots,p-1\right\} $ be the first $p$-adic digit of $\mathfrak{z}$; that is $j=\left[\mathfrak{z}\right]_{p}$. Then, letting $\mathfrak{z}^{\prime}$ denote the $p$-adic integer $\left(\mathfrak{z}-j\right)/p$, we can write: \begin{equation} 0=\chi_{H}\left(\mathfrak{z}\right)=\chi_{H}\left(p\mathfrak{z}^{\prime}+j\right)=H_{j}\left(\chi_{H}\left(\mathfrak{z}^{\prime}\right)\right) \end{equation} Next, suppose $j=0$. Then, since $H\left(0\right)=0$: \begin{equation} 0=H_{0}\left(\chi_{H}\left(\mathfrak{z}^{\prime}\right)\right)=\frac{\mu_{0}}{p}\chi_{H}\left(\mathfrak{z}^{\prime}\right) \end{equation} This forces $\chi_{H}\left(\mathfrak{z}^{\prime}\right)=0$, seeing as $\mu_{0}\neq0$. In this way, if the first $n$ $p$-adic digits of $\mathfrak{z}$ are all $0$, we can remove those digits from $\mathfrak{z}$ to obtain a $p$-adic integer $\mathfrak{z}^{\left(n\right)}\overset{\textrm{def}}{=}p^{-n}\mathfrak{z}$ with the property that $\left\|\mathfrak{z}^{\left(n\right)}\right\|_{p}=1$ (i.e., $\left[\mathfrak{z}^{\left(n\right)}\right]_{p}\neq0$). If \emph{all }of the digits of $\mathfrak{z}$ are $0$, this then makes $\mathfrak{z}=0$, which would contradict the given that $\mathfrak{z}$ was an element of $\mathbb{Z}_{p}\backslash\mathbb{Q}$. So, without loss of generality, we can assume that $j\neq0$. Hence: \begin{align} 0 & =H_{j}\left(\chi_{H}\left(\mathfrak{z}^{\prime}\right)\right)\\ & \Updownarrow\\ \chi_{H}\left(\mathfrak{z}^{\prime}\right) & =-\frac{pH_{j}\left(0\right)}{\mu_{j}} \end{align} Since $H$ is semi-basic, the fact that $j\neq0$ tells us that $\left\|\mu_{j}\right\|_{q_{H}}<1$ and $\left\|p\right\|_{q_{H}}=1$. This means $\left\|H_{j}\left(0\right)\right\|_{q_{H}}<1$; else we would have that $\left\|\chi_{H}\left(\mathfrak{z}^{\prime}\right)\right\|_{q_{H}}>1$, which is impossible seeing as $\chi_{H}\left(\mathbb{Z}_{p}\right)\subseteq\mathbb{Z}_{q_{H}}$ and every $q_{H}$-adic integer has\textbf{ }$q_{H}$-adic absolute value $\leq1$. So, $\left\|H_{j}\left(0\right)\right\|_{q_{H}}<1$\textemdash but, we were given that $\left\|H_{j}\left(0\right)\right\|_{q_{H}}=1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. There's our first contradiction. So, $\mathfrak{z}$ has no non-zero $p$-adic digits, which forces $\mathfrak{z}$ to be $0$. But, once again, $\mathfrak{z}$ was given to be an element of $\mathbb{Z}_{p}\backslash\mathbb{Q}$. There's our second and final contradiction. So, $\chi_{H}\left(\mathfrak{z}\right)$ must be non-zero. Q.E.D. \begin{thm} \label{thm:Divergent trajectories come from irrational z}\index{Hydra map!divergent trajectories}Let $H$ be a proper, integral, contracting, semi-basic $p$-Hydra map fixing $0$ so that $\left\|H_{j}\left(0\right)\right\|_{q_{H}}=1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. Let $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$ be such that $\chi_{H}\left(\mathfrak{z}\right)\in\mathbb{Z}$. Then $\chi_{H}\left(\mathfrak{z}\right)$ belongs to a divergent trajectory of $H$. \end{thm} Proof: Let $H$ and $\mathfrak{z}$ be as given. By \textbf{Proposition \ref{prop:Preparing for application to divergent trajectories}}, $\chi_{H}\left(\mathfrak{z}\right)\neq0$. Now, by way of contradiction, suppose that $\chi_{H}\left(\mathfrak{z}\right)$ did not belong to a divergent trajectory of $H$. By the basic theory of dynamical systems on $\mathbb{Z}$ (\textbf{Theorem \ref{thm:orbit classes partition domain}}), every element of $\mathbb{Z}$ belongs to either a divergent trajectory of $H$, or to an orbit class of $H$ which contains a cycle. Thus, it must be that $\chi_{H}\left(\mathfrak{z}\right)$ is pre-periodic. As such, there is an $n\geq0$ so that $H^{\circ n}\left(\chi_{H}\left(\mathfrak{z}\right)\right)$ is a periodic point of $H$. Using $\chi_{H}$'s functional equations from \textbf{Lemma \ref{lem:Unique rising continuation and p-adic functional equation of Chi_H}}, it then follows that $H^{\circ n}\left(\chi_{H}\left(\mathfrak{z}\right)\right)=\chi_{H}\left(\mathfrak{y}\right)$ where $\mathfrak{y}=m+p^{\lambda_{p}\left(m\right)}\mathfrak{z}$, where $m$ is the unique non-negative integer whose sequence of $p$-adic digits corresponds to the composition sequence of branches of $H$ brought to bear when we apply $H^{\circ n}$ to send $\chi_{H}\left(\mathfrak{z}\right)$ to $\chi_{H}\left(\mathfrak{y}\right)$. Moreover, since $\chi_{H}\left(\mathfrak{z}\right)\neq0$, the fact that $\left\{ 0\right\} $ is an isolated cycle of $H$ guarantees that the periodic point $\chi_{H}\left(\mathfrak{y}\right)$ is non-zero. Since $H$ is proper, we can apply Version 4 of \textbf{Correspondence Principle }(\textbf{Corollary \ref{cor:CP v4}}). This tells us $\mathfrak{y}$ is an element of $\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$. Since $\mathfrak{y}$ is rational, its $p$-adic digits are eventually periodic. However, $\mathfrak{y}=m+p^{\lambda_{p}\left(m\right)}\mathfrak{z}$, where $\mathfrak{z}$ is ``irrational'' ($\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$)\textemdash the $p$-adic digits of $\mathfrak{z}$ are aperiodic, which means the same is true of $\mathfrak{y}$. This is a contradiction! So, our assumption that $\chi_{H}\left(\mathfrak{z}\right)$ did not belong to a divergent trajectory of $H$ forces the digits of $\mathfrak{y}$ (which are not periodic) to be eventually periodic\textemdash a clear impossibility. This proves that $\chi_{H}\left(\mathfrak{z}\right)$ must be a divergent point of $H$. Q.E.D. \vphantom{} This theorem suggests the following: \begin{conjecture}[\textbf{A Correspondence Principle for Divergent Points?}] \label{conj:correspondence theorem for divergent trajectories}Provided that $H$ satisfies certain prerequisites such as the hypotheses of \textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}}, $x\in\mathbb{Z}$ belongs to a divergent trajectory under $H$ if and only if there is a $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$ so that $\chi_{H}\left(\mathfrak{z}\right)\in\mathbb{Z}$. \end{conjecture} \vphantom{} The difficulty of this conjecture is that, unlike the Correspondence Principle for periodic points of $H$, I have yet to find \emph{constructive }method of producing a $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$ for which $\chi_{H}\left(\mathfrak{z}\right)$ is in $\mathbb{Z}$. Finally, we have: \begin{conjecture}[\textbf{Divergence Conjectures for $qx+1$}] Let $\chi_{q}\left(\mathfrak{z}\right)$ denote the numen of the shortened $qx+1$ map\index{$qx+1$ map}, where $q$ is an odd prime. Then, there exists a $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{Q}$ so that $\chi_{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$ if and only if $q=3$. \end{conjecture} \subsection{\label{subsec:2.2.4 Other-Avenues}Other Avenues} \subsubsection{\label{subsec:Relaxing-the-Requirements}Relaxing the Requirements of Monogenicity and Non-Degeneracy} With regard to the convergence of $\chi_{H}$ over the $p$-adics, the key expression is (\ref{eq:Formal rising limit of Chi_H}): \begin{equation} \lim_{m\rightarrow\infty}\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{m}}\right)\overset{\mathbb{Z}_{q}}{=}\sum_{\ell=1}^{\infty}\frac{b_{j_{\ell}\left(\mathfrak{z}\right)}}{d_{j_{\ell}\left(\mathfrak{z}\right)}}M_{H}\left(\mathbf{j}_{\ell-1}\left(\mathfrak{z}\right)\right) \end{equation} where, recall, we write the $p$-adic integer $\mathfrak{z}$ as: \begin{equation} \mathfrak{z}=j_{1}\left(\mathfrak{z}\right)+j_{2}\left(\mathfrak{z}\right)p+j_{3}\left(\mathfrak{z}\right)p^{2}+\cdots \end{equation} and write $\mathbf{j}_{\ell-1}\left(\mathfrak{z}\right)=\left(j_{1}\left(\mathfrak{z}\right),\ldots,j_{\ell-1}\left(\mathfrak{z}\right)\right)\in\left(\mathbb{Z}/p\mathbb{Z}\right)^{\ell-1}$, with the convention that $M_{H}\left(\mathbf{j}_{0}\left(\mathfrak{z}\right)\right)=1$. By the concatenation identity \textbf{Proposition \ref{prop:M_H concatenation identity}}, to obtain $M_{H}\left(\mathbf{j}_{\ell}\left(\mathfrak{z}\right)\right)$, we multiply $M_{H}\left(\mathbf{j}_{\ell-1}\left(\mathfrak{z}\right)\right)$ by a factor of $a_{j_{\ell}\left(\mathfrak{z}\right)}/d_{j_{\ell}\left(\mathfrak{z}\right)}$. The various conditions contained in the notion of semi-basicness were designed so as to ensure that $\left\|a_{j}/d_{j}\right\|_{q_{H}}\geq1$ occurs if and only if $j=0$. This $q_{H}$-adic condition, in turn, is what guarantees that (\ref{eq:Formal rising limit of Chi_H}) will converge in $\mathbb{Z}_{q_{H}}$ for any $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$\textemdash that is, any $\mathfrak{z}$ with infinitely many non-zero $p$-adic digits. For the sake of clarity, rather than continue to work with the encumbrance of our indexing notation and an arbitrary $p$-Hydra map, we will illustrate the obstacles in relaxing the various conditions (monogenicity, (semi)-simplicity, non-degeneracy), by way of an example. \begin{example}[\textbf{A polygenic Hydra map}] \label{exa:Polygenic example, part 1}Consider the $3$-Hydra map $H:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by: \begin{equation} H\left(n\right)\overset{\textrm{def}}{=}\begin{cases} \frac{an}{3} & \textrm{if }n\overset{3}{\equiv}0\\ \frac{bn+b^{\prime}}{3} & \textrm{if }n\overset{3}{\equiv}1\\ \frac{cn+2c^{\prime}}{3} & \textrm{if }n\overset{3}{\equiv}2 \end{cases}\label{eq:Toy model for a polygenic 3-Hydra map} \end{equation} where $a$, $b$, and $c$ are positive integers co-prime to $3$, and where $b^{\prime}$ and $c^{\prime}$ are positive integers so that $b+b^{\prime}\overset{3}{\equiv}0$ and $c+c^{\prime}\overset{3}{\equiv}0$, respectively; $b^{\prime}$ and $c^{\prime}$ guarantee that each of the branches of $H$ outputs a non-negative integer. For this $H$, we have that: \begin{equation} M_{H}\left(\mathbf{j}\right)=\frac{a^{\#_{3:0}\left(\mathbf{j}\right)}\times b^{\#_{3:1}\left(\mathbf{j}\right)}\times c^{\#_{3:2}\left(\mathbf{j}\right)}}{3^{\left\|\mathbf{j}\right\|}} \end{equation} In order for (\ref{eq:Formal rising limit of Chi_H}) to converge, we need $M_{H}\left(\mathbf{j}\right)$ to become small in some sense as $\left\|\mathbf{j}\right\|\rightarrow\infty$ with infinitely many non-zero digits. As such, the value of $a$ is irrelevant to our convergence issue; only the constants $b$ and $c$, associated\textemdash respectively\textemdash to the branches $H_{1}$ and $H_{2}$ are of import, because any $\mathbf{j}$ with only finitely many non-zero entries reduces to an integer. In the monogenic case dealt with in this dissertation, we required both $b$ and $c$ to be multiples of some integer $q_{H}\geq2$. \textbf{Proposition \ref{prop:q-adic behavior of M_H of j as the number of non-zero digits tends to infinity}} showed that this condition guaranteed the $q_{H}$-adic decay of $M_{H}\left(\mathbf{j}\right)$ to $0$ as the number of non-zero digits of $\mathbf{j}$ increased to infinity. On the other hand, if this were the \index{Hydra map!polygenic}\textbf{polygenic }case, then $b$ and $c$ would be co-prime to one another. Note that if one or both of $b$ and $c$ are equal to $1$, then we have a degenerate case, because multiplication by $1$ does not decrease $p$-adic size for any prime $p$. So, we can suppose that $b$ and $c$ are co-prime. For simplicity, let $b$ and $c$ be distinct primes, with neither $b$ nor $c$ being $3$. Then, there are several possibilities for $M_{H}\left(\mathbf{j}\right)$ as $\mathbf{j}\rightarrow\infty$ with $\mathbf{j}$ having infinitely many non-zero entries: \end{example} \begin{enumerate} \item If $\mathbf{j}$ has infinitely many $1$s (this corresponds to multiplications by $b$), then $M_{H}\left(\mathbf{j}\right)$ tends to $0$ in $b$-adic magnitude. \item If $\mathbf{j}$ has infinitely many $2$s (this corresponds to multiplications by $c$), then $M_{H}\left(\mathbf{j}\right)$ tends to $0$ in $c$-adic magnitude. \item If only one of $1$ or $2$ occurs infinitely many times in $\mathbf{j}$, then $M_{H}\left(\mathbf{j}\right)$ will tend to $0$ in either $b$-adic or $c$-adic magnitude, \emph{but not in both}\textemdash \emph{that} occurs if and only if $\mathbf{j}$ contains both infinitely many $1$s \emph{and} infinitely many $2$s. \end{enumerate} \begin{rem} In hindsight, I believe my notion of a \textbf{frame} (see Subsection \ref{subsec:3.3.3 Frames}) might be suited for the polygenic case. Considering \textbf{Example \ref{exa:Polygenic example, part 1}}, we could assign the $b$-adic and $c$-adic convergence, respectively, to the sets of $3$-adic integers whose representative strings $\mathbf{j}$ have infinitely many $1$s and $2$s, respectively. This would then allow for a meaningful definition of $\chi_{H}$, and thereby potentially open the door to applying my methods to polygenic Hydra maps. \end{rem} \subsubsection{\label{subsec:Baker,-Catalan,-and}Baker, Catalan, and Collatz} Any proof of the Weak Collatz Conjecture\footnote{Recall, this is the assertion that the only periodic points of the Collatz map in the positive integers are $1$, $2$, and $4$.} will necessarily entail a significant advancement in\index{transcendental number theory} transcendental number theory. A proof of the Weak Collatz Conjecture would necessarily yield a proof of \textbf{Baker's Theorem} far simpler than any currently known method \cite{Tao Blog}. \index{Baker's Theorem}Baker's Theorem, recall, concerns lower bounds on the absolute values of \textbf{linear forms of logarithms}, which are expressions of the form: \begin{equation} \beta_{1}\ln\alpha_{1}+\cdots+\beta_{N}\ln\alpha_{N}\label{eq:linear form in logarithm} \end{equation} where the $\beta_{n}$s and $\alpha_{n}$s are complex algebraic numbers (with all of the $\alpha_{n}$s being non-zero). Most proofs of Baker's Theorem employ a variant of what is known as Baker's Method, in which one constructs of an analytic function (called an \textbf{auxiliary function})\textbf{ }with a large number of zeroes of specified degree so as to obtain contradictions on the assumption that (\ref{eq:linear form in logarithm}) is small in absolute value. The import of Baker's Theorem is that it allows one to obtain lower bounds on expressions such as $\left\|2^{m}-3^{n}\right\|$, where $m$ and $n$ are positive integers, and it was for applications such as these in conjunction with the study of Diophantine equations that Baker earned the Fields Medal. See \cite{Baker's Transcendental Number Theory} for a comprehensive account of the subject; also, parts of \cite{Cohen Number Theory}. With $\chi_{H}$ and the \textbf{Correspondence Principle }at our disposal, we can get a glimpse at the kinds of advancements in transcendental number theory that might be needed in order to resolve the Weak Collatz Conjecture. To begin, it is instructive to consider the following table of values for $\chi_{H}$ and related functions in the case of the Shortened $qx+1$ maps $T_{q}$, where $q$ is an odd prime. For brevity, we write $\chi_{q}$ to denote $\chi_{T_{q}}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$. \begin{center} \begin{table} \begin{centering} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|} \hline $n$ & $\#_{1}\left(n\right)$ & $\lambda_{2}\left(n\right)$ & $\chi_{q}\left(n\right)$ & $\chi_{q}\left(B_{2}\left(n\right)\right)$ & $\chi_{3}\left(B_{2}\left(n\right)\right)$ & $\chi_{5}\left(B_{2}\left(n\right)\right)$\tabularnewline \hline \hline $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\tabularnewline \hline $1$ & $1$ & $1$ & $\frac{1}{2}$ & $\frac{1}{2-q}$ & $-1$ & $-\frac{1}{3}$\tabularnewline \hline $2$ & $1$ & $2$ & $\frac{1}{4}$ & $\frac{1}{4-q}$ & $1$ & $-1$\tabularnewline \hline $3$ & $2$ & $2$ & $\frac{2+q}{4}$ & $\frac{2+q}{4-q^{2}}$ & $-1$ & $-\frac{1}{3}$\tabularnewline \hline $4$ & $1$ & $3$ & $\frac{1}{8}$ & $\frac{1}{8-q}$ & $\frac{1}{5}$ & $\frac{1}{3}$\tabularnewline \hline $5$ & $2$ & $3$ & $\frac{4+q}{8}$ & $\frac{4+q}{8-q^{2}}$ & $-7$ & $-\frac{9}{17}$\tabularnewline \hline $6$ & $2$ & $3$ & $\frac{2+q}{8}$ & $\frac{2+q}{8-q^{2}}$ & $-5$ & $-\frac{7}{17}$\tabularnewline \hline $7$ & $3$ & $3$ & $\frac{4+2q+q^{2}}{8}$ & $\frac{4+2q+q^{2}}{8-q^{3}}$ & $-1$ & $-\frac{1}{3}$\tabularnewline \hline $8$ & $1$ & $4$ & $\frac{1}{16}$ & $\frac{1}{16-q}$ & $\frac{1}{13}$ & $\frac{1}{11}$\tabularnewline \hline $9$ & $2$ & $4$ & $\frac{8+q}{16}$ & $\frac{8+q}{16-q^{2}}$ & $\frac{11}{7}$ & $-\frac{13}{9}$\tabularnewline \hline $10$ & $2$ & $4$ & $\frac{4+q}{16}$ & $\frac{4+q}{16-q^{2}}$ & $1$ & $-1$\tabularnewline \hline $11$ & $3$ & $4$ & $\frac{8+4q+q^{2}}{16}$ & $\frac{8+4q+q^{2}}{16-q^{3}}$ & $-\frac{29}{11}$ & $-\frac{53}{109}$\tabularnewline \hline $12$ & $2$ & $4$ & $\frac{2+q}{16}$ & $\frac{2+q}{16-q^{2}}$ & $\frac{5}{7}$ & $-\frac{7}{9}$\tabularnewline \hline $13$ & $3$ & $4$ & $\frac{8+2q+q^{2}}{16}$ & $\frac{8+2q+q^{2}}{16-q^{3}}$ & $-\frac{23}{11}$ & $-\frac{43}{109}$\tabularnewline \hline $14$ & $3$ & $4$ & $\frac{4+2q+q^{2}}{16}$ & $\frac{4+2q+q^{2}}{16-q^{3}}$ & $\frac{19}{7}$ & $-\frac{39}{109}$\tabularnewline \hline $15$ & $4$ & $4$ & $\frac{8+4q+2q^{2}+q^{3}}{16}$ & $\frac{8+4q+2q^{2}+q^{3}}{16-q^{4}}$ & $-1$ & $-\frac{1}{3}$\tabularnewline \hline \end{tabular} \par\end{centering} \caption{Values of $\chi_{q}\left(n\right)$ and related functions} \end{table} \par\end{center} \begin{example} \index{Correspondence Principle}As per the Correspondence Principle,\textbf{ }note that the integer values attained by $\chi_{3}\left(B_{2}\left(n\right)\right)$ and $\chi_{5}\left(B_{2}\left(n\right)\right)$ are all periodic points of the maps $T_{3}$ and $T_{5}$, respectively; this includes fixed points at negative integers, as well. Examining $\chi_{q}\left(B_{2}\left(n\right)\right)$, we see certain patterns, such as: \begin{equation} \chi_{q}\left(B_{2}\left(2^{n}-1\right)\right)=\frac{1}{2-q},\textrm{ }\forall n\in\mathbb{N}_{1} \end{equation} More significantly, $\chi_{3}\left(B_{2}\left(n\right)\right)$ appears to be more likely to be positive than negative, whereas the opposite appears to hold true for $q=5$ (and, heuristically, for all $q\geq5$). Of special interest, however, is: \begin{equation} \chi_{q}\left(B_{2}\left(10\right)\right)=\frac{4+q}{16-q^{2}}=\frac{1}{4-q} \end{equation} \end{example} By Version 1 of the \textbf{Correspondence} \textbf{Principle} (\textbf{Theorem \ref{thm:CP v1}}), every cycle $\Omega\subseteq\mathbb{Z}$ of $T_{q}$ with $\left\|\Omega\right\|\geq2$ contains an integer $x$ of the form: \begin{equation} x=\chi_{q}\left(B_{2}\left(n\right)\right)=\frac{\chi_{q}\left(n\right)}{1-\frac{q^{\#_{2:1}\left(n\right)}}{2^{\lambda_{2}\left(n\right)}}}=\frac{2^{\lambda_{2}\left(n\right)}\chi_{q}\left(n\right)}{2^{\lambda_{2}\left(n\right)}-q^{\#_{2:1}\left(n\right)}}\label{eq:Rational expression of odd integer periodic points of ax+1} \end{equation} for some $n\in\mathbb{N}_{1}$. In fact, every periodic point $x$ of $T_{q}$ in the odd integers can be written in this form. As such $\left\|2^{\lambda_{2}\left(n\right)}-q^{\#_{2:1}\left(n\right)}\right\|=1$ is a \emph{sufficient condition }for $\chi_{q}\left(B_{2}\left(n\right)\right)$ to be a periodic point of $T_{q}$. However, as the $n=10$ case shows, this is not\emph{ }a \emph{necessary }condition: there can be values of $n$ where $2^{\lambda_{2}\left(n\right)}-q^{\#_{2:1}\left(n\right)}$ is large in archimedean absolute value, and yet nevertheless divides the numerator on the right-hand side of (\ref{eq:Rational expression of odd integer periodic points of ax+1}), thereby reducing $\chi_{q}\left(B_{2}\left(n\right)\right)$ to an integer. In fact, thanks to P. Mih\u{a}ilescu \index{Mihu{a}ilescu, Preda@Mih\u{a}ilescu, Preda}'s resolution of \index{Catalan's Conjecture}\textbf{ Catalan's Conjecture}, it would seem that Baker's Method-style estimates on the archimedean\emph{ }size of $2^{\lambda_{2}\left(n\right)}-q^{\#_{2:1}\left(n\right)}$ will be of little use in understanding $\chi_{q}\left(B_{2}\left(n\right)\right)$: \begin{thm}[\textbf{Mih\u{a}ilescu's Theorem}\footnote{Presented in \cite{Cohen Number Theory}.}] The only choice of $x,y\in\mathbb{N}_{1}$ and $m,n\in\mathbb{N}_{2}$ for which: \begin{equation} x^{m}-y^{n}=1\label{eq:Mihailescu's Theorem} \end{equation} are $x=3$, $m=2$, $y=2$, $n=3$ (that is, $3^{2}-2^{3}=1$). \end{thm} \vphantom{} With Mih\u{a}ilescu's Theorem, it is easy to see that for any odd integer $q\geq3$, $\left\|q^{\#_{2:1}\left(n\right)}-2^{\lambda_{2}\left(n\right)}\right\|$ will never be equal to $1$ for any $n\geq8$, because the exponent of $2$ (that is, $\lambda_{2}\left(n\right)$) will be $\geq4$ for all $n\geq8$ (any such $n$ has $4$ or more binary digits). Consequently, for any odd prime $q$, if $n\geq8$ makes $\chi_{q}\left(B_{2}\left(n\right)\right)$ into a rational integer (and hence, a periodic point of $T_{q}$), it \emph{must }be that the numerator $2^{\lambda_{2}\left(n\right)}\chi_{q}\left(n\right)$ in (\ref{eq:Rational expression of odd integer periodic points of ax+1}) is a multiple of $2^{\lambda_{2}\left(n\right)}-q^{\#_{2:1}\left(n\right)}$. The set of $p$-Hydra maps for which conclusions of this sort hold can be enlarged in several ways. First, in general, note that we have: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)=\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)}=\frac{p^{\lambda_{p}\left(n\right)}\chi_{H}\left(n\right)}{p^{\lambda_{p}\left(n\right)}\left(1-M_{H}\left(n\right)\right)}=\frac{p^{\lambda_{p}\left(n\right)}\chi_{H}\left(n\right)}{p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}}\label{eq:Chi_H o B functional equation with fraction in simplest form} \end{equation} If $\mu_{0}=1$ and there is an integer $\mu\geq2$ so that $\mu_{1},\ldots,\mu_{p-1}$ are all positive integer powers of $\mu$ (i.e., for each $j\in\left\{ 1,\ldots,p-1\right\} $ there is an $r_{j}\in\mathbb{N}_{1}$ so that $\mu_{j}=\mu^{r_{j}}$) then, when $n\geq1$, the denominator of the right-hand side takes the form $p^{a}-\mu^{b}$ for some positive integers $a$ and $b$ depending solely on $n$. By Mih\u{a}ilescu's Theorem, the only way to have $\left\|p^{a}-\mu^{b}\right\|=1$ is for either $p^{a}-\mu^{b}=3^{2}-2^{3}$ or $p^{a}-\mu^{b}=2^{3}-3^{2}$. As such, for this case, once $n$ is large enough so that both $a$ and $b$ are $\geq3$, the only way for $\chi_{H}\left(B_{p}\left(n\right)\right)$ to be a rational integer for such an $n$ is if $p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}$ is a divisor of the integer $p^{\lambda_{p}\left(n\right)}\chi_{H}\left(n\right)$ of magnitude $\geq2$. In the most general case, if we allow the $\mu_{j}$ to take on different values (while still requiring $H$ to be basic so that the Correspondence Principle applies), in order to replicate the application of Mih\u{a}ilescu's Theorem to the $p^{a}-\mu^{b}$ case, examining the the denominator term $p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}$ from (\ref{eq:Chi_H o B functional equation with fraction in simplest form}) and ``de-parameterizing'' it, the case where the denominator reduces to $1$ or $-1$ can be represented by a Diophantine equation\index{diophantine equation} of the form: \begin{equation} \left\|x^{m}-y_{1}^{n_{1}}y_{2}^{n_{2}}\cdots y_{J}^{n_{J}}\right\|=1\label{eq:Generalized Catalan Diophantine Equation} \end{equation} where $J$ is a fixed integer $\geq2$, and $x$, $y_{1},\ldots,y_{J}$, $m$, and $n_{1},\ldots,n_{J}$ are positive integer variables. For any fixed choice of $m$ and the $n_{j}$s, \textbf{Faltings' Theorem}\index{Faltings' Theorem}\textbf{ }(the resolution of \textbf{Mordell's Conjecture}) guarantees there are only finitely many rational numbers $x,y_{1},\ldots,y_{J}$ for which (\ref{eq:Generalized Catalan Diophantine Equation}) holds true. On the other hand, for any fixed choice of $x,y_{1},\ldots,y_{J}$ (where we identify $x$ with $p$ and the $y_{j}$s with the $\mu_{j}$s), if it can be shown that (\ref{eq:Generalized Catalan Diophantine Equation}) holds for only finitely many choices of $m$ and the $n_{j}$s, it will follow that the corresponding case for $\left\|p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}\right\|$ is not equal to $1$ for all sufficiently large positive integers $n$. In that situation, any cycles of $H$ not accounted for by values of $n$ for which $\left\|p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}\right\|=1$ must occur as a result of $\left\|p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}\right\|$ being an integer $\geq2$ which divides $p^{\lambda_{p}\left(n\right)}\chi_{H}\left(n\right)$. In light of this, rather than the archimedean/euclidean \emph{size }of $p^{\lambda_{p}\left(n\right)}-\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(n\right)}$, it appears we must study its \emph{multiplicative }structure, as well as that of $p^{\lambda_{p}\left(n\right)}\chi_{H}\left(n\right)$\textemdash which is to say, the set of these integers' prime divisors, and how this set depends on $n$. In other words, we ought to study the $p$-adic absolute values of $\left\|\chi_{H}\left(n\right)\right\|_{p}$ and $\left\|1-M_{H}\left(n\right)\right\|_{p}$ for various values of $n\geq1$ and primes $p$. It may also be of interest to use $p$-adic methods (Fourier series, Mellin transform) to study $\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{p}$ as a real-valued function of a $p$-adic integer variable for varying values of $p$. \subsubsection{\label{subsec:Connections-to-Tao}Connections to Tao (2019)} The origins of the present paper lie in work the author did in late 2019 and early 2020. Around the same time, in late 2019, Tao published a paper on the Collatz Conjecture that applied probabilistic techniques in a novel way, by use of what Tao\index{Tao, Terence} called ``\index{Syracuse Random Variables}Syracuse Random Variables'' \cite{Tao Probability paper}. Despite the many substantial differences between these two approaches (Tao's is probabilistic, ours is not), they are linked at a fundamental level, thanks to the function $\chi_{3}$\textemdash our shorthand for $\chi_{T_{3}}$, the $\chi_{H}$ associated to the Shortened Collatz map. Given that both papers have their genesis in an examination of the behavior of different combinations of the iterates of branches of $T_{3}$, it is not entirely surprising that both approaches led to $\chi_{3}$. Tao's approach involves constructing his Syracuse Random Variables and then comparing them to a set-up involving tuples of geometric random variables, the comparison in question being an estimate on the ``distance'' between the Syracuse Random Variables and the geometric model, as measured by the total variation norm for discrete random variables. To attain these results, the central challenge Tao overcomes is obtaining explicit estimates for the decay of the characteristic function of the Syracuse Random Variables. In our terminology, Tao establishes decay estimates for the archimedean absolute value of the function $\varphi_{3}:\hat{\mathbb{Z}}_{3}\rightarrow\mathbb{C}$ defined by: \begin{equation} \varphi_{3}\left(t\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{2}}e^{-2\pi i\left\{ t\chi_{3}\left(\mathfrak{z}\right)\right\} _{3}}d\mathfrak{z},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3}\label{eq:Tao's Characteristic Function} \end{equation} where $\left\{ \cdot\right\} _{3}$ is the $3$-adic fractional part, $d\mathfrak{z}$ is the Haar probability measure on $\mathbb{Z}_{2}$, and $\hat{\mathbb{Z}}_{3}=\mathbb{Z}\left[\frac{1}{3}\right]/\mathbb{Z}=\mathbb{Q}_{3}/\mathbb{Z}_{3}$ is the Pontryagin dual of $\mathbb{Z}_{3}$, identified here with the set of all rational numbers in $\left[0,1\right)$ whose denominators are non-negative integer powers of the prime number $3$. Tao's decay estimate is given in \textbf{Proposition 1.17 }of his paper, where the above integral appears in the form of an expected value \cite{Tao Probability paper}. With this in mind, a natural next step for furthering both this paper and Tao's would be to study the ``characteristic function\index{characteristic function}'' of $\chi_{H}$ for an arbitrary semi-basic $p$-Hydra maps $H$; this is the continuous function $\varphi_{H}:\hat{\mathbb{Z}}_{q_{H}}\rightarrow\mathbb{C}$ defined by: \begin{equation} \varphi_{H}\left(t\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\chi_{H}\left(\mathfrak{z}\right)\right\} _{q_{H}}}d\mathfrak{z},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{q_{H}}\label{eq:Characteristic function of Chi_H} \end{equation} where $\left\{ \cdot\right\} _{q_{H}}$ is the $q_{H}$-adic fractional part, $d\mathfrak{z}$ is the Haar probability measure on $\mathbb{Z}_{p}$, and $\hat{\mathbb{Z}}_{q_{H}}=\mathbb{Z}\left[\frac{1}{q_{H}}\right]/\mathbb{Z}=\mathbb{Q}_{q_{H}}/\mathbb{Z}_{q_{H}}$ is the Pontryagin dual of $\mathbb{Z}_{q_{H}}$, identified here with the set of all rational numbers in $\left[0,1\right)$ whose denominators are non-negative integer powers of $q_{H}$. We can establish functional equations\index{functional equation} can be established for $\varphi_{H}$. \begin{prop} Let $H$ be a semi-basic $p$-Hydra map which fixes $0$, and write $q=q_{H}$. Then, the characteristic function $\varphi_{H}:\hat{\mathbb{Z}}_{q}\rightarrow\mathbb{C}$ defined by \emph{(\ref{eq:Characteristic function of Chi_H})} satisfies the functional equation: \begin{equation} \varphi_{H}\left(t\right)=\frac{1}{p}\sum_{j=0}^{p-1}e^{-2\pi i\left\{ \frac{b_{j}t}{d_{j}}\right\} _{q}}\varphi_{H}\left(\frac{a_{j}t}{d_{j}}\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{q}\label{eq:Phi_H functional equation} \end{equation} \end{prop} \begin{rem} Since $H$ fixes $0$ and is semi-basic, \textbf{Proposition \ref{prop:co-primality of d_j and q_H}} tells us that all of the $d_{j}$s are co-prime to $q$. Thus, the multiplication-by-$d_{j}$ map $t\mapsto d_{j}t$ is a group automorphism of $\hat{\mathbb{Z}}_{q}$, and as such, $b_{j}t/d_{j}$ and $a_{j}t/d_{j}$ denote the images of $b_{j}t$ and $a_{j}t$, respectively, under the inverse of this automorphism. One can remove the denominator terms if desired by writing: \begin{align} d & \overset{\textrm{def}}{=}\textrm{lcm}\left(d_{0},\ldots,d_{p-1}\right)\\ d_{j}^{\prime} & \overset{\textrm{def}}{=}\frac{d}{d_{j}},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \in\mathbb{Z}\\ \alpha_{j} & \overset{\textrm{def}}{=}a_{j}d_{j}^{\prime}\in\mathbb{Z}\\ \beta_{j} & \overset{\textrm{def}}{=}b_{j}d_{j}^{\prime}\in\mathbb{Z} \end{align} and then replacing $t$ with $dt$ in (\ref{eq:Phi_H functional equation}) to obtain: \begin{equation} \varphi_{H}\left(dt\right)=\frac{1}{p}\sum_{j=0}^{p-1}e^{-2\pi i\beta_{j}t}\varphi_{H}\left(\alpha_{j}t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{q} \end{equation} \end{rem} Proof: The proof is a straight-forward computation. Details regarding Fourier analysis on $p$-adic rings can be found in \cite{Automorphic Representations}. \begin{align} \varphi_{H}\left(t\right) & =\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\chi_{H}\left(\mathfrak{z}\right)\right\} _{q}}d\mathfrak{z}\\ & =\sum_{j=0}^{p-1}\int_{j+p\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\chi_{H}\left(\mathfrak{z}\right)\right\} _{q}}d\mathfrak{z}\\ \left(\mathfrak{y}=\frac{\mathfrak{z}-j}{p}\right); & =\frac{1}{p}\sum_{j=0}^{p-1}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\chi_{H}\left(p\mathfrak{y}+j\right)\right\} _{q}}d\mathfrak{y}\\ & =\frac{1}{p}\sum_{j=0}^{p-1}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\frac{a_{j}\chi_{H}\left(\mathfrak{y}\right)+b_{j}}{d_{j}}\right\} _{q}}d\mathfrak{y}\\ & =\frac{1}{p}\sum_{j=0}^{p-1}e^{-2\pi i\left\{ \frac{b_{j}t}{d_{j}}\right\} _{q}}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ \frac{a_{j}t}{d_{j}}\chi_{H}\left(\mathfrak{y}\right)\right\} _{q}}d\mathfrak{y}\\ & =\frac{1}{p}\sum_{j=0}^{p-1}e^{-2\pi i\left\{ \frac{b_{j}t}{d_{j}}\right\} _{q}}\varphi_{H}\left(\frac{a_{j}t}{d_{j}}\right) \end{align} Q.E.D. \vphantom{} The Fourier analysis here naturally leads to probabilistic notions. For any measurable function $f:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$, any $n\in\mathbb{N}_{0}$ and any $m\in\mathbb{Z}$, we write: \begin{equation} \textrm{P}\left(f\overset{q^{n}}{\equiv}m\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\left[f\left(\mathfrak{z}\right)\overset{q^{n}}{\equiv}m\right]d\mathfrak{z}\label{eq:Definition of the probability that f is congruent to k mod q to the n} \end{equation} where $d\mathfrak{z}$ is the Haar probability measure on $\mathbb{Z}_{p}$ and $\left[f\left(\mathfrak{z}\right)\overset{q^{n}}{\equiv}m\right]$ is $1$ if $f\left(\mathfrak{z}\right)$ is congruent to $m$ mod $q^{n}$ and is $0$ otherwise. We adopt the convention that $\mathfrak{x}\overset{1}{\equiv}\mathfrak{y}$ is true for any $q$-adic integers $\mathfrak{x}$ and $\mathfrak{y}$, and hence, that: \begin{equation} \textrm{P}\left(f\overset{q^{0}}{\equiv}m\right)=\textrm{P}\left(f\overset{1}{\equiv}m\right)=\int_{\mathbb{Z}_{p}}\left[f\left(\mathfrak{z}\right)\overset{1}{\equiv}m\right]d\mathfrak{z}=\int_{\mathbb{Z}_{p}}d\mathfrak{z}=1 \end{equation} for any $f$ and any $m$. Fourier relations follow from the identity: \begin{equation} \left[\mathfrak{x}\overset{q^{n}}{\equiv}\mathfrak{y}\right]=\frac{1}{q^{n}}\sum_{k=0}^{q^{n}-1}e^{2\pi i\left\{ k\frac{\mathfrak{x}-\mathfrak{y}}{q^{n}}\right\} _{q}},\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall\mathfrak{x},\mathfrak{y}\in\mathbb{Z}_{q} \end{equation} This gives us the typical Fourier relations between probabilities and characteristic functions: \begin{align} \textrm{P}\left(\chi_{H}\overset{q^{n}}{\equiv}m\right) & =\frac{1}{q^{n}}\sum_{k=0}^{q^{n}-1}e^{-\frac{2\pi ikm}{q^{n}}}\varphi_{H}\left(\frac{k}{q^{n}}\right)\\ \varphi_{H}\left(\frac{m}{q^{n}}\right) & =\sum_{k=0}^{q^{n}-1}e^{\frac{2\pi imk}{q^{n}}}\textrm{P}\left(\chi_{H}\overset{q^{n}}{\equiv}k\right) \end{align} and the Parseval Identity: \begin{equation} \frac{1}{q^{n}}\sum_{k=0}^{q^{n}-1}\left\|\varphi_{H}\left(\frac{k}{q^{n}}\right)\right\|^{2}=\sum_{k=0}^{q^{n}-1}\left(\textrm{P}\left(\chi_{H}\overset{q^{n}}{\equiv}k\right)\right)^{2} \end{equation} Under certain circumstances (such as the case of the $T_{a}$ maps), one can use (\ref{eq:Phi_H functional equation}) to recursively solve for $\varphi_{H}$\textemdash or for $\textrm{P}\left(\chi_{H}\overset{q^{n}}{\equiv}k\right)$, after performing a discrete Fourier transform. Doing so for $H=T_{3}$ yields the recursive formula given by Tao in \textbf{Lemma 1.12 }of his paper \cite{Tao Probability paper}. That being said, it remains to be seen whether or not recursive formulae can be derived for these probabilities and characteristic functions in the case of a general $p$-Hydra map. It may also be of interest to study the expressions $\varphi_{H,p}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}$ given by: \begin{equation} \varphi_{H,p}\left(t\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\chi_{H}\left(\mathfrak{z}\right)\right\} _{p}}d\mathfrak{z}\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}e^{-2\pi i\left\{ t\chi_{H}\left(n\right)\right\} _{p}},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:p-adic characteristic function for Chi_H} \end{equation} where $p$ is an arbitrary prime. Doing so, should, conceivably, advance our understanding of the divisibility of $\chi_{H}\left(n\right)$ by $p$ as $n$ varies. \subsubsection{\label{subsec:Dirichlet-Series-and}Dirichlet Series and Complex-Analytic Methods} Like in most situations, given the Correspondence Principle and the associated identity from \textbf{Lemma \ref{lem:Chi_H o B_p functional equation}}: \begin{equation} \chi_{H}\left(B_{p}\left(n\right)\right)=\frac{\chi_{H}\left(n\right)}{1-M_{H}\left(n\right)} \end{equation} one asks: \emph{what can be done with this?} Prior to stumbling upon the $\left(p,q\right)$-adic approach, my intention was to use classical, complex-analytic techniques of analytic number theory, such as Dirichlet Series\index{Dirichlet series}, Mellin transforms, contour integration, and the Residue Theorem. According to the Correspondence Principle, classifying the periodic points of $H$ in $\mathbb{Z}$ is just a matter of finding those integers $x\in\mathbb{Z}$ and $n\geq0$ so that $x=\chi_{H}\left(B_{p}\left(n\right)\right)$; which is to say: \begin{equation} \left(1-M_{H}\left(n\right)\right)x-\chi_{H}\left(n\right)=0\label{eq:nth term of Dirichlet series} \end{equation} Dividing everything by $\left(n+1\right)^{s}$ and summing over $n\geq0$ gives: \begin{equation} \left(\zeta\left(s\right)-\sum_{n=0}^{\infty}\frac{M_{H}\left(n\right)}{\left(n+1\right)^{s}}\right)x-\sum_{n=0}^{\infty}\frac{\chi_{H}\left(n\right)}{\left(n+1\right)^{s}}=0\label{eq:Dirichlet series} \end{equation} for all complex $s$ for which the Dirichlet series converge. By exploiting $M_{H}$ and $\chi_{H}$'s functional equations, the right-hand side of (\ref{eq:nth term of Dirichlet series}) can be written in terms of a contour integral of the right-hand side of (\ref{eq:Dirichlet series}) using a generalized version of \index{Perron's Formula}\textbf{Perron's Formula} (see\footnote{The series of papers by Flajolet et. al. on applications of the Mellin transform in analytic combinatorics (starting with \cite{Flajolet - Mellin Transforms}) provide a comprehensive and illustrative exposition of the Mellin transform\index{Mellin transform!complex} and its many, \emph{many }applications to combinatorics, number theory, algorithm analysis, and more.} \cite{Flajolet - Digital sums}). As usual, the hope was to shift the contour of integration by first analytically continuing (\ref{eq:Dirichlet series}) to the left half-plane, and then using the Residue Theorem to obtain to obtain a more enlightening expression for (\ref{eq:nth term of Dirichlet series}). Unfortunately, while the Dirichlet series in (\ref{eq:Dirichlet series}) \emph{do }admit analytic continuations to meromorphic functions on $\mathbb{C}$\textemdash where they have half-lattices of poles to the left of their abscissae of absolute convergence\textemdash these continuations exhibit hyper-exponential growth as $\textrm{Re}\left(s\right)\rightarrow-\infty$. This appears to greatly limit the usefulness of this complex analytic approach. I have unpublished work in this vein; a messy rough draft of which (containing much overlap with this dissertation) can be found on arXiv (see \cite{Mellin transform paper}). I intend to pursue these issues more delicately and comprehensively at some later date. That being said, I briefly revisit this approach at the end of Section \ref{sec:4.1 Preparatory-Work--}, so the curious reader can turn there for a first taste of that approach. \chapter{\label{chap:3 Methods-of--adic}Methods of $\left(p,q\right)$-adic Analysis} \includegraphics[scale=0.45]{./PhDDissertationEroica3.png} \begin{quote} \vphantom{} \begin{flushright} $\ldots$many people feel that functions $\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{p}$ are more interesting than functions $\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{q}$, which is understandable. \par\end{flushright} \begin{flushright} \textemdash W. M. Schikhof\footnote{From \emph{Ultrametric Calculus} (\cite{Ultrametric Calculus}), on the bottom of page 97.} \par\end{flushright} \begin{flushright} People will want to know: what is the \emph{point} of $\left(p,q\right)$-adic functions? You need a good answer. \par\end{flushright} \begin{flushright} \textemdash K. Conrad\footnote{<https://mathoverflow.net/questions/409806/in-need-of-help-with-parsing-non-archimedean-function-theory>} \par\end{flushright} \end{quote} \vphantom{} The purpose of this chapter is to present and develop an analytical theory capable of dealing with functions like $\chi_{H}$. Section \ref{sec:3.1 A-Survey-of} presents the ``classical'' theory, much of which I independently re-discovered before learning that Schikhof\index{Schikhof, W. M.} had already done it\footnote{One of the dangers of working in isolation as I did is the high probability of ``re-inventing the wheel'', as one of my professors (Nicolai Haydn) put it.} all\textemdash and more\textemdash in his own PhD dissertation, back in 1967. Prototypically, the ``classical'' material concerns the study of $C\left(\mathbb{Z}_{p},\mathbb{Q}_{q}\right)$\textemdash the space of continuous functions from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{q}$\textemdash and the related interplay between functional analysis and Fourier analysis in that setting. Section \ref{sec:3.1 A-Survey-of} explains these theoretical details, as well as providing a primer on how to do $\left(p,q\right)$-adic computations. The other two sections of this chapter, however, are new; they extend the ``classical'' material to include functions like $\chi_{H}$. Section \ref{sec:3.2 Rising-Continuous-Functions} introduces the notion of \textbf{rising-continuity}, which provides a natural extension of the $C\left(\mathbb{Z}_{p},\mathbb{Q}_{q}\right)$ theory. Section \ref{sec:3.3 quasi-integrability}\textemdash the largest of the three\textemdash is the most important section of Chapter 3, containing the most significant innovations, as well as the greatest complexities. The central focus of Section \ref{sec:3.3 quasi-integrability} is the phenomenon I call \textbf{quasi-integrability}. Classically, the range of $\left(p,q\right)$-adic which can be meaningfully integrated is astonishingly narrow, consisting solely of continuous functions. By re-interpreting discontinuous functions like $\chi_{H}$ as \emph{measures}, we can meaningfully integrate them, as well as integrate their product with any continuous $\left(p,q\right)$-adic function; this is done in Subsection \ref{subsec:3.3.5 Quasi-Integrability}; the quasi-integrable functions are precisely those $\left(p,q\right)$-adic functions for which this method applies. However, in formulating quasi-integrability, we will have to deal with an extremely unorthodox situation: infinite series of a $p$-adic integer variable $\mathfrak{z}$ which converge at every $\mathfrak{z}\in\mathbb{Z}_{p}$, but for which the topology used to sum the series is allowed to vary from point to point. I introduce the concept of a \textbf{frame }in order to bring order and rigor to this risky, messy business (see Subsection \ref{subsec:3.3.3 Frames}). Subsections \ref{subsec:3.3.1 Heuristics-and-Motivations} and \ref{subsec:3.3.2 The--adic-Dirichlet} contain motivational examples and a connection to the Dirichlet kernel of classical Fourier analysis, respectively, and thereby serve to set the stage for frames and quasi-integrability. Subsection \ref{subsec:3.3.4 Toward-a-Taxonomy} introduces some tedious terminology for describing noteworthy types of $\left(p,q\right)$-adic measures. Its most important features are the \textbf{Fourier Resummation Lemmata}, which will be used extensively in our analysis of $\chi_{H}$ in Chapter 4. Subsection \ref{subsec:3.3.6 L^1 Convergence} briefly introduces another hitherto-ignored area of non-archimedean analysis: the Banach space of functions $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ for which the real-valued function $\mathfrak{z}\in\mathbb{Z}_{p}\mapsto\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}\in\mathbb{R}$ is integrable with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$. As will be shown in Subsection \ref{subsec:4.3.2 A-Wisp-of} of Chapter 4, these spaces provide an approachable setting for studying $\chi_{H}$. Whether or not significant progress can be made in that setting remains to be seen. The final subsection of \ref{sec:3.3 quasi-integrability}, Subsection \ref{subsec:3.3.7 -adic-Wiener-Tauberian}, contains the second of the this dissertation's three main results, a $\left(p,q\right)$-adic generalization of \textbf{Wiener's Tauberian Theorem}. I prove two versions, one for continuous $\left(p,q\right)$-adic functions, and another for $\left(p,q\right)$-adic measures. The latter version will be the one used in Chapter 4\textemdash at the end of Subsection \ref{subsec:4.2.2}\textemdash to establish the \textbf{Tauberian Spectral Theorem }for $p$-Hydra maps, my dissertation's third main result. A \emph{sub}-subsection at the end of \ref{subsec:3.3.7 -adic-Wiener-Tauberian} shows how the study of $\chi_{H}$ can be $q$-adically approximated by the study of the eigenvalues of a family of increasingly large Hermitian matrices. \section{\label{sec:3.1 A-Survey-of}A Survey of the ``Classical'' theory} Due to the somewhat peculiar position currently occupied by $\left(p,q\right)$-adic analysis\index{$p,q$-adic!analysis}\textemdash too exotic to be thoroughly explored, yet too specific to merit consideration at length by specialists in non-archimedean analysis\textemdash I have taken the liberty of providing a brief essay on the historical and theoretical context of the subject and its various flavors; this can be found in Subsection \ref{subsec:3.1.1 Some-Historical-and}. Before that, however, I think it will be helpful to point out the most important (and bizarre) results of $\left(p,q\right)$-adic analysis, particularly for the benefit of readers who are used to working with real- or complex-valued functions. The main features are: \begin{itemize} \item The existence of a translation-invariant $\mathbb{C}_{q}$-valued ``measure'' on $\mathbb{Z}_{p}$, unique up to a choice of a normalization constant, and compatible with an algebra of sets, albeit not the Borel $\sigma$-algebra. \index{Borel!sigma-algebra@$\sigma$-algebra}Moreover, this measure (the \textbf{$\left(p,q\right)$-adic Haar measure}) is a continuous linear functional on $C\left(\mathbb{Z}_{p},\mathbb{Q}_{q}\right)$. This is very good news for us, because it means we can integrate and do Fourier analysis in the $\left(p,q\right)$-adic setting. \item The lack of a concept of ``almost everywhere\emph{''}. Although a substantive theory of $\left(p,q\right)$-adic measures exists, it has no ``almost everywhere\emph{''}.\emph{ }In fact, unless the normalization constant of the $\left(p,q\right)$-adic Haar measure is chosen to be $0$, \emph{the only measurable set of measure $0$ is the empty set!} \item Because there is no way to take a difference quotient for a $\left(p,q\right)$-adic function (how do you divide a $q$-adic number by a $p$-adic number?), there is no differential calculus\footnote{However, my methods allow for us to define a $\left(p,q\right)$-adic Mellin transform (see \textbf{Remark \ref{rem:pq adic mellin transform}} on page \pageref{rem:pq adic mellin transform}). Since the Mellin transform is used to formulate differentiation in the distributional sense for functions $f:\mathbb{Q}_{p}\rightarrow\mathbb{C}$, it seems quite likely that that same approach, done in the $\left(p,q\right)$-adic context, can be used to give $\left(p,q\right)$-adic analysis a concept of distributional derivatives, and thereby open the door to analytical investigation of $\left(p,q\right)$-adic differential equations.} in $\left(p,q\right)$-adic analysis. There are also no such things as polynomials, rational functions, or analytic functions. \item \emph{A function is integrable if and only if it is continuous!} This equivalence is intimately connected to the absence of non-empty sets of measure zero. \item We will have to leave the triangle inequality for integrals to wait for us outside by the door; it has no place in $\left(p,q\right)$-adic analysis. In this subject, there is generally no meaningful relationship between $\left\|\int f\right\|_{q}$ and $\int\left\|f\right\|_{q}$. As much as we would want to write $\left\|\int f\right\|_{q}\leq\int\left\|f\right\|_{q}$, it simply isn't justified. \textbf{Example \ref{exa:triangle inequality failure} }on page \pageref{exa:triangle inequality failure} provides the unfortunate details. \end{itemize} A first encounter with the theory of $\left(p,q\right)$-adic Fourier analysis is a surreal experience. Indeed, in the ``classical'' $\left(p,q\right)$-adic theory, given a $\left(p,q\right)$-adic function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ (where, of course, $p\neq q$),\emph{ the following are equivalent}: \begin{enumerate} \item $f$ possesses a well-defined Fourier transform $\hat{f}$. \item $f$ is integrable. \item $f$ is continuous. \item The Fourier series representation of $f$ (using $\hat{f}$) is \emph{absolutely convergent }uniformly over $\mathbb{Z}_{p}$. \end{enumerate} These three wonders\textemdash especially the spectacular equivalence of (1) through (4)\textemdash are, among some other factors, the principal reasons why $\left(p,q\right)$-adic analysis has lain neglected for the over-half-a-century that has elapsed since its inception. The subject's rigid inflexibility and astonishing lack of subtlety goes a long way toward explaining why it has been consigned to eek out a lonely existence in the cabinet of mathematical curiosities. Of course, part of the purpose of this dissertation is to argue that this consignment was premature. Lastly, before freeing the reader to pursue the exposition I have prepared for them, let me first give an exposition for that exposition, which is somewhat non-standard, to the extent that there even \emph{is }a ``standard'' pedagogical approach to $\left(p,q\right)$-adic analysis. One of the main features of non-archimedean analysis as a subject is that the available integration theories will depend on the fields used for the functions' domains and codomains. Of these, the one we will use is what Khrennikov calls the \textbf{Monna-Springer integral} \cite{Quantum Paradoxes}. Schikhof in an appendix of \emph{Ultrametric Analysis} \cite{Ultrametric Calculus} and van Rooij in a chapter of \emph{Non-Archimedean Functional Analysis} \cite{van Rooij - Non-Archmedean Functional Analysis} also give expositions of this theory, though not by that name. For the probability-minded, Khrennikov gives a second, full-throated presentation of the integration theory in his article \cite{Probabilities taking values in non-archimedean fields} on non-archimedean probability theories\index{non-archimedean!probability theory}. As anyone who has gone through a standard course on integration and measures can attest, presentations of measure theory and measure-theoretic integration often lean toward the abstract. The same is true for the non-archimedean case. In certain respect, the non-archimedean abstraction outdoes the archimedean. These presentations typically begin with one of the following: \begin{itemize} \item An explanation of how the set-based approach to measure theory utilized in most university courses is not compatible with the non\textendash archimedean case. Instead, they turn to the functional analysis approach advocated by the Bourbaki group, defining measures as linear functionals satisfying a certain norm condition. \item To get around the fact that Borel sets aren't well-suited to form a set-based theory of measures, the presentation takes as a primitive notion the existence of a collection of sets satisfying most of the properties you would expect, along with a few extra technicalities needed to avoid disaster. \end{itemize} Although we will \emph{eventually }get to this sort of material (see Subsection \ref{subsec:3.1.6 Monna-Springer-Integration}), because of the paramount importance of explicit, specific, \emph{concrete} computations for this dissertation, I have weighted my presentation toward such practical matters. With the exception of \ref{subsec:3.1.6 Monna-Springer-Integration}, I have tried to keep the level of generality to an absolute minimum, so as to allow the basic mechanics to shine through as clearly as possible. After the historical essay in Subsection \ref{subsec:3.1.1 Some-Historical-and}, the exposition begins in Subsection \ref{subsec:3.1.2 Banach-Spaces-over} with a sliver of the more general theory of Banach spaces over a non-archimedean field. Subsection \ref{subsec:3.1.3 The-van-der} introduces the \textbf{van der Put basis} for spaces of functions on $\mathbb{Z}_{p}$, which will be used throughout this section to explicitly construct spaces like $C\left(\mathbb{Z}_{p},\mathbb{Q}_{q}\right)$ and compute integrals and Fourier transforms. After this, we explore the $\left(p,q\right)$-adic Fourier transform of a continuous $\left(p,q\right)$-adic function (Subsection \ref{subsec:3.1.4. The--adic-Fourier}), and then use it to construct $\left(p,q\right)$-adic measures. The initial viewpoint will be of a measure as an object defined by its \textbf{Fourier-Stieltjes transform} (Subsection \ref{subsec:3.1.5-adic-Integration-=00003D000026}). This view of $\left(p,q\right)$-adic measures will be very important in Section \ref{sec:3.3 quasi-integrability} and throughout Chapters \ref{chap:4 A-Study-of} and \ref{chap:6 A-Study-of}. In the process, we will see how to compute continuous $\left(p,q\right)$-adic functions' Fourier transforms\textemdash two different formulae will be given\textemdash along with how to integrate functions with respect to $\left(p,q\right)$-adic measures, convolutions, and the basic\textemdash but fundamental\textemdash change of variable formula for affine transformations ($\mathfrak{y}=\mathfrak{a}\mathfrak{z}+\mathfrak{b}$) in integrals. We will constantly draw from this practical information, knowledge of which is essential for further independent study of $\chi_{H}$ and related matters, We conclude our survey by returning to the abstract with Subsection \ref{subsec:3.1.6 Monna-Springer-Integration}, giving an exposition of Monna-Springer integration theory. This provides the ``correct'' generalization of the explicit, concrete methods of $\left(p,q\right)$-adic integration dealt with in the earlier subsections. As a pedagogical approach, I feel that grounding the Monna-Springer integral in the concrete case of $\left(p,q\right)$-adic Fourier series and $\left(p,q\right)$-adic Fourier-Stieltjes transforms makes the abstract version less intimidating. That being said, my inclusion of the Monna-Springer theory is primarily for completeness' sake. $\left(p,q\right)$-adic analysis is extraordinarily pathological, even by the bizarre standards of non-archimedean analysis. Familiar results such as the Dominated Convergence Theorem or Hlder's Inequality reduce to trivialities in the $\left(p,q\right)$-adic setting. \subsection{\label{subsec:3.1.1 Some-Historical-and}Some Historical and Theoretical Context} In the title of Section \ref{sec:3.1 A-Survey-of}, the word ``classical'' is enclosed in scare quotes. This seems like a fair compromise to me. On the one hand, in $\left(p,q\right)$-adic analysis we have a nearly-sixty-year-old subject ripe for generalization; on the other, to the extent that a well-articulated theory of $\left(p,q\right)$-adic analysis exists, it is too esoteric and (understandably) under-appreciated to truly merit the status of a ``classic''. Until now, the $\left(p,q\right)$-adic analysis chronicled in this Chapter has smoldered in double disregard. Why, indeed, would anyone in their right mind want to investigate a subject which is too exotic and inert for the general community, but also too \emph{insufficiently }general to have earned clear place in the work of the exotic specialists? Somewhat surprisingly, the same could be said for the $p$-adic numbers themselves. Kurt Hensel first published his discovery of $p$-adic numbers in 1897 \cite{Hensel-s original article,Gouvea's p-adic number history slides,Journey throughout the history of p-adic numbers}. At the time, they did not make much of a splash. The eventual inclusion of $p$-adic numbers in the mathematical canon came about purely by serendipity, with Helmut Hasse in the role of the inadvertent midwife. It would make for an interesting scene in a Wes Anderson film: in 1913, Hensel publishes a book on the $p$-adics; sometime between then and 1920, a copy of said book managed to find way to a certain antique shop in Gttingen. In 1920, Hasse\textemdash a then-student at the University of Gttingen\textemdash stumbled across said book in said antique shop while wandering about town as graduate students in Gttingen are wont to do when they feel a hankering for a breath of fresh air. Through this chain of events, Hasse chose to transfer to the University of Marburg, at which Hensel taught, with the express purpose of learning everything he could about Hensel's marvelous, star-crossed innovation. The result? In 1922, Hasse publishes his doctoral thesis, in which he establishes what is now called the \textbf{Hasse-Minkowski Theorem}, otherwise known as the \textbf{local-global principle} \textbf{for quadratic forms }\cite{Gouvea's p-adic number history slides,On the origins of p-adic analysis}. For the uninitiated, the local-global principle is the aspiration that one can find construct a tuple of rational numbers which solve a given polynomial equation in several variables by solving the equation in $\mathbb{R}$ as well as in the $p$-adics for all primes $p$, and then using the Chinese Remainder Theorem to stitch together the real and $p$-adic solutions into a solution over $\mathbb{Q}$. (Conrad's online notes \cite{Conrad - Local Global Principle} give an excellent demonstration of this method in practice.) Fascinatingly, although the Hasse-Minkowski Theorem assures us this approach will work in polynomials for which each variable has an exponent of at most $2$, the local-global principle \emph{does not }apply for polynomial equations of arbitrary degree. A particularly well-known counterexample is due to Selmer \cite{Conrad - Local Global Principle,Local-Global Principle Failure}: \begin{thm}[\textbf{\textit{Selmer's Counterexample}}] The cubic diophantine equation: \begin{equation} 3x^{3}+4y^{3}+5z^{3}=0 \end{equation} has solutions $\left(x,y,z\right)\neq\left(0,0,0\right)$ over $\mathbb{R}$ and over $\mathbb{Q}_{p}$ for every prime $p$, yet its only solution over $\mathbb{Q}$ is $\left(x,y,z\right)=\left(0,0,0\right)$. \end{thm} \vphantom{} Understanding the conditions where it does or does not apply is still an active area of research in arithmetic geometry. Indeed, as Conrad points out, the famous \textbf{Birch and Swinnerton-Dyer Conjecture }concerns, in part, a relation between the rational points of an elliptic curve $E$ over $\mathbb{Q}$ and the points of $E$ over $\mathbb{R}$ and the $p$-adics \cite{Conrad - Local Global Principle}. The local-global philosophy has since had an explosive influence in mathematics, above all in algebraic number theory and adjacent fields, where it opened the door to the introduction of $p$-adic and adlic approaches. A central protagonist of this story is the late, great John Tate (1925 \textendash{} 2019)\index{Tate, John}. In his epochal thesis of 1950 (``\emph{Fourier analysis in number fields and Hecke's zeta functions}''), Tate revolutionized algebraic number theory by demonstrating how the \textbf{Euler Products} for $L$-functions such as the Riemann Zeta Function: \begin{equation} \zeta\left(s\right)\overset{\mathbb{C}}{=}\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}},\textrm{ }\forall\textrm{ }\textrm{Re}\left(s\right)>1\label{eq:Euler Product for the RZF} \end{equation} (where $\mathbb{P}$ denotes the set of all prime numbers) could be understood in the local-global spirit as the product of so-called \textbf{local factors} associated to each prime $p$ \cite{Tate's thesis}. This recontextualization occurred by way of Mellin-transform type integrals over the $p$-adic numbers. Letting $d\mathfrak{z}_{p}$ denote the real-valued Haar probability measure on $\mathbb{Z}_{p}$, the aforementioned local factors of (\ref{eq:Euler Product for the RZF}) can be expressed by the integral: \begin{equation} \frac{1}{1-p^{-s}}\overset{\mathbb{C}}{=}\frac{p}{p-1}\int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}_{p}\right\|_{p}^{s-1}d\mathfrak{z}_{p},\textrm{ }\forall\textrm{Re}\left(s\right)>1 \end{equation} This can also be re-written as: \begin{equation} \frac{1}{1-p^{-s}}\overset{\mathbb{C}}{=}\int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}_{p}\right\|_{p}^{s-1}d^{\times}\mathfrak{z}_{p},\textrm{ }\forall\textrm{Re}\left(s\right)>1\label{eq:RZF local factors} \end{equation} where: \begin{equation} d^{\times}\mathfrak{z}_{p}\overset{\textrm{def}}{=}\frac{p}{p-1}d\mathfrak{z}_{p} \end{equation} is the real-valued Haar probability measure for the group $\left(\mathbb{Z}_{p}^{\times},\times\right)$ of multiplicatively invertible elements of $\mathbb{Z}_{p}$. Taking the product of the local factors (\ref{eq:RZF local factors}) over all primes $p$, the tensor product transforms the product of the integrals into a single integral over the profinite integers, $\check{\mathbb{Z}}$\nomenclature{$\check{\mathbb{Z}}$}{the ring of profinite integers, $\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$ \nopageref}: \begin{equation} \zeta\left(s\right)\overset{\mathbb{C}}{=}\int_{\check{\mathbb{Z}}}\left\|\mathfrak{z}\right\|^{s}d^{\times}\mathfrak{z}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}_{p}\right\|_{p}^{s-1}d^{\times}\mathfrak{z}_{p},\textrm{ }\forall\textrm{Re}\left(s\right)>1 \end{equation} Here: \begin{equation} \check{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}\label{eq:Definition of the Profinite Integers} \end{equation} and so, every $\mathfrak{z}\in\check{\mathbb{Z}}$ is a $\mathbb{P}$-tuple $\left(\mathfrak{z}_{2},\mathfrak{z}_{3},\mathfrak{z}_{5},\ldots\right)$ of $p$-adic integers, one from each $p$. $d^{\times}\mathfrak{z}$, meanwhile, is the Haar probability measure for the multiplicative group of units of $\check{\mathbb{Z}}$. The significance of this new perspective and its impact on subsequent developments in number theory cannot be overstated. Indeed, in hindsight, it is hard to believe that anyone could have ever doubted the value of the $p$-adics to mathematics. But the story does not end here. If anything, Tate's work is like an intermezzo, bridging two worlds by transmuting the one into the other. Starting in the mid-twentieth century, mathematical analysis began to pick up where Hensel left off \cite{Gouvea's p-adic number history slides,Journey throughout the history of p-adic numbers}, exploring the $p$-adic numbers' ultrametric structure and the implications it would have for analysis and its theories in those settings\textemdash continuity, differentiability, integrability, and so on. Although Hensel himself had taken the first steps toward investigating the properties of functions of his $p$-adic numbers, a general programme for analysis on valued fields other than $\mathbb{R}$ or $\mathbb{C}$\textemdash independent of any number-theoretic investigations was first posed by A. F. Monna\index{Monna, A. F.} in a series of papers in 1943 \cite{van Rooij - Non-Archmedean Functional Analysis}. From this grew the subject that would come to be known as \emph{non-archimedean analysis}. In this sense, non-archimedean analysis\textemdash as opposed to the more number-theoretically loaded term ``$p$-adic analysis''\textemdash has its origins in the activities of graduate students, this time Netherlands in the 1960s, when W. M. Schikhof, A. C. M. van Rooij\index{van Rooij, A. C. M.}, M. van der Put\index{van der Put, Marius}, and J. van Tiel established a so-called ``$p$-adics and fine dining'' society, intent on exploring the possibility of analysis\textemdash particularly \emph{functional} analysis\textemdash over non-archimedean valued fields. The subject is also sometimes called \emph{ultrametric analysis}\footnote{In this vein, one of the subject's principal introductory texts is called \emph{Ultrametric Calculus} \cite{Ultrametric Calculus}.}, to emphasize the fundamental distinction between itself and classical analysis. In the grand scheme of things, non-archimedean valued fields aren't as wild as one might think. As was proved by Ostrowski in 1915, the only valued fields of characteristic zero obtainable as metric completions of $\mathbb{Q}$ are $\mathbb{R}$, $\mathbb{C}$, and the $p$-adics. A necessary and immediate consequence of this straight-forward classification is that non-archimedean analysis comes in four different flavors depending on the nature of the fields we use for our domains and co-domains. Let $\mathbb{F}$ and $K$ be non-archimedean valued fields, complete with respect to their absolute values. Moreover, let the residue fields of $\mathbb{F}$ and $K$ have different characteristic; say, let $\mathbb{F}$ be a $p$-adic field, and let $K$ be a $q$-adic field. By combinatorics, note that there are really only four\emph{ }different types of non-archimedean functions we can consider: functions from an archimedean field to a non-archimedean field ($\mathbb{C}\rightarrow\mathbb{F}$); from a non-archimedean field to an archimedean field ($\mathbb{F}\rightarrow\mathbb{C}$); from \emph{and} to the same non-archimedean field ($\mathbb{F}\rightarrow\mathbb{F}$); and those between two archimedean fields with non-equal residue fields ($\mathbb{F}\rightarrow K$). In order, I refer to these as \textbf{$\left(\infty,p\right)$-adic analysis} ($\mathbb{C}\rightarrow\mathbb{F}$), \textbf{$\left(p,\infty\right)$-adic analysis} ($\mathbb{F}\rightarrow\mathbb{C}$), \textbf{$\left(p,p\right)$-adic analysis} ($\mathbb{F}\rightarrow\mathbb{F}$), and \textbf{$\left(p,q\right)$-adic analysis} ($\mathbb{F}\rightarrow K$). In this terminology, real and complex analysis would fall under the label \textbf{$\left(\infty,\infty\right)$-adic analysis}. The first flavor\textemdash $\left(\infty,p\right)$-adic analysis ($\mathbb{C}\rightarrow\mathbb{F}$)\textemdash I must confess, I know little about, and have seen little, if any, work done with it. Hopefully, this will soon be rectified. Each of the remaining three, however, merit deeper discussion. \index{analysis!left(p,inftyright)-adic@$\left(p,\infty\right)$-adic}The second flavor of non-archimedean analysis\textemdash $\left(p,\infty\right)$-adic analysis ($\mathbb{F}\rightarrow\mathbb{C}$) is the \emph{least} exotic of the three. Provided that $\mathbb{F}$ is locally compact (so, an at most finite-degree extension of $\mathbb{Q}_{p}$), this theory is addressed by modern abstract harmonic analysis, specifically through \textbf{Pontryagin duality} and the theory of Fourier analysis on a locally compact abelian group. Tate's Thesis is of this flavor. In addition to Tate's legacy, the past thirty years have seen a not-insignificant enthusiasm in $\left(p,\infty\right)$-adic analysis from theoretical physics, of all places \cite{First 30 years of p-adic mathematical physics,Quantum Paradoxes,p-adic space-time} (with Volovich's 1987 paper \cite{p-adic space-time} being the ``classic'' of this burgeoning subject). This also includes the adlic variant of $\left(p,\infty\right)$-adic analysis, where $\mathbb{F}$ is instead replaced with $\mathbb{A}_{\mathbb{Q}}$, the \textbf{adle ring }of $\mathbb{Q}$; see \cite{Adelic Harmonic Oscillator}, for instance, for an \emph{adlic} harmonic oscillator. This body of work is generally known by the name $p$-adic (mathematical) physics and $p$-adic quantum mechanics, and is as mathematically intriguing as it is conceptually audacious. The hope in the depths of this Pandora's box is the conviction that non-archimedean mathematics might provide a better model for reality at its smallest scales. On the side of pure mathematics, $\left(p,\infty\right)$-adic analysis is a vital tool in representation theory (see, for instance \cite{Automorphic Representations}). In hindsight, this is not surprising: the abstract harmonic analysis used to formulate Fourier theory over locally compact abelian groups must appeal to representation theory in order to formulate Fourier analysis over an arbitrary compact group. Of the many flavors of non-archimedean analysis, I feel that only the third\textemdash \index{analysis!left(p,pright)-adic@$\left(p,p\right)$-adic}$\left(p,p\right)$-adic anlaysis ($\mathbb{F}\rightarrow\mathbb{F}$)\textemdash truly deserves to be called ``\index{analysis!$p$-adic}$p$-adic analysis''. Of course, even this flavor has its own variations. One can study functions $\mathbb{Z}_{p}\rightarrow\mathbb{F}$, as well as functions $\mathbb{F}\rightarrow\mathbb{F}^{\prime}$, where $\mathbb{F}^{\prime}$ is a field extension of $\mathbb{F}$, possibly of infinite degree. The utility, flexibility, adaptability, and number-theoretic import of $\left(p,p\right)$-adic analysis have all contributed to the high esteem and widespread popularity this subject now enjoys. $\left(p,p\right)$-adic analysis can be studied in its own right for a single, fixed prime $p$, or it can be used in conjunction with the local-global philosophy, wherein one does $\left(p,p\right)$-adic analysis for many\textemdash even infinitely many\textemdash different values of $p$ simultaneously. This synergy is on full display in one of $\left(p,p\right)$-adic analysis most famous early triumphs: Bernard Dwork\index{Dwork, Bernard}'s proof (\cite{Dwork Zeta Rationality}) of the rationality of the zeta function for a finite field\textemdash the first of the three parts of the \textbf{Weil Conjectures }to be proven true \cite{Dwork Zeta Rationality,Koblitz's other book,Robert's Book,p-adic proof for =00003D0003C0}. Much like the local-global principle itself, Dwork's application of $p$-adic analysis is a stratospheric elevation of the Fundamental Theorem of Arithmetic: every integer $\geq2$ is uniquely factorable as a product of primes. This is evident in the so-called \textbf{Product Formula} of algebraic number theory, applied to $\mathbb{Q}$: \begin{equation} \left\|a\right\|_{\infty}\cdot\prod_{p\in\mathbb{P}}\left\|a\right\|_{p}=1,\textrm{ }\forall a\in\mathbb{Q}\backslash\left\{ 0\right\} \label{eq:The Product Formula} \end{equation} where $\left\|a\right\|_{p}$ is the $p$-adic absolute value of $a$, and $\left\|a\right\|_{\infty}$ is the usual absolute value (on $\mathbb{R}$) of $a$. Indeed, (\ref{eq:The Product Formula}) is but a restatement of the Fundamental Theorem of Arithmetic. The key tool in Dwork's proof was the following theorem: \begin{thm}[\textbf{Dwork}\footnote{Cited in \cite{p-adic proof for =00003D0003C0}.}] Let $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be a power series in the complex variable $\mathbb{C}$ with coefficients $\left\{ a_{n}\right\} _{n\geq0}$ in $\mathbb{Q}$. Let $S$ be a set of finitely many places\footnote{For the uninitiated, a ``place'' is a fancy term for a prime number. In algebraic number theory, primes\textemdash and, more generally, prime \emph{ideals}\textemdash of a given field or ring are associated with the valuations they induce. Due to a technicality, much like a Haar measure on a group, valuations are not unique. The absolute value $\mathfrak{z}\in\mathbb{Z}_{2}\mapsto3^{-v_{2}\left(\mathfrak{z}\right)}$, for example, defines a topology on $\mathbb{Z}_{2}$ which is equivalent to that of the standard $2$-adic absolute value. The valuation associated to this modified absolute value would be $\mathfrak{z}\mapsto v_{2}\left(\mathfrak{z}\right)\log_{2}3$. A \textbf{place}, then, is an equivalence class of valuations associated to a given prime ideal, so that any two valuations in the class induce equivalent topologies on the ring or field under consideration.} of $\mathbb{Q}$ which contains the archimedean complex place\footnote{Meaning the valuation corresponding to the ordinary absolute value}. If $S$ satisfies: \vphantom{} I. For any\footnote{Here, $p$ represents \emph{both} a prime number and the associated non-archimedean absolute value.} $p\notin S$, $\left\|a_{n}\right\|_{p}\leq1$ for all $n\geq0$ (i.e., $a_{n}\in\mathbb{Z}_{p}$); \vphantom{} II. For any\footnote{Here, $v$ is a absolute value on $\mathbb{Q}$, and $\mathbb{C}_{v}$ is the algebraic closure of the $v$-adic completion of $\mathbb{Q}$.} $p\in S$, there is a disk $D_{p}\subseteq\mathbb{C}_{p}$ of radius $R_{p}$ so that $f\left(z\right)$ extends to a meromorphic function $D_{p}\rightarrow\mathbb{C}_{p}$ and that $\prod_{p\in S}R_{p}>1$; \vphantom{} Then $f\left(z\right)$ is a rational function. \end{thm} \vphantom{} This is a $p$-adic generalization of an observation, first made by Borel, that if a power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ with \emph{integer} coefficients has a radius of convergence strictly greater than $1$, the power series is necessarily\footnote{Proof: let $R$ be the radius of convergence, and suppose $R>1$. Then, there is an $r\in\left(1,R\right)$ for which $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ converges uniformly for $\left\|z\right\|=r$, and hence, by the standard Cauchy estimate: \[ \left\|a_{n}\right\|=\left\|\frac{f^{\left(n\right)}\left(0\right)}{n!}\right\|\leq\frac{1}{r^{n}}\sup_{\left\|z\right\|=r}\left\|f\left(z\right)\right\| \] Since $r>1$, the upper bound will be $<1$ for all sufficiently large $n$. However, if the $a_{n}$s are all integers, then any $n$ for which $\left\|a_{n}\right\|<1$ necessarily forces $a_{n}=0$. Hence $f$ is a polynomial.} a polynomial. It is a testament to the power of Dwork's Theorem\textemdash via a generalization due Bertrandias\textemdash that it can be used to prove the \index{Lindemann-Weierstrass Theorem}\textbf{Lindemann-Weierstrass Theorem}\textemdash the Lindemann-Weierstrass Theorem being the result that proved $\pi$\index{transcendence of pi@transcendence of $\pi$} to be a transcendental number, thereby resolving the millennia-old problem of squaring the circle\textemdash the answer being a resounding \emph{no}. See \cite{p-adic proof for =00003D0003C0} for a lovely exposition of this argument. The fourth, most general flavor of non-archimedean analysis concerns functions between arbitrary non-archimedean valued fields. The subject has gone even further, studying functions $\Omega\rightarrow\mathbb{F}$, where $\Omega$ is any appropriately well-behaved Hausdorff space. Of the different flavors, I feel this is the one most deserving of the title ``\textbf{non-archimedean analysis}\index{analysis!non-archimedean}''.\textbf{ }A good deal of the work done in non-archimedean analysis is in the vein of so-called ``soft analysis'' variety, using the abstract and often heavily algebraified language common to modern real or complex functional analysis\textemdash ``maximal ideals'', ``convex sets'', ``nets'', and the like\textemdash to investigate exotic generalities. Aside from generality's sake, part of the reason for the lack of a significant distinction in the literature between non-archimedean analysis and non-archimedean \emph{functional }analysis is one the central difficulties of non-archimedean analysis: \emph{the absence of a ``nice'' analogue of differentiation in the classical sense.} In the navet of my youth, I dared to dream that calculus in multiple variables\textemdash once I learned it\textemdash would be every bit as elegant and wondrous as its single-variable counterparts. Inevitably, I was disabused of this notion. When it comes to differential calculus, single-variable $\left(p,\infty\right)$-adic analysis (ex: $\mathbb{Z}_{p}\rightarrow\mathbb{C}$) and non-archimedean analysis (such as $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$) manage to surpass multi-variable real analysis in theoretical complexity without needing to draw on even half as much indecency of notation. The main problem in these settings is that there is no way to define a \emph{difference quotient }for functions. The topologies and algebraic operations of functions' inputs and outputs are \emph{completely} different. You can't divide a complex number by a $p$-adic integer, and so on and so forth. On the other hand, the theory of integration rests on much firmer ground. Thanks to Haar measures and Pontryagin Duality, integration in $\left(p,\infty\right)$-adic analysis falls squarely under the jurisdiction of standard measure theory and abstract harmonic analysis. In fact, the parallels between $\left(p,\infty\right)$-adic Fourier analysis and classical Fourier analysis on tori or euclidean space are so nice that, in 1988, Vasilii Vladimirov was able to use Fourier analysis to introduce $\left(p,\infty\right)$-adic differentiation in the sense of distributions. This was done by co-opting the classic \emph{complex-valued }$p$-adic Mellin integral \cite{Vladimirov - the big paper about complex-valued distributions over the p-adics}: \begin{equation} \int_{\mathbb{Q}_{p}}\left\|\mathfrak{z}\right\|_{p}^{s-1}f\left(\mathfrak{z}\right)d\mathfrak{z} \end{equation} where $d\mathfrak{z}$ is the \emph{real-valued} Haar measure on $\mathbb{Q}_{p}$, normalized to be a probability measure on $\mathbb{Z}_{p}$ and where $s$ is a complex variable. Here, $f$ is a complex-valued function on $\mathbb{Q}_{p}$, and\textemdash for safety\textemdash we assume $f$ is compactly supported. The fruit of this approach is the \index{fractional differentiation}\index{Vladimirov operator}\textbf{ Vladimirov (Fractional) Differentiation / Integration operator} of order $\alpha$ (where $\alpha$ is a real number $>0$). For a function $f:\mathbb{Q}_{p}\rightarrow\mathbb{C}$, this is given by the integral: \begin{equation} D^{\alpha}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\frac{1}{\Gamma_{p}\left(-\alpha\right)}\int_{\mathbb{Q}_{p}}\frac{f\left(\mathfrak{z}\right)-f\left(\mathfrak{y}\right)}{\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}^{1+\alpha}}d\mathfrak{y}\label{eq:Definition of the Vladimirov Fractional Differentiation Operator} \end{equation} provided the integral exists. Here, $\Gamma_{p}$ is the physicists' notation for: \begin{equation} \Gamma_{p}\left(-\alpha\right)\overset{\textrm{def}}{=}\frac{p^{\alpha}-1}{1-p^{-1-\alpha}}\label{eq:Physicist's gamma_p} \end{equation} This operator can be extended to negative $\alpha$ by being rewritten so as to produce the $\left(p,\infty\right)$-adic equivalent of the usual, Fourier-transform-based definition of fractional derivatives in fractional-order Sobolev spaces. As for \emph{non-archimedean} analysis (including $\left(p,q\right)$), there is a well developed theory of integration, albeit with a strong functional analytic flavor. Integrals and measures are formulated not as $K$-valued functions of sets, but continuous, $K$-valued linear functionals on the Banach space of continuous functions $X\rightarrow K$, where $X$ is an ultrametric space and $K$ is a metrically complete non-archimedean field. An exposition of this theory is given in Subsection \ref{subsec:3.1.6 Monna-Springer-Integration}. To return to physics for a moment, one of the principal \emph{mathematical }reasons for interest in $p$-adic analysis among quantum physicists has been the hope of realize a theory of probability based on axioms other than Kolmogorov's. The \emph{ide fixe }of this approach is that sequences of rational numbers (particularly those representing frequencies of events) which diverge in $\mathbb{R}$ can be convergent in $\mathbb{Q}_{p}$ for an appropriate choice of $p$. The hope is that this will allow for a resurrection of the pre-Kolmogorov frequency-based probability theory, with the aim of using the frequency interpretation to assign probabilistic meaning to sequences of rational numbers which converge in a $p$-adic topology. Andrei Khrennikov and his associates have spearheaded these efforts; \cite{Measure-theoretic approach to p-adic probability theory} is an excellent (though dense) exposition of such a non-archimedean probability theory, one which strives to establish a more classical measure-based notion of probability. Khrennikov's eclectic monograph \cite{Quantum Paradoxes} also makes for a fascinating read, containing much more information, and\textemdash most importantly\textemdash worked examples from both physics and probability. Now for the bad news. Both $\left(p,\infty\right)$-adic and non-archimedean analysis are effectively incompatible with any notion of analytic functions; there are no power series, nor rational functions. $\left(p,p\right)$-adic analysis \emph{does not }suffer this limitation, which is another reason for its longstanding success. Even so, this theory of $p$-adic analytic functions still managed to get itself bewitched when reality was in the drafting stage: the non-archimedean structure of the $p$-adics makes even the most foundational principles of classical analytic function theory completely untenable in the $p$-adic setting. While one \emph{can} define the derivative of a function, say, $f:\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{p}$ as the limit of difference quotient: \begin{equation} f^{\prime}\left(\mathfrak{z}\right)=\lim_{\mathfrak{y}\rightarrow\mathfrak{z}}\frac{f\left(\mathfrak{y}\right)-f\left(\mathfrak{z}\right)}{\mathfrak{y}-\mathfrak{z}}\label{eq:Naive differentiability} \end{equation} doing so courts disaster. Because there is sense in the world, this notion of ``derivative\index{derivative}'' \emph{does }work for analytic functions. However, it fails \emph{miserably }for everything else. Yes, this kind of differentiability still implies continuity\textemdash that much, we are allowed to keep. Nevertheless, there are simple examples of \emph{differentiable functions $f:\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{p}$ }(in the sense that (\ref{eq:Naive differentiability}) exists at every point)\emph{ which are injective, but whose derivatives are everywhere zero!} \begin{example} \label{exa:p-adic differentiation is crazy}Given\footnote{As a minor spoiler, this example proves to be useful to one of my ideas for studying $\chi_{H}$, because \emph{of course }it would.} any $\mathfrak{z}\in\mathbb{Z}_{p}$, write $\mathfrak{z}$ as $\sum_{n=0}^{\infty}\mathfrak{z}_{n}p^{n}$, where, $\mathfrak{z}_{n}\in\left\{ 0,\ldots,p-1\right\} $ is the $n$th $p$-adic digit of $\mathfrak{z}$, and consider the function\footnote{Amusingly, this function will turn out to be relevant to studying Collatz and the like; see the lead-up to \textbf{Example \ref{exa:L^1 method example}} (page \pageref{exa:L^1 method example}) from Subsection \ref{subsec:4.3.4 Archimedean-Estimates} in Chapter 4 for the details.} $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Q}_{p}$ defined by: \begin{equation} \phi\left(\mathfrak{z}\right)=\sum_{n=0}^{\infty}\mathfrak{z}_{n}p^{2n}\label{eq:Phi for the L^1 method} \end{equation} Because $\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}=1/p^{n+1}$, where $n$ is the largest integer $\geq0$ for which $\mathfrak{z}_{n}=\mathfrak{y}_{n}$ (with the convention that $n=-1$ if no such $n$ exists), we have that: \begin{equation} \left\|\phi\left(\mathfrak{z}\right)-\phi\left(\mathfrak{y}\right)\right\|_{p}=\frac{1}{p^{2\left(n+1\right)}}=\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}^{2} \end{equation} Thus, $\phi$ is continuous, injective, and differentiable, with: \begin{equation} \left\|\phi^{\prime}\left(\mathfrak{z}\right)\right\|_{p}=\lim_{\mathfrak{y}\rightarrow\mathfrak{z}}\left\|\frac{\phi\left(\mathfrak{y}\right)-\phi\left(\mathfrak{z}\right)}{\mathfrak{y}-\mathfrak{z}}\right\|_{p}=\lim_{\mathfrak{y}\rightarrow\mathfrak{z}}\left\|\mathfrak{y}-\mathfrak{z}\right\|_{p}=0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} which is \emph{not} what one would like to see for an injective function! The notion of \textbf{strict differentiability }(see \cite{Robert's Book,Ultrametric Calculus}) is needed to get a non-degenerate notion of differentiability for non-analytic functions. In short, this is the requirement that we take the limit of (\ref{eq:Naive differentiability}) as\emph{ both }$\mathfrak{y}$ and $\mathfrak{z}$ to converge to a single point $\mathfrak{z}_{0}$, and that, regardless of path chosen, the limit exists; this then shows that $f$ is ``strictly differentiable'' at $\mathfrak{z}_{0}$. \end{example} \vphantom{} To continue our tour of the doom and the gloom, the $p$-adic Mean Value Theorem is but a shadow of its real-analytic self; it might as well not exist at all. Unsurprisingly, $p$-adic analysis suffers when it comes to integration. Without a dependable Mean Value Theorem, the relationship between integration and anti-differentiation familiar to us from real and complex analysis fails to hold in the $\left(p,p\right)$-adic context. But the story only gets worse from there. Surely, one of the most beautiful syzygies in modern analysis is the intermingling of measure theory and functional analysis. This is made all the more elegant when Haar measures and harmonic analysis are brought into the mix. Unfortunately, in $\left(p,p\right)$-adic waters, this elegance bubbles, burns, and crumbles into nothingness. Basic considerations of algebra and arithmetic show that any $\left(p,p\right)$-adic ``Haar measure'' (translation-invariant $\mathbb{F}$-valued function of ``nice'' sets in $\mathbb{Z}_{p}$) must assign the rational number $1/p^{n}$ as the measure of sets of the form $k+p^{n}\mathbb{Z}_{p}$, where $k\in\mathbb{Z}$ and $n\in\mathbb{N}_{0}$. Because the $p$-adic absolute value of $1/p^{n}$ tends to $\infty$ as $n\rightarrow\infty$, the only \emph{continuous}, translation-invariant $\mathbb{Q}_{p}$-valued function of sets on $\mathbb{Z}_{p}$ is the constant function $0$\textemdash the zero measure. This same result holds for any ring extension of $\mathbb{Z}_{p}$, as well as for $\mathbb{Q}_{p}$ and any field extension thereof. Fortunately, there ways of getting around this sorry state of affairs, even if the result ends up being completely different than classical integration theory. The first of these is the \textbf{Volkenborn integral}, which\index{integral!Volkenborn} generalizes the classical notion of an integral as the limit of a Riemann sum\index{Riemann sum}. The Volkenborn integral\footnote{Robert covers Volkenborn integration in Section 5 of his chapter on Differentiation in \cite{Robert's Book}, starting at page 263. One can also turn to Borm's dissertation \cite{pp adic Fourier theory}.} of a function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ is defined by the limit: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d_{\textrm{Volk}}\mathfrak{z}\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}f\left(n\right)\label{eq:definition of the volkenborn integral} \end{equation} which\textemdash the reader should note\textemdash is \emph{not }translation invariant (see \cite{Robert's Book})! The Volkenborn integral turns out to be intimately connected with the finite differences and the indefinite summation operator. In all likelihood, the most noteworthy property of the Volkenborn integral is that, for any prime $p$ and any integer $n\geq0$, the integral of the function $\mathfrak{z}^{n}$ is equal to the $n$th Bernoulli number (with $B_{1}=-1/2$). In practice, however, the Volkenborn integral is neglected in favor for the Amice-Iwasawa\index{Amice, Yvette} approach, otherwise known as the theory of \index{$p$-adic!distribution}\textbf{$p$-adic distributions}\footnote{Note, these should \textbf{\emph{not}}\emph{ }be confused with the distributions / ``generalized functions'' dealt with by Vladimirov in \cite{Vladimirov - the big paper about complex-valued distributions over the p-adics}. Amice-Iwasawa ``$p$-adic distributions'' are linear functionals taking values in $\mathbb{Q}_{p}$ or a finite degree field extension thereof. Vladimirov's ``$p$-adic distributions'' are linear functionals taking values in $\mathbb{C}$.} (\cite{p-adic L-functions paper,Iwasawa}). Let $\mathbb{F}$ be a metrically complete field extension of $\mathbb{Q}_{p}$. The idea is to exploit the fact that the space $C\left(\mathbb{Z}_{p},\mathbb{F}\right)$ has as a basis the binomial coefficient polynomials ($\left\{ \binom{\mathfrak{z}}{n}\right\} _{n\geq0}$): \begin{align} \binom{\mathfrak{z}}{0} & =1\\ \binom{\mathfrak{z}}{1} & =\mathfrak{z}\\ \binom{\mathfrak{z}}{2} & =\frac{\mathfrak{z}\left(\mathfrak{z}-1\right)}{2}\\ & \vdots \end{align} That these functions form a basis of $C\left(\mathbb{Z}_{p},\mathbb{F}\right)$ is a classic result of $p$-adic analysis first proved by Kurt Mahler in 1958\footnote{\cite{Mahler Series} is Mahler's original paper, but his result is now standard material in any worthwhile course on $p$-adc analysis \cite{Mahler,Robert's Book,Ultrametric Calculus}.}. This basis is now known as the\textbf{ Mahler basis}, in his honor. From this, one can define $\mathbb{F}$-valued measures and distributions by specifying what they do to $\binom{\mathfrak{z}}{n}$ for each $n$, and then extending by linearity. This construction is of considerable importance in modern algebraic number theory, where it is one of several ways of defining $p$-adic $L$ functions\index{$p$-adic!$L$-function}; the fact that the several different methods of constructing $p$-adic $L$ functions are all equivalent was a major landmark of twentieth century number theory \cite{p-adic L-functions paper,Iwasawa}. So, it's no wonder the Volkenborn integral has gotten the proverbial cold shoulder. Despite all this, the particular sub-discipline this dissertation focuses on remains essentially untouched.\textbf{ }In terms of the four flavor classification given above, \textbf{$\left(p,q\right)$-adic analysis}\index{analysis!left(p,qright)-adic@$\left(p,q\right)$-adic}\textemdash the study of functions from $\mathbb{Z}_{p}$ (or a metrically complete field extension thereof) to $\mathbb{Z}_{q}$ (or a metrically complete field extension thereof), where $p$ and $q$ are distinct primes, is technically a form of non-archimedean analysis. Even so, this dissertation stands in stark contrast to much of the established literature in non-archimedean analysis not only by its unusual content, but also by virtue of its concreteness. One generally needs to turn to works of mathematical physics like \cite{Quantum Paradoxes} to get comparable levels of computational depth. Also significant, unlike several works going by the title ``non-archimedean analysis'' (or some variation thereof), my dissertation has the distinction of actually \emph{being }a work of mathematical analysis, as opposed to algebra going by a false name\footnote{A particularly egregious example of this is \cite{Bosch lying title}, a text on $p$-adic algebraic geometry and Tate's ``rigid analytic geometry'' which has the \emph{nerve} to call itself \emph{Non-archimedean analysis}, despite not containing so much as a single line of bonafide analysis. Other examples of such wolves-in-sheep's-clothing are \cite{Schneider awful book,More Schneider lies}, both due to Peter Schneider.}. And, by mathematical analysis, I mean \emph{hard} analysis: $\epsilon$s and $\delta$s, approximations, asymptotics, and the like, and all the detailed computations entailed. The past twenty or so years have seen many new strides in $p$-adic and non-archimedean analysis, particularly in the vein of generalizing as many of classical analysis' most useful tools, principally to the setting of $\left(p,\infty\right)$-adic analysis, but also to $\left(p,p\right)$ as well. The tools in question include, asymptotic analysis, Tauberian theorems, distributions, oscillatory integrals (a $p$-adic van der Corput lemma), , and many others; see, \cite{p-adic Tauberian}, \cite{Vladimirov - the big paper about complex-valued distributions over the p-adics,Volo - p-adic Wiener}, \cite{p-adic van der Corput lemma,Real and p-Adic Oscillatory integrals}). However, because $\left(p,q\right)$-adic analysis works with functions taking values in non-archimedean fields, the vast majority of these tools and their associated proofs do not extend to the $\left(p,q\right)$-adic case\textemdash at least, not yet. And it is precisely there where we will begin our work. \subsection{\label{subsec:3.1.2 Banach-Spaces-over}Banach Spaces over a Non-Archimedean Field} THROUGHOUT THIS SUBSECTION, $K$ IS A COMPLETE NON-ARCHIMEDEAN VALUED FIELD. \vphantom{} In classical analysis, Banach spaces frequently trot onto the stage when one moves from asking ``biographical'' questions about specific functions, specific equations, and the like, to ``sociological'' ones\textemdash those regarding whole classes of functions. Much like with real or complex analysis, there is quite a lot one can do in $\left(p,p\right)$-adic analysis without needing to appeal to the general theories of Banach spaces and functional analysis. Power series and Mahler's basis for continuous $\left(p,p\right)$-adic functions give $p$-adic analysis enough machinery to make concreteness just as viable as in the real or complex case. In $\left(p,q\right)$-adic analysis, however, the lack of any notions of differentiation, power series, and the non-applicability of the Mahler Basis make for an entirely different set of circumstances. Rather than drawing from the broader perspectives offered by functional analysis as a means of enriching how we approach what we already know, Banach spaces are foundational to non-archimedean analysis precisely because they can be used to determine what is or is not actually \emph{possible} in the subject. As a simple example, consider Hilbert spaces and orthogonal functions. Investigations into Fourier series and differential equations in the nineteenth century motivated subsequent explorations of function spaces as a whole, especially in the abstract. The Cauchy-Schwarz Inequality came before inner product spaces. In non-archimedean analysis, however, it can be shown that the only inner product spaces over non-archimedean fields which are Hilbert spaces in the classical sense\footnote{Specifically, their norm is induced by a bilinear form, and orthogonal projections exist for every closed subspace.} are necessarily finite-dimensional (\textbf{Theorem 4 }from \cite{Schikhof Banach Space Paper}). Sobering information such as this gives non-archimedean analysis an idea of what they should or should not set their hopes on. Most of the exposition given here is taken from Schikhof's excellent article on Banach spaces over a non-archimedean field \cite{Schikhof Banach Space Paper}, which\textemdash unlike van Rooij's book \cite{van Rooij - Non-Archmedean Functional Analysis}\textemdash is still in print. We begin with the basic definitions. \begin{defn}[S\textbf{emi-norms, normed vector spaces, etc.} \cite{Ultrametric Calculus}] \label{def:seminorms etc.}Let $E$ be a vector space over a non-archimedean valued field $K$ (a.k.a., $E$ is a \textbf{$K$-vector space} or \textbf{$K$-linear space}). A\index{non-archimedean!semi-norm} \textbf{(non-archimedean) semi-norm }on $E$ is a function $\rho:E\rightarrow\mathbb{R}$ such that for all $x,y\in E$ and all $\mathfrak{a}\in K$: \vphantom{} I. $\rho\left(x\right)\geq0$; \vphantom{} II. $\rho\left(\mathfrak{a}x\right)=\left\|\mathfrak{a}\right\|_{K}\rho\left(x\right)$; \vphantom{} III. $\rho\left(x+y\right)\leq\max\left\{ \rho\left(x\right),\rho\left(y\right)\right\} $. \vphantom{} Additionally, $\rho$ is said to be a \textbf{(non-archimedean) norm }over\index{non-archimedean!norm} $E$ whenever it satisfies the additional condition: \vphantom{} IV. $\rho\left(x\right)=0$ if and only if $x=0$. \vphantom{} Next, given a non-archimedean norm $\left\Vert \cdot\right\Vert $, the pair $\left(E,\left\Vert \cdot\right\Vert \right)$ is called a \textbf{(non-archimedean) normed vector space}; this is then an ultrametric space with respect to the metric induced by the norm. We say $\left(E,\left\Vert \cdot\right\Vert \right)$ is a \textbf{(non-archimedean) Banach space }when\index{non-archimedean!Banach space}\index{Banach algebra!non-archimedean} it is complete as a metric space. Finally, a \textbf{(non-archimedean) Banach algebra }is\index{non-archimedean!Banach algebra} a Banach space $\left(E,\left\Vert \cdot\right\Vert \right)$ with a multiplication operation $m:E\times E\rightarrow E$ so that $\left\Vert m\left(x,y\right)\right\Vert \leq\left\Vert x\right\Vert \left\Vert y\right\Vert $ for all $x,y\in E$. Also, a $K$-Banach space (resp. algebra) is a Banach space (resp. algebra) over the field $K$. \end{defn} \begin{prop} \label{prop:series characterization of a Banach space}Let $E$ be a non-archimedean normed vector space. Then, $E$ is complete if and only if, for every sequence $\left\{ x_{n}\right\} _{n\geq0}$ in $E$ for which $\lim_{n\rightarrow\infty}\left\Vert x_{n}\right\Vert \overset{\mathbb{R}}{=}0$, the series $\sum_{n=0}^{\infty}x_{n}$ converges to a limit in $E$. \end{prop} Proof: I. Suppose $E$ is complete. Then, $\left(E,\left\Vert \cdot\right\Vert \right)$ is a complete ultrametric space, and as such, an infinite series $\sum_{n=0}^{\infty}x_{n}$ in $E$ converges if and only if $\lim_{n\rightarrow\infty}\left\Vert x_{n}\right\Vert \overset{\mathbb{R}}{=}0$. \vphantom{} II. Conversely, suppose an infinite series $\sum_{n=0}^{\infty}x_{n}$ in $E$ converges if and only if $\lim_{n\rightarrow\infty}\left\Vert x_{n}\right\Vert \overset{\mathbb{R}}{=}0$. Then, letting $\left\{ x_{n}\right\} _{n\geq0}$ be a Cauchy sequence in $E$, we can choose a sequence $n_{1}<n_{2}<\cdots$ so that, for any $j$, $\left\Vert x_{m}-x_{n}\right\Vert <2^{-j}$ holds for all $m,n\geq n_{j}$. Setting $y_{1}=x_{n_{1}}$ and $y_{j}=x_{n_{j}}-x_{n_{j-1}}$ gives: \begin{equation} \lim_{J\rightarrow\infty}x_{n_{J}}\overset{E}{=}\lim_{J\rightarrow\infty}\sum_{j=1}^{J}y_{j} \end{equation} By construction, the Cauchyness of the $x_{n}$s guarantees that $\left\Vert y_{j}\right\Vert \rightarrow0$ as $j\rightarrow\infty$. As such, our assumption ``every infinite series in $E$ converges if and only if its terms tend to zero in $E$-norm'' forces $\lim_{J\rightarrow\infty}\sum_{j=1}^{J}y_{j}$ (and hence, $\lim_{J\rightarrow\infty}x_{n_{J}}$) to converge to a limit in $E$. Since the $x_{n}$s are Cauchy, the existence of a convergent subsequence then forces the entire sequence to converge in $E$, with: \begin{equation} \lim_{n\rightarrow\infty}x_{n}\overset{E}{=}\sum_{j=1}^{\infty}y_{j} \end{equation} Because the $x_{n}$s were arbitrary, this shows that $E$ is complete. Q.E.D. \vphantom{} Next, we give the fundamental examples of non-archimedean Banach spaces. Throughout, we will suppose that $K$ is a complete non-archimedean valued field. We address the finite-dimensional cases first. \begin{defn} Given a vector space $V$ over a non-archimedean field $K$, two non-archimedean norms $\left\Vert \cdot\right\Vert $ and $\left\Vert \cdot\right\Vert ^{\prime}$ defined on $V$ are said to be \textbf{equivalent }whenever there are real constants $0<c\leq C<\infty$ so that: \begin{equation} c\left\Vert \mathbf{x}\right\Vert \leq\left\Vert \mathbf{x}\right\Vert ^{\prime}\leq C\left\Vert \mathbf{x}\right\Vert \end{equation} \end{defn} \cite{Robert's Book} gives the following basic results: \begin{thm} Let $V$ and $W$ be normed vector spaces over $\mathbb{Q}_{p}$. \vphantom{} I. If $V$ is finite-dimensional, then all non-archimedean norms on $V$ are equivalent. \vphantom{} II. If $V$ is locally compact, then $V$ is finite dimensional. \vphantom{} III. If $V$ is locally compact, then $V$ has the Heine-Borel property\textemdash sets in $V$ are compact if and only if they are closed and bounded. \vphantom{} IV. If $V$ and $W$ are finite-dimensional, then every linear map $L:V\rightarrow W$ is continuous. \end{thm} \begin{defn} \ I. In light of the above, we shall use only a single norm on finite-dimensional vector spaces over $\mathbb{Q}_{p}$: the $\ell^{\infty}$-norm (written $\left\Vert \cdot\right\Vert _{p}$), which outputs the maximum of the $p$-adic absolute values of the entries of an element of the vector space. \vphantom{} II. Given integers $\rho,c\geq1$, we write \nomenclature{$K^{\rho,c}$}{$\rho\times c$ matrices with entries in $K$ \nopageref}$K^{\rho,c}$ to denote the set of all $\rho\times c$ matrices with entries in $K$. We make this a non-archimedean normed vector space by equipping it with the $\infty$-norm, which, for any $\mathbf{A}\in K^{\rho,c}$ with entries $\left\{ a_{i,j}\right\} _{1\leq i\leq\rho,1,j\leq c}$, is given by: \nomenclature{$\left\Vert \mathbf{A}\right\Vert _{K}$}{max. of the $K$-adic absolute values of a matrix's entries \nopageref}\nomenclature{$\left\Vert \mathbf{a}\right\Vert _{K}$}{max. of the $K$-adic absolute values of a vector's entries \nopageref} \begin{equation} \left\Vert \mathbf{A}\right\Vert _{K}\overset{\textrm{def}}{=}\max_{1\leq i\leq\rho,1\leq j\leq c}\left\|a_{i,j}\right\|_{K}\label{eq:Definition of the non-archimedean matrix norm} \end{equation} When $c=1$, we identify $K^{\rho,1}$ with the set of all $\rho\times1$ column vectors. \vphantom{} III. Given $d\geq2$, we write $\textrm{GL}_{d}\left(K\right)$\nomenclature{$\textrm{GL}_{d}\left(K\right)$}{set of invertible $d\times d$ matrices with entries in $K$ \nopageref} to denote the set of all invertible $d\times d$ matrices with entries in $K$, made into a non-abelian group by matrix multiplication. \vphantom{} IV. Because we will need it for parts of Chapter 5, we define \nomenclature{$\left\Vert \mathbf{x}\right\Vert _{\infty}$}{$\overset{\textrm{def}}{=}\max\left\{ \left\|x_{1}\right\|,\ldots,\left\|x_{d}\right\|\right\}$}$\left\Vert \cdot\right\Vert _{\infty}$ on $\mathbb{C}^{d}$ (or any subset thereof) by: \begin{equation} \left\Vert \mathbf{x}\right\Vert _{\infty}\overset{\textrm{def}}{=}\max\left\{ \left\|x_{1}\right\|,\ldots,\left\|x_{d}\right\|\right\} \label{eq:Definition of infinity norm} \end{equation} where $\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)$. Here, the absolute values are those on $\mathbb{C}$. \end{defn} \begin{prop} $K^{\rho,c}$ is a non-archimedean Banach space over $K$. If $\rho=c$, it is also a Banach algebra, with the multiplication operation being matrix multiplication. \end{prop} We will not mention matrix spaces until we get to Chapters 5 and 6, where we they will be ubiquitous. Next, we introduce the fundamental examples of \emph{infinite}-dimensional non-archimedean Banach spaces. \begin{defn} Let $X$ be a set and let $K$ be a metrically-complete non-archimedean valued field. \vphantom{} I. A function $f:X\rightarrow K$ is said to be \textbf{bounded }if: \begin{equation} \sup_{x\in X}\left\|f\left(x\right)\right\|_{K}<\infty \end{equation} We get a norm \nomenclature{$\left\Vert f\right\Vert _{X,K}$}{$\sup_{x\in X}\left\|f\left(x\right)\right\|_{K}$} out of this by defining: \begin{equation} \left\Vert f\right\Vert _{X,K}\overset{\textrm{def}}{=}\sup_{x\in X}\left\|f\left(x\right)\right\|_{K}\label{eq:Definition of X,K norm} \end{equation} We call this the $K$\textbf{-supremum norm }\index{norm!$K$ supremum} on $X$. We write \nomenclature{$B\left(X,K\right)$}{Bounded $K$-valued functions on $X$}$B\left(X,K\right)$ to denote the set of all bounded $K$-valued functions on $X$. This is a non-archimedean Banach space under $\left\Vert \cdot\right\Vert _{X,K}$. When $X=\mathbb{Z}_{p}$, we call the norm $\left\Vert \cdot\right\Vert _{X,K}$ the \textbf{$\left(p,K\right)$-adic norm}\index{norm!left(p,Kright)-adic@$\left(p,K\right)$-adic}, and denote it by $\left\Vert \cdot\right\Vert _{p,K}$\nomenclature{$\left\Vert \cdot\right\Vert _{p,K}$}{ }. When $K$ is a $q$-adic field, we call this the \index{norm!left(p,qright)-adic@$\left(p,q\right)$-adic}\index{$p,q$-adic!norm}\textbf{$\left(p,q\right)$-adic norm}, and denote it by \nomenclature{$\left\Vert \cdot\right\Vert _{p,q}$}{ }$\left\Vert \cdot\right\Vert _{p,q}$. In a minor abuse of notation, we also use $\left\Vert \cdot\right\Vert _{p,K}$ and $\left\Vert \cdot\right\Vert _{p,q}$ to denote the $K$ supremum norm on $X$ when $X$ is $\hat{\mathbb{Z}}_{p}$. \vphantom{} II. If $X$ is a compact Hausdorff space, we write \nomenclature{$C\left(X,K\right)$}{set of continuous $K$-valued functions on $X$}$C\left(X,K\right)$ to denote the set of all continuous functions $f:X\rightarrow K$. This is a non-archimedean Banach space\footnote{This construction is of vital importance to us, because it allows us to consider continuous $K$-valued functions defined on a compact ultrametric space $X$ like $\mathbb{Z}_{p}$, even if $K$'s residue field has characteristic $q\neq p$.} under $\left\Vert \cdot\right\Vert _{X,K}$. \vphantom{} III. We write $\text{\ensuremath{\ell^{\infty}\left(K\right)}}$ (or $\ell^{\infty}$, if $K$ is not in question) to denote the set of all bounded sequences in $K$. This is a non-archimedean Banach space under the norm: \begin{equation} \left\Vert \left(a_{1},a_{2},\ldots\right)\right\Vert _{K}\overset{\textrm{def}}{=}\sup_{n\geq1}\left\|a_{n}\right\|_{K}\label{eq:K sequence norm} \end{equation} \vphantom{} IV. We write \nomenclature{$c_{0}\left(K\right)$}{set of $K$-valued sequences converging to $0$}$c_{0}\left(K\right)$ (or $c_{0}$, if $K$ is not in question) to denote the set of all sequences in $K$ which converge to $0$ in $K$'s absolute value. This is a Banach space under (\ref{eq:K sequence norm}) and is a closed subspace of $\ell^{\infty}$. \vphantom{} V. We write\footnote{I've adapted this notation from what van Rooij calls ``$c_{0}\left(X\right)$''. \cite{van Rooij - Non-Archmedean Functional Analysis}} $c_{0}\left(X,K\right)$ to denote the set of all $f\in B\left(X,K\right)$ so that, for every $\epsilon>0$, there are only finitely many $x\in X$ for which $\left\|f\left(x\right)\right\|_{K}\geq\epsilon$. This is a Banach space with respect to $\left\Vert \cdot\right\Vert _{X,K}$, being a closed subspace of $B\left(X,K\right)$. \vphantom{} VI. \nomenclature{$\text{\ensuremath{\ell^{1}\left(K\right)}}$}{set of absolutely summable $K$-valued sequences}We write $\text{\ensuremath{\ell^{1}\left(K\right)}}$ (or $\ell^{1}$, if $K$ is not in question) to denote the set of all sequences $\mathbf{c}:\mathbb{N}_{0}\rightarrow K$ so that: \[ \sum_{n=0}^{\infty}\left\|\mathbf{c}\left(n\right)\right\|_{K}<\infty \] We write: \begin{equation} \left\Vert \mathbf{c}\right\Vert _{1}\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}\left\|\mathbf{c}\left(n\right)\right\|_{K}\label{eq:Definition of ell-1 norm} \end{equation} $\ell^{1}$ is an \emph{archimedean }$K$-Banach space under $\left\Vert \cdot\right\Vert _{1}$\textemdash that is, $\left\Vert \mathbf{a}+\mathbf{b}\right\Vert _{1}\leq\left\Vert \mathbf{a}\right\Vert _{1}+\left\Vert \mathbf{b}\right\Vert _{1}$, however, the strong triangle inequality does not hold on it; that is, there exist $\mathbf{a},\mathbf{b}$ for which $\left\Vert \mathbf{a}+\mathbf{b}\right\Vert _{1}>\max\left\{ \left\Vert \mathbf{a}\right\Vert _{1},\left\Vert \mathbf{b}\right\Vert _{1}\right\} $. This also goes to show that not every Banach space over a non-archimedean field is, itself, non-archimedean. \end{defn} \begin{rem} As a rule, when writing expressions involving $\left\Vert \cdot\right\Vert $, I will use a single subscript (ex: $\left\Vert \cdot\right\Vert _{K}$) whenever only the absolute value of $K$ is being used. Expressions with \emph{two} subscripts (ex: $\left\Vert \cdot\right\Vert _{p,K}$, $\left\Vert \cdot\right\Vert _{p,q}$), meanwhile, indicate that we are using the absolute value indicated by the subscript on the right as well as taking a supremum of an input variable with respect to the absolute value indicated by the subscript on the left. \end{rem} \vphantom{} Although $\ell^{1}$ and its continuous counterpart $L^{1}$ are of fundamental importance in real and complex analysis, their non-archimedean analogues are not nearly as distinguished. The first hint that something is amiss comes from the fact that $\ell^{1}$ is archimedean, rather than non-archimedean.\emph{ Traditionally}\footnote{Schikhof himself says as much at the top of page 6 in \cite{Schikhof Banach Space Paper}, writing ``Absolute summability plays no a role in {[}non-archimedean{]} analysis.''}, $\ell^{1}$ is of no interest in non-archimedean analysts. Unsurprisingly, it turns out that $\ell^{1}$\textemdash specifically, its continuous analogue, $L^{1}$) is pertinent to the study of $\chi_{H}$. This is the subject of Subsection \ref{subsec:3.3.6 L^1 Convergence}. Next, we have linear operators and functionals. \begin{defn} Let $E$ and $F$ be non-archimedean Banach spaces, both over the field $K$. Then one writes $\mathcal{L}\left(E,F\right)$ to denote the set of all continuous linear operators from $E$ to $F$. We equip $\mathcal{L}\left(E,F\right)$ with the \textbf{operator norm}: \begin{equation} \left\Vert T\right\Vert \overset{\textrm{def}}{=}\sup_{x\in E}\frac{\left\Vert T\left(x\right)\right\Vert _{F}}{\left\Vert x\right\Vert _{E}}\label{eq:Definition of operator norm} \end{equation} \end{defn} \begin{prop} $\mathcal{L}\left(E,F\right)$ is a non-archimedean Banach space over $K$ with respect to the operator norm. \end{prop} \begin{prop} A linear operator $T:E\rightarrow F$ between two non-archimedean Banach spaces $E$ and $F$ over the field $K$ is continuous if and only if it has finite operator norm. Consequently, \textbf{bounded linear operators }and \textbf{continuous linear operators }are one and the same, just like in the classical case. \end{prop} \begin{defn} Given a non-archimedean $K$-Banach space $E$, the \textbf{dual}\index{Banach space!dual of} of $E$ (denoted $E^{\prime}$) is, as usual, the $K$-Banach space of all continuous linear functionals from $E$ to $K$. Given an $f\in E^{\prime}$, we write $\left\Vert f\right\Vert _{E^{\prime}}$ to denote the norm of $f$; this is defined by the usual variant of (\ref{eq:Definition of operator norm}): \begin{equation} \left\Vert f\right\Vert _{E^{\prime}}\overset{\textrm{def}}{=}\sup_{x\in E}\frac{\left\|f\left(x\right)\right\|_{K}}{\left\Vert x\right\Vert _{E}}\label{eq:Definition of the norm of a linear functional} \end{equation} \end{defn} \begin{rem} For the case of the space $C\left(\mathbb{Z}_{p},K\right)$, where $K$ is a $q$-adic field, where $p$ and $q$ are distinct primes, note that its dual $C\left(\mathbb{Z}_{p},K\right)^{\prime}$ \nomenclature{$C\left(\mathbb{Z}_{p},K\right)^{\prime}$}{set of continuous $K$-valued linear functionals on $C\left(\mathbb{Z}_{p},K\right)$} will consist of all continuous $K$-valued linear functionals on $C\left(\mathbb{Z}_{p},K\right)$. \end{rem} \begin{fact} \emph{Importantly, because of their proofs' dependence on the }\textbf{\emph{Baire Category Theorem}}\emph{,\index{Uniform Boundedness Principle} the }\textbf{\emph{Uniform Boundedness Principle}}\emph{, the }\textbf{\emph{Closed Graph Theorem}}\emph{, the }\textbf{\emph{Open Mapping Theorem}}\emph{, and the }\textbf{\emph{Banach-Steinhaus Theorem}}\emph{ }all apply to non-archimedean Banach spaces\emph{ \cite{Schikhof Banach Space Paper}. The Hahn-Banach Theorem, however, is more subtle, as is discussed below.} \end{fact} \vphantom{} As we now move to discuss notions of duality, the notion of \index{spherically complete}\textbf{spherical completeness}\footnote{Recall, a valued field $K$ (or, more generally, any metric space) is said to be \textbf{spherically complete} whenever any sequence of nested non-empty balls $B_{1}\supseteq B_{2}\supseteq\cdots$ in $K$ has a non-empty intersection. $\mathbb{Q}_{p}$ and any finite extension thereof are spherically complete; $\mathbb{C}_{p}$, however, is not.} will raise its head in surprising ways. The first instance of this is in the non-archimedean Hahn-Banach Theorem. First, however, another definition, to deal with the fact that non-archimedean fields need not be separable as topological spaces. \begin{defn} A $K$-Banach space $E$ is said\index{Banach space!of countable type} to be \textbf{of countable type }if there is a countable set whose $K$-span is dense in $E$. \end{defn} \begin{thm}[\textbf{Non-Archimedean Hahn-Banach Theorems}\footnote{Theorems 9 and 10 in \cite{Schikhof Banach Space Paper}.}\index{Hahn-Banach Theorem}] Let $E$ be a Banach space over a complete non-archimedean valued field $K$, let $D$ be a subspace of $E$. Let $D^{\prime}$ be the continuous dual of $D$ (continuous linear functionals $D\rightarrow K$) and let $f\in D^{\prime}$. \vphantom{} I. If $K$ is \emph{spherically complete} (for instance, if $K$ is $\mathbb{Q}_{q}$ or a finite, metrically-complete extension thereof), there then exists a continuous linear functional $\overline{f}\in E^{\prime}$ whose restriction to $D$ is equal to $f$, and so that $\left\Vert \overline{f}\right\Vert _{E^{\prime}}=\left\Vert f\right\Vert _{D^{\prime}}$. \vphantom{} II. If $E$ is of countable type, then\textemdash even if $K$ is \textbf{\emph{not }}spherically complete\textemdash for any $\epsilon>0$, there exists a continuous linear functional $\overline{f}:E\rightarrow K$ whose restriction to $D$ is equal to $f$. Moreover, $\left\Vert \overline{f}\right\Vert _{E^{\prime}}\leq\left(1+\epsilon\right)\left\Vert f\right\Vert _{D^{\prime}}$. \end{thm} \begin{rem} The requirement that $E$ be of countable type \emph{cannot} be dropped, nor can $\epsilon$ ever be set to $0$. There exists a two-dimensional non-archimedean Banach space $E$ over a non-spherically-complete $K$ and a subspace $D\subseteq E$ and an $f\in D^{\prime}$ such that no extension $\overline{f}\in E^{\prime}$ exists satisfying $\left\Vert \overline{f}\right\Vert _{E^{\prime}}\leq\left\Vert f\right\Vert _{D^{\prime}}$ \cite{Schikhof Banach Space Paper}. This counterexample can be found on page 68 of van Rooij's book \cite{van Rooij - Non-Archmedean Functional Analysis}. \end{rem} \vphantom{} Spherical completeness is responsible for non-archimedean Banach spaces' most significant deviations from the classical theory, which we now chronicle. The names I give these theorems were taken from \cite{van Rooij - Non-Archmedean Functional Analysis}. \begin{thm}[\textbf{Fleischer's Theorem}\footnote{See \cite{Schikhof Banach Space Paper,van Rooij - Non-Archmedean Functional Analysis}. This theorem is given as \textbf{Theorem}}] Let $K$ be \textbf{spherically complete}. Then, a $K$-Banach space is reflexive\index{Banach space!reflexive} if and only if it is finite-dimensional. \end{thm} \begin{thm}[\textbf{van der Put's Theorem}\footnote{See \cite{Schikhof Banach Space Paper,van Rooij - Non-Archmedean Functional Analysis}. This theorem is a specific part (iii) of \textbf{Theorem 4.21 }on page 118 of \cite{van Rooij - Non-Archmedean Functional Analysis}.}] \label{thm:c_o and ell_infinit are each others duals in a spherically incomplete NA field}Let $K$ be\textbf{\emph{ spherically incomplete}}. Then, for any countable\footnote{To give the reader a taste of the level of generality of van Rooij's text, while I state the theorem for countable $X$, the formulation given in \cite{van Rooij - Non-Archmedean Functional Analysis} states that it holds for any ``small'' set $X$, a definition that involves working with cardinal numbers and arbitrary $\sigma$-additive measures on Boolean rings. On page 31 of the same, van Rooij says that ``obviously'', a set with cardinality $\aleph_{0}$ (that is, a countable set) is ``small''. I choose to believe him, from whence I obtained the version of the theorem given here.} set $X$, the $K$-Banach spaces $c_{0}\left(X,K\right)$ and $B\left(X,K\right)$ are reflexive\index{Banach space!reflexive}, being one another's duals in a natural way. \end{thm} \begin{cor} If $K$ is \textbf{\emph{spherically incomplete}}, then \textbf{every} $K$-Banach space of countable type is reflexive. \cite{van Rooij - Non-Archmedean Functional Analysis} \end{cor} \subsection{\label{subsec:3.1.3 The-van-der}The van der Put Basis} IN THIS SUBSECTION, $p$ IS A PRIME. $K$ IS A METRICALLY COMPLETE VALUED FIELD (NOT NECESSARILY NON-ARCHIMEDEAN) OF CHARACTERISTIC ZERO. \vphantom{} A principal reason for the ``simplicity'' of $\left(p,q\right)$-adic analysis in comparison with more general theories of non-archimedean analysis is that, much like in the $\left(p,p\right)$-adic case, the Banach space of continuous functions $\left(p,q\right)$-adic functions admits a countable basis. This is \textbf{the van der Put basis}. This basis actually works for the space of continuous functions $f:\mathbb{Z}_{p}\rightarrow K$ where $K$ is \emph{any }metrically complete non-archimedean field, unlike the Mahler basis, which only works when $K$ is $\mathbb{Q}_{p}$ or an extension thereof. Because of the van der Put basis, we will be able to explicitly compute integrals and Fourier transforms, which will be of the utmost importance for our analyses of $\chi_{H}$. \begin{defn}[\textbf{The van der Put Basis} \cite{Robert's Book,Ultrametric Calculus}] \label{def:vdP basis, n_minus, vpD coefficients}\ \vphantom{} I. We call the indicator functions $\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(j\right)}}{\equiv}j\right]$ the \index{van der Put!basis}\textbf{van der Put (basis) functions }and refer to the set $\left\{ \left[\mathfrak{z}\overset{p^{\lambda_{p}\left(j\right)}}{\equiv}j\right]\right\} _{j\in\mathbb{N}_{0}}$ as the \textbf{van der Put basis}. \vphantom{} II. Given an integer $n\geq1$, we can express $n$ $p$-adically as: \begin{equation} n=\sum_{k=0}^{\lambda_{p}\left(n\right)-1}n_{k}p^{k} \end{equation} where the $n_{k}$s are the $p$-adic digits of $n$. Following Robert and Schikhof (\cite{Robert's Book,Ultrametric Calculus}), we write \nomenclature{$n_{-}$}{ }$n_{-}$ to denote the integer obtained by deleting the $k=\lambda_{p}\left(n\right)-1$ term from the above series; equivalently, $n_{-}$ is the integer obtained by deleting the right-most non-zero digit in $n$'s $p$-adic representation of $n$. That is: \begin{equation} n_{-}=n-n_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}\label{eq:Definition of n minus} \end{equation} We can also write this as the value of $n$ modulo $p^{\lambda_{p}\left(n\right)-1}$: \begin{equation} n_{-}=\left[n\right]_{p^{\lambda_{p}\left(n\right)-1}} \end{equation} \vphantom{} III. Let $f:\mathbb{Z}_{p}\rightarrow K$ be any function. Then, for all $n\in\mathbb{N}_{0}$, we define the constants $\left\{ c_{n}\left(f\right)\right\} _{n\geq0}\in K$ by: \begin{equation} c_{n}\left(f\right)\overset{\textrm{def}}{=}\begin{cases} f\left(0\right) & \textrm{if }n=0\\ f\left(n\right)-f\left(n_{-}\right) & \textrm{if }n\geq1 \end{cases},\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Def of c_n of f} \end{equation} We call \nomenclature{$c_{n}\left(f\right)$}{$n$th van der Put coefficient of $f$}$c_{n}\left(f\right)$ the $n$th \index{van der Put!coefficients}\textbf{van der Put coefficient }of $f$. \vphantom{} IV. We write \nomenclature{$\textrm{vdP}\left(\mathbb{Z}_{p},K\right)$}{the set of formal van der Put series with coefficients in $K$}$\textrm{vdP}\left(\mathbb{Z}_{p},K\right)$ to denote the $K$-vector space of all \textbf{formal van der Put series}\index{van der Put!series}. The elements of $\textrm{vdP}\left(\mathbb{Z}_{p},K\right)$ are formal sums: \begin{equation} \sum_{n=0}^{\infty}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} where the $\mathfrak{a}_{n}$s are elements of $K$. The \textbf{domain of convergence }of a formal van der Put series is defined as the set of all $\mathfrak{z}\in\mathbb{Z}_{p}$ for which the series converges in $K$. We call $\textrm{vdP}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ the space of \textbf{formal $\left(p,q\right)$-adic van der Put series}. \vphantom{} V. Let \nomenclature{$F\left(\mathbb{Z}_{p},K\right)$}{$K$-valued functions on $\mathbb{Z}_{p}$}$F\left(\mathbb{Z}_{p},K\right)$ denote the $K$-linear space of all $K$-valued functions on $\mathbb{Z}_{p}$. Then, we define the linear operator \nomenclature{$S_{p}$}{van-der-Put-series-creating operator}$S_{p}:F\left(\mathbb{Z}_{p},K\right)\rightarrow\textrm{vdP}\left(\mathbb{Z}_{p},K\right)$ by: \begin{equation} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall f\in F\left(\mathbb{Z}_{p},K\right)\label{eq:Definition of S_p of f} \end{equation} We also define partial sum operators \nomenclature{$S_{p:N}$}{$N$th partial van-der-Put-series-creating operator}$S_{p:N}:F\left(\mathbb{Z}_{p},K\right)\rightarrow C\left(\mathbb{Z}_{p},K\right)$ by: \begin{equation} S_{p:N}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall f\in F\left(\mathbb{Z}_{p},K\right)\label{eq:Definition of S_p N of f} \end{equation} \end{defn} \begin{rem} Fixing $\mathfrak{z}\in\mathbb{N}_{0}$, observe that $\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n$ can hold true for at most finitely many $n$. As such, every formal van der Put series necessarily converges in $K$ at every $\mathfrak{z}\in\mathbb{N}_{0}$. \end{rem} \begin{example}[\textbf{$\lambda$-Decomposition}] Given a sum of the form with coefficients in $\mathbb{C}_{q}$: \begin{equation} \sum_{n=0}^{\infty}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} we can decompose it by splitting the sum into distinct pieces, with $\lambda_{p}\left(n\right)$ taking a single value over each individual piece. In particular, noting that for any $m\in\mathbb{N}_{1}$: \begin{equation} \lambda_{p}\left(n\right)=m\Leftrightarrow p^{m-1}\leq n\leq p^{m}-1 \end{equation} we can write: \begin{equation} \sum_{n=0}^{\infty}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\overset{\mathbb{C}_{q}}{=}\mathfrak{a}_{0}+\sum_{m=1}^{\infty}\sum_{n=p^{m-1}}^{p^{m}-1}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{m}}{\equiv}n\right]\label{eq:Lambda Decomposition} \end{equation} We will use this decomposition frequently enough for it to be worth having its own name: \emph{the Lambda decomposition}, or \emph{$\lambda$-decomposition\index{lambda-decomposition@$\lambda$-decomposition}}, for short. \end{example} \vphantom{} For us, the chief utility of van der Put series is the fact that they tell us how to compute a type of function-limit which we have seen to great effect in Chapter 2. The next proposition gives the details. \begin{prop}[\textbf{\textit{van der Put Identity}}] \label{prop:vdP identity}Let $f\in B\left(\mathbb{Z}_{p},K\right)$. Then, for any $\mathfrak{z}\in\mathbb{Z}_{p}$ which is in the domain of convergence of $S_{p}\left\{ f\right\} $: \begin{equation} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{K}{=}\lim_{k\rightarrow\infty}f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)\label{eq:van der Put identity} \end{equation} We call this the \textbf{van der Put identity}\index{van der Put!identity}. We also have: \begin{equation} \sum_{n=0}^{p^{N}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\overset{\mathbb{F}}{=}f\left(\left[\mathfrak{z}\right]_{p^{N}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:truncated van der Put identity} \end{equation} which we call the (\textbf{$N$th}) \textbf{truncated van der Put identity}. \end{prop} \begin{rem} This is part of Exercise 62.B on page 192 of \cite{Ultrametric Calculus}. \end{rem} Proof: Performing a $\lambda$-decomposition yields: \begin{equation} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{n=0}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\overset{K}{=}c_{0}\left(f\right)+\sum_{k=1}^{\infty}\sum_{n=p^{k-1}}^{p^{k}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{k}}{\equiv}n\right] \end{equation} Fixing $\mathfrak{z}$ and letting $k$ be arbitrary, we note that there is at most one value of $n\in\left\{ p^{k-1},\ldots,p^{k}-1\right\} $ which solves the congruence $\mathfrak{z}\overset{p^{k}}{\equiv}n$; that value is $n=\left[\mathfrak{z}\right]_{p^{k}}$. Thus: \begin{equation} \sum_{n=p^{k-1}}^{p^{k}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{k}}{\equiv}n\right]\overset{K}{=}c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\label{eq:Inner term of vdP lambda decomposition} \end{equation} where the Iverson bracket on the right indicates that the right-hand side is $0$ whenever $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)$ is not equal to $k$. This failure of equality occurs precisely when there is no $n\in\left\{ p^{k-1},\ldots,p^{k}-1\right\} $ which solves the congruence $\mathfrak{z}\overset{p^{k}}{\equiv}n$. Next, writing $\mathfrak{z}$ as $\mathfrak{z}=\sum_{j=0}^{\infty}\mathfrak{z}_{j}p^{j}$, where the $\mathfrak{z}_{j}$s are the $p$-adic digits of $\mathfrak{z}$, we have that: \begin{equation} c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right)\overset{K}{=}f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k}}-\mathfrak{z}_{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-1}p^{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-1}\right) \end{equation} Note that $\left[\mathfrak{z}\right]_{p^{k}}$ can have \emph{at most} $k$ $p$-adic digits. In particular, $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=j$ if and only if the $\mathfrak{z}_{j-1}$ is the right-most non-zero $p$-adic digit of $\mathfrak{z}$. Consequently, letting $j\in\left\{ 0,\ldots,k\right\} $ denote the integer $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)$, we have that $\left[\mathfrak{z}\right]_{p^{k}}=\left[\mathfrak{z}\right]_{p^{j}}$. We can then write: \begin{align} c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right) & \overset{K}{=}f\left(\left[\mathfrak{z}\right]_{p^{j}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{j}}-\mathfrak{z}_{j-1}p^{j-1}\right)\\ & =f\left(\left[\mathfrak{z}\right]_{p^{j}}\right)-f\left(\sum_{i=0}^{j-1}\mathfrak{z}_{i}p^{i}-\mathfrak{z}_{j-1}p^{j-1}\right)\\ & =f\left(\left[\mathfrak{z}\right]_{p^{j}}\right)-f\left(\sum_{i=0}^{j-2}\mathfrak{z}_{i}p^{i}\right)\\ & =f\left(\left[\mathfrak{z}\right]_{p^{j}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{j-1}}\right) \end{align} That is to say: \begin{equation} c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(\chi\right)\overset{K}{=}\chi\left(\left[\mathfrak{z}\right]_{p^{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-1}}\right) \end{equation} for all $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $, and all $k$ large enough so that $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)\geq1$; that is, all $k>v_{p}\left(\mathfrak{z}\right)$. So, we have: \begin{equation} c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\overset{K}{=}\left(f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\label{eq:Iverson Bracket Check for vdP identity} \end{equation} \begin{claim} The Iverson bracket on the right-hand side of (\ref{eq:Iverson Bracket Check for vdP identity}) can be removed. Proof of claim: If $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k$, then the Iverson bracket is $1$, and it disappears on its own without causing us any trouble. So, suppose $\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)\neq k$. Then the right-most digit of $\left[\mathfrak{z}\right]_{p^{k}}$\textemdash i.e., $\mathfrak{z}_{k-1}$\textemdash is $0$. But then, $\left[\mathfrak{z}\right]_{p^{k}}=\left[\mathfrak{z}\right]_{p^{k-1}}$, which makes the right-hand side of (\ref{eq:Iverson Bracket Check for vdP identity}) anyway, because $f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)$ is then equal to $0$. So, \emph{any case} which would cause the Iverson bracket to vanish causes $f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)$ to vanish. This tells us that the Iverson bracket in (\ref{eq:Iverson Bracket Check for vdP identity}) isn't actually needed. As such, we are justified in writing: \[ c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\overset{K}{=}f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right) \] This proves the claim. \end{claim} \vphantom{} That being done, we can write: \begin{align} \sum_{n=0}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] & \overset{K}{=}c_{0}\left(f\right)+\sum_{k=1}^{\infty}\sum_{n=p^{k-1}}^{p^{k}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{k}}{\equiv}n\right]\\ & =c_{0}\left(f\right)+\sum_{k=1}^{\infty}c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(f\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\\ & =c_{0}\left(f\right)+\underbrace{\sum_{k=1}^{\infty}\left(f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right)}_{\textrm{telescoping}}\\ \left(c_{0}\left(f\right)=f\left(\left[\mathfrak{z}\right]_{p^{0}}\right)=f\left(0\right)\right); & \overset{K}{=}f\left(0\right)+\lim_{k\rightarrow\infty}f\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-f\left(0\right)\\ & \overset{K}{=}\lim_{k\rightarrow\infty}f\left(\left[\mathfrak{z}\right]_{p^{k}}\right) \end{align} which is the van der Put identity. If we instead only sum $n$ from $0$ up to $p^{N}-1$, the upper limit of the $k$-sum in the above is changed from $\infty$ to $N$, which yields (\ref{eq:truncated van der Put identity}). Q.E.D. \vphantom{} With the van der Put basis, we can give a satisfying characterization of the $\left(p,K\right)$-adic continuity of a function in terms of the decay of its van der Put coefficients and the uniform convergence of its van der Put series. \begin{thm}[\textbf{van der Put Basis Theorem}\footnote{\cite{Robert's Book} gives this theorem on pages 182 and 183, however, it is stated there only for the case where $K$ is an extension of $\mathbb{Q}_{p}$.}] \label{thm:vdP basis theorem}Let $K$ be non-archimedean, and let $f:\mathbb{Z}_{p}\rightarrow K$ be any function. Then, the following are equivalent: \vphantom{} I. $f$ is continuous. \vphantom{} II. $\lim_{n\rightarrow\infty}\left\|c_{n}\left(f\right)\right\|_{K}=0$. \vphantom{} III. $S_{p:N}\left\{ f\right\} $ converges uniformly to $f$ in $\left(p,K\right)$-adic norm; that is: \begin{equation} \lim_{N\rightarrow\infty}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-\sum_{n=0}^{p^{N}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\right\|_{K}=0 \end{equation} \end{thm} Proof: \textbullet{} ($\textrm{(I)}\Rightarrow\textrm{(II)}$) Suppose $f$ is continuous. Since $n_{-}=n-n_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}$ for all $n\geq1$, we have that: \begin{equation} \left\|n-n_{-}\right\|_{p}=\left\|n_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1}\right\|_{p}=\frac{1}{p^{\lambda_{p}\left(n\right)-1}} \end{equation} Since this tends to $0$ as $n\rightarrow\infty$, the continuity of $f$ guarantees that $\left\|c_{n}\left(f\right)\right\|_{K}=\left\|f\left(n\right)-f\left(n_{-}\right)\right\|_{K}$ tends to $0$ as $n\rightarrow\infty$. \vphantom{} \textbullet{} ($\textrm{(II)}\Rightarrow\textrm{(III)}$) Suppose $\lim_{n\rightarrow\infty}\left\|c_{n}\left(f\right)\right\|_{K}=0$. Then, since $K$ is non-archimedean, the absolute value of the difference between $S_{p}\left\{ f\right\} $ and its $N$th partial sum satisfies: \begin{align} \left\|S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)-\sum_{n=0}^{p^{N}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\right\|_{K} & =\left\|\sum_{n=p^{N}}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\right\|_{K}\\ \left(\textrm{ultrametric ineq.}\right); & \leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\right\|_{K}\\ \left(\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\in\left\{ 0,1\right\} ,\textrm{ }\forall n,\mathfrak{z}\right); & \leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{K}\\ & \leq\sup_{n\geq N}\left\|c_{n}\left(f\right)\right\|_{K} \end{align} Since $\left\|c_{n}\left(f\right)\right\|_{K}$ is given to tend to $0$ as $N\rightarrow\infty$, we then have that $\sup_{n\geq N}\left\|c_{n}\left(f\right)\right\|_{K}\rightarrow0$ as $N\rightarrow\infty$, and hence, that $S_{p}\left\{ f\right\} $ converges uniformly over $\mathbb{Z}_{p}$. The van der Put identity (\ref{eq:van der Put identity}) shows that $f$ is the point-wise limit of $S_{p:N}\left\{ f\right\} $, which in turn shows that $S_{p}\left\{ f\right\} $ converges point-wise to $f$. Since $S_{p}\left\{ f\right\} $ converges uniformly, this proves that $S_{p:N}\left\{ f\right\} $ converges uniformly to $f$. \vphantom{} \textbullet{} ($\textrm{(III)}\Rightarrow\textrm{(I)}$). Suppose $S_{p}\left\{ f\right\} $ converges uniformly to $f$. Since $S_{p}\left\{ f\right\} $ is the limit of a sequence of continuous functions ($\left\{ S_{p:N}\left\{ f\right\} \right\} _{N\geq0}$) which converge uniformly, its limit is necessarily continuous. Thus, $f$ is continuous. Q.E.D. \vphantom{} As an aside, we note the following: \begin{prop} \label{prop:Convergence of real-valued vdP series}Let $f\in C\left(\mathbb{Z}_{p},K\right)$. Then, the van der Put series for the continuous real-valued function $\mathfrak{z}\in\mathbb{Z}_{p}\mapsto\left\|f\left(\mathfrak{z}\right)\right\|_{q}\in\mathbb{R}$ is uniformly convergent. \end{prop} Proof: Since the $q$-adic absolute value is a continuous function from $K$ to $\mathbb{R}$, the continuity of $f:\mathbb{Z}_{p}\rightarrow K$ guarantees that $\left\|f\left(\mathfrak{z}\right)\right\|_{q}$ is a continuous\textemdash in fact, \emph{uniformly }continuous\textemdash real-valued function on $\mathbb{Z}_{p}$. The van der Put coefficients of $\left\|f\right\|_{q}$ are then: \begin{equation} c_{n}\left(\left\|f\right\|_{q}\right)=\begin{cases} \left\|f\left(0\right)\right\|_{q} & \textrm{if }n=0\\ \left\|f\left(n\right)\right\|_{q}-\left\|f\left(n_{-}\right)\right\|_{q} & \textrm{if }n\geq1 \end{cases} \end{equation} As such, (\ref{eq:van der Put identity}) yields: \begin{equation} \sum_{n=0}^{\infty}c_{n}\left(\left\|f\right\|_{q}\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\overset{\mathbb{R}}{=}\lim_{n\rightarrow\infty}\left\|f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q} \end{equation} Since $\left\|f\right\|_{q}$ is continuous, we see that the van der Put series for $\left\|f\right\|_{q}$ then converges point-wise to $\left\|f\right\|_{q}$. Thus, we need only upgrade the convergence from point-wise to uniform. To do this, we show that the sequence $\left\{ \left\|f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}\right\} _{n\geq1}$ is uniformly Cauchy. Let $m$ and $n$ be arbitrary positive integers. Then, by the reverse triangle inequality for the $q$-adic absolute value: \begin{equation} \left\|\left\|f\left(\left[\mathfrak{z}\right]_{p^{m}}\right)\right\|_{q}-\left\|f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}\right\|\leq\left\|f\left(\left[\mathfrak{z}\right]_{p^{m}}\right)-f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q} \end{equation} Since the van der Put series for $f$ converges uniformly in $K$ over $\mathbb{Z}_{p}$, the $f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$s are therefore uniformly Cauchy, and hence, so too are the $\left\|f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}$s. This establishes the desired uniform convergence. Q.E.D. \vphantom{} With the van der Put basis, we can completely characterize $C\left(\mathbb{Z}_{p},K\right)$ when $K$ is non-archimedean. \begin{thm} \label{thm:C(Z_p,K) is iso to c_0 K}If $K$ is non-archimedean, then the Banach space $C\left(\mathbb{Z}_{p},K\right)$ is isometrically isomorphic to $c_{0}\left(K\right)$ (the space of sequences in $K$ that converge to $0$ in $K$). \end{thm} Proof: Let $L:C\left(\mathbb{Z}_{p},K\right)\rightarrow c_{0}\left(K\right)$ be the map which sends every $f\in C\left(\mathbb{Z}_{p},K\right)$ to the sequence $\mathbf{c}\left(f\right)=\left(c_{0}\left(f\right),c_{1}\left(f\right),\ldots\right)$ whose entries are the van der Put coefficients of $f$. Since: \begin{equation} c_{n}\left(\alpha f+\beta g\right)=\alpha f\left(n\right)+\beta g\left(n\right)-\left(\alpha f\left(n_{-}\right)+\beta g\left(n_{-}\right)\right)=\alpha c_{n}\left(f\right)+\beta c_{n}\left(g\right) \end{equation} for all $\alpha,\beta\in K$ and all $f,g\in C\left(\mathbb{Z}_{p},K\right)$, we have that $L$ is linear. By the \textbf{van der Put Basis Theorem} (\textbf{Theorem \ref{thm:vdP basis theorem}}), for any $\mathbf{c}=\left(c_{0},c_{1},\ldots\right)$ in $c_{0}\left(K\right)$, the van der Put series $\sum_{n=0}^{\infty}c_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]$ is an element of $C\left(\mathbb{Z}_{p},K\right)$ which $L$ sends to $\mathbf{c}$; thus, $L$ is surjective. All that remains is to establish norm-preservation; injectivity will then follow automatically. Since the norm on $c_{0}\left(K\right)$ is the supremum norm: \begin{equation} \left\Vert \mathbf{c}\right\Vert _{K}=\sup_{n\geq0}\left\|c_{n}\right\|_{K},\textrm{ }\forall\mathbf{c}\in c_{0}\left(K\right) \end{equation} observe that: \begin{equation} \left\Vert L\left\{ f\right\} \right\Vert _{p,K}=\left\Vert \mathbf{c}\left(f\right)\right\Vert _{K}=\sup_{n\geq0}\left\|c_{n}\left(f\right)\right\|_{K}\geq\underbrace{\left\|\sum_{n=0}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\right\|_{K}}_{\left\|f\left(\mathfrak{z}\right)\right\|_{K}} \end{equation} for all $f\in C\left(\mathbb{Z}_{p},K\right)$ and all $\mathfrak{z}\in\mathbb{Z}_{p}$, and hence: \[ \left\Vert L\left\{ f\right\} \right\Vert _{p,K}\geq\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{K}=\left\Vert f\right\Vert _{p,K},\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},K\right) \] Finally, suppose there is an $f\in C\left(\mathbb{Z}_{p},K\right)$ so that the lower bound is strict: \begin{equation} \left\Vert L\left\{ f\right\} \right\Vert _{p,K}>\left\Vert f\right\Vert _{p,K} \end{equation} Since $\lim_{n\rightarrow\infty}\left\|c_{n}\left(f\right)\right\|_{K}=0$ the principles of ultrametric analysis (see page \pageref{fact:Principles of Ultrametric Analysis}) guarantee the existence of a $\nu\geq0$ so that $\sup_{n\geq0}\left\|c_{n}\left(f\right)\right\|_{K}=\left\|c_{\nu}\left(f\right)\right\|_{K}$, and hence, so that: \begin{equation} \left\Vert L\left\{ f\right\} \right\Vert _{p,K}=\left\|c_{\nu}\left(f\right)\right\|_{K}=\left\|f\left(\nu\right)-f\left(\nu_{-}\right)\right\|_{K} \end{equation} Thus, our assumption $\left\Vert L\left\{ f\right\} \right\Vert _{p,K}>\left\Vert f\right\Vert _{p,K}$ forces: \begin{equation} \sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{K}<\left\|f\left(\nu\right)-f\left(\nu_{-}\right)\right\|_{K} \end{equation} In particular, we have that the right-hand side is strictly greater than both $\left\|f\left(\nu\right)\right\|_{K}$ and $\left\|f\left(\nu_{-}\right)\right\|_{K}$; that is: \begin{equation} \max\left\{ \left\|f\left(\nu\right)\right\|_{K},\left\|f\left(\nu_{-}\right)\right\|_{K}\right\} <\left\|f\left(\nu\right)-f\left(\nu_{-}\right)\right\|_{K} \end{equation} However, the strong triangle inequality allows us to write: \begin{equation} \max\left\{ \left\|f\left(\nu\right)\right\|_{K},\left\|f\left(\nu_{-}\right)\right\|_{K}\right\} <\left\|f\left(\nu\right)-f\left(\nu_{-}\right)\right\|_{K}\leq\max\left\{ \left\|f\left(\nu\right)\right\|_{K},\left\|f\left(\nu_{-}\right)\right\|_{K}\right\} \end{equation} which is impossible. Thus, no such $f$ exists, and it must be that $\left\Vert L\left\{ f\right\} \right\Vert _{p,K}=\left\Vert f\right\Vert _{p,K}$, for all $f\in C\left(\mathbb{Z}_{p},K\right)$, and $L$ is therefore an isometry. Q.E.D. \begin{cor} Every $f\in C\left(\mathbb{Z}_{p},K\right)$ is uniquely representable as a van der Put series. \end{cor} Proof: Because map $L$ from the proof of \ref{thm:C(Z_p,K) is iso to c_0 K} is an isometry, it is necessarily injective. Q.E.D. \begin{cor} If $K$ is spherically incomplete, then $C\left(\mathbb{Z}_{p},K\right)$ is a reflexive Banach space, and its dual, is then isometrically isomorphic to $\ell^{\infty}\left(K\right)$. \end{cor} Proof: Since \textbf{Theorem \ref{thm:C(Z_p,K) is iso to c_0 K}} shows that $C\left(\mathbb{Z}_{p},K\right)$ is isometrically isomorphic to $c_{0}\left(K\right)$, by \textbf{van der Put's Theorem} (\textbf{Theorem \ref{thm:c_o and ell_infinit are each others duals in a spherically incomplete NA field}}, page \pageref{thm:c_o and ell_infinit are each others duals in a spherically incomplete NA field}), the spherical incompleteness of $K$ then guarantees that the Banach space $C\left(\mathbb{Z}_{p},K\right)$ is reflexive, and that its dual is isometrically isomorphic to $\ell^{\infty}\left(K\right)$. Q.E.D. \begin{thm} The span of the van der Put basis is dense in $B\left(\mathbb{Z}_{p},K\right)$, the space of bounded functions $f:\mathbb{Z}_{p}\rightarrow K$. \end{thm} Proof: The span (finite $K$-linear combinations) of the indicator functions: \[ \left\{ \left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]:n\geq0,k\in\left\{ 0,\ldots,p^{n}-1\right\} \right\} \] is precisely the set of locally constant functions $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$. Since each of these indicator functions is continuous, it can be approximated in supremum norm to arbitrary accuracy by the span of the van der Put basis functions. Since the locally constant functions are dense in $B\left(\mathbb{Z}_{p},K\right)$, we have that the $\left\{ \left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]:n\geq0,k\in\left\{ 0,\ldots,p^{n}-1\right\} \right\} $s are dense in $B\left(\mathbb{Z}_{p},K\right)$. This proves the van der Put basis is dense in $B\left(\mathbb{Z}_{p},K\right)$. Q.E.D. \subsection{\label{subsec:3.1.4. The--adic-Fourier}The $\left(p,q\right)$-adic Fourier Transform\index{Fourier!transform}} \begin{defn} A \textbf{$\left(p,q\right)$-adic Fourier series}\index{Fourier!series}\textbf{ }is a sum of the form: \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:General Form of a (p,q)-adic Fourier Series} \end{equation} where $\hat{f}$ is any function $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$, and where $\mathfrak{z}$ is a $p$-adic variable (in $\mathbb{Z}_{p}$, or, more generally, $\mathbb{Q}_{p}$). We say the Fourier series is\textbf{ convergent at $\mathfrak{z}_{0}\in\mathbb{Q}_{p}$ }whenever the sum $f\left(\mathfrak{z}_{0}\right)$ converges in $\mathbb{C}_{q}$. As per our abuse of notation, we interpret $e^{2\pi i\left\{ t\mathfrak{z}_{0}\right\} _{p}}$ as a function of $t\in\hat{\mathbb{Z}}_{p}$ which, for each $t$, outputs a certain $p$-power root of unity in $\mathbb{C}_{q}$ in accordance with a fixed embedding of $\overline{\mathbb{Q}}$ in both $\mathbb{C}_{q}$ and $\mathbb{C}$. Additionally, we say the Fourier series \textbf{converges} \textbf{(point-wise) }on a set $U\subseteq\mathbb{Q}_{p}$ if it converges at each point in $U$. \end{defn} \vphantom{} For us, the most important Fourier series is the one for indicator functions of sets of the form $\mathfrak{a}+p^{n}\mathbb{Z}_{p}$, where $\mathfrak{a}\in\mathbb{Z}_{p}$ and $n\in\mathbb{N}_{0}$ \begin{prop} \label{prop:Indicator for p-adic coset Fourier identity}Fix $\mathfrak{a}\in\mathbb{Z}_{p}$ and $n\in\mathbb{Z}_{0}$. Then, Fourier series of the indicator function $\left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{a}\right]$ is: \begin{equation} \left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{a}\right]\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p^{n}}\sum_{k=0}^{p^{n}-1}e^{2\pi i\left\{ k\frac{\mathfrak{z}-\mathfrak{a}}{p^{n}}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Fourier Identity for indicator functions} \end{equation} \end{prop} \begin{rem} This can also be written as: \begin{equation} \left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{a}\right]\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p^{n}}\sum_{\left\|t\right\|_{p}\leq p^{n}}e^{2\pi i\left\{ t\left(\mathfrak{z}-\mathfrak{a}\right)\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} \end{rem} \begin{rem} As indicated by the $\overline{\mathbb{Q}}$ over the equality, the above identity is valid in $\overline{\mathbb{Q}}$, and hence, in any field extension of $\overline{\mathbb{Q}}$, such as $\mathbb{C}$, or $\mathbb{C}_{q}$ for any prime $q$. \end{rem} Proof: \[ \frac{1}{p^{n}}\sum_{k=0}^{p^{n}-1}e^{2\pi i\left\{ k\frac{\mathfrak{z}-\mathfrak{a}}{p^{n}}\right\} _{p}}=\frac{1}{p^{n}}\sum_{k=0}^{p^{n}-1}e^{2\pi ik\left[\mathfrak{z}-\mathfrak{a}\right]_{p^{n}}/p^{n}}=\begin{cases} 1 & \textrm{if }\left[\mathfrak{z}-\mathfrak{a}\right]_{p^{n}}=0\\ 0 & \textrm{else} \end{cases} \] The formula follows from the fact that $\left[\mathfrak{z}-\mathfrak{a}\right]_{p^{n}}=0$ is equivalent to $\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{a}$. Q.E.D. \begin{prop} A $\left(p,q\right)$-adic Fourier series $\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ converges in $\mathbb{C}_{q}$ uniformly with respect to $\mathfrak{z}\in\mathbb{Q}_{p}$ if and only if $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$; that is, if and only if: \begin{equation} \lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} \end{prop} Proof: Consider an arbitrary $\left(p,q\right)$-adic Fourier series $\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$. Due to the ultrametric topology of $\mathbb{C}_{q}$, the series will converge at any given $\mathfrak{z}\in\mathbb{Q}_{p}$ if and only if: \begin{equation} \lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} In our abuse of notation, we have that $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ is a $p$-power root of unity in $\mathbb{C}_{q}$ for all $t\in\hat{\mathbb{Z}}_{p}$ and all $\mathfrak{z}\in\mathbb{Q}_{p}$; consequently, $\left\|e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}=1$ for all $t$ and $\mathfrak{z}$. Hence: \begin{equation} \lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\overset{\mathbb{R}}{=}\lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)\right\|_{q},\textrm{ }\forall\mathfrak{z}\in\mathbb{Q}_{p} \end{equation} Q.E.D. \vphantom{} Before we prove more on convergence, existence, or uniqueness, let us give the formal definition for what the Fourier coefficients of $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ \emph{should} be, if they happened to exist. \begin{defn} \label{def:pq adic Fourier coefficients}For a function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, for each $t\in\hat{\mathbb{Z}}_{p}$, we define the $t$th $\left(p,q\right)$\textbf{-adic Fourier coefficient }of\index{Fourier!coefficients} $f$, denoted $\hat{f}\left(t\right)$, by the rule: \begin{equation} \hat{f}\left(t\right)\overset{\mathbb{C}_{q}}{=}\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}\label{eq:Definition of (p,q)-adic Fourier Coefficients} \end{equation} provided that the sum on the right is convergent in $\mathbb{C}_{q}$. \end{defn} \vphantom{} The existence criteria for $\hat{f}\left(t\right)$ are extremely simple: \begin{lem} Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be a function. Then, the series defining $\hat{f}\left(t\right)$ will converge in $\mathbb{C}_{q}$ uniformly with respect to $t\in\hat{\mathbb{Z}}_{p}$ if and only if: \begin{equation} \lim_{n\rightarrow\infty}\left\|\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} In particular, the uniform convergence of $\hat{f}\left(t\right)$ for all values of $t$ will occur if and only if $f$ is continuous. \end{lem} Proof: $\mathbb{C}_{q}$'s non-archimedean topology guarantees that: \begin{equation} \hat{f}\left(t\right)\overset{\mathbb{C}_{q}}{=}\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it} \end{equation} converges if and only if: \begin{equation} \lim_{n\rightarrow\infty}\left\|\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}\right\|_{q}\overset{\mathbb{R}}{=}\lim_{n\rightarrow\infty}\left\|\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} Q.E.D. \begin{defn} We write \nomenclature{$c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$}{set of functions $\hat{f}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ for which $\lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)\right\|_{q}\overset{\mathbb{R}}{=}0$}$c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ to denote the $\mathbb{C}_{q}$-vector space of all functions $\hat{f}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ for which: \begin{equation} \lim_{n\rightarrow\infty}\max_{\left\|t\right\|_{p}=p^{n}}\left\|\hat{f}\left(t\right)\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} We make $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ into a non-archimedean Banach space by equipping it with the norm: \begin{equation} \left\Vert \hat{f}\right\Vert _{p,q}\overset{\textrm{def}}{=}\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{f}\left(t\right)\right\|_{q} \end{equation} \end{defn} \begin{thm} \label{thm:formula for Fourier series}Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be a function. Then: \vphantom{} I. $f$ admits a uniformly convergent $\left(p,q\right)$-adic Fourier series if and only if $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. In particular, $f$ is continuous if and only if $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. \vphantom{} II. If $f$ admits a uniformly convergent $\left(p,q\right)$-adic Fourier series, the series is unique, and is given by: \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} \end{thm} Proof: Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be a function. For the moment, let us proceed formally. Using (\ref{eq:Fourier Identity for indicator functions}), we can re-write the crossed van der Put series of $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ as a Fourier series: \begin{align} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}c_{0}\left(f\right)+\sum_{n=1}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\\ & =c_{0}\left(f\right)+\sum_{n=1}^{\infty}c_{n}\left(f\right)\frac{1}{p^{\lambda_{p}\left(n\right)}}\sum_{k=0}^{p^{\lambda_{p}\left(n\right)}-1}\exp\left(2k\pi i\left\{ \frac{\mathfrak{z}-n}{p^{\lambda_{p}\left(n\right)}}\right\} _{p}\right)\\ & =c_{0}\left(f\right)+\sum_{n=1}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\sum_{k=0}^{p^{\lambda_{p}\left(n\right)}-1}\exp\left(-2n\pi i\frac{k}{p^{\lambda_{p}\left(n\right)}}\right)\exp\left(2\pi i\left\{ \frac{k\mathfrak{z}}{p^{\lambda_{p}\left(n\right)}}\right\} _{p}\right) \end{align} Now, observe that for any function $g$: \begin{align} \sum_{k=0}^{p^{m}-1}g\left(\frac{k}{p^{m}}\right) & =\sum_{k=0}^{p^{m-1}-1}g\left(\frac{k}{p^{m-1}}\right)+\sum_{k\in\left(\mathbb{Z}/p^{m}\mathbb{Z}\right)^{\times}}g\left(\frac{k}{p^{m}}\right)\label{eq:Prufer-group sum decomposition - inductive step} \end{align} As such, by induction: \begin{align} \sum_{k=0}^{p^{m}-1}g\left(\frac{k}{p^{m}}\right) & =g\left(0\right)+\sum_{j=1}^{m}\sum_{k\in\left(\mathbb{Z}/p^{j}\mathbb{Z}\right)^{\times}}g\left(\frac{k}{p^{j}}\right),\textrm{ }\forall m\in\mathbb{N}_{1}\label{eq:Prufer-group sum decomposition} \end{align} The sum on the right is taken over all irreducible non-integer fractions whose denominator is a divisor of $p^{m}$. Adding $0$ into the mix, we see that the sum on the right is then exactly the same as summing $g\left(t\right)$ over every element of $\hat{\mathbb{Z}}_{p}$ whose $p$-adic magnitude is $\leq p^{m}$. Hence: \begin{equation} \sum_{k=0}^{p^{m}-1}g\left(\frac{k}{p^{m}}\right)=\sum_{\left\|t\right\|_{p}\leq p^{m}}g\left(t\right)\label{eq:Prufer-group sum decomposition - Re-indexing} \end{equation} and so: \[ \sum_{k=0}^{p^{\lambda_{p}\left(n\right)}-1}\exp\left(-2n\pi i\frac{k}{p^{\lambda_{p}\left(n\right)}}\right)\exp\left(2\pi i\left\{ \frac{k\mathfrak{z}}{p^{\lambda_{p}\left(n\right)}}\right\} _{p}\right)=\sum_{\left\|t\right\|_{p}\leq p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \] and so: \begin{align} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}c_{0}\left(f\right)+\sum_{n=1}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\sum_{\left\|t\right\|_{p}\leq p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =c_{0}\left(f\right)+\sum_{n=1}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left(1+\sum_{0<\left\|t\right\|_{p}\leq p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right)\\ & =\underbrace{c_{0}\left(f\right)+\sum_{n=1}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}}_{\sum_{n=0}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}}+\sum_{n=1}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\sum_{0<\left\|t\right\|_{p}\leq p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} Next, observe that a given $t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} $ will occur in the innermost summand if and only if: \begin{align} p^{\lambda_{p}\left(n\right)} & \geq\left\|t\right\|_{p}\\ & \Updownarrow\\ n & \geq p^{-\nu_{p}\left(t\right)-1}=\frac{1}{p}\left\|t\right\|_{p} \end{align} So, re-indexing in terms of $t$, we have that: \begin{align} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\sum_{n=0}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}+\sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{t\in\hat{\mathbb{Z}}_{p}}\underbrace{\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}}_{\hat{f}\left(t\right)}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} As such, if $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, then: \[ S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \] converges in $\mathbb{C}_{q}$ uniformly over $\mathbb{Z}_{p}$ and is equal to $f$, which is necessarily uniformly continuous. Conversely, if $f$ is continuous (which then makes $f$ uniformly continuous, since its domain is compact), then all of the above formal computations count as grouping and re-ordering of the uniformly convergent series: \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{n=0}^{\infty}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} This proves that: \begin{equation} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} converges in $\mathbb{C}_{q}$ uniformly over $\mathbb{Z}_{p}$, which then forces $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Uniqueness follows from the uniqueness of the van der Put coefficients of a continuous $\left(p,q\right)$-adic function. Q.E.D. \begin{cor} \label{cor:pq adic Fourier transform is an isometric isomorphism}The $\left(p,q\right)$-adic Fourier transform: \begin{equation} f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\mapsto\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right) \end{equation} is an isometric isomorphism of the Banach spaces $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. \end{cor} Proof: \textbf{Theorem \ref{thm:formula for Fourier series} }shows that the Fourier transform is a bijection between $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. The linearity of the Fourier transform then shows that it is an isomorphism of vector space (a.k.a., linear spaces). As for continuity, let $\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ be arbitrary. Then, we can write: \begin{align} \left\Vert \hat{f}\right\Vert _{p,q} & =\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}\right\|_{q}\\ & \leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\sup_{n\geq\frac{1}{p}\left\|t\right\|_{p}}\left\|\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it}\right\|_{q}\\ & =\sup_{t\in\hat{\mathbb{Z}}_{p}}\sup_{n\geq\frac{1}{p}\left\|t\right\|_{p}}\left\|c_{n}\left(f\right)\right\|_{q}\\ & =\sup_{n\geq0}\left\|c_{n}\left(f\right)\right\|_{q} \end{align} For $n\geq1$, $c_{n}\left(f\right)=f\left(n\right)-f\left(n_{-}\right)$. As such, the ultrametric inequality implies that: \[ \left\Vert \hat{f}\right\Vert _{p,q}\leq\sup_{n\geq0}\left\|c_{n}\left(f\right)\right\|_{q}\leq\sup_{n\geq0}\left\|f\left(n\right)\right\|_{q}\leq\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}=\left\Vert f\right\Vert _{p,q} \] On the other hand, if $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, then: \begin{align} \left\Vert f\right\Vert _{p,q} & =\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi it\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\\ & \leq\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{f}\left(t\right)\right\|_{q}\\ & \leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{f}\left(t\right)\right\|_{q}\\ & =\left\Vert \hat{f}\right\Vert _{p,q} \end{align} Since the inequality holds in both directions, this forces $\left\Vert f\right\Vert _{p,q}=\left\Vert \hat{f}\right\Vert _{p,q}$. Thus, the $\left(p,q\right)$-adic Fourier transform is an isometric isomorphism, as desired. Q.E.D. \subsection{\label{subsec:3.1.5-adic-Integration-=00003D000026}$\left(p,q\right)$-adic Integration and the Fourier-Stieltjes Transform} Because we are focusing on the concrete case of functions on $\mathbb{Z}_{p}$ taking values in a non-archimedean field $K$ with residue field of characteristic $q\neq p$, the van der Put basis for $C\left(\mathbb{Z}_{p},K\right)$ allows us to explicitly define and compute integrals and measures in terms of their effects on this basis, and\textemdash equivalently\textemdash in terms of the $\left(p,q\right)$-adic characters $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$. \begin{defn} A\textbf{ $\left(p,q\right)$-adic measure}\index{measure!left(p,qright)-adic@$\left(p,q\right)$-adic}\index{measure}\index{$p,q$-adic!measure} $d\mu$ is a continuous $K$-valued linear functional $C\left(\mathbb{Z}_{p},K\right)$; that is, $d\mu\in C\left(\mathbb{Z}_{p},K\right)^{\prime}$. Given a function $f\in C\left(\mathbb{Z}_{p},K\right)$, we write: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \end{equation} to denote the image of $f$ under the linear functional $d\mu$. We call this the \textbf{integral }of $f$ with respect to $d\mu$. \vphantom{} Like one would expect, we can multiply measures by continuous functions. Given $g\in C\left(\mathbb{Z}_{p},K\right)$ and a measure $d\mu$, we define the measure $d\nu=gd\mu$ by the rule: \begin{equation} \nu\left(f\right)\overset{K}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\nu\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\mu\left(f\times g\right) \end{equation} This definition makes sense since the continuity of $f$ and $g$ guarantees the continuity of their product. \vphantom{} Finally, we say a measure $d\mu$ is\index{translation invariance} \textbf{translation invariant }whenever: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}+\mathfrak{a}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},K\right),\textrm{ }\forall\mathfrak{a}\in\mathbb{Z}_{p} \end{equation} Note that the zero-measure (which sends every function to $0$) is translation-invariant, so this notion is not vacuous. \end{defn} \begin{prop}[\textbf{Integral-Series Interchange}] \label{prop:pq-adic integral interchange rule}Let $d\mu$ be a $\left(p,q\right)$-adic measure, and let $\left\{ F_{N}\right\} _{N\geq1}$ be a convergent sequence in $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \begin{equation} \lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}F_{N}\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}\left(\lim_{N\rightarrow\infty}F_{N}\left(\mathfrak{z}\right)\right)d\mu\left(\mathfrak{z}\right)\label{eq:pq adic integral interchange rule} \end{equation} Equivalently, limits and integrals can be interchanged whenever the limit converges uniformly. \end{prop} Proof: This follows immediately from the definition of $\left(p,q\right)$-adic measures as continuous linear functionals on $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Q.E.D. \begin{prop}[\textbf{Integration term-by-term}] \label{prop:term-by-term evaluation of integrals}Let $d\mu$ be a $\left(p,q\right)$-adic measure, and let $f\in C\left(\mathbb{Z}_{p},K\right)$. I. The integral of $f$ can be evaluated term-by term using its van der Put series \index{van der Put!series}: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\sum_{n=0}^{\infty}c_{n}\left(f\right)\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mu\left(\mathfrak{z}\right)\label{eq:Integration in terms of vdP basis} \end{equation} II. The integral of $f$ can be evaluated term-by term using its Fourier series: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)\int_{\mathbb{Z}_{p}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mu\left(\mathfrak{z}\right),\forall f\in C\left(\mathbb{Z}_{p},K\right)\label{eq:Integration in terms of Fourier basis} \end{equation} \end{prop} Proof: Let $f$ and $d\mu$ be as given. By \textbf{Theorems \ref{thm:vdP basis theorem}} and \textbf{\ref{thm:formula for Fourier series}}, the continuity of $f$ guarantees the uniform convergence of the van der Put and Fourier series of $f$, respectively. \textbf{Proposition \ref{prop:pq-adic integral interchange rule}} then justifies the interchanges of limits and integrals in (I) and (II). Q.E.D. \vphantom{} Because of the van der Put basis, every $\left(p,q\right)$-adic measure is uniquely determined by what it does to the $n$th van der Put basis function. On the other hand, since every $f\in C\left(\mathbb{Z}_{p},K\right)$ can also be represented by a uniformly continuous $\left(p,q\right)$-adic Fourier series, a measure is therefore uniquely determined by what it does to the $\left(p,q\right)$-adic characters $e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$. This leads to Fourier-Stieltjes transform of a $\left(p,q\right)$-adic measure. \begin{defn} Let $d\mu$ be a $\left(p,q\right)$-adic measure. Then, the \textbf{Fourier-Stieltjes transform}\index{Fourier-Stieltjes transform} of $d\mu$, denoted $\hat{\mu}$, is the function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ defined by: \begin{equation} \hat{\mu}\left(t\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mu\left(\mathfrak{z}\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier-Stieltjes transform of a measure} \end{equation} where, for each $t$, $\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mu\left(\mathfrak{z}\right)$ denotes the image of the function $e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=e^{2\pi i\left\{ \left(-t\right)\mathfrak{z}\right\} _{p}}$ under $d\mu$. \end{defn} \begin{rem} \label{rem:parseval-plancherel formula for integration against measures}In this notation, (\ref{eq:Integration in terms of Fourier basis}) can be written as: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)\hat{\mu}\left(-t\right)\overset{K}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\label{eq:Integration in terms of Fourier-Stieltjes coefficients} \end{equation} This formula will be \emph{vital }for studying $\chi_{H}$. \end{rem} \begin{thm} \label{thm:FST is an iso from measures to ell infinity}The Fourier-Stieltjes transform defines an isometric isomorphism from the Banach space $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$ onto the Banach space $B\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ of bounded $\mathbb{C}_{q}$-valued functions on $\hat{\mathbb{Z}}_{p}$. \end{thm} \begin{rem} The more general version of this result is so significant to non-archimedean functional analysis that van Rooij put it on the cover of his book \cite{van Rooij - Non-Archmedean Functional Analysis}! \end{rem} Proof: By \textbf{Corollary \ref{cor:pq adic Fourier transform is an isometric isomorphism}}, Fourier transform is an isometric isomorphism of the Banach spaces $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. By \textbf{Theorem \ref{thm:c_o and ell_infinit are each others duals in a spherically incomplete NA field}}, since $\hat{\mathbb{Z}}_{p}$ is countable and $\mathbb{C}_{q}$ is spherically incomplete, $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$'s dual is then $B\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. The isometric isomorphism the Fourier transform between the base spaces $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ then indices an isometric isomorphism between their duals, as desired. Q.E.D. \begin{prop} \label{prop:Fourier transform of haar measure}Let $d\mu,d\nu\in C\left(\mathbb{Z}_{p},K\right)^{\prime}$ be the measures defined by: \begin{equation} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]d\mu\left(\mathfrak{z}\right)=\frac{1}{p^{n}},\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall k\in\mathbb{Z} \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\nu\left(\mathfrak{z}\right)=\mathbf{1}_{0}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} Then, $d\mu=d\nu$. \end{prop} Proof: Direct computation using the formula: \begin{equation} \left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]=\frac{1}{p^{n}}\sum_{\left\|t\right\|_{p}\leq p^{n}}e^{-2\pi i\left\{ t\left(\mathfrak{z}-k\right)\right\} _{p}} \end{equation} from \textbf{Proposition \ref{prop:Indicator for p-adic coset Fourier identity}}. Q.E.D. \begin{defn} Following \textbf{Proposition \ref{prop:Fourier transform of haar measure}}, I write $d\mathfrak{z}$ to denote the $\left(p,q\right)$-adic measure defined by: \begin{equation} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]d\mathfrak{z}\overset{\textrm{def}}{=}\frac{1}{p^{n}},\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall k\in\mathbb{Z}\label{eq:Definition of (p,q)-adic Haar probability measure of k + p^n Z_p} \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z}\overset{\textrm{def}}{=}\mathbf{1}_{0}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of the Fourier-Stieltjes transform of the (p,q)-adic Haar probability measure} \end{equation} I call $d\mathfrak{z}$ the \textbf{$\left(p,q\right)$-adic Haar (probability) measure}\index{measure!Haar}. Note that this is the \emph{unique} measure satisfying the above two identities. \end{defn} \begin{thm} The set of \index{translation invariance}translation-invariant $\left(p,q\right)$-adic measures is the one-dimensional subspace of $B\left(\hat{\mathbb{Z}}_{p},K\right)$ spanned by $d\mathfrak{z}$. Additionally, $d\mathfrak{z}$ is the unique translation-invariant $\left(p,q\right)$-adic measure satisfying the unit normalization condition: \begin{equation} \int_{\mathbb{Z}_{p}}d\mathfrak{z}\overset{K}{=}1 \end{equation} \end{thm} Proof: First, let $d\mu=\mathfrak{c}d\mathfrak{z}$ for some $\mathfrak{c}\in K$. Then, letting $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $\mathfrak{a}\in\mathbb{Z}_{p}$ be arbitrary: \begin{align} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}+\mathfrak{a}\right)d\mu\left(\mathfrak{z}\right) & =\mathfrak{c}\int_{\mathbb{Z}_{p}}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\left(\mathfrak{z}+\mathfrak{a}\right)\right\} _{p}}d\mathfrak{z}\\ \left(\textrm{\textbf{Proposition }\textbf{\ref{prop:pq-adic integral interchange rule}}}\right); & =\mathfrak{c}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{a}\right\} _{p}}\int_{\mathbb{Z}_{p}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z}\\ & =\mathfrak{c}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{a}\right\} _{p}}\mathbf{1}_{0}\left(-t\right)\\ & =\mathfrak{c}\hat{f}\left(0\right)\\ & =\mathfrak{c}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\\ & =\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \end{align} So, every scalar multiple of $d\mathfrak{z}$ is translation-invariant. Next, let $d\mu$ be translation-invariant. Then, for all $t\in\hat{\mathbb{Z}}_{p}$: \begin{equation} \hat{\mu}\left(t\right)=\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mu\left(\mathfrak{z}\right)=\int_{\mathbb{Z}_{p}}e^{-2\pi i\left\{ t\left(\mathfrak{z}+1\right)\right\} _{p}}d\mu\left(\mathfrak{z}\right)=e^{-2\pi it}\hat{\mu}\left(t\right) \end{equation} Since $\hat{\mu}\left(t\right)\in\mathbb{C}_{q}$ for all $t$, the above forces $\hat{\mu}\left(t\right)=0$ for all $t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} $. As such: \[ \hat{\mu}\left(t\right)=\hat{\mu}\left(0\right)\mathbf{1}_{0}\left(t\right) \] which shows $d\mu\left(\mathfrak{z}\right)=\hat{\mu}\left(0\right)d\mathfrak{z}$. Thus, $d\mu$ is in the span of $d\mathfrak{z}$. Hence, every translation-invariant measure is a scalar multiple of $d\mathfrak{z}$. Finally, since $\int_{\mathbb{Z}_{p}}d\mu\left(\mathfrak{z}\right)=\hat{\mu}\left(0\right)$, we have that $d\mathfrak{z}$ itself is the unique translation-invariant $\left(p,q\right)$-adic measure $d\mu\left(\mathfrak{z}\right)$ satisfying $\int_{\mathbb{Z}_{p}}d\mu\left(\mathfrak{z}\right)=1$. Q.E.D. \begin{lem}[\textbf{Integral Formula for the Fourier Transform}] For\index{Fourier!transform} any $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, its Fourier transform $\hat{f}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ is given by the integral formula: \begin{equation} \hat{f}\left(t\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Integral Formula for the Fourier transform} \end{equation} \end{lem} Proof: Fix an arbitrary $t\in\hat{\mathbb{Z}}_{p}$. Then, by definition, (\ref{eq:Integral Formula for the Fourier transform}) is the image of the continuous $\left(p,q\right)$-adic function $f\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ under the $\left(p,q\right)$-adic Haar measure. Expressing $f$ as a Fourier series, we have that: \[ e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}f\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(s\right)e^{2\pi i\left\{ s\mathfrak{z}\right\} _{p}}=\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(s\right)e^{2\pi i\left\{ \left(s-t\right)\mathfrak{z}\right\} _{p}} \] for all $\mathfrak{z}\in\mathbb{Z}_{p}$. Thus, by linearity, since the Fourier series is uniformly convergent in $\mathbb{C}_{q}$, \textbf{Proposition \ref{prop:term-by-term evaluation of integrals}} justifies integrating term-by-term: \begin{align} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z} & =\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(s\right)\int_{\mathbb{Z}_{p}}e^{2\pi i\left\{ \left(s-t\right)\mathfrak{z}\right\} _{p}}d\mathfrak{z}\\ & =\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(s\right)\underbrace{\mathbf{1}_{0}\left(s-t\right)}_{1\textrm{iff }s=t}\\ & =\hat{f}\left(t\right) \end{align} which gives the desired result. Q.E.D. \vphantom{} With this integral formula, we then get the usual integral formulas for convolution\index{convolution!of left(p,qright)-adic functions@of $\left(p,q\right)$-adic functions}, and they behave exactly as we would expect them to behave. \begin{defn}[\textbf{Convolution}] We define the \textbf{convolution }of functions on $\mathbb{Z}_{p}$ and $\hat{\mathbb{Z}}_{p}$, respectively, by: \begin{equation} \left(fg\right)\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}-\mathfrak{y}\right)g\left(\mathfrak{y}\right)d\mathfrak{y},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\forall f,g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Continuous Convolution Definition} \end{equation} \begin{equation} \left(\hat{f}\hat{g}\right)\left(t\right)\overset{\mathbb{C}_{q}}{=}\sum_{\tau\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t-\tau\right)\hat{g}\left(\tau\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p},\forall\hat{f},\hat{g}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)\label{eq:Discrete Convolution Definition} \end{equation} \end{defn} \begin{thm}[\textbf{The} \textbf{Convolution Theorem}] \label{thm:Convolution Theorem}\index{convolution!theorem}For $f,g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$: \begin{align} \widehat{\left(fg\right)}\left(t\right) & =\hat{f}\left(t\right)\hat{g}\left(t\right)\label{eq:Convolution Theorem 1}\\ \widehat{\left(fg\right)}\left(t\right) & =\left(\hat{f}\hat{g}\right)\left(t\right)\label{eq:Convolution Theorem 2} \end{align} \end{thm} \vphantom{} We can also take convolutions of measures: \begin{defn} Let $d\mu,d\nu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$ be $\left(p,q\right)$-adic measures. Then, the \textbf{convolution }of $d\mu$ and $d\nu$, denoted $d\mud\nu$, is the measure defined by the Fourier-Stieltjes transform: \begin{equation} \widehat{\left(d\mud\nu\right)}\left(t\right)\overset{\textrm{def}}{=}\hat{\mu}\left(t\right)\hat{\nu}\left(t\right)\label{eq:Definition of the convolution of two pq adic measures} \end{equation} That is: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\left(d\mud\nu\right)\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\hat{\nu}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Definition of the action of the convolution of pq adic measures on a function} \end{equation} \end{defn} \vphantom{} Next up, \textbf{Parseval-Plancherel Identity}:\index{Parseval-Plancherel Identity} \begin{thm}[\textbf{The Parseval-Plancherel Identity}] \label{thm:Parseval-Plancherel Identity}Let $f,g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \end{thm} \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)\hat{g}\left(-t\right),\textrm{ }\forall f,g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Parseval-Plancherel Identity} \end{equation} \vphantom{} In all of these formulae, convergence, rearrangements, and interchanges are justified by the $q$-adic decay of $\hat{f}$ and $\hat{g}$ which is guaranteed to occur thanks to the continuity of $f$ and $g$. \begin{prop} Let $g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Viewing $g$ as the measure: \[ f\mapsto\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mathfrak{z} \] we can then define the convolution of $g$ and $d\mu$ as the measure with the $\left(p,q\right)$-adic Fourier-Stieltjes transform: \[ \widehat{\left(gd\mu\right)}\left(t\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{g}\left(t\right)\hat{\mu}\left(t\right) \] The measure is $fd\mu$ is then absolutely continuous with respect to the $\left(p,q\right)$-adic Haar measure $d\mathfrak{z}$, meaning that there is a continuous $\left(p,q\right)$-adic function $g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ so that: \[ \left(fd\mu\right)\left(\mathfrak{z}\right)=g\left(\mathfrak{z}\right)d\mathfrak{z} \] In fact, we have that $\hat{g}\left(t\right)=\hat{f}\left(t\right)\hat{\mu}\left(t\right)$, and hence, we can\footnote{That is, the convolution of a continuous $\left(p,q\right)$-adic function and a $\left(p,q\right)$-adic measure is a continuous function.} view $fd\mu$ as a continuous function $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$. \end{prop} Proof: Since $f$ is continuous, $\left\|\hat{f}\left(t\right)\right\|_{q}\rightarrow0$ as $\left\|t\right\|_{p}\rightarrow\infty$. Since $d\mu$ is a measure, $\left\Vert \hat{\mu}\right\Vert _{p,q}=\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\mu}\left(t\right)\right\|_{q}<\infty$, hence, defining $\hat{g}\left(t\right)\overset{\textrm{def}}{=}\hat{f}\left(t\right)\hat{\mu}\left(t\right)$, we have that $\hat{g}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, which shows that $\hat{g}$ is the Fourier transform of the continuous $\left(p,q\right)$-adic function: \[ g\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{g}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \] Furthermore, for any $h\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$: \begin{align} \int_{\mathbb{Z}_{p}}h\left(\mathfrak{z}\right)\left(fd\mu\right)\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{h}\left(-t\right)\hat{f}\left(t\right)\hat{\mu}\left(t\right)\\ & \overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{h}\left(-t\right)\hat{g}\left(t\right)\\ & \overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}h\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mathfrak{z} \end{align} Hence, $\left(fd\mu\right)\left(\mathfrak{z}\right)$ and $g\left(\mathfrak{z}\right)d\mathfrak{z}$ are identical as measures. Q.E.D. \vphantom{} For doing computations with $\left(p,q\right)$-adic integrals, the following simple change-of-variables\index{change of variables!affine substitutions} formula will be of the utmost import: \begin{lem}[\textbf{Change of Variables \textendash{} Affine substitutions}] \label{lem:Affine substitution change of variable formula}Let $f\in C\left(\mathbb{Z}_{p},K\right)$, where $K$ is a complete ring extension of $\mathbb{Z}_{q}$ which is itself contained in $\mathbb{C}_{q}$. Then: \begin{equation} \int_{\mathfrak{a}\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}=\int_{p^{v_{p}\left(\mathfrak{a}\right)}\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z},\textrm{ }\forall\mathfrak{a}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} \label{eq:Multiplying domain of integration by a} \end{equation} \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{a}\mathfrak{z}+\mathfrak{b}\right)d\mathfrak{z}\overset{K}{=}\frac{1}{\left\|\mathfrak{a}\right\|_{p}}\int_{\mathfrak{a}\mathbb{Z}_{p}+\mathfrak{b}}f\left(\mathfrak{z}\right)d\mathfrak{z},\textrm{ }\forall\mathfrak{a}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} ,\forall\mathfrak{b}\in\mathbb{Z}_{p}\label{eq:Change of variables formula - affine linear substitutions} \end{equation} \end{lem} \begin{rem} When working with these integrals, a key identity is: \[ \int_{p^{n}\mathbb{Z}_{p}+k}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)=\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \] which holds for all $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, all $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, and all $n,k\in\mathbb{Z}$ with $n\geq0$. \end{rem} Proof: (\ref{eq:Multiplying domain of integration by a}) follows from the fact that $\mathfrak{a}\neq0$ implies there are (unique) $m\in\mathbb{N}_{0}$ and $\mathfrak{u}\in\mathbb{Z}_{p}^{\times}$ so that $\mathfrak{a}=p^{m}\mathfrak{u}$. Since multiplication by $\mathfrak{u}$ is a measure-preserving automorphism of the group $\left(\mathbb{Z}_{p},+\right)$, $\mathfrak{a}\mathbb{Z}_{p}=p^{m}\mathfrak{u}\mathbb{Z}_{p}=p^{m}\mathbb{Z}_{p}$. Here, $m=v_{p}\left(\mathfrak{z}\right)$. For (\ref{eq:Change of variables formula - affine linear substitutions}), by the translation-invariance of $d\mathfrak{z}$, it suffices to prove: \[ \int_{\mathbb{Z}_{p}}f\left(\mathfrak{a}\mathfrak{z}\right)d\mathfrak{z}\overset{K}{=}\frac{1}{\left\|\mathfrak{a}\right\|_{p}}\int_{\mathfrak{a}\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z} \] In fact, using the van der Put basis, we need only verify the formula for the indicator functions $f\left(\mathfrak{z}\right)=\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]$. Because the integral of such an $f$ is rational-valued, note that the computation will be the same for the case of the \emph{real-valued }Haar probability measure $d\mathfrak{z}$ on $\mathbb{Z}_{p}$. For any field extension $\mathbb{F}$ of $\mathbb{Q}$, any translation-invariant linear functional on the space of $\mathbb{Q}$-valued functions on $\mathbb{Z}_{p}$ normalized to send the constant function $1$ to the number $1$ necessarily sends the functions $\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]$ to the number $1/p^{n}$. Thus, the proof for the real-valued case, such as can be found in \cite{Bell - Harmonic Analysis on the p-adics} or \cite{Automorphic Representations} applies, and we are done. Q.E.D. \begin{rem} (\ref{eq:Change of variables formula - affine linear substitutions}) is \emph{also }valid in the case where $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ ($\mathbb{C}$, not $\mathbb{C}_{q}$!) is integrable with respect to the \emph{real-valued }Haar probability measure on $\mathbb{Z}_{p}$. \end{rem} \begin{rem} Seeing as we write $\left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{b}\right]$ to denote the indicator function for the set $\mathfrak{b}+p^{n}\mathbb{Z}_{p}$, we then have the identities: \begin{equation} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}\mathfrak{b}\right]f\left(\mathfrak{z}\right)d\mathfrak{z}=\int_{\mathfrak{b}+p^{n}\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}=\frac{1}{p^{n}}\int_{\mathbb{Z}_{p}}f\left(p^{n}\mathfrak{y}+\mathfrak{b}\right)d\mathfrak{y}\label{eq:change of variable trifecta} \end{equation} \end{rem} \vphantom{} Lastly, we have the basic integral inequalities. Alas, the triangle inequality is not among them. \begin{prop}[\textbf{Integral Inequalities}] \index{triangle inequality!left(p,qright)-adic@$\left(p,q\right)$-adic}Let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \vphantom{} I. \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\leq\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}=\left\Vert f\right\Vert _{p,q}\label{eq:Triangle Inequality for the (p,q)-adic Haar measure} \end{equation} \vphantom{} II. For any measure $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right\|_{q}\leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\right\|_{q}\label{eq:Triangle inequality for an arbitrary (p,q)-adic measure (with Fourier)} \end{equation} \vphantom{} III.\index{triangle inequality!van der Put series} \begin{equation} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\leq\sum_{n=0}^{\infty}\frac{\left\|c_{n}\left(f\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\label{eq:Triangle inequality for Integral of q-adic absolute value} \end{equation} \end{prop} \begin{rem} As shown in \textbf{Example} \textbf{\ref{exa:triangle inequality failure}} (page \pageref{exa:triangle inequality failure}), unlike in real or complex analysis, there is not necessarily any relationship between $\left\|\int f\right\|_{q}$ and $\int\left\|f\right\|_{q}$. \end{rem} Proof: (I) is the consequence of the Theorem from page 281 of \emph{Ultrametric Calculus }(\cite{Ultrametric Calculus}) and the Exercise that occurs immediately after it. (II), meanwhile, is an immediate consequence of (\ref{eq:Integration in terms of Fourier-Stieltjes coefficients}). (III) is the only one that requires an argument. For (III), let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\overset{K}{=}\sum_{n=0}^{\infty}c_{n}\left(f\right)\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z}=\sum_{n=0}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}} \end{equation} where the interchange of sum and integral/linear functional is valid due to the convergence of the series, guaranteed by the $q$-adic decay of the $c_{n}\left(f\right)$s. Now, since $f$ is continuous, $\left\|f\left(\mathfrak{z}\right)\right\|_{q}$ is a continuous function from $\mathbb{Z}_{p}$ to $\mathbb{R}$\textemdash uniformly continuous, in fact, by the compactness of $\mathbb{Z}_{p}$. Thus, by \textbf{Proposition \ref{prop:Convergence of real-valued vdP series}}, $\left\|f\left(\mathfrak{z}\right)\right\|_{q}$'s van der Put series: \begin{equation} \sum_{n=0}^{\infty}c_{n}\left(\left\|f\right\|_{q}\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]=\left\|f\left(0\right)\right\|_{q}+\sum_{n=0}^{\infty}\underbrace{\left(\left\|f\left(n\right)\right\|_{q}-\left\|f\left(n_{-}\right)\right\|_{q}\right)}_{c_{n}\left(\left\|f\right\|_{q}\right)}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} is uniformly convergent in $\mathbb{R}$. As such, when integrating it with respect to the real-valued $p$-adic Haar measure, we may interchange integration and summation: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}=\sum_{n=0}^{\infty}c_{n}\left(\left\|f\right\|_{q}\right)\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z}=\sum_{n=0}^{\infty}\frac{c_{n}\left(\left\|f\right\|_{q}\right)}{p^{\lambda_{p}\left(n\right)}} \end{equation} and the infinite series on the right is then necessarily convergent, since $f$'s uniformly continuity guarantees its integrability. Here: \begin{equation} \sum_{n=0}^{\infty}\frac{c_{n}\left(\left\|f\right\|_{q}\right)}{p^{\lambda_{p}\left(n\right)}}=\sum_{n=0}^{\infty}\frac{\left\|f\left(n\right)\right\|_{q}-\left\|f\left(n_{-}\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}} \end{equation} Hence: \begin{align} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & =\left\|\int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\right\|\\ & \leq\sum_{n=0}^{\infty}\frac{\left\|\left\|f\left(n\right)\right\|_{q}-\left\|f\left(n_{-}\right)\right\|_{q}\right\|}{p^{\lambda_{p}\left(n\right)}}\\ \left(\textrm{reverse }\Delta\textrm{-ineq.}\right); & \leq\sum_{n=0}^{\infty}\frac{\left\|f\left(n\right)-f\left(n_{-}\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\\ & =\sum_{n=0}^{\infty}\frac{\left\|c_{n}\left(f\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}} \end{align} as desired. Q.E.D. \vphantom{} The most important practical consequence of this approach to $\left(p,q\right)$-adic integration is that it trivializes most of the basic concerns of classical real analysis. Because of our definition of $\left(p,q\right)$-adic measures as continuous linear functionals on the space of continuous $\left(p,q\right)$-adic functions, the equivalence of continuity with representability by a uniformly convergent Fourier series along with the action of measures on functions in terms of their Fourier series expansions shows that all continuous $\left(p,q\right)$-adic functions are integrable with respect to the $\left(p,q\right)$-adic Haar probability measure. Indeed, the following argument illustrates this quite nicely. \begin{example} \label{exa:end of 3.1.5. example}Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be given by the van der Put series: \begin{equation} f\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{n=0}^{\infty}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} where the series is assumed to converge merely point-wise. Letting: \begin{equation} f_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\mathfrak{a}_{n}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} observe that, like in the classic real-analytical Lebesgue theory, the $f_{N}$s are step functions via (\ref{eq:truncated van der Put identity}): \begin{equation} f_{N}\left(\mathfrak{z}\right)=f\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\sum_{n=0}^{p^{N}-1}f\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{equation} which converge to $f$. Similar to what is done in the theory of Lebesgue integration of real-valued functions on measure spaces, in Subsection \ref{subsec:3.1.6 Monna-Springer-Integration}, we will construct the class of integrable functions by considering limits of sequences of step functions. To that end, observe that: \begin{equation} \int_{\mathbb{Z}_{p}}f_{N}\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{q}}{=}\sum_{n=0}^{p^{N}-1}\mathfrak{a}_{n}\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z}=\sum_{n=0}^{p^{N}-1}\frac{\mathfrak{a}_{n}}{p^{\lambda_{p}\left(n\right)}} \end{equation} Since $p$ and $q$ are co-prime, $\left\|\mathfrak{a}_{n}p^{-\lambda_{p}\left(n\right)}\right\|_{q}=\left\|\mathfrak{a}_{n}\right\|_{q}$, which shows that: \begin{equation} \lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}f_{N}\left(\mathfrak{z}\right)d\mathfrak{z} \end{equation} exists in $\mathbb{C}_{q}$ if and only if $\lim_{n\rightarrow\infty}\left\|\mathfrak{a}_{n}\right\|_{q}\overset{\mathbb{R}}{=}0$. However, as we saw in \textbf{Theorem \ref{thm:vdP basis theorem}}, $\left\|\mathfrak{a}_{n}\right\|_{q}\rightarrow0$ if and only if $f$ is continuous! Schikhof leaves as an exercise in \emph{Ultrametric Calculus}' discussion of the van der Put basis the fact that $f\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ is the best possible approximation of $f$ in terms of the simple functions $\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]$ for $n\in\left\{ 0,\ldots,p^{N}-1\right\} $ \cite{Ultrametric Calculus}. As such, if we are going to use step-function-approximation to construct integrable functions, this analysis of ours tells us that only the continuous functions are going to be integrable. \end{example} \vphantom{} Before we conclude, we need to cover one last special case where integration becomes quite simple. \begin{defn} Let $\mathbb{F}$ be any field, let $f:\mathbb{Z}_{p}\rightarrow\mathbb{F}$, and let $n\in\mathbb{N}_{0}$. We say $f$ is \textbf{constant over inputs modulo $p^{n}$} whenever $f\left(\mathfrak{z}\right)=f\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. \end{defn} \begin{prop} Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{F}$ be constant over inputs modulo $p^{N}$. Then: \begin{equation} f\left(\mathfrak{z}\right)=\sum_{n=0}^{p^{N}-1}f\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\label{eq:representation of functions constant over inputs mod p^N} \end{equation} \end{prop} Proof: Immediate. Q.E.D. \begin{prop} \label{prop:How to integrate locally constant functions}Let $\mathbb{F}$ be a metrically complete valued field, and let $d\mu$ be a translation invariant $\mathbb{F}$-valued probability measure on $\mathbb{Z}_{p}$; that is: \begin{equation} \int_{\mathbb{Z}_{p}}d\mu\left(\mathfrak{z}\right)\overset{\mathbb{F}}{=}1 \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)=\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}+\mathfrak{a}\right)d\mu\left(\mathfrak{z}\right)\in\mathbb{F},\textrm{ }\forall\mathfrak{a}\in\mathbb{Z}_{p},\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{F}\right) \end{equation} If $f$ is constant over inputs modulo $p^{N}$, then: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu=\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}f\left(n\right) \end{equation} \end{prop} Proof: By the translation invariance of the $\mathbb{F}$-valued probability measure $d\mu$, it follows that: \begin{align} 1 & =\int_{\mathbb{Z}_{p}}d\mu\left(\mathfrak{z}\right)\\ & =\int_{\mathbb{Z}_{p}}\sum_{n=0}^{p^{N}-1}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]d\mu\left(\mathfrak{z}\right)\\ & =\sum_{n=0}^{p^{N}-1}\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]d\mu\left(\mathfrak{z}\right)\\ & =p^{N}\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]d\mu\left(\mathfrak{z}\right) \end{align} Hence: \[ \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]d\mu\left(\mathfrak{z}\right)\overset{\mathbb{F}}{=}\frac{1}{p^{N}} \] and so, by translation invariance: \[ \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]d\mu\left(\mathfrak{z}\right)\overset{\mathbb{F}}{=}\frac{1}{p^{N}},\textrm{ }\forall n,N \] By linearity, integrating the formula (\ref{eq:representation of functions constant over inputs mod p^N}) yields the desired formula for the integral of $f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)$. Q.E.D. \vphantom{} Note that this method can be used to integrate both $\left(p,q\right)$-adic functions \emph{and }$\left(p,\infty\right)$-adic functions, seeing as (provided that $p\neq q$), both those cases admit translation invariant probability measures on $\mathbb{Z}_{p}$. \subsection{\label{subsec:3.1.6 Monna-Springer-Integration}Monna-Springer Integration} FOR THIS SUBSECTION, $K$ IS A COMPLETE ULTRAMETRIC FIELD, $p$ AND $q$ ARE DISTINCT PRIMES, AND $X$ IS AN ARBITRARY ULTRAMETRIC SPACE. \vphantom{} The simplicity, directness, practicality and technical clarity of the basis-based approach to $\left(p,q\right)$-adic integration we have covered so far all speak their usefulness in doing $\left(p,q\right)$-adic analysis. Indeed, it will completely suffice for all of our investigations of $\chi_{H}$. That this approach works out so nicely is all thanks to the translation-invariance of the $\left(p,q\right)$-adic Haar measure. At the same time, note that we have failed to mention many of the issues central to most any theory of integration: measurable sets, measurable functions, and the interchanging of limits and integrals. To do these topics justice, we need Monna-Springer integration theory: the method for defining measures and integration of $K$-valued functions on an arbitrary ultrametric space $X$. Presenting Monna-Springer theory means bombarding the reader with many definitions and almost just as many infima and suprema. As such, to maximize clarity, I have supplemented the standard exposition with examples from the $\left(p,q\right)$-case and additional discussion. To begin, it is worth reflecting on the notion of measure defined in Subsection \ref{subsec:3.1.5-adic-Integration-=00003D000026}. Because we defined measure as continuous linear functionals on the space of continuous $\left(p,q\right)$-adic functions, we will be able to say a set in $\mathbb{Z}_{p}$ is $\left(p,q\right)$-adically measurable'' precisely when the indicator function for that set is $\left(p,q\right)$-adically continuous. In classical integration theory\textemdash particularly when done in the Bourbaki-style functional analysis approach\textemdash indicator functions provide a bridge between the world of functions and the world of sets. Indicator functions for measurable subsets of $\mathbb{R}$ can be approximated in $L^{1}$ to arbitrary accuracy by piece-wise continuous functions. What makes non-archimedean integration theory so different from its archimedean counterpart are the restrictions which $\mathbb{C}_{q}$'s ultrametric structure imposes on continuous functions. Because of the equivalence of integrable functions and continuous functions, the only sets in $\mathbb{Z}_{p}$ which we can call $\left(p,q\right)$-adically measurable are those whose indicator functions are $\left(p,q\right)$-adically continuous. Since indicator functions $\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]$ are continuous, it follows that any finite union, intersection, or complement of sets of the form $k+p^{n}\mathbb{Z}_{p}$ ``$\left(p,q\right)$-adically measurable''. Note that all of these sets are compact clopen sets. On the other hand, single points will \emph{not }be measurable. \begin{defn} Fix $\mathfrak{a}\in\mathbb{Z}_{p}$. Then, define the function \nomenclature{$\mathbf{1}_{\mathfrak{a}}$}{ }$\mathbf{1}_{\mathfrak{a}}:\mathbb{Z}_{p}\rightarrow K$ by: \begin{equation} \mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)=\begin{cases} 1 & \textrm{if }\mathfrak{z}=\mathfrak{a}\\ 0 & \textrm{else} \end{cases}\label{eq:Definition of a one-point function} \end{equation} I call any non-zero scalar multiple of (\ref{eq:Definition of a one-point function}) a \textbf{one-point function} supported at $\mathfrak{a}$. More generally, I call any finite linear combination of the form: \begin{equation} \sum_{n=1}^{N}\mathfrak{c}_{n}\mathbf{1}_{\mathfrak{a}_{n}}\left(\mathfrak{z}\right) \end{equation} an\textbf{ $N$-point function}; here, the $\mathfrak{a}_{n}$s in $\mathbb{Z}_{p}$ and the $\mathfrak{c}_{n}$s in $K\backslash\left\{ 0\right\} $. \end{defn} \begin{rem} It is a simple exercise to prove the formulae: \begin{equation} \mathbf{1}_{0}\left(\mathfrak{z}\right)=1-\sum_{n=0}^{\infty}\sum_{k=1}^{p-1}\left[\mathfrak{z}\overset{p^{n+1}}{\equiv}kp^{n}\right]\label{eq:van der Point series for one point function at 0} \end{equation} \begin{equation} \mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)=1-\sum_{n=0}^{\infty}\sum_{k=1}^{p-1}\left[\mathfrak{z}-\mathfrak{a}\overset{p^{n+1}}{\equiv}kp^{n}\right]\label{eq:vdP series for a one-point function} \end{equation} \end{rem} \begin{example} The indicator function of a single point in $\mathbb{Z}_{p}$ is not integrable with respect to the $\left(p,q\right)$-adically Haar probability measure. To see this, note that (\ref{eq:van der Point series for one point function at 0}) is actually a van der Put series expression for $\mathbf{1}_{0}\left(\mathfrak{z}\right)$, and the non-zero van der Put coefficients of that series are either $1$ or $-1$. By \textbf{Example \ref{exa:end of 3.1.5. example}} (see page \pageref{exa:end of 3.1.5. example}), since these coefficients do not tend to zero, the integrals of the step-function-approximations of $\mathbf{1}_{0}\left(\mathfrak{z}\right)$ do not converge $q$-adically to a limit. For the case of a general one-point function, the translation invariance of $d\mathfrak{z}$ allows us to extend the argument for $\mathbf{1}_{0}$ to $\mathbf{1}_{\mathfrak{a}}$ for any $\mathfrak{a}\in\mathbb{Z}_{p}$. \end{example} \vphantom{} Seeing as we will \emph{define }the most general class of $\left(p,q\right)$-adically Haar-measurable subsets of $\mathbb{Z}_{p}$ as being precisely those sets whose indicator functions are the limit of a sequence of $\left(p,q\right)$-adically integrable functions, the failure of one-point functions to be integrable tells us that a set consisting of one point (or, more generally, any set consisting of finitely many points) \emph{is not going to be $\left(p,q\right)$-adically Haar integrable!} This is a \emph{fundamental} contrast with the real case, where the vanishing measure of a single point tells us that integrability is unaffected by changing functions' values at finitely many (or even countably many) points. In $\left(p,q\right)$-adic analysis, therefore, there are no notions of ``almost everywhere'' or ``null sets''. From a more measure-theoretic standpoint, this also shows that it is too much to expect an arbitrary Borel set\index{Borel!set} to be $\left(p,q\right)$-adically measurable. In light of this, Monna-Springer integration instead begins by working with a slightly smaller, more manageable algebra of sets: the compact clopen sets. \begin{defn} We write $\Omega\left(X\right)$ to denote the collection of all compact open subsets of $X$ (equivalently, compact clopen sets). We make $\Omega\left(X\right)$ an algebra of sets by equipping it with the operations of unions, intersections, and complements. \end{defn} \vphantom{} In more abstract treatments of non-archimedean integration (such as \cite{van Rooij - Non-Archmedean Functional Analysis,Measure-theoretic approach to p-adic probability theory,van Rooij and Schikhof "Non-archimedean integration theory"}), one begins by fixing a ring of sets $\mathcal{R}$ to use to construct simple functions, which then give rise to a notion of measure. Here, we have chosen $\mathcal{R}$ to be $\Omega\left(X\right)$. Taking limits of simple functions in $L^{1}$ with respect to a given measure $d\mu$ then enlarges $\mathcal{R}$ to an algebra of sets, denoted $\mathcal{R}_{\mu}$, which is the maximal extension of $\mathcal{R}$ containing subsets of $X$ which admit a meaningful notion of $\mu$-measure. This is, in essence, what we will do here, albeit in slightly more concrete terms. For now, though, let us continue through the definitions. Before proceeding, I should also mention that my exposition here is a hybrid of the treatments given by Schikhof in one of the Appendices of \cite{Ultrametric Calculus} and the chapter on Monna-Springer integration theory given by Khrennikov in his book \cite{Quantum Paradoxes} and in the paper \cite{Measure-theoretic approach to p-adic probability theory}. \begin{defn}[\textbf{Integrals and Measures}\footnote{Taken from \cite{Ultrametric Calculus}.}] \ \vphantom{} I. Elements of the dual space $C\left(X,K\right)^{\prime}$ are \index{integral!Monna-Springer} called \textbf{integrals }on $C\left(X,K\right)$. \vphantom{} II. A \index{measure}\textbf{ measure}\index{measure!non-archimedean} is a function $\mu:\Omega\left(X\right)\rightarrow K$ which satisfies the conditions: \vphantom{} i. (Additivity): $\mu\left(U\cup V\right)=\mu\left(U\right)+\mu\left(V\right)$ for all $U,V\in\Omega\left(X\right)$ with $U\cap V=\varnothing$. \vphantom{} ii. (Boundedness): the real number $\left\Vert \mu\right\Vert $ (the\index{measure!total variation} \textbf{total variation }of $\mu$) defined by: \begin{equation} \left\Vert \mu\right\Vert \overset{\textrm{def}}{=}\sup\left\{ \left\|\mu\left(U\right)\right\|_{K}:U\in\Omega\left(X\right)\right\} \label{eq:Definition of the norm of a measure} \end{equation} is finite. \vphantom{} III. We write $M\left(X,K\right)$ to denote the vector space of $K$-valued measures on $X$. Equipping this vector space with (\ref{eq:Definition of the norm of a measure}) makes it into a non-archimedean normed vector space. \end{defn} \vphantom{} Even though our starting algebra of measurable sets ($\Omega\left(X\right)$) is smaller than its classical counterpart, there is still a correspondence between measures and continuous linear functionals \cite{Ultrametric Calculus}. \begin{thm} For each $\varphi\in C\left(X,K\right)^{\prime}$ (which sends a function $f$ to the scalar $\varphi\left(f\right)$) the map $\mu_{\varphi}:\Omega\left(X\right)\rightarrow K$ defined by: \[ \mu_{\varphi}\left(U\right)\overset{\textrm{def}}{=}\varphi\left(\mathbf{1}_{U}\right),\textrm{ }\forall U\in\Omega\left(X\right) \] (where $\mathbf{1}_{U}$ is the indicator function of $U$) is a measure. Moreover, the map: \[ \varphi\in C\left(X,K\right)^{\prime}\mapsto\mu_{\varphi}\in M\left(X,K\right) \] is a $K$-linear isometry of $C\left(X,K\right)^{\prime}$ onto $M\left(X,K\right)$. We also have that: \begin{equation} \left\|\varphi\left(f\right)\right\|_{K}\leq\left\Vert f\right\Vert _{X,K}\left\Vert \mu_{\varphi}\right\Vert ,\textrm{ }\forall f\in C\left(X,K\right),\forall\varphi\in C\left(X,K\right)^{\prime}\label{eq:Absolute Value of the output of a functional in terms of the associated measure} \end{equation} \end{thm} \begin{notation} In light of the above theorem, we shall now adopt the following notations for $K$-valued measures and elements of $C\left(X,K\right)^{\prime}$: \vphantom{} I. Elements of $C\left(X,K\right)^{\prime}$ will all be written with a $d$ in front of them: $d\mathfrak{z}$, $d\mu$, $d\mu\left(\mathfrak{z}\right)$, $dA_{3}$, etc. \vphantom{} II. Given $d\mu\in C\left(X,K\right)^{\prime}$ and $f\in C\left(X,K\right)$, we denote the image of $f$ under $d\mu$ by: \[ \int_{X}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \] \vphantom{} III. Given $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$ and $U\in\Omega\left(\mathbb{Z}_{p}\right)$, we will denote the $\mu$-measure of $U$ by the symbols $\mu\left(U\right)$ and $\int_{U}d\mu$. \vphantom{} IV. Given any function $g\in C\left(X,K\right)$, the operation of point-wise multiplication: \[ f\in C\left(X,K\right)\mapsto fg\in C\left(X,K\right) \] defines a continuous linear operator on $C\left(X,K\right)$. As such, we can multiply elements of $C\left(X,K\right)^{\prime}$ by continuous functions to obtain new elements of $C\left(X,K\right)^{\prime}$. Given $d\mu\in C\left(X,K\right)^{\prime}$, we write $g\left(\mathfrak{z}\right)d\mu$ and $g\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)$ to denote the measure which first multiplies a function $f$ by $g$ and then integrates the product against $d\mu$: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\left(g\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\left(f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)\right)d\mu\left(\mathfrak{z}\right)\label{eq:Definition of the product of a measure by a continuous function} \end{equation} \vphantom{} V. Given any $f\in C\left(X,K\right)$, any $d\mu\in C\left(X,K\right)^{\prime}$, and any $U\in\Omega\left(X\right)$, we write: \[ \int_{U}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \] to denote: \[ \int_{\mathbb{Z}_{p}}\left(f\left(\mathfrak{z}\right)\mathbf{1}_{U}\left(\mathfrak{z}\right)\right)d\mu\left(\mathfrak{z}\right) \] \end{notation} \vphantom{} Like with the real case, there is also a Riemann-sum\index{Riemann sum}-type formulation of the integrability of a function with respect to a given measure \cite{Ultrametric Calculus}: \begin{prop}[\textbf{$\left(p,q\right)$-adic integration via Riemann sums}] Let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$, and let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, for any $\epsilon>0$, there exists a $\delta>0$, such that for each partition of $\mathbb{Z}_{p}$ into pair-wise disjoint balls $B_{1},\ldots,B_{N}\in\Omega\left(\mathbb{Z}_{p}\right)$ , all of radius $<\delta$, and any choice of $\mathfrak{a}_{1}\in B_{1}$, $\mathfrak{a}_{2}\in B_{2}$,$\ldots$: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)-\sum_{n=1}^{N}f\left(\mathfrak{a}_{n}\right)\mu\left(B_{j}\right)\right\|_{q}<\epsilon\label{eq:Riemann sum criterion for (p,q)-adic integrability} \end{equation} \end{prop} \begin{rem} The pair-wise disjoint balls of radius $<\delta$ must be replaced by pair-wise disjoint open sets of diameter $<\delta$ in the case where $\mathbb{Z}_{p}$ is replaced by a compact ultrametric space $X$ which does not possess the Heine-Borel property. If $X$ possesses the Heine-Borel property, then the open balls may be used instead of the more general open sets. \end{rem} \begin{thm}[\textbf{Fubini's Theorem}\index{Fubini's Theorem} \cite{Ultrametric Calculus}] Let $d\mu,d\nu$ be $K$-valued measures on $C\left(X,K\right)$ and $C\left(Y,K\right)$, respectively, where $X$ and $Y$ are compact ultrametric spaces. Then, for any continuous $f:X\times Y\rightarrow K$: \begin{equation} \int_{X}\int_{Y}f\left(x,y\right)d\mu\left(x\right)d\nu\left(y\right)=\int_{X}\int_{Y}f\left(x,y\right)d\nu\left(y\right)d\mu\left(x\right)\label{eq:Fubini's Theorem} \end{equation} \end{thm} \begin{example}[\textbf{A $\left(p,p\right)$-adic measure of great importance}\footnote{Given as part of an exercise on page 278 in the Appendix of \cite{Ultrametric Calculus}.}] Let $\mathbb{C}_{p}^{+}$ denote the set: \begin{equation} \mathbb{C}_{p}^{+}\overset{\textrm{def}}{=}\left\{ \mathfrak{s}\in\mathbb{C}_{p}:\left\|1-\mathfrak{s}\right\|_{p}<1\right\} \end{equation} Schikhof calls such $\mathfrak{s}$ ``positive'' \cite{Ultrametric Calculus}, hence the superscript $+$. For any $\mathfrak{s}\in\mathbb{C}_{p}\backslash\mathbb{C}_{p}^{+}$, define $d\mu_{\mathfrak{s}}\in M\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ by: \begin{equation} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]d\mu_{\mathfrak{s}}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{p}}{=}\frac{\mathfrak{s}^{k}}{1-\mathfrak{s}^{p^{n}}},\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall k\in\mathbb{Z}/p^{n}\mathbb{Z} \end{equation} This measure satisfies $\left\Vert \mu_{\mathfrak{s}}\right\Vert <1$, and is used by Koblitz (among others) \cite{Koblitz's other book} to define the \textbf{$p$-adic zeta function}\index{$p$-adic!Zeta function}. A modification of this approach is one of the methods used to construct more general $p$-adic $L$-functions\footnote{One of the major accomplishments of $p$-adic analysis ($\left(p,p\right)$, not $\left(p,q\right)$) was to confirm that the measure-based approach to constructing $p$-adic $L$-functions is, in fact, the same as the $p$-adic $L$-functions produced by interpolating the likes of Kummer's famous congruences for the Bernoulli numbers \cite{p-adic L-functions paper}.}\index{$p$-adic!$L$-function}\cite{p-adic L-functions paper}. Note that because $d\mu_{\mathfrak{s}}$ is a $\left(p,p\right)$-adic measure, it has no hope of being translation invariant, seeing as the only translation-invariant $\left(p,p\right)$-adic linear functional is the zero map. \end{example} \vphantom{} It is at this point, however, where we begin to step away from Subsections \ref{subsec:3.1.4. The--adic-Fourier} and \ref{subsec:3.1.5-adic-Integration-=00003D000026} and make our way toward abstraction. The standard construction of a measure-theoretic integral is to first define it for the indicator function of a measurable set, and then for finite linear combinations thereof, and finally taking $\sup$s for the general case. Unfortunately, this does not work well, both due to the shortage of measurable sets, and\textemdash more gallingly\textemdash due to a terrible betrayal by one of analysts' closest companions: the triangle inequality for integrals. \begin{example}[\textbf{Failure of the triangle inequality for $\left(p,q\right)$-adic integrals}] \textbf{\label{exa:triangle inequality failure}}\index{triangle inequality!failure of}Letting $d\mathfrak{z}$ denote the $\left(p,q\right)$-adic Haar measure (which sends the indicator function $\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]$ to the scalar $\frac{1}{p^{n}}\in\mathbb{Q}_{q}$), note that for any $n\geq1$, any distinct $j,k\in\left\{ 0,\ldots,p^{n}-1\right\} $, and any $\mathfrak{a},\mathfrak{b}\in\mathbb{Q}_{q}$: \begin{equation} \left\|\mathfrak{a}\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]+\mathfrak{b}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]\right\|_{q}\overset{\mathbb{R}}{=}\left\|\mathfrak{a}\right\|_{q}\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]+\left\|\mathfrak{b}\right\|_{q}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} because, for any $\mathfrak{z}$, at most one of $\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]$ and $\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]$ can be equal to $1$. So, letting: \begin{equation} f\left(\mathfrak{z}\right)=\mathfrak{a}\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]+\mathfrak{b}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right] \end{equation} we have that: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\overset{\mathbb{R}}{=}\left\|\frac{\mathfrak{a}+\mathfrak{b}}{p^{n}}\right\|_{q} \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}\int_{\mathbb{Z}_{p}}\left(\left\|\mathfrak{a}\right\|_{q}\left[\mathfrak{z}\overset{p^{n}}{\equiv}j\right]+\left\|\mathfrak{b}\right\|_{q}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]\right)d\mathfrak{z}=\frac{\left\|\mathfrak{a}\right\|_{q}+\left\|\mathfrak{b}\right\|_{q}}{p^{n}} \end{equation} where the integral in the middle on the second line is with respect to the standard \emph{real-valued} Haar probability measure on $\mathbb{Z}_{p}$. If we choose $\mathfrak{a}=1$ and $\mathfrak{b}=q^{m}-1$, where $m\in\mathbb{N}_{1}$, then: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\overset{\mathbb{R}}{=}\left\|\frac{q^{m}}{p^{n}}\right\|_{q}=\frac{1}{q^{m}} \end{equation} \begin{equation} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}\frac{\left\|1\right\|_{q}+\left\|q^{m}-1\right\|_{q}}{p^{n}}=\frac{2}{p^{n}} \end{equation} Because of this, the symbol $\sim$ in the expression: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\sim\int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} \end{equation} can be arranged to be any of the symbols $\leq$, $\geq$, $<$, $>$, or $=$ by making an appropriate choice of the values of $m$, $n$, $p$, and $q$. A very distressing outcome, indeed. \end{example} \vphantom{} The \emph{point} of Monna-Springer theory, one could argue, is to find a work-around for this depressing state of affairs. That work-around takes the form of an observation. What we would like to do is to write: \begin{equation} \left\|\int f\right\|\leq\int\left\|f\right\| \end{equation} However, we can't. Instead, we take out the middle man, which, in this case, is the absolute value on the left-hand side. Rather than view the triangle inequality as an operation we apply to an integral already in our possession, we view it as an operation we apply directly to the function $f$ we intend to integrate: \begin{equation} f\mapsto\int\left\|f\right\| \end{equation} where, for the sake of discussion, we view everything as happening on $\mathbb{R}$; $f:\mathbb{R}\rightarrow\mathbb{R}$, integration is with respect to the Lebesgue measure, etc. Because the vanishing of the integral only forces $f$ to be $0$ \emph{almost }everywhere, the above map is not a true norm, but a semi-norm. Classically, remember, $L^{1}$-norm only becomes a norm after we mod out by functions which are zero almost everywhere. More generally, given\textemdash say\textemdash a Radon measure $d\mu$, the map $f\mapsto\int\left\|f\right\|d\mu$ defines a semi-norm which we can use to compute the $\mu$-measure of a set $E$ by applying it to continuous functions which approximate $E$'s indicator function. While we will not be able to realize this semi-norm as the integral of $\left\|f\right\|$ $d\mu$ in the non-archimedean case, we can still obtain a semi-norm by working from the ground up. Given a measure $\mu\in C\left(X,K\right)^{\prime}$, we consider the semi-norm: \begin{equation} f\in C\left(X,Y\right)\mapsto\sup_{g\in C\left(X,K\right)\backslash\left\{ 0\right\} }\frac{1}{\left\Vert g\right\Vert _{X,K}}\left\|\int_{X}f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right\|_{K}\in\mathbb{R} \end{equation} In the appendices of \cite{Ultrametric Calculus}, Schikhof takes this formula as the definition for the semi-norm $N_{\mu}$ induced by the measure $\mu$, which he then defines as a function on $\Omega\left(X\right)$ by defining $N_{\mu}\left(U\right)$ to be $N_{\mu}\left(\mathbf{1}_{U}\right)$, where $\mathbf{1}_{U}$ is the indicator function of $U$. Taking the infimum over all $U\in\Omega\left(X\right)$ containing a given point $\mathfrak{z}$ then allows for $N_{\mu}$ to be extended to a real-valued function on $X$ by way of the formula: \begin{equation} \mathfrak{z}\in X\mapsto\inf_{U\in\Omega\left(X\right):\mathfrak{z}\in U}\sup_{g\in C\left(X,K\right)\backslash\left\{ 0\right\} }\frac{\left\|\int_{X}\mathbf{1}_{U}\left(\mathfrak{y}\right)g\left(\mathfrak{y}\right)d\mu\left(\mathfrak{y}\right)\right\|_{K}}{\left\Vert g\right\Vert _{X,K}}\in\mathbb{R} \end{equation} While this approach is arguably somewhat more intuitive, the above definitions are not very pleasing to look at, and using them is a hassle. Instead, I follow Khrennikov et. al. \cite{Measure-theoretic approach to p-adic probability theory} in giving the following formulas as the definition of $N_{\mu}$: \begin{defn} \label{def:Khrennikov N_mu definition}Let\footnote{Taken from \cite{Measure-theoretic approach to p-adic probability theory}.} $d\mu\in C\left(X,K\right)^{\prime}$. \vphantom{} I. For any $U\in\Omega\left(X\right)$, we define $N_{\mu}\left(U\right)$ by: \begin{equation} N_{\mu}\left(U\right)=\sup_{V\in\Omega\left(X\right):V\subset U}\left\|\mu\left(V\right)\right\|_{K},\textrm{ }\forall U\in\Omega\left(X\right)\label{eq:mu seminorm of a compact clopen set} \end{equation} \vphantom{} II. For any $\mathfrak{z}\in X$, we define $N_{\mu}\left(\mathfrak{z}\right)$ by: \begin{equation} N_{\mu}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\inf_{U\in\Omega\left(X\right):\mathfrak{z}\in U}N_{\mu}\left(\mathbf{1}_{U}\right)\label{eq:Definition of N_mu at a point} \end{equation} \vphantom{} III. Given any $f:X\rightarrow K$ (not necessarily continuous), we define \begin{equation} N_{\mu}\left(f\right)\overset{\textrm{def}}{=}\sup_{\mathfrak{z}\in X}N_{\mu}\left(\mathfrak{z}\right)\left\|f\left(\mathfrak{z}\right)\right\|_{K}\label{eq:Definition of N_mu for an arbitrary f} \end{equation} \end{defn} \vphantom{} In \cite{Ultrametric Calculus}, Schikhof proves that the formulas of \textbf{Definition \ref{def:Khrennikov N_mu definition} }as a theorem. Instead, following Khrennikov et. al. in \cite{Measure-theoretic approach to p-adic probability theory}, I state Schikhof's definitions of $N_{\mu}$ as a theorem (\textbf{Theorem \ref{thm:Schikhof's N_mu definition}}), deducible from \textbf{Definition \ref{def:Khrennikov N_mu definition}}. \begin{thm} \label{thm:Schikhof's N_mu definition}\ \vphantom{} I. For all $f\in C\left(X,K\right)$: \begin{equation} N_{\mu}\left(f\right)=\sup_{g\in C\left(X,K\right)\backslash\left\{ 0\right\} }\frac{1}{\left\Vert g\right\Vert _{X,K}}\left\|\int_{X}f\left(\mathfrak{z}\right)g\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right\|_{K}\label{eq:Schikhof's definition of N_mu} \end{equation} \vphantom{} II. \begin{equation} N_{\mu}\left(\mathfrak{z}\right)=\inf_{U\in\Omega\left(X\right):\mathfrak{z}\in U}\sup_{g\in C\left(X,K\right)\backslash\left\{ 0\right\} }\frac{\left\|\int_{U}g\left(\mathfrak{y}\right)d\mu\left(\mathfrak{y}\right)\right\|_{K}}{\left\Vert g\right\Vert _{X,K}} \end{equation} \end{thm} \begin{rem} The notation Schikhof uses in \cite{Ultrametric Calculus} for the semi-norms is slightly different than the one given here, which I have adopted from Khrennikov's exposition of the Monna-Springer integral in \cite{Quantum Paradoxes}, which is simpler and more consistent. \end{rem} \vphantom{} In order to be able to extend the notion of $\mu$-integrable functions from $C\left(X,K\right)$ to a large space, we need a non-archimedean version of the triangle inequality for integrals. As we saw above, however, this doesn't really exist in the non-archimedean context. This is where $N_{\mu}$ comes in; it is our replacement for what where $\int\left\|f\right\|d\mu$ would be in the archimedean case. \begin{prop}[\textbf{Monna-Springer Triangle Inequality} \cite{Ultrametric Calculus,Quantum Paradoxes}] Let $X$ be a compact ultrametric space, and let $d\mu\in C\left(X,K\right)^{\prime}$. Then: \begin{equation} \left\|\int_{X}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right\|_{K}\leq N_{\mu}\left(f\right)\leq\left\Vert \mu\right\Vert \left\Vert f\right\Vert _{X,K}\label{eq:Non-archimedean triangle inequality} \end{equation} \end{prop} \begin{rem} Unfortunately it is often the case\textemdash especially so for $\left(p,q\right)$-adic analysis\textemdash that this estimate is little better than bounding an integral by the product of the $\sup$ of the integrand and the measure of the domain of integration. \end{rem} \vphantom{} Having done all this, the idea is to use $N_{\mu}$ to define what it means for an arbitrary function $f:X\rightarrow K$ to be $\mu$-integrable. This is done in the way one would expect: identifying $f,g\in C\left(X,K\right)$ whenever $N_{\mu}\left(f-g\right)=0$, and then modding $C\left(X,K\right)$ out by this equivalence relation, so as to upgrade $N_{\mu}$ from a semi-norm to a fully-fledged norm. We then take the completion of $C\left(X,K\right)$ with respect to this norm to obtain our space of $\mu$-integrable functions. \cite{Quantum Paradoxes,Measure-theoretic approach to p-adic probability theory} and \cite{Ultrametric Calculus} give us the definitions that will guide us through this process. \begin{defn} \ \vphantom{} I. Because the indicator function for any set $S\subseteq X$ is then a function $X\rightarrow K$, we can use (\ref{eq:Definition of N_mu for an arbitrary f}) to define $N_{\mu}$ for an arbitrary $S\subseteq X$: \begin{equation} N_{\mu}\left(S\right)\overset{\textrm{def}}{=}\inf_{U\in\Omega\left(X\right):S\subseteq U}N_{\mu}\left(U\right)=\inf_{U\in\Omega\left(X\right):S\subseteq U}\sup_{V\in\Omega\left(X\right):V\subset U}\left\|\mu\left(V\right)\right\|_{K}\label{eq:Definition of N_mu of an arbitrary set} \end{equation} \vphantom{} II.\textbf{ }A function $f:X\rightarrow K$ (not necessarily continuous) is said to be \textbf{$\mu$-integrable }whenever there is a sequence $\left\{ f_{n}\right\} _{n\geq1}$ in $C\left(X,K\right)$ so that $\lim_{n\rightarrow\infty}N_{\mu}\left(f-f_{n}\right)=0$, where we compute $N_{\mu}\left(f-f_{n}\right)$ using (\ref{eq:Definition of N_mu for an arbitrary f}). We then write $\mathcal{L}_{\mu}^{1}\left(X,K\right)$ to denote the vector space of all $\mu$-integrable functions from $X$ to $K$. \vphantom{} III. We say a set $S\subseteq X$ is \textbf{$\mu$-measurable }whenever $N_{\mu}\left(S\right)$ is finite. Equivalently, $S$ is $\mu$-measurable whenever there is a sequence $\left\{ f_{n}\right\} _{n\geq1}$ in $C\left(X,K\right)$ so that $\lim_{n\rightarrow\infty}N_{\mu}\left(\mathbf{1}_{S}-f_{n}\right)=0$. We write $\mathcal{R}_{\mu}$ to denote the collection of all $\mu$-measurable subsets of $X$. \vphantom{} IV. A \textbf{non-archimedean measure space}\index{non-archimedean!measure space}\textbf{ (over $K$) }is a triple $\left(X,d\mu,\mathcal{R}_{\mu}\right)$, where $X$ is an ultrametric space, $d\mu\in C\left(X,K\right)^{\prime}$, and $\mathcal{R}_{\mu}$ is the collection of all $\mu$-measurable subsets of $X$. \vphantom{} V. Given two non-archimedean measure spaces $\left(X_{1},d\mu_{1},\mathcal{R}_{\mu_{1}}\right)$ and $\left(X_{2},d\mu_{2},\mathcal{R}_{\mu_{2}}\right)$ with we say a function $\phi:X_{1}\rightarrow X_{2}$ is \textbf{$\mu$-measurable} whenever $\phi^{-1}\left(V\right)\in\mathcal{R}_{\mu_{1}}$ for all $V\in\mathcal{R}_{\mu_{2}}$. \vphantom{} VI. We define $E_{\mu}$ as the set: \begin{equation} E_{\mu}\overset{\textrm{def}}{=}\left\{ f:X\rightarrow K\textrm{ \& }N_{\mu}\left(f\right)<\infty\right\} \label{eq:Definition of E_mu} \end{equation} Given any $f\in E_{\mu}$, we say $f$ is \textbf{$\mu$-negligible }whenever $N_{\mu}\left(f\right)=0$. Likewise, given any set $S\subseteq X$, we say $S$ is \textbf{$\mu$-negligible} whenever $N_{\mu}\left(S\right)=0$. \vphantom{} VII. Given any real number $\alpha>0$, we define: \begin{equation} X_{\mu:\alpha}\overset{\textrm{def}}{=}\left\{ \mathfrak{z}\in X:N_{\mu}\left(\mathfrak{z}\right)\geq\alpha\right\} \label{eq:Definition of X mu alpha} \end{equation} \begin{equation} X_{\mu:+}\overset{\textrm{def}}{=}\bigcup_{\alpha>0}X_{\mu:\alpha}\label{eq:Definition of X mu plus} \end{equation} \begin{equation} X_{\mu:0}\overset{\textrm{def}}{=}\left\{ \mathfrak{z}\in X:N_{\mu}\left(\mathfrak{z}\right)=0\right\} \label{eq:Definition of X mu 0} \end{equation} Note that $X=X_{\mu:0}\cup X_{\mu:+}$. We omit $\mu$ and just write $X_{\alpha}$, $X_{+}$, and $X_{0}$ whenever there is no confusion as to the measure $\mu$ we happen to be using. \vphantom{} VIII. We write $L_{\mu}^{1}\left(X,K\right)$ to denote the space of equivalence classes of $\mathcal{L}_{\mu}^{1}\left(X,K\right)$ under the relation: \[ f\sim g\Leftrightarrow f-g\textrm{ is }\mu\textrm{-negligible} \] \end{defn} \vphantom{} Using these definitions, we can finally construct an honest-to-goodness space of integrable non-archimedean functions \cite{Ultrametric Calculus}. \begin{thm} Let $d\mu\in C\left(X,K\right)^{\prime}$. Then: \vphantom{} I. $\mathcal{L}_{\mu}^{1}\left(X,K\right)$ contains $C\left(X,K\right)$, as well as all $\mu$-negligible functions from $X$ to $K$. \vphantom{} II. $d\mu$ can be extended from $C\left(X,K\right)$ to $\mathcal{L}_{\mu}^{1}\left(X,K\right)$; denote this extension by $\overline{d\mu}$. Then: \[ \left\|\int_{X}f\left(\mathfrak{z}\right)\overline{d\mu}\left(\mathfrak{z}\right)\right\|_{K}\leq N_{\mu}\left(f\right),\textrm{ }\forall f\in\mathcal{L}_{\mu}^{1}\left(X,K\right) \] \vphantom{} III. $L_{\mu}^{1}\left(X,K\right)$ is a Banach space over $K$ with respect to the norm induced by $N_{\mu}$. \end{thm} \vphantom{} In $\left(p,q\right)$-adic case, however, these constructions turn out to be overkill. \begin{thm} Let $d\mu$ be the $\left(p,q\right)$-adic Haar probability measure $d\mathfrak{z}$. Then $N_{\mu}\left(\mathfrak{z}\right)=1$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. Equivalently, for $X=\mathbb{Z}_{p}$, we have\footnote{This is given as an exercise on Page 281 of \cite{Ultrametric Calculus}.}: \begin{align} X_{0} & =\varnothing\\ X_{+} & =X\\ X_{\alpha} & =\begin{cases} \varnothing & \textrm{if }0<\alpha<1\\ X & \textrm{if }\alpha=1\\ \varnothing & \alpha>1 \end{cases} \end{align} \end{thm} \vphantom{} Put in words, the only set which is negligible with respect to $d\mathfrak{z}$ is the \emph{empty set!} This is the reason why integrability and continuity are synonymous in $\left(p,q\right)$-adic analysis. \begin{cor}[\textbf{The Fundamental Theorem of $\left(p,q\right)$-adic Analysis}\footnote{The name is my own.}] Letting $d\mathfrak{z}$ denote the $\left(p,q\right)$-adic Haar measure, we have the equality (not just an isomorphism, but an\emph{ }\textbf{equality}!): \begin{equation} L_{d\mathfrak{z}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)=C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:in (p,q)-adic analysis, integrable equals continuous} \end{equation} Consequently: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\leq\left\Vert f\right\Vert _{p,q},\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:(p,q)-adic triangle inequality} \end{equation} \end{cor} Proof: Since $N_{d\mathfrak{z}}=1$, $N_{d\mathfrak{z}}$ is then the $\infty$-norm on $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Hence: \[ \mathcal{L}_{d\mathfrak{z}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)=L_{d\mathfrak{z}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)=C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right) \] Q.E.D. \vphantom{} With a little more work, one can prove many of the basic results familiar to us from classical analysis. While these hold for Monna-Springer theory in general, the simplicity of the $\left(p,q\right)$-adic case makes these theorems significantly less useful than their classical counterparts. I include them primarily for completeness' sake. \begin{thm}[\textbf{Non-Archimedean Dominated Convergence Theorem}\footnote{Given in Exercise 7F on page 263 of \cite{van Rooij - Non-Archmedean Functional Analysis}.}] Let\index{Dominated Convergence Theorem} $X$ be a locally compact ultrametric space. Let $g\in L_{\mu}^{1}\left(X,K\right)$ and let $\left\{ f_{n}\right\} _{n\geq0}$ be a sequence in $L_{\mu}^{1}\left(X,K\right)$ such that: I. For every set $E\in\mathcal{R}_{\mu}$, $f_{n}$ converges uniformly to $f$ on $E$; II. $\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{K}\leq\left\|g\left(\mathfrak{z}\right)\right\|_{K}$ for all $\mathfrak{z}\in X$ and all $n$. Then, $f\in L_{\mu}^{1}\left(X,K\right)$, $\lim_{n\rightarrow\infty}N_{\mu}\left(f_{n}-f\right)=0$ and: \begin{equation} \lim_{n\rightarrow\infty}\int_{X}f_{n}\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\int_{X}\lim_{n\rightarrow\infty}f_{n}\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\overset{K}{=}\int_{X}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\label{eq:Non-Archimedean Dominated Convergence Theorem} \end{equation} \end{thm} \begin{rem} For $\left(p,q\right)$-analysis, the Dominated Convergence Theorem is simply the statement that: \begin{equation} \lim_{n\rightarrow\infty}\int f_{n}=\int\lim_{n\rightarrow\infty}f_{n} \end{equation} occurs if and only if $f_{n}\rightarrow f$ uniformly on $\mathbb{Z}_{p}$: \begin{equation} \lim_{n\rightarrow\infty}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-f_{n}\left(\mathfrak{z}\right)\right\|_{q}=0 \end{equation} \end{rem} \begin{thm}[\textbf{Non-Archimedean Hlder's Inequality} \cite{Quantum Paradoxes}] Let $f,g\in C\left(X,K\right)$, and let $d\mu\in C\left(X,K\right)^{\prime}$ Then: \begin{equation} N_{\mu}\left(fg\right)\leq N_{\mu}\left(f\right)\left\Vert g\right\Vert _{X,K}\label{eq:(p,q)-adic Holder inequality} \end{equation} \end{thm} \begin{rem} For\index{Hlder's Inequality} $\left(p,q\right)$-analysis, Hlder's Inequality is extremely crude, being the statement that $\left\|\int fg\right\|_{q}\leq\left\Vert f\right\Vert _{X,K}\left\Vert g\right\Vert _{X,K}$. \end{rem} \vphantom{} Finally\textemdash although we shall not use it beyond the affine case detailed in \textbf{Lemma \ref{lem:Affine substitution change of variable formula}}, we also have a general change of variables formula. First, however, the necessary definitional lead-up. \begin{defn} \label{def:pullback measure}Let $\left(X,d\mu,\mathcal{R}_{\mu}\right)$ and $\left(Y,d\nu,\mathcal{R}_{\nu}\right)$ be non-archimedean measure spaces over $K$. For any measurable $\phi:X\rightarrow Y$, we define the measure $d\mu_{\phi}:\mathcal{R}_{\nu}\rightarrow K$ by: \begin{equation} \mu_{\phi}\left(V\right)\overset{\textrm{def}}{=}\mu\left(\phi^{-1}\left(V\right)\right),\textrm{ }\forall V\in\mathcal{R}_{\nu}\label{eq:Definition of change-of-variables measure} \end{equation} \end{defn} \begin{lem} Let $\left(X,d\mu,\mathcal{R}_{\mu}\right)$ and $\left(Y,d\nu,\mathcal{R}_{\nu}\right)$ be non-archimedean measure spaces over $K$. For any measurable $\phi:X\rightarrow Y$, and for every $\mathcal{R}_{\nu}$ continuous $f:Y\rightarrow K$: \begin{equation} N_{\mu_{\phi}}\left(f\right)\leq N_{\mu}\left(f\circ\phi\right) \end{equation} \end{lem} \begin{thm}[\textbf{Change of Variables \textendash{} Monna-Springer Integration }\cite{Measure-theoretic approach to p-adic probability theory}] Let\index{change of variables} $\left(X,d\mu,\mathcal{R}_{\mu}\right)$ and $\left(Y,d\nu,\mathcal{R}_{\nu}\right)$ be non-archimedean measure spaces over $K$. For any measurable $\phi:X\rightarrow Y$, and for every $\mathcal{R}_{\nu}$ continuous\footnote{As remarked in \cite{Measure-theoretic approach to p-adic probability theory}, it would be desirable if the hypothesis of continuity could be weakened.} $f:Y\rightarrow K$ so that $f\circ\phi\in L_{\mu}^{1}\left(X,K\right)$, the function $f$ is in $L_{\nu}^{1}\left(Y,K\right)$ and: \begin{equation} \int_{X}f\left(\phi\left(\mathfrak{z}\right)\right)d\mu\left(\mathfrak{z}\right)=\int_{Y}f\left(\mathfrak{y}\right)d\mu_{\phi}\left(y\right)\label{eq:Monna-Springer Integral Change-of-Variables Formula} \end{equation} \end{thm} \newpage{} \section{\label{sec:3.2 Rising-Continuous-Functions}Rising-Continuous Functions} IN THIS SECTION, $p$ A PRIME AND $K$ IS A COMPLETE NON-ARCHIMEDEAN VALUED FIELD. \vphantom{} Given the somewhat unusual method we used to construct $\chi_{H}$\textemdash strings and all\textemdash you would think that the $\chi_{H}$s would be unusual functions, located out in the left field of ``mainstream'' $\left(p,q\right)$-adic function theory. Surprisingly, this is not the case. The key property here is the limit: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\lim_{n\rightarrow\infty}\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{equation} \textbf{Rising-continuous functions} are precisely those functions which satisfy this limit. In this section, we shall get to know these functions as a whole and see how they are easily obtained from the smaller class of continuous functions. \subsection{\label{subsec:3.2.1 -adic-Interpolation-of}$\left(p,q\right)$-adic Interpolation of Functions on $\mathbb{N}_{0}$} It is not an understatement to say that the entire theory of rising-continuous functions emerges (or, should I say, ``\emph{rises}''?) out of the van der Put identity ((\ref{eq:van der Put identity}) from \textbf{Proposition \ref{prop:vdP identity} }on page \pageref{prop:vdP identity}): \begin{equation} \sum_{n=0}^{\infty}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\overset{K}{=}\lim_{k\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{k}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} We make two observations: \begin{enumerate} \item The field $K$ can be \emph{any }metrically complete valued field\textemdash archimedean or non-archimedean. \item $\chi$ can be\emph{ any }function $\chi:\mathbb{Z}_{p}\rightarrow K$. \end{enumerate} (1) will be arguably even more important than (2), because it highlights what will quickly become a cornerstone of our approach: the ability to allow the field (and hence, topology)\emph{ }used to sum our series \emph{vary depending on $\mathfrak{z}$}. First, however, some terminology. \begin{defn} Let $p$ be an integer $\geq2$. \vphantom{} I. Recall, we write \nomenclature{$\mathbb{Z}_{p}^{\prime}$}{$\overset{\textrm{def}}{=}\mathbb{Z}_{p}\backslash\left\{ 0,1,2,3,\ldots\right\}$ \nopageref}$\mathbb{Z}_{p}^{\prime}$ to denote the set: \begin{equation} \mathbb{Z}_{p}^{\prime}\overset{\textrm{def}}{=}\mathbb{Z}_{p}\backslash\mathbb{N}_{0}\label{eq:Definition of Z_p prime} \end{equation} \vphantom{} II. A sequence $\left\{ \mathfrak{z}_{n}\right\} _{n\geq0}$ in $\mathbb{Z}_{p}$ is said to be \textbf{($p$-adically) rising }if, as $n\rightarrow\infty$, the number of non-zero $p$-adic digits in $\mathfrak{z}_{n}$ tends to $\infty$. \end{defn} \begin{rem} $\mathbb{Z}_{p}^{\prime}$ is neither open nor closed, and has an empty interior. \end{rem} \begin{rem} Note that for any rising $\left\{ \mathfrak{z}_{n}\right\} _{n\geq0}\subseteq\mathbb{Z}_{p}$, the number of non-zero $p$-adic digits in $\mathfrak{z}_{n}$s tends to $\infty$ as $n\rightarrow\infty$; that is, $\lim_{n\rightarrow\infty}\sum_{j=1}^{p-1}\#_{p:j}\left(\mathfrak{z}_{n}\right)\overset{\mathbb{R}}{=}\infty$. However, the converse of this is not true: there are sequences $\left\{ \mathfrak{z}_{n}\right\} _{n\geq0}\subseteq\mathbb{Z}_{p}$ for which $\lim_{n\rightarrow\infty}\sum_{j=1}^{p-1}\#_{p:j}\left(\mathfrak{z}_{n}\right)\overset{\mathbb{R}}{=}\infty$ but $\lim_{n\rightarrow\infty}\mathfrak{z}_{n}\in\mathbb{N}_{0}$. \end{rem} \begin{example} For any $p$, let $\mathfrak{z}_{n}$'s sequence of $p$-adic digits be $n$ consecutive $0$s followed by infinitely many $1$s; then each $\mathfrak{z}_{n}$ has infinitely many non-zero $p$-adic digits, but the $\mathfrak{z}_{n}$s converge $p$-adically to $0$. \end{example} \vphantom{} In ``classical'' non-archimedean analysis, for a function $\chi$, $\chi$'s continuity is equivalent to the $K$-adic decay of its van der Put coefficients to zero. Rising-continuous functions, in contrast, arise when we relax this requirement of decay. Because we still need for there to be enough regularity to the van der Put coefficients of $\chi$ in order for its van der Put series to converge point-wise, it turns out the ``correct'' definition for rising-continuous functions is the limit condition we established for $\chi_{H}$ in our proof of \textbf{Lemma \ref{lem:Unique rising continuation and p-adic functional equation of Chi_H}} (page \pageref{lem:Unique rising continuation and p-adic functional equation of Chi_H}). \begin{defn} Let $K$ be a metrically complete non-archimedean valued field. A function $\chi:\mathbb{Z}_{p}\rightarrow K$ is said to be \textbf{($\left(p,K\right)$-adically)} \textbf{rising-continuous}\index{rising-continuous!function}\index{rising-continuous!left(p,Kright)-adically@$\left(p,K\right)$-adically}\textbf{ }whenever: \begin{equation} \lim_{n\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\overset{K}{=}\chi\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Definition of a rising-continuous function} \end{equation} We write $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ to denote the $K$-linear space\footnote{Note that this is a subspace of $B\left(\mathbb{Z}_{p},K\right)$.} of all rising-continuous functions $\chi:\mathbb{Z}_{p}\rightarrow K$. We call elements of \nomenclature{$\tilde{C}\left(\mathbb{Z}_{p},K\right)$}{set of $K$-valued rising-continuous functions on $\mathbb{Z}_{p}$}$\tilde{C}\left(\mathbb{Z}_{p},K\right)$ \textbf{rising-continuous functions}; or, more pedantically, \textbf{$\left(p,K\right)$-adic rising-continuous functions}, or $\left(p,q\right)$\textbf{-adic rising-continuous functions}, when $K$ is a $q$-adic field. \end{defn} \begin{rem} The convergence in (\ref{eq:Definition of a rising-continuous function}) only needs to occur \emph{point-wise}. \end{rem} \begin{example} Every continuous $f:\mathbb{Z}_{p}\rightarrow K$ is also rising-continuous, but not every rising-continuous function is continuous. As an example, consider: \begin{equation} f\left(\mathfrak{z}\right)=q^{\textrm{digits}_{p}\left(\mathfrak{z}\right)}\label{eq:Example of a discontinuous rising-continuous function} \end{equation} where $\textrm{digits}_{p}:\mathbb{Z}_{p}\rightarrow\mathbb{N}_{0}\cup\left\{ +\infty\right\} $ outputs the number of non-zero $p$-adic digits in $\mathfrak{z}$. In particular, $\textrm{digits}_{p}\left(\mathfrak{z}\right)=0$ if and only if $\mathfrak{z}=0$, $\textrm{digits}_{p}\left(\mathfrak{z}\right)=+\infty$ if and only if $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. As defined, $f\left(\mathfrak{z}\right)$ then satisfies: \begin{equation} f\left(\mathfrak{z}\right)\overset{K}{=}\begin{cases} q^{\textrm{digits}_{p}\left(\mathfrak{z}\right)} & \textrm{if }\mathfrak{z}\in\mathbb{N}_{0}\\ 0 & \textrm{if }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime} \end{cases} \end{equation} Since $\mathbb{Z}_{p}^{\prime}$ is dense in $\mathbb{Z}_{p}$, if $f$ \emph{were} continuous, the fact that it vanishes on $\mathbb{Z}_{p}$ would force it to be identically zero, which is not the case. Thus, $f$ \emph{cannot} be continuous. \end{example} \begin{example} Somewhat unfortunately, even if $\chi$ is rising continuous, we cannot guarantee that $\chi\left(\mathfrak{z}_{n}\right)$ converges to $\chi\left(\mathfrak{z}\right)$ for every $p$-adically rising sequence $\left\{ \mathfrak{z}_{n}\right\} _{n\geq1}$ with limit $\mathfrak{z}_{n}\rightarrow\mathfrak{z}$. $\chi_{q}$\textemdash the numen of the Shortened $qx+1$ map\textemdash gives us a simple example of this. As we shall soon prove (see \textbf{Theorem \ref{thm:rising-continuability of Generic H-type functional equations}} on page \pageref{thm:rising-continuability of Generic H-type functional equations}), $\chi_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$ is rising-continuous. Now, consider the sequence: \begin{equation} \mathfrak{z}_{n}\overset{\textrm{def}}{=}-2^{n}=\centerdot_{2}\underbrace{0\ldots0}_{n}\overline{1}\ldots \end{equation} where $\overline{1}$ indicates that all the remaining $2$-adic digits of $\mathfrak{z}_{n}$ are $1$s; the first $n$ $2$-adic digits of $\mathfrak{z}_{n}$ are $0$. This is a rising sequence, yet it converges $2$-adically to $0$. Moreover, observe that: \begin{equation} \chi_{q}\left(-2^{n}\right)=\frac{1}{2^{n}}\chi_{q}\left(-1\right)=\frac{1}{2^{n}}\chi_{q}\left(B_{2}\left(1\right)\right)=\frac{1}{2^{n}}\frac{\chi_{q}\left(1\right)}{1-M_{q}\left(1\right)}=\frac{1}{2^{n}}\frac{1}{2-q} \end{equation} Thus, $\chi_{q}\left(-2^{n}\right)$ does not even converge $q$-adically to a limit as $n\rightarrow\infty$, even though $\left\{ -2^{n}\right\} _{n\geq1}$ is a $2$-adically rising sequence which converges $2$-adically to $0$. \end{example} \begin{prop} \label{prop:vdP criterion for rising continuity}Let $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then, the van der Put series of $\chi$ (that is, $S_{p}\left\{ \chi\right\} $) converges at $\mathfrak{z}\in\mathbb{Z}_{p}$ if and only if: \begin{equation} \lim_{n\rightarrow\infty}c_{\left[\mathfrak{z}\right]_{p^{n}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=n\right]\overset{\mathbb{C}_{q}}{=}0\label{eq:vdP criterion for rising-continuity} \end{equation} where $c_{\left[\mathfrak{z}\right]_{p^{n}}}\left(\chi\right)$ is the $\left[\mathfrak{z}\right]_{p^{n}}$th van der Put coefficient of $\chi$. \end{prop} Proof: Using (\ref{eq:Inner term of vdP lambda decomposition}), we can write the van der Put series for $\chi$ as: \begin{equation} S_{p}\left\{ \chi\right\} \left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}c_{0}\left(\chi\right)+\sum_{n=1}^{\infty}c_{\left[\mathfrak{z}\right]_{p^{n}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Lambda-decomposed Chi} \end{equation} The ultrametric properties of $\mathbb{C}_{q}$ tell us that the $q$-adic convergence of this series at any given $\mathfrak{z}\in\mathbb{Z}_{p}$ is equivalent to: \begin{equation} \lim_{k\rightarrow\infty}c_{\left[\mathfrak{z}\right]_{p^{k}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)=k\right]\overset{\mathbb{C}_{q}}{=}0\label{eq:vdP coefficient decay for lemma} \end{equation} Q.E.D. \begin{thm} \label{thm:S_p}The operator $S_{p}$ which sends a function to its formal van der Put series is a isomorphism from the $K$-linear space $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ onto the subspace of $\textrm{vdP}\left(\mathbb{Z}_{p},K\right)$ consisting of all formal van der Put series\index{van der Put!series} which converge at every $\mathfrak{z}\in\mathbb{Z}_{p}$. In particular, we have that for every $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$: \vphantom{} I. $\chi=S_{p}\left\{ \chi\right\} $; \vphantom{} II. $\chi$ is uniquely represented by its van der Put series: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{K}{=}\sum_{n=0}^{\infty}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Chi vdP series} \end{equation} where the convergence is point-wise. \end{thm} Proof: Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ be arbitrary, by the truncated van der Put identity (\ref{eq:truncated van der Put identity}), $\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=S_{p:N}\left\{ \chi\right\} \left(\mathfrak{z}\right)$. Thus, the rising-continuity of $\chi$ guarantees that $S_{p:N}\left\{ \chi\right\} \left(\mathfrak{z}\right)$ converges in $K$ to $\chi\left(\mathfrak{z}\right)$ as $N\rightarrow\infty$ for each $\mathfrak{z}\in\mathbb{Z}_{p}$. By (\ref{eq:vdP criterion for rising-continuity}), this then implies that the van der Put coefficients of $\chi$ satisfy the conditions of \textbf{Proposition \ref{prop:vdP criterion for rising continuity}} for all $\mathfrak{z}\in\mathbb{Z}_{p}$, which then shows that the van der Put series $S_{p}\left\{ \chi\right\} \left(\mathfrak{z}\right)$ converges at every $\mathfrak{z}\in\mathbb{Z}_{p}$, and\textemdash moreover\textemdash that it converges to $\chi\left(\mathfrak{z}\right)$. This proves (I). As for (II), the uniqueness specified therein is equivalent to demonstrating that $S_{p}$ is an isomorphism in the manner described above. The proof of this is like so: \begin{itemize} \item (Surjectivity) Let $V\left(\mathfrak{z}\right)$ be any formal van der Put series which converges $q$-adically at every $\mathfrak{z}\in\mathbb{Z}_{p}$. Then, letting: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}S_{p:N}\left\{ V\right\} \left(\mathfrak{z}\right) \end{equation} we have that $V\left(m\right)=\chi\left(m\right)$, and hence, $V\left(\mathfrak{z}\right)=S_{p}\left\{ \chi\right\} $. Thus, $S_{p:N}\left\{ \chi\right\} \left(\mathfrak{z}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$, and so $\chi\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}S_{p:N}\left\{ V\right\} \left(\mathfrak{z}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ shows that $\chi$ is rising-continuous. This shows that $V=S_{p}\left\{ \chi\right\} $, and thus, that $S_{p}$ is surjective. \item (Injectivity) Let $\chi_{1},\chi_{2}\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ and suppose $S_{p}\left\{ \chi_{1}\right\} =S_{p}\left\{ \chi_{2}\right\} $. Then, by (I): \begin{equation} \chi_{1}\left(\mathfrak{z}\right)\overset{\textrm{(I)}}{=}S_{p}\left\{ \chi_{1}\right\} \left(\mathfrak{z}\right)=S_{p}\left\{ \chi_{2}\right\} \left(\mathfrak{z}\right)\overset{\textrm{(I)}}{=}\chi_{2}\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} which proves $\chi_{1}=\chi_{2}$, which proves the injectivity of $S_{p}$. \end{itemize} Thus, $S_{p}$ is an isomorphism. Q.E.D. \vphantom{} While this shows that rising-continuous functions are equivalent to those van der Put series which converge point-wise at every $\mathfrak{z}\in\mathbb{Z}_{p}$, they are also more than that. As suggested by the name, rising-continuous functions naturally occur when we wish to take a function on $\mathbb{N}_{0}$ and interpolate it to one on $\mathbb{Z}_{p}$. This will provide us with an alternative characterization of rising-continuous functions as being precisely those functions on $\mathbb{Z}_{p}$ whose restrictions to $\mathbb{N}_{0}$ admit interpolations to functions on $\mathbb{Z}_{p}$. I call the process underlying \textbf{rising-continuation}. \begin{defn} \label{def:rising-continuation}Let $\mathbb{F}$ be $\mathbb{Q}$ or a field extension thereof, and let $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}$ be a function. Letting $p,q$ be integers $\geq2$, with $q$ prime, we say $\chi$ has (or ``admits'') a \textbf{$\left(p,q\right)$-adic }\index{rising-continuation}\textbf{rising-continuation} \textbf{(to $K$)} whenever there is a metrically complete $q$-adic field extension $K$ of $\mathbb{F}$ and a rising-continuous function $\chi^{\prime}:\mathbb{Z}_{p}\rightarrow K$ so that $\chi^{\prime}\left(n\right)=\chi\left(n\right)$ for all $n\in\mathbb{N}_{0}$. We call any $\chi^{\prime}$ satisfying this property a \textbf{($\left(p,q\right)$-adic} or \textbf{$\left(p,K\right)$-adic)}\emph{ }\textbf{rising-continuation }of $\chi$ (to $K$). \end{defn} \begin{prop} \label{prop:rising-continuation admission}Let $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}$ be a function admitting a $\left(p,q\right)$-adic rising-continuation to $K$. Then: \vphantom{} I. The rising-continuation of $\chi$ is unique. As such, we will write $\chi^{\prime}$\nomenclature{$\chi^{\prime}$}{the rising-contination of $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}$} to denote the rising continuation of $\chi$. \vphantom{} II. \begin{equation} \chi^{\prime}\left(\mathfrak{z}\right)\overset{K}{=}\lim_{j\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{j}}\right)=\sum_{n=0}^{\infty}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Rising continuation limit formula} \end{equation} \end{prop} Proof: I. Suppose $\chi$ admits two (possibly distinct) rising-continuations, $\chi^{\prime}$ and $\chi^{\prime\prime}$. To see that $\chi^{\prime}$ and $\chi^{\prime\prime}$ must be the same, we note that since the restrictions of both $\chi^{\prime}$ and $\chi^{\prime\prime}$ to $\mathbb{N}_{0}$ are, by definition, equal to $\chi$, $\chi^{\prime}\left(n\right)=\chi^{\prime\prime}\left(n\right)$ for all $n\in\mathbb{N}_{0}$. So, let $\mathfrak{z}$ be an arbitrary element of $\mathbb{Z}_{p}^{\prime}$; necessarily, $\mathfrak{z}$ has infinitely many non-zero $p$-adic digits. Consequently, $\left\{ \left[\mathfrak{z}\right]_{p^{j}}\right\} _{j\geq1}$ is a rising sequence of non-negative integers converging to $\mathfrak{z}$. As such, using rising-continuity of $\chi^{\prime}$ and $\chi^{\prime\prime}$, we can write: \[ \lim_{j\rightarrow\infty}\chi^{\prime}\left(\left[\mathfrak{z}\right]_{p^{j}}\right)\overset{K}{=}\chi^{\prime}\left(\mathfrak{z}\right) \] \[ \lim_{j\rightarrow\infty}\chi^{\prime\prime}\left(\left[\mathfrak{z}\right]_{p^{j}}\right)\overset{K}{=}\chi^{\prime\prime}\left(\mathfrak{z}\right) \] Since the $\left[\mathfrak{z}\right]_{p^{j}}$s are integers, this yields: \[ \chi^{\prime}\left(\mathfrak{z}\right)=\lim_{j\rightarrow\infty}\chi^{\prime}\left(\left[\mathfrak{z}\right]_{p^{j}}\right)=\lim_{j\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{j}}\right)=\lim_{j\rightarrow\infty}\chi^{\prime\prime}\left(\left[\mathfrak{z}\right]_{p^{j}}\right)=\chi^{\prime\prime}\left(\mathfrak{z}\right) \] Since $\mathfrak{z}$ was arbitrary, we have that $\chi^{\prime}\left(\mathfrak{z}\right)=\chi^{\prime\prime}\left(\mathfrak{z}\right)$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. Thus, $\chi^{\prime}$ and $\chi^{\prime\prime}$ are equal to one another on both $\mathbb{Z}_{p}^{\prime}$ and $\mathbb{N}_{0}$, and hence, on all of $\mathbb{Z}_{p}$. This shows that $\chi^{\prime}$ and $\chi^{\prime\prime}$ are in fact the same function, and therefore proves the uniqueness of $\chi$'s rising-continuation. \vphantom{} II. As a rising-continuous function, $\chi^{\prime}\left(\mathfrak{z}\right)$ is uniquely determined by its values on $\mathbb{N}_{0}$. Since $\chi^{\prime}\mid_{\mathbb{N}_{0}}=\chi$, $\chi^{\prime}$ and $\chi$ then have the same van der Put coefficients. Using the van der Put identity (\ref{eq:van der Put identity}) then gives (\ref{eq:Rising continuation limit formula}). Q.E.D. \vphantom{} Now we have our characterization of rising-continuous functions in terms of rising-continuability. \begin{thm} \label{thm:characterization of rising-continuability}Let $\mathbb{F}$ be $\mathbb{Q}$ or a field extension thereof, let $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}$ be a function, let $p,q$ be integers $\geq2$, with $q$ prime, and let $K$ be a metrically complete $q$-adic field extension of $\mathbb{F}$. Then, the following are equivalent: \vphantom{} I. $\chi$ admits a $\left(p,q\right)$-adic rising-continuation to $K$. \vphantom{} II. For each $\mathfrak{z}\in\mathbb{Z}_{p}$, $\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ converges to a limit in $K$ as $n\rightarrow\infty$. \end{thm} Proof: i. Suppose (I) holds. Then, by \textbf{Proposition \ref{prop:rising-continuation admission}}, (\ref{eq:Rising continuation limit formula}) holds, which shows that (II) is true. \vphantom{} ii. Conversely, suppose (II) holds. Then, by the van der Put identity (\textbf{Proposition \ref{prop:vdP identity}}), $S_{p}\left\{ \chi\right\} $ is a rising-continuous function whose restriction to $\mathbb{N}_{0}$ is equal to $\chi$, which means that $S_{p}\left\{ \chi\right\} $ is the rising-continuation of $\chi$, and thus, that $\chi$ is rising-continuable. So, (II) and (I) are equivalent. Q.E.D. \vphantom{} As a consequence of this, we then have that $K$-linear space (under point-wise addition) of all $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}$ which admit $\left(p,q\right)$-adic rising-continuations to $K$ is then isomorphic to $\tilde{C}\left(\mathbb{Z}_{p},K\right)$, with the isomorphism being the act of rising-continuation: $\chi\mapsto\chi^{\prime}$. As such, \textbf{\emph{from now on (for the most part) we will identify $\chi^{\prime}$ and $\chi=\chi^{\prime}\mid_{\mathbb{N}_{0}}$ and treat them as one and the same.}} The exceptions to this convention will be those occasions where we have a $\left(p,q\right)$-adically continuable function $\chi:\mathbb{N}_{0}\rightarrow\mathbb{F}\subseteq K$ and a function $\eta:\mathbb{Z}_{p}\rightarrow K$, and we wish to show that $\mu$ is in fact equal $\chi^{\prime}$. Using $\chi^{\prime}$ in this context will then allow us to distinguish between $\mu$ (our candidate for $\chi^{\prime}$) and the actual ``correct'' rising-continuation of $\chi$. Because many of our rising-continuous functions will emerge as interpolations of solutions of systems of functional equations on $\mathbb{N}_{0}$, our next theorem will be quite the time-saver. \begin{thm} \label{thm:rising-continuability of Generic H-type functional equations}Let $H$ be a semi-basic $p$-Hydra map which fixes $0$, and consider the system of functional equations\index{functional equation}: \begin{equation} f\left(pn+j\right)=\frac{\mu_{j}}{p}f\left(n\right)+c_{j},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} ,\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Generic H-type functional equations} \end{equation} where $\mu_{j}/p=H_{j}^{\prime}\left(0\right)$. Then: \vphantom{} I. There is a unique function $\chi:\mathbb{N}_{0}\rightarrow\overline{\mathbb{Q}}$ such that $f=\chi$ is a solution of \emph{(\ref{eq:Generic H-type functional equations})}. \vphantom{} II. The solution $\chi$ \emph{(\ref{eq:Generic H-type functional equations})} is rising-continuable\index{rising-continuation} to a function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q_{H}}$. Moreover, this continuation satisfies: \begin{equation} \chi\left(p\mathfrak{z}+j\right)=\frac{\mu_{j}}{p}\chi\left(\mathfrak{z}\right)+c_{j},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} ,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Rising-continuation Generic H-type functional equations} \end{equation} \vphantom{} III. The function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q_{H}}$ described in \emph{(III)} is the unique rising-continuous function $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q_{H}}$ satisfying \emph{(\ref{eq:Rising-continuation Generic H-type functional equations})}. \end{thm} Proof: I. Let $f:\mathbb{N}_{0}\rightarrow\overline{\mathbb{Q}}$ be any solution of (\ref{eq:Generic H-type functional equations}). Setting $n=j=0$ yields: \begin{align} f\left(0\right) & =\frac{\mu_{0}}{p}f\left(0\right)+c_{0}\\ & \Updownarrow\\ f\left(0\right) & =\frac{c_{0}}{1-\frac{\mu_{0}}{p}} \end{align} Since $H$ is semi-basic, $\mu_{0}/p\neq1$, so $f\left(0\right)$ is well-defined. Then, we have that: \begin{equation} f\left(j\right)=\frac{\mu_{j}}{p}f\left(0\right)+c_{j},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \end{equation} and, more generally: \begin{equation} f\left(m\right)=\frac{\mu_{\left[m\right]_{p}}}{p}f\left(\frac{m-\left[m\right]_{p}}{p}\right)+c_{\left[m\right]_{p}},\textrm{ }\forall m\in\mathbb{N}_{0}\label{eq:f,m, digit shifting} \end{equation} Since the map $m\mapsto\frac{m-\left[m\right]_{p}}{p}$ sends the non-negative integer: \[ m=m_{0}+m_{1}p+\cdots+m_{\lambda_{p}\left(m\right)-1}p^{\lambda_{p}\left(m\right)-1} \] to the integer: \[ m=m_{1}+m_{2}p+\cdots+m_{\lambda_{p}\left(m\right)-1}p^{\lambda_{p}\left(m\right)-2} \] it follows that $m\mapsto\frac{m-\left[m\right]_{p}}{p}$ eventually iterates every $m\in\mathbb{N}_{0}$ to $0$, and hence (\ref{eq:f,m, digit shifting}) implies that, for every $m\in\mathbb{N}_{0}$, $f\left(m\right)$ is entirely determined by $f\left(0\right)$ and the $c_{j}$s. Since $f\left(0\right)$ is uniquely determined by $c_{0}$ and $H$, the equation (\ref{eq:Generic H-type functional equations}) possesses exactly one solution. Let us denote this solution by $\chi$. \vphantom{} II. Because $H$ is semi-basic, we can repeat for $\chi$ the argument we used to prove the existence of $\chi_{H}$'s rising-continuation in \textbf{Lemma \ref{lem:Unique rising continuation and p-adic functional equation of Chi_H}} (page \pageref{lem:Unique rising continuation and p-adic functional equation of Chi_H}): since any $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ has infinitely many non-zero $p$-adic digits, the product of $\mu_{j}/p$ taken over all the digits of such a $\mathfrak{z}$ will converge $q_{H}$-adically to zero. Then, seeing as (\ref{eq:Generic H-type functional equations}) shows that $\chi\left(\mathfrak{z}\right)$ will be a sum of the form: \begin{equation} \beta_{0}+\alpha_{1}\beta_{1}+\alpha_{1}\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{2}\alpha_{3}\beta_{3}+\cdots \end{equation} (where for each $n$, $\beta_{n}$ is one of the $c_{j}$s and $\alpha_{n}$ is $\mu_{j_{n}}/p$, where $j_{n}$ is the $n$th $p$-adic digit of $\mathfrak{z}$) this sum will converge in $\mathbb{C}_{q_{H}}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ because of the estimate guaranteed by $H$'s semi-basicness: \begin{equation} \left\|\frac{\mu_{j}}{p}\right\|_{q_{H}}<1,\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \end{equation} This proves $\chi$ is rising-continuable. \vphantom{} III. Because $\chi$ admits a rising continuation, its continuation at any $\mathfrak{z}\in\mathbb{Z}_{p}$ is given by the value of $\chi$'s van der Put series at $\mathfrak{z}$. As such: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q_{H}}}{=}S_{p}\left\{ \chi\right\} \left(\mathfrak{z}\right)\overset{\mathbb{C}_{q_{H}}}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{equation} Since $\chi$ satisfies the functional equations (\ref{eq:Generic H-type functional equations}), we can write: \begin{equation} \chi\left(p\left[\mathfrak{z}\right]_{p^{N}}+j\right)\overset{\overline{\mathbb{Q}}}{=}\frac{\mu_{j}}{p}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+c_{j} \end{equation} for all $\mathfrak{z}\in\mathbb{Z}_{p}$, all $j\in\left\{ 0,\ldots,p-1\right\} $, and all $N\geq0$. Letting $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ be arbitrary, we let $N\rightarrow\infty$. This gives us: \begin{equation} \chi\left(p\mathfrak{z}+j\right)\overset{\mathbb{C}_{q_{H}}}{=}\frac{\mu_{j}}{p}\chi\left(\mathfrak{z}\right)+c_{j},\textrm{ }\forall j,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime} \end{equation} Note that these identities automatically hold for $\mathfrak{z}\in\mathbb{N}_{0}$ seeing as those particular cases were governed by (\ref{eq:Generic H-type functional equations}). The uniqueness of $\chi$ as a solution of (\ref{eq:Rising-continuation Generic H-type functional equations}) follows from the fact that any $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q_{H}}$ satisfying (\ref{eq:Rising-continuation Generic H-type functional equations}) has a restriction to $\mathbb{N}_{0}$ which satisfies (\ref{eq:Generic H-type functional equations}). This then forces $f=\chi$. Q.E.D. \vphantom{} A slightly more general version of this type of argument (which will particularly useful in Chapters 4 and 6) is as follows: \begin{lem} \label{lem:rising-continuations preserve functional equations}Fix integers $p,q\geq2$, let $K$ be a metrically complete $q$-adic field, and let $\Phi_{j}:\mathbb{Z}_{p}\times K\rightarrow K$ be continuous for $j\in\left\{ 0,\ldots,p-1\right\} $. If $\chi:\mathbb{N}_{0}\rightarrow K$ is rising-continuable to an element $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, and if: \[ \chi\left(pn+j\right)=\Phi_{j}\left(n,\chi\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \] then: \[ \chi\left(p\mathfrak{z}+j\right)=\Phi_{j}\left(\mathfrak{z},\chi\left(\mathfrak{z}\right)\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \] \end{lem} Proof: Let everything be as given. Then, since $\chi$ is rising continuous, we have that: \[ \chi\left(\mathfrak{z}\right)\overset{K}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \] Since: \[ \chi\left(p\left[\mathfrak{z}\right]_{p^{N}}+j\right)=\Phi_{j}\left(\left[\mathfrak{z}\right]_{p^{N}},\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right) \] holds true for all $\mathfrak{z}\in\mathbb{Z}_{p}$ and all $N\geq0$, the rising-continuity of $\chi$ and the continuity of $\Phi_{j}$ then guarantee that: \begin{align} \chi\left(p\mathfrak{z}+j\right) & \overset{K}{=}\lim_{N\rightarrow\infty}\chi\left(p\left[\mathfrak{z}\right]_{p^{N}}+j\right)\\ & \overset{K}{=}\lim_{N\rightarrow\infty}\Phi_{j}\left(\left[\mathfrak{z}\right]_{p^{N}},\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right)\\ & \overset{K}{=}\Phi_{j}\left(\mathfrak{z},\chi\left(\mathfrak{z}\right)\right) \end{align} as desired. Q.E.D. \subsection{\label{subsec:3.2.2 Truncations-=00003D000026-The}Truncations and The Banach Algebra of Rising-Continuous Functions} RECALL THAT WE WRITE $\left\Vert \cdot\right\Vert _{p,q}$ TO DENOTE THE $\left(p,q\right)$-adic SUPREMUM NORM. \vphantom{} In this section, we will examine the structure of the vector space $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ of rising-continuous functions $\chi:\mathbb{Z}_{p}\rightarrow K$. We will demonstrate that $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ is a Banach algebra over $K$\textemdash with the usual ``point-wise multiplication of functions'' as its multiplication operation. Not only that, we will also see that $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ extends $C\left(\mathbb{Z}_{p},K\right)$, containing it as a proper sub-algebra. Before we can even begin our discussion of Banach algebras, however, we need to introduce a simple construction which will be of extraordinary importance in our analyses of $\chi_{H}$ in Chapter 4 and beyond. To begin, as we saw in Subsection \ref{subsec:3.1.4. The--adic-Fourier}, the Fourier Transform of a continuous $\left(p,q\right)$-adic function $f$ can be computed using $f$'s van der Put coefficients via equation (\ref{eq:Definition of (p,q)-adic Fourier Coefficients}) from \textbf{Definition \ref{def:pq adic Fourier coefficients}} \begin{equation} \hat{f}\left(t\right)\overset{\mathbb{C}_{q}}{=}\sum_{n=\frac{1}{p}\left\|t\right\|_{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2n\pi it},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} Because the convergence of this infinite series \emph{requires }$\lim_{n\rightarrow\infty}\left\|c_{n}\left(f\right)\right\|_{q}=0$, this formula for $\hat{f}$ is not compatible with general rising-continuous functions; the van der Put coefficients need not converge $q$-adically to $0$. As will be shown in Chapter 4, we can get around the non-convergence of (\ref{eq:Definition of (p,q)-adic Fourier Coefficients}) for an arbitrary rising-continuous function $\chi$ by replacing $\chi$ with a locally constant approximation. I call these approximations \textbf{truncations}, and\textemdash as locally constant functions\textemdash they have the highly desirable property of $\left(p,q\right)$-adic continuity. \begin{defn}[\textbf{$N$th truncations}] \label{def:Nth truncation}For any $\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and any $N\in\mathbb{N}_{0}$, the \index{$N$th truncation}\textbf{$N$th truncation of $\chi$}, denoted $\chi_{N}$, is the function $\chi_{N}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ defined by: \begin{equation} \chi_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Definition of Nth truncation} \end{equation} We also extend this notation to negative $N$ by defining $\chi_{N}\left(\mathfrak{z}\right)$ to be identically zero whenever $N<0$. \end{defn} \begin{rem} For any $N\in\mathbb{N}_{0}$: \begin{equation} \chi_{N}\left(\mathfrak{z}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Nth truncation of Chi in terms of Chi} \end{equation} Combining (\ref{eq:Nth truncation of Chi in terms of Chi}) and (\ref{eq:truncated van der Put identity}), we then have that: \begin{equation} \chi_{N}\left(\mathfrak{z}\right)=\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]=\sum_{n=0}^{p^{N}-1}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]=\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\label{eq:Nth truncation and truncated van-der-Put identity compared} \end{equation} Moreover, note that $\chi_{N}\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ for all $N\in\mathbb{N}_{0}$. \end{rem} \begin{rem} Although Schikhof did not make any significant investigations with truncations, he was aware of there existence. There is an exercise in \cite{Ultrametric Calculus} in the section on the van der Put basis which asks the reader to show that the $N$th truncation of $\chi$ is the best possible $\left(p,q\right)$-adic approximation of $\chi$ which is constant with respect to inputs modulo $p^{N}$. \end{rem} \begin{prop} \label{prop:Truncations converge pointwise iff rising-continuous}Let $\chi:\mathbb{Z}_{p}\rightarrow K$ be\textbf{ any}\emph{ }function. Then, $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ if and only if $\chi_{N}$ converges in $K$ to $\chi$ point-wise on $\mathbb{Z}_{p}$ as $N\rightarrow\infty$. \end{prop} Proof: Since $\chi_{N}\left(\mathfrak{z}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$, this proposition is just a restatement of the definition of what it means for $\chi$ to be rising continuous ($\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ converges to $\chi\left(\mathfrak{z}\right)$ point-wise everywhere). Q.E.D. \begin{prop} \label{prop:Unif. convergence of truncation equals continuity}Let $f:\mathbb{Z}_{p}\rightarrow K$ be \textbf{any} function. Then, $f$ is continuous if and only if $f_{N}$ converges in $K$ to $f$ uniformly on $\mathbb{Z}_{p}$ as $N\rightarrow\infty$. \end{prop} Proof: I. If the $f_{N}$s converge uniformly to $f$, $f$ is continuous, seeing as the $f_{N}$ are locally constant\textemdash and hence, continuous\textemdash and seeing as how the uniform limit of continuous functions is continuous. \vphantom{} II. Conversely, if $f$ is continuous then, since $\mathbb{Z}_{p}$ is compact, $f$ is uniformly continuous. So, letting $\epsilon>0$, pick $\delta$ so that $\left\|f\left(\mathfrak{z}\right)-f\left(\mathfrak{y}\right)\right\|_{q}<\epsilon$ for all $\mathfrak{z},\mathfrak{y}\in\mathbb{Z}_{p}$ with $\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}<\delta$. Note that: \[ \left\|\mathfrak{z}-\left[\mathfrak{z}\right]_{p^{N}}\right\|_{p}\leq p^{-N},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall N\in\mathbb{N}_{0} \] So, choose $N$ large enough so that $p^{-N}<\delta$. Then $\left\|\mathfrak{z}-\left[\mathfrak{z}\right]_{p^{N}}\right\|_{p}<\delta$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$, and so: \[ \sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right)\right\|_{q}=\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-f\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}<\epsilon \] which proves that the $f_{N}$s converge uniformly to $f$. Q.E.D. \begin{rem} As is proved in \textbf{Theorem \ref{thm:vdP basis theorem}} (see (\ref{eq:Fourier transform of Nth truncation in terms of vdP coefficients}) on page \pageref{eq:Fourier transform of Nth truncation in terms of vdP coefficients}), the formula (\ref{eq:Definition of (p,q)-adic Fourier Coefficients}) can be applied to: \begin{equation} f_{N}\left(\mathfrak{z}\right)=f\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\sum_{n=0}^{p^{N}-1}c_{n}\left(f\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] \end{equation} whereupon we obtain: \begin{equation} \hat{f}_{N}\left(t\right)=\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{p^{N}-1}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int} \end{equation} where $\hat{f}_{N}$ is the Fourier transform of the $N$th truncation of $f$. While this formula for $\hat{f}_{N}$ can be used to perform the kind of Fourier analysis we shall do in Chapters 4 and 6, it turns out to be much easier to work with the $N$th truncation directly instead of using the van der Put coefficients in this manner, primarily because the formula for $c_{n}\left(\chi_{H}\right)$ becomes unwieldy\footnote{See\textbf{ Proposition \ref{prop:van der Put series for Chi_H}} on page \pageref{prop:van der Put series for Chi_H} for the details.} when $H$ is a $p$-Hydra map for $p\geq3$. Nevertheless, the van der Put coefficient expression for $\hat{f}_{N}\left(t\right)$ is of interest in its own right, and it may be worth investigating how the above formula might be inverted so as to produce expressions for $c_{n}\left(f\right)$ in terms of $\hat{f}_{N}\left(t\right)$. That being said, both of these realizations of a function's $N$th truncation will be of use to us, both now and in the future. \end{rem} \vphantom{} Now we can begin our approach of the Banach algebra of rising-continuous\index{rising-continuous!Banach algebra of} functions. \begin{defn} For any integer $n\geq0$, we define the operator \nomenclature{$\nabla_{p^{n}}$}{ }$\nabla_{p^{n}}:B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\rightarrow B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ by: \begin{equation} \nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\begin{cases} \chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right) & \textrm{if }n\geq1\\ \chi\left(0\right) & \textrm{if }n=0 \end{cases},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Definition of Del p^n of Chi} \end{equation} We then define the \textbf{$\nabla$-norm }(``del norm'') $\left\Vert \cdot\right\Vert _{\nabla}:B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\rightarrow\left[0,\infty\right)$ by: \begin{equation} \left\Vert \chi\right\Vert _{\nabla}\overset{\textrm{def}}{=}\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}=\sup_{n\geq0}\left(\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}\right)\label{eq:Defintion of Del-norm} \end{equation} \end{defn} \begin{prop} Let $\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \begin{equation} S_{p:N}\left\{ \chi\right\} \left(\mathfrak{z}\right)=\sum_{n=0}^{N}\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall N\in\mathbb{N}_{0}\label{eq:Partial vdP series in terms of Del} \end{equation} Moreover, we have the formal identity: \begin{equation} S_{p}\left\{ \chi\right\} \left(\mathfrak{z}\right)=\sum_{n=0}^{\infty}\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\label{eq:vdP series in terms of Del} \end{equation} This identity holds for any $\mathfrak{z}\in\mathbb{Z}_{p}$ for which either the left- or right-hand side converges in $\mathbb{C}_{q}$. In particular, (\ref{eq:vdP series in terms of Del}) holds in $\mathbb{C}_{q}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ whenever $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \end{prop} Proof: Sum the telescoping series. Q.E.D. \begin{prop} Let $\chi,\eta\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, for all $n\in\mathbb{N}_{0}$ and all $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{equation} \nabla_{p^{n}}\left\{ \chi\cdot\eta\right\} \left(\mathfrak{z}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\nabla_{p^{n}}\left\{ \eta\right\} \left(\mathfrak{z}\right)+\eta\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\label{eq:Quasi-derivation identity for del} \end{equation} using the abuse of notation $\chi\left(\left[\mathfrak{z}\right]_{p^{0-1}}\right)$ to denote the zero function in the $n=0$ case. \end{prop} Proof: Using truncation notation: \begin{align} \chi_{n-1}\left(\mathfrak{z}\right)\nabla_{p^{n}}\left\{ \eta\right\} \left(\mathfrak{z}\right)+\eta_{n}\left(\mathfrak{z}\right)\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right) & =\chi_{n-1}\left(\mathfrak{z}\right)\left(\eta_{n}\left(\mathfrak{z}\right)-\eta_{n-1}\left(\mathfrak{z}\right)\right)\\ & +\eta_{n}\left(\mathfrak{z}\right)\left(\chi_{n}\left(\mathfrak{z}\right)-\chi_{n-1}\left(\mathfrak{z}\right)\right)\\ & =\bcancel{\chi_{n-1}\left(\mathfrak{z}\right)\eta_{n}\left(\mathfrak{z}\right)}-\chi_{n-1}\left(\mathfrak{z}\right)\eta_{n-1}\left(\mathfrak{z}\right)\\ & +\eta_{n}\left(\mathfrak{z}\right)\chi_{n}\left(\mathfrak{z}\right)-\cancel{\eta_{n}\left(\mathfrak{z}\right)\chi_{n-1}\left(\mathfrak{z}\right)}\\ & =\chi_{n}\left(\mathfrak{z}\right)\eta_{n}\left(\mathfrak{z}\right)-\chi_{n-1}\left(\mathfrak{z}\right)\eta_{n-1}\left(\mathfrak{z}\right)\\ & =\nabla_{p^{n}}\left\{ \chi\cdot\eta\right\} \left(\mathfrak{z}\right) \end{align} Q.E.D. \begin{prop} Let $\chi\in B\left(\mathbb{Z}_{p},K\right)$. Then, $S_{p}\left\{ \chi\right\} $ is identically zero whenever $\chi$ vanishes on $\mathbb{N}_{0}$. \end{prop} Proof: If $\chi\left(n\right)=0$ for all $n\in\mathbb{N}_{0}$, then $c_{n}\left(\chi\right)=0$ for all $n\in\mathbb{N}_{0}$, and hence $S_{p}\left\{ \chi\right\} $ is identically zero. Q.E.D. \vphantom{} The moral of this proposition is that the operator $S_{p}$ behaves like a projection operator on $B\left(\mathbb{Z}_{p},K\right)$. For example, given any rising-continuous function $\chi$ and any function $f\in B\left(\mathbb{Z}_{p},K\right)$ with $f\left(n\right)=0$ for all $n\in\mathbb{N}_{0}$, we have that: \begin{equation} S_{p}\left\{ \chi+f\right\} =S_{p}\left\{ \chi\right\} =\chi \end{equation} The reason why we say $S_{p}$ behaves merely ``like'' a projection operator on $B\left(\mathbb{Z}_{p},K\right)$\textemdash rather than proving that $S_{p}$ actually \emph{is }a projection operator on $B\left(\mathbb{Z}_{p},K\right)$\textemdash is because there exist $f\in B\left(\mathbb{Z}_{p},K\right)$ for which $S_{p}\left\{ f\right\} \left(\mathfrak{z}\right)$ fails to converge at \emph{any} $\mathfrak{z}\in\mathbb{Z}_{p}$. Case in point: \begin{example} Let $f\in B\left(\mathbb{Z}_{p},K\right)$ be defined by: \begin{equation} f\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\begin{cases} \mathfrak{z} & \textrm{if }\mathfrak{z}\in\mathbb{N}_{0}\\ 0 & \textrm{else} \end{cases}\label{eq:Example of a bounded (p,q)-adic function whose van der Put series is divergent.} \end{equation} Then, we have the formal identity: \begin{align} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right) & =\sum_{n=1}^{\infty}n\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\\ & =\sum_{n=1}^{\infty}\sum_{m=p^{n-1}}^{p^{n}-1}m\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]\\ & =\sum_{n=1}^{\infty}\left(\sum_{m=0}^{p^{n}-1}m\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]-\sum_{m=0}^{p^{n-1}-1}m\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]\right) \end{align} Here: \[ \sum_{m=0}^{p^{n}-1}m\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]=\left[\mathfrak{z}\right]_{p^{n}} \] because $m=\left[\mathfrak{z}\right]_{p^{n}}$ is the unique integer in $\left\{ 0,\ldots,p^{n}-1\right\} $ which is congruent to $\mathfrak{z}$ mod $p^{n}$. On the other hand, for the $m$-sum with $p^{n-1}-1$ as an upper bound, we end up with: \[ \sum_{m=0}^{p^{n-1}-1}m\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]=\left[\mathfrak{z}\right]_{p^{n}}\left[\left[\mathfrak{z}\right]_{p^{n}}<p^{n-1}\right] \] and so: \begin{align} S_{p}\left\{ f\right\} \left(\mathfrak{z}\right) & =\sum_{n=1}^{\infty}\left[\mathfrak{z}\right]_{p^{n}}\left(1-\left[\left[\mathfrak{z}\right]_{p^{n}}<p^{n-1}\right]\right)\\ & =\sum_{n=1}^{\infty}\left[\mathfrak{z}\right]_{p^{n}}\left[\left[\mathfrak{z}\right]_{p^{n}}\geq p^{n-1}\right] \end{align} Consequently: \[ S_{p}\left\{ f\right\} \left(-1\right)=\sum_{n=1}^{\infty}\left(p^{n}-1\right)\left[p^{n}-1\geq p^{n-1}\right]=\sum_{n=1}^{\infty}\left(p^{n}-1\right) \] which does not converge in $\mathbb{C}_{q}$ because $\left\|p^{n}-1\right\|_{q}$ does not tend to $0$ in $\mathbb{R}$ as $n\rightarrow\infty$. \end{example} \vphantom{} The above example also shows that $S_{p}$ is as much a continuation-creating operator as it is a projection operator. Bounded functions which do not behave well under limits of sequences in $\mathbb{Z}_{p}$ will have ill-behaved images under $S_{p}$. This motivates the following definition: \begin{defn} We write \nomenclature{$\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$}{set of $f\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ so that $S_{p}\left\{ f\right\} \in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$.}$\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ to denote the set of all $f\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ so that $S_{p}\left\{ f\right\} \in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ (i.e., $S_{p}\left\{ f\right\} $ converges point-wise in $\mathbb{C}_{q}$). \end{defn} \begin{rem} Note that $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\neq\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. The only function in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which vanishes on $\mathbb{N}_{0}$ is the constant function $0$. Consequently, any function in $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which is supported on $\mathbb{Z}_{p}^{\prime}$ is an element of $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which is not in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. That such a function is in $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is, of course, because its image under $S_{p}$ is identically $0$. \end{rem} \begin{prop} The map: \[ \left(\chi,\eta\right)\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\times B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\mapsto\left\Vert \chi-\eta\right\Vert _{\nabla}\in\left[0,\infty\right) \] defines a non-archimedean semi-norm on $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \end{prop} Proof: I. (Positivity) Observe that $\left\Vert \chi\right\Vert _{\nabla}<\infty$ for all $\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. However, $\left\Vert \cdot\right\Vert _{\nabla}$ is \emph{not }positive-definite; let $\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ be any function supported on $\mathbb{Z}_{p}^{\prime}$. Then, $\chi$ vanishes on $\mathbb{N}_{0}$, and so, $\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=0$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ and all $n\in\mathbb{N}_{0}$. This forces $\left\Vert \chi\right\Vert _{\nabla}$ to be zero; however, $\chi$ is not identically zero. So, $\left\Vert \cdot\right\Vert _{\nabla}$ cannot be a norm on $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ \vphantom{} II. Since $\nabla_{p^{n}}$ is linear for all $n\geq0$, for any $\mathfrak{a}\in\mathbb{C}_{q}$, we have that: \[ \left\Vert \mathfrak{a}\chi\right\Vert _{\nabla}=\sup_{n\geq0}\left\Vert \mathfrak{a}\nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}=\left\|\mathfrak{a}\right\|_{q}\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}=\left\|\mathfrak{a}\right\|_{q}\left\Vert \chi\right\Vert _{\nabla} \] for all $\chi\in B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. As for the ultrametric inequality, observing that: \[ \left\Vert \chi-\eta\right\Vert _{\nabla}=\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi-\eta\right\} \right\Vert _{p,q}=\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} -\nabla_{p^{n}}\left\{ \eta\right\} \right\Vert _{p,q} \] the fact that $d\left(\chi,\eta\right)=\left\Vert \chi-\eta\right\Vert _{p,q}$ defines an ultrametric on $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ lets us write: \begin{align} \left\Vert \chi-\eta\right\Vert _{\nabla} & =\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} -\nabla_{p^{n}}\left\{ \eta\right\} \right\Vert _{p,q}\\ & \leq\sup_{n\geq0}\max\left\{ \left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q},\left\Vert \nabla_{p^{n}}\left\{ \eta\right\} \right\Vert _{p,q}\right\} \\ & =\max\left\{ \left\Vert \chi\right\Vert _{\nabla},\left\Vert \eta\right\Vert _{\nabla}\right\} \end{align} which proves the ultrametric inequality. Thus, $\left\Vert \cdot\right\Vert _{\nabla}$ is a non-archimedean semi-norm on $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. The arguments for $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ are identical. Q.E.D. \vphantom{} Next, we show that $\nabla$-(semi)norm dominates the $\left(p,q\right)$-adic supremum norm. \begin{prop} \label{prop:continuous embeds in rising-continuous}If $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, then: \begin{equation} \left\Vert \chi\right\Vert _{p,q}\leq\left\Vert \chi\right\Vert _{\nabla}\label{eq:Supremum norm is dominated by Del norm} \end{equation} \end{prop} Proof: If $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, then the van der Put series for $\chi$ converges $q$-adically to $\chi$ point-wise over $\mathbb{Z}_{p}$. As such: \begin{align} \left\|\chi\left(\mathfrak{z}\right)\right\|_{q} & =\left\|\sum_{n=0}^{\infty}\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}\\ & \leq\sup_{n\geq0}\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q} \end{align} Taking suprema over $\mathfrak{z}\in\mathbb{Z}_{p}$ gives: \begin{align} \left\Vert \chi\right\Vert _{p,q} & =\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}\\ & \leq\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\sup_{n\geq0}\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}\\ \left(\sup_{m}\sup_{n}x_{m,n}=\sup_{n}\sup_{m}x_{m,n}\right); & \leq\sup_{n\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}\\ & =\left\Vert \chi\right\Vert _{\nabla} \end{align} Q.E.D. \begin{lem} $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ is a non-archimedean normed linear space. In particular, we can identify $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ with the image of $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ under $S_{p}$. Moreover, $S_{p}$ is then an isometry of $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \end{lem} Proof: By the First Isomorphism Theorem from abstract algebra, $S_{p}$ maps the $\mathbb{C}_{q}$-linear space $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ isomorphically onto $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)/\textrm{Ker}\left(S_{p}\right)$. Since $\left\Vert \cdot\right\Vert _{\nabla}$ is a semi-norm on $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, and because the set of non-zero $\chi\in\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ possessing $\left\Vert \chi\right\Vert _{\nabla}=0$ is equal to $\textrm{Ker}\left(S_{p}\right)$, the non-archimedean semi-norm $\left\Vert \cdot\right\Vert _{\nabla}$ is then a norm on the quotient space $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)/\textrm{Ker}\left(S_{p}\right)$. To conclude, we need only show that $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)/\textrm{Ker}\left(S_{p}\right)$ can be identified with $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. To do so, let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ be arbitrary. Then, $\chi$ is an element of $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. As we saw in the previous subsection (viz., \textbf{Theorem \ref{thm:S_p}}), $S_{p}$ is the identity map on $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Consequently, $S_{p}\left\{ \chi\right\} =\chi$, and so, $\chi\in\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. For the other direction, let $\chi$ and $\eta$ be two elements of $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which belong to the same equivalence class in $\tilde{B}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)/\textrm{Ker}\left(S_{p}\right)$. Then, $S_{p}\left\{ \chi-\eta\right\} $ is the zero function. By the linearity of $S_{p}$, this means $S_{p}\left\{ \chi\right\} =S_{p}\left\{ \eta\right\} $. So, $S_{p}\left\{ \chi\right\} $ and $S_{p}\left\{ \eta\right\} $ have the same van der Put series, and that series defines an element of $B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Since this series converges everywhere point-wise, the van der Put identity (\textbf{Proposition \ref{prop:vdP identity}}) shows it is an element of $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Thus, the two spaces are equivalent. Q.E.D. \begin{thm} $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ is a non-archimedean Banach algebra over $\mathbb{C}_{q}$, with point-wise multiplication of functions as the algebra multiplication identity. Additionally, $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ contains $\left(C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ as a sub-algebra. \end{thm} Proof: To see that $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ is a normed algebra, let $\chi,\eta\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Using (\ref{eq:Quasi-derivation identity for del}) and truncation notation, we have that: \begin{align} \left\Vert \chi\cdot\eta\right\Vert _{\nabla} & =\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\cdot\eta\right\} \right\Vert _{p,q}\\ & =\sup_{n\geq0}\left\Vert \chi_{n-1}\cdot\nabla_{p^{n}}\left\{ \eta\right\} +\eta_{n}\cdot\nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}\\ \left(\left\Vert \cdot\right\Vert _{p,q}\textrm{ is non-arch.}\right); & \leq\max\left\{ \sup_{m\geq0}\left\Vert \chi_{m-1}\cdot\nabla_{p^{m}}\left\{ \eta\right\} \right\Vert _{p,q},\sup_{n\geq0}\left\Vert \eta_{n}\cdot\nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}\right\} \end{align} For each $m$ and $n$, the function $\chi_{m-1}$, $\nabla_{p^{m}}\left\{ \eta\right\} $, $\eta_{n}$, and $\nabla_{p^{n}}\left\{ \chi\right\} $ are continuous functions. Then, because $\left(C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{p,q}\right)$ is a Banach algebra, we can write: \begin{align} \left\Vert \chi\cdot\eta\right\Vert _{\nabla} & \leq\max\left\{ \sup_{m\geq0}\left(\left\Vert \chi_{m-1}\right\Vert _{p,q}\left\Vert \nabla_{p^{m}}\left\{ \eta\right\} \right\Vert _{p,q}\right),\sup_{n\geq0}\left(\left\Vert \eta_{n}\right\Vert _{p,q}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}\right)\right\} \end{align} Since $\chi_{m-1}$ is rising-continuous, \textbf{Proposition \ref{prop:continuous embeds in rising-continuous}} tells us that $\left\Vert \chi_{m-1}\right\Vert _{p,q}\leq\left\Vert \chi_{m-1}\right\Vert _{\nabla}$. Moreover, because $\chi_{m-1}$ is the restriction of $\chi$ to inputs mod $p^{m-1}$, we have: \begin{align} \left\Vert \chi_{m-1}\right\Vert _{\nabla} & =\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\nabla_{p^{k}}\left\{ \chi_{m-1}\right\} \left(\mathfrak{z}\right)\right\|_{q}\\ & \leq\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi_{m-1}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-\chi_{m-1}\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right\|_{q}\\ & \leq\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi\left(\left[\left[\mathfrak{z}\right]_{p^{k}}\right]_{p^{m-1}}\right)-\chi_{m-1}\left(\left[\left[\mathfrak{z}\right]_{p^{k-1}}\right]_{p^{m-1}}\right)\right\|_{q} \end{align} Here: \begin{equation} \left[\left[\mathfrak{z}\right]_{p^{k}}\right]_{p^{m-1}}=\left[\mathfrak{z}\right]_{p^{\min\left\{ k,m-1\right\} }} \end{equation} and so: \begin{align} \left\Vert \chi_{m-1}\right\Vert _{\nabla} & =\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\nabla_{p^{k}}\left\{ \chi_{m-1}\right\} \left(\mathfrak{z}\right)\right\|_{q}\\ & \leq\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi_{m-1}\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-\chi_{m-1}\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right\|_{q}\\ & \leq\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\underbrace{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{\min\left\{ k,m-1\right\} }}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{\min\left\{ k-1,m-1\right\} }}\right)\right\|_{q}}_{0\textrm{ when }k\geq m}\\ & \leq\sup_{0\leq k<m}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right\|_{q}\\ & \leq\sup_{k\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{k}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{k-1}}\right)\right\|_{q}\\ & =\left\Vert \chi\right\Vert _{\nabla} \end{align} Applying this argument to the $\eta_{n}$ yields: \begin{equation} \left\Vert \eta_{n}\right\Vert _{p,q}\leq\left\Vert \eta_{n}\right\Vert _{\nabla}\leq\left\Vert \eta\right\Vert _{\nabla} \end{equation} and hence: \begin{align} \left\Vert \chi\cdot\eta\right\Vert _{\nabla} & \leq\max\left\{ \sup_{m\geq0}\left(\left\Vert \chi_{m-1}\right\Vert _{p,q}\left\Vert \nabla_{p^{m}}\left\{ \eta\right\} \right\Vert _{p,q}\right),\sup_{n\geq0}\left(\left\Vert \eta_{n}\right\Vert _{p,q}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}\right)\right\} \\ & \leq\max\left\{ \left\Vert \chi\right\Vert _{\nabla}\sup_{m\geq0}\left\Vert \nabla_{p^{m}}\left\{ \eta\right\} \right\Vert _{p,q},\left\Vert \eta\right\Vert _{\nabla}\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \chi\right\} \right\Vert _{p,q}\right\} \\ & =\max\left\{ \left\Vert \chi\right\Vert _{\nabla}\left\Vert \eta\right\Vert _{\nabla},\left\Vert \eta\right\Vert _{\nabla}\left\Vert \chi\right\Vert _{\nabla}\right\} \\ & =\left\Vert \chi\right\Vert _{\nabla}\left\Vert \eta\right\Vert _{\nabla} \end{align} Thus, $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ is a normed algebra. The inequality from \textbf{Proposition \ref{prop:continuous embeds in rising-continuous}} then shows that $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ contains $\left(C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$ as a sub-algebra. To conclude, we need only establish the completeness of $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$. We do this using \textbf{Proposition \ref{prop:series characterization of a Banach space}}: it suffices to prove that every sequence in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which tends to $0$ in $\nabla$-norm is summable to an element of $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. So, let $\left\{ \chi_{n}\right\} _{n\geq0}$ be a sequence\footnote{This is \emph{not }an instance truncation notation. I apologize for this abuse of notation.} in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which tends to $0$ in $\nabla$-norm. Then, by \textbf{Proposition \ref{prop:series characterization of a Banach space}}, upon letting $M$ and $N$ be arbitrary positive integers with $M\leq N$, we can write: \begin{equation} \left\Vert \sum_{n=M}^{N}\chi_{n}\left(\mathfrak{z}\right)\right\Vert _{p,q}\leq\left\Vert \sum_{n=M}^{N}\chi_{n}\left(\mathfrak{z}\right)\right\Vert _{\nabla}\leq\max_{n\geq M}\left\Vert \chi_{n}\right\Vert _{\nabla} \end{equation} Here, the upper bound tends to $0$ as $M\rightarrow\infty$. Consequently the series $\sum_{n=0}^{\infty}\chi_{n}\left(\mathfrak{z}\right)$ converges $q$-adically to a sum $X\left(\mathfrak{z}\right)$, and, moreover, this convergence is \emph{uniform} in $\mathfrak{z}\in\mathbb{Z}_{p}$. Since the $\chi_{n}$s are rising-continuous, we have: \begin{equation} S_{p}\left\{ \sum_{n=0}^{N}\chi_{n}\right\} \left(\mathfrak{z}\right)=\sum_{n=0}^{N}S_{p}\left\{ \chi_{n}\right\} \left(\mathfrak{z}\right)=\sum_{n=0}^{N}\chi_{n}\left(\mathfrak{z}\right) \end{equation} The uniform convergence of the $n$-sum justifies interchanging limits with $S_{p}$: \[ X\left(\mathfrak{z}\right)=\lim_{N\rightarrow\infty}\sum_{n=0}^{N}\chi_{n}=\lim_{N\rightarrow\infty}S_{p}\left\{ \sum_{n=0}^{N}\chi_{n}\right\} \left(\mathfrak{z}\right)=S_{p}\left\{ \lim_{N\rightarrow\infty}\sum_{n=0}^{N}\chi_{n}\right\} \left(\mathfrak{z}\right)=S_{p}\left\{ X\right\} \left(\mathfrak{z}\right) \] Consequently, $X=S_{p}\left\{ X\right\} $. This proves that $X$ is rising-continuous. So, $\sum_{n=0}^{\infty}\chi_{n}\left(\mathfrak{z}\right)$ converges in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, establishing the completeness of $\left(\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\left\Vert \cdot\right\Vert _{\nabla}\right)$. Q.E.D. \begin{problem} Although we shall not explore it here, I think it would be interesting to investigate the behavior of $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ under linear operators $T_{\phi}:\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\rightarrow B\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ of the form: \begin{equation} T_{\phi}\left\{ \chi\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\chi\left(\phi\left(\mathfrak{z}\right)\right),\textrm{ }\forall\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Definition of T_phi of Chi} \end{equation} where $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$. The simplest $T_{\phi}$s are translation operator\index{translation operator}s: \begin{equation} \tau_{\mathfrak{a}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\chi\left(\mathfrak{z}+\mathfrak{a}\right)\label{eq:Definition of the translation operator} \end{equation} where $\mathfrak{a}$ is a fixed element of $\mathbb{Z}_{p}$. The van der Put series (\ref{eq:van der Point series for one point function at 0}) given for $\mathbf{1}_{0}\left(\mathfrak{z}\right)$ in Subsection \ref{subsec:3.1.6 Monna-Springer-Integration} shows that $\mathbf{1}_{0}\left(\mathfrak{z}\right)\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, and that $\tau_{n}\left\{ \mathbf{1}_{0}\right\} =\mathbf{1}_{n}$ is in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ for all $n\geq0$. However, observe that for any $\mathfrak{a}\in\mathbb{Z}_{p}^{\prime}$, $\mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)$ vanishes on $\mathbb{N}_{0}$, and as a consequence, $S_{p}\left\{ \mathbf{1}_{\mathfrak{a}}\right\} $ is identically $0$; that is to say, $\mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)$ is rising-continuous \emph{if and only if }$\mathfrak{a}\in\mathbb{N}_{0}$. This shows that $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is not closed under translations. Aside from being a curiosity in its own right, studying operators like (\ref{eq:Definition of T_phi of Chi}) is also relevant to our investigations of Hydra maps. Choosing $\phi=B_{p}$, where: \begin{equation} B_{p}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{p}}{=}\begin{cases} 0 & \textrm{if }\mathfrak{z}=0\\ \frac{\mathfrak{z}}{1-p^{\lambda_{p}\left(\mathfrak{z}\right)}} & \textrm{if }\mathfrak{z}\in\mathbb{N}_{1}\\ \mathfrak{z} & \textrm{if }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime} \end{cases} \end{equation} is $B_{p}$ as defined as (\ref{eq:Definition of B projection function}), observe that the functional equation (\ref{eq:Chi_H B functional equation}) can be written as: \[ T_{B_{p}}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right)=\chi_{H}\left(B_{p}\left(\mathfrak{z}\right)\right)=\begin{cases} \chi_{H}\left(0\right) & \textrm{if }\mathfrak{z}=0\\ \frac{\chi_{H}\left(\mathfrak{z}\right)}{1-M_{H}\left(\mathfrak{z}\right)} & \textrm{else} \end{cases} \] where, by the arguments shown in Chapter 2, $M_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q}}{=}0$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ ( every such $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ has infinitely many non-zero $p$-adic digits). As such, understanding the action of the operator $T_{B_{p}}$ may be of use to understanding the periodic points of $H$. \end{problem} \vphantom{} We now move to investigate the units of $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \begin{prop}[\textbf{Unit criteria for} $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$] \label{prop:criteria for being a rising-continuous unit}\ \vphantom{} I. Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. Then, $\chi$ is a unit of $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ whenever: \begin{equation} \inf_{n\in\mathbb{N}_{0}}\left\|\chi\left(n\right)\right\|_{q}>0\label{eq:Sufficient condition for a rising continuous function to have a reciprocal} \end{equation} \vphantom{} II. Let $\chi\in C\left(\mathbb{Z}_{p},K\right)$. Then, $\chi$ is a unit of $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ if and only if $\chi$ has no zeroes. \end{prop} Proof: I. Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. \begin{align} \left\Vert \frac{1}{\chi}\right\Vert _{\nabla} & =\sup_{n\geq0}\left\Vert \nabla_{p^{n}}\left\{ \frac{1}{\chi}\right\} \right\Vert _{\nabla}\\ & =\max\left\{ \left\|\frac{1}{\chi\left(0\right)}\right\|_{q},\sup_{n\geq1}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\frac{1}{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)}-\frac{1}{\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}\right\} \\ & =\max\left\{ \left\|\frac{1}{\chi\left(0\right)}\right\|_{q},\sup_{n\geq1}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\frac{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}\right\} \\ & \leq\max\left\{ \left\|\frac{1}{\chi\left(0\right)}\right\|_{q},\sup_{m\geq0}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\nabla_{p^{m}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|\sup_{k\geq1}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{k}}\right)\right\|_{q}^{-1}\cdot\sup_{n\geq1}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right\|_{q}^{-1}\right\} \\ & \leq\max\left\{ \left\|\frac{1}{\chi\left(0\right)}\right\|_{q},\frac{\left\Vert \chi\right\Vert _{\nabla}}{\inf_{n\geq1}\left\|\chi\left(n\right)\right\|_{q}^{2}}\right\} \end{align} which will be finite whenever $\inf_{n\in\mathbb{N}_{0}}\left\|\chi\left(n\right)\right\|_{q}>0$. \vphantom{} II. If $\chi\in C\left(\mathbb{Z}_{p},K\right)$ has no zeroes, then $1/\chi\in C\left(\mathbb{Z}_{p},K\right)$. Since $C\left(\mathbb{Z}_{p},K\right)\subset\tilde{C}\left(\mathbb{Z}_{p},K\right)$, we then have that $1/\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. Conversely, if $1/\chi\in C\left(\mathbb{Z}_{p},K\right)$ has a zero, then $1/\chi\notin C\left(\mathbb{Z}_{p},K\right)$. If $1/\chi$ were in $\tilde{C}\left(\mathbb{Z}_{p},K\right)$, then since the norm on $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ dominates the norm on $C\left(\mathbb{Z}_{p},K\right)$, $1/\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ force $1/\chi$ to be an element of $C\left(\mathbb{Z}_{p},K\right)$, which would be a contradiction. Q.E.D. \begin{lem} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. Then, $1/\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ if and only if $\chi_{N}^{-1}=1/\chi_{N}$ converges point-wise in $K$ as $N\rightarrow\infty$, in which case, the point-wise limit is equal to $1/\chi$. \end{lem} Proof: Since $\chi_{N}^{-1}$ is the $N$th truncation of $1/\chi_{N}$, we have that $1/\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ if and only if $\chi_{N}^{-1}$ converges point-wise to a limit, in which case that limit is necessarily $1/\chi$. Q.E.D. \vphantom{} This shows that we can simplify the study of units of $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ by considering their truncations, which is very nice, seeing as the $\left(p,q\right)$-adic continuity of an $N$th truncation allows us to employ the techniques $\left(p,q\right)$-adic Fourier analysis. Our next inversion criterion takes the form of a characterization of the rate at which the truncations of a rising continuous function converge in the limit, and has the appeal of being both necessary \emph{and }sufficient. \begin{lem}[\textbf{The Square Root Lemma}\index{Square Root Lemma}] \label{lem:square root lemma}Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, let $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, and let $\mathfrak{c}\in K\backslash\chi\left(\mathbb{N}_{0}\right)$. Then, $\chi\left(\mathfrak{z}\right)=\mathfrak{c}$ if and only if: \begin{equation} \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}<\infty\label{eq:Square Root Lemma} \end{equation} \end{lem} \begin{rem} Since this is a biconditional statement, it is equivalent to its inverse, which is: \emph{ \begin{equation} \chi\left(\mathfrak{z}\right)\neq\mathfrak{c}\Leftrightarrow\lim_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty\label{eq:Square Root Lemma - Inverse} \end{equation} }This is because the $\liminf$ \[ \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty \] forces the true limit to exist and be equal to $\infty$. \end{rem} Proof: Let $\chi$, $\mathfrak{z}$, and $\mathfrak{c}$ be as given. We will prove (\ref{eq:Square Root Lemma - Inverse}). I. Suppose $\chi\left(\mathfrak{z}\right)\neq\mathfrak{c}$. Then, it must be that $\liminf_{n\rightarrow\infty}\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}>0$. On the other hand, since $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, the denominator of (\ref{eq:Square Root Lemma - Inverse}) tends to $0$. This shows (\ref{eq:Square Root Lemma - Inverse}) holds whenever $\chi\left(\mathfrak{z}\right)\neq\mathfrak{c}$. \vphantom{} II. Suppose: \[ \lim_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty \] Now, letting $\mathfrak{y}\in\mathbb{Z}_{p}^{\prime}$ be arbitrary, observe that we have the formal identity: \begin{align} \frac{1}{\chi\left(\mathfrak{y}\right)-\mathfrak{c}} & =\frac{1}{\chi\left(0\right)-\mathfrak{c}}+\sum_{n=1}^{\infty}\left(\frac{1}{\chi\left(\left[\mathfrak{y}\right]_{p^{n}}\right)-\mathfrak{c}}-\frac{1}{\chi\left(\left[\mathfrak{y}\right]_{p^{n-1}}\right)-\mathfrak{c}}\right)\\ & =\frac{1}{\chi\left(0\right)-\mathfrak{c}}-\sum_{n=1}^{\infty}\frac{\overbrace{\chi\left(\left[\mathfrak{y}\right]_{p^{n}}\right)-\chi\left(\left[\mathfrak{y}\right]_{p^{n-1}}\right)}^{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{y}\right)}}{\left(\chi\left(\left[\mathfrak{y}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{y}\right]_{p^{n-1}}\right)-\mathfrak{c}\right)} \end{align} Note that we needn't worry about $1/\left(\chi\left(0\right)-\mathfrak{c}\right)$, since $\mathfrak{c}$ was given to not be of the form $\mathfrak{c}=\chi\left(n\right)$ for any $n\in\mathbb{N}_{0}$. Now, observe that $\chi\left(\mathfrak{z}\right)\neq\mathfrak{c}$ if and only if the above series converges in $K$ at $\mathfrak{y}=\mathfrak{z}$. This, in turn, occurs if and only if: \[ \lim_{n\rightarrow\infty}\left\|\frac{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}{\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)-\mathfrak{c}\right)}\right\|_{q}=0 \] and hence, if and only if: \begin{equation} \lim_{n\rightarrow\infty}\left\|\frac{\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)-\mathfrak{c}\right)}{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}\right\|_{q}=\infty\label{eq:Desired Limit} \end{equation} So, in order to show that (\ref{eq:Square Root Lemma - Inverse}) implies $\chi\left(\mathfrak{z}\right)\neq\mathfrak{c}$, we need only show that (\ref{eq:Square Root Lemma - Inverse}) implies (\ref{eq:Desired Limit}). Here, note that: \[ \chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)=\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right) \] and so: \begin{align} \left\|\frac{\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)-\mathfrak{c}\right)}{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}\right\|_{q} & =\left\|\frac{\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}-\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)}{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}\right\|_{q}\\ & =\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}-\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}\\ & \geq\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}\left\|\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}-\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}\right\| \end{align} Now, since $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, we have $\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}\rightarrow0$ as $n\rightarrow\infty$. On the other hand, for this part of the proof, we assumed that: \begin{equation} \lim_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty \end{equation} Consequently, since $\mathfrak{z}$ is fixed: \begin{align} \lim_{n\rightarrow\infty}\left\|\frac{\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right)\left(\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)-\mathfrak{c}\right)}{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}\right\|_{q} & \geq\lim_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}\left\|\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}-\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}\right\|\\ & =\infty\cdot\left\|\infty-0\right\|\\ & =\infty \end{align} which is (\ref{eq:Desired Limit}). Q.E.D. \vphantom{} This time, the moral is that $\mathfrak{c}\in K\backslash\chi\left(\mathbb{N}_{0}\right)$ is in the image of $\chi$ if and only if there is a subsequence of $n$s along which $\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}$ tends to $0$ \emph{at least as quickly as} $\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}$. \begin{prop} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, let $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, and let $\mathfrak{c}\in K\backslash\chi\left(\mathbb{N}_{0}\right)$. If: \[ \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}<\infty \] then: \[ \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ f\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}>0 \] occurs if and only if, for any strictly increasing sequence $\left\{ n_{j}\right\} _{j\geq1}\subseteq\mathbb{N}_{0}$ so that: \[ \lim_{j\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n_{j}}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}} \] the congruences: \begin{equation} v_{q}\left(\nabla_{p^{n_{j+1}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)\overset{2}{\equiv}v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right) \end{equation} hold for all sufficiently large $j$; that is, the parity of the $q$-adic valuation of $\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)$ becomes constant for all sufficiently large $j$. \end{prop} Proof: Suppose: \[ \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}>0 \] In particular, let us write the value of this $\liminf$ as $q^{C}$, where $C$ is a positive real number. Letting the $n_{j}$s be any sequence as described above, by way of contradiction, suppose there are two disjoint subsequences of $j$s so that $v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)$ is always even for all $j$ in the first subsequence and is always odd for all $j$ in the second subsequence. Now, for \emph{any} $j$, let $v_{j}$ be the positive integer so that $\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n_{j}}}\right)-\mathfrak{c}\right\|_{q}=q^{-v_{j}}$. Then: \begin{align} q^{C}=\lim_{j\rightarrow\infty}q^{\frac{1}{2}v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)}\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n_{j}}}\right)-\mathfrak{c}\right\|_{q} & =\lim_{j\rightarrow\infty}q^{\frac{1}{2}v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)-v_{j}} \end{align} and hence: \begin{equation} C=\lim_{j\rightarrow\infty}\left(\frac{1}{2}v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)-v_{j}\right)\label{eq:Limit formula for K} \end{equation} for \emph{both }subsequences of $j$s. For the $j$s in the first subsequence (where $v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)$ is always even), the limit on the right hand side is of a subsequence in the set $\mathbb{Z}$. For the $j$s in the second subsequence (where $v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)$ is always odd), however, the limit on the right hand side is that of a sequence in $\mathbb{Z}+\frac{1}{2}$. But this is impossible: it forces $C$ to be an element of both $\mathbb{Z}$ and $\mathbb{Z}+\frac{1}{2}$, and those two sets are disjoint. Thus, the parity of $v_{q}\left(\nabla_{p^{n_{j}}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right)$ must become constant for all sufficiently large $j$ whenever $C>-\infty$. Q.E.D. \vphantom{} Given the criteria described in these last few results, it seems natural to introduce the following definitions: \begin{defn} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. A pair $\left(\mathfrak{z},\mathfrak{c}\right)$, where $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ and $\mathfrak{c}\in\chi\left(\mathbb{Z}_{p}\right)\backslash\chi\left(\mathbb{N}_{0}\right)$ is said to be a \index{quick approach}\textbf{quick approach of $\chi$ / quickly approached by $\chi$ }whenever: \[ \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=0 \] We say that the pair is a\textbf{ slow approach of $\chi$ / slowly approached by $\chi$ }whenever the $\liminf$ is finite and non-zero. \end{defn} \begin{question} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. What can be said about the speed of approach of $\chi$ to $\mathfrak{c}$ at any given $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$? Can there be infinitely many (or at most finitely many\textemdash or none?) points of quick approach? Slow approach? Are there other properties we can use to characterize points of quick or slow approach?\footnote{In Chapter 4, we will show that, for every semi-basic, contracting $2$-Hydra map which fixes $0$, every point in $\chi_{H}\left(\mathbb{Z}_{p}\right)\backslash\chi_{H}\left(\mathbb{N}_{0}\right)$ is a point of quick approach $\chi_{H}$. The more general case of a $p$-Hydra map likely also holds, but I did not have sufficient time to investigate it fully.} \end{question} \begin{rem} To share a bit of my research process, although I discovered the notion of quasi-integrability back in the second half of 2020, it wasn't until the 2021 holiday season that I began to unify my $\left(p,q\right)$-adic work in earnest. With the Correspondence Principle on my mind, I sought to find ways to characterize the image of a $\left(p,q\right)$-adic function. In particular, I found myself gravitating toward the question: \emph{given a rising-continuous function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$, when is $1/\chi$ rising-continuous? }Given a $\chi_{H}$, investigating the singular behavior of $1/\left(\chi_{H}\left(\mathfrak{z}\right)-x\right)$ where $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ would seem to be a natural way to exploit the Correspondence Principle, because an $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ would be a periodic point of $H$ if and only if $1/\left(\chi_{H}\left(\mathfrak{z}\right)-x\right)$ had a singularity for some $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. I drew inspiration from W. Cherry's notes on non-archimedean function theory \cite{Cherry non-archimedean function theory notes}, and the existence of a $p$-adic Nevanlinna theory\textemdash Nevanlinna theory being one of analytic function theorists' most powerful tools for keeping track of the poles of meromorphic functions. Due to the (present) lack of a meaningful notion of an \emph{analytic }$\left(p,q\right)$-adic function, however, I had to go down another route to investigate reciprocals of rising-continuous functions. Things began to pick up speed once I realized that the set of rising-continuous functions could be realized as a non-archimedean Banach algebra under point-wise multiplication. In that context, exploring the Correspondence Principle becomes extremely natural: it is the quest to understand those values of $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ for which $\chi_{H}\left(\mathfrak{z}\right)-x$ is not a unit of the Banach algebra of rising-continuous functions. \end{rem} \subsubsection{A Very Brief Foray Into Berkovitch Space\index{Berkovitch space}} To give yet another reason why $\left(p,q\right)$-adic analysis has been considered ``uninteresting'' by the mathematical community at large: their zeroes do not enjoy any special properties. Case in point: \begin{prop} \label{prop:(p,q)-adic functions are "uninteresting"}Let $A$ be any closed subset of $\mathbb{Z}_{p}$. Then, there is an $f\in C\left(\mathbb{Z}_{p},K\right)$ such that $f\left(\mathfrak{z}\right)=0$ if and only if $\mathfrak{z}\in A$. \end{prop} Proof: Let $A$ be a closed subset of $\mathbb{Z}_{p}$. For any $\mathfrak{z}\in\mathbb{Z}_{p}\backslash A$, the Hausdorff distance: \begin{equation} d\left(\mathfrak{z},A\right)\overset{\textrm{def}}{=}\inf_{\mathfrak{a}\in A}\left\|\mathfrak{z}-\mathfrak{a}\right\|_{p}\label{eq:Definition of the distance of a point in Z_p from a set in Z_p} \end{equation} of $\mathfrak{z}$ form $A$ will be of the form $\frac{1}{p^{n}}$ for some $n\in\mathbb{N}_{0}$. Consequently, we can define a function $g:\mathbb{Z}_{p}\backslash A\rightarrow\mathbb{N}_{0}$ so that: \begin{equation} d\left(x,A\right)=p^{-g\left(x\right)},\textrm{ }\forall x\in\mathbb{Z}_{p}\backslash A \end{equation} Then, the function: \begin{equation} f\left(x\right)\overset{\textrm{def}}{=}\left[x\notin A\right]q^{g\left(x\right)}\label{eq:Function with zeroes on an arbitrary closed subset of Z_p} \end{equation} will be continuous, and will have $A$ as its vanishing set. Q.E.D. \vphantom{} In this short sub-subsection, we deal with some topological properties of rising-continuous functions which can be proven using \textbf{Berkovitch spaces}. For the uninitiated\textemdash such as myself\textemdash Berkovitch space is the capstone of a series of turbulent efforts by algebraic and arithmetic geometers in the second half of the twentieth century which had as their \emph{ide-fixe }the hope of circumventing the Achilles' heel of the native theory of $\left(p,p\right)$-adic\footnote{i.e., $f:\mathbb{C}_{p}\rightarrow\mathbb{C}_{p}$.} analytic functions: the abysmal failure of Weierstrass' germ-based notions of analytic continuation\index{analytic continuation}. Because of the $p$-adics' ultrametric structure, every $\mathfrak{z}$ in the disk $\left\{ \mathfrak{z}\in\mathbb{C}_{p}:\left\|\mathfrak{z}-\mathfrak{a}\right\|_{p}<r\right\} $ is, technically, at the center of said disk; here $\mathfrak{a}\in\mathbb{C}_{p}$ and $r>0$. Along with the $p$-adics' stark convergence properties (viz. the principles of ultrametric analysis on \pageref{fact:Principles of Ultrametric Analysis}) this strange property of non-archimedean disks leads to the following unexpected behaviors in a $p$-adic power series $f\left(\mathfrak{z}\right)=\sum_{n=0}^{\infty}\mathfrak{a}_{n}\mathfrak{z}^{n}$. Here, $r$ denotes $f$'s radius of convergence $r>0$. \begin{enumerate} \item Either $f\left(\mathfrak{z}\right)$ converges at every $\mathfrak{z}\in\mathbb{C}_{p}$ with $\left\|\mathfrak{z}\right\|_{p}=r$, or diverges at every such $\mathfrak{z}$. \item Developing $f\left(\mathfrak{z}\right)$ into a power series at any $\mathfrak{z}_{0}\in\mathbb{C}_{p}$ with $\left\|\mathfrak{z}_{0}\right\|_{p}<r$ results in a power series whose radius of convergence is equal to $r$. So, no analytic continuation is possible purely by way of rejiggering $f$'s power series. \end{enumerate} The construction of Berkovitch space provides a larger space containing $\mathbb{C}_{p}$ which is Hausdorff, and\textemdash crucially, unlike the totally disconnected space $\mathbb{C}_{p}$\textemdash is actually arc-wise connected.\footnote{Let $X$ be a topological space. Two points $x,y\in X$ are said to be \textbf{topologically distinguishable }if there exists an open set $U\subseteq X$ which contains one and only one of $x$ or $y$. $X$ is then said to be \textbf{arc-wise connected }whenever, for any topologically distinguishable points $x,y\in X$, there is an embedding $\alpha:\left[0,1\right]\rightarrow X$ (continuous and injective) such that $\alpha\left(0\right)=x$ and $\alpha\left(1\right)=y$; such an $\alpha$ is called an \textbf{arc}.} The idea\footnote{I base my exposition here on what is given in \cite{The Berkovitch Space Paper}.} (a minor bit of trepanation) is to replace points in $\mathbb{C}_{p}$ with \index{semi-norm}\emph{semi-norms}. This procedure holds, in fact, for an arbitrary Banach algebra over a non-archimedean field. Before going forward, let me mention that I include the following definitions (taken from \cite{The Berkovitch Space Paper}) primarily for the reader's edification, provided the reader is the sort of person who finds stratospheric abstractions edifying. Meanwhile, the rest of us mere mortals can safely skip over to the proofs at the end of this sub-subsection. \begin{defn} Let $K$ be an algebraically closed non-archimedean field, complete with respect to its ultrametric. Let $A$ be a commutative $K$-Banach algebra with identity $1_{A}$, and with norm $\left\Vert \cdot\right\Vert $. A function $\rho:A\times A\rightarrow\left[0,\infty\right)$ is an \textbf{algebra semi-norm }if\index{Banach algebra!semi-norm}, in addition to being a semi-norm on $A$ when $A$ is viewed as a $K$-Banach space, we have that: \vphantom{} I. $\rho\left(1_{A}\right)=1$; \vphantom{} II. $\rho\left(f\cdot g\right)\leq\rho\left(f\right)\cdot\rho\left(g\right)$ for all $f,g\in A$. $\rho$ is said to be \textbf{multiplicative} whenever\index{semi-norm!multiplicative} (II) holds with equality for all $f,g\in A$. The \textbf{multiplicative spectrum}\index{Banach algebra!multiplicative spectrum}\textbf{ of $A$}, denoted $\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$, is the set of multiplicative semi-norms on $A$ which are continuous with respect to the topology defined by $\left\Vert \cdot\right\Vert $; i.e., for each $\rho\in\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$, there is a real constant $K\geq0$ so that: \begin{equation} \rho\left(f\right)\leq K\left\Vert f\right\Vert ,\textrm{ }\forall f\in A\label{eq:multiplicative semi-norm thing for Berkovitch space} \end{equation} \end{defn} \vphantom{} Using this bizarre setting, algebraic geometers then construct affine spaces over $K$. \begin{defn} We write $\mathbb{A}_{K}^{1}$ to denote the space of all multiplicative semi-norms on the space $K\left[X\right]$ of formal polynomials in the indeterminate $X$ with coefficients in $K$. $\mathbb{A}_{K}^{1}$ is called the \textbf{affine line over $K$}\index{affine line over K@affine line over \textbf{$K$}}. \end{defn} \begin{fact} Both $\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$ and $\mathbb{\mathbb{A}}_{K}^{1}$ are made into topological spaces with the topology of point-wise convergence. \end{fact} \begin{thm} $\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$ is Hausdorff and compact. $\mathbb{A}_{K}^{1}$ is Hausdorff, locally compact, and arc-wise connected \cite{The Berkovitch Space Paper}. \end{thm} \begin{defn} For each $t\in A$, write $\theta_{t}$ to denote the $K$-algbra homomorphism $\theta_{t}:K\left[X\right]\longrightarrow A$ defined by $\theta_{t}\left(X\right)=t$. For each $t\in A$, the \index{Gelfand transform}\textbf{Gelfand transform of $t$} is then defined to be the map $\mathcal{G}_{t}:\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)\rightarrow\mathbb{A}_{K}^{1}$ given by: \begin{equation} \mathcal{G}_{t}\left\{ \rho\right\} \overset{\textrm{def}}{=}\rho\circ\theta_{t}\label{Gelfand transform of t} \end{equation} $\mathcal{G}_{t}\left\{ \rho\right\} $ is then a multiplicative semi-norm on $K\left[X\right]$ for all $\rho$ and $t$. The map $\mathcal{G}$ which sends $t\in A$ to the map $\mathcal{G}_{t}$ is called the \textbf{Gelfand transform}. \end{defn} \begin{thm} $\mathcal{G}_{t}$ is continuous for every $t\in A$. \end{thm} \vphantom{} The point of this, apparently, is that $\mathcal{G}$ allows any $t\in A$ to be interpreted as the continuous function $\mathcal{G}_{t}$. Now for some more definitions: \begin{defn} Given $t\in A$: \vphantom{} I. We write $\textrm{s}\left(t\right)$ to denote the \textbf{scalar spectrum }of $t$: the set of $\lambda\in K$ for which $t-\lambda1_{A}$ is not invertible in $A$. \vphantom{} II. We write $\sigma\left(t\right)$ to denote the \textbf{spectrum }of $t$: the image of $\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$ under $\mathcal{G}_{t}$. \end{defn} \vphantom{} Since $K$ is a Banach algebra over itself (with $\left\|\cdot\right\|_{K}$ as the norm), $\mathcal{M}\left(K,\left\|\cdot\right\|_{K}\right)$ consists of a single semi-norm: the absolute value on $K$. This allows $K$ to be embedded in $\mathbb{A}_{K}^{1}$ by way of the Gelfand transform $\mathcal{G}$, identifying any $\mathfrak{a}\in K$ with $\mathcal{G}_{\mathfrak{a}}$. Since $\mathcal{M}\left(K,\left\|\cdot\right\|_{K}\right)$ contains only the single element $\left\|\cdot\right\|_{K}$, the image of $\mathcal{G}_{\mathfrak{a}}$ ($\sigma\left(\mathfrak{a}\right)$\textemdash the spectrum of $\mathfrak{a}$) is the semi-norm which sends $P\in K\left[X\right]$ to $\left\|\theta_{\mathfrak{a}}\left(P\right)\right\|_{K}$. That is, for any $\mathfrak{a}\in K$, we we identify with $\mathfrak{a}$ the semi-norm $P\mapsto\left\|\theta_{\mathfrak{a}}\left(P\right)\right\|_{K}$ on $K\left[X\right]$. \begin{defn} The \textbf{closed disk }in $\mathbb{A}_{K}^{1}$ centered at $\mathfrak{a}\in K$ of radius $r>0$ is defined as the closure in $\mathbb{A}_{K}^{1}$ (with respect to point-wise convergence) of the set of semi-norms $P\in K\left[X\right]\mapsto\left\|\theta_{\mathfrak{z}}\left(P\right)\right\|_{K}$ as $\mathfrak{z}$ varies over all elements of $K$ with $\left\|\mathfrak{z}-\mathfrak{a}\right\|_{K}\leq r$. \end{defn} \begin{thm}[\textbf{The First Spectral Radius Formula }\cite{The Berkovitch Space Paper}] Let $A$ be a commutative $K$-Banach algebra with identity $1_{A}$, and with norm $\left\Vert \cdot\right\Vert $, and let $t\in A$. Then: \vphantom{} I. $\sigma\left(t\right)$ is non-empty and compact. \vphantom{} II. The smallest closed disk in $\mathbb{A}_{K}^{1}$ centered at $0$ which contains $\sigma\left(t\right)$ has radius: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert t^{n}\right\Vert ^{1/n} \end{equation} \vphantom{} III. $K\cap\sigma\left(t\right)=s\left(t\right)$, where by $K$, we mean the copy of $K$ embedded in $\mathbb{A}_{K}^{1}$ as described above. \end{thm} \vphantom{} We now recall some basic definitions from topology: \begin{defn} Let $X$ and $Y$ be topological spaces. A map $M:X\rightarrow Y$ is said to be: \vphantom{} I.\textbf{ Proper},\textbf{ }if $M^{-1}\left(C\right)$ is compact in $X$ for all compact sets $C\subseteq Y$. \vphantom{} II. \textbf{Closed}, if $M\left(S\right)$ is closed in $Y$ for all closed sets $S\subseteq X$. \end{defn} \begin{fact} Let $X$ and $Y$ be topological spaces, and let $M:X\rightarrow Y$ be continuous. If $X$ is compact and $Y$ is Hausdorff, then $M$ is both proper and closed. \end{fact} \begin{prop} $s\left(t\right)$ is compact in $K$. \end{prop} Proof: Let $t\in A$ be arbitrary. Since $\mathcal{G}_{t}:\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)\rightarrow\mathbb{A}_{K}^{1}$ is continuous, since $\mathcal{M}\left(A,\left\Vert \cdot\right\Vert \right)$ is compact, and since $\mathbb{A}_{K}^{1}$ is Hausdorff, it then follows that $\mathcal{G}_{t}$ is both proper and closed. Consequently, since $K$ is closed, so is its copy in $\mathbb{A}_{K}^{1}$. Since $\sigma\left(t\right)$ is compact in $\mathbb{A}_{K}^{1}$, this means that $s\left(t\right)=K\cap\sigma\left(t\right)$ is compact in $\mathbb{A}_{K}^{1}\cap K$, and hence in $K$. Q.E.D. \vphantom{} The non-algebraic reader can now resume paying attention. \begin{lem} \label{lem:compactness of the image of a rising-continuous function}Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. Then, $\chi\left(\mathbb{Z}_{p}\right)$ is a compact subset of $K$. \end{lem} Proof: First, we embed $\chi$ in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Since $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is a commutative Banach algebra over the algebraically closed, metrically complete non-archimedean field $\mathbb{C}_{q}$, it follows by the previous proposition that the scalar spectrum of $\chi$ is compact in $\mathbb{C}_{q}$. Since the scalar spectrum is precisely the set of $\mathfrak{c}\in\mathbb{C}_{q}$ for which $\frac{1}{\chi\left(\mathfrak{z}\right)-\mathfrak{c}}\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, it then follows that the scalar spectrum of $\chi$\textemdash a compact set in $\mathbb{C}_{q}$\textemdash is equal to $\chi\left(\mathbb{Z}_{p}\right)$, the image of $\chi$ over $\mathbb{Z}_{p}$. Since $K$ is closed in $\mathbb{C}_{q}$, the set $\chi\left(\mathbb{Z}_{p}\right)\cap K$ (an intersection of a compact set and a closed set) is compact in both $\mathbb{C}_{q}$ and $K$. Since, by definition of $K$, $\chi\left(\mathbb{Z}_{p}\right)=\chi\left(\mathbb{Z}_{p}\right)\cap K$, we have that $\chi\left(\mathbb{Z}_{p}\right)$ is compact in $K$. Q.E.D. \vphantom{} Before we get to our next results, note the following: \begin{fact} Let $U$ be any clopen subset of $\mathbb{Z}_{p}$. Then, the indicator function for $U$ is continuous as a function from $\mathbb{Z}_{p}$ to any topological ring (where the ``$1$'' taken by the function over $U$ is the multiplicative identity element of the ring). \end{fact} \begin{thm} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. Then, $\chi\left(U\right)$ is a compact subset of $K$ for all clopen sets $U\subseteq\mathbb{Z}_{p}$. \end{thm} Proof: Let $U$ be any clopen subset of $\mathbb{Z}_{p}$ with non-empty complement. Then, let $V\overset{\textrm{def}}{=}\mathbb{Z}_{p}\backslash U$ be the complement of $U$, and let $\mathfrak{a}\in\chi\left(U\right)$ be any value attained by $\chi$ on $U$. Then, define: \begin{equation} f\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\left[\mathfrak{x}\in U\right]\chi\left(\mathfrak{z}\right)+\mathfrak{a}\left[\mathfrak{x}\in V\right]\label{eq:Berkovitch space f construction} \end{equation} Since both $V$ and $U$ are clopen sets, their indicator functions $\left[\mathfrak{x}\in U\right]$ and $\left[\mathfrak{x}\in V\right]$ are continuous functions from $\mathbb{Z}_{p}$ to $K$. Since $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ is an algebra, we have that $f\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$. As such, by \textbf{Lemma \ref{lem:compactness of the image of a rising-continuous function}}, $f\left(\mathbb{Z}_{p}\right)$ is compact in $K$. Observing that: \begin{align} f\left(\mathbb{Z}_{p}\right) & =f\left(U\right)\cup f\left(V\right)\\ & =\chi\left(U\right)\cup\left\{ \mathfrak{a}\right\} \\ \left(\mathfrak{a}\in\chi\left(U\right)\right); & =\chi\left(U\right) \end{align} we then have that $\chi\left(U\right)$ is compact in $K$, as desired. Q.E.D. \begin{thm} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, and suppose there is a set $S\subseteq\mathbb{Z}_{p}$ and a real constant $v\geq0$ so that $\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}>q^{-v}$ for all $\mathfrak{z}\in S$. Then, $\chi\left(U\right)$ is compact in $K$ for any set $U\subseteq S$ which is closed in $\mathbb{Z}_{p}$. \end{thm} \begin{rem} In other words, if $\chi$ is bounded away from zero on some set $S$, then the restriction of $\chi$ to $S$ is a proper map. \end{rem} Proof: Since $U$ is closed in $\mathbb{Z}_{p}$, as we saw in \textbf{Proposition \ref{prop:(p,q)-adic functions are "uninteresting"}}, there is a function $g:\mathbb{Z}_{p}\rightarrow\mathbb{N}_{0}$ so that $h:\mathbb{Z}_{p}\rightarrow K$ defined by: \begin{equation} h\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\left[\mathfrak{z}\notin U\right]q^{g\left(\mathfrak{z}\right)}\label{eq:h construction for Berkovitch space} \end{equation} is continuous and vanishes if and only if $\mathfrak{z}\in U$. Then, letting: \begin{equation} f\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\left[\mathfrak{x}\in U\right]\chi\left(\mathfrak{z}\right)+q^{v}h\left(\mathfrak{z}\right)\label{eq:Second f construction for Berkovitch space} \end{equation} we have that $f\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, and that: \begin{equation} f\left(\mathbb{Z}_{p}\right)=\chi\left(U\right)\cup q^{v+g\left(V\right)} \end{equation} is compact in $K$, where $q^{v+g\left(V\right)}=\left\{ q^{v+n}:n\in g\left(V\right)\right\} $. Since $\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}>q^{-v}$ for all $\mathfrak{z}\in S$, we have; \begin{equation} \chi\left(U\right)\subseteq\chi\left(S\right)\subseteq K\backslash q^{v}\mathbb{Z}_{q} \end{equation} As such: \begin{align} \left(K\backslash q^{v}\mathbb{Z}_{q}\right)\cap f\left(\mathbb{Z}_{p}\right) & =\left(\chi\left(U\right)\cup q^{v+g\left(V\right)}\right)\cap\left(K\backslash q^{v}\mathbb{Z}_{q}\right)\\ & =\left(\chi\left(U\right)\cap\left(K\backslash q^{v}\mathbb{Z}_{q}\right)\right)\cup\left(q^{v+g\left(V\right)}\cap K\cap\left(q^{v}\mathbb{Z}_{q}\right)^{c}\right)\\ & =\underbrace{\left(\chi\left(U\right)\cap\left(K\backslash q^{v}\mathbb{Z}_{q}\right)\right)}_{\chi\left(U\right)}\cup\underbrace{\left(q^{v+g\left(V\right)}\cap K\cap\left(q^{v}\mathbb{Z}_{q}\right)^{c}\right)}_{\varnothing}\\ \left(q^{v+g\left(V\right)}\cap\left(q^{v}\mathbb{Z}_{q}\right)^{c}=\varnothing\right); & =\chi\left(U\right) \end{align} Since $K\backslash q^{v}\mathbb{Z}_{q}$ is closed and $f\left(\mathbb{Z}_{p}\right)$ is compact, $\left(K\backslash q^{v}\mathbb{Z}_{q}\right)\cap f\left(\mathbb{Z}_{p}\right)=\chi\left(U\right)$ is compact, as desired. Q.E.D. \begin{lem} $f\in C\left(\mathbb{Z}_{p},K\right)$ is a unit if and only if $f$ has no zeroes. \end{lem} Proof: I. If $f$ is a unit, then $\frac{1}{f}$ is continuous, and hence, $f$ can have no zeroes. \vphantom{} II. Conversely, suppose $f$ has no zeroes. Since $f$ is continuous, its image $f\left(\mathbb{Z}_{p}\right)$ is a compact subset of $K$ (even if $K$ itself is \emph{not} locally compact). Since $K$ is a topological field, the reciprocation map $\iota\left(\mathfrak{y}\right)\overset{\textrm{def}}{=}\frac{1}{\mathfrak{y}}$ is a self-homeomorphism of $K\backslash\left\{ 0\right\} $. Here, we observe that the pre-image of a set $Y$ under $1/f$ is then equal to $f^{-1}\left(\iota^{-1}\left(Y\right)\right)$. So, let $Y$ be an arbitrary closed subset of $K$. Then, the pre-image of $Y$ under $1/f$ is equal to: \begin{equation} f^{-1}\left(\iota^{-1}\left(Y\right)\right)=f^{-1}\left(\iota^{-1}\left(Y\right)\cap f\left(\mathbb{Z}_{p}\right)\right) \end{equation} Since $f\left(\mathbb{Z}_{p}\right)$ is compact and bounded away from zero: \begin{equation} \iota^{-1}\left(Y\right)\cap f\left(\mathbb{Z}_{p}\right) \end{equation} is closed and bounded away from zero, and so, by the continuity of $f$, $f^{-1}\left(\iota^{-1}\left(Y\right)\cap f\left(\mathbb{Z}_{p}\right)\right)$ is closed. Thus, the pre-image of every closed subset of $K$ under $1/f$ is closed; this proves that $1/f$ is continuous whenever $f$ has no zeroes. Q.E.D. \begin{lem} \label{lem:3.17}$\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$ is a unit of $\tilde{C}\left(\mathbb{Z}_{p},K\right)$ whenever the following conditions hold: \vphantom{} I. $\chi$ has no zeroes. \vphantom{} II. For each $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{equation} \sup_{n\geq1}\left\|\frac{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)}{\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}<\infty \end{equation} Note that this bound need not be uniform in $\mathfrak{z}$. \end{lem} Proof: Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, and suppose $0\notin\chi\left(\mathbb{Z}_{p}\right)$. Then, by the van der Put identity, we have that $\frac{1}{\chi}$ will be rising-continuous if and only if: \[ \lim_{n\rightarrow\infty}\left\|\frac{1}{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)}-\frac{1}{\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}=0 \] for all $\mathfrak{z}\in\mathbb{Z}_{p}$, which is the same as: \[ \lim_{n\rightarrow\infty}\left\|\frac{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}=0 \] By the \textbf{Square-Root Lemma} (\textbf{Lemma \ref{lem:square root lemma}}), since $0$ is not in the image of $\chi$, it must be that: \begin{equation} \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty \end{equation} which then forces: \begin{equation} \lim_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}=\infty \end{equation} which forces: \begin{equation} \lim_{n\rightarrow\infty}\frac{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}=0 \end{equation} Now, the condition required for $1/\chi$ to be rising-continuous is: \begin{equation} \lim_{n\rightarrow\infty}\left\|\frac{\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)}{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}=0 \end{equation} This can be re-written as: \[ \lim_{n\rightarrow\infty}\frac{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}\cdot\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right\|_{q}}\cdot\frac{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}=0 \] The assumptions on $\chi$, meanwhile, guarantee both: \begin{equation} \frac{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}}\rightarrow0 \end{equation} and: \begin{equation} \sup_{n\geq1}\left\|\frac{\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)}{\chi\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)}\right\|_{q}<\infty \end{equation} Consequently, the above limit is zero for each $\mathfrak{z}\in\mathbb{Z}_{p}$. This proves $1/\chi$ is rising-continuous. Q.E.D. \begin{thm} Let $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{T}_{q}\right)$ denote the set of all $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ for which $\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}=1$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$, where $\mathbb{T}_{q}$ is the set of all $q$-adic complex numbers with a $q$-adic absolute value of $1$. Then, $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{T}_{q}\right)$ is an abelian group under the operation of point-wise multiplication; the identity element is the constant function $1$. \end{thm} Proof: Every $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{T}_{q}\right)$ satisfies the conditions of \textbf{Lemma \ref{lem:3.17}}, because $\left\|\chi\left(\mathfrak{z}\right)\right\|_{q}=\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}=1$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ and all $n\in\mathbb{N}_{0}$. Q.E.D. \newpage{} \section{\label{sec:3.3 quasi-integrability}$\left(p,q\right)$-adic Measures and Quasi-Integrability} THROUGHOUT THIS SECTION $p$ AND $q$ DENOTE DISTINCT PRIME NUMBERS. \vphantom{} Despite the pronouncements from Schikhof and Konrad used in this chapter's epigraphs, there is more to $\left(p,q\right)$-adic integration than meets the eye. Consider a function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$. Viewing $\overline{\mathbb{Q}}$ as being embedded in $\mathbb{C}_{q}$, observe that if $\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\mu}\left(t\right)\right\|_{q}<\infty$, we can then make sense of $\hat{\mu}$ as the Fourier-Stieltjes transform of the measure $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$ defined by the formula: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right) \end{equation} Now, for any $g\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, the $\left(p,q\right)$-adic measure $d\mu\left(\mathfrak{z}\right)=g\left(\mathfrak{z}\right)d\mathfrak{z}$ satisfies: \begin{equation} g\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:g in terms of mu hat} \end{equation} because the continuity of $g$ guarantees that $\hat{\mu}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. For a measure like $d\mu\left(\mathfrak{z}\right)=g\left(\mathfrak{z}\right)d\mathfrak{z}$, the Fourier series on the right-hand side of (\ref{eq:g in terms of mu hat}) makes perfect sense. On the other hand, for an arbitrary bounded function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$, there is no guarantee that the right-hand side of (\ref{eq:g in terms of mu hat})\textemdash viewed as the limit\footnote{I will, at times, refer to (\ref{eq:The Limit of Interest}) as the ``limit of interest''.} of the partial sums: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:The Limit of Interest} \end{equation} \textemdash will even exist for any given $\mathfrak{z}\in\mathbb{Z}_{p}$. In this section, we will investigate the classes of $\left(p,q\right)$-adic measures $d\mu$ so that (\ref{eq:The Limit of Interest}) makes sense for some\textemdash or even all\textemdash values of $\mathfrak{z}\in\mathbb{Z}_{p}$\textemdash even if $\hat{\mu}\notin c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Because measures are, by definition, ``integrable'', we can enlarge the tent of $\left(p,q\right)$-adically ``integrable'' functions by including those functions which just so happen to be given by (\ref{eq:The Limit of Interest}) for some bounded $\hat{\mu}$. I call these \textbf{quasi-integrable functions}.\textbf{ }Given a quasi-integrable function\index{quasi-integrability} $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, upon identifying $\chi$ with the measure $d\mu$, integration against $\chi$ can then be defined as integration against $d\mu$ by way of the formula: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\chi\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right) \end{equation} I have two points to make before we begin. First, \emph{the reader should be aware that} \emph{this chapter contains a large number of qualitative definitions}. As will be explained in detail throughout, this is primarily because, at present, there does not appear to be a way to get desirable conclusions properties out of generic $\left(p,q\right)$-adic functions without relying on explicit formulae and computations thereof, such as (\ref{eq:Definition of right-ended p-adic structural equations}), (\ref{eq:Radial-Magnitude Fourier Resummation Lemma - p-adically distributed case}), and (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}), and for these explicit computations to be feasible, we need our functions to come pre-equipped with at least \emph{some }concrete algebraic structure. By and by, I will indicate where I have been forced to complicate definitions for the sake of subsequent proof. It is obviously of interest to see how much these definitions can be streamlined and simplified. Secondly and finally, I have to say that there are still many basic technical considerations regarding quasi-integrability and quasi-integrable functions which I have not been able to develop to my satisfaction due to a lack of time and/or cleverness. The relevant questions are stated and discussed on pages \pageref{que:3.3}, \pageref{que:3.2}, pages \pageref{que:3.4} through \pageref{que:3.6}, and page \pageref{que:3.7}. \subsection{\label{subsec:3.3.1 Heuristics-and-Motivations}Heuristics and Motivations} To the extent that I have discussed my ideas with other mathematicians who did not already know me personally, the reactions I have gotten from bringing up frames seems to indicate that my ideas are considered ``difficult'', maybe even radical. But they were difficult for me long before they were difficult for anyone else. It took over a year before I finally understood what was going on. My discoveries of quasi-integrability and the necessity of frames occurred completely by chance. In performing a certain lengthy computation, I seemed to get a different answer every time. It was only after breaking up the computations into multiple steps\textemdash performing each separately and in greater generality\textemdash that I realized what was going on. To that end, rather than pull a Bourbaki, I think it will be best if we first consider the main examples that I struggled with. It's really the natural thing to do. Pure mathematics \emph{is} a natural science, after all. Its empirical data are the patterns we notice and the observations we happen to make. The hypotheses are our latest ideas for a problem, and the experiments are our efforts to see if those ideas happen to bear fruit. In this respect, frames and quasi-integrability should not be seen as platonic ideals, but as my effort to describe and speak clearly about an unusual phenomenon. The examples below are the context in which the reader should first try to understand my ideas, because this was the context in which all of my work was originally done. Instead of saving the best for last, let's go right to heart of the matter and consider the example that caused me the most trouble, by far. First, however, I must introduce the main antagonist to feature in our studies of the shortened $qx+1$ maps. \begin{defn}[$\hat{A}_{q}\left(t\right)$] Let \index{hat{A}{q}@$\hat{A}_{q}$}$q$ be an odd prime. \nomenclature{$\hat{A}_{q}\left(t\right)$}{ }I define $\hat{A}_{q}:\hat{\mathbb{Z}}_{2}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \hat{A}_{q}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 1 & \textrm{if }t=0\\ \prod_{n=0}^{-v_{2}\left(t\right)-1}\frac{1+qe^{-2\pi i\left(2^{n}t\right)}}{4} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Definition of A_q hat} \end{equation} Noting that: \begin{equation} \sup_{t\in\hat{\mathbb{Z}}_{2}}\left\|\hat{A}_{q}\left(t\right)\right\|_{q}=1 \end{equation} it follows that we can define a $\left(2,q\right)$-adic measure \nomenclature{$dA_{q}$}{ }$dA_{q}\in C\left(\mathbb{Z}_{2},\mathbb{C}_{q}\right)^{\prime}$ by way of the formula: \begin{equation} \int_{\mathbb{Z}_{2}}f\left(\mathfrak{z}\right)dA_{q}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{q}}\hat{f}\left(-t\right)\hat{A}_{q}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{2},\mathbb{C}_{q}\right) \end{equation} As defined, $dA_{q}$ has $\hat{A}_{q}$ as its Fourier-Stieltjes transform\index{$dA_{q}$!Fourier-Stieltjes transf.}. \end{defn} \vphantom{} The $\hat{A}_{q}$s will be of \emph{immen}se importance in Chapter 4, where we will see how the behavior of $dA_{q}$ is for $q=3$ is drastically different from all other odd primes. For now, though, let's focus on the $q=3$ case\textemdash the Collatz Case\textemdash seeing as it provides us an example of a particularly pathological $\left(p,q\right)$-adic measure\textemdash what I call a \textbf{degenerate measure}\textemdash as well as a motivation for the topology-mixing inspiration that birthed my concept of a frame. \begin{prop} \label{prop:Nth partial sum of Fourier series of A_3 hat}\index{hat{A}{3}@$\hat{A}_{3}$} \begin{equation} \sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}=\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2},\textrm{ }\forall N\geq0\label{eq:Convolution of dA_3 with D_N} \end{equation} where $\#_{1}\left(m\right)\overset{\textrm{def}}{=}\#_{2:1}\left(m\right)$ is the number of $1$s in the binary expansion of $m$. \end{prop} Proof: As will be proven in \textbf{Proposition \ref{prop:Generating function identities}} (page \pageref{prop:Generating function identities}), we have a generating function identity: \begin{equation} \prod_{m=0}^{n-1}\left(1+az^{2^{m}}\right)=\sum_{m=0}^{2^{n}-1}a^{\#_{1}\left(m\right)}z^{m}\label{eq:Product-to-sum identity for number of 1s digits} \end{equation} which holds for any field $\mathbb{F}$ of characteristic $0$ and any $a,z\in\mathbb{F}$, and any $n\geq1$. Consequently: \begin{equation} \prod_{n=0}^{-v_{2}\left(t\right)-1}\frac{1+3e^{-2\pi i\left(2^{n}t\right)}}{4}=\frac{1}{\left\|t\right\|_{2}^{2}}\sum_{m=0}^{\left\|t\right\|_{2}-1}3^{\#_{1}\left(m\right)}e^{-2\pi imt} \end{equation} As such, for any $n\geq1$: \begin{align} \sum_{\left\|t\right\|_{2}=2^{n}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}} & =\sum_{\left\|t\right\|_{2}=2^{n}}\left(\frac{1}{\left\|t\right\|_{2}^{2}}\sum_{m=0}^{\left\|t\right\|_{2}-1}3^{\#_{1}\left(m\right)}e^{-2\pi imt}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & =\frac{1}{4^{n}}\sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\sum_{\left\|t\right\|_{2}=2^{n}}e^{2\pi i\left\{ t\left(\mathfrak{z}-m\right)\right\} _{2}}\\ & =\frac{1}{4^{n}}\sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left(2^{n}\left[\mathfrak{z}\overset{2^{n}}{\equiv}m\right]-2^{n-1}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right]\right)\\ & =\frac{1}{2^{n}}\sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n}}{\equiv}m\right]-\frac{1}{2^{n+1}}\sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right] \end{align} Here, note that $m=\left[\mathfrak{z}\right]_{2^{n}}$ is the unique integer $m\in\left\{ 0,\ldots,2^{n}-1\right\} $ for which $\mathfrak{z}\overset{2^{n}}{\equiv}m$ holds true. Thus: \begin{equation} \sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n}}{\equiv}m\right]=3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)} \end{equation} Likewise: \begin{align} \sum_{m=0}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right] & =\sum_{m=0}^{2^{n-1}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right]+\sum_{m=2^{n-1}}^{2^{n}-1}3^{\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right]\\ & =3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}+\sum_{m=0}^{2^{n-1}-1}3^{\#_{1}\left(m+2^{n-1}\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m+2^{n-1}\right]\\ & =3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}+\sum_{m=0}^{2^{n-1}-1}3^{1+\#_{1}\left(m\right)}\left[\mathfrak{z}\overset{2^{n-1}}{\equiv}m\right]\\ & =3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}+3\cdot3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}\\ & =4\cdot3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)} \end{align} and so: \[ \sum_{\left\|t\right\|_{2}=2^{n}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}=\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}-\frac{4\cdot3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}}{2^{n+1}}=\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}-\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}}{2^{n-1}} \] Consequently: \begin{align} \sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}} & =\hat{A}_{3}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{2}=2^{n}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & =1+\sum_{n=1}^{N}\left(\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}-\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n-1}}\right)}}{2^{n-1}}\right)\\ \left(\textrm{telescoping series}\right); & =1+\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}-\underbrace{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{0}}\right)}}_{=3^{0}=1}\\ \left(\left[\mathfrak{z}\right]_{2^{0}}=0\right); & =\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}} \end{align} Q.E.D. \vphantom{} With this formula, the reader will see exactly what I when I say the measure $dA_{3}$ is ``\index{$dA_{3}$!degeneracy}degenerate''. \begin{prop} \label{prop:dA_3 is a degenerate measure}We have: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\overset{\mathbb{C}_{3}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}\label{eq:Degeneracy of dA_3 on Z_2 prime} \end{equation} and: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\overset{\mathbb{C}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{N}_{0}\label{eq:Degeneracy of dA_3 on N_0} \end{equation} \end{prop} \begin{rem} That is to say, the upper limit occurs in the topology of $\mathbb{C}_{3}$ while the lower limit occurs in the topology of $\mathbb{C}$. The reason this works is because, for each $N$, the Fourier series is a sum of finitely many elements of $\overline{\mathbb{Q}}$, which we have embedded in both $\mathbb{C}_{3}$ and $\mathbb{C}$, and that this sum is invariant under the action of $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$. \end{rem} Proof: By (\ref{eq:Convolution of dA_3 with D_N}), we have: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}=\lim_{N\rightarrow\infty}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}} \end{equation} When $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$, the number $\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)$ tends to $\infty$ as $N\rightarrow\infty$, thereby guaranteeing that the above limit converges to $0$ in $\mathbb{C}_{3}$. On the other hand, when $\mathfrak{z}\in\mathbb{N}_{0}$, $\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)=\#_{1}\left(\mathfrak{z}\right)$ for all $N\geq\lambda_{2}\left(\mathfrak{z}\right)$, and so, the above limit then tends to $0$ in $\mathbb{C}$. Q.E.D. \begin{prop} We have the integral formula: \begin{equation} \int_{\mathbb{Z}_{2}}f\left(\mathfrak{z}\right)dA_{3}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{3}}{=}\lim_{N\rightarrow\infty}\frac{1}{4^{N}}\sum_{n=0}^{2^{N}-1}f\left(n\right)3^{\#_{1}\left(n\right)},\textrm{ }\forall f\in C\left(\mathbb{Z}_{2},\mathbb{C}_{3}\right)\label{eq:Riemann sum formula for the dA_3 integral of a continuous (2,3)-adic function} \end{equation} \end{prop} Proof: Since: \begin{equation} \left[\mathfrak{z}\overset{2^{N}}{\equiv}n\right]=\frac{1}{2^{N}}\sum_{\left\|t\right\|_{2}\leq2^{N}}e^{2\pi i\left\{ t\left(\mathfrak{z}-n\right)\right\} _{2}}=\frac{1}{2^{N}}\sum_{\left\|t\right\|_{2}\leq2^{N}}e^{-2\pi i\left\{ tn\right\} _{2}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}} \end{equation} we have: \begin{equation} \int_{\mathbb{Z}_{2}}\left[\mathfrak{z}\overset{2^{N}}{\equiv}n\right]dA_{3}\left(\mathfrak{z}\right)=\frac{1}{2^{N}}\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{3}\left(t\right)e^{2\pi i\left\{ tn\right\} _{2}}=\frac{3^{\#_{1}\left(n\right)}}{4^{N}} \end{equation} for all $N\geq0$ and $n\in\left\{ 0,\ldots,2^{N}-1\right\} $. So, letting $f\in C\left(\mathbb{Z}_{2},\mathbb{C}_{3}\right)$ be arbitrary, taking $N$th truncations yields: \begin{equation} \int_{\mathbb{Z}_{2}}f_{N}\left(\mathfrak{z}\right)dA_{3}\left(\mathfrak{z}\right)=\sum_{n=0}^{2^{N}-1}f\left(n\right)\int_{\mathbb{Z}_{2}}\left[\mathfrak{z}\overset{2^{N}}{\equiv}n\right]dA_{3}\left(\mathfrak{z}\right)=\frac{1}{4^{N}}\sum_{n=0}^{2^{N}-1}f\left(n\right)3^{\#_{1}\left(n\right)} \end{equation} Since $f$ is continuous, the $f_{N}$ converge uniformly to $f$ (\textbf{Proposition \ref{prop:Unif. convergence of truncation equals continuity}}). Since $dA_{3}\in C\left(\mathbb{Z}_{2},\mathbb{C}_{3}\right)^{\prime}$, this guarantees: \begin{equation} \int_{\mathbb{Z}_{2}}f\left(\mathfrak{z}\right)dA_{3}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{3}}{=}\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{2}}f_{N}\left(\mathfrak{z}\right)dA_{3}\left(\mathfrak{z}\right)=\lim_{N\rightarrow\infty}\frac{1}{4^{N}}\sum_{n=0}^{2^{N}-1}f\left(n\right)3^{\#_{1}\left(n\right)} \end{equation} Q.E.D. \vphantom{} Thus, for $d\mu=dA_{3}$, our limit of interest (\ref{eq:The Limit of Interest}) converges to $0$, yet $dA_{3}$ is not the zero measure: this is $dA_{3}$'s ``degeneracy''. $dA_{3}$ also places us in the extremely unusual position of resorting to \emph{different topologies} for different inputs in order to guarantee the point-wise existence of our limit for every $\mathfrak{z}\in\mathbb{Z}_{2}$. The purpose of Section \ref{sec:3.3 quasi-integrability} is to demonstrate that this procedure can be done consistently, and, moreover, in a way that leads to useful results. Contrary to what Monna-Springer theory would have us believe, our next examples show that there are discontinuous\textemdash even \emph{singular!}\textemdash $\left(p,q\right)$-adic functions which can be meaningfully integrated. \begin{prop} \label{prop:sum of v_p}Let $p$ be an integer $\geq2$. Then, for each $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1},\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right)\label{eq:Fourier sum of v_p of t} \end{equation} Here, the use of $\overset{\overline{\mathbb{Q}}}{=}$ is to indicate that the equality is one of elements of $\overline{\mathbb{Q}}$. \end{prop} Proof: Fixing $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $, we have: \begin{align} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\left(-n\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{n=1}^{N}\left(-n\right)\left(p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]-p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}0\right]\right) \end{align} Simplifying gives: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=-Np^{N}\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]+\sum_{n=0}^{N-1}p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right] \end{equation} Because $\mathfrak{z}$ is non-zero, $\mathfrak{z}=p^{v_{p}\left(\mathfrak{z}\right)}\mathfrak{u}$ for some $\mathfrak{u}\in\mathbb{Z}_{p}^{\times}$, where $v_{p}\left(\mathfrak{z}\right)$ is a non-negative rational integer. As such, the congruence $\mathfrak{z}\overset{p^{n}}{\equiv}0$ will fail to hold for all $n\geq v_{p}\left(\mathfrak{z}\right)+1$. As such, we can write:: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right],\textrm{ }\forall N\geq v_{p}\left(\mathfrak{z}\right)+1,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} \end{equation} Since $\mathfrak{z}\overset{p^{n}}{\equiv}0$ holds true if and only if $n\in\left\{ 0,\ldots,v_{p}\left(\mathfrak{z}\right)\right\} $, the Iverson brackets in the above formula all evaluate to $1$. This leaves us with a finite geometric series: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]=\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}p^{n}=\frac{p^{v_{p}\left(\mathfrak{z}\right)+1}-1}{p-1} \end{equation} for all $N\geq v_{p}\left(\mathfrak{z}\right)+1$. Noting that $p\left\|\mathfrak{z}\right\|_{p}^{-1}=p^{v_{p}\left(\mathfrak{z}\right)+1}$ then completes the computation. Q.E.D. \vphantom{} In general, there is no way to define (\ref{eq:Fourier sum of v_p of t}) when $\mathfrak{z}=0$. Indeed: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)=\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\left(-n\right)=-\sum_{n=1}^{N}n\varphi\left(p^{n}\right)=-\left(p-1\right)\sum_{n=1}^{N}np^{n-1} \end{equation} where $\varphi$ is the \textbf{Euler Totient Function}. Since $p$ and $q$ are distinct primes, the $q$-adic absolute value of the $n$th term of the partial sum of the series is: \begin{equation} \left\|np^{n-1}\right\|_{q}=\left\|n\right\|_{q} \end{equation} which does not tend to $0$ as $n\rightarrow\infty$. Thus, (\ref{eq:Fourier sum of v_p of t}) does not converge $q$-adically when $\mathfrak{z}=0$. On the other hand, in the topology of $\mathbb{C}$, the series diverges to $\infty$. Nevertheless, because of the $\hat{\mu}$ defined above, we can write: \begin{equation} \frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1}d\mathfrak{z} \end{equation} to denote the measure defined by: \begin{equation} \int_{\mathbb{Z}_{p}}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1}f\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\textrm{def}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\hat{f}\left(-t\right)v_{p}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right) \end{equation} In this sense, the $\left(p,q\right)$-adic function: \begin{equation} \frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1} \end{equation} is ``quasi-integrable'' in that despite having a singularity at $\mathfrak{z}=0$, we can still use the above integral formula to define its integral: \begin{equation} \int_{\mathbb{Z}_{p}}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1}d\mathfrak{z}\overset{\textrm{def}}{=}0 \end{equation} as the image of the constant function $1$ under the associated measure. With regard to our notational formalism, it is then natural to ask: \begin{equation} \int_{\mathbb{Z}_{p}}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1}d\mathfrak{z}\overset{?}{=}\frac{p}{p-1}\int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}\right\|_{p}^{-1}d\mathfrak{z}-\frac{1}{p-1}\underbrace{\int_{\mathbb{Z}_{p}}d\mathfrak{z}}_{1} \end{equation} The answer is \emph{yes}. \begin{prop} Let $p$ be an integer $\geq2$. Then, for all $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $, we have: \begin{equation} \left\|\mathfrak{z}\right\|_{p}^{-1}\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p}+\frac{p-1}{p}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right)\label{eq:reciprocal of the p-adic absolute value of z is quasi-integrable} \end{equation} \end{prop} Proof: Let: \begin{equation} \frac{1}{p-1}d\mathfrak{z} \end{equation} denote the $\left(p,q\right)$-adic measure which is the scalar multiple of the $\left(p,q\right)$-adic Haar probability measure by $1/\left(p-1\right)$. Then, by definition, for any $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$: \begin{align} \int_{\mathbb{Z}_{p}}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1}f\left(\mathfrak{z}\right)d\mathfrak{z}+\int_{\mathbb{Z}_{p}}\frac{f\left(\mathfrak{z}\right)}{p-1}d\mathfrak{z} & =\sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\hat{f}\left(-t\right)v_{p}\left(t\right)+\frac{\hat{f}\left(0\right)}{p-1}\\ & =\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\nu}\left(t\right) \end{align} where $\hat{\nu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ is defined by: \begin{equation} \hat{\nu}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} \frac{1}{p-1} & \textrm{if }t=0\\ v_{p}\left(t\right) & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} Running through the computation of $\tilde{\nu}_{N}\left(\mathfrak{z}\right)$ yields: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\nu\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{Q}}{=}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}}{p-1},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} ,\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right) \end{equation} Hence, $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $ and $N>v_{p}\left(\mathfrak{z}\right)$ imply: \begin{align} \left\|\mathfrak{z}\right\|_{p}^{-1} & \overset{\overline{\mathbb{Q}}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\frac{p-1}{p}\nu\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & \overset{\overline{\mathbb{Q}}}{=}\frac{1}{p}+\frac{p-1}{p}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} as desired. Q.E.D. \begin{example} Letting $\mathfrak{a}\in\mathbb{Z}_{p}$ be arbitrary, for all $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $ and all $N>v_{p}\left(\mathfrak{z}\right)$, we have: \[ \left\|\mathfrak{z}-\mathfrak{a}\right\|_{p}^{-1}\overset{\mathbb{Q}}{=}\frac{1}{p}+\frac{p-1}{p}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{-2\pi i\left\{ t\mathfrak{a}\right\} _{p}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \] Consequently, letting $\left\{ \mathfrak{a}_{n}\right\} _{n\geq0}$ be any sequence of distinct elements of $\mathbb{Z}_{p}$ and letting $\left\{ \mathfrak{b}_{n}\right\} _{n\geq0}$ be any sequence in $\mathbb{C}_{q}$ tending to $0$ in $q$-adic magnitude, the function: \begin{equation} \sum_{n=0}^{\infty}\frac{\mathfrak{b}_{n}}{\left\|\mathfrak{z}-\mathfrak{a}\right\|_{p}}\label{eq:"quasi-integrable" function with infinitely many singularities} \end{equation} is ``quasi-integrable'', because we can get a measure out of it by writing: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\sum_{n=0}^{\infty}\frac{\mathfrak{b}_{n}}{\left\|\mathfrak{z}-\mathfrak{a}\right\|_{p}}d\mathfrak{z} \end{equation} as: \begin{equation} \frac{\hat{f}\left(0\right)}{p}\sum_{n=0}^{\infty}\mathfrak{b}_{n}+\frac{p-1}{p}\sum_{t\neq0}v_{p}\left(t\right)\left(\sum_{n=0}^{\infty}\mathfrak{b}_{n}e^{-2\pi i\left\{ t\mathfrak{a}_{n}\right\} _{p}}\right)\hat{f}\left(-t\right) \end{equation} for any $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Moreover, because of the failure of (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}) to converge in either $\mathbb{C}_{q}$ or $\mathbb{C}$ for $\mathfrak{z}=0$, the Fourier series for (\ref{eq:"quasi-integrable" function with infinitely many singularities}) then fails to converge in either $\mathbb{C}_{q}$ or $\mathbb{C}$ for $\mathfrak{z}=\mathfrak{a}_{n}$ for any $n$. Even then, this is not the most pathological case we might have to deal with. Once again, thanks to the invariance of (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}) under the action of $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$, the point-wise convergence of (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}) on $\mathbb{Z}_{p}\backslash\left\{ 0\right\} $ has the useful property of occurring in \emph{every} field extension of $\overline{\mathbb{Q}}$: for each $\mathfrak{z}$, (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}) is constant for all $N>v_{p}\left(\mathfrak{z}\right)$. On the other hand, suppose the limit of interest (\ref{eq:The Limit of Interest}) yields a rising-continuous function $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which converges in $\mathbb{C}_{q}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ and converges in $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$. Then, translating by some $\mathfrak{a}\in\mathbb{Z}_{p}$, we can shift the sets on which $\chi$ converges $q$-adically by considering the Fourier series for $\chi\left(\mathfrak{z}+\mathfrak{a}\right)$. But then, letting $\mathfrak{a}$ and $\mathfrak{b}$ be distinct $p$-adic integers, how might we make sense of the convergence of the Fourier series of: \begin{equation} \chi\left(\mathfrak{z}+\mathfrak{a}\right)+\chi\left(\mathfrak{z}+\mathfrak{b}\right) \end{equation} In this set-up, there may be a $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$ for which $\chi\left(\mathfrak{z}+\mathfrak{a}\right)$'s Fourier series converges $q$-adically and $\chi\left(\mathfrak{z}+\mathfrak{b}\right)$'s series converges in $\mathbb{C}$ and \emph{not }in $\mathbb{C}_{q}$! Worse yet, there is a very natural reason to \emph{want} to work with linear combinations of the form: \begin{equation} \sum_{n}\mathfrak{b}_{n}\chi\left(\mathfrak{z}+\mathfrak{a}_{n}\right) \end{equation} for $\mathfrak{b}_{n}$s in $\mathbb{C}_{q}$ and $\mathfrak{a}_{n}$s in $\mathbb{Z}_{p}$; these arise naturally when considering $\left(p,q\right)$-adic analogues of the Wiener Tauberian Theorem. Results of this type describe conditions in which (and the extent to which) a quasi-integrable function $\chi\left(\mathfrak{z}\right)$ will have well-defined reciprocal. Since we want to determine the values $x\in\mathbb{Z}$ for which $\chi_{H}\left(\mathfrak{z}\right)-x$ vanishes for some $\mathfrak{z}\in\mathbb{Z}_{p}$, these issues are, obviously, of the utmost import to us. This particular problem will be dealt with by defining vector spaces of $\left(p,q\right)$-adically bounded functions $\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$, so as to guarantee that we can take linear combinations of translates of functions on $\mathbb{Z}_{p}$ without having to worry about getting our topologies in a twist. \end{example} \vphantom{} Whereas the previous two examples were liberating, the next example will be dour and sobering. It tells us that our newly acquired freedoms \emph{do} come at a price. \begin{example} Having shown that: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} ,\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right) \end{equation} upon letting $N\rightarrow\infty$, we can write: \begin{equation} \sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{F}}{=}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1} \end{equation} where $\mathbb{F}$ is any valued field extension of $\overline{\mathbb{Q}}$; note, completeness of $\mathbb{F}$ is \emph{not }required! Now, let $p=2$. Then, we can write: \begin{equation} \sum_{\left\|t\right\|_{2}\leq2^{N}}\left(1-\mathbf{1}_{0}\left(t\right)\right)v_{2}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\overset{\overline{\mathbb{Q}}}{=}2\left\|\mathfrak{z}\right\|_{2}^{-1}-1,\textrm{ }\forall N\geq v_{2}\left(\mathfrak{z}\right) \end{equation} Here, $1-\mathbf{1}_{0}\left(t\right)$ is $0$ when $t\overset{1}{\equiv}0$ and is $1$ for all other $t$. Now, let us add the Fourier series generated by $\hat{A}_{3}$. This gives: \begin{equation} \sum_{\left\|t\right\|_{2}\leq2^{N}}\left(\hat{A}_{3}\left(t\right)+\left(1-\mathbf{1}_{0}\left(t\right)\right)v_{2}\left(t\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\overset{\overline{\mathbb{Q}}}{=}2\left\|\mathfrak{z}\right\|_{2}^{-1}-1+\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}} \end{equation} As $N\rightarrow\infty$, when $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$, the right-hand side converges to $2\left\|\mathfrak{z}\right\|_{2}^{-1}-1$ in $\mathbb{C}_{3}$ ; for $\mathbb{\mathfrak{z}}\in\mathbb{N}_{0}$, meanwhile, the right-hand side converges to $2\left\|\mathfrak{z}\right\|_{2}^{-1}-1$ in $\mathbb{C}$. The same is true if we remove $\hat{A}_{3}\left(t\right)$. As such, we have two different formulas: \begin{equation} f\mapsto\sum_{t\in\hat{\mathbb{Z}}_{2}\backslash\left\{ 0\right\} }\hat{f}\left(t\right)v_{2}\left(-t\right) \end{equation} \begin{equation} f\mapsto\sum_{t\in\hat{\mathbb{Z}}_{2}}\hat{f}\left(t\right)\left(\hat{A}_{3}\left(-t\right)+\left(1-\mathbf{1}_{0}\left(-t\right)\right)v_{2}\left(-t\right)\right) \end{equation} representing two \emph{entirely different }$\left(2,3\right)$-adic measures, and yet, \emph{both }of them constitute perfectly reasonable ways of defining the integral of $\left(2\left\|\mathfrak{z}\right\|_{2}^{-1}-1\right)f\left(\mathfrak{z}\right)$. These are: \begin{equation} \int_{\mathbb{Z}_{2}}\left(2\left\|\mathfrak{z}\right\|_{2}^{-1}-1\right)f\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{3}}{=}\sum_{t\in\hat{\mathbb{Z}}_{2}\backslash\left\{ 0\right\} }\hat{f}\left(t\right)v_{2}\left(-t\right) \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{2}}\left(2\left\|\mathfrak{z}\right\|_{2}^{-1}-1\right)f\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{3}}{=}\sum_{t\in\hat{\mathbb{Z}}_{2}}\hat{f}\left(t\right)\left(\hat{A}_{3}\left(-t\right)+\left(1-\mathbf{1}_{0}\left(-t\right)\right)v_{2}\left(-t\right)\right) \end{equation} Both of these are valid because the $\hat{\mu}$ we are using against $\hat{f}$ are $3$-adically bounded. However, because of $dA_{3}$'s degeneracy, even if $\int fdA_{3}\neq0$, when we try to consider the Fourier series generated by $\hat{\mu}$, the part generated by $\hat{A}_{3}$ vanishes into thin air. So, while quasi-integrability allows us to integrate discontinuous functions, there will be no canonical choice for these integrals' values. Indeed, as we will see in Subsection \ref{subsec:3.3.5 Quasi-Integrability}, the integral of a quasi-integrable function will only be well-defined modulo an arbitrary degenerate measure. \end{example} \vphantom{} Before we begin our study of these matters in earnest, we record the following result, which\textemdash as described below\textemdash can be used to define a ``$p,q$-adic Mellin transform\index{Mellin transform!left(p,qright)-adic@$\left(p,q\right)$-adic}\index{$p,q$-adic!Mellin transform}''. This may be of great interest in expanding the scope of $\left(p,q\right)$-adic analysis beyond what was previously thought possible. \begin{lem} Let $p$ and $q$ be integers $\geq2$ so that $q\mid\left(p-1\right)$. Then, for any non-zero rational number $r$ and any $\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} $: \begin{equation} \left\|\mathfrak{z}\right\|_{p}^{-r}\overset{\mathbb{C}_{q}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\frac{1}{p^{r}}\sum_{m=0}^{\infty}\binom{r}{m}\left(p-1\right)^{m}\sum_{\begin{array}{c} 0<\left\|s_{1}\right\|_{p},\ldots,\left\|s_{m}\right\|_{p}\leq p^{N}\\ s_{1}+\cdots+s_{m}=t \end{array}}\prod_{j=1}^{m}v_{p}\left(s_{j}\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:-rth power of the p-adic absolute value is sometimes quasi-integrable} \end{equation} for all $N>v_{p}\left(\mathfrak{z}\right)$. Here, the condition on $q\mid\left(p-1\right)$ guarantees that the $m$-series converges $q$-adically. Also, the sum over $s_{1},\ldots,s_{m}$ is defined to be $1$ when $m=0$. \emph{Note}: if $r$ is a positive integer, the infinite series reduces to the finite sum $\sum_{m=0}^{r}$, and as such, the condition that $q\mid\left(p-1\right)$ can be dropped. \end{lem} Proof: Letting $p$, $q$, $\mathfrak{z}$, and $N$ be as given, we take (\ref{eq:reciprocal of the p-adic absolute value of z is quasi-integrable}) and raise it to the $r$th power: \begin{align} \left\|\mathfrak{z}\right\|_{p}^{-r} & =\left(\frac{1}{p}+\frac{p-1}{p}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right)^{r}\\ & =\frac{1}{p^{r}}\left(1+\left(p-1\right)\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right)^{r}\\ & =\frac{1}{p^{r}}\sum_{m=0}^{\infty}\binom{r}{m}\left(p-1\right)^{m}\underbrace{\sum_{0<\left\|t_{1}\right\|_{p},\ldots,\left\|t_{m}\right\|_{p}\leq p^{N}}v_{p}\left(t_{1}\right)\cdots v_{p}\left(t_{m}\right)e^{2\pi i\left\{ \left(t_{1}+\cdots+t_{m}\right)\mathfrak{z}\right\} _{p}}}_{1\textrm{ when }m=0} \end{align} Note that the use of the binomial series for $\left(1+\mathfrak{y}\right)^{r}$ is valid here, seeing as: \begin{equation} \left(1+\mathfrak{y}\right)^{r}\overset{\mathbb{C}_{q}}{=}\sum_{m=0}^{\infty}\binom{r}{m}\mathfrak{y}^{m},\textrm{ }\forall\mathfrak{y}\in\mathbb{C}_{q}:\left\|\mathfrak{y}\right\|_{q}<1 \end{equation} and that: \begin{equation} \left\|\left(p-1\right)\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\leq\left\|p-1\right\|_{q}\cdot\underbrace{\max_{0<\left\|t\right\|_{p}\leq p^{N}}\left\|v_{p}\left(t\right)\right\|_{q}}_{\leq1}\leq\left\|p-1\right\|_{q} \end{equation} where the divisibility of $p-1$ by $q$ then guarantees $\left\|p-1\right\|_{q}<1$, and hence, that the series in $m$ converges uniformly in $t$ and $\mathfrak{z}$. We then have: \begin{eqnarray} & \sum_{0<\left\|t_{1}\right\|_{p},\ldots,\left\|t_{m}\right\|_{p}\leq p^{N}}v_{p}\left(t_{1}\right)\cdots v_{p}\left(t_{m}\right)e^{2\pi i\left\{ \left(t_{1}+\cdots+t_{m}\right)\mathfrak{z}\right\} _{p}}\\ & =\\ & \sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\sum_{\begin{array}{c} 0<\left\|s_{1}\right\|_{p},\ldots,\left\|s_{m}\right\|_{p}\leq p^{N}\\ s_{1}+\cdots+s_{m}=t \end{array}}v_{p}\left(s_{1}\right)\cdots v_{p}\left(s_{m}\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{eqnarray} Hence: \begin{align} \left\|\mathfrak{z}\right\|_{p}^{-r} & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\frac{1}{p^{r}}\sum_{m=0}^{\infty}\binom{r}{m}\left(p-1\right)^{m}\underbrace{\sum_{\begin{array}{c} 0<\left\|s_{1}\right\|_{p},\ldots,\left\|s_{m}\right\|_{p}\leq p^{N}\\ s_{1}+\cdots+s_{m}=t \end{array}}\prod_{j=1}^{m}v_{p}\left(s_{j}\right)}_{1\textrm{ when }m=0}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} for all $N>v_{p}\left(\mathfrak{z}\right)$, as desired. Q.E.D. \begin{rem}[\textbf{A $\left(p,q\right)$-adic Mellin transform}] \label{rem:pq adic mellin transform}The above allows us to define a $\left(p,q\right)$-adic Mellin transform, $\mathscr{M}_{p,q}$, by: \begin{equation} \mathscr{M}_{p,q}\left\{ f\right\} \left(r\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}\right\|_{p}^{r-1}f\left(\mathfrak{z}\right)d\mathfrak{z},\textrm{ }\forall r\in\mathbb{Q},\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Definition of (p,q)-adic Mellin transform} \end{equation} whenever $q\mid\left(p-1\right)$. In particular, we then have: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\mathfrak{z}\right\|_{p}^{r}f\left(\mathfrak{z}\right)d\mathfrak{z}=\sum_{t\in\hat{\mathbb{Z}}_{p}}\left(p^{r}\sum_{m=0}^{\infty}\binom{1-r}{m}\left(p-1\right)^{m}\sum_{\begin{array}{c} \mathbf{s}\in\left(\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} \right)^{m}\\ \Sigma\left(\mathbf{s}\right)=t \end{array}}v_{p}\left(\mathbf{s}\right)\right)\hat{f}\left(-t\right)\label{eq:Formula for (p,q)-adic Mellin transform} \end{equation} where the $m$-sum is $1$ when $=0$, and where: \begin{align} \Sigma\left(\mathbf{s}\right) & \overset{\textrm{def}}{=}\sum_{j=1}^{m}s_{j}\\ v_{p}\left(\mathbf{s}\right) & \overset{\textrm{def}}{=}\prod_{j=1}^{m}v_{p}\left(s_{j}\right) \end{align} for all $\mathbf{s}=\left(s_{1},\ldots,s_{m}\right)\in\left(\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} \right)^{m}$. If we allow for $f$ to take values in the universal $q$-adic field $\Omega_{q}$ (the spherical completion of $\mathbb{C}_{q}$, see \cite{Robert's Book} for details), it may be possible to interpret $\mathscr{M}_{p,q}\left\{ f\right\} \left(r\right)$ for an arbitrary real number $r$ (which would mean that $\mathscr{M}_{p,q}\left\{ f\right\} :\mathbb{R}\rightarrow\Omega_{q}$), although care would obviously need to be taken to properly define and interpret (\ref{eq:Formula for (p,q)-adic Mellin transform}) in this case. Furthermore, if we can show that, given a quasi-integrable function $\chi$, the product $\left\|\mathfrak{z}\right\|_{p}^{r-1}\chi\left(\mathfrak{z}\right)$ will be quasi-integrable for $r\in\mathbb{Q}$ under certain circumstances, we could then define the $\left(p,q\right)$-adic Mellin transform of $\chi$. \end{rem} \begin{rem} It is worth noting that this Mellin transform formalism potentially opens the door for a $\left(p,q\right)$-adic notion of differentiability in the sense of distributions. This notion of differentiability is already well-established for the $\left(p,\infty\right)$-adic case (real/complex-valued functions), having originated a 1988 paper by V. Vladimirov, \emph{Generalized functions over the field of $p$-adic numbers} \cite{Vladimirov - the big paper about complex-valued distributions over the p-adics}. This paper is a comprehensive expos of everything one needs to know to work with distributions in the $\left(p,\infty\right)$-adic context. This method, now known as the \textbf{Vladimirov operator}\index{Vladimirov operator}\textbf{ }or \index{$p$-adic!fractional differentiation}\textbf{$p$-adic (fractional) differentiation }has since become standard in the areas of $\left(p,\infty\right)$-adic mathematical theoretical physics (``$p$-adic mathematical physics'') where it is employed; see, for instance, the article \cite{First 30 years of p-adic mathematical physics}, which gives a summary of the first thirty years' worth of developments in the subject. To borrow from \cite{First 30 years of p-adic mathematical physics} (although, it should be noted there is a typographical error in their statement of the definition), for $\alpha\in\mathbb{C}$, the order $\alpha$ $p$-adic fractional derivative of a function $f:\mathbb{Q}_{p}\rightarrow\mathbb{C}$ is defined\footnote{This definition is slightly more general than the one given in Subsection \ref{subsec:3.1.1 Some-Historical-and}, which was only valid for real $\alpha>0$; we re-obtain (\ref{eq:Definition of the Vladimirov Fractional Differentiation Operator}) from (\ref{eq:First formula for the p-adic fractional derivative}) below.} by: \begin{equation} D^{\alpha}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\int_{\mathbb{Q}_{p}}\left\|\mathfrak{y}\right\|_{p}^{\alpha}\hat{f}\left(-\mathfrak{y}\right)e^{2\pi i\left\{ \mathfrak{y}\mathfrak{z}\right\} _{p}}d\mathfrak{y}\label{eq:First formula for the p-adic fractional derivative} \end{equation} where $d\mathfrak{y}$ is the real-valued $p$-adic Haar measure on $\mathbb{Q}_{p}$, normalized to be a probability measure on $\mathbb{Z}_{p}$, and where: \begin{equation} \hat{f}\left(\mathfrak{y}\right)\overset{\textrm{def}}{=}\int_{\mathbb{Q}_{p}}f\left(\mathfrak{x}\right)e^{-2\pi i\left\{ \mathfrak{y}\mathfrak{x}\right\} _{p}}d\mathfrak{x},\textrm{ }\forall\mathfrak{y}\in\mathbb{Q}_{p}\label{eq:Fourier transform over Q_p} \end{equation} is the Fourier transform of $f$. All of these require $f$ to be sufficiently well-behaved\textemdash say, compactly supported. When the support of $f$ lies in $\mathbb{Z}_{p}$, (\ref{eq:Fourier transform over Q_p}) reduces to: \[ \hat{f}\left(t\right)\overset{\mathbb{C}}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{x}\right)e^{-2\pi i\left\{ t\mathfrak{x}\right\} _{p}}d\mathfrak{x} \] and (\ref{eq:First formula for the p-adic fractional derivative}) becomes: \begin{equation} D^{\alpha}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\left\|t\right\|_{p}^{\alpha}\hat{f}\left(-t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:p-adic fractional derivative over Z_p} \end{equation} When $\alpha>0$, (\ref{eq:First formula for the p-adic fractional derivative}) can be written as: \begin{equation} D^{\alpha}\left\{ f\right\} \left(\mathfrak{z}\right)=\frac{1}{\Gamma_{p}\left(-\alpha\right)}\int_{\mathbb{Q}_{p}}\frac{f\left(\mathfrak{z}\right)-f\left(\mathfrak{y}\right)}{\left\|\mathfrak{z}-\mathfrak{y}\right\|_{p}^{\alpha+1}}d\mathfrak{y}\label{eq:Second formula for the p-adic fractional derivative} \end{equation} where, recall, $\Gamma_{p}\left(-\alpha\right)$ is the physicist's notation for the normalization constant: \begin{equation} \Gamma_{p}\left(-\alpha\right)=\frac{p^{\alpha}-1}{1-p^{-1-\alpha}} \end{equation} \cite{First 30 years of p-adic mathematical physics} contains no less than \emph{three-hundred forty two} references, describing innumerable applications of this fractional derivative, including ``spectral properties, operators on bounded regions, analogs of elliptic and parabolic equations, a wave-type equation'', and many more. Given that my $\left(p,q\right)$-adic implementation of the Mellin transform (\ref{eq:Definition of (p,q)-adic Mellin transform}) can then be used to make sense of (\ref{eq:First formula for the p-adic fractional derivative}) for any $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and any $r\in\mathbb{Q}$, this seems like a research direction worth exploring. \end{rem} \subsection{\label{subsec:3.3.2 The--adic-Dirichlet}The $p$-adic Dirichlet Kernel and $\left(p,q\right)$-adic Fourier Resummation Lemmata} In this subsection, we will introduce specific types of $\left(p,q\right)$-adic measures for which (\ref{eq:The Limit of Interest}) is sufficiently well-behaved for us to construct useful spaces of quasi-integrable functions. Consequently, this subsection will be low on concepts and high on computations\textemdash though that's nothing in comparison with what we'll confront in Chapters 4 and 6\textemdash but, I digress. A recurring device in Section \ref{sec:3.3 quasi-integrability} is to start with a $\overline{\mathbb{Q}}$-valued function defined on $\hat{\mathbb{Z}}_{p}$ and then use it to create functions on $\mathbb{Z}_{p}$ by summing the associated Fourier series. Delightfully, this procedure is a direct $\left(p,q\right)$-adic analogue of a fundamental construction of classical Fourier analysis: convolution against the Dirichlet kernel. \begin{defn}[\textbf{The $p$-adic Dirichlet Kernel}] \index{Fourier series!$N$th partial sum}\ \vphantom{} I. Let $\mathbb{F}$ be an algebraically closed field extension of $\mathbb{Q}$, and let $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{F}$. For each $N\in\mathbb{N}_{0}$, we then define \nomenclature{$\tilde{\mu}_{N}$}{$N$th partial sum of Fourier series generated by $\hat{\mu}$}$\tilde{\mu}_{N}:\mathbb{Z}_{p}\rightarrow\mathbb{F}$ by: \begin{equation} \tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Definition of mu_N twiddle} \end{equation} where the sum is taken over all $t\in\hat{\mathbb{Z}}_{p}$ satisfying $\left\|t\right\|_{p}\leq p^{N}$. Note that $\tilde{\mu}_{N}$ is then locally constant. As such, for any valued field $K$ extending $\mathbb{F}$, $\tilde{\mu}_{N}$ will be continuous as a function from $\mathbb{Z}_{p}$ to $K$. We call $\tilde{\mu}_{N}$ the \textbf{$N$th partial sum of the Fourier series generated by $\hat{\mu}$}. \vphantom{} II. For each $N\in\mathbb{N}_{0}$, we define the function $D_{p:N}:\mathbb{Z}_{p}\rightarrow\mathbb{Q}$ by: \begin{equation} D_{p:N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}p^{N}\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]\label{eq:Definition of the p-adic Dirichlet Kernel} \end{equation} We call \nomenclature{$D_{p:N}$}{$N$th $p$-adic Dirichlet kernel}$D_{p:N}$ the $N$th \textbf{$p$-adic Dirichlet kernel}\index{$p$-adic!Dirichlet kernel}. $\left\{ D_{p:N}\right\} _{N\geq0}$ is the \textbf{family of $p$-adic Dirichlet kernels}. Note that since each $D_{p:N}$ is locally constant, each is an element of $C\left(\mathbb{Z}_{p},K\right)$. Moreover, the Fourier transform of $D_{p:N}$ has finite support, with: \begin{equation} \hat{D}_{p:N}\left(t\right)=\mathbf{1}_{0}\left(p^{N}t\right)\overset{\textrm{def}}{=}\begin{cases} 1 & \textrm{if }\left\|t\right\|_{p}\leq p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases}\label{eq:Fourier Transform of the p-adic Dirichlet Kernel} \end{equation} We note here that since the $p$-adic Dirichlet kernels are rational valued, we can also compute their real/complex valued Fourier transforms with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$, and\textemdash moreover\textemdash the resulting $\hat{D}_{p:N}\left(t\right)$ will be \emph{the same} as the one given above. In other words, the $\left(p,q\right)$-adic and $\left(p,\infty\right)$-adic Fourier transforms of $D_{p:N}$ and the procedures for computing them are \emph{formally identical}. I use the term \textbf{$\left(p,\infty\right)$-adic Dirichlet Kernel} when viewing the $D_{p:N}$s as taking values in $\mathbb{C}$; I use the term \textbf{$\left(p,K\right)$-adic Dirichlet Kernel }when viewing the $D_{p:N}$s as taking values in a metrically complete $q$-adic field $K$; I use the term \textbf{$\left(p,q\right)$-adic Dirichlet Kernel} for when the specific field $K$ is either not of concern or is $\mathbb{C}_{q}$. \end{defn} \begin{rem} Regarding the name ``$p$-adic Dirichlet Kernel'', it is worth noting that when one is doing $\left(p,\infty\right)$-adic analysis, the family of $p$-adic Dirichlet Kernels then form an approximate identity, in the sense that for any \nomenclature{$L^{1}\left(\mathbb{Z}_{p},\mathbb{C}\right)$}{set of absolutely integrable $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$}$f\in L^{1}\left(\mathbb{Z}_{p},\mathbb{C}\right)$ (complex-valued function on $\mathbb{Z}_{p}$, integrable with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$), it can be shown that: \begin{equation} \lim_{N\rightarrow\infty}\left(D_{p:N}f\right)\left(\mathfrak{z}\right)\overset{\mathbb{C}}{=}\lim_{N\rightarrow\infty}p^{N}\int_{\mathbb{Z}_{p}}\left[\mathfrak{y}\overset{p^{N}}{\equiv}\mathfrak{z}\right]f\left(\mathfrak{y}\right)d\mathfrak{y}=f\left(\mathfrak{z}\right)\label{eq:p infinity adic Lebesgue differentiation theorem} \end{equation} for \emph{almost} every $\mathfrak{z}\in\mathbb{Z}_{p}$; in particular, the limit holds for all $\mathfrak{z}$ at which $f$ is continuous. Indeed, (\ref{eq:p infinity adic Lebesgue differentiation theorem}) is an instance of the \textbf{Lebesgue Differentiation Theorem}\index{Lebesgue Differentiation Theorem}\footnote{Like in the archimedean case, the theorem is a consequence of a covering lemma and estimates for the ($\left(p,\infty\right)$-adic) Hardy-Littlewood maximal function (see \cite{Geometric Measure Theory}, for instance).}\textbf{ }for real and complex valued functions of a $p$-adic variable. Our use of the $p$-adic Dirichlet Kernels in the $\left(p,q\right)$-adic context therefore falls under the topic of summability kernels and approximate identities\index{approximate identity}. It remains to be seen if convolving other types of kernels with $\left(p,q\right)$-adic measures or $\left(p,q\right)$-adic functions will yield interesting results. \end{rem} \vphantom{} For now, let us establish the identities of import. \begin{prop}[\textbf{$D_{p:N}$ is an approximate identity}] \label{prop:D_p is an approximate identity}\ \vphantom{} I. Let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, for all $N\in\mathbb{N}_{0}$: \begin{equation} \left(D_{p:N}f\right)\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}p^{N}\left[\mathfrak{y}\overset{p^{N}}{\equiv}\mathfrak{z}\right]f\left(\mathfrak{y}\right)d\mathfrak{y}=\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} Moreover, as $N\rightarrow\infty$, $D_{p:N}f$ converges uniformly in $\mathbb{C}_{q}$ to $f$: \begin{equation} \lim_{N\rightarrow\infty}\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-\left(D_{p:N}f\right)\left(\mathfrak{z}\right)\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} Also, this shows that the evaluation map $f\mapsto f\left(\mathfrak{z}_{0}\right)$ is a continuous linear functional on $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \vphantom{} II. Let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Then, for all $N\in\mathbb{N}_{0}$: \begin{equation} \left(D_{p:N}d\mu\right)\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}p^{N}\left[\mathfrak{y}\overset{p^{N}}{\equiv}\mathfrak{z}\right]d\mu\left(\mathfrak{y}\right)=\tilde{\mu}_{N}\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} For fixed $N$, convergence occurs uniformly with respect to $\mathfrak{z}$. Additionally, if $\hat{\mu}$ takes values in $\overline{\mathbb{Q}}$, the the above equality also holds in $\mathbb{C}$, and, for fixed $N$, the convergence there is uniform with respect to $\mathfrak{z}$. \end{prop} Proof: The \textbf{Convolution Theorem }(\textbf{Theorem \ref{thm:Convolution Theorem}}) for the\textbf{ }$\left(p,q\right)$-adic Fourier Transform tells us that: \begin{equation} \left(D_{p:N}f\right)\left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{D}_{p:N}\left(t\right)\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} and: \begin{equation} \left(D_{p:N}d\mu\right)\left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{D}_{p:N}\left(t\right)\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} Using (\ref{eq:Fourier Transform of the p-adic Dirichlet Kernel}) gives the desired results. The uniform convergence of $D_{p:N}f$ to $f$ follows from: \begin{equation} \sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-\left(D_{p:N}f\right)\left(\mathfrak{z}\right)\right\|_{q}=\left\|\sum_{\left\|t\right\|_{p}>p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\leq\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q} \end{equation} with the upper bound tending to $0$ in $\mathbb{R}$ as $N\rightarrow\infty$, since $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\Leftrightarrow\hat{f}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Finally, fixing $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$, the maps $f\mapsto\left(D_{p:N}f\right)\left(\mathfrak{z}_{0}\right)$ are a family of continuous linear functionals on $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ which are indexed by $N$ which converge to the evaluation map $f\mapsto f\left(\mathfrak{z}_{0}\right)$ in supremum norm as $N\rightarrow\infty$. Since the dual space of $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is a Banach space, it is closed under such limits. This proves the evaluation map $f\mapsto f\left(\mathfrak{z}_{0}\right)$ is an element of $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Q.E.D. \vphantom{} With respect to the above, the introduction of frames in Subsection \ref{subsec:3.3.3 Frames} is little more than a backdrop for studying the point-wise behavior of the limit $\left(D_{p:N}d\mu\right)\left(\mathfrak{z}\right)$ as $N\rightarrow\infty$ for various $d\mu$. Since we are potentially going to need to utilize different topologies to make sense of this limit, we might as well come up with a consistent set of terminology for describing our choices of topologies, instead of having to repeatedly specify the topologies \emph{every time }we want to talk about a limit\textemdash this is what frames are for. For now, though, let us continue our investigation of $\tilde{\mu}_{N}=D_{p:N}d\mu$. \begin{prop} \label{prop:Criterion for zero measure in terms of partial sums of Fourier series}Let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Then $d\mu$ is the zero measure if and only if: \begin{equation} \lim_{N\rightarrow\infty}\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q}\overset{\mathbb{R}}{=}0\label{eq:Criterion for a (p,q)-adic measure to be zero} \end{equation} where, recall, $\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q}=\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}$. \end{prop} Proof: I. If $d\mu$ is the zero measure, the $\hat{\mu}$ is identically zero, and hence, so is $\tilde{\mu}_{N}$. \vphantom{} II. Conversely, suppose $\lim_{N\rightarrow\infty}\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q}\overset{\mathbb{R}}{=}0$. Then, let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ be arbitrary. Since $\tilde{\mu}_{N}$ is continuous for each $N$ and has $\mathbf{1}_{0}\left(p^{N}t\right)\hat{\mu}\left(t\right)$ as its Fourier transform, we can write: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\tilde{\mu}_{N}\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{q}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right) \end{equation} Then, by the $\left(p,q\right)$-adic triangle inequality (\ref{eq:(p,q)-adic triangle inequality}): \[ \left\|\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\right\|_{q}\overset{\mathbb{R}}{=}\left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\tilde{\mu}_{N}\left(\mathfrak{z}\right)d\mathfrak{z}\right\|_{q}\leq\left\Vert f\cdot\tilde{\mu}_{N}\right\Vert _{p,q}\leq\left\Vert f\right\Vert _{p,q}\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q} \] Because the upper bound tends to $0$ in $\mathbb{R}$ as $N\rightarrow\infty$, and because: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\overset{\mathbb{C}_{q}}{=}\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right) \end{equation} this shows that: \begin{equation} \left\|\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)d\mu\left(\mathfrak{z}\right)\right\|_{q}\overset{\mathbb{R}}{=}\lim_{N\rightarrow\infty}\left\|\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\right\|_{q}\leq\left\Vert f\right\Vert _{p,q}\cdot\lim_{N\rightarrow\infty}\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q}=0 \end{equation} Since $f$ was arbitrary, we conclude that $d\mu$ is the zero measure. Q.E.D. \vphantom{}Next, we have the \textbf{Fourier resummation lemmata} which\index{resummation lemmata} will synergize with our analysis of quasi-integrable functions. To make these easier to state and remember, here are some definitions describing the conditions which the lemmata will impose on $\hat{\mu}$. \begin{defn} Consider a field $\mathbb{F}$ of characteristic $0$, a function $\kappa:\mathbb{N}_{0}\rightarrow\mathbb{F}$, and let $K$ be a metrically complete valued field extension of $\mathbb{F}$ which contains $\kappa\left(\mathbb{N}_{0}\right)$. \vphantom{} I. We say $\kappa$ is \index{tame}\textbf{$\left(p,K\right)$-adically tame} on a set $X\subseteq\mathbb{Z}_{p}$ whenever $\lim_{n\rightarrow\infty}\left\|\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{K}=0$ for all $\mathfrak{z}\in X$. We do not mention $X$ when $X=\mathbb{Z}_{p}$. If $K$ is a $q$-adic field, we say $\kappa$ is $\left(p,q\right)$-adically tame on $X$; if $K$ is archimedean, we say $\kappa$ is $\left(p,\infty\right)$-adically tame on $X$. \vphantom{} II. For a prime $p$, we say $\kappa$ has \index{$p$-adic!structure}\textbf{$p$-adic structure}\footnote{It would seem that this property is necessary in order for our computations to simplify in interesting ways. Life is much more difficult without it.}\textbf{ }whenever there are constants $a_{0},\ldots,a_{p-1}\in\kappa\left(\mathbb{N}_{0}\right)$ so that: \begin{equation} \kappa\left(pn+j\right)=a_{j}\kappa\left(n\right),\textrm{ }\forall n\geq0,\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \label{eq:Definition of left-ended p-adic structural equations} \end{equation} We call (\ref{eq:Definition of left-ended p-adic structural equations}) the \textbf{left-ended structural equations }of $\kappa$. \end{defn} \begin{rem} Note that the set of functions $\kappa:\mathbb{N}_{0}\rightarrow\mathbb{F}$ with $p$-adic structure is then a linear space over $\mathbb{F}$, as well as over any field extension $K$ of $\mathbb{F}$. \end{rem} \begin{rem} Comparing the above to what we did with $\chi_{H}$, one could distinguish between (\ref{eq:Definition of left-ended p-adic structural equations}) (which could be called \textbf{linear $p$-adic structure}) and a more $\chi_{H}$-like structure: \begin{equation} \kappa\left(pn+j\right)=a_{j}\kappa\left(n\right)+b_{j} \end{equation} which we would call \textbf{affine $p$-adic structure}. For the present purposes however, we shall only concern ourselves with linear $p$-adic structure, so all mentions of ``$p$-adic structure'' in this and any subsequent part of this dissertation means only the \emph{linear }variety of $p$-adic structure defined in (\ref{eq:Definition of left-ended p-adic structural equations}), unless specifically stated otherwise. \end{rem} As defined, functions with $p$-adic structure clearly generalize the systems of functional equations that characterized $\chi_{H}$ in Chapter 2. Because the functional equations satisfied by these functions appear to be crucial when it comes to working with quasi-integrable functions, it is important that we establish their properties as a general class of functions. As for the terminology, I use ``left-ended'' to describe the equations in (\ref{eq:Definition of left-ended p-adic structural equations}) because, given an integer $m=pn+j$ (where $j=\left[m\right]_{p}$), they show what happens when we pull out the left-most $p$-adic digit of $m$. Naturally, there is a ``right-ended'' counterpart to (\ref{eq:Definition of left-ended p-adic structural equations}), obtained by pulling out the right-most $p$-adic digit of $m$. Moreover, as the next proposition shows, $\kappa$ satisfies one of these two systems of functional equations if and only if it satisfies both systems. \begin{prop}[\textbf{Properties of functions with $p$-adic structure}] \label{prop:properties of functions with p-adic structure} Let $\mathbb{F}$ be any field of characteristic zero, and consider a function $\kappa:\mathbb{N}_{0}\rightarrow\mathbb{F}$. Then, $\kappa$ has $p$-adic structure if and only if there are constants $a_{0}^{\prime},\ldots,a_{p-1}^{\prime}$ so that: \begin{equation} \kappa\left(n+jp^{k}\right)=a_{j}^{\prime}\kappa\left(n\right),\textrm{ }\forall n\geq0,\textrm{ }\forall m\in\left\{ 0,\ldots,p^{n}-1\right\} ,\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \label{eq:Definition of right-ended p-adic structural equations} \end{equation} We call \emph{(\ref{eq:Definition of right-ended p-adic structural equations})} the \textbf{right-ended structural equations }of $\kappa$. \end{prop} Proof: Fix an integer $m\in\mathbb{N}_{0}$. Then, we can write $m$ uniquely in $p$-adic form as: \begin{equation} m=\sum_{\ell=0}^{\lambda_{p}\left(m\right)-1}m_{\ell}p^{\ell} \end{equation} where $m_{\ell}\in\left\{ 0,\ldots,p-1\right\} $ for all $\ell$, with $m_{\lambda_{p}\left(m\right)-1}\neq0$. I. Let $\kappa$ satisfy the left-ended structural equations (\ref{eq:Definition of left-ended p-adic structural equations}). Then: \begin{align} \kappa\left(m\right) & =\kappa\left(p\left(\sum_{\ell=1}^{\lambda_{p}\left(m\right)-1}m_{\ell}p^{\ell-1}\right)+m_{0}\right)\\ \left(\textrm{use }(\ref{eq:Definition of left-ended p-adic structural equations})\right); & =a_{m_{0}}\kappa\left(\sum_{\ell=1}^{\lambda_{p}\left(m\right)-1}m_{\ell}p^{\ell-1}\right)\\ & \vdots\\ & =\left(\prod_{\ell=0}^{\lambda_{p}\left(m\right)-1}a_{m_{\ell}}\right)\kappa\left(0\right) \end{align} Pulling out the right-most $p$-adic digit of $m$ gives: \begin{equation} m=\left(\sum_{\ell=0}^{\lambda_{p}\left(m\right)-2}m_{\ell}p^{\ell-1}\right)+m_{\lambda_{p}\left(m\right)-1}p^{\lambda_{p}\left(m\right)-1} \end{equation} Since the above implies: \begin{equation} \kappa\left(\sum_{\ell=0}^{\lambda_{p}\left(m\right)-2}m_{\ell}p^{\ell-1}\right)=\left(\prod_{\ell=0}^{\lambda_{p}\left(m\right)-2}a_{m_{\ell}}\right)\kappa\left(0\right) \end{equation} it follows that: \begin{align} \left(\prod_{\ell=1}^{\lambda_{p}\left(m\right)-1}a_{m_{\ell}}\right)\kappa\left(0\right) & =\kappa\left(m\right)\\ & =\kappa\left(\sum_{\ell=0}^{\lambda_{p}\left(m\right)-2}m_{\ell}p^{\ell-1}+m_{\lambda_{p}\left(m\right)-1}p^{\lambda_{p}\left(m\right)-1}\right)\\ & =\left(\prod_{\ell=0}^{\lambda_{p}\left(m\right)-1}a_{m_{\ell}}\right)\kappa\left(0\right)\\ & =a_{m_{\lambda_{p}\left(m\right)-1}}\left(\prod_{\ell=0}^{\lambda_{p}\left(m\right)-2}a_{m_{\ell}}\right)\kappa\left(0\right)\\ & =a_{m_{\lambda_{p}\left(m\right)-1}}\kappa\left(\sum_{\ell=0}^{\lambda_{p}\left(m\right)-2}m_{\ell}p^{\ell-1}\right) \end{align} This proves (\ref{eq:Definition of right-ended p-adic structural equations}). \vphantom{} II. Supposing instead that $\kappa$ satisfies (\ref{eq:Definition of right-ended p-adic structural equations}), perform the same argument as for (I), but with $a^{\prime}$s instead of $a$s, and pulling out $a_{m_{0}}$ instead of $a_{m_{\lambda_{p}\left(m\right)-1}}$. This yields (\ref{eq:Definition of left-ended p-adic structural equations}). Q.E.D. \begin{prop} \label{prop:value of a p-adic structure at 0}Let $\kappa:\mathbb{N}_{0}\rightarrow\mathbb{F}$ have linear $p$-adic structure, and be non-identically zero. Then the constants $a_{j}$ and $a_{j}^{\prime}$ from the left- and right-ended structural equations for $\kappa$ are: \begin{align} a_{j} & =\kappa\left(j\right)\\ a_{j}^{\prime} & =\frac{\kappa\left(j\right)}{\kappa\left(0\right)} \end{align} \end{prop} Proof: Use the structural equations to evaluate $\kappa\left(j\right)$ for $j\in\left\{ 0,\ldots,p-1\right\} $. Q.E.D. \begin{prop} \label{prop:formula for functions with p-adic structure}Let $\left\{ c_{n}\right\} _{n\geq0}$ be a sequence in $\left\{ 0,\ldots,p-1\right\} $. If $\kappa$ is non-zero and has linear $p$-adic structure: \begin{equation} \kappa\left(\sum_{m=0}^{N-1}c_{n}p^{m}\right)=\frac{1}{\left(\kappa\left(0\right)\right)^{N}}\prod_{n=0}^{N-1}\kappa\left(c_{n}\right)\label{eq:Kappa action in terms of p-adic digits} \end{equation} \end{prop} Proof: Use $\kappa$'s right-ended structural equations along with the formula for the $a_{j}^{\prime}$s given in \textbf{Proposition \ref{prop:value of a p-adic structure at 0}}. Q.E.D. \begin{lem} \label{lem:structural equations uniquely determine p-adic structured functions}Let $\mathbb{F}$ be any field of characteristic zero. Then, every $p$-adically structured $\kappa:\mathbb{N}_{0}\rightarrow\mathbb{F}$ is uniquely determined by its structural equations. Moreover: \vphantom{} I. $\kappa$ is identically zero if and only if $\kappa\left(0\right)=0$. \vphantom{} II. There exists an $n\in\mathbb{N}_{0}$ so that $\kappa\left(n\right)\neq0$ if and only if $\kappa\left(0\right)=1$. \end{lem} Proof: We first prove that $\kappa$ is identically zero if and only if $\kappa\left(0\right)=0$. To do this, we need only show that $\kappa\left(0\right)=0$ implies $\kappa$ is identically zero. So, let $x$ be an arbitrary integer $\geq1$. Observe that every positive integer $x$ can be written in the form $x_{-}+jp^{n}$, where $n=\lambda_{p}\left(x\right)$, where $x_{-}$ is the integer obtained by deleting $j$\textemdash the right-most digit of $x$'s $p$-adic representation. Moreover, by construction, $x_{-}$ is in $\left\{ 0,\ldots,p^{n}-1\right\} $. So, since $\kappa$ has linear $p$-adic structure, the right-ended structural equations yield: \begin{equation} \kappa\left(x\right)=\kappa\left(x_{-}+jp^{n}\right)=c_{j}\kappa\left(x_{-}\right) \end{equation} Noting that the map $n\in\mathbb{N}_{0}\mapsto n_{-}\in\mathbb{N}_{0}$ eventually iterates every non-negative integer to $0$, we can write: \begin{equation} \kappa\left(x\right)=\left(\prod_{j}c_{j}\right)\kappa\left(0\right)=0 \end{equation} Here, the product is taken over finitely many $j$s. In particular, these $j$s are precisely the non-zero $p$-adic digits of $x$. Since $x$ was an arbitrary integer $\geq1$, and since $\kappa$ was given to vanish on $\left\{ 0,\ldots,p-1\right\} $, this shows that $\kappa$ is identically zero whenever $\kappa\left(0\right)=0$. Conversely, if $\kappa$ is identically zero, then, necessarily, $\kappa\left(0\right)=0$. Next, suppose $\kappa$ is $p$-adically structured and is \emph{not} identically zero. Then, by the left-ended structural equations (\ref{eq:Definition of left-ended p-adic structural equations}), we have $\kappa\left(0\right)=a_{0}\kappa\left(0\right)$, where $\kappa\left(0\right)\neq0$. This forces $a_{0}=1$. Applying \textbf{Proposition \ref{prop:value of a p-adic structure at 0}} gives $\kappa\left(0\right)=a_{0}$. So, $\kappa\left(0\right)=1$. Finally, let $\kappa_{1}$ and $\kappa_{2}$ be two $p$-adically structured functions satisfying the same set of structural equations. Then, $\kappa_{1}-\kappa_{2}$ is then $p$-adically structured and satisfies: \begin{equation} \left(\kappa_{1}-\kappa_{2}\right)\left(0\right)=\kappa_{1}\left(0\right)-\kappa_{2}\left(0\right)=1-1=0 \end{equation} which, as we just saw, then forces $\kappa_{1}-\kappa_{2}$ to be identically zero. This proves that $\kappa$ is uniquely determined by its structural equations. Q.E.D. \vphantom{} For us, the most important case will be when $\kappa$ is both $\left(p,K\right)$-adically tame over $\mathbb{Z}_{p}^{\prime}$ \emph{and }$p$-adically structured. \begin{defn} Let $\mathbb{F}$ be a field of characteristic $0$, and let $K$ be a metrically complete valued field extension of $\mathbb{F}$. We say $\kappa$ is \textbf{$\left(p,K\right)$-adically regular} whenever\index{$p,q$-adic!regular function} it has linear $p$-adic structure and is $\left(p,K\right)$-adically tame over $\mathbb{Z}_{p}^{\prime}$. In the case $K$ is a $q$-adic field, we call this \textbf{$\left(p,q\right)$-adic regularity};\textbf{ }when $K$ is $\mathbb{R}$ or $\mathbb{C}$, we call this \textbf{$\left(p,\infty\right)$-adic regularity} when $K=\mathbb{C}$. \end{defn} \vphantom{} This might seem like a lot of definitions, but we are really only just beginning to explore the possibilities. The next few definitions describe various important ``standard forms'' for $\left(p,q\right)$-adic measures in terms of their Fourier-Stieltjes coefficients. \begin{defn} Let $K$ be an algebraically closed field of characteristic zero and consider a function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow K$. \vphantom{} I. We say $\hat{\mu}$ is \textbf{radially symmetric }whenever\index{radially symmetric}: \begin{equation} \hat{\mu}\left(t\right)=\hat{\mu}\left(\frac{1}{\left\|t\right\|_{p}}\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} \label{eq:Definition of a radially symmetric function} \end{equation} That is, the value of $\hat{\mu}\left(t\right)$ depends only on the $p$-adic absolute value of $t$. Similarly, we say a measure $d\mu$ is \textbf{radially symmetric} whenever it has a radially symmetric Fourier-Stieltjes transform. \vphantom{} II. We say $\hat{\mu}$ is \textbf{magnitudinal }whenever\index{magnitudinal} there is a function $\kappa:\mathbb{N}_{0}\rightarrow K$ satisfying: \begin{equation} \hat{\mu}\left(t\right)=\begin{cases} \kappa\left(0\right) & \textrm{if }t=0\\ \sum_{m=0}^{\left\|t\right\|_{p}-1}\kappa\left(m\right)e^{-2\pi imt} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of a Magnitudinal function} \end{equation} Here, we write $e^{-2\pi imt}$ to denote the $m$th power of the particular $\left\|t\right\|_{p}$th root of unity in $K$ indicated by $e^{-2\pi imt}$. We call an expression of the form of the right-hand side of (\ref{eq:Definition of a Magnitudinal function}) a \textbf{magnitudinal series}. Similarly, we say a measure $d\mu$ is \textbf{magnitudinal} whenever it has a magnitude dependent Fourier-Stieltjes transform. Additionally, for $q$ prime or $q=\infty$, we say that such a $\hat{\mu}$ and $d\mu$ are \textbf{$\left(p,q\right)$-adically regular and magnitudinal} whenever $\kappa$ is $\left(p,q\right)$-adically regular. We say $\hat{\mu}$ and $d\mu$ \textbf{have $p$-adic structure }whenever $\kappa$ does. \vphantom{} III. We say a function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{F}$ is \index{radially-magnitudinal}\textbf{radially-magnitudinal }if it is of the form $\hat{\mu}\left(t\right)=\hat{\nu}\left(t\right)\hat{\eta}\left(t\right)$ for a radially symmetric $\hat{\nu}$ and a magnitude\index{$p,q$-adic!regular magnitudinal measure} dependent $\hat{\eta}$. A \textbf{$\left(p,q\right)$-adically regular, radially-magnitudinal }$\hat{\mu}$ is one for which $\hat{\eta}$ is $\left(p,q\right)$-adically magnitudinal. \index{measure!regular, radially-magnitudinal} \end{defn} \begin{lem}[\textbf{Fourier Resummation \textendash{} Radially Symmetric Measures}] \textbf{\label{lem:Rad-Sym Fourier Resum Lemma}}Let\index{resummation lemmata} $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ be radially symmetric. Then: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}\begin{cases} \hat{\mu}\left(0\right)+\left(p-1\right)\sum_{n=1}^{N}p^{n-1}\hat{\mu}\left(\frac{1}{p^{n}}\right) & \textrm{for }\mathfrak{z}=0\textrm{ and }N\geq0\\ \sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)p^{n} & \textrm{for }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\textrm{ \& }N>v_{p}\left(\mathfrak{z}\right) \end{cases}\label{eq:Radial Fourier Resummation Lemma} \end{equation} The lower line can also be written as: \begin{equation} \sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)p^{n}=\hat{\mu}\left(0\right)-\left\|\mathfrak{z}\right\|_{p}^{-1}\hat{\mu}\left(\frac{\left\|\mathfrak{z}\right\|_{p}}{p}\right)+\left(1-\frac{1}{p}\right)\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n}\label{eq:Radial Fourier Resummation - Lower Line Simplification} \end{equation} \end{lem} Proof: When $\mathfrak{z}=0$: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\mu}\left(t\right)\\ \left(\hat{\mu}\textrm{ is radially symm.}\right); & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\mu}\left(\frac{1}{p^{n}}\right)\\ \left(\sum_{\left\|t\right\|_{p}=p^{n}}1=\varphi\left(p^{n}\right)=\left(p-1\right)p^{n-1}\right); & =\hat{\mu}\left(0\right)+\left(p-1\right)\sum_{n=1}^{N}p^{n-1}\hat{\mu}\left(\frac{1}{p^{n}}\right) \end{align} Next, letting $\mathfrak{z}$ be non-zero: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ \left(\hat{\mu}\textrm{ is radially symm.}\right); & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\mu}\left(\frac{1}{\left\|t\right\|_{p}}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\mu}\left(p^{-n}\right)\sum_{\left\|t\right\|_{p}=p^{n}}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\mu}\left(p^{-n}\right)\left(p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]-p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}0\right]\right)\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\mu}\left(p^{-n}\right)p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]-\sum_{n=1}^{N}\hat{\mu}\left(p^{-n}\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}0\right]\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\mu}\left(p^{-n}\right)p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]-\sum_{n=0}^{N-1}\hat{\mu}\left(p^{-n-1}\right)p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]\\ \left(\hat{\mu}\left(t+1\right)=\hat{\mu}\left(t\right)\right); & =p^{N}\hat{\mu}\left(p^{-N}\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]+\sum_{n=0}^{N-1}p^{n}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)\left[\mathfrak{z}\overset{p^{n}}{\equiv}0\right]\\ \left(\mathfrak{z}\overset{p^{n}}{\equiv}0\Leftrightarrow n\leq v_{p}\left(\mathfrak{z}\right)\right); & =p^{N}\hat{\mu}\left(p^{-N}\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]+\sum_{n=0}^{\min\left\{ v_{p}\left(\mathfrak{z}\right)+1,N\right\} -1}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)p^{n} \end{align} Since $\mathfrak{z}\neq0$, $v_{p}\left(\mathfrak{z}\right)$ is an integer, and, as such, $\min\left\{ v_{p}\left(\mathfrak{z}\right)+1,N\right\} =v_{p}\left(\mathfrak{z}\right)+1$ for all sufficiently large $N$\textemdash specifically, all $N\geq1+v_{p}\left(\mathfrak{z}\right)$. For such $N$, we also have that $\mathfrak{z}\overset{p^{N}}{\cancel{\equiv}}0$, and thus, that: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)p^{n},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime},\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right) \end{equation} As for (\ref{eq:Radial Fourier Resummation - Lower Line Simplification}), we can write: \begin{align} \sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\left(\hat{\mu}\left(p^{-n}\right)-\hat{\mu}\left(p^{-n-1}\right)\right)p^{n} & =\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n}-\frac{1}{p}\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n+1}}\right)p^{n+1}\\ & =\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n}-\frac{1}{p}\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)+1}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n}\\ & =\hat{\mu}\left(0\right)-p^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{p^{-v_{p}\left(\mathfrak{z}\right)}}{p}\right)+\left(1-\frac{1}{p}\right)\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n}\\ & =\hat{\mu}\left(0\right)-\left\|\mathfrak{z}\right\|_{p}^{-1}\hat{\mu}\left(\frac{\left\|\mathfrak{z}\right\|_{p}}{p}\right)+\left(1-\frac{1}{p}\right)\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n} \end{align} Q.E.D. \begin{lem}[\textbf{Fourier Resummation \textendash{} Magnitudinal measures}] Let \index{resummation lemmata}$\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ be magnitudinal and not identically $0$. Then, for all $N\in\mathbb{N}_{1}$ and all $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}p^{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\sum_{n=0}^{N-1}\sum_{j=1}^{p-1}p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}+jp^{n}\right)\label{eq:Magnitude Fourier Resummation Lemma} \end{equation} If, in addition, $\hat{\mu}$ has $p$-adic structure, then: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}p^{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\left(\sum_{j=1}^{p-1}\kappa\left(j\right)\right)\sum_{n=0}^{N-1}p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Magnitudinal Fourier Resummation Lemma - p-adically distributed case} \end{equation} \end{lem} Proof: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\left(\sum_{m=0}^{p^{n}-1}\kappa\left(m\right)e^{-2\pi imt}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{m=0}^{p^{n}-1}\kappa\left(m\right)\sum_{\left\|t\right\|_{p}=p^{n}}e^{2\pi i\left\{ t\left(\mathfrak{z}-m\right)\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{m=0}^{p^{n}-1}\kappa\left(m\right)\left(p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]-p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]\right) \end{align} Next, note that as $m$ varies in $\left\{ 0,\ldots,p^{N}-1\right\} $, the only value of $m$ satisfying $\mathfrak{z}\overset{p^{N}}{\equiv}m$ is $m=\left[\mathfrak{z}\right]_{p^{N}}$. Consequently: \begin{equation} \sum_{m=0}^{p^{n}-1}\kappa\left(m\right)p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]=p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{equation} On the other hand: \begin{align} \sum_{m=0}^{p^{n}-1}\kappa\left(m\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right] & =\sum_{m=p^{n-1}}^{p^{n}-1}\kappa\left(m\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]+\sum_{m=0}^{p^{n-1}-1}\kappa\left(m\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]\\ & =\sum_{m=0}^{p^{n}-p^{n-1}-1}\kappa\left(m+p^{n-1}\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m+p^{n-1}\right]+p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\\ & =\sum_{m=0}^{\left(p-1\right)p^{n-1}-1}\kappa\left(m+p^{n-1}\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]+p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\\ & =\sum_{j=1}^{p-1}\sum_{m=\left(j-1\right)p^{n-1}}^{jp^{n-1}-1}\kappa\left(m+p^{n-1}\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]+p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\\ & =\sum_{j=1}^{p-1}\sum_{m=0}^{p^{n-1}-1}\kappa\left(m+jp^{n-1}\right)p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]+p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\\ & =\sum_{j=1}^{p-1}p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right)+p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\\ & =\sum_{j=0}^{p-1}p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right) \end{align} Thus: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{m=0}^{p^{n}-1}\kappa\left(m\right)\left(p^{n}\left[\mathfrak{z}\overset{p^{n}}{\equiv}m\right]-p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]\right)\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\left(p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\sum_{j=0}^{p-1}p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right)\right)\\ & =\hat{\mu}\left(0\right)+\underbrace{\sum_{n=1}^{N}\left(p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right)}_{\textrm{telescoping}}-\sum_{n=1}^{N}\sum_{j=1}^{p-1}p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right)\\ & =\hat{\mu}\left(0\right)-\underbrace{\kappa\left(0\right)}_{\hat{\mu}\left(0\right)}+p^{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\sum_{n=0}^{N-1}\sum_{j=1}^{p-1}p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}+jp^{n}\right) \end{align} as desired. Finally, when $\kappa$ has $p$-adic structure, $\kappa$'s right-ended structural equations let us write: \[ \sum_{n=0}^{N-1}\sum_{j=1}^{p-1}p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}+jp^{n}\right)=\left(\sum_{j=1}^{p-1}\kappa\left(j\right)\right)\sum_{n=0}^{N-1}p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \] Q.E.D. \begin{lem}[\textbf{Fourier Resummation \textendash{} Radially-Magnitudinal measures}] \textbf{\label{lem:Radially-Mag Fourier Resummation Lemma}} Let $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ be a function which is not identically $0$, and suppose that $\hat{\mu}\left(t\right)=\hat{\nu}\left(t\right)\hat{\eta}\left(t\right)$, where $\hat{\nu},\hat{\eta}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ are radially symmetric and magnitude-dependent, respectively. Then, for all $N\in\mathbb{N}_{1}$ and all $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\overline{\mathbb{Q}}}{=}\hat{\nu}\left(0\right)\kappa\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\left(p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right)\label{eq:Radial-Magnitude Fourier Resummation Lemma}\\ & -\sum_{j=1}^{p-1}\sum_{n=1}^{N}p^{n-1}\hat{\nu}\left(p^{-n}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right)\nonumber \end{align} If, in addition, $\kappa$ has $p$-adic structure, then: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\overline{\mathbb{Q}}}{=}p^{N}\hat{\nu}\left(p^{-N}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\label{eq:Radial-Magnitude Fourier Resummation Lemma - p-adically distributed case}\\ & +\sum_{n=0}^{N-1}\left(\hat{\nu}\left(p^{-n}\right)-\left(\sum_{j=0}^{p-1}\kappa\left(j\right)\right)\hat{\nu}\left(p^{-n-1}\right)\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\nonumber \end{align} \end{lem} \begin{rem} In order to make things more compact, we will frequently adopt the notation: \begin{align} A & \overset{\textrm{def}}{=}\sum_{j=1}^{p-1}\kappa\left(j\right)\label{eq:Definition of Big A}\\ v_{n} & \overset{\textrm{def}}{=}\hat{\nu}\left(\frac{1}{p^{n}}\right)\label{eq:Definition of v_n} \end{align} so that (\ref{eq:Radial-Magnitude Fourier Resummation Lemma - p-adically distributed case}) becomes: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}v_{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)p^{N}+\sum_{n=0}^{N-1}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\label{eq:Radial-Magnitude p-adic distributed Fourier Resummation Identity, simplified} \end{equation} \end{rem} Proof: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\nu}\left(t\right)\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ \left(\hat{\nu}\left(t\right)=\hat{\nu}\left(\frac{1}{\left\|t\right\|_{p}}\right)\right); & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} From (\ref{eq:Magnitude Fourier Resummation Lemma}), we have that: \[ \sum_{\left\|t\right\|_{p}=p^{n}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}=p^{n}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\sum_{j=0}^{p-1}p^{n-1}\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right) \] and hence: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\sum_{\left\|t\right\|_{p}=p^{n}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\\ & -\sum_{j=0}^{p-1}\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right)p^{n-1} \end{align} If, in addition, $\hat{\mu}$ is $p$-adically distributed, then: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\hat{\mu}\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\\ & -\left(\sum_{j=0}^{p-1}\kappa\left(j\right)\right)\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)p^{n-1}\\ \left(\hat{\mu}\left(0\right)=\hat{\nu}\left(0\right)\underbrace{\hat{\eta}\left(0\right)}_{\kappa\left(0\right)}\right); & =\underbrace{\hat{\nu}\left(0\right)\kappa\left(0\right)-\hat{\nu}\left(\frac{1}{p}\right)\kappa\left(0\right)\left(\sum_{j=0}^{p-1}\kappa\left(j\right)\right)}_{n=0\textrm{ term of the series on the bottom line}}+p^{N}\hat{\nu}\left(p^{-N}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\\ & +\sum_{n=1}^{N-1}\left(\hat{\nu}\left(p^{-n}\right)-\left(\sum_{j=0}^{p-1}\kappa\left(j\right)\right)\hat{\nu}\left(p^{-n-1}\right)\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n} \end{align} Q.E.D. \begin{prop}[\textbf{Convolution with $v_{p}$}] \label{prop:v_p of t times mu hat sum}Let $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{C}_{q}}{=}-N\tilde{\mu}_{N}\left(\mathfrak{z}\right)+\sum_{n=0}^{N-1}\tilde{\mu}_{n}\left(\mathfrak{z}\right)\label{eq:Fourier sum of v_p times mu-hat} \end{equation} \end{prop} Proof: \begin{align} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\left(-n\right)\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =-\sum_{n=1}^{N}n\left(\sum_{\left\|t\right\|_{p}\leq p^{n}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}-\sum_{\left\|t\right\|_{p}\leq p^{n-1}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right)\\ & =-\sum_{n=1}^{N}n\left(\tilde{\mu}_{n}\left(\mathfrak{z}\right)-\tilde{\mu}_{n-1}\left(\mathfrak{z}\right)\right)\\ & =\sum_{n=1}^{N}n\tilde{\mu}_{n-1}\left(\mathfrak{z}\right)-\sum_{n=1}^{N}n\tilde{\mu}_{n}\left(\mathfrak{z}\right)\\ & =\sum_{n=1}^{N}\left(\tilde{\mu}_{n-1}\left(\mathfrak{z}\right)+\left(n-1\right)\tilde{\mu}_{n-1}\left(\mathfrak{z}\right)\right)-\sum_{n=1}^{N}n\tilde{\mu}_{n}\left(\mathfrak{z}\right)\\ & =\sum_{n=1}^{N}\tilde{\mu}_{n-1}\left(\mathfrak{z}\right)+\sum_{n=0}^{N-1}\underbrace{n\tilde{\mu}_{n}\left(\mathfrak{z}\right)}_{0\textrm{ when }n=0}-\sum_{n=1}^{N}n\tilde{\mu}_{n}\left(\mathfrak{z}\right)\\ & =\sum_{n=0}^{N-1}\tilde{\mu}_{n}\left(\mathfrak{z}\right)-N\tilde{\mu}_{N}\left(\mathfrak{z}\right) \end{align} Q.E.D. \subsection{\label{subsec:3.3.3 Frames}Frames} \begin{rem} In order to be fully comprehensive, this sub-section takes a significant dive into abstraction. This is meant primarily for readers with a good background in non-archimedean analysis, as well as anyone who can handle a significant number of definitions being thrown their way. For readers more interested in how all of this relates to $\chi_{H}$ and Hydra maps, they can safely skip this section so long as the keep the following concepts in mind: Consider a function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$. A sequence of functions $\left\{ f_{n}\right\} _{n\geq1}$ on $\mathbb{Z}_{p}$ is said to \textbf{converge} \textbf{to $f$ with respect to the standard $\left(p,q\right)$-adic frame }if\index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic}: \vphantom{} I. For all $n$, $f_{n}\left(\mathfrak{z}\right)\in\overline{\mathbb{Q}}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. \vphantom{} II. For all $\mathfrak{z}\in\mathbb{N}_{0}$, $f\left(\mathfrak{z}\right)\in\overline{\mathbb{Q}}$ and $\lim_{n\rightarrow\infty}f_{n}\left(\mathfrak{z}\right)\overset{\mathbb{C}}{=}f\left(\mathfrak{z}\right)$ (meaning the convergence is in the topology of $\mathbb{C}$). \vphantom{} III. For all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, $f\left(\mathfrak{z}\right)\in\mathbb{C}_{q}$ and $\lim_{n\rightarrow\infty}f_{n}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}f\left(\mathfrak{z}\right)$ (meaning the convergence is in the topology of $\mathbb{C}_{q}$). \vphantom{} We write $\mathcal{F}_{p,q}$ to denote the standard $\left(p,q\right)$-adic frame and write $\lim_{n\rightarrow\infty}f_{n}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q}}{=}f\left(\mathfrak{z}\right)$ to denote convergence with respect to the standard frame. Also, if the $f_{n}$s are the partial sums of an infinite series, we say that $\lim_{n\rightarrow\infty}f_{n}\left(\mathfrak{z}\right)$ is an \textbf{$\mathcal{F}_{p,q}$-series }for/of\index{mathcal{F}-@$\mathcal{F}$-!series} $f$. \end{rem} \vphantom{} HERE BEGINS THE COMPREHENSIVE ACCOUNT OF FRAMES \vphantom{} To begin, let us recall the modern notion of what a ``function'' actually \emph{is}. Nowadays a ``function'' is merely a rule which to each element of a designated input set $A$ associates a single element of a designated output set, $B$. The first appearance of this type of understanding is generally attributed to Dirichlet. Prior to his work (and, honestly, for a good time after that), functions were often synonymous with what we now call analytic functions\textemdash those which are expressible as a power series, or some combination of functions of that type. This synonymy is understandable: power series are wonderful. Not only do they give us something concrete to manipulate, they even allow us to compute functions' values to arbitrary accuracy\textemdash{} provided that the series converges, of course. For example, suppose we wish to use the geometric series formula: \begin{equation} \sum_{n=0}^{\infty}z^{n}=\frac{1}{1-z} \end{equation} to compute the value of the function $1/\left(1-z\right)$. As we all know, for any real or complex number $z$ with $\left\|z\right\|<1$, the series will converge in $\mathbb{R}$ or $\mathbb{C}$ to the right hand side. However, \emph{what if we plug in $z=3/2$}? There, the topologies of $\mathbb{R}$ and $\mathbb{C}$ do not help us. While we \emph{could} use analytic continuation\textemdash rejiggering the series until we get something which converges at $3/2$\textemdash that particular method doesn't make much sense in non-archimedean analysis, let alone $\left(p,q\right)$-adic analysis, so that route isn't available to us. Instead, we can observe that our obstacle isn't the geometric series formula itself, but rather the topologies of $\mathbb{R}$ and $\mathbb{C}$. As long as we stubbornly insist on living in those archimedean universes and no others, we cannot sum the series as written for any $\left\|z\right\|\geq1$. However, just like with speculative fiction, there is no reason we must confine ourselves to a single universe. If we go through the magic portal to the world of the $3$-adic topology, we can easily compute the true value of $1/\left(1-z\right)$ at $z=3/2$ using the power series formula. There: \begin{equation} \sum_{n=0}^{\infty}\left(\frac{3}{2}\right)^{n}\overset{\mathbb{Z}_{3}}{=}\frac{1}{1-\frac{3}{2}} \end{equation} is a perfectly sensible statement, both ordinary and rigorously meaningful. Most importantly, even though we have \emph{summed }the series in an otherworldly topology, the answer ends up being an ordinary real number, one which agrees with the value of $1/\left(1-z\right)$ at $z=3/2$ in $\mathbb{R}$ or $\mathbb{C}$. More generally, for any rational number $p/q$ where $p$ and $q$ are co-prime integers with $q\neq0$, we can go about computing $1/\left(1-z\right)$ at $z=p/q$ by summing the series: \begin{equation} \sum_{n=0}^{\infty}\left(\frac{p}{q}\right)^{n} \end{equation} in $\mathbb{Z}_{p}$. That this all works is thanks to the universality of the geometric series universality \index{geometric series universality}(see page \pageref{fact:Geometric series universality}). In this way, we can get more mileage out of our series formulae. Even though the geometric series converges in the topology of $\mathbb{C}$ only for complex numbers $\left\|z\right\|<1$, we can make the formula work at every complex algebraic number $\alpha$ of absolute value $>1$ by summing: \begin{equation} \sum_{n=0}^{\infty}\alpha^{n} \end{equation} in the topology of the $\alpha$-adic completion of the field $\mathbb{Q\left(\alpha\right)}$. This then gives us the correct value of the complex-valued function $1/\left(1-z\right)$ at $z=\alpha$. The only unorthodoxy is the route we took to get there. Nevertheless, even this is fully in line with the modern notion of functions: the output a function assigns to a given input should be defined \emph{independently} of any method (if any) we use to explicitly compute it. Treating $1/\left(1-z\right)$ as a function from $\overline{\mathbb{Q}}$ to $\overline{\mathbb{Q}}$, we can use the geometric series to compute the value of $1/\left(1-z\right)$ at any $\alpha\in\overline{\mathbb{Q}}$ by choosing a topological universe (the space of $\alpha$-adic numbers) where the series formula happens to make sense. This discussion of $1/\left(1-z\right)$ contains in a nutshell all the key concepts that come into play when defining frames. In this dissertation, the $\left(p,q\right)$-adic functions like $\chi_{H}$ we desire to study \emph{will not }possess series representations which converge in a single topology at every point of the function's domain. However, if we allow ourselves to vary from point to point the specific topology we use to sum these series, we can arrange things so that the series is meaningful at every possible input value. That this all \emph{works} is thanks to the observation that the values we get by varying the topology in this way are the correct outputs that the function assigns to its inputs, even if\textemdash as stated above\textemdash the route we took to compute them happened to be unusual. So, when working in a certain topology $\mathcal{T}$, instead of admitting defeat and giving up hope when our series become non-convergent\textemdash or, worse, \emph{divergent}\textemdash in $\mathcal{T}$'s topology, we will instead hop over to a \emph{different} topological universe $\mathcal{T}^{\prime}$ in which the series \emph{is} convergent. It almost goes without saying that world-hopping in this way is fraught with pitfalls. As Hensel's famous blunder (see Subsection \ref{subsec:1.3.5 Hensel's-Infamous-Blunder}) perfectly illustrates, just because there exists \emph{some }topology in which a series converges, it does not mean that the sum of the series has meaning from the perspective of a different topology\textemdash the one exception, of course, being the universality of the geometric series. Though cumbersome, the abstractions presented in this subsection are a necessary security measure. We need them to ensure that the series and functions we work with will be compatible with the itinerary of world-hopping we will use to make sense of them. As alluded to in Subsection \ref{subsec:3.3.1 Heuristics-and-Motivations}, the general idea behind a quasi-integrable function $\chi$ on $\mathbb{Z}_{p}$ is that we can express $\chi$ as the limit of interest (\ref{eq:The Limit of Interest}) for some $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$. Moreover, as demonstrated by $\hat{A}_{3}$ and the discussion of the geometric series above, given a function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{K}$, it will be too restrictive to mandate that the partial sums of our Fourier series: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} converge point-wise on $\mathbb{Z}_{p}$ with respect to a \emph{single }topology. Instead, we will only require that at every $\mathfrak{z}\in\mathbb{Z}_{p}$ for which the function $\chi\left(\mathfrak{z}\right)$ takes a finite value, either the sequence of partial sums in the limit of interest (\ref{eq:The Limit of Interest}) equals its limit for all sufficiently large $N$ (as was the case in \textbf{Proposition \ref{prop:sum of v_p}}), or, that there exists\emph{ }a valued field $K_{\mathfrak{z}}$ with absolute value $\left\|\cdot\right\|_{K_{\mathfrak{z}}}$ for which: \begin{equation} \lim_{N\rightarrow\infty}\left\|\chi\left(\mathfrak{z}\right)-\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{K_{\mathfrak{z}}}=0\label{eq:Quasi-integrability in a nutshell} \end{equation} With this procedure in mind, frames will be formulated as a collection of pairs $\left(\mathfrak{z},K_{\mathfrak{z}}\right)$, where $\mathfrak{z}$ is a point in $\mathbb{Z}_{p}$ and $K_{\mathfrak{z}}$ is a field so that (\ref{eq:The Limit of Interest}) converges at $\mathfrak{z}$ in the topology of $K_{\mathfrak{z}}$. That being said, we still have to have \emph{some} limitations. In applying $K_{\mathfrak{z}}$-absolute values to an equation like (\ref{eq:Quasi-integrability in a nutshell}), it is implicitly required that all of the terms the sum are elements of $K_{\mathfrak{z}}$. To surmount this, note that for any $t$ and $\mathfrak{z}$, $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ is some root of unity. Roots of unity exist abstractly in any algebraically closed field. So, we can fit these complex exponentials inside all of the $K_{\mathfrak{z}}$s by requiring the $K_{\mathfrak{z}}$s to be \textbf{algebraically closed}. There is also the issue of $\chi$ itself. Suppose that there are $\mathfrak{a},\mathfrak{b}\in\mathbb{Z}_{p}$ so that $K_{\mathfrak{a}}=\mathbb{C}_{q}$ and $K_{\mathfrak{b}}=\mathbb{C}_{r}$, for distinct primes $p,q,r$? Unless it just so happens that $\chi\left(\mathfrak{a}\right)$ and $\chi\left(\mathfrak{b}\right)$ both lie in $\overline{\mathbb{Q}}$, we will not be able to deal with all of $\chi$ by forcing $\chi$'s co-domain to be a single field. So, we will need to be flexible. Just as we allow for the topology of convergence to vary from point to point, so too will we allow for the co-domain of our functions to be more than just one field. This then suggests the following set up: \begin{itemize} \item A ``big'' space in which the limits of our (Fourier) series shall live. While for us, this will end up always being in $\mathbb{C}_{q}$, in general, it can be more than a single field\footnote{Originally, I required the ``big'' space to be a single field, but, upon contemplating what would need to be done in order to make frames applicable to the study of $\chi_{H}$ in the case of a polygenic Hydra map, I realized I would need to allow for $\chi_{H}$ to take values in more than one field.}. \item A choice of topologies of convergence $\left(\mathfrak{z},K_{\mathfrak{z}}\right)$ for every point in $\mathbb{Z}_{p}$. These shall be topological universes we use to sum our Fourier series so as to arrive at elements of the big space. \item We should require both the big space and the $K_{\mathfrak{z}}$s, to contain $\overline{\mathbb{Q}}$, and will require the coefficients of our Fourier series to be contained \emph{in }$\overline{\mathbb{Q}}$. This way, we will be able to compute partial sums of Fourier series in all of the $K_{\mathfrak{z}}$s simultaneously by working in $\overline{\mathbb{Q}}$, the space they share in common. \end{itemize} To make this work, we will first start with a choice of the pairs $\left(\mathfrak{z},K_{\mathfrak{z}}\right)$. This will get us a ``$p$-adic frame'', denoted $\mathcal{F}$. We will then define $I\left(\mathcal{F}\right)$\textemdash the \textbf{image }of $\mathcal{F}$\textemdash to denote the big space; this will just be the union of all the $K_{\mathfrak{z}}$s. Lastly, we will have $C\left(\mathcal{F}\right)$, the $\overline{\mathbb{Q}}$-linear space of all the functions which are compatible with this set up; the archetypical elements of these spaces will be partial sums of Fourier series generated by functions $\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$. \begin{defn}[\textbf{Frames}] \label{def:Frames}A \textbf{$p$-adic quasi-integrability frame }\index{frame}(or just ``frame'', for short) $\mathcal{F}$ consists of the following pieces of data: \vphantom{} I. A non-empty set \nomenclature{$U_{\mathcal{F}}$}{domain of $\mathcal{F}$}$U_{\mathcal{F}}\subseteq\mathbb{Z}_{p}$, called the \textbf{domain }of $\mathcal{F}$.\index{frame!domain} \vphantom{} II. For every $\mathfrak{z}\in U_{\mathcal{F}}$, a topological field $K_{\mathfrak{z}}$ \nomenclature{$K_{\mathfrak{z}}$}{ } with $K_{\mathfrak{z}}\supseteq\overline{\mathbb{Q}}$. We allow for $K_{\mathfrak{z}}=\overline{\mathbb{Q}}$. Additionally, for any $\mathfrak{z}\in U_{\mathcal{F}}$, we require the topology of $K_{\mathfrak{z}}$ to be either the discrete topology\emph{ or} the topology induced by an absolute value, denoted $\left\|\cdot\right\|_{K_{\mathfrak{z}}}$\nomenclature{$\left\|\cdot\right\|_{K_{\mathfrak{z}}}$}{ }, so that $\left(K_{\mathfrak{z}},\left\|\cdot\right\|_{\mathfrak{z}}\right)$ is then a metrically complete valued field. \end{defn} \vphantom{} The two most important objects associated with a given frame are its image and the space of compatible functions: \begin{defn}[\textbf{Image and Compatible Functions}] \label{def:Frame terminology}Let $\mathcal{F}$ be a $p$-adic frame. \vphantom{} I. The \textbf{image }of $\mathcal{F}$, denoted $I\left(\mathcal{F}\right)$, is the \emph{set }defined by: \begin{equation} I\left(\mathcal{F}\right)\overset{\textrm{def}}{=}\bigcup_{\mathfrak{z}\in U_{\mathcal{F}}}K_{\mathfrak{z}}\label{eq:The image of a frame} \end{equation} \nomenclature{$I\left(\mathcal{F}\right)$}{The image of $\mathcal{F}$}\index{frame!image} \vphantom{} II. A function $\chi:U_{\mathcal{F}}\rightarrow I\left(\mathcal{F}\right)$ is said to be \textbf{$\mathcal{F}$-compatible} / \textbf{compatible }(\textbf{with $\mathcal{F}$}) whenever \index{mathcal{F}-@$\mathcal{F}$-!compatible}\index{frame!compatible functions} $\chi\left(\mathfrak{z}\right)\in K_{\mathfrak{z}}$ for all $\mathfrak{z}\in U_{\mathcal{F}}$. I write $C\left(\mathcal{F}\right)$ \nomenclature{$C\left(\mathcal{F}\right)$}{set of $\mathcal{F}$-compatible functions} to denote the set of all $\mathcal{F}$-compatible functions. \end{defn} \begin{rem} Note that $C\left(\mathcal{F}\right)$ is a linear space over $\overline{\mathbb{Q}}$ with respect to point-wise addition of functions and scalar multiplication. \end{rem} \vphantom{} Next, we have some terminology which is of use when working with a frame. \begin{defn}[\textbf{Frame Terminology}] Let $\mathcal{F}$ be a $p$-adic frame. \vphantom{} I. I call $\mathbb{Z}_{p}\backslash U_{\mathcal{F}}$ the \textbf{set of singularities }of the frame. I say that $\mathcal{F}$ is \textbf{non-singular }whenever $U_{\mathcal{F}}=\mathbb{Z}_{p}$.\index{frame!non-singular} \vphantom{} II. I write $\textrm{dis}$ to denote the discrete topology. I write $\infty$ to denote the topology of $\mathbb{C}$ (induced by $\left\|\cdot\right\|$), and I write $p$ to denote the $p$-adic topology (that of $\mathbb{C}_{p}$, induced by $\left\|\cdot\right\|_{p}$). I write $\textrm{non}$ to denote an arbitrary non-archimedean topology (induced by some non-archimedean absolute value on $\overline{\mathbb{Q}}$). \vphantom{} III. Given a topology $\tau$ (so, $\tau$ could be $\textrm{dis}$, $\infty$, $\textrm{non}$, or $p$), I write\footnote{So, we can write $U_{\textrm{dis}}\left(\mathcal{F}\right)$, $U_{\infty}\left(\mathcal{F}\right)$, $U_{\textrm{non}}\left(\mathcal{F}\right)$, or $U_{p}\left(\mathcal{F}\right)$.} $U_{\tau}\left(\mathcal{F}\right)$ to denote the set of $\mathfrak{z}\in U_{\mathcal{F}}$ so that $K_{\mathfrak{z}}$ has been equipped with $\tau$. I call the $U_{\tau}\left(\mathcal{F}\right)$\textbf{ $\tau$-convergence domain }or \textbf{domain of $\tau$ convergence }of $\mathcal{F}$. In this way, we can speak of the \textbf{discrete convergence domain}, the \textbf{archimedean convergence domain}, the \textbf{non-archimedean convergence domain}, and the \textbf{$p$-adic convergence domain} of a given frame $\mathcal{F}$. \vphantom{} IV. I say $\mathcal{F}$ is \textbf{simple} \index{frame!simple}if either $U_{\textrm{non}}\left(\mathcal{F}\right)$ is empty or there exists a single metrically complete non-archimedean field extension $K$ of $\overline{\mathbb{Q}}$ so that $K=K_{\mathfrak{z}}$ for all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. (That is to say, for a simple frame, we use at most one non-archimedean topology.) \vphantom{} V. I say $\mathcal{F}$ is \textbf{proper }\index{frame!proper}proper whenever $K_{\mathfrak{z}}$ is not a $p$-adic field for any $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. \end{defn} \vphantom{} UNLESS STATED OTHERWISE, ALL FRAMES ARE ASSUMED TO BE PROPER. \vphantom{} Only one frame will ever be used in this dissertation. It is described below. \begin{defn} The \textbf{standard ($\left(p,q\right)$-adic) frame}\index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic}, denoted $\mathcal{F}_{p,q}$, is the frame for which the topology of $\mathbb{C}$ is associated to $\mathbb{N}_{0}$ and the topology of $\mathbb{C}_{q}$ is associated to $\mathbb{Z}_{p}^{\prime}$. \end{defn} \vphantom{} With frames, we can avoid circumlocution when discussing topologies of convergence for $\left(p,q\right)$-adic functions and sequences thereof. \begin{defn}[\textbf{$\mathcal{F}$-convergence}] Given a frame a $\mathcal{F}$ and a function $\chi\in C\left(\mathcal{F}\right)$, we say a sequence $\left\{ \chi_{n}\right\} _{n\geq1}$ in $C\left(\mathcal{F}\right)$ \textbf{converges to $\chi$ over $\mathcal{F}$ }(or \textbf{is $\mathcal{F}$-convergent to $\chi$}) whenever, for each $\mathfrak{z}\in U_{\mathcal{F}}$, we have: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\left(\mathfrak{z}\right)\overset{K_{\mathfrak{z}}}{=}\chi\left(\mathfrak{z}\right) \end{equation} Note that if $K_{\mathfrak{z}}$ has the discrete topology, the limit implies that $\chi_{n}\left(\mathfrak{z}\right)=\chi\left(\mathfrak{z}\right)$ for all sufficiently large $n$.\index{mathcal{F}-@$\mathcal{F}$-!convergence}\index{frame!convergence} The convergence is point-wise with respect to $\mathfrak{z}$. We then call $\chi$ the \index{mathcal{F}-@$\mathcal{F}$-!limit}\textbf{$\mathcal{F}$-limit of the $\chi_{n}$s}. \index{frame!limit}More generally, we say $\left\{ \chi_{n}\right\} _{n\geq1}$ is \textbf{$\mathcal{F}$-convergent / converges over $\mathcal{F}$} whenever there is a function $\chi\in C\left(\mathcal{F}\right)$ so that the $\chi_{n}$s are $\mathcal{F}$-convergent to $\chi$. In symbols, we denote $\mathcal{F}$-convergence by: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\left(\mathfrak{z}\right)\overset{\mathcal{F}}{=}\chi\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in U_{\mathcal{F}}\label{eq:Definition-in-symbols of F-convergence} \end{equation} or simply: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\overset{\mathcal{F}}{=}\chi\label{eq:Simplified version of expressing f as the F-limit of f_ns} \end{equation} \end{defn} \begin{rem} With respect to the standard $\left(p,q\right)$-adic frame, $\mathcal{F}_{p,q}$-convergence means $\chi_{n}\left(\mathfrak{z}\right)$ converges to $\chi\left(\mathfrak{z}\right)$ in the topology of $\mathbb{C}_{q}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ and converges to $\chi\left(\mathfrak{z}\right)$ in the topology of $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$. \end{rem} \begin{rem} In an abuse of notation, we will sometimes write $\mathcal{F}$ convergence as: \begin{equation} \lim_{n\rightarrow\infty}\left\|\chi_{n}\left(\mathfrak{z}\right)-\chi\left(\mathfrak{z}\right)\right\|_{K_{\mathfrak{z}}},\textrm{ }\forall\mathfrak{z}\in U_{\mathcal{F}} \end{equation} The abuse here is that the absolute value $\left\|\cdot\right\|_{K_{\mathfrak{z}}}$ will not exist if $\mathfrak{z}\in U_{\textrm{dis}}\left(\mathcal{F}\right)$. As such, for any $\mathfrak{z}\in U_{\textrm{dis}}\left(\mathcal{F}\right)$, we define: \begin{equation} \lim_{n\rightarrow\infty}\left\|\chi_{n}\left(\mathfrak{z}\right)-\chi\left(\mathfrak{z}\right)\right\|_{K_{\mathfrak{z}}}=0 \end{equation} to mean that, for all sufficiently large $n$, the equality $\chi_{n}\left(\mathfrak{z}\right)=\chi_{n+1}\left(\mathfrak{z}\right)$ holds in $K_{\mathfrak{z}}$. \end{rem} \begin{prop} Let $\mathcal{F}$ be a $p$-adic frame, and let $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$. Then, $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$\textemdash the $N$th partial sum of the Fourier series generated by $\hat{\mu}$\textemdash is an element of $C\left(\mathcal{F}\right)$. \end{prop} Proof: Because $\hat{\mu}$ takes values in $\overline{\mathbb{Q}}$, $\tilde{\mu}_{N}$ is a function from $\mathbb{Z}_{p}$ to $\overline{\mathbb{Q}}$. As such, restricting $\tilde{\mu}_{N}$ to $U_{\mathcal{F}}$ then gives us a function $U_{\mathcal{F}}\rightarrow I\left(\mathcal{F}\right)$, because $\overline{\mathbb{Q}}\subseteq I\left(\mathcal{F}\right)$. Q.E.D. \vphantom{} Now that we have the language of frames at our disposal, we can begin to define classes of $\left(p,q\right)$-adic measures which will be of interest to us. \begin{defn} For any prime $q\neq p$, we say $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$ is\index{measure!algebraic} \textbf{algebraic},\textbf{ }if its Fourier-Stieltjes transform $\hat{\mu}$ takes values in $\overline{\mathbb{Q}}$. \end{defn} \begin{defn}[\textbf{$\mathcal{F}$-measures}] Given a $p$-adic frame $\mathcal{F}$, we write \nomenclature{$M\left(\mathcal{F}\right)$}{set of $\mathcal{F}$-measures}$M\left(\mathcal{F}\right)$ to denote the set of all functions $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ so that $\hat{\mu}\in B\left(\hat{\mathbb{Z}}_{p},K_{\mathfrak{z}}\right)$ for all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$; that is: \begin{equation} \left\Vert \hat{\mu}\right\Vert _{p,K_{\mathfrak{z}}}<\infty,\textrm{ }\forall\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)\label{eq:Definition of an F-measure} \end{equation} where, recall $\left\Vert \hat{\mu}\right\Vert _{p,K_{\mathfrak{z}}}$ denotes $\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\mu}\left(t\right)\right\|_{K_{\mathfrak{z}}}$. Next, observe that for every $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, the map: \begin{equation} f\in C\left(\mathbb{Z}_{p},K_{\mathfrak{z}}\right)\mapsto\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)\hat{\mu}\left(-t\right)\in K_{\mathfrak{z}}\label{eq:Definition of the action of an F-measure} \end{equation} then defines an element of $C\left(\mathbb{Z}_{p},K_{\mathfrak{z}}\right)^{\prime}$. As such, we will identify $M\left(\mathcal{F}\right)$ with elements of: \begin{equation} \bigcap_{\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)}C\left(\mathbb{Z}_{p},K_{\mathfrak{z}}\right)^{\prime} \end{equation} and refer to elements of $M\left(\mathcal{F}\right)$ as \textbf{$\mathcal{F}$-measures}\index{mathcal{F}-@$\mathcal{F}$-!measure}, writing them as $d\mu$. Thus, the statement ``$d\mu\in M\left(\mathcal{F}\right)$'' means that $d\mu$ is an algebraic measure on $C\left(\mathbb{Z}_{p},K_{\mathfrak{z}}\right)$ for all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, which necessarily forces $\hat{\mu}$ to be bounded on $\hat{\mathbb{Z}}_{p}$ in $K_{\mathfrak{z}}$-absolute-value for all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. \end{defn} \begin{rem} If $\mathcal{F}$ is simple and there is a prime $q\neq p$ so that $K_{\mathfrak{z}}=\mathbb{C}_{q}$ for all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, observe that $M\left(\mathcal{F}\right)$ is then precisely the set of all algebraic $\left(p,q\right)$-adic measures. This is also the case for the standard $\left(p,q\right)$-adic frame. \end{rem} \begin{rem} $M\left(\mathcal{F}\right)$ is a linear space over $\overline{\mathbb{Q}}$. \end{rem} \vphantom{} Next, we introduce definitions to link frames with the limits of Fourier series, by way of the convolution of a measure with the $p$-adic Dirichlet Kernels. \begin{defn}[\textbf{$\mathcal{F}$-rising measure}] Given a $p$-adic frame $\mathcal{F}$, we say $d\mu\in M\left(\mathcal{F}\right)$ \textbf{rises over $\mathcal{F}$} / is \textbf{$\mathcal{F}$-rising} whenever\index{measure!mathcal{F}-rising@$\mathcal{F}$-rising} \index{frame!rising measure}the sequence $\left\{ \tilde{\mu}_{N}\right\} _{N\geq0}=\left\{ D_{p:N}d\mu\right\} _{N\geq0}$ is $\mathcal{F}$-convergent: \begin{equation} \lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)\textrm{ exists in }K_{\mathfrak{z}},\textrm{ }\forall\mathfrak{z}\in U_{\mathcal{F}}\label{eq:Definition of an F-rising measure} \end{equation} In the case of the standard $\left(p,q\right)$-adic frame, this means: \vphantom{} I. For every $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ converges in $\mathbb{C}_{q}$. \vphantom{} II. For every $\mathfrak{z}\in\mathbb{N}_{0}$, $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ converges in $\mathbb{C}$. \vphantom{} In all cases, the convergence is point-wise. We then write $M_{\textrm{rise}}\left(\mathcal{F}\right)$\nomenclature{$M_{\textrm{rise}}\left(\mathcal{F}\right)$}{the set of $\mathcal{F}$-rising measures} to denote the set of all $\mathcal{F}$-rising measures. Note that $M_{\textrm{rise}}\left(\mathcal{F}\right)$ forms a linear space over $\overline{\mathbb{Q}}$. Given a $p$-adic frame $\mathcal{F}$ and an $\mathcal{F}$-rising measure $d\mu\in M_{\textrm{rise}}\left(\mathcal{F}\right)$, the \textbf{$\mathcal{F}$-derivative }of \index{measure!mathcal{F}-derivative@$\mathcal{F}$-derivative} $d\mu$ (or simply \textbf{derivative }of $d\mu$), denoted $\tilde{\mu}$, is the function on $U_{\mathcal{F}}$ defined by the $\mathcal{F}$-limit of the $\tilde{\mu}_{N}$s: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)\overset{\mathcal{F}}{=}\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)\label{eq:Definition of mu-twiddle / the derivative of mu} \end{equation} That is, for each $\mathfrak{z}\in U_{\mathcal{F}}$, $\tilde{\mu}\left(\mathfrak{z}\right)$ is defined by the limit of the $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$s in $K_{\mathfrak{z}}$. When $\mathcal{F}$ is the standard $\left(p,q\right)$-adic frame, we have that: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)=\begin{cases} \lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)\textrm{ in }\mathbb{C}_{q} & \textrm{if }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\\ \lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)\textrm{ in }\mathbb{C} & \textrm{if }\mathfrak{z}\in\mathbb{N}_{0} \end{cases}\label{eq:F-limit in the standard frame} \end{equation} We then write $\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)$ to denote the $N$th truncation of the derivative of $d\mu$: \begin{equation} \left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\tilde{\mu}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\label{eq:Definition of the Nth truncation of the derivative of a rising measure} \end{equation} More generally, we have: \begin{equation} \left(\tilde{\mu}_{M}\right)_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\tilde{\mu}_{M}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\label{eq:Nth truncation of the Mth partial sum of mu's derivative} \end{equation} is the $N$th truncation of $\tilde{\mu}_{M}$. \end{defn} \begin{defn} We say $d\mu\in\mathcal{M}_{\textrm{rise}}\left(\mathcal{F}_{p,q}\right)$ is \textbf{rising-continuous }if\index{measure!rising-continuous} it\index{rising-continuous!measure} whenever its $\mathcal{F}_{p,q}$-derivative $\tilde{\mu}\left(\mathfrak{z}\right)$ is a rising-continuous function. \end{defn} \vphantom{} Next up: \emph{degenerate }measures. \begin{defn}[\textbf{$\mathcal{F}$-degenerate measure}] \label{def:degenerate measure}Given a $p$-adic frame $\mathcal{F}$, an $\mathcal{F}$-rising measure $d\mu\in M_{\textrm{rise}}\left(\mathcal{F}\right)$ is said to be\textbf{ degenerate }whenever\index{measure!degenerate} its\index{mathcal{F}-@$\mathcal{F}$-!degenerate measure} $\mathcal{F}$-derivative\index{frame!degenerate measure} is identically zero. We drop the modifier $\mathcal{F}$ when there is no confusion. We write $M_{\textrm{dgen}}\left(\mathcal{F}\right)$\nomenclature{$M_{\textrm{dgen}}\left(\mathcal{F}\right)$}{the set of $\mathcal{F}$-degenerate measures} to denote the set of all $\mathcal{F}$-degenerate measures. Note that $M_{\textrm{dgen}}\left(\mathcal{F}\right)$ is a linear subspace of $M_{\textrm{rise}}\left(\mathcal{F}\right)$. \end{defn} \begin{defn} Given a $p$-adic frame $\mathcal{F}$, an $\mathcal{F}$-rising measure $d\mu\in M_{\textrm{rise}}\left(\mathcal{F}\right)$ is\index{measure!mathcal{F}-proper@$\mathcal{F}$-proper} said\index{mathcal{F}-@$\mathcal{F}$-!proper measure} to be \textbf{proper / $\mathcal{F}$-proper }whenever: \begin{equation} \lim_{N\rightarrow\infty}\left(\tilde{\mu}_{N}-\left(\tilde{\mu}\right)_{N}\right)\overset{\mathcal{F}}{=}0\label{eq:Definition of a proper measure} \end{equation} If this property fails to hold, we call $d\mu$ \textbf{improper / $\mathcal{F}$-improper}.\textbf{ }We then write $M_{\textrm{prop}}\left(\mathcal{F}\right)$ to denote the set of all $\mathcal{F}$-proper measures. Note that $M_{\textrm{prop}}\left(\mathcal{F}\right)$ is a linear subspace of $M_{\textrm{rise}}\left(\mathcal{F}\right)$. \end{defn} \begin{question} \label{que:3.2}Is the notion of a non-proper measure a vacuous concept, or does there exist a frame $\mathcal{F}$ and an $\mathcal{F}$-rising measure $d\mu$ which is improper? \end{question} \vphantom{} Before we move on to the next subsection, we record one last result: \begin{prop} Let $d\mu$ be a rising-continuous $\left(p,q\right)$-adic measure. Then, the derivative $\tilde{\mu}$ is a rising-continuous function if and only if: \vphantom{} I. $d\mu$ is proper. \vphantom{} II. $S_{p}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)$ converges in $\mathbb{C}_{q}$ for every $\mathfrak{z}\in\mathbb{Z}_{p}$. \end{prop} Proof: We begin by defining a function $\mu^{\prime}$ by: \begin{equation} \mu^{\prime}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}S_{p}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{n}\left(\tilde{\mu}\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\label{eq:Chi prime} \end{equation} That is to say, $\mu^{\prime}$ is obtained by computing the values $\tilde{\mu}\left(m\right)$ via the limits of $\tilde{\mu}_{N}\left(m\right)$, and then using those values to construct a van der Put series. On the other hand, $\tilde{\mu}\left(\mathfrak{z}\right)$ is constructed by computing the $\mathcal{F}_{p,q}$-limits of $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ for each $\mathfrak{z}\in\mathbb{Z}_{p}$. To complete the proof, we just need to show that $\tilde{\mu}$ and $\mu^{\prime}$ are the same function. In fact, we need only show that they are equal to one another over $\mathbb{Z}_{p}^{\prime}$; that they are equal over $\mathbb{N}_{0}$ is because $\mu^{\prime}$ is the function represented by the van der Put series of $\tilde{\mu}$, and a van der Put series of a function \emph{always} converges on $\mathbb{N}_{0}$ to said function. To do this, we write: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)-\mu^{\prime}\left(\mathfrak{z}\right)=\tilde{\mu}\left(\mathfrak{z}\right)-\tilde{\mu}_{N}\left(\mathfrak{z}\right)+\tilde{\mu}_{N}\left(\mathfrak{z}\right)-\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)+\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)-\mu^{\prime}\left(\mathfrak{z}\right) \end{equation} Since: \begin{equation} S_{p:N}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)=\tilde{\mu}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\overset{\textrm{def}}{=}\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right) \end{equation} we then have: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)-\mu^{\prime}\left(\mathfrak{z}\right)=\underbrace{\tilde{\mu}\left(\mathfrak{z}\right)-\tilde{\mu}_{N}\left(\mathfrak{z}\right)}_{\textrm{A}}+\underbrace{\tilde{\mu}_{N}\left(\mathfrak{z}\right)-\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)}_{\textrm{B}}+\underbrace{S_{p:N}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)-\mu^{\prime}\left(\mathfrak{z}\right)}_{\textrm{C}}\label{eq:A,B,C,} \end{equation} i. Suppose $d\mu$ is proper and that $S_{p}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}S_{p:N}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)$ converges for every $\mathfrak{z}\in\mathbb{Z}_{p}$. Since $d\mu$ is rising-continuous, $\lim_{N\rightarrow\infty}\left\|\tilde{\mu}\left(\mathfrak{z}\right)-\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}=0$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, which proves that (A) tends to zero $q$-adically. Additionally, because $d\mu$ is proper, the fact that $d\mu$ is proper then guarantees that $\lim_{N\rightarrow\infty}\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)-\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)\right\|_{q}=0$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, seeing as $\mathbb{Z}_{p}^{\prime}$ is the domain of non-archimedean convergence for the standard $\left(p,q\right)$-adic frame. This shows that (B) tends to zero $q$-adically. Finally, (C) tends to zero $q$-adically as $N\rightarrow\infty$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ because firstly, by definition, $\mu^{\prime}$ is the limit of $S_{p:N}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)$ as $N\rightarrow\infty$, and, secondly, because $S_{p:N}\left\{ \tilde{\mu}\right\} \left(\mathfrak{z}\right)$ converges to a limit $q$-adically for every $\mathfrak{z}\in\mathbb{Z}_{p}$. This shows that the right hand side of (\ref{eq:A,B,C,}) is identically zero for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. So, $\tilde{\mu}=\mu^{\prime}$ on $\mathbb{Z}_{p}^{\prime}$, as desired. \vphantom{} ii. Suppose $\tilde{\mu}$ is rising-continuous. Then, $\mu^{\prime}$\textemdash the van der Put series of $\tilde{\mu}$\textemdash is necessarily equal to $\tilde{\mu}$, and converges everywhere, thereby forcing (II) to hold true. Next, by the van der Put identity (\textbf{Proposition \ref{prop:vdP identity}}), we can write: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)=\mu^{\prime}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} Moreover, $\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)=\left(\tilde{\mu}\right)_{N+1}\left(\mathfrak{z}\right)$ for all sufficiently large $N$ whenever $\mathfrak{z}\in\mathbb{N}_{0}$, so $\left(\tilde{\mu}\right)_{N}\left(\mathfrak{z}\right)$ converges to $\tilde{\mu}\left(\mathfrak{z}\right)$ in $\mathbb{C}$ for $\mathfrak{z}\in\mathbb{N}_{0}$ and in $\mathbb{C}_{q}$ for $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. This shows that $\left(\tilde{\mu}\right)_{N}$ is $\mathcal{F}_{p,q}$-convergent to $\tilde{\mu}$. Since, $d\mu$ is rising-continuous, by definition, the $\tilde{\mu}_{N}$s are $\mathcal{F}_{p,q}$-convergent to $\tilde{\mu}$. Since $\left(\tilde{\mu}\right)_{N}$ and $\tilde{\mu}_{N}$ then $\mathcal{F}_{p,q}$-converge to the same function, they then necessarily $\mathcal{F}_{p,q}$-converge to one another. This proves $d\mu$ is proper, and hence, proves (I). Q.E.D. \subsection{\label{subsec:3.3.4 Toward-a-Taxonomy}Toward a Taxonomy of $\left(p,q\right)$-adic Measures} Although frames lubricate our discussion of the convergence behavior of Fourier series generated by the Fourier-Stieltjes transforms of $\left(p,q\right)$-adic measures, we still do not have many practical examples of these measures. In the present subsection, I have gathered the few results about $\left(p,q\right)$-adic measures which I \emph{have }been able to prove, in the hopes of being able to give more substantial descriptions of them. Somewhat vexingly, the methods in this subsection depend nigh-entirely on the structural properties of whichever type of measure we happen to be considering. Without assuming such properties hold, it becomes difficult to say anything meaningful about the measures and the Fourier series they generate\textemdash this is one of the main challenges we have to deal with when doing theory with frames. In order to keep this subsection text from drowning in adjectives, we will use the following abbreviations: \begin{defn}[\textbf{Measure Type Codes}] \ \vphantom{} I.\textbf{ }(\textbf{ED1}) We say \index{measure!elementary type D1}$d\mu\in M_{\textrm{alg}}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is an \textbf{elementary D1 measure }(ED1) whenever it can be written as $d\mu=d\nud\eta$ for $d\nu,d\eta\in M_{\textrm{alg}}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, where $d\nu$ is radially symmetric and where $d\eta$ is $\left(p,q\right)$-adically regular magnitudinal measure. That is to say, $\hat{\nu}\left(t\right)$ is a $\overline{\mathbb{Q}}$-valued function which depends only on $\left\|t\right\|_{p}$, and $\hat{\eta}\left(t\right)$ is of the form: \begin{equation} \hat{\eta}\left(t\right)=\begin{cases} \kappa\left(0\right) & \textrm{if }t=0\\ \sum_{m=0}^{\left\|t\right\|_{p}-1}\kappa\left(m\right)e^{-2\pi imt} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} for some function $\kappa:\mathbb{N}_{0}\rightarrow\overline{\mathbb{Q}}$ such that: \begin{equation} \lim_{n\rightarrow\infty}\left\|\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}=0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime} \end{equation} and, moreover, for which there are constants $a_{1},\ldots,a_{p-1}\in\overline{\mathbb{Q}}$ so that: \[ \kappa\left(\left[\mathfrak{z}\right]_{p^{n}}+jp^{n}\right)=a_{j}\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall n\geq1,\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \] Note that this then implies $\kappa\left(0\right)=1$. \vphantom{} II. (\textbf{D1}) We say $d\mu\in M_{\textrm{alg}}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ a \textbf{type D1 measure }whenever\index{measure!type D1} it is a $\overline{\mathbb{Q}}$-linear combination of finitely many ED1 measures. \vphantom{} III. (\textbf{ED2}) We say $d\mu\in M_{\textrm{alg}}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ an \textbf{elementary D2 measure} whenever it is ED1, with\index{measure!elementary type D2}: \begin{equation} v_{n}=\left(1+A\right)v_{n+1},\textrm{ }\forall n\geq0\label{eq:ED2 measure} \end{equation} where, recall: \begin{equation} A=\sum_{j=1}^{p-1}a_{j}=\sum_{j=1}^{p-1}\kappa\left(j\right) \end{equation} is the sum of the constants from the structural equation of $\kappa$ and $v_{n}$ denotes $\hat{\nu}\left(p^{-n}\right)$. \vphantom{} IV. (\textbf{D2}) We say $d\mu\in M_{\textrm{alg}}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ a \textbf{type D2 measure }whenever it is a $\overline{\mathbb{Q}}$-linear combination of finitely many ED2 measures. \index{measure!type D2} \end{defn} \vphantom{} Obviously, there are many other variants we could consider. For example, w measures of the form: \begin{equation} \hat{\mu}\left(t\right)=\begin{cases} \kappa\left(0\right) & \textrm{if }t=0\\ \sum_{n=0}^{-v_{p}\left(t\right)-1}\kappa\left(n\right)e^{-2\pi int} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} for some $\left(p,q\right)$-adically regular $\kappa:\mathbb{N}_{0}\rightarrow\overline{\mathbb{Q}}$; also, spaces generated by convolutions of measures with this type, or any combination of convolutions of radially-symmetric and/or magnitudinal measures, ED1 measures for which: \begin{equation} \lim_{n\rightarrow\infty}\hat{\nu}\left(p^{-N}\right)p^{N}\overset{\mathbb{C}}{=}0 \end{equation} and so on and so forth. \begin{prop} \label{prop:ED1 measures are zero whenever kappa of 0 is 0}An\textbf{ }ED1\textbf{ }measure is the zero measure whenever $\kappa\left(0\right)=0$. \end{prop} Proof: If $\kappa\left(0\right)=0$, then \textbf{Lemma \ref{lem:structural equations uniquely determine p-adic structured functions}} applies, telling us that $\kappa$ is identically zero. This makes $\hat{\eta}$ identically zero, which makes $\hat{\mu}=\hat{\nu}\cdot\hat{\eta}$ identically zero. This proves that $d\mu$ is the zero measure. Q.E.D. \begin{lem} \label{lem:degenerate rad sym measures are zero}The only degenerate radially symmetric measure is the zero measure. \end{lem} Proof: Let $d\mu$ be a degenerate radially-symmetric measure. Using the Radial Fourier Resummation formula (\ref{eq:Radial Fourier Resummation Lemma}), the degeneracy of $d\mu$ allows us to write: \begin{equation} 0\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\hat{\mu}\left(0\right)-\left\|\mathfrak{z}\right\|_{p}^{-1}\hat{\mu}\left(\frac{\left\|\mathfrak{z}\right\|_{p}}{p}\right)+\left(1-\frac{1}{p}\right)\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\backslash\left\{ 0\right\} \label{eq:Degeneracy Lemma - Algebraic Vanishing} \end{equation} Hence, it must be that the algebraic number: \begin{equation} \hat{\mu}\left(0\right)-\left\|\mathfrak{z}\right\|_{p}^{-1}\hat{\mu}\left(\frac{\left\|\mathfrak{z}\right\|_{p}}{p}\right)+\left(1-\frac{1}{p}\right)\sum_{n=1}^{v_{p}\left(\mathfrak{z}\right)}\hat{\mu}\left(\frac{1}{p^{n}}\right)p^{n} \end{equation} is $0$ for all non-zero $p$-adic integers $\mathfrak{z}$. Letting $\left\|\mathfrak{z}\right\|_{p}=p^{k}$ for $k\geq1$, and letting $\hat{\mu}\left(p^{-n}\right)=c_{n}$, the above becomes: \begin{equation} 0\overset{\overline{\mathbb{Q}}}{=}c_{0}-c_{k+1}p^{k}+\left(1-\frac{1}{p}\right)\sum_{n=1}^{k}c_{n}p^{n},\textrm{ }\forall k\geq1 \end{equation} Next, we subtract the $\left(k-1\right)$th case from the $k$th case. \begin{equation} 0\overset{\overline{\mathbb{Q}}}{=}c_{k}p^{k-1}-c_{k+1}p^{k}+\left(1-\frac{1}{p}\right)c_{k}p^{k},\textrm{ }\forall k\geq2 \end{equation} and hence: \begin{equation} c_{k+1}\overset{\overline{\mathbb{Q}}}{=}c_{k},\textrm{ }\forall k\geq2 \end{equation} Meanwhile, when $k=1$, we get: \begin{equation} c_{2}\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p}c_{0}+\left(1-\frac{1}{p}\right)c_{1} \end{equation} As for the $\mathfrak{z}=0$ case, using the \textbf{Radial Fourier Resummation Lemma }(\textbf{Lemma \ref{lem:Rad-Sym Fourier Resum Lemma}}), the degeneracy of $d\mu$ tell us that: \[ 0\overset{\mathbb{C}}{=}\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(0\right)=c_{0}+\left(p-1\right)\lim_{N\rightarrow\infty}\sum_{k=1}^{N}c_{k}p^{k-1}\overset{\mathbb{C}}{=}c_{0}+\left(1-\frac{1}{p}\right)\sum_{k=1}^{\infty}c_{k}p^{k} \] This then forces $c_{n}p^{n}$ to tend to $0$ in $\mathbb{C}$ as $n\rightarrow\infty$. However, we just showed that $c_{k}=c_{k+1}$ for all $k\geq2$. So, the only way convergence occurs is if $c_{k}=0$ for all $k\geq2$. Consequently, $\hat{\mu}\left(t\right)=0$ for all $\left\|t\right\|_{p}\geq p^{2}$. Using the \textbf{Radial Fourier Resummation Lemma} once more, we obtain: \begin{equation} \tilde{\mu}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\begin{cases} \hat{\mu}\left(0\right)-\hat{\mu}\left(\frac{1}{p}\right) & \textrm{if }\left\|\mathfrak{z}\right\|_{p}=1\\ \hat{\mu}\left(0\right)+\left(p-1\right)\hat{\mu}\left(\frac{1}{p}\right) & \textrm{if }\left\|\mathfrak{z}\right\|_{p}\leq\frac{1}{p} \end{cases} \end{equation} Since $d\mu$ is degenerate, $\tilde{\mu}\left(\mathfrak{z}\right)$ is identically zero. This forces $\hat{\mu}\left(0\right)=\hat{\mu}\left(\frac{1}{p}\right)$. So, we can write: \begin{equation} 0=\hat{\mu}\left(0\right)+\left(p-1\right)\hat{\mu}\left(\frac{1}{p}\right)=p\hat{\mu}\left(0\right) \end{equation} which forces $\hat{\mu}\left(0\right)=\hat{\mu}\left(\frac{1}{p}\right)=\hat{\mu}\left(\frac{1}{p^{2}}\right)=\cdots=0$. This proves that $d\mu$ is the zero measure. Q.E.D. \begin{prop} Let $d\mu=d\nud\eta$ be an ED2 measure. Then, $d\mu$ is degenerate if and only if: \begin{equation} \lim_{N\rightarrow\infty}\hat{\nu}\left(p^{-N}\right)p^{N}\overset{\mathbb{C}}{=}0\label{eq:Degeneracy criterion for elementary D4 measures} \end{equation} \end{prop} Proof: Since $d\mu$ is elementary and non-zero, the fact that every ED2 measure is ED1 means that \textbf{Proposition \ref{prop:ED1 measures are zero whenever kappa of 0 is 0} }applies, guaranteeing that $\kappa\left(0\right)\neq0$. Now, supposing $d\mu$ satisfies (\ref{eq:Degeneracy criterion for elementary D4 measures}), the \textbf{Radially-Magnitudinal Fourier Resummation Lemma }(\textbf{Lemma \ref{lem:Radially-Mag Fourier Resummation Lemma}}) becomes: \begin{equation} \tilde{\mu}_{N}\left(\mathfrak{z}\right)=\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}\hat{\nu}\left(p^{-N}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)p^{N} \end{equation} The $\left(p,q\right)$-adic regularity of $\kappa$ and along with the assumptions then guarantee that $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ tends to $0$ in $\mathbb{C}$ (resp. $\mathbb{C}_{q}$) for $\mathfrak{z}\in\mathbb{N}_{0}$ (resp. $\mathbb{Z}_{p}^{\prime}$) as $N\rightarrow\infty$, which shows that $d\mu$ is then degenerate. Conversely, suppose that the non-zero ED2 measure $d\mu$ is degenerate. Using \textbf{Lemma \ref{lem:Radially-Mag Fourier Resummation Lemma}} once again, we write: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\mathbb{\overline{\mathbb{Q}}}}{=}v_{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)p^{N}+\sum_{n=0}^{N-1}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\label{eq:Radial Magnitudinal identity for the Degeneracy Lemma} \end{align} where, recall, $v_{n}=\hat{\nu}\left(p^{-n}\right)$. Since $d\mu$ is ED2, $v_{n}=Av_{n+1}$ for all $n\geq0$, and hence: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{\overline{\mathbb{Q}}}}{=}v_{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)p^{N} \end{equation} Since $d\mu$ is ED2, it is also ED1, and as such, the assumption that $d\mu$ is non-zero guarantees that $\kappa\left(0\right)\neq0$. So, setting $\mathfrak{z}=0$, the degeneracy of $d\mu$ tells us that: \begin{equation} 0\overset{\mathbb{C}}{=}\lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)\overset{\mathbb{\overline{\mathbb{Q}}}}{=}\kappa\left(0\right)\lim_{N\rightarrow\infty}v_{N}p^{N} \end{equation} Since $\kappa\left(0\right)\neq0$, this forces $\lim_{N\rightarrow\infty}v_{N}p^{N}$ to converge to $0$ in $\mathbb{C}$, which is exactly what we wished to show (equation (\ref{eq:Degeneracy criterion for elementary D4 measures})). Q.E.D. \begin{prop} Let $d\mu=d\nud\eta$ be an ED1 measure. Then: \end{prop} \begin{equation} \tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\hat{\nu}\left(0\right)\kappa\left(0\right)+\sum_{n=1}^{N}\hat{\nu}\left(p^{-n}\right)\left(\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\kappa\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right)p^{n}\label{eq:radial-magnitude Fourier resummation w/ SbP} \end{equation} Proof: Apply summation by parts to equation (\ref{eq:Radial-Magnitude p-adic distributed Fourier Resummation Identity, simplified}) from \textbf{Lemma \ref{lem:Radially-Mag Fourier Resummation Lemma}}. Q.E.D. \begin{prop} \label{prop:non-zero degenerate ED1}Let $d\mu=d\nud\eta$ be a non-zero degenerate ED1 measure. Then $\left\|\kappa\left(j\right)\right\|_{q}<1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. In particular, there is a $j\in\left\{ 1,\ldots,p-1\right\} $ so that $\left\|\kappa\left(j\right)\right\|_{q}>0$. \end{prop} Proof: Suppose that the non-zero ED1\textbf{ }measure $d\mu$ is degenerate. Then, by equation (\ref{eq:Radial Magnitudinal identity for the Degeneracy Lemma}): \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\mathbb{\overline{\mathbb{Q}}}}{=}v_{N}\kappa\left(\left[\mathfrak{z}\right]_{p^{N}}\right)p^{N}+\sum_{n=0}^{N-1}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\label{eq:Radial Magnitudinal identity for the Degeneracy Lemma-1} \end{align} where, again, $v_{n}=\hat{\nu}\left(p^{-n}\right)$. Since $p$ and $q$ are distinct primes, and since $d\nu$ is a $\left(p,q\right)$-adic measure, we have that: \begin{equation} \sup_{N\geq1}\left\|v_{N}p^{N}\right\|_{q}<\infty \end{equation} Also, since $\kappa$ is $\left(p,q\right)$-adically regular, letting $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ be arbitrary, taking the limit of (\ref{eq:Radial Magnitudinal identity for the Degeneracy Lemma-1}) in $\mathbb{C}_{q}$ as $N\rightarrow\infty$ gives: \begin{equation} \sum_{n=0}^{\infty}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)p^{n}\overset{\mathbb{C}_{q}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\label{eq:Degeneracy lemma, equation 1-1} \end{equation} which is equal to $0$, by the presumed degeneracy of $d\mu$. Now, letting $j\in\left\{ 1,\ldots,p-1\right\} $, observe that $\left[-j\right]_{p^{n}}$ has exactly $n$ $p$-adic digits, all of which are $p-j$. That is: \begin{equation} \left[-j\right]_{p^{n+1}}=\left[-j\right]_{p^{n}}+\left(p-j\right)p^{n},\textrm{ }\forall n\geq0 \end{equation} Because $\kappa$ is $\left(p,q\right)$-adically regular and non-zero, we can then write: \[ \kappa\left(\left[-j\right]_{p^{n+1}}\right)=\kappa\left(\left[-j\right]_{p^{n}}+\left(p-j\right)p^{n}\right)=\kappa\left(p-j\kappa\right)\left(\left[-j\right]_{p^{n}}\right),\textrm{ }\forall n\geq0 \] Thus, by induction: \begin{equation} \kappa\left(\left[-j\right]_{p^{n}}\right)=\left(\kappa\left(p-j\right)\right)^{n}\kappa\left(0\right),\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} ,\textrm{ }\forall n\geq0 \end{equation} Moreover, because $\left\|\kappa\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}\rightarrow0$ in $\mathbb{R}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, it must be that: \begin{equation} 0\overset{\mathbb{R}}{=}\lim_{n\rightarrow\infty}\left\|\kappa\left(\left[-j\right]_{p^{n}}\right)\right\|_{q}=\lim_{n\rightarrow\infty}\left\|\kappa\left(p-j\right)\right\|_{q}^{n}\left\|\kappa\left(0\right)\right\|_{q} \end{equation} Since $\kappa\left(0\right)\neq0$, this forces $\left\|\kappa\left(p-j\right)\right\|_{q}<1$ for all $j\in\left\{ 1,\ldots p-1\right\} $. So, for $\mathfrak{z}=-j$, equation (\ref{eq:Degeneracy lemma, equation 1-1}) becomes: \begin{equation} \kappa\left(0\right)\sum_{n=0}^{\infty}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\left(\kappa\left(p-j\right)\right)^{n}p^{n}\overset{\mathbb{C}_{q}}{=}0,\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \end{equation} Since $\kappa\left(0\right)\neq0$, it must be that the series is zero. Re-indexing the $j$s then gives us: \begin{equation} \sum_{n=0}^{\infty}\left(v_{n}-\left(1+A\right)v_{n+1}\right)\left(\kappa\left(j\right)\right)^{n}p^{n}\overset{\mathbb{C}_{q}}{=}0,\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \end{equation} \begin{claim} There exists a $j\in\left\{ 1,\ldots,p-1\right\} $ so that $\left\|\kappa\left(j\right)\right\|_{q}\in\left(0,1\right)$. Proof of claim: We already know that all of the $\kappa\left(j\right)$s have $q$-adic absolute value $<1$. So by way of contradiction, suppose $\kappa\left(j\right)=0$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. This tells us that $\kappa\left(n\right)$ is $1$ when $n=0$ and is $0$ for all $n\geq1$. Consequently, $\hat{\eta}\left(t\right)=\kappa\left(0\right)$ for all $t\in\hat{\mathbb{Z}}_{p}$. With this, we see that the degenerate measure $d\mu$ must be radially-symmetric, and so, $\hat{\mu}\left(t\right)=\kappa\left(0\right)\hat{\nu}\left(t\right)=\hat{\nu}\left(t\right)$ for some radially-symmetric $\hat{\nu}$. Using \textbf{Lemma \ref{lem:degenerate rad sym measures are zero}}, $d\mu$ is then the zero measure, which contradicts the fact that $d\mu$ was given to be non-zero. This proves the claim. \end{claim} \vphantom{} This shows that $\left\|\kappa\left(j\right)\right\|_{q}<1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $, and that there is one such $j$ for which $0<\left\|\kappa\left(j\right)\right\|_{q}<1$. Q.E.D. \begin{prop} Let $d\mu=d\nud\eta$ be a non-zero degenerate ED1 measure. Then $A\neq p-1$, where, recall, $A=\sum_{j=1}^{p-1}\kappa\left(j\right)$. \end{prop} Proof: Let $d\mu$ be as given. By way of contradiction, suppose $A=p-1$. Note that since $d\mu$ is non-zero, we have that $\kappa\left(0\right)=1$. Now, letting $\mathfrak{z}=0$, in (\ref{eq:radial-magnitude Fourier resummation w/ SbP}) we get: \begin{equation} \tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}v_{0}\kappa\left(0\right)+\sum_{n=1}^{N}v_{n}\left(\kappa\left(0\right)-\kappa\left(0\right)\right)p^{n}=v_{0}\kappa\left(0\right) \end{equation} Letting $N\rightarrow\infty$, we have: \begin{equation} 0=v_{0}\kappa\left(0\right) \end{equation} Since $\kappa\left(0\right)=1$, this then implies $v_{0}=0$. Next, by \textbf{Proposition \ref{prop:non-zero degenerate ED1}}, we can fix a $j\in\left\{ 1,\ldots,p-1\right\} $ so that $0<\left\|\kappa\left(j\right)\right\|_{q}<1$. Since $\kappa$ is $\left(p,q\right)$-adically regular, we have: \begin{equation} \kappa\left(\left[j+jp+\cdots+jp^{m-1}\right]_{p^{n}}\right)=\kappa\left(j\right)\kappa\left(\left[j+jp+\cdots+jp^{m-2}\right]_{p^{n}}\right) \end{equation} As such: \begin{equation} \kappa\left(\left[j\frac{p^{m}-1}{p-1}\right]_{p^{n}}\right)=\kappa\left(\left[j+jp+\cdots+jp^{m-1}\right]_{p^{n}}\right)=\left(\kappa\left(j\right)\right)^{\min\left\{ n,m\right\} }\label{eq:kappa of j times (p to the m -1 over p -1)} \end{equation} and so, letting $\mathfrak{z}=j\frac{p^{m}-1}{p-1}\in\mathbb{N}_{1}$ for $m\geq1$, (\ref{eq:radial-magnitude Fourier resummation w/ SbP}) becomes: \begin{align} \tilde{\mu}_{N}\left(j\frac{p^{m}-1}{p-1}\right) & \overset{\overline{\mathbb{Q}}}{=}v_{0}+\sum_{n=1}^{N}v_{n}\left(\kappa\left(\left[j\frac{p^{m}-1}{p-1}\right]_{p^{n}}\right)-\kappa\left(\left[j\frac{p^{m}-1}{p-1}\right]_{p^{n-1}}\right)\right)p^{n}\\ & =v_{0}+\sum_{n=1}^{N}v_{n}\left(\kappa\left(j\right)^{\min\left\{ n,m\right\} }-\left(\kappa\left(j\right)\right)^{\min\left\{ n-1,m\right\} }\right)p^{n}\\ & =v_{0}+\sum_{n=1}^{m-1}v_{n}\left(\kappa\left(j\right)\right)^{n}p^{n}-\sum_{n=1}^{m}v_{n}\left(\kappa\left(j\right)\right)^{n-1}p^{n}\\ & +\left(\kappa\left(j\right)\right)^{m}\left(\sum_{n=m}^{N}v_{n}p^{n}-\sum_{n=m+1}^{N}v_{n}p^{n}\right)\\ & =v_{0}+\sum_{n=1}^{m}v_{n}\left(\left(\kappa\left(j\right)\right)^{n}-\left(\kappa\left(j\right)\right)^{n-1}\right)p^{n} \end{align} Letting $N\rightarrow\infty$ gives: \begin{align} v_{0}+\sum_{n=1}^{m}v_{n}\left(\left(\kappa\left(j\right)\right)^{n}-\left(\kappa\left(j\right)\right)^{n-1}\right)p^{n} & \overset{\overline{\mathbb{Q}}}{=}0,\textrm{ }\forall m\geq1\label{eq:m cases} \end{align} Subtracting the $\left(m-1\right)$th case from the $m$th case for $m\geq2$ leaves us with: \begin{equation} v_{m}\left(\left(\kappa\left(j\right)\right)^{m}-\left(\kappa\left(j\right)\right)^{m-1}\right)p^{m}\overset{\overline{\mathbb{Q}}}{=}0,\textrm{ }\forall m\geq2 \end{equation} Multiplying through by the non-zero algebraic number $\left(\kappa\left(j\right)\right)^{1-m}$ yields: \begin{equation} v_{m}\left(\kappa\left(j\right)-1\right)p^{m}\overset{\overline{\mathbb{Q}}}{=}0,\textrm{ }\forall m\geq2 \end{equation} Here, $0<\left\|\kappa\left(j\right)\right\|_{q}<1$ guarantees that $\left(\kappa\left(j\right)-1\right)p^{m}\neq0$, and hence, it must be that $v_{m}=0$ for all $m\geq2$. So, only the $m=1$ case of (\ref{eq:m cases}) is left: \begin{equation} v_{0}+v_{1}\left(\kappa\left(j\right)-1\right)p\overset{\overline{\mathbb{Q}}}{=}0 \end{equation} Since $v_{0}=0$, this gives: \begin{equation} v_{1}\left(\kappa\left(j\right)-1\right)p\overset{\overline{\mathbb{Q}}}{=}0 \end{equation} Since $0<\left\|\kappa\left(j\right)\right\|_{q}<1$, this then shows that $\left(\kappa\left(j\right)-1\right)p\neq0$, and hence that $v_{1}=0$. So, $v_{n}=0$ for all $n\geq0$. This means the radially symmetric measure $d\nu$ is identically zero. Thus, so is $d\mu$\textemdash but $d\mu$ was given to be non-zero. There's our contradiction. Thus, if $d\mu$ is non-zero, it must be that $A\neq p-1$. Q.E.D. \begin{prop} Let $d\nu\in M_{\textrm{dgen}}\left(\mathbb{Z}_{p},K\right)$. Then $e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}d\nu\left(\mathfrak{z}\right)$ is a degenerate measure for all $\tau\in\hat{\mathbb{Z}}_{p}$. \end{prop} Proof: The Fourier-Stieltjes transform of $d\mu\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}d\nu\left(\mathfrak{z}\right)$ is $\hat{\nu}\left(t-\tau\right)$. Then, letting $N$ be large enough so that $\left\|\tau\right\|_{p}\leq p^{N}$, we have that the map $t\mapsto t-\tau$ is then a bijection of the set $\left\{ t\in\hat{\mathbb{Z}}_{p}:\left\|t\right\|_{p}\leq p^{N}\right\} $. For all such $N$: \begin{align} \tilde{\mu}_{N}\left(\mathfrak{z}\right) & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\nu}\left(t-\tau\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\nu}\left(t\right)e^{2\pi i\left\{ \left(t+\tau\right)\mathfrak{z}\right\} _{p}}\\ & =e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}\tilde{\nu}_{N}\left(\mathfrak{z}\right) \end{align} Since $e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}$ is bounded in $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$, the degeneracy of $d\nu$ guarantees that $\tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\mathbb{C}}{\rightarrow}0$ as $N\rightarrow\infty$ for all $\mathfrak{z}\in\mathbb{N}_{0}$. Likewise, since $e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}$ is bounded in $\mathbb{C}_{q}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, the degeneracy of $d\nu$ guarantees that $\tilde{\mu}_{N}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{\rightarrow}0$ as $N\rightarrow\infty$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. Hence, $d\mu=e^{2\pi i\left\{ \tau\mathfrak{z}\right\} _{p}}d\nu\left(\mathfrak{z}\right)$ is degenerate. Q.E.D. \vphantom{} For our purposes, the following proposition will be of great import. \begin{prop} Let $d\mu$ be a degenerate ED2 measure. Then, either $d\mu$ is the zero measure, or $\hat{\mu}\left(0\right)\neq0$. \end{prop} Proof: Let $d\mu$ be as given. If $d\mu$ is the zero measure, then $\hat{\mu}\left(0\right)=0$. Thus, to complete the proof, we need only show that $\hat{\mu}\left(0\right)\neq0$ whenever $d\mu$ is a non-zero degenerate ED2 measure. First, let $d\mu$ be elementary, and suppose $\hat{\mu}\left(0\right)=0$. Since $\hat{\mu}\left(0\right)=\hat{\nu}\left(0\right)\hat{\eta}\left(0\right)=\hat{\nu}\left(1\right)\kappa\left(0\right)$, there are two possibilities: either $\kappa\left(0\right)=0$ or $\hat{\nu}\left(1\right)=0$. If $\kappa\left(0\right)=0$, \textbf{Lemma \ref{lem:structural equations uniquely determine p-adic structured functions}} shows that $d\mu$ is then the zero measure. Alternatively, suppose $\hat{\nu}\left(1\right)=0$. Then, since $d\nu$ is an ED2 measure, we have that: \begin{equation} \hat{\nu}\left(p^{-n}\right)=\left(1+A\right)\hat{\nu}\left(p^{-\left(n+1\right)}\right),\textrm{ }\forall n\geq0 \end{equation} If $A=-1$, then $\hat{\nu}\left(p^{-n}\right)$ is zero for all $n\geq1$. Since $\hat{\nu}\left(1\right)=\hat{\nu}\left(p^{-0}\right)$ was assumed to be $0$, the radial symmetry of $\hat{\nu}$ forces $\hat{\nu}\left(t\right)=0$ for all $t\in\hat{\mathbb{Z}}_{p}$. Since $\hat{\mu}\left(t\right)=\hat{\nu}\left(t\right)\hat{\eta}\left(t\right)$, this shows that $d\mu$ is the zero measure. On the other hand, if $A\neq-1$, then: \begin{equation} \hat{\nu}\left(p^{-\left(n+1\right)}\right)=\frac{1}{A+1}\hat{\nu}\left(p^{-n}\right),\textrm{ }\forall n\geq0 \end{equation} and thus, the assumption that $\hat{\nu}\left(1\right)=0$ then forces $\hat{\nu}\left(p^{-n}\right)=0$ for all $n\geq0$. Like with the $A=-1$ case, this forces $d\mu$ to be the zero measure. Thus, either $\hat{\mu}\left(0\right)\neq0$ or $d\mu$ is the zero measure. Q.E.D. \begin{question} \label{que:3.3}Although we shall not explore it here, there is also the question of how spaces of $\left(p,q\right)$-adic measures behave under pre-compositions by maps $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$. For example, given a rising-continuous $d\mu$, consider the measure $d\nu$ defined by: \begin{equation} \hat{\nu}\left(t\right)\overset{\textrm{def}}{=}\hat{\mu}\left(t\right)e^{-2\pi i\left\{ t\mathfrak{a}\right\} _{p}} \end{equation} where $\mathfrak{a}\in\mathbb{Z}_{p}$ is a constant (i.e., $d\nu\left(\mathfrak{z}\right)=d\mu\left(\mathfrak{z}-\mathfrak{a}\right)$). Will $d\nu$ still be rising continuous? How will the frame of convergence for $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ be affected by the map $d\mu\left(\mathfrak{z}\right)\mapsto d\mu\left(\mathfrak{z}-\mathfrak{a}\right)$? The most general case of this kind of question would be to consider the map $d\mu\left(\mathfrak{z}\right)\mapsto d\mu_{\phi}\left(\mathfrak{z}\right)$ where $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is any sufficiently nice map, as in \textbf{\emph{Definition \ref{def:pullback measure}}}\emph{ on page \pageref{def:pullback measure}}. These all appear to be worth exploring. \end{question} \subsection{\label{subsec:3.3.5 Quasi-Integrability}Quasi-Integrability} \begin{rem} Like in Subsection \ref{subsec:3.3.3 Frames}, this subsection exists primarily to give a firm foundation for my concept of quasi-integrability. With regard to the analysis of $\chi_{H}$ to be performed in Chapter 4, the only thing the reader needs to know is that I say a function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ is \textbf{quasi-integrable }with\index{quasi-integrability} respect to the standard $\left(p,q\right)$-adic frame whenever there exists a function $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ so that: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q}}{=}\lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} where, recall, the $\mathcal{F}_{p,q}$ means that we interpret the right-hand side in the topology of $\mathbb{C}$ when $\mathfrak{z}\in\mathbb{N}_{0}$ and in the topology of $\mathbb{C}_{q}$ whenever $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. We call the function $\hat{\chi}$ \emph{a }\textbf{Fourier transform}\footnote{The reason we cannot say \emph{the }Fourier transform of $\chi$ is because of the existence of degenerate measures. For those who did not read all of Subsection \ref{subsec:3.3.3 Frames}, a $\left(p,q\right)$-adic measure $d\mu$ is said to \index{measure!degenerate}be \textbf{degenerate }(with respect to the standard frame) whenever its Fourier-Stieltjes transform $\hat{\mu}$ takes values in $\overline{\mathbb{Q}}$ and satisfies the limit condition: \begin{equation} \lim_{N\rightarrow\infty}\underbrace{\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}}_{\tilde{\mu}_{N}\left(\mathfrak{z}\right)}\overset{\mathcal{F}_{p,q}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} As a result, given any Fourier transform $\hat{\chi}$ of the quasi-integrable function $\chi$, the function $\hat{\chi}+\hat{\mu}$ will \emph{also }be a Fourier transform of $\chi$ for any degenerate measure $d\mu$. This prevents $\hat{\chi}$ from being unique.}\textbf{ }of $\chi$. Finally, given a Fourier transform $\hat{\chi}$ of a quasi-integrable function $\chi$, we write $\tilde{\chi}_{N}$ to denote the $N$th partial Fourier series generated by $\hat{\chi}$. Interested readers can continue with this subsection to learn more (although, in that case, they should probably read Subsection \ref{subsec:3.3.3 Frames}, assuming they haven't already done so). Otherwise, the reader can safely proceed to the next subsection. \vphantom{} \end{rem} HERE BEGINS THE DISCUSSION OF QUASI-INTEGRABILITY. \vphantom{} Given the issues discussed in Subsection \ref{subsec:3.3.3 Frames}, it would be a bit unreasonable to expect a comprehensive definition of the phenomenon of quasi-integrability given the current novelty of $\left(p,q\right)$-adic analysis. As such, the definition I am about to give should come with an asterisk attached; it is in desperate need of deeper examination, especially regarding interactions with the many types of measures described in \ref{subsec:3.3.3 Frames}. \begin{defn} Consider a $p$-adic frame $\mathcal{F}$. We say a function $\chi\in C\left(\mathcal{F}\right)$ is\textbf{ quasi-integrable} \textbf{with respect to $\mathcal{F}$} / $\mathcal{F}$\textbf{-quasi-integrable }whenever $\chi:U_{\mathcal{F}}\rightarrow I\left(\mathcal{F}\right)$ is the derivative of some $\mathcal{F}$-rising measure $d\mu\in M_{\textrm{rise}}\left(\mathcal{F}\right)$. That is, there is a measure $d\mu$ whose Fourier-Stieltjes transform $\hat{\mu}$ satisfies: \vphantom{} I. $\hat{\mu}\left(t\right)\in\overline{\mathbb{Q}}$, $\forall t\in\hat{\mathbb{Z}}_{p}$; \vphantom{} II. $\left\Vert \hat{\mu}\right\Vert _{p,K_{\mathfrak{z}}}<\infty$ all $\mathfrak{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$; \vphantom{} II. $\tilde{\mu}_{N}$ $\mathcal{F}$-converges to $\chi$: \begin{equation} \lim_{N\rightarrow\infty}\left\|\chi\left(\mathfrak{z}\right)-\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{K_{\mathfrak{z}}}\overset{\mathbb{R}}{=}0,\textrm{ }\forall\mathfrak{z}\in U_{\mathcal{F}}\label{eq:definition of quasi-integrability} \end{equation} We call any $d\mu$ satisfying these properties an\textbf{ $\mathcal{F}$-quasi-integral}\index{mathcal{F}-@$\mathcal{F}$-!quasi-integral}\textbf{ }of $\chi$, dropping the $\mathcal{F}$ when there is no confusion. More generally, we say a function $\chi$ is quasi-integrable\index{quasi-integrability}\textbf{ }if it is quasi-integrable with respect to some frame $\mathcal{F}$. We then write \nomenclature{$\tilde{L}^{1}\left(\mathcal{F}\right)$}{set of $\mathcal{F}$-quasi-integrable functions}$\tilde{L}^{1}\left(\mathcal{F}\right)$ to denote the set of all $\mathcal{F}$-quasi-integrable functions. (Note that this is a vector space over $\overline{\mathbb{Q}}$.) When $\mathcal{F}$ is the standard $\left(p,q\right)$-adic frame, this definition becomes: \vphantom{} i. $\chi$ is a function from $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ so that $\chi\left(\mathfrak{z}\right)\in\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$ and $\chi\left(\mathfrak{z}\right)\in\mathbb{C}_{q}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. \vphantom{} ii. For each $\mathfrak{z}\in\mathbb{N}_{0}$, as $N\rightarrow\infty$, $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ converges to $\chi\left(\mathfrak{z}\right)$ in the topology of $\mathbb{C}$. \vphantom{} iii. For each $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, as $N\rightarrow\infty$, $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ converges to $\chi\left(\mathfrak{z}\right)$ in the topology of $\mathbb{C}_{q}$. \vphantom{} We then write \nomenclature{$\tilde{L}^{1}\left(\mathcal{F}_{p,q}\right)$}{ }$\tilde{L}^{1}\left(\mathcal{F}_{p,q}\right)$ and $\tilde{L}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ \nomenclature{$\tilde{L}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$}{ } to denote the space of all functions $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ which are quasi-integrable with respect to the standard frame $\left(p,q\right)$-adic frame. \end{defn} \vphantom{} Using quasi-integrability, we can define a generalization of the $\left(p,q\right)$-adic Fourier transform. The proposition which guarantees this is as follows: \begin{prop} Let $\chi\in\tilde{L}^{1}\left(\mathcal{F}\right)$. Then, the quasi-integral of $\chi$ is unique modulo $M_{\textrm{dgen}}\left(\mathcal{F}\right)$. That is, if two measures $d\mu$ and $d\nu$ are both quasi-integrals of $\chi$, then the measure $d\mu-d\nu$ is $\mathcal{F}$-degenerate\index{measure!mathcal{F}-degenerate@$\mathcal{F}$-degenerate}\index{mathcal{F}-@$\mathcal{F}$-!degenerate}. Equivalently, we have an isomorphism of $\overline{\mathbb{Q}}$-linear spaces: \begin{equation} \tilde{L}^{1}\left(\mathcal{F}\right)\cong M_{\textrm{rise}}\left(\mathcal{F}\right)/M_{\textrm{dgen}}\left(\mathcal{F}\right)\label{eq:L1 twiddle isomorphism to M alg / M dgen} \end{equation} where the isomorphism associates a given $\chi\in\tilde{L}^{1}\left(\mathcal{F}\right)$ to the equivalence class of $\mathcal{F}$-rising measures which are quasi-integrals of $\chi$. \end{prop} Proof: Let $\mathcal{F}$, $\chi$, $d\mu$, and $d\nu$ be as given. Then, by the definition of what it means to be a quasi-integral of $\chi$, for the measure $d\eta\overset{\textrm{def}}{=}d\mu-d\nu$, we have that $\tilde{\eta}_{N}\left(\mathfrak{z}\right)$ tends to $0$ in $K_{\mathfrak{z}}$ for all $\mathfrak{z}\in U_{\mathcal{F}}$. Hence, $d\mu-d\nu$ is indeed $\mathcal{F}$-degenerate. Q.E.D. \begin{defn} Let $\chi$ be quasi-integrable with respect to $\mathcal{F}$. Then, \textbf{a} \textbf{Fourier Transform} of $\chi$ is a function $\hat{\mu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ which is the Fourier-Stieltjes transform of some $\mathcal{F}$-quasi-integral $d\mu$ of $\chi$. We write these functions as $\hat{\chi}$ and write the associated measures as $\chi\left(\mathfrak{z}\right)d\mathfrak{z}$. \end{defn} \vphantom{} Because there exist degenerate measures which are not the zero measure, our use of the notation $\chi\left(\mathfrak{z}\right)d\mathfrak{z}$ to denote a given quasi-integral $d\mu\left(\mathfrak{z}\right)$ of $\chi$ is \emph{not} well-defined, being dependent on our particular choice of $\hat{\chi}$. So, provided we are working with a fixed choice of $\hat{\chi}$, this then allows us to define the integral $\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\chi\left(\mathfrak{z}\right)d\mathfrak{z}$ for any $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ by interpreting it as the image of $f$ under integration against the measure $\chi\left(\mathfrak{z}\right)d\mathfrak{z}$: \begin{equation} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)\chi\left(\mathfrak{z}\right)d\mathfrak{z}\overset{\mathbb{C}_{q}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\chi}\left(t\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Integral of a quasi-integrable function} \end{equation} More pedantically, we could say that ``the'' Fourier transform of $\chi$ is an equivalence class of functions $\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ so that the difference of any two functions in the equivalence class is the Fourier-Stieltjes transform of an $\mathcal{F}$-degenerate algebraic measure. Additionally, because the Fourier transform of $\chi$ is unique only modulo degenerate measures, there is (at least at present) no \emph{canonical }choice for what $\hat{\chi}$ should be. In this dissertation, the symbol $\hat{\chi}$ will be used in one of two senses: either to designate an \emph{arbitrary }choice of a Fourier transform for a quasi-integrable function $\chi$\textemdash as is the case in Chapters 3 \& 5\textemdash OR, to designate a \emph{specific }choice of a Fourier transform for a quasi-integrable function $\chi$, as is the case in Chapters 4 \& 6. \begin{defn} Given a choice of Fourier transform $\hat{\chi}$ for \nomenclature{$\tilde{\chi}_{N}\left(\mathfrak{z}\right)$}{$N$th partial sum of Fourier series generated by $\hat{\chi}$.} $\chi\in\tilde{L}^{1}\left(\mathcal{F}\right)$, we write: \begin{equation} \tilde{\chi}_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Definition of Chi_N twiddle} \end{equation} to denote the \textbf{$N$th partial sum of the Fourier series generated by $\hat{\chi}$}. \end{defn} \vphantom{} Working with quasi-integrable functions in the abstract is particularly difficult because it is not at all clear how to actually \emph{prove }anything without being able to draw upon identities like the Fourier Resummation Lemmata. Moreover, for \emph{specific }quasi-integrable functions\textemdash those given by a particular formula\textemdash one can often easily prove many results which seem nearly impossible to address in the abstract. Conversely, those results which \emph{can }be proven in the abstract are generally unimpressive; case in point: \begin{prop} \label{prop:Chi_N and Chi_twiddle N F converge to one another}Let $\chi\in\tilde{L}^{1}\left(\mathcal{F}\right)$. Then, for any Fourier transform $\hat{\chi}$ of $\chi$, the difference $\chi_{N}-\tilde{\chi}_{N}$ (where $\chi_{N}$ is the $N$th truncation of $\chi$) $\mathcal{F}$-converges to $0$. \end{prop} Proof: Immediate from the definitions of $\hat{\chi}$, $\tilde{\chi}_{N}$, and $\chi_{N}$. Q.E.D. \vphantom{} Both because it is not my purpose to do so here, and as a simple matter of time constraints, there are quite a few issues in the theory of quasi-integrable functions that I have not yet been able to resolve to my satisfaction. Below, I have outlined what I feel are the most outstanding issues. \begin{question} \label{que:3.4}Fix a $\left(p,q\right)$-adic frame $\mathcal{F}$. Given a function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, is there a practical criterion for determining whether or not $\chi$ is $\mathcal{F}$-quasi-integrable? Similarly, given a function $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$, is there a practical criterion for determining whether or not $\hat{\chi}$ is the Fourier transform of an $\mathcal{F}$-quasi-integrable function (other than directly summing the Fourier series generated by $\hat{\chi}$)? \end{question} \vphantom{} Question \ref{que:3.4} probably bothers me more than any other outstanding issue\footnote{The verification of when the convolution of quasi-integrable functions is, itself, quasi-integrable comes in at a close second.} in this fledgling theory. At present, the only means I have of verifying quasi-integrability is by performing the rather intensive computational regimen of Chapters 4 and 6 on a given $\chi$ in the hopes of being able to find a function $\hat{\chi}$ which serves as the Fourier transform of $\chi$ with respect to some frame. Arguably even worse is the fact that there currently doesn't appear to be any reasonable criterion for showing that a function is \emph{not }quasi-integrable. In particular, despite being nearly certain that the following conjecture is true, I have no way to prove it: \begin{conjecture} Let $\mathfrak{a}\in\mathbb{Z}_{p}$. Then, the one-point function $\mathbf{1}_{\mathfrak{a}}\left(\mathfrak{z}\right)$ is not quasi-integrable with respect to any $\left(p,q\right)$-adic quasi-integrability frame. \end{conjecture} \vphantom{} The methods of Chapters 4 and 6 do not seem applicable here. These methods are best described as a kind of ``asymptotic analysis'' in which we truncate the function $\chi$ being studied and then explicitly compute the Fourier transform $\hat{\chi}_{N}\left(t\right)$ of the locally constant (and hence, continuous) $\left(p,q\right)$-adic function $\chi_{N}\left(\mathfrak{z}\right)$. Careful computational manipulations allow us to ``untangle'' the dependence of $N$ and $t$ in $\hat{\chi}_{N}\left(t\right)$ and to express $\hat{\chi}_{N}\left(t\right)$ in the form: \begin{equation} \hat{\chi}_{N}\left(t\right)=\hat{F}_{N}\left(t\right)+\mathbf{1}_{0}\left(p^{N}t\right)\hat{G}\left(t\right) \end{equation} where $\hat{F}_{N},\hat{G}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ are $q$-adically bounded, and with the Fourier series generated by $\hat{F}_{N}$ converging to zero in the standard frame as $N\rightarrow\infty$. $\hat{G}\left(t\right)$ can be thought of as the ``fine structure'' hidden beneath the tumult of the divergent $\hat{F}_{N}\left(t\right)$ term. This suggests that $\hat{G}$ should be a Fourier transform of $\hat{\chi}$\textemdash and, with some work, this intuition will be proved correct. Even though the Fourier series generated by $\hat{F}_{N}$ only $\mathcal{F}_{p,q}$-converges to $0$ when $\chi$ satisfies certain arithmetical properties\footnote{For example, of the $\chi_{q}$s (the numina of the shortened $qx+1$ maps), the $\hat{F}_{N}$ term's Fourier series is $\mathcal{F}_{2,q}$-convergent to $0$ if and only if $q=3$.}, we can bootstrap this special case and use it to tackle the more general case where the Fourier series generated by $\hat{F}_{N}$ \emph{doesn't} $\mathcal{F}_{p,q}$-converge to $0$. Applying this procedure to, say, $\mathbf{1}_{0}\left(\mathfrak{z}\right)$, the $N$th truncation of $\mathbf{1}_{0}\left(\mathfrak{z}\right)$ is: \begin{equation} \mathbf{1}_{0,N}\left(\mathfrak{z}\right)=\left[\mathfrak{z}\overset{p^{N}}{\equiv}0\right]\label{eq:Nth truncation of 1_0} \end{equation} which has the Fourier transform: \begin{equation} \hat{\mathbf{1}}_{0,N}\left(t\right)=\frac{1}{p^{N}}\mathbf{1}_{0}\left(p^{N}t\right)=\begin{cases} \frac{1}{p^{N}} & \textrm{if }\left\|t\right\|_{p}\leq p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases}\label{eq:Fourier transform of Nth truncation of 1_0} \end{equation} On the one hand, there appears to be no way to replicate Chapter 4's argument to obtain a candidate for a Fourier transform of $\mathbf{1}_{0}$ and thereby establish its quasi-integrability. But how might we make this \emph{rigorous}? At present, I have no idea. Only time will tell if it will stay that way. The next major issue regards the convolution of quasi-integrable functions. \begin{question} Fix a frame $\mathcal{F}$, and let $\chi,\kappa\in\tilde{L}^{1}\left(\mathcal{F}\right)$. The natural definition for the \textbf{convolution }of $\chi$ and $\kappa$ would be: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)\hat{\kappa}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Limit formula for the convolution of two quasi-integrable functions} \end{equation} for a choice $\hat{\chi}$ and $\hat{\kappa}$ of Fourier transforms of $\chi$ and $\kappa$, respectively. However, this immediately leads to difficulties: (1) As defined, $\hat{\chi}$ and $\hat{\kappa}$ are unique only up to $\mathcal{F}$-degenerate measures. However, as is shown in \emph{(\ref{eq:Limit of Fourier sum of v_p A_H hat}) }from \emph{Chapter 4's} \textbf{\emph{Lemma \ref{lem:v_p A_H hat summation formulae}}} \emph{(page \pageref{lem:v_p A_H hat summation formulae})}, there exists a degenerate measure $d\mu$ and a bounded $\hat{\nu}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ so that: \[ \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\nu}\left(t\right)\hat{\mu}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \] is \textbf{\emph{not}}\emph{ }$\mathcal{F}$-convergent to $0$ everywhere. As a result, it appears that \emph{(\ref{eq:Limit formula for the convolution of two quasi-integrable functions})} will not be well defined modulo $\mathcal{F}$-degenerate measures; that is, replacing $\hat{\chi}$ in \emph{(\ref{eq:Limit formula for the convolution of two quasi-integrable functions})} with $\hat{\chi}+\hat{\mu}$ where $d\mu$ is $\mathcal{F}$-degenerate may cause the $\mathcal{F}$-limit of \emph{(\ref{eq:Limit formula for the convolution of two quasi-integrable functions})} to change. This prompts the question: there a way to \uline{uniquely} define the convolution of two quasi-integrable functions? Or will we have to make everything contingent upon the particular choice of Fourier transforms in order to work with quasi-integrable functions? (2) As an extension of the present lack of a criterion for determining whether or not a given $\chi$ is quasi-integrable (or whether a given $\hat{\chi}$ is a Fourier transform of a quasi-integrable function) are there any useful conditions we can impose on $\chi$, $\kappa$, $\hat{\chi}$, and $\hat{\kappa}$ to ensure that \emph{(\ref{eq:Limit formula for the convolution of two quasi-integrable functions})} exists and is quasi-integrable? The one obvious answer is when one or both of $\hat{\chi}$ or $\hat{\kappa}$ is an element of $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, in which case $\chi\kappa$ will then be a continuous function, due to the $q$-adic decay of $\hat{\chi}\cdot\hat{\kappa}$. However, that would be quite an unsatisfying state of affairs! Worse, the computations done in \emph{Chapter 4} involve convolving quasi-integrable functions; for example \emph{(\ref{eq:Limit of Fourier sum of v_p A_H hat} }from \textbf{\emph{Lemma \ref{lem:v_p A_H hat summation formulae}}} on page \emph{\pageref{lem:v_p A_H hat summation formulae})} arises in the analysis of $\chi_{3}$, in which we convolve $dA_{3}$ with the quasi-integrable function obtained by letting $N\rightarrow\infty$ in \emph{(\ref{eq:Fourier sum of v_p of t})}. So, there are clearly cases where convolution \emph{\uline{is}}\emph{ }valid! \end{question} \begin{question} \label{que:3.6}How, if at all, can we work with functions $\chi$ and $\kappa$ which are quasi-integrable with respect to \uline{different frames}? \end{question} \subsection{\label{subsec:3.3.6 L^1 Convergence}$L^{1}$ Convergence} When doing real or complex analysis, few settings are as fundamental as $L^{p}$ spaces, with $L^{1}$ and $L^{2}$ possessing a mutual claim to primacy. As\index{non-archimedean!$L^{1}$ theory} was mentioned in Subsection \ref{subsec:3.1.2 Banach-Spaces-over}, however, neither $L^{1}$ nor $L^{2}$\textemdash nor any $L^{p}$ space, for that matter\textemdash have much significance when working with functions taking values in a non-archimedean valued field. The magic of the complex numbers is that they are a finite-dimensional field extension of $\mathbb{R}$ which is nonetheless algebraically complete. Because $\mathbb{C}_{q}$ is an infinite dimensional extension of $\mathbb{Q}_{q}$, the Galois-theoretic identity $z\overline{z}=\left\|z\right\|^{2}$ from complex analysis does not hold, and the elegant duality of $L^{2}$ spaces gets lost as a result in the $q$-adic context. From an analytical standpoint, the issue with non-archimedean $L^{1}$ spaces is arguably even more fundamental. For infinite series of complex numbers, we need absolute convergence in order to guarantee the invariance of the sum under groupings and rearrangements of its terms. In the ultrametric setting, however, this is no longer the case: all that matters is that the $n$th term of the series tend to $0$ in non-archimedean absolute value. Because of this, as we saw in our discussion of Monna-Springer Integration in Subsection \ref{subsec:3.1.6 Monna-Springer-Integration}, when working with non-archimedean absolute values, there is no longer a meaningful connection between $\left\|\int f\right\|$ and $\int\left\|f\right\|$. Despite this, it appears that a non-archimedean $L^{1}$ theory \emph{is} implicated in $\left(p,q\right)$-adic analysis. Rather than begin with theory or definitions, let us start with a concrete, Collatz-adjacent example involving $\chi_{5}$\index{chi{5}@$\chi_{5}$}, the numen of \index{$5x+1$ map} the shortened $5x+1$ map. \begin{example} For a given $\mathcal{F}$-quasi-integrable $\chi$, \textbf{Proposition \ref{prop:Chi_N and Chi_twiddle N F converge to one another}} tells us that $\chi_{N}-\tilde{\chi}_{N}$ is $\mathcal{F}$-convergent to zero. Nevertheless, for any given choice of $\hat{\chi}$, the limit of the differences $\hat{\chi}_{N}\left(t\right)-\hat{\chi}\left(t\right)$ as $N\rightarrow\infty$ need not be well-behaved. As we will see in Chapter 4, $\chi_{5}\in\tilde{L}^{1}\left(\mathbb{Z}_{2},\mathbb{C}_{5}\right)$ (that is, $\chi_{5}$ is the derivative of a rising-continuous $\left(2,5\right)$-adic measure); moreover, one possible choice for a Fourier transform of $\chi_{5}$ is: \begin{equation} \hat{\chi}_{5}\left(t\right)=-\frac{1}{4}\mathbf{1}_{0}\left(t\right)-\frac{1}{4}\hat{A}_{5}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Choice of Fourier Transform for Chi_5} \end{equation} where $\hat{A}_{5}$ is as defined in (\ref{eq:Definition of A_q hat}), and where $\mathbf{1}_{0}$ is the indicator function of $0$ in $\hat{\mathbb{Z}}_{2}$. Letting $\hat{\chi}_{5,N}$ denote the Fourier transform of the $N$th truncation of $\chi_{5}$ ($\chi_{5,N}$) it can be shown that for this choice of $\hat{\chi}_{5}$: \begin{equation} \hat{\chi}_{5,N}\left(t\right)-\hat{\chi}_{5}\left(t\right)=\begin{cases} \frac{1}{2}\left(\frac{3}{2}\right)^{N+\min\left\{ v_{2}\left(t\right),0\right\} }\hat{A}_{5}\left(t\right) & \textrm{if }\left\|t\right\|_{2}\leq2^{N}\\ \frac{1}{4}\hat{A}_{5}\left(t\right) & \textrm{if }\left\|t\right\|_{2}>2^{N} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Difference between Chi_5,N hat and Chi_5 hat} \end{equation} For any given $t$, as $N\rightarrow\infty$, the limit of this difference exists neither in $\mathbb{C}_{5}$ nor in $\mathbb{C}$, despite the fact that the difference of $\chi_{5,N}$ and $\tilde{\chi}_{5,N}$ (the $N$th partial sum of the Fourier series generated by $\hat{\chi}_{5}$) is $\mathcal{F}$-convergent to $0$. Nevertheless, going through the motions and applying Fourier inversion to the above (multiplying by $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}$ and summing over $\left\|t\right\|_{2}\leq2^{N}$), we obtain: \begin{equation} \chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)=\frac{5^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N+1}}\label{eq:Fourier inversion of Chi_5,N hat - Chi_5 hat} \end{equation} which, as expected, converges to zero $5$-adically as $N\rightarrow\infty$ for all $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$ and converges to zero in $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$. Note that this convergence is \emph{not }uniform over $\mathbb{Z}_{2}^{\prime}$: we can make the rate of convergence with respect to $N$ arbitrarily slow by choosing a $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{N}_{0}$ whose $1$s digits in its $2$-adic expansion are spaced sufficiently far apart. Despite the lack of the \emph{uniform} convergence of $\chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)$ to zero, we can get $L^{1}$ convergence for certain cases, and in the classical sense, no less. We have: \begin{align} \int_{\mathbb{Z}_{2}}\left\|\chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)\right\|_{5}d\mathfrak{z} & \overset{\mathbb{R}}{=}\int_{\mathbb{Z}_{2}}\left(\frac{1}{5}\right)^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}d\mathfrak{z}\\ & =\frac{1}{2^{N}}\sum_{n=0}^{2^{N}-1}\left(\frac{1}{5}\right)^{\#_{1}\left(n\right)}\\ & =\frac{1}{2^{N}}\underbrace{\prod_{n=0}^{N-1}\left(1+\frac{1}{5}\left(1^{2^{n}}\right)\right)}_{\left(6/5\right)^{N}}\\ & =\left(\frac{3}{5}\right)^{N} \end{align} which tends to $0$ in $\mathbb{R}$ as $N\rightarrow\infty$. By using the non-trivial explicit formula for $\chi_{5}$ and $\chi_{5,N}$ from Subsection \ref{subsec:4.2.2}'s\textbf{ Corollary \ref{cor:F-series for Chi_H for p equals 2}} and \textbf{Theorem \ref{thm:F-series for Nth truncation of Chi_H, alpha is 1}},\textbf{ }we obtain: \begin{equation} \chi_{5}\left(\mathfrak{z}\right)-\chi_{5,N}\left(\mathfrak{z}\right)=-\frac{1}{4}\frac{5^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}+\frac{1}{8}\sum_{n=N}^{\infty}\frac{5^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\label{eq:Difference between Chi_5 and Chi_5,N} \end{equation} where the convergence is in $\mathbb{C}$ for $\mathfrak{z}\in\mathbb{N}_{0}$ and is in $\mathbb{C}_{5}$ for $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$. Again, regardless of the topology, the convergence is point-wise. Nevertheless, using the standard $5$-adic absolute value estimate on the right-hand side of (\ref{eq:Difference between Chi_5 and Chi_5,N}), we have: \begin{align} \int_{\mathbb{Z}_{2}}\left\|\chi_{5}\left(\mathfrak{z}\right)-\chi_{5,N}\left(\mathfrak{z}\right)\right\|_{5}d\mathfrak{z} & \overset{\mathbb{R}}{\leq}\int_{\mathbb{Z}_{2}}\max_{n\geq N}\left(5^{-\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}\right)d\mathfrak{z}\\ & =\int_{\mathbb{Z}_{2}}\left(\frac{1}{5}\right)^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}d\mathfrak{z}\\ & =\left(\frac{3}{5}\right)^{N} \end{align} which, once again, tends to $0$ in $\mathbb{R}$ as $N\rightarrow\infty$. Finally, when working with the $\left(2,5\right)$-adic integral of the difference, we obtain: \begin{align} \int_{\mathbb{Z}_{2}}\left(\chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)\right)d\mathfrak{z} & \overset{\mathbb{C}_{5}}{=}\int_{\mathbb{Z}_{2}}\frac{5^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N+1}}d\mathfrak{z}\\ & \overset{\mathbb{C}_{5}}{=}\frac{1}{2^{N}}\sum_{n=0}^{2^{N}-1}\frac{5^{\#_{1}\left(n\right)}}{2^{N+1}}\\ & \overset{\mathbb{C}_{5}}{=}\frac{1/2}{4^{N}}\prod_{n=0}^{N-1}\left(1+5\left(1^{2^{n}}\right)\right)\\ & \overset{\mathbb{C}_{5}}{=}\frac{1}{2}\left(\frac{3}{2}\right)^{N} \end{align} which, just like $\hat{\chi}_{5,N}\left(t\right)-\hat{\chi}_{5}\left(t\right)$, fails to converge $5$-adically as $N\rightarrow\infty$. Indeed, we actually have that both of these quantities fail to converge $5$-adically in precisely the same way: \[ \left(\frac{2}{3}\right)^{N}\int_{\mathbb{Z}_{2}}\left(\chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)\right)d\mathfrak{z}\overset{\mathbb{C}_{5}}{=}\frac{1}{2},\textrm{ }\forall N\geq1 \] \[ \lim_{N\rightarrow\infty}\left(\frac{2}{3}\right)^{N}\left(\hat{\chi}_{5,N}\left(t\right)-\hat{\chi}_{5}\left(t\right)\right)\overset{\mathbb{C}_{5}}{=}\frac{1}{2}\left(\frac{3}{2}\right)^{N+\min\left\{ v_{2}\left(t\right),0\right\} }\hat{A}_{5}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \] and hence: \[ \lim_{N\rightarrow\infty}\frac{\hat{\chi}_{5,N}\left(t\right)-\hat{\chi}_{5}\left(t\right)}{\int_{\mathbb{Z}_{2}}\left(\chi_{5,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{5,N}\left(\mathfrak{z}\right)\right)d\mathfrak{z}}\overset{\mathbb{C}_{5}}{=}\left(\frac{3}{2}\right)^{\min\left\{ v_{2}\left(t\right),0\right\} }\hat{A}_{5}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \] \end{example} \vphantom{} At present, the stringent restrictions demanded by $\mathcal{F}$-convergence seems to prevent us from equipping spaces such as $M\left(\mathcal{F}\right)$ or $\tilde{L}^{1}\left(\mathcal{F}\right)$ with a norm to induce a topology. Non-archimedean $L^{1}$ theory\index{non-archimedean!$L^{1}$ theory} might very well provided a viable work-around. \begin{defn} We write$\mathcal{L}_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ to denote the set of all functions $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ so that the real-valued function $\mathfrak{z}\in\mathbb{Z}_{p}\mapsto\left\|f\left(\mathfrak{z}\right)\right\|_{q}\in\mathbb{R}$ is integrable with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$. As is customary, we then define an equivalence relation $\sim$ on $\mathcal{L}_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, and say that $f\sim g$ whenever $\left\|f\left(\mathfrak{z}\right)-g\left(\mathfrak{z}\right)\right\|_{q}=0$ for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$. We then write $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$\nomenclature{$L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$}{set of $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ so that $\left\|f\right\|_{q}\in L^{1}\left(\mathbb{Z}_{p},\mathbb{C}\right)$} to denote the space of equivalence classes: \begin{equation} L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\overset{\textrm{def}}{=}\mathcal{L}_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)/\sim\label{eq:Definition of L^1_R} \end{equation} and we make this an\emph{ archimedean}\footnote{That is, $\left\Vert \mathfrak{a}f+\mathfrak{b}g\right\Vert _{1}\leq\left\|\mathfrak{a}\right\|_{q}\left\Vert f\right\Vert _{1}+\left\|\mathfrak{b}\right\|_{q}\left\Vert g\right\Vert _{1}$ for all $f,g\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and all $\mathfrak{a},\mathfrak{b}\in\mathbb{C}_{q}$.}\emph{ }normed linear space over $\mathbb{C}_{q}$ by defining: \begin{equation} \left\Vert f\right\Vert _{1}\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\in\mathbb{R},\textrm{ }\forall f\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\label{eq:Definition of L^1 norm} \end{equation} \end{defn} \begin{lem} \label{lem:Egorov lemma}Let $\left\{ f_{n}\right\} _{n\geq1}$ be a sequence of functions $f_{n}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ so that: \vphantom{} i. The real-valued functions $\mathfrak{z}\mapsto\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}$ are measurable with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$; \vphantom{} ii. There is a constant $C\in\mathbb{R}>0$ so that for each $n$, $\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}<C$ holds almost everywhere. \vphantom{} Then: \vphantom{} I. If $\lim_{n\rightarrow\infty}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}\overset{\mathbb{R}}{=}0$ holds for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$, then $\lim_{n\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0$. \vphantom{} II. If $\lim_{n\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0$ then there is a subsequence $\left\{ f_{n_{j}}\right\} _{j\geq1}$ so that $\lim_{j\rightarrow0}\left\|f_{n_{j}}\left(\mathfrak{z}\right)\right\|_{q}\overset{\mathbb{R}}{=}0$ holds for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$. \end{lem} Proof: I. Suppose the $\left\|f_{n}\right\|_{q}$s converge point-wise to $0$ almost everywhere. Then, letting $\epsilon>0$, by \textbf{Egorov's Theorem}\footnote{Let $\left(X,\Omega,\mu\right)$ be a measure space with $\mu\left(X\right)<\infty$, and let $f$ and $\left\{ f_{N}\right\} _{N\geq0}$ be measurable, complex-valued functions on $X$ so that $f_{N}$ converges to $f$ almost everywhere. Then, for every $\epsilon>0$, there is a measurable set $E\subseteq X$ with $\mu\left(E\right)<\epsilon$ such that $f_{N}\rightarrow f$ uniformly on $X\backslash E$. \cite{Folland - real analysis}}, there is a measurable set $E\subseteq\mathbb{Z}_{p}$ of real-valued Haar measure $<\epsilon$ so that the $\left\|f_{n}\right\|_{q}$s converge uniformly to $0$ on $\mathbb{Z}_{p}\backslash E$. \begin{align} \int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & =\int_{E}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}+\int_{\mathbb{Z}_{p}\backslash E}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\\ \left(\left\|f_{n}\right\|_{q}<C\textrm{ a.e.}\right); & \leq C\epsilon+\int_{\mathbb{Z}_{p}\backslash E}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} \end{align} Since the $\left\|f_{n}\right\|_{q}$s converge uniformly to $0$ on $\mathbb{Z}_{p}\backslash E$, $\int_{\mathbb{Z}_{p}\backslash E}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}$ tends to $0$ in $\mathbb{R}$ as $n\rightarrow\infty$, and we are left with: \begin{equation} \limsup_{n\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\ll\epsilon \end{equation} Since $\epsilon$ was arbitrary, we conclude that: \begin{equation} \lim_{n\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}=0 \end{equation} \vphantom{} II. This is a standard result of measure theory. See Chapter 2.4 in (\cite{Folland - real analysis}) for details. Q.E.D. \begin{thm} \label{thm:L^1_R is an archimedean Banach space}$L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is an archimedean Banach space over $\mathbb{C}_{q}$. \end{thm} Proof: To show that the archimedean normed vector space $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is a Banach space, we use \textbf{Theorem 5.1 }from \cite{Folland - real analysis} (the archimedean analogue of \textbf{Proposition \ref{prop:series characterization of a Banach space}}): it suffices to show that every absolutely convergent series in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ converges to a limit in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. To do this, let $\left\{ f_{n}\right\} _{n\geq0}$ be a sequence in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ so that: \begin{equation} \sum_{n=0}^{\infty}\left\Vert f_{n}\right\Vert _{1}<\infty \end{equation} That is: \begin{equation} \lim_{N\rightarrow\infty}\sum_{n=0}^{N-1}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} \end{equation} exists in $\mathbb{R}$. Consequently: \begin{equation} \lim_{N\rightarrow\infty}\sum_{n=0}^{N-1}\int_{\mathbb{Z}_{p}}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}\int_{\mathbb{Z}_{p}}\lim_{N\rightarrow\infty}\sum_{n=0}^{N-1}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} \end{equation} and hence, the function defined by $\lim_{N\rightarrow\infty}\sum_{n=0}^{N-1}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}$ is in $L^{1}\left(\mathbb{Z}_{p},\mathbb{R}\right)$, and the point-wise limit exists $\sum_{n=0}^{\infty}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}$ for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$. By the ordinary $q$-adic triangle inequality (not the ultrametric inequality, just the ordinary triangle inequality), this then shows that $\sum_{n=0}^{\infty}f_{n}\left(\mathfrak{z}\right)$ converges in $\mathbb{C}_{q}$ for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$. As such, let $F:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be defined by: \begin{equation} F\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\begin{cases} \sum_{n=0}^{\infty}f_{n}\left(\mathfrak{z}\right) & \textrm{if \ensuremath{\sum_{n=0}^{\infty}f_{n}\left(\mathfrak{z}\right)} converges in }\mathbb{C}_{q}\\ 0 & \textrm{if \ensuremath{\sum_{n=0}^{\infty}f_{n}\left(\mathfrak{z}\right)} does not converge in }\mathbb{C}_{q} \end{cases},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} Then: \begin{equation} \left\|F\left(\mathfrak{z}\right)-\sum_{n=0}^{N-1}f_{n}\left(\mathfrak{z}\right)\right\|_{q}=\left\|\sum_{n=N}^{\infty}f_{n}\left(\mathfrak{z}\right)\right\|_{q} \end{equation} holds for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$, and so: \begin{align} \lim_{N\rightarrow\infty}\left\Vert F-\sum_{n=0}^{N-1}f_{n}\right\Vert _{1} & \overset{\mathbb{R}}{=}\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|F\left(\mathfrak{z}\right)-\sum_{n=0}^{N-1}f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\\ & =\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\sum_{n=N}^{\infty}f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\\ & \leq\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\sum_{n=N}^{\infty}\left\|f_{n}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\\ & =\lim_{N\rightarrow\infty}\sum_{n=N}^{\infty}\left\Vert f_{n}\right\Vert _{1}\\ \left(\sum_{n=0}^{\infty}\left\Vert f_{n}\right\Vert _{1}<\infty\right); & =0 \end{align} which shows that $F$ is the limit of the series $\sum_{n=0}^{N-1}f_{n}\left(\mathfrak{z}\right)$ in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Thus, every absolutely convergent series in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ converges, which proves that $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is a Banach space. Q.E.D. \begin{prop} Let $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be a function so that: \begin{equation} \chi\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{equation} holds for almost every $\mathfrak{z}\in\mathbb{Z}_{p}$. Then, $\chi\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ whenever: \begin{equation} \sum_{N=1}^{\infty}\frac{1}{p^{N}}\sum_{n=p^{N-1}}^{p^{N}-1}\left\|c_{n}\left(\chi\right)\right\|_{q}<\infty\label{eq:van der Put criterion for real L^1 integrability} \end{equation} \end{prop} Proof: Using the truncated van der Put identity (\ref{eq:truncated van der Put identity}), we have: \begin{align} \chi_{N}\left(\mathfrak{z}\right)-\chi_{N-1}\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & =\sum_{n=p^{N-1}}^{p^{N}-1}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\\ \left(\lambda_{p}\left(n\right)=N\Leftrightarrow p^{N-1}\leq n\leq p^{N}-1\right); & =\sum_{n=p^{N-1}}^{p^{N}-1}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{align} Now, fixing $N$, note that for any $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{equation} \sum_{n=p^{N-1}}^{p^{N}-1}c_{n}\left(\chi\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\overset{\mathbb{R}}{=}\begin{cases} c_{\left[\mathfrak{z}\right]_{p^{N}}}\left(\chi\right) & \textrm{if }p^{N-1}\leq\left[\mathfrak{z}\right]_{p^{N}}\leq p^{N}-1\\ 0 & \textrm{else} \end{cases} \end{equation} Hence: \begin{align} \left\|\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\chi\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} & \overset{\mathbb{R}}{=}\left\|c_{\left[\mathfrak{z}\right]_{p^{N}}}\left(\chi\right)\right\|_{q}\left[p^{N-1}\leq\left[\mathfrak{z}\right]_{p^{N}}\leq p^{N}-1\right]\\ & =\left\|c_{\left[\mathfrak{z}\right]_{p^{N}}}\left(\chi\right)\right\|_{q}\sum_{n=p^{N-1}}^{p^{N}-1}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\\ & =\sum_{n=p^{N-1}}^{p^{N}-1}\left\|c_{n}\left(\chi\right)\right\|_{q}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{align} Since the integral of $\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]$ is $p^{-N}$, this yields: \begin{equation} \left\Vert \chi_{N}-\chi_{N-1}\right\Vert _{1}\overset{\mathbb{R}}{=}\frac{1}{p^{N}}\sum_{n=p^{N-1}}^{p^{N}-1}\left\|c_{n}\left(\chi\right)\right\|_{q}\label{eq:Real Integral of difference of Nth truncations of Chi} \end{equation} If the right-hand side is summable in $\mathbb{R}$ with respect to $N$, then: \begin{equation} \sum_{N=1}^{\infty}\left\Vert \chi_{N}-\chi_{N-1}\right\Vert _{1}<\infty \end{equation} and hence, the series: \[ \sum_{N=1}^{\infty}\left(\chi_{N}\left(\mathfrak{z}\right)-\chi_{N-1}\left(\mathfrak{z}\right)\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\chi_{N}\left(\mathfrak{z}\right)-\chi\left(0\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\chi\left(0\right) \] converges in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. As such, since $\chi\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ almost everywhere, it then follows that $\chi\left(\mathfrak{z}\right)-\chi\left(0\right)$ is an element of $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, which shows that $\chi\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, since the constant function $\chi\left(0\right)$ is in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Q.E.D. \begin{prop} Let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Then: \begin{equation} \lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0 \end{equation} is a necessary\emph{ }condition for all of the following: \vphantom{} I. For $d\mu$ to be the zero measure; \vphantom{} II. For $d\mu$ to be a degenerate rising-continuous measure; \vphantom{} III. For $d\mu$ to be $\mathcal{F}$-rising, where the non-archimedean domain of $\mathcal{F}$ is a set of full measure in $\mathbb{Z}_{p}$ (that is, its complement has measure zero in $\mathbb{Z}_{p}$). \end{prop} Proof: If $d\mu$ is the zero measure, then, by \textbf{Proposition \ref{prop:Criterion for zero measure in terms of partial sums of Fourier series}}, $\left\Vert \tilde{\mu}_{N}\right\Vert _{p,q}\rightarrow0$, and hence, the $\tilde{\mu}_{N}$s converge $q$-adically to $0$ as $N\rightarrow\infty$ uniformly over $\mathbb{Z}_{p}$. Consequently, they are uniformly bounded in $q$-adic absolute value. Moreover, since the $\tilde{\mu}_{N}$s are $\left(p,q\right)$-adically continuous, the functions $\mathfrak{z}\mapsto\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}$ are continuous, and hence measurable. Thus, by \textbf{Lemma \ref{lem:Egorov lemma}}, $\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0$. So, this limit is necessary for $d\mu$ to be the zero measure. (II) is a more specific case of (III). (III) holds true because if the non-archimedean domain of $\mathcal{F}$ has full measure in $\mathbb{Z}_{p}$, then the continuous (and hence measurable) real-valued functions $\mathfrak{z}\mapsto\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}$ converge to $0$ almost everywhere on $\mathbb{Z}_{p}$. Moreover, because $d\mu$ is a measure, we have that: \[ \left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}\leq\max_{\left\|t\right\|_{p}\leq p^{N}}\left\|\hat{\mu}\left(t\right)\right\|_{q}\leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\mu}\left(t\right)\right\|_{q}<\infty,\textrm{ }\forall N\geq0 \] As such, we can apply\textbf{ Lemma \ref{lem:Egorov lemma}} to obtain $\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0$. Q.E.D. \vphantom{} I find it interesting that everything I would like to have true for quasi-integrable rising-continuous functions is easily verified for \emph{continuous }$\left(p,q\right)$-adic functions. To show this, we need the following formula:\index{van der Put!coefficients!Fourier coeffs.} \begin{prop} \label{prop:3.78}\ \begin{equation} \hat{f}_{N}\left(t\right)=\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{p^{N}-1}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier transform of Nth truncation in terms of vdP coefficients} \end{equation} \end{prop} Proof: Identical to the computation in \textbf{Theorem \ref{thm:formula for Fourier series}}, but with the $N$th partial van der Put series of $f$ ($S_{p:N}\left\{ f\right\} $) used in place of the full van der Put series of $f$ ($S_{p}\left\{ f\right\} $). Q.E.D. \vphantom{}Before proceeding, we note that the above identity also holds for rising-continuous functions. \begin{prop} Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, the Fourier transform of the $N$th truncation of $\chi$ is given by: \end{prop} \begin{equation} \hat{\chi}_{N}\left(t\right)=\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{p^{N}-1}\frac{c_{n}\left(\chi\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier transform of Nth truncation in terms of vdP coefficients for a rising continuous function} \end{equation} where the sum is defined to be $0$ whenever $\frac{\left\|t\right\|_{p}}{p}>p^{N}-1$ (i.e., whenever $\left\|t\right\|_{p}\geq p^{N+1}$). Note that this equality holds in $\overline{\mathbb{Q}}$ whenever $\chi\mid_{\mathbb{N}_{0}}$ takes values in $\overline{\mathbb{Q}}$. Proof: Since $\chi_{N}\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ for all $N$, this immediately follows by \textbf{Proposition \ref{prop:3.78}}. Q.E.D. \vphantom{} Here are the aforementioned properties provable for continuous $\left(p,q\right)$-adic functions: \begin{lem} Let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Recall that $f_{N}$ is the $N$th truncation of $f$, $\hat{f}_{N}$ is the Fourier transform of $f_{N}$, $\hat{f}$ is the Fourier transform of $f$, and write: \begin{equation} \tilde{f}_{N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Definition of f_N twiddle} \end{equation} Then: \vphantom{} I. \begin{equation} \hat{f}\left(t\right)-\hat{f}_{N}\left(t\right)=\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Difference between f-hat and f_N-hat} \end{equation} with: \begin{equation} \left\Vert \hat{f}-\hat{f}_{N}\right\Vert _{p,q}\leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q},\textrm{ }\forall N\geq0\label{eq:Infinit norm of f-hat - f_N-hat} \end{equation} \vphantom{} II. \begin{equation} \tilde{f}_{N}\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right)=p^{N}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall N\geq0\label{eq:Difference between f_N-twiddle and f_N} \end{equation} with: \begin{equation} \left\Vert \tilde{f}_{N}-f_{N}\right\Vert _{p,q}\leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q}\label{eq:Infinity norm of difference between f_N twiddle and f_N} \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\tilde{f}_{N}\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q}\label{eq:L^1 norm of the difference between f_N-twiddle and f_N} \end{equation} \vphantom{} III. \begin{equation} \left\Vert f-\tilde{f}_{N}\right\Vert _{p,q}\leq\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q}\leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q}\label{eq:Supremum norm of difference between f and f_N twiddle} \end{equation} and: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-\tilde{f}_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\leq\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q}\leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q}\label{eq:L^1 norm of difference between f and f_N twiddle} \end{equation} \end{lem} Proof: Since the Banach space $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is isometrically isomorphic to $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ by way of the $\left(p,q\right)$-adic Fourier transform (\ref{eq:Fourier transform of Nth truncation in terms of vdP coefficients}), we have that: \begin{align} \hat{f}\left(t\right)-\hat{f}_{N}\left(t\right) & \overset{\mathbb{C}_{q}}{=}\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int}-\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{p^{N}-1}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int}\\ & \overset{\mathbb{C}_{q}}{=}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int} \end{align} This proves (\ref{eq:Difference between f-hat and f_N-hat}). Taking $\left(p,q\right)$-adic supremum norm over $\hat{\mathbb{Z}}_{p}$ yields (\ref{eq:Infinit norm of f-hat - f_N-hat}). Next, performing Fourier inversion, we have: \begin{align} \tilde{f}_{N}\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right) & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\hat{f}\left(t\right)-\hat{f}_{N}\left(t\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =p^{N}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \end{align} Hence: \[ \tilde{f}_{N}\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right)=p^{N}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right] \] which proves (\ref{eq:Difference between f_N-twiddle and f_N}). Taking $\left(p,q\right)$-adic supremum norm over $\mathbb{Z}_{p}$ yields (\ref{eq:Infinity norm of difference between f_N twiddle and f_N}). On the other hand, taking $q$-adic absolute values, integrating gives us: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\tilde{f}_{N}\left(\mathfrak{z}\right)-f_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & =\int_{\mathbb{Z}_{p}}\left\|p^{N}\sum_{n=p^{N}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\right\|_{q}d\mathfrak{z}\\ & \leq\int_{\mathbb{Z}_{p}}\sup_{n\geq p^{N}}\left\|\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\right\|_{q}d\mathfrak{z}\\ & \leq\int_{\mathbb{Z}_{p}}\sup_{n\geq p^{N}}\left(\left\|c_{n}\left(f\right)\right\|_{q}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\right)d\mathfrak{z}\\ & \leq\left(\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q}\right)\int_{\mathbb{Z}_{p}}d\mathfrak{z}\\ & =\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q} \end{align} which proves (\ref{eq:L^1 norm of the difference between f_N-twiddle and f_N}). Lastly: \begin{align} f\left(\mathfrak{z}\right)-\tilde{f}_{N}\left(\mathfrak{z}\right) & =\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}-\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{\left\|t\right\|_{p}>p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} and so: \begin{align} \left\Vert f-\tilde{f}_{N}\right\Vert _{p,q} & \leq\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q}\\ & \leq\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\sum_{n=\frac{\left\|t\right\|_{p}}{p}}^{\infty}\frac{c_{n}\left(f\right)}{p^{\lambda_{p}\left(n\right)}}e^{-2\pi int}\right\|_{q}\\ & \leq\sup_{\left\|t\right\|_{p}>p^{N}}\sup_{n\geq\frac{\left\|t\right\|_{p}}{p}}\left\|c_{n}\left(f\right)\right\|_{q}\\ & \leq\sup_{\left\|t\right\|_{p}>p^{N}}\sup_{n\geq\frac{p^{N+1}}{p}}\left\|c_{n}\left(f\right)\right\|_{q}\\ & =\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q} \end{align} This proves (\ref{eq:Supremum norm of difference between f and f_N twiddle}). If we instead integrate, we get: \begin{align} \int_{\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}\right)-\tilde{f}_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & =\int_{\mathbb{Z}_{p}}\left\|\sum_{\left\|t\right\|_{p}>p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}d\mathfrak{z}\\ & \leq\int_{\mathbb{Z}_{p}}\left(\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q}\right)d\mathfrak{z}\\ & =\sup_{\left\|t\right\|_{p}>p^{N}}\left\|\hat{f}\left(t\right)\right\|_{q}\\ & \leq\sup_{n\geq p^{N}}\left\|c_{n}\left(f\right)\right\|_{q} \end{align} Q.E.D. \vphantom{} The last result of this subsection is a sufficient condition for a rising-continuous function to be an element of $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, formulated as a decay condition on the $q$-adic absolute value of the function's van der Put coefficient. We need two propositions first. \begin{prop} Let $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then, for integers $N,k\geq1$ the difference between the $\left(N+k\right)$th and $N$th truncations of $\chi$ is: \begin{align} \chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right) & =\sum_{j=1}^{p^{k}-1}\sum_{n=0}^{p^{N}-1}\left(\chi\left(n+jp^{N}\right)-\chi\left(n\right)\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+jp^{N}\right]\label{eq:Difference between N+kth and Nth truncations of Chi} \end{align} \end{prop} Proof: Note that: \begin{equation} \left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]=\sum_{j=0}^{p^{k}-1}\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}j\right],\textrm{ }\forall N,k\geq0,\textrm{ }\forall n\in\left\{ 0,\ldots,p^{N}-1\right\} ,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p} \end{equation} Hence: \begin{align} \chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right) & =\sum_{n=0}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]-\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\\ & =\sum_{n=0}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]-\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\sum_{j=0}^{p^{k}-1}\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}j\right]\\ & =\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\underbrace{\left(\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]-\sum_{j=0}^{p^{k}-1}\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}j\right]\right)}_{j=0\textrm{ term cancels}}\\ & +\sum_{n=p^{N}}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]\\ & =\sum_{n=p^{N}}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]-\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\sum_{j=1}^{p^{k}-1}\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}j\right] \end{align} Here: \begin{align} \sum_{n=p^{N}}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right] & =\sum_{n=0}^{\left(p^{k}-1\right)p^{N}-1}\chi\left(n+p^{N}\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}\right]\\ & =\sum_{j=1}^{p^{k}-1}\sum_{n=\left(j-1\right)p^{N}}^{jp^{N}-1}\chi\left(n+p^{N}\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+p^{N}\right]\\ & =\sum_{j=1}^{p^{k}-1}\sum_{n=0}^{p^{N}-1}\chi\left(n+jp^{N}\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+jp^{N}\right] \end{align} and so: \begin{align} \chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right) & =\sum_{n=p^{N}}^{p^{N+k}-1}\chi\left(n\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n\right]-\sum_{n=0}^{p^{N}-1}\chi\left(n\right)\sum_{j=1}^{p^{k}-1}\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+jp^{N}\right]\\ & =\sum_{j=1}^{p^{k}-1}\sum_{n=0}^{p^{N}-1}\left(\chi\left(n+jp^{N}\right)-\chi\left(n\right)\right)\left[\mathfrak{z}\overset{p^{N+k}}{\equiv}n+jp^{N}\right] \end{align} Q.E.D. \begin{prop} Let $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then, for all integers $N,k\geq1$: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}=\sum_{j=1}^{p^{k}-1}\sum_{n=0}^{p^{N}-1}\frac{\left\|\chi\left(n+jp^{N}\right)-\chi\left(n\right)\right\|_{q}}{p^{N+k}}\label{eq:L^1_R norm of Chi_N+k minus Chi_N} \end{equation} \end{prop} Proof: Let $N$ and $k$ be arbitrary. Note that the brackets in (\ref{eq:Difference between N+kth and Nth truncations of Chi}) have supports which are pair-wise disjoint with respect to $n$ and $j$, and that each of the brackets has Haar measure $1/p^{N+k}$. As such: \[ \int_{\mathbb{Z}_{p}}\left\|\chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}=\sum_{j=1}^{p^{k}-1}\sum_{n=0}^{p^{N}-1}\frac{\left\|\chi\left(n+jp^{N}\right)-\chi\left(n\right)\right\|_{q}}{p^{N+k}} \] Q.E.D. \vphantom{} Here, then, is the $L_{\mathbb{R}}^{1}$ criterion: \begin{lem} Let\index{van der Put!coefficients!L_{mathbb{R}}^{1} criterion@$L_{\mathbb{R}}^{1}$ criterion} $\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. If there exists an $r\in\left(0,1\right)$ so that for all sufficiently large integers $N\geq0$: \begin{equation} \max_{0\leq n<p^{N}}\left\|c_{n+jp^{N}}\left(\chi\right)\right\|_{q}\ll r^{j},\textrm{ }\forall j\geq1\label{eq:van der Put coefficient asymptotic condition for L^1_R integrability} \end{equation} Then $\chi\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. \end{lem} Proof: Letting $\chi$ and $r$ be as given, note that: \begin{equation} c_{n+jp^{N}}\left(\chi\right)=\chi\left(n+jp^{N}\right)-\chi\left(n\right),\textrm{ }\forall n<p^{N},\textrm{ }\forall j\geq1 \end{equation} By (\ref{eq:L^1_R norm of Chi_N+k minus Chi_N}), we have that: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\chi_{N+k}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & \ll\sum_{n=0}^{p^{N}-1}\sum_{j=1}^{p^{k}-1}\frac{r^{j}}{p^{N+k}}\\ & =\sum_{n=0}^{p^{N}-1}\frac{1}{p^{N+k}}\left(-1+\frac{1-r^{p^{k}}}{1-r}\right)\\ & =\frac{1}{p^{k}}\frac{r-r^{p^{k}}}{1-r} \end{align} which tends to $0$ in $\mathbb{R}$ as $k\rightarrow\infty$. Since $k$ was arbitrary, it follows that for any $\epsilon>0$, choosing $N$ and $M=N+k$ sufficiently large, we can guarantee that: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\chi_{M}\left(\mathfrak{z}\right)-\chi_{N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}<\epsilon \end{equation} which shows that the $\chi_{N}$s are Cauchy in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ norm. Since $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ is complete, and since, as a rising-continuous function, the $\chi_{N}$s converge point-wise to $\chi$, it follows that $\chi$ is then an element of $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Q.E.D. \vphantom{} We end this subsection with two questions: \begin{question} \label{que:3.7}Is every function in $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\cap L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ and element of $\tilde{L}^{1}\left(\mathcal{F}_{p,q}\right)$? \end{question} \begin{question} Let $\mathcal{F}$ be a $p$-aduc frame. Is $\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\cap\tilde{L}^{1}\left(\mathcal{F}\right)\subseteq L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$? Is $\tilde{L}^{1}\left(\mathcal{F}\right)\subseteq L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$? \end{question} \subsection{\label{subsec:3.3.7 -adic-Wiener-Tauberian}$\left(p,q\right)$-adic Wiener Tauberian Theorems} One of the great mathematical accomplishments of the twentieth century was the transfiguration of the study of divergent series from the realm of foolish fancy to one of serious, rigorous study. Asymptotic analysis\textemdash be it Karamata's theory of functions of regular variation\index{regular variation}\footnote{Even though it has little to do with the topic at hand, I cannot give too strong of a recommendation of Bingham, Goldie, and Teugels' book \emph{Regular Variation} \cite{Regular Variation}. It should be required reading for anyone working in analysis. It is filled with a thousand and one useful beauties that you never knew you wanted to know.}, or Abelian or Tauberian summability theorems\textemdash are powerful and widely useful in number theory, probability theory, and nearly everything in between. The \index{Wiener!Tauberian Theorem}Wiener Tauberian Theorem (WTT) is a capstone of these investigations. This theorem has a protean diversity of forms and restatements, all of which are interesting in their own particular way. Jacob Korevaar\index{Korevaar, Jacob}'s book on Tauberian theory gives an excellent exposition of most of them \cite{Korevaar}. The version of the WTT that matters to us sits at the boarder between Fourier analysis and Functional analysis, where it characterizes the relationship between a $1$-periodic function $\phi:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{C}$ (where $\mathbb{R}/\mathbb{Z}$ is identified with $\left[0,1\right)$ with addition modulo $1$), its Fourier coefficients $\hat{\phi}\left(n\right)$, and its reciprocal $1/\phi$. \begin{defn} The \index{Wiener!algebra}\textbf{Wiener algebra (on the circle)}, denote $A\left(\mathbb{T}\right)$, is the $\mathbb{C}$-linear space of all absolutely convergent Fourier series: \begin{equation} A\left(\mathbb{T}\right)\overset{\textrm{def}}{=}\left\{ \sum_{n\in\mathbb{Z}}\hat{\phi}\left(n\right)e^{2\pi int}:\hat{\phi}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)\right\} \label{eq:Definition of the Wiener algebra} \end{equation} where: \begin{equation} \ell^{1}\left(\mathbb{Z},\mathbb{C}\right)\overset{\textrm{def}}{=}\left\{ \hat{\phi}:\mathbb{Z}\rightarrow\mathbb{C}:\sum_{n\in\mathbb{Z}}\left\|\hat{\phi}\left(n\right)\right\|<\infty\right\} \label{eq:Definition of ell 1 of Z, C} \end{equation} $A\left(\mathbb{T}\right)$ is then made into a Banach space with the norm: \begin{equation} \left\Vert \phi\right\Vert _{A}\overset{\textrm{def}}{=}\left\Vert \hat{\phi}\right\Vert _{1}\overset{\textrm{def}}{=}\sum_{n\in\mathbb{Z}}\left\|\hat{\phi}\left(n\right)\right\|,\textrm{ }\forall\phi\in A\left(\mathbb{T}\right)\label{eq:Definition of Wiener algebra norm} \end{equation} \end{defn} \vphantom{} As defined, observe that $A\left(\mathbb{T}\right)$ is then isometrically isomorphic to the Banach space $\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$. Because $\left(\mathbb{Z},+\right)$ is a locally compact abelian group, $\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ can be made into a Banach algebra (the \textbf{group algebra}\index{group algebra}\textbf{ }of $\mathbb{Z}$) by equipping it with the convolution operation: \begin{equation} \left(\hat{\phi}\hat{\psi}\right)\left(n\right)\overset{\textrm{def}}{=}\sum_{m\in\mathbb{Z}}\hat{\phi}\left(n-m\right)\hat{\psi}\left(m\right)\label{eq:ell 1 convolution} \end{equation} In this way, $A\left(\mathbb{T}\right)$ becomes a Banach algebra under point-wise multiplication, thanks to \textbf{Young's Convolution Inequality} and the fact that the Fourier transform turns multiplication into convolution: \begin{equation} \left\Vert \phi\cdot\psi\right\Vert _{A}=\left\Vert \widehat{\phi\cdot\psi}\right\Vert _{1}=\left\Vert \hat{\phi}\hat{\psi}\right\Vert _{1}\overset{\textrm{Young}}{\leq}\left\Vert \hat{\phi}\right\Vert _{1}\left\Vert \hat{\psi}\right\Vert _{1}=\left\Vert \phi\right\Vert _{A}\left\Vert \psi\right\Vert _{A} \end{equation} With this terminology, we can state three versions of the WTT. \begin{thm}[\textbf{Wiener's Tauberian Theorem, Ver. 1}] Let $\phi\in A\left(\mathbb{T}\right)$. Then, $1/\phi\in A\left(\mathbb{T}\right)$ if and only if\footnote{That is to say, the units of the Wiener algebra are precisely those functions $\phi\in A\left(\mathbb{T}\right)$ which have no zeroes.} $\phi\left(t\right)\neq0$ for all $t\in\mathbb{R}/\mathbb{Z}$. \end{thm} At first glance, this statement might seem almost tautological, until you realize that we are requiring more than just the well-definedness of $1/\phi$: we are also demanding it to have an absolutely convergent Fourier series. Because the Fourier transform turns multiplication into convolution, the existence of a multiplicative inverse for $\phi\in A\left(\mathbb{T}\right)$ is equivalent to the existence of a \index{convolution!inverse}\textbf{convolution inverse}\emph{ }for $\hat{\phi}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$, by which I mean a function $\hat{\phi}^{-1}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ so that: \begin{equation} \left(\hat{\phi}\hat{\phi}^{-1}\right)\left(n\right)=\left(\hat{\phi}^{-1}\hat{\phi}\right)\left(n\right)=\mathbf{1}_{0}\left(n\right)=\begin{cases} 1 & \textrm{if }n=0\\ 0 & \textrm{else} \end{cases}\label{eq:Definition of a convolution inverse} \end{equation} because the function $\mathbf{1}_{0}\left(n\right)$ which is $1$ at $n=0$ and $0$ for all other $n$ is identity element of the convolution operation on $\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$. This gives us a second version of the WTT. \begin{thm}[\textbf{Wiener's Tauberian Theorem, Ver. 2}] Let $\hat{\phi}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$. Then $\hat{\phi}$ has a convolution inverse $\hat{\phi}^{-1}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ if and only if $\phi\left(t\right)\neq0$ for all $t\in\mathbb{R}/\mathbb{Z}$. \end{thm} \vphantom{} To get a more analytic statement of the theorem, observe that the existence of a convolution inverse $\hat{\phi}^{-1}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ for $\hat{\phi}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ means that for any $\hat{\psi}\in\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$, we can convolve $\hat{\phi}$ with a function in $\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ to obtain $\hat{\psi}$: \begin{equation} \hat{\phi}\left(\hat{\phi}^{-1}\hat{\psi}\right)=\hat{\psi} \end{equation} Writing $\hat{\chi}$ to denote $\hat{\phi}^{-1}\hat{\psi}$, observe the limit: \begin{equation} \left(\hat{\phi}\hat{\chi}\right)\left(n\right)\overset{\mathbb{C}}{=}\lim_{M\rightarrow\infty}\sum_{\left\|m\right\|\leq M}\hat{\chi}\left(m\right)\hat{\phi}\left(n-m\right) \end{equation} where, the right-hand side is a linear combination of translates of $\hat{\phi}$. In fact, the convergence is in $\ell^{1}$-norm: \begin{equation} \lim_{M\rightarrow\infty}\sum_{n\in\mathbb{Z}}\left\|\hat{\psi}\left(n\right)-\sum_{\left\|m\right\|\leq M}\hat{\chi}\left(m\right)\hat{\phi}\left(n-m\right)\right\|\overset{\mathbb{C}}{=}0 \end{equation} This gives us a third version of the WTT: \begin{thm}[\textbf{Wiener's Tauberian Theorem, Ver. 3}] Let $\phi\in A\left(\mathbb{T}\right)$. Then, the translates of $\hat{\phi}$ are dense in $\ell^{1}\left(\mathbb{Z},\mathbb{C}\right)$ if and only if $\phi\left(t\right)\neq0$ for any $t\in\mathbb{R}/\mathbb{Z}$. \end{thm} \vphantom{} Because the \textbf{Correspondence Principle }tells us that periodic points of a Hydra map $H$ are precisely those $x\in\mathbb{Z}$ which lie in the image of $\chi_{H}$, a $\left(p,q\right)$-adic analogue of WTT will allow us to study the multiplicative inverse of $\mathfrak{z}\mapsto\chi_{H}\left(\mathfrak{z}\right)-x$ (for any fixed $x$) by using Fourier transforms of $\chi_{H}$. And while this hope is eventually borne out, it does not come to pass without incident. In this subsection, I establish two realizations of a $\left(p,q\right)$-adic WTT: one for continuous functions; another for measures $d\mu$. First, however, a useful notation for translates: \begin{defn} For $s\in\hat{\mathbb{Z}}_{p}$, $\mathfrak{a}\in\mathbb{Z}_{p}$, $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, and $\hat{f}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, we write: \begin{equation} \tau_{s}\left\{ \hat{f}\right\} \left(t\right)\overset{\textrm{def}}{=}\hat{f}\left(t+s\right)\label{eq:definition of the translate of f hat} \end{equation} and: \begin{equation} \tau_{\mathfrak{a}}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}f\left(\mathfrak{z}+\mathfrak{a}\right)\label{eq:Definition of the translate of f} \end{equation} \end{defn} \begin{thm}[\textbf{Wiener Tauberian Theorem for Continuous $\left(p,q\right)$-adic Functions}] \index{Wiener!Tauberian Theorem!left(p,qright)-adic@$\left(p,q\right)$-adic}\label{thm:pq WTT for continuous functions}Let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, the following are equivalent: \vphantom{} I. $\frac{1}{f}\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$; \vphantom{} II. $\hat{f}$ has a convolution inverse in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$; \vphantom{} III. $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{s}\left\{ \hat{f}\right\} \left(t\right):s\in\hat{\mathbb{Z}}_{p}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$; \vphantom{} IV. $f$ has no zeroes. \end{thm} Proof: \textbullet{} ($\textrm{I}\Rightarrow\textrm{II}$) Suppose $\frac{1}{f}$ is continuous. Then, since the $\left(p,q\right)$-adic Fourier transform is an isometric isomorphism of the Banach algebra $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ onto the Banach algebra $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, it follows that: \begin{align} f\left(\mathfrak{z}\right)\cdot\frac{1}{f\left(\mathfrak{z}\right)} & =1,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\\ \left(\textrm{Fourier transform}\right); & \Updownarrow\\ \left(\hat{f}\widehat{\left(\frac{1}{f}\right)}\right)\left(t\right) & =\mathbf{1}_{0}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{align} where both $\hat{f}$ and $\widehat{\left(1/f\right)}$ are in $c_{0}$. $\hat{f}$ has a convolution inverse in $c_{0}$, and this inverse is $\widehat{\left(1/f\right)}$. \vphantom{} \textbullet{} ($\textrm{II}\Rightarrow\textrm{III}$) Suppose $\hat{f}$ has a convolution inverse $\hat{f}^{-1}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Then, letting $\hat{g}\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ be arbitrary, we have that: \begin{equation} \left(\hat{f}\left(\hat{f}^{-1}\hat{g}\right)\right)\left(t\right)=\left(\left(\hat{f}\hat{f}^{-1}\right)\hat{g}\right)\left(t\right)=\left(\mathbf{1}_{0}\hat{g}\right)\left(t\right)=\hat{g}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} In particular, letting $\hat{h}$ denote $\hat{f}^{-1}\hat{g}$, we have that: \begin{equation} \hat{g}\left(t\right)=\left(\hat{f}\hat{h}\right)\left(t\right)\overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right) \end{equation} Since $\hat{f}$ and $\hat{h}$ are in $c_{0}$, note that: \begin{align} \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)-\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q} & \leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\sup_{\left\|s\right\|_{p}>p^{N}}\left\|\hat{h}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}\\ \left(\left\|\hat{f}\right\|_{q}<\infty\right); & \leq\sup_{\left\|s\right\|_{p}>p^{N}}\left\|\hat{h}\left(s\right)\right\|_{q} \end{align} and hence: \begin{equation} \lim_{N\rightarrow\infty}\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)-\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}\overset{\mathbb{R}}{=}\lim_{N\rightarrow\infty}\sup_{\left\|s\right\|_{p}>p^{N}}\left\|\hat{h}\left(s\right)\right\|_{q}\overset{\mathbb{R}}{=}0 \end{equation} which shows that the $q$-adic convergence of $\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)$ to $\hat{g}\left(t\right)=\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)$ is uniform in $t$. Hence: \begin{equation} \lim_{N\rightarrow\infty}\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{g}\left(t\right)-\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}=0 \end{equation} which is precisely the definition of what it means for the sequence $\left\{ \sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{h}\left(s\right)\hat{f}\left(t-s\right)\right\} _{N\geq0}$ to converge in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ to $\hat{g}$. Since $\hat{g}$ was arbitrary, and since this sequence is in the span of the translates of $\hat{f}$ over $\mathbb{C}_{q}$, we see that said span is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. \vphantom{} \textbullet{} ($\textrm{III}\Rightarrow\textrm{IV}$) Suppose $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{s}\left\{ \hat{f}\right\} \left(t\right):s\in\hat{\mathbb{Z}}_{p}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Since $\mathbf{1}_{0}\left(t\right)\in c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, given any $\epsilon\in\left(0,1\right)$, we can then choose constants $\mathfrak{c}_{1},\ldots,\mathfrak{c}_{M}\in\mathbb{C}_{q}$ and $t_{1},\ldots,t_{M}\in\hat{\mathbb{Z}}_{p}$ so that: \begin{equation} \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{f}\left(t-t_{m}\right)\right\|_{q}<\epsilon \end{equation} Now, letting $N\geq\max\left\{ -v_{p}\left(t_{1}\right),\ldots,-v_{p}\left(t_{M}\right)\right\} $ be arbitrary (note that $-v_{p}\left(t\right)=-\infty$ when $t=0$), the maps $t\mapsto t+t_{m}$ are then bijections of the set $\left\{ t\in\hat{\mathbb{Z}}_{p}:\left\|t\right\|_{p}\leq p^{N}\right\} $. Consequently: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{f}\left(t-t_{m}\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\sum_{m=1}^{M}\mathfrak{c}_{m}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ \left(t+t_{m}\right)\mathfrak{z}\right\} _{p}}\\ & =\left(\sum_{m=1}^{M}\mathfrak{c}_{m}e^{2\pi i\left\{ t_{m}\mathfrak{z}\right\} _{p}}\right)\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{f}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} Letting $N\rightarrow\infty$, we obtain: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{f}\left(t-t_{m}\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{C}_{q}}{=}g_{m}\left(\mathfrak{z}\right)f\left(\mathfrak{z}\right) \end{equation} where $g_{m}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ is defined by: \begin{equation} g_{m}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{m=1}^{M}\mathfrak{c}_{m}e^{2\pi i\left\{ t_{m}\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Definition of g_m} \end{equation} Moreover, this convergence is uniform with respect to $\mathfrak{z}$. Consequently: \begin{align} \left\|1-g_{m}\left(\mathfrak{z}\right)f\left(\mathfrak{z}\right)\right\|_{q} & \overset{\mathbb{R}}{=}\lim_{N\rightarrow\infty}\left\|\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{f}\left(t-t_{m}\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|_{q}\\ & \leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{f}\left(t-t_{m}\right)\right\|_{q}\\ & <\epsilon \end{align} for all $\mathfrak{z}\in\mathbb{Z}_{p}$. If $f\left(\mathfrak{z}_{0}\right)=0$ for some $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$, we would then have: \begin{equation} \epsilon>\left\|1-g_{m}\left(\mathfrak{z}_{0}\right)\cdot0\right\|_{q}=1 \end{equation} which would contradict the fact that $\epsilon\in\left(0,1\right)$. As such, $f$ cannot have any zeroes whenever the span of $\hat{f}$'s translates are dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. \vphantom{} \textbullet{} ($\textrm{IV}\Rightarrow\textrm{I}$) Suppose $f$ has no zeroes. Since $\mathfrak{y}\mapsto\frac{1}{\mathfrak{y}}$ is a continuous map on $\mathbb{C}_{q}\backslash\left\{ 0\right\} $, and since compositions of continuous maps are continuous, to show that $1/f$ is continuous, it suffices to show that $\left\|f\left(\mathfrak{z}\right)\right\|_{q}$ is bounded away from $0$. To see this, suppose by way of contradiction that there was a sequence $\left\{ \mathfrak{z}_{n}\right\} _{n\geq0}\subseteq\mathbb{Z}_{p}$ such that for all $\epsilon>0$, $\left\|f\left(\mathfrak{z}_{n}\right)\right\|_{q}<\epsilon$ holds for all sufficiently large $n$\textemdash say, for all $n\geq N_{\epsilon}$. Since $\mathbb{Z}_{p}$ is compact, the $\mathfrak{z}_{n}$s have a subsequence $\mathfrak{z}_{n_{k}}$ which converges in $\mathbb{Z}_{p}$ to some limit $\mathfrak{z}_{\infty}$. The continuity of $f$ forces $f\left(\mathfrak{z}_{\infty}\right)=0$, which contradicts the hypothesis that $f$ was given to have no zeroes. Thus, if $f$ has no zeroes, $\left\|f\left(\mathfrak{z}\right)\right\|_{q}$ is bounded away from zero. This prove that $1/f$ is $\left(p,q\right)$-adically continuous. Q.E.D. \vphantom{} The proof of WTT for $\left(p,q\right)$-adic measures is significantly more involved than the continuous case. While proving the non-vanishing of the limit of $\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ when $\hat{\mu}$'s translates have dense span is as simple as in the continuous case, the other direction requires a more intricate argument. The idea is this: by way of contradiction, suppose there is a $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$ so that $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)$ converges in $\mathbb{C}_{q}$ to zero, yet the span of the translates of $\hat{\mu}$ \emph{are} dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. We get a contradiction by showing that, with these assumptions, we can construct two functions $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, both of which produce interesting results when convolved with $\hat{\mu}$. The first function is constructed so as to give us something close (in the sense of $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$'s norm, the $\left(p,q\right)$-adic sup norm on $\hat{\mathbb{Z}}_{p}$) to the constant function $0$ when we convolve it with $\hat{\mu}$, whereas the second is constructed so as to give us something close to $\mathbf{1}_{0}$ upon convolution with $\hat{\mu}$. Applications of the ultrametric inequality (a.k.a., the strong triangle inequality) show that these two estimates tear each other to pieces, and thus cannot \emph{both }be true; this yields the desired contradiction. While the idea is straightforward, the approach is somewhat technical, because our argument will require restricting our attention to bounded neighborhoods of zero in $\hat{\mathbb{Z}}_{p}$. The contradictory part of estimates we need follow from delicate manipulations of convolutions in tandem with these domain restrictions. As such, it will be very helpful to have a notation for the supremum norm of a function $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ taken over a bounded neighborhood of zero, rather than all of $\hat{\mathbb{Z}}_{p}$. \begin{defn} \label{def:norm notation definition}For any integer $n\geq1$ and any function $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$, we write $\left\Vert \hat{\chi}\right\Vert _{p^{n},q}$ to denote the non-archimedean norm: \begin{equation} \left\Vert \hat{\chi}\right\Vert _{p^{n},q}\overset{\textrm{def}}{=}\sup_{\left\|t\right\|_{p}\leq p^{n}}\left\|\hat{\chi}\left(t\right)\right\|_{q}\label{eq:Definition of truncated norm} \end{equation} \end{defn} \begin{rem} \textbf{WARNING }\textendash{} For the statement and proof of \textbf{Theorem \ref{thm:pq WTT for measures}} and \textbf{Lemma \ref{lem:three convolution estimate}}, we will write $\left\Vert \cdot\right\Vert _{p^{\infty},q}$ to denote the $\left(p,q\right)$-adic supremum norm for functions $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$. writing ``$\left\Vert \cdot\right\Vert _{p,q}$'' would potentially cause confusion with $\left\Vert \cdot\right\Vert _{p^{1},q}$. \end{rem} \begin{thm}[\textbf{Wiener Tauberian Theorem for $\left(p,q\right)$-adic Measures}] \index{Wiener!Tauberian Theorem!left(p,qright)-adic@$\left(p,q\right)$-adic}\label{thm:pq WTT for measures}Let $d\mu\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)^{\prime}$. Then, $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{s}\left\{ \hat{\mu}\right\} \left(t\right):s\in\hat{\mathbb{Z}}_{p}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ if and only if, for any $\mathfrak{z}\in\mathbb{Z}_{p}$ for which the limit $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}\right)$ converges in $\mathbb{C}_{q}$, said limit is necessarily non-zero. \end{thm} \vphantom{} While much of the intricate business involving the intermingling of restrictions and convolutions occurs in the various claims that structure our proof of \textbf{Theorem \ref{thm:pq WTT for measures}}, one particular result in this vein\textemdash the heart of \textbf{Theorem \ref{thm:pq WTT for measures}}'s proof\textemdash is sufficiently non-trivial as to merit separate consideration, so as to avoid cluttering the flow of the argument. This is detailed in the following lemma: \begin{lem} \label{lem:three convolution estimate}Let $M,N\in\mathbb{N}_{0}$ be arbitrary, and let $\hat{\phi}\in B\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ be supported on $\left\|t\right\|_{p}\leq p^{N}$. Then, for any $\hat{f},\hat{g}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ where $\hat{g}$ is supported on $\left\|t\right\|_{p}\leq p^{M}$, we have: \begin{equation} \left\Vert \hat{\phi}\hat{f}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\phi}\right\Vert _{p^{\infty},q}\left\Vert \hat{f}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\label{eq:phi_N hat convolve f hat estimate-1} \end{equation} \begin{equation} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\phi}\hat{f}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\label{eq:phi_N hat convolve f hat convolve g hat estimate-1} \end{equation} \end{lem} Proof: (\ref{eq:phi_N hat convolve f hat estimate-1}) is the easier of the two: \begin{align} \left\Vert \hat{\phi}\hat{f}\right\Vert _{p^{M},q} & =\sup_{\left\|t\right\|_{p}\leq p^{M}}\left\|\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{\phi}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}\\ \left(\hat{\phi}\left(s\right)=0,\textrm{ }\forall\left\|s\right\|_{p}>p^{N}\right); & \leq\sup_{\left\|t\right\|_{p}\leq p^{M}}\left\|\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\phi}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}\\ \left(\textrm{ultrametric ineq.}\right); & \leq\sup_{\left\|t\right\|_{p}\leq p^{M}}\sup_{\left\|s\right\|_{p}\leq p^{N}}\left\|\hat{\phi}\left(s\right)\hat{f}\left(t-s\right)\right\|_{q}\\ & \leq\sup_{\left\|s\right\|_{p}\leq p^{N}}\left\|\hat{\phi}\left(s\right)\right\|_{q}\sup_{\left\|t\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\hat{f}\left(t\right)\right\|_{q}\\ & =\left\Vert \hat{\phi}\right\Vert _{p^{\infty},q}\cdot\left\Vert \hat{f}\right\Vert _{p^{\max\left\{ M,N\right\} },q} \end{align} and we are done. Proving (\ref{eq:phi_N hat convolve f hat convolve g hat estimate-1}) is similar, but more involved. We start by writing out the convolution of $\hat{\phi}\hat{f}$ and $\hat{g}$: \begin{align} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q} & =\sup_{\left\|t\right\|_{p}\leq p^{M}}\left\|\sum_{s\in\hat{\mathbb{Z}}_{p}}\left(\hat{\phi}\hat{f}\right)\left(t-s\right)\hat{g}\left(s\right)\right\|_{q}\\ & \leq\sup_{\left\|t\right\|_{p}\leq p^{M}}\sup_{s\in\hat{\mathbb{Z}}_{p}}\left\|\left(\hat{\phi}\hat{f}\right)\left(t-s\right)\hat{g}\left(s\right)\right\|_{q}\\ \left(\hat{g}\left(s\right)=0,\textrm{ }\forall\left\|s\right\|_{p}>p^{M}\right); & \leq\sup_{\left\|t\right\|_{p}\leq p^{M}}\sup_{\left\|s\right\|_{p}\leq p^{M}}\left\|\left(\hat{\phi}\hat{f}\right)\left(t-s\right)\hat{g}\left(s\right)\right\|_{q}\\ & \leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}}\left\|\left(\hat{\phi}\hat{f}\right)\left(t-s\right)\right\|_{q}\\ \left(\textrm{write out }\hat{\phi}\hat{f}\right); & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}}\left\|\sum_{\tau\in\hat{\mathbb{Z}}_{p}}\hat{\phi}\left(t-s-\tau\right)\hat{f}\left(\tau\right)\right\|_{q}\\ \left(\textrm{let }u=s+\tau\right); & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}}\left\|\sum_{u-s\in\hat{\mathbb{Z}}_{p}}\hat{\phi}\left(t-u\right)\hat{f}\left(u-s\right)\right\|_{q}\\ \left(s+\hat{\mathbb{Z}}_{p}=\hat{\mathbb{Z}}_{p}\right); & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}}\left\|\sum_{u\in\hat{\mathbb{Z}}_{p}}\hat{\phi}\left(t-u\right)\hat{f}\left(u-s\right)\right\|_{q} \end{align} Now we use the fact that $\hat{\phi}\left(t-u\right)$ vanishes for all $\left\|t-u\right\|_{p}>p^{N}$. Because $t$ is restricted to $\left\|t\right\|_{p}\leq p^{M}$, observe that for $\left\|u\right\|_{p}>p^{\max\left\{ M,N\right\} }$, the ultrametric inequality allows us to write: \begin{equation} \left\|t-u\right\|_{p}=\max\left\{ \left\|t\right\|_{p},\left\|u\right\|_{p}\right\} >p^{\max\left\{ M,N\right\} }>p^{N} \end{equation} So, for $\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}$, the summand $\hat{\phi}\left(t-u\right)\hat{f}\left(u-s\right)$ vanishes whenever $\left\|u\right\|_{p}>p^{\max\left\{ M,N\right\} }$. This gives us: \begin{equation} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q}\leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{M}}\left\|\sum_{\left\|u\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\phi}\left(t-u\right)\hat{f}\left(u-s\right)\right\|_{q} \end{equation} Next, we expand the range of $\left\|s\right\|_{p}$ and $\left\|t\right\|_{p}$ from $\leq p^{M}$ in $p$-adic absolute value to $\leq p^{\max\left\{ M,N\right\} }$ in $p$-adic absolute value: \begin{equation} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q}\leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\sum_{\left\|u\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\phi}\left(t-u\right)\hat{f}\left(u-s\right)\right\|_{q} \end{equation} In doing so, we have put $s$, $t$, and $u$ all in the same $p$-adic neighborhood of $0$: \begin{equation} \left\{ x\in\hat{\mathbb{Z}}_{p}:\left\|x\right\|_{p}\leq p^{\max\left\{ M,N\right\} }\right\} \end{equation} We do this because this set is closed under addition: for any $\left\|s\right\|_{p}\leq p^{\max\left\{ M,N\right\} }$, our $u$-sum is invariant under the change of variables $u\mapsto u+s$, and we obtain: \begin{align} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\sum_{\left\|u\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\phi}\left(t-\left(u+s\right)\right)\hat{f}\left(u\right)\right\|_{q}\\ & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\sum_{\left\|u\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\phi}\left(t-s-u\right)\hat{f}\left(u\right)\right\|_{q} \end{align} Finally, observing that: \begin{equation} \left\{ t-s:\left\|t\right\|_{p},\left\|s\right\|_{p}\leq p^{\max\left\{ M,N\right\} }\right\} =\left\{ t:\left\|t\right\|_{p}\leq p^{\max\left\{ M,N\right\} }\right\} \end{equation} we can write: \begin{align} \left\Vert \hat{\phi}\hat{f}\hat{g}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\sum_{\left\|u\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\phi}\left(t-u\right)\hat{f}\left(u\right)\right\|_{q}\\ & \leq\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\underbrace{\sum_{u\in\hat{\mathbb{Z}}_{p}}\hat{\phi}\left(t-u\right)\hat{f}\left(u\right)}_{\hat{\phi}\hat{f}}\right\|_{q}\\ & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\sup_{\left\|t\right\|_{p}\leq p^{\max\left\{ M,N\right\} }}\left\|\left(\hat{\phi}\hat{f}\right)\left(u\right)\right\|_{q}\\ \left(\textrm{by definition}\right); & =\left\Vert \hat{g}\right\Vert _{p^{\infty},q}\left\Vert \hat{\phi}\hat{f}\right\Vert _{p^{\max\left\{ M,N\right\} },q} \end{align} This proves the desired estimate, and with it, the rest of the Lemma. Q.E.D. \vphantom{} \textbf{Proof} \textbf{of Theorem \ref{thm:pq WTT for measures}}: We start with the simpler of the two directions. I. Suppose the span of the translates of $\hat{\mu}$ are dense. Just as in the proof of the WTT for continuous $\left(p,q\right)$-adic functions, we let $\epsilon\in\left(0,1\right)$ and then choose $\mathfrak{c}_{m}$s and $t_{m}$s so that: \begin{equation} \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{\mu}\left(t-t_{m}\right)\right\|_{q}<\epsilon \end{equation} Picking sufficiently large $N$, we obtain: \begin{align} \left\|1-\left(\sum_{m=1}^{M}\mathfrak{c}_{m}e^{2\pi i\left\{ t_{m}\mathfrak{z}\right\} _{p}}\right)\tilde{\mu}_{N}\left(\mathfrak{z}\right)\right\|_{q} & \leq\max_{\left\|t\right\|_{p}\leq p^{N}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{\mu}\left(t-t_{m}\right)\right\|_{q}\\ & \leq\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{m=1}^{M}\mathfrak{c}_{m}\hat{\mu}\left(t-t_{m}\right)\right\|_{q}\\ & <\epsilon \end{align} Now, let $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$ be a point so that $\mathfrak{L}\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)$ converges in $\mathbb{C}_{q}$. We need to show $\mathfrak{L}\neq0$. To do this, plugging $\mathfrak{z}=\mathfrak{z}_{0}$ into the above yields: \[ \epsilon>\lim_{N\rightarrow\infty}\left\|1-\left(\sum_{m=1}^{M}\mathfrak{c}_{m}e^{2\pi i\left\{ t_{m}\mathfrak{z}_{0}\right\} _{p}}\right)\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)\right\|_{q}=\left\|1-\left(\sum_{m=1}^{M}\mathfrak{c}_{m}e^{2\pi i\left\{ t_{m}\mathfrak{z}_{0}\right\} _{p}}\right)\cdot\mathfrak{L}\right\|_{q} \] If $\mathfrak{L}=0$, the right-most expression will be $1$, and hence, we get $\epsilon>1$, but this is impossible; $\epsilon$ was given to be less than $1$. So, if $\lim_{N\rightarrow\infty}\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)$ converges in $\mathbb{C}_{q}$ to $\mathfrak{L}$, $\mathfrak{L}$ must be non-zero. \vphantom{} II. Let $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$ be a zero of $\tilde{\mu}\left(\mathfrak{z}\right)$ such that $\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)\rightarrow0$ in $\mathbb{C}_{q}$ as $N\rightarrow\infty$. Then, by way of contradiction, suppose the span of the translates of $\hat{\mu}$ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, despite the zero of $\tilde{\mu}$ at $\mathfrak{z}_{0}$. Thus, we can use linear combinations of translates of $\hat{\mu}$ approximate any function in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$'s sup norm. In particular, we choose to approximate $\mathbf{1}_{0}$: letting $\epsilon\in\left(0,1\right)$ be arbitrary, there is then a choice of $\mathfrak{c}_{k}$s in $\mathbb{C}_{q}$ and $t_{k}$s in $\hat{\mathbb{Z}}_{p}$ so that: \begin{equation} \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\sum_{k=1}^{K}\mathfrak{c}_{k}\hat{\mu}\left(t-t_{k}\right)\right\|_{q}<\epsilon \end{equation} Letting: \begin{equation} \hat{\eta}_{\epsilon}\left(t\right)\overset{\textrm{def}}{=}\sum_{k=1}^{K}\mathfrak{c}_{k}\mathbf{1}_{t_{k}}\left(t\right)\label{eq:Definition of eta_epsilon hat} \end{equation} we can express the above linear combination as the convolution: \begin{equation} \left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\left(t\right)=\sum_{\tau\in\hat{\mathbb{Z}}_{p}}\hat{\mu}\left(t-\tau\right)\sum_{k=1}^{K}\mathfrak{c}_{k}\mathbf{1}_{t_{k}}\left(\tau\right)=\sum_{k=1}^{K}\mathfrak{c}_{k}\hat{\mu}\left(t-t_{k}\right) \end{equation} So, we get: \begin{equation} \left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}\overset{\textrm{def}}{=}\sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\left(t\right)\right\|_{q}<\epsilon\label{eq:Converse WTT - eq. 1} \end{equation} Before proceeding any further, it is vital to note that we can (and must) assume that $\hat{\eta}_{\epsilon}$ is not identically\footnote{This must hold whenever $\epsilon\in\left(0,1\right)$, because\textemdash were $\hat{\eta}_{\epsilon}$ identically zero\textemdash the supremum: \[ \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\mathbf{1}_{0}\left(t\right)-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\left(t\right)\right\|_{q} \] would then be equal to $1$, rather than $<\epsilon$.} $0$. That being done, equation (\ref{eq:Converse WTT - eq. 1}) shows the assumption we want to contradict: the existence of a function which produces something close to $\mathbf{1}_{0}$ after convolution with $\hat{\mu}$. We will arrive at our contradiction by showing that the zero of $\tilde{\mu}$ at $\mathfrak{z}_{0}$ allows us to construct a second function which yields something close to $0$ after convolution with $\hat{\mu}$. By convolving \emph{both} of these functions with $\hat{\mu}$, we will end up with something which is close to both $\mathbf{1}_{0}$ \emph{and }close to $0$, which is, of course, impossible. Our make-things-close-to-zero-by-convolution function is going to be: \begin{equation} \hat{\phi}_{N}\left(t\right)\overset{\textrm{def}}{=}\mathbf{1}_{0}\left(p^{N}t\right)e^{-2\pi i\left\{ t\mathfrak{z}_{0}\right\} _{p}},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of Phi_N hat} \end{equation} Note that $\hat{\phi}_{N}\left(t\right)$ is only supported for $\left\|t\right\|_{p}\leq p^{N}$. Now, as defined, we have that: \begin{align} \left(\hat{\mu}\hat{\phi}_{N}\right)\left(\tau\right) & =\sum_{s\in\hat{\mathbb{Z}}_{p}}\hat{\mu}\left(\tau-s\right)\hat{\phi}_{N}\left(s\right)\\ \left(\hat{\phi}_{N}\left(s\right)=0,\textrm{ }\forall\left\|s\right\|_{p}>p^{N}\right); & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\mu}\left(\tau-s\right)\hat{\phi}_{N}\left(s\right)\\ & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\mu}\left(\tau-s\right)e^{-2\pi i\left\{ s\mathfrak{z}_{0}\right\} _{p}} \end{align} Fixing $\tau$, observe that the map $s\mapsto\tau-s$ is a bijection of the set $\left\{ s\in\hat{\mathbb{Z}}_{p}:\left\|s\right\|_{p}\leq p^{N}\right\} $ whenever $\left\|\tau\right\|_{p}\leq p^{N}$. So, for any $N\geq-v_{p}\left(\tau\right)$, we obtain: \begin{align} \left(\hat{\mu}\hat{\phi}_{N}\right)\left(\tau\right) & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\mu}\left(\tau-s\right)e^{-2\pi i\left\{ s\mathfrak{z}_{0}\right\} _{p}}\\ & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\mu}\left(s\right)e^{-2\pi i\left\{ \left(\tau-s\right)\mathfrak{z}_{0}\right\} _{p}}\\ & =e^{-2\pi i\left\{ \tau\mathfrak{z}_{0}\right\} _{p}}\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\mu}\left(s\right)e^{2\pi i\left\{ s\mathfrak{z}_{0}\right\} _{p}}\\ & =e^{-2\pi i\left\{ \tau\mathfrak{z}_{0}\right\} _{p}}\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right) \end{align} Since this holds for all $N\geq-v_{p}\left(\tau\right)$, upon letting $N\rightarrow\infty$, $\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)$ converges to $0$ in \emph{$\mathbb{C}_{q}$}, by our assumption. So, for any $\epsilon^{\prime}>0$, there exists an $N_{\epsilon^{\prime}}$ so that $\left\|\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)\right\|_{q}<\epsilon^{\prime}$ for all $N\geq N_{\epsilon^{\prime}}$. Combining this with the above computation (after taking $q$-adic absolute values), we have established the following: \begin{claim} \label{claim:phi_N hat claim}Let $\epsilon^{\prime}>0$ be arbitrary. Then, there exists an $N_{\epsilon^{\prime}}\geq0$ (depending only on $\hat{\mu}$ and $\epsilon^{\prime}$) so that, for all $\tau\in\hat{\mathbb{Z}}_{p}$: \begin{equation} \left\|\left(\hat{\mu}\hat{\phi}_{N}\right)\left(\tau\right)\right\|_{q}=\left\|e^{-2\pi i\left\{ \tau\mathfrak{z}_{0}\right\} _{p}}\tilde{\mu}_{N}\left(\mathfrak{z}_{0}\right)\right\|_{q}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},-v_{p}\left(\tau\right)\right\} ,\tau\in\hat{\mathbb{Z}}_{p}\label{eq:WTT - First Claim} \end{equation} \end{claim} \vphantom{} As stated, the idea is to convolve $\hat{\mu}\hat{\phi}_{N}$ with $\hat{\eta}_{\epsilon}$ so as to obtain a function (via the associativity of convolution) which is both close to $0$ and close to $\mathbf{1}_{0}$. However, our present situation is less than ideal because the lower bound on $N$ in \textbf{Claim \ref{claim:phi_N hat claim}} depends on $\tau$, and so, the convergence of $\left\|\left(\hat{\mu}\hat{\phi}_{N}\right)\left(\tau\right)\right\|_{q}$ to $0$ as $N\rightarrow\infty$ will \emph{not }be uniform in $\tau$. This is where the difficulty of this direction of the proof lies. To overcome this obstacle, instead of convolving $\hat{\mu}\hat{\phi}_{N}$ with $\hat{\eta}_{\epsilon}$, we will convolve $\hat{\mu}\hat{\phi}_{N}$ with a truncated version of $\hat{\eta}_{\epsilon}$, whose support has been restricted to a finite subset of $\hat{\mathbb{Z}}_{p}$. This is the function $\hat{\eta}_{\epsilon,M}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ given by: \begin{equation} \hat{\eta}_{\epsilon,M}\left(t\right)\overset{\textrm{def}}{=}\mathbf{1}_{0}\left(p^{M}t\right)\hat{\eta}_{\epsilon}\left(t\right)=\begin{cases} \hat{\eta}_{\epsilon}\left(t\right) & \textrm{if }\left\|t\right\|_{p}\leq p^{M}\\ 0 & \textrm{else} \end{cases}\label{eq:Definition of eta epsilon M hat} \end{equation} where $M\geq0$ is arbitrary. With this, we can attain the desired ``close to both $0$ and $\mathbf{1}_{0}$'' contradiction for $\hat{\mu}\hat{\phi}_{N}\hat{\eta}_{\epsilon,M}$. The proof will be completed upon demonstrating that this contradiction will remain even as $M\rightarrow\infty$. The analogue of the estimate (\ref{eq:Converse WTT - eq. 1}) for this truncated case is: \begin{align} \left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\left(t\right) & =\sum_{\tau\in\hat{\mathbb{Z}}_{p}}\hat{\mu}\left(t-\tau\right)\mathbf{1}_{0}\left(p^{M}\tau\right)\sum_{k=1}^{K}\mathfrak{c}_{k}\mathbf{1}_{t_{k}}\left(\tau\right)\\ & =\sum_{\left\|\tau\right\|_{p}\leq p^{M}}\hat{\mu}\left(t-\tau\right)\sum_{k=1}^{K}\mathfrak{c}_{k}\mathbf{1}_{t_{k}}\left(\tau\right)\\ & =\sum_{k:\left\|t_{k}\right\|_{p}\leq p^{M}}\mathfrak{c}_{k}\hat{\mu}\left(t-t_{k}\right) \end{align} where, \emph{note}, instead of summing over all the $t_{k}$s that came with $\hat{\eta}_{\epsilon}$, we only sum over those $t_{k}$s in the set $\left\{ t\in\hat{\mathbb{Z}}_{p}:\left\|t\right\|_{p}\leq p^{M}\right\} $. Because the $t_{k}$s came with the un-truncated $\hat{\eta}_{\epsilon}$, observe that we can make $\hat{\mu}\hat{\eta}_{\epsilon,M}$ \emph{equal} to $\hat{\mu}\hat{\eta}_{\epsilon}$ by simply choosing $M$ to be large enough so that all the $t_{k}$s lie in $\left\{ t\in\hat{\mathbb{Z}}_{p}:\left\|t\right\|_{p}\leq p^{M}\right\} $. The lower bound on such $M$s is given by: \begin{equation} M_{0}\overset{\textrm{def}}{=}\max\left\{ -v_{p}\left(t_{1}\right),\ldots,-v_{p}\left(t_{K}\right)\right\} \label{eq:WTT - Choice for M_0} \end{equation} Then, we have: \begin{equation} \left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\left(t\right)=\sum_{k=1}^{K}\mathfrak{c}_{k}\hat{\mu}\left(t-t_{k}\right)=\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\left(t\right),\textrm{ }\forall M\geq M_{0},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Effect of M bigger than M0} \end{equation} So, applying $\left\Vert \cdot\right\Vert _{p^{M},q}$ norm: \begin{align} \left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q} & =\sup_{\left\|t\right\|_{p}\leq p^{M}}\left\|\mathbf{1}_{0}\left(t\right)-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\left(t\right)\right\|_{q}\\ \left(\textrm{if }M\geq M_{0}\right); & =\sup_{\left\|t\right\|_{p}\leq p^{M}}\left\|\mathbf{1}_{0}\left(t\right)-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\left(t\right)\right\|_{q}\\ & \leq\left\Vert \mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\right\Vert _{p^{\infty},q}\\ & <\epsilon \end{align} Finally, note that\emph{ $M_{0}$ }depends on\emph{ only $\hat{\eta}_{\epsilon}$ }and\emph{ $\epsilon$}. \textbf{Claim \ref{claim:truncated convolution estimates for 1d WTT}}, given below, summarizes these findings: \begin{claim} \label{claim:truncated convolution estimates for 1d WTT}Let $\epsilon>0$ be arbitrary. Then, there exists an integer $M_{0}$ depending only on $\hat{\eta}_{\epsilon}$ and $\epsilon$, so that: I. $\hat{\mu}\hat{\eta}_{\epsilon,M}=\hat{\mu}\hat{\eta}_{\epsilon},\textrm{ }\forall M\geq M_{0}$. \vphantom{} II. $\left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{\infty},q}<\epsilon,\textrm{ }\forall M\geq M_{0}$. \vphantom{} III. $\left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}<\epsilon,\textrm{ }\forall M\geq M_{0}$. \end{claim} \vphantom{} The next step is to refine our ``make $\hat{\mu}$ close to zero'' estimate by taking into account $\left\Vert \cdot\right\Vert _{p^{m},q}$. \begin{claim} \label{claim:p^m, q norm of convolution of mu-hat and phi_N hat }Let $\epsilon^{\prime}>0$. Then, there exists $N_{\epsilon^{\prime}}$ depending only on $\epsilon^{\prime}$ and $\hat{\mu}$ so that: \begin{equation} \left\Vert \hat{\phi}_{N}\hat{\mu}\right\Vert _{p^{m},q}<\epsilon^{\prime},\textrm{ }\forall m\geq1,\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},m\right\} \label{eq:WTT - eq. 4} \end{equation} Proof of claim: Let $\epsilon^{\prime}>0$. \textbf{Claim \ref{claim:phi_N hat claim}} tells us that there is an $N_{\epsilon^{\prime}}$ (depending on $\epsilon^{\prime}$, $\hat{\mu}$) so that: \begin{equation} \left\|\left(\hat{\phi}_{N}\hat{\mu}\right)\left(\tau\right)\right\|_{q}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},-v_{p}\left(\tau\right)\right\} ,\textrm{ }\forall\tau\in\hat{\mathbb{Z}}_{p} \end{equation} So, letting $m\geq1$ be arbitrary, note that $\left\|\tau\right\|_{p}\leq p^{m}$ implies $-v_{p}\left(\tau\right)\leq m$. As such, we can make the result of \textbf{Claim \ref{claim:phi_N hat claim}} hold for all $\left\|\tau\right\|_{p}\leq p^{m}$ by choosing $N\geq\max\left\{ N_{\epsilon^{\prime}},m\right\} $: \begin{equation} \underbrace{\sup_{\left\|\tau\right\|_{p}\leq p^{m}}\left\|\left(\hat{\phi}_{N}\hat{\mu}\right)\left(\tau\right)\right\|_{q}}_{\left\Vert \hat{\mu}\hat{\phi}_{N}\right\Vert _{p^{m},q}}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},m\right\} \end{equation} This proves the claim. \end{claim} \vphantom{} Using \textbf{Lemma \ref{lem:three convolution estimate}}, we can now set up the string of estimates we need to arrive at the desired contradiction. First, let us choose an $\epsilon\in\left(0,1\right)$ and a function $\hat{\eta}_{\epsilon}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ which is not identically zero, so that: \begin{equation} \left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}<\epsilon \end{equation} Then, by \textbf{Claim \ref{claim:truncated convolution estimates for 1d WTT}}, we can choose a $M_{\epsilon}$ depending only on $\epsilon$ and $\hat{\eta}_{\epsilon}$ so that: \begin{equation} M\geq M_{\epsilon}\Rightarrow\left\Vert \mathbf{1}_{0}-\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}<\epsilon\label{eq:Thing to contradict} \end{equation} This shows that $\hat{\mu}$ can be convolved to become close to $\mathbf{1}_{0}$. The contradiction will follow from convolving this with $\hat{\phi}_{N}$, where $N$, at this point, is arbitrary: \begin{equation} \left\Vert \hat{\phi}_{N}\left(\mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q}=\left\Vert \hat{\phi}_{N}-\left(\hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right\Vert _{p^{M},q}\label{eq:WTT - Target of attack} \end{equation} Our goal here is to show that (\ref{eq:WTT - Target of attack}) is close to both $0$ and $1$ simultaneously. First, writing: \begin{equation} \left\Vert \hat{\phi}_{N}\left(\mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q} \end{equation} the fact that $\hat{\phi}_{N}\left(t\right)$ is supported on $\left\|t\right\|_{p}\leq p^{N}$ allows us to apply equation (\ref{eq:phi_N hat convolve f hat estimate}) from \textbf{Lemma \ref{lem:three convolution estimate}}. This gives the estimate: \begin{equation} \left\Vert \hat{\phi}_{N}\left(\mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q}\leq\underbrace{\left\Vert \hat{\phi}_{N}\right\Vert _{p^{\infty},q}}_{1}\left\Vert \mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right\Vert _{p^{\max\left\{ M,N\right\} },q} \end{equation} for all $M$ and $N$. Letting $M\geq M_{\epsilon}$, we can apply \textbf{Claim \ref{claim:truncated convolution estimates for 1d WTT}} and write $\hat{\mu}\hat{\eta}_{\epsilon,M}=\hat{\mu}\hat{\eta}_{\epsilon}$. So, for $M\geq M_{\epsilon}$ and arbitrary $N$, we have: \begin{align} \left\Vert \hat{\phi}_{N}\left(\mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q} & \leq\left\Vert \mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\right\Vert _{p^{\max\left\{ M,N\right\} },q}\\ & \leq\left\Vert \mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon}\right)\right\Vert _{p^{\infty},q}\\ \left(\textrm{\textbf{Claim \ensuremath{\ref{claim:truncated convolution estimates for 1d WTT}}}}\right); & <\epsilon \end{align} Thus, the \emph{left-hand side} of (\ref{eq:WTT - Target of attack}) is $<\epsilon$. This is the first estimate. Keeping $M\geq M_{\epsilon}$ and $N$ arbitrary, we will obtain the desired contradiction by showing that the \emph{right-hand side} of (\ref{eq:WTT - Target of attack}) is $>\epsilon$. Since $\left\Vert \cdot\right\Vert _{p^{M},q}$ is a non-archimedean norm, it satisfies the ultrametric inequality. Applying this to the right-hand side of (\ref{eq:WTT - Target of attack}) yields: \begin{equation} \left\Vert \hat{\phi}_{N}-\left(\hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right\Vert _{p^{M},q}\leq\max\left\{ \left\Vert \hat{\phi}_{N}\right\Vert _{p^{M},q},\left\Vert \hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}\right\} \label{eq:WTT - Ultrametric inequality} \end{equation} Because $\hat{\eta}_{\epsilon,M}\left(t\right)$ and $\hat{\phi}_{N}$ are supported on $\left\|t\right\|_{p}\leq p^{M}$ and $\left\|t\right\|_{p}\leq p^{N}$, respectively, we can apply (\ref{eq:phi_N hat convolve f hat convolve g hat estimate}) from \textbf{Lemma \ref{lem:three convolution estimate}} and write: \begin{equation} \left\Vert \hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\phi}_{N}\hat{\mu}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\cdot\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}\label{eq:Ready for epsilon prime} \end{equation} Since $\hat{\eta}_{\epsilon}$ was given to \emph{not }be identically zero, the quantity $\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}$ must be positive. Consequently, for our given $\epsilon\in\left(0,1\right)$, let us define $\epsilon^{\prime}$ by: \begin{equation} \epsilon^{\prime}=\frac{\epsilon}{2\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}} \end{equation} So far, $N$ is still arbitrary. By \textbf{Claim \ref{claim:p^m, q norm of convolution of mu-hat and phi_N hat }}, for this $\epsilon^{\prime}$, we know there exists an $N_{\epsilon^{\prime}}$ (depending only on $\epsilon$, $\hat{\eta}_{\epsilon}$, and $\hat{\mu}$) so that:\textbf{ \begin{equation} \left\Vert \hat{\phi}_{N}\hat{\mu}\right\Vert _{p^{m},q}<\epsilon^{\prime},\textrm{ }\forall m\geq1,\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},m\right\} \end{equation} }Choosing $m=N_{\epsilon^{\prime}}$ gives us: \begin{equation} N\geq N_{\epsilon^{\prime}}\Rightarrow\left\Vert \hat{\phi}_{N}\hat{\mu}\right\Vert _{p^{N},q}<\epsilon^{\prime} \end{equation} So, choose $N\geq\max\left\{ N_{\epsilon^{\prime}},M\right\} $. Then, $\max\left\{ M,N\right\} =N$, and so (\ref{eq:Ready for epsilon prime}) becomes: \begin{equation} \left\Vert \hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\phi}_{N}\hat{\mu}\right\Vert _{p^{N},q}\cdot\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}<\frac{\epsilon}{2\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}}\cdot\left\Vert \hat{\eta}_{\epsilon}\right\Vert _{p^{\infty},q}=\frac{\epsilon}{2} \end{equation} Since $\left\Vert \hat{\phi}_{N}\right\Vert _{p^{M},q}=1$ for all $M,N\geq0$, this shows that: \begin{equation} \left\Vert \hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}<\frac{\epsilon}{2}<1=\left\Vert \hat{\phi}_{N}\right\Vert _{p^{M},q} \end{equation} Now comes the hammer: by the ultrametric inequality, (\ref{eq:WTT - Ultrametric inequality}) is an \emph{equality }whenever one of $\left\Vert \hat{\phi}_{N}\right\Vert _{p^{M},q}$ or $\left\Vert \hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right\Vert _{p^{M},q}$ is strictly greater than the other. Having proved that to be the case, (\ref{eq:WTT - Target of attack}) becomes: \begin{equation} \epsilon>\left\Vert \hat{\phi}_{N}\left(\mathbf{1}_{0}-\left(\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q}=\left\Vert \hat{\phi}_{N}-\left(\hat{\phi}_{N}\hat{\mu}\hat{\eta}_{\epsilon,M}\right)\right\Vert _{p^{M},q}=1>\epsilon \end{equation} for all $M\geq M_{\epsilon}$ and all $N\geq\max\left\{ N_{\epsilon}^{\prime},M\right\} $. The left-hand side is our first estimate, and the right-hand side is our second. Since $\epsilon<1$, this is clearly impossible. This proves that the existence of the zero $\mathfrak{z}_{0}$ precludes the translates of $\hat{\mu}$ from being dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Q.E.D. \subsubsection{A Matter of \label{subsec:A-Matter-of}Matrices} Along with the \textbf{Correspondence Principle }and the derivation of a Formula for the Fourier transforms of the $\chi_{H}$s, \textbf{Theorem \ref{thm:pq WTT for measures}} (and its multi-dimensional analogue in Subsection \ref{subsec:5.4.4More-Fourier-Resummation}) is one of this dissertation's central results. Combined, these three theorems can (and will) be used to show that an integer $x$ is a periodic point of a contracting, semi-basic $p$-Hydra map $H$ if and only if the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are dense in $c_{0}\left(\mathbb{Z}_{p},\mathbb{C}_{q_{H}}\right)$. Aside from the mere aesthetic gratification of this equivalence, it also appears to be fertile ground for future investigations into the periodic points of Hydra maps, thanks to the algebraic structure of $\hat{\mathbb{Z}}_{p}$ and a spoonful of linear algebra. Identifying the group $\mathbb{Z}/p^{N}\mathbb{Z}$ with the set: \begin{equation} \left\{ 0,\frac{1}{p^{N}},\frac{2}{p^{N}}\ldots,\frac{p^{N}-1}{p^{N}}\right\} \end{equation} equipped with addition modulo $1$, it is a standard fact of higher algebra that $\hat{\mathbb{Z}}_{p}$ is the so-called \index{direct limit}\textbf{direct limit }of the $\mathbb{Z}/p^{N}\mathbb{Z}$s. In terms of functions, this is just the fact that any function $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ can be restricted to a function $\mathbb{Z}/p^{N}\mathbb{Z}\rightarrow\mathbb{C}_{q}$ by multiplication by $\mathbf{1}_{0}\left(p^{N}t\right)$. This makes it feasible to study functions $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ by considering functions $\mathbb{Z}/p^{N}\mathbb{Z}\rightarrow\mathbb{C}_{q}$ as $N\rightarrow\infty$. This is particularly nice, seeing as the question of dense translations over $\mathbb{Z}/p^{N}\mathbb{Z}$ reduces to a matter of matrices. \begin{defn} \label{def:Conv inv}Let $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$. Then, a \index{convolution!inverse}\textbf{convolution inverse }of $\hat{\chi}$ is a function $\hat{\chi}^{-1}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ so that: \begin{equation} \underbrace{\lim_{N\rightarrow\infty}\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(t-s\right)\hat{\chi}^{-1}\left(s\right)}_{\overset{\textrm{def}}{=}\left(\hat{\chi}\hat{\chi}^{-1}\right)\left(t\right)}\overset{\mathbb{C}_{q}}{=}\mathbf{1}_{0}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of Convolution Inverse} \end{equation} Note that $\mathbf{1}_{0}$ is the identity element with respect to convolution of functions $\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$. Given $N\geq0$, we say that $\hat{\chi}^{-1}$ is an \textbf{$N$th partial convolution inverse }of $\hat{\chi}$ if: \begin{equation} \sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(t-s\right)\hat{\chi}^{-1}\left(s\right)\overset{\mathbb{C}_{q}}{=}\mathbf{1}_{0}\left(t\right),\textrm{ }\forall\left\|t\right\|_{p}\leq p^{N}\label{eq:Definition of Nth Partial Convolution Inverse} \end{equation} \end{defn} \begin{prop} Let $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$, and suppose there is a function $\hat{\chi}^{-1}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ which is an $N$th partial convolution inverse of $\hat{\chi}$ for all sufficiently large $N$. Then, $\hat{\chi}^{-1}$ is a convolution inverse of $\hat{\chi}$. \end{prop} Proof: Take limits of (\ref{eq:Definition of Nth Partial Convolution Inverse}) as $N\rightarrow\infty$. (Note: the convergence is only guaranteed to be point-wise with respect to $t$.) Q.E.D. \vphantom{} Now, given any $N$, observe that the equation (\ref{eq:Definition of Nth Partial Convolution Inverse}) defining an $N$th partial convolution inverse of $\hat{\chi}$ is actually a system of linear equations with the values of $\hat{\chi}^{-1}\left(s\right)$ on $\left\|s\right\|_{p}\leq p^{N}$ as its unknowns: \begin{align} 1 & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(0-s\right)\hat{\chi}^{-1}\left(s\right)\\ 0 & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(\frac{1}{p^{N}}-s\right)\hat{\chi}^{-1}\left(s\right)\\ 0 & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(\frac{2}{p^{N}}-s\right)\hat{\chi}^{-1}\left(s\right)\\ & \vdots\\ 0 & =\sum_{\left\|s\right\|_{p}\leq p^{N}}\hat{\chi}\left(\frac{p^{N}-1}{p^{N}}-s\right)\hat{\chi}^{-1}\left(s\right) \end{align} We can express this linear system as a matrix. In fact, we can do so in two noteworthy ways. \begin{defn} Let $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ and let $N\geq1$. \vphantom{} I. We write $\mathbf{M}_{N}\left(\hat{\chi}\right)$ to denote the $p^{N}\times p^{N}$matrix: \begin{equation} \mathbf{M}_{N}\left(\hat{\chi}\right)\overset{\textrm{def}}{=}\left(\begin{array}{ccccc} \hat{\chi}\left(0\right) & \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}\right) & \cdots & \hat{\chi}\left(\frac{2}{p^{N}}\right) & \hat{\chi}\left(\frac{1}{p^{N}}\right)\\ \hat{\chi}\left(\frac{1}{p^{N}}\right) & \hat{\chi}\left(0\right) & \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}\right) & & \hat{\chi}\left(\frac{2}{p^{N}}\right)\\ \vdots & \hat{\chi}\left(\frac{1}{p^{N}}\right) & \hat{\chi}\left(0\right) & \ddots & \vdots\\ \hat{\chi}\left(\frac{p^{N}-2}{p^{N}}\right) & & \ddots & \ddots & \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}\right)\\ \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}\right) & \hat{\chi}\left(\frac{p^{N}-2}{p^{N}}\right) & \cdots & \hat{\chi}\left(\frac{1}{p^{N}}\right) & \hat{\chi}\left(0\right) \end{array}\right)\label{eq:Definition of bold M_N of Chi hat} \end{equation} \vphantom{} II. By the \textbf{radial enumeration of $\mathbb{Z}/p^{N}\mathbb{Z}$}, we mean the sequence: \begin{equation} 0,\frac{1}{p},\ldots,\frac{p-1}{p},\frac{1}{p^{2}},\ldots,\frac{p-1}{p^{2}},\frac{p+1}{p^{2}},\ldots,\frac{p^{2}-1}{p^{2}},\frac{1}{p^{3}},\ldots,\frac{p^{N}-1}{p^{N}}\label{eq:Definition of the radial enumeration of Z mod p^N Z} \end{equation} that is to say, the radial enumeration of $\mathbb{Z}/p^{N}\mathbb{Z}$ starts with $0$, which is then followed by the list of all $t\in\hat{\mathbb{Z}}_{p}$ with $\left\|t\right\|_{p}=p$ in order of increasing value in the numerator, which is then followed by the list of all $t\in\hat{\mathbb{Z}}_{p}$ with $\left\|t\right\|_{p}=p^{2}$ in order of increasing value in the numerator, so on and so forth, until we have listed all $t$ with $\left\|t\right\|_{p}\leq p^{N}$. By the \textbf{standard enumeration of }$\mathbb{Z}/p^{N}\mathbb{Z}$, on the other hand, we mean the enumeration: \begin{equation} 0,\frac{1}{p^{N}},\ldots,\frac{p^{N}-1}{p^{N}}\label{eq:Definition of the standard enumeration of Z mod p^N Z} \end{equation} \vphantom{} III. We write $\mathbf{R}_{N}\left(\hat{\chi}\right)$ to denote the $p^{N}\times p^{N}$ matrix: \[ \mathbf{R}_{N}\left(\hat{\chi}\right)\overset{\textrm{def}}{=}\left(\begin{array}{cccccc} \hat{\chi}\left(0-0\right) & \hat{\chi}\left(0-\frac{1}{p}\right) & \cdots & \hat{\chi}\left(0-\frac{p-1}{p}\right) & \cdots & \hat{\chi}\left(0-\frac{p^{N}-1}{p^{N}}\right)\\ \hat{\chi}\left(\frac{1}{p}-0\right) & \hat{\chi}\left(0\right) & \cdots & \hat{\chi}\left(\frac{1}{p}-\frac{p-1}{p}\right) & \cdots & \hat{\chi}\left(\frac{1}{p}-\frac{p^{N}-1}{p^{N}}\right)\\ \vdots & \vdots & \ddots & \vdots & & \vdots\\ \hat{\chi}\left(\frac{p-1}{p}-0\right) & \hat{\chi}\left(\frac{p-1}{p}-\frac{1}{p}\right) & \cdots & \hat{\chi}\left(0\right) & \cdots & \hat{\chi}\left(\frac{p-1}{p}-\frac{p^{N}-1}{p^{N}}\right)\\ \vdots & \vdots & & \vdots & \ddots & \vdots\\ \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}-0\right) & \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}-\frac{1}{p}\right) & \cdots & \hat{\chi}\left(\frac{p^{N}-1}{p^{N}}-\frac{p-1}{p}\right) & \cdots & \hat{\chi}\left(0\right) \end{array}\right) \] That is, the entries of the $n$th row of $\mathbf{R}_{N}\left(\hat{\chi}\right)$, where $n\in\left\{ 0,\ldots,p^{N}-1\right\} $ are: \[ \hat{\chi}\left(t_{n}-s_{0}\right),\hat{\chi}\left(t_{n}-s_{1}\right),\hat{\chi}\left(t_{n}-s_{2}\right)\ldots \] where $t_{n}$ and $s_{k}$ are the $n$th and $k$th elements of the radial enumeration of $\mathbb{Z}/p^{N}\mathbb{Z}$, respectively ($s_{0}=t_{0}=0$, $s_{1}=t_{1}=1/p$, $s_{2}=t_{2}=2/p$, etc.). \end{defn} \begin{rem} If $\hat{\chi}$ is a Fourier transform of some function $\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, we will also write $\mathbf{M}_{N}\left(\chi\right)$ and $\mathbf{R}_{N}\left(\chi\right)$ to denote $\mathbf{M}_{N}\left(\hat{\chi}\right)$ and $\mathbf{R}_{N}\left(\hat{\chi}\right)$, respectively. If $\chi$ and/or $\hat{\chi}$ are not in question, we will just write $\mathbf{M}_{N}$ and $\mathbf{R}_{N}$. \end{rem} \begin{rem} As defined, we have that (\ref{eq:Definition of Nth Partial Convolution Inverse}) can be written in matrix form as: \begin{equation} \mathbf{M}_{N}\left(\hat{\chi}\right)\left(\begin{array}{c} \hat{\chi}^{-1}\left(0\right)\\ \hat{\chi}^{-1}\left(\frac{1}{p^{N}}\right)\\ \vdots\\ \hat{\chi}^{-1}\left(\frac{p^{N}-1}{p^{N}}\right) \end{array}\right)=\underbrace{\left(\begin{array}{c} 1\\ 0\\ \vdots\\ 0 \end{array}\right)}_{\mathbf{e}_{1}}\label{eq:Matrix Equation - Circulant Form} \end{equation} and as: \begin{equation} \mathbf{R}_{N}\left(\hat{\chi}\right)\left(\begin{array}{c} \hat{\chi}^{-1}\left(0\right)\\ \hat{\chi}^{-1}\left(\frac{1}{p}\right)\\ \vdots\\ \hat{\chi}^{-1}\left(\frac{p-1}{p}\right)\\ \vdots\\ \hat{\chi}^{-1}\left(\frac{p^{N}-1}{p^{N}}\right) \end{array}\right)=\mathbf{e}_{1}\label{eq:Matrix Equation - Radial Form} \end{equation} where the column of $\hat{\chi}^{-1}$ is listed according to the radial enumeration of $\hat{\mathbb{Z}}_{p}$. \end{rem} \begin{prop} \label{prop:permutation similarity}For each $N$ for which at least one of $\mathbf{R}_{N}\left(\hat{\chi}\right)$ and $\mathbf{M}_{N}\left(\hat{\chi}\right)$ is invertible, there exists a permutation matrix $\mathbf{E}_{N}$ (depending only on $N$ and $p$) so that: \begin{equation} \mathbf{R}_{N}\left(\hat{\chi}\right)=\mathbf{E}_{N}\mathbf{M}_{N}\left(\hat{\chi}\right)\mathbf{E}_{N}^{-1}\label{eq:Permutation Similarity of M and R} \end{equation} That is to say, $\mathbf{M}_{N}\left(\hat{\chi}\right)$ and $\mathbf{R}_{N}\left(\hat{\chi}\right)$ are \textbf{permutation similar}. \end{prop} Proof: Let $\mathbf{E}_{N}$ be the permutation matrix which sends the standard enumeration of $\mathbb{Z}/p^{N}\mathbb{Z}$ to the radial enumeration of $\mathbb{Z}/p^{N}\mathbb{Z}$: \begin{equation} \mathbf{E}_{N}\left(\begin{array}{c} 0\\ \frac{1}{p^{N}}\\ \vdots\\ \frac{p^{N}-1}{p^{N}} \end{array}\right)=\left(\begin{array}{c} 0\\ \frac{1}{p}\\ \vdots\\ \frac{p-1}{p}\\ \vdots\\ \frac{p^{N}-1}{p^{N}} \end{array}\right)\label{eq:Definition of bold E_N} \end{equation} Consequently, $\mathbf{E}_{N}^{-1}$ sends the radial enumeration to the standard enumeration. Now, without loss of generality, suppose $\mathbf{M}_{N}\left(\hat{\chi}\right)$ is invertible. The proof for when $\mathbf{E}_{N}\left(\hat{\chi}\right)$ is nearly identical, just replace every instance of $\mathbf{E}_{N}^{-1}$ below with $\mathbf{E}_{N}$, and vice-versa, and replace every standard enumeration with the radial enumeration, and vice-versa. Then, taking (\ref{eq:Matrix Equation - Radial Form}), letting $\mathbf{x}$ denote the column vector with $\hat{\chi}^{-1}$ in the standard enumeration, and letting $\mathbf{y}$ denote the column vector with $\hat{\chi}^{-1}$ in the radial enumeration, observe that: \begin{equation} \mathbf{R}_{N}\left(\hat{\chi}\right)\mathbf{y}=\mathbf{R}_{N}\left(\hat{\chi}\right)\mathbf{E}_{N}\mathbf{x}=\mathbf{e}_{1} \end{equation} Left-multiplying both sides by $\mathbf{E}_{N}^{-1}$ yields: \begin{equation} \mathbf{E}_{N}^{-1}\mathbf{R}_{N}\left(\hat{\chi}\right)\mathbf{E}_{N}\mathbf{x}=\mathbf{E}_{N}^{-1}\mathbf{e}_{1}=\mathbf{e}_{1} \end{equation} where the right-most equality follows from the fact that the $1$ at the top of $\mathbf{e}_{1}$ is fixed by the permutation encoded by $\mathbf{E}_{N}^{-1}$. Since $\mathbf{M}_{N}\left(\hat{\chi}\right)$ was assumed to be invertible, (\ref{eq:Matrix Equation - Circulant Form}) shows that $\mathbf{M}_{N}\left(\hat{\chi}\right)$ is the \emph{unique }matrix so that: \[ \mathbf{M}_{N}\left(\hat{\chi}\right)\mathbf{x}=\mathbf{e}_{1} \] The above shows that $\left(\mathbf{E}_{N}^{-1}\mathbf{R}_{N}\left(\hat{\chi}\right)\mathbf{E}_{N}\right)\mathbf{x}=\mathbf{e}_{1}$, which then forces $\mathbf{E}_{N}^{-1}\mathbf{R}_{N}\left(\hat{\chi}\right)\mathbf{E}_{N}=\mathbf{M}_{N}\left(\hat{\chi}\right)$, and hence, (\ref{eq:Permutation Similarity of M and R}). Q.E.D. \vphantom{} Observe that $\mathbf{M}_{N}\left(\hat{\chi}\right)$ is a matrix of the form: \begin{equation} \left(\begin{array}{ccccc} c_{0} & c_{p^{N}-1} & \cdots & c_{2} & c_{1}\\ c_{1} & c_{0} & c_{p^{N}-1} & & c_{2}\\ \vdots & c_{1} & c_{0} & \ddots & \vdots\\ c_{p^{N}-2} & & \ddots & \ddots & c_{p^{N}-1}\\ c_{p^{N}-1} & c_{p^{N}-2} & \cdots & c_{1} & c_{0} \end{array}\right)\label{eq:Bold M_N as a p^N by p^N Circulant Matrix} \end{equation} Our analysis is greatly facilitated by the elegant properties of properties satisfied by matrices of the form (\ref{eq:Bold M_N as a p^N by p^N Circulant Matrix}), which are called \textbf{circulant matrices}\index{circulant matrix}. A standard reference for this subject is \cite{Circulant Matrices}. \begin{defn}[\textbf{Circulant matrices}] Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$, and let $N$ be an integer $\geq1$. Then, an\textbf{ $N\times N$ circulant matrix with entries in $\mathbb{F}$} is a matrix $\mathbf{M}$ of the form: \begin{equation} \left(\begin{array}{ccccc} c_{0} & c_{N-1} & \cdots & c_{2} & c_{1}\\ c_{1} & c_{0} & c_{N-1} & & c_{2}\\ \vdots & c_{1} & c_{0} & \ddots & \vdots\\ c_{N-2} & & \ddots & \ddots & c_{N-1}\\ c_{N-1} & c_{N-2} & \cdots & c_{1} & c_{0} \end{array}\right)\label{eq:Definition of a Circulant Matrix-1} \end{equation} where $c_{0},\ldots,c_{N-1}\in\mathbb{F}$. The polynomial $f_{\mathbf{M}}:\mathbb{F}\rightarrow\mathbb{F}$ defined by: \begin{equation} f_{\mathbf{M}}\left(x\right)\overset{\textrm{def}}{=}\sum_{n=0}^{N-1}c_{n}x^{n}\label{eq:Definition of the associated polynomial of a circulant matrix} \end{equation} is called the \textbf{associated polynomial}\index{circulant matrix!associated polynomial}\textbf{ }of the matrix $\mathbf{M}$. \end{defn} \begin{rem} $\mathbf{M}_{N}\left(\hat{\chi}\right)$ is a circulant matrix for all $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$. \end{rem} \begin{defn} Let $\mathbb{F}$ be a field of characteristic $0$. For a function $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{F}$, we then write $f_{\hat{\chi},N}:\mathbb{F}\rightarrow\mathbb{F}$ to denote the associated polynomial of $\mathbf{M}_{N}\left(\hat{\chi}\right)$: \begin{equation} f_{\hat{\chi},N}\left(z\right)=\sum_{n=0}^{p^{N}-1}\hat{\chi}\left(\frac{n}{p^{N}}\right)z^{n}\label{eq:Definition/Notation for the associated polynomial of bold M_N of Chi hat} \end{equation} \end{defn} \begin{rem} Like with $\mathbf{M}_{N}$ and $\mathbf{R}_{N}$, if $\hat{\chi}$ is a Fourier transform of some function $\chi$, we will write $f_{\chi,N}$ to denote $f_{\hat{\chi},N}$. \end{rem} \vphantom{} The chief properties of circulant matrices stem from their intimate connection with convolution inverses and the finite Fourier transform. \begin{thm} \label{thm:3.52}Let $\mathbf{M}$ be an $N\times N$ circulant matrix\index{circulant matrix!determinant} with coefficients in an algebraically closed field $\mathbb{F}$, and let $\omega\in\mathbb{F}$ be any primitive $N$th root of unity. Then: \vphantom{} I. \begin{equation} \det\mathbf{M}=\prod_{n=0}^{N-1}f_{\mathbf{M}}\left(\omega^{n}\right)\label{eq:Determinant of a Circulant Matrix} \end{equation} \vphantom{} II. The\index{circulant matrix!eigenvalues} eigenvalues of $\mathbf{M}$ are $f_{\mathbf{M}}\left(\omega^{n}\right)$ for $n\in\left\{ 0,\ldots,N-1\right\} $. \end{thm} \vphantom{} We employ this result to compute the characteristic polynomial of $\mathbf{M}_{N}\left(\hat{\chi}\right)$ for any $\hat{\chi}$. \begin{prop} \label{prop:3.71}Let $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then: \begin{equation} \det\left(\mathbf{M}_{N}\left(\hat{\chi}\right)-\lambda\mathbf{I}_{p^{N}}\right)\overset{\mathbb{C}_{q}}{=}\prod_{n=0}^{p^{N}-1}\left(\tilde{\chi}_{N}\left(n\right)-\lambda\right),\textrm{ }\forall\lambda\in\mathbb{C}_{q}\label{eq:Characteristic polynomial of bold M_N of Chi hat} \end{equation} where, $\mathbf{I}_{p^{N}}$ is a $p^{N}\times p^{N}$ identity matrix, and where, as usual, $\tilde{\chi}_{N}\left(\mathfrak{z}\right)=\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$. \end{prop} Proof: First, observe that: \begin{equation} \mathbf{M}_{N}\left(\hat{\chi}\right)-\lambda\mathbf{I}_{p^{N}}=\mathbf{M}_{N}\left(\hat{\chi}-\lambda\mathbf{1}_{0}\right) \end{equation} where: \begin{equation} \hat{\chi}\left(t\right)-\lambda\mathbf{1}_{0}\left(t\right)=\begin{cases} \hat{\chi}\left(0\right)-\lambda & \textrm{if }t=0\\ \hat{\chi}\left(t\right) & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} Here, $\mathbf{M}_{N}\left(\hat{\chi}-\lambda\mathbf{1}_{0}\right)$ is a circulant matrix with the associated polynomial: \begin{align} f_{\hat{\chi}-\lambda\mathbf{1}_{0},N}\left(\mathfrak{z}\right) & =\sum_{n=0}^{p^{N}-1}\left(\hat{\chi}\left(\frac{n}{p^{N}}\right)-\lambda\mathbf{1}_{0}\left(\frac{n}{p^{N}}\right)\right)\mathfrak{z}^{n}\\ \left(\mathbf{1}_{0}\left(\frac{n}{p^{N}}\right)=\begin{cases} 1 & \textrm{if }n=0\\ 0 & \textrm{else} \end{cases}\right); & =-\lambda+\sum_{n=0}^{p^{N}-1}\hat{\chi}\left(\frac{n}{p^{N}}\right)\mathfrak{z}^{n}\\ & =f_{\hat{\chi},N}\left(\mathfrak{z}\right)-\lambda \end{align} By (\ref{eq:Determinant of a Circulant Matrix}), since $\mathbf{M}_{N}\left(\hat{\chi}-\lambda\mathbf{1}_{0}\right)=\mathbf{M}_{N}\left(\hat{\chi}\right)-\lambda\mathbf{I}_{p^{N}}$ is an $p^{N}\times p^{N}$ circulant matrix, we have that: \begin{align} \det\left(\mathbf{M}_{N}\left(\hat{\chi}\right)-\lambda\mathbf{I}_{p^{N}}\right) & \overset{\mathbb{C}_{q}}{=}\prod_{n=0}^{p^{N}-1}\left(f_{\hat{\chi},N}\left(e^{2\pi in/p^{N}}\right)-\lambda\right)\\ & \overset{\mathbb{C}_{q}}{=}\prod_{n=0}^{p^{N}-1}\left(\sum_{k=0}^{p^{N}-1}\hat{\chi}\left(\frac{k}{p^{N}}\right)e^{\frac{2\pi ikn}{p^{N}}}-\lambda\right)\\ & \overset{\mathbb{C}_{q}}{=}\prod_{n=0}^{p^{N}-1}\left(\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\chi}\left(t\right)e^{2\pi itn}-\lambda\right)\\ \left(e^{2\pi itn}=e^{2\pi i\left\{ tn\right\} _{p}}\right); & =\prod_{n=0}^{p^{N}-1}\left(\tilde{\chi}_{N}\left(n\right)-\lambda\right) \end{align} Q.E.D. \begin{thm}[\textbf{$\chi_{H}$ as an eigenvalue problem}] \label{thm:Eigenvalue problem}Let $\hat{\chi}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}_{q}$ be any function. Then: \vphantom{} I. For any $\lambda\in\mathbb{C}_{q}$, $\mathbf{M}_{N}-\lambda\mathbf{I}_{p^{N}}$ is invertible if and only if $\tilde{\chi}_{N}\left(n\right)\neq\lambda$ for any $n\in\left\{ 0,\ldots,p^{N}-1\right\} $. \vphantom{} II. The eigenvalues\index{matrix!eigenvalues} of $\mathbf{M}_{N}$ are $\left\{ \tilde{\chi}_{N}\left(n\right)\right\} _{0\leq n\leq p^{N}-1}$. \vphantom{} III. Let $N\geq1$ satisfy $\tilde{\chi}_{N}\left(n\right)\neq0$ for any $n\in\left\{ 0,\ldots,p^{N}-1\right\} $. Then $\mathbf{M}_{N}$ and $\mathbf{R}_{N}$ are permutation similar, and (\ref{eq:Permutation Similarity of M and R}) holds true. \end{thm} Proof: Use \textbf{Proposition \ref{prop:permutation similarity}}, \textbf{Theorem \ref{thm:3.52}}, and \textbf{Proposition \ref{prop:3.71}}. Q.E.D. \chapter{\label{chap:4 A-Study-of}A Study of $\chi_{H}$ - The One-Dimensional Case} \includegraphics[scale=0.45]{./PhDDissertationEroica4.png} \vphantom{} THROUGHOUT THIS CHAPTER, UNLESS STATED OTHERWISE, WE ASSUME $p$ IS A PRIME NUMBER, AND THAT $H$ IS A CONTRACTING, SEMI-BASIC $p$-HYDRA MAP WHICH FIXES $0$. \vphantom{} Assuming the reader hasn't already figured it out, the musical epigrams accompanying each of this dissertation's chapters are excerpts from the fourth movement of Beethoven's Symphony No. 3\textemdash the \emph{Eroica}. The excerpt for Chapter 4 is from one of the movement's fugal sections. For the uninitiated, a fugue is a form of western classical composition based around strict, rigorous, academic applications of imitative counterpoint among multiple interacting musical voices, and it is the perfect analogy for the work we are about to undertake. It is this chapter that we will finally take the techniques recounted or developed in Chapter 3 and deploy them to analyze $\chi_{H}$. The main goal is to obtain a meaningful formula for a Fourier transform ($\hat{\chi}_{H}$) of $\chi_{H}$, and thereby show that $\chi_{H}$ is quasi-integrable. With a formula for $\hat{\chi}_{H}$, we can utilize the Correspondence Principle in conjunction with our $\left(p,q\right)$-adic generalization of Wiener's Tauberian Theorem (WTT) to reformulate questions about periodic points and divergent points of $H$ into Fourier-analytic questions about $\hat{\chi}_{H}$. Roughly speaking, the equivalences are as follows: \begin{eqnarray} & x\in\mathbb{Z}\backslash\left\{ 0\right\} \textrm{ is a periodic point or divergent point of }H\\ & \Updownarrow\\ & \chi_{H}\left(\mathfrak{z}\right)-x\textrm{ vanishes at some }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\\ & \Updownarrow\\ & \textrm{translates of }\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)\textrm{ are not dense in }c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right) \end{eqnarray} where the upper $\Updownarrow$ is the Correspondence Principle and the lower $\Updownarrow$ is the WTT. I conjecture that \emph{both} of the $\Updownarrow$s are, in fact, rigorous if-and-only-if equivalences. At present, however, each $\Updownarrow$ is only three-quarters true, and requires that $H$ satisfy the hypotheses of \textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}} (page \pageref{thm:Divergent trajectories come from irrational z}). For such an $H$, the actual implications of the Correspondence Principal are: \begin{eqnarray} & x\in\mathbb{Z}\backslash\left\{ 0\right\} \textrm{ is a periodic point of }H\\ & \Updownarrow\\ & \chi_{H}\left(\mathfrak{z}\right)-x\textrm{ vanishes at some }\mathfrak{z}\in\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime} \end{eqnarray} and: \begin{eqnarray} & x\in\mathbb{Z}\backslash\left\{ 0\right\} \textrm{ is a divergent point of }H\\ & \Uparrow\\ & \chi_{H}\left(\mathfrak{z}\right)-x\textrm{ vanishes at some }\mathfrak{z}\in\mathbb{Z}_{p}\backslash\mathbb{Q}_{p} \end{eqnarray} With the WTT and the help of a a condition on $H$ called \textbf{non-singularity }we will prove the \textbf{Tauberian Spectral Theorem }for $\chi_{H}$ (\textbf{Theorem \ref{thm:Periodic Points using WTT}}, page \pageref{thm:Periodic Points using WTT}) then yields: \begin{eqnarray} & \chi_{H}\left(\mathfrak{z}\right)-x\textrm{ vanishes at some }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\\ & \Updownarrow\\ & \textrm{translates of }\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)\textrm{ are not dense in }c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right) \end{eqnarray} To keep the computations in this chapter as simple and manageable as possible, Section \ref{sec:4.1 Preparatory-Work--} contains some additional notational conventions (definitions of functions, constants, etc.) which will streamline the presentation. Section \ref{sec:4.1 Preparatory-Work--} also includes some identities and asymptotic estimates which will be used throughout the rest of the chapter. \section{\label{sec:4.1 Preparatory-Work--}Preparatory Work \textendash{} Conventions, Identities, etc.} We begin by introducing some new notations. \begin{defn} \label{def:alpha, beta, gamma 1D}We define \nomenclature{$\alpha_{H}\left(t\right)$}{ }\nomenclature{$\beta_{H}\left(t\right)$}{ }$\alpha_{H},\beta_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \alpha_{H}\left(t\right)\overset{\textrm{def}}{=}\frac{1}{p}\sum_{j=0}^{p-1}H_{j}^{\prime}\left(0\right)e^{-2\pi ijt}=\frac{1}{p}\sum_{j=0}^{p-1}\frac{\mu_{j}}{p}e^{-2\pi ijt}=\frac{1}{p}\sum_{j=0}^{p-1}\frac{a_{j}}{d_{j}}e^{-2\pi ijt}\label{eq:Definition of alpha_H} \end{equation} \begin{equation} \beta_{H}\left(t\right)\overset{\textrm{def}}{=}\frac{1}{p}\sum_{j=0}^{p-1}H_{j}\left(0\right)e^{-2\pi ijt}=\frac{1}{p}\sum_{j=0}^{p-1}\frac{b_{j}}{d_{j}}e^{-2\pi ijt}\label{eq:Definition of beta_H} \end{equation} where, recall, $H_{j}$ denotes the $j$th branch of $H$. We also adopt the notation: \nomenclature{$\gamma_{H}\left(t\right)$}{ } \begin{equation} \gamma_{H}\left(t\right)\overset{\textrm{def}}{=}\frac{\beta_{H}\left(t\right)}{\alpha_{H}\left(t\right)}\label{eq:Definition of gamma_H} \end{equation} Additionally, we write $\sigma_{H}$ to denote the constant: \begin{equation} \sigma_{H}\overset{\textrm{def}}{=}\log_{p}\left(\sum_{j=0}^{p-1}H_{j}^{\prime}\left(0\right)\right)=\log_{p}\left(\frac{1}{p}\sum_{j=0}^{p-1}\mu_{j}\right)=1+\log_{p}\left(\alpha_{H}\left(0\right)\right)\label{eq:Definition of sigma_H} \end{equation} where $\log_{p}x=\frac{\ln x}{\ln p}$. Equivalently: \begin{equation} \alpha_{H}\left(0\right)=p^{\sigma_{H}-1}\label{eq:alpha_H of 0 in terms of sigma_H} \end{equation} Finally, we define the function \nomenclature{$\kappa_{H}\left(n\right)$}{ }$\kappa_{H}:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ by:\index{alpha{H}left(tright)@$\alpha_{H}\left(t\right)$}\index{beta{H}left(tright)@$\beta_{H}\left(t\right)$}\index{gamma{H}left(tright)@$\gamma_{H}\left(t\right)$} \begin{equation} \kappa_{H}\left(n\right)\overset{\textrm{def}}{=}M_{H}\left(n\right)\left(\frac{\mu_{0}}{p}\right)^{-\lambda_{p}\left(n\right)}\label{eq:Definition of Kappa_H} \end{equation} \end{defn} \begin{defn} We say $H$ is \textbf{non-singular }whenever\index{Hydra map!non-singular} $\alpha_{H}\left(j/p\right)\neq0$ for any $j\in\mathbb{Z}/p\mathbb{Z}$. \end{defn} \vphantom{} The most important identities for this Chapter involve generating functions for the $\#_{p:j}\left(n\right)$; recall that these functions tell us the number of $j$s in the $p$-adic digits of $n$. \begin{prop}[\textbf{A Generating Function Identity}] \label{prop:Generating function identities}Let $\mathbb{F}$ be a field of characteristic zero, let $p$ be an integer $\geq2$, and let $c_{0},\ldots,c_{p-1}$ be any non-zero elements of $\mathbb{F}$. Then, for all $z\in\mathbb{F}$, we have the identities: \vphantom{} I. For all $n\geq1$: \begin{equation} \prod_{m=0}^{n-1}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right)=c_{0}^{n}\sum_{k=0}^{p^{n}-1}c_{0}^{-\lambda_{p}\left(k\right)}\left(\prod_{j=0}^{p-1}c_{j}^{\#_{p:j}\left(k\right)}\right)z^{k}\label{eq:M_H partial sum generating identity} \end{equation} \vphantom{} II. If $c_{0}=1$, for all we have the identity: \begin{equation} \sum_{k=p^{n-1}}^{p^{n}-1}\left(\prod_{j=0}^{p-1}c_{j}^{\#_{p:j}\left(k\right)}\right)z^{k}=\left(\sum_{k=1}^{p-1}c_{k}z^{kp^{n-1}}\right)\prod_{m=0}^{n-2}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right)\label{eq:Lambda-restricted partial M_H sum} \end{equation} The $n$-product is defined to be $1$ whenever $n=1$. \end{prop} Proof: I. Each term on the right of (\ref{eq:M_H partial sum generating identity}) is obtained by taking a product of the form: \begin{equation} \prod_{\ell=0}^{n-1}c_{j_{\ell}}z^{j_{\ell}p^{\ell}}=\left(\prod_{h=0}^{n-1}c_{j_{h}}\right)z^{\sum_{\ell=0}^{n-1}j_{\ell}p^{\ell}} \end{equation} for some choice of $j_{1},\ldots,j_{n-1}\in\mathbb{Z}/p\mathbb{Z}$. Note, however, that if there is an $m\in\left\{ 0,\ldots,n-1\right\} $ so that $j_{\ell}=0$ for all $\ell\in\left\{ m,\ldots,n-1\right\} $, the $c_{j_{\ell}}$s will still keep contributing to the product even for those values of $\ell$. So, in order to use our $\#_{p:j}\left(k\right)$ notation\textemdash which stops counting the $p$-adic digits of $k$ after the last \emph{non-zero} digit in $k$'s $p$-adic expansion\textemdash we need to modify the above product to take into account the number of extra $0$s that occur past the last non-zero digit in $k$'s $p$-adic expansion. So, let $k\geq1$, and let: \begin{equation} k=\sum_{\ell=0}^{n-1}j_{\ell}p^{\ell} \end{equation} Then, $j_{\ell}=0$ for all $\ell\geq\lambda_{p}\left(k\right)$. As such: \begin{equation} \prod_{h=0}^{n-1}c_{j_{h}}=c_{0}^{\left\|\left\{ h\in\left\{ 0,\ldots,n-1\right\} :j_{h}=0\right\} \right\|}\times\prod_{\ell=1}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)} \end{equation} Now, $\#_{p:0}\left(k\right)$ is the number of $\ell\leq\lambda_{p}\left(k\right)-1$ for which $j_{\ell}=0$. Since $j_{\ell}=0$ for all $\ell\geq\lambda_{p}\left(k\right)$, and because there are $n-\lambda_{p}\left(k\right)$ values of $h\in\left\{ 0,\ldots,n-1\right\} $ which are in the interval $\left[\lambda_{p}\left(k\right),n-1\right]$, we can write: \begin{equation} \left\|\left\{ h\in\left\{ 0,\ldots,n-1\right\} :j_{h}=0\right\} \right\|=\#_{p:0}\left(k\right)+n-\lambda_{p}\left(k\right) \end{equation} So: \begin{equation} \prod_{h=0}^{n-1}c_{j_{h}}=c_{0}^{\#_{p:0}\left(k\right)+n-\lambda_{p}\left(k\right)}\times\prod_{\ell=1}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)}=c_{0}^{n-\lambda_{p}\left(k\right)}\prod_{\ell=0}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)} \end{equation} Hence, every term on the right hand side of (\ref{eq:M_H partial sum generating identity}) is of the form: \begin{equation} \prod_{\ell=0}^{n-1}c_{j_{\ell}}z^{j_{\ell}p^{\ell}}=\left(\prod_{h=0}^{n-1}c_{j_{h}}\right)z^{\sum_{\ell=0}^{n-1}j_{\ell}p^{\ell}}=c_{0}^{n-\lambda_{p}\left(k\right)}\left(\prod_{\ell=0}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)}\right)z^{k} \end{equation} where: \begin{equation} k=\sum_{\ell=0}^{n-1}j_{\ell}p^{\ell}\leq\sum_{\ell=0}^{n-1}\left(p-1\right)p^{\ell}=p^{n}-1 \end{equation} Summing over $k\in\left\{ 0,\ldots,p^{n}-1\right\} $ then proves the identity in (I). \vphantom{} II. Suppose $c_{0}=1$. Then (I) can be written as: \[ \prod_{m=0}^{n-1}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right)=\sum_{k=0}^{p^{n}-1}\left(\prod_{\ell=0}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)}\right)z^{k} \] Thus, subtracting the $\left(n-2\right)$nd case from the $\left(n-1\right)$st case gives: \begin{align} \sum_{k=p^{n-1}}^{p^{n}-1}\left(\prod_{\ell=0}^{p-1}c_{\ell}^{\#_{p:\ell}\left(k\right)}\right)z^{k} & =\prod_{m=0}^{n-1}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right)-\prod_{m=0}^{n-2}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right)\\ & =\left(\sum_{k=0}^{p-1}c_{k}z^{kp^{n-1}}-1\right)\prod_{m=0}^{n-2}\left(\sum_{j=0}^{p-1}c_{j}z^{jp^{m}}\right) \end{align} Note that the $k=0$ term is $c_{0}z^{0\cdot p^{n-1}}=1\cdot1=1$; this cancels out the $-1$, thereby proving (II). Q.E.D. \vphantom{} Next, we will need some additional formulae, properties, and estimates for $\kappa_{H}$. \begin{prop}[\textbf{Properties of} $\kappa_{H}$] \label{prop:Properties of Kappa_H}\ \vphantom{} I. \index{kappa{H}@$\kappa_{H}$!properties} \begin{equation} \kappa_{H}\left(n\right)=\prod_{k=1}^{p-1}\left(\frac{\mu_{k}}{\mu_{0}}\right)^{\#_{p:k}\left(n\right)},\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Kappa_H explicit formula} \end{equation} \vphantom{} II. $\kappa_{H}$ is the unique function $\mathbb{N}_{0}\rightarrow\mathbb{Q}$ satisfying the functional equations \index{kappa{H}@$\kappa_{H}$!functional equation}: \begin{equation} \kappa_{H}\left(pn+j\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(n\right),\textrm{ }\forall n\geq0,\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Kappa_H functional equations} \end{equation} subject to the initial condition $\kappa_{H}\left(0\right)=1$. \vphantom{} III. $\kappa_{H}$ has $p$-adic structure, with: \begin{equation} \kappa_{H}\left(m+jp^{n}\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(m\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z},\textrm{ }\forall m\in\mathbb{Z}/p^{n}\mathbb{Z}\label{eq:Kappa_H is rho-adically distributed} \end{equation} \vphantom{} IV. If $H$ is semi-basic, then $\kappa_{H}$ is $\left(p,q_{H}\right)$-adically regular, with: \begin{equation} \lim_{n\rightarrow\infty}\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q_{H}}\overset{\mathbb{R}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\label{eq:Semi-basic q-adic decay for Kappa_H} \end{equation} \vphantom{} V. If $H$ is contracting\index{Hydra map!contracting} ($\mu_{0}/p<1$), then: \begin{equation} \lim_{N\rightarrow\infty}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\overset{\mathbb{R}}{=}0,\textrm{ }\forall\mathfrak{z}\in\mathbb{N}_{0}\label{eq:Kappa_H decay when H is contracting} \end{equation} If $H$ is expanding ($\mu_{0}/p>1$), then: \begin{equation} \lim_{N\rightarrow\infty}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\overset{\mathbb{R}}{=}+\infty,\textrm{ }\forall\mathfrak{z}\in\mathbb{N}_{0}\label{eq:Kappa_H growth when H is expanding} \end{equation} \end{prop} Proof: I. Using \textbf{Proposition \ref{prop:Explicit Formulas for M_H} }(page \pageref{prop:Explicit Formulas for M_H}), we have: \begin{equation} \prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}=p^{\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\frac{M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}{p^{-\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}} \end{equation} Multiplying through by: \begin{equation} \left(\frac{\mu_{0}}{p}\right)^{N}\mu_{0}^{-\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)} \end{equation} makes the right-hand side into: \begin{equation} \left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{equation} So: \begin{align} \left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) & =\left(\frac{\mu_{0}}{p}\right)^{N}\mu_{0}^{-\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\\ & =\left(\frac{\mu_{0}}{p}\right)^{N}\mu_{0}^{\#_{p:0}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\lambda_{p}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\prod_{j=1}^{p-1}\mu_{j}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\\ \left(\lambda_{p}\left(k\right)=\sum_{j=0}^{p-1}\#_{p:j}\left(k\right)\right); & =\left(\frac{\mu_{0}}{p}\right)^{N}\mu_{0}^{-\sum_{j=1}^{p-1}\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\prod_{j=1}^{p-1}\mu_{j}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\\ & =\left(\frac{\mu_{0}}{p}\right)^{N}\prod_{j=1}^{p-1}\left(\frac{\mu_{j}}{\mu_{0}}\right)^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)} \end{align} Now, divide by $\left(\mu_{0}/p\right)^{N}$, pick $\mathfrak{z}=n\in\mathbb{N}_{0}$, and let $N\geq\lambda_{p}\left(n\right)$. This yields (I). \vphantom{} II \& III. Let $n\geq1$ and let $j\in\mathbb{Z}/p\mathbb{Z}$. Then: \begin{align} \lambda_{p}\left(pn+j\right) & =\lambda_{p}\left(n\right)+1\\ M_{H}\left(pn+j\right) & =\frac{\mu_{j}}{p}M_{H}\left(n\right) \end{align} Consequently, we have: \begin{align} \kappa_{H}\left(pn+j\right) & =\left(\frac{p}{\mu_{0}}\right)^{\lambda_{p}\left(pn+j\right)}M_{H}\left(pn+j\right)\\ & =\frac{p}{\mu_{0}}\cdot\frac{\mu_{j}}{p}\cdot\left(\frac{p}{\mu_{0}}\right)^{\lambda_{p}\left(n\right)}M_{H}\left(n\right)\\ & =\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(n\right) \end{align} Next: \begin{align} \kappa_{H}\left(j\right) & =\begin{cases} \left(\frac{p}{\mu_{0}}\right)^{\lambda_{p}\left(0\right)}M_{H}\left(0\right) & \textrm{if }j=0\\ \left(\frac{p}{\mu_{0}}\right)^{\lambda_{p}\left(pn+j\right)}M_{H}\left(j\right) & \textrm{if }j\in\left\{ 1,\ldots,p-1\right\} \end{cases}\\ & =\begin{cases} 1 & \textrm{if }j=0\\ \frac{\mu_{j}}{\mu_{0}} & \textrm{if }j\in\left\{ 1,\ldots,p-1\right\} \end{cases} \end{align} So, $\kappa_{H}\left(0\right)=1$. Consequently \begin{equation} \kappa_{H}\left(j\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(0\right),\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z} \end{equation} Combining this with the other cases computed above proves (\ref{eq:Kappa_H functional equations}). As for (III), replace $n$ in (\ref{eq:Kappa_H explicit formula}) with $m+jp^{n}$, where $m\in\left\{ 0,\ldots,p^{n}-1\right\} $. This gives us: \begin{align} \kappa_{H}\left(m+jp^{n}\right) & =\prod_{k=1}^{p-1}\left(\frac{\mu_{k}}{\mu_{0}}\right)^{\#_{p:k}\left(m+jp^{n}\right)}\\ \left(\#_{p:k}\left(m+jp^{n}\right)=\#_{p:k}\left(m\right)+\left[j=k\right]\right); & =\frac{\mu_{j}}{\mu_{0}}\prod_{k=1}^{p-1}\left(\frac{\mu_{k}}{\mu_{0}}\right)^{\#_{p:k}\left(m\right)}\\ & =\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(m\right) \end{align} So: \[ \kappa_{H}\left(pn+j\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(n\right),\textrm{ }\forall n\geq0 \] \[ \kappa_{H}\left(jp^{n}+m\right)=\frac{\mu_{m}}{\mu_{0}}\kappa_{H}\left(n\right),\textrm{ }\forall n\geq0 \] Finally, by \textbf{Proposition \ref{prop:formula for functions with p-adic structure}}, we then have that $\kappa_{H}\left(n\right)$ is uniquely determined by the $\mu_{j}$s and $\kappa_{H}\left(0\right)$. This proves that any function $\mathbb{N}_{0}\rightarrow\mathbb{Q}$ satisfying (\ref{eq:Kappa_H functional equations}) and $\kappa_{H}\left(0\right)=1$ is necessarily equal to $\kappa_{H}$. This shows the desired uniqueness, thereby proving the rest of (II). \vphantom{} IV. Let $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. If $H$ is semi-basic, we have that $\mu_{0}$ and $p$ are both co-prime to $q_{H}$. Taking $q$-adic absolute values of (I) yields: \begin{equation} \left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}=\left\|\prod_{j=1}^{p-1}\left(\frac{\mu_{j}}{\mu_{0}}\right)^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}\right\|_{q}=\prod_{j=1}^{p-1}\left\|\mu_{j}\right\|_{q}^{\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)} \end{equation} With $H$ being semi-basic, $\mu_{j}$ is a multiple of $q_{H}$ for all non-zero $j$. As such, by the pigeonhole principle, since $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, there is at least one $j\in\left\{ 1,\ldots,p-1\right\} $ so that infinitely many of $\mathfrak{z}$'s $p$-adic digits are $j$. This makes $\#_{p:j}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\rightarrow\infty$ in $\mathbb{R}$ as $N\rightarrow\infty$, and demonstrates that (\ref{eq:Semi-basic q-adic decay for Kappa_H}) holds. This proves (IV). \vphantom{} V. Let $\mathfrak{z}\in\mathbb{N}_{0}$. Then, $\left[\mathfrak{z}\right]_{p^{N}}=\mathfrak{z}$ for all $N\geq\lambda_{p}\left(\mathfrak{z}\right)$, and so: \begin{equation} \left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\mathfrak{z}\right),\textrm{ }\forall N\geq\lambda_{p}\left(\mathfrak{z}\right) \end{equation} where $\kappa_{H}\left(\mathfrak{z}\right)\neq0$ is then a non-zero rational constant $c$. Thus, as $N\rightarrow\infty$ the limit in $\mathbb{R}$ of the left-hand side will be $0$ if and only if $H$ is contracting ($\mu_{0}/p<1$) and will be $+\infty$ if and only if $H$ is expanding ($\mu_{0}/p>1$). This proves (V). Q.E.D. \vphantom{} Next, we compute the value of the summatory function\index{chi{H}@$\chi_{H}$!summatory function} $\sum_{n=0}^{N}\chi_{H}\left(n\right)$ via a recursive method. \begin{lem}[\textbf{Summatory function of} $\chi_{H}$] \label{lem:Summatory function of Chi_H}Let $H$ be a $p$-Hydra map (we \emph{do not }require that $\mu_{0}=1$). Then: \begin{equation} \sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)=\begin{cases} \beta_{H}\left(0\right)Np^{N} & \textrm{if }\sigma_{H}=1\\ \frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(p^{\sigma_{H}N}-p^{N}\right) & \textrm{else} \end{cases},\textrm{ }\forall N\geq1\label{eq:rho-N minus 1 sum of Chi_H} \end{equation} \end{lem} Proof: For any $N\geq0$, define: \begin{equation} S_{H}\left(N\right)\overset{\textrm{def}}{=}\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)\label{eq:Definition of S_H of N} \end{equation} Letting $N\geq1$, note that: \begin{align} p^{N}S_{H}\left(N\right) & =\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)\\ & =\sum_{j=0}^{p-1}\sum_{n=0}^{p^{N-1}-1}\chi_{H}\left(pn+j\right)\\ \left(\textrm{use }\chi_{H}\textrm{'s functional eqs.}\right); & =\sum_{j=0}^{p-1}\sum_{n=0}^{p^{N-1}-1}\frac{a_{j}\chi_{H}\left(n\right)+b_{j}}{d_{j}}\\ & =\sum_{j=0}^{p-1}\left(\frac{a_{j}}{d_{j}}\sum_{n=0}^{p^{N-1}-1}\chi_{H}\left(n\right)+\frac{b_{j}}{d_{j}}p^{N-1}\right)\\ & =\left(\sum_{j=0}^{p-1}\frac{a_{j}}{d_{j}}\right)p^{N-1}S_{H}\left(N-1\right)+\left(\sum_{j=0}^{p-1}\frac{b_{j}}{d_{j}}\right)p^{N-1} \end{align} Dividing through by $p^{N}$, note that: \begin{align} \alpha_{H}\left(0\right) & =\frac{1}{p}\sum_{j=0}^{p-1}\frac{a_{j}}{d_{j}}=\frac{1}{p}\left(\frac{1}{p}\sum_{j=0}^{p-1}\mu_{j}\right)=p^{\sigma_{H}-1}\\ \beta_{H}\left(0\right) & =\frac{1}{p}\sum_{j=0}^{p-1}\frac{b_{j}}{d_{j}} \end{align} As such, we can write: \begin{equation} S_{H}\left(N\right)=p^{\sigma_{H}-1}S_{H}\left(N-1\right)+\beta_{H}\left(0\right)\label{eq:Recursive formula for S_H} \end{equation} With this, we see that $S_{H}\left(N\right)$ is the image of $S_{H}\left(0\right)$ under $N$ iterates of the affine linear map: \begin{equation} x\mapsto p^{\sigma_{H}-1}x+\beta_{H}\left(0\right)\label{eq:Affine linear map generating S_H} \end{equation} This leaves us with two cases to investigate: \vphantom{} i. Suppose $\sigma_{H}=1$. Then, (\ref{eq:Affine linear map generating S_H}) reduces to a translation: \begin{equation} x\mapsto x+\beta_{H}\left(0\right) \end{equation} in which case: \begin{equation} S_{H}\left(N\right)=S_{H}\left(0\right)+\beta_{H}\left(0\right)N\label{eq:Explicit formula for S_H of N when sigma_H =00003D00003D 1} \end{equation} \vphantom{} ii. Suppose $\sigma_{H}\neq1$. Then, (\ref{eq:Affine linear map generating S_H}) is not a translation, and so, using the explicit formula for this $N$th iterate, we obtain: \begin{equation} S_{H}\left(N\right)=p^{\left(\sigma_{H}-1\right)N}S_{H}\left(0\right)+\frac{p^{\left(\sigma_{H}-1\right)N}-1}{p^{\sigma_{H}-1}-1}\beta_{H}\left(0\right)\label{eq:Explicit formula for S_H of N} \end{equation} Noting that: \begin{align} S_{H}\left(0\right) & =\sum_{n=0}^{p^{0}-1}\chi_{H}\left(n\right)=\chi_{H}\left(0\right)=0 \end{align} we then have: \begin{equation} \frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)=S_{H}\left(N\right)=\begin{cases} \beta_{H}\left(0\right)N & \textrm{if }\sigma_{H}=1\\ \frac{p^{\left(\sigma_{H}-1\right)N}-1}{p^{\sigma_{H}-1}-1}\beta_{H}\left(0\right) & \textrm{else} \end{cases} \end{equation} Multiplying through on all sides by $p^{N}$ and re-writing $p^{\sigma_{H}-1}-1$ as $\alpha_{H}\left(0\right)-1$ then yields: \begin{equation} \sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)=\begin{cases} \beta_{H}\left(0\right)Np^{N} & \textrm{if }\sigma_{H}=1\\ \frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(p^{\sigma_{H}N}-p^{N}\right) & \textrm{else} \end{cases} \end{equation} which proves (\ref{eq:rho-N minus 1 sum of Chi_H}). Q.E.D. \vphantom{} Although not of any particular use (at least at the time of writing), we can use the formula for the summatory function to obtain simple upper and lower (archimedean) bounds. We do this with help of the following trick: \begin{prop}[\textbf{Replacing $N$ with }$\log_{p}\left(N+1\right)$] Let $\left\{ a_{n}\right\} _{n\geq0}$ be a sequence of non-negative real numbers, and let $p$ be an integer $\geq2$. Suppose there is a non-decreasing function $A:\left[0,\infty\right)\rightarrow\left[0,\infty\right)$ such that: \[ \sum_{n=0}^{p^{N}-1}a_{n}=A\left(N\right),\textrm{ }\forall N\in\mathbb{N}_{1} \] Then: \begin{equation} A\left(\log_{p}N\right)\leq\sum_{n=0}^{N}a_{n}\leq2A\left(\log_{p}\left(N\right)+1\right)-A\left(\log_{p}\left(N\right)-1\right),\textrm{ }\forall N\geq1\label{eq:Replacement Lemma} \end{equation} \end{prop} Proof: As just remarked, for each $k\in\mathbb{N}_{1}$, $\left\{ p^{k-1},p^{k-1}+1,\ldots,p^{k}-1\right\} $ is the set of all positive integers $n$ for which $\lambda_{p}\left(n\right)=k$. As such, letting $N\in\mathbb{N}_{1}$ we have that $p^{\lambda_{p}\left(N\right)-1}$ is the smallest positive integer whose $p$-adic representation has the same number of digits as the $p$-adic representation of $N$. Consequently, we can write: \begin{equation} \sum_{n=0}^{N}a_{n}=\sum_{n=0}^{p^{\lambda_{p}\left(N\right)-1}-1}a_{n}+\sum_{n=p^{\lambda_{p}\left(N\right)-1}}^{N}a_{n}=A\left(\lambda_{p}\left(N\right)\right)+\sum_{n=p^{\lambda_{p}\left(N\right)-1}}^{N}a_{n}\label{eq:Lambda decomposition of Nth partial sum} \end{equation} Since the $a_{n}$s are non-negative, this then yields the lower bound: \begin{equation} \sum_{n=0}^{N}a_{n}\geq A\left(\lambda_{p}\left(N\right)\right) \end{equation} Since: \begin{equation} \lambda_{p}\left(N\right)=1+\left\lfloor \log_{p}N\right\rfloor \geq\log_{p}N \end{equation} and since $A$ is a non-decreasing function, this then implies: \begin{equation} \sum_{n=0}^{N}a_{n}\geq A\left(\log_{p}N\right) \end{equation} As for an upper bound, we note that the non-negativity of the $a_{n}$s allows us to write: \begin{align} \sum_{n=p^{\lambda_{p}\left(N\right)-1}}^{N}a_{n} & \leq\sum_{n=p^{\lambda_{p}\left(N\right)-1}}^{p^{\lambda_{p}\left(N\right)}-1}a_{n}\\ & =\sum_{n=0}^{p^{\lambda_{p}\left(N\right)}-1}a_{n}-\sum_{n=0}^{p^{\lambda_{p}\left(N\right)-1}-1}a_{n}\\ & =A\left(\lambda_{p}\left(N\right)\right)-A\left(\lambda_{p}\left(N\right)-1\right) \end{align} seeing as $p^{\lambda_{p}\left(N\right)}-1$ is the largest positive integer with the same number of $p$-adic digits as $N$. With this, (\ref{eq:Lambda decomposition of Nth partial sum}) becomes: \begin{align} \sum_{n=0}^{N}a_{n} & \leq2A\left(\lambda_{p}\left(N\right)\right)-A\left(\lambda_{p}\left(N\right)-1\right) \end{align} Since $A$ is non-decreasing, and since: \begin{equation} \log_{p}N\leq1+\left\lfloor \log_{p}N\right\rfloor =\lambda_{p}\left(N\right)=1+\left\lfloor \log_{p}N\right\rfloor \leq1+\log_{p}N \end{equation} we then have that: \begin{align} \sum_{n=0}^{N}a_{n} & \leq2A\left(\log_{p}\left(N\right)+1\right)-A\left(\log_{p}\left(N\right)-1\right) \end{align} Q.E.D. \begin{prop}[\textbf{Archimedean Bounds for $\chi_{H}$'s Summatory Function}] \label{prop:archimedean bounds on Chi_H summatory function}Let $H$ be a $p$-Hydra map (we \emph{do not }require that $\mu_{0}=1$). Then\index{chi{H}@$\chi_{H}$!summatory function!estimates}: \vphantom{} I. If $\sigma_{H}=1$: \begin{equation} \beta_{H}\left(0\right)N\log_{p}N\leq\sum_{n=0}^{N}\chi_{H}\left(n\right)\leq C_{H,1}N+\frac{2p^{2}-1}{p}\beta_{H}\left(0\right)N\log_{p}N\label{eq:Chi_H partial sum asympotitcs for when sigma_H is 1} \end{equation} where: \begin{equation} C_{H,1}\overset{\textrm{def}}{=}\frac{2p^{2}+1}{p}\beta_{H}\left(0\right) \end{equation} \vphantom{} II. If $\sigma_{H}>1$: \begin{equation} C_{H,2}N^{\sigma_{H}}-\gamma_{H}\left(0\right)N\leq\sum_{n=0}^{N}\chi_{H}\left(n\right)\leq\frac{2p^{2\sigma_{H}}-1}{p^{\sigma_{H}}}C_{H,2}N^{\sigma_{H}}-\frac{2p^{2}-1}{p}\gamma_{H}\left(0\right)N\label{eq:Chi_H partial sum asympotitcs for when sigma_H is greater than 1} \end{equation} where: \begin{equation} C_{H,2}\overset{\textrm{def}}{=}\gamma_{H}\left(0\right) \end{equation} \begin{equation} \gamma_{H}\left(0\right)\overset{\textrm{def}}{=}\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1} \end{equation} \end{prop} Proof: When $\sigma_{H}\geq1$, the formulae in (\ref{eq:rho-N minus 1 sum of Chi_H}) are non-decreasing functions of $N$. As such, we may use (\ref{eq:Replacement Lemma}) which, upon simplification, yields (\ref{eq:Chi_H partial sum asympotitcs for when sigma_H is 1}) and (\ref{eq:Chi_H partial sum asympotitcs for when sigma_H is greater than 1}). Q.E.D. \vphantom{} Finally, we have the truncations of $\chi_{H}$: \begin{notation} We write $q$ to denote $q_{H}$. We write $\chi_{H,N}:\mathbb{Z}_{p}\rightarrow\mathbb{Q}\subset\mathbb{Q}_{q}$ to denote the $N$th truncation\index{chi{H}@$\chi_{H}$!$N$th truncation} of $\chi_{H}$: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall N\in\mathbb{N}_{0}\label{eq:Definition of Nth truncation of Chi_H} \end{equation} Recall that: \begin{equation} \chi_{H,N}\left(n\right)=\chi_{H}\left(n\right),\textrm{ }\forall n\in\left\{ 0,\ldots,p^{N}-1\right\} \end{equation} In particular: \begin{equation} \chi_{H}\left(n\right)=\chi_{H,\lambda_{p}\left(n\right)}\left(n\right),\textrm{ }\forall n\in\mathbb{N}_{0} \end{equation} Because $\chi_{H,N}$ is a rational-valued function which is a linear combination of finitely many indicator functions for clopen subsets of $\mathbb{Z}_{p}$, $\chi_{H,N}$ is continuous both as a function $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$ and as a function $\mathbb{Z}_{p}\rightarrow\mathbb{C}$. As a result of this, in writing $\hat{\chi}_{H,N}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ to denote the Fourier Transform of $\chi_{H,N}$: \begin{equation} \hat{\chi}_{H,N}\left(t\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\chi_{H,N}\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of the Fourier Coefficients of Chi_H,N} \end{equation} observe that this integral is convergent in both the topology of $\mathbb{C}$ and the topology of $\mathbb{C}_{q}$. In fact, because $\chi_{H,N}$ is locally constant, the integral will reduce to a finite sum (see (\ref{eq:Chi_H,N hat transform formula - sum form})): \begin{equation} \hat{\chi}_{H,N}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)e^{-2\pi int},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} where: \nomenclature{$\mathbf{1}_{0}\left(t\right)$}{ } \begin{equation} \mathbf{1}_{0}\left(p^{N}t\right)=\left[\left\|t\right\|_{p}\leq p^{N}\right],\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} As such, all of the computations and formal manipulations we will perform hold simultaneously in $\mathbb{C}$ and in $\mathbb{C}_{q}$; they both occur in $\overline{\mathbb{Q}}\subset\mathbb{C}\cap\mathbb{C}_{q}$. The difference between the archimedean and non-archimedean topologies will only emerge when we consider what happens as $N$ tends to $\infty$. Additionally, note that because of $\chi_{H,N}$'s local constant-ness and the finitude of its range, we have that, for each $N$, $\hat{\chi}_{H,N}$ has finite support ($\left\|t\right\|_{p}\geq p^{N+1}$): \begin{equation} \hat{\chi}_{H,N}\left(t\right)=0,\textrm{ }\forall\left\|t\right\|_{p}\geq p^{N+1},\textrm{ }\forall N\in\mathbb{N}_{0}\label{eq:Vanishing of Chi_H,N hat for all t with sufficiently large denominators} \end{equation} and thus, that the Fourier series: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{\chi}_{H,N}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Fourier series for Chi_H,N} \end{equation} will be uniformly convergent in both $\mathbb{C}$ and $\mathbb{C}_{q}$ with respect to $\mathfrak{z}\in\mathbb{Z}_{p}$, reducing to a finite sum in all cases. Finally, with regard to algebraic-number-theoretic issues of picking embeddings of $\overline{\mathbb{Q}}$ in $\mathbb{C}$ and $\mathbb{C}_{q}$, there is no need to worry. This is because all of our work hinges on the Fourier series identity: \begin{equation} \left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\overset{\overline{\mathbb{Q}}}{=}\frac{1}{p^{N}}\sum_{\left\|t\right\|_{p}\leq p^{N}}e^{2\pi i\left\{ t\left(\mathfrak{z}-n\right)\right\} _{p}} \end{equation} which\textemdash \emph{crucially}\textemdash is invariant under the action of elements of the Galois group $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$, seeing as the sum evaluates to a rational number (the indicator function, which is either $0$ or $1$). As such, all finite operations (sums, multiplication, series rearrangements) involving this identity are \emph{independent }of our choices of embeddings of $\overline{\mathbb{Q}}$ in $\mathbb{C}$ and $\mathbb{C}_{q}$. \end{notation} \vphantom{} Lastly, we need to make is the following lemma regarding the interaction between truncation and functional equations. This result will play a critical role in actually \emph{proving }that the functions $\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ which we will derive are, in fact, Fourier transforms of $\chi_{H}$. \begin{lem} \label{lem:Functional equations and truncation}Let $\chi:\mathbb{N}_{0}\rightarrow\mathbb{Q}$, and suppose that for $j\in\left\{ 0,\ldots,p-1\right\} $ there are functions $\Phi_{j}:\mathbb{N}_{0}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that: \begin{equation} \chi\left(pn+j\right)=\Phi_{j}\left(n,\chi\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Relation between truncations and functional equations - Hypothesis} \end{equation} Then, the $N$th truncations $\chi_{N}$ satisfy the functional equations\index{functional equation}: \begin{equation} \chi_{N}\left(pn+j\right)=\Phi_{j}\left(\left[n\right]_{p^{N-1}},\chi_{N-1}\left(n\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Relation between truncations and functional equations, version 1} \end{equation} Equivalently: \begin{equation} \chi\left(\left[pn+j\right]_{p^{N}}\right)=\Phi_{j}\left(\left[n\right]_{p^{N-1}},\chi\left(\left[n\right]_{p^{N-1}}\right)\right),\textrm{ }\forall n\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Relation between truncations and functional equations, version 2} \end{equation} \end{lem} Proof: Fix $N\geq0$, $n\in\mathbb{N}_{0}$, and $j\in\mathbb{Z}/p\mathbb{Z}$. Then: \begin{align} \chi\left(\left[pn+j\right]_{p^{N}}\right) & =\chi_{N}\left(pn+j\right)\\ & =\sum_{m=0}^{p^{N}-1}\chi\left(m\right)\left[pn+j\overset{p^{N}}{\equiv}m\right]\\ \left(\textrm{split }m\textrm{ mod }p\right); & =\sum_{\ell=0}^{p^{N-1}-1}\sum_{k=0}^{p-1}\chi\left(p\ell+k\right)\left[pn+j\overset{p^{N}}{\equiv}p\ell+k\right]\\ & =\sum_{\ell=0}^{p^{N-1}-1}\sum_{k=0}^{p-1}\Phi_{k}\left(\ell,\chi\left(\ell\right)\right)\underbrace{\left[n\overset{p^{N-1}}{\equiv}\ell+\frac{k-j}{p}\right]}_{0\textrm{ }\forall n\textrm{ if }k\neq j}\\ & =\sum_{\ell=0}^{p^{N-1}-1}\Phi_{j}\left(\ell,\chi\left(\ell\right)\right)\left[n\overset{p^{N-1}}{\equiv}\ell\right]\\ & =\Phi_{j}\left(\left[n\right]_{p^{N-1}},\chi\left(\left[n\right]_{p^{N-1}}\right)\right) \end{align} Q.E.D. \begin{rem} As an aside, it is worth mentioning that $\sigma_{H}$ is implicated in the abscissa of convergence of the Dirichlet series\index{Dirichlet series}\index{chi{H}@$\chi_{H}$!Dirichlet series}: \begin{equation} \zeta_{M_{H}}\left(s\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}\frac{M_{H}\left(n\right)}{\left(n+1\right)^{s}} \end{equation} and: \begin{equation} \zeta_{\chi_{H}}\left(s\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}\frac{\chi_{H}\left(n\right)}{\left(n+1\right)^{s}} \end{equation} For instance, using \textbf{Abel's Summation Formula }in the standard way yields: \begin{equation} \sum_{n=0}^{p^{N}-1}\frac{\chi_{H}\left(n\right)}{\left(1+n\right)^{s}}=\frac{1}{p^{Ns}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)+s\int_{0}^{p^{N}-1}\frac{\sum_{n=0}^{\left\lfloor x\right\rfloor }\chi_{H}\left(n\right)}{\left(1+x\right)^{s+1}}dx \end{equation} So, by \textbf{Proposition \ref{lem:Summatory function of Chi_H}}, when $\sigma_{H}=1$, we obtain: \[ \sum_{n=0}^{p^{N}-1}\frac{\chi_{H}\left(n\right)}{\left(1+n\right)^{s}}=\frac{N\beta_{H}\left(0\right)}{p^{N\left(s-1\right)}}+s\int_{0}^{p^{N}-1}\frac{\sum_{n=0}^{\left\lfloor x\right\rfloor }\chi_{H}\left(n\right)}{\left(1+x\right)^{s+1}}dx \] whereas for $\sigma_{H}\neq1$, we obtain: \[ \sum_{n=0}^{p^{N}-1}\frac{\chi_{H}\left(n\right)}{\left(1+n\right)^{s}}=\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(\frac{1}{p^{N\left(s-\sigma_{H}\right)}}-\frac{1}{p^{N\left(s-1\right)}}\right)+s\int_{0}^{p^{N}-1}\frac{\sum_{n=0}^{\left\lfloor x\right\rfloor }\chi_{H}\left(n\right)}{\left(1+x\right)^{s+1}}dx \] which then gives: \begin{equation} \zeta_{\chi_{H}}\left(s\right)=s\int_{0}^{\infty}\frac{\sum_{n=0}^{\left\lfloor x\right\rfloor }\chi_{H}\left(n\right)}{\left(1+x\right)^{s+1}}dx,\textrm{ }\forall\textrm{Re}\left(s\right)>\max\left\{ 1,\sigma_{H}\right\} \end{equation} The (archimedean) bounds on $\chi_{H}$'s summatory function given in \textbf{Proposition \ref{prop:archimedean bounds on Chi_H summatory function} }then show that $\zeta_{\chi_{H}}\left(s\right)$ converges absolutely for all $\textrm{Re}\left(s\right)>\sigma_{H}$ for all $H$ with $\sigma_{H}\geq1$. As mentioned at the end of Chapter 2, this Dirichlet series can be analytically continued to meromorphic function on $\mathbb{C}$ with a half-lattice of poles in the half-plane $\textrm{Re}\left(s\right)\leq\sigma_{H}$, however, its growth rate as $\textrm{Re}\left(s\right)\rightarrow-\infty$ is hyper-exponential, making it seemingly ill-suited for analysis via techniques of contour integration. \end{rem} \newpage{} \section{\label{sec:4.2 Fourier-Transforms-=00003D000026}Fourier Transforms and Quasi-Integrability} THROUGHOUT THIS SECTION, WE ALSO ASSUME $H$ IS NON-SINGULAR. \vphantom{} This section is the core of our analysis of $\chi_{H}$. Understandably, it is also the most computationally intensive portion of this dissertation\textemdash at least until Chapter 6. Our work will be broken up into three parts. In Subsection \ref{subsec:4.2.1}, we use the functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}) to establish a recursion relation between the Fourier transforms of the $N$th truncations of $\chi_{H}$. Solving this yields an explicit formula for $\hat{\chi}_{H,N}\left(t\right)$ (\ref{eq:Explicit formula for Chi_H,N hat}) which is a sum of products of $\beta_{H}$ and a partial product involving $\alpha_{H}$. The partial products involving $\alpha_{H}$ depend on both $N$ and $t$, and by carefully manipulating the product, we can extract from the product the part that depends on $N$. Doing this yields a function $\hat{A}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ (\vref{eq:Definition of A_H hat}) whose properties we then study in detail, establishing various formulae along the way. Subsection \ref{subsec:4.2.1} concludes with greatly simplified expressions for $\hat{\chi}_{H,N}\left(t\right)$ given by\textbf{ Theorem} \textbf{\ref{thm:(N,t) asymptotic decomposition of Chi_H,N hat}}. These can be considered explicit ``asymptotic formulae'' for $\hat{\chi}_{H,N}\left(t\right)$ in terms of $t$ and $N$ as $N\rightarrow\infty$. The asymptotic formulae of \textbf{Theorem} \textbf{\ref{thm:(N,t) asymptotic decomposition of Chi_H,N hat}} demonstrate the sensitive dependence of $\hat{\chi}_{H,N}\left(t\right)$ on $p$ and, most of all, on $\alpha_{H}\left(0\right)$. The dependence on $p$ is a consequence of the fact that the multiplicative group $\left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}$ of multiplicatively invertible integers modulo $p$ satisfies $\left\|\left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}\right\|=1$ if and only if $p=2$. Much more significant, however, is the dependence on $\alpha_{H}\left(0\right)$. The essence of this dependence was already on display in the formula for $\chi_{H}$'s summatory function as given in \textbf{Lemma \ref{lem:Summatory function of Chi_H}}, where we saw two distinct behaviors, one for $\sigma_{H}=1$ (equivalently, for $\alpha_{H}\left(0\right)=1$) and one for all other values. This dependency will appear again in our analysis of $\hat{A}_{H}$, where it yields two cases: $\alpha_{H}\left(0\right)=1$ and $\alpha_{H}\left(0\right)\neq1$. In Subsection \ref{subsec:4.2.2}, we first show that $\chi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame in the $\alpha_{H}\left(0\right)=1$ case. We will then exploit the functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}) to extend this result to arbitrary values of $\alpha_{H}\left(0\right)$. As a consequence, we will be able to provide Fourier transforms for $\chi_{H}$\textemdash thereby proving $\chi_{H}$'s quasi-integrability\textemdash along with interesting non-trivial explicit series expressions for $\chi_{H}$ itself. \subsection{\label{subsec:4.2.1}$\hat{\chi}_{H,N}$ and $\hat{A}_{H}$} We begin by computing a recursive formula\index{hat{chi}{H,N}@$\hat{\chi}_{H,N}$!recursive formula} for $\hat{\chi}_{H,N}$ in terms of $\hat{\chi}_{H,N-1}$. Solving this produces an explicit formula for $\hat{\chi}_{H,N}$ in terms of $t$ and $N$. \begin{prop}[\textbf{Formulae for }$\hat{\chi}_{H,N}$] \label{prop:Computation of Chi_H,N hat}\ \vphantom{} I. \begin{equation} \hat{\chi}_{H,N}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)e^{-2\pi int}\label{eq:Chi_H,N hat transform formula - sum form} \end{equation} \vphantom{} II. \begin{equation} \hat{\chi}_{H,N}\left(t\right)=\mathbf{1}_{0}\left(p^{N}t\right)\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)+\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right),\textrm{ }\forall N\geq1,\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Chi_H,N hat functional equation} \end{equation} the nesting of which yields: \begin{equation} \hat{\chi}_{H,N}\left(t\right)=\sum_{n=0}^{N-1}\mathbf{1}_{0}\left(p^{n+1}t\right)\beta_{H}\left(p^{n}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)\label{eq:Explicit formula for Chi_H,N hat} \end{equation} where the $m$-product is defined to be $1$ when $n=0$. In particular, note that $\hat{\chi}_{H,N}\left(t\right)=0$ for all $t\in\hat{\mathbb{Z}}_{p}$ with $\left\|t\right\|_{p}>p^{N}$. \end{prop} Proof: We use the Fourier series for $\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]$ to re-write $\chi_{H,N}$ as a Fourier series: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right) & =\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]\\ & =\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)\frac{1}{p^{N}}\sum_{\left\|t\right\|_{p}\leq p^{N}}e^{2\pi i\left\{ t\left(\mathfrak{z}-n\right)\right\} _{p}}\\ & =\sum_{\left\|t\right\|_{p}\leq p^{N}}\left(\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)e^{-2\pi int}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} which is the Fourier series representation of $\chi_{H,N}$, with $\hat{\chi}_{H,N}\left(t\right)$ being the coefficient of $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ in the series. As such: \[ \hat{\chi}_{H,N}\left(t\right)=\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{n=0}^{p^{N}-1}\chi_{H}\left(n\right)e^{-2\pi int} \] which proves (I). Next, we split the $n$-sum modulo $p$ and utilize $\chi_{H}$'s functional equation (here $\left\|t\right\|_{p}\leq p^{N}$): \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{j=0}^{p-1}\sum_{n=0}^{p^{N-1}-1}\chi_{H}\left(pn+j\right)e^{-2\pi i\left(pn+j\right)t}\\ & =\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{j=0}^{p-1}\sum_{n=0}^{p^{N-1}-1}\frac{a_{j}\chi_{H}\left(n\right)+b_{j}}{d_{j}}e^{-2\pi i\left(pn+j\right)t}\\ & =\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N}}\sum_{n=0}^{p^{N-1}-1}\left(\sum_{j=0}^{p-1}\frac{a_{j}\chi_{H}\left(n\right)+b_{j}}{d_{j}}e^{-2\pi ijt}\right)e^{-2\pi in\left(pt\right)} \end{align} The right-hand side can be re-written like so: \[ \frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}\left(\underbrace{\frac{1}{p}\sum_{j=0}^{p-1}\frac{a_{j}}{d_{j}}e^{-2\pi ijt}}_{\alpha_{H}\left(t\right)}\chi_{H}\left(n\right)+\underbrace{\frac{1}{p}\sum_{j=0}^{p-1}\frac{b_{j}}{d_{j}}e^{-2\pi ijt}}_{\beta_{H}\left(t\right)}\right)e^{-2\pi in\left(pt\right)} \] which leaves us with: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\frac{\mathbf{1}_{0}\left(p^{N}t\right)}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}\left(\alpha_{H}\left(t\right)\chi_{H}\left(n\right)+\beta_{H}\left(t\right)\right)e^{-2\pi in\left(pt\right)}\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\left(\underbrace{\frac{\alpha_{H}\left(t\right)}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}\chi_{H}\left(n\right)e^{-2\pi in\left(pt\right)}}_{\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)}+\frac{\beta_{H}\left(t\right)}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}e^{-2\pi in\left(pt\right)}\right) \end{align} and so: \begin{equation} \hat{\chi}_{H,N}\left(t\right)=\mathbf{1}_{0}\left(p^{N}t\right)\left(\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)+\frac{\beta_{H}\left(t\right)}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}e^{-2\pi in\left(pt\right)}\right)\label{eq:Chi_H N hat, almost ready to nest} \end{equation} Simplifying the remaining $n$-sum, we find that: \begin{align} \frac{1}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}e^{-2\pi in\left(pt\right)} & =\begin{cases} \frac{1}{p^{N-1}}\sum_{n=0}^{p^{N-1}-1}1 & \textrm{if }\left\|t\right\|_{p}\leq p\\ \frac{1}{p^{N-1}}\frac{\left(e^{-2\pi i\left(pt\right)}\right)^{p^{N-1}}-1}{e^{-2\pi i\left(pt\right)}-1} & \textrm{if }\left\|t\right\|_{p}\geq p^{2} \end{cases}\\ & =\begin{cases} 1 & \textrm{if }\left\|t\right\|_{p}\leq p\\ 0 & \textrm{if }\left\|t\right\|_{p}\geq p^{2} \end{cases}\\ & =\mathbf{1}_{0}\left(pt\right) \end{align} Hence: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\mathbf{1}_{0}\left(p^{N}t\right)\left(\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)+\beta_{H}\left(t\right)\mathbf{1}_{0}\left(pt\right)\right)\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)+\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right) \end{align} This proves (II). With (II) proven, we can then nest (\ref{eq:Chi_H,N hat functional equation}) to derive an explicit formula for $\hat{\chi}_{H,N}\left(t\right)$: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\mathbf{1}_{0}\left(p^{N}t\right)\alpha_{H}\left(t\right)\hat{\chi}_{H,N-1}\left(pt\right)+\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right)\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\alpha_{H}\left(t\right)\left(\mathbf{1}_{0}\left(p^{N-1}pt\right)\alpha_{H}\left(pt\right)\hat{\chi}_{H,N-2}\left(p^{2}t\right)+\mathbf{1}_{0}\left(p^{2}t\right)\beta_{H}\left(pt\right)\right)+\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right)\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\alpha_{H}\left(t\right)\alpha_{H}\left(pt\right)\hat{\chi}_{H,N-2}\left(p^{2}t\right)+\mathbf{1}_{0}\left(p^{2}t\right)\alpha_{H}\left(t\right)\beta_{H}\left(pt\right)+\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right)\\ & \vdots\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\left(\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}t\right)\right)\hat{\chi}_{H,1}\left(p^{N-1}t\right)+\sum_{n=0}^{N-2}\beta_{H}\left(p^{n}t\right)\mathbf{1}_{0}\left(p^{n+1}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) \end{align} where the $m$-product is $1$ whenever $n=0$. Finally: \begin{align} \hat{\chi}_{H,1}\left(t\right) & =\frac{\mathbf{1}_{0}\left(pt\right)}{p}\sum_{j=0}^{p-1}\chi_{H}\left(j\right)e^{-2\pi ijt} \end{align} Since $\chi_{H}\left(0\right)=0$, $\chi_{H}$'s functional equation gives: \[ \chi_{H}\left(j\right)=\chi_{H}\left(p\cdot0+j\right)=\frac{a_{j}\chi_{H}\left(0\right)+b_{j}}{d_{j}}=\frac{b_{j}}{d_{j}} \] So: \begin{align} \hat{\chi}_{H,1}\left(t\right) & =\frac{\mathbf{1}_{0}\left(pt\right)}{p}\sum_{j=0}^{p-1}\chi_{H}\left(j\right)e^{-2\pi ijt}=\frac{\mathbf{1}_{0}\left(pt\right)}{p}\sum_{j=0}^{p-1}\frac{b_{j}}{d_{j}}e^{-2\pi ijt}=\mathbf{1}_{0}\left(pt\right)\beta_{H}\left(t\right) \end{align} Consequently: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\mathbf{1}_{0}\left(p^{N}t\right)\left(\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}t\right)\right)\hat{\chi}_{H,1}\left(p^{N-1}t\right)+\sum_{n=0}^{N-2}\mathbf{1}_{0}\left(p^{n+1}t\right)\beta_{H}\left(p^{n}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)\\ & =\mathbf{1}_{0}\left(p^{N}t\right)\beta_{H}\left(p^{N-1}t\right)\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}t\right)+\sum_{n=0}^{N-2}\mathbf{1}_{0}\left(p^{n+1}t\right)\beta_{H}\left(p^{n}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)\\ & =\sum_{n=0}^{N-1}\mathbf{1}_{0}\left(p^{n+1}t\right)\beta_{H}\left(p^{n}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) \end{align} This proves (\ref{eq:Explicit formula for Chi_H,N hat}). Q.E.D. \vphantom{} Because $\chi_{H}$ is rising-continuous, rather than continuous, for any fixed $t\in\hat{\mathbb{Z}}_{p}$, the limit of $\hat{\chi}_{H,N}\left(t\right)$ as $N\rightarrow\infty$ need not converge in $\mathbb{C}_{q}$. To overcome this, we perform a kind of asymptotic analysis, excising as much as possible the dependence of (\ref{eq:Explicit formula for Chi_H,N hat}) on $N$. Instead, we \emph{reverse} the dependence of $t$ and $n$ or $N$ in (\ref{eq:Explicit formula for Chi_H,N hat}). That this can be done hinges on the observation that, for any fixed $t\in\hat{\mathbb{Z}}_{p}$, the congruence $p^{m}t\overset{1}{\equiv}0$ will be satisfied for all sufficiently large $m$\textemdash namely, all $m\geq-v_{p}\left(t\right)$. So, as $n\rightarrow\infty$, the $m$th term of the product: \begin{equation} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) \end{equation} will be $\alpha_{H}\left(0\right)$ for all $m\geq-v_{p}\left(t\right)+1$. This suggests that we can untangle the dependency of this product on $n$ by breaking it up into the partial product from $m=0$ to $m=-v_{p}\left(t\right)-1$, and a remainder product which will be independent of $t$. This partial product is precisely $\hat{A}_{H}\left(t\right)$. \begin{defn}[$\hat{A}_{H}\left(t\right)$] We\index{hat{A}{H}left(tright)@$\hat{A}_{H}\left(t\right)$} define the function \nomenclature{$\hat{A}_{H}\left(t\right)$}{ }$\hat{A}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \hat{A}_{H}\left(t\right)\overset{\textrm{def}}{=}\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Definition of A_H hat} \end{equation} where the $m$-product is defined to be $1$ whenever $t=0$; that is, $\hat{A}_{H}\left(0\right)\overset{\textrm{def}}{=}1$. \end{defn} \vphantom{} Our next proposition actually does the work of untangling $\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)$. \begin{prop}[\textbf{$\alpha_{H}$-product in terms of $\hat{A}_{H}$}] \label{prop:alpha product in terms of A_H hat}Fix $t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} $. Then: \begin{align} \mathbf{1}_{0}\left(p^{n+1}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & =\begin{cases} 0 & \textrm{if }n\leq-v_{p}\left(t\right)-2\\ \frac{\hat{A}_{H}\left(t\right)}{\alpha_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)} & \textrm{if }n=-v_{p}\left(t\right)-1\\ \left(\alpha_{H}\left(0\right)\right)^{n+v_{p}\left(t\right)}\hat{A}_{H}\left(t\right) & \textrm{if }n\geq-v_{p}\left(t\right) \end{cases}\label{eq:alpha product in terms of A_H hat} \end{align} \end{prop} Proof: Fix $t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} $. Then, we can write: \begin{align} \mathbf{1}_{0}\left(p^{n+1}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & =\begin{cases} 0 & \textrm{if }\left\|t\right\|_{p}\geq p^{n+2}\\ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & \textrm{if }\left\|t\right\|_{p}\leq p^{n+1} \end{cases}\\ & =\begin{cases} 0 & \textrm{if }n\leq-v_{p}\left(t\right)-2\\ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n\geq-v_{p}\left(t\right)-1 \end{cases} \end{align} As remarked above $m\geq-v_{p}\left(t\right)$ makes the congruence $p^{m}t\overset{1}{\equiv}0$ hold true, guaranteeing that $\alpha_{H}\left(p^{m}t\right)=\alpha_{H}\left(0\right)$ for all $m\geq-v_{p}\left(t\right)$. So, for fixed $t$, we prepare our decomposition like so: \[ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)=\begin{cases} \prod_{m=0}^{-v_{p}\left(t\right)-2}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n=-v_{p}\left(t\right)-1\\ \prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n=-v_{p}\left(t\right)\\ \prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right)\times\prod_{k=-v_{p}\left(t\right)}^{n-1}\alpha_{H}\left(p^{k}t\right) & \textrm{if }n\geq-v_{p}\left(t\right)+1 \end{cases} \] Since $\alpha_{H}\left(p^{k}t\right)=\alpha_{H}\left(0\right)$ for all terms of the $k$-product on the bottom line, we can write: \begin{align} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & =\begin{cases} \prod_{m=0}^{-v_{p}\left(t\right)-2}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n=-v_{p}\left(t\right)-1\\ \left(\alpha_{H}\left(0\right)\right)^{n+v_{p}\left(t\right)}\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n\geq-v_{p}\left(t\right) \end{cases}\\ \left(\times\&\div\textrm{ by }\alpha_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)\right); & =\begin{cases} \frac{\hat{A}_{H}\left(t\right)}{\alpha_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)} & \textrm{if }n=-v_{p}\left(t\right)-1\\ \left(\alpha_{H}\left(0\right)\right)^{n+v_{p}\left(t\right)}\hat{A}_{H}\left(t\right) & \textrm{if }n\geq-v_{p}\left(t\right) \end{cases} \end{align} which gives us the desired formula. Q.E.D. \vphantom{} Now, we learn how to express the $\alpha_{H}$ product as a series. \begin{prop}[$\alpha_{H}$\textbf{ series formula}] \label{prop:alpha product series expansion} \end{prop} \begin{equation} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)=\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{-2\pi imt},\textrm{ }\forall n\geq0,\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:alpha_H product expansion} \end{equation} Proof: We start by writing: \[ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right)=\prod_{m=0}^{n-1}\left(\sum_{j=0}^{p-1}\frac{\mu_{j}}{p^{2}}e^{-2\pi ij\left(p^{m}t\right)}\right) \] and then apply (\ref{eq:M_H partial sum generating identity}) from \textbf{Proposition \ref{prop:Generating function identities}} with $c_{j}=\mu_{j}/p^{2}$ and $z=e^{-2\pi it}$: \begin{align} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & =\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\left(\frac{\mu_{0}}{p^{2}}\right)^{-\lambda_{p}\left(m\right)}\left(\prod_{j=0}^{p-1}\left(\frac{\mu_{j}}{p^{2}}\right)^{\#_{p:j}\left(m\right)}\right)e^{-2\pi imt}\\ \left(\sum_{j=0}^{p-1}\#_{p:j}\left(m\right)=\lambda_{p}\left(m\right)\right); & =\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\left(\frac{\mu_{0}}{p^{2}}\right)^{-\lambda_{p}\left(m\right)}\left(\frac{\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(m\right)}}{p^{2\lambda_{p}\left(m\right)}}\right)e^{-2\pi imt}\\ \left(M_{H}\left(m\right)=\frac{\prod_{j=0}^{p-1}\mu_{j}^{\#_{p:j}\left(m\right)}}{p^{\lambda_{p}\left(m\right)}}\right); & =\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\left(\frac{\mu_{0}}{p^{2}}\right)^{-\lambda_{p}\left(m\right)}\frac{M_{H}\left(m\right)}{p^{\lambda_{p}\left(m\right)}}e^{-2\pi imt}\\ & =\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\left(\frac{p}{\mu_{0}}\right)^{\lambda_{p}\left(m\right)}M_{H}\left(m\right)e^{-2\pi imt}\\ & =\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{-2\pi imt} \end{align} Q.E.D. \vphantom{} This formula reveals $\hat{A}_{H}$ as being a radially-magnitudinal function. A simple estimate shows that that $\hat{A}_{H}$ is the Fourier-Stieltjes transform of a $\left(p,q\right)$-adic measure. \begin{prop} $\hat{A}_{H}\left(t\right)$ is the Fourier-Stieltjes transform of a $\left(p,q\right)$-adic measure. \end{prop} Proof: Let $H$ be semi-basic. Then, $p$ and the $d_{j}$s are co-prime to both $q$ and the $a_{j}$s. So, we get: \[ \left\|\alpha_{H}\left(t\right)\right\|_{q}=\left\|\sum_{j=0}^{p-1}\frac{\mu_{j}}{p^{2}}e^{-2\pi ijt}\right\|_{q}=\left\|\sum_{j=0}^{p-1}\frac{a_{j}}{pd_{j}}e^{-2\pi ijt}\right\|_{q}\leq\max_{0\leq j\leq p-1}\left\|\frac{a_{j}}{pd_{j}}\right\|_{q}\leq1 \] Hence: \begin{align} \sup_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{A}_{H}\left(t\right)\right\|_{q} & \leq\max\left\{ 1,\sup_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\prod_{n=0}^{-v_{p}\left(t\right)-1}\left\|\alpha_{H}\left(p^{n}t\right)\right\|_{q}\right\} \\ & \leq\max\left\{ 1,\sup_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\prod_{n=0}^{-v_{p}\left(t\right)-1}1\right\} \\ & =1 \end{align} This shows $\hat{A}_{H}$ is then in $B\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. As such, the map: \begin{equation} f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)\mapsto\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{A}_{H}\left(t\right)\in\mathbb{C}_{q} \end{equation} is a continuous linear functional on $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$\textemdash that is, a $\left(p,q\right)$-adic measure. Q.E.D. \vphantom{} Here, we introduce the notation for the partial sums of the Fourier series generated by $\hat{A}_{H}$. \begin{defn} We\index{$dA_{H}$} write \nomenclature{$dA_{H}$}{ }$dA_{H}$ to denote the $\left(p,q\right)$-adic measure whose Fourier-Stieltjes transform is $\hat{A}_{H}\left(t\right)$. We then have \nomenclature{$\tilde{A}_{H,N}\left(\mathfrak{z}\right)$}{$N$th partial Fourier series generated by $\hat{A}_{H}\left(t\right)$}: \begin{equation} \tilde{A}_{H,N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Definition of A_H,N twiddle} \end{equation} \end{defn} \vphantom{} The theorem given below summarizes the main properties of $dA_{H}$: \begin{thm}[\textbf{Properties of $dA_{H}$}] \label{thm:Properties of dA_H}\index{$dA_{H}$!properties of} \ \vphantom{} I. ($dA_{H}$ is radially-magnitudinal and $\left(p,q\right)$-adically regular) \begin{equation} \hat{A}_{H}\left(t\right)=\begin{cases} 1 & \textrm{if }t=0\\ \frac{1}{\left\|t\right\|_{p}^{2-\log_{p}\mu_{0}}}\sum_{m=0}^{\left\|t\right\|_{p}-1}\kappa_{H}\left(m\right)e^{-2\pi imt} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:A_H hat as the product of radially symmetric and magnitude-dependent measures} \end{equation} \vphantom{} II. (Formula for $\tilde{A}_{H,N}\left(\mathfrak{z}\right)$) For any $N\in\mathbb{N}_{0}$ and $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{equation} \tilde{A}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\left(1-\alpha_{H}\left(0\right)\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Convolution of dA_H and D_N} \end{equation} \vphantom{} III. ($q$-adic convergence of $\tilde{A}_{H,N}\left(\mathfrak{z}\right)$) As $N\rightarrow\infty$, \emph{(\ref{eq:Convolution of dA_H and D_N})} converges in $\mathbb{C}_{q}$ (in fact, $\mathbb{Z}_{q}$) for all $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, with: \begin{equation} \lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}\left(1-\alpha_{H}\left(0\right)\right)\sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\label{eq:Derivative of dA_H on Z_rho prime} \end{equation} This convergence is point-wise. \vphantom{} IV. (Convergence of $\tilde{A}_{H,N}\left(\mathfrak{z}\right)$ in $\mathbb{C}$) As $N\rightarrow\infty$, \emph{(\ref{eq:Convolution of dA_H and D_N})} either converges in $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$, which occurs if and only if $H$ is contracting ($\mu_{0}/p<1$); or diverges in $\mathbb{C}$ for all $\mathfrak{z}\in\mathbb{N}_{0}$, which occurs if and only if $H$ is expanding ($\mu_{0}/p>1$). The convergence/divergence in both cases is point-wise. Moreover, if convergence occurs, the limit in $\mathbb{C}$ is given by \begin{equation} \lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(m\right)\overset{\mathbb{C}}{=}\left(1-\alpha_{H}\left(0\right)\right)\sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right),\textrm{ }\forall m\in\mathbb{N}_{0}\label{eq:derivative of dA_H on N_0} \end{equation} \vphantom{} V. (Characterization of degeneracy)\index{$dA_{H}$!degeneracy} If $H$ is contracting, then $dA_{H}$ is degenerate if and only if $\alpha_{H}\left(0\right)=1$. \end{thm} Proof: I. By \textbf{Proposition \ref{prop:alpha product series expansion}}, we have that: \begin{equation} \hat{A}_{H}\left(t\right)=\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right)=\left(\frac{\mu_{0}}{p^{2}}\right)^{-v_{p}\left(t\right)}\sum_{m=0}^{p^{-v_{p}\left(t\right)}-1}\kappa_{H}\left(m\right)e^{-2\pi imt} \end{equation} and hence: \begin{equation} \hat{A}_{H}\left(t\right)=\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right)=\frac{1}{\left\|t\right\|_{p}^{2-\log_{p}\mu_{0}}}\sum_{m=0}^{\left\|t\right\|_{p}-1}\kappa_{H}\left(m\right)e^{-2\pi imt} \end{equation} \vphantom{} II. Since: \[ \hat{A}_{H}\left(t\right)=\prod_{n=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{n}t\right)=\left(\frac{\mu_{0}}{p^{2}}\right)^{-v_{p}\left(t\right)}\sum_{m=0}^{\left\|t\right\|_{p}-1}\kappa_{H}\left(m\right)e^{-2\pi imt} \] we can use the Radial-Magnitudinal Fourier Resummation Lemma (\textbf{Lemma \ref{lem:Radially-Mag Fourier Resummation Lemma}}) to write: \begin{align} \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\overline{\mathbb{Q}}}{=}\hat{A}_{H}\left(0\right)+\sum_{n=1}^{N}\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\left(p^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-p^{n-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\right)\\ & -\sum_{j=1}^{p-1}\sum_{n=1}^{N}p^{n-1}\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}+jp^{n-1}\right) \end{align} By the $\left(p,q\right)$-adic regularity of $\kappa_{H}$, we then have: \[ \sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}1+\sum_{n=1}^{N}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\underbrace{\left(\frac{1}{p}\sum_{j=0}^{p-1}\frac{\mu_{j}}{p}\right)}_{\alpha_{H}\left(0\right)}\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \] and hence (since $\kappa_{H}\left(0\right)=1$): \begin{equation} \tilde{A}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\left(1-\alpha_{H}\left(0\right)\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{equation} as desired. \vphantom{} III. Fix $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. Then, since $H$ is semi-basic, we have: \begin{equation} \left\|\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}=\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q} \end{equation} So, by \textbf{Proposition \ref{prop:Properties of Kappa_H}}, $\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ converges $q$-adically to $0$ as $n\rightarrow\infty$.The ultrametric structure of $\mathbb{C}_{q}$ then guarantees the convergence of (\ref{eq:Derivative of dA_H on Z_rho prime}). \vphantom{} IV. Let $\mathfrak{z}\in\mathbb{N}_{0}$. Then, $\kappa_{H}\left(\mathfrak{z}\right)=\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n+1}}\right)\in\mathbb{Q}\backslash\left\{ 0\right\} $ for all $n\geq\lambda_{p}\left(\mathfrak{z}\right)$. So, let $c$ denote the value of $M_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ for all such $n$. Examining the tail of the $n$-series in (\ref{eq:Convolution of dA_H and D_N}), $N\geq\lambda_{p}\left(\mathfrak{z}\right)+1$ implies: \begin{align} \sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) & \overset{\mathbb{R}}{=}\sum_{n=0}^{\lambda_{p}\left(\mathfrak{z}\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)+\sum_{n=\lambda_{p}\left(\mathfrak{z}\right)}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & \overset{\mathbb{R}}{=}O\left(1\right)+\kappa_{H}\left(\mathfrak{z}\right)\sum_{n=\lambda_{p}\left(\mathfrak{z}\right)}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n} \end{align} The series on the bottom line is a \emph{geometric} series, and hence, is convergent in $\mathbb{C}$ if and only if $\mu_{0}/p<1$. Likewise, by \textbf{Proposition \ref{prop:Properties of Kappa_H}},\textbf{ }$\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ tends to $0$ as $N\rightarrow\infty$ if and only if $\mu_{0}/p<1$, and tends to $\infty$ if and only if $\mu_{0}/p>1$. \vphantom{} V. Letting $H$ be semi-basic and contracting, (\ref{eq:Derivative of dA_H on Z_rho prime}) shows that $\lim_{N\rightarrow\infty}\tilde{A}_{H,N}$ will be identically zero if and only if $\alpha_{H}\left(0\right)=1$. Q.E.D. \vphantom{} Using the above, we can give the formulae for integrating $\left(p,q\right)$-adic functions against $dA_{H}$. \begin{cor}[\textbf{$dA_{H}$ Integration Formulae}] \ \vphantom{} I. \index{$dA_{H}$!integration}($dA_{H}$ is a probability measure\footnote{In the sense of \cite{Probabilities taking values in non-archimedean fields,Measure-theoretic approach to p-adic probability theory}.}): \begin{equation} \int_{\mathbb{Z}_{p}}dA_{H}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q}}{=}1\label{eq:dA_H is a probabiity measure} \end{equation} Also, for all $n\in\mathbb{N}_{1}$ and all $k\in\left\{ 0,\ldots,p^{n}-1\right\} $: \begin{align} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]dA_{H}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(k\right)+\left(1-\alpha_{H}\left(0\right)\right)\sum_{m=0}^{n-1}\left(\frac{\mu_{0}}{p}\right)^{m}\kappa_{H}\left(\left[k\right]_{p^{m}}\right)\label{eq:integration of indicator functions dA_H} \end{align} \vphantom{} II. Let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then: \begin{align} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)dA_{H}\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\left(\left(\frac{\mu_{0}}{p^{2}}\right)^{N}\sum_{n=0}^{p^{N}-1}\kappa_{H}\left(n\right)f\left(n\right)\right.\label{eq:Truncation-based formula for integral of f dA_H}\\ & +\left.\frac{1-\alpha_{H}\left(0\right)}{p^{N}}\sum_{n=0}^{p^{N}-1}\sum_{m=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{m}\kappa_{H}\left(\left[n\right]_{p^{m}}\right)f\left(n\right)\right)\nonumber \end{align} \end{cor} \begin{rem} One can also compute $\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)dA_{H}\left(\mathfrak{z}\right)$ by expressing $f$ as a van der Put series and then integrating term-by-term; this yields formulae nearly identical to the ones given above, albeit in terms of $c_{n}\left(f\right)$ (the $n$th van der Put coefficient of $f$), rather than $f\left(n\right)$. \end{rem} Proof: (I) follows by using (\ref{eq:Convolution of dA_H and D_N}) along with the fact that, by definition: \begin{equation} \int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}k\right]dA_{H}\left(\mathfrak{z}\right)=\frac{1}{p^{n}}\sum_{\left\|t\right\|_{p}\leq p^{n}}\hat{A}_{H}\left(t\right)e^{-2\pi i\left\{ tk\right\} _{p}} \end{equation} and that $\left[k\right]_{p^{n}}=k$ if and only if $k\in\left\{ 0,\ldots,p^{n}-1\right\} $. As for (II), let $f\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ be arbitrary. Then, the truncations $f_{N}$ converge $q$-adically to $f$ uniformly over $\mathbb{Z}_{p}$. Consequently: \begin{align} \int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)dA_{H}\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\sum_{n=0}^{p^{N}-1}f\left(n\right)\int_{\mathbb{Z}_{p}}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n\right]dA_{H}\left(\mathfrak{z}\right)\\ \left(\textrm{Use (I)}\right); & =\lim_{N\rightarrow\infty}\sum_{n=0}^{p^{N}-1}f\left(n\right)\left(\frac{1}{p^{N}}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(n\right)\right)\\ & +\lim_{N\rightarrow\infty}\sum_{n=0}^{p^{N}-1}f\left(n\right)\frac{1-\alpha_{H}\left(0\right)}{p^{N}}\sum_{m=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{m}\kappa_{H}\left(\left[n\right]_{p^{m}}\right) \end{align} Q.E.D. \vphantom{} Given the import of the shortened $qx+1$ map\index{$qx+1$ map}s, let me give the details for what happens with their associated $A_{H}$s. \begin{cor}[\textbf{$dA_{H}$ for $qx+1$}] \label{cor:Formula for Nth partial sum of the Fourier series generated by A_q hat}Let $q$ be an odd integer $\geq3$, and let $A_{q}$ denote $A_{H}$ where $H=T_{q}$ is the shortened $qx+1$ map. Finally, write: \begin{equation} \tilde{A}_{q,N}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{q}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}=\left(D_{2:N}dA_{q}\right)\left(\mathfrak{z}\right)\label{eq:Definition of A_q,n twiddle} \end{equation} Then: \begin{equation} \tilde{A}_{q,N}\left(\mathfrak{z}\right)=\frac{q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}+\left(1-\frac{q+1}{4}\right)\sum_{n=0}^{N-1}\frac{q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\label{eq:Convolution of dA_q and D_N} \end{equation} where, for any integer $m\geq0$, $\#_{1}\left(m\right)$ is the number of $1$s in the binary representation of $m$ (a shorthand for $\#_{2:1}\left(m\right)$). Integration against $dA_{q}$ is then given by: \begin{align} \int_{\mathbb{Z}_{2}}f\left(\mathfrak{z}\right)dA_{q}\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q}}{=}\lim_{N\rightarrow\infty}\frac{1}{4^{N}}\sum_{n=0}^{2^{N}-1}f\left(n\right)q^{\#_{1}\left(n\right)}\\ & +\frac{3-q}{4}\lim_{N\rightarrow\infty}\frac{1}{2^{N}}\sum_{n=0}^{2^{N}-1}f\left(n\right)\sum_{m=0}^{N-1}\frac{q^{\#_{1}\left(\left[n\right]_{2^{m}}\right)}}{2^{m}}\nonumber \end{align} In particular, for $q=3$ (the Shortened Collatz map\index{Collatz!map}), we have: \begin{equation} \tilde{A}_{3,N}\left(\mathfrak{z}\right)=\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}\label{eq:A_3 measure of a coset in Z_2} \end{equation} and hence: \begin{equation} \int_{\mathbb{Z}_{2}}f\left(\mathfrak{z}\right)dA_{3}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{3}}{=}\lim_{N\rightarrow\infty}\frac{1}{4^{N}}\sum_{n=0}^{2^{N}-1}3^{\#_{1}\left(n\right)}f\left(n\right),\textrm{ }\forall f\in C\left(\mathbb{Z}_{2},\mathbb{C}_{3}\right)\label{eq:Integral of f dA_3} \end{equation} \end{cor} Proof: Use (\ref{eq:Truncation-based formula for integral of f dA_H}). Q.E.D. \vphantom{} By using $\hat{A}_{H}$ and \textbf{Proposition \ref{prop:alpha product in terms of A_H hat}}, we can re-write our explicit formula for $\hat{\chi}_{H,N}$\textemdash equation (\ref{eq:Explicit formula for Chi_H,N hat}) from \textbf{Proposition \ref{prop:Computation of Chi_H,N hat}}\textemdash in a way that separates the dependence of $t$ and $N$. Like in previous computations, the formulae for \index{hat{chi}{H,N}@$\hat{\chi}_{H,N}$!left(N,tright) asymptotic decomposition@$\left(N,t\right)$ asymptotic decomposition}$\hat{\chi}_{H,N}$ come in two forms, based on whether or not $\alpha_{H}\left(0\right)=1$. \begin{thm}[\textbf{$\left(N,t\right)$-Asymptotic Decomposition of $\hat{\chi}_{H,N}$}] \label{thm:(N,t) asymptotic decomposition of Chi_H,N hat}Recall that we write $\gamma_{H}\left(t\right)$ to denote $\beta_{H}\left(t\right)/\alpha_{H}\left(t\right)$. \vphantom{} I. If $\alpha_{H}\left(0\right)=1$, then: \begin{equation} \hat{\chi}_{H,N}\left(t\right)=\begin{cases} \beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right) & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)\left(N+v_{p}\left(t\right)\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fine Structure Formula for Chi_H,N hat when alpha is 1} \end{equation} \vphantom{} II. If $\alpha_{H}\left(0\right)\neq1$. Then: \begin{equation} \hat{\chi}_{H,N}\left(t\right)=\begin{cases} \beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N}-1}{\alpha_{H}\left(0\right)-1}\hat{A}_{H}\left(t\right) & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N+v_{p}\left(t\right)}-1}{\alpha_{H}\left(0\right)-1}\right)\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fine Structure Formula for Chi_H,N hat when alpha is not 1} \end{equation} \end{thm} Proof: For brevity, we write: \begin{equation} \hat{A}_{H,n}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 1 & \textrm{if }n=0\\ \mathbf{1}_{0}\left(p^{n+1}t\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}t\right) & \textrm{if }n\geq1 \end{cases}\label{eq:Definition of A_H,n+1 hat} \end{equation} so that (\ref{eq:Explicit formula for Chi_H,N hat}) becomes: \[ \hat{\chi}_{H,N}\left(t\right)=\sum_{n=0}^{N-1}\beta_{H}\left(p^{n}t\right)\hat{A}_{H,n}\left(t\right) \] Then, letting $t\in\hat{\mathbb{Z}}_{p}$ be non-zero and satisfy $\left\|t\right\|_{p}\leq p^{N}$, we use \textbf{Proposition \ref{prop:alpha product in terms of A_H hat}} to write: \begin{equation} \hat{\chi}_{H,N}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\sum_{n=0}^{N-1}\beta_{H}\left(p^{n}t\right)\hat{A}_{H,n}\left(t\right)=\sum_{n=-v_{p}\left(t\right)-1}^{N-1}\beta_{H}\left(p^{n}t\right)\hat{A}_{H,n}\left(t\right)\label{eq:Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis} \end{equation} Here, we have used the fact that: \begin{equation} \hat{A}_{H,n}\left(t\right)=0,\textrm{ }\forall n\leq-v_{p}\left(t\right)-2 \end{equation} because $\mathbf{1}_{0}\left(p^{n+1}t\right)$ vanishes whenever $n\leq-v_{p}\left(t\right)-2$. This leaves us with two cases. \vphantom{} I. First, suppose $\left\|t\right\|_{p}=p^{N}$. Then $N=-v_{p}\left(t\right)$, and so, (\ref{eq:Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis}) becomes: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\sum_{n=-v_{p}\left(t\right)-1}^{N-1}\beta_{H}\left(p^{n}t\right)\hat{A}_{H,n}\left(t\right)\\ & =\beta_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)\hat{A}_{H,-v_{p}\left(t\right)-1}\left(t\right)\\ & =\beta_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)\cdot\mathbf{1}_{0}\left(p^{-v_{p}\left(t\right)-1+1}t\right)\prod_{m=0}^{-v_{p}\left(t\right)-1-1}\alpha_{H}\left(p^{m}t\right)\\ & =\beta_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)\cdot\frac{\mathbf{1}_{0}\left(p^{-v_{p}\left(t\right)}t\right)}{\alpha_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)}\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right) \end{align} Since $p^{-v_{p}\left(t\right)}t=\left\|t\right\|_{p}t$ is the integer in the numerator of $t$, $\mathbf{1}_{0}\left(p^{-v_{p}\left(t\right)}t\right)=1$, and we are left with: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\frac{\beta_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)}{\alpha_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)}\underbrace{\prod_{m=0}^{-v_{p}\left(t\right)-1}\alpha_{H}\left(p^{m}t\right)}_{\hat{A}_{H}\left(t\right)}\\ & =\frac{\beta_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)}{\alpha_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)}\hat{A}_{H}\left(t\right)\\ & =\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) \end{align} \vphantom{} II. Second, suppose $0<\left\|t\right\|_{p}<p^{N}$. Then $-v_{p}\left(t\right)-1$ is strictly less than $N-1$, and so $p^{n}t\overset{1}{\equiv}0$ for all $n\geq-v_{p}\left(t\right)$. With this, (\ref{eq:Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis}) becomes: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\beta_{H}\left(p^{-v_{p}\left(t\right)-1}t\right)\hat{A}_{H,-v_{p}\left(t\right)-1}\left(t\right)+\sum_{n=-v_{p}\left(t\right)}^{N-1}\beta_{H}\left(p^{n}t\right)\hat{A}_{H,n}\left(t\right)\\ \left(p^{n}t\overset{1}{\equiv}0\textrm{ }\forall n\geq-v_{p}\left(t\right)\right); & =\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)+\beta_{H}\left(0\right)\sum_{n=-v_{p}\left(t\right)}^{N-1}\hat{A}_{H,n}\left(t\right) \end{align} Using \textbf{Proposition \ref{prop:alpha product in terms of A_H hat}}, we get: \begin{equation} \hat{A}_{H,n}\left(t\right)=\left(\alpha_{H}\left(0\right)\right)^{n+v_{p}\left(t\right)}\hat{A}_{H}\left(t\right),\textrm{ }\forall n\geq-v_{p}\left(t\right) \end{equation} and hence: \begin{align} \hat{\chi}_{H,N}\left(t\right) & =\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)+\beta_{H}\left(0\right)\sum_{n=-v_{p}\left(t\right)}^{N-1}\left(\alpha_{H}\left(0\right)\right)^{n+v_{p}\left(t\right)}\hat{A}_{H}\left(t\right)\\ & =\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)+\beta_{H}\left(0\right)\left(\sum_{n=0}^{N+v_{p}\left(t\right)-1}\left(\alpha_{H}\left(0\right)\right)^{n}\right)\hat{A}_{H}\left(t\right)\\ & =\begin{cases} \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)+\beta_{H}\left(0\right)\left(N+v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }\alpha_{H}\left(0\right)=1\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)+\beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N+v_{p}\left(t\right)}-1}{\alpha_{H}\left(0\right)-1}\hat{A}_{H}\left(t\right) & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases} \end{align} Finally, when $t=0$: \begin{align} \hat{\chi}_{H,N}\left(0\right) & =\sum_{n=0}^{N-1}\beta_{H}\left(0\right)\hat{A}_{H,n}\left(0\right)\\ & =\beta_{H}\left(0\right)+\sum_{n=1}^{N-1}\beta_{H}\left(0\right)\mathbf{1}_{0}\left(0\right)\prod_{m=0}^{n-1}\alpha_{H}\left(0\right)\\ & =\beta_{H}\left(0\right)+\sum_{n=1}^{N-1}\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{n}\\ & =\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\alpha_{H}\left(0\right)\right)^{n}\\ & =\begin{cases} \beta_{H}\left(0\right)N & \textrm{if }\alpha_{H}\left(0\right)=1\\ \beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N}-1}{\alpha_{H}\left(0\right)-1} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases} \end{align} Since $\hat{A}_{H}\left(0\right)=1$, we have that $\beta_{H}\left(0\right)N=\beta_{H}\left(0\right)N\hat{A}_{H}\left(0\right)$ for the $\alpha_{H}\left(0\right)=1$ case and $\beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N}-1}{\alpha_{H}\left(0\right)-1}=\beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N}-1}{\alpha_{H}\left(0\right)-1}\hat{A}_{H}\left(0\right)$ for the $\alpha_{H}\left(0\right)\neq1$ case. Q.E.D. \vphantom{} Subtracting $\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right)$ (which is $0$ for all $\left\|t\right\|_{p}\geq p^{N}$) from the $\alpha_{H}\left(0\right)=1$ case (equation (\ref{eq:Fine Structure Formula for Chi_H,N hat when alpha is 1}))\index{hat{chi}{H,N}@$\hat{\chi}_{H,N}$!fine structure} we obtain: \begin{equation} \hat{\chi}_{H,N}\left(t\right)-\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right)=\begin{cases} 0 & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases}\label{eq:Fine structure of Chi_H,N hat when alpha is 1} \end{equation} Even though the ranges of $t$ assigned to the individual pieces of this formula depends on $N$, note that the actual \emph{values} of the pieces on the right-hand side are independent of $N$ for all $\left\|t\right\|_{p}<p^{N}$. So, while the Fourier transform $\hat{\chi}_{H}\left(t\right)$ might not exist in a natural way, by having subtracted off the ``divergent'' $\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right)$ term from $\hat{\chi}_{H,N}\left(t\right)$, we have discovered fine-scale behavior of $\hat{\chi}_{H,N}\left(t\right)$ which was hidden beneath the tumult. Indeed, for fixed $t$, the $q$-adic limit of the left-hand side of (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) as $N\rightarrow\infty$ is: \begin{equation} \begin{cases} 0 & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}>0 \end{cases}\label{eq:Point-wise limit of the left-hand side of the fine structure formula when alpha is 1} \end{equation} Moreover, for any given $t$, the left-hand side of (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) is actually \emph{equal} to the above limit provided that $N>-v_{p}\left(t\right)$. This strongly suggests that (\ref{eq:Point-wise limit of the left-hand side of the fine structure formula when alpha is 1}) is, or is very close to a ``valid'' formula for a Fourier transform of $\chi_{H}$. The next subsection is dedicated to making this intuition rigorous. \subsection{\label{subsec:4.2.2}$\hat{\chi}_{H}$ and $\tilde{\chi}_{H,N}$} We begin by computing the Fourier series generated by $v_{p}\left(t\right)\hat{A}_{H}\left(t\right)$. \begin{lem}[\textbf{$v_{p}\hat{A}_{H}$ summation formulae}] \label{lem:v_p A_H hat summation formulae}\ \vphantom{} I. \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}-N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\sum_{n=0}^{N-1}\left(\alpha_{H}\left(0\right)\left(n+1\right)-n\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Fourier sum of A_H hat v_p} \end{equation} \vphantom{} II. If $H$ is semi-basic and contracting: \begin{equation} \sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }v_{p}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{C}_{q}}{=}\sum_{n=0}^{\infty}\left(\alpha_{H}\left(0\right)\left(n+1\right)-n\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\label{eq:Limit of Fourier sum of v_p A_H hat} \end{equation} where the convergence is point-wise. \end{lem} Proof: For (I), use (\ref{eq:Convolution of dA_H and D_N}) from \textbf{Theorem \ref{thm:Properties of dA_H}}, along with \textbf{Proposition \ref{prop:v_p of t times mu hat sum}}. If $H$ is semi-basic and contracting, the decay estimates on $\kappa_{H}$ and $\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ given by \textbf{Proposition \ref{prop:Properties of Kappa_H}} then guarantee the convergence of (I) to (II) in $\mathbb{C}_{q}$ for $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ as $N\rightarrow\infty$. Q.E.D. \vphantom{} Next, we need to deal with how $p$ affects the situation. The primary complication in our computations comes from $\gamma_{H}\left(t\left\|t\right\|_{p}/p\right)$, whose behavior bifurcates at $p=2$. The function: \begin{equation} t\in\hat{\mathbb{Z}}_{p}\mapsto\frac{t\left\|t\right\|_{p}}{p}\in\hat{\mathbb{Z}}_{p}\label{eq:Projection onto 1 over rho} \end{equation} is a projection which sends every non-zero element $t=k/p^{n}$ in $\hat{\mathbb{Z}}_{p}$ and outputs the fraction $\left[k\right]_{p}/p$ obtained by reducing $k$ mod $p$ and then sticking it over $p$. When $p=2$, $\left[k\right]_{p}=1$ for all $t\in\hat{\mathbb{Z}}_{2}$, making (\ref{eq:Projection onto 1 over rho}) constant on $\hat{\mathbb{Z}}_{2}\backslash\left\{ 0\right\} $, where it takes the value $1/2$. When $p\geq3$, however, (\ref{eq:Projection onto 1 over rho}) is no longer constant on $\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} $, which is going to significantly complicate our computations. This can be made much more manageable, provided the reader will put up with another bit of notation. \begin{defn}[$\varepsilon_{n}\left(\mathfrak{z}\right)$] \nomenclature{$\varepsilon_{n}\left(\mathfrak{z}\right)$}{ }For each $n\in\mathbb{N}_{0}$, we define $\varepsilon_{n}:\mathbb{Z}_{p}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \varepsilon_{n}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}e^{\frac{2\pi i}{p^{n+1}}\left(\left[\mathfrak{z}\right]_{p^{n+1}}-\left[\mathfrak{z}\right]_{p^{n}}\right)}=e^{2\pi i\left\{ \frac{\mathfrak{z}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathfrak{z}}{p^{n}}\right\} _{p}}\label{eq:Definition of epsilon_n} \end{equation} \end{defn} \vphantom{} As with nearly every other noteworthy function in this dissertation, the $\varepsilon_{n}$s satisfy functional equations, which we record below. \begin{prop}[\textbf{Properties of $\varepsilon_{n}$}] \ \vphantom{} I. \begin{equation} \varepsilon_{0}\left(\mathfrak{z}\right)=e^{2\pi i\left\{ \frac{\mathfrak{z}}{p}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Epsilon 0 of z} \end{equation} \begin{equation} \varepsilon_{n}\left(j\right)=1,\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z},\textrm{ }\forall n\geq1\label{eq:Epsilon_n of j} \end{equation} \vphantom{} II. \begin{equation} \varepsilon_{n}\left(pm+j\right)=\begin{cases} \varepsilon_{0}\left(j\right) & \textrm{if }n=0\\ \varepsilon_{n-1}\left(m\right) & \textrm{if }n\geq1 \end{cases},\textrm{ }\forall m\in\mathbb{N}_{0},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z},\textrm{ }\forall n\geq1\label{eq:epsilon_n functional equations} \end{equation} \vphantom{} III. Let $\mathfrak{z}\neq0$. Then $\varepsilon_{n}\left(\mathfrak{z}\right)=1\textrm{ }$ for all $n<v_{p}\left(\mathfrak{z}\right)$. \end{prop} Proof: I. The identity (\ref{eq:Epsilon 0 of z}) is immediate from the definition of $\varepsilon_{n}$. As for the other identity, note that for $j\in\mathbb{Z}/p\mathbb{Z}$, $\left[j\right]_{p^{n}}=j$ for all $n\geq1$. Hence, for $n\geq1$: \begin{equation} \varepsilon_{n}\left(j\right)=e^{\frac{2\pi i}{p^{n+1}}\left(\left[j\right]_{p^{n+1}}-\left[j\right]_{p^{n}}\right)}=e^{\frac{2\pi i}{p^{n+1}}\cdot0}=1 \end{equation} \vphantom{} II. \begin{align} \varepsilon_{n}\left(pm+j\right) & =e^{2\pi i\left\{ \frac{pm+j}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{pm+j}{p^{n}}\right\} _{p}}\\ & =e^{2\pi i\left\{ \frac{j}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{j}{p^{n}}\right\} _{p}}\cdot e^{2\pi i\left\{ \frac{m}{p^{n}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{m}{p^{n-1}}\right\} _{p}}\\ & =\varepsilon_{n}\left(j\right)\varepsilon_{n-1}\left(m\right)\\ \left(\textrm{by (I)}\right); & =\begin{cases} \varepsilon_{0}\left(j\right) & \textrm{if }n=0\\ \varepsilon_{n-1}\left(m\right) & \textrm{if }n\geq1 \end{cases} \end{align} \vphantom{} III. Let $\mathfrak{z}$ be non-zero. When $n<v_{p}\left(\mathfrak{z}\right)$, we have that $p^{-n}\mathfrak{z}$ and $p^{-\left(n+1\right)}\mathfrak{z}$ are then $p$-adic integers, and hence: \begin{equation} \varepsilon_{n}\left(\mathfrak{z}\right)=e^{2\pi i\left\{ \frac{\mathfrak{z}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathfrak{z}}{p^{n}}\right\} _{p}}=e^{0}\cdot e^{-0}=1 \end{equation} Q.E.D. \vphantom{} Now we compute the sum of the Fourier series generated by $\gamma_{H}\left(t\left\|t\right\|_{p}/p\right)\hat{A}_{H}\left(t\right)$. \begin{lem}[\textbf{$\gamma_{H}\hat{A}_{H}$ summation formulae}] \label{lem:1D gamma formula}Let $p\geq2$. Then: \vphantom{} I. \begin{align} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \overset{\overline{\mathbb{Q}}}{=}\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Gamma formula} \end{align} \vphantom{} II. If $H$ is a contracting semi-basic $p$-Hydra map, then, as $N\rightarrow\infty$, \emph{(\ref{eq:Gamma formula})} is $\mathcal{F}_{p,q_{H}}$ convergent to: \begin{equation} \sum_{t\in\hat{\mathbb{Z}}_{p}\backslash\left\{ 0\right\} }\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:F limit of Gamma_H A_H hat Fourier series when alpha is 1} \end{equation} whenever $H$ is semi-basic and contracting. \end{lem} \begin{rem} Note that the non-singularity of $H$ ($\alpha_{H}\left(j/p\right)\neq0$ for all $j\in\mathbb{Z}/p\mathbb{Z}$) is \emph{essential} for this result. \end{rem} Proof: I. Note that the map $t\in\hat{\mathbb{Z}}_{p}\mapsto\frac{t\left\|t\right\|_{p}}{p}\in\hat{\mathbb{Z}}_{p}$ takes fractions $k/p^{n}$ and sends them to $\left[k\right]_{p}/p$. Now, for brevity, let: \begin{align} \gamma_{j} & \overset{\textrm{def}}{=}\gamma_{H}\left(\frac{j}{p}\right)\\ F_{N}\left(\mathfrak{z}\right) & \overset{\textrm{def}}{=}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} Observing that: \begin{equation} \left\{ t\in\hat{\mathbb{Z}}_{p}:\left\|t\right\|_{p}=p^{n}\right\} =\left\{ \frac{pk+j}{p^{n}}:k\in\left\{ 0,\ldots,p^{n-1}-1\right\} ,\textrm{ }j\in\left\{ 1,\ldots,p-1\right\} \right\} \label{eq:Decomposition of level sets} \end{equation} we can then express $F_{N}$ as a sum involving the $\gamma_{j}$s, like so: \begin{align} F_{N}\left(\mathfrak{z}\right) & =\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}=p^{n}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ \left(\textrm{use }(\ref{eq:Decomposition of level sets})\right); & =\sum_{n=1}^{N}\sum_{j=1}^{p-1}\sum_{k=0}^{p^{n-1}-1}\gamma_{H}\left(\frac{j}{p}\right)\hat{A}_{H}\left(\frac{pk+j}{p^{n}}\right)e^{2\pi i\left\{ \frac{pk+j}{p^{n}}\mathfrak{z}\right\} _{p}} \end{align} Using the formal identity: \begin{equation} \sum_{k=0}^{p^{n-1}-1}f\left(\frac{pk+j}{p^{n}}\right)=\sum_{\left\|t\right\|_{p}\leq p^{n-1}}f\left(t+\frac{j}{p^{n}}\right) \end{equation} we can then write: \begin{align} F_{N}\left(\mathfrak{z}\right) & =\sum_{n=1}^{N}\sum_{\left\|t\right\|_{p}\leq p^{n-1}}\sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\hat{A}_{H}\left(t+\frac{j}{p^{n}}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Halfway through the gamma computation} \end{align} To deal with the $j$-sum, we express $\hat{A}_{H}$ in product form, changing: \begin{equation} \sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\hat{A}_{H}\left(t+\frac{j}{p^{n}}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} into: \begin{equation} \sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\left(t+\frac{j}{p^{n}}\right)\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} Using\textbf{ Proposition \ref{prop:alpha product series expansion}} to write the $\alpha_{H}$-product out as a series, the above becomes: \begin{equation} \sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\left(\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{-2\pi im\left(t+\frac{j}{p^{n}}\right)}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} Hence: \begin{equation} \sum_{j=1}^{p-1}\gamma_{j}\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m\right)}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ t\left(\mathfrak{z}-m\right)\right\} _{p}} \end{equation} Summing over $\left\|t\right\|_{p}\leq p^{n-1}$, and using: \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{n-1}}e^{2\pi i\left\{ t\left(\mathfrak{z}-m\right)\right\} _{p}}=p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right] \end{equation} we obtain: \begin{equation} \sum_{j=1}^{p-1}\gamma_{j}\left(\frac{\mu_{0}}{p^{2}}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m\right)}{p^{n}}\right\} _{p}}p^{n-1}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right] \end{equation} In summary, so far, we have; \begin{eqnarray} & \sum_{\left\|t\right\|_{p}\leq p^{n-1}}\sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\hat{A}_{H}\left(t+\frac{j}{p^{n}}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\nonumber \\ & =\label{eq:2/3rds of the way through the gamma computation}\\ & \sum_{j=1}^{p-1}\frac{\gamma_{j}}{p}\left(\frac{\mu_{0}}{p}\right)^{n}\sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m\right)}{p^{n}}\right\} _{p}}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]\nonumber \end{eqnarray} Next, using the formal identity: \begin{equation} \sum_{m=0}^{p^{n}-1}f\left(m\right)=\sum_{k=0}^{p-1}\sum_{m=0}^{p^{n-1}-1}f\left(m+kp^{n-1}\right)\label{eq:rho to the n formal identity} \end{equation} and the functional equation identity for $\kappa_{H}$ (equation (\ref{eq:Kappa_H functional equations}) from \textbf{Proposition \ref{prop:Properties of Kappa_H}}), we have: \begin{eqnarray} & \sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m\right)}{p^{n}}\right\} _{p}}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right]\\ & =\\ & \sum_{k=0}^{p-1}\sum_{m=0}^{p^{n-1}-1}\frac{\mu_{k}}{\mu_{0}}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m-kp^{n-1}\right)}{p^{n}}\right\} _{p}}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right] \end{eqnarray} Here, note that $\left[\mathfrak{z}\right]_{p^{n-1}}$ is the unique integer $m\in\left\{ 0,\ldots,p^{n-1}-1\right\} $ satisfying $\mathfrak{z}\overset{p^{n-1}}{\equiv}m$. This leaves us with: \begin{align} \sum_{m=0}^{p^{n}-1}\kappa_{H}\left(m\right)e^{2\pi i\left\{ \frac{j\left(\mathfrak{z}-m\right)}{p^{n}}\right\} _{p}}\left[\mathfrak{z}\overset{p^{n-1}}{\equiv}m\right] & =\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\left(\varepsilon_{n-1}\left(\mathfrak{z}\right)\right)^{j}\overbrace{\sum_{k=0}^{p-1}\frac{\mu_{k}}{\mu_{0}}e^{-2\pi i\frac{jk}{p}}}^{\frac{p^{2}}{\mu_{0}}\alpha_{H}\left(\frac{j}{p}\right)}\\ & =\frac{p^{2}}{\mu_{0}}\alpha_{H}\left(\frac{j}{p}\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right)\left(\varepsilon_{n-1}\left(\mathfrak{z}\right)\right)^{j} \end{align} Returning with this to (\ref{eq:2/3rds of the way through the gamma computation}), the equation \begin{equation} \sum_{\left\|t\right\|_{p}\leq p^{n-1}}\sum_{j=1}^{p-1}\gamma_{j}e^{2\pi i\left\{ \frac{j\mathfrak{z}}{p^{n}}\right\} _{p}}\hat{A}_{H}\left(t+\frac{j}{p^{n}}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{equation} transforms into: \begin{equation} \sum_{j=1}^{p-1}\frac{\gamma_{j}}{p}\left(\frac{\mu_{0}}{p}\right)^{n}\frac{p^{2}}{\mu_{0}}\alpha_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n-1}\left(\mathfrak{z}\right)\right)^{j}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right) \end{equation} Almost finished, recall that: \begin{equation} \gamma_{j}=\gamma_{H}\left(\frac{j}{p}\right)=\frac{\beta_{H}\left(\frac{j}{p}\right)}{\alpha_{H}\left(\frac{j}{p}\right)} \end{equation} With this, we can write the previous line as: \begin{equation} \left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n-1}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right) \end{equation} Applying this to the right-hand side of (\ref{eq:Halfway through the gamma computation}) gives us: \begin{align} F_{N}\left(\mathfrak{z}\right) & =\sum_{n=1}^{N}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n-1}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n-1}}\right) \end{align} Re-indexing $n$ by a shift of $1$ produces equation (\ref{eq:Gamma formula}). \vphantom{} II. For each $\mathfrak{z}\in\mathbb{Z}_{p}$, the algebraic number $\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}$ is uniformly bounded with respect to $n\in\mathbb{N}_{0}$ in both $\mathbb{C}$ and $\mathbb{C}_{q}$. Like in the $p=2$ case, since $H$ is semi-basic, the sequence $\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ is then $\mathcal{F}_{p,q_{H}}$-convergent to $0$. The $q$-adic convergence to $0$ over $\mathbb{Z}_{p}^{\prime}$ guarantees the $q$-adic convergence of (\ref{eq:Gamma formula}) as $N\rightarrow\infty$ for $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. On the other hand, for $\mathfrak{z}\in\mathbb{N}_{0}$, as we have seen: \begin{equation} \left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\mathfrak{z}\right),\textrm{ }\forall n\geq\lambda_{p}\left(\mathfrak{z}\right) \end{equation} and the fact that $H$ is contracting then guarantees (by the ratio test, no less) that (\ref{eq:Gamma formula}) is convergent in $\mathbb{C}$ for $\mathfrak{z}\in\mathbb{N}_{0}$. This proves (II). Q.E.D. \vphantom{} With these formulae, we can sum the Fourier series generated by (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) to obtain a non-trivial formula for $\chi_{H,N}$. \begin{thm} \label{thm:F-series for Nth truncation of Chi_H, alpha is 1}Suppose\index{chi{H}@$\chi_{H}$!$N$th truncation} $\alpha_{H}\left(0\right)=1$. Then, for all $N\geq1$ and all $\mathfrak{z}\in\mathbb{Z}_{p}$: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\sum_{n=0}^{N-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Chi_H,N when alpha is 1 and rho is arbitrary} \end{align} In particular, when $p=2$: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\gamma_{H}\left(\frac{1}{2}\right)\left(\frac{\mu_{0}}{2}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)\label{eq:Chi_H,N when alpha is 1 and rho is 2} \end{equation} \end{thm} Proof: We start by multiplying (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) by $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ and then summing over all $\left\|t\right\|_{p}\leq p^{N}$. The left-hand side of (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) becomes: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\left(D_{p:N-1}dA_{H}\right)\left(\mathfrak{z}\right) \end{equation} whereas the right-hand side is: \begin{align} \sum_{0<\left\|t\right\|_{p}\leq p^{N-1}}\left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ +\sum_{\left\|t\right\|_{p}=p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} Simplifying produces: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)N\left(D_{p:N-1}dA_{H}\right)\left(\mathfrak{z}\right)\label{eq:Chi_H,N rho not equal to 2, ready to simplify}\\ & +\beta_{H}\left(0\right)\sum_{0<\left\|t\right\|_{p}\leq p^{N-1}}v_{p}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\nonumber \\ & +\sum_{0<\left\|t\right\|_{p}\leq p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\nonumber \end{align} Now we call upon our legion of formulae: (\ref{eq:Convolution of dA_H and D_N}), \textbf{Lemma \ref{lem:v_p A_H hat summation formulae}}, and \textbf{Lemma \ref{lem:1D gamma formula}}. Using them (whilst remembering that $\alpha_{H}\left(0\right)=1$) makes (\ref{eq:Chi_H,N rho not equal to 2, ready to simplify}) into: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & -\beta_{H}\left(0\right)\left(N-1\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & +\beta_{H}\left(0\right)\sum_{n=0}^{N-2}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & \sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & \overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} The bottom two lines combine to form a single sum by noting that the upper most of the two is the $j=0$ case of the bottom-most of the two. Finally, when $p=2$, rather than simplify (\ref{eq:Chi_H,N when alpha is 1 and rho is arbitrary}), it will actually be easier to compute it from scratch all over again. Multiplying (\ref{eq:Fine structure of Chi_H,N hat when alpha is 1}) by $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}$ and summing over all $\left\|t\right\|_{2}\leq2^{N}$ gives: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\left(D_{2:N-1}dA_{H}\right)\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\gamma_{H}\left(\frac{1}{2}\right)\sum_{0<\left\|t\right\|_{2}\leq2^{N-1}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & +\beta_{H}\left(0\right)\sum_{0<\left\|t\right\|_{2}\leq2^{N-1}}v_{2}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & +\gamma_{H}\left(\frac{1}{2}\right)\sum_{\left\|t\right\|_{2}=2^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}} \end{align} This simplifies to: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\left(D_{2:N-1}dA_{H}\right)\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\gamma_{H}\left(\frac{1}{2}\right)\sum_{0<\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & +\beta_{H}\left(0\right)\sum_{0<\left\|t\right\|_{2}\leq2^{N-1}}v_{2}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}\\ & \overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)\hat{A}_{H}\left(0\right)\\ & +\gamma_{H}\left(\frac{1}{2}\right)\underbrace{\sum_{\left\|t\right\|_{2}\leq2^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}}}_{\left(D_{2:N}dA_{H}\right)\left(\mathfrak{z}\right)}\\ & +\beta_{H}\left(0\right)\sum_{0<\left\|t\right\|_{2}\leq2^{N-1}}v_{2}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{2}} \end{align} Applying (\ref{eq:Convolution of dA_H and D_N}) and \textbf{Lemma \ref{lem:v_p A_H hat summation formulae}}, the above becomes: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{2}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N-1}}\right) & \overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)\\ & +\gamma_{H}\left(\frac{1}{2}\right)\left(\frac{\mu_{0}}{2}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)\\ & -\beta_{H}\left(0\right)\left(N-1\right)\left(\frac{\mu_{0}}{2}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N-1}}\right)\\ & +\beta_{H}\left(0\right)\sum_{n=0}^{N-2}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right) \end{align} Simplifying yields: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\gamma_{H}\left(\frac{1}{2}\right)\left(\frac{\mu_{0}}{2}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right) \end{equation} Q.E.D. \begin{cor}[\textbf{$\mathcal{F}$-series for $\chi_{H}$ when $\alpha_{H}\left(0\right)=1$}] \label{cor:Chi_H, F-convergent formula}Suppose\index{chi{H}@$\chi_{H}$!mathcal{F}-series@$\mathcal{F}$-series} For the given $p$-Hydra map $H$ (semi-basic, contracting, non-singular, fixes $0$), suppose $\alpha_{H}\left(0\right)=1$. Then: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary} \end{equation} with the special case: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,q_{H}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Explicit Formula for Chi_H when alpha is 1 and rho is 2} \end{equation} for when $p=2$. In both cases, as indicated, the topology of convergence of the series on the right-hand side is that of the standard $\left(p,q_{H}\right)$-adic frame: the topology of $\mathbb{C}$ when $\mathfrak{z}\in\mathbb{N}_{0}$, and the topology of $\mathbb{C}_{q_{H}}$ when $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. \end{cor} Proof: The given properties of $H$ tell us that $\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ has $0$ as a $\mathcal{F}_{p,q_{H}}$-limit. Appealing to the limits formulae in \textbf{Lemmata \ref{lem:1D gamma formula}} and \textbf{\ref{lem:v_p A_H hat summation formulae}} then allows us to take the $\mathcal{F}_{p,q_{H}}$-limits of (\ref{eq:Chi_H,N when alpha is 1 and rho is arbitrary}) and (\ref{eq:Chi_H,N when alpha is 1 and rho is 2}) as $N\rightarrow\infty$, which produces (\ref{eq:Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary}) and (\ref{eq:Explicit Formula for Chi_H when alpha is 1 and rho is 2}), respectively. Q.E.D. \vphantom{} Next, we sum in $\mathbb{C}$ for $\mathfrak{z}\in\mathbb{N}_{0}$. \begin{cor} \label{cor:Chi_H explicit formula on N_0}If $\alpha_{H}\left(0\right)=1$, then: \begin{equation} \chi_{H}\left(n\right)\overset{\mathbb{C}}{=}\sum_{k=0}^{\lambda_{p}\left(n\right)-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{k}\left(n\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{k}\kappa_{H}\left(\left[n\right]_{p^{k}}\right),\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:archimedean Chi_H when rho is arbitrary and alpha_H of 0 is 1} \end{equation} with the special case: \begin{equation} \chi_{H}\left(n\right)\overset{\mathbb{C}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\frac{2\beta_{H}\left(0\right)}{2-\mu_{0}}M_{H}\left(n\right)+\beta_{H}\left(0\right)\sum_{k=0}^{\lambda_{2}\left(n\right)-1}\left(\frac{\mu_{0}}{2}\right)^{k}\kappa_{H}\left(\left[n\right]_{2^{k}}\right),\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:archimedean Chi_H when rho is 2 and alpha_H of 0 is 1} \end{equation} when $p=2$. Regardless of the value of $p$, the $k$-sums are defined to be $0$ when $n=0$. \end{cor} Proof: Let $n\in\mathbb{N}_{0}$. Since $\kappa_{H}\left(\left[n\right]_{p^{k}}\right)=\kappa_{H}\left(n\right)$ and $\varepsilon_{k}\left(n\right)=1$ for all $k\geq\lambda_{p}\left(n\right)$, the equation (\ref{eq:Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary}) becomes: \begin{align} \chi_{H}\left(n\right) & \overset{\mathbb{C}}{=}\sum_{k=0}^{\lambda_{p}\left(n\right)-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{k}\left(n\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{k}\kappa_{H}\left(\left[n\right]_{p^{k}}\right)\\ & +\left(\sum_{k=\lambda_{p}\left(n\right)}^{\infty}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\cdot1\right)\left(\frac{\mu_{0}}{p}\right)^{k}\right)\kappa_{H}\left(n\right) \end{align} Since $H$ is contracting, the geometric series: \begin{equation} \sum_{k=\lambda_{p}\left(n\right)}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{k} \end{equation} converges to a limit in $\mathbb{C}$. However, this does not matter much, because: \[ \sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)=\sum_{j=0}^{p-1}\frac{1}{p}\sum_{\ell=0}^{p-1}\frac{b_{\ell}}{d_{\ell}}e^{-2\pi i\ell\frac{j}{p}}=\sum_{\ell=0}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{j=0}^{p-1}\frac{1}{p}e^{-2\pi i\ell\frac{j}{p}}=\frac{b_{\ell}}{d_{\ell}}\left[\ell\overset{p}{\equiv}0\right]=\frac{b_{0}}{d_{0}} \] Since $b_{0}=H\left(0\right)=0$, we are then left with: \begin{align} \chi_{H}\left(n\right) & \overset{\mathbb{C}}{=}\sum_{k=0}^{\lambda_{p}\left(n\right)-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{k}\left(n\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{k}\kappa_{H}\left(\left[n\right]_{p^{k}}\right) \end{align} which is the desired formula. As for the $p=2$ case, applying the same argument given above to (\ref{eq:Explicit Formula for Chi_H when alpha is 1 and rho is 2}) yields: \begin{align} \chi_{H}\left(n\right) & \overset{\mathbb{C}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\beta_{H}\left(0\right)\sum_{k=0}^{\lambda_{2}\left(n\right)-1}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[n\right]_{2^{k}}\right)\\ & +\beta_{H}\left(0\right)\kappa_{H}\left(n\right)\sum_{k=\lambda_{2}\left(n\right)}^{\infty}\left(\frac{\mu_{0}}{2}\right)^{k}\\ \left(H\textrm{ is contracting}\right); & \overset{\mathbb{C}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\beta_{H}\left(0\right)\sum_{k=0}^{\lambda_{2}\left(n\right)-1}\left(\frac{\mu_{0}}{2}\right)^{k}\kappa_{H}\left(\left[n\right]_{2^{k}}\right)\\ & +\beta_{H}\left(0\right)\frac{\left(\frac{\mu_{0}}{2}\right)^{\lambda_{2}\left(n\right)}\kappa_{H}\left(n\right)}{1-\frac{\mu_{0}}{2}} \end{align} Finally, since: \[ \kappa_{H}\left(n\right)=\left(\frac{2}{\mu_{0}}\right)^{\lambda_{2}\left(n\right)}M_{H}\left(n\right) \] we then obtain: \begin{align} \chi_{H}\left(n\right) & \overset{\mathbb{C}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\beta_{H}\left(0\right)\frac{M_{H}\left(n\right)}{1-\frac{\mu_{0}}{2}}+\beta_{H}\left(0\right)\sum_{k=0}^{\lambda_{2}\left(n\right)-1}\left(\frac{\mu_{0}}{2}\right)^{k}\kappa_{H}\left(\left[n\right]_{2^{k}}\right) \end{align} Q.E.D. \vphantom{} Taken together, these two corollaries then establish the quasi-integrability of $\chi_{H}$\index{chi{H}@$\chi_{H}$!quasi-integrability} for $p\geq2$ and $\alpha_{H}\left(0\right)=1$. \begin{cor}[\textbf{Quasi-Integrability of $\chi_{H}$ when $\alpha_{H}\left(0\right)=1$}] \label{cor:Quasi-integrability of Chi_H for alpha equals 1}If $\alpha_{H}\left(0\right)=1$, then, $\chi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame. In\index{chi{H}@$\chi_{H}$!Fourier transform} particular, when $p=2$, the function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{2}\rightarrow\overline{\mathbb{Q}}$ defined by: \begin{equation} \hat{\chi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} -\gamma_{H}\left(\frac{1}{2}\right) & \textrm{if }t=0\\ \beta_{H}\left(0\right)v_{2}\left(t\right)\hat{A}_{H}\left(t\right) & \textrm{else } \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Formula for Chi_H hat when rho is 2 and alpha is 1} \end{equation} is then a Fourier transform of $\chi_{H}$. In this case, the function defined by \emph{(\ref{eq:Point-wise limit of the left-hand side of the fine structure formula when alpha is 1})} is also a Fourier transform of $\chi_{H}$, differing from the $\hat{\chi}_{H}$ given above by $\gamma_{H}\left(\frac{1}{2}\right)\hat{A}_{H}\left(t\right)$, which, by \textbf{\emph{Theorem \ref{thm:Properties of dA_H}}}, is a degenerate measure, seeing as $\alpha_{H}\left(0\right)=1$. For $p\geq3$, we can obtain a Fourier transform for $\chi_{H}$ by defining a function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \hat{\chi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 0 & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{else} \end{cases}\label{eq:Chi_H hat when rho is not 2 and when alpha is 1} \end{equation} \end{cor} Proof: Corollaries\textbf{ \ref{cor:Chi_H explicit formula on N_0}} and\textbf{ \ref{cor:Chi_H, F-convergent formula}} show that the $N$th partial sums of the Fourier series generated by (\ref{eq:Point-wise limit of the left-hand side of the fine structure formula when alpha is 1}) are $\mathcal{F}_{p,q_{H}}$-convergent to (\ref{eq:Formula for Chi_H hat when rho is 2 and alpha is 1}) and (\ref{eq:Chi_H hat when rho is not 2 and when alpha is 1}) for $p=2$ and $p\geq3$, respectively, thereby establishing the quasi-integrability of $\chi_{H}$ with respect to the standard $\left(p,q_{H}\right)$-adic frame. Finally, letting $\hat{\chi}_{H}^{\prime}\left(t\right)$ denote (\ref{eq:Point-wise limit of the left-hand side of the fine structure formula when alpha is 1}), observe that when $\alpha_{H}\left(0\right)=1$ and $p=2$: \begin{equation} \hat{\chi}_{H}^{\prime}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\begin{cases} 0 & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{1}{2}\right)+\beta_{H}\left(0\right)v_{2}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{2}>0 \end{cases} \end{equation} Since $\hat{A}_{H}\left(0\right)=1$, we have that: \begin{equation} \hat{\chi}_{H}^{\prime}\left(t\right)-\gamma_{H}\left(\frac{1}{2}\right)\hat{A}_{H}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\begin{cases} -\gamma_{H}\left(\frac{1}{2}\right) & \textrm{if }t=0\\ \beta_{H}\left(0\right)v_{2}\left(t\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{2}>0 \end{cases}=\hat{\chi}_{H}\left(t\right) \end{equation} which shows that $\hat{\chi}_{H}^{\prime}\left(t\right)$ and $\hat{\chi}_{H}\left(t\right)$ differ by a factor of $\gamma_{H}\left(\frac{1}{2}\right)\hat{A}_{H}\left(t\right)$, which is a degenerate measure because $\alpha_{H}\left(0\right)=1$. Q.E.D. \vphantom{} This completes the $\alpha_{H}\left(0\right)=1$ case. By using $\chi_{H}$'s functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}), we can extend all of the above work to cover all $p$-Hydra maps, regardless of the value of $\alpha_{H}\left(0\right)$. The key to this are \textbf{Corollary \ref{cor:Chi_H explicit formula on N_0} }and \textbf{Lemma \ref{lem:Functional equations and truncation}}. First, however, a definition: \begin{defn}[\textbf{Little Psi-$H$ \& Big Psi-$H$}] \ \vphantom{} I. We define \nomenclature{$\psi_{H}\left(m\right)$}{ }$\psi_{H}:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ (``Little Psi-$H$'') by: \begin{equation} \psi_{H}\left(m\right)\overset{\textrm{def}}{=}\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right),\textrm{ }\forall m\in\mathbb{N}_{0}\label{eq:Definition of Little Psi_H} \end{equation} where the $n$-sum is defined to be $0$ when $m=0$. \vphantom{} II. We define \nomenclature{$\Psi_{H}\left(m\right)$}{ }$\Psi_{H}:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ (``Big Psi-$H$'') by: \begin{equation} \Psi_{H}\left(m\right)\overset{\textrm{def}}{=}-\beta_{H}\left(0\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(m\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\label{eq:Definition of Big Psi_H} \end{equation} \end{defn} \vphantom{} As a simple computation will immediately verify, (\ref{eq:Definition of Little Psi_H}) and (\ref{eq:Definition of Big Psi_H}) are merely the sums in $\mathbb{C}$ of (\ref{eq:Limit of Fourier sum of v_p A_H hat}) and (\ref{eq:F limit of Gamma_H A_H hat Fourier series when alpha is 1}) from \textbf{Lemmata \ref{lem:v_p A_H hat summation formulae}} and \textbf{\ref{lem:1D gamma formula}}, respectively, for $\mathfrak{z}\in\mathbb{N}_{0}$. To derive Fourier transforms and quasi-integrability for $\chi_{H}$ when $\alpha_{H}\left(0\right)\neq1$, we will first show that $\psi_{H}$ and $\Psi_{H}$ are rising-continuable to standard-frame-quasi-integrable $\left(p,q_{H}\right)$-adic functions, and that these continuations are characterized by systems of functional equations very to near the ones satisfied by $\chi_{H}$ (\ref{eq:Functional Equations for Chi_H over the rho-adics}). With a bit of linear algebra, we can then \emph{solve }for the appropriate linear combinations of $\psi_{H}$ and $\Psi_{H}$ to needed to obtain $\chi_{H}$, regardless of the value of $\alpha_{H}\left(0\right)$. \begin{lem}[\textbf{Rising-Continuability and Functional Equations for $\psi_{H}$ \& $\Psi_{H}$}] \label{lem:Rising continuation and uniquenss of the psi_Hs}For $H$ as given at the start of \emph{Section \pageref{sec:4.2 Fourier-Transforms-=00003D000026}}: \vphantom{} I. $\psi_{H}$ is rising-continuable to a $\left(p,q_{H}\right)$-adic function $\psi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$ given by: \begin{equation} \psi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Rising-continuation of Little Psi_H} \end{equation} Moreover, $\psi_{H}\left(\mathfrak{z}\right)$ is the unique rising-continuous $\left(p,q_{H}\right)$-adic function satisfying the system of \index{functional equation!psi_{H}@$\psi_{H}$}functional equations: \begin{equation} \psi_{H}\left(p\mathfrak{z}+j\right)=\frac{\mu_{j}}{p}\psi_{H}\left(\mathfrak{z}\right)+1,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\textrm{ \& }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Little Psi_H functional equations} \end{equation} \vphantom{} II. \index{functional equation!Psi_{H}@$\Psi_{H}$}$\Psi_{H}$ is rising-continuable to a $\left(p,q_{H}\right)$-adic function $\Psi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q_{H}}$ given by: \begin{equation} \Psi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Rising-continuation of Big Psi_H} \end{equation} Moreover, $\Psi_{H}\left(\mathfrak{z}\right)$ is the unique rising-continuous $\left(p,q_{H}\right)$-adic function satisfying the system of functional equations: \begin{equation} \Psi_{H}\left(p\mathfrak{z}+j\right)=\frac{\mu_{j}}{p}\Psi_{H}\left(\mathfrak{z}\right)+H_{j}\left(0\right)-\beta_{H}\left(0\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\textrm{ \& }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Big Psi_H functional equations} \end{equation} \end{lem} \begin{rem} (\ref{eq:Little Psi_H functional equations}) is the most significant part of this result. It shows that $\psi_{H}$ is a shifted form of $\chi_{H}$, and\textemdash crucially\textemdash that this relation between $\chi_{H}$ and $\psi_{H}$ is \emph{independent }of the value of $\alpha_{H}\left(0\right)$. Indeed, it is solely a consequence of the properties of the expression (\ref{eq:Definition of Little Psi_H}). \end{rem} Proof: For both parts, we use (\ref{eq:Relation between truncations and functional equations, version 2}) from \textbf{Lemma \ref{lem:Functional equations and truncation}}. With this, observe that the functional equations (\ref{eq:Kappa_H functional equations}): \begin{equation} \kappa_{H}\left(pm+j\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(m\right) \end{equation} then imply that: \begin{equation} \kappa_{H}\left(\left[pm+j\right]_{p^{n}}\right)=\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(\left[m\right]_{p^{n-1}}\right),\textrm{ }\forall m\in\mathbb{N}_{0},\textrm{ }\forall n\in\mathbb{N}_{1},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:functional equation truncation Lemma applied to kappa_H} \end{equation} where the function $\Phi_{j}$ from (\ref{eq:Relation between truncations and functional equations, version 1}) is, in this case: \begin{equation} \Phi_{j}\left(m,n\right)=\frac{\mu_{j}}{\mu_{0}}n \end{equation} The rising-continuability of $\psi_{H}$ and $\psi_{H}$ to the given series follow by the givens on $H$, which guarantee that, for each $\mathfrak{z}\in\mathbb{Z}_{p}$, $M_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)$ tends to $0$ in the standard $\left(p,q_{H}\right)$-adic frame as $n\rightarrow\infty$, and convergence is then guaranteed by the same arguments used for (\ref{eq:Limit of Fourier sum of v_p A_H hat}) and (\ref{eq:F limit of Gamma_H A_H hat Fourier series when alpha is 1}) from \textbf{Lemmata \ref{lem:v_p A_H hat summation formulae}} and \textbf{\ref{lem:1D gamma formula}}, respectively. All that remains is to verify the functional equations. \textbf{Theorem \ref{thm:rising-continuability of Generic H-type functional equations}} from Subsection \ref{subsec:3.2.1 -adic-Interpolation-of} then guarantees the uniqueness of $\psi_{H}$ and $\Psi_{H}$ as rising-continuous solutions of their respective systems of functional equations. \vphantom{} I. We pull out the $n=0$ term from $\psi_{H}\left(pm+j\right)$: \begin{align} \psi_{H}\left(pm+j\right) & =\frac{\mu_{j}}{p}\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\overbrace{\kappa_{H}\left(0\right)}^{1}+\sum_{n=1}^{\lambda_{p}\left(m\right)}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[pm+j\right]_{p^{n}}\right)\\ & =\frac{\mu_{j}}{p}\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+1+\sum_{n=1}^{\lambda_{p}\left(m\right)}\left(\frac{\mu_{0}}{p}\right)^{n}\frac{\mu_{j}}{p}\kappa_{H}\left(\left[m\right]_{p^{n-1}}\right)\\ & =1+\frac{\mu_{j}}{p}\left(\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=1}^{\lambda_{p}\left(m\right)}\left(\frac{\mu_{0}}{p}\right)^{n-1}\kappa_{H}\left(\left[m\right]_{p^{n-1}}\right)\right)\\ & =1+\frac{\mu_{j}}{p}\underbrace{\left(\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\right)}_{\psi_{H}\left(m\right)}\\ & =1+\frac{\mu_{j}}{p}\psi_{H}\left(m\right) \end{align} Consequently: \begin{equation} \psi_{H}\left(pm+j\right)=\frac{\mu_{j}}{p}\psi_{H}\left(m\right)+1,\textrm{ }\forall m\geq0\textrm{ \& }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:Little Psi_H functional equation on the integers} \end{equation} shows that (\ref{eq:Little Psi_H functional equation on the integers}) extends to hold for the rising-continuation of $\psi_{H}$, and that this rising-continuation is the \emph{unique }$\left(p,q_{H}\right)$-adic function satisfying (\ref{eq:Little Psi_H functional equations}). Finally, letting $m\in\mathbb{N}_{0}$ and setting $\mathfrak{z}=m$, the right-hand side of (\ref{eq:Rising-continuation of Little Psi_H}) becomes: \begin{align} \sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right) & =\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)+\sum_{n=\lambda_{p}\left(m\right)}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(m\right)\\ & \overset{\mathbb{C}}{=}\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)+\frac{\kappa_{H}\left(m\right)\left(\frac{\mu_{0}}{p}\right)^{\lambda_{p}\left(m\right)}}{1-\frac{\mu_{0}}{p}}\\ & =\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\\ & =\psi_{H}\left(m\right) \end{align} Hence, (\ref{eq:Rising-continuation of Little Psi_H}) converges to $\psi_{H}$ in the standard frame. \vphantom{} II. Pulling out $n=0$ from (\ref{eq:Definition of Big Psi_H}) yields: \begin{align} \Psi_{H}\left(m\right) & =-\beta_{H}\left(0\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{0}^{k}\left(m\right)\\ & +\sum_{n=1}^{\lambda_{p}\left(m\right)-1}\left(\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{n}^{k}\left(m\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right) \end{align} Then: \begin{align} \sum_{k=0}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{0}^{k}\left(m\right) & =\sum_{k=0}^{p-1}\left(\frac{1}{p}\sum_{j=0}^{p-1}H_{j}\left(0\right)e^{-\frac{2\pi ijk}{p}}\right)\left(e^{\frac{2\pi i}{p}\left(\left[m\right]_{p}-\left[m\right]_{p^{0}}\right)}\right)^{k}\\ & =\sum_{j=0}^{p-1}H_{j}\left(0\right)\frac{1}{p}\sum_{k=0}^{p-1}e^{-\frac{2\pi ijk}{p}}e^{\frac{2\pi ikm}{p}}\\ & =\sum_{j=0}^{p-1}H_{j}\left(0\right)\frac{1}{p}\sum_{k=0}^{p-1}e^{\frac{2\pi ik\left(m-j\right)}{p}}\\ & =\sum_{j=0}^{p-1}H_{j}\left(0\right)\left[j\overset{p}{\equiv}m\right]\\ & =H_{\left[m\right]_{p}}\left(0\right) \end{align} and so: \begin{equation} \sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{0}^{k}\left(m\right)=H_{\left[m\right]_{p}}\left(0\right)-\beta_{H}\left(0\right)\label{eq:Fourier sum (but not with 0) of beta_H} \end{equation} Consequently: \begin{align} \Psi_{H}\left(m\right) & =-\beta_{H}\left(0\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+H_{\left[m\right]_{p}}\left(0\right)-\beta_{H}\left(0\right)\\ & +\sum_{n=1}^{\lambda_{p}\left(m\right)-1}\left(\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{n}^{k}\left(m\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right) \end{align} Replacing $m$ with $pm+j$ (where at least one of $m$ and $j$ is non-zero), we use (\ref{eq:functional equation truncation Lemma applied to kappa_H}) and the functional equations for $M_{H}$, $\varepsilon_{n}$, and $\lambda_{p}$ to obtain: \begin{align} \Psi_{H}\left(m\right) & =-\beta_{H}\left(0\right)\frac{\mu_{j}}{p}\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+H_{j}\left(0\right)-\beta_{H}\left(0\right)\\ & +\sum_{n=1}^{\lambda_{p}\left(m\right)}\left(\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{n-1}^{k}\left(m\right)\varepsilon_{n}^{k}\left(j\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(\left[m\right]_{p^{n-1}}\right) \end{align} Because $j\in\left\{ 0,\ldots,p-1\right\} $, note that $\left[j\right]_{p^{n}}=j$ for all $n\geq1$. As such: \[ \varepsilon_{n}\left(j\right)=e^{\frac{2\pi i}{p^{n+1}}\left(\left[j\right]_{p^{n+1}}-\left[j\right]_{p^{n}}\right)}=e^{\frac{2\pi i}{p^{n+1}}\left(j-j\right)}=1,\textrm{ }\forall n\geq1 \] Re-indexing the $n$-sum then gives us: \begin{align} \Psi_{H}\left(m\right) & =-\beta_{H}\left(0\right)\frac{\mu_{j}}{p}\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+H_{j}\left(0\right)-\beta_{H}\left(0\right)\\ & +\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{n}^{k}\left(m\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\frac{\mu_{0}}{p}\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\\ & =H_{j}\left(0\right)-\beta_{H}\left(0\right)\\ & +\frac{\mu_{j}}{p}\underbrace{\left(-\beta_{H}\left(0\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)-1}\left(\sum_{k=1}^{p-1}\beta_{H}\left(\frac{k}{p}\right)\varepsilon_{n}^{k}\left(m\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\right)}_{\Psi_{H}\left(m\right)}\\ & =\frac{\mu_{j}}{p}\Psi_{H}\left(m\right)+H_{j}\left(0\right)-\beta_{H}\left(0\right) \end{align} Finally, letting $\mathfrak{z}=m$ where $m\in\mathbb{N}_{0}$, the right-hand side of (\ref{eq:Rising-continuation of Big Psi_H}) becomes: \begin{align} \sum_{n=0}^{\lambda_{p}\left(m\right)}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(m\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\\ +\sum_{n=\lambda_{p}\left(m\right)}^{\infty}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(m\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(m\right) \end{align} Here: \begin{equation} \varepsilon_{n}\left(m\right)=e^{\frac{2\pi i}{p^{n+1}}\left(\left[m\right]_{p^{n+1}}-\left[m\right]_{p^{n}}\right)}=e^{\frac{2\pi i}{p^{n+1}}\left(m-m\right)}=1,\textrm{ }\forall n\geq\lambda_{p}\left(m\right) \end{equation} So: \begin{align} \sum_{n=0}^{\lambda_{p}\left(m\right)}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(m\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\\ +\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\right)\sum_{n=\lambda_{p}\left(m\right)}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(m\right) \end{align} Summing this in the topology of $\mathbb{C}$ yields: \begin{align} \sum_{n=0}^{\lambda_{p}\left(m\right)}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(m\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right)\\ +\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}} \end{align} Using (\ref{eq:Fourier sum (but not with 0) of beta_H}) with $m=0$ (which makes $\varepsilon_{0}^{k}\left(m\right)$ identically equal to $1$) gives us: \begin{equation} \sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)=H_{0}\left(0\right)-\beta_{H}\left(0\right)=-\beta_{H}\left(0\right) \end{equation} and so, we are left with: \begin{equation} -\beta_{H}\left(0\right)\frac{M_{H}\left(m\right)}{1-\frac{\mu_{0}}{p}}+\sum_{n=0}^{\lambda_{p}\left(m\right)}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(m\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[m\right]_{p^{n}}\right) \end{equation} This, of course, is precisely $\Psi_{H}\left(m\right)$. This proves that (\ref{eq:Rising-continuation of Big Psi_H}) converges to $\Psi_{H}$ in the standard frame. Q.E.D. \begin{prop}[\textbf{Quasi-Integrability of $\psi_{H}$ \& $\Psi_{H}$}] Let $H$ be as given at the start of \emph{Section \pageref{sec:4.2 Fourier-Transforms-=00003D000026}}. Then: \vphantom{} I. \begin{equation} \psi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\begin{cases} \lim_{N\rightarrow\infty}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\frac{\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)}e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Little Psi_H as an F-limit} \end{equation} As such, $\psi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic quasi-integrability frame, and the function $\hat{\psi}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ defined by: \begin{equation} \hat{\psi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} \begin{cases} 0 & \textrm{if }t=0\\ v_{p}\left(t\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \frac{\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier Transform of Little Psi_H} \end{equation} is a Fourier transform of $\psi_{H}$. Hence: \begin{equation} \hat{\psi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}-N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Little Psi H N twiddle when alpha is 1} \end{equation} when $\alpha_{H}\left(0\right)=1$ and: \begin{equation} \tilde{\psi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\left(\frac{\mu_{0}}{p}\right)^{N}\frac{\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)}{1-\alpha_{H}\left(0\right)}+\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Little Psi H N twiddle when alpha is not 1} \end{equation} when $\alpha_{H}\left(0\right)\neq1$. \vphantom{} II. \begin{equation} \Psi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\lim_{N\rightarrow\infty}\sum_{0<\left\|t\right\|_{p}\leq p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Big Psi_H as an F-limit} \end{equation} As such, $\Psi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic quasi-integrability frame, and the function $\hat{\Psi}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ defined by: \begin{equation} \hat{\Psi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 0 & \textrm{if }t=0\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier Transform of Big Psi_H} \end{equation} is a Fourier transform of $\Psi_{H}$. Hence: \begin{equation} \tilde{\Psi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Big Psi H N twiddle} \end{equation} \end{prop} Proof: I. When $\alpha_{H}\left(0\right)=1$, (\ref{eq:Little Psi_H as an F-limit}) follows from the limit (\ref{eq:Limit of Fourier sum of v_p A_H hat}) from \textbf{Lemma \ref{lem:v_p A_H hat summation formulae}}. So, the $\alpha_{H}\left(0\right)=1$ case of (\ref{eq:Fourier Transform of Little Psi_H}) is then indeed a Fourier transform of $\psi_{H}$. This establishes the $\mathcal{F}_{p,q_{H}}$-quasi-integrability of $\psi_{H}$ when $\alpha_{H}\left(0\right)=1$. When $\alpha_{H}\left(0\right)\neq1$, comparing (\ref{eq:Convolution of dA_H and D_N}) and (\ref{eq:Rising-continuation of Little Psi_H}) we observe that $\left(1-\alpha_{H}\left(0\right)\right)\psi_{H}\left(\mathfrak{z}\right)$ then takes the values: \[ \lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \] at every $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. Since $H$ is semi-basic, applying our standard argument involving the $q$-adic decay of: \[ \left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \] to (\ref{eq:Little Psi_H functional equation on the integers}) shows that $\psi_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ converges in $\mathbb{Z}_{q_{H}}$ as $N\rightarrow\infty$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$. This establishes the rising-continuity of $\psi_{H}$. Consequently, the function $A_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q_{H}}$ defined by: \begin{equation} A_{H}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\begin{cases} \left(1-\alpha_{H}\left(0\right)\right)\psi_{H}\left(\mathfrak{z}\right) & \textrm{if }\mathfrak{z}\in\mathbb{N}_{0}\\ \left(1-\alpha_{H}\left(0\right)\right)\lim_{N\rightarrow\infty}\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & \textrm{if }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime} \end{cases}\label{eq:definition of A_H} \end{equation} is then rising-continuous, and its restriction to $\mathbb{N}_{0}$ is equal to $\left(1-\alpha_{H}\left(0\right)\right)\psi_{H}\left(\mathfrak{z}\right)$. By (\ref{eq:Little Psi_H as an F-limit}), it then follows that $\hat{A}_{H}$ is a Fourier transform of $A_{H}$ with respect to the standard $\left(p,q_{H}\right)$-adic frame, and thus, that $A_{H}$ is $\mathcal{F}_{p,q_{H}}$-quasi-integrable. So, when $\alpha_{H}\left(0\right)\neq1$, $\psi_{H}\left(\mathfrak{z}\right)=\left(1-\alpha_{H}\left(0\right)\right)A_{H}\left(\mathfrak{z}\right)$ is also $\mathcal{F}_{p,q_{H}}$-quasi-integrable, and the $\alpha_{H}\left(0\right)\neq1$ case of (\ref{eq:Fourier Transform of Little Psi_H}) is then a Fourier transform of $\psi_{H}$. As for formulae (\ref{eq:Little Psi H N twiddle when alpha is 1}) and (\ref{eq:Little Psi H N twiddle when alpha is not 1}), these are nothing more than restatements of \textbf{Lemma \ref{lem:v_p A_H hat summation formulae}} and (\ref{eq:Convolution of dA_H and D_N}), respectively. \vphantom{} II. (\ref{eq:Big Psi_H as an F-limit}) and (\ref{eq:Fourier Transform of Big Psi_H}) are exactly what we proved in \textbf{Lemma \ref{lem:1D gamma formula}}; (\ref{eq:Big Psi H N twiddle}) is just (\ref{eq:Gamma formula}). Q.E.D. \vphantom{} Now that we know that $\psi_{H}$ and $\Psi_{H}$ are quasi-integrable \emph{regardless} of the value of $\alpha_{H}\left(0\right)$ (or $p$, for that matter), we can then reverse-engineer formulae for the Fourier transform of $\chi_{H}$ \emph{regardless }of the value of $\alpha_{H}\left(0\right)$; we need only solve a system of linear equations to find the constants which relate $\psi_{H}$, $\Psi_{H}$ and $\chi_{H}$. We do the $p=2$ and $p\geq3$ cases separately. Both of the following theorems can then be immediately applied to obtain formulae for $\hat{\chi}_{H}$, $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$ \emph{and} $\chi_{H}\left(\mathfrak{z}\right)$. \begin{thm}[\textbf{Quasi-Integrability of $\chi_{H}$ for a $2$-Hydra map}] \label{thm:Quasi-integrability of an arbitrary 2-Hydra map}Let $H$ be any contracting non-singular semi-basic $2$-Hydra map which fixes $0$. Then:\index{chi{H}@$\chi_{H}$!quasi-integrability} \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,q_{H}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right)\psi_{H}\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Chi_H in terms of Little Psi_H for a 2-Hydra map} \end{equation} In particular, by \emph{(\ref{eq:Fourier Transform of Little Psi_H})}, this shows that $\chi_{H}$ is then quasi-integrable with respect to the standard $\left(2,q_{H}\right)$-adic quasi-integrability frame, with the function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{2}\rightarrow\overline{\mathbb{Q}}$ defined by: \index{chi{H}@$\chi_{H}$!Fourier transform} \begin{equation} \hat{\chi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} \begin{cases} -\gamma_{H}\left(\frac{1}{2}\right) & \textrm{if }t=0\\ \left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right)v_{2}\left(t\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)=1\\ -\gamma_{H}\left(\frac{1}{2}\right)\mathbf{1}_{0}\left(t\right)+\gamma_{H}\left(\frac{1}{2}\right)\frac{1-H^{\prime}\left(0\right)}{1-\alpha_{H}\left(0\right)}\hat{A}_{H}\left(t\right) & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Fourier Transform of Chi_H for a contracting semi-basic 2-Hydra map} \end{equation} being a Fourier transform of $\chi_{H}$. \end{thm} Proof: Let $A,B$ be undetermined constants, and let $\psi\left(\mathfrak{z}\right)=A\chi_{H}\left(\mathfrak{z}\right)+B$. Then, multiplying $\chi_{H}$'s functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}) by $A$ and adding $B$ gives: \begin{align} A\chi_{H}\left(2\mathfrak{z}+j\right)+B & =\frac{Aa_{j}\chi_{H}\left(\mathfrak{z}\right)+Ab_{j}}{d_{j}}+B\\ & \Updownarrow\\ \psi\left(2\mathfrak{z}+j\right) & =\frac{a_{j}\left(\psi\left(\mathfrak{z}\right)-B\right)+Ab_{j}}{d_{j}}+B\\ & =\frac{\mu_{j}}{2}\psi\left(\mathfrak{z}\right)+A\frac{b_{j}}{d_{j}}+B\left(1-\frac{a_{j}}{d_{j}}\right) \end{align} for $j\in\left\{ 0,1\right\} $. Since: \[ \psi_{H}\left(2\mathfrak{z}+j\right)=\frac{\mu_{j}}{2}\psi\left(\mathfrak{z}\right)+1 \] it then follows that to make $\psi=\psi_{H}$, we need for $A$ and $B$ to satisfy: \begin{align} \frac{b_{0}}{d_{0}}A+\left(1-\frac{a_{0}}{d_{0}}\right)B & =1\\ \frac{b_{1}}{d_{1}}A+\left(1-\frac{a_{1}}{d_{1}}\right)B & =1 \end{align} Now, since: \begin{align} \psi_{H}\left(\mathfrak{z}\right) & =A\chi_{H}\left(\mathfrak{z}\right)+B\\ & \Updownarrow\\ \chi_{H}\left(\mathfrak{z}\right) & =\frac{1}{A}\psi_{H}\left(\mathfrak{z}\right)-\frac{B}{A} \end{align} upon letting $C=\frac{1}{A}$ and $D=-\frac{B}{A}$, we have that $A=\frac{1}{C}$, $B=-\frac{D}{C}$, and hence, our linear system becomes: \begin{align} d_{0}C+\left(d_{0}-a_{0}\right)D & =b_{0}\\ d_{1}C+\left(d_{1}-a_{1}\right)D & =b_{1} \end{align} where: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)=C\psi_{H}\left(\mathfrak{z}\right)+D\label{eq:Chi_H =00003D00003D C Psi_H + D} \end{equation} The determinant of the coefficient matrix is: \begin{equation} a_{0}d_{1}-a_{1}d_{0} \end{equation} So, as long as $a_{0}d_{1}\neq a_{1}d_{0}$, the system admits a unique solution, which is easily computed to be: \begin{align} C & =\frac{a_{0}b_{1}-a_{1}b_{0}+b_{0}d_{1}-b_{1}d_{0}}{a_{0}d_{1}-a_{1}d_{0}}\\ D & =\frac{b_{1}d_{0}-d_{1}b_{0}}{a_{0}d_{1}-a_{1}d_{0}} \end{align} Finally, note that: \begin{align} a_{0}d_{1} & \neq a_{1}d_{0}\\ & \Updownarrow\\ 0 & \neq\frac{a_{0}}{d_{0}}-\frac{a_{1}}{d_{1}}\\ & =2\alpha_{H}\left(\frac{1}{2}\right) \end{align} and so: \[ D=\frac{b_{1}d_{0}-d_{1}b_{0}}{a_{0}d_{1}-a_{1}d_{0}}=-\frac{\frac{1}{2}\left(\frac{b_{0}}{d_{0}}-\frac{b_{1}}{d_{1}}\right)}{\frac{1}{2}\left(\frac{a_{0}}{d_{0}}-\frac{a_{1}}{d_{1}}\right)}=-\frac{\beta_{H}\left(\frac{1}{2}\right)}{\alpha_{H}\left(\frac{1}{2}\right)}=-\gamma_{H}\left(\frac{1}{2}\right) \] As for $C$, note that our requirement that $H\left(0\right)=0$ then forces $b_{0}=0$, and so: \begin{align} C & =\frac{a_{0}b_{1}-a_{1}b_{0}+b_{0}d_{1}-b_{1}d_{0}}{a_{0}d_{1}-a_{1}d_{0}}\\ & =\frac{a_{0}-d_{0}}{d_{0}}\frac{\frac{b_{1}}{d_{1}}}{\frac{a_{0}}{d_{0}}-\frac{a_{1}}{d_{1}}}\\ \left(\begin{array}{c} \alpha_{H}\left(\frac{1}{2}\right)=\frac{1}{2}\left(\frac{a_{0}}{a_{0}}-\frac{a_{1}}{a_{1}}\right)\\ \beta_{H}\left(\frac{1}{2}\right)=\frac{1}{2}\left(\frac{b_{0}}{d_{0}}-\frac{b_{1}}{d_{1}}\right)=-\frac{1}{2}\frac{b_{1}}{d_{1}} \end{array}\right); & =\frac{a_{0}-d_{0}}{d_{0}}\frac{-2\beta_{H}\left(\frac{1}{2}\right)}{2\alpha_{H}\left(\frac{1}{2}\right)}\\ & =\frac{d_{0}-a_{0}}{d_{0}}\gamma_{H}\left(\frac{1}{2}\right)\\ \left(H^{\prime}\left(0\right)=\frac{a_{0}}{d_{0}}\right); & =\left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right) \end{align} Plugging these values for $C$ and $D$ into (\ref{eq:Chi_H =00003D00003D C Psi_H + D}) produces (\ref{eq:Chi_H in terms of Little Psi_H for a 2-Hydra map}). Q.E.D. \vphantom{} Next up, our non-trivial formulae for $\tilde{\chi}_{H,N}$. \begin{cor} \label{cor:Non-trivial formula for Chi_H,N twiddle for a 2-hydra map, alpha arbitrary}Let\index{chi{H}@$\chi_{H}$!$N$th partial Fourier series} $H$ be a $2$-Hydra map like in \textbf{\emph{Theorem \ref{thm:Quasi-integrability of an arbitrary 2-Hydra map}}}, with $\hat{\chi}_{H}$ being as defined by \emph{(\ref{eq:Fourier Transform of Chi_H for a contracting semi-basic 2-Hydra map})}. Then: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)-\left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right)N\left(\frac{\mu_{0}}{2}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)\label{eq:Explicit Formula for Chi_H,N twiddle when rho is 2 and alpha is 1}\\ & +\left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)\nonumber \end{align} if $\alpha_{H}\left(0\right)=1$, and: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\gamma_{H}\left(\frac{1}{2}\right)\frac{1-H^{\prime}\left(0\right)}{1-\alpha_{H}\left(0\right)}\left(\frac{\mu_{0}}{2}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)\label{eq:Explicit Formula for Chi_H,N twiddle when rho is 2 and alpha is not 1}\\ & +\gamma_{H}\left(\frac{1}{2}\right)\left(1-H^{\prime}\left(0\right)\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)\nonumber \end{align} if $\alpha_{H}\left(0\right)\neq1$. \end{cor} Proof: Take (\ref{eq:Fourier Transform of Chi_H for a contracting semi-basic 2-Hydra map}) and use formulas from \textbf{Lemma \ref{lem:1D gamma formula}}, \textbf{Lemma (\ref{lem:v_p A_H hat summation formulae}}, and \textbf{Theorem \ref{thm:Properties of dA_H}}. Q.E.D. \begin{cor}[\textbf{$\mathcal{F}$-series for $\chi_{H}$ when $H$ is a $2$-Hydra map}] Let \label{cor:F-series for Chi_H for p equals 2}\index{chi{H}@$\chi_{H}$!mathcal{F}-series@$\mathcal{F}$-series}$H$ be as given in \textbf{\emph{Theorem \ref{thm:Quasi-integrability of an arbitrary 2-Hydra map}}}. Then: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,q_{H}}}{=}-\gamma_{H}\left(\frac{1}{2}\right)+\left(1-H^{\prime}\left(0\right)\right)\gamma_{H}\left(\frac{1}{2}\right)\sum_{n=0}^{\infty}\left(\frac{\mu_{0}}{2}\right)^{n}\left(\frac{a_{1}/d_{1}}{a_{0}/d_{0}}\right)^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Chi_H non-trivial series formula, rho =00003D00003D 2} \end{equation} \end{cor} Proof: Write out (\ref{eq:Chi_H in terms of Little Psi_H for a 2-Hydra map}), then use (\ref{eq:Definition of Kappa_H}) and $M_{H}$'s explicit formula (\textbf{Proposition \ref{prop:Explicit Formulas for M_H}}) to write: \begin{align} \left(\frac{\mu_{0}}{2}\right)^{n}\kappa_{H}\left(m\right) & =\left(\frac{\mu_{0}}{2}\right)^{n}\left(\frac{2}{\mu_{0}}\right)^{\lambda_{2}\left(m\right)}M_{H}\left(m\right)\\ & =\left(\frac{\mu_{0}}{2}\right)^{n}\left(\frac{2}{\mu_{0}}\right)^{\lambda_{2}\left(m\right)}\frac{\mu_{0}^{\#_{2:0}\left(m\right)}\mu_{1}^{\#_{1}\left(m\right)}}{2^{\lambda_{2}\left(m\right)}}\\ \left(\#_{2:0}\left(m\right)=\lambda_{2}\left(m\right)-\#_{1}\left(m\right)\right); & =\left(\frac{\mu_{0}}{2}\right)^{n}\left(\frac{\mu_{1}}{\mu_{0}}\right)^{\#_{1}\left(m\right)}\\ & =\left(\frac{\mu_{0}}{2}\right)^{n}\left(\frac{a_{1}/d_{1}}{a_{0}/d_{0}}\right)^{\#_{1}\left(m\right)} \end{align} Q.E.D. \vphantom{} Now we move to the general case. \begin{thm}[\textbf{Quasi-Integrability of $\chi_{H}$ for a $p$-Hydra map}] \index{chi{H}@$\chi_{H}$!mathcal{F}-series@$\mathcal{F}$-series}\label{thm:F-series for an arbitrary 1D Chi_H}Let $H$ be an arbitrary contracting, non-singular, semi-basic $p$-Hydra map which fixes $0$, where $p\geq2$. Suppose additionally that $H_{j}\left(0\right)\neq0$ for any $j\in\left\{ 1,\ldots,p-1\right\} $. Then\footnote{I call the following identity for $\chi_{H}$ the \textbf{$\mathcal{F}$-series} of $\chi_{H}$.}: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\beta_{H}\left(0\right)\psi_{H}\left(\mathfrak{z}\right)+\Psi_{H}\left(\mathfrak{z}\right),\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Chi_H in terms of Little Psi_H and Big Psi_H for arbitrary rho} \end{equation} In particular, using \emph{(\ref{eq:Fourier Transform of Little Psi_H})}, this shows that $\chi_{H}$ is then quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic quasi-integrability frame, with the function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ defined by: \index{chi{H}@$\chi_{H}$!Fourier transform} \begin{equation} \hat{\chi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} \begin{cases} 0 & \textrm{if }t=0\\ \left(\beta_{H}\left(0\right)v_{p}\left(t\right)+\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \frac{\beta_{H}\left(0\right)\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)}+\begin{cases} 0 & \textrm{if }t=0\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Fourier Transform of Chi_H for a contracting semi-basic rho-Hydra map} \end{equation} being a Fourier transform of $\chi_{H}$. \end{thm} Proof: We make the ansatz: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)=A\psi_{H}\left(\mathfrak{z}\right)+B\Psi_{H}\left(\mathfrak{z}\right) \end{equation} for constants $A$ and $B$. Using the functional equations (\ref{eq:Little Psi_H functional equations}) and (\ref{eq:Big Psi_H functional equations}), we then have that: \begin{align} \chi_{H}\left(p\mathfrak{z}+j\right) & =A\psi_{H}\left(p\mathfrak{z}+j\right)+B\Psi_{H}\left(p\mathfrak{z}+j\right)\\ & =A\frac{\mu_{j}}{p}\psi_{H}\left(\mathfrak{z}\right)+B\frac{\mu_{j}}{p}\Psi_{H}\left(\mathfrak{z}\right)+A+B\left(H_{j}\left(0\right)-\beta_{H}\left(0\right)\right) \end{align} and: \[ \chi_{H}\left(p\mathfrak{z}+j\right)=\frac{\mu_{j}}{p}\chi_{H}\left(\mathfrak{z}\right)+H_{j}\left(0\right)=A\frac{\mu_{j}}{p}\psi_{H}\left(\mathfrak{z}\right)+B\frac{\mu_{j}}{p}\Psi_{H}\left(\mathfrak{z}\right)+H_{j}\left(0\right) \] Combining these two formulae for $\chi_{H}\left(p\mathfrak{z}+j\right)$ yields: \[ A\frac{\mu_{j}}{p}\psi_{H}\left(\mathfrak{z}\right)+B\frac{\mu_{j}}{p}\Psi_{H}\left(\mathfrak{z}\right)+A+BH_{j}\left(0\right)-B\beta_{H}\left(0\right)=A\frac{\mu_{j}}{p}\psi_{H}\left(\mathfrak{z}\right)+B\frac{\mu_{j}}{p}\Psi_{H}\left(\mathfrak{z}\right)+H_{j}\left(0\right) \] and hence: \begin{equation} A+\left(H_{j}\left(0\right)-\beta_{H}\left(0\right)\right)B=H_{j}\left(0\right),\textrm{ }\forall j\in\left\{ 0,\ldots,p-1\right\} \label{eq:Seemingly over-determined system} \end{equation} Note that this is a system of $j$ linear equations in \emph{two }unknowns ($A$ and $B$). Most such systems are usually overdetermined and admit no solution. Not here, however. Since $H$ fixes $0$, $H_{0}\left(0\right)=0$, and the $j=0$ equation becomes: \begin{equation} A-\beta_{H}\left(0\right)B=0 \end{equation} Letting $j\in\left\{ 1,\ldots,p-1\right\} $, we plug $A=\beta_{H}\left(0\right)B$ into (\ref{eq:Seemingly over-determined system}) to obtain: \begin{align} \beta_{H}\left(0\right)B+\left(H_{j}\left(0\right)-\beta_{H}\left(0\right)\right)B & =H_{j}\left(0\right)\\ & \Updownarrow\\ H_{j}\left(0\right)B & =H_{j}\left(0\right)\\ \left(\textrm{if }H_{j}\left(0\right)\neq0\textrm{ for any }j\in\left\{ 1,\ldots,p-1\right\} \right); & \Updownarrow\\ B & =1 \end{align} Hence, our condition on the $H_{j}\left(0\right)$s guarantees that (\ref{eq:Seemingly over-determined system}) has $A=\beta_{H}\left(0\right)$ and $B=1$ as its unique solution. Using (\ref{eq:Fourier Transform of Little Psi_H}) and (\ref{eq:Fourier Transform of Big Psi_H}) for the Fourier transforms of $\psi_{H}$ and $\Psi_{H}$ yields (\ref{eq:Chi_H in terms of Little Psi_H and Big Psi_H for arbitrary rho}). Q.E.D. \begin{cor} \label{cor:Chi_H,N twiddle explicit formula, arbitrary p, arbitrary alpha}Let $H$ be as given in \textbf{\emph{Theorem \ref{thm:F-series for an arbitrary 1D Chi_H}}}, with $p\geq2$. Then: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\sum_{n=0}^{N-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Explicit Formula for Chi_H,N twiddle for arbitrary rho and alpha equals 1} \end{align} if $\alpha_{H}\left(0\right)=1$, and: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\frac{\beta_{H}\left(0\right)}{1-\alpha_{H}\left(0\right)}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\sum_{n=0}^{N-1}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Explicit Formula for Chi_H,N twiddle for arbitrary rho and alpha not equal to 1} \end{align} if $\alpha_{H}\left(0\right)\neq1$. \end{cor} Proof: Take (\ref{eq:Fourier Transform of Chi_H for a contracting semi-basic rho-Hydra map}) and use the formulas from \textbf{Lemmata \ref{lem:1D gamma formula}} and \textbf{\ref{lem:v_p A_H hat summation formulae}}. Q.E.D. \begin{cor}[\textbf{$\mathcal{F}$-series for $\chi_{H}$}] \label{cor:F-series for Chi_H, arbitrary p and alpha}Let\index{chi{H}@$\chi_{H}$!mathcal{F}-series@$\mathcal{F}$-series} $H$ be as given in \textbf{\emph{Theorem \ref{thm:F-series for an arbitrary 1D Chi_H}}}. Then: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\label{eq:Chi_H non-trivial series formula, rho not equal to 2} \end{equation} \end{cor} Proof: Take the $\mathcal{F}_{p,q_{H}}$-limit of the formulae in \textbf{Corollary \ref{cor:Chi_H,N twiddle explicit formula, arbitrary p, arbitrary alpha}}. Q.E.D. \vphantom{} Finally, using the Wiener Tauberian Theorem for $\left(p,q\right)$-adic measures (\textbf{Theorem \ref{thm:pq WTT for measures}}), we can prove the third chief result of this dissertation: the \textbf{Tauberian Spectral Theorem }for $p$-Hydra maps. \begin{thm}[\textbf{Tauberian Spectral Theorem for $p$-Hydra Maps}] \index{Hydra map!periodic points}\index{Hydra map!Tauberian Spectral Theorem}\label{thm:Periodic Points using WTT}\index{Wiener!Tauberian Theorem!left(p,qright)-adic@$\left(p,q\right)$-adic}\index{Hydra map!divergent trajectories}Let $H$ be as given in \textbf{\emph{Theorem \ref{thm:F-series for an arbitrary 1D Chi_H}}}. Let $x\in\mathbb{Z}\backslash\left\{ 0\right\} $, and let: \begin{equation} \hat{\chi}_{H}\left(t\right)=\begin{cases} \begin{cases} 0 & \textrm{if }t=0\\ \left(\beta_{H}\left(0\right)v_{p}\left(t\right)+\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \frac{\beta_{H}\left(0\right)\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)}+\begin{cases} 0 & \textrm{if }t=0\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \end{equation} \vphantom{} I. If $x$ is a periodic point of $H$, then the translates of the function $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are \emph{not }dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q_{H}}\right)$. \vphantom{} II. Suppose in addition that $H$ is integral\footnote{Propriety is defined alongside Hydra maps themselves on \pageref{def:p-Hydra map}; all the shortened $qx+1$ maps are integral.}, and that $\left\|H_{j}\left(0\right)\right\|_{q_{H}}=1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. If the translates of the function $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are \emph{not }dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q_{H}}\right)$, then either $x$ is a periodic point of $H$ or $x$ belongs to a divergent trajectory of $H$. When $\alpha_{H}\left(0\right)=1$ and $p=2$, we can also work with: \begin{equation} \hat{\chi}_{H}\left(t\right)=\begin{cases} -\gamma_{H}\left(\frac{1}{2}\right) & \textrm{if }t=0\\ \beta_{H}\left(0\right)v_{2}\left(t\right)\hat{A}_{H}\left(t\right) & \textrm{else } \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \end{equation} \end{thm} Proof: Let $H$ and $\hat{\chi}_{H}$ be as given. In particular, since $H$ is semi-basic, $\chi_{H}$ exists and the \index{Correspondence Principle} \textbf{Correspondence Principle }applies. Moreover, since $H$ is contracting and $H_{j}\left(0\right)\neq0$ for any $j\in\left\{ 1,\ldots,p-1\right\} $, $\chi_{H}$ is quasi-integrable with respect to the standard $\left(p,q\right)$-adic frame, and $\hat{\chi}_{H}$ is a Fourier transform of $\chi_{H}$. Note that the alternative formula for $\hat{\chi}_{H}$ when $p=2$ and $\alpha_{H}\left(0\right)=1$ follows from the fact that it and the first formula differ by the Fourier-Stieltjes transform of a degenerate measure, as was shown in \textbf{Corollary \ref{cor:Quasi-integrability of Chi_H for alpha equals 1}}. Now, let $x\in\mathbb{Z}\backslash\left\{ 0\right\} $. Since the constant function $x$ has $x\mathbf{1}_{0}\left(t\right)$ as its Fourier transform, the quasi-integrability of $\chi_{H}$ tells us that the difference $\chi_{H}\left(\mathfrak{z}\right)-x$ is quasi-integrable, and that $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ is then a Fourier transform of $\chi_{H}\left(\mathfrak{z}\right)-x$. So, $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ is then the Fourier-Stieltjes transform of a $\left(p,q\right)$-adic measure, and the Fourier series it generates converges in $\mathbb{C}_{q_{H}}$ (to $\chi_{H}\left(\mathfrak{z}\right)-x$) if and only if $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. As such, the \textbf{Wiener Tauberian Theorem for $\left(p,q\right)$-adic measures} (\textbf{Theorem \ref{thm:pq WTT for measures}}) guarantees that the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ will be dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q_{H}}\right)$ if and only if $\chi_{H}\left(\mathfrak{z}\right)-x\neq0$ for every $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. So: \vphantom{} I. Suppose $x$ is a periodic point of $H$. By the \textbf{Correspondence Principle} (specifically, \textbf{Corollary \ref{cor:CP v4}}), there exists a $\mathfrak{z}_{0}\in\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$ so that $\chi_{H}\left(\mathfrak{z}_{0}\right)=x$. Then, the WTT tells us that the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are \emph{not }dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. \vphantom{} II. Let $H$ be integral and let $\left\|H_{j}\left(0\right)\right\|_{q_{H}}=1$ for all $j\in\left\{ 1,\ldots,p-1\right\} $. Since $H$ is integral, this implies $H$ is proper (\textbf{Lemma \ref{lem:integrality lemma}} from page \pageref{lem:integrality lemma}). Next, suppose the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ are not dense in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$. Then, there is a $\mathfrak{z}_{0}\in\mathbb{Z}_{p}^{\prime}$ so that $\chi_{H}\left(\mathfrak{z}_{0}\right)=x$. If the $p$-adic digits of $\mathfrak{z}_{0}$ are eventually periodic (i.e., if $\mathfrak{z}_{0}\in\mathbb{Q}\cap\mathbb{Z}_{p}^{\prime}$), since $H$ is proper, \textbf{Corollary \ref{cor:CP v4}} guarantees that $x$ is a periodic point of $H$. If the $p$-adic digits of $\mathfrak{z}_{0}$ are not eventually periodic ($\mathfrak{z}_{0}\in\mathbb{Z}_{p}\backslash\mathbb{Q}$), \textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}}\textemdash applicable because of the hypotheses placed on $H$\textemdash then guarantees that $x$ belongs to a divergent trajectory of $H$. Q.E.D. \vphantom{} Finally, for the reader's benefit, here are the formulae associated to $\chi_{3}$\index{chi{3}@$\chi_{3}$!mathcal{F}-series@$\mathcal{F}$-series}. \index{chi{3}@$\chi_{3}$}We have its $\mathcal{F}$-series: \begin{equation} \chi_{3}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,3}}{=}-\frac{1}{2}+\frac{1}{4}\sum_{k=0}^{\infty}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{k}}\right)}}{2^{k}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Explicit formula for Chi_3} \end{equation} Plugging $n\in\mathbb{N}_{0}$ in for $\mathfrak{z}$ and summing the resultant geometric series in $\mathbb{R}$ yields a formula for $\chi_{3}$ on the integers: \begin{equation} \chi_{3}\left(n\right)=-\frac{1}{2}+\frac{1}{4}\frac{3^{\#_{1}\left(n\right)}}{2^{\lambda_{2}\left(n\right)}}+\frac{1}{4}\sum_{k=0}^{\lambda_{2}\left(n\right)}\frac{3^{\#_{1}\left(\left[n\right]_{2^{k}}\right)}}{2^{k}},\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Explicit Formula for Chi_3 on the integers} \end{equation} A nice choice of Fourier transform for $\chi_{3}$ is\index{chi{3}@$\chi_{3}$!Fourier transform}: \begin{equation} \hat{\chi}_{3}\left(t\right)=\begin{cases} -\frac{1}{2} & \textrm{if }t=0\\ \frac{1}{4}v_{2}\left(t\right)\hat{A}_{3}\left(t\right) & \textrm{if }t\neq0 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Formula for Chi_3 hat} \end{equation} Since it is the most natural case for comparison, here are corresponding formulae for $\chi_{5}$:\index{chi{5}@$\chi_{5}$} \begin{equation} \hat{\chi}_{5}\left(t\right)=\begin{cases} -\frac{1}{2} & \textrm{if }t=0\\ -\frac{1}{4}\hat{A}_{5}\left(t\right) & \textrm{if }t\neq0 \end{cases}\label{eq:Chi_5 Fourier transform} \end{equation} \index{chi{5}@$\chi_{5}$!Fourier transform}\index{chi{5}@$\chi_{5}$!mathcal{F}-series@$\mathcal{F}$-series} \begin{equation} \chi_{5}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,5}}{=}-\frac{1}{4}+\frac{1}{8}\sum_{k=0}^{\infty}\frac{5^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{k}}\right)}}{2^{k}}\label{eq:Chi_5 F-series} \end{equation} \begin{equation} \chi_{5}\left(n\right)=-\frac{1}{4}+\frac{1}{4}\frac{5^{\#_{1}\left(n\right)}}{2^{\lambda_{2}\left(n\right)}}+\frac{1}{8}\sum_{k=0}^{\lambda_{2}\left(n\right)}\frac{5^{\#_{1}\left(\left[n\right]_{2^{k}}\right)}}{2^{k}},\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Explicit Formula for Chi_5 on the integers} \end{equation} \index{chi{q}@$\chi_{q}$}\index{chi{q}@$\chi_{q}$!Fourier transform}\index{chi{q}@$\chi_{q}$!mathcal{F}-series@$\mathcal{F}$-series} More generally, for any odd prime $q\geq5$, letting $\chi_{q}$ be the numen of the Shortened $qx+1$ map, we have that: \begin{equation} \hat{\chi}_{q}\left(t\right)=\begin{cases} -\frac{1}{q-3} & \textrm{if }t=0\\ -\frac{2\hat{A}_{q}\left(t\right)}{\left(q-1\right)\left(q-3\right)} & \textrm{if }t\neq0 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2}\label{eq:Fourier transform of the numen of the shortened qx+1 map} \end{equation} with the $\mathcal{F}$-series: \begin{equation} \chi_{q}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,q}}{=}-\frac{1}{q-1}+\frac{1}{2\left(q-1\right)}\sum_{k=0}^{\infty}\frac{q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{k}}\right)}}{2^{k}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:F-series for Chi_q} \end{equation} and: \begin{equation} \chi_{q}\left(n\right)\overset{\mathbb{Q}}{=}-\frac{1}{q-1}+\frac{1}{2\left(q-1\right)}\frac{q^{\#_{1}\left(n\right)}}{2^{\lambda_{2}\left(n\right)}}+\frac{1}{2\left(q-1\right)}\sum_{k=0}^{\lambda_{2}\left(n\right)}\frac{q^{\#_{1}\left(\left[n\right]_{2^{k}}\right)}}{2^{k}},\textrm{ }\forall n\in\mathbb{N}_{0}\label{eq:Chi_q on N_0} \end{equation} Even though (\ref{eq:Fourier transform of the numen of the shortened qx+1 map}) is not the correct formula for $\hat{\chi}_{q}$ when $q=3$, it seems impossible that it is merely coincidental that the right-hand side of (\ref{eq:Fourier transform of the numen of the shortened qx+1 map}) is singular when $q\in\left\{ 1,3\right\} $. When $q=1$, it is easy to show that the shortened $qx+1$ map iterates every positive integer to $1$. Meanwhile, when $q=3$, it is conjectured that every positive integer goes to $1$. This suggests that for a given family $\mathcal{H}$ of Hydra maps with quasi-integrable numina $\hat{\chi}_{H}$ for each $H\in\mathcal{H}$, we have something along the lines of ``a given map $T\in\mathcal{H}$ has finitely many orbit classes (possibly all of which contain cycles) in $\mathbb{Z}$ if and only if the map $H\mapsto\hat{\chi}_{H}$ is 'singular' at $T$.'' \newpage{} \section{\label{sec:4.3 Salmagundi}Salmagundi} FOR THIS SUBSECTION, UNLESS STATED OTHERWISE, $H$ IS ALSO ASSUMED TO BE INTEGRAL AND NON-SINGULAR. \vphantom{} The title of this section is a wonderful word which my dictionary informs me, means either ``a dish of chopped meat, anchovies, eggs, onions, and seasoning'', or ``a general mixture; a miscellaneous collection''. It is the latter meaning that we are going to invoke here. This section contains a general mixture of results on $\chi_{H}$. Subsection \ref{subsec:4.3.1 p-adic-Estimates} demonstrates how the $\mathcal{F}$-series formula for $\chi_{H}$ (\textbf{Corollaries \ref{cor:F-series for Chi_H for p equals 2}} and \textbf{\ref{cor:F-series for Chi_H, arbitrary p and alpha}}) can be used to obtain a crude lower bound on the $q_{H}$-adic absolute value of $\chi_{H}\left(\mathfrak{z}\right)-\lambda$. It would be interesting to see if (and, if, then \emph{how}) this approach to estimation could be refined. \ref{subsec:4.3.2 A-Wisp-of} takes us back to the content of Subsection \ref{subsec:3.3.6 L^1 Convergence}\textemdash $L^{1}$ integrability of $\left(p,q\right)$-adic functions\textemdash and shows that the real-valued function $\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{q_{H}}$ will be integrable with respect to the real-valued Haar probability measure on $\mathbb{Z}_{p}$ whenever $\gcd\left(\mu_{j},q_{H}\right)>1$ for at least one $j\in\left\{ 0,\ldots,p-1\right\} $ (\textbf{Theorem \ref{thm:L^1 criterion for Chi_H}}). I also show that: \[ \lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q_{H}}d\mathfrak{z}\overset{\mathbb{R}}{=}0 \] whenever $H$ is semi-basic and contracting (\textbf{Theorem \ref{thm:L^1 convergence of Chi_H,N minus Chi_H,N twiddle}}). In Subsection \ref{subsec:4.3.3 Quick-Approach-of}, we revisit the circle of ideas around the \textbf{Square Root Lemma }(page \pageref{lem:square root lemma}). Finally, Subsection \ref{subsec:4.3.4 Archimedean-Estimates} establishes what I have termed the ``$L^{1}$-method'', a means of using geometric series universality to straddle the chasm between archimedean and non-archimedean topologies so as to obtain upper bounds on the ordinary, archimedean absolute value of $\chi_{H}\left(\mathfrak{z}\right)$ for appropriately chosen values of $\mathfrak{z}$ at which $\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$ converges to $\chi_{H}\left(\mathfrak{z}\right)$ in $\mathbb{Z}_{q_{H}}$ \emph{and }$\mathbb{R}$ as $N\rightarrow\infty$ (\textbf{Theorem \ref{thm:The L^1 method}}). I also sketch a sample application of this method to $\chi_{3}$. \subsection{\label{subsec:4.3.1 p-adic-Estimates}$p$-adic Estimates} While it is my belief that the most fruitful investigations into $\chi_{H}$ will come from studying $\hat{\chi}_{H}$ in the context of the $\left(p,q\right)$-adic Wiener Tauberian Theorem\textemdash assuming, of course, that a proper converse for the case of $\left(p,q\right)$-adic measures / quasi-integrable functions can be formulated and proven\textemdash the explicit series formulae from \textbf{Corollaries \ref{cor:F-series for Chi_H for p equals 2}} and \textbf{\ref{cor:F-series for Chi_H, arbitrary p and alpha}} may be of use in their own right. With sufficient knowledge of the $p$-adic digits of a given $\mathfrak{z}$\textemdash equivalently, with sufficient knowledge on the pattern of applications of branches of $H$ needed to ensure that a trajectory cycles\textemdash one can use the explicit series formulae to obtain $\left(p,q\right)$-adic estimates for $\chi_{H}$ which can be used to rule out periodic points. The more we know about $\mathfrak{z}$, the more refined the estimates can be. In that respect, even though the estimates detailed below are relatively simple, the method may be of use, so I might as well give the details. First, two notations to make our life easier: \begin{defn}[$u_{p}\left(\mathfrak{z}\right)$ \textbf{and} $K_{H}$] \ \vphantom{} I. We write \nomenclature{$u_{p}\left(\mathfrak{z}\right)$}{$\mathfrak{z}\left\|\mathfrak{z}\right\|_{p}$ }$u_{p}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}^{\times}\cup\left\{ 0\right\} $ to denote the function: \begin{equation} u_{p}\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\mathfrak{z}\left\|\mathfrak{z}\right\|_{p}\label{eq:Definition of u_p} \end{equation} \vphantom{} II. Define:\nomenclature{$K_{H}$}{ } \begin{equation} K_{H}\overset{\textrm{def}}{=}\max_{j\in\left\{ 1,\ldots,p-1\right\} }\left\|\kappa_{H}\left(j\right)\right\|_{q_{H}}=\max_{j\in\left\{ 1,\ldots,p-1\right\} }\left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q_{H}}\label{eq:Definition of K_H} \end{equation} Note that since $H$ is semi-basic, $\mu_{j}/\mu_{0}$ is a $q_{H}$-adic integer with $q_{H}$-adic absolute value $\leq1/q_{H}$. \end{defn} \begin{prop} We have: \begin{equation} \left\|\kappa_{H}\left(n\right)\right\|_{q_{H}}\leq K_{H}\leq\frac{1}{q_{H}}<1,\textrm{ }\forall n\in\mathbb{N}_{1}\label{eq:Kappa_H K_H bound} \end{equation} \end{prop} Proof: Use (\ref{eq:Kappa_H is rho-adically distributed}). Q.E.D. \vphantom{} Using the explicit formulae from \textbf{Proposition \ref{prop:formula for functions with p-adic structure}} we can now establish the promised estimates. There isn't much thought behind these estimates; it's mostly a matter of computation. So, to keep things from getting too unwieldy, I think it appropriate to show one concrete example\textemdash say, for $\chi_{3}$\textemdash before diving into the details. \begin{example} Using \textbf{Corollary \ref{cor:F-series for Chi_H for p equals 2}}, for the case of $\chi_{3}$, we have that: \begin{equation} \tilde{\chi}_{3,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}-\frac{1}{2}+\frac{N}{4}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}+\frac{1}{4}\sum_{n=0}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}},\textrm{ }\forall N\in\mathbb{N}_{1},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Explicit formula for Chi_3,N twiddle} \end{equation} Now, observe that $\left[\mathfrak{z}\right]_{2^{n}}=0$ for all $n\leq v_{2}\left(\mathfrak{z}\right)$; ex. $\left[8\right]_{2}=\left[8\right]_{4}=\left[8\right]_{8}=0$. As such, $n=v_{2}\left(\mathfrak{z}\right)+1$ is the smallest non-negative integer for which $\left[\mathfrak{z}\right]_{2^{n}}>0$, and hence, is the smallest integer for which $\kappa_{3}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)=3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}$ is $>1$. Indeed, $\kappa_{3}\left(\left[\mathfrak{z}\right]_{2^{v_{2}\left(\mathfrak{z}\right)+1}}\right)=3$ for all $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0\right\} $. So, $3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}$ will en be a multiple of $3$ for all $n\geq v_{2}\left(\mathfrak{z}\right)+1$. This tells us we can pull out terms from the $n$-sum which have a $3$-adic magnitude of $1$: \begin{align} \sum_{n=0}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}} & =\sum_{n=0}^{v_{2}\left(\mathfrak{z}\right)}\frac{1}{2^{n}}+\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\\ \left(3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}=1\textrm{ }\forall n\leq v_{2}\left(\mathfrak{z}\right)\right); & =\sum_{n=0}^{v_{2}\left(\mathfrak{z}\right)}\frac{1}{2^{n}}+\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\\ & =\frac{1-\frac{1}{2^{v_{2}\left(\mathfrak{z}\right)+1}}}{1-\frac{1}{2}}+\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\\ \left(\frac{1}{2^{v_{2}\left(\mathfrak{z}\right)}}=\left\|\mathfrak{z}\right\|_{2}\right); & =2-\left\|\mathfrak{z}\right\|_{2}+\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}} \end{align} which, for any $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0\right\} $, holds for all $N-1\geq v_{2}\left(\mathfrak{z}\right)+1$. So, (\ref{eq:Explicit formula for Chi_3,N twiddle}) becomes: \begin{equation} \tilde{\chi}_{3,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\frac{N}{4}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{N}}\right)}}{2^{N}}-\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}+\frac{1}{4}\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{N-1}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}\label{eq:Explicit formula for Chi_3,N twiddle with z pulled out} \end{equation} In terms of strings and composition sequence of trajectories, if $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}\cap\mathbb{Q}$ made $\chi_{3}\left(\mathfrak{z}\right)$ into a periodic point (belonging to the cycle $\Omega$) of the shortened Collatz map $T_{3}$ , then $v_{2}\left(\mathfrak{z}\right)$ would be the number of times we had to divide by $2$ to get from the penultimate element of $\Omega$ and return to $\chi_{3}\left(\mathfrak{z}\right)$. In this light, (\ref{eq:Explicit formula for Chi_3,N twiddle with z pulled out}) shows how we can obtain greater detail by using our knowledge of the number of times we divide by $2$ on the final ``fall'' in a given cycle. Channeling the Correspondence Principle, let $\mathfrak{z}\in\mathbb{Q}\cap\mathbb{Z}_{2}^{\prime}$. Then, (\ref{eq:Explicit formula for Chi_3,N twiddle with z pulled out}) converges in $\mathbb{Z}_{3}$ to $\chi_{3}\left(\mathfrak{z}\right)$ as $N\rightarrow\infty$, and we obtain: \begin{equation} \chi_{3}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{3}}{=}-\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}+\frac{1}{4}\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{\infty}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}^{\prime} \end{equation} If we knew that the first $1$ in the sequence of $\mathfrak{z}$'s $2$-adic digits was followed by $m$ consecutive zeroes, we would then be able to pull out all of the terms of the $n$-sum of $3$-adic magnitude $1/3$, and so on and so forth\textemdash but doing already takes us beyond the simple estimate I had in mind. \emph{That }(admittedly crude) estimate consists of taking the above, subtracting $\lambda\in\mathbb{Z}$, and then applying the $3$-adic absolute value: \[ \left\|\chi_{3}\left(\mathfrak{z}\right)-\lambda\right\|_{3}\overset{\mathbb{Z}_{3}}{=}\left\|\lambda+\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}-\frac{1}{4}\underbrace{\sum_{n=v_{2}\left(\mathfrak{z}\right)+1}^{\infty}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}}}_{\textrm{has }\left\|\cdot\right\|_{3}\leq1/3}\right\|_{3} \] By the $3$-adic ultrametric inequality, observe that $\left\|\lambda+\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}\right\|_{3}=1$ then forces: \[ \left\|\chi_{3}\left(\mathfrak{z}\right)-\lambda\right\|_{3}=\left\|\lambda+\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}\right\|_{3}=1 \] So, we have that $\chi_{3}\left(\mathfrak{z}\right)\neq\lambda$ for any $\lambda\in\mathbb{Z}$ and $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$ such that $\left\|\lambda+\frac{1}{4}\left\|\mathfrak{z}\right\|_{2}\right\|_{3}=1$. For example, if $\left\|\mathfrak{z}\right\|_{2}=2^{-n}$, then $\left\|2^{n+2}\lambda+1\right\|_{3}=1$ implies $\chi_{3}\left(\mathfrak{z}\right)\neq\lambda$. Hence, $2^{n+2}\lambda+1\overset{3}{\equiv}0$ is necessary in order for $\lambda$ to be a non-zero periodic point of Collatz with $\chi_{3}\left(\mathfrak{z}\right)=\lambda$ where $\left\|\mathfrak{z}\right\|_{2}=2^{-n}$. \end{example} \vphantom{} The following Corollary of the explicit formulae from Section \ref{sec:4.2 Fourier-Transforms-=00003D000026} gives the above result for all of the kinds of Hydra maps we considered in that section. \begin{cor} \label{cor:Non-archimedean estimates of Chi_H}Let $\lambda\in\mathbb{Z}_{q_{H}}$ and $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$.\index{chi{H}@$\chi_{H}$!non-archimedean estimates} Then: \begin{equation} \left\|\chi_{H}\left(\mathfrak{z}\right)-\lambda\right\|_{q_{H}}=\left\|\chi_{H}\left(\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}\right)\left\|\mathfrak{z}\right\|_{p}^{1-\log_{p}\mu_{0}}-\lambda\right\|_{q_{H}}>0\label{eq:Crude Estimate} \end{equation} whenever: \begin{equation} \left\|\chi_{H}\left(\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}\right)\left\|\mathfrak{z}\right\|_{p}^{1-\log_{p}\mu_{0}}-\lambda\right\|_{q_{H}}>K_{H}\sup_{n>v_{p}\left(\mathfrak{z}\right)}\left\|\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right\|_{q_{H}}\label{eq:Crude Estimate Prerequisite} \end{equation} In particular, $\chi_{H}\left(\mathfrak{z}\right)\neq\lambda$ whenever \emph{(\ref{eq:Crude Estimate Prerequisite})} holds true. \end{cor} Proof: Fix $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$ and recall that: \begin{equation} \kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)=\kappa_{H}\left(0\right)=1,\textrm{ }\forall n\leq v_{p}\left(\mathfrak{z}\right) \end{equation} \begin{equation} \varepsilon_{n}\left(\mathfrak{z}\right)=e^{2\pi i\left\{ \frac{\mathfrak{z}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathfrak{z}}{p^{n}}\right\} _{p}}=1,\textrm{ }\forall n<v_{p}\left(\mathfrak{z}\right) \end{equation} Now, let $N-1\geq v_{p}\left(\mathfrak{z}\right)+1$. Then, there are two cases. \vphantom{} I. Suppose $\alpha_{H}\left(0\right)=1$. Then, (\ref{eq:Explicit Formula for Chi_H,N twiddle for arbitrary rho and alpha equals 1}) becomes: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(0\right)\\ & +\sum_{n=0}^{v_{p}\left(\mathfrak{z}\right)-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(1\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(0\right)\\ & +\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{v_{p}\left(\mathfrak{z}\right)}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}\kappa_{H}\left(0\right)\\ & +\beta_{H}\left(0\right)\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} Here: \begin{align} \sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right) & =\sum_{j=0}^{p-1}\left(\frac{1}{p}\sum_{k=0}^{p-1}H_{k}\left(0\right)e^{-\frac{2\pi ijk}{p}}\right)\\ & =\sum_{k=0}^{p-1}H_{k}\left(0\right)\frac{1}{p}\sum_{j=0}^{p-1}e^{-\frac{2\pi ijk}{p}}\\ & =\sum_{k=0}^{p-1}H_{k}\left(0\right)\left[k\overset{p}{\equiv}0\right]\\ & =H_{0}\left(0\right)\\ & =0 \end{align} So: \begin{equation} \sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)=-\beta_{H}\left(0\right)+\underbrace{\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)}_{0}=-\beta_{H}\left(0\right)\label{eq:Sum of beta H of j/p for j in 1,...,p-1} \end{equation} On the other hand: \begin{align} \sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{v_{p}\left(\mathfrak{z}\right)}^{j}\left(\mathfrak{z}\right) & =\sum_{j=0}^{p-1}\left(\frac{1}{p}\sum_{k=0}^{p-1}H_{k}\left(0\right)e^{-\frac{2\pi ijk}{p}}\right)\left(e^{2\pi i\left\{ \frac{\mathfrak{z}}{p^{v_{p}\left(\mathfrak{z}\right)+1}}\right\} _{p}-\frac{2\pi i}{p}\left\{ \frac{\mathfrak{z}}{p^{v_{p}\left(\mathfrak{z}\right)}}\right\} _{p}}\right)^{j}\\ & =\sum_{j=0}^{p-1}\left(\frac{1}{p}\sum_{k=0}^{p-1}H_{k}\left(0\right)e^{-\frac{2\pi ijk}{p}}\right)e^{2\pi i\frac{j\left[\left\|\mathfrak{z}\right\|_{p}\mathfrak{z}\right]_{p}}{p}}\\ & =\frac{1}{p}\sum_{k=0}^{p-1}H_{k}\left(0\right)\sum_{j=0}^{p-1}e^{\frac{2\pi ij}{p}\left(\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}-k\right)}\\ & =\sum_{k=0}^{p-1}H_{k}\left(0\right)\left[\left\|\mathfrak{z}\right\|_{p}\mathfrak{z}\overset{p}{\equiv}k\right]\\ & =H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right) \end{align} and so: \begin{equation} \sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{v_{p}\left(\mathfrak{z}\right)}^{j}\left(\mathfrak{z}\right)=-\beta_{H}\left(0\right)+\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{v_{p}\left(\mathfrak{z}\right)}^{j}\left(\mathfrak{z}\right)=H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)-\beta_{H}\left(0\right)\label{eq:Sum of beta H of j/p times the epsilon for j in 1,...,p-1} \end{equation} With this, $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$ becomes: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\\ & +\beta_{H}\left(0\right)\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} Letting $N\rightarrow\infty$, the above converges in $\mathbb{C}_{q_{H}}$ to: \begin{align} \chi_{H}\left(\mathfrak{z}\right) & \overset{\mathbb{C}_{q_{H}}}{=}H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}+\beta_{H}\left(0\right)\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{\infty}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} The $n$-sum on the top line is the $j=0$th term of the sum on the bottom line. This leaves us with: \[ \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{C}_{q_{H}}}{=}H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}+\sum_{n=v_{p}\left(\mathfrak{z}\right)+1}^{\infty}\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \] Because $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, observe that $\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q_{H}}\leq K_{H}$ for all $n>v_{p}\left(\mathfrak{z}\right)$. As such, subtracting $\lambda\in\mathbb{Z}_{q_{H}}$ from both sides, we can apply the $q_{H}$-adic ultrametric inequality. This gives us the equality: \begin{equation} \left\|\chi_{H}\left(\mathfrak{z}\right)-\lambda\right\|_{q_{H}}=\left\|H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}-\lambda\right\|_{q_{H}} \end{equation} provided that the condition: \begin{equation} \left\|H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}-\lambda\right\|_{q_{H}}>K_{H}\sup_{n>v_{p}\left(\mathfrak{z}\right)}\left\|\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right\|_{q_{H}} \end{equation} is satisfied. Finally, writing: \begin{equation} \left(\frac{\mu_{0}}{p}\right)^{v_{p}\left(\mathfrak{z}\right)}=\left(p^{-v_{p}\left(\mathfrak{z}\right)}\right)^{1-\log_{p}\mu_{0}}=\left\|\mathfrak{z}\right\|_{p}^{1-\log_{p}\mu_{0}} \end{equation} we obtain (I) by noting that: \begin{equation} H_{\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}}\left(0\right)=\chi_{H}\left(\left[u_{p}\left(\mathfrak{z}\right)\right]_{p}\right) \end{equation} \vphantom{} II. Suppose $\alpha_{H}\left(0\right)\neq1$. Then (\ref{eq:Explicit Formula for Chi_H,N twiddle for arbitrary rho and alpha not equal to 1}) becomes: \begin{align} \tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\frac{\beta_{H}\left(0\right)}{1-\alpha_{H}\left(0\right)}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} The only term in this equation that differs from the $\alpha_{H}\left(0\right)=1$ case is the initial term: \begin{equation} \frac{\beta_{H}\left(0\right)}{1-\alpha_{H}\left(0\right)}\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{equation} The corresponding term of the $\alpha_{H}\left(0\right)=1$ case was: \begin{equation} -\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{equation} Since both of these terms converge $q_{H}$-adically to zero as $N\rightarrow\infty$ whenever $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, all of the computations done for the $\alpha_{H}\left(0\right)=1$ case will apply here\textemdash the only term that differs between the two cases ends up vanishing as $N\rightarrow\infty$. Q.E.D. \subsection{\label{subsec:4.3.2 A-Wisp-of}A Wisp of $L^{1}$\index{non-archimedean!$L^{1}$ theory}} One of my goals for future research is to pursue $\left(p,q\right)$-adic analysis in the setting of $L_{\mathbb{R}}^{1}$. To that end, in this subsection, a simple sufficient condition is establish for determining when $\chi_{H}\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q_{H}}\right)$. We begin by computing the van der Put series for $\chi_{H}$. When $p=2$, this is easily done. For $p\geq3$, we will have to settle on a mutated sort of van der Put series, seeing as the van der Put coefficients of $\chi_{H}$ are not as well-behaved for $p\geq3$ as they are for $p=2$. \begin{prop}[\textbf{``van der Put'' series for $\chi_{H}$}] \index{chi{H}@$\chi_{H}$!van der Put series}\label{prop:van der Put series for Chi_H}When $p=2$: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\frac{b_{1}}{a_{1}}\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\mathfrak{z}\overset{2^{\lambda_{2}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2}\label{eq:Simplified vdP series for Chi_H when rho is 2} \end{equation} More generally, for any $p$: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right)+\sum_{n=1}^{\infty}\left(\sum_{j=1}^{p-1}H_{j}\left(0\right)\left[u_{p}\left(\mathfrak{z}-n\right)\overset{p}{\equiv}j\right]\right)M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Quasi vdP series for Chi__H} \end{equation} \end{prop} Proof: Let $n\in\mathbb{N}_{1}$ be arbitrary, and let $\mathbf{j}=\left(j_{1},\ldots,j_{\lambda_{p}\left(n\right)}\right)$ be the shortest string in $\textrm{String}\left(p\right)$ representing $n$. Since $n\neq0$, observe that $j_{\lambda_{p}\left(n\right)}\in\left\{ 1,\ldots,p-1\right\} $. Using \textbf{Proposition \ref{prop:Explicit formula for Chi_H of bold j} }(page \pageref{prop:Explicit formula for Chi_H of bold j}), we have that: \begin{equation} \chi_{H}\left(n\right)=\sum_{m=1}^{\lambda_{p}\left(n\right)}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m} \end{equation} Note that $n_{-}$ is the integer represented by the string $\mathbf{j}^{\prime}=\left(j_{1},\ldots,j_{\lambda_{p}\left(n\right)-1}\right)$. Nevertheless, since $H\left(0\right)=0$, $\chi_{H}\left(\mathbf{j}^{\prime}\right)=\chi_{H}\left(n_{-}\right)$, since removing zeroes at the right end of a string does not affect the value of $\chi_{H}$. Consequently: \begin{equation} \chi_{H}\left(n_{-}\right)=\sum_{m=1}^{\lambda_{p}\left(n\right)-1}\frac{b_{j_{m}}}{a_{j_{m}}}\left(\prod_{k=1}^{m}\mu_{j_{k}}\right)p^{-m} \end{equation} Thus: \begin{align} c_{n}\left(\chi_{H}\right) & =\chi_{H}\left(n\right)-\chi_{H}\left(n_{-}\right)=\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}\left(\prod_{k=1}^{\lambda_{p}\left(n\right)}\mu_{j_{k}}\right)p^{-\lambda_{p}\left(n\right)}=\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right) \end{align} which tends to $0$ $q_{H}$-adically as the number of non-zero $p$-adic digits in $n$ tends to $\infty$, which gives another proof of the rising-continuity of $\chi_{H}$; our first proof of this fact was implicit in \textbf{Lemma \ref{lem:Unique rising continuation and p-adic functional equation of Chi_H}}. Since $\chi_{H}$ is rising-continuous, it is represented everywhere by its van der Put series: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\sum_{n=1}^{\infty}\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}\label{eq:Unsimplified van der Put series for Chi_H} \end{equation} where the $n=0$ term is $0$ because $c_{0}\left(\chi_{H}\right)=\chi_{H}\left(0\right)=0$. When $p=2$, note that $j_{\lambda_{p}\left(n\right)}=1$ for all $n\geq1$; that is to say, the right-most $2$-adic digit of every non-zero integer is $1$. This allows us to write: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\frac{b_{1}}{a_{1}}\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\mathfrak{z}\overset{2^{\lambda_{2}\left(n\right)}}{\equiv}n\right],\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2} \end{equation} When $p\geq3$, we use the formal identity: \begin{equation} \sum_{n=1}^{p^{N}-1}f\left(n\right)=\sum_{n=1}^{p^{N-1}-1}f\left(n\right)+\sum_{\ell=1}^{p-1}\sum_{n=1}^{p^{N-1}-1}f\left(n+\ell p^{N-1}\right) \end{equation} When applying this to $\chi_{H}$'s van der Put series, observe that $j_{\lambda_{p}\left(n+\ell p^{N-1}\right)}$, the right-most $p$-adic digit of $n+\ell p^{N-1}$, is then equal to $\ell$. Also: \begin{equation} M_{H}\left(n+\ell p^{N-1}\right)=\frac{\mu_{\ell}}{p}M_{H}\left(n\right) \end{equation} and: \begin{equation} \left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n+\ell p^{N-1}\right)}}{\equiv}n+\ell p^{N-1}\right]=\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right] \end{equation} These hold for all $n\in\left\{ 1,\ldots,p^{N-1}-1\right\} $ and all $\ell\in\left\{ 1,\ldots,p-1\right\} $. As such: \begin{align} \sum_{n=1}^{p^{N}-1}\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right] & =\sum_{n=1}^{p^{N-1}-1}\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\\ & +\sum_{\ell=1}^{p-1}\sum_{n=1}^{p^{N-1}-1}\frac{b_{\ell}}{a_{\ell}}\frac{\mu_{\ell}}{p}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right]\\ & =\sum_{n=1}^{p^{N-1}-1}\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\\ & +\sum_{\ell=1}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{n=1}^{p^{N-1}-1}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right] \end{align} Noting that: \begin{equation} \sum_{n=1}^{p^{N}-1}\frac{b_{j_{\lambda_{p}\left(n\right)}}}{a_{j_{\lambda_{p}\left(n\right)}}}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]=S_{p:N}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right) \end{equation} we can simplify the previous equation to: \[ S_{p:N}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right)=S_{p:N-1}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right)+\sum_{\ell=1}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{n=1}^{p^{N-1}-1}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right] \] Next, we apply the truncated van der Put identity (\textbf{Proposition \ref{prop:vdP identity}}) to note that: \begin{equation} S_{p:N}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right)-S_{p:N-1}\left\{ \chi_{H}\right\} \left(\mathfrak{z}\right)=\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right) \end{equation} So: \begin{align} \chi_{H}\left(\left[\mathfrak{z}\right]_{p^{K}}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right) & =\sum_{N=2}^{K}\left(\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right)\\ & =\sum_{N=2}^{K}\sum_{\ell=1}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{n=1}^{p^{N-1}-1}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right]\\ & =\sum_{\ell=1}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{N=1}^{K-1}\sum_{n=1}^{p^{N}-1}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N+1}}{\equiv}n+\ell p^{N}\right]\\ & =\sum_{\ell=1}^{p-1}\frac{b_{\ell}}{d_{\ell}}\sum_{n=1}^{p^{K-1}-1}\sum_{N=\lambda_{p}\left(n\right)}^{K-1}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{N+1}}{\equiv}n+\ell p^{N}\right]\\ \left(H_{\ell}\left(0\right)=\frac{b_{\ell}}{d_{\ell}}\right); & =\sum_{\ell=1}^{p-1}H_{\ell}\left(0\right)\sum_{n=1}^{p^{K-1}-1}M_{H}\left(n\right)\sum_{N=\lambda_{p}\left(n\right)+1}^{K}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+\ell p^{N-1}\right] \end{align} Letting $K\rightarrow\infty$ yields: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\sum_{N=\lambda_{p}\left(n\right)+1}^{\infty}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right]\label{eq:Nearly finished with van-der-put computation for Chi_H} \end{equation} \[ \chi_{H}\left(\mathfrak{z}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right)\overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\sum_{N=\lambda_{p}\left(n\right)+1}^{\infty}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right] \] Now, fix $n$ and $j$. Then: \[ \sum_{N=\lambda_{p}\left(n\right)+1}^{\infty}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right]=\sum_{N=\lambda_{p}\left(n\right)+1}^{\infty}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right]\left[\mathfrak{z}\overset{p^{N-1}}{\equiv}n\right] \] Since $j$ is co-prime to $p$, observe that the congruence $\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}$ is equivalent to the following pair of conditions: i. $\mathfrak{z}\overset{p^{N-1}}{\equiv}n$; ii. $v_{p}\left(\frac{\mathfrak{z}-n}{p^{N}}-\frac{j}{p}\right)\geq0$. So: \begin{equation} \left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right]=\left[\mathfrak{z}\overset{p^{N-1}}{\equiv}n\right]\left[v_{p}\left(\frac{\mathfrak{z}-n}{p^{N}}-\frac{j}{p}\right)\geq0\right] \end{equation} Here: \begin{align} v_{p}\left(\frac{\mathfrak{z}-n}{p^{N}}-\frac{j}{p}\right) & \geq0\\ \left(1+v_{p}\left(\mathfrak{y}\right)=v_{p}\left(p\mathfrak{y}\right)\right); & \Updownarrow\\ v_{p}\left(\frac{\mathfrak{z}-n}{p^{N-1}}-j\right) & \geq1\\ & \Updownarrow\\ \left\|\frac{\mathfrak{z}-n}{p^{N-1}}-j\right\|_{p} & \leq\frac{1}{p} \end{align} Because $\left\|j\right\|_{p}=1$, observe that if $\left\|\frac{\mathfrak{z}-n}{p^{N-1}}\right\|_{p}<1$, we can then use $p$-adic ultrametric inequality to write: \begin{equation} \frac{1}{p}\geq\left\|\frac{\mathfrak{z}-n}{p^{N-1}}-j\right\|_{p}=\max\left\{ \left\|\frac{\mathfrak{z}-n}{p^{N-1}}\right\|_{p},\left\|j\right\|_{p}\right\} =1 \end{equation} Of course, this equality is impossible. So, if we have $\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}$, it \emph{must} be that $\left\|\frac{\mathfrak{z}-n}{p^{N-1}}\right\|_{p}=1$. This latter condition occurs if and only if $N=v_{p}\left(\mathfrak{z}-n\right)+1$. Putting it all together, (\ref{eq:Nearly finished with van-der-put computation for Chi_H}) becomes: \begin{align} \chi_{H}\left(\mathfrak{z}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right) & \overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\sum_{N=\lambda_{p}\left(n\right)+1}^{\infty}\left[\mathfrak{z}\overset{p^{N}}{\equiv}n+jp^{N-1}\right]\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\lambda_{p}\left(n\right)\leq v_{p}\left(\mathfrak{z}-n\right)\right]\left[\mathfrak{z}\overset{p^{v_{p}\left(\mathfrak{z}-n\right)+1}}{\equiv}n+jp^{v_{p}\left(\mathfrak{z}-n\right)}\right]\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\left\|\mathfrak{z}-n\right\|_{p}\leq p^{-\lambda_{p}\left(n\right)}\right]\left[\mathfrak{z}\overset{p^{v_{p}\left(\mathfrak{z}-n\right)+1}}{\equiv}n+jp^{v_{p}\left(\mathfrak{z}-n\right)}\right]\\ & \overset{\mathbb{Z}_{q_{H}}}{=}\sum_{j=1}^{p-1}H_{j}\left(0\right)\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\left[\mathfrak{z}\overset{p^{v_{p}\left(\mathfrak{z}-n\right)+1}}{\equiv}n+jp^{v_{p}\left(\mathfrak{z}-n\right)}\right] \end{align} Here: \begin{align} \mathfrak{z} & \overset{p^{v_{p}\left(\mathfrak{z}-n\right)+1}}{\equiv}n+jp^{v_{p}\left(\mathfrak{z}-n\right)}\\ & \Updownarrow\\ \mathfrak{z}-n & \in jp^{v_{p}\left(\mathfrak{z}-n\right)}+p^{v_{p}\left(\mathfrak{z}-n\right)+1}\mathbb{Z}_{p}\\ & \Updownarrow\\ \left(\mathfrak{z}-n\right)\left\|\mathfrak{z}-n\right\|_{p} & \in j+p\mathbb{Z}_{p}\\ & \Updownarrow\\ u_{p}\left(\mathfrak{z}-n\right) & \overset{p}{\equiv}j \end{align} and so: \begin{align} \chi_{H}\left(\mathfrak{z}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right) & \overset{\mathbb{Z}_{q_{H}}}{=}\sum_{n=1}^{\infty}M_{H}\left(n\right)\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]\sum_{j=1}^{p-1}H_{j}\left(0\right)\left[u_{p}\left(\mathfrak{z}-n\right)\overset{p}{\equiv}j\right] \end{align} Q.E.D. \vphantom{} With these formulae, we can directly estimate the $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q_{H}}\right)$ norm of $\chi_{H}$. \begin{prop} \label{prop:L_R^1 norm of Chi_H}\index{chi{H}@$\chi_{H}$!L{mathbb{R}}{1}-norm@$L_{\mathbb{R}}^{1}$-norm} \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{q_{H}}d\mathfrak{z}\leq\frac{1}{p}\left(\sum_{j=1}^{p-1}\left\|\chi_{H}\left(j\right)\right\|_{q_{H}}\right)\sum_{n=0}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q_{H}}}{p^{\lambda_{p}\left(n\right)}}\label{eq:real L^1 estimate for Chi_H} \end{equation} \end{prop} Proof: For brevity, let $q=q_{H}$. We use \textbf{Proposition \ref{prop:van der Put series for Chi_H}}. This gives: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & \leq\sum_{n=1}^{\infty}\sum_{j=1}^{p-1}\left\|H_{j}\left(0\right)M_{H}\left(n\right)\right\|_{q}\int_{\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{z}-n\right)\overset{p}{\equiv}j\right]\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z}\\ & +\int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right)\right\|_{q}d\mathfrak{z} \end{align} Using the translation invariance of $d\mathfrak{z}$, we can write: \begin{align} \int_{\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{z}-n\right)\overset{p}{\equiv}j\right]\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z} & \overset{\mathbb{R}}{=}\int_{\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{z}\right)\overset{p}{\equiv}j\right]\left[\mathfrak{z}+n\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]d\mathfrak{z}\\ & =\int_{\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{z}\right)\overset{p}{\equiv}j\right]\left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}0\right]d\mathfrak{z}\\ & =\int_{p^{\lambda_{p}\left(n\right)}\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{z}\right)\overset{p}{\equiv}j\right]d\mathfrak{z}\\ \left(\mathfrak{y}=\frac{\mathfrak{z}}{p^{\lambda_{p}\left(n\right)}}\right); & =\frac{1}{p^{\lambda_{p}\left(n\right)}}\int_{\mathbb{Z}_{p}}\left[u_{p}\left(p^{\lambda_{p}\left(n\right)}\mathfrak{y}\right)\overset{p}{\equiv}j\right]d\mathfrak{z}\\ \left(u_{p}\left(p\mathfrak{y}\right)=u_{p}\left(\mathfrak{y}\right)\right); & =\frac{1}{p^{\lambda_{p}\left(n\right)}}\int_{\mathbb{Z}_{p}}\left[u_{p}\left(\mathfrak{y}\right)\overset{p}{\equiv}j\right]d\mathfrak{z}\\ \left(j\in\left\{ 1,\ldots,p-1\right\} :u_{p}\left(\mathfrak{y}\right)\overset{p}{\equiv}j\Leftrightarrow\mathfrak{y}\overset{p}{\equiv}j\right); & =\frac{1}{p^{\lambda_{p}\left(n\right)}}\int_{\mathbb{Z}_{p}}\left[\mathfrak{y}\overset{p}{\equiv}j\right]d\mathfrak{z}\\ & =\frac{1}{p^{\lambda_{p}\left(n\right)+1}} \end{align} Thus: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & \leq\sum_{n=1}^{\infty}\sum_{j=1}^{p-1}\frac{\left\|H_{j}\left(0\right)M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)+1}}+\int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\left[\mathfrak{z}\right]_{p}\right)\right\|_{q}d\mathfrak{z}\\ & =\sum_{n=1}^{\infty}\sum_{j=1}^{p-1}\frac{\left\|H_{j}\left(0\right)M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)+1}}+\frac{1}{p}\sum_{j=0}^{p-1}\left\|\chi_{H}\left(j\right)\right\|_{q}\\ \left(\chi_{H}\left(0\right)=0\right); & =\frac{1}{p}\sum_{j=1}^{p-1}\left(\left\|\chi_{H}\left(j\right)\right\|_{q}+\sum_{n=1}^{\infty}\left\|H_{j}\left(0\right)\right\|_{q}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\right) \end{align} Finally, since $\chi_{H}\left(j\right)=H_{j}\left(0\right)$ for all $j\in\left\{ 0,\ldots,p-1\right\} $, we have: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\chi_{H}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & \leq\frac{1}{p}\sum_{j=1}^{p-1}\left(\left\|\chi_{H}\left(j\right)\right\|_{q}+\sum_{n=1}^{\infty}\left\|H_{j}\left(0\right)\right\|_{q}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\right)\\ & =\frac{1}{p}\left(\sum_{j=1}^{p-1}\left\|\chi_{H}\left(j\right)\right\|_{q}\right)\left(1+\sum_{n=1}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\right)\\ \left(M_{H}\left(0\right)=1\right); & =\frac{1}{p}\left(\sum_{j=1}^{p-1}\left\|\chi_{H}\left(j\right)\right\|_{q}\right)\sum_{n=0}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}} \end{align} Q.E.D. \vphantom{} With this, we can now state the sufficient condition for $\chi_{H}$ to be in $L_{\mathbb{R}}^{1}$. \begin{thm}[\textbf{Criterion for $\chi_{H}\in L_{\mathbb{R}}^{1}$}] \label{thm:L^1 criterion for Chi_H}$\chi_{H}$ is an element of $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q_{H}}\right)$ whenever: \begin{equation} \sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q_{H}}<p\label{eq:Criterion for Chi_H to be real L^1 integrable} \end{equation} In particular, since the $\mu_{j}$s are integers, this shows that $\chi_{H}\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q_{H}}\right)$ occurs whenever $\gcd\left(\mu_{j},q_{H}\right)>1$ for at least one $j\in\left\{ 0,\ldots,p-1\right\} $. \end{thm} Proof: For brevity, let $q=q_{H}$. By \textbf{Proposition \ref{prop:L_R^1 norm of Chi_H}}, to show that $\chi_{H}$ has finite $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ norm, it suffices to prove that: \begin{equation} \sum_{n=0}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}<\infty \end{equation} To do this, we use a lambda decomposition: \begin{equation} \sum_{n=0}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}}\overset{\mathbb{R}}{=}1+\sum_{m=1}^{\infty}\frac{1}{p^{m}}\sum_{n=p^{m-1}}^{p^{m}-1}\left\|M_{H}\left(n\right)\right\|_{q}\label{eq:M_H q-adic absolute value lambda decomposition} \end{equation} Once again, we proceed recursively: \begin{equation} S_{m}\overset{\textrm{def}}{=}\sum_{n=0}^{p^{m}-1}\left\|M_{H}\left(n\right)\right\|_{q} \end{equation} Then, using $M_{H}$'s functional equations: \begin{align} \sum_{n=0}^{p^{m}-1}\left\|M_{H}\left(n\right)\right\|_{q} & =\sum_{j=0}^{p-1}\left\|M_{H}\left(j\right)\right\|_{q}+\sum_{j=0}^{p-1}\sum_{n=1}^{p^{m-1}-1}\left\|M_{H}\left(pn+j\right)\right\|_{q}\\ & =\sum_{j=0}^{p-1}\left\|M_{H}\left(j\right)\right\|_{q}+\sum_{j=0}^{p-1}\sum_{n=1}^{p^{m-1}-1}\left\|\frac{\mu_{j}}{p}M_{H}\left(n\right)\right\|_{q}\\ \left(\gcd\left(p,q\right)=1\right); & =\sum_{j=0}^{p-1}\left\|M_{H}\left(j\right)\right\|_{q}+\sum_{n=1}^{p^{m-1}-1}\left(\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q}\right)\left\|M_{H}\left(n\right)\right\|_{q}\\ \left(M_{H}\left(j\right)=\mu_{j}\textrm{ }\forall j\in\left\{ 1,\ldots,p-1\right\} \right); & =1+\sum_{j=1}^{p-1}\left\|\mu_{j}\right\|_{q}+\left(\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q}\right)\sum_{n=1}^{p^{m-1}-1}\left\|M_{H}\left(n\right)\right\|_{q}\\ & =1-\left\|\mu_{0}\right\|_{q}+\left(\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q}\right)\left(1+\sum_{n=1}^{p^{m-1}-1}\left\|M_{H}\left(n\right)\right\|_{q}\right)\\ & =1-\left\|\mu_{0}\right\|_{q}+\left(\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q}\right)\sum_{n=0}^{p^{m-1}-1}\left\|M_{H}\left(n\right)\right\|_{q} \end{align} So: \begin{equation} S_{m}=1-\left\|\mu_{0}\right\|_{q}+\left(\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q}\right)S_{m-1},\textrm{ }\forall m\geq1 \end{equation} where: \begin{equation} S_{0}=\left\|M_{H}\left(0\right)\right\|_{q}=1 \end{equation} Finally, defining $A$ and $B$ by: \begin{align} A & \overset{\textrm{def}}{=}1-\left\|\mu_{0}\right\|_{q}\\ B & \overset{\textrm{def}}{=}\sum_{j=0}^{p-1}\left\|\mu_{j}\right\|_{q} \end{align} we can write: \begin{align} S_{m} & =A+BS_{m-1}\\ & =A+AB+B^{2}S_{m-2}\\ & =A+AB+AB^{2}+B^{3}S_{m-3}\\ & \vdots\\ & =A\sum_{k=0}^{m-1}B^{k}+B^{m}S_{0}\\ & =B^{m}+A\sum_{k=0}^{m-1}B^{k} \end{align} Using this, our lambda decomposition (\ref{eq:M_H q-adic absolute value lambda decomposition}) becomes: \begin{align} \sum_{n=0}^{\infty}\frac{\left\|M_{H}\left(n\right)\right\|_{q}}{p^{\lambda_{p}\left(n\right)}} & \overset{\mathbb{R}}{=}1+\sum_{m=1}^{\infty}\frac{1}{p^{m}}\left(\sum_{n=0}^{p^{m}-1}\left\|M_{H}\left(n\right)\right\|_{q}-\sum_{n=0}^{p^{m-1}-1}\left\|M_{H}\left(n\right)\right\|_{q}\right)\\ & =1+\sum_{m=1}^{\infty}\frac{1}{p^{m}}\left(S_{m}-S_{m-1}\right)\\ & =1+\sum_{m=1}^{\infty}\frac{S_{m}}{p^{m}}-p\sum_{m=1}^{\infty}\frac{S_{m-1}}{p^{m-1}}\\ & =1-p\frac{S_{0}}{p^{0}}+\sum_{m=1}^{\infty}\frac{S_{m}}{p^{m}}-p\sum_{m=1}^{\infty}\frac{S_{m}}{p^{m}}\\ & =1-p+\left(1-p\right)\sum_{m=1}^{\infty}\frac{S_{m}}{p^{m}}\\ & =1-p+\left(1-p\right)\sum_{m=1}^{\infty}\frac{\left(B^{m}+A\sum_{k=0}^{m-1}B^{k}\right)}{p^{m}}\\ & \overset{\mathbb{R}}{=}\begin{cases} 1-p+\left(1-p\right)\sum_{m=1}^{\infty}\frac{Am+1}{p^{m}} & \textrm{if }B=1\\ 1-p+\left(1-p\right)\sum_{m=1}^{\infty}\frac{B^{m}+A\frac{B^{m}-1}{B-1}}{p^{m}} & \textrm{else} \end{cases} \end{align} The $B=1$ case is finite since $p\geq2$. As for $B\neq1$, summing the geometric series yields: \begin{equation} \sum_{m=1}^{\infty}\frac{B^{m}+A\frac{B^{m}-1}{B-1}}{p^{m}}\overset{\mathbb{R}}{=}-\frac{A}{B-1}\frac{1}{p-1}+\frac{A+B-1}{B-1}\sum_{m=1}^{\infty}\left(\frac{B}{p}\right)^{m} \end{equation} Hence, $0\leq B<p$ is sufficient to guarantee convergence in the topology of $\mathbb{R}$. Q.E.D. \vphantom{} We end our wisp of $L_{\mathbb{R}}^{1}$ by proving in general what was shown for $\chi_{5}$ in Subsection \ref{subsec:3.3.6 L^1 Convergence}: that $\tilde{\chi}_{H,N}-\chi_{H,N}$ converges to $0$ in $L_{\mathbb{R}}^{1}$ as $N\rightarrow\infty$. The first step is another computation\textemdash this time outsourced to previous work. \begin{prop} \label{prop:Convolution of alpha and A_H hat}Suppose $\alpha_{H}\left(0\right)\notin\left\{ 0,1\right\} $. Then: \begin{equation} \sum_{0<\left\|t\right\|_{p}\leq p^{N}}\left(\alpha_{H}\left(0\right)\right)^{v_{p}\left(t\right)}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}}{=}\left(\frac{\mu_{0}}{p\alpha_{H}\left(0\right)}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-1\label{eq:Partial Fourier Sum of A_H hat times alpha to the v_p(t)} \end{equation} \end{prop} Proof: Use (\ref{eq:Convolution of dA_H and D_N}) from \textbf{Theorem \ref{thm:Properties of dA_H}}. Q.E.D. \vphantom{} With this, we can compute simple closed-form expressions for $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)$. It should be noted that the formulae given below only hold for the indicated choice of $\hat{\chi}_{H}$. Choosing a different Fourier transform of $\hat{\chi}_{H}$ will alter the formula for $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)$. \begin{thm}[\textbf{$\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$}] \label{thm:Chi_H,N / Chi_H,N twiddle error}Choose $\hat{\chi}_{H}\left(t\right)$ to be the Fourier Transform of $\chi_{H}$ defined by \emph{(\ref{eq:Fourier Transform of Chi_H for a contracting semi-basic rho-Hydra map})}: \[ \hat{\chi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} \begin{cases} 0 & \textrm{if }t=0\\ \left(\beta_{H}\left(0\right)v_{p}\left(t\right)+\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \frac{\beta_{H}\left(0\right)\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)}+\begin{cases} 0 & \textrm{if }t=0\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \] Then: \vphantom{} I. If $\alpha_{H}\left(0\right)=1$: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\left(\frac{2\mu_{0}N}{p}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\left(N-1\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right)\label{eq:Chi_H,N / Chi_H,N twiddle identity for alpha equals 1} \end{equation} \vphantom{} II. If $\alpha_{H}\left(0\right)\neq1$: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(\frac{\mu_{0}}{p}\right)^{N-1}\left(\frac{2\mu_{0}}{p}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\alpha_{H}\left(0\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right)\label{eq:Chi_H,N / Chi_H,N twiddle identity for alpha is not 1} \end{equation} \end{thm} Proof: I. Let $\alpha_{H}\left(0\right)=1$. Then, we have that: \begin{equation} \hat{\chi}_{H}\left(t\right)\overset{\overline{\mathbb{Q}}}{=}\begin{cases} 0 & \textrm{if }t=0\\ \left(\beta_{H}\left(0\right)v_{p}\left(t\right)+\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} \end{equation} Turning to \textbf{Theorem \ref{thm:(N,t) asymptotic decomposition of Chi_H,N hat}}), the associated ``fine-structure'' formula for $\hat{\chi}_{H,N}$ in this case is: \[ \hat{\chi}_{H,N}\left(t\right)-\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right)=\begin{cases} 0 & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)v_{p}\left(t\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases} \] Using use our chosen $\hat{\chi}_{H}$ (and remembering that $\hat{A}_{H}\left(0\right)=1$), this can be rewritten as: \begin{align} \hat{\chi}_{H,N}\left(t\right)-\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right) & \overset{\overline{\mathbb{Q}}}{=}\mathbf{1}_{0}\left(p^{N-1}t\right)\hat{\chi}_{H}\left(t\right)\label{eq:Chi_H,N / Chi_H,N twiddle proof: eq 1}\\ & +\left[\left\|t\right\|_{p}=p^{N}\right]\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right),\textrm{ }\forall\left\|t\right\|_{p}\leq p^{N}\nonumber \end{align} which is to say: \begin{align} \hat{\chi}_{H,N}\left(t\right)-\beta_{H}\left(0\right)N\hat{A}_{H}\left(t\right)\mathbf{1}_{0}\left(p^{N-1}t\right) & \overset{\overline{\mathbb{Q}}}{=}\mathbf{1}_{0}\left(p^{N}t\right)\hat{\chi}_{H}\left(t\right)\label{eq:Chi_H,N / Chi_H,N twiddle proof: eq 2}\\ & +\left[\left\|t\right\|_{p}=p^{N}\right]\left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)-\hat{\chi}_{H}\left(t\right)\right),\textrm{ }\forall\left\|t\right\|_{p}\leq p^{N}\nonumber \end{align} Multiplying through by $e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ and summing over $\left\|t\right\|_{p}\leq p^{N}$ yields: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\tilde{A}_{H,N-1}\left(\mathfrak{z}\right) & \overset{\overline{\mathbb{Q}}}{=}\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\label{eq:Chi_H,N / Chi_H,N twiddle proof: eq 3}\\ & +\sum_{\left\|t\right\|_{p}=p^{N}}\left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)-\hat{\chi}_{H}\left(t\right)\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\nonumber \end{align} Using \textbf{Lemma \ref{lem:1D gamma formula}}, we have: \begin{align} \sum_{\left\|t\right\|_{p}=p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\sum_{0<\left\|t\right\|_{p}\leq p^{N}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & -\sum_{0<\left\|t\right\|_{p}\leq p^{N-1}}\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & =\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & -\sum_{n=0}^{N-2}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & =\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{N-1}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right) \end{align} Using (\ref{eq:Explicit Formula for Chi_H,N twiddle for arbitrary rho and alpha equals 1}) from \textbf{Corollary \ref{cor:Chi_H,N twiddle explicit formula, arbitrary p, arbitrary alpha}} gives us: \begin{align} \sum_{\left\|t\right\|_{p}=p^{N}}\hat{\chi}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} & =\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N-1}\left(\mathfrak{z}\right)\\ & \overset{\overline{\mathbb{Q}}}{=}-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)+\beta_{H}\left(0\right)\sum_{n=0}^{N-1}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=0}^{N-1}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\beta_{H}\left(0\right)\left(N-1\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & -\beta_{H}\left(0\right)\sum_{n=0}^{N-2}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & -\sum_{n=0}^{N-2}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{n}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & \overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)\left(N-1\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & -\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\\ & +\left(\sum_{j=0}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\varepsilon_{N-1}^{j}\left(\mathfrak{z}\right)\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right) \end{align} Finally, employing our familiar formula for for $\tilde{A}_{H,N}\left(\mathfrak{z}\right)$ (equation (\ref{eq:Convolution of dA_H and D_N}) from \textbf{Theorem \ref{thm:Properties of dA_H}}) and remembering that $\alpha_{H}\left(0\right)=1$, (\ref{eq:Chi_H,N / Chi_H,N twiddle proof: eq 3}) becomes: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) & \overset{\overline{\mathbb{Q}}}{=}\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\\ & -\beta_{H}\left(0\right)\left(N-1\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\\ & +\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right) \end{align} and hence: \[ \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\overset{\overline{\mathbb{Q}}}{=}\beta_{H}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\left(\frac{2\mu_{0}N}{p}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\left(N-1\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right) \] which proves (I). \vphantom{} II. Suppose $\alpha_{H}\left(0\right)\neq1$. The ``fine-structure'' equation from our $\left(N,t\right)$-asymptotic analysis of $\hat{\chi}_{H,N}$ ((\ref{eq:Fine Structure Formula for Chi_H,N hat when alpha is not 1}) from \textbf{Theorem \ref{thm:(N,t) asymptotic decomposition of Chi_H,N hat}}) is: \[ \hat{\chi}_{H,N}\left(t\right)=\begin{cases} \beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N}-1}{\alpha_{H}\left(0\right)-1} & \textrm{if }t=0\\ \left(\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)+\beta_{H}\left(0\right)\frac{\left(\alpha_{H}\left(0\right)\right)^{N+v_{p}\left(t\right)}-1}{\alpha_{H}\left(0\right)-1}\right)\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ 0 & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p} \] Meanwhile, $\hat{\chi}_{H}\left(t\right)$ is: \[ \hat{\chi}_{H}\left(t\right)=\begin{cases} \frac{\beta_{H}\left(0\right)}{1-\alpha_{H}\left(0\right)} & \textrm{if }t=0\\ \left(\frac{\beta_{H}\left(0\right)}{1-\alpha_{H}\left(0\right)}+\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }t\neq0 \end{cases} \] Subtracting yields: \begin{equation} \hat{\chi}_{H,N}\left(t\right)-\hat{\chi}_{H}\left(t\right)=\begin{cases} \frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N}}{\alpha_{H}\left(0\right)-1} & \textrm{if }t=0\\ \frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N+v_{p}\left(t\right)}}{\alpha_{H}\left(0\right)-1}\hat{A}_{H}\left(t\right) & \textrm{if }0<\left\|t\right\|_{p}<p^{N}\\ \frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}=p^{N}\\ \left(\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}-\gamma_{H}\left(\frac{t\left\|t\right\|_{p}}{p}\right)\right)\hat{A}_{H}\left(t\right) & \textrm{if }\left\|t\right\|_{p}>p^{N} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{p}\label{eq:Chi_H,N / Chi_H,N twiddle proof: eq 4} \end{equation} Since $\hat{\chi}_{H,N}\left(t\right)$ is supported on $\left\|t\right\|_{p}\leq p^{N}$, when summing our Fourier series, we need only sum over $\left\|t\right\|_{p}\leq p^{N}$. Doing so yields: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & =\frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N}}{\alpha_{H}\left(0\right)-1}+\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\sum_{\left\|t\right\|_{p}=p^{N}}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\\ & +\frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N}}{\alpha_{H}\left(0\right)-1}\sum_{0<\left\|t\right\|_{p}<p^{N}}\left(\alpha_{H}\left(0\right)\right)^{v_{p}\left(t\right)}\hat{A}_{H}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}} \end{align} Using (\ref{eq:Convolution of dA_H and D_N}) and \textbf{Proposition \ref{prop:Convolution of alpha and A_H hat}}, we obtain: \begin{align} \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & =\frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N}}{\alpha_{H}\left(0\right)-1}\\ & +\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\alpha_{H}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right)\\ & +\frac{\beta_{H}\left(0\right)\left(\alpha_{H}\left(0\right)\right)^{N}}{\alpha_{H}\left(0\right)-1}\left(\left(\frac{\mu_{0}}{p\alpha_{H}\left(0\right)}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-1\right)\\ & =\frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(\frac{\mu_{0}}{p}\right)^{N-1}\left(\frac{2\mu_{0}}{p}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\alpha_{H}\left(0\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right) \end{align} Q.E.D. \vphantom{} The result just proved is of independent interest, seeing as it provides us with an explicit form for the error between $\chi_{H,N}\left(\mathfrak{z}\right)$ and $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$. A significant issue can be seen by examining (\ref{eq:Chi_H,N / Chi_H,N twiddle identity for alpha equals 1}) and (\ref{eq:Chi_H,N / Chi_H,N twiddle identity for alpha is not 1}) for $\mathfrak{z}=0$. Since $\chi_{H,N}\left(0\right)=\chi_{H}\left(\left[0\right]_{p^{N}}\right)=0$ for all $N\geq0$, we obtain: \begin{equation} \tilde{\chi}_{H,N}\left(0\right)\overset{\overline{\mathbb{Q}}}{=}\begin{cases} \beta_{H}\left(0\right)\left(\left(\frac{2\mu_{0}}{p}-1\right)N+1\right)\left(\frac{\mu_{0}}{p}\right)^{N-1} & \textrm{if }\alpha_{H}\left(0\right)=1\\ \frac{\beta_{H}\left(0\right)}{\alpha_{H}\left(0\right)-1}\left(\frac{2\mu_{0}}{p}-\alpha_{H}\left(0\right)\right)\left(\frac{\mu_{0}}{p}\right)^{N-1} & \textrm{if }\alpha_{H}\left(0\right)\neq1 \end{cases},\textrm{ }\forall N\geq1\label{eq:Formula for Chi_H,N twiddle at 0} \end{equation} When $\alpha_{H}\left(0\right)=1$, observe that $\tilde{\chi}_{H,N}\left(0\right)$ is non-zero for all $N\geq1$, with it tending to $0$ in $\mathbb{R}$ as $N\rightarrow\infty$. The same is true when $\alpha_{H}\left(0\right)\neq1$, unless $2\mu_{0}=p\alpha_{H}\left(0\right)$. Because the $N$th truncation $\chi_{H,N}\left(\mathfrak{z}\right)$ is continuous and vanishes for $\mathfrak{z}\overset{p^{N}}{\equiv}0$, the WTT for continuous $\left(p,q\right)$-adic functions tells us that $\hat{\chi}_{H,N}\left(t\right)$ does not have a convolution inverse. However, this statement need not be true for $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$. That being said, using the previous proposition, we can establish the $L_{\mathbb{R}}^{1}$ convergence of $\tilde{\chi}_{H,N}$ to $\chi_{H,N}$ and vice-versa. We just need one last computation under our belts: the $L_{\mathbb{R}}^{1}$-norm of $\kappa_{H,N}\left(\mathfrak{z}\right)$ \begin{prop}[\textbf{$L_{\mathbb{R}}^{1}$-norm of $\kappa_{H,N}$}] \label{prop:L^1_R norm of kapp_H,N}\index{kappa{H}@$\kappa_{H}$!truncation and L{mathbb{R}}{1}-norm@truncation and $L_{\mathbb{R}}^{1}$-norm}Let $H$ be a contracting semi-basic $p$-Hydra map, and let $q=q_{H}$. Then: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\kappa_{H,N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}=\int_{\mathbb{Z}_{p}}\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}d\mathfrak{z}\leq\left(\frac{p+q-1}{pq}\right)^{N},\textrm{ }\forall N\geq1\label{eq:L^1_R norm estimate of Nth truncation of Kappa_H} \end{equation} In particular, since $p+q-1<pq$ for all $p,q\geq2$, we have that the $N$th truncations of $\kappa_{H}$ converge to $0$ in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ as $N\rightarrow\infty$. \end{prop} Proof: Since $\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}$ is a locally constant real valued function of $\mathfrak{z}$ whose output is determined by the value of $\mathfrak{z}$ mod $p^{N}$, we have that: \begin{equation} \int_{\mathbb{Z}_{p}}\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}d\mathfrak{z}=\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}\left\|\kappa_{H}\left(n\right)\right\|_{q} \end{equation} Once again, we proceed recursively. Let: \begin{equation} S_{N}=\frac{1}{p^{N}}\sum_{n=0}^{p^{N}-1}\left\|\kappa_{H}\left(n\right)\right\|_{q} \end{equation} where, note, $S_{0}=\left\|\kappa_{H}\left(0\right)\right\|_{q}=\left\|1\right\|_{q}=1$. Then, splitting the $n$-sum modulo $p$: \begin{align} S_{N} & =\frac{1}{p^{N}}\sum_{n=0}^{p^{N-1}-1}\sum_{j=0}^{p-1}\left\|\kappa_{H}\left(pn+j\right)\right\|_{q}\\ & =\frac{1}{p^{N}}\sum_{n=0}^{p^{N-1}-1}\sum_{j=0}^{p-1}\left\|\frac{\mu_{j}}{\mu_{0}}\kappa_{H}\left(n\right)\right\|_{q}\\ & =\left(\sum_{j=0}^{p-1}\left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q}\right)\frac{1}{p^{N}}\sum_{n=0}^{p^{N-1}-1}\left\|\kappa_{H}\left(n\right)\right\|_{q}\\ & =\left(\frac{1}{p}\sum_{j=0}^{p-1}\left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q}\right)S_{N-1} \end{align} Hence: \begin{equation} S_{N}=\left(\frac{1}{p}\sum_{j=0}^{p-1}\left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q}\right)^{N}S_{0}=\left(\frac{1}{p}\sum_{j=0}^{p-1}\left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q}\right)^{N} \end{equation} Since $H$ is given to be semi-basic, we have that: \begin{equation} \left\|\frac{\mu_{j}}{\mu_{0}}\right\|_{q}\leq\frac{1}{q},\textrm{ }\forall j\in\left\{ 1,\ldots,q-1\right\} \end{equation} and so: \[ S_{N}\leq\left(\frac{1}{p}\left(1+\sum_{j=1}^{p-1}\frac{1}{q}\right)\right)^{N}=\left(\frac{p+q-1}{pq}\right)^{N} \] Q.E.D. \vphantom{} Here, then, is the desired theorem: \begin{thm} \label{thm:L^1 convergence of Chi_H,N minus Chi_H,N twiddle}Let $H$ be a contracting semi-basic $p$-Hydra map. Then: \begin{equation} \lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}}\left\|\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z}\overset{\mathbb{R}}{=}0\label{eq:L^1 converges of the difference between Chi_H,N and Chi_H,N twiddle} \end{equation} \end{thm} Proof: Regardless of whether or not $\alpha_{H}\left(0\right)=1$, the co-primality of $p$ and $q$ along with an application of the \emph{ordinary} $q$-adic triangle inequality, (\ref{eq:Chi_H,N / Chi_H,N twiddle identity for alpha equals 1}) and (\ref{eq:Chi_H,N / Chi_H,N twiddle identity for alpha is not 1}) from \textbf{Theorem \ref{thm:Chi_H,N / Chi_H,N twiddle error}} both become: \begin{align} \int_{\mathbb{Z}_{p}}\left\|\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q}d\mathfrak{z} & \ll\int_{\mathbb{Z}_{p}}\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}d\mathfrak{z}+\int_{\mathbb{Z}_{p}}\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}d\mathfrak{z} \end{align} By \textbf{Proposition \ref{prop:L^1_R norm of kapp_H,N}}, both of the terms in the upper bound go to $0$ as $N\rightarrow\infty$. Q.E.D. \subsection{\label{subsec:4.3.3 Quick-Approach-of}\index{quick approach}Quick Approach of $\chi_{H}$} In Subsection \ref{subsec:3.2.2 Truncations-=00003D000026-The}, we proved the \textbf{Square Root Lemma }(page \pageref{eq:Square Root Lemma}), which\textemdash recall\textemdash asserted that for any $\chi\in\tilde{C}\left(\mathbb{Z}_{p},K\right)$, any $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, and any $\mathfrak{c}\in K\backslash\chi\left(\mathbb{N}_{0}\right)$, the equality $\chi\left(\mathfrak{z}\right)=\mathfrak{c}$ held if and only if: \begin{equation} \liminf_{n\rightarrow\infty}\frac{\left\|\chi\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\mathfrak{c}\right\|_{q}}{\left\|\nabla_{p^{n}}\left\{ \chi\right\} \left(\mathfrak{z}\right)\right\|_{q}^{1/2}}<\infty \end{equation} With that in mind, we then said the pair $\left(\mathfrak{z},\mathfrak{c}\right)$ was quickly (resp. slowly) approached by $\chi$ whenever the above $\liminf$ was $0$ (resp. $\in\left(0,\infty\right)$). Below, we prove that the pair $\left(\mathfrak{z},\chi_{H}\left(\mathfrak{z}\right)\right)$ is approached quickly by $\chi_{H}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ whenever $H$ is a contracting, semi-basic $p$-Hydra map which fixes $0$. \begin{thm}[\textbf{\textit{Quick Approach of $\chi_{H}$ on $\mathbb{Z}_{p}^{\prime}$}}] Let $H$ be a contracting, semi-basic $p$-Hydra map which fixes $0$, and write $q=q_{H}$. Then: \begin{equation} \lim_{n\rightarrow\infty}\frac{\left\|\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)-\chi_{H}\left(\mathfrak{z}\right)\right\|_{q}}{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\right\|_{q}^{1/2}}=0,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\label{eq:Quickness of a p-Hydra map on Z_p prime} \end{equation} \end{thm} Proof: Suppose that $\alpha_{H}\left(0\right)=1$, and fix $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. We wish to investigate $\chi_{H}\left(\mathfrak{z}\right)-\chi_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)$, or\textemdash which is the same\textemdash $\chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)$. To do this, we write: \begin{equation} \chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)=\left(\chi_{H}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right)-\left(\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right) \end{equation} Subtracting \textbf{Corollary \ref{cor:Chi_H,N twiddle explicit formula, arbitrary p, arbitrary alpha}}'s formula for $\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)$) from \textbf{Corollary \ref{cor:F-series for Chi_H, arbitrary p and alpha}}'s $\mathcal{F}$-series formula for $\chi_{H}\left(\mathfrak{z}\right)$) gives: \begin{align} \chi_{H}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right) & \overset{\mathcal{F}_{p,q_{H}}}{=}\beta_{H}\left(0\right)N\left(\frac{\mu_{0}}{p}\right)^{N}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\\ & +\beta_{H}\left(0\right)\sum_{n=N}^{\infty}\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\\ & +\sum_{n=N}^{\infty}\left(\sum_{j=1}^{p-1}\beta_{H}\left(\frac{j}{p}\right)\left(\varepsilon_{n}\left(\mathfrak{z}\right)\right)^{j}\right)\left(\frac{\mu_{0}}{p}\right)^{n}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right) \end{align} Since $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, the topology of convergence as $N\rightarrow\infty$ is the $q$-adic topology, and as such, estimating with the $q$-adic absolute value gives: \begin{equation} \left\|\chi_{H}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q}\ll\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q} \end{equation} On the other hand, because $\alpha_{H}\left(0\right)=1$,\textbf{ Theorem \ref{thm:Chi_H,N / Chi_H,N twiddle error}} tells us that: \begin{equation} \chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)=\beta_{H}\left(0\right)\left(\frac{\mu_{0}}{p}\right)^{N-1}\left(\frac{2\mu_{0}N}{p}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)-\left(N-1\right)\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right) \end{equation} and so: \begin{equation} \left\|\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q}\ll\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} \end{equation} Since $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$, $\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}\geq\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}$, which leaves us with: \[ \max\left\{ \left\|\chi_{H}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q},\left\|\chi_{H,N}\left(\mathfrak{z}\right)-\tilde{\chi}_{H,N}\left(\mathfrak{z}\right)\right\|_{q}\right\} \ll\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} \] Hence, by the ultrametric inequality: \begin{equation} \left\|\chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)\right\|_{q}\ll\left\|\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} \end{equation} Since: \begin{equation} \kappa_{H}\left(m\right)=M_{H}\left(m\right)\left(\frac{\mu_{0}}{p}\right)^{-\lambda_{p}\left(m\right)} \end{equation} we have that: \begin{equation} \left\|\kappa_{H}\left(m\right)\right\|_{q}=\left\|M_{H}\left(m\right)\left(\frac{\mu_{0}}{p}\right)^{-\lambda_{p}\left(m\right)}\right\|_{q}=\left\|M_{H}\left(m\right)\right\|_{q} \end{equation} and so: \begin{equation} \left\|\chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)\right\|_{q}\ll\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} \end{equation} Dividing by $\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}^{1/2}$ gives: \begin{equation} \frac{\left\|\chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)\right\|_{q}}{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}^{1/2}}\ll\frac{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}}{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}^{1/2}} \end{equation} Since the string of $p$-adic digits of $\left[\mathfrak{z}\right]_{p^{N}}$ is either equal to that of $\left[\mathfrak{z}\right]_{p^{N-1}}$ or differs by a single digit on the right, $M_{H}$'s functional equations (\textbf{Proposition \ref{prop:M_H concatenation identity}}) tell us that: \begin{equation} M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)=\begin{cases} M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right) & \textrm{if }\left[\mathfrak{z}\right]_{p^{N-1}}=\left[\mathfrak{z}\right]_{p^{N}}\\ M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\cdot M_{H}\left(\textrm{coefficient of }p^{N}\textrm{ in }\mathfrak{z}\right) & \textrm{if }\left[\mathfrak{z}\right]_{p^{N-1}}\neq\left[\mathfrak{z}\right]_{p^{N}} \end{cases} \end{equation} Seeing $\left[\mathfrak{z}\right]_{p^{N-1}}\neq\left[\mathfrak{z}\right]_{p^{N}}$ occurs if and only if the coefficient of $p^{N}$ in $\mathfrak{z}$ is non-zero, the semi-basicness of $H$ then guarantees that: \begin{equation} \left\|M_{H}\left(\textrm{coefficient of }p^{N}\textrm{ in }\mathfrak{z}\right)\right\|_{p}\leq\max_{1\leq j\leq p-1}\left\|M_{H}\left(j\right)\right\|_{p}=p^{-\nu} \end{equation} occurs for some integer constant $\nu\geq1$. Hence: \begin{equation} \left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}\geq p^{-\nu}\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q} \end{equation} and so: \begin{align} \frac{\left\|\chi_{H}\left(\mathfrak{z}\right)-\chi_{H,N}\left(\mathfrak{z}\right)\right\|_{q}}{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}^{1/2}} & \ll\frac{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}}{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N}}\right)\right\|_{q}^{1/2}}\\ & \leq\frac{\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}}{p^{-\frac{\nu}{2}}\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}^{1/2}}\\ & =p^{\nu/2}\left\|M_{H}\left(\left[\mathfrak{z}\right]_{p^{N-1}}\right)\right\|_{q}^{1/2} \end{align} which tends to $0$ as $N\rightarrow\infty$, since $\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}$. The case where $\alpha_{H}\left(0\right)\neq1$ works in almost exactly the same way. Q.E.D. \subsection{\label{subsec:4.3.4 Archimedean-Estimates}Archimedean Estimates} As we discussed in Chapter 1, mixing up convergence in $\mathbb{Q}_{p}$ or $\mathbb{C}_{p}$ with convergence in $\mathbb{R}$ or $\mathbb{C}$ is, quite literally, the oldest mistake in the big book of $p$-adic analysis. As a rule, mixing topologies in this way is fraught with danger. So far, however, the formalism of frames has been a helpful guard rail, keeping us on track. However, in this subsection, we are going to simply dive off the edge of the cliff head-first. Let $\hat{\eta}:\hat{\mathbb{Z}}_{p}\rightarrow\overline{\mathbb{Q}}$ be an element of $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$, so that the $\left(p,q\right)$-adic Fourier series: \begin{equation} \eta\left(\mathfrak{z}\right)=\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\label{eq:Getting Ready to Jump off the Cliff} \end{equation} converges in $\mathbb{C}_{q}$ uniformly with respect to $\mathfrak{z}\in\mathbb{Z}_{p}$. In particular, for any $\mathfrak{z}$, the ultrametric structure of $\mathbb{C}_{q}$ allows us to freely group and rearrange the terms of (\ref{eq:Getting Ready to Jump off the Cliff}) however we please without affecting the convergence. Because every term of the series is, technically, an element of $\overline{\mathbb{Q}}$, note that the $N$th partial sums of this series are perfectly well-defined\emph{ }complex-valued functions (complex, not $q$-adic complex) on $\mathbb{Z}_{p}$. Moreover, it may just so happen that there are $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$ so that the partial sums: \begin{equation} \tilde{\eta}_{N}\left(\mathfrak{z}_{0}\right)=\sum_{\left\|t\right\|_{p}\leq p^{N}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}_{0}\right\} _{p}}\label{eq:Jumping off the Cliff} \end{equation} converge to a limit in $\mathbb{C}$ as $N\rightarrow\infty$, in addition to the known limit to which they converge in $\mathbb{C}_{q}$. As we saw when discussing Hensel's error, \textbf{\emph{we cannot, in general, assume that the limit of }}(\ref{eq:Jumping off the Cliff})\textbf{\emph{ in $\mathbb{C}$ has, in any way, shape, or form, a meaningful relation to its limit in $\mathbb{C}_{q}$}}\textemdash \textbf{except}, of course, when (\ref{eq:Jumping off the Cliff}) happens to be a \textbf{\emph{geometric series}}, thanks to the \emph{universality }of the geometric series\index{geometric series universality} (page \pageref{fact:Geometric series universality}). Specifically, for any $\mathfrak{z}_{0}$, we can characterize the limiting behavior in $\mathbb{C}$ (not $\mathbb{C}_{q}$!) of (\ref{eq:Jumping off the Cliff}) as $N\rightarrow\infty$ like so: \begin{fact} Let $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$. Then, exactly one of the following occurs for \emph{(\ref{eq:Jumping off the Cliff})} as $N\rightarrow\infty$: \vphantom{} I. \emph{(\ref{eq:Jumping off the Cliff})} fails to converge to a limit in $\mathbb{C}$. \vphantom{} II. (\emph{\ref{eq:Jumping off the Cliff})} converges to a limit in $\mathbb{C}$, and \emph{(\ref{eq:Getting Ready to Jump off the Cliff})}\textbf{ }\textbf{\emph{can}} be rearranged into a geometric series of the form $\sum_{n=0}^{\infty}c_{n}r^{n}$ for $c_{n},r\in\overline{\mathbb{Q}}$, where $r$ has archimedean absolute value $<1$. \vphantom{} III. \emph{(\ref{eq:Jumping off the Cliff})} converges to a limit in $\mathbb{C}$, but \emph{(\ref{eq:Getting Ready to Jump off the Cliff})}\textbf{ }\textbf{\emph{cannot}} be rearranged into a geometric series of the form $\sum_{n=0}^{\infty}c_{n}r^{n}$ for $c_{n},r\in\overline{\mathbb{Q}}$, where $r$ has archimedean absolute value $<1$. \end{fact} \vphantom{} By the universality of the geometric series, (II) is \emph{precisely }the case where we \emph{can }conclude that the limit of (\ref{eq:Jumping off the Cliff}) in $\mathbb{C}$ as $N\rightarrow\infty$ is the same number as the limit of (\ref{eq:Jumping off the Cliff}) in $\mathbb{C}_{q}$ as $N\rightarrow\infty$. For any $\mathfrak{z}$ satisfying (II), the value of (\ref{eq:Getting Ready to Jump off the Cliff}) is the same regardless of the topology (archimedean or $q$-adic) we use to make sense of it. This then leads to the following observation: let $U$ be the set of all $\mathfrak{z}\in\mathbb{Z}_{p}$ for which either (II) or (III) holds true. Then, we can define a function $\eta^{\prime}:U\rightarrow\mathbb{C}$ by the limit of (\ref{eq:Jumping off the Cliff}) in $\mathbb{C}$ (not $\mathbb{C}_{q}$!). For any $\mathfrak{z}_{0}$ in $U$ satisfying (III), there is no reason for the complex number $\eta^{\prime}\left(\mathfrak{z}_{0}\right)$ to have anything to do with the $q$-adic complex number $\eta\left(\mathfrak{z}_{0}\right)$ defined by the limit of (\ref{eq:Jumping off the Cliff}) in $\mathbb{C}_{q}$. However$\ldots$ for $\mathfrak{z}_{0}\in U$ satisfying (II), the universality of the geometric series \emph{guarantees} that $\eta^{\prime}\left(\mathfrak{z}_{0}\right)=\eta\left(\mathfrak{z}_{0}\right)$, where the equality holds \emph{simultaneously }in $\mathbb{C}$ and in $\mathbb{C}_{q}$. The general implementation of this idea is as following. First\textemdash of course\textemdash some terminology: \begin{notation} We write $L^{1}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}\right)$ to denote the Banach space of all complex-valued functions (not $q$-adic complex valued!) $\hat{\eta}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}$ with finite $L^{1}$ norm: \begin{equation} \left\Vert \hat{\eta}\right\Vert _{L^{1}}\overset{\textrm{def}}{=}\sum_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\eta}\left(t\right)\right\|\label{eq:Definition of complex L^1 norm on Z_p hat} \end{equation} For $r\in\left[1,2\right]$, we write $L^{r}\left(\mathbb{Z}_{p},\mathbb{C}\right)$ to denote the Banach space of all complex-valued functions (not $q$-adic complex valued!) $\eta:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ with finite $L^{r}$ norm: \begin{equation} \left\Vert \eta\right\Vert _{L^{r}}\overset{\textrm{def}}{=}\left(\int_{\mathbb{Z}_{p}}\left\|\eta\left(\mathfrak{z}\right)\right\|^{r}d\mathfrak{z}\right)^{1/r}\label{eq:Definition of complex L^r norm on Z_p} \end{equation} \end{notation} \begin{defn} Let $\chi\in\tilde{C}$$\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, and suppose that $\chi\left(\mathbb{N}_{0}\right)\subseteq\overline{\mathbb{Q}}$ (so that every van der Put coefficient of $\chi$ is in $\overline{\mathbb{Q}}$). Let $\mathfrak{z}_{0}\in\mathbb{Z}_{p}$. \vphantom{} I. We\index{double convergence} say $\chi$ is \textbf{doubly convergent }at $\mathfrak{z}_{0}$ whenever the van der Put series $S_{p}\left\{ \chi\right\} \left(\mathfrak{z}_{0}\right)$ can be rearranged into a geometric series of the form $\sum_{n=0}^{\infty}c_{n}r^{n}$ where $r$ and the $c_{n}$s are elements of $\overline{\mathbb{Q}}$ so that both the archimedean and $q$-adic absolute values of $r$ are $<1$. We call $\mathfrak{z}_{0}$ a \textbf{double convergence point }of $\chi$ whenever $\chi$ is doubly convergent at $\mathfrak{z}_{0}$. \vphantom{} II. We say $\chi$ is \textbf{spurious }at $\mathfrak{z}_{0}$ whenever $S_{p}\left\{ \chi\right\} \left(\mathfrak{z}_{0}\right)$ converges in $\mathbb{C}$, yet cannot be rearranged into a geometric series which is $q$-adically convergent. Likewise, we call $\mathfrak{z}_{0}$ a \textbf{spurious point }of $\chi$ whenever $\chi$ is spurious at $\mathfrak{z}_{0}$. \vphantom{} III. We say $\chi$ is \textbf{mixed }at $\mathfrak{z}_{0}$ whenever $\chi$ is either doubly convergent or spurious at $\mathfrak{z}_{0}$. Likewise, we call $\mathfrak{z}_{0}$ a \textbf{mixed point }of $\chi$ whenever $\chi$ is either doubly convergent or spurious at $\mathfrak{z}_{0}$. \end{defn} \begin{rem} Observe that the complex number value attained by $S_{p}\left\{ \chi\right\} \left(\mathfrak{z}_{0}\right)$ at a mixed point $\mathfrak{z}_{0}$ is equal to the $q$-adic value attained by $\chi$ at $\mathfrak{z}_{0}$ \emph{if and only if} $\mathfrak{z}_{0}$ is a point of double convergence. \end{rem} \vphantom{} The next proposition shows that, however audacious this approach might be, it is nevertheless sound in mind and body. \begin{prop} \label{prop:measurability proposition}$\chi\in\tilde{C}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, suppose that $\chi\left(\mathbb{N}_{0}\right)\subseteq\overline{\mathbb{Q}}$, and let $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ be a function which is measurable with respect to the \textbf{\emph{real-valued}}\emph{ }Haar probability measure on $\mathbb{Z}_{p}$ so that every $\mathfrak{z}\in\phi\left(\mathbb{Z}_{p}\right)$ is a mixed point of $\chi$. Then, the function $\eta:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ given by: \begin{equation} \eta\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right) \end{equation} is measurable with respect to the \textbf{\emph{real-valued}}\emph{ }Haar probability measure on $\mathbb{Z}_{p}$. Moreover, we have that $\eta\left(\mathfrak{z}\right)=\chi\left(\phi\left(\mathfrak{z}\right)\right)$ for all $\mathfrak{z}$ for which $\chi$ is doubly convergent at $\phi\left(\mathfrak{z}\right)$. \end{prop} Proof: Let $\chi$, $\phi$, and $\eta$ be as given. Then, for each $N$, the $N$th partial van der Put series $S_{p,N}\left\{ \chi\right\} \left(\mathfrak{z}\right)$ is a locally-constant $\overline{\mathbb{Q}}$-valued function on $\mathbb{Z}_{p}$, and, as such, is a \emph{continuous }$\overline{\mathbb{Q}}$-valued function on $\mathbb{Z}_{p}$. Since $\phi$ is measurable, this means that $\mathfrak{z}\mapsto S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right)$ is a measurable $\overline{\mathbb{Q}}$-valued function on $\mathbb{Z}_{p}$. Since $\phi\left(\mathfrak{z}\right)$ is a mixed point of $\chi$ for every $\mathfrak{z}$, the van der Put series $S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right)$ converge point-wise in $\mathbb{C}$ to limit, and this limit is, by definition, $\eta\left(\mathfrak{z}\right)$. Since $\eta$ is the point-wise limit of a sequence of measurable complex-valued functions on $\mathbb{Z}_{p}$, $\eta$ itself is then measurable. Finally, note that for any $\mathfrak{z}$ for which $\phi\left(\mathfrak{z}\right)$ is a double convergent point of $\chi$, the limit of $S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right)$ is a geometric series which converges simultaneously in $\mathbb{C}$ and $\mathbb{C}_{q}$. The universality of the geometric series then guarantees that the limit of the series in $\mathbb{C}$ is equal to the limit of the series in $\mathbb{C}_{q}$. Since $\chi$ is rising-continuous, the $N$th partial van der Put series $S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right)$ converges in $\mathbb{C}_{q}$ to $\chi\left(\phi\left(\mathfrak{z}\right)\right)$ for all $\mathfrak{z}$. So, $\eta\left(\mathfrak{z}\right)=\chi\left(\phi\left(\mathfrak{z}\right)\right)$ for all $\mathfrak{z}$ for which $\phi\left(\mathfrak{z}\right)$ is a double convergent point of $\chi$. Q.E.D. \begin{thm}[\textbf{$L^{1}$ Method for Archimedean Estimates}] \label{thm:The L^1 method} Let\index{$L^{1}$ method} $\chi\in\tilde{C}$$\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$, suppose that $\chi\left(\mathbb{N}_{0}\right)\subseteq\overline{\mathbb{Q}}$, and let $\phi:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ be a function which is measurable with respect to the \textbf{\emph{real-valued}}\emph{ }Haar probability measure on $\mathbb{Z}_{p}$ so that: \vphantom{} I. Every $\mathfrak{z}\in\phi\left(\mathbb{Z}_{p}\right)$ is a mixed point of $\chi$. \vphantom{} II. The measurable function $\eta:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ defined by: \begin{equation} \eta\left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}S_{p,N}\left\{ \chi\right\} \left(\phi\left(\mathfrak{z}\right)\right) \end{equation} is in $L^{r}\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$ for some $r\in\left[1,2\right]$, so that the Fourier transform $\hat{\eta}:\hat{\mathbb{Z}}_{p}\rightarrow\mathbb{C}$: \begin{equation} \hat{\eta}\left(t\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}}\eta\left(\mathfrak{z}\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z} \end{equation} is an element of $L^{1}\left(\mathbb{\hat{Z}}_{p},\mathbb{C}_{q}\right)$. Then: \begin{equation} \left\|\chi\left(\phi\left(\mathfrak{z}\right)\right)\right\|\leq\left\Vert \hat{\eta}\right\Vert _{L^{1}}\label{eq:The L^1 Method} \end{equation} for all $\mathfrak{z}\in\mathbb{Z}_{p}$ so that $\chi$ is doubly convergent at $\phi\left(\mathfrak{z}\right)$. \end{thm} \begin{rem} For any $\mathfrak{z}\in\mathbb{Z}_{p}$ so that $\chi$ is doubly convergent at $\phi\left(\mathfrak{z}\right)$, it necessarily follows that the value of $\chi\left(\mathfrak{y}\right)$ at $\mathfrak{y}=\phi\left(\mathfrak{z}\right)$ is then an element of $\mathbb{C}_{q}\cap\overline{\mathbb{Q}}$. As such, the $L^{1}$ method allows us to extract \emph{archimedean upper bounds }on outputs of $\chi$ that happen to be both $q$-adic complex numbers and ordinary algebraic numbers. \end{rem} Proof: Let everything be as given. By \textbf{Proposition \ref{prop:measurability proposition}}, $\eta$ is measurable. It is a well-known fact that the Fourier transform of $\eta$ (a complex-valued function on the compact abelian group $\mathbb{Z}_{p}$) is defined whenever $\eta$ is in $L^{r}\left(\mathbb{Z}_{p},\mathbb{C}\right)$ for some $r\in\left[1,2\right]$ (see \cite{Folland - harmonic analysis}, for example). Consequently, the hypothesis that $\hat{\eta}$ be in $L^{1}$ tells us that $\eta\left(\mathfrak{z}\right)$'s Fourier series is absolutely convergent, and hence, that: \begin{equation} \left\|\eta\left(\mathfrak{z}\right)\right\|=\left\|\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{\eta}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\right\|\leq\sum_{t\in\hat{\mathbb{Z}}_{p}}\left\|\hat{\eta}\left(t\right)\right\|=\left\Vert \hat{\eta}\right\Vert _{L^{1}} \end{equation} Since $\eta\left(\mathfrak{z}\right)=\chi\left(\phi\left(\mathfrak{z}\right)\right)$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ for which $\chi$ is doubly convergent at $\phi\left(\mathfrak{z}\right)$, the result follows. Q.E.D. \vphantom{} Originally, I employed this method to obtain archimedean bounds on $\chi_{q}\left(\mathfrak{z}\right)$ for odd $q\geq3$. However, the bounds were not very good, in that my choice of $\phi$ was relatively poor\textemdash the exact meaning of this will be clear in a moment. Like with most of my original work in this dissertation, the bounds followed only after a lengthy, involved Fourier analytic computation. As such, rather than continue to draw out this three-hundred-plus-page-long behemoth, I hope the reader will forgive me for merely sketching the argument. The idea for $\phi$ comes from how the $2$-adic digits of a given $\mathfrak{z}\in\mathbb{Z}_{2}$ affect the value of $\chi_{q}\left(\mathfrak{z}\right)$. Consider, for instance the explicit formula for $\chi_{3}\left(\mathfrak{z}\right)$ (equation (\ref{eq:Explicit formula for Chi_3})): \begin{equation} \chi_{3}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{2,3}}{=}-\frac{1}{2}+\frac{1}{4}\sum_{n=0}^{\infty}\frac{3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}}{2^{n}},\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{2} \end{equation} For fixed $\mathfrak{z}\in\mathbb{Z}_{2}^{\prime}$, as $n$ increases to $n+1$, $3^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}$ will increase only if the coefficient of $2^{n}$ in the $2$-adic expansion of $\mathfrak{z}$ is $1$. If that coefficient is $0$, the denominator $2^{n}$ in the geometric series will increase, but the numerator will not increase. A simple analysis shows that, for $\chi_{3}$, we can guarantee that the series will become a convergent geometric series for $\mathfrak{z}\in\mathbb{Q}\cap\mathbb{Z}_{2}^{\prime}$ by requiring that any two $1$s in the $2$-adic digits of $\mathfrak{z}$ are separated by at least one $0$. There is, in fact, a function $\phi:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{2}$ which can guarantee this for us, and\textemdash amusingly\textemdash we have actually already encountered this particular $\phi$ before: it was the function from \textbf{Example \ref{exa:p-adic differentiation is crazy}} (page \pageref{exa:p-adic differentiation is crazy}) which altered the $2$-adic expansion of a given $2$-adic integer like so: \begin{equation} \phi_{3}\left(\sum_{n=0}^{\infty}c_{n}2^{n}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{n}2^{2n}\label{eq:Definition of Phi_2} \end{equation} As we saw in Subsection \ref{subsec:3.1.1 Some-Historical-and}, $\phi_{2}$ is injective, continuous (and hence, measurable), differentiable, and in fact \emph{continuously} differentiable, and its derivative is identically $0$. Observe that for a $\mathfrak{z}$ with a $2$-adic digit sequence of $\mathfrak{z}=\centerdot c_{0}c_{1}c_{2}\ldots$, we have: \begin{align} \phi_{2}\left(\mathfrak{z}\right) & =c_{0}+c_{1}2^{2}+c_{2}2^{4}+\cdots\\ & =c_{0}+0\cdot2^{1}+c_{1}2^{2}+0\cdot2^{3}+c_{2}2^{4}+\cdots\\ & =\centerdot c_{0}0c_{1}0c_{2}0\ldots \end{align} Thus, $\phi_{2}\left(\mathfrak{z}\right)$ is a mixed point of $\chi_{3}$ for all $\mathfrak{z}\in\mathbb{Z}_{2}$. To use the $L^{1}$ method, like with nearly everything else in this dissertation, we once again appeal to functional equations\index{functional equation}. Specifically: \begin{prop} Let $d$ be an integer $\geq2$, and let $\phi_{d}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ be defined by: \begin{equation} \phi_{d}\left(\sum_{n=0}^{\infty}c_{n}p^{n}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}c_{n}p^{dn}\label{eq:Definition of phi_d} \end{equation} for all $p$-adic integers $\sum_{n=0}^{\infty}c_{n}p^{n}$. Then: \begin{equation} \phi_{d}\left(p\mathfrak{z}+j\right)=p^{d}\phi_{d}\left(\mathfrak{z}\right)+j,\textrm{ }\forall\mathfrak{z}\in\mathbb{Z}_{p},\textrm{ }\forall j\in\mathbb{Z}/p\mathbb{Z}\label{eq:phi_d functional equations} \end{equation} \end{prop} Proof: Let $j\in\left\{ 0,\ldots,p-1\right\} $, and let $\mathfrak{z}=\sum_{n=0}^{\infty}c_{n}p^{n}$. Then: \begin{align} \phi_{d}\left(p\mathfrak{z}+j\right) & =\phi_{d}\left(j+\sum_{n=0}^{\infty}c_{n}p^{n+1}\right)\\ & =j+\sum_{n=0}^{\infty}c_{n}p^{d\left(n+1\right)}\\ & =j+p^{d}\sum_{n=0}^{\infty}c_{n}p^{dn}\\ & =j+p^{d}\phi_{d}\left(\mathfrak{z}\right) \end{align} Q.E.D. \vphantom{} The $\phi_{d}$s are just one example of a possible choice of $\phi$ for which we can apply the $L^{1}$ method. The goal is to choose a $\phi$ satisfying desirable functional equations like these which will then allow us to exploit $\chi_{H}$'s functional equation to show that $\eta=\chi\circ\phi$ satisfies a functional equation as well. This will allow us to recursively solve for an explicit formula of $\hat{\text{\ensuremath{\eta}}}$ whose $L^{1}$ convergence can then be proven by direct estimation. \begin{example}[\textbf{A sample application of the $L^{1}$ method to $\chi_{3}$}] \label{exa:L^1 method example}Let $\eta=\chi_{3}\circ\phi_{2}$. Then, combining the functional equations of $\chi_{3}$ and $\phi_{2}$ yields: \begin{equation} \eta\left(2\mathfrak{z}\right)=\frac{1}{4}\eta\left(\mathfrak{z}\right) \end{equation} \begin{equation} \eta\left(2\mathfrak{z}+1\right)=\frac{3\eta\left(\mathfrak{z}\right)+2}{4} \end{equation} Since $\eta$ will be integrable over $\mathbb{Z}_{2}$ whenever $\hat{\eta}$ is in $L^{1}$, we can proceed by formally computing $\hat{\eta}$ and then showing that it is in $L^{1}$. Formally: \begin{align} \hat{\eta}\left(t\right) & \overset{\mathbb{C}}{=}\int_{\mathbb{Z}_{2}}\eta\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{2}}d\mathfrak{z}\\ & =\int_{2\mathbb{Z}_{2}}\eta\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{2}}d\mathfrak{z}+\int_{2\mathbb{Z}_{2}+1}\eta\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{2}}d\mathfrak{z}\\ \left(\mathfrak{y}=\frac{\mathfrak{z}}{2},\frac{\mathfrak{z}-1}{2}\right); & =\frac{1}{2}\int_{\mathbb{Z}_{2}}\eta\left(2\mathfrak{y}\right)e^{-2\pi i\left\{ 2t\mathfrak{y}\right\} _{2}}d\mathfrak{y}+\frac{1}{2}\int_{\mathbb{Z}_{2}}\eta\left(2\mathfrak{y}+1\right)e^{-2\pi i\left\{ t\left(2\mathfrak{y}+1\right)\right\} _{2}}d\mathfrak{y}\\ & =\frac{1}{8}\int_{\mathbb{Z}_{2}}\eta\left(\mathfrak{y}\right)e^{-2\pi i\left\{ 2t\mathfrak{y}\right\} _{2}}d\mathfrak{y}+\frac{1}{8}\int_{\mathbb{Z}_{2}}\left(3\eta\left(\mathfrak{z}\right)+2\right)e^{-2\pi i\left\{ t\left(2\mathfrak{y}+1\right)\right\} _{2}}d\mathfrak{y}\\ & =\frac{1}{8}\hat{\eta}\left(2t\right)+\frac{3e^{-2\pi i\left\{ t\right\} _{2}}}{8}\hat{\eta}\left(2t\right)+\frac{e^{-2\pi i\left\{ t\right\} _{2}}}{4}\mathbf{1}_{0}\left(2t\right) \end{align} and hence: \begin{equation} \hat{\eta}\left(t\right)\overset{\mathbb{C}}{=}\frac{1+3e^{-2\pi i\left\{ t\right\} _{2}}}{8}\hat{\eta}\left(2t\right)+\frac{e^{-2\pi i\left\{ t\right\} _{2}}}{4}\mathbf{1}_{0}\left(2t\right) \end{equation} Just like we did with $\hat{\chi}_{H,N}\left(t\right)$, nesting this formula will lead to an explicit formula for $\hat{\eta}\left(t\right)$, which can then be used to prove $\left\Vert \hat{\eta}\right\Vert _{L^{1}}<\infty$ and thereby obtain a bound on $\left\|\chi_{3}\left(\phi_{2}\left(\mathfrak{z}\right)\right)\right\|$ via \textbf{Theorem \ref{thm:The L^1 method}}. The reason why this particular choice of $\phi$ was relatively poor is because by inserting a $0$ between \emph{every }two consecutive $2$-adic digits of $\mathfrak{z}$, it forces us to consider $\mathfrak{z}$s associated to composition sequences of the branches of the Shortened Collatz Map where every application of $\frac{3x+1}{2}$ is followed by at least one application of $\frac{x}{2}$. By experimenting with more flexible choices of $\phi$ (say, those satisfying approximate functional equations / inequalities, rather than \emph{exact} ones), it may be possible to obtain non-trivial archimedean bounds on $\chi_{3}$\textemdash and, on $\chi_{H}$s in general. \end{example} \chapter{\label{chap:5 The-Multi-Dimensional-Case}The Multi-Dimensional Case} \includegraphics[scale=0.45]{./PhDDissertationEroica5.png} \vphantom{} As stated in the Introduction (Chapter 1), Chapter 5 is essentially a compactification of the contents of Chapters 2 and 3, presenting most of their contents as their occur in the context of multi-dimensional Hydra maps and the techniques of multi-variable $\left(p,q\right)$-adic analysis employed to study them. Section \ref{sec:5.1 Hydra-Maps-on} introduces multi-dimensional Hydra maps and the two equivalent contexts we use to study them: $\mathbb{Z}^{d}$ and $\mathcal{O}_{\mathbb{F}}$. \section{\label{sec:5.1 Hydra-Maps-on}Hydra Maps on Lattices and Number Rings} When attempting to generalize Hydra maps to ``higher-dimensional'' spaces, there are two possible approaches. The first is to view the higher-dimensional analogue of $\mathbb{Z}$ as being the ring of integers (a.k.a. \textbf{number ring}) of a given \textbf{number field }$\mathbb{F}$\textemdash a degree $d<\infty$ extension of $\mathbb{Q}$. As is traditional in algebraic number theory, we write $\mathcal{O}_{\mathbb{F}}$ to denote the number ring of integers of $\mathbb{F}$ (``$\mathbb{F}$-integers''). The second approach is to work with the lattice $\mathbb{Z}^{d}$ instead of $\mathcal{O}_{\mathbb{F}}$. In theory, these two approaches are ultimately equivalent, with $\mathcal{O}_{\mathbb{F}}$ being $\mathbb{Z}$-module-isomorphic to $\mathbb{Z}^{d}$ after making a choice of a basis. In practice, however, the way of number rings introduces an irksome complication by way of the \textbf{Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain}. Before we begin, it is informative to consider how we might translate the one-dimensional case into the number ring setting. In the one-dimensional case, a $p$-Hydra map $H$ was obtained by stitching together a finite number of affine-linear maps $H_{0},\ldots,H_{p-1}$ on $\mathbb{Q}$ by defining $H\left(n\right)$ to be the image of the integer $n$ under $H_{j}$, where $j$ was the value of $n$ modulo $p$. The essential features of this construction were: \begin{itemize} \item $\mathbb{F}$, a number field; previously, this was $\mathbb{Q}$. \item $\mathcal{O}_{\mathbb{F}}$, the ring of integers of $\mathbb{F}$; previously, this was $\mathbb{Z}$. \item \nomenclature{$\mathfrak{I}$}{a non-zero proper ideal of $\mathcal{O}_{\mathbb{F}}$}A proper ideal\index{ideal!of mathcal{O}{mathbb{F}}@of $\mathcal{O}_{\mathbb{F}}$} of $\mathcal{O}_{\mathbb{F}}$; previously, this was $p\mathbb{Z}$. \item The different co-sets of $\mathfrak{I}$ in $\mathcal{O}_{\mathbb{F}}$; previously, these were the different equivalence classes modulo $p$. \item For each co-set of $\mathfrak{I}$, an affine linear map which sends elements of that particular co-set to some (possibly different) co-set of $\mathfrak{I}$; previously, these were the $H_{j}$s. \item Congruences which tell us the co-set of $\mathfrak{I}$ to which a given element of $\mathcal{O}_{\mathbb{F}}$ belongs; previously, these were $\overset{p}{\equiv}j$ for $j\in\mathbb{Z}/p\mathbb{Z}$. \end{itemize} \begin{example} \label{exa:root 2 example}Let $\mathbb{F}=\mathbb{Q}\left(\sqrt{2}\right)$, $\mathcal{O}_{\mathbb{F}}=\mathbb{Z}\left[\sqrt{2}\right]$, and consider the ideal $\mathfrak{I}=\left\langle \sqrt{2}\right\rangle $ generated by $\sqrt{2}$ along with the quotient ring $\mathbb{Z}\left[\sqrt{2}\right]/\left\langle \sqrt{2}\right\rangle $. We $\left\{ 1,\sqrt{2}\right\} $ as our basis for $\mathbb{Z}\left[\sqrt{2}\right]$. Since $\left\langle 2\right\rangle $ is a proper ideal of $\left\langle \sqrt{2}\right\rangle $, it follows that everything in $\mathbb{Z}\left[\sqrt{2}\right]$ which is congruent to $0$ mod $2$ will also be congruent to $0$ mod $\sqrt{2}$. Additionally, since $2\in\mathfrak{I}$, it follows that $a+b\sqrt{2}\overset{\sqrt{2}}{\equiv}a$ for all $a,b\in\mathbb{Z}$. If $a$ is even, then $a\overset{\sqrt{2}}{\equiv}0$. If $a$ is odd, then $a\overset{2}{\equiv}1$. Moreover, since $\sqrt{2}$ divides $2$, note that $a\overset{2}{\equiv}1$ forces $a\overset{\sqrt{2}}{\equiv}1$. Consequently, given any number $a+b\sqrt{2}\in\mathbb{Z}\left[\sqrt{2}\right]$, the equivalence class in $\mathbb{Z}\left[\sqrt{2}\right]/\left\langle \sqrt{2}\right\rangle $ to which belongs\textemdash a.k.a, the value of $a+b\sqrt{2}$ ``mod $\sqrt{2}$''\textemdash is then uniquely determined by the value of $a$ mod $2$. In particular, every $a+b\sqrt{2}\in\mathbb{Z}\left[\sqrt{2}\right]$ is congruent to either $0$ or $1$ mod $\sqrt{2}$, with: \[ a+b\sqrt{2}\overset{\sqrt{2}}{\equiv}0\Leftrightarrow a\overset{2}{\equiv}0 \] \[ a+b\sqrt{2}\overset{\sqrt{2}}{\equiv}1\Leftrightarrow a\overset{2}{\equiv}1 \] As such, if we write $\mathbb{Z}\left[\sqrt{2}\right]$ in $\left\{ 1,\sqrt{2}\right\} $-coordinates, the $2$-tuple $\mathbf{m}=\left(a,b\right)\in\mathbb{Z}^{2}$ corresponding to $a+b\sqrt{2}\in\mathbb{Z}\left[\sqrt{2}\right]$ represents a number congruent to $0$ (resp. $1$) mod $\sqrt{2}$ if and only if the first component of $\mathbf{m}$ ($a$) is congruent to $0$ (resp. $1$) mod $2$. \end{example} \vphantom{} The reason \textbf{Example \ref{exa:root 2 example}} works out so nicely is because quotienting the number ring $\mathbb{Z}\left[\sqrt{2}\right]$ by the ideal $\left\langle \sqrt{2}\right\rangle $ yields a ring isomorphic to $\mathbb{Z}/2\mathbb{Z}$, which is cyclic as an additive group. However, as any algebraic number theorist will readily tell you, for a general number field $\mathbb{F}$ and an arbitrary non-zero proper ideal $\mathfrak{I}$ of $\mathcal{O}_{\mathbb{F}}$, there is no guarantee that the ring $\mathcal{O}_{\mathbb{F}}/\mathfrak{I}$ will be cyclic as an additive group. Rather, the Structure Theorem for Finitely-Generated Modules over a Principal Ideal Domain tells us that the most we can expect is for there to be an isomorphism of additive groups: \begin{equation} \mathcal{O}_{\mathbb{F}}/\mathfrak{I}\cong\mathfrak{C}_{p_{1}}\times\cdots\times\mathfrak{C}_{p_{r}}\label{eq:Direct Product Representation of O_F / I} \end{equation} where $\mathfrak{C}_{p_{n}}$ is the cyclic group of order $p_{n}$ \nomenclature{$\mathfrak{C}_{n}$}{cyclic group of order $n$ \nopageref}, where $r\in\left\{ 1,\ldots,\left\|\mathcal{O}_{\mathbb{F}}/\mathfrak{I}\right\|\right\} $, and where the $p_{n}$s are integers $\geq2$ so that $p_{n}\mid p_{n+1}$ for all $n\in\left\{ 1,\ldots,r-1\right\} $. In analogy to the one-dimensional case, $\mathcal{O}_{\mathbb{F}}/\mathfrak{I}$ was $\mathbb{Z}/p\mathbb{Z}$, the set to which the branch-determining parameter $j$ belonged. As (\ref{eq:Direct Product Representation of O_F / I}) shows, however, for the multi-dimensional case, we cannot assume that the branch-determining parameter will take integer values like in \textbf{Example \ref{exa:root 2 example}}, where the parameter was $j\in\left\{ 0,1\right\} $. Instead, the branch-determining parameter we be an \textbf{$r$-tuple of integers }$\mathbf{j}\in\left(\mathbb{Z}/p_{1}\mathbb{Z}\right)\times\cdots\times\left(\mathbb{Z}/p_{r}\mathbb{Z}\right)$. In the one-dimensional case, we obtained an extension of $\chi_{H}$ by considering infinitely long lists (strings) of parameters. In the multi-dimensional case, an infinitely long list of $r$-tuples $\mathbf{j}\in\left(\mathbb{Z}/p_{1}\mathbb{Z}\right)\times\cdots\times\left(\mathbb{Z}/p_{r}\mathbb{Z}\right)$ would then be identified with an $r$-tuple $\mathbf{z}\in\mathbb{Z}_{p_{1}}\times\cdots\times\mathbb{Z}_{p_{r}}$ whose $n$th entry was a $p_{n}$-adic integer corresponding to the string whose elements are the $n$th entries of the $\mathbf{j}$s. Although this situation creates no problems when performing Fourier analysis on multi-dimensional $\chi_{H}$, it causes trouble for our proof of the multi-dimensional analogue of the Correspondence Principle, because it potentially makes it impossible for us to define a $p$-adic extension of the multi-dimensional Hydra map $H$. As such, for Subsection \ref{sec:5.2 The-Numen-of}, we will need to restrict to the case where every element of $P$ is a single prime, $p$. \subsection{\label{subsec:5.1.1. Algebraic-Conventions}Algebraic Conventions} To minimize any cumbersome aspects of notation, we introduce the following conventions\footnote{The one exception to this will be for multi-dimensional analogues of functions from the one-dimensional case, such as $\chi_{H}$, $\hat{A}_{H}$, $\alpha_{H}$, and the rest. For them, I retain as much of the one-dimensional notation as possible to emphasize the parallels between the two cases. Indeed, most of the computations will be repeated nearly verbatim.}: \begin{itemize} \item \textbf{Bold, lower case letters }(ex: $\mathbf{j}$, $\mathbf{a}$, $\mathbf{n}$, $\mathbf{x}$, $\mathbf{z}$, etc.) are reserved to denote tuples of finite length. In computations involving matrices, such tuples will always be treated as column vectors. Row vectors are written with a superscript $T$ to denote the transpose of column vector (ex: \textbf{$\mathbf{j}^{T}$}). \item \textbf{BOLD, UPPER CASE LETTERS }(ex: $\mathbf{A},\mathbf{D},\mathbf{P}$, etc.) are reserved to denote $d\times d$ matrices. In particular, $\mathbf{D}$ will be used for \textbf{diagonal matrices}, while $\mathbf{P}$ will be used for \textbf{permutation matrices}\textemdash matrices (whose entries are $0$s and $1$s) that give the so-called ``defining representation'' of the symmetric group $\mathfrak{S}_{d}$ acting on $\mathbb{R}^{d}$. \end{itemize} It is not an exaggeration to say that the primary challenge of the multi-dimensional case is its notation. As a rule of thumb, whenever vectors are acting on vectors, reader should assume that the operations are being done entry-wise, unless state otherwise. That being said, there is quite a lot of notation we will have to cover. In order to avoid symbol overload, the notation will only be introduced as needed. The primary chunks of notation occur in the segment given below, and then again at the start of Subsections \ref{subsec:5.3.1 Tensor-Products} and \ref{subsec:5.4.1 Multi-Dimensional--adic-Fourier}. \begin{notation}[\textbf{Multi-Dimensional Notational Conventions} \index{multi-dimensional!notation}] \label{nota:First MD notation batch}Let $d$ be an integer, let $\mathbb{F}$ be a field, let $s\in\mathbb{F}$, and let $\mathbf{a}=\left(a_{1},\ldots,a_{d}\right)$, $\mathbf{b}=\left(b_{1},\ldots,b_{d}\right)$, and $\mathbf{c}=\left(c_{1},\ldots,c_{d}\right)$ be elements of $\mathbb{F}^{d}$. Then:\index{multi-dimensional!notation} \vphantom{} I. $\mathbf{a}+\mathbf{b}\overset{\textrm{def}}{=}\left(a_{1}+b_{1},\ldots,a_{d}+b_{d}\right)$ \vphantom{} II. $\mathbf{a}\mathbf{b}\overset{\textrm{def}}{=}\left(a_{1}b_{1},\ldots,a_{d}b_{d}\right)$ \vphantom{} III. $\frac{\mathbf{a}}{\mathbf{b}}\overset{\textrm{def}}{=}\left(\frac{a_{1}}{b_{1}},\ldots,\frac{a_{d}}{b_{d}}\right)$ \vphantom{} IV. $s\mathbf{a}\overset{\textrm{def}}{=}\left(sa_{1},\ldots,sa_{d}\right)$ \vphantom{} V. $\mathbf{a}+s\overset{\textrm{def}}{=}\left(a_{1}+s,\ldots,a_{d}+s\right)$ \vphantom{} VI. \[ \sum_{\mathbf{k}=\mathbf{a}}^{\mathbf{b}}\overset{\textrm{def}}{=}\sum_{k_{1}=a_{1}}^{b_{1}}\cdots\sum_{k_{d}=a_{d}}^{b_{d}} \] \vphantom{} VII. $\mathbf{I}_{d}$ \nomenclature{$\mathbf{I}_{d}$}{$d\times d$ identity matrix \nopageref} denotes the $d\times d$ identity matrix. $\mathbf{O}_{d}$\nomenclature{$\mathbf{O}_{d}$}{$d\times d$ zero matrix \nopageref} denotes the $d\times d$ zero matrix. $\mathbf{0}$\nomenclature{$\mathbf{0}$}{zero vector \nopageref} denotes a column vector of $0$s (a.k.a., the zero vector). Unfortunately, the length of this $\mathbf{0}$ will usually depend on context. \vphantom{} VIII. Given $d\times d$ matrices $\mathbf{A}$ and $\mathbf{B}$ with entries in a field $\mathbb{F}$, if $\mathbf{B}$ is invertible, we write: \nomenclature{$\frac{\mathbf{A}}{\mathbf{B}}$}{$\mathbf{B}^{-1}\mathbf{A}$ \nopageref} \begin{equation} \frac{\mathbf{A}}{\mathbf{B}}\overset{\textrm{def}}{=}\mathbf{B}^{-1}\mathbf{A}\label{eq:Matrix fraction convention} \end{equation} \end{notation} \begin{defn} We write\nomenclature{$P$}{$\left(p_{1},\ldots,p_{r}\right)$}: \begin{equation} P\overset{\textrm{def}}{=}\left(p_{1},\ldots,p_{r}\right)\label{eq:Definition of Big P} \end{equation} be to denote $r$-tuple of integers (all of which are $\geq2$), where $r$ is a positive integer, and where $p_{n}\mid p_{n+1}$ for all $n\in\left\{ 1,\ldots,r-1\right\} $. We then write\nomenclature{$\mathbb{Z}^{r}/P\mathbb{Z}^{r}$}{$\overset{\textrm{def}}{=}\prod_{m=1}^{r}\left(\mathbb{Z}/p_{m}\mathbb{Z}\right)$ } to denote the direct product of rings: \begin{equation} \mathbb{Z}^{r}/P\mathbb{Z}^{r}\overset{\textrm{def}}{=}\prod_{m=1}^{r}\left(\mathbb{Z}/p_{m}\mathbb{Z}\right) \end{equation} That is, $\mathbb{Z}^{r}/P\mathbb{Z}^{r}$ is set of all $r$-tuples $\mathbf{j}=\left(j_{1},\ldots,j_{r}\right)$ so that $j_{m}\in\mathbb{Z}/p_{m}\mathbb{Z}$ for all $m$, equipped with component-wise addition and multiplication operations on the $\mathbb{Z}/p_{m}\mathbb{Z}$s. We also write $\mathbb{Z}^{r}/p\mathbb{Z}^{r}\overset{\textrm{def}}{=}\left(\mathbb{Z}/p\mathbb{Z}\right)^{r}$ to denote the case where $p_{n}=p$ (for some integer $p\geq2$) for all $n\in\left\{ 1,\ldots,r\right\} $. Given a $\mathbf{j}=\left(j_{1},\ldots,j_{r}\right)\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$ and any tuple $\mathbf{x}=\left(x_{1},\ldots,x_{\left\|\mathbf{x}\right\|}\right)$ of length $\geq1$, we write: \begin{align} \mathbf{x} & \overset{P}{\equiv}\mathbf{j}\\ & \Updownarrow\\ x_{n} & \overset{p_{n}}{\equiv}j_{n},\textrm{ }\forall n\in\left\{ 1,\ldots,\min\left\{ r,\left\|\mathbf{x}\right\|\right\} \right\} \end{align} When there is an integer $p\geq2$ so that $p_{n}=p$ for all $n$, we write: \begin{align} \mathbf{x} & \overset{p}{\equiv}\mathbf{j}\\ & \Updownarrow\\ x_{n} & \overset{p}{\equiv}j_{n},\textrm{ }\forall n\in\left\{ 1,\ldots,\min\left\{ r,\left\|\mathbf{x}\right\|\right\} \right\} \end{align} \end{defn} \subsection{Multi-Dimensional Hydra Maps \label{subsec:5.1.2 Co=00003D0000F6rdinates,-Half-Lattices,-and}} \begin{defn} Fix a number field $\mathbb{F}$ of degree $d$ over $\mathbb{Q}$, and a non-zero proper ideal $\mathfrak{I}\subset\mathcal{O}_{\mathbb{F}}$. \nomenclature{$\iota$}{$\left\|\mathcal{O}_{\mathbb{F}}/\mathfrak{I}\right\|$}We write: \begin{equation} \iota\overset{\textrm{def}}{=}\left\|\mathcal{O}_{\mathbb{F}}/\mathfrak{I}\right\|\label{eq:definition of iota_I} \end{equation} In algebraic terminology, $\iota$ is the \textbf{index }of\index{ideal!index of a} $\mathfrak{I}$ in\emph{ $\mathcal{O}_{\mathbb{F}}$. }As discussed above, there is an isomorphism of additive groups: \begin{equation} \mathcal{O}_{\mathbb{F}}/\mathfrak{I}\cong\mathfrak{C}_{p_{1}}\times\cdots\times\mathfrak{C}_{p_{r}} \end{equation} where $\mathfrak{C}_{p_{i}}$ is the cyclic group of order $p_{i}$, where $r\in\left\{ 1,\ldots,\iota\right\} $, where $p_{n}\mid p_{n+1}$ for all $n\in\left\{ 1,\ldots,r-1\right\} $ and: \begin{equation} \prod_{n=1}^{r}p_{n}=\iota\label{eq:Relation between the rho_is and iota_I} \end{equation} I call $r$ the \textbf{depth}\footnote{From an algebraic perspective, the decomposition (\ref{eq:Direct Product Representation of O_F / I-1}) is relatively unremarkable, being a specific case of the structure theorem for finitely generated modules over a principal ideal domain. To an algebraist, $r_{\mathfrak{I}}$ would be ``the number of invariant factors of the $\mathbb{Z}$-module $\mathcal{O}_{\mathbb{F}}/\mathfrak{I}$''; ``depth'', though not standard terminology, is certainly pithier.}\textbf{ }of\index{ideal!depth of a} $\mathfrak{I}$. In addition to the above, note that there is a set $\mathcal{B}\subset\mathcal{O}_{\mathbb{F}}$ which is a $\mathbb{Z}$-basis of the group $\left(\mathcal{O}_{\mathbb{F}},+\right)$, so that, for any $d$-tuple $\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{Z}^{d}$ representing an element $z\in\mathcal{O}_{\mathbb{F}}$, the equivalence class of $\mathcal{O}_{\mathbb{F}}/\mathfrak{I}$ to which $z$ belongs is then completely determined by the values $x_{1}$ mod $p_{1}$, $x_{2}$ mod $p_{2}$, $\ldots$, $x_{r_{\mathfrak{I}}}$ mod $p_{r}$. \emph{From here on out, we will work with a fixed choice of such a basis $\mathcal{B}$}. We write $\mathbf{R}$ to denote the $d\times d$ diagonal matrix: \begin{equation} \mathbf{R}\overset{\textrm{def}}{=}\left[\begin{array}{cccccc} p_{1}\\ & \ddots\\ & & p_{r}\\ & & & 1\\ & & & & \ddots\\ & & & & & 1 \end{array}\right]\label{eq:Definition of bold R} \end{equation} where there are $d-r$ $1$s after $p_{r}$. Lastly, when adding vectors of unequal length, we do everything from left to right: \begin{equation} \mathbf{v}+\mathbf{j}\overset{\textrm{def}}{=}\left(v_{1}+j_{1},\ldots,v_{r}+j_{r},v_{r+1}+0,\ldots,v_{d}+0\right),\textrm{ }\forall\mathbf{v}\in\mathbb{Z}^{d},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}\label{eq:Definition of the sum of bold_m and bold_j} \end{equation} \end{defn} \begin{defn} Letting everything be as described above, enumerate the elements of the basis $\mathcal{B}$ as $\left\{ \gamma_{1},\ldots,\gamma_{d}\right\} $. We then write $\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}$ to denote the \textbf{half-lattice}\index{half-lattice}\textbf{ of }$\mathcal{B}$\textbf{-positive $\mathbb{F}$-integers} (or ``positive half-lattice''), defined by: \begin{equation} \mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\overset{\textrm{def}}{=}\left\{ \sum_{\ell=1}^{d}x_{\ell}\gamma_{\ell}:x_{1},\ldots,x_{d}\in\mathbb{N}_{0}\right\} \label{eq:Def of O-plus_F,B} \end{equation} The \textbf{half-lattice of $\mathcal{B}$-negative $\mathbb{F}$-integers }(or ``negative half-lattice''), denoted $\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}$, is defined by: \begin{equation} \mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\overset{\textrm{def}}{=}\left\{ -\sum_{\ell=1}^{d}x_{\ell}\gamma_{\ell}:x_{1},\ldots,x_{d}\in\mathbb{N}_{0}\right\} \label{eq:Def of O_F,B minus} \end{equation} Finally, we write \nomenclature{$\varphi_{\mathcal{B}}$}{ }$\varphi_{\mathcal{B}}:\mathbb{Q}^{d}\rightarrow\mathbb{F}$ to denote the vector space isomorphism that sends each $\mathbf{x}\in\mathbb{Q}^{d}$ to the unique element of $\mathbb{F}$ whose representation in $\mathcal{B}$-cordinates is $\mathbf{x}$, and which satisfies the property that the restriction $\varphi_{\mathcal{B}}\mid_{\mathbb{Z}^{d}}$ of $\varphi_{\mathcal{B}}$ to $\mathbb{Z}^{d}$ is an isomorphism of the groups $\left(\mathbb{Z}^{d},+\right)$ and $\left(\mathcal{O}_{\mathbb{F}},+\right)$. We write $\varphi_{\mathcal{B}}^{-1}$ to denote the inverse of $\varphi_{\mathcal{B}}$; $\varphi_{\mathcal{B}}^{-1}$ outputs the $d$-tuple $\mathcal{B}$-cordinate representation of the inputted $z\in\mathbb{F}$. We then write $\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+}$ \nomenclature{$\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+}$}{$\varphi_{\mathcal{B}}\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)$} and $\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{-}$ \nomenclature{$\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{-}$}{$\varphi_{\mathcal{B}}\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\right)$} to denote $\varphi_{\mathcal{B}}\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)\subset\mathbb{Z}^{d}$ and $\varphi_{\mathcal{B}}\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\right)\subset\mathbb{Z}^{d}$, respectively. Note that, as subsets of $\mathbb{Z}^{d}$, we have: \begin{align} \mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+} & =\mathbb{N}_{0}^{d}\\ \mathbb{Z}_{\mathbb{F},\mathcal{B}}^{-} & =-\mathbb{N}_{0}^{d} \end{align} \end{defn} \begin{defn}[\textbf{$\left(\mathbb{F},\mathfrak{I},\mathcal{B}\right)$-Hydra maps}] \label{def:(F,I,B)-Hydra map}Let $\mathbb{F}$ be a number field of dimension $d$, let $\mathfrak{I}$ be a non-zero proper ideal of $\mathcal{O}_{\mathbb{F}}$ of index $\iota$, and let $\mathcal{B}$ be a basis as discussed above. We write $\mathfrak{I}_{0},\ldots,\mathfrak{I}_{\iota}$ to denote the co-sets of $\mathfrak{I}$ in $\mathcal{O}_{\mathbb{F}}$, with $\mathfrak{I}_{0}$ denoting $\mathfrak{I}$ itself. Then, a \textbf{$\left(\mathbb{F},\mathfrak{I},\mathcal{B}\right)$-Hydra map} \textbf{on $\mathcal{O}_{\mathbb{F}}$} is a surjective map $\tilde{H}:\mathcal{O}_{\mathbb{F}}\rightarrow\mathcal{O}_{\mathbb{F}}$ of the form: \begin{equation} \tilde{H}\left(z\right)=\begin{cases} \frac{a_{0}z+b_{0}}{d_{0}} & \textrm{if }z\in\mathfrak{I}_{0}\\ \vdots & \vdots\\ \frac{a_{\iota-1}z+b_{\iota-1}}{d_{\iota-1}} & \textrm{if }z\in\mathfrak{I}_{\iota} \end{cases}\label{eq:Definition of I-hydra map} \end{equation} We call $r$ the \textbf{depth }of $\tilde{H}$. Here, the $a_{j}$s, $b_{j}$s, and $d_{j}$s are elements of $\mathcal{O}_{\mathbb{F}}$ so that: \vphantom{} I. $a_{j},d_{j}\neq0$ for all $j\in\left\{ 0,\ldots,\iota-1\right\} $. \vphantom{} II. $\gcd\left(a_{j},d_{j}\right)=1$ for all $j\in\left\{ 0,\ldots,\iota-1\right\} $. \vphantom{} III. $H\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)\subseteq H\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)$ and $H\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\backslash\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)\subseteq H\left(\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\backslash\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right)$, where $\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}\backslash\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}$ is the set of all elements of $\mathcal{O}_{\mathbb{F},\mathcal{B}}^{-}$ which are not elements of $\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}$. \vphantom{} IV. For all $j\in\left\{ 0,\ldots,\iota-1\right\} $, the ideal $\left\langle d_{j}\right\rangle _{\mathcal{O}_{\mathbb{F}}}$ in $\mathcal{O}_{\mathbb{F}}$ generated by $d_{j}$ is contained in $\mathfrak{I}$. \vphantom{} V. For all $j\in\left\{ 0,\ldots,\iota-1\right\} $, the the matrix representation in $\mathcal{B}$-coordinates on $\mathbb{Q}^{d}$ of the ``multiplication by $a_{j}/d_{j}$'' map on $\mathbb{F}$ is of the form: \begin{equation} \frac{\mathbf{A}}{\mathbf{D}}\overset{\textrm{def}}{=}\mathbf{D}^{-1}\mathbf{A}\label{eq:A / D notation} \end{equation} where $\mathbf{A},\mathbf{D}$ are invertible $d\times d$ matrices so that: \vphantom{} V-i. $\mathbf{A}=\tilde{\mathbf{A}}\mathbf{P}$, where $\mathbf{P}$ is a permutation matrix\footnote{That is, a matrix of $0$s and $1$s which is a representation of the action on $\mathbb{Z}^{d}$ of an element of the symmetric group on $d$ objects by way of a permutation of the coordinate entries of the $d$-tuples in $\mathbb{Z}^{d}$.} and where $\mathbf{\tilde{\mathbf{A}}}$ is a diagonal matrix whose non-zero entries are positive integers. \vphantom{} V-ii. $\mathbf{D}$ is a diagonal matrix whose non-zero entries are positive integers such that every entry on the diagonal of $\mathbf{R}\mathbf{D}^{-1}$ is a positive integer. Note that this forces the $\left(r+1\right)$th through $d$th diagonal entries of $\mathbf{D}$ to be equal to $1$. Moreover, for each $\ell\in\left\{ 1,\ldots,r\right\} $, this forces the $\ell$th entry of the diagonal of $\mathbf{D}$ to be a divisor of $p_{\ell}$. \vphantom{} V-iii. Every non-zero element of $\mathbf{A}$ is co-prime to every non-zero element of $\mathbf{D}$. \vphantom{} Additionally, we say that $\tilde{H}$ is \textbf{integral }if it satisfies\index{Hydra map!integral}: \vphantom{} VI. For all $j\in\left\{ 0,\ldots,\iota-1\right\} $, the number $\frac{a_{j}z+b_{j}}{d_{j}}$ is an element of $\mathcal{O}_{\mathbb{F}}$ if and only if $z\in\mathcal{O}_{\mathbb{F}}\cap\mathfrak{I}_{j}$. \vphantom{} If $\tilde{H}$ does not satisfy (VI), we say $\tilde{H}$ is \textbf{non-integral}. \index{Hydra map!non-integral} \end{defn} \begin{defn}[\textbf{$P$-Hydra maps}] \label{def:P-Hydra map}Let $\mathbb{F}$ be a number field of dimension $d$, let $\mathfrak{I}$ be a non-zero proper ideal of $\mathcal{O}_{\mathbb{F}}$ of index $\iota$, and let $\mathcal{B}$ be a basis as discussed above. A $P$\textbf{-Hydra map} \textbf{on $\mathbb{Z}^{d}$}\index{$P$-Hydra map}\textbf{ }is a map $H:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ so that: \begin{equation} H=\varphi_{\mathcal{B}}\circ\tilde{H}\circ\varphi_{\mathcal{B}}^{-1}\label{eq:Definition of a Hydra map on Zd} \end{equation} for some $\left(\mathbb{F},\mathfrak{I},\mathcal{B}\right)$-Hydra map $\tilde{H}$ on $\mathcal{O}_{\mathbb{F}}$ for which $P=\left(p_{1},\ldots,p_{r}\right)$. We\index{Hydra map!field analogue} \index{Hydra map!lattice analogue}call $\tilde{H}$ the \textbf{field analogue }of $H$, and call $H$ the \textbf{lattice analogue }of $\tilde{H}$. We say $H$ has \textbf{depth} $r$ whenever $\tilde{H}$ has depth $r$.\index{Hydra map!depth} \end{defn} \begin{prop}[Formula for $P$-Hydra maps] Let $\tilde{H}:\mathcal{O}_{\mathbb{F}}\rightarrow\mathcal{O}_{\mathbb{F}}$ be an $\left(\mathbb{F},\mathfrak{I},\mathcal{B}\right)$-Hydra map, and let $H$ be its lattice analogue (a $P$-Hydra map). Then, there is a unique collection of integer-entry matrices: \[ \left\{ \mathbf{A}_{\mathbf{j}}\right\} _{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}},\left\{ \mathbf{D}_{\mathbf{j}}\right\} _{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\subseteq\textrm{GL}_{d}\left(\mathbb{Q}\right) \] satisfying conditions \emph{(V-i)}, \emph{(V-ii), and (V-iii) }from\textbf{ }\emph{(\ref{def:(F,I,B)-Hydra map})} for each $\mathbf{j}$, respectively, and a unique collection of $d\times1$ column vectors $\left\{ \mathbf{b}_{\mathbf{j}}\right\} _{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\subseteq\mathbb{Z}^{d}$ so that: \nomenclature{$\mathbf{A}_{\mathbf{j}}$}{ } \nomenclature{$\mathbf{D}_{\mathbf{j}}$}{ } \nomenclature{$\mathbf{b}_{\mathbf{j}}$}{ } \index{Hydra map}\index{Hydra map!on mathbb{Z}{d}@on $\mathbb{Z}^{d}$} \index{$P$-Hydra map} \index{multi-dimensional!Hydra map} \begin{equation} H\left(\mathbf{x}\right)=\sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\left[\mathbf{x}\overset{P}{\equiv}\mathbf{j}\right]\frac{\mathbf{A}_{\mathbf{j}}\mathbf{x}+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\textrm{ }\forall\mathbf{x}\in\mathbb{Z}^{d}\label{eq:MD Hydra Map Formula} \end{equation} In particular, for all $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, the $d$-tuple $\mathbf{D}_{\mathbf{j}}^{-1}\left(\mathbf{A}_{\mathbf{j}}\mathbf{x}+\mathbf{b}_{\mathbf{j}}\right)$ will be an element of $\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+}$ for all $\mathbf{x}\in\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+}$ for which $\mathbf{x}\overset{P}{\equiv}\mathbf{j}$. \end{prop} \begin{rem} Recall that the congruence $\mathbf{x}\overset{P}{\equiv}\mathbf{j}$ means that for each $n\in\left\{ 1,\ldots,r\right\} $, the $n$th entry of $\mathbf{x}$ is congruent mod $p_{n}$ to the $n$th entry of $\mathbf{j}$. Moreover, $\mathbf{x}\overset{P}{\equiv}\mathbf{j}$ is completely independent of the $\left(r+1\right)$th through $d$th entries of $\mathbf{x}$. \end{rem} Proof: Let everything be as described above. Fix $z=\sum_{n=1}^{d}c_{n}\gamma_{n}\in\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}$, where the $c_{n}$s are non-negative integers. Letting $\mathfrak{I}_{j}$ denote the unique equivalence class of $\mathfrak{I}$ in $\mathcal{O}_{\mathbb{F}}$ to which $z$ belongs, it follows by definition of $\tilde{H}$ that: \begin{equation} \tilde{H}\left(z\right)=\frac{a_{j}z+b_{j}}{d_{j}} \end{equation} Now, consider $\mathbb{F}$ as a $d$-dimensional linear space over $\mathbb{Q}$, equipped with the coordinate system given by the basis $\mathcal{B}=\left\{ \gamma_{1},\ldots,\gamma_{d}\right\} $. In these coordinates, the ``multiplication by $a_{j}/d_{j}$'' map on $\mathbb{F}$ can be uniquely represented as left-multiplication by some $\mathbf{C}\in\textrm{GL}_{d}\left(\mathbb{Q}\right)$ with rational entries. So, letting $\mathbf{x}$ denote the coordinate $d$-tuple representing $z$ (that is, $\mathbf{x}=\varphi_{\mathcal{B}}^{-1}\left(z\right)$), it follows that: \begin{equation} \varphi_{\mathcal{B}}^{-1}\left(\frac{a_{j}z}{d_{j}}\right)=\mathbf{C}\mathbf{x} \end{equation} Letting $\mathbf{k}=\varphi_{\mathcal{B}}^{-1}\left(b_{j}/d_{j}\right)$ be unique the coordinate $d$-tuple representing $b_{j}/d_{j}$, we then have that: \begin{equation} \varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right)=\varphi_{\mathcal{B}}^{-1}\left(\frac{a_{j}z+b_{j}}{d_{j}}\right)=\varphi_{\mathcal{B}}^{-1}\left(\frac{a_{j}z}{d_{j}}\right)+\varphi_{\mathcal{B}}^{-1}\left(\frac{b_{j}}{d_{j}}\right)=\mathbf{C}\mathbf{x}+\mathbf{k} \end{equation} where we used the fact that, as defined, $\varphi_{\mathcal{B}}$ is an isomorphism of the linear spaces $\mathbb{F}$ and $\mathbb{Q}^{d}$. By (\ref{eq:A / D notation}), we can write $\mathbf{C}$ as: \begin{equation} \mathbf{C}=\mathbf{D}^{-1}\mathbf{A} \end{equation} for $\mathbf{D},\mathbf{A}$ as described in (V) of (\ref{def:(F,I,B)-Hydra map}). We can make $\mathbf{D}$ and $\mathbf{A}$ unique by choosing them so as to make their non-zero entires (all of which are positive integers) as small as possible. Consequently, the vector: \begin{equation} \mathbf{b}\overset{\textrm{def}}{=}\mathbf{D}\mathbf{k} \end{equation} will have integer entries, and we can express the action of the $j$th branch of $\tilde{H}$ on an arbitrary $\mathbf{x}$ by: \begin{equation} \varphi_{\mathcal{B}}^{-1}\left(\frac{a_{j}z+b_{j}}{d_{j}}\right)=\frac{\mathbf{A}\mathbf{x}+\mathbf{b}}{\mathbf{D}} \end{equation} Since $\mathcal{O}_{\mathbb{F}}/\mathfrak{I}\cong\mathbb{Z}^{r}/P\mathbb{Z}^{r}$ for each $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, by the argument given in the previous paragraph, there are unique invertible $\mathbf{A}_{\mathbf{j}},\mathbf{D}_{\mathbf{j}}$ satisfying (V-i), (V-ii), and (V-iii) respectively and a unique $d\times1$ column vector $\mathbf{b}_{\mathbf{j}}$ with integer entries so that: \begin{equation} \varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right)=\frac{\mathbf{A}_{\mathbf{j}}\varphi_{\mathcal{B}}^{-1}\left(z\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}} \end{equation} holds for all $z\in\mathcal{O}_{\mathbb{F}}$ that are congruent to $\varphi_{\mathcal{B}}\left(\mathbf{j}\right)$ mod $\mathfrak{I}$. Multiplying by the Iverson bracket $\left[\varphi_{\mathcal{B}}^{-1}\left(z\right)\overset{P}{\equiv}\mathbf{j}\right]$ and summing over all $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$ gives: \begin{eqnarray} \sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\left[\varphi_{\mathcal{B}}^{-1}\left(z\right)\overset{P}{\equiv}\mathbf{j}\right]\left(\frac{\mathbf{A}_{\mathbf{j}}\varphi_{\mathcal{B}}^{-1}\left(z\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\right) & = & \sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\left[\varphi_{\mathcal{B}}^{-1}\left(z\right)\overset{P}{\equiv}\mathbf{j}\right]\varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right)\\ & = & \sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\underbrace{\left[z\overset{\mathfrak{I}}{\equiv}\varphi_{\mathcal{B}}\left(\mathbf{j}\right)\right]}_{\in\left\{ 0,1\right\} }\varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right)\\ & = & \varphi_{\mathcal{B}}^{-1}\left(\sum_{\mathbf{j}\in\mathcal{O}_{\mathbb{F}}/\mathfrak{I}}\left[z\overset{\mathfrak{I}}{\equiv}\varphi_{\mathcal{B}}\left(\mathbf{j}\right)\right]\tilde{H}\left(z\right)\right)\\ \left(\varphi_{\mathcal{B}}\left(\mathbf{j}\right)\textrm{s partition }\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+}\right); & = & \varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right) \end{eqnarray} Hence: \begin{equation} \varphi_{\mathcal{B}}^{-1}\left(\tilde{H}\left(z\right)\right)=\sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\left[\varphi_{\mathcal{B}}^{-1}\left(z\right)\overset{P}{\equiv}\mathbf{j}\right]\frac{\mathbf{A}_{\mathbf{j}}\varphi_{\mathcal{B}}^{-1}\left(z\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\forall z\in\mathcal{O}_{\mathbb{F},\mathcal{B}}^{+} \end{equation} Replacing $z$ with $\varphi_{\mathcal{B}}\left(\mathbf{x}\right)$ (where $\mathbf{x}\in\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+}$) then gives the desired formula for $H\left(\mathbf{x}\right)$: \begin{equation} \underbrace{\left(\varphi_{\mathcal{B}}^{-1}\circ\tilde{H}\circ\varphi_{\mathcal{B}}\right)\left(\mathbf{x}\right)}_{H\left(\mathbf{x}\right)}=\sum_{\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}}\left[\mathbf{x}\overset{P}{\equiv}\mathbf{j}\right]\frac{\mathbf{A}_{\mathbf{j}}\mathbf{x}+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\forall\mathbf{x}\in\mathbb{Z}_{\mathbb{F},\mathcal{B}}^{+} \end{equation} Q.E.D. \vphantom{} Finally, we introduce notation to take on the roles played by $\mu_{j}$ and $p$ in the one-dimensional case. \begin{defn} Let $H$ be a $P$-Hydra map on $\mathbb{Z}^{d}$. \vphantom{} I. For each $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, we write: \begin{equation} \mathbf{M}_{\mathbf{j}}\overset{\textrm{def}}{=}\mathbf{R}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\overset{\textrm{def}}{=}\mathbf{R}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\label{eq:Definition of bold M bold j} \end{equation} \vphantom{} II. \nomenclature{$H_{\mathbf{j}}\left(\mathbf{x}\right)$}{$\mathbf{j}$th branch of a multi-dimensional Hydra map}For each $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, we call the affine linear map $H_{\mathbf{j}}:\mathbb{Q}^{d}\rightarrow\mathbb{Q}^{d}$ defined by: \begin{equation} H_{\mathbf{j}}\left(\mathbf{x}\right)\overset{\textrm{def}}{=}\frac{\mathbf{A}_{\mathbf{j}}\mathbf{x}+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\label{eq:Definition of the bold jth branch of H} \end{equation} the \textbf{$\mathbf{j}$th branch }of $H$. Note that this overrides the definition used for $H_{\mathbf{j}}$ in Chapter 2. Using $\mathbf{M}_{\mathbf{j}}$, we can write: \begin{equation} H_{\mathbf{j}}\left(\mathbf{x}\right)=\frac{\mathbf{M}_{\mathbf{j}}}{\mathbf{R}}\mathbf{x}+H_{\mathbf{j}}\left(\mathbf{0}\right)\label{eq:bold jth branch of H in terms of bold M bold j} \end{equation} We then define $H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)$ as: \begin{equation} H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\overset{\textrm{def}}{=}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}=\frac{\mathbf{M}_{\mathbf{j}}}{\mathbf{R}}\label{eq:Definition of H_bold j prime of bold 0} \end{equation} Also, note then that $H^{\prime}\left(\mathbf{0}\right)=H_{\mathbf{0}}^{\prime}\left(\mathbf{0}\right)$. \vphantom{} III. We say $H$ is integral (resp. non-integral) if its field analogue is integral (resp. non-integral). \end{defn} \begin{example} Consider the map $\tilde{H}:\mathbb{Z}\left[\sqrt{3}\right]\rightarrow\mathbb{Z}\left[\sqrt{3}\right]$ defined by: \begin{equation} \tilde{H}\left(z\right)\overset{\textrm{def}}{=}\begin{cases} \frac{z}{\sqrt{3}} & \textrm{if }z=0\mod\sqrt{3}\\ \frac{z-1}{\sqrt{3}} & \textrm{if }z=1\mod\sqrt{3}\\ \frac{4z+1}{\sqrt{3}} & \textrm{if }z=2\mod\sqrt{3} \end{cases} \end{equation} Here $\gamma_{1}=1$ and $\gamma_{2}=\sqrt{3}$, $\mathfrak{I}=\left\langle \sqrt{3}\right\rangle $. Since: \begin{equation} \tilde{H}\left(a+b\sqrt{3}\right)=\begin{cases} b+\frac{a}{3}\sqrt{3} & \textrm{if }a=0\mod3\\ b+\frac{a-1}{3}\sqrt{3} & \textrm{if }a=1\mod3\\ 4b+\frac{4a+1}{3}\sqrt{3} & \textrm{if }a=2\mod3 \end{cases} \end{equation} we have: \begin{align} H\left(\left[\begin{array}{c} v_{1}\\ v_{2} \end{array}\right]\right) & =\begin{cases} \left[\begin{array}{c} v_{2}\\ \frac{v_{1}}{3} \end{array}\right] & \textrm{if }v_{1}=0\mod3\\ \left[\begin{array}{c} v_{2}\\ \frac{v_{1}-1}{3} \end{array}\right] & \textrm{if }v_{1}=1\mod3\\ \left[\begin{array}{c} 4v_{2}\\ \frac{4v_{1}+1}{3} \end{array}\right] & \textrm{if }v_{1}=2\mod3 \end{cases}\\ & =\begin{cases} \left[\begin{array}{cc} 0 & 1\\ \frac{1}{3} & 0 \end{array}\right]\left[\begin{array}{c} v_{1}\\ v_{2} \end{array}\right] & \textrm{if }v_{1}=0\mod3\\ \left[\begin{array}{cc} 0 & 1\\ \frac{1}{3} & 0 \end{array}\right]\left[\begin{array}{c} v_{1}\\ v_{2} \end{array}\right]+\left[\begin{array}{c} 0\\ -\frac{1}{3} \end{array}\right] & \textrm{if }v_{1}=1\mod3\\ \left[\begin{array}{cc} 0 & 4\\ \frac{4}{3} & 0 \end{array}\right]\left[\begin{array}{c} v_{1}\\ v_{2} \end{array}\right]+\left[\begin{array}{c} 0\\ \frac{1}{3} \end{array}\right] & \textrm{if }v_{1}=2\mod3 \end{cases} \end{align} and so: \begin{align} H\left(\mathbf{v}\right) & =\left[\mathbf{v}\overset{3}{\equiv}\left[\begin{array}{c} 0\\ 0 \end{array}\right]\right]\left[\begin{array}{cc} 0 & 1\\ \frac{1}{3} & 0 \end{array}\right]\mathbf{v}+\left[\mathbf{v}\overset{3}{\equiv}\left[\begin{array}{c} 1\\ 0 \end{array}\right]\right]\left(\left[\begin{array}{cc} 0 & 1\\ \frac{1}{3} & 0 \end{array}\right]\mathbf{v}+\left[\begin{array}{c} 0\\ -\frac{1}{3} \end{array}\right]\right)\\ & +\left[\mathbf{v}\overset{3}{\equiv}\left[\begin{array}{c} 2\\ 0 \end{array}\right]\right]\left(\left[\begin{array}{cc} 0 & 4\\ \frac{4}{3} & 0 \end{array}\right]\mathbf{v}+\left[\begin{array}{c} 0\\ \frac{1}{3} \end{array}\right]\right) \end{align} where, in all three branches, the permutation is: \begin{equation} \left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right] \end{equation} \end{example} \vphantom{} Lastly, we will need to distinguish those $P$-Hydra maps for which $p_{n}=p$ for all $n\in\left\{ 1,\ldots,r\right\} $ for some prime $p$. \begin{defn} \index{ideal!smooth}I say a non-zero proper ideal $\mathfrak{I}$ of $\mathcal{O}_{\mathbb{F}}$ is \textbf{smooth }if the $p_{n}$s from the factorization: \[ \mathcal{O}_{\mathbb{F}}/\mathfrak{I}\cong\mathfrak{C}_{p_{1}}\times\cdots\times\mathfrak{C}_{p_{r}} \] are all equal, with there being some integer $p\geq2$ so that $p_{n}=p$ for all $n\in\left\{ 1,\ldots,r\right\} $. I call a Multi-Dimensional Hydra map \textbf{smooth}\index{Hydra map!smooth}\index{Hydra map!$p$-smooth}\textbf{ }(or, more specifically, \textbf{$p$-smooth})\textbf{ }whenever either of the equivalent conditions is satisfied: \vphantom{} i $H$ is an $\left(\mathbb{F},\mathcal{B},\mathfrak{I}\right)$-Hydra map and $\mathfrak{I}$ is smooth. \vphantom{} ii. $H$ is a $d$-dimensional $P$-Hydra map of depth $r$, and there is an integer $p\geq2$ so that $p_{n}=p$ for all $n\in\left\{ 1,\ldots,r\right\} $. In this case, I refer to $H$ is a \textbf{$d$-dimensional $p$-Hydra map of depth $r$}. \vphantom{} Finally, I say $H$ (equivalently, $\tilde{H}$) is \textbf{prime }whenever it is $p$-smooth for some prime $p$. \end{defn} \newpage{} \section{\label{sec:5.2 The-Numen-of}The Numen of a Multi-Dimensional Hydra Map} THROUGHOUT THIS SECTION, UNLESS STATED OTHERWISE, $H$ DENOTES A PRIME ($p$-SMOOTH) $d$-DIMENSIONAL HYDRA MAP OF DEPTH $r$ WHICH FIXES $\mathbf{0}$. \vphantom{} In this section, we will do for multi-dimensional Hydra maps what we did for their one-dimensional cousins in Chapter 2. The exposition here will be more briskly paced than in that earlier chapter. Throughout, we will keep the one-dimensional case as a guiding analogy. Of course, we will pause as needed to discuss deviations particular to the multi-dimensional setting. \subsection{\label{subsec:5.2.1 Notation-and-Preliminary}Notation and Preliminary Definitions} Much like as in the one-dimensional case, we will restrict our attention to $\mathbf{m}\in\mathbb{N}_{0}^{d}$, the multi-dimensional analogue of $\mathbb{N}_{0}$. Whereas our composition sequences in the one-dimensional case were of the form: \begin{equation} H_{j_{1}}\circ H_{j_{2}}\circ\cdots \end{equation} for integers $j_{1},j_{2},\ldots\in\mathbb{Z}/p\mathbb{Z}$, in the multi-dimensional case, our composition sequences will be of the form: \begin{equation} H_{\mathbf{j}_{1}}\circ H_{\mathbf{j}_{2}}\circ\cdots \end{equation} where $\mathbf{j}_{1},\mathbf{j}_{2},\ldots\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ are \emph{tuples}, rather than integers.\emph{ }What we denoted by $H_{\mathbf{j}}$ in the one-dimensional case will now be written as $H_{\mathbf{J}}$. As indicated by the capital letter, $\mathbf{J}$ denotes a \emph{matrix} whose columns are $\mathbf{j}_{1},\mathbf{j}_{2},\ldots$. The particular definitions are as follows: \begin{defn}[\textbf{Block strings}] \ \vphantom{} I. A \textbf{$p$-block string }of \textbf{depth }$r$ is a\index{block string}\index{block string!depth} finite tuple $\mathbf{J}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}\right)^{T}$, where: \vphantom{} i. \nomenclature{$\left\|\mathbf{J}\right\|$}{Length of the block string $\mathbf{J}$}$\left\|\mathbf{J}\right\|$ is an integer $\geq1$ which we call the \textbf{length }of $\mathbf{J}$. \vphantom{} ii. For each $m\in\left\{ 1,\ldots,\left\|\mathbf{J}\right\|\right\} $, $\mathbf{j}_{m}$ is an $r$-tuple: \begin{equation} \mathbf{j}_{m}=\left(j_{m,1},j_{m,2},\ldots,j_{m,r}\right)\in\mathbb{Z}^{r}/p\mathbb{Z}^{r} \end{equation} That is, for each $m$, the $n$th entry of $\mathbf{j}_{m}$ is an element of $\mathbb{Z}/p\mathbb{Z}$. We can also view $\mathbf{J}$ as a $\left\|\mathbf{J}\right\|\times r$ matrix, where the $m$th row of $\mathbf{J}$ consists of the entries of $\mathbf{j}_{m}$: \begin{equation} \mathbf{J}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}\right)^{T}=\left(\begin{array}{cccc} j_{1,1} & j_{1,2} & \cdots & j_{1,r}\\ j_{2,1} & j_{2,2} & \cdots & j_{2,r}\\ \vdots & \vdots & \ddots & \vdots\\ j_{\left\|\mathbf{J}\right\|,1} & j_{\left\|\mathbf{J}\right\|,2} & \cdots & j_{\left\|\mathbf{J}\right\|,r} \end{array}\right)\label{eq:J as a matrix} \end{equation} \vphantom{} II. We write $\textrm{String}^{r}\left(p\right)$\nomenclature{$\textrm{String}^{r}\left(p\right)$}{ } to denote the set of all finite length $p$-block strings of depth $r$. \vphantom{} III. Given any $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$, the \textbf{composition}\index{composition sequence!multi-dimensional}\textbf{ sequence }$H_{\mathbf{J}}:\mathbb{Q}^{d}\rightarrow\mathbb{Q}^{d}$ \nomenclature{$H_{\mathbf{J}}\left(\mathbf{x}\right)$}{$\overset{\textrm{def}}{=}\left(H_{\mathbf{j}_{1}}\circ\cdots\circ H_{\mathbf{j}_{\left\|\mathbf{J}\right\|}}\right)\left(\mathbf{x}\right) $}is the map: \begin{equation} H_{\mathbf{J}}\left(\mathbf{x}\right)\overset{\textrm{def}}{=}\left(H_{\mathbf{j}_{1}}\circ\cdots\circ H_{\mathbf{j}_{\left\|\mathbf{J}\right\|}}\right)\left(\mathbf{x}\right),\textrm{ }\forall\mathbf{J}\in\textrm{String}^{r}\left(p\right),\textrm{ }\forall\mathbf{x}\in\mathbb{R}^{d}\label{eq:Def of composition sequence-1} \end{equation} \vphantom{} IV. We write $\textrm{String}_{\infty}^{r}\left(p\right)$ \nomenclature{$\textrm{String}_{\infty}^{r}\left(p\right)$}{ }to denote the set of all $p$-block strings of finite or infinite length. We also include the empty set as an element of $\textrm{String}_{\infty}^{r}\left(p\right)$, and call it the \textbf{empty $p$-block string}. \end{defn} \begin{rem} As $\left\|\mathbf{J}\right\|\rightarrow\infty$, we will have that the sequence consisting of the $m$th entry of $\mathbf{j}_{1}$ followed by the $m$th entry of $\mathbf{j}_{2}$, followed by $m$th entry of $\mathbf{j}_{3}$, and so on will be the digits of a $p_{m}$-adic integer. \end{rem} \begin{defn}[\textbf{Concatenation}] The\index{block string!concatenation} \textbf{concatenation operator} $\wedge:\textrm{String}^{r}\left(p\right)\times\textrm{String}_{\infty}^{r}\left(p\right)\rightarrow\textrm{String}_{\infty}^{r}\left(p\right)$ is defined by: \index{concatenation!operation!multi-dimensional} \begin{equation} \mathbf{J}\wedge\mathbf{K}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}\right)\wedge\left(\mathbf{k}_{1},\ldots,\mathbf{k}_{\left\|\mathbf{K}\right\|}\right)\overset{\textrm{def}}{=}\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|},\mathbf{k}_{1},\ldots,\mathbf{k}_{\left\|\mathbf{K}\right\|}\right)\label{eq:Definition of Concatenation-1} \end{equation} for all $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ and $\mathbf{K}\in\textrm{String}_{\infty}^{r}\left(p\right)$. Also, for any integer $m\geq1$, if $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$, we write $\mathbf{J}^{\wedge m}$ to denote the concatenation of $m$ copies of $\mathbf{J}$: \begin{equation} \mathbf{J}^{\wedge m}\overset{\textrm{def}}{=}\left(\underbrace{\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|},\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|},\ldots,\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}}_{m\textrm{ times}}\right)^{T}\label{eq:Definition of block string concatenation} \end{equation} We shall only use this notation when $\mathbf{J}$ has finite length. Note that the empty block string is the identity element with respect to $\wedge$: \begin{equation} \mathbf{J}\wedge\varnothing=\varnothing\wedge\mathbf{J}=\mathbf{J} \end{equation} \end{defn} \begin{defn} We write $\left\Vert \cdot\right\Vert _{p}$ to denote the non-archimedean norm:\nomenclature{$\left\Vert \mathbf{z}\right\Vert _{p}$}{$\max\left\{ \left\|\mathfrak{z}_{1}\right\|_{p},\ldots,\left\|\mathfrak{z}_{r}\right\|_{p}\right\}$ } \begin{equation} \left\Vert \mathbf{z}\right\Vert _{p}\overset{\textrm{def}}{=}\max\left\{ \left\|\mathfrak{z}_{1}\right\|_{p},\ldots,\left\|\mathfrak{z}_{r}\right\|_{p}\right\} ,\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:Definition of p norm} \end{equation} which outputs the maximum of the $p$-adic absolute values of the entries of $\mathbf{z}$. Here \begin{equation} \left\Vert \mathbf{z}-\mathbf{w}\right\Vert _{p}\leq\max\left\{ \left\Vert \mathbf{z}\right\Vert _{p},\left\Vert \mathbf{w}\right\Vert _{p}\right\} \label{eq:p norm ultrametric inequality} \end{equation} with equality whenever $\left\Vert \mathbf{z}\right\Vert _{p}\neq\left\Vert \mathbf{w}\right\Vert _{p}$, and: \begin{equation} \left\Vert \mathbf{z}\mathbf{w}\right\Vert _{p}\leq\left\Vert \mathbf{z}\right\Vert _{p}\left\Vert \mathbf{w}\right\Vert _{p}\label{eq:Product property of p norm} \end{equation} \end{defn} \vphantom{} Next, we have notations for projections mod $p^{k}$ and for the multi-dimensional analogue of $\mathbb{Z}_{p}^{\prime}\backslash\mathbb{N}_{0}$. \begin{defn} For any $\mathbf{z}=\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\in\mathbb{Z}_{p}^{r}$, we write: \nomenclature{$\left[\mathbf{z}\right]_{p^{k}}$}{$\overset{\textrm{def}}{=}\left(\left[\mathfrak{z}_{1}\right]_{p^{k}},\ldots,\left[\mathfrak{z}_{r}\right]_{p^{k}}\right)$ \nopageref} \begin{equation} \left[\mathbf{z}\right]_{p^{k}}\overset{\textrm{def}}{=}\left(\left[\mathfrak{z}_{1}\right]_{p^{k}},\ldots,\left[\mathfrak{z}_{r}\right]_{p^{k}}\right)\label{eq:definition of z-bar mod P^k} \end{equation} We also use the notation:\nomenclature{$\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$}{$\overset{\textrm{def}}{=}\mathbb{Z}_{p}^{r}\backslash\mathbb{N}_{0}^{r}$ \nopageref} \begin{equation} \left(\mathbb{Z}_{p}^{r}\right)^{\prime}\overset{\textrm{def}}{=}\mathbb{Z}_{p}^{r}\backslash\mathbb{N}_{0}^{r}\label{eq:Definition of Z_P prime} \end{equation} \end{defn} \begin{defn} Given $\mathbf{J}\in\textrm{String}_{\infty}^{r}\left(p\right)$ and $\mathbf{z}=\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\in\mathbb{Z}_{p}^{r}$ we say $\mathbf{J}$ \textbf{represents }(or \textbf{is} \textbf{associated to})\textbf{ }$\mathbf{z}$ and vice-versa\textemdash written $\mathbf{J}\sim\mathbf{z}$ or $\mathbf{z}\sim\mathbf{J}$\textemdash whenever the entries of the $k$th column of $\mathbf{J}$ represent the $p$-adic integer $\mathfrak{z}_{k}$ for all $k\in\left\{ 1,\ldots,r\right\} $ in the sense of Chapter 2: that is, for each $k$, the entries of the $k$th column of $\mathbf{J}$ are the sequence of the $p$-adic digits of $\mathfrak{z}_{k}$: \begin{equation} \mathbf{J}\sim\mathbf{z}\Leftrightarrow\mathbf{z}\sim\mathbf{J}\Leftrightarrow\mathfrak{z}_{k}=j_{1,k}+j_{2,k}p+j_{3,k}p^{2}+\cdots\textrm{ }\forall k\in\left\{ 1,\ldots,r\right\} \label{eq:Definition of n-bold-j correspondence.-1} \end{equation} We extend the relation $\sim$ to a binary relation on $\textrm{String}_{\infty}^{r}\left(p\right)$ by writing $\mathbf{J}\sim\mathbf{K}$ if and only if both $\mathbf{J}$ and $\mathbf{K}$ represent the same $d$-tuple $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, and write \nomenclature{$\textrm{String}_{\infty}^{r}\left(p\right)/\sim$}{ } $\textrm{String}_{\infty}^{r}\left(p\right)/\sim$ to denote the set of all equivalence classes of $\textrm{String}_{\infty}^{r}\left(p\right)$ with respect to $\sim$. In particular, we identify any $p$-block string $\mathbf{J}$ of depth $r$ whose entries are all $0$s (such a string is called a \textbf{zero $p$-block string}) with the empty set, and view this particular equivalence class of $\textrm{String}_{\infty}^{r}\left(p\right)/\sim$ as representing the zero vector in $\mathbb{Z}_{p}^{r}$. \end{defn} \vphantom{} Similar to what we did in Chapter 2, every function on $\mathbb{Z}_{p}^{r}$ can be defined as a function of block strings. \begin{defn} For each $\mathbf{k}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$, we define $\#_{\mathfrak{I}:\mathbf{k}}:\textrm{String}^{r}\left(p\right)\rightarrow\mathbb{N}_{0}$ by: \begin{equation} \#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{J}\right)\overset{\textrm{def}}{=}\left\|\left\{ \mathbf{j}\in\mathbf{J}:\mathbf{j}=\mathbf{k}\right\} \right\|,\textrm{ }\forall\mathbf{J}\in\textrm{String}^{r}\left(p\right)\label{eq:Definition of pound of big bold j} \end{equation} We also define $\#_{\mathfrak{I}:\mathbf{j}}$ as a function on $\mathbb{N}_{0}^{r}$ by writing: \begin{equation} \#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}\#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{J}\right)\label{eq:Definition of pound bold n} \end{equation} where $\mathbf{J}$ is the shortest element of $\textrm{String}^{r}\left(p\right)$ representing $\mathbf{n}$. \end{defn} \begin{defn} We define the function \nomenclature{$\lambda_{p}\left(\mathbf{n}\right)$}{ } $\lambda_{p}:\mathbb{N}_{0}^{r}\rightarrow\mathbb{N}_{0}$ by: \begin{equation} \lambda_{p}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}\max_{1\leq\ell\leq r}\lambda_{p}\left(n_{\ell}\right)\label{eq:Definition of lambda P of bold n} \end{equation} Note that $\lambda_{p}\left(\mathbf{n}\right)$ is then the length of the shortest $p$-block string representing $\mathbf{n}$. \end{defn} \begin{prop} For any $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ there exists a unique $\mathbf{J}\in\textrm{String}_{\infty}^{r}\left(p\right)$ of minimal length which represents $\mathbf{z}$. This $\mathbf{J}$ has finite length if and only if $\mathbf{z}=\mathbf{n}\in\mathbb{N}_{0}^{r}$, in which case, $\left\|\mathbf{J}\right\|=\min\left\{ 1,\lambda_{p}\left(\mathbf{n}\right)\right\} $ (the minimum is necessary to account for the case when $\mathbf{n}=\mathbf{0}$). \end{prop} Proof: We begin with the following special case: \begin{claim} Let $\mathbf{n}\in\mathbb{N}_{0}^{r}$. Then, there is a unique $\mathbf{J}\in\textrm{String}_{\infty}^{r}\left(p\right)$ of minimal length which represents $\mathbf{n}$. Proof of claim:\textbf{ }If $\mathbf{n}$ is the zero vector, then we can choose $\mathbf{J}$ to have length $1$ and contain only $0$s. So, suppose $\mathbf{n}$ is not the zero vector. Then $\lambda_{p}\left(\mathbf{n}\right)=\max_{1\leq k\leq r}\lambda_{p}\left(n_{k}\right)$ is positive. Using this, we construct the desired $\mathbf{J}$ like so: for any $k$ for which $\lambda_{p}\left(n_{k}\right)=L$, we write out in the $k$th column of $\mathbf{J}$ (starting from the top, proceeding downward) all of the $p$-adic digits of $n_{k}$. Next, for any $k$ for which $\lambda_{p}\left(n_{k}\right)<L$, we have that $j_{\ell,k}=0$ for all $\ell\in\left\{ \lambda_{p}\left(n_{k}\right)+1,\ldots,L\right\} $; that is, after we have written out all of the $p$-adic digits of $n_{k}$ in the $k$th column of $\mathbf{J}$, if there are still spaces further down in that column without entries, we fill those spaces with $0$s. As constructed, this $\mathbf{J}$ represents $\mathbf{n}$. Moreover, its length is minimal: if $\mathbf{J}^{\prime}$ is a block string of length $\left\|\mathbf{J}\right\|-1=L-1$, then, for any $k\in\left\{ 1,\ldots,r\right\} $ for which $\lambda_{p}\left(n_{k}\right)=L$, the $k$th column of $\mathbf{J}^{\prime}$ will fail to represent $n_{k}$. This also establishes $\mathbf{J}$'s uniqueness. This proves the claim. \vphantom{} \end{claim} Having proved the claim, using the density of $\mathbb{N}_{0}^{r}$ in $\mathbb{Z}_{p}^{r}$, given any $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, let $\mathbf{J}$ be the limit (in the projective sense) of the sequence $\mathbf{J}_{1},\mathbf{J}_{2},\ldots\in\textrm{String}_{\infty}^{r}\left(p\right)$, where $\mathbf{J}_{m}$ is the unique minimal-length block string representing $\left[\mathbf{z}\right]_{p^{m}}$, the $r$-tuple whose entries are $\left[\mathfrak{z}_{k}\right]_{p^{m}}$. $\mathbf{J}$ will then be the unique block string representing $\mathbf{z}$. Q.E.D. \vphantom{} We can restate the above as: \begin{prop} The map which sends each $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ to the unique string $\mathbf{J}\in\textrm{String}_{\infty}^{r}\left(p\right)$ of minimal length which represents $\mathbf{z}$ is a bijection from $\mathbb{Z}_{p}^{r}$ onto $\textrm{String}_{\infty}^{r}\left(p\right)/\sim$. \end{prop} \begin{rem} We identify the zero vector in $\mathbb{Z}_{p}^{r}$ with the empty string. On the other hand, for any non-zero vector $\mathbf{z}=\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)$ so that $\mathfrak{z}_{k}=0$ for some $k$, we have that the $k$th column of the unique $\mathbf{J}$ representing $\mathbf{z}$ will contain only $0$s as entries. \end{rem} \begin{prop} \label{prop:MD lambda and digit-number functional equations}$\lambda_{p}$ and the $\#_{\mathfrak{I}:\mathbf{k}}$s satisfy the \index{functional equation!lambda{p}@$\lambda_{p}$}functional equations: \begin{equation} \lambda_{p}\left(p\mathbf{n}+\mathbf{j}\right)=\lambda_{p}\left(\mathbf{n}\right)+1,\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:lambda_P functional equation} \end{equation} \begin{equation} \#_{\mathfrak{I}:\mathbf{k}}\left(p\mathbf{n}+\mathbf{j}\right)=\begin{cases} \#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{n}\right)+1 & \textrm{if }\mathbf{j}=\mathbf{k}\\ \#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{n}\right) & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:Pound of bold n functional equation} \end{equation} More generally, for any $k\geq1$, any $\mathbf{m},\mathbf{n}\in\mathbb{N}_{0}^{r}$ with $\lambda_{p}\left(\mathbf{n}\right)\leq k$: \begin{equation} \lambda_{p}\left(\mathbf{m}+p^{k}\mathbf{n}\right)=\lambda_{p}\left(\mathbf{m}\right)+\lambda_{p}\left(\mathbf{n}\right)\label{eq:lambda_P pseudoconcatenation identity} \end{equation} \begin{equation} \#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{m}+p^{k}\mathbf{n}\right)=\#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{m}\right)+\#_{\mathfrak{I}:\mathbf{k}}\left(\mathbf{n}\right)\label{eq:Pound of bold n pseudoconcatenation identity} \end{equation} \end{prop} \subsection{\label{subsec:5.2.2 Construction-of-the}Construction of the Numen} For any affine linear map $L:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ defined by: \begin{equation} L\left(\mathbf{x}\right)=\mathbf{A}\mathbf{x}+\mathbf{b} \end{equation} for a $d\times d$ matrix $\mathbf{A}$ and a $d\times1$ column vector $\mathbf{b}$ (both with real entries), we will define the derivative of $L$ to be the matrix $\mathbf{A}$. Thus, given any such $L$, we will write $L^{\prime}\left(\mathbf{0}\right)$ to denote the derivative of $L$. $L^{\prime}\left(\mathbf{0}\right)$ is then the unique $d\times d$ matrix so that: \begin{equation} L\left(\mathbf{x}\right)=L^{\prime}\left(\mathbf{0}\right)\mathbf{x}+\mathbf{b} \end{equation} In fact, we can write: \begin{equation} L\left(\mathbf{x}\right)=L^{\prime}\left(\mathbf{0}\right)\mathbf{x}+L\left(\mathbf{0}\right) \end{equation} Applying this formalism to a composition sequence $H_{\mathbf{J}}$\textemdash which is valid, since, as a composition of a sequence of the affine linear maps $H_{\mathbf{j}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$\textemdash we see that $H_{\mathbf{J}}$ is an affine linear map on $\mathbb{R}^{d}$, with: \begin{equation} H_{\mathbf{J}}\left(\mathbf{x}\right)=H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\mathbf{x}+H_{\mathbf{J}}\left(\mathbf{0}\right),\textrm{ }\forall\mathbf{J}\in\textrm{String}^{r}\left(p\right)\label{eq:ax+b formula for H_bold_J} \end{equation} If $\mathbf{J}$ is a $p$-block string for which $H_{\mathbf{J}}\left(\mathbf{x}\right)=\mathbf{x}$, this becomes: \begin{equation} \mathbf{x}=H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\mathbf{x}+H_{\mathbf{J}}\left(\mathbf{0}\right)\label{eq:affine formula for bold x} \end{equation} So: \begin{equation} \mathbf{x}=\left(\mathbf{I}_{d}-H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\right)^{-1}H_{\mathbf{J}}\left(\mathbf{0}\right)\label{eq:Formula for bold x in terms of bold J} \end{equation} where $\mathbf{I}_{d}$ is a $d\times d$ identity matrix. Now for the definitions: \begin{defn}[\textbf{Multi-Dimensional $\chi_{H}$}] \label{def:MD Chi_H} \index{chi{H}@$\chi_{H}$!multi-dimensional}\nomenclature{$\chi_{H}\left(\mathbf{J}\right)$}{ }\nomenclature{$\chi_{H}\left(\mathbf{n}\right)$}{ }\ \vphantom{} I. We define $\chi_{H}:\textrm{String}^{r}\left(p\right)\rightarrow\mathbb{Q}^{d}$ by: \begin{equation} \chi_{H}\left(\mathbf{J}\right)\overset{\textrm{def}}{=}H_{\mathbf{J}}\left(\mathbf{0}\right),\textrm{ }\forall\mathbf{J}\in\textrm{String}^{r}\left(p\right)\label{eq:MD Definition of Chi_H of bold J} \end{equation} \vphantom{} II. For any $\mathbf{n}\in\mathbb{N}_{0}^{r}$, we write: \begin{equation} \chi_{H}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}\chi_{H}\left(\mathbf{J}\right)=\begin{cases} H_{\mathbf{J}}\left(\mathbf{0}\right) & \textrm{if }\mathbf{n}\neq\mathbf{0}\\ \mathbf{0} & \textrm{else} \end{cases}\label{eq:MD Definition of Chi_H of n} \end{equation} where $\mathbf{J}$ is any element of $\textrm{String}^{r}\left(p\right)$ which represents $\mathbf{n}$. \end{defn} \begin{rem} Much like in the one-dimensional case, since $H_{\mathbf{0}}\left(\mathbf{0}\right)=\mathbf{0}$, it then follows that $\chi_{H}$ is then a well-defined function from $\textrm{String}^{r}\left(p\right)/\sim$ (equivalently, $\mathbb{N}_{0}^{r}$) to $\mathbb{Q}^{d}$. \end{rem} \begin{defn}[\textbf{Multi-Dimensional $M_{H}$}] We define $M_{H}:\mathbb{N}_{0}^{r}\rightarrow\textrm{GL}_{d}\left(\mathbb{Q}\right)$ by: \nomenclature{$M_{H}\left(\mathbf{J}\right)$}{$\overset{\textrm{def}}{=}H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)$ }\nomenclature{$M_{H}\left(\mathbf{n}\right)$}{ }\index{$M_{H}$!multi-dimensional} \begin{equation} M_{H}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}M_{H}\left(\mathbf{J}\right)\overset{\textrm{def}}{=}H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} \label{eq:MD Definition of M_H} \end{equation} where $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ is the unique minimal length block string representing $\mathbf{n}$. We define $M_{H}\left(\mathbf{0}\right)$ and $M_{H}\left(\varnothing\right)$ to be the $d\times d$ identity matrix $\mathbf{I}_{d}$, where $\mathbf{0}$ is the $r\times1$ zero vector and where $\varnothing$ is the empty block string. \end{defn} \vphantom{} In the one-dimensional case, the $q$-adic convergence of $\chi_{H}$ over $\mathbb{Z}_{p}$ was due to the $q$-adic decay of $M_{H}\left(\mathbf{j}\right)$ as $\left\|\mathbf{j}\right\|\rightarrow\infty$ with infinitely many non-zero entries. We will do the same in the multi-dimensional case, except with matrices. Unlike the one-dimensional case, however, the multi-dimensional case has to deal with the various ways in which the $\mathbf{A}_{\mathbf{j}}$s might interact with respect to multiplication. Like in the one-dimensional case, $M_{H}$ and $\chi_{H}$ will be characterized by functional equations and concatenation identities. \begin{prop}[\textbf{Functional Equations for Multi-Dimensional $M_{H}$}] \label{prop:MD M_H functional equations}\index{functional equation!$M_{H}$!multi-dimensional}\ \begin{equation} M_{H}\left(p\mathbf{n}+\mathbf{j}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}M_{H}\left(\mathbf{n}\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}:p\mathbf{n}+\mathbf{j}\neq\mathbf{0}\label{eq:MD M_H functional equation} \end{equation} In terms of block strings and concatenations, we have: \begin{equation} M_{H}\left(\mathbf{J}\wedge\mathbf{K}\right)=M_{H}\left(\mathbf{J}\right)M_{H}\left(\mathbf{K}\right),\textrm{ }\forall\mathbf{J},\mathbf{K}\in\textrm{String}^{r}\left(p\right)\label{eq:MD M_H concatenation identity} \end{equation} \end{prop} \begin{rem} Because $M_{H}$ is matrix-valued, $M_{H}\left(\mathbf{J}\right)M_{H}\left(\mathbf{K}\right)$ is not generally equal to $M_{H}\left(\mathbf{K}\right)M_{H}\left(\mathbf{J}\right)$. However, if $H$ is $\mathbf{A}$-scalar, then: \begin{equation} M_{H}\left(\mathbf{J}\right)M_{H}\left(\mathbf{K}\right)=M_{H}\left(\mathbf{K}\right)M_{H}\left(\mathbf{J}\right) \end{equation} does hold for all $\mathbf{J},\mathbf{K}$. Consequently, \emph{when $H$ is $\mathbf{A}$-scalar, permuting the rows of $\mathbf{J}$ (a $\left\|\mathbf{J}\right\|\times r$ matrix) does not change the value of $M_{H}\left(\mathbf{J}\right)$}.\index{concatenation!identities!multi-dimensional} \end{rem} \begin{lem}[\textbf{Functional Equations for Multi-Dimensional $\chi_{H}$}] \label{lem:MD Chi_H functional equations and characterization}$\chi_{H}$ is the unique function $\mathbb{N}_{0}^{r}\rightarrow\mathbb{Q}^{d}$ satisfying the system of functional equations\index{functional equation!chi_{H}@$\chi_{H}$!multi-dimensional}: \begin{equation} \chi_{H}\left(p\mathbf{n}+\mathbf{j}\right)=\frac{\mathbf{A_{j}}\chi_{H}\left(\mathbf{n}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Chi_H functional equation} \end{equation} Equivalently: \begin{equation} \chi_{H}\left(p\mathbf{n}+\mathbf{j}\right)=H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{n}\right)\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Chi_H functional equation, alternate statement} \end{equation} In terms of concatenations of block strings, the above are written as: \begin{equation} \chi_{H}\left(\mathbf{J}\wedge\mathbf{K}\right)=H_{\mathbf{J}}\left(\chi_{H}\left(\mathbf{K}\right)\right),\textrm{ }\forall\mathbf{J},\mathbf{K}\in\textrm{String}^{r}\left(p\right)\label{eq:MD Chi_H concatenation identity} \end{equation} \end{lem} Proof: For once, I'll leave this to the reader as an exercise. Q.E.D. \vphantom{} Like in the one-dimensional case, we will also need explicit formulae for $M_{H}$ and $\chi_{H}$. \begin{prop} \label{prop:H_bold J in terms of M_H and Chi_H, explicitly}For all $\mathbf{x}\in\mathbb{N}_{0}^{d}$ and all $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$: \begin{equation} H_{\mathbf{J}}\left(\mathbf{x}\right)=\left(\prod_{k=1}^{\left\|\mathbf{J}\right\|}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right)\mathbf{x}+\sum_{n=1}^{\left\|\mathbf{J}\right\|}\left(\prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right)H_{\mathbf{j}_{k}}\left(\mathbf{0}\right)\label{eq:MD H composition sequence formula} \end{equation} where the products are defined to be $\mathbf{I}_{d}$ whenever their upper limits of multiplication are $0$. \end{prop} Proof: Induction. Q.E.D. \vphantom{} Next up, the explicit, string-based formulae for $M_{H}$ and $\chi_{H}$. \begin{prop} \label{prop:Bold J formulae for M_H and Chi_H}\ \vphantom{} I. For all non-zero $\mathbf{n}\in\mathbb{N}_{0}^{r}$ and any $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ representing $\mathbf{n}$: \begin{equation} M_{H}\left(\mathbf{n}\right)=M_{H}\left(\mathbf{J}\right)=\prod_{k=1}^{\left\|\mathbf{J}\right\|}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}=\prod_{k=1}^{\left\|\mathbf{J}\right\|}\frac{\mathbf{A}_{\mathbf{j}_{k}}}{\mathbf{D}_{\mathbf{j}_{k}}}\label{eq:MD M_H formula} \end{equation} \vphantom{} II. For all $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$: \begin{equation} \chi_{H}\left(\mathbf{J}\right)=\sum_{n=1}^{\left\|\mathbf{J}\right\|}\left(\prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right)H_{\mathbf{j}_{n}}\left(\mathbf{0}\right)\label{eq:MD Chi_H formula} \end{equation} \end{prop} \begin{rem} All the above equalities hold in $\mathbb{Q}^{d,d}$, the space of $d\times d$ matrices with rational entries. \end{rem} Proof: Use (\ref{eq:MD H composition sequence formula}) along with the fact that: \begin{equation} H_{\mathbf{J}}\left(\mathbf{x}\right)=H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\mathbf{x}+H_{\mathbf{J}}\left(\mathbf{0}\right)=M_{H}\left(\mathbf{J}\right)\mathbf{x}+\chi_{H}\left(\mathbf{J}\right) \end{equation} Q.E.D. \vphantom{} Now we deal with the rising-continuation of $\chi_{H}$. Like in the one-dimensional case, $\chi_{H}$ will end up being a rising-continuous function, albeit multi-dimensionally so. The specifics of multi-dimensional rising-continuous functions are dealt with in Section \ref{sec:5.3 Rising-Continuity-in-Multiple}. Our current goal is to state the multi-dimensional generalizations of conditions like semi-basicness and basicness which were needed to guaranteed the rising-continuability of $\chi_{H}$ in the one-dimensional case. To motivate these, let us examine (\ref{eq:MD Chi_H formula}). Recall that the \emph{columns }of $\mathbf{J}$ are going to represent $p$-adic integers as $\left\|\mathbf{J}\right\|\rightarrow\infty$; that is: \begin{equation} \mathbf{J}\rightarrow\left.\left[\begin{array}{cccc} \mid & \mid & & \mid\\ \mathfrak{z}_{1} & \mathfrak{z}_{2} & \cdots & \mathfrak{z}_{r}\\ \mid & \mid & & \mid \end{array}\right]\right\} \textrm{ infinitely many rows} \end{equation} where the $m$th column on the right contains the digits of the $p$-adic integer $\mathfrak{z}_{m}$; the $k$th row on the right, recall, is $\mathbf{j}_{k}$. In particular, the length of $\mathbf{J}$ is infinite if and only if at least one of the $\mathfrak{z}_{m}$s is an element of $\mathbb{Z}_{p}^{\prime}$. This, in turn, will occur if and only if said $\mathfrak{z}_{m}$ has \emph{infinitely many non-zero $p$-adic digits}, which occurs if and only if $\lambda_{p}\left(\mathbf{J}\right)\rightarrow\infty$. Applying $\chi_{H}$ to $\mathbf{J}$, we see that: \begin{equation} \chi_{H}\left(\left[\begin{array}{cccc} \mid & \mid & & \mid\\ \mathfrak{z}_{1} & \mathfrak{z}_{2} & \cdots & \mathfrak{z}_{r}\\ \mid & \mid & & \mid \end{array}\right]\right)=\sum_{n=1}^{\left\|\mathbf{J}\right\|}\left(\prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right)H_{\mathbf{j}_{k}}\left(\mathbf{0}\right) \end{equation} So, $\chi_{H}$ will extend to a function on $\mathbb{Z}_{p}^{r}$, with the convergence of the series on the right occurring because at least one of the $\mathfrak{z}_{m}$s has infinitely many non-zero $p$-adic digits. To deal with the convergence of sums of products of matrices, let briefly recall how convergence issues like this work in spaces of matrices. \begin{prop}[\textbf{Convergence for Matrix Sums and Matrix Products}] Let $K$ be a metrically complete non-archimedean valued field, let $d\geq2$, and let $\left\{ \mathbf{M}_{n}\right\} _{n\geq1}$ be a sequence in $K^{d,d}$ ($d\times d$ matrices with entries in $K$). Then, the partial products: \begin{equation} \prod_{n=1}^{N}\mathbf{M}_{n} \end{equation} converge in $K^{d,d}$ to the $d\times d$ zero matrix $\mathbf{O}_{d}$ whenever $\prod_{n=1}^{N}\left\Vert \mathbf{M}_{n}\right\Vert _{K}\rightarrow0$ in $\mathbb{R}$ as $N\rightarrow\infty$, where $\left\Vert \mathbf{M}_{n}\right\Vert _{K}$ is the maximum of the $K$-absolute-values of the entries of $\mathbf{M}_{n}$.\index{matrix!infinite product}\index{series!of matrices}\index{matrix!infinite series} Likewise, since $K$ is non-archimedean, we have that a matrix series: \begin{equation} \sum_{n=1}^{\infty}\mathbf{M}_{n} \end{equation} converges entry-wise in $K$ to an element of $K^{d,d}$ if and only if $\left\Vert \mathbf{M}_{n}\right\Vert _{K}\rightarrow0$. \end{prop} Proof: $\left(K^{d,d},\left\Vert \cdot\right\Vert _{K}\right)$ is a finite-dimensional Banach space over $K$. Q.E.D. \vphantom{} To cleanly state the multi-dimensional analogue of what it means for $H$ to be contracting, we will need to use the proposition given below. Recall that we write $\left\Vert \cdot\right\Vert _{\infty}$ to denote the maximum of the ordinary, complex absolute value of the entries of a matrix or vector. \begin{prop} \label{prop:MD contracting proposition}Let $\mathbf{D}$ and $\mathbf{P}$ be invertible $d\times d$ matrices with rational entries, with $\mathbf{D}$ being diagonal and $\mathbf{P}$ being a permutation matrix, representing a permutation of order $\omega$. Then, $\lim_{n\rightarrow\infty}\left\Vert \left(\mathbf{DP}\right)^{n}\right\Vert _{\infty}=0$ whenever $\left\Vert \left(\mathbf{D}\mathbf{P}\right)^{\omega}\right\Vert _{\infty}<1$. \end{prop} Proof: As given, $\omega$ is the smallest positive integer so that $\mathbf{P}^{\omega}=\mathbf{I}_{d}$. Now, letting $\mathbf{v}\in\mathbb{Q}^{d}$ be arbitrary observe that there will be an invertible diagonal matrix $\mathbf{D}_{\omega}$ so that: \begin{equation} \left(\mathbf{D}\mathbf{P}\right)^{\omega}\mathbf{v}=\mathbf{D}_{\omega}\mathbf{v} \end{equation} This is because each time we apply $\mathbf{D}\mathbf{P}$ to $\mathbf{v}$, we permute the entries of $\mathbf{v}$ by $\mathbf{P}$ and then multiply by $\mathbf{D}$. Since $\mathbf{P}$ has order $\omega$, if we label the entries of $\mathbf{v}$ as $v_{1},\ldots,v_{d}$, the positions of the $v_{j}$s in $\mathbf{v}$ will return to $v_{1},\ldots,v_{d}$ only after we have multiplied $\mathbf{v}$ by $\mathbf{P}$ $\omega$ times. In each application of $\mathbf{D}\mathbf{P}$, the $v_{j}$s accrue a non-zero multiplicative factor\textemdash{} courtesy of $\mathbf{D}$. So, after $\omega$ applications of $\mathbf{D}\mathbf{P}$, all of the entries of $\mathbf{v}$ will be returned to the original places, albeit with their original values having been multiplied by constants chosen from the diagonal entries of $\mathbf{D}$. These constants are then precisely the diagonal entries of $\mathbf{D}_{\omega}$. Since $\mathbf{v}$ was arbitrary, we conclude that $\left(\mathbf{D}\mathbf{P}\right)^{\omega}=\mathbf{D}_{\omega}$. Now, letting the integer $n\geq0$ be arbitrary, we can write $n$ as $\omega\left\lfloor \frac{n}{\omega}\right\rfloor +\left[n\right]_{\omega}$, where $\left\lfloor \frac{n}{\omega}\right\rfloor $ is the largest integer $\leq n/\omega$. As such: \begin{equation} \left\Vert \left(\mathbf{D}\mathbf{P}\right)^{n}\right\Vert _{\infty}\leq\left\Vert \underbrace{\left(\mathbf{D}\mathbf{P}\right)^{\omega}}_{\mathbf{D}_{\omega}}\right\Vert _{\infty}^{\left\lfloor \frac{n}{\omega}\right\rfloor }\left\Vert \mathbf{D}\right\Vert _{\infty}^{\left[n\right]_{\omega}}\underbrace{\left\Vert \mathbf{P}\right\Vert _{\infty}^{\left[n\right]_{\omega}}}_{1} \end{equation} Since $\left[n\right]_{\omega}$ only takes values in $\left\{ 0,\ldots,\omega-1\right\} $, the quantity $\left\Vert \mathbf{D}\right\Vert _{\infty}^{\left[n\right]_{\omega}}$ is uniformly bounded with respect to $n$, we can write: \begin{equation} \left\Vert \left(\mathbf{D}\mathbf{P}\right)^{n}\right\Vert _{\infty}\ll\left\Vert \mathbf{D}_{\omega}\right\Vert _{\infty}^{\left\lfloor \frac{n}{\omega}\right\rfloor } \end{equation} The upper bounds tends to zero as $n\rightarrow\infty$ whenever $\left\Vert \mathbf{D}_{\omega}\right\Vert _{\infty}<1$. Q.E.D. \vphantom{} Below, we formulate the analogues of properties such as monogenicity, degeneracy, simplicity, and the like for multi-dimensional Hydra maps. \begin{defn}[\textbf{Qualitative Terminology for Multi-Dimensional Hydra Maps}] \label{def:qualitative definitions for MD hydra maps}Let $H$ be a $P$-Hydra map, where $P=\left(p_{1},\ldots,p_{r}\right)$ where the $p_{\ell}$s are possibly distinct, and possibly non-prime. \vphantom{} I. The \textbf{ground }of $H$ is\index{Hydra map!ground} the set: \begin{equation} \textrm{Gr}\left(H\right)\overset{\textrm{def}}{=}\left\{ p\textrm{ prime}:p\mid p_{m}\textrm{ for some }m\in\left\{ 1,\ldots,r\right\} \right\} \label{eq:Definition of the Ground of H} \end{equation} \vphantom{} II. I say $H$ is \textbf{rough }if $H$ is not smooth. In particular, $H$ is called: i. \textbf{Weakly rough }if $H$ is rough and $\left\|\textrm{Gr}\left(H\right)\right\|=1$; \vphantom{} ii. \textbf{Strongly rough }if $H$ is rough and $\left\|\textrm{Gr}\left(H\right)\right\|\geq2$. \vphantom{} III.\index{Hydra map!degenerate} We say $H$ is \textbf{degenerate }if $\mathbf{A}_{\mathbf{j}}$ is a permutation matrix (including possibly the identity matrix) for some $\mathbf{j}\in\left(\mathbb{Z}^{r}/P\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. We say $H$ is \textbf{non-degenerate }whenever it is not degenerate. \vphantom{} IV. We say $H$ is \textbf{semi-simple }\index{Hydra map!semi-simple}whenever, for any $\mathbf{j},\mathbf{k}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, every non-zero entry in $\mathbf{A}_{\mathbf{j}}$ is co-prime to every non-zero entry in $\mathbf{D}_{\mathbf{k}}$. If, in addition: \begin{equation} \mathbf{D}_{\mathbf{j}}=\left(\begin{array}{cccccc} p_{1}\\ & \ddots\\ & & p_{r}\\ & & & 1\\ & & & & \ddots\\ & & & & & 1 \end{array}\right),\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}\label{eq:MD Definition of simplicity} \end{equation} we then say $H$ is \textbf{simple}. \vphantom{} V. We say $H$ is \textbf{monogenic }\index{Hydra map!monogenic}if there exists a prime $q\notin\textrm{Gr}\left(H\right)$ so that $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q}\leq1/q$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/P\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. We write $q_{H}$ to denote the smallest such prime. We say $H$ \index{Hydra map!polygenic} is \textbf{polygenic }if $H$ is not monogenic, but there exists a set $Q$ primes, with $Q\cap\textrm{Gr}\left(H\right)=\varnothing$ so that for each $\mathbf{j}\in\left(\mathbb{Z}^{r}/P\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $, there exists a $q\in Q$ for which $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q}\leq1/q$. \vphantom{} VI.\footnote{This definition will not be relevant until Chapter 6.} For each $\mathbf{j}\in\mathbb{Z}^{r}/P\mathbb{Z}^{r}$, we write $\omega_{\mathbf{j}}$ to denote the order of the permutation represented by the permutation matrix $\mathbf{P}_{\mathbf{j}}$, where $\mathbf{A}_{\mathbf{j}}=\tilde{\mathbf{A}}_{\mathbf{j}}\mathbf{P}_{\mathbf{j}}$, where $\tilde{\mathbf{A}}_{\mathbf{j}}$ is diagonal. We say $H$ \index{Hydra map!contracting} is \textbf{contracting }whenever $\lim_{n\rightarrow\infty}\left\Vert \left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\right\Vert _{\infty}=0$; recall that $H^{\prime}\left(\mathbf{0}\right)=\mathbf{D}_{\mathbf{0}}^{-1}\mathbf{A}_{\mathbf{0}}$. By \textbf{Proposition \ref{prop:MD contracting proposition}}, it follows that $H$ is contracting whenever $\left\Vert \left(H^{\prime}\left(\mathbf{0}\right)\right)^{\omega_{\mathbf{0}}}\right\Vert _{\infty}<1$. \vphantom{} VII. We say $H$ \index{Hydra map!basic}is \textbf{basic }if $H$ is simple, non-degenerate, and monogenic. \vphantom{} VIII. We say $H$ is \textbf{semi-basic }if \index{Hydra map!semi-basic}$H$ is semi-simple, non-degenerate, and monogenic. \end{defn} \begin{rem} For brevity, we will sometimes write $q$ to denote $q_{H}$. More generally, in any context where a Hydra map is being worked with\textemdash unless explicitly stated otherwise\textemdash $q$ will \emph{always }denote $q_{H}$. \end{rem} \begin{rem} Like in the one-dimensional case, everything given in this subsection holds when $q_{H}$ is replaced by any prime $q\neq p$ such that $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q}\leq1/q$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. \end{rem} \begin{rem} Like with polygenicity in the one-dimensional case, I would like to believe that frames can be used to provide a setting where we can speak meaningfully about the $p$-adic interpolation of $\chi_{H}$ in the case of a $p$-smooth polygenic multi-dimensional Hydra map. \end{rem} \begin{prop} \label{prop:MD Contracting H proposition}If $H$ is contracting, the matrix $\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)$ is invertible. \end{prop} Proof: If $H$ is contracting, there is an integer $\omega_{\mathbf{0}}\geq1$ so that the entries of the matrix $\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\omega_{\mathbf{0}}}$ are all rational numbers with archimedean absolute value $<1$. Consequently, letting $\mathbf{X}$ denote $H^{\prime}\left(\mathbf{0}\right)$: \begin{align} \sum_{n=0}^{\infty}\mathbf{X}^{n} & =\sum_{k=0}^{\omega_{\mathbf{0}}-1}\sum_{n=0}^{\infty}\mathbf{X}^{\omega_{\mathbf{0}}n+k}\\ & =\sum_{k=0}^{\omega_{\mathbf{0}}-1}\mathbf{X}^{k}\sum_{n=0}^{\infty}\mathbf{X}^{\omega_{\mathbf{0}}n}\\ \left(\left\Vert \mathbf{X}^{\omega_{\mathbf{0}}}\right\Vert _{\infty}<1\right); & \overset{\mathbb{R}^{d,d}}{=}\sum_{k=0}^{\omega_{\mathbf{0}}-1}\mathbf{X}^{k}\left(\mathbf{I}_{d}-\mathbf{X}^{\omega_{\mathbf{0}}}\right)^{-1} \end{align} As such, the series $\sum_{n=0}^{\infty}\mathbf{X}^{n}$ converges in $\mathbb{R}^{d,d}$, and therefore defines a matrix $\mathbf{M}\in\mathbb{R}^{d,d}$. Since: \[ \left(\mathbf{I}_{d}-\mathbf{X}\right)\mathbf{M}=\mathbf{M}\left(\mathbf{I}_{d}-\mathbf{X}\right)=\mathbf{I}_{d} \] this proves that $\mathbf{M}=\left(\mathbf{I}_{d}-\mathbf{X}\right)^{-1}$, showing that $\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)$ is invertible. Q.E.D. \vphantom{} Like in the one-dimensional case, monogenicity will be needed to ensure the existence of a rising-continuation of $\chi_{H}$. \begin{prop} \label{prop:MD M_H decay estimate}Let $H$ be monogenic. Then: \textup{ \begin{equation} \left\Vert M_{H}\left(\mathbf{J}\right)\right\Vert _{q_{H}}=\left\Vert \prod_{k=1}^{\left\|\mathbf{J}\right\|}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right\Vert _{q_{H}}\leq q_{H}^{-\#\left(\mathbf{J}\right)},\textrm{ }\forall\mathbf{J}\in\textrm{String}^{r}\left(p\right)\label{eq:Decay estimate on M_H of big bold j} \end{equation} }This result also holds when we replace $\mathbf{J}$ by the unique $\mathbf{n}\in\mathbb{N}_{0}^{r}$ for which $\mathbf{J}\sim\mathbf{n}$. \end{prop} Proof: Since $\mathbf{M}_{\mathbf{j}}=\mathbf{R}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}$, and since the entries of $\mathbf{R}$ and $\mathbf{D}_{\mathbf{j}}$ are co-prime to $q_{H}$, we have that: \begin{equation} \left\Vert \frac{\mathbf{M}_{\mathbf{j}}}{\mathbf{R}}\right\Vert _{q_{H}}\leq\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q_{H}} \end{equation} Since $H$ is monogenic, $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q_{H}}\leq1/q_{H}$ whenever $\mathbf{j}\neq\mathbf{0}$. Since $M_{H}\left(\mathbf{J}\right)=\prod_{k=1}^{\left\|\mathbf{J}\right\|}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}$, (\ref{eq:Decay estimate on M_H of big bold j}) then holds. Q.E.D. \vphantom{} Now, the all-important characterization of $\chi_{H}$'s interpolation as the unique rising-continuous solution of a certain system of functional equations. \begin{lem}[\textbf{$\left(p,q\right)$-adic Characterization of $\chi_{H}$}] \label{lem:MD rising-continuation and functional equations of Chi_H}Let $H$ be monogenic. Then, the limit:\nomenclature{$\chi_{H}\left(\mathbf{z}\right)$}{ } \begin{equation} \chi_{H}\left(\mathbf{z}\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{n\rightarrow\infty}\chi_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\label{eq:MD Rising Continuity Formula for Chi_H} \end{equation} exists for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, and thereby defines a continuation of $\chi_{H}$ to a function $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{Z}_{q_{H}}^{d}$. Moreover:\index{functional equation!chi_{H}@$\chi_{H}$!multi-dimensional} \vphantom{} I. \begin{equation} \chi_{H}\left(p\mathbf{z}+\mathbf{j}\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\frac{\mathbf{A_{j}}\chi_{H}\left(\mathbf{z}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Functional Equations for Chi_H over the rho-bar-adics} \end{equation} \vphantom{} II. \emph{(\ref{eq:MD Rising Continuity Formula for Chi_H})} is the unique function $\mathbf{f}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{Z}_{q_{H}}^{d}$ satisfying both\emph{: \begin{equation} \mathbf{f}\left(p\mathbf{z}+\mathbf{j}\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\frac{\mathbf{A_{j}}\mathbf{f}\left(\mathbf{z}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD extension Lemma for Chi_H - functional equation} \end{equation} and: \begin{equation} \mathbf{f}\left(\mathbf{z}\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{n\rightarrow\infty}\mathbf{f}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\label{eq:MD extension Lemma for Chi_H - rising continuity} \end{equation} } \end{lem} Proof: Let $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ and $N\geq0$ be arbitrary. Then, choose: \begin{equation} \mathbf{J}=\left[\begin{array}{ccc} \mathfrak{z}_{1,0} & \cdots & \mathfrak{z}_{r,0}\\ \mathfrak{z}_{1,1} & \cdots & \mathfrak{z}_{r,1}\\ \vdots & & \vdots\\ \mathfrak{z}_{1,N-1} & \cdots & \mathfrak{z}_{r,N-1} \end{array}\right]=\left[\begin{array}{ccc} \mid & & \mid\\ \left[\mathfrak{z}_{1}\right]_{p^{N}} & \cdots & \left[\mathfrak{z}_{r}\right]_{p^{N}}\\ \mid & & \mid \end{array}\right]\sim\left[\mathbf{z}\right]_{^{N}} \end{equation} where the $m$th column's entries are the first $N$ $p$-adic digits of $\mathfrak{z}_{m}$. Then, taking (\ref{eq:MD Chi_H formula}) from \textbf{Proposition \ref{prop:Bold J formulae for M_H and Chi_H}}: \begin{equation} \chi_{H}\left(\mathbf{J}\right)=\sum_{n=1}^{\left\|\mathbf{J}\right\|}\left(\prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right)H_{\mathbf{j}_{k}}\left(\mathbf{0}\right) \end{equation} we obtain: \begin{equation} \chi_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)=\sum_{n=1}^{N}\left(\prod_{k=1}^{n-1}\frac{\mathbf{M}_{\left(\mathfrak{z}_{1,k},\ldots,\mathfrak{z}_{r,k}\right)}}{\mathbf{R}}\right)H_{\left(\mathfrak{z}_{1,n},\ldots,\mathfrak{z}_{r,n}\right)}\left(\mathbf{0}\right)\label{eq:MD Chi_H formula for truncation of z-arrow} \end{equation} If $\mathbf{z}\in\mathbb{N}_{0}^{r}$, then $\left(\mathfrak{z}_{1,n},\ldots,\mathfrak{z}_{r,n}\right)$ must be a tuple of $0$s for all sufficiently large $n$. Since $H$ fixes $\mathbf{0}$, we know that $\mathbf{b}_{\mathbf{0}}=\mathbf{0}$. As such, the $n$th terms of (\ref{eq:MD Chi_H formula for truncation of z-arrow}) are then $\mathbf{0}$ for all sufficiently large $n$, which shows that (\ref{eq:MD Rising Continuity Formula for Chi_H}) holds for $\mathbf{z}\in\mathbb{N}_{0}^{r}$. On the other hand, suppose at least one entry of $\mathbf{z}$ is not in $\mathbb{N}_{0}$. Then, as $n\rightarrow\infty$, in the matrix product: \begin{equation} \prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}} \end{equation} there will be infinitely many values of $k$ for which the $r$-tuple $\mathbf{j}=\left(\mathfrak{z}_{1,k},\ldots,\mathfrak{z}_{r,k}\right)$ has a non-zero entry. Since $H$ is monogenic, the decay estimate given by \textbf{Proposition \ref{prop:MD M_H decay estimate}} guarantees that: \begin{equation} \left\Vert \prod_{k=1}^{n-1}\frac{\mathbf{M}_{\mathbf{j}_{k}}}{\mathbf{R}}\right\Vert _{q_{H}}\leq q_{H}^{-\left\|\left\{ k\in\left\{ 1,\ldots,n-1\right\} :\mathbf{j}_{k}\neq0\right\} \right\|} \end{equation} which tends to $0$ as $n\rightarrow\infty$. This shows that (\ref{eq:MD Chi_H formula for truncation of z-arrow}) converges point-wise in $\mathbb{Z}_{q_{H}}^{d}$ to a limit. Since (\ref{eq:MD Functional Equations for Chi_H over the rho-bar-adics}) holds for all $\mathbf{z}\in\mathbb{N}_{0}^{r}$, the limit (\ref{eq:MD Functional Equations for Chi_H over the rho-bar-adics}) then shows that (\ref{eq:MD Functional Equations for Chi_H over the rho-bar-adics}) in fact holds for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, thereby proving (I). Finally, \textbf{Lemma \ref{lem:MD Chi_H functional equations and characterization} }shows that any solution $\mathbf{f}$ of (\ref{eq:MD extension Lemma for Chi_H - functional equation}) must be equal to $\chi_{H}$ on $\mathbb{N}_{0}^{r}$. If $\mathbf{f}$ also satisfies (\ref{eq:MD extension Lemma for Chi_H - rising continuity}), this then shows that, just like $\chi_{H}$'s extension to $\mathbb{Z}_{p}^{r}$, the extension of $\mathbf{f}$ to $\mathbb{Z}_{p}^{r}$ is uniquely determined by $\mathbf{f}$'s values on $\mathbb{N}_{0}^{r}$. Since $\mathbf{f}=\chi_{H}$ on $\mathbb{N}_{0}^{r}$, this then forces $\mathbf{f}=\chi_{H}$ on $\mathbb{Z}_{p}^{r}$, proving the uniqueness of $\chi_{H}$. Q.E.D. \subsection{\label{subsec:5.2.3 The-Correspondence-Principle}The Correspondence Principle Revisited} THROUGHOUT THIS SUBSECTION, WE ALSO ASSUME $H$ IS INTEGRAL, UNLESS STATED OTHERWISE. \vphantom{} Like in the one-dimensional case, we start with a self-concatenation identity. \begin{prop} \label{prop:MD self-concatenation identity}Let $\mathbf{n}\in\mathbb{N}_{0}^{r}$ and let $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ be the shortest block string representing. Then, for all $k\in\mathbb{N}_{1}$: \begin{equation} \left(\mathbf{j}_{1}^{\wedge k},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}^{\wedge k}\right)^{T}=\mathbf{J}^{\wedge k}\sim\frac{1-p^{k\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n}\overset{\textrm{def}}{=}\left(n_{1}\frac{1-p^{k\lambda_{p}\left(n_{1}\right)}}{1-p^{\lambda_{p_{}}\left(n_{1}\right)}},\ldots,n_{r}\frac{1-p^{k\lambda_{p}\left(n_{r}\right)}}{1-p^{\lambda_{p}\left(n_{r}\right)}}\right)\label{eq:MD Self-Concatentation Identity} \end{equation} \end{prop} Proof: Apply \textbf{Proposition \ref{prop:Concatenation exponentiation}} to each row of $\mathbf{J}^{\wedge k}$. Q.E.D. \vphantom{}Next come the multi-dimensional analogues of $B_{p}$ and the functional equation satisfied by $\chi_{H}\circ B_{p}$. \begin{defn}[$\mathbf{B}_{p}$] We define $\mathbf{B}_{p}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{Z}_{p}^{r}$ by:\nomenclature{$\mathbf{B}_{p}\left(\mathbf{z}\right)$}{$\overset{\textrm{def}}{=}\left(B_{p}\left(\mathfrak{z}_{1}\right),\ldots,B_{p}\left(\mathfrak{z}_{m}\right)\right)$} \begin{equation} \mathbf{B}_{p}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\left(B_{p}\left(\mathfrak{z}_{1}\right),\ldots,B_{p}\left(\mathfrak{z}_{m}\right)\right)\label{eq:MD Definition of B projection function} \end{equation} where, recall: \begin{equation} B_{p}\left(\mathfrak{z}\right)=\begin{cases} \mathfrak{z} & \textrm{if }\mathfrak{z}\in\mathbb{Z}_{p}^{\prime}\\ \frac{\mathfrak{z}}{1-p^{\lambda_{p}\left(\mathfrak{z}\right)}} & \textrm{if }\mathfrak{z}\in\mathbb{N}_{0} \end{cases} \end{equation} \end{defn} \begin{lem}[\textbf{Functional Equation for $\chi_{H}\circ\mathbf{B}_{p}$}] \label{lem:MD Chi_H o B functional equation}Let\index{functional equation!chi{H}circmathbf{B}{p}@$\chi_{H}\circ\mathbf{B}_{p}$} $H$ be monogenic. Then: \begin{equation} \chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} \label{eq:MD Chi_H B functional equation} \end{equation} Moreover, both sides of the above identity are $d\times1$ column vectors with entries in $\mathbb{Q}$. \end{lem} Proof: Let $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $, and let $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ be the shortest block string representing $\mathbf{n}$. Then: \begin{equation} H_{\mathbf{J}^{\wedge k}}\left(\mathbf{m}\right)=H_{\mathbf{J}}^{\circ k}\left(\mathbf{m}\right),\textrm{ }\forall\mathbf{m}\in\mathbb{Z}^{d},\textrm{ }\forall k\geq1 \end{equation} Since: \begin{equation} H_{\mathbf{J}}\left(\mathbf{m}\right)=M_{H}\left(\mathbf{J}\right)\mathbf{m}+\chi_{H}\left(\mathbf{J}\right) \end{equation} observe that: \begin{align} H_{\mathbf{J}^{\wedge k}}\left(\mathbf{m}\right) & \overset{\mathbb{Q}^{d}}{=}H_{\mathbf{J}}^{\circ k}\left(\mathbf{m}\right)\\ & =\left(M_{H}\left(\mathbf{J}\right)\right)^{k}\mathbf{m}+\left(\sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{J}\right)\right)^{\ell}\right)\chi_{H}\left(\mathbf{J}\right)\\ & =\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m}+\left(\sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell}\right)\chi_{H}\left(\mathbf{n}\right) \end{align} Since $\mathbf{n}\neq\mathbf{0}$, there is an $m\in\left\{ 1,\ldots,r\right\} $ so that the $m$th entry of $\mathbf{n}$ has at least one non-zero $p$-adic digit. Because $H$ is monogenic, we then have that at least \emph{one} of the matrices in the product $M_{H}\left(\mathbf{n}\right)$ has $q_{H}$-adic norm $<1$. Because all the matrices in that product have $q$-adic norm $\leq1$, this shows that $\left\Vert M_{H}\left(\mathbf{n}\right)\right\Vert _{q_{H}}<1$. Consequently, the series: \begin{equation} \sum_{\ell=0}^{\infty}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell} \end{equation} converges in $\mathbb{Q}_{q_{H}}^{d,d}$, thereby defining the inverse of $\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)$. In fact, because $\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)$ has entries in $\mathbb{Q}$, note that that $\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\in\mathbb{Q}_{q_{H}}^{d,d}$. Moreover, we can use the geometric series formula to write: \begin{equation} \left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)\sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell}=\sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell}-\sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell+1}=\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k} \end{equation} So: \begin{equation} \sum_{\ell=0}^{k-1}\left(M_{H}\left(\mathbf{n}\right)\right)^{\ell}=\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\overset{\textrm{def}}{=}\left(\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\right)^{-1}\left(\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\right) \end{equation} Consequently: \begin{equation} H_{\mathbf{J}^{\wedge k}}\left(\mathbf{m}\right)\overset{\mathbb{Q}^{d}}{=}\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m}+\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right) \end{equation} and so: \begin{equation} M_{H}\left(\mathbf{J}^{\wedge k}\right)\mathbf{m}+\chi_{H}\left(\mathbf{J}^{\wedge k}\right)\overset{\mathbb{Q}^{d}}{=}\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m}+\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right) \end{equation} Now, using $M_{H}$'s concatenation identity (\textbf{Proposition \ref{prop:MD M_H functional equations}}): \begin{equation} M_{H}\left(\mathbf{J}^{\wedge k}\right)\mathbf{m}=\left(M_{H}\left(\mathbf{J}\right)\right)^{k}\mathbf{m}=\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m} \end{equation} we have: \begin{equation} \left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m}+\chi_{H}\left(\mathbf{J}^{\wedge k}\right)\overset{\mathbb{Q}^{d}}{=}\left(M_{H}\left(\mathbf{n}\right)\right)^{k}\mathbf{m}+\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right) \end{equation} and so: \begin{equation} \chi_{H}\left(\mathbf{J}^{\wedge k}\right)=\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right)\label{eq:MD Chi_H B functional equation, ready to take limits} \end{equation} Using \textbf{Proposition \ref{prop:MD self-concatenation identity}} gives us: \begin{align} \chi_{H}\left(\mathbf{J}^{\wedge k}\right) & =\chi_{H}\left(\mathbf{n}\frac{1-p^{k\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\right)\\ & =\chi_{H}\left(n_{1}\frac{1-p^{k\lambda_{p}\left(n_{1}\right)}}{1-p^{\lambda_{p}\left(n_{1}\right)}},\ldots,n_{r}\frac{1-p^{k\lambda_{p}\left(n_{r}\right)}}{1-p^{\lambda_{p}\left(n_{r}\right)}}\right) \end{align} $\chi_{H}$'s rising-continuity property (\ref{eq:MD Rising Continuity Formula for Chi_H}) allows us to take the limit as $k\rightarrow\infty$: \begin{equation} \lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{J}^{\wedge k}\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(\frac{n_{1}}{1-p^{\lambda_{p}\left(n_{1}\right)}},\ldots,\frac{n_{r}}{1-p^{\lambda_{p}\left(n_{r}\right)}}\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \end{equation} On the other hand, (\ref{eq:MD Chi_H B functional equation, ready to take limits}) yields; \begin{align} \lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{J}^{\wedge k}\right) & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}\frac{\mathbf{I}_{d}-\left(M_{H}\left(\mathbf{n}\right)\right)^{k}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right)\\ \left(\left\Vert M_{H}\left(\mathbf{n}\right)\right\Vert _{q}<1\right); & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\frac{\mathbf{I}_{d}-\mathbf{O}_{d}}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\chi_{H}\left(\mathbf{n}\right)\\ & =\frac{\chi_{H}\left(\mathbf{n}\right)}{\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)}\\ & =\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right) \end{align} Putting the two together gives: \begin{equation} \chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right) \end{equation} Lastly, because the right-hand side is the product of $\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}$ (a $d\times d$ matrix with entries in $\mathbb{Q}$) and $\chi_{H}\left(\mathbf{n}\right)$ (a $d\times1$ column vector with entries in $\mathbb{Q}$), we see that both sides of (\ref{eq:MD Chi_H B functional equation}) are $d\times1$ column vectors with entries in $\mathbb{Q}$. Q.E.D. \vphantom{}Next up: the multi-dimensional notion of a ``wrong value''. But first, let us establish the $p$-adic extendibility of $H$. \begin{prop} \label{prop:p-adic extension of H-1}Let $H:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ be any $p$-Hydra map of depth $r$, not necessarily integral. Then, $H$ admits an extension to a continuous map $\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}$ defined by: \begin{equation} H\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}}\left[\mathbf{z}\overset{p}{\equiv}\mathbf{j}\right]\mathbf{D}_{\mathbf{j}}^{-1}\left(\mathbf{A}_{\mathbf{j}}\mathbf{z}+\mathbf{b}_{\mathbf{j}}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD p-adic extension of H} \end{equation} Moreover, the function $\mathbf{f}:\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}$ defined by: \begin{equation} \mathbf{f}\left(\mathbf{z}\right)=\sum_{\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}}\left[\mathbf{z}\overset{p}{\equiv}\mathbf{j}\right]\mathbf{D}_{\mathbf{j}}^{-1}\left(\mathbf{A}_{\mathbf{j}}\mathbf{z}+\mathbf{b}_{\mathbf{j}}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} is the unique continuous function $\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}$ whose restriction to $\mathbb{Z}^{d}$ is equal to $H$. \end{prop} Proof: Immediate, just like the one-dimensional case. Q.E.D. \begin{defn}[\textbf{Wrong Values and Propriety}] \label{def:MD Wrong values and propriety}\index{Hydra map!wrong value}\index{Hydra map!proper}Let $H$ be any $p$-Hydra map, not necessarily integral. \vphantom{} I. We say $\mathbf{y}=\left(\mathfrak{y}_{1},\ldots,\mathfrak{y}_{d}\right)\in\mathbb{Q}_{p}^{d}$ is a \textbf{wrong value for $H$ }whenever there are $\mathbf{z}\in\mathbb{Z}_{p}^{d}$ and $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ so that $\mathbf{y}=H_{\mathbf{J}}\left(\mathbf{z}\right)$ and $H_{\mathbf{J}}\left(\mathbf{z}\right)\neq H^{\circ\left\|\mathbf{J}\right\|}\left(\mathbf{z}\right)$. We call $\mathbf{z}$ a \textbf{seed }of $\mathbf{y}$. \vphantom{} II.\index{Hydra map!proper} We say $H$ is \textbf{proper }whenever every wrong value of $H$ is an element of $\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$; that is, every wrong value of $H$ is a $d$-tuple of $p$-adic rational numbers at least one of which is \emph{not} a $p$-adic integer. \end{defn} \vphantom{} We now repeat the lead-up to the Correspondence Principle from the one-dimensional case. \begin{prop} \label{prop:MD Q_p / Z_p prop}Let $H$ be any $p$-Hydra map of dimension $d$ and depth $r$. Then, $H_{\mathbf{j}}\left(\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}\right)\subseteq\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$ for all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. \end{prop} \begin{rem} Here, we are viewing the $H_{\mathbf{j}}$s as functions $\mathbb{Q}_{p}^{d}\rightarrow\mathbb{Q}_{p}^{d}$. \end{rem} Proof: Let $\mathbf{y}\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$ and $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ be arbitrary. Note that as a $p$-Hydra map, $p$ divides every element of $\mathbf{D}_{\mathbf{j}}$ which is not $1$. Moreover, as a multi-dimensional $p$-Hydra map, every non-zero element of $\mathbf{A}_{\mathbf{j}}$ is co-prime to every non-zero element of $\mathbf{D}_{\mathbf{j}}$. Consequently, $p$ does not divide any non-zero element of $\mathbf{A}_{\mathbf{j}}$, and so, $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{p}=1$. Thus: \begin{equation} \left\Vert \mathbf{A}_{\mathbf{j}}\mathbf{y}\right\Vert _{p}=\left\Vert \mathbf{y}\right\Vert _{p}>1 \end{equation} So, $\mathbf{A}_{\mathbf{j}}\mathbf{y}\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. Since every entry of the $d$-tuple $\mathbf{b}_{\mathbf{j}}$ is a rational integer, the $p$-adic ultrametric inequality guarantees that $\left\Vert \mathbf{A}_{\mathbf{j}}\mathbf{y}+\mathbf{b}_{\mathbf{j}}\right\Vert >1$, and thus, $\mathbf{A}_{\mathbf{j}}\mathbf{y}+\mathbf{b}_{\mathbf{j}}$ is in $\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. Finally, $p$ dividing the diagonal elements of $\mathbf{D}_{\mathbf{j}}$ which are \emph{not }$1$ implies $\left\Vert D_{\mathbf{j}}\right\Vert _{p}<1$, and so: \begin{equation} \left\Vert H_{\mathbf{j}}\left(\mathbf{y}\right)\right\Vert _{p}=\left\Vert \frac{\mathbf{A}_{\mathbf{j}}\mathbf{y}+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\right\Vert _{p}\geq\left\Vert \mathbf{A}_{\mathbf{j}}\mathbf{y}+\mathbf{b}_{\mathbf{j}}\right\Vert _{p}>1 \end{equation} which shows that $H_{\mathbf{j}}\left(\mathbf{y}\right)\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. Q.E.D. \begin{lem} \label{lem:MD integrality lemma}Let $H$ be a semi-basic $p$-Hydra map of dimension $d$ and depth $r$, where $p$ is prime; we do not require $H$ to be integral. Then, $H$ is proper if and only if $H$ is integral. \end{lem} Proof: Let $H$ be semi-basic. I. (Proper implies integral) Suppose $H$ is proper, and let $\mathbf{n}\in\mathbb{Z}^{d}$ be arbitrary. Then, clearly, when $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ satisfies $\mathbf{j}=\left[\mathbf{n}\right]_{p}$ (recall, this means the first $r$ entries of $\mathbf{n}$ are congruent mod $p$ to the first $r$ entries of $\mathbf{j}$), we have that $H_{\mathbf{j}}\left(\mathbf{n}\right)\in\mathbb{Z}^{d}$. So, suppose instead that $\mathbf{j}\neq\left[\mathbf{n}\right]_{p}$. Since $H$ is proper, the fact that $\mathbf{n}\in\mathbb{Z}^{d}\subseteq\mathbb{Z}_{p}^{d}$ tells us that $\left\Vert H_{\mathbf{j}}\left(\mathbf{n}\right)\right\Vert _{p}>1$. So, at least one entry of $H_{\mathbf{j}}\left(\mathbf{n}\right)$ is not a $p$-adic integer, and thus, is not a rational integer, either. This proves $H$ is integral. \vphantom{} II. (Integral implies proper) Suppose $H$ is integral, and\textemdash by way of contradiction\textemdash suppose $H$ is \emph{not}\textbf{ }proper. Then, there is a $\mathbf{z}\in\mathbb{Z}_{p}^{d}$ and a $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ with $\mathbf{j}\neq\left[\mathbf{z}\right]_{p}$ (again, only the first $r$ entries are affected) so that $\left\Vert H_{\mathbf{j}}\left(\mathbf{z}\right)\right\Vert _{p}\leq1$. Now, writing \[ \mathbf{z}=\sum_{m=1}^{d}\sum_{n=0}^{\infty}c_{m,n}p^{n}\mathbf{e}_{m} \] where $\mathbf{e}_{m}$ is the $m$th standard basis vector in $\mathbb{Z}_{p}^{d}$: \begin{align} H_{\mathbf{j}}\left(\mathbf{z}\right) & =\frac{\mathbf{A}_{\mathbf{j}}\left(\sum_{m=1}^{d}\sum_{n=0}^{\infty}c_{m,n}p^{n}\mathbf{e}_{m}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\\ & =\frac{\mathbf{A}_{\mathbf{j}}\left(\sum_{m=1}^{d}c_{m,0}\mathbf{e}_{m}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}+\sum_{m=1}^{d}\sum_{n=1}^{\infty}c_{m,n}p^{n-1}\left(p\mathbf{D}_{\mathbf{j}}^{-1}\right)\mathbf{e}_{m}\\ \left(\sum_{m=1}^{d}c_{m,0}\mathbf{e}_{m}=\left[\mathbf{z}\right]_{p}\right); & =H_{\mathbf{j}}\left(\left[\mathbf{z}\right]_{p}\right)+\sum_{m=1}^{d}\sum_{n=1}^{\infty}c_{m,n}p^{n-1}\left(p\mathbf{D}_{\mathbf{j}}^{-1}\right)\mathbf{e}_{m} \end{align} Because $H$ is a $p$-Hydra map, every diagonal entry of $\mathbf{D}_{\mathbf{j}}$ which is not $1$ must be a divisor of $p$$d_{j}$ must be a divisor of $p$. The primality of $p$ then forces $d_{j}$ to be either $1$ or $p$. In either case, we have that $p\mathfrak{y}/d_{j}$ is an element of $\mathbb{Z}_{p}$. Our contradictory assumption $\left\|H_{j}\left(\mathfrak{z}\right)\right\|_{p}\leq1$ tells us that $H_{j}\left(\mathfrak{z}\right)$ is also in $\mathbb{Z}_{p}$, and so: \begin{equation} H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)=H_{j}\left(\mathfrak{z}\right)-\frac{p\mathfrak{y}}{d_{j}}\in\mathbb{Z}_{p} \end{equation} Since $H$ was given to be integral, $j\neq\left[\mathfrak{z}\right]_{p}$ implies $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)\notin\mathbb{Z}$. As such, $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)$ is a $p$-adic integer which is \emph{not }a rational integer. Since $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)$ is a non-integer rational number which is a $p$-adic integer, the denominator of $H_{j}\left(\left[\mathfrak{z}\right]_{p}\right)=\frac{a_{j}\left[\mathfrak{z}\right]_{p}+b_{j}}{d_{j}}$ must be divisible by some prime $q\neq p$, and hence, $q\mid d_{j}$. However, we saw that $p$ being prime forced $d_{j}\in\left\{ 1,p\right\} $\textemdash this is impossible! Thus, it must be that $H$ is proper. Q.E.D. \vphantom{} Next, we use \textbf{Proposition \ref{prop:MD Q_p / Z_p prop} }to establish a result about the wrong values of $H$ when $H$ is proper. \begin{lem} \label{lem:MD wrong values lemma}Let $H$ be any proper $p$-Hydra map, not necessarily integral or semi-basic. Then, all wrong values of $H$ are elements of $\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. \end{lem} Proof: Let $H$ be as given, let $\mathbf{z}\in\mathbb{Z}_{p}^{d}$, and let $\mathbf{i}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ be such that $\left[\mathbf{z}\right]_{p}\neq\mathbf{i}$. Then, by definition of properness, the quantity: \begin{equation} H_{\mathbf{i}}\left(\mathbf{z}\right)=\frac{\mathbf{A}_{\mathbf{i}}\mathbf{z}+\mathbf{b}_{\mathbf{i}}}{\mathbf{D}_{\mathbf{i}}} \end{equation} has $p$-adic norm $>1$. By \textbf{Proposition \ref{prop:MD Q_p / Z_p prop}}, this then forces $H_{\mathbf{J}}\left(H_{\mathbf{i}}\left(\mathbf{z}\right)\right)$ to be an element of $\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$ for all $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$. Since every wrong value with seed $\mathbf{z}$ is of the form $H_{\mathbf{J}}\left(H_{\mathbf{i}}\left(\mathbf{z}\right)\right)$ for some $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$, some $\mathbf{z}\in\mathbb{Z}_{p}^{d}$, and some $\mathbf{i}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ for which $\left[\mathbf{z}\right]_{p}\neq\mathbf{i}$, this shows that every wrong value of $H$ is in $\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. So, $H$ is proper. Q.E.D. \begin{lem} \label{lem:MD properness lemma}Let $H$ be any proper $p$-Hydra map, not necessarily semi-basic or integral. Let $\mathbf{z}\in\mathbb{Z}_{p}^{d}$, and let $\mathbf{J}\in\mathrm{String}^{r}\left(p\right)$. If $H_{\mathbf{J}}\left(\mathbf{z}\right)=\mathbf{z}$, then $\ensuremath{H^{\circ\left\|\mathbf{J}\right\|}\left(\mathbf{z}\right)=\mathbf{z}}$. \end{lem} Proof: Let $H$, $\mathbf{z}$, and $\mathbf{J}$ be as given. By way of contradiction, suppose $H^{\circ\left\|\mathbf{J}\right\|}\left(\mathbf{z}\right)\neq\mathbf{z}$. But then $\mathbf{z}=H_{\mathbf{J}}\left(\mathbf{z}\right)$ implies $H^{\circ\left\|\mathbf{J}\right\|}\left(\mathbf{z}\right)\neq H_{\mathbf{J}}\left(\mathbf{z}\right)$. Hence, $H_{\mathbf{J}}\left(\mathbf{z}\right)$ is a wrong value of $H$ with seed $\mathbf{z}$. \textbf{Lemma} \ref{lem:MD wrong values lemma} then forces $H_{\mathbf{J}}\left(\mathbf{z}\right)\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$. However, $H_{\mathbf{J}}\left(\mathbf{z}\right)=\mathbf{z}$, and $\mathbf{z}$ was given to be in $\mathbb{Z}_{p}^{d}$. This is impossible! Consequently, $H_{\mathbf{J}}\left(\mathbf{z}\right)=\mathbf{z}$ implies $H^{\circ\left\|\mathbf{J}\right\|}\left(\mathbf{z}\right)=\mathbf{z}$. Q.E.D. \vphantom{} Now comes the multi-dimensional Correspondence Principle. Like in the one-dimensional case, we start by proving a correspondence between cycles of $H$ and rational-integer values attained by $\chi_{H}$. The task of refining this correspondence to one involving the individual periodic points of $H$ will be dealt with in a Corollary. \begin{thm}[\textbf{Multi-Dimensional Correspondence Principle, Ver. 1}] \label{thm:MD CP v1}Let $H$ be a semi-basic $p$-Hydra map of dimension $d$ and depth $r$ that fixes $\mathbf{0}$. \index{Correspondence Principle} Then: \vphantom{} I. Let $\Omega$ be any cycle of $H$ in $\mathbb{Z}^{d}$, with $\left\|\Omega\right\|\geq2$. Then, there exist $\mathbf{x}\in\Omega$ and an $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $ so that: \begin{equation} \chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\mathbf{x} \end{equation} \vphantom{} II. Suppose also that $H$ is integral, and let $\mathbf{n}\in\mathbb{N}_{0}^{r}$. If: \begin{equation} \chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\in\mathbb{Z}_{q_{H}}^{d}\cap\mathbb{Z}^{d} \end{equation} then $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ is a periodic point of $H$ in $\mathbb{Z}^{d}$. \end{thm} Proof: I. Let $\Omega$ be a cycle of $H$ in $\mathbb{Z}^{d}$ with $\left\|\Omega\right\|\geq2$. Observe that for any $\mathbf{x}\in\Omega$ and any $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ for which $\left\|\mathbf{J}\right\|=\left\|\Omega\right\|\geq2$ and $H_{\mathbf{J}}\left(\mathbf{x}\right)=\mathbf{x}$, it must be that $\mathbf{J}$ contains at least one non-zero $r$-tuple; that is, in writing $\mathbf{J}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}\right)$, there is at least one $\mathbf{j}_{k}$ which is not $\mathbf{0}$. As in the one-dimensional case, we can assume without loss of generality that we have chosen $\mathbf{x}\in\Omega$ for which the known non-zero $r$-tuple of $\mathbf{J}$ occurs in the bottom-most row of $\mathbf{J}$ (that is, $\mathbf{j}_{\left\|\mathbf{J}\right\|}\neq\mathbf{0}$), viewing $\mathbf{J}$ as a matrix. Writing the affine map $H_{\mathbf{J}}$ in affine form gives: \begin{equation} \mathbf{x}=H_{\mathbf{J}}\left(\mathbf{x}\right)=H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\mathbf{x}+H_{\mathbf{J}}\left(\mathbf{0}\right)=M_{H}\left(\mathbf{J}\right)\mathbf{x}+\chi_{H}\left(\mathbf{J}\right) \end{equation} where all the equalities are in $\mathbb{Q}^{d}$. Then, noting the implication: \begin{equation} H_{\mathbf{J}}\left(\mathbf{x}\right)=\mathbf{x}\Rightarrow H_{\mathbf{J}^{\wedge m}}\left(\mathbf{x}\right)=\mathbf{x} \end{equation} we have: \begin{equation} \mathbf{x}=H_{\mathbf{J}^{\wedge m}}\left(\mathbf{x}\right)=M_{H}\left(\mathbf{J}^{\wedge m}\right)\mathbf{x}+\chi_{H}\left(\mathbf{J}^{\wedge m}\right)=\left(M_{H}\left(\mathbf{J}\right)\right)^{m}\mathbf{x}+\chi_{H}\left(\mathbf{J}^{\wedge m}\right) \end{equation} Since $\mathbf{J}$ contains a tuple which is \emph{not} the zero tuple, the decay estimate from \textbf{Proposition \ref{prop:MD M_H decay estimate}} tells us that that $\left\Vert \left(M_{H}\left(\mathbf{J}\right)\right)^{m}\right\Vert _{q_{H}}$ tends to $0$ in $\mathbb{R}$ as $m\rightarrow\infty$. Moreover, the $\mathbf{n}\in\mathbb{N}_{0}^{r}$ represented by $\mathbf{J}$ is necessarily non-$\mathbf{0}$. So, we can apply \textbf{Proposition \ref{prop:MD self-concatenation identity}}: \begin{equation} \mathbf{J}^{\wedge m}\sim\frac{1-p^{m\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n} \end{equation} and, like in the proof of \textbf{Lemma \ref{lem:MD Chi_H o B functional equation}}, we can write: \begin{align} \lim_{m\rightarrow\infty}\chi_{H}\left(\mathbf{J}^{\wedge m}\right) & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{m\rightarrow\infty}\chi_{H}\left(\frac{1-p^{m\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n}\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(\frac{\mathbf{n}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\right)\\ & =\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \end{align} Finally, letting $m\rightarrow\infty$ in the equation: \begin{equation} \mathbf{x}=\left(M_{H}\left(\mathbf{J}\right)\right)^{m}\mathbf{x}+\chi_{H}\left(\mathbf{J}^{\wedge m}\right)=\left(M_{H}\left(\mathbf{J}\right)\right)^{m}\mathbf{x}+\chi_{H}\left(\frac{1-p^{m\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n}\right) \end{equation} we obtain the equality: \begin{equation} \mathbf{x}\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(\frac{\mathbf{n}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \end{equation} This proves the existence of the desired $\mathbf{x}$ and $\mathbf{n}$. \vphantom{} II. Let $H$ be integral; then, by \textbf{Lemma \ref{lem:MD integrality lemma}}, $H$ is proper. Next, let $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $ be such that $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ is in $\mathbb{Z}_{q_{H}}^{d}\cap\mathbb{Z}^{d}$, and let $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ be the shortest block string representing $\mathbf{n}$. Then, by $\chi_{H}$'s concatenation identity (equation \ref{eq:MD Chi_H concatenation identity} from \textbf{Lemma \ref{lem:MD Chi_H functional equations and characterization}}), we can write: \begin{equation} \chi_{H}\left(\frac{1-p^{k\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n}\right)=\chi_{H}\left(\mathbf{J}^{\wedge k}\right)=H\left(\chi_{H}\left(\mathbf{J}^{\wedge\left(k-1\right)}\right)\right)=\cdots=H^{\circ\left(k-1\right)}\left(\chi_{H}\left(\mathbf{J}\right)\right)\label{eq:MD self-concatenation identity for Chi_H} \end{equation} Letting $k\rightarrow\infty$, \textbf{Lemma \ref{lem:MD Chi_H functional equations and characterization}} along with the convergence of $\frac{1-p^{k\lambda_{p}\left(\mathbf{n}\right)}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\mathbf{n}$ to $\frac{\mathbf{n}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}$ in $\mathbb{Z}_{p}^{r}$ then yields the $q$-adic equality: \begin{equation} \lim_{k\rightarrow\infty}H_{\mathbf{J}}^{\circ\left(k-1\right)}\left(\chi_{H}\left(\mathbf{n}\right)\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(\frac{\mathbf{n}}{1-p^{\lambda_{p}\left(\mathbf{n}\right)}}\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\label{eq:Iterating H_bold-J on Chi_H} \end{equation} Because $H$ is semi-basic, the non-zero entries of any $\mathbf{A}_{\mathbf{j}}$ are co-prime to the non-zero entries of any $\mathbf{D}_{\mathbf{k}}$. Moreover, since $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q_{H}}\leq1/q_{H}$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $, this guarantees that for any finite length block string such as our $\mathbf{J}$, the matrix $H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)$ and the $d$-tuple $H_{\mathbf{J}}\left(\mathbf{0}\right)$ have entries in $\mathbb{Q}$ whose $q_{H}$-adic absolute values are $\leq1$; that is, their entries are in $\mathbb{Q}\cap\mathbb{Z}_{q_{H}}$. Consequently, the affine linear map: \begin{equation} \mathbf{w}\in\mathbb{Z}_{q_{H}}^{d}\mapsto H_{\mathbf{J}}\left(\mathbf{w}\right)=H_{\mathbf{J}}^{\prime}\left(\mathbf{0}\right)\mathbf{w}+H_{\mathbf{J}}\left(\mathbf{0}\right)\in\mathbb{Z}_{q_{H}}^{d} \end{equation} defines a continuous self-map of $\mathbb{Z}_{q_{H}}^{d}$. With this, we can then write: \begin{align} H_{\mathbf{J}}\left(\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\right) & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}H_{\mathbf{j}}\left(\lim_{k\rightarrow\infty}H_{\mathbf{J}}^{\circ\left(k-1\right)}\left(\chi_{H}\left(\mathbf{n}\right)\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{J}}^{\circ k}\left(\chi_{H}\left(\mathbf{n}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{J}}^{\circ\left(k-1\right)}\left(\chi_{H}\left(\mathbf{n}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \end{align} This proves that the rational $d$-tuple $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ is a fixed point of the composition sequence $H_{\mathbf{J}}$ as an element of $\mathbb{Z}_{q_{H}}^{d}$. Now, since we assumed that $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ was in $\mathbb{Z}^{d}$, the fact that $H_{\mathbf{J}}$ is a map on $\mathbb{Q}^{d}$, tells us that the equality $H_{\mathbf{J}}\left(\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$\textemdash nominally occurring in $\mathbb{Z}_{q_{H}}^{d}$\textemdash is actually valid in $\mathbb{Q}^{d}$. Consequently, this equality is also valid in $\mathbb{Q}_{p}^{d}$, and so, $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ is a fixed point of the map $H_{\mathbf{J}}:\mathbb{Q}_{p}^{d}\rightarrow\mathbb{Q}_{p}^{d}$. Next, because the assumed integrality of $H$ makes $H$ proper, we can apply \textbf{Lemma \ref{lem:MD properness lemma}} to conclude that $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\in\mathbb{Z}^{d}\subseteq\mathbb{Z}_{p}^{d}$, as a fixed point of $H_{\mathbf{J}}$, is in fact a fixed point of $H^{\circ\left\|\mathbf{J}\right\|}$: \begin{equation} H^{\circ\left\|\mathbf{J}\right\|}\left(\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \end{equation} Lastly, because $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ is in $\mathbb{Z}^{d}$, the fact that $H:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{d}$ is equal to the restriction to $\mathbb{Z}^{d}$ of $H:\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}$ necessarily forces $\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right)$ to be a periodic point of $H$ in $\mathbb{Z}^{d}$, as desired. Q.E.D. \vphantom{} Like with the one-dimensional case, we now have several alternative versions. \begin{cor}[\textbf{Multi-Dimensional Correspondence Principle, Ver. 2}] \label{cor:MD CP v2}Suppose that $H$ is semi-basic\index{Correspondence Principle} $p$-Hydra map of dimension $d$ and depth $r$ that fixes $\mathbf{0}$. Then: \vphantom{} I. For every cycle $\Omega\subseteq\mathbb{Z}^{d}$ of $H$, viewing $\Omega\subseteq\mathbb{Z}^{d}$ as a subset of $\mathbb{Z}_{q_{H}}^{d}$, the intersection $\chi_{H}\left(\mathbb{Z}_{p}^{r}\right)\cap\Omega$ is non-empty. Moreover, for every $\mathbf{x}\in\chi_{H}\left(\mathbb{Z}_{p}^{r}\right)\cap\Omega$, there is an $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $ so that: \[ \mathbf{x}=\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \] where the equality occurs in $\mathbb{Q}^{d}$. \vphantom{} II. Suppose in addition that $H$ is integral. Let $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $. If the tuple $\mathbf{x}\in\mathbb{Z}_{q_{H}}^{d}$ given by: \[ \mathbf{x}=\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right)=\chi_{H}\left(\mathbf{B}_{p}\left(\mathbf{n}\right)\right) \] is an element of $\mathbb{Z}^{d}$, then $\mathbf{x}$ is necessarily a periodic point of $H$. \end{cor} Proof: Re-write the results of \textbf{Theorem \ref{thm:MD CP v1}} using \textbf{Lemma \ref{lem:MD Chi_H o B functional equation}}. Q.E.D. \begin{cor}[\textbf{Multi-Dimensional Correspondence Principle, Ver. 3}] \label{cor:MD CP v3}Suppose that $H$ is an integral semi-basic \index{Correspondence Principle}$p$-Hydra map of dimension $d$ and depth $r$ that fixes $\mathbf{0}$. Then, the set of non-zero periodic points of $H$ is equal to $\mathbb{Z}^{d}\cap\chi_{H}\left(\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\right)$ (where, recall, $\left(\mathbb{Z}_{p}^{r}\right)^{\prime}=\mathbb{Z}_{p}^{\prime}\backslash\mathbb{N}_{0}^{r}$). \end{cor} \begin{rem} The implication ``If $\mathbf{x}\in\mathbb{Z}^{d}\backslash\left\{ \mathbf{0}\right\} $ is a periodic point, then $\mathbf{x}\in\chi_{H}\left(\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\right)$'' \emph{does not }require $H$ to be integral. \end{rem} Proof: Before we do anything else, note that because $H$ is integral and semi-basic, \textbf{Lemma \ref{lem:MD integrality lemma} }guarantees $H$ will be proper. I. Let $\mathbf{x}$ be a non-zero periodic point of $H$, and let $\Omega$ be the unique cycle of $H$ in $\mathbb{Z}$ which contains $\mathbf{x}$. By Version 1 of the Correspondence Principle (\textbf{Theorem \ref{thm:MD CP v1}}), there exists a $\mathbf{y}\in\Omega$ and a $\mathbf{z}=B_{p}\left(\mathbf{n}\right)\subset\mathbb{Z}_{p}^{r}$ (for some $\mathbf{n}\in\mathbb{N}_{0}^{r}$ containing at least one non-zero entry) so that $\chi_{H}\left(\mathbf{z}\right)=\mathbf{y}$. Because $\mathbf{y}$ is in $\Omega$, there is an $k\geq1$ so that $\mathbf{x}=H^{\circ k}\left(\mathbf{y}\right)$. As such, there is a \emph{unique} length $k$ block-string $\mathbf{I}\in\textrm{String}^{r}\left(p\right)$ so that $H_{\mathbf{I}}\left(\mathbf{y}\right)=H^{\circ k}\left(\mathbf{y}\right)=\mathbf{x}$. Now, let $\mathbf{J}\in\textrm{String}_{\infty}^{r}\left(p\right)$ represent $\mathbf{z}$; note that $\mathbf{J}$ is infinite and that its columns are periodic. Using $\chi_{H}$'s concatenation identity (\textbf{Lemma \ref{eq:MD Chi_H concatenation identity}}), we have: \begin{equation} \mathbf{x}=H_{\mathbf{I}}\left(\mathbf{y}\right)=H_{\mathbf{I}}\left(\chi_{H}\left(\mathbf{z}\right)\right)=\chi_{H}\left(\mathbf{I}\wedge\mathbf{J}\right) \end{equation} Next, let $\mathbf{w}\in\mathbb{Z}_{p}^{r}$ be the $r$-tuple of $p$-adic integers represented by the infinite block-string $\mathbf{I}\wedge\mathbf{J}$; note that $\mathbf{w}$ is \emph{not} an element of $\mathbb{N}_{0}^{r}$. By the above, $\chi_{H}\left(\mathbf{w}\right)=\mathbf{x}$, and hence that $\mathbf{x}\in\mathbb{Z}^{d}\cap\chi_{H}\left(\mathbb{Z}_{p}^{r}\right)$. Finally, since $\mathbf{z}=B_{p}\left(\mathbf{n}\right)$ and $\mathbf{n}\neq\mathbf{0}$, the $p$-adic digits of $\mathbf{z}$'s entries are periodic, which forces $\mathbf{z}$ to be an element of $\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. So, letting $\mathbf{m}$ be the rational integer $r$-tuple represented by length-$k$ block-string $\mathbf{I}$, we have: \begin{equation} \mathbf{w}\sim\mathbf{I}\wedge\mathbf{J}\sim\mathbf{m}+p^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{z} \end{equation} This shows that $\mathbf{w}\in\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$, and hence, that: \[ \mathbf{x}=\chi_{H}\left(\mathbf{w}\right)\in\mathbb{Z}^{d}\cap\chi_{H}\left(\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\right) \] \vphantom{} II. Suppose $\mathbf{x}\in\mathbb{Z}^{d}\cap\chi_{H}\left(\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\right)$, with $\mathbf{x}=\chi_{H}\left(\mathbf{z}\right)$ for some $\mathbf{z}\in\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. As a $r$-tuple of rational numbers are all $p$-adic integers which are \emph{not} elements of $\mathbb{N}_{0}$, the $p$-adic digits of the entries of $\mathbf{z}$ are all eventually periodic. As such, there are $\mathbf{m},\mathbf{n}\in\mathbb{N}_{0}^{r}$ (with $\mathbf{n}\neq\mathbf{0}$) so that: \begin{equation} \mathbf{z}=\mathbf{m}+p^{\lambda_{p}\left(\mathbf{m}\right)}B_{p}\left(\mathbf{n}\right) \end{equation} Here, the $p$-adic digits of the entries of $\mathbf{n}$ generate the periodic parts of the digits of $\mathbf{z}$'s entries. On the other hand, the $p$-adic digits of $\mathbf{m}$'s entries are the finite-length sequences in the digits of the entries of $\mathbf{z}$ that occur before the periodicity sets in. Now, let $\mathbf{I}$ be the finite block string representing $\mathbf{m}$, and let $\mathbf{J}$ be the infinite block-string representing $B_{p}\left(\mathbf{n}\right)$. Then, $\mathbf{z}=\mathbf{I}\wedge\mathbf{J}$. Using \textbf{Lemmata \ref{eq:MD Chi_H concatenation identity}} and \textbf{\ref{lem:MD Chi_H o B functional equation}} (the concatenation identity and $\chi_{H}\circ\mathbf{B}_{p}$ functional equation, respectively), we have: \begin{equation} \mathbf{x}=\chi_{H}\left(\mathbf{I}\wedge\mathbf{J}\right)=H_{\mathbf{I}}\left(\chi_{H}\left(\mathbf{J}\right)\right)=H_{\mathbf{I}}\left(\chi_{H}\left(B_{p}\left(\mathbf{n}\right)\right)\right)=H_{\mathbf{I}}\left(\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right)\right) \end{equation} Here, let $\mathbf{y}\overset{\textrm{def}}{=}\left(\mathbf{I}_{d}-M_{H}\left(\mathbf{n}\right)\right)^{-1}\chi_{H}\left(\mathbf{n}\right)$ is a $d$-tuple of rational numbers. \begin{claim} $\left\Vert \mathbf{y}\right\Vert _{p}\leq1$. Proof of claim: First, since $\mathbf{y}\in\mathbb{Q}^{d}$, it also lies in $\mathbb{Q}_{p}^{d}$. So, by way of contradiction, suppose $\left\Vert \mathbf{y}\right\Vert _{p}>1$. Since $H$ is proper, \textbf{Lemma \ref{lem:MD wrong values lemma}} tells us that no matter which branches of $H$ are specified by $\mathbf{I}$, the composition sequence $H_{\mathbf{I}}$ maps $\mathbf{y}\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$ to $H_{\mathbf{I}}\left(\mathbf{y}\right)\in\mathbb{Q}_{p}^{d}\backslash\mathbb{Z}_{p}^{d}$, and so $\left\Vert H_{\mathbf{i}}\left(\mathbf{y}\right)\right\Vert _{p}>1$. However, $H_{\mathbf{i}}\left(\mathbf{y}\right)=\mathbf{x}$, and $\mathbf{x}\in\mathbb{N}_{0}^{d}$; hence, $1<\left\Vert H_{\mathbf{i}}\left(\mathbf{y}\right)\right\Vert _{p}=\left\Vert \mathbf{x}\right\Vert _{p}\leq1$. This is impossible! So, it must be that $\left\Vert \mathbf{y}\right\Vert _{p}\leq1$. This proves the claim. \end{claim} \begin{claim} \label{claim:5}$\mathbf{x}=H^{\circ\left\|\mathbf{I}\right\|}\left(\mathbf{y}\right)$ Proof of claim: Suppose the equality failed. Then $H_{\mathbf{I}}\left(\mathbf{y}\right)=\mathbf{x}\neq H^{\circ\left\|\mathbf{I}\right\|}\left(\mathbf{y}\right)$, and so $\mathbf{x}=H_{\mathbf{I}}\left(\mathbf{y}\right)$ is a wrong value of $H$ with seed $\mathbf{y}$. Because $H$ is proper,\textbf{ Lemma \ref{lem:MD wrong values lemma}} forces $\left\Vert \mathbf{x}\right\Vert _{p}=\left\Vert H_{\mathbf{I}}\left(\mathbf{y}\right)\right\Vert _{p}>1$. However, like in the one-dimensional case, this is just as impossible as it was in the previous paragraph, seeing as $\left\Vert \mathbf{x}\right\Vert _{p}\leq1$. This proves the claim. \end{claim} \vphantom{} Finally, let $\mathbf{V}$ be the shortest block-string representing $\mathbf{n}$, so that $\mathbf{J}$ (the block-string representing $B_{p}\left(\mathbf{n}\right)$) is obtained by concatenating infinitely many copies of $\mathbf{v}$. Because $q_{H}$ is co-prime to all the non-zero entries of $\mathbf{D}_{\mathbf{j}}$ for all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$, \emph{note that $H_{\mathbf{V}}$ is continuous on $\mathbb{Z}_{q_{H}}^{d}$}. As such: \begin{equation} \chi_{H}\left(B_{p}\left(\mathbf{n}\right)\right)\overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{V}^{\wedge k}\right) \end{equation} implies: \begin{align} H_{\mathbf{V}}\left(\chi_{H}\left(B_{p}\left(\mathbf{n}\right)\right)\right) & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}H_{\mathbf{V}}\left(\chi_{H}\left(\mathbf{V}^{\wedge k}\right)\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\lim_{k\rightarrow\infty}\chi_{H}\left(\mathbf{V}^{\wedge\left(k+1\right)}\right)\\ & \overset{\mathbb{Z}_{q_{H}}^{d}}{=}\chi_{H}\left(B_{p}\left(\mathbf{n}\right)\right) \end{align} Hence, $H_{\mathbf{V}}\left(\mathbf{y}\right)=\mathbf{y}$. Since we showed that $\left\Vert \mathbf{y}\right\Vert _{p}\leq1$, the propriety of $H$ allows us to apply \textbf{Lemma \ref{lem:properness lemma}} and conclude that $H_{\mathbf{V}}\left(\mathbf{y}\right)=H^{\circ\left\|\mathbf{V}\right\|}\left(\mathbf{y}\right)$. Thus, $\mathbf{y}$ is a periodic point of $H:\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}$. By \textbf{Claim \ref{claim:5}}, $H$ iterates $\mathbf{y}$ to $\mathbf{x}$. Since $\mathbf{y}$ is a periodic point of $H$ in $\mathbb{Z}_{p}^{d}$, this forces $\mathbf{x}$ and $\mathbf{y}$ to belong to the same cycle of $H$ in $\mathbb{Z}_{p}^{d}$, with $\mathbf{x}$ being a periodic point of $H$. As such, just as $H$ iterates $\mathbf{y}$ to $\mathbf{x}$, so too does $H$ iterate $\mathbf{x}$ to $\mathbf{y}$. Finally, since $\mathbf{x}\in\mathbb{N}_{0}^{d}$, so too is $\mathbf{y}$. integer as well. Thus, $\mathbf{x}$ belongs to a cycle of $H$ in $\mathbb{Z}^{d}$, as desired. Q.E.D. \vphantom{} Like in the one-dimensional case, the fact that the orbit classes of $H$'s attracting cycles, isolated cycles, and divergent trajectories constitute a partition of $\mathbb{Z}^{d}$ allows us to use the Correspondence Principle to make conclusions about the divergent trajectories. \begin{prop} \label{prop:MD Preparing for application to divergent trajectories}Let $H$ be as given in \textbf{\emph{Corollary \ref{cor:MD CP v3}}}. Additionally, suppose that $\left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}}=1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. Then $\chi_{H}\left(\mathbf{z}\right)\neq\mathbf{0}$ for any $\mathbf{z}\in\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$. \end{prop} Proof: Let $H$ as given. By way of contradiction, suppose that $\chi_{H}\left(\mathbf{z}\right)=\mathbf{0}$ for some $\mathbf{z}=\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\in\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$. Let $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ denote $\left[\mathbf{z}\right]_{p}$, and let $\mathbf{z}^{\prime}$ denote: \begin{equation} \mathbf{z}^{\prime}\overset{\textrm{def}}{=}\frac{\mathbf{z}-\mathbf{j}}{p}=\left(\frac{\mathfrak{z}_{1}-j_{1}}{p},\ldots,\frac{\mathfrak{z}_{r}-j_{r}}{p}\right) \end{equation} Then, we can write: \[ \mathbf{0}=\chi_{H}\left(\mathbf{z}\right)=\chi_{H}\left(p\mathbf{z}^{\prime}+\mathbf{j}\right)=H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{z}^{\prime}\right)\right) \] Next, suppose $\mathbf{j}=\mathbf{0}$. Because $H\left(\mathbf{0}\right)=\mathbf{0}$, we have: \[ \mathbf{0}=H_{\mathbf{0}}\left(\chi_{H}\left(\mathbf{z}^{\prime}\right)\right)=\mathbf{D}_{\mathbf{0}}^{-1}\mathbf{A}_{\mathbf{0}}\chi_{H}\left(\mathbf{z}^{\prime}\right) \] Since $\mathbf{A}_{\mathbf{0}}$ and $\mathbf{D}_{\mathbf{0}}$ are invertible, this forces $\chi_{H}\left(\mathbf{z}^{\prime}\right)=\mathbf{0}$. In this way, if the the first $n$ $p$-adic digits of $\mathfrak{z}_{m}$ are $0$ for all $m\in\left\{ 1,\ldots,r\right\} $, we can pull those digits out from $\mathbf{z}$ to obtain a $p$-adic integer tuple of unit $p$-adic norm ($\left\Vert \cdot\right\Vert _{p}$). If \emph{all }of the digits of all the $\mathfrak{z}_{m}$s are $0$, this then makes $\mathbf{z}=\mathbf{0}$, contradicting that $\mathbf{z}$ was given to be an element of $\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$. So, without loss of generality, we can assume that $\mathbf{j}\neq\mathbf{0}$. Hence: \begin{align} \mathbf{0} & =H_{\mathbf{j}}\left(\chi_{H}\left(\mathbf{z}^{\prime}\right)\right)\\ & =H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\chi_{H}\left(\mathbf{z}^{\prime}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)\\ \left(H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\right); & \Updownarrow\\ \mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\chi_{H}\left(\mathbf{z}^{\prime}\right) & =-H_{\mathbf{j}}\left(\mathbf{0}\right) \end{align} Because $H$ is semi-basic, $\mathbf{j}\neq\mathbf{0}$ tells us that $\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q_{H}}<1$ and $\left\Vert \mathbf{D}_{\mathbf{j}}\right\Vert _{q_{H}}=1$. So: \begin{align} \left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}} & =\left\Vert \mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\chi_{H}\left(\mathbf{z}^{\prime}\right)\right\Vert _{q_{H}}\\ & \leq\underbrace{\left\Vert \mathbf{D}_{\mathbf{j}}^{-1}\right\Vert _{q_{H}}\left\Vert \mathbf{A}_{\mathbf{j}}\right\Vert _{q_{H}}}_{<1}\left\Vert \chi_{H}\left(\mathbf{z}^{\prime}\right)\right\Vert _{q_{H}}\\ & <\left\Vert \chi_{H}\left(\mathbf{z}^{\prime}\right)\right\Vert _{q_{H}} \end{align} Since $\chi_{H}\left(\mathbb{Z}_{p}^{r}\right)\subseteq\mathbb{Z}_{q_{H}}^{d}$, we see that $\left\Vert \chi_{H}\left(\mathbf{z}^{\prime}\right)\right\Vert _{q_{H}}\leq1$, which forces $\left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}}<1$ for our non-zero $\mathbf{j}$. However, we were given that $\left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}}=1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. This is a contradiction! So, for each $m$, $\mathfrak{z}_{m}$ has no non-zero $p$-adic digits. This forces all of the $\mathfrak{z}_{m}$s to be $0$, which forces $\mathbf{z}=\mathbf{0}$. But $\mathbf{z}$ was given to be an element of $\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$, yet $\mathbf{0}\in\mathbb{Q}^{r}$\textemdash and that's our contradiction. So, it must be that $\chi_{H}\left(\mathbf{z}\right)\neq\mathbf{0}$. Q.E.D. \begin{thm} \label{thm:MD Divergent trajectories come from irrational z}Let\index{Hydra map@\emph{Hydra map!divergent trajectories}} $H$ be as given in \textbf{\emph{Corollary \ref{cor:MD CP v3}}}. Additionally, suppose that: \vphantom{} I. $H$ is contracting; \vphantom{} II. $\left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}}=1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. \vphantom{} Under these hypotheses, let $\mathbf{z}\in\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$. If $\chi_{H}\left(\mathbf{z}\right)\in\mathbb{Z}^{d}$, then $\chi_{H}\left(\mathbf{z}\right)$ belongs to a divergent orbit class of $H$. \end{thm} Proof: Let $H$ and $\mathbf{z}$ be as given. By \textbf{Proposition \ref{prop:MD Preparing for application to divergent trajectories}}, $\chi_{H}\left(\mathbf{z}\right)\neq\mathbf{0}$. Now, by way of contradiction, suppose that $\chi_{H}\left(\mathbf{z}\right)$ did not belong to a divergent orbit class of $H$. By \textbf{Theorem \ref{thm:orbit classes partition domain}}, every element of $\mathbb{Z}^{d}$ belongs to either a divergent orbit class of $H$, or to an orbit class of $H$ which contains a cycle of $H$. This forces $\chi_{H}\left(\mathbf{z}\right)$ to be a pre-periodic point of $H$. As such, there is an $n\geq0$ so that $H^{\circ n}\left(\chi_{H}\left(\mathbf{z}\right)\right)$ is a periodic point of $H$. Using the functional equations of $\chi_{H}$ (\textbf{Lemma \ref{lem:MD Chi_H functional equations and characterization}}), we can write $H^{\circ n}\left(\chi_{H}\left(\mathbf{z}\right)\right)=\chi_{H}\left(\mathbf{y}\right)$ where $\mathbf{y}=\mathbf{m}+p^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{z}$, and where $\mathbf{m}\in\mathbb{N}_{0}^{r}$ is the unique $r$-tuple of non-negative integers which represents the composition sequence of branches used in iterating $H$ $n$ times at $\chi_{H}\left(\mathbf{z}\right)$ to produce $\chi_{H}\left(\mathbf{y}\right)$. Because $\chi_{H}\left(\mathbf{z}\right)\neq\mathbf{0}$, we can use the integrality of $H$ along with the fact that $\mathbf{0}=H\left(\mathbf{0}\right)=H_{\mathbf{0}}\left(\mathbf{0}\right)$ to conclude that $\left\{ \mathbf{0}\right\} $ is an isolated cycle of $H$. This then guarantees that the periodic point $\chi_{H}\left(\mathbf{y}\right)$ is non-zero. Consequently, the multi-dimensional \textbf{Correspondence Principle }tells us that $\mathbf{y}$ must be an element of $\mathbb{Q}^{r}\cap\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. So, \emph{for each} $\ell\in\left\{ 1,\ldots,r\right\} $, the $\ell$th entry of $\mathbf{y}$ is a rational number; therefore, its sequence of $p$-adic digits is eventually periodic. However, since $\mathbf{y}=\mathbf{m}+p^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{z}$ where $\mathbf{z}\in\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$, there is at least one $\ell\in\left\{ 1,\ldots,r\right\} $ so that $\mathfrak{z}_{\ell}$'s $p$-adic digits are \emph{aperiodic}. This is a contradiction! So, it must be that $\chi_{H}\left(\mathbf{z}\right)$ is a divergent point of $H$. Q.E.D. \vphantom{} Like with the one-dimensional case, it would be highly desirable to prove the following conjecture true: \begin{conjecture}[\textbf{Correspondence Principle for Multi-Dimensional Divergent Points?}] \label{conj:MD correspondence theorem for divergent trajectories}Provided that $H$ satisfies certain prerequisites such as the hypotheses of \textbf{\emph{Theorem \ref{thm:MD Divergent trajectories come from irrational z}}}, $\mathbf{x}\in\mathbb{Z}^{d}$ belongs to a divergent trajectory under $H$ if and only if there is a $\mathbf{z}\in\mathbb{Z}_{p}^{r}\backslash\mathbb{Q}^{r}$ so that $\chi_{H}\left(\mathbf{z}\right)\in\mathbb{Z}^{d}$. \end{conjecture} \newpage{} \section{\label{sec:5.3 Rising-Continuity-in-Multiple}Rising-Continuity in Multiple Dimensions} IN THIS SECTION, WE FIX INTEGERS $r,d\geq1$ (WITH $r\leq d$) ALONG WITH TWO DISTINCT PRIME NUMBERS $p$ AND $q$. UNLESS SAID OTHERWISE, $\mathbb{K}$ DENOTES A METRICALLY COMPLETE VALUED FIELD, POSSIBLY ARCHIMEDEAN; $K$, MEANWHILE, DENOTES A METRICALLY COMPLETE VALUED \emph{NON-ARCHIMEDEAN} FIELD, UNLESS STATED OTHERWISE. \subsection{\label{subsec:5.3.1 Tensor-Products}Multi-Dimensional Notation and Tensor Products} \begin{notation} \index{multi-dimensional!notation}\ \vphantom{} I. Like with $\left\Vert \mathbf{z}\right\Vert _{p}$ for $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, for $\hat{\mathbb{Z}}_{p}^{r}$, we write: \begin{equation} \left\Vert \mathbf{t}\right\Vert _{p}\overset{\textrm{def}}{=}\max\left\{ p^{-v_{p}\left(t_{1}\right)},\ldots,p^{-v_{p}\left(t_{m}\right)}\right\} ,\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:Definition of bold t p norm} \end{equation} \begin{equation} v_{p}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\min\left\{ v_{p}\left(t_{1}\right),\ldots,v_{p}\left(t_{m}\right)\right\} ,\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:Definition of v_P of bold t} \end{equation} so that $\left\Vert \mathbf{t}\right\Vert _{p}=p^{-v_{p}\left(\mathbf{t}\right)}$. In this notation, $\left\{ \mathbf{t}:\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}\right\} $ (equivalently $\left\{ \mathbf{t}:v_{p}\left(\mathbf{t}\right)=-n\right\} $) tells us that for each $\mathbf{t}$, the inequality $\left\|t_{m}\right\|_{p}\leq p^{n}$ occurs for all $m$, with equality actually being achieved for \emph{at least} one $m\in\left\{ 1,\ldots,r\right\} $. $\left\{ \mathbf{t}:\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}\right\} $, meanwhile, tells us that, for each $\mathbf{t}$, $\left\|t_{m}\right\|_{p}\leq p^{N}$ occurs for all $m$. In particular: \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}=\sum_{\left\|t_{1}\right\|_{p}\leq p^{N}}\cdots\sum_{\left\|t_{r}\right\|_{p}\leq p^{N}}\label{eq:Definition of sum of bold t norm _p lessthanorequalto p^N} \end{equation} That being said, the reader should take care to notice that: \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\neq\sum_{\left\|t_{1}\right\|_{p}=p^{n}}\cdots\sum_{\left\|t_{r}\right\|_{p}=p^{n}}\label{eq:Warning about summing bold t norm _p equal to p^N} \end{equation} \vphantom{} II. We write: \begin{equation} \sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\overset{\textrm{def}}{=}\sum_{n_{1}=0}^{p^{N}-1}\cdots\sum_{n_{r}=0}^{p^{N}-1}\label{eq:Definition of sum from bold n eq zero to P^N minus 1} \end{equation} and define: \begin{equation} \mathbf{n}\leq p^{m}-1\label{eq:Definition of bold n lessthanoreqto P^m minus 1} \end{equation} to indicate that $0\leq n_{\ell}\leq p^{m}-1$ for all $\ell\in\left\{ 1,\ldots,r\right\} $. Note then that: \begin{equation} \sum_{\mathbf{j}=\mathbf{0}}^{p-1}=\sum_{\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}}\label{eq:Definition of sum from bold j is 0 to P minus 1} \end{equation} and also: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\overset{\textrm{def}}{=}\sum_{\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} }\label{eq:Definition of sum from bold j greaterthan 0 to P minus 1} \end{equation} Consequently, $\forall\mathbf{j}\in\mathbb{Z}^{r}/\mathbb{Z}^{r}$ and $\forall\mathbf{j}\leq p-1$ denote the same sets of $\mathbf{j}$s. \vphantom{} III. $\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)$\nomenclature{$\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)$}{ } denotes the function on $\hat{\mathbb{Z}}_{p}^{r}$ which is $1$ if $\mathbf{t}\overset{1}{\equiv}\mathbf{0}$ and is $0$ otherwise. For $\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}$, I write $\mathbf{1}_{\mathbf{s}}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}-\mathbf{s}\right)$. \vphantom{} IV. For a matrix $\mathbf{A}$ with entries in $\mathbb{Z}_{q}$, recall we write $\left\Vert \mathbf{A}\right\Vert _{q}$ to denote the maximum of the $q$-adic absolute values of $\mathbf{A}$'s entries. When $\mathbf{A}$ has entries in $\overline{\mathbb{Q}}$, we write $\left\Vert \mathbf{A}\right\Vert _{\infty}$ to denote the maximum of the complex absolute values of $\mathbf{A}$'s entries. \vphantom{} V. For any $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$, we write \nomenclature{$\left\|\mathbf{t}\right\|_{p}$}{$\left(p^{-v_{p}\left(\mathbf{t}\right)},\ldots,p^{-v_{p}\left(\mathbf{t}\right)}\right)$}$\left\|\mathbf{t}\right\|_{p}$ to denote the $r$\emph{-tuple}: \begin{equation} \left\|\mathbf{t}\right\|_{p}\overset{\textrm{def}}{=}\left(p^{-v_{p}\left(\mathbf{t}\right)},\ldots,p^{-v_{p}\left(\mathbf{t}\right)}\right)\label{eq:Definition of single bars of bold t _P} \end{equation} Consequently: \begin{equation} \frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}=p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}=\left(p^{-v_{p}\left(\mathbf{t}\right)-1}t_{1},\ldots,p^{-v_{p}\left(\mathbf{t}\right)-1}t_{r}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:Definition of bold t projection onto first Z_P hat level set} \end{equation} \end{notation} \vphantom{} Recall our earlier notation for writing $\mathbb{K}^{\rho,c}$ to denote the $\mathbb{K}$-linear space of all $\rho\times c$ matrices with entries in $\mathbb{K}$ for integers $\rho,c\geq1$. When $c=1$, we identify $\mathbb{K}^{\rho,1}$ with $\mathbb{K}^{\rho}$. \begin{defn} \label{nota:Second batch}For a function $\mathbf{F}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{K}^{\rho,c}$, we write \nomenclature{$\left\Vert \mathbf{F}\right\Vert _{p,\mathbb{K}}$}{ } $\left\Vert \mathbf{F}\right\Vert _{p,\mathbb{K}}$ to denote: \begin{equation} \left\Vert \mathbf{F}\right\Vert _{p,\mathbb{K}}\overset{\textrm{def}}{=}\sup_{\mathbf{z}\in\mathbb{Z}_{p}^{r}}\left\Vert \mathbf{F}\left(\mathbf{z}\right)\right\Vert _{\mathbb{K}}\label{eq:Definition of P,K norm} \end{equation} Here, $\left\Vert \mathbf{F}\left(\mathbf{z}\right)\right\Vert _{\mathbb{K}}$ is the maximum of the $\mathbb{K}$ absolute values of the entries of the matrix $\mathbf{F}\left(\mathbf{z}\right)$. We write $\left\Vert \cdot\right\Vert _{p,q}$ in the case where $\mathbb{K}$ is a $q$-adic field, and write $\left\Vert \cdot\right\Vert _{p,\infty}$ when $\mathbb{K}$ is $\mathbb{C}$. We \emph{also} use $\left\Vert \cdot\right\Vert _{p,\mathbb{K}}$ to denote the corresponding norm of a matrix-valued function on $\hat{\mathbb{Z}}_{p}^{r}$. \end{defn} \begin{defn} Let\index{matrix!Hadamard product} $\mathbf{A}=\left\{ a_{j,k}\right\} _{1\leq j\leq\rho,1\leq k\leq c}$ and $\mathbf{B}=\left\{ b_{j,k}\right\} _{1\leq j\leq\rho,1\leq k\leq c}$ be elements of $\mathbb{K}^{\rho,c}$. Then, we write \nomenclature{$\mathbf{A}\odot\mathbf{B}$}{the Hadamard product of matrices}$\mathbf{A}\odot\mathbf{B}$ to denote the matrix: \begin{equation} \mathbf{A}\odot\mathbf{B}\overset{\textrm{def}}{=}\left\{ a_{j,k}\cdot b_{j,k}\right\} _{1\leq j\leq\rho,1\leq k\leq c}\label{eq:Definition of the Hadamard product of two matrices} \end{equation} This is called the \textbf{Hadamard product }of $\mathbf{A}$ and $\mathbf{B}$. Given $\mathbf{A}_{1},\ldots,\mathbf{A}_{N}$, where: \begin{equation} \mathbf{A}_{n}=\left\{ a_{j,k}\left(n\right)\right\} _{1\leq j\leq\rho,1\leq k\leq c} \end{equation} for each $n$, we then define write: \begin{equation} \bigodot_{n=1}^{N}\mathbf{A}_{n}\overset{\textrm{def}}{=}\left\{ \prod_{n=1}^{N}a_{j,k}\left(n\right)\right\} _{1\leq j\leq\rho,1\leq k\leq c}\label{eq:Definition of the tensor product of n matrices} \end{equation} Note that $\odot$ reduces to standard multiplication when $\rho=c=1$. \end{defn} \begin{defn} We say a function $\mathbf{F}$ (resp. $\hat{\mathbf{F}}$) defined on $\mathbb{Z}_{p}^{r}$ (resp. $\hat{\mathbb{Z}}_{p}^{r}$) taking values in $\mathbb{K}^{\rho,c}$ is \textbf{elementary }if\index{elementary!function} there are functions $\mathbf{F}_{1},\ldots,\mathbf{F}_{r}$ (resp. $\hat{\mathbf{F}}_{1},\ldots,\hat{\mathbf{F}}_{r}$) defined on $\mathbb{Z}_{p}$ (resp. $\hat{\mathbb{Z}}_{p}$) so that: \begin{equation} \mathbf{F}\left(\mathbf{z}\right)=\bigodot_{m=1}^{r}\mathbf{F}\left(\mathfrak{z}_{m}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} and: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)=\bigodot_{m=1}^{r}\hat{\mathbf{F}}\left(t_{m}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} respectively. We write \nomenclature{$E\left(\mathbb{Z}_{p}^{r},\mathbb{K}^{\left(\rho,c\right)}\right)$}{set of elementary tensors $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{K}^{\rho,c}$}$E\left(\mathbb{Z}_{p}^{r},\mathbb{K}^{\rho,c}\right)$ (resp. \nomenclature{$E\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{K}^{\left(\rho,c\right)}\right)$}{set of elementary tensors $\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{K}^{\rho,c}$}$E\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{K}^{\rho,c}\right)$) to denote the $\mathbb{K}$-span of all elementary functions (a.k.a., \textbf{elementary tensors}\index{elementary!tensors}) on $\mathbb{Z}_{p}^{r}$ (resp. $\hat{\mathbb{Z}}_{p}^{r}$). Note that the Hadamard products reduce to entry-wise products of vectors when $c=1$. \end{defn} \begin{defn}[\textbf{Multi-Dimensional Function Spaces}] \ \vphantom{} I. We write \nomenclature{$B\left(\mathbb{Z}_{p}^{r},K\right)$}{set of bounded functions $\mathbb{Z}_{p}^{r}\rightarrow K$}$B\left(\mathbb{Z}_{p}^{r},K\right)$ to denote the $K$-linear space of all bounded functions $f:\mathbb{Z}_{p}^{r}\rightarrow K$. This is a non-archimedean Banach space under the norm: \begin{equation} \left\Vert f\right\Vert _{p,K}\overset{\textrm{def}}{=}\sup_{\mathbf{z}\in\mathbb{Z}_{p}^{r}}\left\|f\left(\mathbf{z}\right)\right\|_{K}=\sup_{\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\in\mathbb{Z}_{p}}\left\|f\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\right\|_{K}\label{eq:Definition of scalar valued C Z_P norm} \end{equation} We write \nomenclature{$C\left(\mathbb{Z}_{p}^{r},K\right)$}{set of continuous functions $\mathbb{Z}_{p}^{r}\rightarrow K$ }$C\left(\mathbb{Z}_{p}^{r},K\right)$ to denote the space of all continuous $f:\mathbb{Z}_{p}^{r}\rightarrow K$. \vphantom{} II. We write \nomenclature{$B\left(\mathbb{Z}_{p}^{r},K^{\left(\rho,c\right)}\right)$}{set of bounded functions $\mathbb{Z}_{p}^{r}\rightarrow K^{\rho,c}$ }$B\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$ to denote the $K$-linear space of all bounded functions $\mathbf{F}:\mathbb{Z}_{p}^{r}\rightarrow K^{\rho,c}$, where: \begin{equation} \mathbf{F}\left(\mathbf{z}\right)=\left\{ F_{j,k}\left(\mathbf{z}\right)\right\} _{1\leq j\leq\rho,1\leq k\leq c}\in K^{\rho,c},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} where, for each $\left(j,k\right)$, $F_{j,k}\in B\left(\mathbb{Z}_{p}^{r},K\right)$. $B\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$ becomes a non-archimedean Banach space under the norm $\left\Vert \mathbf{F}\right\Vert _{p,K}$ (the maximum of the supremum-over-$\mathbb{Z}_{p}^{r}$ of the $K$-adic absolute values of each entry of $\mathbf{F}$). We write \nomenclature{$C\left(\mathbb{Z}_{p}^{r},K^{\left(\rho,c\right)}\right)$}{set of continuous functions $\mathbb{Z}_{p}^{r}\rightarrow K^{\rho,c}$ }$C\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$ to denote the space of all continuous functions $\mathbf{F}:\mathbb{Z}_{p}^{r}\rightarrow K^{\rho,c}$. \vphantom{} III. We write \nomenclature{$B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$}{set of bounded functions $\hat{\mathbb{Z}}_{p}^{r}\rightarrow K$}$B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$ to denote the $K$-linear space of all bounded functions $\hat{f}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow K$. This is a non-archimedean Banach space under the norm: \begin{equation} \left\Vert \hat{f}\right\Vert _{p,K}\overset{\textrm{def}}{=}\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\|\hat{f}\left(\mathbf{t}\right)\right\|_{K}=\sup_{t_{1},\ldots,t_{r}\in\hat{\mathbb{Z}}_{p}}\left\|\hat{f}\left(\hat{t}_{1},\ldots,\hat{t}_{r}\right)\right\|_{K}\label{eq:Definition of scalar valued C Z_P hat norm} \end{equation} We write $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$\nomenclature{$c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$}{set of $f\in B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$ so that $\lim_{\left\Vert \mathbf{t}\right\Vert _{p}\rightarrow\infty}\left\|\hat{f}\left(\mathbf{t}\right)\right\|_{K}=0$} to denote the subspace of $B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$ consisting of all $\hat{f}$ for which: \begin{equation} \lim_{\left\Vert \mathbf{t}\right\Vert _{p}\rightarrow\infty}\left\|\hat{f}\left(\mathbf{t}\right)\right\|_{K}=0 \end{equation} \vphantom{} IV. We write \nomenclature{$B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\left(\rho,c\right)}\right)$}{set of bounded functions $\hat{\mathbb{Z}}_{p}^{r}\rightarrow K^{\rho,c}$}$B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$ to denote the $K$-linear space of all bounded functions $\hat{\mathbf{F}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow K^{\rho,c}$, where: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)=\left\{ \hat{F}_{j,k}\left(\mathbf{t}\right)\right\} _{1\leq j\leq\rho,1\leq k\leq c}\in K^{\rho,c},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} where, for each $\left(j,k\right)$, $\hat{F}_{m}\in B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$. $B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$ becomes a non-archimedean Banach space under the norm $\left\Vert \hat{\mathbf{F}}\right\Vert _{p,K}^{r}$. We then write \nomenclature{$c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K^{\left(\rho,c\right)}\right)$}{set of bounded functions $\hat{\mathbb{Z}}_{p}^{r}\rightarrow K^{\rho,c}$ so that $\lim_{\left\Vert \mathbf{t}\right\Vert _{p}^{r}\rightarrow\infty}\left\Vert \hat{\mathbf{F}}\left(\mathbf{t}\right)\right\Vert _{K}=0$}$c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$ to denote the Banach subspace of $B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$ consisting of all those $\hat{\mathbf{F}}$ so that: \begin{equation} \lim_{\left\Vert \mathbf{t}\right\Vert _{p}^{r}\rightarrow\infty}\left\Vert \hat{\mathbf{F}}\left(\mathbf{t}\right)\right\Vert _{K}=0 \end{equation} \end{defn} \begin{rem} The Fourier transform will send functions $\mathbb{Z}_{p}^{r}\rightarrow K$ to functions $\hat{\mathbb{Z}}_{p}^{r}\rightarrow K$. Fourier transforms on $C\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$ are done component-wise. \end{rem} \vphantom{} Now, we need to introduce the tensor product. Fortunately, in all of the cases we shall be working with, the tensor product is nothing more than point-wise multiplication of functions. A reader desirous for greater abstract detail can refer to \cite{van Rooij - Non-Archmedean Functional Analysis} to slake their thirst. The fourth chapter of van Rooij's book has an \emph{entire section} dedicated to the tensor product\index{tensor!product} ($\otimes$\nomenclature{$\otimes$}{tensor product}). \begin{thm} \cite{van Rooij - Non-Archmedean Functional Analysis} Let $E$, $F$, and $G$ be Banach spaces over metrically complete non-archimedean fields. Then: \begin{equation} \left\Vert f\otimes g\right\Vert =\left\Vert f\right\Vert _{E}\left\Vert g\right\Vert _{F},\textrm{ }\forall f\in E,\textrm{ }\forall g\in F\label{eq:Norm of a tensor product} \end{equation} Additionally, for any linear operator $T:E\otimes F\rightarrow G$, the operator norm\index{operator norm} of $T$ is given by: \begin{equation} \left\Vert T\right\Vert \overset{\textrm{def}}{=}\sup_{\left\Vert f\right\Vert _{E}\leq1,\left\Vert g\right\Vert _{F}\leq1}\frac{\left\Vert T\left\{ f\otimes g\right\} \right\Vert }{\left\Vert f\right\Vert _{E}\left\Vert g\right\Vert _{F}}\label{eq:Definition of the norm of a linear operator on a tensor product} \end{equation} \end{thm} \begin{thm} \cite{van Rooij - Non-Archmedean Functional Analysis} Let $E_{1},F_{1},E_{2},F_{2}$ be Banach spaces over metrically complete non-archimedean fields. Given continuous linear operators $T_{j}:E_{j}\rightarrow F_{j}$ for $j\in\left\{ 1,2\right\} $, the operator $T_{1}\otimes T_{2}:E_{1}\otimes F_{1}\rightarrow E_{2}\otimes F_{2}$ is given by: \begin{equation} \left(T_{1}\otimes T_{2}\right)\left\{ f\otimes g\right\} =T_{1}\left\{ f\right\} \otimes T_{2}\left\{ g\right\} \label{eq:Action of tensor products of linear operators} \end{equation} \end{thm} \begin{cor} \cite{van Rooij - Non-Archmedean Functional Analysis} Let $E$ and $F$ be Banach spaces over a metrically complete non-archimedean field $K$. Then, the dual of $E\otimes F$ is $E^{\prime}\otimes F^{\prime}$, with elementary tensors $\mu\otimes\nu\in E^{\prime}\otimes F^{\prime}$ acting on elementary tensors $f\otimes g\in E\otimes F$ by: \begin{equation} \left(\mu\otimes\nu\right)\left(f\otimes g\right)\overset{K}{=}\mu\left(f\right)\cdot\nu\left(g\right)\label{eq:action of tensor product of linear functionals} \end{equation} \end{cor} \begin{thm} \cite{van Rooij - Non-Archmedean Functional Analysis}\ \vphantom{} I. \begin{equation} B\left(\mathbb{Z}_{p}^{r},K\right)=\bigotimes_{m=1}^{r}B\left(\mathbb{Z}_{p},K\right) \end{equation} \index{tensor!elementary}Elementary tensors in $B\left(\mathbb{Z}_{p}^{r},K\right)$ are of the form: \begin{equation} \left(\bigotimes_{m=1}^{r}f_{m}\right)\left(\mathbf{z}\right)=\left(\bigotimes_{m=1}^{r}f_{m}\right)\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\overset{\textrm{def}}{=}\prod_{m=1}^{r}f_{m}\left(\mathfrak{z}_{m}\right) \end{equation} This also holds for $C\left(\mathbb{Z}_{p}^{r},K\right)$. \vphantom{} II. \begin{equation} B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)=\bigotimes_{m=1}^{r}B\left(\hat{\mathbb{Z}}_{p},K\right) \end{equation} Elementary tensors in $B\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$ are of the form: \begin{equation} \left(\bigotimes_{m=1}^{r}\hat{f}_{m}\right)\left(\mathbf{t}\right)=\left(\bigotimes_{m=1}^{r}\hat{f}_{m}\right)\left(t_{1},\ldots,t_{r}\right)\overset{\textrm{def}}{=}\prod_{m=1}^{r}\hat{f}_{m}\left(t_{m}\right) \end{equation} This also holds for $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K\right)$. \vphantom{} III. \begin{equation} B\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)=\bigotimes_{m=1}^{r}B\left(\mathbb{Z}_{p},K^{\rho,c}\right) \end{equation} where an elementary tensor in $B\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$ is of the form: \begin{equation} \mathbf{F}\left(\mathbf{z}\right)=\bigodot_{m=1}^{r}\mathbf{F}_{m}\left(\mathfrak{z}_{m}\right) \end{equation} where $\mathbf{F}_{m}$ is in $B\left(\mathbb{Z}_{p},K^{\rho,c}\right)$ for each $m$. This also holds for $C\left(\mathbb{Z}_{p}^{r},K^{\rho,c}\right)$. \vphantom{} IV. \begin{equation} B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)=\bigotimes_{m=1}^{r}B\left(\hat{\mathbb{Z}}_{p},K^{\rho,c}\right) \end{equation} where an elementary tensor in $B\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$ is of the form: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)=\bigodot_{m=1}^{r}\hat{\mathbf{F}}_{m}\left(t_{m}\right) \end{equation} where $\hat{\mathbf{F}}_{m}$ is in $B\left(\hat{\mathbb{Z}}_{p},K^{\rho,c}\right)$ for each $m$. This also holds for $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},K^{\rho,c}\right)$. \end{thm} \begin{rem} Every function in $C\left(\mathbb{Z}_{p}^{r},K\right)$ can be written as a linear combination of (possibly infinitely many) elementary functions. If the linear combination is infinite, then elements of the linear combination can be enumerated in a sequence which converges to $0$ in norm, so as to guarantee the convergence of the resulting infinite sum. \end{rem} \begin{defn}[\textbf{Multi-Dimensional van der Put Basis}] Given a prime $p$, we write $\mathcal{B}_{p}$ to denote the set of van der Put basis functions for $\mathbb{Z}_{p}$: \begin{equation} \mathcal{B}_{p}\overset{\textrm{def}}{=}\left\{ \left[\mathfrak{z}\overset{p^{\lambda_{p}\left(n\right)}}{\equiv}n\right]:n\in\mathbb{N}_{0}\right\} \label{eq:Definition of the van der Put basis} \end{equation} Next, for any $\mathbf{n}\in\mathbb{N}_{0}^{r}$, we write: \begin{equation} \left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\overset{\textrm{def}}{=}\prod_{\ell=1}^{r}\left[\mathfrak{z}_{\ell}\overset{p^{\lambda_{p}\left(n_{\ell}\right)}}{\equiv}n_{\ell}\right],\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:Notation for elementary P-adic van der Put tensors} \end{equation} We call (\ref{eq:Notation for elementary P-adic van der Put tensors}) an \textbf{elementary $p$-adic van der Put tensor}. \nomenclature{$\mathcal{B}_{p}^{\otimes r}$}{$\left\{ \left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]:\mathbf{n}\in\mathbb{N}_{0}^{r}\right\}$}We then define the \textbf{$p$-adic van der Put basis of depth $r$ }as the set: \begin{equation} \mathcal{B}_{p}^{\otimes r}\overset{\textrm{def}}{=}\left\{ \left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]:\mathbf{n}\in\mathbb{N}_{0}^{r}\right\} \label{eq:definition of elementary tensor van der Put basis} \end{equation} \end{defn} \begin{thm} $\mathcal{B}_{p}^{\otimes r}$ is a basis for $C\left(\mathbb{Z}_{p}^{r},K\right)$. \end{thm} Proof: $\mathcal{B}_{p}$ is a basis for $C\left(\mathbb{Z}_{p},K\right)$, so, tensoring the $\mathcal{B}_{p}$s to get $\mathcal{B}_{p}^{\otimes r}$ yields a basis for $C\left(\mathbb{Z}_{p}^{r},K\right)$. Q.E.D. \begin{defn}[\textbf{Multi-Dimensional van der Put Series}] We write \nomenclature{$\textrm{vdP}\left(\mathbb{Z}_{p}^{r},K\right)$}{the set of formal $\left(p,K\right)$-adic van der Put series of depth $r$ } to denote the $K$-linear space of all\index{van der Put!series!formal} \textbf{formal $\left(p,K\right)$-adic van der Put series of depth $r$}. The elements of $\textrm{vdP}\left(\mathbb{Z}_{p}^{r},K\right)$ are formal sums: \begin{equation} \sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}\mathfrak{a}_{\mathbf{n}}\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\label{eq:Definition of a formal (P,K)-adic van der Put series} \end{equation} where the $\mathfrak{a}_{\mathbf{n}}$s are elements of $K$. The \textbf{domain of convergence }of a formal van der Put series is the set of all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ for which the series converges\footnote{Note: since $\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}$ can hold true for at most finitely many $\mathbf{n}$ whenever $\mathbf{z}\in\mathbb{N}_{0}^{r}$, observe that every formal van der Put series necessarily converges in $K$ at every $\mathbf{z}\in\mathbb{N}_{0}^{r}$.} in $K$. We call $\textrm{vdP}\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$ the space of \textbf{formal $\left(p,q\right)$-adic van der Put series of depth $r$}. \end{defn} \begin{defn} Recall that for an integer $n\geq0$ and a prime $p$, we write: \begin{equation} n_{-}=\begin{cases} 0 & \textrm{if }n=0\\ n-n_{\lambda_{p}\left(n\right)-1}p^{\lambda_{p}\left(n\right)-1} & \textrm{if }n\geq1 \end{cases} \end{equation} where $n_{\lambda_{p}\left(n\right)-1}$ is the coefficient of the largest power of $p$ present in the $p$-adic expansion of $n$. \vphantom{} I. Given $\mathbf{n}\in\mathbb{N}_{0}^{d}$, we write \nomenclature{$\mathbf{n}_{-}$}{multi-dimensional analogue of $n_{-}$}$\mathbf{n}_{-}$ to denote the $d$-tuple obtained by applying the subscript $-$ operation to each entry of $\mathbf{n}$: $\mathbf{n}_{-}\overset{\textrm{def}}{=}\left(\left(n_{1}\right)_{-},\ldots,\left(n_{d}\right)_{-}\right)$. \vphantom{} II. Let $f:\mathbb{Z}_{p}^{r}\rightarrow K$ be an elementary function, with $f=\prod_{m=1}^{r}f_{m}$. Then, we define the \textbf{van der Put coefficients}\index{van der Put!coefficients}\textbf{ }\nomenclature{$c_{\mathbf{n}}\left(f\right)$}{$\mathbf{n}$th van der Put coefficient of $f$}$c_{\mathbf{n}}\left(f\right)$ of $f$ by: \begin{equation} c_{\mathbf{n}}\left(f\right)\overset{\textrm{def}}{=}\prod_{m=1}^{r}c_{n_{m}}\left(f_{m}\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\label{eq:Def of the van der put coefficients of an elementary function} \end{equation} \end{defn} \begin{prop} \label{prop:vdP series of an elementary function}Every elementary\index{elementary!function} $f\in C\left(\mathbb{Z}_{p}^{r},K\right)$ is uniquely representable as a $\left(p,K\right)$-adic van der Put series of depth $r$: \begin{equation} f\left(\mathbf{z}\right)\overset{K}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right],\textrm{ }\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} Moreover, the $K$-convergence of this series is uniform over $\mathbb{Z}_{p}^{r}$. \end{prop} Proof: Let $f$ be elementary. Then, $f$ is a product of the form: \begin{equation} f\left(\mathbf{z}\right)=f\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)=\prod_{m=1}^{r}f_{m}\left(\mathfrak{z}_{m}\right) \end{equation} for $f_{m}\in C\left(\mathbb{Z}_{p},K\right)$. As such, each $f_{m}$ is uniquely representable by a uniformly convergent van der Put series: \begin{equation} f_{m}\left(\mathfrak{z}_{m}\right)=\sum_{n_{m}=0}^{\infty}c_{n_{m}}\left(f_{m}\right)\left[\mathfrak{z}_{m}\overset{p^{\lambda_{p}\left(n_{m}\right)}}{\equiv}n_{m}\right] \end{equation} with $\left\|c_{n_{m}}\left(f_{m}\right)\right\|_{K}\rightarrow0$ as $n_{m}\rightarrow\infty$. So: \begin{align} f\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right) & =\prod_{m=1}^{r}\left(\sum_{n_{m}=0}^{\infty}c_{n_{m}}\left(f_{m}\right)\left[\mathfrak{z}_{m}\overset{p^{\lambda_{p}\left(n_{m}\right)}}{\equiv}n_{m}\right]\right)\\ & =\sum_{n_{1}=0}^{\infty}\cdots\sum_{n_{r}=0}^{\infty}\left(\prod_{m=1}^{r}c_{n_{m}}\left(f_{m}\right)\left[\mathfrak{z}_{m}\overset{p^{\lambda_{p}\left(n_{m}\right)}}{\equiv}n_{m}\right]\right)\\ & =\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right] \end{align} Q.E.D. \begin{prop} Let $g_{1},\ldots g_{N}$ be elementary functions in $C\left(\mathbb{Z}_{p}^{r},K\right)$. Then, for any $\alpha_{1},\ldots,\alpha_{N}\in K$, the linear combination $\sum_{k=1}^{N}\alpha_{k}g_{k}$ is represented by a uniformly convergent $\left(p,K\right)$-adic van der Put series of depth $r$, the coefficients of which are given by: \begin{equation} c_{\mathbf{n}}\left(\sum_{k=1}^{N}\alpha_{k}g_{k}\right)=\sum_{k=1}^{N}\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)\label{eq:Linear extension of MD van der Put coefficients} \end{equation} where $c_{\mathbf{n}}\left(g_{k}\right)$ is computed by \emph{(\ref{eq:Def of the van der put coefficients of an elementary function})}. This result also extends to infinite linear combinations, provided that they converge in norm. \end{prop} Proof: Each $g_{k}$ is representable by a uniformly convergent $\left(p,K\right)$-adic van der Put series of depth $r$. The proposition follows from the fact that the space of van der Put series is closed under linear combinations. Q.E.D. \begin{rem} Consequently, we can use (\ref{eq:Linear extension of MD van der Put coefficients}) to compute the van der Put coefficients of \emph{any }function in $C\left(\mathbb{Z}_{p}^{r},K\right)$, seeing as the $K$-span of $C\left(\mathbb{Z}_{p}^{r},K\right)$'s elementary functions is dense in $C\left(\mathbb{Z}_{p}^{r},K\right)$. \end{rem} \begin{lem} \label{lem:van der Put series for an arbitrary K-valued function on Z_P}Let $f\in C\left(\mathbb{Z}_{p}^{r},K\right)$. Then, there are unique constants $\left\{ c_{\mathbf{n}}\left(f\right)\right\} _{\mathbf{n}\in\mathbb{N}_{0}^{r}}$ so that: \begin{equation} f\left(\mathbf{z}\right)\overset{K}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\label{eq:van der Put series for an arbitrary K-valued function on Z_P} \end{equation} where the convergence is uniform in $\mathbf{z}$; specifically: \begin{equation} \lim_{\left\Vert \mathbf{n}\right\Vert _{\infty}\rightarrow\infty}\left\|c_{\mathbf{n}}\left(f\right)\right\|_{K}=0 \end{equation} where, recall (see \emph{(\ref{eq:Definition of infinity norm})} on page \emph{\pageref{eq:Definition of infinity norm}}): \begin{equation} \left\Vert \mathbf{n}\right\Vert _{\infty}=\max\left\{ n_{1},\ldots,n_{r}\right\} \end{equation} In particular, letting $\left\{ g_{k}\right\} _{k\geq1}$ and $\left\{ \alpha_{k}\right\} _{k\geq1}$ be any sequences of \index{elementary!function}elementary functions in $C\left(\mathbb{Z}_{p}^{r},K\right)$ and scalars in $K$, respectively, so that: \begin{equation} \lim_{N\rightarrow\infty}\left\Vert f-\sum_{k=1}^{N}\alpha_{k}g_{k}\right\Vert _{p,q}=0 \end{equation} then: \begin{equation} c_{\mathbf{n}}\left(f\right)\overset{K}{=}\lim_{N\rightarrow\infty}c_{\mathbf{n}}\left(\sum_{k=1}^{N}\alpha_{k}g_{k}\right)=\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)\label{eq:Limit formula for the van der Put coefficients of an abitrary continuous (P,K)-adic function} \end{equation} We call the $c_{\mathbf{n}}\left(f\right)$s the \index{van der Put!coefficients}\textbf{van der Put coefficients }of $f$. \end{lem} Proof: I. (Existence) Since the $C\left(\mathbb{Z}_{p}^{r},K\right)$-elementary functions are dense in $C\left(\mathbb{Z}_{p}^{r},K\right)$, given any $f\in C\left(\mathbb{Z}_{p}^{r},K\right)$, we can choose $\left\{ g_{k}\right\} _{k\geq1}$ and $\left\{ \alpha_{k}\right\} _{k\geq1}$ as described above so that the norm convergence occurs. Writing: \begin{equation} \sum_{k=1}^{N}\alpha_{k}g_{k}\left(\mathbf{z}\right)\overset{K}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}\left(\sum_{k=1}^{N}\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right] \end{equation} \textemdash note, the convergence is uniform in $\mathbf{z}$\textemdash the assumed norm convergence allows us to interchange limits and sums: \begin{equation} f\left(\mathbf{z}\right)\overset{K}{=}\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\alpha_{k}g_{k}\left(\mathbf{z}\right)\overset{K}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}\underbrace{\left(\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)\right)}_{c_{\mathbf{n}}\left(f\right)}\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right] \end{equation} \vphantom{} II. (Uniqueness) Let $\left\{ g_{k}\right\} _{k\geq1}$ and $\left\{ \alpha_{k}\right\} _{k\geq1}$ be one choice of $g$s and $\alpha$s whose linear combinations converge in norm to $f$; let $\left\{ g_{k}^{\prime}\right\} _{k\geq1}$ and $\left\{ \alpha_{k}^{\prime}\right\} _{k\geq1}$ be another choice of $g$s and $\alpha$s satisfying the same. Then: \begin{align} 0 & \overset{K}{=}f\left(\mathbf{z}\right)-f\left(\mathbf{z}\right)\\ & \overset{K}{=}\lim_{N\rightarrow\infty}\left(\sum_{k=1}^{N}\left(\alpha_{k}g_{k}\left(\mathbf{z}\right)-\alpha_{k}^{\prime}g_{k}^{\prime}\left(\mathbf{z}\right)\right)\right)\\ & \overset{K}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}\lim_{N\rightarrow\infty}\left(\sum_{k=1}^{N}\left(\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)-\alpha_{k}^{\prime}c_{\mathbf{n}}\left(g_{k}^{\prime}\right)\right)\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right] \end{align} Since $0$ is an elementary function, by \textbf{Proposition \ref{prop:vdP series of an elementary function},} it is uniquely represented by the van der Put series whose coefficients are all $0$. This forces: \begin{equation} \lim_{N\rightarrow\infty}\sum_{k=1}^{N}\alpha_{k}c_{\mathbf{n}}\left(g_{k}\right)\overset{K}{=}\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\alpha_{k}^{\prime}c_{\mathbf{n}}\left(g_{k}^{\prime}\right) \end{equation} which shows that $c_{\mathbf{n}}\left(f\right)$ is, indeed, unique. Q.E.D. \begin{defn} \label{def:MD S_p}We write $F\left(\mathbb{Z}_{p}^{r},K\right)$\nomenclature{$F\left(\mathbb{Z}_{p}^{r},K\right)$}{$K$-valued functions on $\mathbb{Z}_{p}^{r}$}, to denote the space of all $K$-valued functions on $\mathbb{Z}_{p}^{r}$. We then define the linear operator $S_{p}:F\left(\mathbb{Z}_{p}^{r},K\right)\rightarrow\textrm{vdP}\left(\mathbb{Z}_{p}^{r},K\right)$ by: \begin{equation} S_{p}\left\{ f\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right],\textrm{ }\forall f\in F\left(\mathbb{Z}_{p}^{r},K\right)\label{eq:MD Definition of S_p of f} \end{equation} We also define partial sum operators: $S_{p:N}:F\left(\mathbb{Z}_{p}^{r},K\right)\rightarrow C\left(\mathbb{Z}_{p}^{r},K\right)$ by: \begin{equation} S_{p:N}\left\{ f\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right],\textrm{ }\forall f\in F\left(\mathbb{Z}_{p}^{r},K\right)\label{eq:MD Definition of S_p N of f} \end{equation} Recall, the sum here is taken over all $\mathbf{n}\in\mathbb{N}_{0}^{r}$ so that, for each $m\in\left\{ 1,\ldots,r\right\} $, $0\leq n_{m}\leq p^{N}-1$. \end{defn} \begin{prop} We can extend $S_{p}$ from $F\left(\mathbb{Z}_{p}^{r},K\right)$ to $B\left(\mathbb{Z}_{p}^{r},K\right)$ by taking limits in $F\left(\mathbb{Z}_{p}^{r},K\right)$ with respect to $B\left(\mathbb{Z}_{p}^{r},K\right)$'s norm. We can then compute $c_{\mathbf{n}}\left(f\right)$ for any $f\in B\left(\mathbb{Z}_{p}^{r},K\right)$ by using the construction given in \textbf{\emph{Lemma \ref{lem:van der Put series for an arbitrary K-valued function on Z_P}}}, albeit with elementary functions in $B\left(\mathbb{Z}_{p}^{r},K\right)$, rather than in $C\left(\mathbb{Z}_{p}^{r},K\right)$. \end{prop} Proof: Essentially the same as the proof of \textbf{Lemma \ref{lem:van der Put series for an arbitrary K-valued function on Z_P}}. Q.E.D. \begin{lem}[\textbf{Multi-Dimensional van der Put Identities}] \label{lem:MD vdP identities}\ \vphantom{} I. For any $f:\mathbb{Z}_{p}^{r}\rightarrow K$ and any $N\in\mathbb{N}_{0}$: \begin{equation} S_{p:N}\left\{ f\right\} \left(\mathbf{z}\right)=\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\overset{K}{=}f\left(\left[\mathbf{z}\right]_{p^{N}}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD truncated vdP identity} \end{equation} \vphantom{} II. For any $f\in B\left(\mathbb{Z}_{p}^{r},K\right)$: \begin{equation} S_{p}\left\{ f\right\} \left(\mathbf{z}\right)=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(f\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\overset{K}{=}\lim_{k\rightarrow\infty}f\left(\left[\mathbf{z}\right]_{p^{k}}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD vdP identity} \end{equation} \end{lem} Proof: I. Let $f:\mathbb{Z}_{p}^{r}\rightarrow K$ be elementary, with $f=\prod_{m=1}^{r}f_{m}$. Then, by the one-dimensional truncated van der Put identity (\ref{eq:truncated van der Put identity}): \begin{align} f\left(\left[\mathbf{z}\right]_{p^{N}}\right) & =\prod_{m=1}^{r}f_{m}\left(\left[\mathfrak{z}_{m}\right]_{p^{N}}\right)\\ & =\prod_{m=1}^{r}S_{p:N}\left\{ f_{m}\right\} \left(\mathfrak{z}_{m}\right)\\ \left(\textrm{Computation from \textbf{Prop. \ref{prop:vdP series of an elementary function}})}\right); & =S_{p:N}\left\{ \prod_{m=1}^{r}f_{m}\right\} \left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)\\ & =S_{p:N}\left\{ f\right\} \left(\mathbf{z}\right) \end{align} Using the linearity of $S_{p:N}$ then gives the desired result. \vphantom{} II. Take limits with (I). Q.E.D. \begin{thm}[\textbf{Multi-Dimensional van der Put Basis Theorem}] \label{thm:MD vdP basis theorem}$\mathcal{B}_{p}^{\otimes r}$ is a basis for $C\left(\mathbb{Z}_{p}^{r},K\right)$. In particular, for any $f:\mathbb{Z}_{p}^{r}\rightarrow K$, the following are equivalent: \vphantom{} I. $f$ is continuous. \vphantom{} II. $\lim_{\left\Vert \mathbf{n}\right\Vert _{\infty}\rightarrow\infty}\left\|c_{\mathbf{n}}\left(f\right)\right\|_{K}=0$. \vphantom{} III. $\lim_{N\rightarrow\infty}\left\Vert f-S_{p:N}\left\{ f\right\} \right\Vert _{p,K}=0$ \end{thm} Proof: Formally equivalent to its one-dimensional analogue\textemdash \textbf{Theorem \ref{thm:vdP basis theorem}}. Q.E.D. \begin{cor} If $K$ is non-archimedean, then the Banach space $C\left(\mathbb{Z}_{p}^{r},K\right)$ is isometrically isomorphic to $c_{0}\left(\mathbb{N}_{0}^{r},K\right)$ (the space of sequences $\left\{ c_{\mathbf{n}}\right\} _{\mathbf{n}\in\mathbb{N}_{0}^{r}}$ in $K$ that converge to $0$ in $K$ as $\left\Vert \mathbf{n}\right\Vert _{\infty}\rightarrow\infty$). \end{cor} Proof: Use the tensor product along with the isometric isomorphism $C\left(\mathbb{Z}_{p},K\right)\cong c_{0}\left(K\right)$ proven in \textbf{Theorem \ref{thm:C(Z_p,K) is iso to c_0 K}} (page \pageref{thm:C(Z_p,K) is iso to c_0 K}). Q.E.D. \begin{rem} Before we move on to the next subsection, note that everything we have done so far extends in the obvious way to vector\emph{-} or matrix-valued\emph{ }functions on $\mathbb{Z}_{p}^{r}$ by working component-wise. To that end, for $\mathbf{F}:\mathbb{Z}_{p}^{r}\rightarrow K^{\rho,c}$ given by $\mathbf{F}\left(\mathbf{z}\right)=\left\{ F_{j,k}\right\} _{j,k}$, \nomenclature{$c_{\mathbf{n}}\left(\mathbf{F}\right)$}{$\mathbf{n}$th van der Put coefficient of $\mathbf{F}$} we write: \begin{equation} c_{\mathbf{n}}\left(\mathbf{F}\right)\overset{\textrm{def}}{=}\left\{ c_{\mathbf{n}}\left(F_{j,k}\right)\right\} _{j,k} \end{equation} \begin{equation} S_{p:N}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\left\{ S_{p:N}\left\{ F_{j,k}\right\} \left(\mathbf{z}\right)\right\} _{j,k} \end{equation} \end{rem} \subsection{\label{subsec:5.3.2. Interpolation-Revisited}Interpolation Revisited} We begin with rising sequences and rising-continuous functions. \begin{defn} A sequence $\left\{ \mathbf{z}_{n}\right\} _{n\geq0}$ in $\mathbb{Z}_{p}^{r}$ (where $\mathbf{z}_{n}=\left(\mathfrak{z}_{n,1},\ldots,\mathfrak{z}_{n,r}\right)$) is said to be \textbf{($p$-adically) rising }if there exists an $\ell\in\left\{ 1,\ldots,r\right\} $ so that the number of non-zero $p$-adic digits in $\mathfrak{z}_{n,\ell}$ tends to $\infty$ as $n\rightarrow\infty$. \end{defn} \begin{defn} \label{def:MD rising-continuity}A function $\chi:\mathbb{Z}_{p}^{r}\rightarrow K^{d}$ is said to be \textbf{($\left(p,K\right)$-adically)} \textbf{rising-continuous}\index{rising-continuous!left(p,Kright)-adically@$\left(p,K\right)$-adically}\textbf{ }\index{rising-continuous!function}whenever: \begin{equation} \lim_{n\rightarrow\infty}\chi\left(\left[\mathbf{z}\right]_{p^{n}}\right)\overset{K^{d}}{=}\chi\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Definition of a rising-continuous function} \end{equation} where the convergence is point-wise. We write \nomenclature{$\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$}{set of $K^{d}$-valued rising-continuous functions on $\mathbb{Z}_{p}^{r}$ }$\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ to denote the $K$-linear space of all rising-continuous functions $\mathbb{Z}_{p}^{r}\rightarrow K^{d}$. \end{defn} \begin{prop} \label{prop:MD vdP criterion for rising continuity}Let $\chi\in B\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ be any function. Then, $S_{p}\left\{ \chi\right\} $ (the van der Put series\index{van der Put!series} of $\chi$) converges at $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ if and only if: \begin{equation} \lim_{k\rightarrow\infty}c_{\left[\mathbf{z}\right]_{p^{k}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=k\right]\overset{\mathbb{C}_{q}^{d}}{=}\mathbf{0}\label{eq:MD vdP criterion for rising-continuity} \end{equation} where $c_{\left[\mathbf{z}\right]_{p^{k}}}\left(\chi\right)$ is the $\left[\mathbf{z}\right]_{p^{k}}=\left(\left[\mathfrak{z}_{1}\right]_{p^{k}},\ldots,\left[\mathfrak{z}_{r}\right]_{p^{k}}\right)$ th van der Put coefficient of $\chi$. \end{prop} Proof: We start by writing: \begin{align} S_{p}\left\{ \chi\right\} \left(\mathbf{z}\right) & \overset{\mathbb{C}_{q}^{d}}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(\chi\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right]\\ & =c_{\mathbf{0}}\left(\chi\right)+\sum_{k=1}^{\infty}\sum_{\mathbf{n}:\lambda_{p}\left(\mathbf{n}\right)=k}c_{\mathbf{n}}\left(\chi\right)\left[\mathbf{z}\overset{p^{k}}{\equiv}\mathbf{n}\right] \end{align} For each $\mathbf{n}$, observe that if $\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=k$, then: \[ \mathbf{n}\overset{\textrm{def}}{=}\left[\mathbf{z}\right]_{p^{k}}=\left(\left[\mathfrak{z}_{1}\right]_{p^{k}},\ldots,\left[\mathfrak{z}_{r}\right]_{p^{k}}\right) \] is the unique element of $\mathbb{N}_{0}^{r}$ satisfying both $\lambda_{p}\left(\mathbf{n}\right)=k$ and $\mathbf{z}\overset{p^{k}}{\equiv}\mathbf{n}$. Hence: \begin{align} S_{p}\left\{ \chi\right\} \left(\mathbf{z}\right) & \overset{\mathbb{C}_{q}^{d}}{=}c_{\mathbf{0}}\left(\chi\right)+\sum_{k=1}^{\infty}c_{\left[\mathbf{z}\right]_{p^{k}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=k\right] \end{align} The ultrametric properties of $\mathbb{C}_{q}^{d}$ tell us that the $q$-adic convergence of this series at any given $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ is equivalent to: \begin{equation} \lim_{k\rightarrow\infty}c_{\left[\mathbf{z}\right]_{p^{k}}}\left(\chi\right)\left[\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=k\right]\overset{\mathbb{C}_{q}^{d}}{=}\mathbf{0} \end{equation} Q.E.D. \begin{thm} The operator $S_{p}$ which sends a function to its formal van der Put series is a isomorphism of $K$-linear spaces. This isomorphism map $\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ onto the subspace of $\textrm{vdP}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ consisting of all van der Put series which converge at every $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. Additionally, for every $\chi\in\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$: \vphantom{} I. $\chi=S_{p}\left\{ \chi\right\} $; \vphantom{} II. $\chi$ is uniquely represented by its van der Put series: \begin{equation} \chi\left(\mathbf{z}\right)\overset{K^{d}}{=}\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(\chi\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right],\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Chi vdP series} \end{equation} where the convergence is point-wise. \end{thm} Proof: Let $\chi\in\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ be arbitrary. By the truncated van der Put identity (\ref{eq:MD truncated vdP identity}), $\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right)=S_{p:N}\left\{ \chi\right\} \left(\mathbf{z}\right)$. Here, the rising-continuity of $\chi$ guarantees the point-wise convergence of $S_{p:N}\left\{ \chi\right\} \left(\mathbf{z}\right)$ in $K^{d}$ to $\chi\left(\mathbf{z}\right)$ as $N\rightarrow\infty$. By \textbf{Proposition \ref{prop:MD vdP criterion for rising continuity}}, this implies the van der Put coefficients of $\chi$ satisfy (\ref{eq:MD vdP criterion for rising-continuity}) for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. Consequently, the van der Put series $S_{p}\left\{ \chi\right\} \left(\mathbf{z}\right)$ converges at every $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, where it is equal to $\chi\left(\mathbf{z}\right)$. This proves (I). As for (II), the uniqueness specified therein is equivalent to demonstrating that $S_{p}$ is an isomorphism in the manner described above. We do this below: \begin{itemize} \item (Surjectivity) Let $V\left(\mathbf{z}\right)\in\textrm{vdP}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ be any formal van der Put series which converges $q$-adically at every $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. Letting: \begin{equation} \chi\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}S_{p:N}\left\{ V\right\} \left(\mathbf{z}\right) \end{equation} we have $V\left(\mathbf{m}\right)=\chi\left(\mathbf{m}\right)$ for all $\mathbf{m}\in\mathbb{N}_{0}^{r}$, and hence, $V\left(\mathbf{z}\right)=S_{p}\left\{ \chi\right\} $. Thus, $S_{p:N}\left\{ \chi\right\} \left(\mathbf{z}\right)=\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right)$. Since $\chi\left(\mathbf{z}\right)$ is defined by $\lim_{N\rightarrow\infty}S_{p:N}\left\{ V\right\} \left(\mathbf{z}\right)$, this gives: \begin{equation} \chi\left(\mathbf{z}\right)=\lim_{N\rightarrow\infty}S_{p:N}\left\{ V\right\} \left(\mathbf{z}\right)=\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right) \end{equation} which establishes the rising-continuity of $\chi$. This proves $V=S_{p}\left\{ \chi\right\} $, and thus, that $S_{p}$ is surjective. \item (Injectivity) Let $\chi_{1},\chi_{2}\in\tilde{C}\left(\mathbb{Z}_{p}^{r},K^{d}\right)$ and suppose $S_{p}\left\{ \chi_{1}\right\} =S_{p}\left\{ \chi_{2}\right\} $. Then, by (I): \[ \chi_{1}\left(\mathbf{z}\right)\overset{\textrm{(I)}}{=}S_{p}\left\{ \chi_{1}\right\} \left(\mathbf{z}\right)=S_{p}\left\{ \chi_{2}\right\} \left(\mathbf{z}\right)\overset{\textrm{(I)}}{=}\chi_{2}\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \] This proves $\chi_{1}=\chi_{2}$, which establishes $S_{p}$'s injectivity. \end{itemize} Thus, $S_{p}$ is an isomorphism. Q.E.D. \begin{defn} Let $\mathbb{F}$ be $\mathbb{Q}$ or a field extension thereof, and let $\chi:\mathbb{N}_{0}^{r}\rightarrow\mathbb{F}^{d}$ be a function. We say $\chi$ has (or ``admits'') a \textbf{$\left(p,q\right)$-adic rising-continuation} \textbf{(to $K^{d}$)} whenever\index{rising-continuation} there is a metrically complete $q$-adic field extension $K$ of $\mathbb{F}$ and a rising-continuous function $\chi^{\prime}:\mathbb{Z}_{p}^{r}\rightarrow K^{d}$ so that $\chi^{\prime}\left(\mathbf{n}\right)=\chi\left(\mathbf{n}\right)$ for all $\mathbf{n}\in\mathbb{N}_{0}^{r}$. We call any $\chi^{\prime}$ satisfying this property a \textbf{($\left(p,q\right)$-adic)}\emph{ }\textbf{rising-continuation }of $\chi$ (to $K^{d}$). \end{defn} \begin{prop} \label{prop:Uniqueness of rising-continuation, MD}Let $\chi:\mathbb{N}_{0}^{r}\rightarrow\mathbb{F}^{d}$ be a function admitting a $\left(p,q\right)$-adic rising-continuation to $K^{d}$. Then: \vphantom{} I. The rising-continuation of $\chi$ is unique. Consequently, we write $\chi^{\prime}$ to denote the rising continuation of $\chi$. \vphantom{} II. \begin{equation} \chi^{\prime}\left(\mathbf{z}\right)\overset{K^{d}}{=}\lim_{k\rightarrow\infty}\chi\left(\left[\mathbf{z}\right]_{p^{k}}\right)=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{r}}c_{\mathbf{n}}\left(\chi\right)\left[\mathbf{z}\overset{p^{\lambda_{p}\left(\mathbf{n}\right)}}{\equiv}\mathbf{n}\right],\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Rising continuation limit formula} \end{equation} \end{prop} Proof: I. Suppose $\chi$ admits two (possibly distinct) rising-continuations, $\chi^{\prime}$ and $\chi^{\prime\prime}$. To see that $\chi^{\prime}$ and $\chi^{\prime\prime}$ must be the same, we note that since the restrictions of both $\chi^{\prime}$ and $\chi^{\prime\prime}$ to $\mathbb{N}_{0}^{r}$ are, by definition, equal to $\chi$, these restrictions must be equal to one another. So, let $\mathbf{z}$ be an arbitrary element of $\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. Then, we note that at least one entry of $\mathbf{z}$ (say, the $\ell$th entry) necessarily has infinitely many non-zero $p$-adic digits. Consequently, $\left\{ \left[\mathbf{z}\right]_{p^{k}}\right\} _{k\geq1}$ is a rising sequence of non-negative integer $r$-tuples converging to $\mathbf{z}$. As such, by the rising-continuity of $\chi^{\prime}$ and $\chi^{\prime\prime}$: \begin{equation} \lim_{k\rightarrow\infty}\chi^{\prime}\left(\left[\mathbf{z}\right]_{p^{k}}\right)\overset{K^{d}}{=}\chi^{\prime}\left(\mathbf{z}\right) \end{equation} \begin{equation} \lim_{k\rightarrow\infty}\chi^{\prime\prime}\left(\left[\mathbf{z}\right]_{p^{k}}\right)\overset{K^{d}}{=}\chi^{\prime\prime}\left(\mathbf{z}\right) \end{equation} Because the $\left[\mathbf{z}\right]_{p^{k}}$s are $r$-tuples of integers, we can then write: \begin{equation} \chi^{\prime}\left(\mathbf{z}\right)=\lim_{k\rightarrow\infty}\chi^{\prime}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=\lim_{k\rightarrow\infty}\chi\left(\left[\mathbf{z}\right]_{p^{k}}\right)=\lim_{k\rightarrow\infty}\chi^{\prime\prime}\left(\left[\mathbf{z}\right]_{p^{k}}\right)=\chi^{\prime\prime}\left(\mathbf{z}\right) \end{equation} Since $\mathbf{z}$ was arbitrary, we conclude $\chi^{\prime}\left(\mathbf{z}\right)=\chi^{\prime\prime}\left(\mathbf{z}\right)$ for all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. Thus, $\chi^{\prime}$ and $\chi^{\prime\prime}$ are equal to one another on both $\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$ and $\mathbb{N}_{0}^{r}$, which is all of $\mathbb{Z}_{p}^{r}$. So, $\chi^{\prime}$ and $\chi^{\prime\prime}$ are, in fact, the same function. This proves the uniqueness of $\chi$'s rising-continuation. \vphantom{} II. As a rising-continuous function, $\chi^{\prime}\left(\mathbf{z}\right)$ is uniquely determined by its values on $\mathbb{N}_{0}^{r}$. Since $\chi^{\prime}\mid_{\mathbb{N}_{0}^{r}}=\chi$, $\chi^{\prime}$ and $\chi$ then have the same van der Put coefficients. Applying the van der Put identity (\ref{eq:van der Put identity}) then yields (\ref{eq:MD Rising continuation limit formula}). Q.E.D. \begin{thm} Let $\mathbb{F}$ be $\mathbb{Q}$ or a field extension thereof, let $\chi:\mathbb{N}_{0}^{r}\rightarrow\mathbb{F}^{d}$ be a function, and let $K$ be a metrically complete $q$-adic field extension of $\mathbb{F}$, where $q$ is prime. Then, the following are equivalent: \vphantom{} I. $\chi$ admits a $\left(p,q\right)$-adic rising-continuation to $K^{d}$. \vphantom{} II. For each $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, $\chi\left(\left[\mathbf{z}\right]_{p^{n}}\right)$ converges to a limit in $K^{d}$ as $n\rightarrow\infty$. \end{thm} Proof: i. Suppose (I) holds. Then, by \textbf{Proposition \ref{prop:Uniqueness of rising-continuation, MD}} we have that (\ref{eq:MD Rising continuation limit formula}) holds, which shows that (II) is true. \vphantom{} ii. Conversely, suppose (II) holds. Then, by the van der Put identity (\ref{eq:van der Put identity}), $S_{p}\left\{ \chi\right\} $ is a rising-continuous function whose restriction to $\mathbb{N}_{0}^{r}$ is equal to $\chi$. So, $S_{p}\left\{ \chi\right\} $ is the rising-continuation of $\chi$, and hence, $\chi$ is rising-continuable. This shows the equivalence of (II) and (I). Q.E.D. \begin{rem} Like in the one-dimensional case, we will now identify a function $\chi:\mathbb{N}_{0}^{r}\rightarrow K^{d}$ with its rising-continuation $\chi^{\prime}$. \end{rem} \vphantom{} Finally, we have the analogues of the functional equation results from the end of Subsection \ref{subsec:3.2.1 -adic-Interpolation-of}. \begin{thm} \label{thm:generic Chi_type functional equations, MD}Let $H$ be a semi-basic $p$-smooth $d$-dimensional depth-$r$ Hydra map which fixes $\mathbf{0}$, and consider the system of functional equations\index{functional equation!rising-continuability}: \begin{equation} \mathbf{f}\left(p\mathbf{n}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\mathbf{f}\left(\mathbf{n}\right)+\mathbf{c}_{\mathbf{j}},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\label{eq:MD Generic H-type functional equations} \end{equation} where $\left\{ \mathbf{c}_{\mathbf{j}}\right\} _{\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}}$ are vector constants in $\overline{\mathbb{Q}}^{d}$. Additionally, suppose $\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)$ is invertible. Then: \vphantom{} I. There is a unique function $\chi:\mathbb{N}_{0}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ such that $\mathbf{f}=\chi$ is a solution of \emph{(\ref{eq:MD Generic H-type functional equations})}. \vphantom{} II. The solution $\chi$ \emph{(\ref{eq:MD Generic H-type functional equations})} is rising-continuable to a function $\chi:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q_{H}}^{d}$ which satisfies: \begin{equation} \chi\left(p\mathbf{z}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\chi\left(\mathbf{n}\right)+\mathbf{c}_{\mathbf{j}},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Rising-continuation Generic H-type functional equations} \end{equation} \vphantom{} III. The function $\chi:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q_{H}}^{d}$ described in \emph{(III)} is the unique rising-continuous function $\mathbf{f}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q_{H}}^{d}$ satisfying: \begin{equation} \mathbf{f}\left(p\mathbf{z}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\mathbf{f}\left(\mathbf{n}\right)+\mathbf{c}_{\mathbf{j}},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} \end{thm} Proof: I. Let $\mathbf{f}:\mathbb{N}_{0}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ be any solution of (\ref{eq:MD Generic H-type functional equations}). Setting $\mathbf{n}=\mathbf{j}=\mathbf{0}$ yields: \begin{align} \mathbf{f}\left(\mathbf{0}\right) & =H^{\prime}\left(\mathbf{0}\right)\mathbf{f}\left(\mathbf{0}\right)+\mathbf{c}_{\mathbf{0}}\\ & \Updownarrow\\ \mathbf{f}\left(\mathbf{0}\right) & =\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\mathbf{c}_{\mathbf{0}} \end{align} which is well-defined since $\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)$ was given to be invertible. Then, we have that: \begin{equation} \mathbf{f}\left(\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\mathbf{f}\left(\mathbf{0}\right)+\mathbf{c}_{\mathbf{j}},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r} \end{equation} and, more generally: \begin{equation} \mathbf{f}\left(\mathbf{m}\right)=H_{\left[\mathbf{m}\right]_{p}}^{\prime}\left(\mathbf{0}\right)\mathbf{f}\left(\frac{\mathbf{m}-\left[\mathbf{m}\right]_{p}}{p}\right)+\mathbf{c}_{\left[\mathbf{m}\right]_{p}},\textrm{ }\forall\mathbf{m}\in\mathbb{N}_{0}^{r}\label{eq:MD f,m, digit shifting} \end{equation} Next, because the map $\mathbf{m}\mapsto\frac{\mathbf{m}-\left[\mathbf{m}\right]_{p}}{p}$ sends the non-negative integer tuple: \begin{equation} \mathbf{m}=\left(\sum_{k=0}^{\lambda_{p}\left(m_{1}\right)-1}m_{1,k}p^{k},\ldots,\sum_{k=0}^{\lambda_{p}\left(m_{r}\right)-1}m_{r,k}p^{k}\right) \end{equation} to the integer tuple: \begin{equation} \mathbf{m}=\left(\sum_{k=0}^{\lambda_{p}\left(m_{1}\right)-2}m_{1,k}p^{k},\ldots,\sum_{k=0}^{\lambda_{p}\left(m_{r}\right)-2}m_{r,k}p^{k}\right) \end{equation} it follows that $\mathbf{m}\mapsto\frac{\mathbf{m}-\left[\mathbf{m}\right]_{p}}{p}$ eventually iterates every $\mathbf{m}\in\mathbb{N}_{0}^{r}$ to $\mathbf{0}$. So, (\ref{eq:MD f,m, digit shifting}) implies that, for every $\mathbf{m}\in\mathbb{N}_{0}^{r}$, $\mathbf{f}\left(\mathbf{m}\right)$ is entirely determined by $\mathbf{f}\left(\mathbf{0}\right)$ and the $\mathbf{c}_{\mathbf{j}}$s. Since $\mathbf{f}\left(\mathbf{0}\right)$ is uniquely determined by $\mathbf{c}_{\mathbf{0}}$ and $H$, (\ref{eq:MD Generic H-type functional equations}) then possesses exactly one solution, which we shall denote by $\chi$. \vphantom{} II. Rehashing the argument for existence of the numen $\chi_{H}$ when $H$ is semi-basic and fixes $\mathbf{0}$, because $H$ is here semi-basic, any $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$ will have at least one entry infinitely many non-zero $p$-adic digits, and hence, the product of $H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)$ taken over all the digits of such a $\mathbf{z}$ will converge $q_{H}$-adically to zero. Then, using (\ref{eq:MD Generic H-type functional equations}), we see that $\chi\left(\mathbf{z}\right)$ will be a sum of the form: \begin{equation} \beta_{0}+\alpha_{1}\beta_{1}+\alpha_{1}\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{2}\alpha_{3}\beta_{3}+\cdots \end{equation} where, for each $n$, $\beta_{n}$ is one of the $\mathbf{c}_{\mathbf{j}}$s and $\alpha_{n}$ is $H_{\mathbf{j}_{n}}^{\prime}\left(\mathbf{0}\right)$, where $\mathbf{j}_{n}$ is the $r$-tuple of the $n$th $p$-adic digit of $\mathfrak{z}_{1}$ through the $n$th $p$-adic digit of $\mathfrak{z}_{r}$, where $\mathbf{z}=\left(\mathfrak{z}_{1},\ldots,\mathfrak{z}_{r}\right)$. Consequently, this series converges in $\mathbb{C}_{q_{H}}^{d}$ for all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$, seeing as $\left\Vert H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\right\Vert _{q_{H}}<1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $ because $H$ is semi-basic. This guarantees the rising-continuability of $\chi$. \vphantom{} III. Because $\chi$ admits a rising continuation, this continuation is given at every $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ by the van der Put series $S_{p}\left\{ \chi\right\} \left(\mathbf{z}\right)$ . As such: \begin{equation} \chi\left(\mathbf{z}\right)\overset{\mathbb{C}_{q_{H}}^{d}}{=}S_{p}\left\{ \chi\right\} \left(\mathbf{z}\right)\overset{\mathbb{C}_{q_{H}}^{d}}{=}\lim_{N\rightarrow\infty}\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right) \end{equation} Because $\chi$ satisfies (\ref{eq:MD Generic H-type functional equations}), we can write: \begin{equation} \chi\left(p\left[\mathbf{z}\right]_{p^{N}}+\mathbf{j}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right)+\mathbf{c}_{\mathbf{j}} \end{equation} for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$, and all $N\geq0$. So, for $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$, letting $N\rightarrow\infty$ yields: \begin{equation} \chi\left(p\mathbf{z}+\mathbf{j}\right)\overset{\mathbb{C}_{q_{H}}^{d}}{=}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\chi\left(\mathbf{z}\right)+\mathbf{c}_{\mathbf{j}},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime} \end{equation} Note that these identities hold automatically for $\mathbf{z}\in\mathbb{N}_{0}^{r}$, because those are the cases governed by (\ref{eq:MD Generic H-type functional equations}). The uniqueness of $\chi$ as a solution of (\ref{eq:MD Rising-continuation Generic H-type functional equations}) occurs because any $\mathbf{f}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q_{H}}^{d}$ satisfying (\ref{eq:MD Rising-continuation Generic H-type functional equations}) has a restriction to $\mathbb{N}_{0}^{r}$ which satisfies (\ref{eq:MD Generic H-type functional equations}), thereby forcing $\mathbf{f}=\chi$. Q.E.D. \vphantom{}A slightly more general version of this type of argument is as follows: \begin{lem} Fix an integer $q\geq2$, let $K$ be a metrically complete $q$-adic field, and let $\Phi_{\mathbf{j}}:\mathbb{Z}_{p}^{r}\times K^{d}\rightarrow K^{d}$ be continuous for $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. Suppose $\chi:\mathbb{N}_{0}^{r}\rightarrow K^{d}$ is $\left(p,K\right)$-adically rising-continuable. If $\chi$ satisfies the functional equations: \begin{equation} \chi\left(p\mathbf{n}+\mathbf{j}\right)=\Phi_{\mathbf{j}}\left(\mathbf{n},\chi\left(\mathbf{n}\right)\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r} \end{equation} then the rising-continuation of $\chi$ satisfies: \begin{equation} \chi\left(p\mathbf{z}+\mathbf{j}\right)=\Phi_{\mathbf{j}}\left(\mathbf{z},\chi\left(\mathbf{z}\right)\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r} \end{equation} \end{lem} Proof: Let everything be as given. Since: \begin{equation} \chi\left(p\left[\mathbf{z}\right]_{p^{N}}+\mathbf{j}\right)=\Phi_{\mathbf{j}}\left(\left[\mathbf{z}\right]_{p^{N}},\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right)\right) \end{equation} holds true for all $\mathbf{z}\in\mathbb{Z}_{p}$ and all $N\geq0$, the rising-continuability of $\chi$ and the continuity of $\Phi_{\mathbf{j}}$ guarantee that: \begin{align} \chi\left(p\mathbf{z}+\mathbf{j}\right) & \overset{K^{d}}{=}\lim_{N\rightarrow\infty}\chi\left(p\left[\mathbf{z}\right]_{p^{N}}+\mathbf{j}\right)\\ & \overset{K^{d}}{=}\lim_{N\rightarrow\infty}\Phi_{\mathbf{j}}\left(\left[\mathbf{z}\right]_{p^{N}},\chi\left(\left[\mathbf{z}\right]_{p^{N}}\right)\right)\\ & \overset{K^{d}}{=}\Phi_{\mathbf{j}}\left(\mathbf{z},\chi\left(\mathbf{z}\right)\right) \end{align} as desired. Q.E.D. \vphantom{} \begin{rem} Like in Subsection \ref{subsec:5.3.2. Interpolation-Revisited}, all of the above holds when functions taking values in $\mathbb{C}_{q}^{d}$ (vectors) are replaced with functions taking values in $\mathbb{C}_{q}^{\rho,c}$ (matrices). \end{rem} \newpage{} \section{\label{sec:5.4. Quasi-Integrability-in-Multiple}Quasi-Integrability in Multiple Dimensions} I see little reason to cover the multi-dimensional analogue of the results on the Banach algebra of rising-continuous functions, seeing as we will not use them in our analysis of the multi-dimensional $\chi_{H}$. As such, we will instead proceed directly to the multi-dimensional $\left(p,q\right)$-adic Fourier transform, and from there to frames and quasi-integrability. \subsection{Multi-Dimensional $\left(p,q\right)$-adic Fourier Analysis and Thick Measures\label{subsec:5.4.1 Multi-Dimensional--adic-Fourier}} With regard to multi-dimensional Fourier analysis, I will be less pedantic than the reader has probably come to expect me to be. So long as the reader understands how to do $\left(p,q\right)$-adic Fourier analysis in one dimension, the formalism for Fourier analysis of scalar or vector valued functions over $\mathbb{R}^{n}$ will provide enough information for the reader to do multi-dimensional $\left(p,q\right)$-adic Fourier analysis\index{multi-dimensional!Fourier transform}, with the help of the notation provided below, of course. \begin{notation}[\textbf{Formalism for Multi-Dimensional $\left(p,q\right)$-adic Fourier Analysis}] \label{nota:third batch}\ \vphantom{} I. Given $\mathbf{x}\in\mathbb{Q}_{p}^{r}$, where $\mathbf{x}=\left(\mathfrak{x}_{1},\ldots,\mathfrak{x}_{r}\right)$, we write: \begin{equation} \left\{ \mathbf{x}\right\} _{p}\overset{\textrm{def}}{=}\sum_{m=1}^{r}\left\{ \mathfrak{x}_{m}\right\} _{p}\label{eq:Definition of P-adic fractional part (MD)} \end{equation} Consequently, for $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$ and $\mathbf{z}\in\mathbb{Z}_{p}^{r}$: \begin{equation} \left\{ \mathbf{t}\cdot\mathbf{z}\right\} _{p}=\left\{ \mathbf{t}\mathbf{z}\right\} _{p}=\sum_{m=1}^{r}\left\{ t_{m}\mathfrak{z}_{m}\right\} _{p}\label{eq:P-adic fractional part of bold t times bold z (MD)} \end{equation} \vphantom{} II. Let $f\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$ be elementary, with $f\left(\mathbf{z}\right)=\prod_{m=1}^{r}f_{m}\left(\mathfrak{z}_{m}\right)$. Then, the Fourier transform of $f$ is defined by: \begin{equation} \hat{f}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}^{r}}f\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z}=\prod_{m=1}^{r}\int_{\mathbb{Z}_{p}^{r}}f\left(\mathfrak{z}_{m}\right)e^{-2\pi i\left\{ t_{m}\mathfrak{z}_{m}\right\} _{p}}d\mathfrak{z}_{m}\label{eq:Definition of Fourier transform of an elementary scalar function on Z_P} \end{equation} We then define $\hat{f}\left(\mathbf{t}\right)$ for any $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$ by extending via linearity. \vphantom{} III. Let $\mathbf{f}\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}^{d}\right)$, with $\mathbf{f}\left(\mathfrak{z}\right)=\left(f_{1}\left(\mathfrak{z}\right),\ldots,f_{d}\left(\mathfrak{z}\right)\right)$ for $f_{1},\ldots,f_{d}\in C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$. Then, we define $\hat{\mathbf{f}}\left(t\right)$ as the $d$-tuple of the Fourier transforms of the $f_{m}$s: \begin{equation} \hat{\mathbf{f}}\left(t\right)\overset{\textrm{def}}{=}\left(\int_{\mathbb{Z}_{p}}f_{1}\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z},\ldots,\int_{\mathbb{Z}_{p}}f_{d}\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z}\right)\label{eq:Definition of the Fourier transform of a vector-valued function on Z_p} \end{equation} \vphantom{} IV. Let $\mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, with $\mathbf{f}\left(\mathbf{z}\right)=\left(f_{1}\left(\mathbf{z}\right),\ldots,f_{d}\left(\mathbf{z}\right)\right)$ with $f_{1},\ldots,f_{d}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$. Then, we define: \begin{equation} \hat{\mathbf{f}}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\left(\int_{\mathbb{Z}_{p}^{r}}f_{1}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z},\ldots,\int_{\mathbb{Z}_{p}^{r}}f_{d}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z}\right)\label{eq:Definition of the Fourier transform of a vector-valued function on Z_P} \end{equation} This will also be written as: \begin{equation} \hat{\mathbf{f}}\left(\mathbf{t}\right)=\int_{\mathbb{Z}_{p}^{r}}\mathbf{f}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z}\label{eq:Non-tuple formula for the Fourier transform of a vector-valued function on Z_P} \end{equation} \vphantom{} V. Most generally, let $\mathbf{F}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{\rho,c}\right)$, with $\mathbf{F}\left(\mathbf{z}\right)=\left\{ F_{j,k}\left(\mathbf{z}\right)\right\} _{1\leq j\leq\rho,1\leq k\leq c}$ where each $F_{j,k}$ is in $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$. Then: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\left\{ \int_{\mathbb{Z}_{p}^{r}}F_{j,k}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z}\right\} _{1\leq j\leq\rho,1\leq k\leq c}\label{eq:Definition of the Fourier transform of a matrix-valued function on Z_P} \end{equation} This will also be written as: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)=\int_{\mathbb{Z}_{p}^{r}}\mathbf{F}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z}\label{eq:Non-tuple formula for the Fourier transform of a matrix-valued function on Z_P} \end{equation} \end{notation} Like in the one-dimensional case, the most important identity for us is the Fourier series for an indicator function for the clopen set $\mathbf{n}+p^{N}\mathbb{Z}_{p}^{r}$: \begin{prop}[\textbf{Fourier Series for the Multi-Dimensional Indicator Function}] \label{prop:Multi-Dimensional indicator function Fourier Series}For $r$-tuples $\mathbf{z}$ and $\mathbf{n}$: \begin{equation} \left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right]=\frac{1}{p^{Nr}}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{n}\right)\right\} _{p}}\label{eq:MD Fourier series for indicator function} \end{equation} \end{prop} Proof: \begin{align} \left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right] & =\prod_{\ell=1}^{r}\left[\mathfrak{z}_{\ell}\overset{p^{N}}{\equiv}n_{\ell}\right]\\ & =\prod_{\ell=1}^{r}\frac{1}{p^{N}}\sum_{k_{\ell}=0}^{p^{N}-1}e^{2\pi i\left\{ \frac{k_{\ell}}{p^{N}}\left(\mathfrak{z}_{\ell}-n_{\ell}\right)\right\} _{p}}\\ & =\frac{1}{p^{Nr}}\prod_{\ell=1}^{r}\sum_{\left\|t_{\ell}\right\|_{p}\leq p^{N}}e^{2\pi i\left\{ t_{\ell}\left(\mathfrak{z}_{\ell}-n_{\ell}\right)\right\} _{p}}\\ & =\frac{1}{p^{Nr}}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{n}\right)\right\} _{p}} \end{align} Q.E.D. \vphantom{} Just to be extra cautious, let us prove that the following formal summation identities \emph{do }in fact hold true: \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}f\left(\mathbf{t}\right)=f\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}f\left(\mathbf{t}\right)\label{eq:First MD level set summation identity} \end{equation} \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}f\left(\mathbf{t}\right)=\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}}f\left(\mathbf{t}\right)-\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}f\left(\mathbf{t}\right)\label{eq:Second MD level set summation identity} \end{equation} so as to justify all this hullabaloo over notation. \begin{prop} For all $n\geq1$: \begin{equation} \left\{ \left\Vert \mathbf{t}\right\Vert _{p}=p^{n}\right\} =\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}\right\} \backslash\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}\right\} \label{eq:MD Level set decomposition} \end{equation} \end{prop} Proof: I. If $\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}$, then $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}$. However, there is at least one $m\in\left\{ 1,\ldots,r\right\} $ so that $-v_{p}\left(t_{m}\right)=n>n-1$. As such, $\left\Vert \mathbf{t}\right\Vert _{p}>p^{n-1}$, and so: \begin{equation} \left\{ \left\Vert \mathbf{t}\right\Vert _{p}=p^{n}\right\} \subseteq\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}\right\} \backslash\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}\right\} \end{equation} \vphantom{} II. If $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}$ and $\left\Vert \mathbf{t}\right\Vert _{p}\nleq p^{n-1}$, then $-v_{p}\left(t_{m}\right)\leq n$ occurs for all $m\in\left\{ 1,\ldots,r\right\} $. Moreover, there is at least one such $m$ for which $-v_{p}\left(t_{m}\right)=n$. This means $\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}$, and so: \begin{equation} \left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}\right\} \backslash\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}\right\} \subseteq\left\{ \left\Vert \mathbf{t}\right\Vert _{p}=p^{n}\right\} \end{equation} Hence, the two sets are equal. Q.E.D. \vphantom{}Consequently, we have: \begin{prop} \label{prop:MD ramanujan sum} \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}=p^{rn}\left[\mathbf{z}\overset{p^{n}}{\equiv}\mathbf{0}\right]-p^{r\left(n-1\right)}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{0}\right],\textrm{ }\forall n\geq1,\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Ramanujan sum} \end{equation} \end{prop} Proof: Use (\ref{eq:Second MD level set summation identity}) and \textbf{Proposition \ref{prop:Multi-Dimensional indicator function Fourier Series}}. Q.E.D. \begin{thm} Let $\mathbf{F}\left(\mathbf{t}\right)=\bigodot_{m=1}^{r}\mathbf{F}_{m}\left(t_{m}\right)$ be an elementary function in $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{\rho,c}\right)$. Then: \begin{equation} \hat{\mathbf{F}}\left(\mathbf{t}\right)=\hat{\mathbf{F}}\left(t_{1},\ldots,t_{r}\right)=\bigodot_{m=1}^{r}\hat{\mathbf{F}}_{m}\left(t_{m}\right)\label{eq:Fourier transform mixes with tensoring} \end{equation} \end{thm} Proof: The complex exponentials turn multiplication into addition. Q.E.D. \begin{thm} The $\mathbb{C}_{q}$-valued functions $\left\{ \mathbf{z}\mapsto e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\right\} _{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}$ form a basis of $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}\right)$. More generally, the vectors: \[ \left[\begin{array}{c} e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ 0\\ \vdots\\ 0 \end{array}\right],\left[\begin{array}{c} 0\\ e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ \vdots\\ 0 \end{array}\right],\ldots,\left[\begin{array}{c} 0\\ 0\\ \vdots\\ e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{array}\right] \] form a basis of $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. \end{thm} Proof: $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ is the closure in $\left\Vert \cdot\right\Vert _{p,q}$-norm of the set of linear combinations of elementary continuous functions $\mathbf{f}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$. By the properties of the tensor product and the Fourier transform, every such elementary function $\mathbf{f}\left(\mathbf{z}\right)=\prod_{m=1}^{r}\mathbf{f}_{m}\left(\mathfrak{z}_{m}\right)$ is then uniquely expressible as: \begin{align} \mathbf{f}\left(\mathbf{z}\right) & =\prod_{m=1}^{r}\sum_{t_{m}\in\hat{\mathbb{Z}}_{p}}\hat{\mathbf{f}}\left(t_{m}\right)e^{2\pi i\left\{ t_{m}\mathfrak{z}_{m}\right\} _{p}}\\ & =\sum_{t_{1}\in\hat{\mathbb{Z}}_{p}}\cdots\sum_{t_{r}\in\hat{\mathbb{Z}}_{p}}\left(\prod_{m=1}^{r}\hat{\mathbf{f}}\left(t_{m}\right)\right)e^{2\pi i\sum_{\ell=1}^{r}\left\{ t_{\ell}\mathfrak{z}_{\ell}\right\} _{p}}\\ & =\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}}\hat{\mathbf{f}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{align} The theorem follows by the density of these elementary functions in $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Q.E.D. \begin{thm} The Fourier transform $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{\rho,c}\right)\rightarrow c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{\rho,c}\right)$ is an isometric isomorphism of non-archimedean Banach spaces. \end{thm} Proof: Use linearity and tensors to extend the one-dimensional case (\textbf{Corollary \ref{cor:pq adic Fourier transform is an isometric isomorphism}}). Q.E.D. \begin{defn} Given a function $\hat{\mathbf{F}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\rho,c}$ and an $N\geq0$, we write $\tilde{\mathbf{F}}_{N}$ to denote the function \nomenclature{$\tilde{\mathbf{F}}_{N}$}{$N$th partial Fourier series generated by $\hat{\mathbf{F}}\left(\mathbf{t}\right)$}$\tilde{\mathbf{F}}_{N}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\rho,c}$ defined by: \begin{equation} \tilde{\mathbf{F}}_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{F}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\label{eq:Definition of bold F_N twiddle} \end{equation} We call $\tilde{\mathbf{F}}_{N}$ the \textbf{$N$th partial Fourier series }generated by $\hat{\mathbf{F}}$. \end{defn} \vphantom{} At this point, were we in the one-dimensional case, we would introduce measures. However, the situation is not as simple in the multi-dimensional case. Let us review our set-up so far. $\chi_{H}$\textemdash the function we want to study\textemdash goes from $\mathbb{Z}_{p}^{r}$ to $\mathbb{Z}_{q}^{d}$. As such, our setting will be functions $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$. The goal is to interpret integration of $\chi_{H}$ as the action of \emph{some sort }of linear map acting on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. In the one-dimensional case, this linear map was a continuous, \emph{scalar-valued} linear operator $C\left(\mathbb{Z}_{p},K\right)\rightarrow K$ (a.k.a. a \emph{measure}), which acted upon continuous functions by way of the formula: \begin{equation} f\in C\left(\mathbb{Z}_{p},K\right)\mapsto\sum_{t\in\hat{\mathbb{Z}}_{p}}\hat{f}\left(-t\right)\hat{\mu}\left(t\right)\in K \end{equation} Because $\chi_{H}$ is vector-valued, in order to replicate this Parseval-Plancherel identity, we will need to find a way for the vector $\hat{\chi}_{H}\left(\mathbf{t}\right)$ to act on $\hat{\mathbf{f}}\left(\mathbf{t}\right)\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. However, because we want to use this action to integrate $\chi_{H}\left(\mathbf{z}\right)$ entry-by-entry, the \emph{result }of the action cannot be scalar: it needs to be a vector. Thus, our linear operator will not send $\hat{\mathbf{f}}\left(\mathbf{t}\right)$ to a scalar, but to a \emph{vector}. So, instead of linear functionals, we will work with linear operators on finite-dimensional vector spaces. This is where a subtlety emerges: \emph{we will need to distinguish between different realizations of linear operators}.\emph{ }For example, we could have a linear operator which sends $\mathbf{f}\left(\mathbf{z}\right)\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ to the sum: \begin{equation} \sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{f}}\left(-\mathbf{t}\right)\hat{\mathbf{m}}\left(\mathbf{t}\right) \end{equation} where $\hat{\mathbf{m}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ is a column-vector valued function on $\hat{\mathbb{Z}}_{p}^{r}$ and the juxtaposition $\hat{\mathbf{f}}\left(-\mathbf{t}\right)\hat{\mathbf{m}}\left(\mathbf{t}\right)$ denotes the entry-wise product of $\hat{\mathbf{f}}\left(-\mathbf{t}\right)$ and $\mathbf{\hat{m}}\left(\mathbf{t}\right)$. Alternatively, we could have a linear operator which sends $\mathbf{f}\left(\mathbf{z}\right)$ to: \begin{equation} \sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right) \end{equation} where $\hat{\mathbf{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ is a $d\times d$-matrix valued function and the juxtaposition $\hat{\mathbf{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)$ denotes ordinary left-multiplication of the vector $\hat{\mathbf{f}}\left(-\mathbf{t}\right)$ by the matrix $\hat{\mathbf{M}}\left(\mathbf{t}\right)$. Note that $\hat{\mathbf{f}}\left(-\mathbf{t}\right)\hat{\mathbf{m}}\left(\mathbf{t}\right)$ could also be written as left-multiplication of $\hat{\mathbf{f}}\left(-\mathbf{t}\right)$ by the $d\times d$ diagonal matrix whose diagonal entries are precisely the entries of the vector $\hat{\mathbf{m}}\left(-\mathbf{t}\right)$. However, the object we would obtain by summing the Fourier series generated by said matrix would be a $d\times d$ matrix rather than a $d\times1$ vector. With all of this in mind, instead of forcing the reader to keep different context-based identifications in mind, I feel it will be simpler to distinguish between those linear operators which are represented by entry-wise multiplication against a vector and those which are represented by left-multiplication by a matrix. These considerations motivate our next definition, along with those elaborated in the examples that follow. \begin{defn}[\textbf{Thick measures}] \index{measure!thick}\index{thick measure}A\textbf{ $\left(p,q\right)$-adic} \textbf{thick measure of depth $r$ }is a continuous linear operator $\mathcal{M}:C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\rightarrow\mathbb{C}_{q}^{d}$. I write\footnote{In this way, we can reserve the notation $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)^{\prime}$ to denote the dual space of $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$\textemdash continuous \emph{scalar}-valued linear operators on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$.} \nomenclature{$C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)^{}$}{Thick measures on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$}$C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)^{}$ to denote the space of vector measures on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Note that this is a $\mathbb{C}_{q}$-Banach space under operator norm. Finally, we will write $\mathcal{M}\left(\mathbf{f}\right)$ to denote the $d\times1$ vector obtained by applying $\mathcal{M}$ to $\mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. \end{defn} \begin{rem} I use the asterisk $$ for the space of thick measures rather than a $\prime$, because, for the sake of consistency, the space denoted $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)^{\prime}$ should be the dual of $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, which is the space of continuous \emph{scalar-valued }linear maps on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, as opposed to $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)^{}$, which is the space of continuous \emph{vector-valued }linear maps on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. \end{rem} \begin{example}[\textbf{Thick measures of matrix type}] \label{exa:thick matrix measures}Consider a bounded $d\times d$-matrix-valued function $\hat{\mathbf{M}}\in B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d,d}\right)$. Then, the map: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)\in\mathbb{C}_{q}^{d}\label{eq:Action of a matrix-valued multiplier} \end{equation} defines a continuous linear operator $\mathcal{M}:C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\rightarrow\mathbb{C}_{q}^{d}$, and hence, a $\mathbb{C}_{q}^{d}$-valued thick measure. I call such an $\mathcal{M}$ a \textbf{thick measure of} \textbf{matrix type}\index{thick measure!matrix type}.\textbf{ }Consequently, we define $\hat{\mathbf{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ as the \textbf{Fourier-Stieltjes transform} of $\mathcal{M}$, and write: \begin{equation} \hat{\mathcal{M}}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{\mathbf{M}}\left(\mathbf{t}\right)\label{eq:Fourier-Stieltjes transform of a matrix type thick measure} \end{equation} \index{Fourier-Stieltjes transform} \end{example} \begin{example}[\textbf{Thick measures of vector type}] \label{exa:thick vector measures}Consider a bounded $d\times1$-vector-valued function $\hat{\mathbf{m}}\in B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, as well as the map: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{m}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)\in\mathbb{C}_{q}^{d}\label{eq:Action of a vector-valued multiplier} \end{equation} Here $\hat{\mathbf{m}}\left(\mathbf{t}\right)=\left(\hat{m}_{1}\left(\mathbf{t}\right),\ldots,\hat{m}_{d}\left(\mathbf{t}\right)\right)$ and we write: \begin{equation} \hat{\mathbf{m}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)=\left(\begin{array}{c} \hat{m}_{1}\left(\mathbf{t}\right)\hat{f}_{1}\left(-\mathbf{t}\right)\\ \vdots\\ \hat{m}_{d}\left(\mathbf{t}\right)\hat{f}_{d}\left(-\mathbf{t}\right) \end{array}\right)\in\mathbb{C}_{q}^{d} \end{equation} to denote the entry-wise product of the column vectors $\hat{\mathbf{m}}\left(\mathbf{t}\right)$ and $\hat{\mathbf{f}}\left(-\mathbf{t}\right)$. As written, (\ref{eq:Action of a vector-valued multiplier}) defines a a continuous linear operator $\mathcal{M}:C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\rightarrow\mathbb{C}_{q}^{d}$, and hence, a $\mathbb{C}_{q}^{d}$-valued Banach measure. I call such an $\mathcal{M}$ a \textbf{thick measure of vector type}\index{thick measure!vector type}.\textbf{ }The \textbf{Fourier-Stieltjes transform} of $\mathcal{M}$ is given by: \begin{equation} \hat{\mathcal{M}}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{\mathbf{m}}\left(\mathbf{t}\right)\label{eq:Fourier transform of a vector-type multiplier} \end{equation} \end{example} \begin{defn} Given a thick measure $\mathcal{M}:C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\rightarrow\mathbb{C}_{q}^{d}$ of matrix or vector type, the \textbf{Fourier-Stieltjes transform }of $\mathcal{M}$ is the function $\hat{\mathcal{M}}$ defined in the above two examples, respectively. We say a thick measure $\mathcal{M}$ is \textbf{algebraic }whenever the entries of $\hat{\mathcal{M}}\left(\mathbf{t}\right)$ are elements of $\overline{\mathbb{Q}}$ for all $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$. \end{defn} \begin{defn} We write $M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ and $M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d,d}\right)$ \nomenclature{$M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$}{thick vector-type measures} \nomenclature{$M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{\left(d,d\right)}\right)$}{thick matrix-type measures} to denote the sets (in fact, $\mathbb{C}_{q}$-linear spaces) of vector- and matrix-type thick measures on $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, respectively. By definition, $M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ and $M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d,d}\right)$ are the operators induced by Fourier-Stieltjes transforms in $B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ and $B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d,d}\right)$, respectively, in the manner described in \textbf{Examples \ref{exa:thick vector measures}} and \textbf{\ref{exa:thick matrix measures}}. \end{defn} \begin{defn}[\textbf{$N$th partial kernel of a thick measure}] Consider a thick $\left(p,q\right)$-adic measure $\mathcal{M}$ with Fourier-Stieltjes transform $\hat{\mathcal{M}}\left(\mathbf{t}\right)$. For each $N\geq0$, we write \nomenclature{$\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)$}{$N$th partial kernel of $\mathcal{M}$}$\tilde{\mathcal{M}}_{N}$ to denote the \textbf{$N$th partial kernel }of\index{thick measure!partial kernel} $\mathcal{M}$, which is defined as the continuous (in fact, locally constant)\emph{ }function: \begin{equation} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:Concrete Definition of the Nth Partial Kernel of a Fourier Multiplier} \end{equation} This function is vector-valued when $\mathcal{M}$ is vector-type and is matrix-valued when $\mathcal{M}$ is matrix-type. \end{defn} \subsection{\label{subsec:5.4.2 Multi-Dimensional-Frames}Multi-Dimensional Frames} \begin{rem} Like with \ref{subsec:3.3.3 Frames}, the present subsection is going to be heavy on abstract definitions. Practical-minded readers can safely skip this section so long as the keep the following concepts in mind: Consider a function $\mathbf{F}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\rho,c}$. A sequence of functions $\left\{ \mathbf{F}_{n}\right\} _{n\geq1}$ on $\mathbb{Z}_{p}^{r}$ is said to \textbf{converge} \textbf{to $f$ with respect to the standard $\left(p,q\right)$-adic frame }if\index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic}: \vphantom{} I. For all $n$, $\mathbf{F}_{n}\left(\mathbf{z}\right)\in\overline{\mathbb{Q}}^{\rho,c}$ for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. \vphantom{} II. For all $\mathbf{z}\in\mathbb{N}_{0}^{r}$, $\mathbf{F}\left(\mathbf{z}\right)\in\overline{\mathbb{Q}}^{\rho,c}$ and $\lim_{n\rightarrow\infty}\mathbf{F}_{n}\left(\mathbf{z}\right)\overset{\mathbb{C}^{\rho,c}}{=}\mathbf{F}\left(\mathbf{z}\right)$ (meaning the convergence is in the topology of $\mathbb{C}$). \vphantom{} III. For all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$, $\mathbf{F}\left(\mathbf{z}\right)\in\mathbb{C}_{q}^{\rho,c}$ and $\lim_{n\rightarrow\infty}\mathbf{F}_{n}\left(\mathbf{z}\right)\overset{\mathbb{C}_{q}^{\rho,c}}{=}\mathbf{F}\left(\mathbf{z}\right)$ (meaning the convergence is in the topology of $\mathbb{C}_{q}$). \vphantom{} We write $\mathcal{F}_{p,q}$ to denote the standard $\left(p,q\right)$-adic frame. In particular, we write: \vphantom{} i. $\mathcal{F}_{p,q}^{d}$, to denote the case where $\mathbf{F}\left(\mathbf{z}\right)$ is a $d\times1$-column-vector-valued function (the $d$-dimensional standard frame\index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic!$d$-dimensional}); \vphantom{} ii. $\mathcal{F}_{p,q}^{d,d}$ to denote the case where $\mathbf{F}\left(\mathbf{z}\right)$ is a $d\times d$-matrix-valued function (the $d,d$-dimensional standard frame \index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic!left(d,dright) -dimensional@$d,d$-dimensional}). \vphantom{} We then write $\lim_{n\rightarrow\infty}\mathbf{f}_{n}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q}^{d}}{=}\mathbf{f}\left(\mathbf{z}\right)$ and $\lim_{n\rightarrow\infty}\mathbf{F}_{n}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q}^{d,d}}{=}\mathbf{F}\left(\mathbf{z}\right)$ to denote convergence with respect to the $d$-dimensional and $d,d$-dimensional standard frames, respectively. \end{rem} \vphantom{} HERE BEGINS THE DISCUSSION OF MULTI-DIMENSIONAL FRAMES \vphantom{} The set-up for the multi-dimensional case will be a tad bit more involved than the one-dimensional case. The most significant difference is that because our functions can now take vector or matrix values, instead of assigning topological fields to each point of the domain, we will assign topological \emph{vector spaces}, all of which contain a finite-dimensional $\overline{\mathbb{Q}}$-vector-space of an appropriate dimension. These will come in two flavors: those with the discrete topology, and those which are Banach spaces with respect to a norm induced by an absolute value. \begin{defn}[\textbf{Multi-Dimensional Frames}] \label{def:MD frame}Fix an integer $d\geq1$ . A $d$-dimensional $p$-adic \index{frame!$p$-adic}\index{frame}(or just ``frame'', for short) $\mathcal{F}$ of depth $r$ is the following collection of data: \vphantom{} I. A set $U_{\mathcal{F}}\subseteq\mathbb{Z}_{p}^{r}$, called the \textbf{domain }of $\mathcal{F}$. \vphantom{} II. For each $\mathbf{z}\in U_{\mathcal{F}}$, a topological field $K_{\mathbf{z}}$ containing $\overline{\mathbb{Q}}$ and a $d$-dimensional topological vector space $V_{\mathbf{z}}$ over $K_{\mathbf{z}}$. \nomenclature{$K_{\mathbf{z}}^{d}$}{ }\nomenclature{$V_{\mathbf{z}}$}{ } We allow for $K_{\mathbf{z}}=\overline{\mathbb{Q}}$. In particular, for any $\mathbf{z}\in U_{\mathcal{F}}$, we require the topology of $K_{\mathbf{z}}$ to be either the discrete topology\emph{ or} the topology induced by an absolute value,\nomenclature{$K_{\mathbf{z}}$}{ }, denoted \nomenclature{$\left\|\cdot\right\|_{K_{\mathbf{z}}}$}{ }$\left\|\cdot\right\|_{K_{\mathbf{z}}}$. For any $\mathbf{z}$ for which $K_{\mathbf{z}}$ is given the discrete topology, we endow $V_{\mathbf{z}}$ with the discrete topology as well. If $K_{\mathbf{z}}$ is given the topology induced by an absolute value, we require $\left(K_{\mathbf{z}},\left\|\cdot\right\|_{K_{\mathbf{z}}}\right)$ to be a complete metric space, and then endow $V_{\mathbf{z}}$ with the topology induced by the norm $\left\Vert \cdot\right\Vert _{K_{\mathbf{z}}}$\nomenclature{$\left\Vert \cdot\right\Vert _{K_{\mathbf{z}}}$}{ }, defined here as outputting the maximum of the $K_{\mathbf{z}}$-absolute values of the entries of a given element of $V_{\mathbf{z}}$. This makes $V_{\mathbf{z}}$ into a finite-dimensional Banach space. \end{defn} \begin{defn} I adopt the convention of writing $\mathcal{F}^{d}$ when I mean that the $V_{\mathbf{z}}$s are spaces of $d\times1$ column vectors. In that case, I write $V_{\mathbf{z}}$ as $K_{\mathbf{z}}^{d}$. On the other hand, when working with $\rho\times c$ matrices (where $d=\rho c$) instead of $d\times1$ column vectors, I write $\mathcal{F}^{\rho,c}$ instead of $\mathcal{F}$, and say that this frame has dimension $\rho,c$. For this case, I write $K_{\mathbf{z}}^{\rho,c}$ instead of $V_{\mathbf{z}}$. \end{defn} \vphantom{} The two most important objects associated with a given frame are its image and the space of compatible functions, defined below. \begin{defn}[\textbf{Image and Compatible Functions}] \label{def:MD Frame terminology}Let $\mathcal{F}$ be a $p$-adic frame. \vphantom{} I. The \textbf{image }of $\mathcal{F}$, denoted $I\left(\mathcal{F}\right)$, is the \emph{set }defined by: \begin{equation} I\left(\mathcal{F}\right)\overset{\textrm{def}}{=}\bigcup_{\mathbf{z}\in U_{\mathcal{F}}}V_{\mathbf{z}}\label{eq:MD The image of a frame} \end{equation} \index{frame!image} \vphantom{} II. A function $\chi:U_{\mathcal{F}}\rightarrow I\left(\mathcal{F}\right)$ is said to be \textbf{$\mathcal{F}$-compatible} / \textbf{compatible }(\textbf{with $\mathcal{F}$}) whenever \index{mathcal{F}-@$\mathcal{F}$-!compatible}\index{frame!compatible functions} $\chi\left(\mathfrak{z}\right)\in V_{\mathbf{z}}$ for all $\mathbf{z}\in U_{\mathcal{F}}$. I write $C\left(\mathcal{F}\right)$ to denote the set of all $\mathcal{F}$-compatible functions. \end{defn} \begin{rem} Note that $C\left(\mathcal{F}\right)$ is a vector space over $\overline{\mathbb{Q}}$ with respect to point-wise addition of functions and scalar multiplication. \end{rem} \vphantom{} In addition to this, we also have the following bits of terminology: \begin{defn}[\textbf{Frame Terminology}] Let $\mathcal{F}$ be a $p$-adic frame of dimension $d$ and depth $r$. \vphantom{} I. I call $\mathbb{Z}_{p}^{r}\backslash U_{\mathcal{F}}$ the \textbf{set of singularities }of the frame. I say that $\mathcal{F}$ is \textbf{non-singular }whenever $U_{\mathcal{F}}=\mathbb{Z}_{p}^{r}$.\index{frame!non-singular} \vphantom{} II. Given a topology $\tau$ (so, $\tau$ could be $\textrm{dis}$, $\infty$, $\textrm{non}$, or $p$), I writ $U_{\tau}\left(\mathcal{F}\right)$ to denote the set of $\mathbf{z}\in U_{\mathcal{F}}$ so that $K_{\mathbf{z}}$ has been equipped with $\tau$. I call the $U_{\tau}\left(\mathcal{F}\right)$\textbf{ $\tau$-convergence domain }or \textbf{domain of $\tau$ convergence }of $\mathcal{F}$. In this way, we can speak of the \textbf{discrete convergence domain}, the \textbf{archimedean convergence domain}, the \textbf{non-archimedean convergence domain}, and the \textbf{$p$-adic convergence domain} of a given frame $\mathcal{F}$. \vphantom{} IV. I say $\mathcal{F}$ is \textbf{simple} \index{frame!simple}if either $U_{\textrm{non}}\left(\mathcal{F}\right)$ is empty or there exists a single metrically complete non-archimedean field extension $K$ of $\overline{\mathbb{Q}}$ so that $K=K_{\mathbf{z}}$ for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. (That is to say, for a simple frame, we use at most one non-archimedean topology.) \vphantom{} V. I say $\mathcal{F}$ is \textbf{proper }\index{frame!proper}proper whenever $K_{\mathbf{z}}$ is not a $p$-adic field for any $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. \end{defn} \vphantom{} UNLESS STATED OTHERWISE, ALL FRAMES ARE ASSUMED TO BE PROPER. \vphantom{} Like in the one-dimensional case, only one $p$-adic frame will ever be used in this dissertation: the standard one. \begin{defn} The \textbf{standard $d$-dimensional ($\left(p,q\right)$-adic) frame}\index{frame!standard left(p,qright)-adic@standard $\left(p,q\right)$-adic}, denoted $\mathcal{F}_{p,q}$, is the frame for which the topology of $\mathbb{C}$ is associated to $\mathbb{N}_{0}^{d}$ and the topology of $\mathbb{C}_{q}$ is associated to $\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. In particular, I write $\mathcal{F}_{p,q}^{d}$ to denote the case where the topological vector spaces are spaces of $d\times1$ column vectors; I write $\mathcal{F}_{p,q}^{\rho,c}$ to denote the case where the topological vector spaces are spaces of $\rho\times c$ column vectors. \end{defn} \vphantom{} Next we have the all-important notion of $\mathcal{F}$-convergence. \begin{defn}[\textbf{$\mathcal{F}$-convergence}] Given a frame a $\mathcal{F}$ and a function $\chi\in C\left(\mathcal{F}\right)$, we say a sequence $\left\{ \chi_{n}\right\} _{n\geq1}$ in $C\left(\mathcal{F}\right)$ \textbf{converges to $\chi$ over $\mathcal{F}$ }(or \textbf{is $\mathcal{F}$-convergent to $\chi$}) whenever, for each $\mathbf{z}\in U_{\mathcal{F}}$, we have: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\left(\mathbf{z}\right)\overset{V_{\mathbf{z}}}{=}\chi\left(\mathbf{z}\right) \end{equation} Note that if $K_{\mathbf{z}}$ has the discrete topology, the limit implies that $\chi_{n}\left(\mathbf{z}\right)=\chi\left(\mathbf{z}\right)$ for all sufficiently large $n$.\index{mathcal{F}-@$\mathcal{F}$-!convergence}\index{frame!convergence} On the other hand, if $K_{\mathbf{z}}$ has the topology of a valued field, we then require: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert \chi\left(\mathbf{z}\right)-\chi_{n}\left(\mathbf{z}\right)\right\Vert _{K_{\mathbf{z}}}\overset{\mathbb{R}}{=}0 \end{equation} I call $\chi$ the \index{mathcal{F}-@$\mathcal{F}$-!limit}\textbf{$\mathcal{F}$-limit of the $\chi_{n}$s}. \index{frame!limit}More generally, I say $\left\{ \chi_{n}\right\} _{n\geq1}$ is \textbf{$\mathcal{F}$-convergent / converges over $\mathcal{F}$} whenever there is a function $\chi\in C\left(\mathcal{F}\right)$ so that the $\chi_{n}$s are $\mathcal{F}$-convergent to $\chi$. In symbols, I denote $\mathcal{F}$-convergence by: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\left(\mathbf{z}\right)\overset{\mathcal{F}}{=}\chi\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in U_{\mathcal{F}}\label{eq:MD Definition-in-symbols of F-convergence} \end{equation} or simply: \begin{equation} \lim_{n\rightarrow\infty}\chi_{n}\overset{\mathcal{F}}{=}\chi\label{eq:MD Simplified version of expressing f as the F-limit of f_ns} \end{equation} \end{defn} \begin{rem} In an abuse of notation, we will sometimes write $\mathcal{F}$ convergence as: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert \chi_{n}\left(\mathbf{z}\right)-\chi\left(\mathbf{z}\right)\right\Vert _{K_{\mathbf{z}}},\textrm{ }\forall\mathfrak{z}\in U_{\mathcal{F}} \end{equation} The abuse here is that the norm $\left\Vert \cdot\right\Vert _{K_{\mathbf{z}}}$ will not exist if $\mathbf{z}\in U_{\textrm{dis}}\left(\mathcal{F}\right)$. As such, for any $\mathbf{z}\in U_{\textrm{dis}}\left(\mathcal{F}\right)$, I define the expression: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert \chi_{n}\left(\mathbf{z}\right)-\chi\left(\mathbf{z}\right)\right\Vert _{K_{\mathbf{z}}}=0 \end{equation} as meaning that for all sufficiently large $n$, the equality $\chi_{n}\left(\mathbf{z}\right)=\chi_{n+1}\left(\mathbf{z}\right)$ holds in $K_{\mathbf{z}}^{d}$. \end{rem} \vphantom{} Now that we have the language of frames at our disposal, we can begin to define classes of $\left(p,q\right)$-adic measures which will be of interest to us. \begin{defn}[\textbf{Thick} \textbf{$\mathcal{F}$-measures}] Given a $d$-dimensional depth $r$ $p$-adic frame $\mathcal{F}$, let $V=\overline{\mathbb{Q}}^{d}$ if $\mathcal{F}=\mathcal{F}^{d}$ and let $V=\overline{\mathbb{Q}}^{\rho,c}$ if $\mathcal{F}=\mathcal{F}^{\rho,c}$. We write \nomenclature{$M\left(\mathcal{F}\right)$}{$\mathcal{F}$-measures}$M\left(\mathcal{F}\right)$ to denote the set of all functions $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow V$ so that $\hat{\mathcal{M}}\in B\left(\hat{\mathbb{Z}}_{p}^{r},V_{\mathbf{z}}\right)$ for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$; that is: \begin{equation} \left\Vert \hat{\mathcal{M}}\right\Vert _{p,K_{\mathbf{z}}}<\infty,\textrm{ }\forall\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)\label{eq:Definition of a thick F-measure} \end{equation} where: \[ \left\Vert \hat{\mathcal{M}}\right\Vert _{p,K_{\mathbf{z}}}=\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \hat{\mathcal{M}}\left(\mathbf{t}\right)\right\Vert _{K_{\mathbf{z}}} \] where $\left\Vert \cdot\right\Vert _{K_{\mathbf{z}}}$, recall, is the norm on $V_{\mathbf{z}}$, being the maximum of the $K_{\mathbf{z}}$-absolute-values of the entries of $\hat{\mathcal{M}}\left(\mathbf{t}\right)$. \vphantom{} Next observe that for every $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, the map: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},K_{\mathbf{z}}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)\in K_{\mathbf{z}}\label{eq:Definition of the action of a thick F-measure} \end{equation} then defines an element of $C\left(\mathbb{Z}_{p}^{r},K_{\mathbf{z}}^{d}\right)^{}$, the space of all continuous $K_{\mathbf{z}}^{d}$-valued linear maps on the space of continuous functions $\mathbf{f}:\mathbb{Z}_{p}^{r}\rightarrow K_{\mathbf{z}}^{d}$. As such, we will identify $M\left(\mathcal{F}\right)$ with elements of $\bigcap_{\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)}C\left(\mathbb{Z}_{p}^{r},K_{\mathbf{z}}\right)^{}$, and refer to elements of $M\left(\mathcal{F}\right)$ as \textbf{thick} \textbf{$\mathcal{F}$-measures}\index{mathcal{F}-@$\mathcal{F}$-!thick measure}\index{measure!thick}. Note that the specific type of multiplication signified by the juxtaposition of $\hat{\mathcal{M}}\left(\mathbf{t}\right)$ and $\hat{\mathbf{f}}\left(-\mathbf{t}\right)$ in (\ref{eq:Definition of the action of a thick F-measure}) depends on whether $\hat{\mathcal{M}}$ is of vector type or matrix type. We write $M\left(\mathcal{F}^{d}\right)$\nomenclature{$M\left(\mathcal{F}^{d}\right)$}{ } and $M\left(\mathcal{F}^{\rho,c}\right)$\nomenclature{$M\left(\mathcal{F}^{\rho,c}\right)$}{ } to denote the vector type and matrix type cases, respectively. Regardless, we then identify $\hat{\mathcal{M}}$ with the Fourier-Stieltjes transform of a thick measure $\mathcal{M}$. Thus, the statement ``$\mathcal{M}\in M\left(\mathcal{F}^{d}\right)$'' means that, for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, $\mathcal{M}$ is an element of $C\left(\mathbb{Z}_{p}^{r},K_{\mathbf{z}}\right)^{}$ whose action on a function is given by $\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)$, where $\hat{\mathcal{M}}\in B\left(\hat{\mathbb{Z}}_{p}^{r},K_{\mathbf{z}}^{d}\right)$. ``$\mathcal{M}\in M\left(\mathcal{F}^{\rho,c}\right)$'', meanwhile, means that, for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$, $\mathcal{M}$ is an element of $C\left(\mathbb{Z}_{p}^{r},K_{\mathbf{z}}\right)^{}$ whose action on a function is given by $\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)$, where $\hat{\mathcal{M}}\in B\left(\hat{\mathbb{Z}}_{p}^{r},K_{\mathbf{z}}^{d,d}\right)$. Both cases force $\left\Vert \hat{\mathcal{M}}\right\Vert _{p,K_{\mathbf{z}}}$ to be finite for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$. \end{defn} \begin{prop} Let $\hat{\mathcal{M}}\in M\left(\mathcal{F}\right)$. Then, the $N$th partial sum of the Fourier series generated by $\hat{\mathcal{M}}$: \begin{equation} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} is an element of $C\left(\mathcal{F}\right)$. \end{prop} Proof: A straight-forward multi-dimensional analogue of the one-dimensional case. Q.E.D. \begin{defn}[\textbf{$\mathcal{F}$-rising thick measure}] Given a $p$-adic frame $\mathcal{F}$ of dimension $d$ and depth $r$, we say $\mathcal{M}\in M\left(\mathcal{F}\right)$ \index{thick measure!mathcal{F}-rising@$\mathcal{F}$-rising} \textbf{rises over $\mathcal{F}$} / is \textbf{$\mathcal{F}$-rising} whenever the sequence $\left\{ \tilde{\mathcal{M}}_{N}\right\} _{N\geq0}$ is $\mathcal{F}$-convergent. That is: \begin{equation} \tilde{\mathcal{M}}\left(\mathbf{z}\right)\overset{\mathcal{F}}{=}\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in U_{\mathcal{F}} \end{equation} which, recall, means: \vphantom{} I. For every $\mathbf{z}\in U_{\textrm{dis}}\left(\mathcal{F}\right)$, $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)=\tilde{\mathcal{M}}_{N+1}\left(\mathbf{z}\right)$ for all sufficiently large $N$. \vphantom{} II. For every $\mathbf{z}\in U_{\textrm{arch}}\left(\mathcal{F}\right)\cup U_{\textrm{non}}\left(\mathcal{F}\right)$, $\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)$ converges to a limit in the topology of $V_{\mathbf{z}}$. \vphantom{} We write $\mathcal{M}_{\textrm{rise}}\left(\mathcal{F}\right)$ to denote the space of thick $\mathcal{F}$-rising measures, with $\mathcal{M}_{\textrm{rise}}\left(\mathcal{F}^{d}\right)$ and $\mathcal{M}_{\textrm{rise}}\left(\mathcal{F}^{\rho,c}\right)$ being used to remind the reader that the thick measures in question are of vector type and matrix type, respectively. Additionally, we call a thick measure \textbf{rising-continuous }whenever it rises over the standard frame. \index{thick measure!rising-continuous} \end{defn} \begin{defn}[\textbf{Kernel of an $\mathcal{F}$-rising measure}] Given a $p$-adic frame $\mathcal{F}$ of dimension $d$ and depth $r$ along with an $\mathcal{F}$-rising thick measure $\mathcal{M}$, the\index{mathcal{F}-@$\mathcal{F}$-!kernel} \textbf{kernel }of $\mathcal{M}$\textemdash denoted\index{thick measure!mathcal{F}-kernel@$\mathcal{F}$-kernel} \nomenclature{$\tilde{\mathcal{M}}\left(\mathbf{z}\right)$}{kernel of $\mathcal{M}$}\textemdash is the vector- or matrix-valued function on $U_{\mathcal{F}}$ defined by the $\mathcal{F}$-limit of the $\tilde{\mathcal{M}}_{N}$s: \begin{equation} \tilde{\mathcal{M}}\left(\mathbf{z}\right)\overset{\mathcal{F}}{=}\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in U_{\mathcal{F}}\label{eq:Definition of M-twiddle / the kernel of M} \end{equation} We write $\left(\tilde{\mathcal{M}}\right)_{N}\left(\mathbf{z}\right)$\nomenclature{$\left(\tilde{\mathcal{M}}\right)_{N}\left(\mathbf{z}\right)$}{$N$th truncation of the kernel of $\mathcal{M}$} to denote the $N$th truncation of the kernel of $\mathcal{M}$: \begin{equation} \left(\tilde{\mathcal{M}}\right)_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\tilde{\mathcal{M}}\left(\mathbf{n}\right)\left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right]\label{eq:Definition of the Nth truncation of the full kernel of a rising multiplier} \end{equation} More generally, we write: \begin{equation} \left(\tilde{\mathcal{M}}_{M}\right)_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\tilde{\mathcal{M}}_{M}\left(\mathbf{n}\right)\left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right]\label{eq:Nth truncation of the Mth partial Kernel of M} \end{equation} to denote the $N$th truncation of $\tilde{\mathcal{M}}_{M}$. \end{defn} \begin{rem} In this terminology, observe that for $\mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, we can express the image of $\mathbf{f}$ under $\mathcal{M}$ as: \begin{equation} \mathcal{M}\left(\mathbf{f}\right)\overset{\mathbb{C}_{q}^{d}}{=}\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{p}^{r}}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)\mathbf{f}\left(\mathbf{z}\right)d\mathbf{z}=\lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right) \end{equation} \end{rem} \begin{defn}[\textbf{$\mathcal{F}$-degenerate thick measure}] For a $p$-adic frame $\mathcal{F}$, a thick $\mathcal{F}$-rising measure \index{thick measure!mathcal{F}-degenerate@$\mathcal{F}$-degenerate}$\mathcal{M}$ is said to be \textbf{($\mathcal{F}$-)degenerate }whenever its kernel is identically zero. We write $M_{\textrm{dgen}}\left(\mathcal{F}\right)$ to denote the set of $\mathcal{F}$-degenerate thick measures. We write this as $M_{\textrm{dgen}}\left(\mathcal{F}^{d}\right)$ and $M_{\textrm{dgen}}\left(\mathcal{F}^{\rho,c}\right)$ when we wish to single out vector-type and matrix-type thick measures, respectively. \end{defn} \subsection{\label{subsec:5.4.2 Quasi-Integrability-Revisited}Quasi-Integrability and Thick Measures} \begin{rem} Like in \ref{subsec:3.3.5 Quasi-Integrability}, the purpose of Subsection \ref{subsec:5.4.3 -adic-Wiener-Tauberian} is to provide a solid foundation for understanding and discussing quasi-integrability in its multi-dimensional incarnation. As it regards Chapter \ref{chap:6 A-Study-of}'s analysis of multi-dimensional $\chi_{H}$, the main thing the reader needs to know is that I say a function $\chi:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ is \textbf{quasi-integrable }with\index{quasi-integrability} respect to the standard $\left(p,q\right)$-adic frame whenever there exists a function $\hat{\chi}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ so that: \begin{equation} \chi\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q}^{d}}{=}\lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\chi}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r} \end{equation} We call the function $\hat{\chi}$ a\emph{ }\textbf{Fourier transform }of $\chi$. Once again, this Fourier transform is only unique modulo the Fourier-Stieltjes transform of a thick measure of vector type which is degenerate with respect to the standard $\left(p,q\right)$-adic frame. Like in the one-dimensional case, given a Fourier transform $\hat{\chi}$ of a quasi-integrable function $\chi$, we write $\tilde{\chi}_{N}$ to denote the $N$th partial Fourier series generated by $\hat{\chi}$. \end{rem} \vphantom{} HERE BEGINS THE DISCUSSION OF MULTI-DIMENSIONAL QUASI-INTEGRABILITY \vphantom{} \begin{defn} Consider a $d$-dimensional $p$-adic frame $\mathcal{F}$ of depth $r$. \vphantom{} We say a function $\chi\in C\left(\mathcal{F}\right)$ is\textbf{ quasi-integrable} \textbf{with respect to $\mathcal{F}$} / $\mathcal{F}$\textbf{-quasi-integrable }whenever $\chi:U_{\mathcal{F}}\rightarrow I\left(\mathcal{F}\right)$ is the kernel of some $\mathcal{F}$-rising measure $\mathcal{M}\in M\left(\mathcal{F}^{d}\right)$ (that is, of \emph{vector }type). In other words: \vphantom{} I. There is a vector-type thick measure $\mathcal{M}$ with a Fourier-Stieltjes transform $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ which is bounded in $\left\Vert \cdot\right\Vert _{p,K_{\mathbf{z}}}$-norm for all $\mathbf{z}\in U_{\textrm{non}}\left(\mathcal{F}\right)$; \vphantom{} II. $\tilde{\mathcal{M}}_{N}$ $\mathcal{F}$-converges to $\chi$ as $N\rightarrow\infty$: \begin{equation} \lim_{N\rightarrow\infty}\left\Vert \chi\left(\mathbf{z}\right)-\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)\right\Vert _{K_{\mathbf{z}}}\overset{\mathbb{R}}{=}0,\textrm{ }\forall\mathbf{z}\in U_{\mathcal{F}}\label{eq:MD definition of quasi-integrability} \end{equation} \vphantom{} I call any $\mathcal{M}$ satisfying these properties an\textbf{ $\mathcal{F}$-quasi-integral}\index{mathcal{F}-@$\mathcal{F}$-!quasi-integral}. I write $\tilde{L}^{1}\left(\mathcal{F}\right)$ to denote the set of all $\mathcal{F}$-quasi-integrable functions. (Note that $\tilde{L}^{1}\left(\mathcal{F}\right)$ is then a vector space over $\overline{\mathbb{Q}}$.) We write $\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$ when we wish to remind our audience that the elements of $\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$ are $d\times1$-vector-valued functions, rather than scalar valued functions. When $\mathcal{F}$ is the standard $d$-dimensional depth $r$ $\left(p,q\right)$-adic frame, this definition becomes: \vphantom{} i. $\chi$ is a function from $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ so that $\chi\left(\mathbf{z}\right)\in\mathbb{C}^{d}$ for all $\mathbf{z}\in\mathbb{N}_{0}^{r}$ and $\chi\left(\mathbf{z}\right)\in\mathbb{C}_{q}^{d}$ for all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$; \vphantom{} ii. For each $\mathbf{z}\in\mathbb{N}_{0}^{r}$, as $N\rightarrow\infty$, $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)$ converges to $\chi\left(\mathbf{z}\right)$ in $\mathbb{C}^{d}$. \vphantom{} iii. For each $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$, as $N\rightarrow\infty$, $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)$ converges to $\chi\left(\mathbf{z}\right)$ in $\mathbb{C}_{q}^{d}$. \vphantom{} We then write \nomenclature{$\tilde{L}^{1}\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$}{ }$\tilde{L}^{1}\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ and \nomenclature{$\tilde{L}^{1}\left(\mathcal{F}_{p,q}^{d}\right)$}{ }$\tilde{L}^{1}\left(\mathcal{F}_{p,q}^{d}\right)$ to denote the space of all functions $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ which are quasi-integrable with respect to the standard frame. \end{defn} \vphantom{} Once again, we will end up with a generalization of the Fourier transform of a function $\chi:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ by treating $\chi$ as the kernel of some thick vector-type measure. \begin{defn} Let $\chi$ be $\mathcal{F}$-quasi-integrable. Then, \textbf{a} \textbf{Fourier Transform} of $\chi$ is a function $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ which is the Fourier-Stieltjes transform of some $\mathcal{F}$-quasi-integral $\mathcal{M}$ of $\chi$. We write these functions as $\hat{\chi}$ and write the associated thick measure as $\chi\left(\mathbf{z}\right)d\mathbf{z}$. Given $\mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, we then define: \begin{equation} \int_{\mathbb{Z}_{p}^{r}}\mathbf{f}\left(\mathbf{z}\right)\chi\left(\mathbf{z}\right)d\mathbf{z}\overset{\textrm{def}}{=}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{f}}\left(-\mathbf{t}\right)\hat{\chi}\left(\mathbf{t}\right)\label{eq:MD Definition of the integral of a quasi-integrable function against a continuous function} \end{equation} Note that because $\mathcal{M}$ is a thick measure of vector type, its Fourier-Stieltjes transform $\hat{\mathcal{M}}\left(\mathbf{t}\right)=\hat{\chi}\left(\mathbf{t}\right)$ is a $d\times1$ column vector with entries in $\overline{\mathbb{Q}}$. The above integral is then the same thing as the image of $\mathbf{f}$ under $\mathcal{M}$: \begin{equation} \mathcal{M}\left(\mathbf{f}\right)\overset{\mathbb{C}_{q}^{d}}{=}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{f}}\left(-\mathbf{t}\right)\hat{\mathcal{M}}\left(\mathbf{t}\right)\overset{\mathbb{C}_{q}^{d}}{=}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right) \end{equation} where the order of juxtaposition of the $d$-tuples $\hat{\mathbf{f}}$ and $\hat{\mathcal{M}}=\hat{\chi}$ is irrelevant because the juxtaposition signifies \emph{entry-wise} multiplication\textemdash a commutative operation. \end{defn} \begin{defn} Given a choice of Fourier transform $\hat{\chi}$ for $\chi\in\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$, we write: \begin{equation} \tilde{\chi}_{N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\chi}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Definition of Chi_N twiddle} \end{equation} to denote the \textbf{$N$th partial sum of the Fourier series generated by $\hat{\chi}$}. \end{defn} \vphantom{} Just like with the one-dimensional case, Fourier transforms of quasi-integrable functions are only unique modulo the Fourier-Stieltjes transform of a degenerate thick measure. \begin{prop} Let $\chi\in\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$. The quasi-integral of $\chi$ is unique modulo $M_{\textrm{dgen}}\left(\mathcal{F}^{d}\right)$. That is, if two thick measures $\mathcal{M}$ and $\mathcal{N}$ are both quasi-integrals of $\chi$, then the thick measure $\mathcal{M}-\mathcal{N}$ is $\mathcal{F}$-degenerate\index{thick measure!mathcal{F}-degenerate@$\mathcal{F}$-degenerate}\index{mathcal{F}-@$\mathcal{F}$-!degenerate}. Equivalently, we have an isomorphism of $\overline{\mathbb{Q}}$-linear spaces: \begin{equation} \tilde{L}^{1}\left(\mathcal{F}^{d}\right)\cong M_{\textrm{rise}}\left(\mathcal{F}^{d}\right)/M_{\textrm{dgen}}\left(\mathcal{F}^{d}\right)\label{eq:MD L1 twiddle isomorphism to M alg / M dgen} \end{equation} where the isomorphism associates a given $\chi\in\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$ to the equivalence class of $\mathcal{F}$-rising thick measures which are quasi-integrals of $\chi$. \end{prop} Proof: Let $\mathcal{F}$, $\chi$, $\mathcal{M}$, and $\mathcal{N}$ be as given. Then, by the definition of what it means to be a quasi-integral of $\chi$, for the thick vector-type measure $\mathcal{P}\overset{\textrm{def}}{=}\mathcal{M}-\mathcal{N}$, we have that $\tilde{\mathcal{P}}_{N}\left(\mathbf{z}\right)$ tends to $\mathbf{0}$ in $K_{\mathbf{z}}^{d}$ for all $\mathbf{z}\in U_{\mathcal{F}}$. Hence, $\mathcal{P}=\mathcal{M}-\mathcal{N}$ is indeed $\mathcal{F}$-degenerate. Q.E.D. \begin{prop} Let $\chi\in\tilde{L}^{1}\left(\mathcal{F}^{d}\right)$. Then, for any Fourier transform $\hat{\chi}$ of $\chi$, the difference $\chi_{N}-\tilde{\chi}_{N}$ (where $\chi_{N}$ is the $N$th truncation of $\chi$) $\mathcal{F}$-converges to $\mathbf{0}$. \end{prop} Proof: Immediate from the definitions of $\hat{\chi}$, $\tilde{\chi}_{N}$, and $\chi_{N}$. Q.E.D. \begin{rem} Note that all of the main unresolved issues regarding one-dimensional quasi-integrable functions (convolutions, a criterion for determining non-quasi-integrability, etc.) also apply to the multi-dimensional case. \end{rem} \subsection{\label{subsec:5.4.3 -adic-Wiener-Tauberian}$\left(p,q\right)$-adic Wiener Tauberian Theorems Revisited} Here, we quickly establish the multi-dimensional analogues of the results of Subsection \ref{subsec:3.3.7 -adic-Wiener-Tauberian}. \begin{defn} For \nomenclature{$\tau_{\mathbf{s}}$}{Multi-dimensional translation operator}$\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}$, and for a thick $\left(p,q\right)$-adic measure $\mathcal{M}$ with Fourier-Stieltjes transform $\hat{\mathcal{M}}$, we write: \begin{equation} \tau_{\mathbf{s}}\left\{ \hat{\mathcal{M}}\right\} \left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{\mathcal{M}}_{\mathbf{t}+\mathbf{s}},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:definition of the translate of M hat} \end{equation} We also write: \begin{equation} \tau_{\mathbf{s}}\left\{ \hat{\mathbf{f}}\right\} \left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{\mathbf{f}}\left(\mathbf{t}+\mathbf{s}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Definition of the translate of f hat} \end{equation} \end{defn} \begin{defn} We write $\left(\mathbb{C}_{q}^{d}\right)^{\times}$\nomenclature{$\left(\mathbb{C}_{q}^{d}\right)^{\times}$}{$d\times1$ vectors whose entries are non-zero $q$-adic complex numbers } to denote the set of all $d\times1$ vectors whose entries are non-zero $q$-adic complex numbers. We make $\left(\mathbb{C}_{q}^{d}\right)^{\times}$ an abelian group with respect to the operation of entry-wise multiplication; i.e.: \begin{equation} \left(\begin{array}{c} \mathfrak{a}_{1}\\ \mathfrak{a}_{2} \end{array}\right)\left(\begin{array}{c} \mathfrak{b}_{1}\\ \mathfrak{b}_{2} \end{array}\right)=\left(\begin{array}{c} \mathfrak{a}_{1}\mathfrak{b}_{1}\\ \mathfrak{a}_{2}\mathfrak{b}_{2} \end{array}\right) \end{equation} We write \nomenclature{$\mathbf{1}$}{$d\times 1$ vector whose every entry is $1$}$\mathbf{1}$ to denote the identity element of $\left(\mathbb{C}_{q}^{d}\right)^{\times}$, the $d\times1$ vector whose entries are all $1$s. We also write $\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)$\nomenclature{$\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)$}{function which is $\mathbf{1}$ when $\mathbf{t}=\mathbf{0}$ and which is $\mathbf{0}$ otherwise.} to denote the function $\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ which is equal to $\mathbf{1}$ when $\mathbf{t}=\mathbf{0}$ and which is equal to $\mathbf{0}$ for all other $\mathbf{t}$. Note that for any $\hat{\mathbf{f}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$: \begin{equation} \left(\hat{\mathbf{f}}\vec{\mathbf{1}}_{\mathbf{0}}\right)\left(\mathbf{t}\right)=\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{s}\right)=\hat{\mathbf{f}}\left(\mathbf{t}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} so $\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)$ is the identity element with respect to the convolution operation on $B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Incidentally, this also proves that the constant function $\mathbf{z}\mapsto\mathbf{1}$ and the function $\mathbf{t}\mapsto\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)$ are a Fourier transform pair. More generally, we write $\vec{\mathbf{1}}_{\mathbf{s}}\left(\mathbf{t}\right)$ to denote $\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}-\mathbf{s}\right)$, which is $\mathbf{1}$ for $\mathbf{t}\overset{\mathbf{1}}{\equiv}\mathbf{s}$ and $\mathbf{0}$ otherwise. The reason for introducing the vector notation is to avoid confusing the vector-valued indicator function for the point $\left\{ \mathbf{s}\right\} $ ($\vec{\mathbf{1}}_{\mathbf{s}}\left(\mathbf{t}\right)$) and the \emph{scalar}-valued indicator function for the point $\left\{ \mathbf{s}\right\} $ ($\mathbf{1}_{\mathbf{s}}\left(\mathbf{t}\right)$). \end{defn} \vphantom{} As in the one-dimensional case, we establish two WTTs: one for continuous functions, and another for thick measures. \begin{thm}[\textbf{Multi-Dimensional Wiener Tauberian Theorem for Continuous $\left(p,q\right)$-adic Functions}] \label{thm:MD pq-adic WTT for continuous functions}Let\index{Wiener!Tauberian Theorem!left(p,qright)-adic@$\left(p,q\right)$-adic} $\mathbf{f}=\left(f_{1}\left(\mathbf{z}\right),\ldots,f_{d}\left(\mathbf{z}\right)\right)\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, and let: \begin{equation} \mathbf{f}^{-1}\left(\mathbf{z}\right)=\left(\frac{1}{f_{1}\left(\mathbf{z}\right)},\ldots,\frac{1}{f_{d}\left(\mathbf{z}\right)}\right) \end{equation} denote the multiplicative inverse\footnote{If $\mathbf{F}$ is matrix-valued, then $\mathbf{F}^{-1}$ is the matrix inverse; if $\mathbf{F}$ is scalar-valued, then $\mathbf{F}^{-1}$ is the reciprocal of $\mathbf{F}$; if $\mathbf{F}$ is vector-valued, then $\mathbf{F}^{-1}$ is the vector whose entries are the reciprocals of the entries of $\mathbf{F}$.} of $\mathbf{f}$, if it exists. Then, the following are equivalent: \vphantom{} I. $\mathbf{f}^{-1}$ exists and is an element of $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$; \vphantom{} II. $\hat{\mathbf{f}}$ has a convolution inverse in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$; \vphantom{} III. $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{\mathbf{s}}\left\{ \hat{\mathbf{f}}\right\} \left(\mathbf{t}\right):\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$; \vphantom{} IV. $\mathbf{f}\left(\mathbf{z}\right)\in\left(\mathbb{C}_{q}^{d}\right)^{\times}$ for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, . \end{thm} Proof: \textbullet{} ($\textrm{I}\Rightarrow\textrm{II}$) Suppose $\mathbf{f}^{-1}$ exists and is continuous. Because the $\left(p,q_{H}\right)$-adic Fourier transform isometrically and isomorphically maps the Banach algebra $C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ onto the Banach algebra $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, we can write: \begin{align} \mathbf{f}\left(\mathbf{z}\right)\cdot\mathbf{f}^{-1}\left(\mathbf{z}\right) & =\mathbf{1},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\\ \left(\textrm{Fourier transform}\right); & \Updownarrow\\ \left(\hat{\mathbf{f}}\widehat{\mathbf{f}^{-1}}\right)\left(\mathbf{t}\right) & =\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{align} where both $\hat{\mathbf{f}}$ and $\widehat{\mathbf{f}^{-1}}$ are in $c_{0}$. This shows that $\widehat{\mathbf{f}^{-1}}$ is $\hat{\mathbf{f}}^{-1}$\textemdash the convolution inverse of $\hat{\mathbf{f}}$. \vphantom{} \textbullet{} ($\textrm{II}\Rightarrow\textrm{III}$) Suppose $\hat{\mathbf{f}}$ has a convolution inverse $\hat{\mathbf{f}}^{-1}\in c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Then, letting $\hat{\mathbf{g}}\in c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ be arbitrary, we can write: \begin{equation} \left(\hat{\mathbf{f}}\left(\hat{\mathbf{f}}^{-1}\hat{\mathbf{g}}\right)\right)\left(\mathbf{t}\right)=\left(\left(\hat{\mathbf{f}}\hat{\mathbf{f}}^{-1}\right)\hat{\mathbf{g}}\right)\left(\mathbf{t}\right)=\left(\mathbf{1}_{\mathbf{0}}\hat{\mathbf{g}}\right)\left(\mathbf{t}\right)=\hat{\mathbf{g}}\left(\mathbf{t}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} Letting $\hat{\mathbf{h}}$ denote $\hat{\mathbf{f}}^{-1}\hat{\mathbf{g}}$, we then have: \begin{equation} \hat{\mathbf{g}}\left(\mathbf{t}\right)=\left(\hat{\mathbf{f}}\hat{\mathbf{h}}\right)\left(\mathbf{t}\right)\overset{\mathbb{C}_{q}^{d}}{=}\lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right) \end{equation} Since $\hat{\mathbf{f}}$ and $\hat{\mathbf{h}}$ are in $c_{0}$, note that: \begin{align} \sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)-\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q} & \leq\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\sup_{\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}}\left\Vert \hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q}\\ \left(\left\Vert \hat{\mathbf{f}}\right\Vert _{p,q}<\infty\right); & \leq\sup_{\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}}\left\Vert \hat{\mathbf{h}}\left(\mathbf{s}\right)\right\Vert _{q} \end{align} and hence: \[ \lim_{N\rightarrow\infty}\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)-\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q} \] is equal to $\lim_{N\rightarrow\infty}\sup_{\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}}\left\Vert \hat{\mathbf{h}}\left(\mathbf{s}\right)\right\Vert _{q}$, which tends to $0$. This shows that the $q$-adic convergence of $\sum_{\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)$ to $\hat{\mathbf{g}}\left(\mathbf{t}\right)=\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)$ is uniform in $\mathbf{t}$. Hence: \begin{equation} \lim_{N\rightarrow\infty}\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \hat{\mathbf{g}}\left(\mathbf{t}\right)-\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q}=0 \end{equation} which is precisely the definition of what it means for the sequence: \begin{equation} \left\{ \sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{h}}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\} _{N\geq0} \end{equation} to converge in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ to $\hat{\mathbf{g}}$. Because our sequence converging to the arbitrary $\hat{\mathbf{g}}\in c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ is an element of: \begin{equation} \textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{\mathbf{s}}\left\{ \hat{\mathbf{f}}\right\} \left(\mathbf{t}\right):\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}\right\} \end{equation} we have proven the density of $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{\mathbf{s}}\left\{ \hat{\mathbf{f}}\right\} \left(\mathbf{t}\right):\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}\right\} $ in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. \vphantom{} \textbullet{} ($\textrm{III}\Rightarrow\textrm{IV}$) Suppose $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{\mathbf{s}}\left\{ \hat{\mathbf{f}}\right\} \left(\mathbf{t}\right):\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Now, let $\epsilon>0$. Because $\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)\in c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$, the assumed density of the span of translates guarantees the existence of constant vectors $\mathbf{c}_{1},\ldots,\mathbf{c}_{M}\in\mathbb{C}_{q}^{d}$ and points $\mathbf{t}_{1},\ldots,\mathbf{t}_{M}\in\hat{\mathbb{Z}}_{p}^{r}$ so that: \begin{equation} \sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right\Vert _{q}<\epsilon \end{equation} Here, note that $\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)$ is the entry-wise product of the $d\times1$ vectors $\mathbf{c}_{m}$ and $\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)$. Next, letting $N\geq\max\left\{ -v_{p}\left(\mathbf{t}_{1}\right),\ldots,-v_{p}\left(\mathbf{t}_{M}\right)\right\} $ be arbitrary\footnote{Note that this choice of $N$ is larger than the negative $p$-adic valuations of all of the entries of all of the $\mathbf{t}_{m}$s.}, we have that for all $m$, the map $\mathbf{t}\mapsto\mathbf{t}+\mathbf{t}_{m}$ is a bijection of $\left\{ \mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}:\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}\right\} $. Consequently: \begin{align} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & =\sum_{m=1}^{M}\mathbf{c}_{m}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{f}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \left(\mathbf{t}+\mathbf{t}_{m}\right)\mathbf{z}\right\} _{p}}\\ & =\left(\sum_{m=1}^{M}\mathbf{c}_{m}e^{2\pi i\left\{ \mathbf{t}_{m}\mathbf{z}\right\} _{p}}\right)\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathbf{f}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{align} Since $N$ was arbitrary, we may let it tend to $\infty$. This gives: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\mathbb{C}_{q}^{d}}{=}\mathbf{g}_{m}\left(\mathbf{z}\right)\mathbf{f}\left(\mathbf{z}\right) \end{equation} where $\mathbf{g}_{m}=\left(g_{m,1},\ldots,g_{m,d}\right):\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ is defined by: \begin{equation} \mathbf{g}_{m}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{m=1}^{M}\mathbf{c}_{m}e^{2\pi i\left\{ \mathbf{t}_{m}\mathbf{z}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Definition of G_m} \end{equation} Moreover, the convergence of the $N$-limit is uniform in $\mathbf{z}$. Consequently: \begin{align} \left\Vert \mathbf{1}-\mathbf{g}_{m}\left(\mathbf{z}\right)\mathbf{f}\left(\mathbf{z}\right)\right\Vert _{q} & \overset{\mathbb{R}}{=}\lim_{N\rightarrow\infty}\left\Vert \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\right\Vert _{q}\\ & \leq\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right\Vert _{q}\\ & <\epsilon \end{align} for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. Finally, by way of contradiction, suppose there is an $\ell\in\left\{ 1,\ldots,d\right\} $ so that $f_{\ell}\left(\mathbf{z}_{0}\right)=0$. Then, the $\ell$th entry of the $d\times1$ vector $\mathbf{1}-\mathbf{g}_{m}\left(\mathbf{z}_{0}\right)\mathbf{f}\left(\mathbf{z}_{0}\right)$ is: \[ 1-g_{m,\ell}\left(\mathbf{z}_{0}\right)f_{\ell}\left(\mathbf{z}_{0}\right)=1-g_{m,\ell}\left(\mathbf{z}_{0}\right)\cdot0=1 \] Because $\left\Vert \mathbf{1}-\mathbf{g}_{m}\left(\mathbf{z}_{0}\right)\mathbf{f}\left(\mathbf{z}_{0}\right)\right\Vert _{q}\geq\left\Vert 1-g_{m,k}\left(\mathbf{z}_{0}\right)f_{k}\left(\mathbf{z}_{0}\right)\right\Vert _{q}$ for all $k\in\left\{ 1,\ldots,d\right\} $, we then have: \begin{equation} \epsilon>\left\Vert \mathbf{1}-\mathbf{g}_{m}\left(\mathbf{z}_{0}\right)\mathbf{f}\left(\mathbf{z}_{0}\right)\right\Vert _{q}\geq\left\Vert 1-g_{m,\ell}\left(\mathbf{z}_{0}\right)\cdot0\right\Vert _{q}=1 \end{equation} However, $\epsilon$ was given to be in $\left(0,1\right)$. This is a contradiction! As such, the entries of $\mathbf{f}$ cannot have any zeroes whenever the span of $\hat{\mathbf{f}}$'s translates are dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. \vphantom{} \textbullet{} ($\textrm{IV}\Rightarrow\textrm{I}$) Suppose $\mathbf{f}\left(\mathbf{z}\right)\in\left(\mathbb{C}_{q}^{d}\right)^{\times}$ for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$; that is, suppose no entry of $\mathbf{f}\left(\mathbf{z}\right)$ is zero for any $\mathbf{z}\in\mathbb{Z}_{p}^{r}$. Because the map which sends $\mathbf{y}=\left(\mathfrak{y}_{1},\ldots,\mathfrak{y}_{d}\right)\in\left(\mathbb{C}_{q}^{d}\right)^{\times}$ to: \[ \frac{1}{\mathbf{y}}=\left(\frac{1}{\mathfrak{y}_{1}},\ldots,\frac{1}{\mathfrak{y}_{d}}\right)\in\left(\mathbb{C}_{q}^{d}\right)^{\times} \] is continuous on $\left(\mathbb{C}_{q}^{d}\right)^{\times}$, the continuity of compositions of continuous maps tells us that we can show $1/\mathbf{f}$ is continuous by establishing that show that $\inf_{\mathbf{z}\in\mathbb{Z}_{p}^{r}}\left\|f_{k}\left(\mathbf{z}\right)\right\|_{q}>0$ for each $k\in\left\{ 1,\ldots,d\right\} $. So, by way of contradiction, suppose there is a $k\in\left\{ 1,\ldots,d\right\} $ so that $\inf_{\mathbf{z}\in\mathbb{Z}_{p}^{r}}\left\|f_{k}\left(\mathbf{z}\right)\right\|_{q}=0$. Then, there exists a sequence $\left\{ \mathbf{z}_{n}\right\} _{n\geq0}\subseteq\mathbb{Z}_{p}^{r}$ so that $\lim_{n\rightarrow\infty}\left\|f_{k}\left(\mathbf{z}_{n}\right)\right\|_{q}=0$. By the compactness of $\mathbb{Z}_{p}^{r}$, the $\mathbf{z}_{n}$s have a subsequence $\mathbf{z}_{n_{\ell}}$ which converges in $\mathbb{Z}_{p}^{r}$ to some limit $\mathbf{z}_{\infty}\in\mathbb{Z}_{p}^{r}$. Because $\mathbf{f}$ is continuous, so is $f_{k}$; this forces $f_{k}\left(\mathbf{z}_{\infty}\right)=0$. Hence, $\mathbf{z}_{\infty}$ is an element of $\mathbb{Z}_{p}^{r}$ for which $\mathbf{f}\left(\mathbf{z}_{\infty}\right)$ has a non-zero entry\textemdash which contradicts our given hypothesis on $\mathbf{f}$. Thus, if none of the entries of $\mathbf{f}$ are ever zero on $\mathbb{Z}_{p}^{r}$, $\left\|f_{k}\left(\mathbf{z}\right)\right\|_{q}$ is bounded away from zero for all $k$, which then shows that $1/\mathbf{f}$ is $\left(p,q\right)$-adically continuous. Q.E.D. \begin{thm}[\textbf{Wiener Tauberian Theorem for Vector-Type Thick $\left(p,q\right)$-adic Measures}] \label{thm:MD WTT for pq adic vector type thick measures}Let $\mathcal{M}\in M\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ be a $\left(p,q\right)$-adic thick measure of vector type, with Fourier-Stieltjes transform $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$. Then, $\textrm{span}_{\mathbb{C}_{q}}\left\{ \tau_{\mathbf{s}}\left\{ \hat{\mathcal{M}}\right\} \left(\mathbf{t}\right):\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}\right\} $ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ if and only if for any $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ for which the limit $\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)$ converges in $\mathbb{C}_{q}^{d}$, said limit is necessarily an element of $\left(\mathbb{C}_{q}^{d}\right)^{\times}$. \index{Wiener!Tauberian Theorem!left(p,qright)-adic@$\left(p,q\right)$-adic} \end{thm} \begin{rem} \textbf{WARNING \textendash{} }Like in the one-dimensional case (\textbf{Theorem \ref{thm:pq WTT for measures}}), for the proof of this theorem\textemdash and only this theorem\textemdash we will need to modify our notation for the $\left(p,q\right)$-adic norm. Instead of writing $\left\Vert \cdot\right\Vert _{p,q}$ to denote the supremum over $\hat{\mathbb{Z}}_{p}^{r}$ of a vector-valued function with entries in $\mathbb{C}_{q}$, we write that norm as $\left\Vert \cdot\right\Vert _{p^{\infty},q}$. \end{rem} Proof: I. Suppose the span of the translates of $\hat{\mathcal{M}}$ are dense. Like in the one-dimensional case, we let $\epsilon\in\left(0,1\right)$ and then, using the assumed density, choose constant vectors $\mathbf{c}_{m}\in\mathbb{C}_{q}^{d}$ and $\mathbf{t}_{m}\in\hat{\mathbb{Z}}_{p}^{r}$ so that: \begin{equation} \sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right\Vert _{q}<\epsilon \end{equation} When $N$ is sufficiently large, we obtain: \begin{align} \left\Vert \mathbf{1}-\left(\sum_{m=1}^{M}\mathbf{c}_{m}e^{2\pi i\left\{ \mathbf{t}_{m}\mathbf{z}\right\} _{p}}\right)\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)\right\Vert _{q} & \leq\max_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right\Vert _{q}\\ & \leq\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{m=1}^{M}\mathbf{c}_{m}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{m}\right)\right\Vert _{q}\\ & <\epsilon \end{align} So, let $\mathbf{z}_{0}\in\mathbb{Z}_{p}^{r}$ be a point so that $\mathfrak{L}=\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)$ converges in $\mathbb{C}_{q}^{d}$. Then, plugging $\mathbf{z}=\mathbf{z}_{0}$ in the above ends yields: \[ \epsilon>\lim_{N\rightarrow\infty}\left\Vert \mathbf{1}-\left(\sum_{m=1}^{M}\mathbf{c}_{m}e^{2\pi i\left\{ \mathbf{t}_{m}\mathbf{z}_{0}\right\} _{p}}\right)\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)\right\Vert _{q}=\left\Vert \mathbf{1}-\left(\sum_{m=1}^{M}\mathbf{c}_{m}e^{2\pi i\left\{ \mathbf{t}_{m}\mathbf{z}_{0}\right\} _{p}}\right)\cdot\mathfrak{L}\right\Vert _{q} \] If $\mathfrak{L}$ had an entry which is $0$ (i.e., if $\mathfrak{L}\in\mathbb{C}_{q}^{d}\backslash\left(\mathbb{C}_{q}^{d}\right)^{\times}$), then we end up with $\epsilon>1$, which contradicts the fact that $\epsilon\in\left(0,1\right)$. Thus, if $\lim_{N\rightarrow\infty}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)$ converges in $\mathbb{C}_{q}^{d}$, it must converge to an element of $\left(\mathbb{C}_{q}^{d}\right)^{\times}$. \vphantom{} II. Let $\mathbf{z}_{0}\in\mathbb{Z}_{p}^{r}$ be such that $\tilde{\mathcal{M}}\left(\mathbf{z}_{0}\right)\in\mathbb{C}_{q}^{d}\backslash\left(\mathbb{C}_{q}^{d}\right)^{\times}$ and that $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)\rightarrow\tilde{\mathcal{M}}\left(\mathbf{z}_{0}\right)$ in $\mathbb{C}_{q}^{d}$ as $N\rightarrow\infty$; in an abuse of terminology, let us call this $\mathbf{z}_{0}$ a ``zero'' of $\tilde{\mathcal{M}}$. Next, by way of contradiction, suppose the span of the translates of $\hat{\mathcal{M}}$ is dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$ in spite of the zero of $\tilde{\mathcal{M}}$ at $\mathbf{z}_{0}$. So, letting $\epsilon\in\left(0,1\right)$ be arbitrary, by the assumed density,, there exists a linear combination of translates of $\hat{\mathcal{M}}$ which approximates $\vec{\mathbf{1}}_{\mathbf{0}}$ in sup norm: \[ \sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\sum_{k=1}^{K}\mathbf{c}_{k}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{k}\right)\right\Vert _{q}<\epsilon \] where the $\mathbf{c}_{k}$s and $\mathbf{t}_{k}$s are constants in $\mathbb{C}_{q}^{d}$ and $\hat{\mathbb{Z}}_{p}^{r}$, respectively. Writing: \begin{equation} \hat{\mathbf{h}}_{\epsilon}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\sum_{k=1}^{K}\mathbf{c}_{k}\vec{\mathbf{1}}_{\mathbf{t}_{k}}\left(\mathbf{t}\right)\label{eq:Definition of bold h_epsilon hat} \end{equation} we can represent our approximating linear combination as a convolution: \[ \left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\left(\mathbf{t}\right)=\sum_{\mathbf{u}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{u}\right)\sum_{k=1}^{K}\mathbf{c}_{k}\vec{\mathbf{1}}_{\mathbf{t}_{k}}\left(\mathbf{u}\right)=\sum_{k=1}^{K}\mathbf{c}_{k}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{k}\right) \] and so: \begin{equation} \left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}\overset{\textrm{def}}{=}\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\left(\mathbf{t}\right)\right\Vert _{q}<\epsilon\label{eq:MD Converse WTT - eq. 1} \end{equation} Before we proceed, exactly like in the one-dimensional case, it is \emph{vital }to note that we can and \emph{must} assume that $\hat{\mathbf{h}}_{\epsilon}$ is \emph{not identically zero}. In fact, we can assume this must hold for any $\epsilon\in\left(0,1\right)$. Indeed, if there was an $\epsilon\in\left(0,1\right)$ for which $\hat{\mathbf{h}}_{\epsilon}$ was identically zero, the supremum $\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\left(\mathbf{t}\right)\right\Vert _{q}$ would be equal to $1$, which is quite impossible, given that the supremum is, by construction, less than $\epsilon$. Exactly like before, we will obtain a contradiction by showing that we can convolve $\hat{\mathcal{M}}$ by two different functions; this first makes the resultant vector-valued function close to $\vec{\mathbf{1}}_{\mathbf{0}}$; the second makes one of the entries of the resultant vector-valued function close to $0$. Our close-to-zero convolving function is going to be: \begin{equation} \hat{\phi}_{N}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\vec{\mathbf{1}}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}_{0}\right\} _{p}}\label{eq:MD Definition of Phi_N hat} \end{equation} Note that we can then write: \begin{align} \left(\hat{\mathcal{M}}\hat{\phi}_{N}\right)\left(\mathbf{u}\right) & =\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathcal{M}}\left(\mathbf{u}-\mathbf{s}\right)\hat{\phi}_{N}\left(\mathbf{s}\right)\\ \left(\hat{\phi}_{N}\left(\mathbf{s}\right)=\mathbf{0},\textrm{ }\forall\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}\right); & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{u}-\mathbf{s}\right)\hat{\phi}_{N}\left(\mathbf{s}\right)\\ & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{u}-\mathbf{s}\right)e^{-2\pi i\left\{ \mathbf{s}\mathbf{z}_{0}\right\} _{p}} \end{align} Now, fixing $\mathbf{u}$ with $\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{N}$, observe that the map $\mathbf{s}\mapsto\mathbf{u}-\mathbf{s}$ is a bijection of the set $\left\{ \mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}:\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}\right\} $; equivalently, this map is a bijection whenever $N\geq-v_{p}\left(\mathbf{u}\right)$. So, picking $N\geq-v_{p}\left(\mathbf{u}\right)$, we can write: \begin{align} \sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{u}-\mathbf{s}\right)e^{-2\pi i\left\{ \mathbf{s}\mathbf{z}_{0}\right\} _{p}} & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{s}\right)e^{-2\pi i\left\{ \left(\mathbf{u}-\mathbf{s}\right)\mathbf{z}_{0}\right\} _{p}}\\ & =e^{-2\pi i\left\{ \mathbf{u}\mathbf{z}_{0}\right\} _{p}}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{s}\right)e^{2\pi i\left\{ \mathbf{s}\mathbf{z}_{0}\right\} _{p}}\\ & =e^{-2\pi i\left\{ \mathbf{u}\mathbf{z}_{0}\right\} _{p}}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right) \end{align} Denoting the entries of $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)$ as $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)=\left(\tilde{\mathcal{M}}_{N,1}\left(\mathbf{z}_{0}\right),\ldots,\tilde{\mathcal{M}}_{N,d}\left(\mathbf{z}_{0}\right)\right)$, the given assumption about the zero at $\mathbf{z}_{0}$ tells us there is an $\ell\in\left\{ 1,\ldots,d\right\} $ so that $\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)$'s $\ell$th entry ($\tilde{\mathcal{M}}_{N,\ell}\left(\mathbf{z}_{0}\right)$) converges to $0$ in\emph{ $\mathbb{C}_{q}$} as $N\rightarrow\infty$. Thus, for any $\epsilon^{\prime}>0$, there exists an $N_{\epsilon^{\prime}}$ so that $\left\|\tilde{\mathcal{M}}_{N,\ell}\left(\mathbf{z}_{0}\right)\right\|_{q}<\epsilon^{\prime}$ for all $N\geq N_{\epsilon^{\prime}}$. Next, let $\mathbf{e}_{\ell}$ denote the $\ell$th standard basis vector of $\mathbb{C}_{q}^{d}$. Then, for any $\hat{\mathbf{f}}=\left(\hat{f}_{1},\ldots,\hat{f}_{d}\right):\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$, the entry-wise product $\mathbf{e}_{\ell}\hat{\mathbf{f}}$ is: \begin{equation} \left(\mathbf{e}_{\ell}\hat{\mathbf{f}}\right)\left(\mathbf{t}\right)=\left(0,\ldots,0,\hat{f}_{\ell}\left(\mathbf{t}\right),0,\ldots,0\right)\label{eq:e_ell f hat} \end{equation} Moreover, since the convolution of two vector-valued functions on $\hat{\mathbb{Z}}_{p}^{r}$ is done entry-wise: \begin{align} \left(\hat{\mathbf{f}}\hat{\mathbf{g}}\right)\left(\mathbf{t}\right) & =\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\hat{\mathbf{g}}\left(\mathbf{s}\right)\\ & =\left(\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{f}_{1}\left(\mathbf{t}-\mathbf{s}\right)\hat{g}_{1}\left(\mathbf{s}\right),\ldots,\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{f}_{d}\left(\mathbf{t}-\mathbf{s}\right)\hat{g}_{d}\left(\mathbf{s}\right)\right) \end{align} we obtain: \begin{equation} \left(\left(\mathbf{e}_{\ell}\hat{\mathbf{f}}\right)\hat{\mathbf{g}}\right)\left(\mathbf{t}\right)=\left(\mathbf{e}_{\ell}\left(\hat{\mathbf{f}}\hat{\mathbf{g}}\right)\right)\left(\mathbf{t}\right)=\left(\left(\hat{f}_{1}\hat{g}_{1}\right)\left(\mathbf{t}\right),0,\ldots,0\right)\label{eq:e_ell f hat convolve with g hat} \end{equation} With this notation, we can write: \begin{align} \left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\phi}_{N}\right)\left(\mathbf{u}\right) & =\mathbf{e}_{\ell}\left(\hat{\mathcal{M}}\hat{\phi}_{N}\right)\left(\mathbf{u}\right)\\ & =\mathbf{e}_{\ell}\left(e^{-2\pi i\left\{ \mathbf{u}\mathbf{z}_{0}\right\} _{p}}\tilde{\mathcal{M}}_{N}\left(\mathbf{z}_{0}\right)\right)\\ & =e^{-2\pi i\left\{ \mathbf{u}\mathbf{z}_{0}\right\} _{p}}\left(\mathbf{e}_{\ell}\tilde{\mathcal{M}}_{N}\right)\left(\mathbf{z}_{0}\right)\\ & =\left(0,\ldots,0,e^{-2\pi i\left\{ \mathbf{u}\mathbf{z}_{0}\right\} _{p}}\tilde{\mathcal{M}}_{N,\ell}\left(\mathbf{z}_{0}\right),0,\ldots,0\right) \end{align} Taking $q$-adic absolute values, we have thereby proven the following claim: \begin{claim} \label{claim:5.11}For any $\epsilon^{\prime}>0$, there is an $N_{\epsilon^{\prime}}\geq0$ (depending only on $\hat{\mathcal{M}}$, $\ell$ and $\epsilon^{\prime}$) so that, for any $\mathbf{u}\in\hat{\mathbb{Z}}_{p}$: \end{claim} \begin{equation} \left\Vert \left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\phi}_{N}\right)\left(\mathbf{u}\right)\right\Vert _{p^{\infty},q}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},-v_{p}\left(\mathbf{u}\right)\right\} \label{eq:MD WTT - First Claim} \end{equation} \vphantom{} For our next step, we restrict the support of $\hat{\mathbf{h}}_{\epsilon}$ by defining the function $\hat{\mathbf{h}}_{\epsilon,M}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ as: \begin{equation} \hat{\mathbf{h}}_{\epsilon,M}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\vec{\mathbf{1}}_{\mathbf{0}}\left(p^{M}\mathbf{t}\right)\hat{\mathbf{h}}_{\epsilon}\left(\mathbf{t}\right)=\begin{cases} \hat{\mathbf{h}}_{\epsilon}\left(\mathbf{t}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}\\ \mathbf{0} & \textrm{else} \end{cases}\label{eq:MD Definition of h epsilon M hat} \end{equation} Here $M$ is an arbitrary integer $\geq0$. Then: \begin{align} \left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\left(\mathbf{t}\right) & =\sum_{\mathbf{u}\in\mathbb{Z}_{p}}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{u}\right)\vec{\mathbf{1}}_{\mathbf{0}}\left(p^{M}\mathbf{u}\right)\sum_{k=1}^{K}\mathbf{c}_{k}\vec{\mathbf{1}}_{\mathbf{t}_{k}}\left(\mathbf{u}\right)\\ & =\sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{M}}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{u}\right)\sum_{k=1}^{K}\mathbf{c}_{k}\vec{\mathbf{1}}_{\mathbf{t}_{k}}\left(\mathbf{u}\right)\\ & =\sum_{k:\left\Vert \mathbf{t}_{k}\right\Vert _{p}\leq p^{M}}\mathbf{c}_{k}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{k}\right) \end{align} Since there are finitely many $\mathbf{t}_{k}$s, we can choose the non-negative integer: \begin{equation} M_{\epsilon}\overset{\textrm{choose}}{=}\max\left\{ -v_{p}\left(\mathbf{t}_{1}\right),\ldots,-v_{p}\left(\mathbf{t}_{K}\right)\right\} \label{eq:MD WTT - Choice for M_e} \end{equation} In particular, note that\emph{ $M_{\epsilon}$ depends only on $\mathbf{h}_{\epsilon}$ and $\epsilon$. }Then: \begin{equation} \left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\left(\mathbf{t}\right)=\sum_{k=1}^{K}\mathbf{c}_{k}\hat{\mathcal{M}}\left(\mathbf{t}-\mathbf{t}_{k}\right)=\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\left(\mathbf{t}\right),\textrm{ }\forall M\geq M_{\epsilon}\label{eq:MD Effect of M bigger than M0} \end{equation} Next, for any $\hat{\mathbf{f}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ and any $n\in\mathbb{N}_{0}$, let us write: \nomenclature{$\left\Vert \hat{\mathbf{f}}\right\Vert _{p^{n},q}$}{ } \begin{equation} \left\Vert \hat{\mathbf{f}}\right\Vert _{p^{n},q}\overset{\textrm{def}}{=}\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n}}\left\Vert \hat{\mathbf{f}}\left(\mathbf{t}\right)\right\Vert _{q}\label{eq:MD Definition of infinity,p^n norm} \end{equation} Then: \begin{align} \left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q} & =\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\left(\mathbf{t}\right)\right\Vert _{q}\\ \left(\textrm{if }M\geq M_{\epsilon}\right); & =\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}\left(\mathbf{t}\right)-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\left(\mathbf{t}\right)\right\Vert _{q}\\ & \leq\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right)\right\Vert _{p^{\infty},q}\\ & <\epsilon \end{align} In summary, we have shown: \begin{claim} \label{claim:5.12}Let $\epsilon>0$ be arbitrary. Then, there exists an $M_{\epsilon}$ (depending only on $\hat{\mathcal{M}}$ and $\epsilon$ so that: \vphantom{} I. $\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}=\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon},\textrm{ }\forall M\geq M_{\epsilon}$ \vphantom{} II. $\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{\infty},q}<\epsilon,\textrm{ }\forall M\geq M_{\epsilon}$ \vphantom{} III. $\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}<\epsilon,\textrm{ }\forall M\geq M_{\epsilon}$ \end{claim} \vphantom{} We now modify these estimates to take into account the norm $\left\Vert \cdot\right\Vert _{p^{m},q}$. \begin{claim} \label{claim:5.13}Let $\epsilon^{\prime}>0$ and $m\in\mathbb{N}_{1}$ be arbitrary. Then, there exists $N_{\epsilon^{\prime}}$ (depending only on $\epsilon^{\prime}$ and $\hat{\mathcal{M}}$) so that: \begin{equation} \left\Vert \left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\right)\hat{\phi}_{N}\right\Vert _{p^{m},q}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon^{\prime}},m\right\} \label{eq:MD WTT - eq. 4} \end{equation} Proof of claim: For arbitrary $\epsilon^{\prime}>0$, \textbf{Claim \ref{claim:5.11}} tells us there is an $N_{\epsilon^{\prime}}$ with the stated dependencies so that, for any $\mathbf{u}\in\hat{\mathbb{Z}}_{p}^{r}$: \begin{equation} \left\Vert \left(\left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\right)\hat{\phi}_{N}\right)\left(\mathbf{u}\right)\right\Vert _{p^{\infty},q}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon},-v\left(\mathbf{u}\right)\right\} \end{equation} So, letting $m\geq1$ be arbitrary, note that $\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{m}$ implies $-v_{p}\left(\mathbf{u}\right)\leq m$. As such, by choosing $N\geq\max\left\{ N_{\epsilon},m\right\} $, we can make the result of \textbf{Claim \ref{claim:5.11}} hold for all $\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{m}$: \begin{equation} \underbrace{\sup_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{m}}\left\Vert \left(\left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\right)\hat{\phi}_{N}\right)\left(\mathbf{u}\right)\right\Vert _{q}}_{\left\Vert \left(\mathbf{e}_{\ell}\hat{\mathcal{M}}\right)\hat{\phi}_{N}\right\Vert _{p^{m},q}}<\epsilon^{\prime},\textrm{ }\forall N\geq\max\left\{ N_{\epsilon},m\right\} \end{equation} This proves the claim. \end{claim} \vphantom{} We need two more estimates before we can wrap up the argument. \begin{claim} \label{claim:5.14}Let $M,N\in\mathbb{N}_{0}$ be arbitrary. Then, for any $\hat{\varphi},\hat{\mathbf{f}},\hat{\mathbf{g}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ satisfying: \vphantom{} i. $\hat{\varphi}\left(\mathbf{t}\right)=\mathbf{0}$ for all $\left\Vert \mathbf{t}\right\Vert _{p}>p^{N}$; \vphantom{} ii. $\left\Vert \hat{\varphi}\right\Vert _{p^{\infty},q}\leq1$; \vphantom{} iii. $\hat{\mathbf{g}}\left(\mathbf{t}\right)=\mathbf{0}$ for all $\left\Vert \mathbf{t}\right\Vert _{p}>p^{M}$; \vphantom{} the following estimates hold: \begin{equation} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\mathbf{f}}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\label{eq:MD phi_N hat convolve f hat estimate} \end{equation} \begin{equation} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\left\Vert \hat{\varphi}\hat{\mathbf{f}}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\label{eq:MD phi_N hat convolve f hat convolve g hat estimate} \end{equation} Proof of claim: As in the one-dimensional case, we start with the top-most inequality: \begin{align} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\right\Vert _{p^{M},q} & =\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\left\|\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\varphi}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\|_{q}\\ \left(\hat{\varphi}\left(\mathbf{s}\right)=\mathbf{0},\textrm{ }\forall\left\Vert \mathbf{s}\right\Vert _{p}>p^{N}\right); & \leq\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\hat{\varphi}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q}\\ \left(\textrm{ultrametric ineq.}\right); & \leq\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\sup_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\left\Vert \hat{\varphi}\left(\mathbf{s}\right)\hat{\mathbf{f}}\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q}\\ & \leq\sup_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{N}}\left\Vert \hat{\varphi}\left(\mathbf{s}\right)\right\Vert _{q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \hat{\mathbf{f}}\left(\mathbf{t}\right)\right\Vert _{q}\\ & =\left\Vert \hat{\varphi}\right\Vert _{p^{\infty},q}\cdot\left\Vert \hat{\mathbf{f}}\right\Vert _{p^{\max\left\{ M,N\right\} },q}\\ \left(\left\Vert \hat{\varphi}\right\Vert _{p^{\infty},q}\leq1\right); & \leq\left\Vert \hat{\mathbf{f}}\right\Vert _{p^{\max\left\{ M,N\right\} },q} \end{align} This proves (\ref{eq:MD phi_N hat convolve f hat estimate}). Next, we deal with (\ref{eq:MD phi_N hat convolve f hat convolve g hat estimate}). For that, we start by writing out the convolution of $\hat{\varphi}\hat{\mathbf{f}}$ with $\hat{\mathbf{g}}$: \begin{align} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q} & =\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\left(\hat{\varphi}\hat{\mathbf{f}}\right)\left(\mathbf{t}-\mathbf{s}\right)\hat{\mathbf{g}}\left(\mathbf{s}\right)\right\Vert _{q}\\ \left(\textrm{ultrametric ineq.}\right); & \leq\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\sup_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \left(\hat{\varphi}\hat{\mathbf{f}}\right)\left(\mathbf{t}-\mathbf{s}\right)\hat{\mathbf{g}}\left(\mathbf{s}\right)\right\Vert _{q}\\ \left(\hat{\mathbf{g}}\left(\mathbf{s}\right)=\mathbf{0},\textrm{ }\forall\left\Vert \mathbf{s}\right\Vert _{p}>p^{M}\right); & \leq\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}}\sup_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \left(\hat{\varphi}\hat{\mathbf{f}}\right)\left(\mathbf{t}-\mathbf{s}\right)\hat{\mathbf{g}}\left(\mathbf{s}\right)\right\Vert _{q}\\ & \leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \left(\hat{\varphi}\hat{\mathbf{f}}\right)\left(\mathbf{t}-\mathbf{s}\right)\right\Vert _{q} \end{align} Next, we write out the convolution $\hat{\varphi}\hat{\mathbf{f}}$. This gives us: \begin{align} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\mathbf{v}\in\hat{\mathbb{Z}}_{p}}\hat{\varphi}\left(\mathbf{t}-\mathbf{s}-\mathbf{v}\right)\hat{\mathbf{f}}\left(\mathbf{v}\right)\right\Vert _{q}\\ \left(\textrm{let }\mathbf{u}=\mathbf{s}+\mathbf{v}\right); & =\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\mathbf{u}-\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}-\mathbf{s}\right)\right\Vert _{q}\\ \left(\mathbf{s}+\hat{\mathbb{Z}}_{p}^{r}=\hat{\mathbb{Z}}_{p}^{r}\right); & =\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\mathbf{u}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}-\mathbf{s}\right)\right\Vert _{q} \end{align} We now invoke the vanishing of $\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)$ for all $\left\Vert \mathbf{t}-\mathbf{u}\right\Vert _{p}>p^{N}$. Indeed, because $\mathbf{t}$ is restricted to $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{M}$, observe that when $\left\Vert \mathbf{u}\right\Vert _{p}>p^{\max\left\{ M,N\right\} }$, the ultrametric inequality yields: \begin{equation} \left\Vert \mathbf{t}-\mathbf{u}\right\Vert _{p}=\max\left\{ \left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{u}\right\Vert _{p}\right\} >p^{\max\left\{ M,N\right\} }>p^{N} \end{equation} Hence, for $\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}$, the summand $\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}-\mathbf{s}\right)$ vanishes whenever $\left\Vert \mathbf{u}\right\Vert >p^{\max\left\{ M,N\right\} }$. This gives us: \begin{equation} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{M}}\left\Vert \sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}-\mathbf{s}\right)\right\Vert _{q} \end{equation} Next, we enlarge things slightly by expanding the range of $\left\Vert \mathbf{s}\right\Vert _{p}$ and $\left\Vert \mathbf{t}\right\Vert _{p}$ from $\leq p^{M}$ to $\leq p^{\max\left\{ M,N\right\} }$: \begin{equation} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q}\leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}-\mathbf{s}\right)\right\Vert _{q} \end{equation} Like in the one-dimensional case, everything is now on the same level: $\mathbf{t}$, $\mathbf{s}$, and $\mathbf{u}$ are all bounded in $p$-adic norm by $p^{\max\left\{ M,N\right\} }$. Exploiting the closure of the set: \[ \left\{ \mathbf{x}\in\hat{\mathbb{Z}}_{p}^{r}:\left\Vert \mathbf{x}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }\right\} \] under addition allows us to note that our $\mathbf{u}$-sum is invariant under the change of variables $\mathbf{u}\mapsto\mathbf{u}+\mathbf{s}$ for any $\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }$. Making this change of variables then gives us: \begin{align} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\varphi}\left(\mathbf{t}-\left(\mathbf{u}+\mathbf{s}\right)\right)\hat{\mathbf{f}}\left(\mathbf{u}\right)\right\Vert _{q}\\ & =\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\varphi}\left(\mathbf{t}-\mathbf{s}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}\right)\right\Vert _{q} \end{align} Because: \begin{equation} \left\{ \mathbf{t}-\mathbf{s}:\left\Vert \mathbf{t}\right\Vert _{p},\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }\right\} =\left\{ \mathbf{t}:\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }\right\} \end{equation} we then have: \begin{align} \left\Vert \hat{\varphi}\hat{\mathbf{f}}\hat{\mathbf{g}}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \sum_{\left\Vert \mathbf{u}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}\right)\right\Vert _{q}\\ & \leq\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \sum_{\mathbf{u}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\varphi}\left(\mathbf{t}-\mathbf{u}\right)\hat{\mathbf{f}}\left(\mathbf{u}\right)\right\Vert _{q}\\ & =\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\sup_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{\max\left\{ M,N\right\} }}\left\Vert \left(\hat{\varphi}\hat{\mathbf{f}}\right)\left(\mathbf{u}\right)\right\Vert _{q}\\ \left(\textrm{by definition}\right); & =\left\Vert \hat{\mathbf{g}}\right\Vert _{p^{\infty},q}\left\Vert \hat{\varphi}\hat{\mathbf{f}}\right\Vert _{p^{\max\left\{ M,N\right\} },q} \end{align} which proves the desired estimate. \end{claim} \vphantom{} Now, let's line everything up: first, we choose $\epsilon\in\left(0,1\right)$ and a non-identically-zero $\hat{\mathbf{h}}_{\epsilon}$ so that: \begin{equation} \left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}<\epsilon \end{equation} Then, by \textbf{Claim \ref{claim:5.12}}, we can choose an integer $M_{\epsilon}$ (depending only on $\epsilon$ and $\hat{\mathbf{h}}_{\epsilon}$) so that: \begin{equation} \left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}<\epsilon,\textrm{ }\forall M\geq M_{\epsilon} \end{equation} Since the norm is the maximum of the components, multiplying the left-hand side entry-wise by the standard basis vector $\mathbf{e}_{\ell}$ can only decrease the norm: \begin{equation} \left\Vert \mathbf{e}_{\ell}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right\Vert _{p^{M},q}\leq\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}<\epsilon,\textrm{ }\forall M\geq M_{\epsilon} \end{equation} Next, we observe that: \begin{equation} \left\Vert \mathbf{e}_{\ell}\left(\hat{\phi}_{N}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right)\right\Vert _{p^{M},q}=\left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\mathbf{e}_{\ell}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q}\label{eq:MD WTT - Target of attack} \end{equation} Like in the one-dimensional case, we will estimate this in two ways. First, we apply (\ref{eq:MD phi_N hat convolve f hat estimate}) from \textbf{Claim \ref{claim:5.14}} with $\hat{\varphi}=\mathbf{e}_{\ell}\hat{\phi}_{N}$ and $\hat{\mathbf{f}}=\mathbf{e}_{\ell}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)$. This gives us: \begin{align} \left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\mathbf{e}_{\ell}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q} & \leq\left\Vert \mathbf{e}_{\ell}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right\Vert _{p^{\max\left\{ M,N\right\} },q}\\ & \leq\left\Vert \vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right\Vert _{p^{\max\left\{ M,N\right\} },q}\\ \left(M\geq M_{\epsilon}\Rightarrow\textrm{use \textbf{Claim \ref{claim:5.12}}}\right); & <\epsilon \end{align} So, the \emph{left-hand side} of (\ref{eq:MD WTT - Target of attack}) is $<\epsilon$. To contradict this, we use the ultrametric inequality for the non-archimedean norm $\left\Vert \cdot\right\Vert _{p^{M},q}$ on the \emph{right-hand side} of (\ref{eq:MD WTT - Target of attack}). This yields: \begin{equation} \max\left\{ \left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\right\Vert _{p^{M},q},\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}\right\} \label{eq:WTT - Ultrametric inequality-1} \end{equation} as an upper bound for: \begin{equation} \left\Vert \mathbf{e}_{\ell}\left(\hat{\phi}_{N}-\left(\hat{\phi}_{N}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q} \end{equation} (Note that we could have also written $\mathbf{e}_{\ell}$ next to $\hat{\phi}_{N}$ and $\hat{\mathbf{h}}_{\epsilon,M}$, though it would have been redundant.) Using (\ref{eq:MD phi_N hat convolve f hat convolve g hat estimate}) from \textbf{Claim \ref{claim:5.14}}, we get: \begin{align} \left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\right\Vert _{p^{M},q}\left\Vert \hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}\\ \left(M\geq M_{\epsilon}\right); & \leq\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\right\Vert _{p^{M},q}\cdot\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q} \end{align} Now, let $\epsilon^{\prime}=\frac{\epsilon}{2\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}}$; we are allowed to do this, thanks to our assumption that $\hat{\mathbf{h}}_{\epsilon}$ was not identically zero. Seeing as $N$ is still arbitrary, let $N$ be larger than both $N_{\epsilon^{\prime}}$ and our choice for $M$; \textbf{Claim \ref{claim:5.13}} tells us that $\left\Vert \mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\phi}_{N}\right\Vert _{p^{M},q}<\epsilon^{\prime}$. Since the size of the $\ell$th component of $\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\phi}_{N}$ is: \begin{equation} \left\Vert \mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\phi}_{N}\right\Vert _{p^{M},q}<\epsilon^{\prime} \end{equation} we can write: \begin{align} \left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q} & \leq\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\right\Vert _{p^{M},q}\cdot\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}\\ & <\epsilon^{\prime}\cdot\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}\\ & =\frac{\epsilon}{2\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}}\cdot\left\Vert \hat{\mathbf{h}}_{\epsilon}\right\Vert _{p^{\infty},q}\\ & =\frac{\epsilon}{2} \end{align} for our large choice of $N$. Because $\left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\right\Vert _{p^{M},q}=1$ for all $M,N\geq0$, this shows that: \begin{equation} \left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\right\Vert _{p^{M},q}=1>\frac{\epsilon}{2}>\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q} \end{equation} Hence, by the ultrametric inequality, the upper bound (\ref{eq:WTT - Ultrametric inequality-1}) is in fact an equality: \begin{align} \left\Vert \mathbf{e}_{\ell}\left(\hat{\phi}_{N}-\left(\hat{\phi}_{N}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right\Vert _{p^{M},q} & =\max\left\{ \left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}\right\Vert _{p^{M},q},\left\Vert \hat{\phi}_{N}\mathbf{e}_{\ell}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right\Vert _{p^{M},q}\right\} \\ & =1 \end{align} With this, (\ref{eq:MD WTT - Target of attack}) becomes: \begin{align} \epsilon & >\left\Vert \mathbf{e}_{\ell}\left(\hat{\phi}_{N}\left(\vec{\mathbf{1}}_{\mathbf{0}}-\left(\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right)\right)\right\Vert _{p^{M},q}\\ & =\left\Vert \mathbf{e}_{\ell}\hat{\phi}_{N}-\mathbf{e}_{\ell}\left(\hat{\phi}_{N}\hat{\mathcal{M}}\hat{\mathbf{h}}_{\epsilon,M}\right)\right\Vert _{p^{M},q}\\ & =1 \end{align} But $\epsilon<1$!\textemdash we have arrived at our contradiction. Consequently, the existence of the ``zero'' $\mathbf{z}_{0}$ precludes the translates of $\hat{\mathcal{M}}$ from being dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d}\right)$. Q.E.D. \begin{rem} Versions of \textbf{Theorem \ref{thm:MD pq-adic WTT for continuous functions}} likely hold for arbitrary thick $\left(p,q\right)$-adic measures, but I will not pursue the matter here. \end{rem} \subsection{\emph{\label{subsec:5.4.4More-Fourier-Resummation}More} Fourier Resummation Lemmata} As in the one-dimensional case, we will need to avail ourselves of various Fourier Resummation Lemmata. First, of course, we need the multi-dimensional definitions: \begin{defn} A $\left(p,q\right)$-adic function $\hat{\mu}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\rho,c}$ is said to be: \vphantom{} I. \textbf{Radial} whenever:\index{thick measure!radial} \[ \hat{\mu}\left(\mathbf{t}\right)=\hat{\mu}\left(\left\|t_{1}\right\|_{p_{1}}^{-1},\ldots,\left\|t_{r}\right\|_{p_{r}}^{-1}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} \] \vphantom{} II. \textbf{Magnitudinal} \index{thick measure!magnitudinal}whenever there is a function $\kappa:\mathbb{N}_{0}^{r}\rightarrow\mathbb{C}_{q}^{\rho,c}$ so that: \[ \hat{\mu}\left(\mathbf{t}\right)=\begin{cases} \kappa\left(\mathbf{0}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)} & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \] \vphantom{} III. \textbf{Radially-Magnitudinal} \index{thick measure!radial-magnitudinal} whenever there is a radial function $\hat{\nu}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\left(r,c\right)}$ and a magnitudinal function $\hat{\eta}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{\left(\rho,r\right)}$ so that: \begin{equation} \hat{\mu}\left(\mathbf{t}\right)=\hat{\eta}\left(\mathbf{t}\right)\hat{\nu}\left(\mathbf{t}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} where the right-hand side, as a product of a $\rho\times r$ matrix-valued function and an $r\times c$ matrix-valued function is a $\rho\times c$ matrix-valued function. \end{defn} \begin{defn} We say a thick measure $\mathcal{M}$ is radial (resp. magnitudinal, radially-magnitudinal) whenever $\hat{\mathcal{M}}\left(\mathbf{t}\right)$ is radial (resp. magnitudinal, radially-magnitudinal). \end{defn} \begin{defn} Consider a function $\kappa:\mathbb{N}_{0}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$. \vphantom{} I. We say $\kappa$ is \textbf{$p$-adically structured} / \textbf{has $p$-adic structure }whenever:\index{$p$-adic!structure} \begin{equation} \kappa\left(p\mathbf{n}+\mathbf{j}\right)=\mathbf{M}_{\mathbf{j}}\kappa_{H}\left(\mathbf{n}\right)\mathbf{M}_{\mathbf{0}}^{-1},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\label{eq:Definition of P-adic structure} \end{equation} where $\left\{ \mathbf{M}_{\mathbf{j}}\right\} _{\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}}$ are invertible matrices in $\mathbb{C}_{q}^{d,d}$. \vphantom{} II. We say $\kappa$\textbf{ }is \index{tame}\textbf{$\left(p,K\right)$-adically tame} on a set $X\subseteq\mathbb{Z}_{p}^{r}$ whenever $\lim_{n\rightarrow\infty}\left\Vert \kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{K}=0$ for all $\mathbf{z}\in X$. We do not mention $X$ when $X=\mathbb{Z}_{p}^{r}$. If $K$ is a $q$-adic field, we say $\kappa$ is $\left(p,q\right)$-adically tame on $X$; if $K$ is archimedean, we say $\kappa$ is $\left(p,\infty\right)$-adically tame on $X$. \end{defn} \begin{prop} \label{prop:MD p-adic structure prop}A function $\kappa:\mathbb{N}_{0}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ is $p$-adically structured if and only if, for every $m\geq0$: \begin{equation} \kappa\left(\mathbf{n}+\mathbf{j}p^{m}\right)=\mathbf{M}_{\mathbf{n}}\kappa_{H}\left(\mathbf{j}\right)\mathbf{M}_{\mathbf{0}}^{-1},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{n}\leq p^{m}-1 \end{equation} where $\mathbf{M}_{\mathbf{n}}=\mathbf{M}_{\mathbf{j}_{1}}\times\cdots\times\mathbf{M}_{\mathbf{j}_{\lambda_{p}\left(\mathbf{n}\right)}}$, where $\mathbf{J}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\lambda_{p}\left(\mathbf{n}\right)}\right)\in\textrm{String}^{r}\left(p\right)$ is the shortest block string representing $\mathbf{n}$. \end{prop} Proof: Write: \[ \mathbf{n}=\left(n_{1},\ldots,n_{r}\right) \] \[ \mathbf{j}=\left(j_{1},\ldots,j_{r}\right) \] where: \[ n_{\ell}=n_{\ell,0}+n_{\ell,1}p+\cdots+n_{\ell,\lambda_{p}\left(n_{\ell}\right)-1}p^{\lambda_{p}\left(n_{\ell}\right)-1},\textrm{ }\forall\ell\in\left\{ 1,\ldots,r\right\} \] and: \[ \mathbf{n}+\mathbf{j}p^{m}=\left(n_{1}+j_{1}p^{m},\ldots,n_{r}+j_{r}p^{m}\right) \] where, for each $\ell$, $\lambda_{p}\left(n_{\ell}\right)-1<m$. The rest is just like the one-dimensional case. Q.E.D. \vphantom{}Next up, the multi-dimensional resummation lemmata. \begin{lem} \label{lem:MD magnitude F Resum Lemma}Let $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ be the Fourier-Stieltjes transform of a magnitudinal thick measure, and let $\kappa$ have $p$-adic structure. Then, for all $N\geq0$ and all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =p^{rN}\kappa\left(\left[\mathbf{z}\right]_{p^{N}}\right)-\sum_{n=0}^{N-1}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n+1}\right)\label{eq:MD magnitudinal resummation formula} \end{align} \end{lem} Proof: We begin with: \[ \hat{\mathcal{M}}\left(\mathbf{t}\right)=\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)} \] Consequently: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{m}\right)\right\} _{p}}\\ & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)\left(p^{rn}\left[\mathbf{z}\overset{p^{n}}{\equiv}\mathbf{m}\right]-p^{r\left(n-1\right)}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right]\right)\\ & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\left(p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-p^{r\left(n-1\right)}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right]\right) \end{align} Because $\kappa$ has $p$-adic structure, we can then write: \begin{align} \sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}\kappa\left(\mathbf{m}+\mathbf{j}p^{n}\right)\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}+\mathbf{j}p^{n}\right]\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n-1}}+\mathbf{j}p^{n}\right) \end{align} and so: \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\mathbf{t}\cdot\mathbf{m}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}=p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-p^{r\left(n-1\right)}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n-1}}+\mathbf{j}p^{n}\right)\label{eq:MD Level set Fourier sum of a magnitudinal multiplier} \end{equation} Consequently: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\left(p^{rn}\kappa\left(\left[\mathbf{z}\right]_{^{n}}\right)-p^{r\left(n-1\right)}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right]\right)\\ & =\kappa\left(\mathbf{0}\right)+\sum_{n=1}^{N}\left(p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-p^{r\left(n-1\right)}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n-1}}+\mathbf{j}p^{n}\right)\right)\\ & =\sum_{n=0}^{N}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-\sum_{n=0}^{N-1}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n+1}\right)\\ & =\sum_{n=0}^{N}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-\sum_{n=0}^{N-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-\sum_{n=0}^{N-1}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n+1}\right)\\ & =p^{rN}\kappa\left(\left[\mathbf{z}\right]_{p^{N}}\right)-\sum_{n=0}^{N-1}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n+1}\right) \end{align} Q.E.D. \begin{lem} \label{lem:MD radial-magnitude F Resum Lemma}Let $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ be the Fourier-Stieltjes transform of a radially-magnitudinal thick measure, and let $\kappa$ have $p$-adic structure. Then, for all $N\geq0$ and all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =p^{rN}\kappa\left(\left[\mathbf{z}\right]_{p^{N}}\right)\hat{\nu}\left(\frac{1}{p^{N}}\right)+\sum_{n=0}^{N-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\hat{\nu}\left(\frac{1}{p^{n}}\right)-\hat{\nu}\left(\frac{1}{p^{n+1}}\right)\right)\label{eq:MD radially-magnitudinal resummation formula}\\ & -\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\hat{\nu}\left(\frac{1}{p^{n+1}}\right)\nonumber \end{align} \end{lem} Proof: Write: \[ \hat{\mathcal{M}}\left(\mathbf{t}\right)=\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa\left(\mathbf{m}\right)\hat{\nu}\left(\mathbf{t}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)} \] Then: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa\left(\mathbf{m}\right)\hat{\nu}\left(\mathbf{t}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =\kappa\left(\mathbf{0}\right)\hat{\nu}\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa\left(\mathbf{m}\right)\hat{\nu}\left(\mathbf{t}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ \left(\textrm{use }(\ref{eq:First MD level set summation identity})\right); & =\kappa\left(\mathbf{0}\right)\hat{\nu}\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\hat{\nu}\left(\frac{1}{p^{n}}\right) \end{align} Using (\ref{eq:MD Level set Fourier sum of a magnitudinal multiplier}) from the proof of \textbf{Lemma \ref{lem:MD magnitude F Resum Lemma}} gives us: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =\kappa\left(\mathbf{0}\right)\hat{\nu}\left(\mathbf{0}\right)+\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\hat{\nu}\left(\frac{1}{p^{n}}\right)\\ & =\kappa\left(\mathbf{0}\right)\hat{\nu}\left(\mathbf{0}\right)+\sum_{n=1}^{N}\left(p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)-p^{r\left(n-1\right)}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n-1}}+\mathbf{j}p^{n-1}\right)\right)\hat{\nu}\left(\frac{1}{p^{n}}\right)\\ & =\sum_{n=0}^{N}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\hat{\nu}\left(\frac{1}{p^{n}}\right)-\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\hat{\nu}\left(\frac{1}{p^{n+1}}\right)\\ & =\sum_{n=0}^{N}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\hat{\nu}\left(\frac{1}{p^{n}}\right)-\sum_{n=0}^{N-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\hat{\nu}\left(\frac{1}{p^{n+1}}\right)\\ & -\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\hat{\nu}\left(\frac{1}{p^{n+1}}\right) \end{align} and so: \begin{align} \tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right) & =p^{rN}\kappa\left(\left[\mathbf{z}\right]_{p^{N}}\right)\hat{\nu}\left(\frac{1}{p^{N}}\right)+\sum_{n=0}^{N-1}p^{rn}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\hat{\nu}\left(\frac{1}{p^{n}}\right)-\hat{\nu}\left(\frac{1}{p^{n+1}}\right)\right)\\ & -\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\hat{\nu}\left(\frac{1}{p^{n+1}}\right) \end{align} Q.E.D. \begin{prop} \label{prop:MD convolution with v_p identity}Let $\hat{\mathcal{M}}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$ be any function. Then: \begin{equation} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\mathbb{C}_{q}^{d,d}}{=}-N\tilde{\mathcal{M}}_{N}\left(\mathbf{z}\right)+\sum_{n=0}^{N-1}\tilde{\mathcal{M}}_{n}\left(\mathbf{z}\right)\label{eq:MD Fourier sum of v_p times mu-hat} \end{equation} \end{prop} Proof: \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} } & =\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}v_{p}\left(\mathbf{t}\right)\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =-\sum_{n=1}^{N}n\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\hat{\mathcal{M}}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =-\sum_{n=1}^{N}n\left(\tilde{\mathcal{M}}_{n}\left(\mathbf{z}\right)-\tilde{\mathcal{M}}_{n-1}\left(\mathbf{z}\right)\right) \end{align} The rest of the proof is formally identical to its one-dimensional counterpart (\textbf{Proposition \ref{prop:v_p of t times mu hat sum}}). Q.E.D. \chapter{\label{chap:6 A-Study-of}A Study of $\chi_{H}$ - The Multi-Dimensional Case} \includegraphics[scale=0.45]{./PhDDissertationEroica6.png} \vphantom{} IN THIS CHAPTER, UNLESS STATED OTHERWISE, WE ASSUME $p$ IS A PRIME AND THAT $H$ IS A CONTRACTING, SEMI-BASIC $p$-SMOOTH $d$-DIMENSIONAL DEPTH $r$ HYDRA MAP WHICH FIXES $\mathbf{0}$. \vphantom{} The layout of Chapter 6 is much like that of its one-dimensional predecessor\textemdash Chapter 4. After an initial string of notational definitions and preparatory work, we investigate $\chi_{H}$ by squeezing out explicit ``asymptotic formulae'' for the Fourier transform of the $N$th truncation of $\chi_{H}$; this is the content of Subsection \ref{subsec:6.2.1 -and-}, with the formulae themselves occurring in \textbf{Theorem \ref{thm:MD N,t asympotics for Chi_H,N hat}}. Just as $1-\alpha_{H}\left(0\right)$ was the key determinative quantity for the one-dimensional case, so too will $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ be a key quantity for the multi-dimensional case, where $\alpha_{H}$ (defined at the start of Section \ref{sec:6.1. Preparatory-Work-(Again)}) is the multi-dimensional generalization of the one-dimensional $\alpha_{H}$. However, the non-commutativity of matrix multiplication will, unfortunately, complicate matters somewhat. As a result, at the beginning of \ref{sec:6.1. Preparatory-Work-(Again)}, I introduce a qualitative notion of the \textbf{commutativity }of a $p$-Hydra map $H$. This condition turns out to be exactly what is needed in order to minimize the computational differences between the one-dimensional case and the multi-dimensional case currently under consideration. The primary consequence of this issue is that I have only been able to establish quasi-integrability results and formulae for Fourier transforms of the multi-dimensional $\chi_{H}$ in the case where $H$ is commutative. A treatment of this case is given in Subsection \ref{subsec:6.2.3 Multi-Dimensional--=00003D000026}, following the $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ case dealt with in \ref{subsec:6.2.2 Multi-Dimensional--=00003D000026}. That being said, as a headache-preventing prophylaxis for future explorers of this subject, \emph{I have taken the liberty of doing all the major computations without relying on the assumption that $H$ is commutative}. For ease of readability, alongside these non-commutative cases, I also state the simpler commutative cases for all significant formulae. Subsection \ref{subsec:6.2.4 Multi-Dimensional--=00003D000026} contains additional computations leading all the way up to the doorstep of a proof of the quasi-integrability of $\chi_{H}$ for non-commutative $H$, leaving the final step as a conjecture to be tackled in future work. \section{\label{sec:6.1. Preparatory-Work-(Again)}Preparatory Work (Again)} This section is just a multi-dimensional copy of its predecessor in Section \pageref{sec:4.1 Preparatory-Work--}. \begin{defn} We define the functions $\alpha_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\textrm{GL}_{d}\left(\overline{\mathbb{Q}}\right)$ and $\beta_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ by:\nomenclature{$\alpha_{H}\left(\mathbf{t}\right)$}{ }\nomenclature{$\beta_{H}\left(\mathbf{t}\right)$}{ } \begin{equation} \alpha_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\label{eq:MD definition of alpha_H} \end{equation} \begin{equation} \beta_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\label{eq:MD definition of beta_H} \end{equation} We then write $\gamma_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ to denote:\nomenclature{$\gamma_{H}\left(\mathbf{t}\right)$}{ } \begin{equation} \gamma_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\left(\alpha_{H}\left(\mathbf{t}\right)\right)^{-1}\beta_{H}\left(\mathbf{t}\right)\label{eq:MD definition of gamma_H} \end{equation} \end{defn} \begin{defn} We say $H$ is \textbf{non-singular }whenever the matrix $\alpha_{H}\left(\mathbf{j}/p\right)$ is invertible for all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. \index{Hydra map!non-singular} \end{defn} \vphantom{} The primary distinction between the work we will do here and the one-dimensional case comes from the non-commutativity of matrix multiplication. Despite this, there is a simple\textemdash and not unreasonable\textemdash qualitative condition we can place on $H$ to make the computational distinctions between the one-dimensional and multi-dimensional cases all-but-trivial: \begin{defn} We say $H$ is\index{Hydra map!commutative} \textbf{commutative }whenever $\alpha_{H}\left(\mathbf{0}\right)$ commutes with $H^{\prime}\left(\mathbf{0}\right)$: \begin{equation} \alpha_{H}\left(\mathbf{0}\right)H^{\prime}\left(\mathbf{0}\right)=H^{\prime}\left(\mathbf{0}\right)\alpha_{H}\left(\mathbf{0}\right)\label{eq:Definition of H commutativity} \end{equation} \end{defn} \vphantom{} Next, we introduce the multi-dimensional analogue of $\kappa_{H}$, here a matrix-valued function. \begin{defn} \nomenclature{$\kappa_{H}\left(\mathbf{n}\right)$}{ }We define the function $\kappa_{H}:\mathbb{N}_{0}^{r}\rightarrow\textrm{GL}_{d}\left(\mathbb{Q}\right)$ by: \begin{equation} \kappa_{H}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}M_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{n}\right)},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\label{eq:MD definition of Kappa_H} \end{equation} \end{defn} \vphantom{} \textbf{Proposition \ref{prop:MD generating function}}, given below, contains the multi-dimensional analogue of the generating function identities from Chapter \ref{chap:4 A-Study-of}'s \textbf{Proposition \ref{prop:Generating function identities}}. \begin{prop}[\textbf{A Generating Function Identity}] \label{prop:MD generating function}For all $n\geq1$, all scalars $z$, and all $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$: \begin{equation} \prod_{m=0}^{n-1}\left(\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}z^{\mathbf{j}\cdot p^{m}\mathbf{t}}\right)=\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n-\lambda_{p}\left(\mathbf{m}\right)}z^{\mathbf{m}\cdot\mathbf{t}}\label{eq:MD M_H partial sum generating identity} \end{equation} \end{prop} Proof: Each term on the right of (\ref{eq:MD M_H partial sum generating identity}) is obtained by taking a product of the form: \begin{equation} \prod_{\ell=0}^{n-1}\frac{\mathbf{A}_{\mathbf{j}_{\ell}}}{\mathbf{D}_{\mathbf{j}_{\ell}}}z^{\mathbf{j}_{\ell}\cdot p^{\ell}\mathbf{t}}=\left(\prod_{\ell=0}^{n-1}\frac{\mathbf{A}_{\mathbf{j}_{\ell}}}{\mathbf{D}_{\mathbf{j}_{\ell}}}\right)z^{\mathbf{t}\cdot\sum_{\ell=0}^{n-1}p^{\ell}\mathbf{j}_{\ell}} \end{equation} for some choice of $\mathbf{j}_{1},\ldots,\mathbf{j}_{n-1}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. Here, writing $\mathbf{j}_{\ell}=\left(j_{\ell,1},\ldots,j_{\ell,r}\right)$ we have: \begin{equation} \sum_{\ell=0}^{n-1}\mathbf{j}_{\ell}p^{\ell}=\left(\sum_{\ell=0}^{n-1}j_{\ell,1}p^{\ell},\ldots,\sum_{\ell=0}^{n-1}j_{\ell,r}p^{\ell}\right) \end{equation} So, writing $\mathbf{J}$ to denote the block string $\left(\mathbf{j}_{0},\ldots,\mathbf{j}_{n-1}\right)$, we define $\mathbf{m}\in\mathbb{N}_{0}^{r}$ by: \begin{equation} \mathbf{m}\overset{\textrm{def}}{=}\sum_{\ell=0}^{n-1}\mathbf{j}_{\ell}p^{\ell}\label{eq:definition of bold m} \end{equation} so that $\mathbf{m}$ is then represented by $\mathbf{J}$. As such: \begin{equation} \prod_{m=0}^{n-1}\left(\sum_{\mathbf{j}=\mathbf{0}}^{-1}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}z^{\mathbf{j}\cdot\left(p^{m}\mathbf{t}\right)}\right)=\sum_{\begin{array}{c} \mathbf{J}\in\textrm{String}^{r}\left(p\right)\\ \left\|\mathbf{J}\right\|=n \end{array}}\left(\frac{\mathbf{A}_{\mathbf{j}_{0}}}{\mathbf{D}_{\mathbf{j}_{0}}}\times\cdots\times\frac{\mathbf{A}_{\mathbf{j}_{n-1}}}{\mathbf{D}_{\mathbf{j}_{n-1}}}\right)z^{\mathbf{J}\cdot\mathbf{t}} \end{equation} As an illustrative example, let $n=4$, consider: \begin{equation} \mathbf{J}=\left(\mathbf{j}_{0},\mathbf{j}_{1},\mathbf{0},\mathbf{0}\right) \end{equation} and let $\mathbf{m}$ be the tuple represented by this $\mathbf{J}$. Note that this $\mathbf{J}$ is the unique \emph{length $4$ }block string representing $\mathbf{m}$; any other length $4$ block string will have a non-zero tuple in either the $\mathbf{j}_{2}$ slot or the $\mathbf{j}_{3}$ slot. More generally, there will be a bijection between the set of all length $n$ block strings and the set of all $r$-tuples of integers \emph{representable }by said block strings. This set of $r$-tuples of integers is precisely: \begin{equation} \left\{ \left(m_{1},\ldots,m_{r}\right):0\leq m_{\ell}\leq p^{n}-1\textrm{ }\forall\ell\in\left\{ 1,\ldots,r\right\} \right\} \end{equation} More compactly, this is exactly the range of tuples indicated by the summation notation: \begin{equation} \sum_{\mathbf{m}=\mathbf{0}}^{p^{N}-1} \end{equation} The problem here is that any terminal $\mathbf{0}$-tuples in $\mathbf{J}$ they affect the matrix product associated to $\mathbf{J}$: \begin{equation} \frac{\mathbf{A}_{\mathbf{j}_{0}}}{\mathbf{D}_{\mathbf{j}_{0}}}\times\cdots\times\frac{\mathbf{A}_{\mathbf{j}_{n-1}}}{\mathbf{D}_{\mathbf{j}_{n-1}}} \end{equation} even though those $\mathbf{0}$-tuples do \emph{not} affect $z^{\mathbf{J}\cdot\mathbf{t}}$. For example, for $\mathbf{J}=\left(\mathbf{j}_{0},\mathbf{j}_{1},\mathbf{0},\mathbf{0}\right)$, the product would be: \begin{align} \frac{\mathbf{A}_{\mathbf{j}_{0}}}{\mathbf{D}_{\mathbf{j}_{0}}}\times\frac{\mathbf{A}_{\mathbf{j}_{1}}}{\mathbf{D}_{\mathbf{j}_{1}}}\times\frac{\mathbf{A}_{\mathbf{0}}}{\mathbf{D}_{\mathbf{0}}}\times\frac{\mathbf{A}_{\mathbf{0}}}{\mathbf{D}_{\mathbf{0}}} & =M_{H}\left(\left(\mathbf{j}_{0},\mathbf{j}_{1}\right)\right)\times\left(\frac{\mathbf{A}_{\mathbf{0}}}{\mathbf{D}_{\mathbf{0}}}\right)^{2}\\ \left(\mathbf{m}\textrm{ is represented by }\left(\mathbf{j}_{0},\mathbf{j}_{1}\right)\right); & =M_{H}\left(\mathbf{m}\right)\times\left(H^{\prime}\left(\mathbf{0}\right)\right)^{2} \end{align} So, we need to modify the product to take into account terminal $\mathbf{0}$s in $\mathbf{J}$. To do this, consider the worst-case scenario where $\mathbf{J}$ is a length-$n$ block string whose every $\mathbf{j}$ is $\mathbf{0}$. Letting $\mathbf{m}$ be the unique $r$-tuple of integers represented by an arbitrary length-$n$ $\mathbf{J}$, observe that $\lambda_{p}\left(\mathbf{m}\right)$ is the number of tuples in $\mathbf{J}$ which are \emph{not }part of a possible run of consecutive terminal $\mathbf{0}$s. This is because $\lambda_{p}\left(\mathbf{m}\right)$ is the length of the shortest block string representing $\mathbf{m}$. Consequently, for each $\mathbf{J}$, we have: \begin{equation} \frac{\mathbf{A}_{\mathbf{j}_{0}}}{\mathbf{D}_{\mathbf{j}_{0}}}\times\cdots\times\frac{\mathbf{A}_{\mathbf{j}_{n-1}}}{\mathbf{D}_{\mathbf{j}_{n-1}}}=M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n-\lambda\left(\mathbf{m}\right)} \end{equation} which leaves us with: \begin{align} \prod_{m=0}^{n-1}\left(\sum_{\mathbf{j}=\mathbf{0}}^{-1}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}z^{\mathbf{j}\cdot\left(^{m}\mathbf{t}\right)}\right) & =\sum_{\begin{array}{c} \mathbf{J}\in\textrm{String}^{r}\left(p\right)\\ \left\|\mathbf{J}\right\|=n \end{array}}\left(\frac{\mathbf{A}_{\mathbf{j}_{0}}}{\mathbf{D}_{\mathbf{j}_{0}}}\times\cdots\times\frac{\mathbf{A}_{\mathbf{j}_{n-1}}}{\mathbf{D}_{\mathbf{j}_{n-1}}}\right)z^{\mathbf{J}\cdot\mathbf{t}}\\ & =\sum_{\mathbf{m}=\mathbf{0}}^{^{n}-1}M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n-\lambda\left(\mathbf{m}\right)}z^{\mathbf{m}\cdot\mathbf{t}} \end{align} which is, of course, our sought-after identity. Q.E.D. \vphantom{} In the process of writing this chapter, I took pains to devise a notation which would make multi-dimensional case's computations as near as possible to verbatim repeats of their one-dimensional predecessors. The principal difficulty our notation must overcome is the non-commutativity of matrix multiplication. This manifests most strongly in the functional equations for the multi-dimensional $\kappa_{H}$. In order to keep our computations from spilling out into the margins of the page\textemdash or beyond\textemdash we will need to introduce a notation for the linear operator which conjugates $d\times d$ matrices by the $n$th power of $H^{\prime}\left(\mathbf{0}\right)$. \begin{defn} For any $n\in\mathbb{N}_{0}$ and any $\mathbf{A}\in\textrm{GL}_{d}\left(\overline{\mathbb{Q}}\right)$, $\textrm{GL}_{d}\left(\mathbb{C}\right)$, or $\textrm{GL}_{d}\left(\mathbb{C}_{q}\right)$, we define: \nomenclature{$\mathcal{C}_{H}\left(\mathbf{A}:n\right)$}{$\overset{\textrm{def}}{=}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\mathbf{A}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}$} \begin{equation} \mathcal{C}_{H}\left(\mathbf{A}:n\right)\overset{\textrm{def}}{=}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\mathbf{A}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\label{eq:Definition of script C_H} \end{equation} \end{defn} \begin{rem} The most important thing to note is that: \begin{equation} \mathcal{C}_{H}\left(\mathbf{I}_{d}:n\right)=\mathbf{I}_{d}\label{eq:Script C_H of I_d} \end{equation} This property is responsible for the impact had on the dynamics of $H$ when $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ (the multi-dimensional analogue of the ``$\alpha_{H}\left(0\right)=1$'' case.). Additionally, note that $\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):n\right)=\alpha_{H}\left(\mathbf{0}\right)$ whenever $H$ is commutative. \end{rem} \begin{lem}[\textbf{Properties of Multi-Dimensional $\kappa_{H}$}] \label{lem:properties of MD kappa_H}\ \vphantom{} I. \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{j}\right)=\alpha_{H}\left(\mathbf{0}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}-\mathbf{I}_{d}\label{eq:MD kappa H sum in terms of MD alpha} \end{equation} \vphantom{} II. $\kappa_{H}$ is the unique function $\mathbb{N}_{0}^{r}\rightarrow\textrm{GL}_{d}\left(\mathbb{Q}\right)$ satisfying the functional equations: \begin{equation} \kappa_{H}\left(p\mathbf{n}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\kappa_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{d},\textrm{ }\forall\mathbf{j}\leq p-1\label{eq:MD Kappa_H functional equations} \end{equation} subject to the initial condition $\kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. \vphantom{} III. $\kappa_{H}$ satisfies: \begin{equation} \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right)=\kappa_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\label{eq:MD Kappa_H has P-adic structure} \end{equation} for all $k\in\mathbb{N}_{1}$, all $\mathbf{m}\leq p^{k}-1$, and all $\mathbf{j}\leq p-1$. Equivalently, for these parameters: \begin{equation} \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right)=\kappa_{H}\left(\mathbf{m}\right)\mathcal{C}_{H}\left(\kappa_{H}\left(\mathbf{j}\right):\lambda_{p}\left(\mathbf{m}\right)\right)\label{eq:MD Kappa_H has P-adic structure, with script C_H} \end{equation} \vphantom{} IV. If $H$ is semi-basic, then $\kappa_{H}$ is $\left(p,q_{H}\right)$-adically regular, with: \begin{equation} \lim_{n\rightarrow\infty}\left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{q_{H}}\overset{\mathbb{R}}{=}0,\textrm{ }\forall\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\label{eq:MD Semi-basic q-adic decay for Kappa_H} \end{equation} \vphantom{} V. If $H$ is contracting, then: \begin{equation} \lim_{N\rightarrow\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\overset{\mathbb{R}^{d,d}}{=}\mathbf{O}_{d},\textrm{ }\forall\mathbf{z}\in\mathbb{N}_{0}^{r}\label{eq:MD Kappa_H decay when H is contracting} \end{equation} \end{lem} Proof: I. \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{j}\right)=\sum_{\mathbf{j}>\mathbf{0}}^{p-1}M_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{j}\right)} \end{equation} Since $\lambda_{p}\left(\mathbf{j}\right)=1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $, and since: \begin{equation} M_{H}\left(\mathbf{j}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}} \end{equation} we have: \begin{align} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{j}\right) & =\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ & =\frac{1}{p^{r}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}\\ \left(\mathbf{D}_{\mathbf{0}}^{-1}\mathbf{A}_{\mathbf{0}}=H^{\prime}\left(\mathbf{0}\right)\right); & =\frac{1}{p^{r}}\left(-H^{\prime}\left(\mathbf{0}\right)+\sum_{\mathbf{j}=\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}\\ & =-\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}+\underbrace{\left(\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\right)}_{\alpha_{H}\left(\mathbf{0}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}\\ & =-\mathbf{I}_{d}+\alpha_{H}\left(\mathbf{0}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1} \end{align} \vphantom{} II. Let $\mathbf{n}\in\mathbb{N}_{0}^{r}\backslash\left\{ \mathbf{0}\right\} $ and let $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. Then: \begin{align} \lambda_{p}\left(p\mathbf{n}+\mathbf{j}\right) & =\lambda_{p}\left(\mathbf{n}\right)+1\\ M_{H}\left(p\mathbf{n}+\mathbf{j}\right) & =M_{H}\left(\mathbf{n}\right)\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}=M_{H}\left(\mathbf{n}\right)\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}} \end{align} As such: \begin{align} \kappa_{H}\left(p\mathbf{n}+\mathbf{j}\right) & =M_{H}\left(p\mathbf{n}+\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(p\mathbf{n}+\mathbf{j}\right)}\\ & =\left(\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}M_{H}\left(\mathbf{n}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{n}\right)-1}\\ & =\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\overbrace{\left(M_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{n}\right)}\right)}^{\kappa_{H}\left(\mathbf{n}\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ \left(H^{\prime}\left(\mathbf{0}\right)=\mathbf{D}_{\mathbf{0}}^{-1}\mathbf{A}_{\mathbf{0}}\right); & =\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\kappa_{H}\left(\mathbf{n}\right)\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}} \end{align} Next: \begin{align} \kappa_{H}\left(\mathbf{j}\right) & =\begin{cases} M_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-0} & \textrm{if }\mathbf{j}=\mathbf{0}\\ M_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1} & \textrm{if }\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} \end{cases}\\ & =\begin{cases} \mathbf{I}_{d} & \textrm{if }\mathbf{j}=\mathbf{0}\\ \frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\frac{\mathbf{A}_{\mathbf{0}}}{\mathbf{D}_{\mathbf{0}}} & \textrm{if }\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} \end{cases} \end{align} where: \begin{equation} \frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\frac{\mathbf{A}_{\mathbf{0}}}{\mathbf{D}_{\mathbf{0}}}=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\mathbf{D}_{\mathbf{0}}^{-1}\mathbf{A}_{\mathbf{0}} \end{equation} Since this shows $\kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, we can then write: \begin{equation} \kappa_{H}\left(\mathbf{j}\right)=\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\kappa_{H}\left(\mathbf{0}\right),\textrm{ }\forall\mathbf{j}\leq p-1 \end{equation} Combining this last equation with the other cases computed above yields (\ref{eq:MD Kappa_H functional equations}). The uniqueness follows by an inductive argument, showing that the initial condition $\kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ uniquely determines solutions of the functional equation (\ref{eq:MD Kappa_H functional equations}) on $\mathbb{N}_{0}^{r}$. Finally, we put things in the desired form by recalling that $H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}$ for all $\mathbf{j}$, and that $H^{\prime}\left(\mathbf{0}\right)=H_{\mathbf{0}}^{\prime}\left(\mathbf{0}\right)$. \vphantom{} III. Let $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ and $k\geq1$ be arbitrary, and let $\mathbf{m}\in\mathbb{N}_{0}^{r}$ satisfy $\lambda\left(\mathbf{m}\right)\leq k$ By definition of multi-dimensional $\kappa_{H}$ (equation (\ref{eq:MD definition of Kappa_H}) and using the functional equations for $M_{H}$ (\textbf{Proposition \ref{prop:MD M_H functional equations}}) and $\lambda_{p}$ (\textbf{Proposition \ref{prop:MD lambda and digit-number functional equations}}) \begin{align} \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right) & =M_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}+p^{k}\mathbf{j}\right)}\\ & =M_{H}\left(\mathbf{m}\right)M_{H}\left(p^{k}\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)-\lambda_{p}\left(\mathbf{j}\right)}\\ \left(M_{H}\left(p^{k}\mathbf{j}\right)=M_{H}\left(\mathbf{j}\right)\right); & =M_{H}\left(\mathbf{m}\right)M_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)-\lambda_{p}\left(\mathbf{j}\right)}\\ & =M_{H}\left(\mathbf{m}\right)\underbrace{M_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{j}\right)}}_{\kappa_{H}\left(\mathbf{j}\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\\ & =M_{H}\left(\mathbf{m}\right)\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)} \end{align} Inserting $\mathbf{I}_{d}=\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}$ in between $M_{H}\left(\mathbf{m}\right)$ and $\kappa_{H}\left(\mathbf{j}\right)$ yields: \begin{align} \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right) & =\underbrace{M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}}_{\kappa_{H}\left(\mathbf{m}\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\\ & =\kappa_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)} \end{align} as desired. \vphantom{} IV. Applying $q$-adic norm to (\ref{eq:MD Kappa_H has P-adic structure}) gives us: \begin{align} \left\Vert \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right)\right\Vert _{q} & \leq\left\Vert \kappa_{H}\left(\mathbf{m}\right)\right\Vert _{q}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}^{\lambda_{p}\left(\mathbf{m}\right)}\left\Vert \kappa_{H}\left(\mathbf{j}\right)\right\Vert _{q}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}^{-\lambda_{p}\left(\mathbf{m}\right)}\\ & =\left\Vert \kappa_{H}\left(\mathbf{m}\right)\right\Vert _{q}\left\Vert \kappa_{H}\left(\mathbf{j}\right)\right\Vert _{q}\\ & =\left\Vert M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\right\Vert _{q}\left\Vert M_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{j}\right)}\right\Vert _{q}\\ & \leq\left\Vert M_{H}\left(\mathbf{m}\right)\right\Vert _{q}\left\Vert M_{H}\left(\mathbf{j}\right)\right\Vert _{q}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}^{-\lambda_{p}\left(\mathbf{m}\right)-\lambda_{p}\left(\mathbf{j}\right)} \end{align} Since $H$ is semi-basic, $\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}=1$, and so: \begin{equation} \left\Vert \kappa_{H}\left(\mathbf{m}+p^{k}\mathbf{j}\right)\right\Vert _{q}\leq\left\Vert M_{H}\left(\mathbf{m}\right)\right\Vert _{q}\left\Vert M_{H}\left(\mathbf{j}\right)\right\Vert _{q} \end{equation} From this, it follows that for a block string $\mathbf{J}$, the $q$-adic norm $\left\Vert \kappa_{H}\left(\mathbf{J}\right)\right\Vert _{q}$ tends to $0$ as $\mathbf{J}$ converges ($\left\|\mathbf{J}\right\|\rightarrow\infty$) to any infinite block string representing an element of $\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. This proves (IV). \vphantom{} V. Let $\mathbf{z}\in\mathbb{N}_{0}^{r}$. Then, $\left[\mathbf{z}\right]_{p^{N}}=\mathbf{z}$ for all $N\geq\lambda_{p}\left(\mathbf{z}\right)$, and so: \begin{equation} \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}=\kappa_{H}\left(\mathbf{z}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N},\textrm{ }\forall N\geq\lambda_{p}\left(\mathbf{z}\right) \end{equation} the right-hand side of which tends to the zero matrix $\mathbf{O}_{d}$ in the topology of $\mathbb{R}^{d,d}$ as $N\rightarrow\infty$, seeing as $H$ is contracting. This proves (V). Q.E.D. \begin{prop}[\textbf{Summatory Function of $\chi_{H}$}] \label{prop:MD summatory function formula for Chi_H} \begin{equation} \sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)=\begin{cases} \beta_{H}\left(\mathbf{0}\right)Np^{rN} & \textrm{if }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\\ \frac{\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}-1}{\alpha_{H}\left(\mathbf{0}\right)-1}\beta_{H}\left(\mathbf{0}\right)p^{rN} & \textrm{else} \end{cases}\label{eq:MD Summatory formula for Chi_H} \end{equation} \end{prop} Proof: Let: \[ S\left(N\right)=\frac{1}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right) \] Then, using $\chi_{H}$'s functional equation (\textbf{Lemma \ref{lem:MD Chi_H functional equations and characterization}}): \begin{align} p^{rN}S\left(N\right) & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\chi_{H}\left(p\mathbf{n}+\mathbf{j}\right)\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\frac{\mathbf{A_{j}}\chi_{H}\left(\mathbf{n}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\left(p^{r\left(N-1\right)}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\frac{1}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\chi_{H}\left(\mathbf{n}\right)+p^{r\left(N-1\right)}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}}\right)\\ & =p^{r\left(N-1\right)}\left(\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\right)S\left(N-1\right)+p^{rN}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}} \end{align} Dividing through by $p^{rN}$ gives us: \begin{equation} S_{H}\left(N\right)=\alpha_{H}\mathbf{\left(0\right)}S_{H}\left(N-1\right)+\beta_{H}\left(\mathbf{0}\right)\label{eq:MD Recursive formula for S_H} \end{equation} Much like the one-dimensional case, this shows that $S_{H}\left(N\right)$ is the image of $S_{H}\left(0\right)$ under $N$ iterates of an affine linear map (this time, on $\mathbb{Q}^{d}$); specifically: \begin{equation} \mathbf{x}\mapsto\alpha_{H}\left(\mathbf{0}\right)\mathbf{x}+\beta_{H}\left(\mathbf{0}\right)\label{eq:Affine linear map generating S_H-1} \end{equation} This leaves us with two cases: \vphantom{} i. Suppose $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Then, (\ref{eq:Affine linear map generating S_H-1}) reduces to a translation: \begin{equation} \mathbf{x}\mapsto\mathbf{x}+\beta_{H}\left(\mathbf{0}\right) \end{equation} in which case: \begin{equation} S_{H}\left(N\right)=S_{H}\left(0\right)+\beta_{H}\left(\mathbf{0}\right)N\label{eq:MD Explicit formula for S_H of N when alpha =00003D00003D 1} \end{equation} \vphantom{} ii. Suppose $\alpha_{H}\left(\mathbf{0}\right)\neq\mathbf{I}_{d}$, so that (\ref{eq:Affine linear map generating S_H-1}) is not a translation. Using the explicit formula for this $N$th iterate, we obtain: \begin{equation} S_{H}\left(N\right)=\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}S_{H}\left(0\right)+\left(\alpha_{H}\left(\mathbf{0}\right)-1\right)^{-1}\left(\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}-1\right)\beta_{H}\left(\mathbf{0}\right)\label{eq:MD Explicit formula for S_H of N} \end{equation} Noting that: \begin{align} S_{H}\left(0\right) & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{0}-1}\chi_{H}\left(\mathbf{n}\right)=\chi_{H}\left(\mathbf{0}\right)=\mathbf{0} \end{align} we then have: \begin{equation} \frac{1}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)=S_{H}\left(N\right)=\begin{cases} \beta_{H}\left(\mathbf{0}\right)N & \textrm{if }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\\ \frac{\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}-1}{\alpha_{H}\left(\mathbf{0}\right)-1}\beta_{H}\left(\mathbf{0}\right) & \textrm{else} \end{cases} \end{equation} Q.E.D. \begin{notation} Just as a reminder, we write $q$ to denote $q_{H}$. Like in the one-dimensional case, we write $\chi_{H,N}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{Q}^{d}\subset\mathbb{Q}_{q}^{d}$ to denote the $N$th truncation of $\chi_{H}$: \begin{equation} \chi_{H,N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)\left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right],\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r},\textrm{ }\forall N\in\mathbb{N}_{0}\label{eq:MD Definition of Nth truncation of Chi_H} \end{equation} Recall that: \begin{equation} \chi_{H,N}\left(\mathbf{n}\right)\overset{\mathbb{Q}^{d}}{=}\chi_{H}\left(\mathbf{n}\right),\textrm{ }\forall\mathbf{n}\leq p^{N}-1 \end{equation} In particular: \begin{equation} \chi_{H}\left(\mathbf{n}\right)=\chi_{H,\lambda_{p}\left(\mathbf{n}\right)}\left(\mathbf{n}\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r} \end{equation} Here, $\chi_{H,N}$ is a $\mathbb{Q}^{d}$-valued function which is a linear combination of finitely many indicator functions for clopen subsets of $\mathbb{Z}_{p}^{r}$. As such, $\chi_{H,N}$ is continuous both as a function $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d}$ and as a function $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}^{d}$; in fact, $\chi_{H,N}$ is continuous on $K^{d}$ for any topological field $K$ containing $\mathbb{Q}$. As a result, of this, in writing $\hat{\chi}_{H,N}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ to denote the Fourier Transform of $\chi_{H,N}$: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\int_{\mathbb{Z}_{p}^{r}}\chi_{H,N}\left(\mathbf{z}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Definition of the Fourier Coefficients of Chi_H,N} \end{equation} observe that this integral is convergent in both the topology of $\mathbb{C}$ \emph{and} the topology of $\mathbb{C}_{q}$. As before, because $\chi_{H,N}$ is locally constant, it will reduce to a finite sum (see \textbf{Proposition \ref{prop:MD Recursive formula for Chi_H,N hat}} in Subsection \ref{subsec:6.2.1 -and-}, below)): \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} where: \begin{equation} \mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)=\left[\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}\right]=\prod_{\ell=1}^{r}\left[\left\|t_{\ell}\right\|_{p}\leq p^{N}\right],\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} is the \emph{scalar-valued }indicator function for the set $\left\{ \left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}\right\} $. Once more, all of the computations and formal manipulations we will perform hold simultaneously in the topologies of $\mathbb{C}$ and in $\mathbb{C}_{q}$; they both occur in $\overline{\mathbb{Q}}\subset\mathbb{C}\cap\mathbb{C}_{q}$. The difference between the archimedean and non-archimedean topologies will only emerge when we consider what happens as $N$ tends to $\infty$. Much like the one-dimensional case, the reason this all works because of the invariance of the Fourier series formula for $\left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right]$ under the action of elements of the Galois group $\textrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)$. Additionally, note that because of $\chi_{H,N}$'s local constant-ness and the finitude of its range, we have that, for each $N$, $\hat{\chi}_{H,N}$ has finite support: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\mathbf{0},\textrm{ }\forall\left\Vert \mathbf{t}\right\Vert _{p}\geq p^{N+1},\textrm{ }\forall N\in\mathbb{N}_{0}\label{eq:MD Vanishing of Chi_H,N hat for all t with sufficiently large denominators} \end{equation} Consequently, the Fourier series: \begin{equation} \chi_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{\chi}_{H,N}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\label{eq:MD Fourier series for Chi_H,N} \end{equation} will converge in both $\mathbb{C}^{d}$ and $\mathbb{C}_{q}^{d}$ uniformly with respect to $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, reducing to a finite sum in all cases. \end{notation} Lastly, like in the one-dimensional case, we will need to express the interaction between $p$-adic structure functional equations and $N$th truncations. For full generality (matrix-type and vector-type), this result is given in terms of functions taking values in an arbitrary linear space $V$ over $\mathbb{Q}$. \begin{lem} \label{lem:MD functional equations and truncation lemma}Let $V$ be a $\mathbb{Q}$-linear space, and consider a function $\chi:\mathbb{N}_{0}^{r}\rightarrow V$. Suppose that for all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$ there are functions $\Phi_{\mathbf{j}}:\mathbb{N}_{0}^{r}\times V\rightarrow V$ so that: \begin{equation} \chi\left(p\mathbf{n}+\mathbf{j}\right)\overset{V}{=}\Phi_{\mathbf{j}}\left(\mathbf{n},\chi\left(\mathbf{n}\right)\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Relation between truncations and functional equations - Hypothesis} \end{equation} Then, the $N$th truncations $\chi_{N}$ satisfy the functional equations: \begin{equation} \chi_{N}\left(p\mathbf{n}+\mathbf{j}\right)\overset{V}{=}\Phi_{\mathbf{j}}\left(\left[\mathbf{n}\right]_{p^{N-1}},\chi_{N-1}\left(\mathbf{n}\right)\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Relation between truncations and functional equations, version 1} \end{equation} Equivalently: \begin{equation} \chi\left(\left[p\mathbf{n}+\mathbf{j}\right]_{p^{N}}\right)\overset{V}{=}\Phi_{\mathbf{j}}\left(\left[\mathbf{n}\right]_{p^{N-1}},\chi\left(\left[\mathbf{n}\right]_{p^{N-1}}\right)\right),\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Relation between truncations and functional equations, version 2} \end{equation} \end{lem} Proof: Fix $N\geq0$, $\mathbf{n}\in\mathbb{N}_{0}^{r}$, and $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. Then: \begin{align} \chi\left(\left[p\mathbf{n}+\mathbf{j}\right]_{p^{N}}\right) & \overset{V}{=}\chi_{N}\left(p\mathbf{n}+\mathbf{j}\right)\\ & =\sum_{\mathbf{m}=\mathbf{0}}^{p^{N}-1}\chi\left(\mathbf{m}\right)\left[p\mathbf{n}+\mathbf{j}\overset{p^{N}}{\equiv}\mathbf{m}\right]\\ \left(\textrm{split }\mathbf{m}\textrm{ mod }p\right); & =\sum_{\mathbf{m}=\mathbf{0}}^{p^{N-1}-1}\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\chi\left(p\mathbf{m}+\mathbf{k}\right)\left[p\mathbf{n}+\mathbf{j}\overset{p^{N}}{\equiv}p\mathbf{m}+\mathbf{k}\right]\\ & =\sum_{\mathbf{m}=\mathbf{0}}^{p^{N-1}-1}\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\Phi_{\mathbf{k}}\left(\mathbf{m},\chi\left(\mathbf{m}\right)\right)\underbrace{\left[\mathbf{n}\overset{p^{N-1}}{\equiv}\mathbf{m}+\frac{\mathbf{k}-\mathbf{j}}{p}\right]}_{0\textrm{ }\forall\mathbf{n}\textrm{ if }\mathbf{k}\neq\mathbf{j}}\\ & =\sum_{\mathbf{m}=\mathbf{0}}^{p^{N-1}-1}\Phi_{\mathbf{j}}\left(\mathbf{m},\chi\left(\mathbf{m}\right)\right)\left[\mathbf{n}\overset{p^{N-1}}{\equiv}\mathbf{m}\right]\\ & =\Phi_{\mathbf{j}}\left(\left[\mathbf{n}\right]_{p^{N-1}},\chi\left(\left[\mathbf{n}\right]_{p^{N-1}}\right)\right) \end{align} Q.E.D. \newpage{} \section{\label{sec:6.2 Fourier-Transforms-and}Fourier Transforms and Quasi-Integrability (Again)} IN THIS SECTION, WE ASSUME THAT $H$ IS NON-SINGULAR. \vphantom{} Section \pageref{sec:6.2 Fourier-Transforms-and} is mostly the same as its one-dimensional predecessor in Section \pageref{sec:4.2 Fourier-Transforms-=00003D000026}. As mentioned earlier, the primary distinction of the multi-dimensional case is the non-commutativity of matrix multiplication. This eventually leads to three distinct cases: those where $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, those where $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible and $H$ is commutative, and those where $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible and $H$ is \emph{not }commutative. Although an $\mathcal{F}$-series representation of $\chi_{H}$ is obtained for all three cases (\textbf{Theorem \ref{thm:MD F-series for Chi_H}}), it is only for the first two cases that this series can be used to deduce a closed-form expression for a Fourier transform of $\hat{\chi}_{H}$. Subsection \pageref{subsec:6.2.4 Multi-Dimensional--=00003D000026} takes the reader right up to the computational obstacle that appears in the case of a non-commutative $H$. That being said, for the two cases where we \emph{can }deduce formulae for $\hat{\chi}_{H}$, our multi-dimensional $\left(p,q\right)$-adic Wiener Tauberian Theorem can then be applied to yield a \textbf{Tauberian Spectral Theorem }for suitable multi-dimensional Hydra maps, exactly like in the one-dimensional case. \subsection{\label{subsec:6.2.1 -and-}$\hat{\chi}_{H,N}$ and $\hat{A}_{H}$ (Again)} We begin by computing a recursive formula for $\hat{\chi}_{H,N}$ in terms of $\hat{\chi}_{H,N-1}$, which we then solve to obtain an explicit formula for $\hat{\chi}_{H,N}$. \begin{prop}[\textbf{Formulae for Multi-Dimensional} $\hat{\chi}_{H,N}$] \label{prop:MD Recursive formula for Chi_H,N hat}\ \vphantom{} I. \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\label{eq:MD Chi_H,N hat transform formula - sum form} \end{equation} \vphantom{} II. \begin{equation} \mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\hat{\chi}_{H,N-1}\left(p\mathbf{t}\right)+\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right),\textrm{ }\forall N\geq1,\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Chi_H,N hat functional equation} \end{equation} the nesting of which yields: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\sum_{n=0}^{N-1}\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\label{eq:MD Explicit formula for Chi_H,N hat} \end{equation} where the $m$-product is defined to be $1$ when $n=0$. In particular, note that $\hat{\chi}_{H,N}\left(\mathbf{t}\right)=\mathbf{0}$ for all $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$ with $\left\Vert \mathbf{t}\right\Vert _{p}>p^{N}$. \end{prop} Proof: Using \textbf{Proposition \ref{prop:Multi-Dimensional indicator function Fourier Series}}, we have: \begin{align} \chi_{H,N}\left(\mathbf{z}\right) & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)\left[\mathbf{z}\overset{p^{N}}{\equiv}\mathbf{n}\right]\\ & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)\frac{1}{p^{rN}}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{n}\right)\right\} _{p}}\\ & =\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\frac{1}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{align} This is the Fourier series representation of $\chi_{H,N}$; as such, the coefficient of $e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}$ in the series is precisely the formula for $\hat{\chi}_{H,N}\left(\mathbf{t}\right)$: \[ \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{rN}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N}-1}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}} \] This proves (I). To prove (II), just like in the one-dimensional case, we split the $\mathbf{n}$-sum modulo $p$ and utilize $\chi_{H}$'s functional equation. Here\textbf{ }$\mathbf{t}$ satisfies $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}$. \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{rN}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\chi_{H}\left(p\mathbf{n}+\mathbf{j}\right)e^{-2\pi i\left(p\mathbf{n}+\mathbf{j}\right)\cdot\mathbf{t}}\\ & =\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{rN}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\frac{\mathbf{A}_{\mathbf{j}}\chi_{H}\left(\mathbf{n}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}e^{-2\pi i\left(p\mathbf{n}+\mathbf{j}\cdot\mathbf{t}\right)} \end{align} Interchanging the $\mathbf{j}$ and $\mathbf{n}$ sums and breaking things up yields \begin{align} \frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\underbrace{\left(\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\right)}_{\alpha_{H}\left(\mathbf{t}\right)}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\left(p\mathbf{n}\right)\cdot\mathbf{t}}\\ +\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\underbrace{\left(\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\right)}_{\beta_{H}\left(\mathbf{t}\right)}e^{-2\pi i\left(p\mathbf{n}\right)\cdot\mathbf{t}} \end{align} Hence: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}\left(\alpha_{H}\left(\mathbf{t}\right)\chi_{H}\left(\mathbf{n}\right)+\beta_{H}\left(\mathbf{t}\right)\right)e^{-2\pi i\mathbf{n}\cdot\left(p\mathbf{t}\right)} \end{equation} Re-writing the right-hand side in terms of $\chi_{H,N-1}$ yields: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\hat{\chi}_{H,N-1}\left(p\mathbf{t}\right)+\beta_{H}\left(\mathbf{t}\right)\frac{\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}e^{-2\pi i\mathbf{n}\cdot\left(p\mathbf{t}\right)}\label{eq:Chi_H N hat, almost ready to nest-1} \end{equation} Simplifying the remaining $n$-sum, we obtain: \begin{align} \frac{1}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}e^{-2\pi i\mathbf{n}\cdot\left(p\mathbf{t}\right)} & =\begin{cases} \frac{1}{p^{r\left(N-1\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{N-1}-1}1 & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\leq p\\ \frac{1}{p^{r\left(N-1\right)}}\prod_{m=1}^{r}\frac{\left(e^{-2\pi ipt_{m}}\right)^{p^{N-1}}-1}{e^{-2\pi ipt_{m}}-1} & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\geq p^{2} \end{cases}\\ & =\begin{cases} 1 & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\leq p\\ 0 & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\geq p^{2} \end{cases}\\ & =\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right) \end{align} Hence: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{t}\right)\hat{\chi}_{H,N-1}\left(p\mathbf{t}\right)+\beta_{H}\left(\mathbf{t}\right)\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\right)\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\hat{\chi}_{H,N-1}\left(p\mathbf{t}\right)+\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right) \end{align} which proves (II). With (II) proven, we can then nest (\ref{eq:MD Chi_H,N hat functional equation}) to derive an explicit formula for $\hat{\chi}_{H,N}\left(\mathbf{t}\right)$: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\hat{\chi}_{H,N-1}\left(p\mathbf{t}\right)+\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right)\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\left(\mathbf{1}_{\mathbf{0}}\left(p^{N-1}p\mathbf{t}\right)\alpha_{H}\left(p\mathbf{t}\right)\hat{\chi}_{H,N-2}\left(p^{2}\mathbf{t}\right)+\mathbf{1}_{0}\left(p^{2}\mathbf{t}\right)\beta_{H}\left(p\mathbf{t}\right)\right)\\ & +\mathbf{1}_{0}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right)\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\alpha_{H}\left(p\mathbf{t}\right)\hat{\chi}_{H,N-2}\left(p^{2}\mathbf{t}\right)\\ & +\mathbf{1}_{\mathbf{0}}\left(p^{2}\mathbf{t}\right)\alpha_{H}\left(\mathbf{t}\right)\beta_{H}\left(p\mathbf{t}\right)+\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right)\\ & \vdots\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\left(\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}\mathbf{t}\right)\right)\hat{\chi}_{H,1}\left(p^{N-1}\mathbf{t}\right)\\ & +\sum_{n=0}^{N-2}\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right) \end{align} where the $m$-product is $1$ whenever $n=0$. Finally, using (\ref{eq:MD Chi_H,N hat transform formula - sum form}) to compute $\hat{\chi}_{H,1}\left(\mathbf{t}\right)$, we have: \begin{align} \hat{\chi}_{H,1}\left(\mathbf{t}\right) & =\frac{\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\chi_{H}\left(\mathbf{j}\right)e^{-2\pi i\mathbf{j}\cdot\mathbf{t}} \end{align} Since $\chi_{H}\left(\mathbf{0}\right)=\mathbf{0}$, $\chi_{H}$'s functional equation gives: \begin{equation} \chi_{H}\left(\mathbf{j}\right)=\chi_{H}\left(p\mathbf{0}+\mathbf{j}\right)=\frac{\mathbf{A}_{\mathbf{j}}\chi_{H}\left(\mathbf{0}\right)+\mathbf{b}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}} \end{equation} and so: \begin{align} \hat{\chi}_{H,1}\left(\mathbf{t}\right) & =\frac{\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\chi_{H}\left(\mathbf{j}\right)e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\\ & =\frac{\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{b}_{\mathbf{j}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\\ & =\mathbf{1}_{\mathbf{0}}\left(p\mathbf{t}\right)\beta_{H}\left(\mathbf{t}\right) \end{align} Consequently: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\mathbf{1}_{\mathbf{0}}\left(p^{N}\mathbf{t}\right)\left(\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}\mathbf{t}\right)\right)\hat{\chi}_{H,1}\left(p^{N-1}\mathbf{t}\right)\\ & +\sum_{n=0}^{N-2}\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{N}t\right)\beta_{H}\left(p^{N-1}\mathbf{t}\right)\prod_{n=0}^{N-2}\alpha_{H}\left(p^{n}\mathbf{t}\right)\\ & +\sum_{n=0}^{N-2}\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\\ & =\sum_{n=0}^{N-1}\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\left(\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{n}\mathbf{t}\right) \end{align} which proves (\ref{eq:MD Explicit formula for Chi_H,N hat}). Q.E.D. \vphantom{} Next, we introduce our multi-dimensional analogue $\hat{A}_{H}$. \begin{defn}[\textbf{Multi-Dimensional $\hat{A}_{H}$}] \nomenclature{$\hat{A}_{H}\left(\mathbf{t}\right)$}{ }\textbf{ }We define the function $\hat{A}_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\textrm{GL}_{d}\left(\overline{\mathbb{Q}}\right)$ by: \begin{equation} \hat{A}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right),\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Definition of A_H hat} \end{equation} where the $m$-product is defined to be $\mathbf{I}_{d}$ whenever $\mathbf{t}=\mathbf{0}$\textemdash so, $\hat{A}_{H}\left(\mathbf{0}\right)\overset{\textrm{def}}{=}\mathbf{I}_{d}$. \end{defn} \begin{prop}[\textbf{Matrix $\alpha_{H}$ product in terms of $\hat{A}_{H}$}] \label{prop:MD alpha_H A_H hat product} Fix $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} $. Then: \begin{align} \mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & =\begin{cases} \mathbf{O}_{d} & \textrm{if }n\leq-v_{p}\left(\mathbf{t}\right)-2\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right)^{-1} & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)} & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right) \end{cases}\label{eq:MD alpha product in terms of A_H hat} \end{align} \end{prop} Proof: Fix $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} $. Then, we can write: \begin{align} \mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & =\begin{cases} \mathbf{O}_{d} & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\geq p^{n+2}\\ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n+1} \end{cases}\\ & =\begin{cases} \mathbf{O}_{d} & \textrm{if }\exists\ell:n\leq-v_{p}\left(t_{\ell}\right)-2\\ \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }n\geq-v_{p}\left(t_{\ell}\right)-1,\textrm{ }\forall\ell \end{cases} \end{align} So, we get the zero matrix whenever $n+1$ is not greater than the negatives of the valuations of each of $\mathbf{t}$s components. Fixing $\mathbf{t}$, we have that $p^{m}\mathbf{t}\overset{1}{\equiv}\mathbf{0}$ only when $m$ is larger than the negative valuations of each of the components of $\mathbf{t}$: \begin{equation} m\geq\max_{1\leq\ell\leq r}\left(-v_{p}\left(t_{\ell}\right)\right)=-\min_{1\leq\ell\leq r}v_{p}\left(t_{\ell}\right)=-v_{p}\left(\mathbf{t}\right) \end{equation} Hence, $\alpha_{H}\left(p^{m}\mathbf{t}\right)=\alpha_{H}\left(\mathbf{0}\right)$ for all $m\geq-v_{p}\left(\mathbf{t}\right)$. Thus, for fixed $\mathbf{t}$: \begin{equation} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)=\begin{cases} \prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-2}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)\\ \prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\times\prod_{k=-v_{p}\left(\mathbf{t}\right)}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right) & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right)+1 \end{cases} \end{equation} Because $\alpha_{H}\left(p^{k}\mathbf{t}\right)=\alpha_{H}\left(\mathbf{0}\right)$ for all $k\geq-v_{p}\left(\mathbf{t}\right)$, this can be simplified to: \begin{align} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & =\begin{cases} \prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-2}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \left(\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)} & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right) \end{cases}\\ \left(\times\&\div\textrm{ by }\alpha_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\right); & =\begin{cases} \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\right)^{-1} & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)} & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right) \end{cases} \end{align} which gives us the desired formula. Q.E.D. \vphantom{} Next, we express the $\alpha_{H}$ product as a series. \begin{prop}[\textbf{Matrix $\alpha_{H}$ series formula}] \label{prop:MD alpha H series} \end{prop} \begin{equation} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)},\textrm{ }\forall n\geq0,\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD alpha_H product expansion} \end{equation} Proof: We start by writing: \begin{equation} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)=\prod_{m=0}^{n-1}\left(\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\frac{1}{p^{r}}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}e^{-2\pi i\mathbf{j}\cdot\left(p^{m}\mathbf{t}\right)}\right) \end{equation} and then apply (\ref{eq:MD M_H partial sum generating identity}) from \textbf{Proposition \ref{prop:MD generating function} }with $z^{\mathbf{j}\cdot\mathbf{t}}$ replaced by $e^{-2\pi i\left(\mathbf{j}\cdot\mathbf{t}\right)}$: \begin{align} \prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & =\prod_{m=0}^{n-1}\left(\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\frac{1}{p^{r}}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}e^{-2\pi i\mathbf{j}\cdot\left(p^{m}\mathbf{t}\right)}\right)\\ & =\frac{1}{p^{rn}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n-\lambda_{p}\left(\mathbf{m}\right)}e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\\ \left(\kappa_{H}\left(\mathbf{m}\right)=M_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\right); & =\frac{1}{p^{rn}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\\ & =\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)} \end{align} Q.E.D. \vphantom{} Just like the one-dimensional case, we now use this formula to exhibit $\hat{A}_{H}$ as a radially-magnitudinal function, and then\textemdash with the help of our multi-dimensional Fourier Resummation Lemmata\textemdash show that $\hat{A}_{H}$ is the Fourier-Stieltjes transform of a rising-continuous $\left(p,q\right)$-adic thick measure of matrix-type. \begin{prop} Let $H$ be semi-basic. Then, $\hat{A}_{H}\left(\mathbf{t}\right)$ is a depth-$r$ $\left(p,q\right)$-adic thick measure of matrix type. \end{prop} Proof: Let $H$ be semi-basic. Then, $p$ divides the diagonal entires of the $\mathbf{D}_{\mathbf{j}}$s which are not $1$, and the entries of any $\mathbf{D}_{\mathbf{j}}$ are co-prime to the entries of every $\mathbf{A}_{\mathbf{j}}$. As such, taking $q_{H}$-adic matrix norms yields: \begin{equation} \left\Vert \alpha_{H}\left(\mathbf{t}\right)\right\Vert _{q_{H}}=\left\Vert \frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\right\Vert _{q_{H}}\leq\max_{\mathbf{j}\leq p-1}\left\|\frac{1}{p^{r}}\frac{\mathbf{A}_{\mathbf{j}}}{\mathbf{D}_{\mathbf{j}}}\right\|_{q_{H}}\leq1 \end{equation} Hence: \begin{align} \sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left\Vert \hat{A}_{H}\left(\mathbf{t}\right)\right\Vert _{q_{H}} & \leq\max\left\{ 1,\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} }\left\Vert \prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right\Vert _{q_{H}}\right\} \\ & \leq\max\left\{ 1,\sup_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} }\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}1\right\} \\ & =1 \end{align} and so, $\hat{A}_{H}$ is then in $B\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q}^{d,d}\right)$. As such, the map: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{A}_{H}\left(\mathbf{t}\right)\hat{\mathbf{f}}\left(-\mathbf{t}\right)\in\mathbb{C}_{q}^{d} \end{equation} is then a continuous, defining a $\left(p,q\right)$-adic thick measure of matrix type. Q.E.D. \vphantom{} Next, some obligatory abbreviations: \begin{defn} \nomenclature{$\tilde{A}_{H,N}\left(\mathbf{z}\right)$}{$N$th partial sum of the Fourier series generated by $\hat{A}_{H}\left(\mathbf{t}\right)$.}For each $N\in\mathbb{N}_{0}$, we write $\tilde{A}_{H,N}:\mathbb{Z}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d,d}$ to denote the matrix-valued function: \begin{equation} \tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\label{eq:MD Definition of A_H,N twiddle} \end{equation} \end{defn} \begin{defn} We write $\mathbf{I}_{H,n}\left(n\right)$ \nomenclature{$\mathbf{I}_{H,n}\left(n\right)$}{$\overset{\textrm{def}}{=}\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}$ \nopageref}to denote the $d\times d$ matrix: \begin{align} \mathbf{I}_{H}\left(n\right) & \overset{\textrm{def}}{=}\mathbf{I}_{d}-\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):n\right)\label{eq:Definition of I_H}\\ & =\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\nonumber \end{align} \end{defn} \begin{rem} Note that $\mathbf{I}_{H}\left(n\right)=\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ for all $n\in\mathbb{N}_{0}$ whenever $H$ is commutative. Also, $\mathbf{I}_{H}\left(n\right)=\mathbf{O}_{d}$ for all $n\in\mathbb{N}_{0}$ whenever $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. \end{rem} Our next theorem details the properties of $A_{H}$: \begin{thm}[\textbf{Properties of Multi-Dimensional $\hat{A}_{H}$}] \label{thm:MD properties of A_H hat}\ \vphantom{} I. ($\hat{A}_{H}$ is radially-magnitudinal and $\left(p,q\right)$-adically regular) \begin{equation} \hat{A}_{H}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\begin{cases} \mathbf{I}_{d} & \textrm{if }\mathbf{t}=0\\ \left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa_{H}\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{t}\right)} & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:A_H hat as the product of radially symmetric and magnitude-dependent measures-1} \end{equation} Here, in an abuse of notation, we write: \[ \left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{0}\right)}\overset{\textrm{def}}{=}\mathbf{I}_{d} \] \vphantom{} II. (Formula for $\tilde{A}_{H,N}\left(\mathbf{z}\right)$) For any $N\in\mathbb{N}_{0}$ and $\mathbf{z}\in\mathbb{Z}_{p}^{r}$: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=} & \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Convolution of dA_H and D_N} \end{align} In particular, if $H$ is commutative: \begin{equation} \tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\label{eq:MD Convolution of dA_H and D_N when H is commutative} \end{equation} \vphantom{} III. As $N\rightarrow\infty$, \emph{(\ref{eq:MD Convolution of dA_H and D_N})} converges in the standard $\left(p,q_{H}\right)$-adic frame to: \begin{equation} \lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Fourier Limit of A_H,N twiddle in standard frame} \end{equation} There are two particular cases: i If $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, then: \begin{equation} \lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\mathbf{O}_{d},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:Full kernel of dA_H on when alpha is 1} \end{equation} ii. If $H$ is commutative, then: \begin{equation} \lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Fourier Limit of A_H,N twiddle in standard frame, commutative} \end{equation} \end{thm} Proof: I. By (\ref{eq:MD alpha_H product expansion}), we have that: \[ \hat{A}_{H}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)=\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{t}\right)}-1}\kappa_{H}\left(\mathbf{m}\right)e^{-2\pi i\left(\mathbf{m}\cdot\mathbf{t}\right)}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{t}\right)} \] \vphantom{} II. By (\ref{eq:MD radially-magnitudinal resummation formula}), we have: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =p^{rN}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(p^{-N}\right)}\\ & +\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(p^{-n}\right)}-\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(p^{-\left(n+1\right)}\right)}\right)\\ & -\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(p^{-\left(n+1\right)}\right)} \end{align} Here, $-v_{p}\left(p^{-n}\right)=n$. Since we are using the convention: \begin{equation} \left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\frac{1}{p^{0}}\right)}=\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{0}\right)}\overset{\textrm{def}}{=}\mathbf{I}_{d} \end{equation} and since $H^{\prime}\left(\mathbf{0}\right)/p^{r}$ is an invertible $d\times d$ matrix, we have that: \begin{equation} \left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(1/p^{n}\right)}=\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n},\textrm{ }\forall n\in\mathbb{N}_{0} \end{equation} Consequently: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =p^{rN}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{N}\\ & +\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}-\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1}\right)\\ & -\sum_{n=0}^{N-1}p^{rn}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}+\mathbf{j}p^{n}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1} \end{align} Here, (\ref{eq:MD Kappa_H has P-adic structure}) from \textbf{Lemma \ref{lem:properties of MD kappa_H}} yields: \begin{align} \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}+p^{n}\mathbf{j}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\\ & =M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\kappa_{H}\left(\mathbf{j}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)} \end{align} As such: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}-\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1}\right)\\ & -\sum_{n=0}^{N-1}p^{rn}M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{j}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1} \end{align} Using (\ref{eq:MD kappa H sum in terms of MD alpha}) gives: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}-\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1}\right)\\ & -\sum_{n=0}^{N-1}p^{rn}M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-1}-\mathbf{I}_{d}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1} \end{align} Re-writing things to make some cancellations evident: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}-\overbrace{\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1}}^{\textrm{these two}}\\ & -\sum_{n=0}^{N-1}p^{rn}M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}\\ & +\underbrace{\sum_{n=0}^{N-1}p^{rn}\overbrace{M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}}^{\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n+1}}_{\textrm{cancel out}} \end{align} we end up with: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}\\ & -\sum_{n=0}^{N-1}p^{rn}M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n} \end{align} Now we utilize our $\mathcal{C}_{H}$ notation to make the lower line manageable. First, recall that: \begin{equation} M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)=\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)} \end{equation} With this, the expression: \begin{equation} M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)} \end{equation} becomes: \begin{equation} \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)} \end{equation} which, using $\mathcal{C}_{H}$, is just: \begin{equation} \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right) \end{equation} Finally, writing out the formula for $\tilde{A}_{H,N}\left(\mathbf{z}\right)$, we have: \begin{align} \tilde{A}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}\\ & -\sum_{n=0}^{N-1}p^{rn}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{n}\\ & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\underbrace{\left(\mathbf{I}_{d}-\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)}_{\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{align} which is the desired identity. \vphantom{} III. Let $H$ be semi-basic, and fix $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. Then: \begin{equation} \left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\right\Vert _{q}\leq\left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\right\Vert _{q} \end{equation} Applying \textbf{Lemma \ref{lem:properties of MD kappa_H}}, $\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)$ converges $q$-adically to $\mathbf{O}_{d}$ as $N\rightarrow\infty$. By the ultrametric structure of $\mathbb{C}_{q}$, this then guarantees the convergence of (\ref{eq:MD Fourier Limit of A_H,N twiddle in standard frame}). Next, let $\mathfrak{\mathbf{z}}=\mathbf{m}\in\mathbb{N}_{0}^{r}$. Then, for all $n\geq\lambda_{p}\left(\mathbf{m}\right)$: \begin{equation} \kappa_{H}\left(\mathbf{m}\right)=\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)=\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n+1}}\right)\in\textrm{Gl}_{d}\left(\mathbb{Q}\right)\backslash\left\{ \mathbf{O}_{d}\right\} \end{equation} Likewise, keeping $n\geq\lambda_{p}\left(\mathbf{m}\right)$, we have: \begin{align} \mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\right) & =\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)}\\ & =\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{m}\right)}\\ & =\mathbf{I}_{d}-\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\mathbf{m}\right)\right)\\ & =\mathbf{I}_{H}\left(\lambda_{p}\left(\mathbf{m}\right)\right) \end{align} Examining the tail of the $n$-series in (\ref{eq:MD Convolution of dA_H and D_N}), note that: \[ \sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\mathbf{m}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\overset{\overline{\mathbb{Q}}^{d,d}}{=}O\left(\mathbf{I}_{d}\right)+\kappa_{H}\left(\mathbf{m}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\mathbf{m}\right)\right)\sum_{n=\lambda_{p}\left(\mathbf{m}\right)}^{N-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \] holds whenever $N\geq\lambda_{p}\left(\mathbf{m}\right)+1$. Since the $n$-series on the right is geometric in the matrix $H^{\prime}\left(\mathbf{0}\right)$, it will converge in the topology of in $\mathbb{C}$ as $N\rightarrow\infty$ if and only if $H$ is contracting. Likewise, by \textbf{Lemma \ref{lem:properties of MD kappa_H}},\textbf{ }$\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}$ will tend to $\mathbf{O}_{d}$ in the topology of $\mathbb{C}$ as $N\rightarrow\infty$ if and only if $H$ is contracting, and will diverge in the topology of $\mathbb{C}$ if and only if $H$ is expanding. This shows that (\ref{eq:MD Fourier Limit of A_H,N twiddle in standard frame}) holds true. Q.E.D. \vphantom{} Now we can show that $\hat{A}_{H}$ is the Fourier-Stieltjes transform of a thick measure of matrix type. \begin{cor}[\textbf{$\hat{A}_{H}$ Induces a Matrix-Type Thick Measure}] \ \vphantom{} I. The matrix-valued function $\hat{A}_{H}\left(\mathbf{t}\right)$ is the Fourier-Stieltjes transform of a $\left(p,q_{H}\right)$-adic matrix-type thick measure of depth $r$. This measure acts on functions by way of the formula: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{A}_{H}\left(-\mathbf{t}\right)\hat{\mathbf{f}}\left(\mathbf{t}\right)\in\mathbb{C}_{q}^{d} \end{equation} Moreover, this thick measure is degenerate with respect to the standard $\left(p,q_{H}\right)$-adic frame if and only if $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. \vphantom{} II. For any $\mathbf{v}\in\mathbb{C}_{q}^{d}$, the vector-valued function $\hat{A}_{H}\left(\mathbf{t}\right)\mathbf{v}$ is the Fourier-Stieltjes transform of a $\left(p,q_{H}\right)$-adic thick measure of vector type. This measure acts on functions by way of the formula: \begin{equation} \mathbf{f}\in C\left(\mathbb{Z}_{p}^{r},\mathbb{C}_{q}^{d}\right)\mapsto\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\left(\hat{A}_{H}\left(-\mathbf{t}\right)\mathbf{v}\right)\hat{\mathbf{f}}\left(\mathbf{t}\right)\in\mathbb{C}_{q}^{d} \end{equation} Moreover, this thick measure is degenerate with respect to the standard $\left(p,q_{H}\right)$-adic frame if and only if $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. \end{cor} Proof: I. Follows by (II) and (III) of \textbf{Theorem \ref{thm:MD properties of A_H hat}}. \vphantom{} II. (I) shows that the matrix-type thick measure associated to $\hat{A}_{H}\left(\mathbf{t}\right)$ is degenerate if and only if $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Noting that the vector-valued Fourier series generated by $\hat{A}_{H}\left(\mathbf{t}\right)\mathbf{v}$ has an $\mathcal{F}_{p,q}$ limit of: \begin{equation} \lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\hat{A}_{H}\left(\mathbf{t}\right)\mathbf{v}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{equation} we can write this as: \begin{equation} \left(\lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\right)\mathbf{v} \end{equation} Note that (I) is exactly the statement that the limit in parenthesis (taken with respect to $\mathcal{F}_{p,q}$) is $\mathbf{O}_{d}$ for every $\mathbf{z}\in\mathbb{Z}_{p}^{r}$ if and only if $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Hence: \begin{align} \lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\left(\hat{A}_{H}\left(\mathbf{t}\right)\mathbf{v}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & =\left(\lim_{N\rightarrow\infty}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\right)\mathbf{v}\\ \left(\textrm{iff }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\right); & =\mathbf{O}_{d}\mathbf{v}\\ & =\mathbf{0} \end{align} This shows that the vector-type thick measure induced by $\hat{A}_{H}\left(\mathbf{t}\right)\mathbf{v}$ is degenerate if and only if $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Q.E.D. \begin{defn}[$\tilde{A}_{H}\left(\mathbf{z}\right)$] We write $\tilde{A}_{H}\left(\mathbf{z}\right)$ to denote the kernel of the matrix-type thick measure induced by $\hat{A}_{H}$; that is, $\tilde{A}_{H}\left(\mathbf{z}\right)$ is the $\mathcal{F}_{p,q_{H}}^{d,d}$-limit of $\tilde{A}_{H,N}\left(\mathbf{z}\right)$ as $N\rightarrow\infty$. \end{defn} \vphantom{} Like in the one-dimensional case, we now can write out our asymptotic decomposition of $\hat{\chi}_{H,N}$. \begin{thm}[\textbf{$\left(N,\mathbf{t}\right)$ Asymptotic Decomposition of $\hat{\chi}_{H,N}$}] \label{thm:MD N,t asympotics for Chi_H,N hat}\ \vphantom{} I. If $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, then: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\begin{cases} N\hat{A}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\left(N+v_{p}\left(\mathbf{t}\right)\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right) & \textrm{if }0<\left\Vert \mathbf{t}\right\Vert _{p}<p^{N}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}=p^{N}\\ \mathbf{0} & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}>p^{N} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Formula for Chi_H,N hat when alpha is 1} \end{equation} \vphantom{} II. If $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, then: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\begin{cases} \hat{A}_{H}\left(\mathbf{t}\right)\frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N+v_{p}\left(\mathbf{t}\right)}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right)\right) & \textrm{if }0<\left\Vert \mathbf{t}\right\Vert _{p}<p^{N}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}=p^{N}\\ \mathbf{0} & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}>p^{N} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Formula for Chi_H,N hat when alpha is not 1} \end{equation} \end{thm} Proof: For brevity, we write: \begin{equation} \hat{A}_{H,n}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\begin{cases} \mathbf{I}_{d} & \textrm{if }n=0\\ \mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)\prod_{m=0}^{n-1}\alpha_{H}\left(p^{m}\mathbf{t}\right) & \textrm{if }n\geq1 \end{cases}\label{eq:MD Definition of A_H,n+1 hat} \end{equation} so that (\ref{eq:MD Chi_H,N hat functional equation}) becomes: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\sum_{n=0}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(p^{n}\mathbf{t}\right) \end{equation} Now, letting $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$ be non-zero and satisfy $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}$, we use \textbf{Proposition \ref{prop:MD alpha_H A_H hat product}} to write: \begin{equation} \hat{A}_{H,n}\left(\mathbf{t}\right)=\begin{cases} \mathbf{O}_{d} & \textrm{if }n\leq-v_{p}\left(\mathbf{t}\right)-2\\ \frac{\hat{A}_{H}\left(\mathbf{t}\right)}{\alpha_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)} & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)}\hat{A}_{H}\left(\mathbf{t}\right) & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right) \end{cases} \end{equation} So: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\sum_{n=0}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(p^{n}\mathbf{t}\right)=\sum_{n=-v_{p}\left(\mathbf{t}\right)-1}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\label{eq:MD Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis} \end{equation} Note that in obtaining (\ref{eq:MD Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis}), we have used the fact: \begin{equation} \hat{A}_{H,n}\left(\mathbf{t}\right)=\mathbf{O}_{d},\textrm{ }\forall n\leq-v_{p}\left(\mathbf{t}\right)-2 \end{equation} which is, itself, a consequence of the vanishing of $\mathbf{1}_{\mathbf{0}}\left(p^{n+1}\mathbf{t}\right)$ whenever $n\leq-v_{p}\left(\mathbf{t}\right)-2$. With that done, we are left with two cases. \vphantom{} \textbullet{} First, suppose $\left\Vert \mathbf{t}\right\Vert _{p}=p^{N}$. Then $N=-v_{p}\left(\mathbf{t}\right)$, and so, (\ref{eq:MD Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis}) becomes: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\sum_{n=-v_{p}\left(\mathbf{t}\right)-1}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\\ & =\hat{A}_{H,-v_{p}\left(\mathbf{t}\right)-1}\left(\mathbf{t}\right)\beta_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\\ & =\left(\mathbf{1}_{\mathbf{0}}\left(p^{-v_{p}\left(\mathbf{t}\right)-1+1}\mathbf{t}\right)\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\beta_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\\ & =\left(\mathbf{1}_{\mathbf{0}}\left(p^{-v_{p}\left(\mathbf{t}\right)}\mathbf{t}\right)\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\left(\alpha_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\right)^{-1}\beta_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right) \end{align} Because $p^{-v_{p}\left(\mathbf{t}\right)}\mathbf{t}$ contains only integer entries, note that $\mathbf{1}_{\mathbf{0}}\left(p^{-v_{p}\left(\mathbf{t}\right)}\mathbf{t}\right)=1$. This leaves us with: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\left(\prod_{m=0}^{-v_{p}\left(\mathbf{t}\right)-1}\alpha_{H}\left(p^{m}\mathbf{t}\right)\right)\left(\alpha_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\right)^{-1}\beta_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)\\ & =\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) \end{align} \vphantom{} \textbullet{} Second, suppose $0<\left\Vert \mathbf{t}\right\Vert _{p}<p^{N}$. Then $-v_{p}\left(\mathbf{t}\right)-1<N-1$, and so $p^{n}\mathbf{t}\overset{1}{\equiv}\mathbf{0}$ for all $n\geq-v_{p}\left(\mathbf{t}\right)$. Consequently, (\ref{eq:MD Chi_H,N hat as Beta_H plus A_H,n hat - ready for t,n analysis}) becomes: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\hat{A}_{H,-v_{p}\left(\mathbf{t}\right)-1}\left(\mathbf{t}\right)\beta_{H}\left(p^{-v_{p}\left(\mathbf{t}\right)-1}\mathbf{t}\right)+\sum_{n=-v_{p}\left(\mathbf{t}\right)}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(p^{n}\mathbf{t}\right)\\ \left(p^{n}\mathbf{t}\overset{1}{\equiv}\mathbf{0}\textrm{ }\forall n\geq-v_{p}\left(\mathbf{t}\right)\right); & =\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\sum_{n=-v_{p}\left(\mathbf{t}\right)}^{N-1}\hat{A}_{H,n}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) \end{align} Now, using \textbf{Proposition \ref{prop:MD alpha_H A_H hat product}}, we write: \begin{equation} \hat{A}_{H,n}\left(\mathbf{t}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)},\textrm{ }\forall n\geq-v_{p}\left(\mathbf{t}\right) \end{equation} and hence: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{t}\right) & =\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\sum_{n=-v_{p}\left(\mathbf{t}\right)}^{N-1}\hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)}\beta_{H}\left(\mathbf{0}\right)\\ & =\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\hat{A}_{H}\left(\mathbf{t}\right)\sum_{n=0}^{N+v_{p}\left(\mathbf{t}\right)-1}\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right) \end{align} When $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, this gives: \begin{equation} \hat{\chi}_{H,N}\left(\mathbf{t}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\left(N+v_{p}\left(\mathbf{t}\right)\right)\hat{A}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)\label{eq:MD asymptotic analysis, nearly done} \end{equation} On the other hand, suppose that $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. Then, since: \begin{align} \left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\sum_{m=0}^{M-1}\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{m} & =\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{M}=\sum_{m=0}^{M-1}\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \end{align} it follows that: \begin{align} \sum_{m=0}^{M-1}\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{m} & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\left(\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{M}\right)\\ & =\left(\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{M}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1} \end{align} With this, the right-hand side of (\ref{eq:MD asymptotic analysis, nearly done}) becomes: \[ \begin{cases} \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\left(N+v_{p}\left(\mathbf{t}\right)\right)\hat{A}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)+\hat{A}_{H}\left(\mathbf{t}\right)\frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N+v_{p}\left(\mathbf{t}\right)}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\det\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\neq0 \end{cases} \] Finally, when $\mathbf{t}=\mathbf{0}$: \begin{align} \hat{\chi}_{H,N}\left(\mathbf{0}\right) & =\sum_{n=0}^{N-1}\hat{A}_{H,n}\left(\mathbf{0}\right)\beta_{H}\left(\mathbf{0}\right)\\ & =\beta_{H}\left(\mathbf{0}\right)+\sum_{n=1}^{N-1}\mathbf{1}_{\mathbf{0}}\left(\mathbf{0}\right)\left(\prod_{m=0}^{n-1}\alpha_{H}\left(\mathbf{0}\right)\right)\beta_{H}\left(\mathbf{0}\right)\\ & =\beta_{H}\left(\mathbf{0}\right)+\sum_{n=1}^{N-1}\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)\\ & =\begin{cases} \beta_{H}\left(\mathbf{0}\right)N & \textrm{if }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\\ \frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\det\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\neq0 \end{cases} \end{align} Since $\hat{A}_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, we have that $\beta_{H}\left(\mathbf{0}\right)N=\hat{A}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)N$ when $\mathbf{t}=\mathbf{0}$ (in the $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ case) and: \[ \frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\frac{\mathbf{I}_{d}-\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{N}}{\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)}\beta_{H}\left(\mathbf{0}\right) \] when $\mathbf{t=0}$ in the $\det\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\neq0$ case. Q.E.D. \vphantom{} When $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, we have that \[ \hat{\chi}_{H,N}\left(\mathbf{t}\right)-N\mathbf{1}_{\mathbf{0}}\left(p^{N-1}\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) \] is given by: \begin{equation} \begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(v_{p}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right) & \textrm{if }0<\left\Vert \mathbf{t}\right\Vert _{p}<p^{N}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}=p^{N}\\ \mathbf{0} & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{p}>p^{N} \end{cases}\label{eq:MD Fine structure of Chi_H,N hat when alpha is 1} \end{equation} \subsection{\label{subsec:6.2.2 Multi-Dimensional--=00003D000026}Multi-Dimensional $\hat{\chi}_{H}$ and $\tilde{\chi}_{H,N}$ for $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$} We begin by computing the Fourier series generated by $v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)$. \begin{lem}[\textbf{$v_{p}\hat{A}_{H}$ Summation Formulae}] \label{lem:MD v_p A_H hat summation formulae}\ \vphantom{} I. \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}^{d,d}}{=} & -N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\label{eq:MD Fourier sum of A_H hat v_rho}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\nonumber \\ & -\sum_{n=0}^{N-1}\left(n+1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\nonumber \end{align} Additionally, if $H$ is commutative: \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\overline{\mathbb{Q}}^{d,d}}{=} & -N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\label{eq:MD Fourier sum of A_H hat v_rho, commutative}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\nonumber \\ & -\sum_{n=0}^{N-1}\left(n+1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\nonumber \end{align} \vphantom{} II. \begin{align} \sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} }v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\mathcal{F}_{p,q}^{d,d}}{=} & \sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Limit of Fourier sum of v_rho A_H hat}\\ & -\sum_{n=0}^{\infty}\left(n+1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\nonumber \end{align} for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, where the convergence is point-wise. The right-hand side can also be written as: \begin{equation} \sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(n+1\right)\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)-n\mathbf{I}_{d}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Limit of Fourier sum of v_rho A_H hat, Alt} \end{equation} When $H$ is commutative, we get: \begin{equation} \sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\backslash\left\{ \mathbf{0}\right\} }v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\overset{\mathcal{F}_{p,q}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(\left(n+1\right)\alpha_{H}\left(\mathbf{0}\right)-n\mathbf{I}_{d}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Limit of Fourier sum of v_rho A_H hat, Alt, commutative} \end{equation} \end{lem} Proof: For (I), using (\ref{eq:MD Convolution of dA_H and D_N}) gives: \begin{align} \sum_{0<p\left\Vert \mathbf{t}\right\Vert \leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & =\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & -N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & -N\sum_{m=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\\ & +\sum_{n=1}^{N-1}\sum_{m=0}^{n-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m} \end{align} where we used the fact that $\tilde{A}_{H,0}\left(\mathbf{z}\right)=\hat{A}_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Next, we note the formal identity (provable by summation by parts): \begin{equation} \sum_{n=1}^{N-1}\sum_{m=0}^{n-1}f\left(m\right)=\sum_{m=0}^{N-2}\sum_{n=m+1}^{N-1}f\left(m\right)=\sum_{m=0}^{N-2}\left(N-1-m\right)f\left(m\right) \end{equation} Applying this to our sum yields: \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & =\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & -N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & -N\sum_{m=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\\ & +\sum_{m=0}^{N-2}\left(N-1-m\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m} \end{align} and hence: \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & =-N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & -\sum_{n=0}^{N-1}\left(n+1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{align} The $\mathcal{F}_{p,q}^{d,d}$-convergence of this sum as $N\rightarrow\infty$ in the case where $H$ is semi-basic and contracting follows from the decay properties ($q$-adic and archimedean, respectively) of $\kappa_{H}$ and $\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}$, as well as the boundedness of $\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)$ in both topologies with respect to $n$. Finally, writing: \begin{equation} \mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)=\mathbf{I}_{d}-\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right) \end{equation} yields (\ref{eq:MD Limit of Fourier sum of v_rho A_H hat, Alt}). Q.E.D. \vphantom{} Like in the one-dimensional case, we now introduce $\varepsilon_{n}$. \begin{defn}[\textbf{Multi-Dimensional $\varepsilon_{n}$}] For each $n\in\mathbb{N}_{0}$, \nomenclature{$\varepsilon_{n}\left(\mathbf{z}\right)$}{ }we define $\varepsilon_{n}:\mathbb{Z}_{p}^{r}\rightarrow\overline{\mathbb{Q}}$ by: \begin{equation} \varepsilon_{n}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}e^{\frac{2\pi i}{p^{n+1}}\left(\left[\mathbf{z}\right]_{p^{n+1}}-\left[\mathbf{z}\right]_{p^{n}}\right)}=e^{2\pi i\left\{ \frac{\mathbf{z}}{p^{n+1}}\right\} _{p}}\cdot e^{-\frac{2\pi i}{p}\left\{ \frac{\mathbf{z}}{p^{n}}\right\} _{p}}\label{eq:MD Definition of epsilon_n} \end{equation} \end{defn} Once again, we have functional equations, among other things. \begin{prop}[\textbf{Properties of Multi-Dimensional $\varepsilon_{n}$}] \label{prop:properties of MD epsilon n}\ \vphantom{} I. \begin{equation} \varepsilon_{0}\left(\mathbf{z}\right)=e^{2\pi i\left\{ \frac{\mathbf{z}}{p}\right\} _{p}},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Epsilon 0 of z} \end{equation} \begin{equation} \varepsilon_{n}\left(\mathbf{j}\right)=1,\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall n\geq1\label{eq:MD Epsilon_n of j} \end{equation} \vphantom{} II. \begin{equation} \varepsilon_{n}\left(p\mathbf{m}+\mathbf{j}\right)=\begin{cases} \varepsilon_{0}\left(\mathbf{j}\right) & \textrm{if }n=0\\ \varepsilon_{n-1}\left(\mathbf{m}\right) & \textrm{if }n\geq1 \end{cases},\textrm{ }\forall\mathbf{m}\in\mathbb{N}_{0}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall n\geq1\label{eq:MD epsilon_n functional equations} \end{equation} \vphantom{} III. Let $\mathbf{z}\neq\mathbf{0}$. Then $\varepsilon_{n}\left(\mathbf{z}\right)=1\textrm{ }$ for all $n<v_{p}\left(\mathbf{z}\right)$. \end{prop} Proof: I. The identity (\ref{eq:MD Epsilon 0 of z}) is immediate from the definition of $\varepsilon_{n}$. As for the other one, note that $\left[\mathbf{j}\right]_{p^{n}}=\mathbf{j}$ for all $n\geq1$ and all $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$. So, letting $n\geq1$, we get: \begin{equation} \varepsilon_{n}\left(\mathbf{j}\right)=e^{\frac{2\pi i}{p^{n+1}}\left(\left[\mathbf{j}\right]_{p^{n+1}}-\left[\mathbf{j}\right]_{p^{n}}\right)}=e^{\frac{2\pi i}{p^{n+1}}\cdot\mathbf{0}}=1 \end{equation} as desired. \vphantom{} II. \begin{align} \varepsilon_{n}\left(p\mathbf{m}+\mathbf{j}\right) & =e^{2\pi i\left\{ \frac{p\mathbf{m}+\mathbf{j}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{p\mathbf{m}+\mathbf{j}}{p^{n}}\right\} _{p}}\\ & =e^{2\pi i\left\{ \frac{\mathbf{j}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathbf{j}}{p^{n}}\right\} _{p}}\cdot e^{2\pi i\left\{ \frac{\mathbf{m}}{p^{n}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathbf{m}}{p^{n-1}}\right\} _{p}}\\ & =\varepsilon_{n}\left(\mathbf{j}\right)\varepsilon_{n-1}\left(\mathbf{m}\right)\\ \left(\textrm{by (I)}\right); & =\begin{cases} \varepsilon_{0}\left(\mathbf{j}\right) & \textrm{if }n=0\\ \varepsilon_{n-1}\left(\mathbf{m}\right) & \textrm{if }n\geq1 \end{cases} \end{align} \vphantom{} III. Let $\mathbf{z}$ be non-zero. When $n<v_{p}\left(\mathbf{z}\right)$, we have that $p^{-n}\mathbf{z}$ and $p^{-\left(n+1\right)}\mathbf{z}$ are then $p$-adic integer tuples. As such: \begin{equation} \varepsilon_{n}\left(\mathbf{z}\right)=e^{2\pi i\left\{ \frac{\mathbf{z}}{p^{n+1}}\right\} _{p}}e^{-\frac{2\pi i}{p}\left\{ \frac{\mathbf{z}}{p^{n}}\right\} _{p}}=e^{0}\cdot e^{-0}=1 \end{equation} Q.E.D. \vphantom{}Now we compute the sum of the Fourier series generated by $\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)$. Once again, the non-singularity of $H$ will be necessary for this to work out, as will $H$'s semi-basicness and contracting-ness. \begin{lem}[\textbf{Multi-Dimensional $\gamma_{H}\hat{A}_{H}$ Summation Formulae}] \label{lem:MD gamma formulae}\ \vphantom{} I. The column vector: \begin{equation} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\in\overline{\mathbb{Q}}^{d}\label{eq:Left Hand Side of MD Gamma Formulae (partial Fourier sum)} \end{equation} is given by: \begin{equation} \sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Gamma formula} \end{equation} \vphantom{} II. As $N\rightarrow\infty$, \emph{(\ref{eq:Left Hand Side of MD Gamma Formulae (partial Fourier sum)})} is $\mathcal{F}_{p,q}^{d}$ convergent to: \begin{equation} \sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:F limit of MD Gamma_H A_H hat Fourier series} \end{equation} since $H$ is semi-basic and contracting. \end{lem} Proof: Throughout this proof, we write $\mathbf{X}$ to denote $H^{\prime}\left(\mathbf{0}\right)$. \vphantom{} I. Like in the one-dimensional case, we first note that the map: \begin{equation} \mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\mapsto\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}=\left(\frac{t_{1}\left\|t_{1}\right\|_{p}}{p},\ldots,\frac{t_{r}\left\|t_{r}\right\|_{p}}{p}\right)\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} takes tuples: \begin{equation} \mathbf{t}=\left(\frac{k_{1}}{p^{n_{1}}},\ldots,\frac{k_{r}}{p^{n_{r}}}\right) \end{equation} and outputs: \begin{equation} \frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}=\left(\frac{\left[k_{1}\right]_{p}}{p},\ldots,\frac{\left[k_{r}\right]_{p}}{p}\right) \end{equation} Now, for brevity, let: \begin{align} \gamma_{\mathbf{j}} & \overset{\textrm{def}}{=}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)=\gamma_{H}\left(\frac{j_{1}}{p},\ldots,\frac{j_{r}}{p}\right),\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\\ F_{N}\left(\mathbf{z}\right) & \overset{\textrm{def}}{=}\sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{align} Finally, for each $\mathbf{k}\leq p^{n-1}-1$, each $\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}$, and each $n\geq1$, we write: \begin{equation} \frac{p\mathbf{k}+\mathbf{j}}{p^{n}}=\left(\frac{pk_{1}+j_{1}}{p^{n}},\ldots,\frac{pk_{r}+j_{r}}{p^{n}}\right) \end{equation} \begin{claim} In the above notation, we have: \begin{equation} \left\{ \mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}:\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}\right\} =\left\{ \frac{p\mathbf{k}+\mathbf{j}}{p^{n}}:\mathbf{k}\leq p^{n-1}-1,\textrm{ }\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} \right\} \label{eq:MD Decomposition of level sets} \end{equation} Proof of claim: \vphantom{} I. Let $\mathbf{t}=\frac{p\mathbf{k}+\mathbf{j}}{p^{n}}$. Then, for each $m\in\left\{ 1,\ldots,r\right\} $, $t_{m}=\frac{pk_{m}+j_{m}}{p^{n}}$ where $0\leq k_{m}\leq p^{n-1}-1$ and $0\leq j_{m}\leq p-1$; moreover, there exists an $\ell\in\left\{ 1,\ldots,r\right\} $ so that $j_{\ell}>0$. Since $pk_{\ell}+j_{\ell}$ is then co-prime to $p_{\ell}$, it follows that $\left\|t_{\ell}\right\|_{p}=p^{n}$. Since $\left\|t_{m}\right\|_{p}\leq p^{n}$ for all $m$, this shows that $\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}$. \vphantom{} II. Let $\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}$. Then, for every $m\in\left\{ 1,\ldots,r\right\} $, we can write $t_{m}=\frac{\nu_{m}}{p^{n}}$, where $\nu_{m}\in\left\{ 0,\ldots,p^{n}-1\right\} $. Moreover, there is an $\ell\in\left\{ 1,\ldots,r\right\} $ so that $\nu_{\ell}$ is co-prime to $p$, so as to guarantee that $\left\|t_{\ell}\right\|_{p}=p^{n}$ Each $\nu_{m}$ can be written as $pk_{m}-j_{m}$, where we have chosen $j_{m}=\left[\nu_{m}\right]_{p}\leq p-1$ and $k_{m}=\left(\nu_{m}-j_{m}\right)/p$. Since $\nu_{m}\leq p^{n}-1$, this forces $k_{m}\leq p^{n-1}-1$. Consequently, for these $j_{m}$s, the tuple $\mathbf{j}=\left(j_{1},\ldots,j_{r}\right)$ is then an element of $\mathbb{Z}^{r}/p\mathbb{Z}^{r}$, and $\mathbf{k}$ is a tuple $\leq p^{n-1}-1$ so that $\mathbf{t}=\left(p\mathbf{k}+\mathbf{j}\right)/p^{n}$. Finally, since $\nu_{\ell}$ is co-prime to $p$, $j_{\ell}\neq0$, which shows that $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. \vphantom{} This proves the claim. \vphantom{} \end{claim} Consequently, we can express $F_{N}$ as a sum involving the $\gamma_{\mathbf{j}}$s like so: \begin{align} F_{N}\left(\mathbf{z}\right) & =\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{n}}\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ \left(\textrm{use }(\ref{eq:MD Decomposition of level sets})\right); & =\sum_{n=1}^{N}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\sum_{\mathbf{k}=\mathbf{0}}^{p^{n-1}-1}\hat{A}_{H}\left(\frac{p\mathbf{k}+\mathbf{j}}{p^{n}}\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)e^{2\pi i\left\{ \frac{p\mathbf{k}+\mathbf{j}}{p^{n}}\mathbf{z}\right\} _{p}} \end{align} Using the formal identity: \begin{equation} \sum_{\mathbf{k}=\mathbf{0}}^{p^{n-1}-1}f\left(\frac{p\mathbf{k}+\mathbf{j}}{p^{n}}\right)=\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}f\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right) \end{equation} we can then write: \begin{align} F_{N}\left(\mathbf{z}\right) & =\sum_{n=1}^{N}\sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\label{eq:MD 1/3rd of the way through the gamma computation} \end{align} To deal with the $\mathbf{j}$-sum, we express $\hat{A}_{H}$ in product form, changing: \[ \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \] into: \[ \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left(\prod_{\mathbf{m}=\mathbf{0}}^{n-1}\alpha_{H}\left(p^{m}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\right)\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \] Using \textbf{Proposition \ref{prop:MD alpha H series}} to write the $\alpha_{H}$-product out as a series gives: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(\frac{\mathbf{X}}{p^{r}}\right)^{n}e^{-2\pi i\mathbf{m}\cdot\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{equation} and hence: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(\frac{\mathbf{X}}{p^{r}}\right)^{n}\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{m}\right)\right\} _{p}} \end{equation} Summing over $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}$ and using the Fourier series for $\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right]$: \begin{equation} \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}e^{2\pi i\left\{ \mathbf{t}\left(\mathbf{z}-\mathbf{m}\right)\right\} _{p}}=p^{r\left(n-1\right)}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] \end{equation} we obtain: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)\left(\frac{\mathbf{X}}{p^{r}}\right)^{n}\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}p^{r\left(n-1\right)}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] \end{equation} In summary: \begin{eqnarray} & \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\nonumber \\ & =\label{eq:MD 2/3rds of the way through the gamma computation}\\ & \frac{1}{p^{r}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left(\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right]\right)\mathbf{X}^{n}\gamma_{\mathbf{j}}\nonumber \end{eqnarray} Next, using the formal summation identity: \begin{equation} \sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}f\left(\mathbf{m}\right)=\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}f\left(\mathbf{m}+p^{n-1}\mathbf{k}\right)\label{eq:MD rho to the n formal identity} \end{equation} the expression: \begin{equation} \sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\kappa_{H}\left(\mathbf{m}\right)e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] \end{equation} becomes: \begin{equation} \sum_{\mathbf{k}=\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}\kappa_{H}\left(\mathbf{m}+p^{n-1}\mathbf{k}\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] \end{equation} Applying the functional equation identity for $\kappa_{H}$ (equation (\ref{eq:MD Kappa_H has P-adic structure}) from \textbf{Lemma \ref{lem:properties of MD kappa_H}}) yields: \begin{equation} \sum_{\mathbf{k}=\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}\kappa_{H}\left(\mathbf{m}\right)\mathcal{C}_{H}\left(\kappa_{H}\left(\mathbf{k}\right):\lambda_{p}\left(\mathbf{m}\right)\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\mathbf{m}\right)}{p^{n}}\right\} _{p}}\left[\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}\right] \end{equation} Here, note that $\left[\mathbf{z}\right]_{p^{n-1}}$ is the unique integer $r$-tuple $\mathbf{m}$ so that $\mathbf{m}\leq p^{n-1}-1$ and $\mathbf{z}\overset{p^{n-1}}{\equiv}\mathbf{m}$. This leaves us with: \begin{equation} \sum_{\mathbf{k}=\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathcal{C}_{H}\left(\kappa_{H}\left(\mathbf{k}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}e^{2\pi i\left\{ \frac{\mathbf{j}\left(\mathbf{z}-\left[\mathbf{z}\right]_{p^{n-1}}\right)}{p^{n}}\right\} _{p}} \end{equation} which is: \begin{equation} \sum_{\mathbf{k}=\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathcal{C}_{H}\left(\kappa_{H}\left(\mathbf{k}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right)\label{eq:gamma proof - the above} \end{equation} Next, observe that for fixed $\left[\mathbf{z}\right]_{p^{n-1}}$, the map $\mathbf{A}\mapsto\mathcal{C}_{H}\left(\mathbf{A}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)$ is linear. So, (\ref{eq:gamma proof - the above}) can be written as: \begin{equation} \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathcal{C}_{H}\left(\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right) \end{equation} With this, (\ref{eq:MD 2/3rds of the way through the gamma computation}) becomes: \begin{eqnarray} & \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\nonumber \\ & =\label{eq:MD 2/3rds and 1/4th of the way through the gamma computation}\\ & \frac{1}{p^{r}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathcal{C}_{H}\left(\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right)\mathbf{X}^{n}\gamma_{\mathbf{j}}\nonumber \end{eqnarray} Our next step is to simplify the expression with $\mathcal{C}_{H}$. Here, we use $\kappa_{H}$'s functional equation ((\ref{eq:MD Kappa_H functional equations}) from \textbf{Lemma \ref{lem:properties of MD kappa_H}}), setting $\mathbf{j}=\mathbf{k}$ and $\mathbf{n}=\mathbf{0}$ in it. This gives us: \begin{align} \sum_{\mathbf{k}=\mathbf{0}}^{p-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}} & =\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{k}}^{-1}\mathbf{A}_{\mathbf{k}}\kappa_{H}\left(\mathbf{0}\right)\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}}e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}\\ \left(\kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\right); & =\left(\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{k}}^{-1}\mathbf{A}_{\mathbf{k}}e^{-2\pi i\frac{\mathbf{j}\cdot\mathbf{k}}{p}}\right)\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}}\\ & =p^{r}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}}\\ & =p^{r}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{X}^{-1} \end{align} Hence, (\ref{eq:MD 2/3rds and 1/4th of the way through the gamma computation}) becomes: \begin{eqnarray} & \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & =\\ & \frac{1}{p^{r}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathcal{C}_{H}\left(p\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{X}^{-1}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\underbrace{\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right)}_{\textrm{scalar}}\mathbf{X}^{n}\gamma_{\mathbf{j}}\\ & =\\ & \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right)\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{X}^{-1}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\mathbf{X}^{n}\gamma_{\mathbf{j}} \end{eqnarray} Next, we note the following formal identity for matrices $\mathbf{A}$ and $\mathbf{X}$ and integers $m,n\geq0$: \begin{align} \mathcal{C}_{H}\left(\mathbf{A}\mathbf{X}^{-1}:m\right)\mathbf{X}^{n} & =\left(\mathbf{X}^{m}\mathbf{A}\mathbf{X}^{-1}\mathbf{X}^{-m}\right)\mathbf{X}^{n}\\ & =\mathbf{X}^{m}\mathbf{A}\mathbf{X}^{-m+n-1}\\ & =\mathbf{X}^{-n+1}\mathbf{X}^{m-n+1}\mathbf{A}\mathbf{X}^{-\left(m-n+1\right)}\\ & =\mathbf{X}^{-n+1}\mathcal{C}_{H}\left(\mathbf{A}:m-n+1\right) \end{align} As such: \begin{eqnarray} & \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\cdot\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\cdot\mathbf{z}\right\} _{p}}\\ & =\\ & \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\underbrace{\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right)}_{\textrm{a scalar}}\mathbf{X}^{-n+1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)-n+1\right)\gamma_{\mathbf{j}}\\ & =\\ & \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\mathbf{X}^{-n+1}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)-n+1\right)\gamma_{\mathbf{j}} \end{eqnarray} Since: \begin{align} \mathbf{X}^{-n+1}\mathcal{C}_{H}\left(\mathbf{A}:m-\left(n-1\right)\right) & =\mathbf{X}^{-n+1}\left(\mathbf{X}^{m-\left(n-1\right)}\mathbf{A}\mathbf{X}^{-m+n-1}\right)\\ & =\mathbf{X}^{-n+1}\mathbf{X}^{-\left(n-1\right)}\left(\mathbf{x}^{m}\mathbf{A}\mathbf{x}^{-m}\right)\mathbf{X}^{n-1}\\ & =\mathcal{C}_{H}\left(\mathbf{A}:m\right)\mathbf{X}^{n-1} \end{align} we then have: \begin{eqnarray} & \sum_{\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{n-1}}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\hat{A}_{H}\left(\mathbf{t}+\frac{\mathbf{j}}{p^{n}}\right)\gamma_{\mathbf{j}}e^{2\pi i\left\{ \frac{\mathbf{j}\mathbf{z}}{p^{n}}\right\} _{p}}e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\nonumber \\ & =\label{eq:MD 2/3rds and 1/4th and a bit of the way through the gamma computation}\\ & \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\mathbf{X}^{n-1}\gamma_{\mathbf{j}}\nonumber \end{eqnarray} Finally, returning with this to (\ref{eq:MD 1/3rd of the way through the gamma computation}), we get: \begin{align} F_{N}\left(\mathbf{z}\right) & =\sum_{n=1}^{N}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n-1}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n-1}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n-1}\gamma_{\mathbf{j}} \end{align} which gives (\ref{eq:MD Gamma formula}) after re-indexing $n$ by a shift of $1$. \vphantom{} II. Taking the $n$th term from (\ref{eq:MD Gamma formula}), we first apply $q$-adic norm to get and upper bound of: \begin{equation} \left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{q}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}^{n}\max_{\mathbf{0}<\mathbf{j}\leq p-1}\left\Vert \beta_{H}\left(\frac{\mathbf{j}}{p}\right)\right\Vert _{q} \end{equation} In this, we used: \begin{equation} \gamma_{\mathbf{j}}=\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)=\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\right)^{-1}\beta_{H}\left(\frac{\mathbf{j}}{p}\right) \end{equation} Since the $q$-adic bound on the $\beta_{H}$s is independent of $n$, and since $\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{q}=1$, the fact that $\left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{q}$ tends to $0$ as $n\rightarrow\infty$ for all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$ then guarantees the convergence of (\ref{eq:F limit of MD Gamma_H A_H hat Fourier series}) for $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. For $\mathbf{z}\in\mathbb{N}_{0}^{r}$, applying the archimedean complex norm to the $n$th term of (\ref{eq:MD Gamma formula}) gives an upper bound of: \begin{equation} \sum_{n=0}^{N-1}\left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{\infty}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{\infty}^{n}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left\Vert \beta_{H}\left(\frac{\mathbf{j}}{p}\right)\right\Vert _{\infty} \end{equation} Here, $\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left\Vert \beta_{H}\left(\frac{\mathbf{j}}{p}\right)\right\Vert _{\infty}$ is uniformly bounded with respect to $n$. Moreover, since $\mathbf{z}\in\mathbb{N}_{0}^{r}$, we have that $\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)=\kappa_{H}\left(\mathbf{z}\right)$ for all sufficiently large $n$. So, if we let $N\rightarrow\infty$, we obtain: \begin{equation} \sum_{n=0}^{N-1}\left\Vert \kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right\Vert _{\infty}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{\infty}^{n}\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\left\Vert \beta_{H}\left(\frac{\mathbf{j}}{p}\right)\right\Vert _{\infty}\ll O\left(1\right)+\sum_{n}\left\Vert H^{\prime}\left(\mathbf{0}\right)\right\Vert _{\infty}^{n} \end{equation} Here, the upper bound is a convergent geometric series because $H$ is contracting. This then guarantees the convergence of (\ref{eq:F limit of MD Gamma_H A_H hat Fourier series}) for $\mathbf{z}\in\mathbb{N}_{0}^{r}$. Q.E.D. \vphantom{} With these formulae, we can now sum the Fourier series generated by (\ref{eq:MD Fine structure of Chi_H,N hat when alpha is 1}) to obtain a non-trivial formula for $\chi_{H,N}$. \begin{thm} Suppose $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Then, for all $N\geq1$ and all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$: \begin{align} \chi_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Chi_H,N when alpha is 1 and rho is arbitrary} \end{align} In particular, when $p=2$: \begin{equation} \chi_{H,N}\left(\mathbf{z}\right)=-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)\label{eq:MD Chi_H,N when alpha is 1 and every prime in P is 2} \end{equation} where: \begin{equation} \gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\overset{\textrm{def}}{=}\gamma_{H}\left(\left(\frac{1}{2},\ldots,\frac{1}{2}\right)\right)\label{eq:Definition of MD gamma_H of 1/2} \end{equation} \end{thm} Proof: We start by multiplying (\ref{eq:MD Fine structure of Chi_H,N hat when alpha is 1}) by $e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}$ and then summing over all $\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}$. The left-hand side of (\ref{eq:MD Fine structure of Chi_H,N hat when alpha is 1}) becomes: \begin{equation} \chi_{H,N}\left(\mathbf{z}\right)-N\tilde{A}_{H,N-1}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right) \end{equation} while the right-hand side becomes: \begin{align} & \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N-1}}\hat{A}_{H}\left(\mathbf{t}\right)\left(v_{p}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\\ & +\sum_{\left\Vert \mathbf{t}\right\Vert _{p}=p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} \end{align} Simplifying produces: \begin{align} \chi_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}N\tilde{A}_{H,N-1}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)\label{eq:MD Chi_H,N rho not equal to 2, ready to simplify}\\ & +\sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N-1}}\hat{A}_{H}\left(\mathbf{t}\right)v_{p}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\nonumber \\ & +\sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}\nonumber \end{align} Once again, we call upon our legion of formulae: (\ref{eq:MD Convolution of dA_H and D_N}), \textbf{Lemmata \ref{lem:MD v_p A_H hat summation formulae}}, and \textbf{\ref{lem:MD gamma formulae}}. Using them (while remembering that $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ makes $\mathbf{I}_{H}$ identically $\mathbf{O}_{d}$) turns (\ref{eq:MD Chi_H,N rho not equal to 2, ready to simplify}) into: \begin{align} \chi_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N-1}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N-1}\beta_{H}\left(\mathbf{0}\right)\\ & -\left(N-1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N-1}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N-1}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{n=0}^{N-2}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Simplifying yields: \begin{align} \chi_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Now, because $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, the vector: \begin{equation} \mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{0}}{p}\right)\varepsilon_{n}\left(\mathbf{0}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{0}}{p}\right) \end{equation} becomes: \begin{equation} \underbrace{\mathcal{C}_{H}\left(\mathbf{I}_{d}:\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)}_{\mathbf{I}_{d}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)=\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right) \end{equation} So, the two lines on the right-hand side of the above formula for $\chi_{H,N}\left(\mathbf{z}\right)$ combine to form a single sum because the upper line is the $\mathbf{j}=\mathbf{0}$ case of the bottom line. Finally, when $p=2$, we can compute everything directly from (\ref{eq:MD Fine structure of Chi_H,N hat when alpha is 1}) by multiplying by $e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}$ and summing over all $\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N}$. This gives: \begin{align} \chi_{H,N}\left(\mathbf{z}\right)-N\tilde{A}_{H,N-1}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)= & \sum_{0<\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N-1}}\hat{A}_{H}\left(\mathbf{t}\right)v_{2}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{0<\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right) \end{align} which simplifies to: \begin{align} \chi_{H,N}\left(\mathbf{z}\right)= & N\tilde{A}_{H,N-1}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\\ & +\sum_{0<\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N-1}}\hat{A}_{H}\left(\mathbf{t}\right)v_{2}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\beta_{H}\left(\mathbf{0}\right) \end{align} Applying (\ref{eq:MD Convolution of dA_H and D_N}) and \textbf{Lemma \ref{lem:MD v_p A_H hat summation formulae}}\textemdash and, again remembering that $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ makes $\mathbf{I}_{H}$ identically $\mathbf{O}_{d}$\textemdash the above becomes: \begin{align} \chi_{H,N}\left(\mathbf{z}\right)= & N\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N-1}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N-1}\beta_{H}\left(\mathbf{0}\right)-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\\ & +\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\\ & -\left(N-1\right)\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N-1}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N-1}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{n=0}^{N-2}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right) \end{align} Simplifying then gives us what we wanted: \[ \chi_{H,N}\left(\mathbf{z}\right)=-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right) \] Q.E.D. \begin{cor}[\textbf{$\mathcal{F}$-Series for $\chi_{H}$ when $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$}] \label{cor:MD alpha is 1 case, F-series}Suppose $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Then: \begin{equation} \chi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary} \end{equation} for all $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, with the special case: \begin{equation} \chi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{2,q_{H}}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\beta_{H}\left(\mathbf{0}\right)\label{eq:MD Explicit Formula for Chi_H when alpha is 1 and every prime in P is 2} \end{equation} for all $\mathbf{z}\in\mathbb{Z}_{2}$ when $p=2$. \end{cor} Proof: Same as in the one-dimensional case, but with vector norms. Q.E.D. \vphantom{} Next, we sum in $\mathbb{C}$ for $\mathbf{z}\in\mathbb{N}_{0}^{r}$. \begin{cor} \label{cor:MD alpha is 1, F-series on N_0^r}Suppose that $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Then: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD archimedean Chi_H when rho is arbitrary and alpha_H of 0 is 1}\\ & +M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} for all $\mathbf{n}\in\mathbb{N}_{0}^{r}$, with the special case: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+M_{H}\left(\mathbf{n}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)\label{eq:MD archimedean Chi_H when every prime in P is 2 and alpha_H of 0 is 1}\\ & +\sum_{k=0}^{\lambda_{2}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{2^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\beta_{H}\left(\mathbf{0}\right)\chi_{H}\left(\mathbf{n}\right) \end{align} when $p=2$. Regardless of the value of $p$, the $k$-sums are defined to be $\mathbf{0}$ when $\mathbf{n}=\mathbf{0}$. \end{cor} Proof: Let $\mathbf{n}\in\mathbb{N}_{0}^{r}$. Since $\left[\mathbf{n}\right]_{p^{k}}=\mathbf{n}$ and $\varepsilon_{k}\left(\mathbf{n}\right)=1$ for all $k\geq\lambda_{p}\left(\mathbf{n}\right)$, (\ref{eq:MD Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary}) becomes: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ & +\kappa_{H}\left(\mathbf{n}\right)\sum_{k=\lambda_{p}\left(\mathbf{n}\right)}^{\infty}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right):\lambda_{p}\left(\mathbf{n}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Here: \[ \sum_{k=\lambda_{p}\left(\mathbf{n}\right)}^{\infty}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right):\lambda_{p}\left(\mathbf{n}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \] becomes: \[ \sum_{k=\lambda_{p}\left(\mathbf{n}\right)}^{\infty}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{n}\right)}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k-\lambda_{p}\left(\mathbf{n}\right)}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \] which simplifies to : \[ \sum_{\mathbf{j}=\mathbf{0}}^{p-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{n}\right)}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\sum_{k=0}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \] Since $H$ is contracting, we get a $\mathbb{C}$-convergent geometric series: \[ \sum_{k=0}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\overset{\mathbb{C}}{=}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \] and so, we are left with: \[ \sum_{\mathbf{j}=\mathbf{0}}^{p-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{n}\right)}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \] Consequently: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ & +\kappa_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{n}\right)}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Since: \[ \kappa_{H}\left(\mathbf{n}\right)=M_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\mathbf{n}\right)} \] the above can be simplified to: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ & +M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} which is the desired formula. As for the case where $p=2$, applying the above argument to (\ref{eq:MD Explicit Formula for Chi_H when alpha is 1 and every prime in P is 2}) given produces: \begin{align} \chi_{H}\left(\mathbf{n}\right) & \overset{\mathbb{C}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{k=0}^{\lambda_{2}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{2^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{k=\lambda_{2}\left(\mathbf{n}\right)}^{\infty}\kappa_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\beta_{H}\left(\mathbf{0}\right)\\ \left(H\textrm{ is contracting}\right); & \overset{\mathbb{C}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{k=0}^{\lambda_{2}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{2^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\beta_{H}\left(\mathbf{0}\right)\\ & +\underbrace{\kappa_{H}\left(\mathbf{n}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{2}\left(\mathbf{n}\right)}}_{M_{H}\left(\mathbf{n}\right)}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right) \end{align} Q.E.D. \vphantom{} Taken together, these two corollaries establish the quasi-integrability of $\chi_{H}$ for arbitrary $p$, provided that $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$: \begin{cor}[\textbf{Quasi-Integrability of $\chi_{H}$ When $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$}] If $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, then, $\chi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame. In particular, when $p=2$, the function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{2}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ defined by: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\begin{cases} -\gamma_{H}\left(\mathbf{\frac{1}{2}}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)v_{2}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) & \textrm{else } \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{2}\label{eq:MD Formula for Chi_H hat when every prime in P is 2 and alpha is 1} \end{equation} is then a Fourier transform of $\chi_{H}$. In this case, the function defined by: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(v_{2}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right) & \textrm{if }\textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{2}^{r}\label{eq:MD Formula for Chi_H hat when every prime in P is 2 and alpha is 1, alt} \end{equation} is also a Fourier transform of $\chi_{H}$, differing from the $\hat{\chi}_{H}$ given above by $\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)$. \emph{By}\textbf{\emph{ Theorem \ref{thm:MD properties of A_H hat}}},\textbf{ }since $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, the function $\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)$ is then the Fourier-Stieltjes transform of a degenerate thick measure of vector type. For odd primes $p$, we can obtain a Fourier transform for $\chi_{H}$ by defining a function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ by: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(v_{p}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right) & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Chi_H hat when rho is not 2 and when alpha is 1} \end{equation} \end{cor} Proof: \textbf{Corollaries \ref{cor:MD alpha is 1 case, F-series}} and \textbf{\ref{cor:MD alpha is 1, F-series on N_0^r}} show that the $N$th partial sums of the Fourier series generated by (\ref{eq:MD Formula for Chi_H hat when every prime in P is 2 and alpha is 1}) and (\ref{eq:MD Chi_H hat when rho is not 2 and when alpha is 1}) are $\mathcal{F}_{p,q_{H}}$-convergent to (\ref{eq:MD Explicit Formula for Chi_H when alpha is 1 and every prime in P is 2}) and (\ref{eq:MD Explicit Formula for Chi_H when alpha is 1 and rho is arbitrary}) for the case where $p=2$ and the case $p\geq3$, respectively, thereby establishing the quasi-integrability of $\chi_{H}$ with respect to the standard $\left(p,q_{H}\right)$-adic frame. Finally, letting $\hat{\chi}_{H}^{\prime}\left(\mathbf{t}\right)$ denote (\ref{eq:MD Formula for Chi_H hat when every prime in P is 2 and alpha is 1, alt}), observe that when $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ and $p=2$: \begin{equation} \hat{\chi}_{H}^{\prime}\left(\mathbf{t}\right)-\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\begin{cases} -\gamma_{H}\left(\mathbf{\frac{1}{2}}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)v_{2}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) & \textrm{if }\left\Vert \mathbf{t}\right\Vert _{2}>0 \end{cases}=\hat{\chi}_{H}\left(\mathbf{t}\right) \end{equation} which shows that $\hat{\chi}_{H}^{\prime}\left(\mathbf{t}\right)$ and $\hat{\chi}_{H}\left(\mathbf{t}\right)$ differ by a factor of $\hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)$, which is the Fourier-Stieltjes transform of a degenerate vector-type thick measure, by \textbf{Theorem \ref{thm:MD properties of A_H hat}}, since $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. Q.E.D. \subsection{\label{subsec:6.2.3 Multi-Dimensional--=00003D000026}Multi-Dimensional $\hat{\chi}_{H}$ and $\tilde{\chi}_{H,N}$ \textendash{} The Commutative Case} Like in the one-dimensional case, we now introduce $\psi_{H}$ and $\Psi_{H}$, which we will use in conjunction with $\chi_{H}$'s functional equations (\ref{eq:Functional Equations for Chi_H over the rho-adics}) to extend what we have done to cover all \textbf{\emph{commutative}} $p$-Hydra maps, regardless of the value of $\alpha_{H}\left(\mathbf{0}\right)$. \begin{defn}[\textbf{Multi-Dimensional Little Psi-$H$ \& Big Psi-$H$}] \ \vphantom{} I. We define $\psi_{H}:\mathbb{N}_{0}^{r}\rightarrow\overline{\mathbb{Q}}^{d,d}$ (``Little Psi-$H$'') by: \begin{equation} \psi_{H}\left(\mathbf{n}\right)\overset{\textrm{def}}{=}M_{H}\left(\mathbf{n}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}+\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\label{eq:MD Definition of Little Psi_H} \end{equation} \vphantom{} II. We define $\Psi_{H}:\mathbb{N}_{0}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ (``Big Psi-$H$'') by: \begin{align} \Psi_{H}\left(\mathbf{n}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Definition of Big Psi_H}\\ & \sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} In both cases, the $k$-sums are defined to be $\mathbf{0}$ when $\mathbf{n}=\mathbf{0}$. \end{defn} \begin{lem}[\textbf{Rising-Continuability and Functional Equations for Multi-Dimensional $\psi_{H}$ \& $\Psi_{H}$}] \label{lem:MD Rising-continuability of Psi_Hs}\ \vphantom{} I. $\psi_{H}$ is rising-continuable to a $d\times d$-matrix-valued $\left(p,q_{H}\right)$-adic function $\psi_{H}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{Z}_{q_{H}}^{d,d}$ given by: \begin{equation} \psi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n},\textrm{ }\forall\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}\label{eq:MD Rising-continuation of Little Psi_H} \end{equation} Moreover, $\psi_{H}\left(\mathbf{z}\right)$ is the unique rising-continuous $d\times d$-matrix-valued $\left(p,q_{H}\right)$-adic function satisfying the system of functional equations: \begin{equation} \psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)\overset{\mathbb{C}_{q_{H}}^{d,d}}{=}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\psi_{H}\left(\mathbf{z}\right)+\mathbf{I}_{d},\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\textrm{ \& }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Little Psi_H functional equations} \end{equation} \index{functional equation!psi_{H}@$\psi_{H}$!multi-dimensional} \vphantom{} II. $\Psi_{H}$ is rising-continuable to a $d\times1$-vector-valued $\left(p,q_{H}\right)$-adic function $\Psi_{H}:\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q_{H}}^{d}$ given by:\index{functional equation!Psi_{H}@$\Psi_{H}$!multi-dimensional} \begin{equation} \Psi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Rising-continuation of Big Psi_H} \end{equation} for all $\mathbf{z}\in\left(\mathbb{Z}_{p}^{r}\right)^{\prime}$. Moreover, $\Psi_{H}\left(\mathbf{z}\right)$ is the unique rising-continuous $d\times1$-vector-valued $\left(p,q_{H}\right)$-adic function satisfying the system of functional equations: \begin{equation} \Psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)\overset{\mathbb{C}_{q_{H}}^{d}}{=}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\Psi_{H}\left(\mathbf{z}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Big Psi_H functional equations} \end{equation} \end{lem} Proof: For both parts, we use (\ref{eq:MD Relation between truncations and functional equations, version 2}) and (\ref{eq:MD Kappa_H functional equations}) from \textbf{Lemma \ref{lem:MD functional equations and truncation lemma}}. With these, observe that $\kappa_{H}$'s functional equations: \begin{equation} \kappa_{H}\left(p\mathbf{m}+\mathbf{j}\right)=\underbrace{\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}}_{H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)}\kappa_{H}\left(\mathbf{m}\right)\underbrace{\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}}}_{\left(H_{\mathbf{j}}\left(\mathbf{0}\right)\right)^{-1}} \end{equation} imply that: \begin{equation} \kappa_{H}\left(\left[p\mathbf{m}+\mathbf{j}\right]_{p^{n}}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n-1}}\right)\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}},\textrm{ }\forall n\in\mathbb{N}_{1},\textrm{ }\forall\mathbf{j}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r},\textrm{ }\forall\mathbf{m}\in\mathbb{N}_{0}^{r}\label{eq:MD functional equation truncation Lemma applied to kappa_H} \end{equation} In this case, the function $\Phi_{\mathbf{j}}$ from (\ref{eq:MD Relation between truncations and functional equations, version 1}) is, in this case: \begin{equation} \Phi_{\mathbf{j}}\left(\mathbf{m},\mathbf{X}\right)=\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\mathbf{X}\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}} \end{equation} where $\mathbf{X}$ is a $d\times d$ matrix. The rising-continuability of $\psi_{H}$ and $\psi_{H}$ to the given series follow by the givens on $H$, which guarantee that, for each $\mathbf{z}\in\mathbb{Z}_{p}^{r}$, $M_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)=\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(n\right)}$ tends to $0$ in the standard $\left(p,q_{H}\right)$-adic frame as $n\rightarrow\infty$, and convergence is then guaranteed by the same arguments used for \textbf{Lemma \ref{lem:MD v_p A_H hat summation formulae}} and equation (\ref{eq:F limit of MD Gamma_H A_H hat Fourier series}). All that remains is to verify the functional equations; \textbf{Theorem \ref{thm:rising-continuability of Generic H-type functional equations}} from Subsection \ref{subsec:5.3.2. Interpolation-Revisited} then guarantees the uniqueness of $\psi_{H}$ and $\Psi_{H}$ as rising-continuous solutions of their respective systems of functional equations. In what follows, recall that since we have assumed at the start of this chapter that $H$ is contracting, \textbf{Proposition \ref{prop:MD Contracting H proposition}} (page \pageref{prop:MD Contracting H proposition}) shows that the matrix $\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)$ is invertible. \vphantom{} I. We pull out the $k=0$ term from $\psi_{H}\left(p\mathbf{n}+\mathbf{j}\right)$: \begin{align} \psi_{H}\left(p\mathbf{n}+\mathbf{j}\right) & \overset{\overline{\mathbb{Q}}^{d,d}}{=}M_{H}\left(p\mathbf{n}+\mathbf{j}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\mathbf{I}_{d}\\ & +\sum_{k=1}^{\lambda_{p}\left(p\mathbf{n}+\mathbf{j}\right)-1}\kappa_{H}\left(\left[p\mathbf{n}+\mathbf{j}\right]_{p^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\\ & =\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}M_{H}\left(\mathbf{n}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}+\mathbf{I}_{d}\\ & +\sum_{k=1}^{\lambda_{p}\left(\mathbf{n}\right)}\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k-1}}\right)\underbrace{\mathbf{A}_{\mathbf{0}}^{-1}\mathbf{D}_{\mathbf{0}}}_{\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\\ & =\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}\underbrace{\left(M_{H}\left(\mathbf{n}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}+\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\right)}_{\psi_{H}\left(\mathbf{n}\right)}+\mathbf{I}_{d} \end{align} Consequently: \begin{equation} \psi_{H}\left(p\mathbf{n}+\mathbf{j}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\underbrace{\mathbf{D}_{\mathbf{j}}^{-1}\mathbf{A}_{\mathbf{j}}}_{H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)}\psi_{H}\left(\mathbf{n}\right)+\mathbf{I}_{d},\textrm{ }\forall\mathbf{n}\in\mathbb{N}_{0}^{r}\textrm{ \& }\forall j\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}\label{eq:MD Little Psi_H functional equation on the integer tuples} \end{equation} then shows that (\ref{eq:MD Little Psi_H functional equation on the integer tuples}) extends to hold for the rising-continuation of $\psi_{H}$, and that this rising-continuation is then the \emph{unique }$\left(p,q_{H}\right)$-adic function satisfying (\ref{eq:MD Little Psi_H functional equations}). Finally, letting $\mathbf{m}\in\mathbb{N}_{0}^{r}$, setting $\mathbf{z}=\mathbf{m}$, the right-hand side of (\ref{eq:MD Rising-continuation of Little Psi_H}) becomes: \begin{align} \sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} & \overset{\mathbb{C}^{d}}{=}\sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & +\kappa_{H}\left(\mathbf{m}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\mathbf{m}\right)}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & =M_{H}\left(\mathbf{n}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}+\sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & =\psi_{H}\left(\mathbf{m}\right) \end{align} Hence, (\ref{eq:MD Rising-continuation of Little Psi_H}) converges to $\psi_{H}$ in the standard frame. \vphantom{} II. Pulling out $k=0$ from (\ref{eq:MD Definition of Big Psi_H}) yields: \begin{align} \Psi_{H}\left(\mathbf{n}\right) & =M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ & +\kappa_{H}\left(\mathbf{0}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right):0\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ & +\sum_{k=1}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Here: \begin{equation} \kappa_{H}\left(\mathbf{0}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(0\right)\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{equation} is: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right) \end{equation} Now: \begin{align} \sum_{\mathbf{j}=\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right) & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\beta_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right)\\ & =\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{k}}^{-1}\mathbf{b}_{\mathbf{k}}\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}e^{2\pi i\frac{\mathbf{j}\cdot\left(\mathbf{n}-\mathbf{k}\right)}{p}}\\ & =\sum_{\mathbf{k}=\mathbf{0}}^{p-1}\mathbf{D}_{\mathbf{k}}^{-1}\mathbf{b}_{\mathbf{k}}\left[\mathbf{n}\overset{p}{\equiv}\mathbf{k}\right]\\ & =\mathbf{D}_{\left[\mathbf{n}\right]_{p}}^{-1}\mathbf{b}_{\left[\mathbf{n}\right]_{p}}\\ & =H_{\left[\mathbf{n}\right]_{p}}\left(\mathbf{0}\right) \end{align} and so: \begin{equation} \sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{0}\left(\mathbf{j}\mathbf{n}\right)=H_{\left[\mathbf{n}\right]_{p}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right) \end{equation} Consequently: \begin{align} \Psi_{H}\left(\mathbf{n}\right) & =M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)+H_{\left[\mathbf{n}\right]_{p}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{k=1}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Now, replacing $\mathbf{n}$ with $p\mathbf{n}+\mathbf{i}$ (where at least one of $\mathbf{n}$ and $\mathbf{i}$ is not the zero tuple), we use (\ref{eq:MD functional equation truncation Lemma applied to kappa_H}) and the functional equations for $M_{H}$, $\varepsilon_{n}$, and $\lambda_{p}$ (along with using (\ref{eq:MD Relation between truncations and functional equations, version 2}) for $\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)$) and the definition of $\mathcal{C}_{H}$ to obtain: \begin{align} & \mathbf{D}_{\mathbf{i}}^{-1}\mathbf{A}_{\mathbf{i}}M_{H}\left(\mathbf{n}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)+H_{\mathbf{i}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right)\\ & +\mathbf{D}_{\mathbf{i}}^{-1}\mathbf{A}_{\mathbf{i}}\sum_{k=0}^{\lambda_{p}\left(\mathbf{n}\right)-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{k}\left(\mathbf{j}\mathbf{n}\right):\lambda_{p}\left(\left[\mathbf{n}\right]_{p^{k}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{k}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} as the formula for $\Psi_{H}\left(p\mathbf{n}+\mathbf{i}\right)$. This is: \[ \Psi_{H}\left(p\mathbf{n}+\mathbf{i}\right)=\underbrace{\mathbf{D}_{\mathbf{i}}^{-1}\mathbf{A}_{\mathbf{i}}}_{H_{\mathbf{i}}^{\prime}\left(\mathbf{0}\right)}\Psi_{H}\left(p\mathbf{n}+\mathbf{i}\right)+H_{\mathbf{i}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right) \] Finally, letting $\mathbf{X}$ denote $H^{\prime}\left(\mathbf{0}\right)$ and letting $\mathbf{m}\in\mathbb{N}_{0}^{r}$, we have that for $\mathbf{z}=\mathbf{m}$, the right-hand side of (\ref{eq:MD Rising-continuation of Big Psi_H}) becomes: \begin{align} \sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right):\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\right)\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ +\sum_{n=\lambda_{p}\left(\mathbf{m}\right)}^{\infty}\kappa_{H}\left(\mathbf{m}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right):\lambda_{p}\left(\mathbf{m}\right)\right)\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Here: \[ \varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right)=e^{\frac{2\pi i}{p^{n+1}}\left(\left[\mathbf{j}\mathbf{m}\right]_{p^{n+1}}-\left[\mathbf{j}\mathbf{m}\right]_{p^{n}}\right)}=e^{\frac{2\pi i}{p^{n+1}}\left(\mathbf{j}\mathbf{m}-\mathbf{j}\mathbf{m}\right)}=1,\textrm{ }\forall n\geq\lambda_{p}\left(\mathbf{m}\right) \] and so: \begin{align} \sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right):\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\right)\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ +\sum_{n=\lambda_{p}\left(\mathbf{m}\right)}^{\infty}\kappa_{H}\left(\mathbf{m}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{X}^{-\lambda_{p}\left(\mathbf{m}\right)}\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Summed in the topology of $\mathbb{C}$, this becomes: \begin{align} \sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right):\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\right)\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ +\kappa_{H}\left(\mathbf{m}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\mathbf{X}^{-\lambda_{p}\left(\mathbf{m}\right)}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\left(\mathbf{I}_{d}-\mathbf{X}\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} which simplifies to: \begin{align} \sum_{n=0}^{\lambda_{p}\left(\mathbf{m}\right)-1}\kappa_{H}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{m}\right):\lambda_{p}\left(\left[\mathbf{m}\right]_{p^{n}}\right)\right)\mathbf{X}^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\\ +M_{H}\left(\mathbf{m}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\left(\mathbf{I}_{d}-\mathbf{X}\right)^{-1}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} which is precisely $\Psi_{H}\left(\mathbf{m}\right)$ as given in (\ref{eq:MD Definition of Big Psi_H}). Hence, (\ref{eq:MD Rising-continuation of Big Psi_H}) converges to $\Psi_{H}$ in the standard frame. Q.E.D. \begin{prop}[\textbf{Quasi-Integrability of $\psi_{H}$ and $\Psi_{H}$}] \label{prop:quasi-integrability of MD Psis}Let $H$ be commutative. Then: \vphantom{} I. $\psi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame whenever either $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ or the matrix $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. For these conditions, the function $\hat{\psi}_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d,d}$ defined by: \begin{equation} \hat{\psi}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\begin{cases} \begin{cases} \mathbf{O}_{d} & \textrm{if }\mathbf{t}=\mathbf{0}\\ v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right) & \textrm{if }\mathbf{t}\neq\mathbf{0} \end{cases} & \textrm{if }\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1} & \textrm{if }\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\textrm{ is invertible} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Fourier Transform of Little Psi_H, commutative} \end{equation} is then a Fourier transform of $\psi_{H}$. Hence: \begin{equation} \hat{\psi}_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}-N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Little Psi H N twiddle when alpha is 1, commutative} \end{equation} when $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ and: \begin{equation} \tilde{\psi}_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Little Psi H N twiddle when 1 minus alpha is invertible, commutative} \end{equation} when $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. \vphantom{} II. $\Psi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame for all $\alpha_{H}\left(\mathbf{0}\right)$, and the function $\hat{\Psi}_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ defined by: \begin{equation} \hat{\Psi}_{H}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 0 & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{if }\mathbf{t}\neq\mathbf{0} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Fourier Transform of Big Psi_H} \end{equation} is a Fourier transform of $\Psi_{H}$. Hence: \begin{equation} \tilde{\Psi}_{H,N}\left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d}}{=}\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\label{eq:MD Big Psi H N twiddle} \end{equation} \end{prop} Proof: I. When $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, (\ref{eq:MD Fourier Transform of Little Psi_H, commutative}) follows from \textbf{Lemma \ref{lem:MD v_p A_H hat summation formulae}}, and hence, the $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$ case of (\ref{eq:MD Fourier Transform of Little Psi_H, commutative}) is then indeed a Fourier transform of $\psi_{H}$, thus proving the $\mathcal{F}_{p,q_{H}}$-quasi-integrability of $\psi_{H}$ when $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$. When $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, comparing (\ref{eq:MD Fourier Limit of A_H,N twiddle in standard frame, commutative}): \begin{align} \tilde{A}_{H}\left(\mathbf{z}\right) & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \end{align} and (\ref{eq:MD Rising-continuation of Little Psi_H}): \[ \psi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \] we observe that: \[ \tilde{A}_{H}\left(\mathbf{z}\right)=\psi_{H}\left(\mathbf{z}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \] Since \textbf{Theorem \ref{thm:MD properties of A_H hat}} shows that $\tilde{A}_{H}\left(\mathbf{z}\right)$ is $\mathcal{F}_{p,q_{H}}$-quasi-integrable when $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible (which necessarily forces $\alpha_{H}\left(\mathbf{0}\right)\neq\mathbf{I}_{d}$), it then follows that: \[ \psi_{H}\left(\mathbf{z}\right)=\tilde{A}_{H}\left(\mathbf{z}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1} \] and hence, that: \[ \hat{\psi}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1} \] is a Fourier transform of $\psi_{H}$. This proves that $\psi_{H}$ is quasi-integrable with respect to the standard frame when $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, and that (\ref{eq:MD Fourier Transform of Little Psi_H, commutative}) is then a Fourier transform of $\psi_{H}$ in this case. Finally, since $H$ is commutative, for the case $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, (\ref{eq:MD Fourier sum of A_H hat v_rho, commutative}) becomes: \begin{align} \sum_{0<\left\Vert \mathbf{t}\right\Vert _{p}\leq p^{N}}v_{p}\left(\mathbf{t}\right)\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}} & \overset{\overline{\mathbb{Q}}^{d,d}}{=}-N\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{align} the left-hand side of which is exactly (\ref{eq:MD Little Psi H N twiddle when alpha is 1, commutative}). On the other hand, for the case where $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, the commutativity of $H$ tells us that (\ref{eq:MD Convolution of dA_H and D_N when H is commutative}) holds: \[ \tilde{A}_{H,N}\left(\mathbf{z}\right)=\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \] Right-multiplying by $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ then yields (\ref{eq:MD Little Psi H N twiddle when 1 minus alpha is invertible, commutative}). \vphantom{} II. (\ref{eq:MD Fourier Transform of Big Psi_H}) is exactly what we proved in \textbf{Lemma \ref{lem:MD Rising-continuability of Psi_Hs}}. (\ref{eq:MD Big Psi H N twiddle}) is precisely what we proved in (\ref{eq:MD Gamma formula}). Q.E.D. \vphantom{} Like in the one-dimensional case, we can bootstrap a Fourier transform for $\chi_{H}$ for the case where $H$ is commutative. However, this follows from a more general result which tells us how to express $\chi_{H}$ in terms of $\psi_{H}$ and $\Psi_{H}$: \begin{thm}[\textbf{$\mathcal{F}$-Series for Multi-Dimensional $\chi_{H}$}] \label{thm:MD F-series for Chi_H}Let $H$ be a $p$-smooth $d$-dimensional depth $r$ Hydra map dimension $d$ which is contracting, semi-basic, non-singular, and fixes $\mathbf{0}$. Regardless of: \vphantom{} I. Whether or not $\alpha_{H}\mathbf{\left(0\right)}=\mathbf{I}_{d}$; \vphantom{} II. Whether or not $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible; \vphantom{} III. Whether or not $H$ is commutative; \vphantom{} the numen $\chi_{H}$ admits the $\mathcal{F}$-series representation: \begin{equation} \chi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d}}{=}\psi_{H}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)+\Psi_{H}\left(\mathbf{z}\right),\textrm{ }\forall\mathbf{z}\in\mathbb{Z}_{p}^{r}\label{eq:MD Chi_H in terms of Little Psi_H and Big Psi_H} \end{equation} \end{thm} Proof: Let $f\left(\mathbf{z}\right)$ denote the function $\psi_{H}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)+\Psi_{H}\left(\mathbf{z}\right)$. Then, using the functional equations for $\psi_{H}$ and $\Psi_{H}$: \begin{equation} \psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\psi_{H}\left(\mathbf{z}\right)+\mathbf{I}_{d} \end{equation} \begin{equation} \Psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)=H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\Psi_{H}\left(\mathbf{z}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right) \end{equation} we have that: \begin{align} f\left(p\mathbf{z}+\mathbf{j}\right) & =\psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)\beta_{H}\left(\mathbf{0}\right)+\Psi_{H}\left(p\mathbf{z}+\mathbf{j}\right)\\ & =\left(H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\psi_{H}\left(\mathbf{z}\right)+\mathbf{I}_{d}\right)\beta_{H}\left(\mathbf{0}\right)+\left(H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\Psi_{H}\left(\mathbf{z}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)-\beta_{H}\left(\mathbf{0}\right)\right)\\ & =H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\psi_{H}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)+H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\Psi_{H}\left(\mathbf{z}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)\\ & =H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\left(\psi_{H}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)+\Psi_{H}\left(\mathbf{z}\right)\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)\\ & =H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)f\left(\mathbf{z}\right)+H_{\mathbf{j}}\left(\mathbf{0}\right)\\ & =H_{\mathbf{j}}\left(f\left(\mathbf{z}\right)\right) \end{align} Thus, $f$ satisfies $\chi_{H}$'s functional equation. Since $\chi_{H}$ is the unique $\left(p,q_{H}\right)$-adic function satisfying its functional equation asa described in \textbf{Lemma \ref{lem:MD rising-continuation and functional equations of Chi_H}}, this forces $f=\chi_{H}$. Q.E.D. \begin{cor}[\textbf{Quasi-Integrability of $\chi_{H}$ When $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is Invertible}] \label{cor:MD Quasi-integrability of Chi_H for I minus alpha invertible}Let $H$ be as given in \textbf{\emph{Theorem \ref{thm:MD F-series for Chi_H}}}. In addition, suppose $H$ is commutative, and that $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. Then, $\chi_{H}$ is quasi-integrable with respect to the standard $\left(p,q_{H}\right)$-adic frame, and the function $\hat{\chi}_{H}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d}$ defined by: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}\label{eq:MD Chi_H hat for a contracting semi-basic commutative P Hydra map where alpha minus I is invertible} \end{equation} is a Fourier transform of $\chi_{H}$. \end{cor} Proof: The given hypotheses tell us that both $\psi_{H}$ and $\Psi_{H}$ are quasi-integrable. As such, by the linearity of the Fourier transform, (\ref{eq:MD Chi_H in terms of Little Psi_H and Big Psi_H}) becomes: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\hat{\psi}_{H}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\hat{\Psi}_{H}\left(\mathbf{t}\right) \end{equation} Using (\ref{eq:MD Fourier Transform of Little Psi_H, commutative}) and (\ref{eq:MD Fourier Transform of Big Psi_H}) from \textbf{Proposition \ref{prop:quasi-integrability of MD Psis}} to express the right-hand side of this yields (\ref{eq:MD Chi_H hat for a contracting semi-basic commutative P Hydra map where alpha minus I is invertible}). Q.E.D. \begin{cor}[\textbf{Formulae for $\tilde{\chi}_{H,N}$}] Let $H$ and $\hat{\chi}_{H}$ be as given in \textbf{\emph{Corollary \ref{cor:MD Quasi-integrability of Chi_H for I minus alpha invertible}}}. \vphantom{} I. If $p=2$, then: \begin{align} \tilde{\chi}_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\left(\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right)\label{eq:MD Chi_H,N twiddle formula when P is 2 and alpha minus 1 is invertible, commutative}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\beta_{H}\left(\mathbf{0}\right)+\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right)\nonumber \end{align} Hence, passing to the $\mathcal{F}_{2,q_{H}}$-limit as $N\rightarrow\infty$: \begin{equation} \chi_{H}\left(\mathbf{z}\right)\overset{\mathcal{F}_{2,q_{H}}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\beta_{H}\left(\mathbf{0}\right)+\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right)\label{eq:MD Chi_H explicit formula when P is 2 and alpha minus 1 is invertible, commutative} \end{equation} \vphantom{} II. If $p$ is odd: \begin{align} \tilde{\chi}_{H,N}\left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\label{eq:MD Chi_H,N twiddle formula for arbitrary P and alpha minus 1 is invertible, commutative}\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\nonumber \end{align} Hence, passing to the $\mathcal{F}_{p,q_{H}}$-limit as $N\rightarrow\infty$: \begin{align} \chi_{H}\left(\mathbf{z}\right) & \overset{\mathcal{F}_{p,q_{H}}^{d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\label{eq:MD Chi_H explicit formula for arbitrary P and alpha minus 1 is invertible, commutative}\\ & +\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right)\nonumber \end{align} \end{cor} Proof: I. Suppose $p=2$. Then, by \textbf{Corollary \ref{cor:MD Quasi-integrability of Chi_H for I minus alpha invertible}}, we have that: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right) & \textrm{else} \end{cases} \end{equation} Hence, by (\ref{eq:MD Convolution of dA_H and D_N when H is commutative}) from\textbf{ Theorem \ref{thm:MD properties of A_H hat}}: \begin{align} \tilde{\chi}_{H,N}\left(\mathbf{z}\right) & =\sum_{\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)\\ & +\sum_{0<\left\Vert \mathbf{t}\right\Vert _{2}\leq2^{N}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{2}}\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\\ & =-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}\left(\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{1}{2}\right)\right)\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\beta_{H}\left(\mathbf{0}\right)+\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right) \end{align} Letting $N\rightarrow\infty$, the above $\mathcal{F}_{2,q_{H}}^{d}$-converges to: \begin{align} \chi_{H}\left(\mathbf{z}\right) & \overset{\mathcal{F}_{2,q_{H}}^{d}}{=}-\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)+\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{2^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\beta_{H}\left(\mathbf{0}\right)+\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\gamma_{H}\left(\mathbf{\frac{1}{2}}\right)\right) \end{align} \vphantom{} II. Letting $p\geq3$, \textbf{Corollary \ref{cor:MD Quasi-integrability of Chi_H for I minus alpha invertible}} gives us: \[ \hat{\chi}_{H}\left(\mathbf{t}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{else} \end{cases} \] Using (\ref{eq:MD Convolution of dA_H and D_N when H is commutative}) and \textbf{Lemma \ref{lem:MD gamma formulae}}, we then have: \begin{align} \tilde{\chi}_{H,N}\left(\mathbf{z}\right) & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{N}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{N}+\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\\ & +\sum_{n=0}^{N-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\sum_{\mathbf{j}>\mathbf{0}}^{p-1}\mathcal{C}_{H}\left(\alpha_{H}\left(\frac{\mathbf{j}}{p}\right)\varepsilon_{n}\left(\mathbf{j}\mathbf{z}\right):\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\gamma_{H}\left(\frac{\mathbf{j}}{p}\right) \end{align} Taking limits as $N\rightarrow\infty$ gives the desired formula. Q.E.D. \begin{thm}[\textbf{Tauberian Spectral Theorem for Multi-Dimensional $p$-Hydra Maps}] \index{Hydra map!Tauberian Spectral Theorem}\label{thm:MD Periodic Points using WTT} \index{$p,q$-adic!Wiener Tauberian Theorem}\index{Hydra map!divergent trajectories}Let $H$ be as given in \textbf{\emph{Theorem \ref{thm:MD F-series for Chi_H}}}. If $\alpha_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d}$, let: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\begin{cases} -\gamma_{H}\left(\mathbf{\frac{1}{2}}\right) & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)v_{2}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right) & \textrm{else } \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{2}^{r} \end{equation} if every $p=2$. If $p\geq3$, let: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(v_{p}\left(\mathbf{t}\right)\beta_{H}\left(\mathbf{0}\right)+\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right) & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} Alternatively, if $\alpha_{H}\left(\mathbf{0}\right)\neq\mathbf{I}_{d}$, suppose $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible and that $H$ is commutative. Then, let: \begin{equation} \hat{\chi}_{H}\left(\mathbf{t}\right)=\hat{A}_{H}\left(\mathbf{t}\right)\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)^{-1}\beta_{H}\left(\mathbf{0}\right)+\begin{cases} \mathbf{0} & \textrm{if }\mathbf{t}=\mathbf{0}\\ \hat{A}_{H}\left(\mathbf{t}\right)\gamma_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right) & \textrm{else} \end{cases},\textrm{ }\forall\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r} \end{equation} Finally, let $\mathbf{x}\in\mathbb{Z}^{d}\backslash\left\{ \mathbf{0}\right\} $. Then, for the formula for $\hat{\chi}_{H}$ chosen according to the situations described above: \vphantom{} I. If $\mathbf{x}$ is a periodic point of $H$, the translates of the function $\hat{\chi}_{H}\left(\mathbf{t}\right)-\mathbf{x}\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)$ are \emph{not }dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q_{H}}^{d}\right)$. \vphantom{} II. Suppose further that $H$ is integral, and that $\left\Vert H_{\mathbf{j}}\left(\mathbf{0}\right)\right\Vert _{q_{H}}=1$ for all $\mathbf{j}\in\left(\mathbb{Z}^{r}/p\mathbb{Z}^{r}\right)\backslash\left\{ \mathbf{0}\right\} $. If the translates of the function $\hat{\chi}_{H}\left(\mathbf{t}\right)-\mathbf{x}\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)$ are \emph{not }dense in $c_{0}\left(\hat{\mathbb{Z}}_{p}^{r},\mathbb{C}_{q_{H}}^{d}\right)$, then either $\mathbf{x}$ is a periodic point of $H$ or $\mathbf{x}$ belongs to a divergent trajectory of $H$. \end{thm} Proof: Essentially identical to the one-dimensional case (\textbf{Theorem \ref{thm:MD pq-adic WTT for continuous functions}}). Q.E.D. \subsection{\label{subsec:6.2.4 Multi-Dimensional--=00003D000026}Multi-Dimensional $\hat{\chi}_{H}$ and $\tilde{\chi}_{H,N}$ \textendash{} The Non-Commutative Case} Suppose that $H$ is non-commutative. Then, the multi-dimensional analogue of the factor $1-\alpha_{H}\left(0\right)$ is now: \begin{equation} \mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)=\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)} \end{equation} In the multi-dimensional case, the $\mathcal{F}_{p,q_{H}}$-limit of the Fourier series generated by $\hat{A}_{H}\left(t\right)$ is: \begin{equation} \tilde{A}_{H}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\lim_{N\rightarrow\infty}\tilde{A}_{H,N}\left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:MD Definition of A_H twiddle} \end{equation} (equation (\ref{eq:MD Fourier Limit of A_H,N twiddle in standard frame}) from \textbf{Theorem \ref{thm:MD properties of A_H hat}}). To speak of the multi-dimensional case for a moment using the terminology of the one-dimensional case, because $\psi_{H}\left(\mathfrak{z}\right)$ satisfied: \begin{equation} \psi_{H}\left(\mathfrak{z}\right)\overset{\mathcal{F}_{p,q_{H}}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathfrak{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(0\right)\right)^{n} \end{equation} (\textbf{Lemma \ref{eq:Rising-continuation of Little Psi_H}}) the method of the one-dimensional case was to exploit the fact that the one-dimensional counterpart of $\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)$ reduced to a constant which could then be factored out of the $n$-sum. To overcome this obstacle in the $\alpha_{H}\left(0\right)\neq1$ case, we ended up writing: \begin{align} \hat{\psi}_{H}\left(t\right) & =\frac{\hat{A}_{H}\left(t\right)}{1-\alpha_{H}\left(0\right)}\\ \psi_{H}\left(\mathfrak{z}\right) & =\frac{A_{H}\left(\mathfrak{z}\right)}{1-\alpha_{H}\left(0\right)} \end{align} the right-hand sides of which were defined solely because $\alpha_{H}\left(0\right)\neq1$. In particular, the non-vanishing of $\alpha_{H}\left(0\right)$ makes multiplication by $1/\left(1-\alpha_{H}\left(0\right)\right)$ into an \emph{invertible linear transformation}, with the linear transformation: \begin{equation} \mathcal{L}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\left(1-\alpha_{H}\left(0\right)\right)f\left(\mathfrak{z}\right) \end{equation} then satisfying $\mathcal{L}\left\{ \psi_{H}\right\} \left(\mathfrak{z}\right)=A_{H}\left(\mathfrak{z}\right)$. Turning now to the multi-dimensional case, this suggests that to replicate this argument, we will need to find an invertible linear transformation which sends $\psi_{H}\left(\mathbf{z}\right)$ to $\tilde{A}_{H}\left(\mathbf{z}\right)$. In theory, by computing the effects of this linear transformation on the Fourier transforms of $\psi_{H}$ and $\tilde{A}_{H}$, we can then reverse-engineer a formula for $\hat{\psi}_{H}$ from our formula for $\hat{A}_{H}$. \begin{defn} We define the following operators on the space of functions $\mathbb{Z}_{p}^{r}\rightarrow\mathbb{C}_{q}^{d,d}$: \vphantom{} I. For all $n\geq0$: \begin{equation} \mathcal{T}_{n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathbf{F}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\label{eq:Definition of the nth MD truncation operator} \end{equation} \vphantom{} II. For all $n\geq0$: \begin{equation} \mathcal{L}_{H,1,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathcal{T}_{n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\label{eq:Definition of L_H,1,n} \end{equation} \vphantom{} III. For all $n\geq1$: \begin{equation} \mathcal{L}_{H,2,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathcal{L}_{H,1,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)-\mathcal{L}_{H,1,n-1}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\label{eq:Definition of L_H,2,n} \end{equation} \vphantom{} IV. \begin{equation} \mathcal{E}_{H}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathbf{F}\left(\mathbf{0}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\label{eq:Definition of E_H} \end{equation} \vphantom{} V. For all $n\geq0$: \begin{equation} \mathcal{L}_{H,3,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\begin{cases} \mathcal{E}_{0}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right) & \textrm{if }n=0\\ \mathcal{E}_{0}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)+\sum_{m=1}^{n}\mathcal{L}_{H,2,m}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right) & \textrm{if }n\geq1 \end{cases}\label{eq:Definition of L_H,3,n} \end{equation} \vphantom{} VI. For all $n\geq0$: \begin{equation} \mathcal{L}_{H,4,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathcal{L}_{H,3,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\left(\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\label{eq:Definition of L_H,4,n} \end{equation} \vphantom{} VII. \begin{equation} \mathcal{L}_{H}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}\mathcal{L}_{H,4,n}\left\{ \mathbf{F}\right\} \left(\mathbf{z}\right)\label{eq:Definition of L_H} \end{equation} \end{defn} \begin{rem} Note that all of these operators are \emph{linear}. \end{rem} \begin{prop} If $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, then $\mathbf{I}_{H}\left(n\right)$ is invertible for all $n\geq0$. \end{prop} Proof: \begin{align} \mathbf{I}_{H}\left(n\right) & =\mathbf{I}_{d}-\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\alpha_{H}\left(\mathbf{0}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\\ & =\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n} \end{align} Since $H$ is a $p$-Hydra map, $H^{\prime}\left(\mathbf{0}\right)$ is invertible, and hence: \[ \det\mathbf{I}_{H}\left(n\right)=\det\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \] which is non-zero if and only if $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. Q.E.D. \begin{prop} Suppose $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. Then: \begin{equation} \mathcal{L}_{H}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\psi_{H}\left(\mathbf{z}\right)\label{eq:L_H sends A_H twiddle to Little Psi_H} \end{equation} where, as indicated, we compute the value of either side by summing the infinite series on either side in accordance with the standard $\left(p,q_{H}\right)$-adic frame. \end{prop} \begin{rem} The linear operator which sends $\psi_{H}$ to $\tilde{A}_{H}$ is nearly identical to $\mathcal{L}_{H}$; all you do is change the multiplication by $\left(I_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}$ in $\mathcal{L}_{H,4,n}$ to multiplication by $I_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)$. \end{rem} Proof: First, we observe that: \begin{align} \mathcal{T}_{m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & =\tilde{A}_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\\ & \overset{\mathbb{C}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\left[\mathbf{z}\right]_{p^{m}}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\left[\mathbf{z}\right]_{p^{m}}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & \overset{\mathbb{C}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\sum_{n=m}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & +\sum_{n=0}^{m-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{align} Summing the geometric series yields: \begin{align} \mathcal{T}_{m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & \overset{\mathbb{C}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ & +\sum_{n=0}^{m-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{align} Thus: \begin{align} \mathcal{L}_{H,1,m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & \overset{\overline{\mathbb{Q}}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)\\ & +\sum_{n=0}^{m-1}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-m} \end{align} and so: \begin{equation} \mathcal{L}_{H,2,m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\overset{\overline{\mathbb{Q}}^{d,d}}{=}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m}}\right)\right)-\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{m-1}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{m-1}}\right)\right) \end{equation} Next, because: \begin{equation} \kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d} \end{equation} and: \begin{equation} \mathbf{I}_{H}\left(0\right)=\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right) \end{equation} we have: \[ \sum_{m=1}^{n}\mathcal{L}_{H,2,m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)=\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)-\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right) \] Now: \begin{equation} \mathcal{E}_{H}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)=\tilde{A}_{H}\left(\mathbf{0}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right) \end{equation} Since $\alpha_{H}\left(\mathbf{0}\right)\neq\mathbf{I}_{d}$: \begin{align} \tilde{A}_{H}\left(\mathbf{0}\right) & =\sum_{n=0}^{\infty}\kappa_{H}\left(\mathbf{0}\right)\mathbf{I}_{H}\left(0\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\sum_{n=0}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{align} and so: \begin{align} \mathcal{E}_{H}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & =\tilde{A}_{H}\left(\mathbf{0}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & =\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right) \end{align} So, for $n\geq1$: \begin{align} \mathcal{L}_{H,3,n}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & =\mathcal{E}_{H}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)+\sum_{m=1}^{n}\mathcal{L}_{H,2,m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\\ & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)+\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)-\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\\ & =\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right) \end{align} Finally: \begin{align} \mathcal{L}_{H}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\mathcal{L}_{H,4,n}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\\ & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\mathcal{L}_{H,3,n}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\left(\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\left(\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\sum_{n=0}^{\infty}\kappa_{H}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n}\\ & \overset{\mathcal{F}_{p,q_{H}}^{d,d}}{=}\psi_{H}\left(\mathbf{z}\right) \end{align} where $\left(\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}$ is defined for all $\mathbf{z}$ and all $n$ because the invertibility of $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ guarantees the invertibility of $\mathbf{I}_{H}\left(m\right)$ for all $m\geq0$. Q.E.D. \vphantom{} So, to obtain a Fourier transform for $\psi_{H}$, we just need only compute the effect $\mathcal{L}_{H}$ has on $\tilde{A}_{H}$'s Fourier transform ($\hat{A}_{H}$). This, however, is easier said than done. I have broken down the computation into several steps; to begin, we need to following definition: \begin{defn} For each $m\in\mathbb{N}_{0}$, let $\hat{K}_{m}:\hat{\mathbb{Z}}_{p}^{r}\rightarrow\mathbb{C}_{q}$ be defined by: \begin{equation} \hat{K}_{m}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\label{eq:Definition of K_m hat} \end{equation} \end{defn} \begin{prop} \label{prop:A_H hat convolve K_H hat MD}Let $m\geq0$. Then, for all $\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}$: \begin{equation} \left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{t}\right)=\sum_{n=0}^{m-1}\mathbf{1}_{\mathbf{0}}\left(p^{n}\mathbf{t}\right)\prod_{k=0}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)+\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\label{eq:Convolution of A_H hat and K_m hat} \end{equation} where the $k$ product is defined to be $\mathbf{I}_{d}$ whenever its upper limit is $<0$. \end{prop} Proof: We begin with the definition of $\hat{A}_{H}\hat{K}_{m}$: \begin{equation} \left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{t}\right)=\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{A}_{H}\left(\mathbf{s}\right)\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{s}\right)\right\} _{p}} \end{equation} We then pull out the $\mathbf{s}=\mathbf{0}$ term: \begin{align} \hat{A}_{H}\left(\mathbf{0}\right)\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{0}\right)\right\} _{p}} & =\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\mathbf{t}\right\} _{p}}\\ & =\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \frac{\mathbf{n}}{p^{m}}p^{m}\mathbf{t}\right\} _{p}}\\ \left(\left\{ \mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}:\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}\right\} =\left\{ \frac{\mathbf{n}}{p^{m}}:\mathbf{n}\leq p^{m}-1\right\} \right); & =\frac{1}{p^{rm}}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}}e^{-2\pi i\left\{ \mathbf{s}p^{m}\mathbf{t}\right\} _{p}}\\ \left(\textrm{\textbf{proposition \ref{prop:Multi-Dimensional indicator function Fourier Series}}}\right); & =\left[p^{m}\mathbf{t}\overset{p^{m}}{\equiv}\mathbf{0}\right] \end{align} Since congruence $p^{m}\mathbf{t}\overset{p^{m}}{\equiv}\mathbf{0}$ means that $p^{m}t_{\ell}\overset{p^{m}}{\equiv}0$ for all $\ell\in\left\{ 1,\ldots,r\right\} $, note that this occurs if and only if $\left\|t_{\ell}\right\|_{p}\leq p^{m}$. As such, our $\mathbf{s}=\mathbf{0}$ term becomes: \[ \hat{A}_{H}\left(\mathbf{0}\right)\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{0}\right)\right\} _{p}}=\left[p^{m}\mathbf{t}\overset{p^{m}}{\equiv}\mathbf{0}\right]=\mathbf{1}_{\mathbf{0}}\left(p^{m}\mathbf{t}\right) \] Next, using \textbf{Proposition \ref{prop:MD alpha H series}}, we express $\hat{A}_{H}\left(\mathbf{s}\right)$ as a series: \begin{equation} \hat{A}_{H}\left(\mathbf{s}\right)=\prod_{m=0}^{-v_{p}\left(\mathbf{s}\right)-1}\alpha_{H}\left(p^{m}\mathbf{s}\right)=\sum_{\mathbf{k}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{s}\right)}-1}\kappa_{H}\left(\mathbf{k}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{s}\right)}e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{s}\right)} \end{equation} and so: \begin{equation} \left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{t}\right)=\sum_{\mathbf{s}\in\hat{\mathbb{Z}}_{p}^{r}}\sum_{\mathbf{k}=\mathbf{0}}^{p^{-v_{p}\left(\mathbf{s}\right)}-1}\kappa_{H}\left(\mathbf{k}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{-v_{p}\left(\mathbf{s}\right)}\frac{e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{s}\right)}}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{s}\right)\right\} _{p}} \end{equation} Like many times before, we now split the $\mathbf{s}$-sum into a sum over level sets $\left\Vert \mathbf{s}\right\Vert _{p}=p^{h}$. This gives: \begin{equation} \mathbf{1}_{\mathbf{0}}\left(p^{m}\mathbf{t}\right)+\sum_{h=1}^{\infty}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}=p^{h}}\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\kappa_{H}\left(\mathbf{k}\right)\left(\frac{H^{\prime}\left(\mathbf{0}\right)}{p^{r}}\right)^{h}\frac{e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{s}\right)}}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{s}\right)\right\} _{p}} \end{equation} as our expression for $\left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{t}\right)$. Pulling the $\left\Vert \mathbf{s}\right\Vert _{p}$-sum to the far right, we have: \begin{align} e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{s}\right)}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}=p^{h}}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{t}-\mathbf{s}\right)\right\} _{p}} & =e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}=p^{h}}e^{2\pi i\left(\mathbf{n}-\mathbf{k}\right)\cdot\mathbf{s}}\\ & =p^{rh}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\left[\mathbf{k}\overset{p^{h}}{\equiv}\mathbf{n}\right] \end{align} and so: \begin{align} \left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{t}\right) & =\mathbf{1}_{\mathbf{0}}\left(p^{m}\mathbf{t}\right)+\sum_{h=1}^{\infty}\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\mathbf{k}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\left[\mathbf{k}\overset{p^{h}}{\equiv}\mathbf{n}\right]\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{m}\mathbf{t}\right)+\sum_{h=1}^{\infty}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{h}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\\ & =\sum_{h=0}^{\infty}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{h}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}} \end{align} Because $m$ is finite, $\mathbf{n}\leq p^{m}-1$ tells us that $\left[\mathbf{n}\right]_{p^{h}}=\mathbf{n}$ for all $h\geq m$. Consequently: \begin{align} \left(\hat{A}_{H}K_{m}\right)\left(\mathbf{t}\right) & =\sum_{h=0}^{m-1}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{h}}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\\ & +\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\sum_{h=m}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h} \end{align} Since $H$ is contracting, by \textbf{Proposition \ref{prop:MD Contracting H proposition}} (page \pageref{prop:MD Contracting H proposition}) the series $\sum_{h=m}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}$ converges in $\mathbb{R}$ to the matrix: \[ \sum_{h=m}^{\infty}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}\overset{\mathbb{R}^{d,d}}{=}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \] which has entries in $\mathbb{Q}$. Thus: \begin{align} \left(\hat{A}_{H}K_{m}\right)\left(\mathbf{t}\right) & =\sum_{h=0}^{m-1}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{h}}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}\\ & +\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{align} Next, we split the $\mathbf{n}$-sum on the upper line modulo $p^{h}$: \begin{align} \sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\left[\mathbf{n}\right]_{p^{h}}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}} & =\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}\kappa_{H}\left(\left[p^{h}\mathbf{n}+\mathbf{k}\right]_{p^{h}}\right)e^{-2\pi i\left(p^{h}\mathbf{n}+\mathbf{k}\right)\cdot\mathbf{t}}\\ & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}\left(\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\mathbf{k}\cdot\mathbf{t}}\right)e^{-2\pi i\mathbf{n}\cdot p^{h}\mathbf{t}} \end{align} and so: \begin{align} \left(\hat{A}_{H}K_{m}\right)\left(\mathbf{t}\right) & =\sum_{h=0}^{m-1}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}\left(\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\mathbf{k}\cdot\mathbf{t}}\right)e^{-2\pi i\mathbf{n}\cdot p^{h}\mathbf{t}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}\label{eq:MD Halfway done with K_m convolution computation}\\ & +\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\kappa_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\mathbf{t}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\nonumber \end{align} To finish, we yet again apply the familiar recursive evaluation technique. First, define: \begin{equation} S_{h}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{t}\right)}\label{eq:S_h for MD K_m convolution computation} \end{equation} Then: \begin{align} S_{h}\left(\mathbf{t}\right) & =\sum_{\mathbf{k}=\mathbf{0}}^{p^{h}-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\left(\mathbf{k}\cdot\mathbf{t}\right)}\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{k}=\mathbf{0}}^{p^{h-1}-1}\kappa_{H}\left(p\mathbf{k}+\mathbf{j}\right)e^{-2\pi i\left(\left(p\mathbf{k}+\mathbf{j}\right)\cdot\mathbf{t}\right)}\\ & =\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{k}=\mathbf{0}}^{p^{h-1}-1}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)\kappa_{H}\left(\mathbf{k}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}e^{-2\pi i\left(\left(p\mathbf{k}+\mathbf{j}\right)\cdot\mathbf{t}\right)}\\ & =\left(\sum_{\mathbf{j}=\mathbf{0}}^{p-1}H_{\mathbf{j}}^{\prime}\left(\mathbf{0}\right)e^{-2\pi i\mathbf{j}\cdot\mathbf{t}}\right)\sum_{\mathbf{k}=\mathbf{0}}^{p^{h-1}-1}\kappa_{H}\left(\mathbf{k}\right)e^{-2\pi i\left(\mathbf{k}\cdot p\mathbf{t}\right)}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ & =p^{r}\alpha_{H}\left(\mathbf{t}\right)S_{h-1}\left(p\mathbf{t}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{align} This yields the recursion relation: \begin{equation} S_{h}\left(\mathbf{t}\right)=p^{r}\alpha_{H}\left(\mathbf{t}\right)S_{h-1}\left(p\mathbf{t}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\label{eq:S_h recursion relation for K_m convolution computation} \end{equation} and so: \begin{equation} S_{h}\left(\mathbf{t}\right)=p^{rh}\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)S_{0}\left(p^{h}\mathbf{t}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-h} \end{equation} Here: \begin{equation} S_{0}\left(\mathbf{t}\right)=\kappa_{H}\left(\mathbf{0}\right)=\mathbf{I}_{d} \end{equation} which leaves us with: \begin{equation} S_{h}\left(\mathbf{t}\right)=p^{hr}\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-h}\label{eq:Explicit formula for S_h in K_m convolution computation} \end{equation} Returning to (\ref{eq:MD Halfway done with K_m convolution computation}), we have: \begin{align} \left(\hat{A}_{H}K_{m}\right)\left(\mathbf{t}\right) & =\sum_{h=0}^{m-1}\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}\left(p^{rh}\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-h}\right)e^{-2\pi i\mathbf{n}\cdot p^{h}\mathbf{t}}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{h}\\ & +\frac{1}{p^{rm}}p^{rm}\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-m}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ & =\sum_{h=0}^{m-1}\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\frac{1}{p^{r\left(m-h\right)}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}e^{-2\pi i\mathbf{n}\cdot p^{h}\mathbf{t}}\\ & +\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{align} Here: \begin{align} \sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}e^{-2\pi i\mathbf{n}\cdot p^{h}\mathbf{t}} & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{m-h}-1}e^{-2\pi i\frac{\mathbf{n}}{p^{m-h}}\cdot p^{m}\mathbf{t}}\\ & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m-h}}e^{-2\pi i\left(\mathbf{s}\cdot p^{m}\mathbf{t}\right)}\\ & =p^{r\left(m-h\right)}\left[p^{m}\mathbf{t}\overset{p^{m-h}}{\equiv}\mathbf{0}\right] \end{align} The congruence says that $p^{m}\mathbf{t}\in p^{m-h}\mathbb{Z}^{r}$. This implies $p^{h}\mathbf{t}\in\mathbb{Z}^{r}$, which means: \begin{equation} \left[p^{m}\mathbf{t}\overset{p^{m-h}}{\equiv}\mathbf{0}\right]=\left[p^{h}\mathbf{t}\overset{1}{\equiv}\mathbf{0}\right]=\mathbf{1}_{\mathbf{0}}\left(p^{h}\mathbf{t}\right) \end{equation} Hence: \begin{align} \left(\hat{A}_{H}K_{m}\right)\left(\mathbf{t}\right) & =\sum_{h=0}^{m-1}\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\frac{1}{p^{r\left(m-h\right)}}p^{m-h}\mathbf{1}_{\mathbf{0}}\left(p^{h}\mathbf{t}\right)\\ & +\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\\ & =\sum_{h=0}^{m-1}\mathbf{1}_{\mathbf{0}}\left(p^{h}\mathbf{t}\right)\left(\prod_{k=0}^{h-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)+\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{align} Q.E.D. \begin{lem} \label{lem:T_m hat of A_H hat}Let $m\in\mathbb{N}_{0}$. Then: \begin{equation} \hat{\mathcal{T}}_{m}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\sum_{n=0}^{m-1}\mathbf{1}_{\mathbf{0}}\left(p^{n}\mathbf{t}\right)\prod_{k=0}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)+\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)^{-1}\label{eq:T_m hat of A_H hat} \end{equation} where the $k$ product is defined to be $\mathbf{I}_{d}$ whenever its upper limit is $<0$, and where the $m$-sum is defined to be $\mathbf{O}_{d}$ when $m=0$. \end{lem} Proof: Note that: \begin{align} \mathcal{T}_{m}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right) & =\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\tilde{A}_{H}\left(\mathbf{n}\right)\left[\mathbf{z}\overset{p^{m}}{\equiv}\mathbf{n}\right]\\ & =\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\tilde{A}_{H}\left(\mathbf{n}\right)\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}}e^{2\pi i\left\{ \mathbf{s}\left(\mathbf{z}-\mathbf{n}\right)\right\} _{p}}\\ & =\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{A}_{H}\left(\mathbf{t}\right)e^{2\pi i\left\{ \mathbf{t}\mathbf{n}\right\} _{p}}\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}}e^{2\pi i\left\{ \mathbf{s}\left(\mathbf{z}-\mathbf{n}\right)\right\} _{p}}\\ & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}}\sum_{\mathbf{t}\in\hat{\mathbb{Z}}_{p}^{r}}\hat{A}_{H}\left(\mathbf{t}\right)\frac{1}{p^{rm}}\sum_{\mathbf{n}=\mathbf{0}}^{p^{m}-1}e^{-2\pi i\left\{ \mathbf{n}\left(\mathbf{s}-\mathbf{t}\right)\right\} _{p}}e^{2\pi i\left\{ \mathbf{s}\mathbf{z}\right\} _{p}}\\ & =\sum_{\left\Vert \mathbf{s}\right\Vert _{p}\leq p^{m}}\left(\hat{A}_{H}\hat{K}_{m}\right)\left(\mathbf{s}\right)e^{2\pi i\left\{ \mathbf{s}\mathbf{z}\right\} _{p}} \end{align} Then use \textbf{Proposition \ref{prop:A_H hat convolve K_H hat MD}}. Q.E.D. \vphantom{} The last ingredient is the proposition given below. However, we will need another bit of notation to help us along the way: \begin{notation} Let $\mathbf{J}=\left(\mathbf{j}_{1},\ldots,\mathbf{j}_{\left\|\mathbf{J}\right\|}\right)\in\textrm{String}^{r}\left(p\right)$ be a $p$-block string. Then, we define: \begin{equation} \mathbf{t}\cdot\mathbf{J}\overset{\textrm{def}}{=}\mathbf{t}\cdot\sum_{k=1}^{\left\|\mathbf{J}\right\|}\mathbf{j}_{k}p^{k-1}\label{eq:Definition of t dot big bold J} \end{equation} Writing each $\mathbf{j}_{k}$ as $\left(j_{k,1},\ldots,j_{k,r}\right)$, we have that: \begin{equation} \sum_{k=1}^{\left\|\mathbf{J}\right\|}\mathbf{j}_{k}p^{k-1}=\left(j_{1,1}+\cdots+j_{\left\|\mathbf{J}\right\|,1}p^{\left\|\mathbf{J}\right\|-1},\ldots,j_{1,r}+\cdots+j_{\left\|\mathbf{J}\right\|,r}p^{\left\|\mathbf{J}\right\|-1}\right) \end{equation} In particular, $\sum_{k=1}^{\left\|\mathbf{J}\right\|}\mathbf{j}_{k}p^{k-1}=\mathbf{n}$, where $\mathbf{n}\in\mathbb{N}_{0}^{r}$ is the $r$-tuple represented by $\mathbf{J}$. Additionally, we write: \begin{equation} \sum_{\left\|\mathbf{J}\right\|=n}=\sum_{\mathbf{j}_{1},\ldots,\mathbf{j}_{n}\in\mathbb{Z}^{r}/p\mathbb{Z}^{r}} \end{equation} to denote a sum taken over all $\mathbf{J}\in\textrm{String}^{r}\left(p\right)$ of length exactly $n$. \end{notation} \begin{prop} Let $n\geq0$. Then, the Fourier transform of the locally constant $\overline{\mathbb{Q}}^{d,d}$-valued function: \begin{equation} \mathbf{z}\mapsto\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right) \end{equation} is given by: \begin{align} \int_{\mathbb{Z}_{p}^{r}}\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z} & =\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\mathbf{I}_{d}-\mathcal{C}_{H}\left(\alpha_{H}\left(\mathbf{0}\right):n\right)\hat{K}_{n}\left(\mathbf{t}\right)\label{eq:Fourier transform of I_H of lambda_P of z mod P^n} \end{align} \end{prop} Proof: To keep the computations from running off the right side of the page, let us write $\mathbf{A}=\alpha_{H}\left(\mathbf{0}\right)$ and $\mathbf{X}=H^{\prime}\left(\mathbf{0}\right)$. Then, the integral: \begin{equation} \int_{\mathbb{Z}_{p}^{r}}\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z} \end{equation} is: \begin{equation} \int_{\mathbb{Z}_{p}^{r}}\left(\mathbf{I}_{d}-\mathbf{X}^{\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)}\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z} \end{equation} Because the main part of the integrand is a constant with respect to values of $\mathbf{z}$ modulo $p^{n}$, the integral reduces to a finite sum: \begin{equation} \mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\mathbf{I}_{d}-\frac{1}{p^{rn}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(\mathbf{m}\right)}e^{-2\pi i\mathbf{t}\cdot\mathbf{\mathbf{m}}} \end{equation} Hopefully, the reader should not be surprised that we evaluate the $\mathbf{m}$-sum using the recursive method. This time around, we define: \begin{equation} S_{n}\left(\mathbf{t}\right)\overset{\textrm{def}}{=}\frac{1}{p^{rn}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n}-1}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(\mathbf{m}\right)}e^{-2\pi i\mathbf{t}\cdot\mathbf{m}}\label{eq:Definition of S_n for I_H computation} \end{equation} Then: \begin{align} S_{n}\left(\mathbf{t}\right) & =\frac{1}{p^{rn}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}\mathbf{X}^{\lambda_{p}\left(p\mathbf{m}+\mathbf{j}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(p\mathbf{m}+\mathbf{j}\right)}e^{-2\pi i\mathbf{t}\cdot\left(p\mathbf{m}+\mathbf{j}\right)}\\ & =\mathbf{X}\underbrace{\left(\frac{1}{p^{r\left(n-1\right)}}\sum_{\mathbf{m}=\mathbf{0}}^{p^{n-1}-1}\mathbf{X}^{\lambda_{p}\left(\mathbf{m}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(\mathbf{m}\right)}e^{-2\pi i\left(\mathbf{m}\cdot p\mathbf{t}\right)}\right)}_{S_{n-1}\left(p\mathbf{t}\right)}\mathbf{X}^{-1}\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}e^{-2\pi i\mathbf{t}\cdot\mathbf{j}} \end{align} So: \begin{equation} S_{n}\left(\mathbf{t}\right)=\mathbf{X}\left(S_{n-1}\left(p\mathbf{t}\right)\right)\mathbf{X}^{-1}\frac{1}{p^{r}}\sum_{\mathbf{j}=\mathbf{0}}^{p-1}e^{-2\pi i\mathbf{t}\cdot\mathbf{j}}\label{eq:S_n recursion identity for I_H computation} \end{equation} Nesting this yields: \begin{equation} S_{n}\left(\mathbf{t}\right)=\mathbf{X}^{n}\left(S_{0}\left(p^{n}\mathbf{t}\right)\right)\mathbf{X}^{-n}\frac{1}{p^{rn}}\sum_{\mathbf{j}_{1},\ldots,\mathbf{j}_{n}\leq p-1}e^{-2\pi i\mathbf{t}\cdot\sum_{k=1}^{n}\mathbf{j}_{k}p^{k}} \end{equation} Because: \begin{equation} S_{0}\left(\mathbf{s}\right)=\mathbf{X}^{\lambda_{p}\left(\mathbf{0}\right)}\mathbf{A}\mathbf{X}^{-\lambda_{p}\left(\mathbf{0}\right)}e^{-2\pi i\mathbf{s}\cdot\mathbf{0}}=\mathbf{A} \end{equation} we are then left with: \begin{equation} S_{n}\left(\mathbf{t}\right)=\frac{1}{p^{rn}}\mathbf{X}^{n}\mathbf{A}\mathbf{X}^{-n}\sum_{\left\|\mathbf{J}\right\|=n}e^{-2\pi i\mathbf{t}\cdot\mathbf{J}} \end{equation} Finally, recall that the set of all $p$-block strings of length $n$ is in a bijective correspondence with the set of all $r$-tuples $\mathbf{k}\in\mathbb{N}_{0}^{r}$ satisfying $\mathbf{k}\leq p^{n}-1$. As such, we can write: \begin{equation} \sum_{\left\|\mathbf{J}\right\|=n}e^{-2\pi i\mathbf{t}\cdot\mathbf{J}}=\sum_{\mathbf{k}=\mathbf{0}}^{p^{n}-1}e^{-2\pi i\mathbf{t}\cdot\mathbf{k}} \end{equation} and hence: \begin{equation} S_{n}\left(\mathbf{t}\right)=\frac{1}{p^{rn}}\mathbf{X}^{n}\mathbf{A}\mathbf{X}^{-n}\sum_{\mathbf{k}=\mathbf{0}}^{p^{n}-1}e^{-2\pi i\mathbf{t}\cdot\mathbf{k}}\label{eq:Explicit formula for S_n for I_H computation} \end{equation} Putting everything together yields: \begin{align} \int_{\mathbb{Z}_{p}^{r}}\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)e^{-2\pi i\left\{ \mathbf{t}\mathbf{z}\right\} _{p}}d\mathbf{z} & =\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\mathbf{I}_{d}-\mathbf{X}^{n}\mathbf{A}\mathbf{X}^{-n}\frac{1}{p^{rn}}\sum_{\mathbf{k}=\mathbf{0}}^{p^{n}-1}e^{-2\pi i\mathbf{t}\cdot\mathbf{k}}\\ & =\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\mathbf{I}_{d}-\mathbf{X}^{n}\mathbf{A}\mathbf{X}^{-n}\hat{K}_{n}\left(\mathbf{t}\right)\\ & =\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\mathbf{I}_{d}-\mathcal{C}_{H}\left(\mathbf{A}:n\right)\hat{K}_{n}\left(\mathbf{t}\right) \end{align} Q.E.D. \vphantom{} The way forward is then like so: we start our computation with: \begin{equation} \hat{\mathcal{L}}_{H,1,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\hat{\mathcal{T}}_{n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n} \end{equation} Using \textbf{Lemma \ref{lem:T_m hat of A_H hat}}, we get: \begin{align} \hat{\mathcal{L}}_{H,1,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right) & =\sum_{m=0}^{n-1}\mathbf{1}_{\mathbf{0}}\left(p^{m}\mathbf{t}\right)\left(\prod_{k=0}^{m-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & +\left(\prod_{k=0}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n} \end{align} Next: \begin{equation} \hat{\mathcal{L}}_{H,2,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\mathcal{L}_{H,1,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)-\mathcal{L}_{H,1,n-1}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-1} \end{equation} and so: \begin{align} \hat{\mathcal{L}}_{H,2,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right) & =\mathbf{1}_{\mathbf{0}}\left(p^{n-1}\mathbf{t}\right)\left(\prod_{k=0}^{n-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & +\left(\prod_{k=0}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}-\left(\prod_{k=0}^{n-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\\ & =\mathbf{1}_{\mathbf{0}}\left(p^{n-1}\mathbf{t}\right)\left(\prod_{k=0}^{n-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & +\left(\prod_{k=0}^{n-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\alpha_{H}\left(p^{n-1}\mathbf{t}\right)-\mathbf{I}_{d}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-n} \end{align} Then: \begin{equation} \hat{\mathcal{L}}_{H,3,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\hat{\mathcal{E}}_{0}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)+\sum_{m=1}^{n}\hat{\mathcal{L}}_{H,2,m}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right) \end{equation} where the $m$-sum is $\mathbf{O}_{d}$ when $n=0$. Recalling that: \begin{equation} \mathcal{E}_{0}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)=\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right) \end{equation} we then have: \begin{equation} \hat{\mathcal{E}}_{0}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right) \end{equation} which gives: \begin{align} \hat{\mathcal{L}}_{H,3,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right) & =\left(\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)\right)\mathbf{1}_{\mathbf{0}}\left(\mathbf{t}\right)\\ & +\sum_{m=1}^{n}\mathbf{1}_{\mathbf{0}}\left(p^{m-1}\mathbf{t}\right)\left(\prod_{k=0}^{m-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-m}\left(\mathbf{I}_{d}-H^{\prime}\left(\mathbf{0}\right)\right)\\ & +\sum_{m=1}^{n}\left(\prod_{k=0}^{m-2}\alpha_{H}\left(p^{k}\mathbf{t}\right)\right)\left(\alpha_{H}\left(p^{m-1}\mathbf{t}\right)-\mathbf{I}_{d}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{-m} \end{align} where the $k$-products are $\mathbf{I}_{d}$ whenever $m=1$. \begin{equation} \prod_{k=0}^{n-1}\alpha_{H}\left(p^{k}\mathbf{t}\right)=\begin{cases} \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\frac{\mathbf{t}\left\|\mathbf{t}\right\|_{p}}{p}\right)\right)^{-1} & \textrm{if }n=-v_{p}\left(\mathbf{t}\right)-1\\ \hat{A}_{H}\left(\mathbf{t}\right)\left(\alpha_{H}\left(\mathbf{0}\right)\right)^{n+v_{p}\left(\mathbf{t}\right)} & \textrm{if }n\geq-v_{p}\left(\mathbf{t}\right) \end{cases} \end{equation} Finally, note that: \begin{equation} \mathcal{L}_{H,4,n}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)=\mathcal{L}_{H,3,n}\left\{ \tilde{A}_{H}\right\} \left(\mathbf{z}\right)\left(\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\right)^{-1}\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{equation} Since the Fourier transform turns multiplication into convolution, we get: \begin{equation} \hat{\mathcal{L}}_{H,4,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\left(\hat{\mathcal{L}}_{H,3,n}\left\{ \hat{A}_{H}\right\} \hat{\mathbf{G}}_{n}^{-1}\right)\left(\mathbf{t}\right)\left(H^{\prime}\left(\mathbf{0}\right)\right)^{n} \end{equation} where $\hat{\mathbf{G}}_{n}^{-1}$ denotes the convolution inverse of the Fourier transform of: \begin{equation} \mathbf{G}_{n}\left(\mathbf{z}\right)\overset{\textrm{def}}{=}\mathbf{I}_{H}\left(\lambda_{p}\left(\left[\mathbf{z}\right]_{p^{n}}\right)\right)\label{eq:Definition of bold G_n} \end{equation} It is here that we hit a rocky obstacle. Because $\hat{\mathbf{G}}_{n}^{-1}\left(\mathbf{t}\right)$ does not appear to have a manageably simple explicit formula, determining whether or not the sum: \begin{equation} \hat{\mathcal{L}}_{H}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)=\sum_{n=0}^{\infty}\hat{\mathcal{L}}_{H,4,n}\left\{ \hat{A}_{H}\right\} \left(\mathbf{t}\right)\label{eq:L_H hat of A_H hat} \end{equation} converges with respect to $\mathcal{F}_{p,q_{H}}^{d,d}$ becomes problematic. Without an explicit formula with which convergence of the right-hand side of (\ref{eq:L_H hat of A_H hat}) could be directly ascertained, it would seem we are at the mercy of foremost weakness of fledgling frame theory: the lack of a method of \emph{indirectly }verifying convergence with respect to a given frame by way of estimates, approximation arguments, and the like. So, while it is intuitively clear that (\ref{eq:L_H hat of A_H hat}) should then furnish a formula for $\hat{\psi}_{H}\left(\mathbf{t}\right)$ in the case of non-commutative $H$ for which $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible, at the time of this writing\textemdash the third day of Putin's invasion of Ukraine (26 February, 2022)\textemdash I do not have an argument to prove this rigorously. If and when such an argument is obtained, it would immediately follow by \textbf{Theorem \ref{thm:MD F-series for Chi_H}}'s formula $\chi_{H}\left(\mathbf{z}\right)=\psi_{H}\left(\mathbf{z}\right)\beta_{H}\left(\mathbf{0}\right)+\Psi_{H}\left(\mathbf{z}\right)$ that $\chi_{H}$ would be quasi-integrable. As such, I end with a conjecture: \begin{conjecture}[\textbf{Quasi-Integrability of $\chi_{H}$ in the Non-Commutative Case}] \label{conj:MD Non-Commutative case conjecture}Let $H$ be as given in \textbf{\emph{Theorem \ref{thm:MD F-series for Chi_H}}}, and suppose that $\mathbf{I}_{d}-\alpha_{H}\left(\mathbf{0}\right)$ is invertible. Then,\emph{ (\ref{eq:L_H hat of A_H hat})} gives a formula $\hat{\mathbb{Z}}_{p}^{r}\rightarrow\overline{\mathbb{Q}}^{d,d}$ for a Fourier transform of $\psi_{H}$ with respect to the standard $\left(p,q_{H}\right)$-adic frame. \end{conjecture} \chapter{Coda} \pagestyle{fancy}\fancyfoot{}\fancyhead[L]{\sl CODA}\fancyhead[R]{\thepage}\addcontentsline{toc}{chapter}{Coda} \includegraphics[scale=0.4]{./PhDDissertationEroicaCoda.png} \section{Conclusion - A Call to Arms} \addcontentsline{toc}{section}{Conclusion - A Call to Arms} Even though insight drives mathematical progress, the utility and efficacy of mathematics is only knowable in hindsight. Will the methods I have presented in my dissertation be of any use in solving the Collatz Conjecture once and for all? The answer\textemdash if it is to ever be found\textemdash lies in posterity. I would very much like to meet it, if I can. In writing this dissertation, the two biggest surprises were the connection to eigenvalues of matrices in Sub-subsection \ref{subsec:A-Matter-of}\textemdash although, in hindsight this feels obvious (of course)\textemdash and, most of all, the result that divergent trajectories of $H$ were associated with values in $\mathbb{Z}$ attained by $\chi_{H}$ over $\mathbb{Z}_{p}\backslash\mathbb{Q}$ (\textbf{Theorem \ref{thm:Divergent trajectories come from irrational z}} on page \pageref{thm:Divergent trajectories come from irrational z}, and \textbf{Theorem \ref{thm:MD Divergent trajectories come from irrational z}} on page \pageref{thm:MD Divergent trajectories come from irrational z}). I only discovered $\chi_{H}$'s involvement in divergent trajectories in February 2022 while I was beginning my polishing and editing runs on this monograph. Prior to that, I was firmly convinced that my methods had nothing to say about divergent trajectories. For once, I was \emph{thrilled} to be proven wrong. Regarding the theory of Hydra maps as I have presented it, the most significant outstanding issue are the \textbf{Conjectures} \textbf{\ref{conj:correspondence theorem for divergent trajectories}} (page \pageref{conj:correspondence theorem for divergent trajectories}) and \textbf{\ref{conj:MD correspondence theorem for divergent trajectories}} (page \pageref{conj:MD correspondence theorem for divergent trajectories}) regarding a Correspondence Principle for divergent trajectories. If these Conjectures could be proven true, the \textbf{Tauberian Spectral Theorems} (pages \pageref{thm:Periodic Points using WTT} \& \pageref{thm:MD Periodic Points using WTT}) for $\chi_{H}$ would then \emph{completely} characterize the dynamics of the Hydra maps under consideration\textemdash that is, both periodic points \emph{and }divergent trajectories. The entire matter of the Collatz Conjecture (not to mention most any comparable conjectures for other Hydra maps) would then be reduced to the study of the density of translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ in $c_{0}\left(\hat{\mathbb{Z}}_{p},\mathbb{C}_{q}\right)$ and its multi-dimensional analogue. At a lesser level, there are the matters of one-dimensional $p^{n}$-Hydra maps (where $p$ is a prime and $n\in\mathbb{N}_{1}$) and multi-dimensional $P$-Hydra maps where $P=\left(p^{n_{1}},\ldots,p^{n_{r}}\right)$ where $p$ is prime and $n_{1}\leq\ldots\leq n_{r}$ is a non-decreasing sequence of positive integers. While I certainly \emph{like} to think that my Tauberian Spectral reformulation of Hydra maps' dynamics is aesthetically pleasing, I have yet to explore whether or not it is actually \emph{useful}. The question of the density of the translates of $\hat{\chi}_{H}\left(t\right)-x\mathbf{1}_{0}\left(t\right)$ might very well turn out to be just as intractable as the original question of whether or not $x$ is a periodic point or divergent trajectory of $H$. Although, obviously, I am biased in favor of believing that my work \emph{will} turn out to be useful, I cannot help but feel that the distinctions between, say, Fourier transforms of the Shortened $3x+1$ map (for which $\alpha_{H}\left(0\right)=1$) and those of the Shortened $5x+1$ map (for which $\alpha_{H}\left(0\right)\neq1$) have the look of a smoking gun. For reference, valid choices of Fourier transforms for these numina are: \begin{equation} \hat{\chi}_{3}\left(t\right)=\begin{cases} -\frac{1}{2} & \textrm{if }t=0\\ \frac{1}{4}v_{2}\left(t\right)\hat{A}_{3}\left(t\right) & \textrm{if }t\neq0 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \end{equation} \begin{equation} \hat{\chi}_{5}\left(t\right)=\begin{cases} -\frac{1}{2} & \textrm{if }t=0\\ -\frac{1}{4}\hat{A}_{5}\left(t\right) & \textrm{if }t\neq0 \end{cases}=-\frac{1}{4}\mathbf{1}_{0}\left(t\right)-\frac{1}{4}\hat{A}_{5}\left(t\right),\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \end{equation} Because $v_{2}\left(t\right)$ takes values in $\left\{ -1,-2,-3,\ldots\right\} $ for $t\in\hat{\mathbb{Z}}_{2}\backslash\left\{ 0\right\} $, the $3$-adic absolute value of $\hat{\chi}_{3}\left(t\right)$ will become arbitrarily small infinitely often: \begin{equation} \liminf_{\left\|t\right\|_{2}\rightarrow\infty}\left\|\hat{\chi}_{3}\left(t\right)\right\|_{3}=0 \end{equation} whereas: \begin{equation} \left\|\hat{\chi}_{5}\left(t\right)\right\|_{5}=1,\textrm{ }\forall t\in\hat{\mathbb{Z}}_{2} \end{equation} Unlike the probabilistic heuristics usually given to argue that $3x+1$ sends all positive integers to $1$ or that $5x+1$ sends almost all positive integers to $\infty$, this observation about the behaviors of $\hat{\chi}_{3}$ and $\hat{\chi}_{5}$ holds with \emph{absolute} certainty. This is but one reason for my my conviction that $\hat{\chi}_{3}\left(t\right)$'s failure to be $3$-adically bounded away from $0$ \emph{must }be a key feature in any proof of the Collatz Conjecture, should one arise. To that end, it would be wonderful if we could establish something along the lines of: \begin{conjecture} \label{conj:Implication of Tauberian Spectral Theorem}Let $H:\mathbb{Z}\rightarrow\mathbb{Z}$ be a contracting, semi-basic $p$-Hydra map, and let $\hat{\chi}_{H}$ be a Fourier transform of $\chi_{H}$. \vphantom{} I. $H$ has finitely many periodic points if and only if $\liminf_{\left\|t\right\|_{p}\rightarrow\infty}\left\|\hat{\chi}_{H}\left(t\right)\right\|_{q_{H}}=0$. \vphantom{} II. If $\liminf_{\left\|t\right\|_{p}\rightarrow\infty}\left\|\hat{\chi}_{H}\left(t\right)\right\|_{q_{H}}>0$, then $H$ has a divergent trajectory. \end{conjecture} \vphantom{} While the particulars of the hypotheses and conclusions of this conjecture might very well need to be tinkered with, my hope in making it will be that questions about the dynamics of $H$ can be reduced to number-theoretic ($q$-adic) properties of the formulae for $\hat{\chi}_{H}$. \begin{example}[\textbf{Matthews' Map, Revisited}] \index{Matthews' Conjecture}Of special interest is the $3$-Hydra map obtained by conjugating the map featured in \textbf{Matthews' Conjecture} (\textbf{Conjecture \ref{conj:Matthews conjecture}} on page \pageref{conj:Matthews conjecture}); specifically, the conjugated version $\tilde{M}$ defined in equation (\ref{eq:Matthews' Conjecture Map, conjugated}) on page \pageref{exa:Matthews' map}: \begin{equation} \tilde{M}\left(n\right)=\begin{cases} \frac{n}{3} & \textrm{if }n\overset{3}{\equiv}0\\ 7n-3 & \textrm{if }n\overset{3}{\equiv}1\\ \frac{7n-2}{3} & \textrm{if }n\overset{3}{\equiv}2 \end{cases} \end{equation} The reason this map is of interest, recall, is because it is easily proved that every integer congruent to $1$ mod $3$ belongs to a divergent trajectory of $\tilde{M}$ (a simple modification of page \pageref{prop:Matthews' map}'s \textbf{Proposition \ref{prop:Matthews' map}}). Though \emph{non-integral}, this $3$-Hydra is nevertheless semi-basic (with $\mu_{0}=1$, $\mu_{1}=21$, $\mu_{2}=7$, and $q=7$) and so it possesses a numen $\chi_{\tilde{M}}$, and every periodic point of $\tilde{M}$ is of the form $\chi_{\tilde{M}}\left(B_{3}\left(n\right)\right)$ for some $n\in\mathbb{N}_{1}$. However, because $\tilde{M}$ is non-integrable, the versions of the Correspondence Principal established in this dissertation do not apply to conclude that every value in $\mathbb{Z}$ attained by $\chi_{\tilde{M}}$ over $\mathbb{Q}\cap\mathbb{Z}_{3}^{\prime}$ is a periodic point of $\tilde{M}$, or that $\mathfrak{z}\in\mathbb{Z}_{3}\backslash\mathbb{Q}$ for which $\chi_{\tilde{M}}\left(\mathfrak{z}\right)\in\mathbb{Z}$ necessarily make $\chi_{\tilde{M}}\left(\mathfrak{z}\right)$ into a divergent point of $\tilde{M}$. Nevertheless, I have not explored this issue in detail\textemdash my goal in my dissertation was to obtain as broad a theory as possible, rather than whittle away at any particular Hydra map\textemdash so there may still be a way to make these results applicable to $\tilde{M}$. Despite these obstacles, nothing prevents us from going through with Chapter 4's Fourier analysis of $\chi_{\tilde{M}}$. Doing so, we find that: \begin{equation} \alpha_{\tilde{M}}\left(t\right)\overset{\textrm{def}}{=}\frac{1+21e^{-2\pi it}+7e^{-4\pi it}}{9},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3}\label{eq:alpha for conjugated matthews map} \end{equation} \begin{equation} \beta_{\tilde{M}}\left(t\right)\overset{\textrm{def}}{=}e^{-2\pi it}-\frac{2}{9}e^{-4\pi it},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3}\label{eq:Beta for conjugated Matthews map} \end{equation} This gives: \begin{align} \alpha_{\tilde{M}}\left(0\right) & =\frac{29}{9}\\ \beta_{\tilde{M}}\left(0\right) & =\frac{7}{9} \end{align} with: \begin{equation} \hat{A}_{\tilde{M}}\left(t\right)\overset{\textrm{def}}{=}\begin{cases} 1 & \textrm{if }t=0\\ \prod_{n=0}^{-v_{3}\left(t\right)-1}\frac{1+21e^{-2\pi i3^{m}t}+7e^{-4\pi i3^{m}t}}{9} & \textrm{else} \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3}\label{eq:A_H-hat for Matthews map} \end{equation} and: \begin{equation} \kappa_{\tilde{M}}\left(n\right)=\left(21\right)^{\#_{3:1}\left(n\right)}7^{\#_{3:2}\left(n\right)}\label{eq:Kappa_H for matthews map} \end{equation} Using \textbf{Theorem \ref{thm:F-series for an arbitrary 1D Chi_H}} (page \pageref{thm:F-series for an arbitrary 1D Chi_H}), a Fourier transform for $\chi_{\tilde{M}}:\mathbb{Z}_{3}\rightarrow\mathbb{Z}_{7}$ is given by: \begin{equation} \hat{\chi}_{\tilde{M}}\left(t\right)=\begin{cases} -\frac{7}{20} & \textrm{if }t=0\\ \left(\frac{9e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}-2e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}{1+21e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}+7e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}-\frac{7}{20}\right)\hat{A}_{\tilde{M}}\left(t\right) & \textrm{if }t\neq0 \end{cases}\label{eq:Fourier Transform for Chi_H for Matthews map} \end{equation} Here, number theory comes into play. Noting that: \begin{equation} \alpha_{\tilde{M}}\left(t\right)=\frac{1+\overbrace{21e^{-2\pi it}+7e^{-4\pi it}}^{\textrm{a multiple of }7}}{9}\in\frac{1}{9}+7\mathbb{C}_{7},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3} \end{equation} the $7$-adic ultrametric inequality yields: \begin{equation} \left\|\alpha_{\tilde{M}}\left(t\right)\right\|_{7}=1,\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3} \end{equation} As such: \begin{equation} \left\|\hat{A}_{\tilde{M}}\left(t\right)\right\|_{7}=\prod_{n=0}^{-v_{3}\left(t\right)-1}\left\|\alpha_{\tilde{M}}\left(3^{n}t\right)\right\|_{7}=1,\textrm{ }\forall t\in\hat{\mathbb{Z}}_{3} \end{equation} So, the $7$-adic absolute value of $\hat{\chi}_{\tilde{M}}\left(t\right)$ for non-zero $t$ is entirely determined by the function: \begin{equation} \frac{9e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}-2e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}{1+21e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}+7e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}-\frac{7}{20}\label{eq:7-adic magnitude determinator for Chi_H for Matthews map} \end{equation} Because $t\left\|t\right\|_{3}/3$ is congruent mod $1$ to either $1/3$ (if the numerator of $t$ is $1$ mod $3$) or $2/3$ (if the numerator of $t$ is $2$ mod $3$) for any $t\in\hat{\mathbb{Z}}_{3}\backslash\left\{ 0\right\} $, (\ref{eq:7-adic magnitude determinator for Chi_H for Matthews map}) takes precisely two values on $\hat{\mathbb{Z}}_{3}\backslash\left\{ 0\right\} $. Letting $\xi$ denote $e^{2\pi i/6}$, and letting $\zeta$ denote $\xi^{2}$, these are: \begin{equation} \frac{9e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}-2e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}{1+21e^{-2\pi i\frac{t\left\|t\right\|_{3}}{3}}+7e^{-4\pi i\frac{t\left\|t\right\|_{3}}{3}}}-\frac{7}{20}=\begin{cases} \frac{17-47\zeta^{2}}{1580} & \textrm{if }t\left\|t\right\|_{3}\overset{3}{\equiv}1\\ \frac{17-47\zeta}{1580} & \textrm{if }t\left\|t\right\|_{3}\overset{3}{\equiv}2 \end{cases} \end{equation} Recalling that the torsion subgroup of $\mathbb{Z}_{7}^{\times}$ is isomorphic to $\mathbb{Z}/6\mathbb{Z}$, all $6$th roots of unity are occur naturally as $7$-adic integers. As such, to compute the $7$-adic absolute values of these expressions, we will need to fall back on our convention that, for an odd prime $p$, $\xi\in\mathbb{Z}_{p}$ denotes the unique primitive $\left(p-1\right)$th root of unity whose first $p$-adic digit is the smallest primitive root of unity in $\mathbb{Z}/p\mathbb{Z}$. Since $p=7$, that digit is $3$. Consequently, $\zeta=\xi^{2}$ has $3^{2}\overset{7}{\equiv}2$ as its first $7$-adic digit and $\zeta^{2}=\xi^{4}$ has $3^{4}\overset{7}{\equiv}4$ as its first $7$-adic digit. Observing the identity: \begin{equation} \left(17-47\zeta^{2}\right)\left(17-47\zeta\right)=\left(17\right)^{2}+\left(47\right)^{2}+17\cdot47=3297=3\cdot7\cdot157 \end{equation} we have that: \begin{equation} 17-47\zeta^{2}\overset{7}{\equiv}17-\left(47\right)\left(4\right)\overset{7}{\equiv}3-\left(5\right)\left(2\right)=-17\overset{7}{\equiv}4 \end{equation} So, $\left\|17-47\zeta^{2}\right\|_{7}=1$. As such: \begin{equation} \left\|\left(17-47\zeta^{2}\right)\left(17-47\zeta\right)\right\|_{7}=\frac{1}{7} \end{equation} then forces: \begin{equation} \left\|17-47\zeta\right\|_{7}=\frac{1}{7} \end{equation} Finally, noting that: \begin{equation} 1580=2^{2}\cdot5\cdot79 \end{equation} we have: \begin{align} \left\|\frac{17-47\zeta}{1580}\right\|_{7} & =\frac{1}{7}\\ \left\|\frac{17-47\zeta^{2}}{1580}\right\|_{7} & =1 \end{align} Since $\left\|\hat{\chi}_{\tilde{M}}\left(0\right)\right\|_{7}=\left\|-7/20\right\|_{7}=1/7$, we then conclude that: \begin{equation} \left\|\hat{\chi}_{\tilde{M}}\left(t\right)\right\|_{7}=\begin{cases} \frac{1}{7} & \textrm{if }t=0\\ 1 & \textrm{if }t\left\|t\right\|_{3}\overset{3}{\equiv}1\\ \frac{1}{7} & \textrm{if }t\left\|t\right\|_{3}\overset{3}{\equiv}2 \end{cases},\textrm{ }\forall t\in\hat{\mathbb{Z}}_{7}\label{eq:7-adic absolute value of the fourier transform of the numen of Matthews map} \end{equation} With this, we see that $\chi_{\tilde{M}}$ is more like $\chi_{q}$ for $q\geq5$, in that its $7$-adic absolute value is bounded away from $0$. This is consistent with \textbf{Conjecture \ref{conj:Implication of Tauberian Spectral Theorem}}\textemdash or, at least, with this particular formulation of the Conjecture. As mentioned above, the precise statement of that Conjecture is still subject to revision; more specific cases should be investigated to help see what the ``correct'' conjecture might be. Subject to the verification of \textbf{Conjecture \ref{conj:Implication of Tauberian Spectral Theorem}}, the method described above\textemdash checking $\hat{\chi}_{H}$ and the values of $\gamma_{H}\left(j/p\right)$ for $j\in\left\{ 1,\ldots,p-1\right\} $ over the $q$-adics\textemdash would then be an extremely elegant means for determining the dynamical properties of $H$. \end{example} \vphantom{} The two other most pressing issues I can think of are the \textbf{exploration of Hydra maps on fields of positive characteristic}, such as those presented near the end of Matthews' slides \cite{Matthews' slides}, and the \textbf{exploration of the polygenic case} as discussed in \textbf{Example \ref{exa:Polygenic example, part 1}} on page \pageref{exa:Polygenic example, part 1}. As mentioned in the remark given after \textbf{Example \ref{exa:Polygenic example, part 1}}, I believe it will be possible to use frames to put $\chi_{H}$ on rigorous footing in the polygenic case. Because the $N$th truncations of $\chi_{H}$ take values in $\mathbb{Q}$, the asymptotic analysis of $\chi_{H,N}$\textemdash untangling the link between $N$ and $t$\textemdash will yield Fourier transforms for polygenic $\chi_{H}$. The only possible issue I can foresee is in verifying that the resultant formulae for $\hat{\chi}_{H}$ form continuous linear functionals\textemdash although, there, the question becomes \emph{on what space?} Provided this can be answered in a straight-forward matter, the $\left(p,q\right)$-adic Wiener Tauberian Theorem for measures will almost certainly apply as in the monogenic case, thereby furnishing a Tauberian Spectral Theorem for polygenic $\chi_{H}$, seeing as how they already depend on the points at which the measure's Fourier series converge $q$-adically. Aside from these obvious next steps, it also seems worthwhile to explore if $L_{\mathbb{R}}^{1}$ can, in fact, be used as a ``base of operations'' for analytic investigations of quasi-integrability and frames. \subsection{A Soapbox Moment} Before I begin the obligatory bibliographic essay, I would like to make a ``call to arms'' for fellow travelers in mathematical analysis and its many subspecialties. Although I am pleased to have broken new ground in non-archimedean analysis\textemdash without it, I would not have attained my hard-earned PhD\textemdash it troubles me that some of the phenomena I have discovered appear to haven't already been discovered, especially considering their astonishing simplicity. The identity (\ref{eq:Fourier sum of v_p of t}) (page \pageref{eq:Fourier sum of v_p of t}): \[ \sum_{0<\left\|t\right\|_{p}\leq p^{N}}v_{p}\left(t\right)e^{2\pi i\left\{ t\mathfrak{z}\right\} _{p}}\overset{\mathbb{Q}}{=}\frac{p\left\|\mathfrak{z}\right\|_{p}^{-1}-1}{p-1},\textrm{ }\forall N>v_{p}\left(\mathfrak{z}\right) \] requires no extraordinary machinery to prove. The computation is as near as mindless as can be, yet\textemdash to my bemusement\textemdash no one seems to have noticed it until now. The mathematical community's failure to notice this identity, as well as the implication it has for $\left(p,q\right)$-adic integration ($\left(p,q\right)$-adic Mellin transforms, distributional derivatives, etc.) cannot be due to a lack of genius or ingenuity; it can only be attributed to a lack of curiosity and an aversion to simplicity. It frustrates me to no end that this state of affairs is allowed to stand. I have abiding empathy for Vladimir Arnold's protests against contemporary mathematical pedagogy \cite{Arnold}. As the venerable Russian analyst put it: \begin{quote} Attempts to create ``pure'' deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. \end{quote} I could not agree with this sentiment more strongly. The deductive-axiomatic scheme is a powerful organizational tool, but I feel it horribly misrepresents the reality of mathematical inquiry. In the research that led to my dissertation, I proceeded very much like a physicist of old, testing different ideas and formulas in the hopes of finding something that managed to hold water and say something useful about the objects of my investigations. Make no mistake: I am not a mindless iconoclast. The trend toward abstraction has and always will be a vital part of mathematical inquiry. In clearing the road of debris and obstacles, they make it easier to discern deep patterns and unexpected symmetries. But, it is my conviction that flights of fancy such as these need to be matched and tempered by good old-fashioned concrete play. We need to be able to get our hands dirty and loamy with tedious, self-limiting specificity. Euler spent a decade trying to prove his famous (and beautiful) generating function identity for pentagonal numbers. If only us moderns had the tenacity to give simple, specific questions that same level of consideration. Progress comes about when people look in places they shouldn't, and when ideas take root far from home. Theories stagnate when all their voices sound the same. Practical-minded analysts should not let topics like non-archimedean analysis fall by the wayside and become sole purview of the algebraists. The existence of the topological, open-set-based definition of continuity does not diminish the $\epsilon$-$\delta$ definition, nor render it obsolete. The same ought to be true of non-archimedean analysis and classical analysis. Just because a thing can be tied to the cross of commutative algebra does not mean there is nothing to be gained from an independent examination of the thing from a purely concrete, analytical perspective. Even if this dissertation of mine ends up being a mere tangent on the long road to Collatz, if it can succeed at enticing other scholars to consider topics or settings they might not have bothered to ever explore, I can say I will be happy with it, with my efforts, and with myself. \vphantom{} \begin{quote} \emph{Oh me! Oh life! of the questions of these recurring,} \emph{Of the endless trains of the faithless, of cities fill\textquoteright d with the foolish,} \emph{Of myself forever reproaching myself, (for who more foolish than I, and who more faithless?)} \emph{Of eyes that vainly crave the light, of the objects mean, of the struggle ever renew\textquoteright d,} \emph{Of the poor results of all, of the plodding and sordid crowds I see around me,} \emph{Of the empty and useless years of the rest, with the rest me intertwined,} \emph{The question, O me! so sad, recurring\textemdash What good amid these, O me, O life?} \begin{center} Answer. \par\end{center} \emph{That you are here\textemdash that life exists and identity,} \emph{That the powerful play goes on, and you may contribute a verse.} \textemdash Walt Whitman (1892) \end{quote} \newpage{} \section{Bibliographic Essay} \addcontentsline{toc}{section}{Bibliographic Essay}As explained in the Preface, the $\left(p,q\right)$-adic analytic methods used in this dissertation were not the first path of attack I tried on the Collatz Conjecture, and I doubt they will be the last. To that end, I have not exercised much restraint with regard to the references catalogued in the Bibliography below; it is my hope that others will find the listed sources as interesting as I did. Like most bibliographic essays, the next few paragraphs are intended to assist the reader in assessing the resources I have left for them. For ease of access, I have organized my discussion by topic. \subsubsection{Collatz Studies} At the time of writing, ``Collatz studies'' has yet to become a well-established mathematical discipline, and the scattered, amorphous state of the literature reflects this. Lagarias' \emph{Ultimate Challenge }book \cite{Ultimate Challenge} is an wonderful start, as are his annotated bibliographies of Collatz research \cite{Collatz Biography}. As of the first quarter of the twenty-first century, Lagarias is generally considered the top authority in Collatz studies, and to that end, all of his publications on the topic are worth reading (\cite{Applegate and Lagarias - Trees,Applegate and Lagarias - Difference inequalities,3x+1 semigroup,natural boundaries,Lagarias-Kontorovich Paper,Lagarias' Survey,Lagarias - rational cycles for 3x+1}). There are doubtless others to be found, should the reader spend the time searching for them. Matthews' \href{http://www.numbertheory.org/3x\%2B1/}{Collatz webpage}\footnote{http://www.numbertheory.org/3x+1/} website contains links to his slides and related articles (\cite{Matthews' Conjecture,Matthews' Leigh Article,Matthews' slides,Matthews and Watts}); these focus primarily on the Markov-chain approach. Also of note, the late Meinardus and Berg \cite{Meinardus and Berg,Meinardus} attempted analytic approaches using functional equations, as did their colleague G. Opfer (\cite{Berg =00003D000026 Opfer,Opfer}), with Opfer's 2011 paper \cite{Opfer} being notable as having been thought to have actually\emph{ proved }Collatz until a gap was discovered (see, for instance, \cite{de Weger on Opfer}). There is also my own (albeit somewhat messy) work in this vein \cite{Dreancatchers for Hydra Maps}. My bibliography only scratches the surface of the extant literature. Eric Roosendaal maintains a $3x+1$ website dedicated to chronicling various computational phenomena and statistics thereof found in the iteration of the Collatz map \cite{Roosendaal's Website}. Then, of course, there is Tao's recent work \cite{Tao Probability paper}. The reader should be aware that most Collatz studies tend to focus solely on the $3x+1$ map itself. One notable exception to this trend is \cite{RCWAG}, which is part of a computational package for studying more general Collatz-type maps\textemdash the ungainly (though accurately) termed ``residue-class-wise affine maps'' (RCWA)\textemdash all due to Stephan Kohl. \cite{RCWAG} is a freely available computational software for studying Collatz type maps, written using the GAP programming language. It strikes me as the sort of thing a person might fool around with\footnote{Particularly if it can be used to create attractive, eye-catching visualizations. As they say, a picture is worth a thousand words. It also makes for a very effective tool for marketing and recruitment.} while lounging about at home, either on a rainy day or a golden Sunday afternoon. \subsubsection{Non-Archimedean Analysis (Including $\left(p,q\right)$)} W. M. Schikhof PhD dissertation \cite{Schikhof's Thesis} establishes the fundamentals of harmonic analysis in the general non-archimedean setting, including the $\left(p,q\right)$-adic case. A particularly valuable\textemdash and accessible!\textemdash introduction to non-archimedean analysis is Schikhof's \emph{Ultrametric Calculus }\cite{Ultrametric Calculus}, a classic of the subject. The bulk of it is focused on $\left(p,p\right)$-adic analysis, though. A treatment of the Monna-Springer integral (though not by that name) is given in one of \emph{Ultrametric Calculus'} appendices. A \emph{much }more exhaustive collection of non-archimedean material can be found in van Rooij's \emph{Non-Archimedean Functional Analysis }\cite{van Rooij - Non-Archmedean Functional Analysis}, though\textemdash as I have mentioned elsewhere\textemdash this book is out of print\footnote{It \emph{can}, however, be pirated.}. van Rooij's vantage is quite general, to the point that his book is a bit bewildering to work with. Those without photographic memories are advised to keep a glossary of his definitions and notations. Khrennikov's writings (especially the wild ride of a book in \cite{Quantum Paradoxes}) are very much worth the read, both for sheer entertainment value, and because of their tendency to be more concrete than van Rooij's presentation or that of other scholars: Aguayo (\cite{Aguayo 1,Aguayo 2,Aguayo and Moraga radon-nikodym theorem paper}), and\textemdash most egregiously\textemdash Ludkovsky terribly dense writing (see \cite{Ludkovsky on non-archimedean measures}, for example). Khrennikov's co-authored paper \cite{Measure-theoretic approach to p-adic probability theory} on non-archimedean probability theory is also an excellent, self-contained introduction to the Monna-Springer integration technique presented here, supplemented with additional considerations regarding the formulation of measure-theoretic approaches to probability theory for probabilities taking values in non-archimedean fields. As a rule of thumb, much of the literature in non-archimedean analysis (as opposed to $p$-adic analysis proper) is couched in the language of abstract functional analysis. Some may find that register of presentation elegant or high-brow; I find it frustratingly un-welcoming, especially for newcomers. This is particularly important for the reader to keep in mind, seeing as the nature of the work I have done in my dissertation is foundationally \emph{concrete}. As mentioned in Subsection \ref{subsec:3.1.1 Some-Historical-and}, a would-be non-archimedean analyst should be aware of 'false friends' like \cite{Bosch lying title,Schneider awful book,More Schneider lies} which are works of algebraic geometry prancing about with analysis-sounding titles. \subsubsection{$\left(p,\infty\right)$-Adic Analysis} An excellent reference for the methods of Fourier analysis for complex-valued functions on local fields is Taibleson's book \cite{Taibleson - Fourier Analysis on Local Fields}. I also recommend perusing the matter of harmonic analysis on locally compact groups. Jordan Bell's lecture notes \cite{Bell - Harmonic Analysis on the p-adics,Bell - Pontryagin Duals of Q/Z and Q} are a great resource for anyone who share my ``just shut up and tell me how to compute stuff!'' attitude and the lack of patience implicit therein. Folland's text on abstract harmonic analysis \cite{Folland - harmonic analysis} makes for an excellent chaser to introduce the theory as a whole\textemdash Pontryagin duality, connections to representation theory, and all. And although \cite{Automorphic Representations} is a text on the representation theory of the general linear group, it nevertheless presents a thorough account of the tools of $\left(p,\infty\right)$-adic analysis\textemdash Fourier transform \emph{and }Mellin transform\textemdash in its opening chapters. Tate's thesis \cite{Tate's thesis}, is of course, a classic, and a beautiful display of these analytical methods at work in a number-theoretic application. The connections with $p$-adic mathematical physics are also worth exploring, particularly for the reader interested in learning more about the nitty-gritty of integration and the theory of distributions. Vladimirov's article \cite{Vladimirov - the big paper about complex-valued distributions over the p-adics} contains a comprehensive account of $\left(p,\infty\right)$-adic distributions and the accompanying notion of distributional derivatives, going so far as to even solve some differential equations by this method. \cite{Real and p-Adic Oscillatory integrals,p-adic van der Corput lemma} deal with $\left(p,\infty\right)$-adic oscillatory integrals. \subsubsection{$\left(p,p\right)$-Adic Analysis} After reading the first chapter of Gouvea's undergraduate-level text \cite{Gouvea's introudction to p-adic numbers book} in $\left(p,p\right)$-adic analysis, I used Robert's \emph{A course in $p$-adic analysis }(\cite{Robert's Book}) to acquaint myself with the ins and outs of the titular subject, along with a little help from the seminar in algebraic number theory taught at the University of Southern California by my eventual co-advisor\footnote{Along with Nicolai Haydn}, Prof. Sheldon Kamienny. Robert's book gives an excellent, balanced treatment of the subject; I never had the feeling that he had to tie and gag his inner number theorist or algebraic geometer to keep it from trying to monopolize the discussion. Curious undergraduates should probably peruse \cite{Gouvea's introudction to p-adic numbers book} before attempting Robert, if only for the pleasure of reading Gouvea's beautiful, lucid explanations. Koblitz's books \cite{Koblitz's book,Koblitz's other book} are more biased toward number theoretical concerns, but are still very accessible. More advanced\footnote{In other words, more \emph{algebraic}.} references include Kedlaya's book on $p$-adic differential equations \cite{Kedlaya}, and\textemdash diving straight in to the algebra and number theory\textemdash Iwasawa's classic treatise on $p$-adic $L$-functions \cite{Iwasawa} or Colmez's exposition of the same \cite{p-adic L-functions paper}, the chapter on $p$-adic distributions in Washington's \emph{Introduction to Cyclotomic Fields} \cite{Cyclotomic fields}. Readers with a desire to force-feed themselves the theory of Berkovitch spaces need only turn to \cite{Berkovich spaces and applications} and \cite{Bosch lying title} for algebraic geometry in all its haunting, malefic glory. On the analytical side, Anashi has a free e-text on $p$-adic ergodic theory \cite{Anashi - p-adic ergodic theory} and Adams' work on $p$-adic transcendental number theory contains an exposition of the $p$-adic analogue of contour integration (the \index{integral!Shnirelmann}\textbf{Shnirelmann integral}) \cite{Adams' paper on p-adic transcendental numbers with an appendix on Schnirelman integration}\textemdash though, to be clear, this is \emph{not }the same as the $p$-adic line integral used in rigid analytic spaces. Cherry's notes \cite{Cherry non-archimedean function theory notes} give an excellent introduction to the theory of analytic and meromorphic $\left(p,p\right)$-adic functions, especially for readers who don't have the patience for Robert's more completionist presentation, or Escassut more advanced, specialized treatment of the topic in \cite{value distribution in p-adic analysis}. George Brom's thesis \cite{pp adic Fourier theory} covers Fourier analysis in the $\left(p,p\right)$-adic setting, a fascinating oddity, seeing as\textemdash recall\textemdash there is no non-trivial translation-invariant $p$-adic-valued linear functional on the space of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}$. \subsubsection{Miscellany} For completeness' sake, I have included in the bibliography much of the material that I had on mind from 2017 to 2019, back when I was still trying to use complex analytic methods and their $\left(p,\infty\right)$-adic extensions to study Hydra maps. To reiterate, the guiding light of behind my investigations of those two years was the functional equation approach which I discovered independently of Meinardus and Berg. This line of research had me trying to learn as much as I could about holomorphic functions on the open unit disk and the boundary values of power series. Garnett's\footnote{My undergraduate analysis professor!} book \cite{Bounded analytic functions} gives a comprehensive account of harmonic analysis on the disk (and also half-plane), including\textemdash but not limited to\textemdash the Poisson Kernel and Hardy spaces. \cite{Ross et al} is an expository text on the Cauchy transform; \cite{Fractional Cauchy transforms} is a textbook on the Fractional Cauchy transform. Pavlovi\'{c}'s book\footnote{Not to be confused with the less-advanced but similarly-titled \cite{Baby Pavlovic}, also by Pavlovi\'{c}.} \cite{function classes on the unit disc} is an \emph{encyclopedic} account of spaces of complex-valued functions on the open unit disk in $\mathbb{C}$. One of the central themes of my complex-analytic investigations was the study of generating functions I called \textbf{set-series}\index{set-series}; these are of the form: \begin{equation} \varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}\label{eq:Definition of a set-series} \end{equation} where $V\subseteq\mathbb{N}_{0}$ is a set of interest. For my work, $V$ was an orbit class (or union thereof) of a given Hydra map on $\mathbb{Z}$. H. S. Wilf's delightfully titled \emph{generatingfunctionology }\cite{generatingfunctionology} is a thrilling read, as is Flajolet and Sedgewick's \emph{Analytic Combinatorics} \cite{analytic combinatorics}. For a thorough accounting of Tauberian methods used to study the coefficients of functions represented by power series or integrals, I cannot give too strong of a recommendation of Korevaar's book \cite{Korevaar} or Bingham, Goldie, and Teugels' \emph{magisterial }tome, the \emph{Regular Variation} \cite{Regular Variation}. For a more general background in (divergent) infinite series, Hardy's \emph{Divergent Series} \cite{Hardy - Divergent Series} is a classic. Flajolet's fascinating series of expository articles on the Mellin transform (\cite{Flajolet - Mellin Transforms,Flajolet - Digital sums}, etc.) provide powerful analytical tools for anyone wishing to study generating functions and power series asymptotics. It is also noteworthy that generating functions have a fascinating connection to transcendental number theory all their own. The study of this connection often goes by the name of \textbf{Mahler Theory}\footnote{Kurt, not Gustav.}\textbf{ }\index{Mahler Theory}\index{Mahler, Kurt}, after the German-Australian mathematician Kurt Mahler (of $p$-adic Mahler basis fame). Letting $d$ be an integer $\geq2$, the function\footnote{The notation $\phi_{d}$ is my own.}: \begin{equation} \phi_{d}\left(z\right)\overset{\textrm{def}}{=}\sum_{n=0}^{\infty}z^{d^{n}}\label{eq:Definition of Mahler's phi_d} \end{equation} is a \textbf{lacunary}\footnote{Meaning ``gap''.}\textbf{ series }\index{series!lacunary} with a natural boundary on the unit circle in $\mathbb{C}$. This function satisfies the functional equation: \begin{equation} \phi_{d}\left(z^{d}\right)=\phi_{d}\left(z\right)-z\label{eq:Mahler functional equation} \end{equation} By using this equation, Mahler was able to prove that the complex number $\phi_{d}\left(\alpha\right)$ is transcendental for any algebraic number $\alpha$ with complex absolute value $0<\left\|\alpha\right\|<1$ \cite{Mahler theory}. Mahler Theory uses functions characterized by polynomial functional equations like (\ref{eq:Mahler functional equation}) to show that said functions' take transcendental values at algebraic numbers \cite{mahler functions,Loxton and van der Poorten - arithemtic properties of functional equations}. Mahler theory is also connected with the theory of automata and automatic sequences\textemdash see \cite{mahler functions} and the sources cited therein.\newpage{} \pagestyle{fancy}\fancyfoot{}\fancyhead[L]{\sl LIST OF SYMBOLS}\fancyhead[R]{\thepage}\printnomenclature{} \begin{thebibliography}{100} \bibitem{mahler functions}\addcontentsline{toc}{chapter}{Bibliography} \pagestyle{fancy}\fancyfoot{}\fancyhead[L]{\sl BIBLIOGRAPHY}\fancyhead[R]{\thepage}Adamczewski, Boris, and Jason Bell. ``A problem around Mahler functions.'' (2012). <https://hal.archives-ouvertes.fr/hal-00798303/document> \bibitem{Adams' paper on p-adic transcendental numbers with an appendix on Schnirelman integration}Adams, William Wells. ``Transcendental numbers in the $p$-adic domain''. Columbia University, 1964. \bibitem{Aguayo 1}Aguayo, J. ``Vector measures and integral operators''. Ultrametric Functional Analysis, Contemp. Math., vol. 384, Amer. Math. Soc., Providence, RI (2005), pp. 1-13. \bibitem{Aguayo 2}Aguayo, J. and T. E. Gilsdorf. ``Non-Archimedean vector measures and integral operators'' $p$-Adic Functional Analysis, Lect. Notes Pure Appl. Math., vol. 222, Marcel Dekker, New York (2001), pp. 1-11. \bibitem{Aguayo and Moraga radon-nikodym theorem paper}Jos Aguayo and Mirta Moraga. ``A radon nikodym theorem in the non-archimedean setting.'' Proyecciones (Antofagasta) 20.3 (2001): 263-279. \bibitem{key-10}Akin, Ethan (2004), ``Why is the $3x+1$ Problem Hard?'', In: Chapel Hill Ergodic Theory Workshops (I. Assani, Ed.), Contemp. Math. vol 356, Amer. Math. Soc. 2004, pp. 1\textendash 20. (MR 2005f:37031). \bibitem{Amice}Amice, Yvette. \emph{Les nombres $p$-adiques}. Vol. 14. Presses universitaires de France, 1975. \bibitem{Anashi - p-adic ergodic theory}Anashi, Vladimir Sergeevich (2014). \emph{The p-adic ergodic theory and applications}. <https://www.researchgate.net/publication/269571423\_The\_p-adic\_ergodic\_theory\_and\_applications>/ \bibitem{Andaloro - first paper on sufficient sets}Andaloro, Paul. ``On total stopping times under $3x+1$ iteration'', Fibonacci Quart. 38 (2000), no. 1, 73\textendash 78. MR 1738650. \bibitem{Applegate and Lagarias - Trees}Applegate, David, and Jeffrey C. Lagarias. ``Density bounds for the $3x+1$ problem. I. Tree-search method.'' mathematics of computation 64.209 (1995): 411-426. \bibitem{Applegate and Lagarias - Difference inequalities}Applegate, David, and Jeffrey C. Lagarias. ``Density bounds for the $3x+1$ problem. II. Krasikov inequalities.'' Mathematics of computation 64.209 (1995): 427-438. \bibitem{3x+1 semigroup}David Applegate, Jeffrey C. Lagarias, ``The $3x+1$ semigroup'', Journal of Number Theory, Volume 117, Issue 1, 2006, Pages 146-159, ISSN 0022-314X. <https://www.sciencedirect.com/science/article/pii/S0022314X05001459> \bibitem{Arens =00003D000026 Singer - Generalized Analytic Functions}Arens, Richard and I. M. Singer (1956). ``Generalized Analytic Functions''. Transactions of the American Mathematical Society, Vol. 81, No. 2 (Mar., 1956), pp. 379-393. Published by: American Mathematical Society. <http://www.jstor.org/stable/1992923>. \bibitem{Arnold}Arnold, V. I., ``On teaching mathematics''. Translated from the original Russian by A.V. Goryunov. Published in: Uspekhi Mat. Nauk 53 (1998), no. 1, 229-234; English translation: Russian Math. Surveys 53 (1998), no. 1, 229-236. <http://www.ceremade.dauphine.fr/\textasciitilde msfr/articles/arnold/PRE\_anglais.ps> \bibitem{key-2-1}Axelsson, Ekaterina Yurova. ``On the sub-coordinate representation of $p$-adic functions''. \emph{Advances in Ultrametric Analysis}, edited by Alain Escassut, et al., American Mathematical Society, 2018. ProQuest Ebook Central, <https://ebookcentral.proquest.com/lib/socal/detail.action?docID=5347081>. \bibitem{Baker's Transcendental Number Theory}Baker, Alan (1990). \emph{Transcendental number theory}. Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171. \bibitem{p-adic proof for =00003D0003C0}Baker, Matt. ``A p-adic proof that pi is transcendental''. \emph{Matt Baker's Math Blog}. 20 March. 2015. <https://mattbaker.blog/2015/03/20/a-p-adic-proof-that-pi-is-transcendental/> \bibitem{Simon =00003D000026 Breuer - Natural Boundaries =00003D000026 Spectral Theory}Barry Simon and Jonathan Breuer (2011).\emph{ }``Natural Boundaries and Spectral Theory''. \bibitem{natural boundaries}Bell, Jason P., and Jeffrey C. Lagarias. ``$3x+1$ inverse orbit generating functions almost always have natural boundaries.'' arXiv preprint arXiv:1408.6884 (2014). \bibitem{Bell - Harmonic Analysis on the p-adics}Bell, Jordan. (2016) ``Harmonic analysis on the $p$-adic numbers''.\emph{ }<https://pdfs.semanticscholar.org/9d4f/f3efad867fe867331bb00402807e0856170c.pdf> \bibitem{Bell - Pontryagin Duals of Q/Z and Q}Bell, Jordan. 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MR 1612686\newpage{} \end{thebibliography} \addcontentsline{toc}{chapter}{Index}\pagestyle{fancy}\fancyfoot{}\fancyhead[L]{\sl INDEX}\fancyhead[R]{\thepage} \printindex \end{document}
2412.03135v1	http://arxiv.org/abs/2412.03135v1	A new look at the classiffication of the tri-covectors of a 6-dimensional symplectic space	\documentclass[leqno,a4paper]{article} \usepackage{amssymb,amsmath,amsthm,verbatim} \title{\textbf{A new look at the classification }\\ \textbf{of the tri-covectors of a} $6$\textbf{-dimensional symplectic space }} \author{\textsc{J. Mu\~{n}oz Masqu\'e, L. M. Pozo Coronado}} \date{} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \begin{document} \maketitle \begin{abstract} Let $\mathbb{F}$ be a field of characteristic $\neq 2$ and $3$, let $V$ be a $\mathbb{F}$-vector space of dimension $6$, and let $\Omega \in \wedge ^2V^\ast $ be a non-degenerate form. A system of generators for polynomial invariant functions under the tensorial action of the group $Sp(\Omega )$ on $\wedge ^3 V^\ast $, is given explicitly. Applications of these results to the normal forms of De Bruyn-Kwiatkowski and Popov are given. \end{abstract} \bigskip \noindent\emph{Mathematics Subject Classification 2010:\/} Primary: 15A21; Secondary: 15A63, 15A75, 20G15 \medskip \noindent\emph{Key words and phrases:\/} Algebraic invariant function, derivation, linear representation, normal forms, symplectic group. \section{Symplectic invariants\label{symplectic}} Below, $\mathbb{F}$ denotes a field of characteristic $\neq 2$ and $3$, and $V$ is a $\mathbb{F}$-vector space of dimension $6$. Notations and elementary properties of algebraic sets and groups have been taken from Fogarty's book \cite{Fogarty}. The group $GL(V)$ acts on $\otimes ^rV^\ast $ by \[ (A\cdot \xi )(x_1,\dotsc,x_r)=\xi (A^{-1}x_1,\dotsc,A^{-1}x_r), \] $\forall\xi \in \otimes ^rV^\ast ,\;\forall x_1,\dotsc,x_r\in V$, and $GL(V)$ acts on $\wedge ^rV^\ast $ by the same formula. In particular, the $\mathbb{F}$-homomorphism induced by the action of $GL(V)$ on $\wedge ^3 V^\ast $ is denoted by $\rho\colon GL(V)\to GL(\wedge ^3V^\ast )$. We denote by $\rho^{\prime}\colon Sp(\Omega ) \to GL(\wedge ^3 V^\ast )$ the restriction of $\rho $ to the symplectic group of a non-degenerate $2$-covector $\Omega \in \wedge ^2V^\ast $. Furthermore, $GL(V)$ acts on $\otimes ^rV^\ast \otimes V$ by $(A\cdot \eta )(x_1,\dotsc,x_r) =A[\eta (A^{-1}x_1,\dotsc,A^{-1}x_r)]$, for all $\eta $ in $\otimes ^rV^\ast \otimes V$, and all $x_1,\dotsc,x_r\in V $. If $(v_i)_{i=1}^6$ is a basis of $V$ such that $\Omega =\sum _{i=1}^6v^i\wedge v^{i+3}$ and $(v^i)_{i=1}^6$ is the dual basis, then we define a system of coordinate functions $y_{abc}$, $1\leq a<b<c\leq6$, on $\wedge ^3 V^\ast $ by setting \begin{equation} \theta =\sum\nolimits_{1\leq a<b<c\leq 6}y_{abc}(\theta ) \left( v^a\wedge v^b\wedge v^c\right) \in \wedge ^3 V^\ast . \end{equation} For every $A\in GL(V)$ and every $1\leq a<b<c\leq 6$ we have \begin{align} A\cdot \left( v^a\wedge v^b\wedge v^c\right) & =(A^{-1})^\ast v^a\wedge (A^{-1})^\ast v^b \wedge (A^{-1})^\ast v^c \nonumber \\ & =\left( v^a\circ A^{-1}\right) \wedge \left( v^b\circ A^{-1}\right) \wedge \left( v^{c}\circ A^{-1}\right) \nonumber \\ & =\left( \lambda _{ah}v^h\right) \wedge \left( \lambda _{bi}v^i\right) \wedge \left( \lambda _{cj}v^{j}\right) \nonumber \\ & =\sum\nolimits_{1\leq h<i<j\leq 6}\left\vert \begin{array} [c]{ccc} \lambda _{ah} & \lambda _{bh} & \lambda _{ch}\\ \lambda _{ai} & \lambda _{bi} & \lambda _{bi}\\ \lambda _{aj} & \lambda _{bj} & \lambda _{cj} \end{array} \right\vert v^h\wedge v^i\wedge v^j, \nonumber \end{align} where $(\lambda _{ij})_{i,j=1}^6$ is the matrix of $(A^{-1})^T$ in the basis $(v_i)_{i=1}^6$. Therefore, we have $\mathbb{F}[\wedge ^3 V^\ast ] =\mathbb{F}[y_{abc}]_{1\leq a<b<c\leq 6}$, and hence, $\mathbb{F}(\wedge ^3V^\ast ) =\mathbb{F}(y_{abc})_{1\leq a<b<c\leq 6}$. A function $I\in\mathbb{F}[\wedge ^3 V^\ast ]$ (resp.\ $I\in \mathbb{F}(\wedge ^3 V^\ast )$) is $Sp(\Omega )$-invariant if \[ I\left( A\cdot \theta \right) =I(\theta ),\quad \forall \theta \in \wedge ^3 V^\ast, \; \forall A\in Sp(\Omega ). \] \section{The basic invariants defined\label{invariants}} \subsection{$I_1$ defined\label{I_1}} For every $\theta \in \wedge ^3 V^\ast $ there exists a unique $J^\theta \in\wedge ^2V^\ast \otimes V$ such that \begin{equation} \theta (x,y,z)=\Omega (J^\theta (x,y),z), \quad \forall x,y,z\in V.\label{J} \end{equation} Given $A\in GL(V)$ and replacing $\theta $ by $A\cdot \theta $ in the formula \eqref{J}, we have \[ (A\cdot \theta )(x,y,z)=\Omega (J^{A\cdot \theta }(x,y),z). \] Expanding on the right-hand side, we deduce \begin{align} (A\cdot \theta )(x,y,z) & =\theta (A^{-1}x,A^{-1}y,A^{-1}z)\\ & =\Omega (J^\theta (A^{-1}x,A^{-1}y),A^{-1}z)\\ & =(A\cdot \Omega)(A[J^\theta (A^{-1}x,A^{-1}y)],z))\\ & =(A\cdot \Omega)((A\cdot J^\theta )(x,y),z). \end{align} Furthermore, if $A\in Sp(\Omega )$, then $A\cdot \Omega =\Omega $ and consequently \[ (A\cdot \theta )(x,y,z)=\Omega ((A\cdot J^\theta )(x,y),z) =\Omega (J^{A\cdot \theta }(x,y),z), \quad \forall x,y,z\in V. \] Hence $J^{A\cdot \theta }=A\cdot J^\theta , \quad \forall A\in Sp(\Omega )$. If \begin{equation} \theta =\sum _{1\leq i<j<k\leq 6} \lambda _{ijk}v^i\wedge v^{j}\wedge v^k \label{theta} \end{equation} and $J^\theta =\sum_{b<c}\mu _{bc}^av^b\wedge v^c\otimes v_a$, then by letting $x=v_i$, $y=v_j$, $z=z^kv_k$ in the formula \eqref{J} for every pair $1\leq i<j\leq 5$, it follows: \[ \sum _{k\neq i,j}\theta \left( v_i,v_j,v_k\right) z^k =-\mu_{ij}^4z^1-\mu_{ij}^5z^2-\mu_{ij}^6z^3+\mu_{ij}^1z^4 +\mu_{ij}^2z^5+\mu_{ij}^3z^6, \] and comparing the coefficients of $z^1,\dotsc,z^6$ in both sides, we deduce \[ \begin{array} [c]{lll} \mu_{ij}^1=\lambda _{ij4}, & \mu_{ij}^2=\lambda _{ij5}, & \mu_{ij}^3=\lambda _{ij6}, \medskip\\ \mu_{ij}^4=-\lambda _{ij1}, & \mu_{ij}^{5}=-\lambda _{ij2}, & \mu_{ij}^6=-\lambda _{ij3}, \end{array} \] with the usual agreement: $\lambda _{ijk}=\varepsilon _\sigma \lambda _{abc}$, where $a<b<c$, $\{ a,b,c\} =\{ i,j,k\} $, $\sigma $ being the permutation $a\mapsto i$, $b\mapsto j$, $c\mapsto k$. Let $\Omega =\sum _{i=1}^6v^i\wedge v^{i+3}$ be as in section \ref{symplectic}. Each $\theta \in \wedge ^3 V^\ast $ determines a vector $v_\theta \in V$ defined by the following equation: \begin{equation} i_{v_\theta }(\Omega \wedge \Omega \wedge \Omega ) =\theta\wedge \Omega .\label{v_theta} \end{equation} Transforming the equation \eqref{v_theta} by $A\in Sp(\Omega )$ and recalling that $A\cdot \Omega =\Omega $, we obtain $A\cdot \left[ i_{v_\theta }(\Omega \wedge \Omega \wedge \Omega ) \right] =(A\cdot \theta )\wedge \Omega =i_{v_{A\cdot \theta }}(\Omega \wedge \Omega \wedge \Omega )$. Moreover, for every system $x_1,\dotsc,x_5\in V$ one has: \[ \begin{array} [c]{rl} \left( A\cdot \left[ i_{v_\theta }(\Omega \wedge \Omega \wedge \Omega ) \right] \right) \left( x_1,\dotsc,x_5\right) =\! & \! \left[ i_{v_\theta }(\Omega \wedge \Omega \wedge \Omega) \right] \left( A^{-1}x_1,\dotsc,A^{-1}x_5\right) \\ =\! & \!(\Omega \wedge \Omega \wedge \Omega ) \left( A^{-1}Av_\theta ,A^{-1} x_1,\dotsc,A^{-1}x_5\right) \\ =\! & \!\! \left( i_{Av_\theta }\left[ A\cdot (\Omega \wedge \Omega \wedge \Omega )\right] \right) \left( x_1,\dotsc,x_5\right) . \end{array} \] Hence $A\cdot \left[ i_{v_\theta }(\Omega \wedge \Omega \wedge \Omega ) \right] =i_{Av_\theta } \left[ A\cdot (\Omega \wedge \Omega \wedge \Omega ) \right] =i_{Av_\theta }(\Omega \wedge \Omega\wedge \Omega)$. Accordingly: $v_{A\cdot \theta }=Av_\theta $. Letting $x=v_\theta $ in \eqref{J}, it follows: \[ \Omega \left[ \left( i_{v_\theta }J^\theta \right) (y),z\right] =\Omega\left[ J^\theta (v_\theta ,y),z\right] =\theta (v_\theta ,y,z), \] and replacing $\theta $ by $A\cdot \theta $, $A\in Sp(\Omega )$, we have \begin{align} \Omega \left[ \left( i_{v_{A\cdot \theta }}J^{A\cdot \theta } \right) (y),z \right] & =\Omega \left[ J^{A\cdot \theta }(v_{A\cdot \theta },y),z \right] =(A\cdot \theta )(v_{A\cdot \theta },y,z)\\ & =\theta \left( A^{-1}v_{A\cdot \theta },A^{-1}y,A^{-1}z \right) =\theta (v_\theta ,A^{-1}y,A^{-1}z)\\ & =\Omega \left[ J^\theta (v_\theta ,A^{-1}y),A^{-1}z \right] =\Omega \left[ (i_{v_\theta }J^\theta )(A^{-1}y),A^{-1}z \right] \\ & =\left( A\cdot \Omega \right) \left[ (i_{v_\theta }J^\theta )(y),z \right] \\ & =\Omega \left[ (i_{v_\theta }J^\theta )(y),z \right] . \end{align} Hence $i_{v_{A\cdot \theta }}J^{A\cdot \theta } =i_{v_\theta }J^\theta $ for all $A\in Sp(\Omega )$. Accordingly, the endomorphism $L_\theta =i_{v_\theta }J^\theta \in V^\ast \otimes V$ is $Sp(\Omega )$-invariant. If $\theta $ is as in \eqref{theta} and $v_\theta =\sum _{h=1}^6x^hv_h$, then, as a computation shows, we have \begin{equation} \begin{array} [c]{lll} x^1=-\tfrac{1}{6}(\lambda _{245}+\lambda _{346}), & x^2=\tfrac{1}{6}(\lambda _{145}-\lambda _{356}), & x^3=\tfrac{1}{6}(\lambda _{146}+\lambda _{256}), \medskip\\ x^4=-\tfrac{1}{6}(\lambda _{125}+\lambda _{136}), & x^5=-\tfrac{1}{6}(\lambda _{236}-\lambda _{124}), & x^6=\tfrac{1}{6}(\lambda _{134}+\lambda _{235}). \end{array} \end{equation} Again writing $J^\theta =\sum _{b<c}\mu _{bc}^av^b\wedge v^c\otimes v_a$, we obtain \begin{equation} \begin{array} [c]{lll} \mu_{ij}^1=\lambda _{ij4}, & \mu _{ij}^2=\lambda _{ij5}, & \mu _{ij}^3=\lambda _{ij6},\medskip\\ \mu _{ij}^4=-\lambda _{ij1}, & \mu _{ij}^{5}=-\lambda _{ij2}, & \mu_{ij}^6=-\lambda _{ij3}, \end{array} \end{equation} or equivalently $\mu_{ij}^{h}=\lambda _{ij,h+1}$, $\mu_{ij}^{h+3}=-\lambda _{ij,h}$ for $1\leq h\leq3$. Therefore \[ L_\theta =i_{v_\theta }J^\theta =\sum\nolimits_{b<c}\mu_{bc}^a\left( x^bv^c-x^cv^b\right) \otimes v_a. \] Hence, the matrix $M=(M_{ia})_{i,a=1}^6$ of $L_\theta $ in the basis $v_1,\dotsc,v_6$ is given by $M_{ia}=\sum\nolimits_{b=1}^6\mu_{bi}^ax^b$, and as a computation shows, the characteristic polynomial of $L_\theta $ is written as follows: \[ \det \left( xI-L_\theta \right) =x^6+c_4(\theta)x^4+c_2(\theta )x^2, \] where \[ \begin{array} [c]{ll} c_2(\theta )=\frac{1}{2^4\cdot 3^4}(I_1)^2, & c_4(\theta )=-\frac{1}{2\cdot3^2}I_1, \end{array} \] and the explicit expression of $I_1$ is given in the Appendix. \subsection{$I_2$ defined\label{I_2}} For every $\theta \in \wedge ^3 V^\ast $ let $J^\theta \barwedge J^\theta \in\wedge ^3 V^\ast \otimes V$ be the tensor defined as follows: \[ \begin{array} [c]{ll} \left( J^\theta \barwedge J^\theta \right) \left( x_1,x_2,x_3\right) = & J^\theta \left( J^\theta \left( x_1,x_2\right) ,x_3\right) +J^\theta \left( J^\theta \left( x_2,x_3\right) ,x_1\right) \\ \multicolumn{1}{r}{} & \multicolumn{1}{r}{+J^\theta \left( J^\theta \left( x_3,x_1\right) ,x_2\right) ,}\\ \multicolumn{1}{r}{} & \multicolumn{1}{r}{\forall x_1,x_2,x_3\in V.} \end{array} \] As $J^\theta \left( v_i,v_j\right) =\mu_{ij}^hv_h$, we have $J^\theta \left( J^\theta \left( v_i,v_j\right) ,v_k\right) =\mu_{ij}^hJ^\theta \left( v_h,v_k\right) =\mu_{ij}^h\mu _{hk}^lv_l$. Hence \[ \left( J^\theta \barwedge J^\theta \right) \left( v_i,v_j,v_k\right) =\left( \mu_{ij}^h\mu_{hk}^l+\mu_{jk}^h\mu_{hi}^l +\mu_{ki}^h\mu_{hj}^l\right) v_l. \] Let $\Omega ^\theta \in\wedge ^6V^\ast $ be defined by $\Omega ^\theta =\tfrac{1}{3!^2}\operatorname{alt} \left[ \tilde{\Omega }\circ \left( J^\theta \barwedge J^\theta \otimes J^\theta \barwedge J^\theta \right) \right] $, where $\tilde{\Omega }\colon V\otimes V\to \mathbb{F}$ is the linear map attached to $\Omega $; i.e., $\tilde{\Omega }(x\otimes y)=\Omega (x,y)$, $\forall x,y\in V$, and $(J^\theta \barwedge J^\theta ) \otimes (J^\theta \barwedge J^\theta ) \colon \wedge ^3 V\otimes\wedge ^3V\to V\otimes V$ is the tensor product of $J^\theta \barwedge J^\theta $ and itself, $J^\theta \barwedge J^\theta $ being understood as a linear map $\wedge ^3 V\to V$, via the canonical isomorphism $\wedge ^3 V^\ast \otimes V=\operatorname{Hom} \left( \wedge ^3 V,V\right) $. Letting $\xi _{ijk}^l=\mu_{ij}^h\mu_{hk}^l+\mu_{jk}^h\mu_{hi}^l +\mu_{ki}^h\mu_{hj}^l$, we have \[ J^\theta \barwedge J^\theta =\sum_{i<j<k}\xi _{ijk}^lv^i\wedge v^j\wedge v^k\otimes v_l. \] Hence $\Omega ^\theta =\sum_{i<j<k}\sum_{a<b<c}\xi _{ijk}^l\xi _{abc}^d \left( v^i\wedge v^j\wedge v^k\right) \wedge \left( v^a\wedge v^b\wedge v^c\right) \Omega (v_l,v_d)$. The product $\left( v^i\wedge v^j\wedge v^k\right) \wedge \left( v^a\wedge v^b\wedge v^c\right) $ does not vanish if and only if the indices $i,j,k,a,b,c$ are pairwise distinct, i.e., $\{ i,j,k,a,b,c\} =\{ 1,\dotsc,6\} $, and in that case, $v^i\wedge v^j\wedge v^k\wedge v^a\wedge v^b\wedge v^c =\varepsilon _\sigma v^1\wedge v^2\wedge v^3\wedge v^4\wedge v^5\wedge v^6$, $\sigma $ being the permutation $\sigma (1)=i$, $\sigma (2)=j$, $\sigma (3)=k$, $\sigma (4)=a$, $\sigma (5)=b$, $\sigma (6)=c$. Therefore $\Omega ^\theta =I_2(\theta)\Omega \wedge \Omega \wedge \Omega $, where \begin{align} I_2(\theta )\!& =\! -\tfrac{1}{6}\sum _{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^l \xi _{\sigma (4)\sigma (5)\sigma (6)}^{d}\Omega (v_{l},v_{d}) \\ \!& =\! \tfrac{1}{6}\sum _{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^4 \xi _{\sigma (4)\sigma (5)\sigma (6)}^{1} -\tfrac{1}{6}\sum _{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^1 \xi _{\sigma (4)\sigma (5)\sigma (6)}^4 \\ & \!\! +\tfrac{1}{6}\sum_{\sigma \in S_{6}} \varepsilon _{\sigma }\xi _{\sigma (1)\sigma (2)\sigma (3)}^5 \xi _{\sigma (4)\sigma (5)\sigma (6)}^2 -\tfrac{1}{6}\sum_{\sigma \in S_6} \varepsilon _{\sigma }\xi _{\sigma (1)\sigma (2)\sigma (3)}^2 \xi _{\sigma (4)\sigma (5)\sigma (6)}^5 \\ & \!\! +\tfrac{1}{6}\sum _{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^6 \xi _{\sigma (4)\sigma (5)\sigma (6)}^3 -\tfrac{1}{6}\sum_{\sigma \in S_{6}} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^3 \xi _{\sigma (4)\sigma (5)\sigma (6)}^6. \end{align} If $\sigma _0$ is the permutation $1\mapsto 4$, $2\mapsto 5$, $3\mapsto 6$, $4\mapsto 1$, $5\mapsto 2$, $6\mapsto3$, then letting $\sigma ^\prime =\sigma\circ\sigma_{0}$, we have $\sigma _0\circ \sigma _0=1$, $\varepsilon _{\sigma ^\prime} =\varepsilon _\sigma \varepsilon _{\sigma _0} =-\varepsilon _\sigma $, \[ \begin{array} [c]{lll} \sigma ^\prime (1)=\sigma (4), & \sigma ^\prime (2) =\sigma (5), & \sigma ^\prime (3)=\sigma (6),\\ \sigma ^\prime (4)=\sigma (1), & \sigma ^\prime (5) =\sigma (2), & \sigma ^\prime (6)=\sigma (3). \end{array} \] and from the previous formula for $I_2(\theta )$ we obtain \begin{equation} \begin{array} [c]{rl} I_2(\theta)= & -\tfrac{1}{3}\sum _{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^1 \xi _{\sigma (4)\sigma (5)\sigma (6)} ^4\smallskip\\ & -\tfrac{1}{3}\sum_{\sigma\in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^2 \xi _{\sigma (4)\sigma (5)\sigma (6)}^5 \smallskip\\ & -\tfrac{1}{3}\sum_{\sigma \in S_6} \varepsilon _\sigma \xi _{\sigma (1)\sigma (2)\sigma (3)}^3 \xi _{\sigma (4)\sigma (5)\sigma (6)}^6. \end{array} \end{equation} The explicit expression for $I_2$ is also given in the Appendix. \section{Infinitesimal criterion of invariance} The derivations \[ \begin{array} [c]{l} \tfrac{\partial }{\partial y_{abc}}\colon \mathbb{F}[\wedge ^3 V^\ast] \to \mathbb{F}[\wedge ^3 V^\ast ]\quad(\mathrm{resp.}\;\tfrac {\partial}{\partial y_{abc}}\colon\mathbb{F}(\wedge ^3 V^\ast ) \to \mathbb{F}(\wedge ^3 V^\ast )),\\ 1\leq a<b<c\leq 6, \end{array} \] are a basis of the $\mathbb{F}[\wedge ^3 V^\ast ]$-module (resp.\ $\mathbb{F}(\wedge ^3 V^\ast )$-vector space) $\operatorname{Der}\nolimits_{\mathbb{F}}\mathbb{F}[\wedge ^3 V^\ast ]$ (resp.\ $\operatorname{Der} \nolimits_{\mathbb{F}}\mathbb{F}(\wedge ^3 V^\ast )$). The subrings of $Sp(\Omega )$-invariant functions are denoted by $\mathbb{F}[\wedge ^3 V^\ast ]^{Sp(\Omega )}$ and $\mathbb{F}(\wedge ^3 V^\ast )^{Sp(\Omega )}$, respectively. \begin{lemma} \label{lemma_bis} Assume $V$ is a $\mathbb{F}$-vector space of dimension $6$ and let $\Omega\in\wedge ^2V^\ast $ be a non-degenerate $2$-covector. If $I\in\mathbb{F}[\wedge ^3 V^\ast ]$ (resp.\ $I\in\mathbb{F}(\wedge ^3V^\ast )$) is a $Sp(\Omega )$-invariant function, then $I$ is a common first integral of the following derivations: \begin{equation} \begin{array} [c]{l} U^\ast =\sum\nolimits_{1\leq h<i<j\leq 6} \left( \sum\nolimits_{1\leq a<b<c\leq 6}U_{hij}^{abc}y_{abc} \right) \tfrac{\partial}{\partial y_{hij}},\\ \forall U=(u_{ij})_{i,j=1}^6\in \mathfrak{sp}(6,\mathbb{F}), \end{array} \label{U^ast} \end{equation} where the functions $U_{hij}^{abc}$ are defined by \begin{equation} U_{hij}^{abc}=-\left\vert \begin{array} [c]{ccc} u_{ha} & \delta_{hb} & \delta_{hc}\\ u_{ia} & \delta_{ib} & \delta_{ic}\\ u_{ja} & \delta_{jb} & \delta_{jc} \end{array} \right\vert -\left\vert \begin{array} [c]{ccc} \delta_{ha} & u_{hb} & \delta_{hc}\\ \delta_{ia} & u_{ib} & \delta_{ic}\\ \delta_{ja} & u_{jb} & \delta_{jc} \end{array} \right\vert -\left\vert \begin{array} [c]{ccc} \delta_{ha} & \delta_{hb} & u_{hc}\\ \delta_{ia} & \delta_{ib} & u_{ic}\\ \delta_{ja}^{{}} & \delta_{jb} & u_{jc} \end{array} \right\vert , \end{equation} and $\delta$ denotes the Kronecker delta. \end{lemma} \begin{proof} We first observe that the derivations in \eqref{U^ast} are the image of the $\mathbb{F}$-homomorphism of Lie algebras $\rho _\ast ^\prime \colon \mathfrak{sp}(6,\mathbb{F})\to \operatorname{Der} \nolimits_{\mathbb{F}}\mathbb{F}[\wedge ^3 V^\ast ]$ induced by $\rho ^\prime $. Accordingly, we only need to show that $U^\ast (I)=0$ for the elements $U$ in a basis of the algebra $\mathfrak{sp}(6,\mathbb{F})$. If $(E_{ij})_{i,j=1}^6$ is the standard basis of $\mathfrak{gl}(6,\mathbb{F})$, then the matrices \begin{equation} \begin{array} [c]{llll} E_{11}+E_{14}-E_{41}-E_{44}, & E_{41}, & E_{52}, & E_{14},\\ E_{22}-E_{25}+E_{52}-E_{55}, & E_{42}+E_{51}, & E_{53}+E_{62}, & E_{25},\\ E_{33}-E_{66}+E_{36}-E_{63}, & E_{43}+E_{61}, & E_{63}, & E_{36},\\ E_{12}-E_{54}, & E_{31}-E_{46}, & E_{15}+E_{24}, & E_{26}+E_{35},\\ E_{21}-E_{45}, & E_{23}-E_{65}, & E_{16}+E_{34}, & E_{13}-E_{64},\\ E_{32}-E_{56}, & & & \end{array} \label{basis} \end{equation} are a basis $\mathcal{B}$ of $\mathfrak{sp}(6,\mathbb{F})$ with the following property: For every $U\in\mathcal{B}$, we have $U^2=0$, as a simple computation proves. Hence, for every $U\in\mathcal{B}$ and $t\in\mathbb{F}$, the endomorphism $I+tU$ ($I$ denoting the identity map of $V$) is symplectic. In fact, if $M_\Omega $, $M_U$ are the matrices of $\Omega $, $U$, respectively, then \begin{align} \left( M_{I+tU}\right) ^{T}M_{\Omega}M_{I+tU} & =\left( I+t\left( M_U\right) ^T\right) M_\Omega \left( I+tM_U\right) \\ & =M_\Omega +t\left[ \left( M_U\right) ^TM_\Omega +M_\Omega M_U\right] +t^2\left( M_U\right) ^TM_\Omega M_U, \end{align} but on one hand, we have $\left( M_U\right) ^TM_\Omega +M_\Omega M_{U}=0$, as $U\in\mathfrak{sp}(6,\mathbb{F})$, and on the other: $\left( M_U\right) ^TM_\Omega M_U=-M_\Omega M_{U^2}=0$. Hence $\left( M_{I+tU}\right) ^TM_\Omega M_{I+tU} =M_\Omega $, thus proving that $I+tU\in Sp(\Omega )$, $\forall t\in \mathbb{F}$. Therefore $I\left( (I+tU)\cdot \theta \right) =I(\theta )$, $\forall t\in\mathbb{F}$, $\forall U=(u_{ij})\in \mathcal{B}$ and $\forall t\in \mathbb{F} $. If $\Lambda(t)=(\lambda _{ij}(t))_{i,j=1}^6=I-tU^T$, then \[ I\Bigl( \sum\nolimits_{\substack{1\leq a<b<c\leq 6 \\1\leq h<i<j\leq 6}}~y_{abc} \left\vert \begin{array} [c]{ccc} \lambda _{ah}(t) & \lambda _{bh}(t) & \lambda _{ch}(t)\\ \lambda _{ai}(t) & \lambda _{bi}(t) & \lambda _{ci}(t)\\ \lambda _{aj}(t) & \lambda _{bj}(t) & \lambda _{cj}(t) \end{array} \right\vert v^{h}\wedge v^i\wedge v^{j} \Bigr) =I(\theta ), \] and taking derivatives at $t=0$, we obtain \[ 0=\sum\nolimits_{1\leq a<b<c\leq 6,1\leq h<i<j\leq6} U_{hij}^{abc}y_{abc} \tfrac{\partial I}{\partial y_{hij}}(\theta). \] \end{proof} \begin{definition} Let $F$ be the field of fractions of an entire ring $R$. The \emph{generic rank} of a finitely-generated $R$-module $\mathcal{M}$ is the dimension of the $F$-vector space $F\otimes_r\mathcal{M}$. \end{definition} \begin{theorem} \label{theorem} The generic rank of the $\mathbb{F}[\wedge ^3 V^\ast ]$-module $\mathcal{M}$ spanned by the derivations in the formula \emph{\eqref{U^ast}} of \emph{Lemma \ref{lemma_bis}} is $18$. \end{theorem} \begin{proof} If $U=(u_{ij})_{i,j=1}^6\in\mathfrak{sp}(\Omega)$ then \[ \begin{array} [c]{lllll} u_{24}=u_{15}, & u_{34}=u_{16}, & u_{35}=u_{26}, & u_{51}=u_{42}, & u_{61}=u_{43},\\ u_{62}=u_{53}, & u_{44}=-u_{11}, & u_{45}=-u_{21}, & u_{46}=-u_{31}, & u_{54}=-u_{12},\\ \multicolumn{1}{r}{u_{55}=-u_{22},} & \multicolumn{1}{r}{u_{56}=-u_{32},} & \multicolumn{1}{r}{u_{64}=-u_{13},} & \multicolumn{1}{r}{u_{65}=-u_{23},} & \multicolumn{1}{r}{u_{66}=-u_{33},} \end{array} \] and the following $21$ functions are coordinates on the symplectic algebra $\mathfrak{sp}(\Omega )$: $u_{11}$, $u_{12}$, $u_{13}$, $u_{14}$, $u_{15}$, $u_{16}$, $u_{21}$, $u_{22}$, $u_{23}$, $u_{25}$, $u_{26}$, $u_{31}$, $u_{32}$, $u_{33}$, $u_{36}$, $u_{41}$, $u_{42}$, $u_{43}$, $u_{52}$, $u_{53}$, $u_{63}$. Hence we can write: \[ \begin{array} [c]{rl} U^\ast \! = & \!\!\!\! u_{11}Z_{11}+u_{12}Z_{12}+u_{13}Z_{13}+u_{14}Z_{14} +u_{15}Z_{15}+u_{16}Z_{16}+u_{21}Z_{21}\\ \! & \multicolumn{1}{r}{\!\!\!\! +u_{22}Z_{22}+u_{23}Z_{23}+u_{25}Z_{25}+u_{26}Z_{26} +u_{31}Z_{31}+u_{32}Z_{32}+u_{33}Z_{33}}\\ \! & \multicolumn{1}{r}{\!\!\!\!+u_{36}Z_{36} +u_{41}Z_{41}+u_{42}Z_{42}+u_{43}Z_{43} +u_{52}Z_{52}+u_{53}Z_{53}+u_{63}Z_{63},} \end{array} \] where \begin{equation} \begin{array}{rc} Z_{11}= & -y_{123}Y_{123}-y_{125}Y_{125}-y_{126}Y_{126} -y_{135}Y_{135}-y_{136}Y_{136}-y_{156}Y_{156} \\ & +y_{234}Y_{234}+y_{245}Y_{245}+y_{246}Y_{246} +y_{345}Y_{345}+y_{346}Y_{346}+y_{456}Y_{456}, \\ Z_{12}= & y_{124}Y_{125}-y_{234}Y_{134}+y_{134}Y_{135} -y_{235}Y_{135}-y_{236}Y_{136}-y_{245}Y_{145} \\ & -y_{246}Y_{146}+y_{146}Y_{156}-y_{256}Y_{156} +y_{234}Y_{235}+y_{246}Y_{256}+y_{346}Y_{356}, \\ Z_{13}= & y_{234}Y_{124}+y_{235}Y_{125}+y_{124}Y_{126} +y_{236}Y_{126}+y_{134}Y_{136}-y_{345}Y_{145} \\ & -y_{346}Y_{146}-y_{145}Y_{156}-y_{356}Y_{156} +y_{234}Y_{236}-y_{245}Y_{256}-y_{345}Y_{356}, \\ Z_{14}= & -y_{234}Y_{123}+y_{245}Y_{125}+y_{246}Y_{126} +y_{345}Y_{135}+y_{346}Y_{136}-y_{456}Y_{156}, \\ Z_{15}= & y_{134}Y_{123}-y_{235}Y_{123}-y_{245}Y_{124} -y_{145}Y_{125}-y_{146}Y_{126}+y_{256}Y_{126} \\ & -y_{345}Y_{134}+y_{356}Y_{136}+y_{456}Y_{146} +y_{345}Y_{235}+y_{346}Y_{236}-y_{456}Y_{256}, \end{array} \end{equation} \begin{equation} \begin{array}{rc} Z_{16}= & -y_{124}Y_{123}-y_{236}Y_{123}-y_{246}Y_{124} -y_{256}Y_{125}-y_{346}Y_{134}-y_{145}Y_{135} \\ & -y_{356}Y_{135}-y_{146}Y_{136}-y_{456}Y_{145} -y_{245}Y_{235}-y_{246}Y_{236}-y_{456}Y_{356}, \\ Z_{21}= & y_{125}Y_{124}+y_{135}Y_{134}+y_{156}Y_{146} -y_{134}Y_{234}+y_{235}Y_{234}-y_{135}Y_{235} \\ & -y_{136}Y_{236}-y_{145}Y_{245}-y_{146}Y_{246} +y_{256}Y_{246}-y_{156}Y_{256}+y_{356}Y_{346}, \\ Z_{22}= & -y_{123}Y_{123}-y_{124}Y_{124}-y_{126}Y_{126} +y_{135}Y_{135}+y_{145}Y_{145}+y_{156}Y_{156} \\ & -y_{234}Y_{234}-y_{236}Y_{236}-y_{246}Y_{246} +y_{345}Y_{345}+y_{356}Y_{356}+y_{456}Y_{456}, \\ Z_{23}= & -y_{134}Y_{124}-y_{135}Y_{125}+y_{125}Y_{126} -y_{136}Y_{126}+y_{135}Y_{136}+y_{145}Y_{146} \\ & +y_{235}Y_{236}-y_{345}Y_{245}+y_{245}Y_{246} -y_{346}Y_{246}-y_{356}Y_{256}+y_{345}Y_{346}, \\ Z_{25}= & y_{135}Y_{123}+y_{145}Y_{124}-y_{156}Y_{126} -y_{345}Y_{234}+y_{356}Y_{236}+y_{456}Y_{246}, \\ Z_{26}= & -y_{125}Y_{123}+y_{136}Y_{123}+y_{146}Y_{124} +y_{156}Y_{125}+y_{145}Y_{134}-y_{156}Y_{136} \\ & +y_{245}Y_{234}-y_{346}Y_{234}-y_{356}Y_{235} -y_{256}Y_{236}-y_{456}Y_{245}+y_{456}Y_{346}, \\ Z_{31}= & y_{126}Y_{124}+y_{136}Y_{134}-y_{156}Y_{145} +y_{124}Y_{234}+y_{236}Y_{234}+y_{125}Y_{235} \\ & +y_{126}Y_{236}-y_{256}Y_{245}-y_{145}Y_{345} -y_{356}Y_{345}-y_{146}Y_{346}-y_{156}Y_{356}, \end{array} \end{equation} \[ \begin{array} [c]{rl} Z_{32}= & y_{126}Y_{125}-y_{124}Y_{134}-y_{125}Y_{135} +y_{136}Y_{135}-y_{126}Y_{136}+y_{146}Y_{145}\\ & \multicolumn{1}{r}{+y_{236}Y_{235}+y_{246}Y_{245}-y_{245}Y_{345} +y_{346}Y_{345}-y_{246}Y_{346}-y_{256}Y_{356},}\\ Z_{33}= & -y_{123}Y_{123}+y_{126}Y_{126}-y_{134}Y_{134} -y_{135}Y_{135}+y_{146}Y_{146}+y_{156}Y_{156}\\ & \multicolumn{1}{r}{-y_{234}Y_{234}-y_{235}Y_{235}+y_{246}Y_{246} +y_{256}Y_{256}-y_{345}Y_{345}+y_{456}Y_{456},}\\ Z_{36}= & -y_{126}Y_{123}+y_{146}Y_{134}+y_{156}Y_{135}+y_{246}Y_{234} +y_{256}Y_{235}-y_{456}Y_{345},\\ Z_{41}= & -y_{123}Y_{234}+y_{125}Y_{245}+y_{126}Y_{246}+y_{135}Y_{345} +y_{136}Y_{346}-y_{156}Y_{456},\\ Z_{42}= & y_{123}Y_{134}-y_{125}Y_{145}-y_{126}Y_{146}-y_{123}Y_{235} -y_{124}Y_{245}+y_{126}Y_{256}\\ & \multicolumn{1}{r}{-y_{134}Y_{345}+y_{235}Y_{345}+y_{236}Y_{346} +y_{136}Y_{356}+y_{146}Y_{456}-y_{256}Y_{456},}\\ Z_{43}= & -y_{123}Y_{124}-y_{135}Y_{145}-y_{136}Y_{146}-y_{123}Y_{236} -y_{235}Y_{245}-y_{124}Y_{246}\\ & \multicolumn{1}{r}{-y_{236}Y_{246}-y_{125}Y_{256}-y_{134}Y_{346} -y_{135}Y_{356}-y_{145}Y_{456}-y_{356}Y_{456},}\\ Z_{52}= & y_{123}Y_{135}+y_{124}Y_{145}-y_{126}Y_{156}-y_{234}Y_{345} +y_{236}Y_{356}+y_{246}Y_{456},\\ Z_{53}= & -y_{123}Y_{125}+y_{123}Y_{136}+y_{134}Y_{145}+y_{124}Y_{146} +y_{125}Y_{156}-y_{136}Y_{156}\\ & +y_{234}Y_{245}-y_{236}Y_{256}-y_{234}Y_{346}-y_{235}Y_{356}-y_{245} Y_{456}+y_{346}Y_{456},\\ Z_{63}= & -y_{123}Y_{126}+y_{134}Y_{146}+y_{135}Y_{156}+y_{234}Y_{246} +y_{235}Y_{256}-y_{345}Y_{456}, \end{array} \] where $Y_{abc}=\frac{\partial }{\partial y_{abc}}$, $1\leq a<b<c\leq6$, is the standard basis of derivations. Moreover, the invariant functions $I_1$ and $I_2$ are algebraically independent. In fact, by using the formulas for $I_1$ and $I_2$ in the Appendix, after a computation, it follows that the determinant \[ \left\vert \begin{array} [c]{cc} dI_1(Y_{123}) & dI_1(Y_{126})\\ dI_2(Y_{123}) & dI_2(Y_{126}) \end{array} \right\vert \] at the point \[ \begin{array} [c]{ccccccc} \lambda _{123}=1, & \lambda _{124}=0, & \lambda _{125}=1, & \lambda _{126}=0, & \lambda _{134}=1, & \lambda _{135}=0, & \lambda _{136}=0,\\ \lambda _{145}=0, & \lambda _{146}=0, & \lambda _{156}=0, & \lambda _{234}=0, & \lambda _{235}=0, & \lambda _{236}=0, & \lambda _{245}=0,\\ \lambda _{246}=0, & \lambda _{256}=0, & \lambda _{345}=0, & \lambda _{346}=0, & \lambda _{356}=0, & \lambda _{456}=1, & \end{array} \] takes the value $2^4\cdot 3^2$. As the differentials $dI_1$ and $dI_2$ vanish over all the vector fields $Z_{11},\dotsc,Z_{63}$, we deduce that the generic rank of $\mathcal{M}$ is $\leq 18$. Moreover, it is easy to obtain values of the variables $y_{abc}$, $1\leq a<b<c\leq 6$, for which the matrix of $Z_{11},\dotsc,Z_{63}$ in the basis $Y_{abc}$, $1\leq a<b<c\leq6$, is $18$, and thus we can conclude the proof. \end{proof} \begin{corollary} If $\mathbb{F}$ is an algebraically closed field of characteristic zero, then $\mathbb{F}[\wedge ^3 V^\ast ]^{Sp(\Omega )} =\mathbb{F}[I_1,I_2]$. \end{corollary} \begin{proof} The result is a direct consequence of Theorem \ref{theorem} and \cite[\textsc{Th\'eor\`{e}me} 1-I]{KPV}. \end{proof} \section{Normal forms and invariants} In the series of papers \cite{BK1}, \cite{BK2}, \cite{BK3}, \cite{BK4} and \cite{BK5}, the normal forms for equivalence classes of trivectors over a $6$-dimensional vector space over an arbitrary field $\mathbb{F}$ of characteristic distinct from $2$ and $3$ under the symplectic group, is given. According to \cite[Theorem 2.1]{BK5} every non-zero trivector of $V$ is equivalent with (at least) one of the following trivectors: \[ \begin{array} [c]{ll} \chi _{A_1}=\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast , & \chi _{A_2}=\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast , \end{array} \] \[ \begin{array} [c]{rl} \chi _{B_1}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{B_2}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast,\\ \chi _{B_3}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{B_4}(\lambda )= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\lambda\bar{e}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{B_5}(\lambda )= & \lambda\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge (\bar{e}_2^\ast -\bar{e}_3^\ast )\wedge (\bar{f}_2^\ast +\bar{f}_3^\ast ), \end{array} \] \[ \begin{array} [c]{rl} \chi _{C_1}(\lambda )= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\lambda\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{C_2}(\lambda)= & \bar{f}_1^\ast \wedge (\bar{e}_2^\ast +\bar{e}_3^\ast ) \wedge (\bar{f}_2^\ast -\bar{f}_3^\ast ) +\lambda \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{C_3}(\lambda)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge\bar{f}_2^\ast +\lambda\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge\bar{f}_3^\ast ,\\ \chi _{C_4}(\lambda)= & \bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge(\bar{e}_2^\ast +\bar{f}_3^\ast )+\lambda\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{C_5}(\lambda )= & \bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge (\bar{f}_2^\ast +\bar{f}_3^\ast ) +\lambda \bar{e}_2^\ast \wedge \bar{f}_3^\ast \wedge (\bar{f}_1^\ast +\bar{e}_3^\ast ),\\ \chi _{C_6}(\lambda,\varepsilon)= & \bar{f}_1^\ast \wedge (\bar{e}_2^\ast +\bar{e}_3^\ast ) \wedge (\bar{f}_2^\ast +\varepsilon \bar{f}_3^\ast) +\lambda \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast , \end{array} \] \[ \begin{array} [c]{rl} \chi _{D_1}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_2^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_1^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{D_2}(\lambda)= & \lambda \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\bar{e}_2^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_1^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{D_3}(\lambda _1,\lambda _2)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\lambda _1\bar{e}_2^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_1^\ast +\lambda _2\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{D_4}(\lambda _1,\lambda _2)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\lambda _1\bar{e}_2^\ast \wedge \bar{e}_3^\ast \wedge (\bar{f}_1^\ast +\bar{f}_3^\ast ) +\lambda _2\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{D_5}(\lambda)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\lambda \bar{e}_2^\ast \wedge \bar{e}_3^\ast \wedge (\bar{f}_1^\ast +\bar{f}_2^\ast +\bar{f}_3^\ast ) -\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{D_6}= & -\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_2^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_1^{\ast }+\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_3^\ast , \end{array} \] \[ \begin{array} [c]{rl} \chi _{E_1}(a,b,h_1,h_2,h_3) = & 2\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast \\ & \multicolumn{1}{r}{+a\left( h_1\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +h_2\bar{e}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast +h_3\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast \right) }\\ & \multicolumn{1}{r}{+(a^2+2b)\left( h_1h_2\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast +h_1h_3\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast \right. }\\ & \multicolumn{1}{r}{\left. +h_2h_3\bar{e}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast \right) +h_1h_2h_3(a^2+3b)\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast ,} \end{array} \] \[ \begin{array} [c]{rl} \chi _{E_2}(a,b,k)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \\ & \multicolumn{1}{r}{+k\left( \bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast -b\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast +a\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \right) ,\medskip}\\ \chi _{E_3}(a,b,k,h)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast +k\left( \bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast -b\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast \right. \\ & \multicolumn{1}{r}{\left. +a\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \right) +h\bar{e}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast ,} \end{array} \] \[ \begin{array} [c]{rl} \chi _{E_4}(a,b,k,h_1,h_2)= & \left[ 1-h_1h_2(a^2+4b)\right] \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast \\ & +\left[ 1+h_1h_2(a^2+4b)\right] \bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \\ & +k\left[ \bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast -b(1-h_1h_2(a^2+4b)) \bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast \right. \\ & \left. +a\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \right] +h_1\left[ 1-h_1h_2(a^2+4b)\right] \bar{e}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast \\ & \multicolumn{1}{r}{+(a^2+4b)h_2\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast ,}\\ & h_1h_2(a^2+4b)\neq 1, \end{array} \] \[ \begin{array} [c]{rl} \chi _{E_5}(a,b,k)= & \bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +2\bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_2^\ast -a\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast \\ & \multicolumn{1}{r}{+a\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast +a\bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +(a^2+b)\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{e}_3^\ast }\\ & \multicolumn{1}{r}{+k\left( a\bar{e}_1^\ast \wedge \bar{f}_2^{\ast }\wedge \bar{e}_3^\ast -\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \right) ,} \end{array} \] (As we are assuming that $\mathbb{F}$ has characteristic distinct from $2$ and $3$, we do not consider the trivectors of types $(E1^\prime )$, $(E2^\prime )$, and $(E3^\prime )$, because they only exist when the characteristic is $2$; indeed, for such trivectors one needs separable quadratic extensions of $\mathbb{F}$, which can only exist in the case of characteristic $2$.) With the same notations as above, we have $\bar{e}_1^\ast =v^1$, $\bar{e}_2^\ast =v^2$, $\bar{e}_3^\ast =v^3$, $\bar{f}_1^\ast =v^4$, $\bar{f}_2^\ast =v^{5}$, $\bar{f}_3^\ast =v^6$. In this section we study the evaluation map $(I_1,I_2)\colon \wedge ^3V^\ast /GL(V) \to \mathbb{F}^2$, by using the normal forms given in \cite{BK5}. This throws light on the quotient space. We have \[ \begin{array}{llll} I_j(\chi _{A_h})=0, \quad I_j(\chi _{B_i})=0, & h,j=1,2, & 1\leq i\leq 5,\\ I_1(\chi _{C_h})=0, & I_2(\chi _{C_h})=-2^3\cdot 3^2\lambda ^2, & 1\leq h\leq 2, & \\ I_1(\chi _{C_h})=\lambda ^2, & I_2(\chi _{C_h}) =2^4\cdot 3\lambda ^2, & 3\leq h\leq 4, & \\ I_1(\chi _{C_5})=0, & I_2(\chi _{C_5}) =-2^3\cdot 3^2\lambda ^2, & & \\ I_1(\chi _{C_6})=\lambda ^2 \varepsilon (\varepsilon +1), & I_2(\chi _{C_6})=2^3\cdot 3\lambda ^2 \varepsilon (2\varepsilon +5), & \varepsilon \neq 0,-1, & \\ I_j(\chi _{D_h})=0, & h\in \{ 1,3,4,5,6\} , & 1\leq j\leq 2, & \\ I_1(\chi _{D_2})=\lambda , & I_2(\chi _{D_2}) =2^3\cdot 3\cdot 5\lambda , & & \\ I_1(\chi _{E_1})=0, & & & \end{array} \] \begin{align} I_2(\chi _{E_1})&=-2^3\cdot 3^2(h_1h_2h_3)^2 \left( 2^2 (a^2+3b)^2(1-2a)+(a^2+2b)^2(5a^2+2^4 b) \right) , \quad \\ I_1(\chi _{E_{j}})&=k^2(a^2+4 b), \qquad \qquad I_2(\chi _{E_j}) =2^4\cdot 3k^2(a^2+4b), \quad 2\leq j\leq 3,\\ I_1(\chi _{E_4}) & =k^2 (a^2+4b)\left( 1-(a^2+4 b) h_1 h_2 \right) , \qquad \qquad \qquad \qquad\\ I_2(\chi _{E_4}) & =2^3\cdot 3 k^2 (a^2+4 b) \left( 1-(a^2+4 b)h_1 h_2 \right) (2+3 (a^2 + 4 b) h_1 h_2), \\ I_1(\chi _{E_5})&=0, \qquad I_2(\chi _{E_5})=-2^3 \cdot 3^2 k^2 (a^2+4 b) \qquad \qquad \qquad \end{align} When $\mathbb{F}$ is an algebraically closed field of characteristic distinct from $2$, Popov \cite{P} gave another alternative for the normal forms for equivalence classes of trivectors over a $6$-dimensional vector space over $\mathbb{F}$ under the symplectic group. We list these normal forms, following the notations in \cite[Theorem 2.1]{BK5}, with $q,p\in\mathbb{F}^\ast $: \[ \begin{array} [c]{rl} \chi _{P_1}= & 0,\\ \chi _{P_2}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_3}(q)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +q\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_4}(q)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +q\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_5}(q)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e} _3^\ast +q\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \\ & +\bar{f}_2^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast +\bar{f}_2^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_6}(q,p)= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +q\bar{f}_1^\ast \wedge \bar{f}_2^\ast \wedge \bar{f}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast \\ & +p\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +p\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_{7}}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast ,\\ \chi _{P_{8}}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_{9}}= & \bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{e} _3^\ast +\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast+\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast , \end{array} \] \[ \begin{array} [c]{rl} \chi _{P_{10}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_{11}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast \\ & +\bar{e}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_{12}}(q)= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e} _3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +q\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast \\ & \multicolumn{1}{r}{+q\bar{e}_3^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,}\\ \chi _{P_{13}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +\bar{f}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast \\ & \multicolumn{1}{r}{+\bar{f}_1^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast ,}\\ \chi _{P_{14}}(q)= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar {e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast +q\bar{f}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast \\ & \multicolumn{1}{r}{+q\bar{f}_3^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,} \end{array} \] \[ \begin{array} [c]{rl} \chi _{P_{15}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast ,\\ \chi _{P_{16}}(q)= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +q\bar{f}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast +q\bar{f}_3^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{P_{17}}(q)= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +q\bar{e}_3^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast +q\bar{e}_3^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast ,\\ \chi _{P_{18}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast ,\\ \chi _{P_{19}}= & \bar{e}_1^\ast \wedge \bar{f}_1^\ast \wedge \bar{e}_3^\ast +\bar{f}_2^\ast \wedge \bar{e}_2^\ast \wedge \bar{e}_3^{\ast }+\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_2^\ast +\bar{e}_1^\ast \wedge \bar{e}_2^\ast \wedge \bar{f}_3^\ast \\ & +\bar{e}_2^\ast \wedge \bar{e}_1^\ast \wedge \bar{f}_1^\ast +\bar{e}_2^\ast \wedge \bar{e}_3^\ast \wedge \bar{f}_3^\ast . \end{array} \] Then, we obtain \[ \begin{array} [c]{lll} I_{j}(\chi _{P_h})=0, & 1\leq j\leq2, & h\in \{ 2,7,8,9,10,11,12,13,15,17,18,19\} , \end{array} \] \[ \begin{array} [c]{lll} I_1(\chi _{P_h})=0, & I_2(\chi _{P_h})=-2^3\cdot 3^2q^2, & h\in\{ 3,4,5\} ,\\ I_1(\chi _{P_6})=-2^2pq, & I_2(\chi _{P_6}) =-2^3\cdot 3q(3q+2^3p), & \\ I_1(\chi _{P_h})=2^2q^2, & I_2(\chi _{P_h})=2^6\cdot 3 q^2, & h\in \{ 14,16\} . \end{array} \] \subsubsection{Acknowledgment} The authors wish to thank the unknown referee for his careful and thorough read of our manuscript and his valuable comments. \section{Appendix} \begin{align} I_1 & =y_{135}y_{234}y_{256}^2-y_{126}y_{234}y_{356}^2-y_{134} y_{136}y_{236}y_{456}-y_{126}y_{134}y_{145}y_{346}\\ & +2y_{126}y_{135}y_{245}y_{346}+y_{124}y_{135}y_{245}y_{256}-y_{126} y_{145}y_{235}y_{245}\\ & -3y_{126}y_{145}y_{235}y_{346}+y_{124}y_{135}y_{146}y_{346}+3y_{124} y_{135}y_{256}y_{346}\\ & -y_{124}y_{135}y_{146}y_{245}+y_{145}y_{156}y_{234}y_{236}+2y_{135} y_{146}y_{234}y_{256}\\ & +y_{124}y_{156}y_{234}y_{356}+y_{156}y_{235}y_{236}y_{346}-y_{156} y_{235}y_{236}y_{245}\\ & -y_{146}y_{156}y_{234}y_{235}+2y_{126}y_{145}y_{234}y_{356}-y_{156} y_{234}y_{236}y_{356}\\ & +y_{136}y_{156}y_{234}y_{346}-y_{156}y_{234}y_{235}y_{256}+y_{136} y_{156}y_{234}y_{245}\\ & -y_{124}y_{156}y_{235}y_{245}+y_{146}^2y_{235}^2+y_{145}^2y_{236} ^2+y_{124}^2y_{356}^2\\ & +2y_{124}y_{156}y_{236}y_{345}-3y_{124}y_{156}y_{235}y_{346}-y_{125} y_{126}y_{345}y_{346}\\ & +y_{123}y_{136}y_{346}y_{456}-y_{123}y_{145}y_{236}y_{456}-y_{125} y_{126}y_{245}y_{345}\\ & +2y_{125}y_{146}y_{235}y_{346}-2y_{136}y_{146}y_{235}y_{245}-2y_{136} y_{145}y_{236}y_{245}\\ & -3y_{123}y_{146}y_{245}y_{356}-2y_{145}y_{146}y_{235}y_{236}-3y_{125} y_{134}y_{236}y_{456}\\ & +2y_{125}y_{136}y_{234}y_{456}+2y_{123}y_{156}y_{245}y_{346}+2y_{123} y_{145}y_{246}y_{356}\\ & -3y_{125}y_{146}y_{236}y_{345}-y_{123}y_{145}y_{146}y_{346}+y_{123} y_{145}y_{146}y_{245}\\ & -y_{123}y_{145}y_{245}y_{256}-y_{123}y_{146}y_{346}y_{356}-y_{125} y_{234}y_{256}y_{356}\\ & +y_{134}y_{136}y_{145}y_{246}+y_{134}y_{136}y_{246}y_{356}-y_{125} y_{145}y_{234}y_{256}\\ & -3y_{125}y_{146}y_{234}y_{356}-3y_{123}y_{145}y_{256}y_{346}+y_{125} y_{145}y_{146}y_{234}\\ & +y_{123}y_{256}y_{346}y_{356}-y_{123}y_{245}y_{256}y_{356}+y_{123} y_{136}y_{245}y_{456}\\ & +y_{123}y_{134}y_{256}y_{456}+y_{135}y_{236}y_{245}y_{256}+2y_{134} y_{136}y_{245}y_{256}\\ & +y_{136}^2y_{245}^2+y_{125}^2y_{346}^2-y_{125}y_{236}y_{256} y_{345}-y_{135}y_{236}y_{256}y_{346}\\ & +3y_{135}y_{146}y_{236}y_{245}+y_{123}y_{235}y_{256}y_{456}-y_{125} y_{235}y_{236}y_{456}\\ & +y_{135}y_{146}y_{236}y_{346}+y_{123}y_{236}y_{356}y_{456}-3y_{124} y_{136}y_{235}y_{456}\\ & +y_{124}y_{125}y_{134}y_{456}+2y_{124}y_{136}y_{245}y_{356}-2y_{124} y_{145}y_{236}y_{356}\\ & -2y_{124}y_{134}y_{256}y_{356}+y_{125}y_{235}y_{246}y_{356}-y_{125} y_{134}y_{145}y_{246}\\ & +3y_{125}y_{134}y_{246}y_{356}+2y_{124}y_{135}y_{236}y_{456}-3y_{124} y_{136}y_{256}y_{345}\\ & +y_{125}y_{145}y_{235}y_{246}+y_{123}y_{146}y_{235}y_{456}+2y_{123} y_{146}y_{256}y_{345}\\ & +y_{123}y_{125}y_{346}y_{456}-y_{124}y_{136}y_{146}y_{345}+y_{123} y_{134}y_{146}y_{456}\\ & +y_{136}y_{236}y_{256}y_{345}+y_{136}y_{235}y_{236}y_{456}-y_{124} y_{126}y_{145}y_{345}\\ & +y_{125}y_{156}y_{234}y_{245}-y_{125}y_{135}y_{246}y_{346}+y_{126} y_{235}y_{346}y_{356}\\ & -y_{125}y_{135}y_{245}y_{246}-2y_{125}y_{136}y_{245}y_{346}-2y_{134} y_{146}y_{235}y_{256} \end{align} \begin{align} & +y_{134}^2y_{256}^2+2y_{134}y_{145}y_{236}y_{256}-y_{135}y_{235} y_{246}y_{256}+y_{136}y_{234}y_{256}y_{356}\\ & -y_{136}y_{145}y_{146}y_{234}-y_{126}y_{235}y_{245}y_{356}-y_{136} y_{146}y_{236}y_{345}\\ & -y_{136}y_{146}y_{234}y_{356}-3y_{136}y_{145}y_{234}y_{256}+y_{125} y_{156}y_{234}y_{346}\\ & +y_{126}y_{145}y_{236}y_{345}-y_{126}y_{136}y_{245}y_{345}+y_{126} y_{134}y_{146}y_{345} \end{align} \begin{align} & +2y_{125}y_{136}y_{246}y_{345}+y_{126}y_{134}y_{256}y_{345}+y_{126} y_{235}y_{256}y_{345}\\ & -y_{134}y_{135}y_{246}y_{256}-y_{126}y_{236}y_{345}y_{356}-y_{124} y_{135}y_{246}y_{356}\\ & -y_{124}y_{145}y_{156}y_{234}+y_{126}y_{146}y_{235}y_{345}-y_{126} y_{136}y_{345}y_{346}\\ & +y_{124}y_{135}y_{145}y_{246}-y_{124}y_{134}y_{156}y_{346}-y_{134} y_{156}y_{234}y_{256}\\ & +2y_{134}y_{156}y_{235}y_{246}+2y_{125}y_{145}y_{236}y_{346}-2y_{125} y_{134}y_{256}y_{346}\\ & +2y_{124}y_{146}y_{235}y_{356}-3y_{134}y_{156}y_{236}y_{245}-y_{134} y_{135}y_{146}y_{246}\\ & -y_{135}y_{136}y_{245}y_{246}-y_{134}y_{146}y_{156}y_{234}-y_{135} y_{145}y_{236}y_{246}\\ & -y_{135}y_{136}y_{246}y_{346}-y_{134}y_{156}y_{236}y_{346}+2y_{126} y_{134}y_{235}y_{456}\\ & -2y_{124}y_{125}y_{346}y_{356}+3y_{136}y_{145}y_{235}y_{246}+y_{126} y_{134}y_{145}y_{245}\\ & -3y_{126}y_{134}y_{245}y_{356}-y_{126}y_{134}y_{346}y_{356}-y_{124} y_{134}y_{136}y_{456}\\ & -y_{124}y_{125}y_{235}y_{456}+y_{124}y_{125}y_{146}y_{345}-y_{124} y_{125}y_{256}y_{345}\\ & -y_{136}y_{235}y_{246}y_{356}+y_{123}y_{124}y_{145}y_{456}-y_{123} y_{124}y_{356}y_{456}\\ & +y_{125}^2y_{246}y_{345}+y_{123}y_{256}^2y_{345}+y_{123}y_{156} y_{245}^2+y_{126}y_{135}y_{245}^2\\ & -y_{135}y_{236}^2y_{456}+y_{135}y_{146}^2y_{234}+y_{125}^2 y_{234}y_{456}-y_{123}y_{145}^2y_{246}\\ & -y_{156}y_{236}^2y_{345}+y_{156}y_{235}^2y_{246}+y_{134}^2 y_{156}y_{246}+y_{123}y_{156}y_{346}^2\\ & +y_{126}y_{135}y_{346}^2-y_{124}^2y_{156}y_{345}+y_{123}y_{146} ^2y_{345}+y_{126}y_{235}^2y_{456}\\ & +y_{126}y_{134}^2y_{456}-y_{124}^2y_{135}y_{456}+y_{136}^2 y_{246}y_{345}-y_{123}y_{246}y_{356}^2\\ & -y_{126}y_{145}^2y_{234}+y_{136}^2y_{234}y_{456}-y_{135}y_{146} y_{235}y_{246}\\ & +y_{123}y_{125}y_{245}y_{456}+y_{124}y_{134}y_{156}y_{245}+y_{135} y_{236}y_{246}y_{356}\\ & +y_{124}y_{126}y_{345}y_{356}. \end{align} \begin{align} I_2 & =24\left( -5y_{124}y_{136}y_{146}y_{345}-5y_{126}y_{145} y_{235}y_{245}-5y_{124}y_{156}y_{235}y_{245}\right. \\ & -4y_{136}y_{146}y_{235}y_{245}-5y_{136}y_{146}y_{236}y_{345}-5y_{136} y_{146}y_{234}y_{356}\\ & -5y_{124}y_{125}y_{256}y_{345}-5y_{125}y_{126}y_{245}y_{345}-2y_{125} y_{136}y_{246}y_{345}\\ & +y_{125}y_{126}y_{345}y_{346}-5y_{124}y_{135}y_{146}y_{245}+5y_{124} y_{135}y_{245}y_{256}\\ & -5y_{125}y_{135}y_{245}y_{246}+y_{135}y_{136}y_{245}y_{246}-2y_{126} y_{135}y_{245}y_{346}\\ & +4y_{124}y_{146}y_{235}y_{356}-5y_{126}y_{235}y_{245}y_{356}-5y_{136} y_{235}y_{246}y_{356}\\ & +6y_{123}y_{156}y_{234}y_{456}+6y_{135}y_{156}y_{234}y_{246}+6y_{126} y_{156}y_{234}y_{345}\\ & +5y_{126}y_{134}y_{146}y_{345}+6y_{123}y_{126}y_{345}y_{456}+6y_{126} y_{135}y_{246}y_{345}\\ & -y_{123}y_{136}y_{245}y_{456}+4y_{124}y_{136}y_{245}y_{356}+y_{126} y_{136}y_{245}y_{345}\\ & -4y_{134}y_{146}y_{235}y_{256}+12y_{123}y_{156}y_{246}y_{345} +12y_{126}y_{135}y_{234}y_{456}\\ & +5y_{123}y_{146}y_{145}y_{245}-5y_{123}y_{145}y_{245}y_{256}-2y_{126} y_{134}y_{235}y_{456}\\ & +5y_{136}y_{235y_{236}}y_{456}-3y_{124}y_{136}y_{235}y_{456}-5y_{125} y_{235}y_{236}y_{456}\\ & -2y_{123}y_{146}y_{256}y_{345}-3y_{123}y_{146}y_{245}y_{356}-5y_{123} y_{245}y_{256}y_{356}\\ & -5y_{134}y_{136}y_{236}y_{456}+5y_{123}y_{256}y_{346}y_{356}-5y_{124} y_{134}y_{136}y_{456}\\ & -2y_{124}y_{135}y_{236}y_{456}-2y_{123}y_{145}y_{246}y_{356}+5y_{124} y_{125}y_{134}y_{456}\\ & -3y_{123}y_{145}y_{256}y_{346}-3y_{125}y_{134}y_{236}y_{456}-2y_{123} y_{156}y_{245}y_{346}\\ & -2y_{125}y_{136}y_{234}y_{456}-5y_{124}y_{125}y_{456}y_{235}+5y_{123} y_{125}y_{456}y_{245}\\ & -4y_{124}y_{134}y_{256}y_{356}-5y_{123}y_{145}y_{146}y_{346}-5y_{123} y_{146}y_{346}y_{356}\\ & +5y_{123}y_{136}y_{456}y_{346}-5y_{126}(y_{145})^2y_{234}-5y_{156} (y_{236})^2y_{345}\\ & -5(y_{124})^2y_{156}y_{345}-5y_{126}y_{234}(y_{356})^2+5y_{135} (y_{146})^2y_{234}\\ & +\!5y_{156}(y_{235})^2y_{246}+5(y_{134})^2y_{156}y_{246}+2(y_{124} )^2(y_{356})^2\\ & -4(y_{156})^2(y_{234})^2-3(y_{126})^2(y_{345})^2-3(y_{123} )^2(y_{456})^2\\ & -3(y_{135})^2(y_{246})^2+2(y_{136})^2(y_{245})^2+5y_{135} y_{234}(y_{256})^2\\ & +5y_{125}^2y_{345}y_{246}+5y_{135}y_{245}^2y_{126}+5y_{126}y_{346} ^2y_{135}\\ & +5(y_{136})^2y_{246}y_{345}+5y_{123}y_{156}(y_{245})^2+5(y_{136} )^2y_{234}y_{456}\\ & +5y_{126}(y_{235})^2y_{456}+5y_{123}(y_{256})^2y_{345}-5y_{123} y_{246}(y_{356})^2\\ & -5y_{135}(y_{236})^2y_{456}-5y_{123}(y_{145})^2y_{246}-5(y_{124} )^2y_{135}y_{456}\\ & +5(y_{125})^2y_{234}y_{456}+5y_{126}(y_{134})^2y_{456}+5y_{123} (y_{146})^2y_{345}\\ & +5y_{123}y_{156}(y_{346})^2+5y_{123}y_{235}y_{256}y_{456}+5y_{126} y_{256}y_{235}y_{345}\\ & +5y_{123}y_{134}y_{146}y_{456}+y_{123}y_{145}y_{236}y_{456}+y_{123} y_{124}y_{356}y_{456} \end{align} \begin{align} & +6y_{123}y_{135}y_{246}y_{456}+y_{135}y_{145}y_{236}y_{246}+y_{124} y_{135}y_{246}y_{356}\\ & +4y_{125}y_{145}y_{236}y_{346}-y_{123}y_{125}y_{346}y_{456}+y_{125} y_{135}y_{246}y_{346}\\ & -5y_{125}y_{134}y_{145}y_{246}+5y_{125}y_{145}y_{235}y_{246}-5y_{125} y_{145}y_{234}y_{256}\\ & -5y_{134}y_{156}y_{236}y_{346}+5y_{135}y_{146}y_{236}y_{346}-4y_{125} y_{134}y_{256}y_{346}\\ & +5y_{124}y_{135}y_{145}y_{246}-4y_{124}y_{145}y_{236}y_{356}+5y_{123} y_{236}y_{356}y_{456}\\ & +5y_{135}y_{236}y_{246}y_{356}+5y_{123}y_{124}y_{145}y_{456}-y_{123} y_{146}y_{235}y_{456}\\ & -y_{123}y_{134}y_{256}y_{456}-y_{126}y_{146}y_{235}y_{345}-y_{126} y_{134}y_{256}y_{345}\\ & -5y_{126}y_{134}y_{145}y_{346}-5y_{126}y_{136}y_{345}y_{346}-5y_{126} y_{134}y_{346}y_{356}\\ & -5y_{135}y_{136}y_{246}y_{346}+5y_{134}y_{136}y_{246}y_{356}+4y_{134} y_{145}y_{236}y_{256}\\ & -5y_{125}y_{236}y_{256}y_{345}-5y_{135}y_{236}y_{256}y_{346}+5y_{125} y_{156}y_{234}y_{245}\\ & -y_{136}y_{156}y_{234}y_{245}-y_{125}y_{156}y_{234}y_{346}+5y_{136} y_{156}y_{234}y_{346}\\ & +5y_{125}y_{145}y_{146}y_{234}-y_{145}y_{156}y_{234}y_{236}-5y_{124} y_{145}y_{156}y_{234}\\ & -3y_{136}y_{145}y_{234}y_{256}-2y_{126}y_{145}y_{234}y_{356}-3y_{125} y_{146}y_{236}y_{345}\\ & -y_{126}y_{145}y_{236}y_{345}-2y_{124}y_{156}y_{236}y_{345}+5y_{136} y_{236}y_{345}y_{256}\\ & -5y_{126}y_{236}y_{345}y_{356}+4y_{125}y_{146}y_{235}y_{346}-3y_{126} y_{145}y_{235}y_{346}\\ & +5y_{156}y_{236}y_{235}y_{346}-3y_{124}y_{156}y_{235}y_{346}+5y_{126} y_{235}y_{346}y_{356}\\ & +5y_{245}y_{134}y_{126}y_{145}-3y_{245}y_{134}y_{236}y_{156}+5y_{245} y_{134}y_{156}y_{124}\\ & +4y_{245}y_{134}y_{136}y_{256}-3y_{245}y_{134}y_{126}y_{356}+5y_{345} y_{124}y_{146}y_{125}\\ & -5y_{345}y_{124}y_{126}y_{145}-3y_{345}y_{124}y_{136}y_{256}-y_{345} y_{124}y_{126}y_{356}\\ & -3y_{356}y_{234}y_{146}y_{125}-5y_{356}y_{234}y_{236}y_{156}-y_{356} y_{234}y_{156}y_{124}\\ & +5y_{356}y_{234}y_{136}y_{256}-5y_{146}y_{234}y_{136}y_{145}+y_{146} y_{234}y_{156}y_{235}\\ & -5y_{146}y_{234}y_{156}y_{134}-2y_{146}y_{234}y_{256}y_{135}+y_{235} y_{246}y_{135}y_{146}\\ & +3y_{235}y_{246}y_{136}y_{145}-2y_{235}y_{246}y_{156}y_{134}-5y_{235} y_{246}y_{256}y_{135}\\ & +5y_{125}y_{235}y_{246}y_{356}+3y_{135}y_{146}y_{236}y_{245}-4y_{136} y_{145}y_{236}y_{245}\\ & -5y_{156}y_{235}y_{236}y_{245}+5y_{135}y_{236}y_{245}y_{256}-5y_{134} y_{135}y_{146}y_{246}\\ & +5y_{134}y_{136}y_{145}y_{246}+y_{134}y_{135}y_{246}y_{256}+3y_{125} y_{134}y_{246}y_{356}\\ & +5y_{124}y_{135}y_{146}y_{346}-5y_{124}y_{134}y_{156}y_{346}+3y_{124} y_{135}y_{346}y_{256}\\ & -4y_{346}y_{124}y_{356}y_{125}-5y_{256}y_{234}y_{156}y_{235}+y_{134} y_{156}y_{234}y_{256}\\ & -5y_{125}y_{234}y_{256}y_{356}-4y_{145}y_{146}y_{236}y_{235}-4y_{125} y_{136}y_{245}y_{346}\\ & \left. +2(y_{146})^2(y_{235})^2+2(y_{145})^2(y_{236})^2 +2(y_{134})^2(y_{256}^2)+2(y_{125})^2(y_{346})^2\right) . \end{align*} \begin{thebibliography}{9} \bibitem {BK1} B. De Bruyn, M. Kwiatkowski, \emph{On the trivectors of a }$6$\emph{-dimensional symplectic vector space}, Linear Alg.\ Appl.\ \textbf{435} (2011), 289--306. \bibitem {BK2} B. De Bruyn, M. Kwiatkowski, \emph{On the trivectors of a }$6$\emph{-dimensional symplectic vector space. II}, Linear Alg.\ Appl.\ \textbf{437} (2012), 1215--1233. \bibitem {BK3} B. De Bruyn, M. Kwiatkowski, \emph{On the trivectors of a }$6$\emph{-dimensional symplectic vector space. III}, Linear Alg.\ Appl.\ \textbf{438} (2013), 374--398. \bibitem {BK4} B. De Bruyn, M. Kwiatkowski, \emph{On the trivectors of a }$6$\emph{-dimensional symplectic vector space. IV}, Linear Alg.\ Appl.\ \textbf{438} (2013), 2405--2429. \bibitem {BK5} B. De Bruyn, M. Kwiatkowski, \emph{The classification of the trivectors of a }$6$\emph{-dimensional symplectic space: Summary, consequences and connections}, Linear Alg.\ Appl.\ \textbf{438} (2013), 3516--3529. \bibitem {Fogarty} J. Fogarty, \emph{Invariant theory}, W. A. Benjamin, Inc., New York, Amsterdam, 1969. \bibitem {KPV} V. G. Kac, V. L. Popov, E. B. Vinberg, \emph{Sur les groupes lin\'eaires alg\'ebriques dont l'alg\`{e}bre des invariants est libre}, C. R. Acad. Sci. Paris S\'er. A-B \textbf{283} (1976), no. 12, Ai, A875--A878. \bibitem {P} V. L. Popov, \emph{Classification of spinors of dimension fourteen}, Trans.\ Mosc.\ Math. Soc.\ \textbf{1} (1980), 181--232. \end{thebibliography} \noindent\textbf{Authors' addresses} \smallskip \noindent(J.M.M.) \textsc{Instituto de Tecnolog\'{\i}as F\'{\i}sicas y de la Informaci\'on, CSIC, C/ Serrano 144, 28006-Madrid, Spain.} \noindent\emph{E-mail:\/} \texttt{[email protected]} \medskip \noindent(L.M.P.C.) \textsc{Departamento de Matem\'atica Aplicada a las T.I.C., E.T.S.I. Sistemas Inform\'aticos, Universidad Polit\'ecnica de Madrid, Carrete\-ra de Valencia, Km.\ 7, 28031-Madrid, Spain.} \noindent\emph{E-mail:\/} \texttt{[email protected]} \end{document}
2412.03164v1	http://arxiv.org/abs/2412.03164v1	Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence	\documentclass[12pt]{article} \setlength{\bigskipamount}{5ex plus1.5ex minus 2ex} \setlength{\textheight}{24cm} \setlength{\textwidth}{16cm} \setlength{\hoffset}{-1.3cm} \setlength{\voffset}{-1.8cm} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{definition}{Definition} \newtheorem{coro}{Corollary} \newtheorem{remark}{Remark} \newtheorem{example}{Example} \newtheorem{problem}{Problem} \newtheorem{construction}{Construction} \newenvironment{proof}{\begin{trivlist} \item[\hskip\labelsep{\it Proof.}]}{$\hfill\Box$\end{trivlist}} \newcommand{\RR}{{\Bbb R}} \newcommand{\NN}{{\Bbb N}} \newcommand{\ZZ}{{\Bbb Z}} \newcommand{\wal}{{\rm wal}} \newcommand{\qed} {\hfill \Box \vspace{0.5cm}} \newcommand{\lt}{L_{2,N}} \newcommand{\disc}{D_N^{\ast}} \newcommand{\rd}{\,{\rm d}} \usepackage{latexsym,amsfonts,amsmath,amssymb} \title{Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence} \author{Josef Dick and Friedrich Pillichshammer} \date{} \begin{document} \maketitle \begin{abstract} In this short note we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important role in approximation theory, and on the other hand there is the star discrepancy of the van der Corput sequence that plays a prominent role in uniform distribution theory. Over the decades, these two quantities have been examined in great detail independently of each other and important results have been proven. Work in these areas has been carried out independently, but as we show here, they actually coincide. Interestingly, many theorems have been discovered in both areas independently, but some results have only been known in one area but not in the other. \end{abstract} \paragraph{Lebesgue Constants.} For a given system of orthonormal functions $\Phi=\{\phi_k \ : \ k =0,1,\ldots\}$ in $L_2([a,b])$ the $n$-th Lebesgue function is defined as \begin{equation} L_n^{\Phi}(u):=\int_a^b \left\|\sum_{k=0}^{n-1} \phi_k(x) \phi_k(u) \right\| \rd x\qquad \mbox{for $u \in [a,b]$.} \end{equation} It follows easily from the fact $S_n(f)(u)=\int_a^b \left(\sum_{k=0}^{n-1} \phi_k(x) \phi_k(u)\right) f(u)\rd u$ that \begin{equation} L_n^{\Phi}(u)=\sup_{\\|f\\|_{C([a,b])} \le 1} \|S_n(f)(u)\|\qquad \mbox{for $u \in [a,b]$,} \end{equation} where $\\|\cdot \\|_{C([a,b])}$ is the uniform norm and $S_n(f)(u)=\sum_{k=0}^{n-1} \widehat{f}(k) \phi_k(u)$ is the $n$-th partial sum of the Fourier expansion of $f$ with respect to the system $\Phi$. If the $n$-th Lebesgue function $L_n^{\Phi}$ is constant over $[a,b]$, then its function value is called the $n$-th Lebesgue constant, which is denoted by $L_n^{\Phi}$ or simply by $L_n$. Lebesgue functions/constants are a fundamental tool in approximation theory; see, e.g., \cite{AT,KS,SWS,zyg}. In this note we consider as an orthonormal system the Walsh functions and report a connection of the corresponding Lebesgue functions to the star discrepancy of the van der Corput sequence, which is an important quantity in uniform distribution and discrepancy theory. Both Lebesgue functions for the Walsh system and star discrepancy of van der Corput sequence are well-studied objects in literature, but so far it has not been observed that both quantities coincide. This is, at least for us, a very interesting connection between concepts from different branches of mathematics that allows to transfer results for one measure to the other and vice versa. \paragraph{Walsh functions.} For a nonnegative integer $k$ with base $2$ representation $k=k_0+k_1 2+\cdots+k_m 2^m$ the $k$-th {\it Walsh function} $\wal_k:\RR \rightarrow \RR$, periodic with period one, is defined by \begin{eqnarray} \wal_k(x):=(-1)^{x_1 k_0 +x_2 k_1+\cdots + x_{m+1} k_m},\nonumber \end{eqnarray} when $x \in [0,1)$ has (canonical) base $2$ representation $x=\frac{x_1}{2}+\frac{x_2}{2^2}+\cdots$. The system $\mathcal{W}=\{\wal_k \ : \ k=0,1,\ldots \}$ is a complete orthonormal system in $L_2([0,1])$. For information about Walsh functions see \cite{fine,SWS} or \cite[Appendix~A]{DP10}. In \cite[Section~5]{fine} Fine introduced the Walsh Dirichlet kernel $$D_n(x,u):= \sum_{k=0}^{n-1} \wal_k(x) \wal_k(u)$$ and the so-called Lebesgue functions $$L_n(x):=\int_0^1\|D_n(x,u)\| \rd u$$ (for simplicity we write $L_n(x)$ rather than $L_n^{\mathcal{W}}(x)$) for the Walsh systems and proved several interesting results. In the first place he proved that the $L_n(x)$ are in fact independent of $x$ (see \cite[p.386]{fine}) and for this reason they coincide with their Lebesgue constants $L_n$. In \cite[Theorem~IX]{fine} several properties of the Lebesgue constants $L_n$ are summarized. \begin{theorem}[Fine]\label{thm:fine} The Lebesgue constants $L_n$ of the Walsh system satisfy \begin{equation}\label{fo:LCW} L_n=\nu-\sum_{1 \le j < i \le \nu} 2^{n_i-n_{j}}, \end{equation} where $n$ is of the form $n=2^{n_1}+2^{n_2}+\cdots+2^{n_{\nu}}$ with integer exponents $n_1 > n_2 > \cdots > n_{\nu}\ge 0$. Furthermore, the following properties hold: \begin{enumerate} \item $L_{2n}=L_n$ and $L_{2n+1}=(1+L_n+L_{n+1})/2$. \item $L_n=O(\log n)$. \item $\frac{1}{n} \sum_{k=1}^n L_k =\frac{\log n}{4 \log 2}+O(1)$. \item $\limsup_{n \rightarrow \infty} \left[L_n-\left(\frac{4}{9}+\frac{\log 3}{3 \log 2}+\frac{\log n}{3 \log 2}\right)\right]=0$. \item the generating function for $L_n$ is $\sum_{n=1}^{\infty} L_n z^n =\frac{1}{2} \frac{z}{(1-z)^2} \sum_{k=0}^{\infty}\frac{1}{2^k} \frac{1-z^{2^k}}{1+z^{2^k}}$ for $\|z\|<1$. \end{enumerate} \end{theorem} Further results on the Lebesgue constants for the Walsh system are shown in \cite{AS1,AS2}. \paragraph{Star discrepancy.} For a sequence $X=(x_k)_{k \ge 0}$ in $[0,1)$ the $n$-th discrepancy function is defined as $$\Delta_{X,n}(t) :=\frac{\#\{k \in \NN_0\ : \ k < n,\ x_k \in [0,t)\}}{n}-t \qquad \mbox{for $t \in [0,1]$}.$$ The star discrepancy of the initial $n$ terms of $X$ is then defined as $$D_n^(X):=\sup_{t \in [0,1]}\|\Delta_{X,n}(t)\|.$$ This is a quantitative measure for the irregularity of distribution of the sequence $X$ in $[0,1)$. The sequence $X$ is uniformly distributed in the sense of Weyl \cite{weyl} iff $\lim_{n \rightarrow \infty}D_n^{\ast}(X)=0$. The smaller $D_n^{\ast}(X)$, the better the $n$ initial terms of $X$ are distributed in $[0,1]$. For an introduction to uniform distribution and discrepancy, see, e.g., \cite{kuinie}. \paragraph{Star discrepancy of the van der Corput sequence.} The prototype of a uniformly distributed sequence is the van der Corput sequence (see \cite{vdc35} or \cite{FKP}) whose construction is based on the reflection of the binary digits of nonnegative integers $k$ around the binary point. If $k$ has binary expansion $k=k_0+k_1 2+k_2 2^2+\cdots$ (which is of course finite) with $k_j \in \{0,1\}$ for $j=0,1,2,\ldots$, then the $k$-th element $y_k$ of the van der Corput sequence $Y^{{\rm vdC}}=(y_k)_{k \ge 0}$ is given by \begin{equation} y_k := \frac{k_0}{2}+\frac{k_1}{2^2}+\frac{k_2}{2^3}+\cdots. \end{equation} The star discrepancy of the van der Corput sequence is intensively studied in literature, see, e.g., \cite{befa,befa77,dlp04,fau1990,hab1966,spi}. See also the survey \cite{FKP} for many other references. Consider the star discrepancy $D_n^{\ast}(Y^{{\rm vdC}})$ of the first $n$ terms of the van der Corput sequence and set $d_n:=n D_n^{\ast}(Y^{{\rm vdC}})$ to be the non-normalized star discrepancy. \begin{theorem}\label{thm2} The $n$-th Lebesgue constant $L_n$ for the Walsh system is exactly the nonnormalized star discrepancy $d_n$ of the first $n$ terms of the van der Corput sequence, that is $$d_n=L_n.$$ \end{theorem} \begin{proof} Let $n \in \NN$ be of the form $n=2^{n_1}+2^{n_2}+\cdots+2^{n_{\nu}}$ with integer exponents $n_1 > n_2 > \cdots > n_{\nu}\ge 0$. For the initial $n$ terms of $Y^{\rm vdC}$ we have \begin{align} \{y_0,y_1,\ldots,y_{n-1}\} = & \bigcup_{i=1}^{\nu}\left\{y_{\ell}\ : \ \ell \in \{2^{n_1}+\cdots+2^{n_{i-1}},\ldots, 2^{n_1}+\cdots+2^{n_{i-1}}+2^{n_i}-1\}\right\}\\ = & \bigcup_{i=1}^{\nu} \left\{\frac{k}{2^{n_i}}+\frac{1}{2^{n_{i-1}+1}}+\frac{1}{2^{n_{i-2}+1}}+\cdots +\frac{1}{2^{n_1+1}} \ : \ k \in \{0,1,\ldots,2^{n_i}-1\}\right\}, \end{align} where we put $2^{n_1}+\cdots+2^{n_{i-1}}:=0$ if $i=1$. It is well known that the star discrepancy of the van der Corput sequence is nonnegative (see \cite[Remark~3]{p04}) and twice the $L_1([0,1])$-norm of the discrepancy function (see, for example, \cite{PA}). Hence \begin{align}\label{stl1} d_n = & 2 n \int_0^1 \Delta_{Y^{\rm vdc},n}(t) \rd t = 2 \int_0^1 \sum_{\ell=0}^{n-1} (\boldsymbol{1}_{[0,t)}(y_{\ell}) -t) \rd t\\ = & 2 \sum_{\ell=0}^{n-1} \left(\frac{1}{2}-y_{\ell}\right) = 2 \sum_{i=1}^{\nu} \sum_{k=0}^{2^{n_i}-1} \left(\frac{1}{2}-\frac{k}{2^{n_i}}- \sum_{j=1}^{i-1} \frac{1}{2^{n_j+1}}\right)\nonumber\\ = & 2 \left(\frac{\nu}{2}-\sum_{i=1}^{\nu}2^{n_i} \sum_{j=1}^{i-1} \frac{1}{2^{n_j+1}} \right) = \nu-\sum_{1\le j < i \le \nu}2^{n_i-n_j} =L_n,\nonumber \end{align} where the last equality is \eqref{fo:LCW} from Theorem~\ref{thm:fine}. \end{proof} The representation $d_n=\nu-\sum_{1\le j < i \le \nu}2^{n_i-n_j}$ for the star discrepancy of $Y^{\rm vdC}$ is also mentioned (in an equivalent form) in \cite[p.~61]{spi}. \paragraph{Applications.} Theorem~\ref{thm2} can now be used to transfer results for the star discrepancy of the van der Corput sequence to results for the Lebesgue constants for the Walsh system and vice versa. In particular Items~1-4 from Theorem~\ref{thm:fine} by Fine have also been proven in the context of discrepancy of the van der Corput sequence by B\'{e}jian and Faure \cite{befa,befa77}. Therein, the estimate $d_n \le \frac{\log n}{3 \log 2} +1$ from \cite[Th\'eor\`eme~3]{befa77} yields a refinement of item~{\it 2} in Theorem~\ref{thm:fine}. Furthermore, it is known (see \cite[Lemme~4.1]{fau1990}) that for every $r \in \mathbb{N}$ we have $$\max_{n \in [2^{r-1},2^r]} d_n = \frac{r}{3}+\frac{7}{9}+\frac{(-1)^r}{9 \cdot 2^{r-1}}$$ and the maximum is attained for $n=(2^{r+1}+(-1)^r)/3$. Thus, for all $n$ of this form we have $d_n \ge \frac{\log n}{3 \log 2} +O(1)$. This result has been proven (in a slightly weaker form) for the Lebesgue constants for the Walsh system in \cite[Theorem~2]{AS2}. From Theorem~\ref{thm2} we also obtain new representations of the Lebesgue constant or star discrepancy of the van der Corput sequence that have not been known so far. For instance, the definition of $L_n$ leads to a new representation of $D_n^{\ast}(Y^{{\rm vdC}})$ via \begin{equation} D_n^{\ast}(Y^{{\rm vdC}})=\frac{d_n}{n}=\frac{L_n}{n}=\int_0^1 \left\|\frac{1}{n}\sum_{k=0}^{n-1} \wal_k(x)\right\| \rd x. \end{equation} Using this formula, we can derive another new formula for the star discrepancy of the van der Corput sequence. We have \begin{align} D_n^{\ast}(Y^{{\rm vdC}}) = & \sum_{m=0}^{2^{n_1+1}-1} \int_{m 2^{-n_1-1}}^{(m+1) 2^{-n_1-1}} \left\|\frac{1}{n} \sum_{k=0}^{n-1} \wal_k(x) \right\| \rd x. \end{align} For $x \in [m 2^{-n_1-1}, (m+1) 2^{-n_1-1})$ we can write $\wal_k(x) = \wal_{m'}(y_k)$, where $m'=m'(m)=m_{n_1}+m_{n_1-1} 2 +\cdots+m_0 2^{n_1}$ for $m$ with binary expansion $m=m_0+m_1 2+\cdots+m_{n_1} 2^{n_1}$. Then \begin{align} D^\ast_n(Y^{{\rm vdC}}) = & \frac{1}{2^{n_1+1}} \sum_{m'=0}^{2^{n_1+1}-1} \left\| \frac{1}{n} \sum_{k=0}^{n-1} \wal_{m'}(y_k) \right\|. \end{align*} On the other hand, a well known representation for the star discrepancy of the van der Corput sequence according to \cite[Th\'eor\`eme~1]{befa77} leads to a new representation for $L_n$ via $$L_n=d_n=n D_n^{\ast}(Y^{{\rm vdC}}) =\sum_{r=1}^{\infty} \left\\|\frac{n}{2^r}\right\\| = \sum_{r=1}^m \left\\|\frac{n}{2^r}\right\\|+\frac{n}{2^m}\ \ \ \mbox{ whenever $1 \le n \le 2^m$},$$ where $\\|x\\|=\min(\{x\},1-\{x\})$ is the distance of a real $x$ to the nearest integer. This formula immediately implies the recursion in Item~1 of Theorem~\ref{thm:fine} (note that $d_1=L_1=1$), which was already known for $d_n$ before (see~\cite[p.~13-07]{befa77}). Also the generating function for $L_n$ (see Item~5 in Theorem~\ref{thm:fine}) was unknown in terms of discrepancy, which can now be formulated as $$\sum_{n=1}^{\infty} d_n z^n =\frac{1}{2} \frac{z}{(1-z)^2} \sum_{k=0}^{\infty}\frac{1}{2^k} \frac{1-z^{2^k}}{1+z^{2^k}} \qquad \mbox{for $\|z\|<1$.}$$ For the star discrepancy of the van der Corput sequence we know a central limit theorem from \cite{dlp04}, which may now be formulated in terms of Lebesgue constants for the Walsh system. Accordingly, for every real $y$ and for $N \rightarrow \infty$ we have $$\frac{1}{N} \# \left\{ n < N \ : \ L_n \le \frac{\log n}{4 \log 2} + \frac{y}{4} \sqrt{ \frac{\log n}{3 \log 2}} \right\} = \Phi(y)+ o(1),$$ where $\Phi(y) = \frac1{\sqrt{2\pi}} \int_{-\infty}^y \exp(-t^2/2) \rd t $ is the Gaussian cumulative distribution function. That is, the Lebesgue constants for the Walsh system satisfy a central limit theorem. As the final example of this note we reformulate another result for the Lebesgue constants to obtain a so far unknown result for the star discrepancy of the van der Corput sequence. For $t \in [0,1]$ and $m \in \mathbb{N}$ let $n_t(m):= \lfloor 2^m(1+t)\rfloor$. Then \cite[Theorem~5]{AS2} in terms of discrepancy of the van der Corput sequence states: \begin{enumerate} \item For almost all $t \in [0,1]$ we have $$\lim_{m \rightarrow \infty} \frac{d_{n_t(m)}}{\log n_t(m) }=\frac{1}{4 \log 2}.$$ \item For all dyadic rational $t \in [0,1]$ we have $$\lim_{m \rightarrow \infty} \frac{d_{n_t(m)}}{\log n_t(m)}=0.$$ \item There exists a dense subset $A \subseteq [0,1]$ such that $$\liminf_{m \rightarrow \infty} \frac{d_{n_t(m)}}{\log n_t(m)}=0 \quad \mbox{ and }\quad \limsup_{m \rightarrow \infty} \frac{d_{n_t(m)}}{\log n_t(m)}=\frac{1}{3 \log 2}$$ for all $t \in A$. \end{enumerate} \begin{thebibliography}{10} \bibitem{AS1} S.V.~Astashkin, E.M.~Semenov: Lebesgue constants of a Walsh system. (Russian) Dokl. Akad. Nauk 462(5): 509--511, 2015; translation in Dokl. Math. 91(3): 344–346, 2015. \bibitem{AS2} S.V.~Astashkin, E.M.~Semenov: Lebesgue constants of a Walsh system and Banach limits. (Russian) Sibirsk. Mat. Zh. 57(3): 512--526, 2016; translation in Sib. Math. J. 57(3): 398--410, 2016. \bibitem{AT} A.P.~Austin, L.N.~Trefethen: Trigonometric interpolation and quadrature in perturbed points. SIAM J. Numer. Anal. 55(5): 2113--2122, 2017. \bibitem{befa} R.~B\'{e}jian, H.~Faure: Discr\'{e}pance de la suite de van der Corput. (French) C. R. Acad. Sci., Paris, S\'{e}r. A 285: 313--316, 1977. \bibitem{befa77} R.~B\'{e}jian, H.~Faure: Discr\'{e}pance de la suite de van der Corput. (French) S\'{e}minaire Delange-Pisot-Poitou (Th\'{e}orie des nombres) 13: 1--14,1977/78. \bibitem{DP10} J.~Dick, F. Pillichshammer: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010. \bibitem{dlp04} M.~Drmota, G.~Larcher, F.~Pillichshammer: Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118: 11--41, 2005. \bibitem{fau1990} H.~Faure: Discr\'{e}pance quadratique de la suite de van der Corput et de sa sym\'{e}trique. (French) Acta Arith. 55: 333--350, 1990. \bibitem{FKP} H.~Faure, P.~Kritzer, F.~Pillichshammer: From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules. Indag. Math. 26: 760--822, 2015. \bibitem{fine} N.J.~Fine: On the Walsh functions. Trans. Amer. Math. Soc. 65: 372--414, 1949. \bibitem{hab1966} S.~Haber: On a sequence of points of interest for numerical quadrature. J. Res. Nat. Bur. Standards Sect. B70: 127--136, 1966. \bibitem{KS} S.~Kaczmarz, H.~Steinhaus: Theorie der Orthogonalreihen. (German) Monografie Matematyczne, Chelsea Publishing Company, New York, 1951. \bibitem{kuinie} L.~Kuipers, H.~Niederreiter: Uniform Distribution of Sequences. John Wiley, New York, 1974. \bibitem{p04} F.~Pillichshammer: On the discrepancy of $(0,1)$-sequences. J. Number Theory 104: 301--314, 2004. \bibitem{PA} P.D.~Pro\u{\i}nov, E.Y.~Atanassov: On the distribution of the van der Corput generalized sequences. C. R. Acad. Sci. Paris S\'er. I Math. 307(18): 895--900, 1988. \bibitem{SWS} F.~Schipp, W.R.~Wade, P.~Simon: Walsh Series. An Introduction to Dyadic Harmonic Analysis. Adam Hilger Ltd., Bristol, 1990. \bibitem{spi} L.~Spiegelhofer: Discrepancy results for the van der Corput sequence. Unif. Distrib. Theory 13(2): 57--69, 2018. \bibitem{vdc35} J.G.~van der Corput: Verteilungsfunktionen I-II. Proc. Akad. Amsterdam 38: 813--821, 1058--1066, 1935. \bibitem{weyl} H.~Weyl: \"Uber die Gleichverteilung mod. Eins. (German) Math. Ann. 77: 313--352, 1916. \bibitem{zyg} A.~Zygmund: Trigonometric Series. Cambridge University Press, New York, 1959. \end{thebibliography} \end{document}
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\DeclareMathOperator{\triv}{triv} \DeclareMathOperator{\rightoverleft}{\parbox{2em}{\centerline{$\longrightarrow$}\vskip -6pt\centerline{$\longleftarrow$}}} \begin{document} \title{${\rm PGL}_2(\C)$-character stacks and Langlands duality over finite fields} \author{ Emmanuel Letellier \\ {\it Universit\'e Paris Cit\'e, IMJ-PRG CNRS UMR 7586} \\{\tt [email protected] } \and Tommaso Scognamiglio \\ {\it University of Heidelberg} \\ {\tt [email protected]} } \pagestyle{myheadings} \maketitle \begin{abstract}In this paper we study the mixed Poincar\'e polynomials of generic $\PGl_2(\C)$-character stacks with coefficients in some local systems arising from the conjugacy class of $\PGl_2(\C)$ which has a non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that these conjectured formulas interpolate the structure coefficients of the two following based rings: $$ \left(\mathcal{C}(\PGl_2(\F_q)),Loc(\PGl_2),\right),\hspace{1cm}\left(\mathcal{C}(\Sl_2(\F_q)), CS(\Sl_2),\cdot\right) $$ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\PGl_2)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\PGl_2$ and $CS(\Sl_2)$ the basis of characteristic functions of Lusztig's character-sheaves on $\Sl_2$. Our result reminds us of a non-abelian Fourier transform. \end{abstract} \tableofcontents \maketitle \section{Introduction} \bigskip Given a finite group $H$ we denote by $\mathcal{C}(H)$ the $\C$-vector space of $\C$-valued class functions on $H$. It is equipped with the basis $Cl(H)$ of characteristic functions of the conjugacy classes of $H$ and the basis $\widehat{H}$ of irreducible characters of $H$. We then consider the two rings $$ \mathcal{A}=(\mathcal{C}(H),Cl(H),),\hspace{.5cm}\text{and }\hspace{.5cm}\mathcal{B}=(\mathcal{C}(H),\widehat{H},\cdot) $$ where $$ is the convolution product and $\cdot$ the pointwise multiplication. The first ring can be identified with the center of the group algebra of $H$ and the second one is the character ring of $H$. \bigskip These rings are determined by their structure coefficients $$ \left\langle 1_{C_1}1_{C_2},1_{C_3}\right\rangle_H,\hspace{1cm}\left\langle\chi_1\otimes\chi_2,\chi_3\right\rangle_H $$ where $\langle\,,\,\rangle_H$ is the usual inner product on $\mathcal{C}(H)$ making the basis $\widehat{H}$ orthonormal and where $C_1, C_2, C_3$ run over the set of conjugacy classes of $H$ and $\chi_1,\chi_2,\chi_3$ run over $\widehat{H}$. \bigskip When $H$ is abelian, these two rings are isomorphic, namely there is a (non-canonical) automorphism of $\mathcal{C}(H)$ mapping $Cl(H)$ to $\widehat{H}$ and transforming $$ into $\cdot$ (which can be thought as a Fourier transform). \bigskip We are interested in the case where $H$ is the group $G(\F_q)$ of rational points of a connected reductive group $G$ defined over a finite field $\F_q$. \bigskip For that purpose we need a base other than $Cl(G^F)$ and $\widehat{G^F}$ : \bigskip Denote by $F:G\rightarrow G$ the geometric Frobenius (then $G^F=G(\F_q)$). \bigskip Instead of the basis $Cl(G^F)$ we consider the basis $Loc(G)^F$ formed by the characteristic functions of the intermediate extensions of the irreducible $F$-equivariant $G$-equivariant local systems on the $F$-stable conjugacy classes of $G$, and instead of the basis $\widehat{G^F}$ we consider the basis $CS(G)^F$ formed by the characteristic functions of the $F$-equivariant Lusztig's character-sheaves on $G$ (which are closely related to irreducible characters and coincide often with them). \bigskip We are interested in the relationship between the structure coefficients of the two based rings $$ \mathcal{A}_G=\left(\mathcal{C}(G^F),Loc(G)^F,\right),\hspace{1cm}\mathcal{B}_{G^}=\left(\mathcal{C}(G^{^F}), CS(G^)^F,\cdot\right) $$ where $(G^,F)$ is the dual group of $(G,F)$ in the sense of Deligne-Lusztig \cite{DL}. \bigskip \subsection{Review of the case $G=\mathrm{GL}_n$} \bigskip In this case $G=G^$ and the basis $CS(G)^F$ coincides with $\widehat{G^F}$. The stabilisers of the elements of $G$ being all connected, the irreducible $G$-equivariant local systems on the conjugacy classes are the constant sheaf. Therefore, the elements of $Loc(G)^F$ and $Cl(G^F)$ attached to semisimple conjugacy classes coincide. \bigskip Given a \emph{generic} $k$-tuple $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$ of conjugacy classes of $G(\C)$ we consider the quotient stack (possibly empty) $$ \mathcal{M}_{\overline{\mathcal{C}}}=\left[\{(x_1,\dots,x_k)\in\overline{\mathcal{C}_1}\times\cdots\times\overline{\mathcal{C}_k}\,\|\, x_1\cdots x_k=1\}/G\right]. $$ The genericity assumption ensures that the diagonal action of $G$ induces a free action of $\PGl_n$. In particular, the stack $\mathcal{M}_{\overline{\mathcal{C}}}$ is a $\mathbb{G}_m$-gerbe over the smooth variety $M_{\overline{\mathcal{C}}}$ obtained by taking the quotient by $\PGl_n$. We still take the quotient by $G=\Gl_n$ in the definition since otherwise we would have to to correct some formulas (like Formula (\ref{eq-intro}) by a factor $q-1$. \bigskip The stack $\mathcal{M}_{\overline{\mathcal{C}}}$, called a $G$-character stack, is the moduli stack of $G$-local systems on the Riemann sphere $\mathbb{P}^1_{\C}$ minus $k$ points $a_1,\dots,a_k$ with local monodromies around $a_i$ in $\overline{\mathcal{C}_i}$. \bigskip We consider the compactly supported intersection cohomology $IH_c^i(\mathcal{M}_{\overline{\mathcal{C}}},\C)$ of $\mathcal{M}_{\overline{\mathcal{C}}}$. \bigskip It is equipped with a weight filtration (increasing) $W^i_\bullet$ from which we define the mixed Poincar\'e series $$ IH_c(\mathcal{M}_{\overline{\mathcal{C}}};q,t)=\sum_{i,r}{\rm dim}\left(W^i_r/W^i_{r-1}\right)q^{r/2}t^i. $$ Our conjugacy classes are defined over a subring $R$ of $\C$ which is finitely generated as a $\mathbb{Z}$-algebra. We can choose this subring $R$ such that for any finite field $\F_q$ and any ring homomorphism $\varphi:R\rightarrow \F_q$, the conjugacy classes of $G(\overline{\F}_q)$ obtained from $\mathcal{C}_1,\dots,\mathcal{C}_k$ by base change are of same Jordan type and form a generic $k$-tuple. To ease the notation we denote them again by $\mathcal{C}_1,\dots,\mathcal{C}_k$. \bigskip Then we have \cite[Theorem 4.13]{L} \begin{align} E(\mathcal{M}_{\overline{\mathcal{C}}};q):&=IH_c(\mathcal{M}_{\overline{\mathcal{C}}};q,-1)\\ &=\left\langle {\bf Y}_{\mathcal{C}_1}\cdots{\bf Y}_{\mathcal{C}_k},1_{\{1\}}\right\rangle_{G(\F_q)}\label{eq-intro} \end{align} where ${\bf Y}_\mathcal{C}:G(\F_q)\rightarrow \overline{\Q}_\ell\simeq\C$ denotes the characteristic function of the intersection cohomology complex of $\overline{\mathcal{C}}$ (in the $\ell$-adic setting). \bigskip In \cite[Conjecture 4.5]{L} we give a conjecture formula for $IH_c(\mathcal{M}_{\overline{\mathcal{C}}};q,t)$ in terms of Macdonald symmetric functions. When the conjugacy classes are semisimple, this was first conjectured in \cite{HA} (see \S \ref{GL2} for a review in the semisimple case). The conjecture is known to be true in some non-trivial examples where we can compute explicitly the weight filtration (see \cite[\S 1.5.3]{HA}). \bigskip This conjectural formula is proved after the specialisation $t\mapsto -1$ (see \cite[Theorem 4.8]{L} and \cite[Theorem 1.2.3]{HA} in the semisimple case). It is also proved after the specialisation $q\mapsto 1$ which gives the Poincar\'e series. In the semisimple case this is due to by A. Mellit \cite{Mellit} who followed a strategy used by O. Schiffmann to compute the Poincar\'e polynomial of the moduli space of semistable Higgs bundles over a smooth projective curve \cite{schiffmann}. In the general case, this is due to M. Ballandras \cite{ballandras}. \bigskip We define the "pure part" of $IH_c(\mathcal{M}_{\overline{\mathcal{C}}};q,t)$ as $$ PIH_c(\mathcal{M}_{\overline{\mathcal{C}}};q)=\sum_i{\rm dim}\, \left(W^i_i/W^i_{i-1}\right) q^{i/2}. $$ The conjectural formula for $IH_c(\mathcal{M}_{\overline{\mathcal{C}}};q,t)$ implies a conjectural formula for $PIH_c(\mathcal{M}_{\overline{\mathcal{C}}};q)$ in terms of Hall-Littlewood symmetric functions. In \cite[Theorem 6.10.1]{letellier2} we prove that the conjectured formula for the pure part coincides with the inner product $$ \left\langle \chi_1\otimes\cdots\otimes\chi_k,1\right\rangle_{G(\F_q)} $$ where $(\chi_1,\dots,\chi_k)$ is a generic $k$-tuple of irreducible characters of $G(\F_q)$ of same Jordan type (in the sense of Lusztig) as the conjugacy classes $\mathcal{C}_1,\dots,\mathcal{C}_k$. \bigskip Therefore, it follows from the main conjecture for the mixed Poincar\'e series of character stacks that the latter interpolate (generically) the structure coefficients of the two rings $\mathcal{A}_{\Gl_n}$ and $\mathcal{B}_{\Gl_n}$. \bigskip The non-generic situation is more complicated and will not be discussed in this paper. However we refer to \cite{scognamiglio1}\cite{scognamiglio2} for results in the non-generic semisimple case. \bigskip Finally to conclude this part on the $G=\Gl_n$ case, notice that there is a notion of character-sheaves on the Lie algebra of $G$ due to Lusztig \cite{Lu-Fourier} (see also \cite{Let}). The Lie algebra analogues $\mathcal{A}^{Lie}$ and $\mathcal{B}^{Lie}$ of these two rings $\mathcal{A}$ and $\mathcal{B}$ are then isomorphic by an arithmetic Fourier transform (the character-sheaves are precisely the Deligne-Fourier transforms of the intersection cohomology complexes on the Zariski closures of the adjoint orbits of the Lie algebra). The generic structure coefficients of $\mathcal{A}^{Lie}$ and $\mathcal{B}^{Lie}$ are thus equal. Moreover we can prove that the generic structure coefficients of $\mathcal{B}^{Lie}$ and $\mathcal{B}$ are also equal \cite[Theorem 6.9.1]{letellier2} and coincide \cite{HA}\cite{letellier2} with the Poincar\'e series (for intersection cohomology) of $$ \mathcal{Q}=\left[\{(x_1,\dots,x_k)\in\overline{\mathcal{O}_1}\times\dots\times\overline{\mathcal{O}_k}\,\|\, x_1+\cdots+x_k=0\}/G\right] $$ where $(\mathcal{O}_1,\dots,\mathcal{O}_k)$ runs over the set of \emph{generic} $k$-tuple of adjoint orbits of $\mathfrak{gl}_n(\C)$ (unlike the multiplicative case, generic $k$-tuples of adjoint orbits of given Jordan types do not always exist). Finally, unlike character stacks, the intersection cohomology of $\mathcal{Q}$ is pure \cite[Proposition 2.2.6]{HA}\cite[Theorem 7.3.2]{letellier2} and its Poincar\'e series is conjectured to be the pure part of the mixed Poincar\'e series of character stacks. This latter conjecture is also a consequence of the conjectural formula for the mixed Poincar\'e series of generic character stacks. \bigskip \subsection{The case where $G=\mathrm{PGL}_2$} \bigskip The aim of this paper is to verify for a group $G\neq G^$ the assumption that the mixed Poincar\'e series of $G$-character stacks interpolate structure coefficients of the two rings $\mathcal{A}_G$ and $\mathcal{B}_{G^}$. We are especially interested in the situation where local monodromies are in conjugacy classes which carry non-trivial $G$-equivariant local systems. \bigskip We now assume that $G=\PGl_2$. \bigskip When $K$ is an algebraically closed field of characteristic $\neq 2$, there is only one conjugacy class of $G(K)$ which has a disconnected centraliser. This is the $G(K)$-conjugacy class $\mathcal{C}_{-1}$ of the image in $\PGl_2(K)$ of the diagonal matrix with coefficients $1$ and $-1$. There are thus two irreducible (up to isomorphism) $G(K)$-equivariant local systems on $\mathcal{C}_{-1}$ which are naturally parametrised by the set $\widehat{\Z/2\Z}=\{{\rm Id},\epsilon\}$ of irreducible characters of the group of connected components of the centraliser. We will denote by $\mathcal{L}_\epsilon$ the non-trivial equivariant irreducible local systems on $\mathcal{C}_{-1}$ (when the characteristic of $K$ is positive, the local systems considered are in the $\ell$-adic setting). \bigskip We will call $\mathcal{C}_{-1}$ the \emph{degenerate} conjugacy class. The other conjugacy classes (non-degenerate) have all a connected centraliser and so the constant sheaf is the only irreducible $G(K)$-equivariant local system on them. \bigskip {\bf $\PGl_2$-character stacks} \bigskip Consider a \emph{generic} $k$-tuple $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$ of semisimple regular conjugacy classes of $G(\C)$ (see Definition \ref{definitiongenericity}) and consider the $G(\C)$-character stack $$ \mathcal{M}_\mathcal{C}=\mathcal{M}_\mathcal{C}(\C):=[\{(x_1,\dots,x_k)\in\mathcal{C}_1\times\cdots\times\mathcal{C}_k\,\|\, x_1\cdots x_k=1\}/G]. $$ Choose on each $\mathcal{C}_i$ an irreducible $G$-equivariant local system $\mathcal{L}_i$ (it is the trivial local system if $\mathcal{C}_i$ is non-degenerate and can be either the trivial local system or $\mathcal{L}_\epsilon$ if $\mathcal{C}_i=\mathcal{C}_{-1}$). This defines a local system $\mathcal{L}_\mathcal{C}$ on $\mathcal{M}_\mathcal{C}$. \bigskip We consider the mixed Poincar\'e polynomial $$ H_c(\mathcal{M}_\mathcal{C},\mathcal{L}_\mathcal{C};q,t)=\sum_{i,r}{\rm dim}\left(W^i_r/W^i_{r-1}\right)q^{r/2}t^i $$ where $W^i_\bullet$ is the weight filtration on $H_c^i(\mathcal{M}_\mathcal{C},\mathcal{L}_\mathcal{C})$. \bigskip In this paper we give an explicit conjectural formula for $H_c(\mathcal{M}_\mathcal{C},\mathcal{L}_\mathcal{C};q,t)$ (see Conjecture \ref{conjEnondegenerate} for the non-degenerate case and Conjecture \ref{conjmhslocalsystems} when at least one conjugacy class is degenerate) and we prove that it is true after the specialisation $t\mapsto -1$ (see Theorem \ref{theoremEnondegenerate} and Theorem \ref{Epolynomialgenericdegenerate}). \bigskip Our conjectural formula for $H_c(\mathcal{M}_\mathcal{C},\mathcal{L}_\mathcal{C};q,t)$ is a consequence of the conjectural formula for the mixed Poincar\'e polynomials of $\Gl_n$-character varieties for which we have many evidences. \bigskip {\bf Langlands duality over finite fields (Lusztig) when $G=\PGl_2$} \bigskip Assume that $K=\overline{\F}_q$ with odd $q$ and that $F:G\rightarrow G$ is the standard Frobenius ($G(\F_q)=G^F$). \bigskip We now explain the correspondence between $F$-stable local systems on $F$-stable semisimple conjugacy classes of $\PGl_2(K)$ and $F$-stable (semisimple) character-sheaves on $\Sl_2(K)$. \bigskip For an $F$-stable $G$-equivariant local system $\mathcal{E}$ on an $F$-stable semisimple conjugacy class $\mathcal{C}$ of $G$, we denote by ${\bf Y}_{(\mathcal{C},\mathcal{E})}:G^F\rightarrow\overline{\Q}_\ell$ the characteristic function of $\mathcal{E}$ defined by $$ {\bf Y}_{(\mathcal{C},\mathcal{E})}(x)=\tr(Frob, \mathcal{E}_x).$$ (1) If $\mathcal{E}=\overline{\Q}_\ell$, then ${\bf Y}_{(\mathcal{C},\overline{\Q}_\ell)}$ is the function $1_{\mathcal{C}^F}$ that takes the value $1$ on $\mathcal{C}^F$ and $0$ elsewhere. \bigskip (2) If $(\mathcal{C},\mathcal{E})=(\mathcal{C}_{-1},\mathcal{L}_\epsilon)$, then $$ {\bf Y}_{(\mathcal{C},\mathcal{E})}=1_{\mathcal{O}_{e}}-1_{\mathcal{O}_{\sigma}} $$ where $$ (\mathcal{C}_{-1})^F=\mathcal{O}_{e}\sqcup\mathcal{O}_{\sigma} $$ is the decomposition into $G^F$-conjugacy classes (see Remark \ref{rmdecomp}). \bigskip Denote by $T$ and $T^$ the maximal tori respectively of $\PGl_2$ and $\Sl_2$ of diagonal matrices. We have a non-canonical group isomorphism $$ T^F\simeq\F_q^\overset{\sim}{\longrightarrow}\widehat{\F_q^}\simeq\widehat{T^{^F}} $$ under which $-1$, the non-trivial square of $1$, corresponds to the non-trivial square $\alpha_o$ of the identity character of $T^{^F}$. The character $\alpha_o$ takes the value $1$ at squares elements and $-1$ at non-square elements. \bigskip Now the group of linear characters of $T^{^F}$ is isomorphic with the group of isomorphism classes of the $F$-stable Kummer local systems on $T^$ (see Proposition \ref{bijectionkummer}). \bigskip We thus have an isomorphism $x\mapsto \mathcal{E}_x$ from $\F_q^\simeq T^F$ to the group of isomorphism classes of $F$-stable Kummer local systems on $T^$. \bigskip Denote by $\mathcal{I}_{T^}^{G^}:\mathcal{M}(T^;F)\rightarrow\mathcal{M}(G^;F)$ the geometric induction functor between categories of $F$-equivariant perverse sheaves (see \S \ref{sectionspringerresolution}). \bigskip Let $\mathcal{E}_x$ be an $F$-stable Kummer local system on $T^$. We have the following two situations : \bigskip (1) If $x\neq -1,1$, then $\mathcal{I}_{T^}^{G^}(\mathcal{E}_x)$ is an $F$-stable irreducible $G^$-equivariant perverse sheaf on $G^$ which is an example of a character-sheaf (its characteristic function is an irreducible character of $G^{^F}$). \bigskip (2) If $x=-1$, then $\mathcal{I}_{T^}^{G^}(\mathcal{E}_x)$ is an $F$-stable $G^$-equivariant semisimple perverse sheaf whose simple direct factors are naturally parametrised by $\widehat{\Z/2\Z}=\{{\rm Id},\epsilon\}$ : $$ \mathcal{I}_{T^}^{G^}(\mathcal{E}_x)=\mathcal{X}_{{\rm Id}}[3]\oplus\mathcal{X}_\epsilon[3]. $$ The perverse sheaves $\mathcal{X}_{{\rm Id}}[3]$ and $\mathcal{X}_\epsilon[3]$ are other examples of character-sheaves. However unlike the ones above their characteristic functions are not irreducible characters (see the end of \S \ref{CS-SL}). \bigskip We thus have a bijection between irreducible equivariant local systems on semisimple conjugcy classes of $G$ and certain character sheaves on $G^$ : $$ \Phi:\left\{(\mathcal{C}_x,\overline{\Q}_\ell)\,\|\, x\neq 1,-1\right\}\sqcup\{(\mathcal{C}_{-1},\mathcal{L}_r)\,\|\, r={\rm Id},\epsilon\}\simeq\left\{\mathcal{I}_{T^}^{G^}(\mathcal{E}_x)\,\|\, x\neq 1,-1\right\}\sqcup\left\{\mathcal{X}_r[3]\,\|\, r={\rm Id},\epsilon\right\} $$ where $\mathcal{C}_x$ is the (semisimple) $G$-conjugacy class of $x\in\F_q^\simeq T^F$, and where $\mathcal{L}_{{\rm Id}}$ is the constant sheaf on $\mathcal{C}_{-1}$. \bigskip Notice that on the left hand side we are missing the unipotent conjugacy classes and also the non-split ones (i.e. the ones with eigenvalues in $\F_{q^2}\backslash\F_q$) and on the right hand side we are missing the unipotent character-sheaves (the simple direct factors of $\mathcal{I}_{T^}^{G^}(\overline{\Q}_\ell)$ whose characteristic functions are the irreducible unipotent characters of $G^{^F}$) and the two cuspidal character-sheaves (see \S \ref{CS-SL}). \bigskip {\bf Main result} \bigskip Let $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$ be a generic $k$-tuple of semisimple regular conjugacy classes of $G(\C)$. Choose for each $i$ an irreducible $G(\C)$-equivariant local system $\mathcal{L}_i$ on $\mathcal{C}_ i$ and denote by $\mathcal{L}_\mathcal{C}$ the resulting local system on $\mathcal{M}_\mathcal{C}(\C)$. \bigskip Choose a generic $k$-tuple of $F$-stable semisimple regular conjugacy classes of $G(\overline{\F}_q)$ whose $i$-th coordinate is degenerate (i.e. equals to $\mathcal{C}_{-1}$) if and only if $\mathcal{C}_i$ is degenerate. To ease the notation we will denote this $k$-tuple again by $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$. On each conjugacy class we choose the same local system as in the complex case and denote it again by $\mathcal{L}_i$. We then denote by ${\bf Y}_i:G^F\rightarrow\overline{\Q}_\ell\simeq\C$ the characteristic function ${\bf Y}_{(\mathcal{C}_i,\mathcal{L}_i)}$ and by ${\bf X}_i$ the characteristic function of the character-sheaf of $G^$ which corresponds to $(\mathcal{C}_i,\mathcal{L}_i)$ under the isomorphism $\Phi$. \bigskip Our main theorem is the following one (see \S \ref{statement}): \begin{teorema} We have (1) $$H_c(\mathcal{M}_\mathcal{C}(\mathbb{C}),\mathcal{L}_\mathcal{C};q,-1)=\left\langle {\bf Y}_1\cdots{\bf Y}_k,1_{\{1\}}\right\rangle_{G^F}.$$ (2) The conjectured formula for the pure part $$ PH_c(\mathcal{M}_\mathcal{C}(\C),\mathcal{L}_\mathcal{C};q):=\sum_i{\rm dim}\,\left(W^i_i/W^i_{i-1}\right)\, q^{i/2} $$ equals $$ q^{k-3}\left\langle{\bf X}_1\cdots {\bf X}_k,1\right\rangle_{G^{^F}}. $$ \end{teorema} Our result reminds us of a Fourier transform. \bigskip \section{$\mathrm{GL}_2$-character varieties } \subsection{Preliminaries on cohomology}\label{polynomialcount-paragraph} In the following, $K$ is an algebraically closed field of $\ch \neq 2$, which is either $\C$ or $\overline{\F}_q$ and $X$ an algebraic variety over $K$. We denote by $H^_c(X)$ its compactly supported cohomology, with $\C$ coefficients if $K=\C$ and $\overline{\Q}_{\ell}$-coefficients if $K=\overline{\F}_q$. If $K=\C$, each cohomology group $H^k_c(X)$ is equipped with the weight filtration $W^i_{\bullet}$, introduced by Deligne in \cite{HodgeIII}. If $K=\overline{\F}_q$ and if we assume given a geometric Frobenius $F:X\rightarrow X$ (or, equivalently, an $\F_q$-scheme $X_o$ such that $X=X_o\times_{\F_q}\overline{\F}_q$) we have a weight filtration $W^k_{\bullet}H^k_c(X)$, where $W^k_{m}H^k_c(X)$ is the subspace on which the eigenvalues of the Frobenius $F$ are of absolute value $\leq q^{\frac{m}{2}}$. \vspace{4 pt} In both cases, we define the mixed Poincar\'e polynomial $H_c(X;q,t) \in \Z[\sqrt{q},t]$ $$ H_c(X;q,t)\coloneqq\sum_{i,k}{\rm dim}(W^k_i/W^k_{i-1})q^{i/2}t^k. $$ Then $ H_c(X;1,t)=\sum_k{\rm dim}\, H_c^k(X)\, t^k$ is the compactly supported Poincar\'e polynomial and $H_c(X;q,-1)$ is the so-called $E$-polynomial denoted by $E(X;q)$. In our cases, $H_c(X,q,t)$ and $E(X,q)$ will be actual polynomial in $q$, i.e. if $W^k_i/W^k_{i-1} \neq 0$ then $i$ is even. We also define the pure part $PH_c(X;q)$ as $$PH_c(X;q)\coloneqq \sum_{k}\dim(W^k_k/W^k_{k-1})q^{k/2}.$$ \bigskip Given a variety $X/_{\overline{\F}_q}$ with Frobenius $F$, we say that $X/_{\overline{\F}_q}$ has polynomial count if there exists $P_{X}(t) \in \Z[t]$ such that $$P_{X}(q^m)=\|X^{F^m}\|=\|X_o(\F_{q^m})\|$$ for all integer $m>0$. In this case, we have the following \cite[Theorem 2.2.6]{LRV}. \begin{teorema} \label{countingfq} If $X/_{\overline{\F}_q}$ has polynomial count with counting polynomial $P_{X}(t)$, we have $$P_{X}(q)=E(X;q) .$$ \end{teorema} \vspace{4 pt} \begin{esempio} The variety $X=\overline{\F}_q^$, with the Frobenius $F(z)=z^q$ has polynomial count with counting polynomial $P_{X}(t)=t-1$. Indeed, in this case we have $$\|X^{F^m}\|=\|\F_{q^m}^\|=q^m-1 .$$ Notice that $H_c(X;q,t)=qt^2+t$ and thus $E(X;q)=P_X(q)$. \end{esempio} \bigskip For a variety $X/_{\C}$, we say that $X$ has polynomial count with counting polynomial $P_X(t)$ if there exists a finitely generated $\Z$-subalgebra $R \subseteq \C$ and an $R$-scheme $X_R$ such that $$X_R \times_R \C \cong X ,$$and such that, for any ring homomorphism $\phi:R \to \F_q$, the variety $X^{\phi}=X_R\times_R\overline{\F}_q$ has polynomial count with counting polynomial $P_X(t)$. We have the following result. \begin{teorema}\cite[Appendix by Katz]{HRV} If $X/_\C$ has polynomial count with counting polynomial $P_X(t)$, then \begin{equation} \label{katzformula} E(X;q)=P_X(q). \end{equation} \label{Katz}\end{teorema} Katz's formulation of the above theorem gives also information on the Hodge filtration. If we do not bother about the Hodge filtration (as it is the case in this paper) we can prove the above theorem in a simple way as follows. \vspace{4 pt} Fix an isomorphism $\C \cong \overline{\Q}_{\ell}$ and identify $H^_c(X,\C) \cong H^_c(X,\overline{\Q}_{\ell})$ through this isomorphism. Recall that, if $\ch(\overline{\F}_q)$ is sufficiently large, we have a natural isomorphism \begin{equation} \label{comparisonisom} H^_c(X^{\phi}) \cong H^_c(X). \end{equation} Indeed, let $f:X_R \to \spec(R)$ be the structural morphism. The complex $Rf_!\overline{\Q}_{\ell}$ is constructible, see for instance \cite[Chapter 2]{Deligne-etale}. In particular, there exists a non-empty open $U \subseteq \spec(R)$ such that $Rf_!\overline{\Q}_{\ell}$ is constant. Denote by $\xi:\spec(\C) \to \spec(R)$ the (geometric) generic point of $\spec(R)$ coming from the embedding $R \subseteq \C$ and, for any $\phi:R \to \F_q$, denote by $\xi_{\phi}:\spec(\overline{\F}_q) \to \spec(R)$ the corresponding geometric point. If $\Imm(\xi_{\phi}) \in U$, from the fact that $Rf_!\overline{\Q}_{\ell}$ is constant and from the proper base change theorem, we have the following chain of isomorphisms: \begin{equation} \label{isomorphism-cohomology-constant} H^_c(X^{\phi}) \cong (Rf_!\overline{\Q}_{\ell})_{\xi_{\phi}} \cong (Rf_!\overline{\Q}_{\ell})_{\xi} \cong H^_c(X) .\end{equation} The results of \cite[Theorem 14]{Deligne} show that the isomorphism (\ref{comparisonisom}) preserves the weight filtration on both sides. In particular, we have an equality \begin{equation} \label{comparisonmhp} H_c(X;q,t)=H_c(X^{\phi};q,t). \end{equation} We deduce thus Formula (\ref{katzformula}) from Theorem \ref{countingfq} and formula (\ref{comparisonmhp}). \subsection{Review on cohomology of $\mathrm{GL}_n$-character varieties}\label{GL2} \label{paragraphcohomologyGl2characterstacks} To alleviate the notation we will put $\Gl_n=\Gl_n(K)$. If $K=\overline{\F}_q$, we denote by $F:\Gl_n \to \Gl_n$ the standard Frobenius $F((a_{i,j}))=(a_{i,j}^q)$. \bigskip Let $C$ be a $k$-tuple $C=(C_1,\dots,C_k)$ of semisimple conjugacy classes of $\Gl_n$. If $K=\overline{\F}_q$, we assume that each $C_i$ is $F$-stable and split, i.e. its eigenvalues belong to $\F_q^$. We define the corresponding $\Gl_n$-character stack $\mathcal{M}_C$ as the quotient stack $$ \mathcal{M}_C=\left[\{(x_1,\dots,x_k)\in C_1\times\cdots\times C_k\,\|\, x_1\cdots x_k=1\}/\PGl_n\right] $$ for the diagonal conjugation action of $\PGl_n$. \bigskip Following \cite[Definition 2.1.1]{HA}, we have the following definition of a \emph{generic} $k$-tuple $C$. \begin{definizione} The $k$-tuple $C$ is generic if \begin{equation} \prod_{i=1}^k\det(C_i)=1, \label{det}\end{equation} and for any integer $1\leq r<n$, if we select $r$ eigenvalues of $C_i$ (for each $i=1,\dots,k$), the product of the $nr$ selected eigenvalues is different from $1$. \end{definizione} \bigskip For each $i$ let $\mu^i$ be the partition of $n$ whose parts are given by the multiplicities of the eigenvalues of $C_i$. We say that $C_i$ is of type $\mu^i$ and that the $k$-tuple $C$ is of type ${\bm \mu}=(\mu^1,\dots,\mu^k)$. \bigskip \begin{remark}Recall \cite[Lemma 2.1.2]{HA} that for any $k$-tuple ${\bm \mu}=(\mu^1,\dots,\mu^k)$ of partitions of $n$, if $\ch(K)$ does not divide $\gcd(\mu^j_i)_{i,j}$, there always exists a generic $k$-tuple of semisimple conjugacy classes of $\Gl_n$ of type $\bm\mu$. \end{remark} Recall the following result \cite[Proposition 2.1.4, Theorem 2.1.5]{HA} \cite[Theorem 1.1.1]{HA1}. \begin{teorema}If $C=(C_1,\dots,C_k)$ is generic, then, if not empty, $\mathcal{M}_C$ is a nonsingular irreducible affine algebraic variety of dimension $$ d_{\bm\mu}:=(k-2)n ^2-\sum_{i,j}(\mu^i_j)^2+2 $$ where $\mu^i=(\mu^i_1,\mu^i_2,\dots)$. \label{irreducible}\end{teorema} \bigskip In \cite[Section 2.3]{HA} the authors defined a rational function $\mathbb{H}_{\bm\mu}(z,w)$ which conjecturally computes the polynomial $H_c(\mathcal{M}_C;q,t)$. It is defined as follows. Consider $k$ sets ${\bm x}_1,\dots,{\bm x}_k$ of infinitely many independent variables and let $\Lambda({\bm x}_1,\dots,{\bm x}_k)$ be the ring of functions separately symmetric in each of the set of variables. We then put $$ \Omega(z,w):=\sum_{\lambda\in\mathcal{P}}\mathcal{H}_\lambda(z,w)\prod_{i=1}^k\tilde{H}_\lambda({\bm x}_i;z^2,w^2) $$ where $\mathcal{P}$ is the set of all partitions, $\tilde{H}_\lambda({\bm x}_i;z,w)$ are the modified Macdonald polynomials and $$ \mathcal{H}_\lambda(z,w):=\prod_s\frac{1}{(z^{2a(s)+2}-w^{2l(s)})(z^{2a(s)}-w^{2l(s)+2})} $$ where $s$ runs over the boxes of the Young diagram of $\lambda$ with $a(s)$ and $l(s)$ its arm and leg length. Then $$ \mathbb{H}_{\bm\mu}(z,w):=(z^2-1)(1-w^2)\left\langle {\rm Log}\,\Omega(z,w),h_{\bm\mu}\right\rangle $$ where $h_{\bm \mu}:=h_{\mu^1}({\bm x}_1)\cdots h_{\mu^k}({\bm x}_k)\in\Lambda({\bm x}_1,\dots,{\bm x}_k)$ is the product of complete symmetric functions and ${\rm Log}$ is the plethystic logarithm. \bigskip It is proved in \cite{MellitD} that these rational functions $\mathbb{H}_{\bf \mu}(z,w)$ are polynomials in $z$, $w$ with integer coefficients as it was conjectured in \cite{HA}. \bigskip We have the following conjecture \cite[Conjecture 1.2.1]{HA}. \begin{conjecture} \label{conjecturemhsgln} The polynomial $H_c(\mathcal{M}_C;q,t)$ depends only on the type ${\bm\mu}$ of $C$ and not the choice of generic eigenvalues and \begin{equation}H_c(\mathcal{M}_C;q,t)=(t\sqrt{q})^{d_{\bm\mu}}\mathbb{H}_{\bm\mu}\left(\frac{1}{\sqrt{q}},-t\sqrt{q}\right). \label{conj}\end{equation} In particular, \begin{equation} PH_c(\mathcal{M}_C;q)=q^{d_{\bm \mu}/2}\mathbb{H}_{\bm \mu}(0,\sqrt{q}). \label{pure}\end{equation} \end{conjecture} \bigskip Notice that since we are working with the Riemann sphere, the sign in the second coordinate could have been omitted because in this case $\mathbb{H}_{\bm\mu}(z,w)$ is a function of $z^2$ and $w^2$ (which is not the case in higher genus). We decided to keep the formulas as there are in \cite{HA} for the convenience of the reader. \bigskip The Euler specialization $t\mapsto -1$ of Formula (\ref{conj}) is proved \cite[Theorem 1.2.3]{HA} by counting points of $\mathcal{M}_C$ over finite fields and using Theorem \ref{Katz}, more precisely we have the following result. \begin{teorema} $$ E(\mathcal{M}_C;q)=q^{d_{\bm\mu}/2}\mathbb{H}_{\bm\mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right). $$ In particular the polynomial $E(\mathcal{M}_C;q)$ depends only on the type ${\bm\mu}$ of $C$ and not the choice of generic eigenvalues. \label{E-poly}\end{teorema} \bigskip Let us now write a more explicit formula when $n=2$. \begin{prop}Assume that $k\geq 1$ and ${\bm\mu}=((1^2),\dots,(1^2))$, then $$ \mathbb{H}_{\bm\mu}(z,w)=\frac{(w^2+1)^k}{(z^2-w^2)(1-w^4)}+\frac{(z^2+1)^k}{(z^4-1)(z^2-w^2)}-\frac{2^{k-1}}{(z^2-1)(1-w^2)}$$ \label{H}\end{prop} \begin{remark} As mentioned earlier, we know from \cite{MellitD} that these rational functions are in fact polynomials in $z$ and $w$ with integer coefficients which a priori is not obvious. \end{remark} \bigskip \begin{proof}[Proof of Proposition \ref{H}] The conjugacy classes of $\Gl_n(\mathbb{F}_q)$ are parametrised by the set of maps from the set of Frobenius-orbits of $\overline{\mathbb{F}}_q^$ to the set of all partitions. The type of a such a map consists of the collection of pairs $(d,\lambda)$ with $d$ a positive integer and $\lambda$ a non-zero partition which is the image of a Frobenius-orbit of size $d$. There are 4 types when $n=2$, i.e. $(1,1)^2,(2,1), (1,(1^2)), (1,(2^1))$. The first one is the type of a conjugacy class with two distincts eigenvalues in $\mathbb{F}_q^$, the second one is the type of a conjugacy class whose characteristic polynomial is irreducible over $\mathbb{F}_q$ (i.e. it has two distinct eigenvalues $x, x^q$ in $\mathbb{F}_{q^2}^$), and the last two types are the types respectively of central matrices and matrices with one Jordan block. Then from \cite[Formula (2.3.9)]{HA} we have $$ \mathbb{H}_{\bm \mu}(z,w)=(z^2-1)(1-w^2)\left\langle \sum_{\omega}C_\omega^o\mathcal{H}_\omega(z,w)\prod_{i=1}^k\tilde{H}_\omega({\bm x}_i;z^2,w^2),h_{\bm \mu}\right\rangle $$ where the sum runs over the types of size $2$. A direct calculation shows that $$ C_{(2^1)}^o=C_{(1^2)}^o=1,\hspace{1cm}C_{(2,1)}^o=C_{(1,1)^2}^o=-\frac{1}{2}$$ and \begin{align} \mathcal{H}_{(1^2)}(z,w)&=\frac{1}{(z^2-w^2)(1-w^4)(z^2-1)(1-w^2)}\\ \mathcal{H}_{(2^1)}(z,w)&=\frac{1}{(z^4-1)(z^2-w^2)(z^2-1)(1-w^2)}\\ \mathcal{H}_{(1,1)^2}(z,w)&=\frac{1}{(z^2-1)^2(1-w^2)^2}\\ \mathcal{H}_{(2,1)}(z,w)&=\frac{1}{(z^4-1)(1-w^4)} \end{align} To compute the pairing we need also to decompose the modified Macdonald symmetric functions into the basis of monomial symmetric functions $\{m_\lambda({\bm x})\}_\lambda$ which is orthogonal to the basis of complete symmetric functions $\{h_\lambda({\bm x})\}_\lambda$ : \begin{align} \tilde{H}_{(1^2)}({\bm x};z,w)&=(w+1)m_{(1^2)}({\bm x})+m_{(2^1)}({\bm x})\\ \tilde{H}_{(2^1)}({\bm x};z,w)&=(z+1)m_{(1^2)}({\bm x})+m_{(2^1)}({\bm x})\\ \tilde{H}_{(1,1)^2}({\bm x};z,w)&=m_{(2^1)}({\bm x})+2m_{(1^2)}({\bm x})\\ \tilde{H}_{(2,1)}({\bm x};z,w)&=m_{(2^1)}({\bm x}). \end{align} \end{proof} Hence in the case $n=2$, Conjecture \ref{conj} reads. \begin{conjecture}For $k\geq 1$ and ${\bm\mu}=((1^2),\dots,(1^2))$ we have $$ H_c(\mathcal{M}_C;q,t)=\frac{(q^2t^4+qt^2)^k}{q^2t^6(q^2t^2-1)(q^2t^4-1)}+\frac{t^{2k-6}(q+1)^k}{(q^2-1)(q^2t^2-1)}-\frac{(2qt^2)^{k-1}}{qt^4(q-1)(qt^2-1)}. $$ \end{conjecture} \bigskip \subsection{Twisted mixed Poincar\'e polynomials}\label{twisted} Assume that $X$ is a $K$-variety endowed with an action of a finite group $W$. If $K=\overline{\F}_q$, we assume that the action of $W$ commutes with $F$. The group action preserves the weight filtration on $H_c^\bullet(X)$ and for $w\in W$, we define the $w$-twisted mixed Poincar\'e polynomial as $$ H_c^w(X;q,t):=\sum_{i,k}{\rm Tr}\left(w\,\|\, W^k_i/W^k_{i-1}\right)\, q^{i/2}t^k. $$ If $K=\overline{\F}_q,$ we say that the pair $(X/_{\overline{\F}_q}, W)$ has polynomial count with (twisted) counting polynomials $\{P_w(t)\}_{w\in W}$ in $\mathbb{Z}[t]$ if $$ \|X^{wF^r}\|=P_w(q^r). $$ for any integer $r \geq 1$. We have the following twisted analogue of Theorem \ref{countingfq}. \begin{teorema} \label{theoremtwistedEFq} If $(X/_{\overline{\F}_q},W)$ has polynomial count with (twisted) counting polynomials $\{P_w(t)\}_{w\in W}$ for any $w\in W$ we have \begin{equation} \label{katztwistedformula} E^w(X;q)=P_w(q). \end{equation} where $E^{w}(X;q)\coloneqq H^w_c(X;q,-1)$. \end{teorema} The proof of this Theorem is very similar to that of \cite[Theorem 2.8]{LRV}. We give it below after Theorem \ref{theoremtwistedE}. \vspace{4 pt} \begin{oss} Given $(X/_{\overline{\F}_q},W)$ as above, for any $w \in W$, the map $wF:X \to X$ is a Frobenius morphism and gives thus another $\F_q$-structure of the $\overline{\F}_q$-variety $X$. In particular, there exists an $\F_q$-scheme $X_w$ with an isomorphism $X_w \times_{\F_q} \overline{\F}_q \cong X$ such that, through this isomorphism, the geometric Frobenius $F_w$ of $X_w$ is identified with $wF$. In general, the polynomial $P_w(t)$ is not the counting polynomial of $X_w$, i.e. it is not true that $\|X_w(\F_{q^r})\|=P_w(q^r)$. Notice indeed that the LHS of the latter equality is $\|X^{(wF)^r}\|$, while the RHS is $\|X^{wF^r}\|$. \vspace{2 pt} For a concrete example, consider , $X=\overline{\F}_q^$ with $F(z)=z^q$ and $W=\bm \mu_2=\{1,\sigma\}$, with the action $\sigma \cdot x=x^{-1}$. The pair $(\overline{\F}_q^,\bm \mu_2)$ has polynomial count with counting polynomials $P_1(t)=(t-1)$ and $P_{\sigma}(t)=(t+1)$. Indeed, we have $\sigma F(x)=x^{-q}$ and thus $X^{\sigma F}=\bm \mu_{q+1}$. Notice that, if $4$ does not divide $q+1$, i.e. if $-1$ is not a square in $\F_q^$, we can consider $X_{\sigma}=\spec(\F_q[s,t]/(s^2+t^2=1))$, with the isomorphism $X_{\sigma} \times_{\F_q}\overline{\F}_q \to \overline{\F}_q^$ given by $(s,t) \to s+it$. We have that $$\#X_{\sigma}(\F_{q^r})=\begin{cases} q^r+1 \text{ if } r \text{ is odd}\\ q^r-1 \text{ if } r \text{ is even} \end{cases} $$ \end{oss} \bigskip If $K=\C$, we say that $(X/_{\C},W)$ has polynomial count with (twisted) counting polynomials $\{P_w(t)\}_{w\in W}$ if there exists a finitely generated $\mathbb{Z}$-subalgebra $R$ of $\C$, a separated $R$-scheme $X_R$ equipped with a $W$-action which gives back $X$ with its $W$-action after scalar extensions from $R$ to $\C$, such that for any ring homomorphism $\varphi: R \to \mathbb{F}_q$, $(X^{\phi},W)$ has polynomial counting with (twisted) counting polynomials $\{P_{w}(t)\}_{w \in W}$. \vspace{2 pt} \bigskip We have the following twisted version of Theorem \ref{Katz}. \begin{teorema} \label{theoremtwistedE} Assume that $(X/_{\C},W)$ has polynomial count with (twisted) counting polynomials $\{P_w(t)\}_{w\in W}$. Then for any $w\in W$ we have \begin{equation} \label{katztwistedformulaC} E^w(X;q)=P_w(q). \end{equation} \end{teorema} \vspace{2 pt} Theorem \ref{theoremtwistedE} above can be deduced from Theorem \ref{theoremtwistedEFq} as follows. Consider a variety $X/_{\C}$ and $R$ as above and $U \subseteq \spec(R)$ as in \cref{polynomialcount-paragraph}. Since the action of $W$ is defined over $R$, the isomorphism (\ref{isomorphism-cohomology-constant}) is compatible with the $W$-action. In particular, since the action of $W$ on $H^_c(X)$ is defined over the rationals, through the isomorphism (\ref{comparisonisom}), we have $$\tr\left(\,w\,\|\, W_{i}^kH^k_c(X)/W^k_{i-1}H^k_c(X)\right)=\tr\left(w\,\|\, W_{i}^kH^k_c(X^{\phi})/W^k_{i-1}H^k_c(X^{\phi})\right)$$ and so $$H^{w}_c(X;q,t)=H^{w}_c(X^{\phi};q,t) $$ from which we get $$E(X;q)=E(X^{\phi};q) .$$ \bigskip \begin{proof}[Proof of Theorem \ref{theoremtwistedEFq}] \label{prooftwistedEFq} By Grothendieck's trace formula, for any $r$, we have \begin{equation} \label{grothendiecktrace} P_w(q^r)=\|X^{wF^r}\|=\sum_{k}(-1)^k \tr\left(wF^r\,\|\,H^k_c(X,\overline{\Q}_{\ell})\right) .\end{equation} Let $\lambda_{i,k,1}q^{\frac{i}{2}},\dots,\lambda_{i,k,s_{k,i}}q^{\frac{i}{2}}$ and $\alpha^w_{i,k,1},\dots,\alpha^w_{i,k,s_{k,i}}$ be the eigenvalues, counted with multiplicities, of $F$ and $w$ on $W^k_i/W^k_{i-1}$. Since $w$ and $F$ commute, up to reordering, we can assume that, for any $r \geq 1$, $$\tr\left(wF^r\,\|\,W^k_{i}/W^k_{i-1}\right)=\sum_{h=1}^{s_{k,i}}\alpha^w_{i,k,h}\lambda^r_{i,k,h}q^{\frac{ri}{2}}$$ and thus $$\sum_{k}(-1)^k \tr\left(wF^r\,\|\,H^k_c(X)\right)=\sum_{i}\left(\sum_k (-1)^k \sum_{h=1}^{s_{k,i}}\alpha^w_{i,k,h}\lambda^r_{i,k,h}\right)q^{\frac{ir}{2}} .$$ If $\displaystyle P_w(t)=\sum_{i}t^i c_{w,i}$, from Formula (\ref{grothendiecktrace}) we deduce that \begin{equation} \sum_k (-1)^k \sum_{h=1}^{s_{k,i}}\alpha^w_{i,k,h}\lambda^r_{i,k,h}=\begin{cases} c_{w,\frac{i}{2}} \text{ if } i \text{ is even }\\ 0 \text{ otherwise} \end{cases}. \end{equation} From \cite[Lemma 2.9]{LRV}, we deduce that we have $$\sum_k (-1)^k \sum_{h=1}^{s_{k,i}}\alpha^w_{i,k,h}=\begin{cases} c_{w,\frac{i}{2}} \text{ if } i \text{ is even }\\ 0 \text{ otherwise} \end{cases}.$$ Since $\displaystyle \sum_k (-1)^k \sum_{h=1}^{s_{k,i}}\alpha^w_{i,k,h}=\sum_k (-1)^k \tr(w\|W^{k}_{i}/W^k_{i-1})$, we have the desired equality (\ref{katztwistedformula}). \end{proof} We keep the notation of the previous section with $n=2$. We will apply the above theorem to the following situation. We assume that $C=(C_1,\dots,C_k)$ is a generic $k$-tuple of semisimple regular conjugacy classes of $\Gl_2$ where the $m$-th first conjugacy classes $C_1,\dots,C_m$ (for some integer $m<k$) are the conjugacy class with eigenvalues $1$ and $-1$ and we let $H_m$ be the kernel of the morphism $$ \epsilon_m:({\bm\mu}_2)^m\longrightarrow \C^,\hspace{1cm}(y_1,\dots,y_m)\mapsto y_1\cdots y_m $$ where ${\bm\mu}_2$ is the multiplicative group $\{1,-1\}$. \bigskip The group $H_m$ acts on the character variety $\mathcal{M}_C$ by multiplication on the first $m$-th coordinates. \bigskip Put $\bm\mu=((1^2),\dots,(1^2))$ and fix $y=(y_1,\dots,y_m)\in H_m$. For each $i=1,\dots,m$, define $$ h^{y_i}_{(1^2)}({\bm x}_i)=\begin{cases}h_{(1^2)}({\bm x}_i)&\text{ if }y_i=1,\\ p_{2}({\bm x}_i)& \text{ otherwise}\end{cases} $$ where $p_2({\bm x}_i)=x_{i1}^2+x_{i2}^2+\cdots$ is the power sum symmetric function in the variable ${\bm x}_i=\{x_{i1},x_{i2},\dots\}$ and put $$ \mathbb{H}^y_{\bm\mu}(z,w):=(z^2-1)(1-w^2)\left\langle{\rm Log}\,\Omega(z,w),h^y_{\bm\mu}\right\rangle $$ where $$ h^y_{\bm\mu}:=h^{y_1}_{(1^2)}({\bm x}_1)\cdots h^{y_m}_{(1^2)}({\bm x}_m)h_{(1^2)}({\bm x}_{m+1})\cdots h_{(1^2)}({\bm x}_k)\in\Lambda({\bm x}_1,\dots,{\bm x}_k). $$ \bigskip \begin{prop}Let $r=r(y):=\#\{i=1,\dots,m\,\|\, y_i=-1\}>0$, then $$ \mathbb{H}^y_{\bm\mu}(z,w)=\frac{(w^2+1)^{k-r}(1-w^2)^r}{(z^2-w^2)(1-w^4)}+\frac{(1-z^2)^r(z^2+1)^{k-r}}{(z^4-1)(z^2-w^2)}. $$ \label{H-twisted}\end{prop} \bigskip \begin{proof}Similar calculation as for the proof of Proposition \ref{H}, noticing that $$ \langle \tilde{H}_{(1^2)}({\bm x};z,w),p_2({\bm x})\rangle=\langle \tilde{H}_{(2,1)}({\bm x};z,w),h_{(1^2)}({\bm x})\rangle=0 $$ which explains why we have only two terms. \end{proof} We make the following conjecture. \begin{conjecture}We have $$ H_c^y(\mathcal{M}_C;q,t)=(t\sqrt{q})^{d_{\bm\mu}}\mathbb{H}_{\bm \mu}^y\left(\frac{1}{\sqrt{q}},-t\sqrt{q}\right). $$ \label{conj-twisted}\end{conjecture} We prove that the above conjecture is true under the Euler specialization $t\mapsto -1$. \begin{teorema} We have $$ E^y(\mathcal{M}_C;q)=q^{d_{\bm\mu}/2}\mathbb{H}^y_{\bm \mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right). $$ In particular, the polynomial $E^y(\mathcal{M}_C;q)$ depends only on the type ${\bm\mu}$ of $C$ and not the choice of generic eigenvalues. \label{Ey}\end{teorema} \begin{proof} From Formula \ref{comparisonmhp}, it is enough to show the case of $K=\overline{\F}_q$. Take thus a finite field $\mathbb{F}_q$ of characteristic $\neq 2$ and consider a generic $k$-tuple $ C=( C_1,\dots, C_k)$ of semisimple regular conjugacy classes of $\Gl_2(\overline{\mathbb{F}}_q)$ with eigenvalues in $\mathbb{F}_q^$ and such that the first $m$ conjugacy classes are the conjugacy class with eigenvalues $1,-1$. \bigskip Then the group $H_m$ acts on $\mathcal{M}_{ C}$ by multiplication coordinate by coordinate on the first $m$-th coordinates. By Theorem \ref{katztwistedformula} we need to show that $$ \# (\mathcal{M}_{C})^{yF}=q^{d_{\bm\mu}/2}\mathbb{H}^y_{\bm \mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right) $$ where $F$ is the standard Frobenius that raises matrix coefficients to their $q$-th power. Fix $i=1,\dots,m$, let $\alpha_i\in\overline{\mathbb{F}}_q^$ be defined as $\alpha_i=1$ if $y_i=1$ and as an element of $\mathbb{F}_{q^2}^$ such that $\alpha_i^q=-\alpha_i$ if $y_i=-1$. We then let ${O}_i$ be the conjugacy class of $\Gl_2(\overline{\mathbb{F}}_q)$ with eigenvalues $\alpha_i,-\alpha_i$ (it is the conjugacy class ${ C}_i$ when $y_i=1$). Notice that the standard Frobenius $F$ leaves stable the conjugacy classes ${O}_i$ and that we have the following commutative diagram $$ \xymatrix{{C}_i\ar[d]_{y_iF}\ar[rr]^{f_i}&&{ O}_i\ar[d]^F\\ { C}_i\ar[rr]^{f_i}&&{ O}_i} $$ where $f_i$ is the multiplication by the scalar $\alpha_i$. Therefore via $\prod_i f_i$, the pair $(\mathcal{M}_{ C},yF)$ can be identified with $(\mathcal{M}_{ O},F)$ where ${ O}$ is the $k+1$-tuple of conjugacy classes $$ {O}=\left({ O}_1,\dots,{O}_k, \{(\alpha_1\cdots\alpha_m)^{-1} I_2\}\right)$$ where ${ O}_i={ C}_i$ for $i>m$. Notice that $\alpha_1\cdots\alpha_m\in\mathbb{F}_q^$ as there is an even number of non-trivial $y_i$. We thus have $$ \#(\mathcal{M}_{ C})^{yF}=\#(\mathcal{M}_{ O})^F. $$ Now for an $F$-stable conjugacy class ${ O}$ of $\Gl_2(\overline{\mathbb{F}}_q)$ denote by $1_{ O}:\Gl_2(\mathbb{F}_q)\rightarrow \C^$ the function that takes the value $1$ at ${O}^F$ and $0$ elsewhere. Then $$ \#(\mathcal{M}_{ O})^F=\frac{1}{\|\PGl_n(\mathbb{F}_q)\|}\left(1_{{ O}_1}\cdots1_{{ O}_k}1_{(\alpha_1\cdots\alpha_k)^{-1} I_2}\right)(1) $$ It follows from \cite[Theorem 4.14]{L} that the RHS equals to $$ q^{d_{\bm\mu}/2}\mathbb{H}^y_{\bm \mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right). $$ Indeed, since the conjugacy classes ${ O}_i$ are semisimple, the function ${\bm X}_{\overline{ O}_i}$ in \cite[Theorem 4.14]{L} is our function $1_{{ O}_i}$. In loc. cit. we introduced the notion of type of a conjugacy class of $\Gl_n(\mathbb{F}_q)$ (see \cite[\S 4.1.2]{L}) which we could avoid here as we are working with semisimple conjugacy classes of $\Gl_2$. The type of a split semisimple regular conjugacy class of $\Gl_2(\mathbb{F}_q)$ is $\omega=(1,1)(1,1)$ and the symmetric function $s_\omega$ (see \cite[Remark 4.1]{L}) is our symmetric function $h_{(1^2)}$. The type of a non-split semisimple regular conjugacy class of $\Gl_2(\mathbb{F}_q)$ like our conjugacy class ${ O}_i$ with $y_i=-1$ (it has eigenvalues in $\mathbb{F}_{q^2}-\mathbb{F}_q$) is $\omega=(2,1)$ and $s_\omega$ is our symmetric function $p_2$. Finally in the formula for $\mathbb{H}_{\bm\omega}(z,w)$ in \cite[Above Lemma 4.2]{L}, there is a sign which in our case is always $1$ as we have an even number of twisted conjugacy classes. \end{proof} \section{$\mathrm{PGL}_2$-character stacks } In this section we work over an algebraically closed field $K$ of characteristic $\neq 2$ (for us $K$ will be either $\mathbb{C}$ or $\overline{\F}_q$). If $K=\overline{\F}_q$ we will work with $\ell$-adic sheaves and if $K=\mathbb{C}$ we can work either with $\ell$-adic sheaves or complex sheaves for the analytic topology. We thus let $\kappa$ be the field $\C$ or $\overline{\Q}_\ell$ depending on the context and use the same letter to denote the constant sheaf. \subsection{Local systems on semisimple conjugacy classes of $\mathrm{PGL}_2(K)$} \label{premilinariesconjclassesC} We denote by $$ p:\Gl_2(K) \to \PGl_2(K) $$ the canonical projection map. Let $T \subseteq \Gl_2(K)$ and $\mathcal{T}\subseteq\PGl_2(K)$ be the maximal tori of diagonal matrices. Identify $W_{\Gl_2}(T)=\Z/2\Z$ and $W_{\PGl_2}(\mathcal{T})=\Z/2\Z$ in the classical way, through permutation matrices. Notice that $p(W_{\Gl_2}(T))=W_{\PGl_2}(\mathcal{T})$. We now describe the irreducible $\PGl_2$-equivariant local systems on the regular semisimple conjugacy classes of $\PGl_2(K)$. \vspace{6 pt} For $x \in K^\setminus{1}$, let $g_x$ be the matrix $$ g_x=\begin{pmatrix}1 &0 \\ 0 &x \end{pmatrix}. $$ The centraliser $C_{\Gl_2}(g_x)$ of $g_x$ in $\Gl_2(K)$ is $T$ and so in particular it is connected. Therefore the only irreducible $\Gl_2(K)$-equivariant local system on the $\Gl_2(K)$-conjugacy class $C_x$ of $g_x$ is the constant sheaf $\kappa$. \bigskip $\bullet$ If $x \neq -1$, then $C_{\PGl_2(K)}(p(g_x))=\mathcal{T}$ is connected and so the only irreducible $\PGl_2(K)$-equivariant local system on the $\PGl_2(K)$-conjugacy class $\mathcal{C}_x$ of $p(g_x)$ is the constant sheaf. Notice that $p$ restricts to an isomorphism $$ p:C_x \to \mathcal{C}_x. $$ $\bullet$ If $x=-1$, then $$ C_{\PGl_2(K)}(p(g_{-1}))=\mathcal{T} \rtimes W_{\PGl_2}(\mathcal{T})=\mathcal{T} \rtimes \Z/2\Z $$ has two connected components and so there are exactly two (up to isomorphism) irreducible $\PGl_2(K)$-equivariant local systems on the $\PGl_2(K)$-conjugacy class $\mathcal{C}_{-1}$ of $p(g_{-1})$. The non-trivial local system $\mathcal{L}_\epsilon$ is obtained from the $2$-covering $$ p:C_{-1} \to \mathcal{C}_{-1} $$ as $$ p_(\kappa) \cong \kappa \oplus \mathcal{L}_\epsilon. $$ The corresponding non-trivial covering automorphism $\sigma: C_{-1} \to C_{-1}$ is given by $\sigma(x)=-x$. The local system $\mathcal{L}_\epsilon$ has thus the following explicit description. For an open $U \subseteq \mathcal{C}_{-1} $, we have \begin{equation} \label{explicitwritinglocalsystem} \mathcal{L}_\epsilon(U)=\{f \in \kappa(p^{-1}(U)) \ \| \ f \circ \sigma=-f\}. \end{equation} We will call the conjugacy classes $\mathcal{C}_x$, with $x\neq -1$, \emph{non-degenerate} and $\mathcal{C}_{-1}$ \emph{degenerate}. \subsection{Geometry of $\mathrm{PGL}_2$-character stacks} \label{paragraphgeometry} Let $p^k:(\Gl_2)^k \to (\PGl_2)^k$ where $p:\Gl_2\rightarrow\PGl_2$ is the quotient map. Fix a $k$-tuple $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$ of regular semisimple conjugacy classes of $\PGl_2(K)$ and consider a $k$-tuple $C=(C_1,\dots,C_k)$ of conjugacy classes of $\Gl_2(K)$ such that $p(C_i)=\mathcal{C}_i$ for each $i=1,\dots,k$. Fix a square root $$ \lambda_C=\sqrt{\prod_{i=1}^k \det(C_i)} $$ and consider the following affine algebraic varieties $$ X_{\mathcal{C}}\coloneqq \left\{(x_1,\dots,x_k) \in \mathcal{C}_1 \times \cdots \times \mathcal{C}_k \ \| \ x_1 \cdots x_k=1\right\} $$ $$X^+_C\coloneqq \{(x_1,\dots,x_k) \in C_1 \times \cdots \times C_k \ \| \ x_1 \cdots x_k=\lambda_C I_2\} $$ $$X^-_C\coloneqq \{(x_1,\dots,x_k) \in C_1 \times \cdots \times C_k \ \| \ x_1 \cdots x_k=-\lambda_C I_2\} .$$ We have a decomposition $$(p^k)^{-1}(X_{\mathcal{C}})= X^+_{\mathcal{C}} \bigsqcup X^-_{\mathcal{C}} $$ Notice that $\PGl_2$ acts diagonally by conjugation on each of the above varieties. We consider the following GIT quotients $$ M^{\pm}_C\coloneqq X^{\pm}_C /\!/ \PGl_2$$ $$ M_{\mathcal{C}}\coloneqq X_{\mathcal{C}} /\!/ \PGl_2 $$ and the quotient stacks $$\mathcal{M}^{\pm}_C\coloneqq [X^{\pm}_C /\PGl_2]. $$ $$\mathcal{M}_{\mathcal{C}}\coloneqq [X_{\mathcal{C}} /\PGl_2] $$ We have the following Proposition. \begin{prop} \label{propdescriptiongeometry} (1) If the $\Gl_2$-conjugacy classes $C_1,\dots,C_k$ are all non-degenerate, we have isomorphisms: \begin{equation} \label{isom1} M_{\mathcal{C}} \cong M^+_C \bigsqcup M^-_C \end{equation} and \begin{equation} \label{isom2} \mathcal{M}_{\mathcal{C}}=\mathcal{M}^{+}_C\bigsqcup \mathcal{M}^{-}_C \end{equation} (2) Suppose that there are $m \geq 1$ degenerate conjugacy classes among $C_1,\dots,C_k$. We have isomorphisms \begin{equation}\label{isom3} M_C^+ \cong M_C^-\end{equation} \begin{equation}\label{isom4} \mathcal{M}_C^+ \cong \mathcal{M}^-_C. \end{equation} Moreover, denote by $p^+:\mathcal{M}_C^+ \to \mathcal{M}_{\mathcal{C}}$ the map obtained by restricting $p^k$ to $X_C^+$. The map $p^+$ is a Galois covering with Galois group $H_m={\rm Ker}(\epsilon_m)$ (see \S \ref{twisted}). \end{prop} \begin{proof} We have a Cartesian diagram \begin{center} \begin{tikzcd} X^+_C \bigsqcup X^-_C \arrow[r," "] \arrow[d," "] & C_1 \times \cdots \times C_k \arrow[d," p^k"] \\ X_{\mathcal{C}} \arrow[r," "] & \mathcal{C}_1 \times \cdots \times\mathcal{C}_k. \end{tikzcd} \end{center} \item From the description of the conjugacy classes given in \cref{premilinariesconjclassesC}, if each the conjugacy classes $\mathcal{C}_i$ are all non-degenerate, $p^k$ is an isomorphism and \begin{equation} X^+_{\mathcal{C}} \bigsqcup X^-_{\mathcal{C}} \cong X_{\mathcal{C}} \end{equation} Taking the quotient by $\PGl_2$, we obtain the isomorphisms (\ref{isom1}) and (\ref{isom2}). \item In the rest of proof, we assume that $\mathcal{C}_1,\dots,\mathcal{C}_m$ are the degenerate classes. From the description of the conjugacy classes given in \cref{premilinariesconjclassesC}, if there are $m$ degenerate conjugacy classes, we see that $p^k$ is a Galois covering with group $({\bm \mu}_2)^m$. The $({\bm\mu}_2)^m$-action is defined as follows. For $y=(y_1,\dots,y_m) \in ({\bm\mu}_2)^m$ and $(x_1,\dots,x_k) \in C_1 \times \cdots \times C_k$, we have $$y \cdot (x_1,\dots,x_k)=(y_1x_1,y_2x_2,\dots,y_mx_m,x_{m+1},\dots,x_k) .$$ By pullback, we see that the map $$X_C^+ \sqcup X_C^- \to X_{\mathcal{C}}$$ is a $({\bm\mu}_2)^m$-Galois covering too. Consider the element $\tau=(-1,1,\dots,1) \in ({\bm\mu}_2)^m$ and let $H_m'$ be the subgroup of $({\bm\mu}_2)^m$ generated by $\tau$. Notice that \begin{equation} \label{splittinggroup} ({\bm\mu}_2)^m \cong H_m \oplus H_m' \end{equation} The automorphism associated to $\tau$ restricts to an isomorphism $$\tau:X_C^+ \to X_C^-$$ $$(x_1,\dots,x_k) \to (-x_1,\dots,x_k) .$$ It is $\PGl_2$-equivariant from which we get the isomorphisms (\ref{isom3}) and (\ref{isom4}). \vspace{2 pt} From the splitting (\ref{splittinggroup}), we see that $$X_C^+ \to X_{\mathcal{C}}$$ is a Galois covering with Galois group $H_m$. Since the Galois covering structure is $\PGl_2$-equivariant, we deduce that the quotient map $p^+$ is also a Galois covering with Galois group $H_m$. \end{proof} \begin{remark} Consider the case in which $m=1$, i.e. there is exactly one degenerate class among $\mathcal{C}_1,\dots,\mathcal{C}_k$. From Proposition \ref{propdescriptiongeometry}, we see that we have an isomorphism $$\mathcal{M}^+_C \cong \mathcal{M}_{\mathcal{C}}.$$ \end{remark} Consider the $k+1$-tuples of conjugacy classes $$ C^+=(C_1,\dots,C_k,\lambda_C^{-1} I_2),\,\,\,\,C^-=(C_1,\dots,C_k,-\lambda_C^{-1} I_2) $$ of $\Gl_2$. \bigskip \begin{definizione} \label{definitiongenericity} The $k$-tuple $\mathcal{C}$ of conjugacy classes of $\PGl_2$ is said to be \emph{generic} if the $(k+1)$-tuples $C^+,C^-$ are both generic in the sense of \S \ref{GL2}. \end{definizione} \vspace{2 pt} \begin{oss} Notice that, if at least one of the conjugacy class $\mathcal{C}_1,\dots,\mathcal{C}_k$ is degenerate (i.e. if $m \geq 1$), then $C^+$ is generic if and only if $C^-$ is generic. \end{oss} \vspace{2 pt} \begin{oss} In \cite[Definition 9]{KNWG}, the authors give a definition of a generic $k$-tuple of conjugacy classes for any reductive group $G$. It is not hard to see that their definition agrees with our definition in the case of $\PGl_2$. \end{oss} From \cite[Theorem 2.1.5]{HA} and Proposition \ref{propdescriptiongeometry} we deduce the following Theorem. \begin{teorema} \label{genericity} (1) If $\mathcal{C}$ is generic and $\mathcal{C}_1,\dots,\mathcal{C}_k$ are non degenerate, the $\PGl_2$-character stack $\mathcal{M}_{\mathcal{C}}$ is a smooth algebraic variety (i.e. the canonical morphism $\mathcal{M}_{\mathcal{C}}\rightarrow M_{\mathcal{C}}$ is an isomorphism of stacks) of dimension $2k-6$. (2) If $\mathcal{C}$ is generic and there are $m$ degenerate classes among $\mathcal{C}_1,\dots,\mathcal{C}_k$ with $m \geq 1$, the $\PGl_2$-character stack $\mathcal{M}_{\mathcal{C}}$ is a smooth Deligne-Mumford stack of dimension $2k-6$. More precisely, we have an isomorphism $\mathcal{M}_{\mathcal{C}} \cong [\mathcal{M}_{C}^+/H_m]$. \end{teorema} \vspace{4 pt} \begin{oss} If there are $m$ degenerate classes between $\mathcal{C}_1,\dots,\mathcal{C}_k$ with $m \geq 1$, it is not always true that the $\PGl_2$-action on $X_{\mathcal{C}}$ is free, i.e. that the canonical morphism $\mathcal{M}_{\mathcal{C}} \to M_{\mathcal{C}}$ is an isomorphism. Consider for example the case of $k=5$, $m=4$ and where $\mathcal{C}_5$ is the $\PGl_2$-conjugacy class of $p(g_4)$. Pick $\lambda_{\mathcal{C}}=2$. It is not hard to see that the $5$-tuple $\mathcal{C}$ is generic. We have that $$(X_1,\dots,X_5)=\left(\begin{pmatrix}1 &0\\ 0 &-1 \end{pmatrix},\begin{pmatrix}0 &\frac{1}{2}\\ 2 &0 \end{pmatrix},\begin{pmatrix}-1 &0\\ 0 &1 \end{pmatrix},\begin{pmatrix}0 &1\\ 1 &0 \end{pmatrix},\begin{pmatrix}4 &0\\ 0 &1 \end{pmatrix}\right) \in X^+_C $$ and therefore $(p(X_1),\dots,p(X_5)) \in X_{\mathcal{C}}$. Notice that \begin{equation} g_{-1}\cdot (X_1,\dots,X_5)=(X_1,-X_2,X_3,-X_4,X_5) \end{equation} from which we deduce that $p(g_{-1}) \cdot (p(X_1),\dots,p(X_5))=(p(X_1),\dots,p(X_5))$, i.e. $p(g_{-1}) \in \Stab_{\PGl_2}((p(X_1),\dots,p(X_5)))$. \end{oss} \subsection{Local systems on $\mathrm{PGL}_2$-character stacks} \label{sectionlocalsystemscharacterstacks} We keep the notation of the previous section and we fix a non-negative integer $m<k$ and we assume that the first $m$ conjugacy classes of $\mathcal{C}_1,\dots,\mathcal{C}_k$ are degenerate and the last $k-m$ conjugacy classes are non-degenerate. From Proposition \ref{propdescriptiongeometry}, the map $p^+:\mathcal{M}^+_C \to \mathcal{M}_{\mathcal{C}}$ is a Galois covering with Galois group $H_m$. We have therefore a decomposition \begin{equation} \label{decompositionsplitting} (p^+)_\kappa \cong \bigoplus_{\chi \in \widehat{H_m}} \mathcal{F}_{\chi}, \end{equation} where $\widehat{H_m}$ denotes the set of irreducible characters of $H_m$ and where each $\mathcal{F}_{\chi}$ is an irreducible local system. We have for any open subset $U$ of $\mathcal{M}_{\mathcal{C}}$ \begin{equation} \label{descriptionlocalsystemgalois} \mathcal{F}_{\chi}(U)=\{f \in \kappa((p^+)^{-1}(U)) \ \| \ y^(f)=\chi(h)f \ \ \text{ for each }y \in H_m\} \end{equation} where, for $y \in H_m$, we denote by $y:\mathcal{M}^+_C \to \mathcal{M}^+_C$ the corresponding covering automorphism. \bigskip Let us now describes the set of irreducible characters of $H_m$. For each subset $A \subseteq \{1,\dots,m\}$, define $\chi_A \in \widehat{({\bm\mu}_2)^m}$ as $$\chi_A:({\bm\mu}_2)^m \to \C^\times$$ $$(y_1,\dots,y_m) \mapsto \prod_{j \in A}y_j .$$ \begin{esempio} With the notation just introduced, we have that $\chi_{\emptyset}=1$ and $\chi_{\{1,\dots,m\}}=\epsilon_m$. \end{esempio} \vspace{4 pt} Notice that $\widehat{({\bm\mu}_2)^{m}}=\{\chi_A\}_{A \subseteq \{1,\dots,m\}}$ and we have the following Proposition. \begin{prop} The restriction map $$\Res_{H_m}:\widehat{({\bm\mu}_2)^{m}} \to \widehat{H_m} $$ is surjective and we have $\Res_{H_m}(\chi_A)=\Res_{H_m}(\chi_B)$ if and only if $\chi_A=\chi_B$ or $\chi_A=\epsilon_m \chi_B$, i.e. if and only if $A=B$ or $B=A^c$. \end{prop} For $B\subseteq\{2,\dots,m\}$ we denote $\Res_{H_m}(\chi_{B\cup\{1\}})$ by $\overline{\chi_B}$. We have therefore the following bijection $$\widehat{H_m} \longleftrightarrow \{\overline{\chi_B} \ \| \ B \subseteq \{2,\dots,m\}\} .$$ \bigskip The decomposition of (\ref{decompositionsplitting}) can therefore be rewritten as \begin{equation} \label{decompositionsplitting1} (p^+)_\kappa \cong \bigoplus_{B \subseteq \{2,\dots,m\}} \mathcal{F}_{\overline{\chi_B}}. \end{equation} We now give a concrete description of the local systems $\mathcal{F}_{\overline{\chi_B}}$. Consider the closed embedding $$i_{\mathcal{C}}:X_{\mathcal{C}} \to \mathcal{C}_1 \times \cdots \times\mathcal{C}_k. $$ \vspace{2 pt} For any $B \subseteq \{2,\dots,m\}$, put $$\mathcal{L}_B \coloneqq i_{\mathcal{C}}^\left(\mathcal{E}_1\boxtimes\mathcal{E}_2\boxtimes\cdots\boxtimes\mathcal{E}_k \right)$$ where $$ \mathcal{E}_i=\begin{cases}\mathcal{L}_\epsilon&\text{ if }i\in \{1\}\cup B,\\\kappa&\text{ otherwise}.\end{cases} $$ Notice that $\mathcal{L}_B$ is $\PGl_2$-equivariant and defines therefore a local system $\mathcal{F}_B$ on $\mathcal{M}_{\mathcal{C}}$. From (\ref{explicitwritinglocalsystem}) and (\ref{descriptionlocalsystemgalois}), we see that for any $B \subseteq \{2,\dots,m\}$, we have an equality \begin{equation} \label{equalitylocalsytems} \mathcal{F}_B=\mathcal{F}_{\overline{\chi_B}}. \end{equation} The decomposition (\ref{decompositionsplitting1}) can therefore be rewritten as \begin{equation} \label{decompositionsplitting2} (p^+)_\kappa \cong \bigoplus_{B \subseteq \{2,\dots,m\}} \mathcal{F}_B. \end{equation} \begin{remark}Notice that if $m=1$, then $B$ is necessarily the empty set and $$ \mathcal{L}_B=(i_{\mathcal{C}})^(\mathcal{L}_\epsilon\boxtimes\kappa\boxtimes\cdots\boxtimes\kappa) $$ is the constant sheaf. \end{remark} \subsection{Cohomology of generic $\mathrm{PGL}_2$-character stacks}\label{cohomology} In this section, we give our main results concerning the cohomology of $\PGl_2$-character stacks. \vspace{2 pt} If $K=\overline{\F}_q$, we pick a generic $k$-tuple $\mathcal{C}$ of regular semisimple conjugacy classes of $\PGl_2(\overline{\F}_q)$ which are $F$-stable and split, i.e. each $\mathcal{C}_i$ is the conjugacy class of $p(g_{x_i})$ with $x_i \in \F_q^$. Notice that, for such a $\mathcal{C}$, we can find a $k$-tuple $C$ of semisimple conjugacy classes of $\Gl_2(\overline{\F}_q)$ which are $F$-stable and split and such that $p(C_i)=\mathcal{C}_i$. For instance we can take $C_i$ to be the conjugacy class of $g_{x_i}$ for each $i$. In this case, we assume in what follows that $\lambda_{C} \in \F_q^$, or, equivalently, that $$\prod_{i} \det(C_i) \in (\F_q^)^2 ,$$ where $(\F_q^)^2 \subseteq \F_q^$ is the subgroup of squares. Notice that, under these assumptions, the constructions of \cref{paragraphgeometry} and \cref{sectionlocalsystemscharacterstacks} are all compatible with $F$. \begin{remark} If $\lambda_C \not\in \F_q^$, then $\|\mathcal{M}_{\mathcal{C}}^F\|=0$, see \cref{eulerspecializationproof} for more details. \end{remark} \subsubsection{The non-degenerate case} Let us first discuss the simpler case where the conjugacy classes $\mathcal{C}_1,\dots,\mathcal{C}_k$ are all non-degenerate. From Proposition \ref{propdescriptiongeometry} and Theorem \ref{E-poly}, we deduce the following Theorem. \begin{teorema} \label{theoremEnondegenerate} If $\mathcal{C}$ is a generic $k$-tuple of non-degenerate regular semisimple conjugacy classes of $\PGl_2$, we have \begin{equation} E(\mathcal{M}_{\mathcal{C}};q)=2q^{k-3}\mathbb{H}_{\bm \mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right) \end{equation} \end{teorema} \begin{remark}We know from Theorem \ref{irreducible} and Theorem \ref{E-poly} that the coefficient of the highest power of $q$ in $\mathbb{H}_{\bm \mu}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right)$ equals $1$. Therefore the coefficient of the highest power of $q$ in $E(\mathcal{M}_{\mathcal{C}};q)$ equals $2$ which is also the number of connected components of the center of the dual group $\Sl_2$ of $\PGl_2$. This has been previously observed for an arbitrary connected reductive group \cite[Remark 3 (iii)]{KNWG}. \end{remark} The following conjecture is a consequence of the conjectural formula (\ref{conj}) and Proposition \ref{propdescriptiongeometry}. \begin{conjecture} \label{conjEnondegenerate} If $\mathcal{C}$ is a generic $k$-tuple of non-degenerate regular semisimple conjugacy classes of $\PGl_2$, we have \begin{equation} H_c(\mathcal{M}_{\mathcal{C}};q,t)=2(qt^2)^{k-3}\mathbb{H}_{\bm \mu}\left(\frac{1}{\sqrt{q}},-t\sqrt{q}\right) \end{equation} In particular, the pure part is given by the formula $$ PH_c(\mathcal{M}_\mathcal{C};q)=2q^{k-3}\mathbb{H}_{\bm \mu}(0,\sqrt{q}).$$ \end{conjecture} \bigskip \subsubsection{The degenerate case} Fix now $1\leq m<k$ and consider the case where the first $m$ conjugacy classes of $\mathcal{C}_1,\dots,\mathcal{C}_k$ are degenerate. \bigskip Taking cohomology in the decomposition (\ref{decompositionsplitting2}) we have the following isomorphism of $H_m$-representations: \begin{equation} \label{decompositioncohomologyirred} H_c^\bullet(\mathcal{M}^+_C) \cong \bigoplus_{B\subseteq\{2,\dots,m\}}H_c^\bullet(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B) \otimes \overline{\chi_B}. \end{equation} i.e. $H^\bullet_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B)$ is the subspace of $H^\bullet_c(\mathcal{M}_C^+)$ on which the group $H_m$ acts by the character $\overline{\chi_B}$. \bigskip The decomposition (\ref{decompositioncohomologyirred}) respects the weight filtrations on both sides and so we have an equality of twisted mixed Poincar\'e series (see \S \ref{twisted}), i.e. for any $y\in H_m$, \begin{equation} \label{directequalityequivariant} H_c^y(\mathcal{M}^+_C;q,t)=\sum_{B\subseteq\{2,\dots,m\}} H_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B;q,t)\,\overline{\chi_B}(y), \end{equation} where for a local system $\mathcal{L}$ on $X$ we put $$ H_c(X,\mathcal{L};q,t):=\sum_{i,k}{\rm dim}(W^k_i/W^k_{i-1})q^{i/2}t^k $$ with $W^k_\bullet$ the weight filtration on $H^k_c(X,\mathcal{L})$. \vspace{2 pt} Given the orthogonality relations in the character ring of $H_m$, we can "invert" the isomorphism (\ref{decompositioncohomologyirred}) and obtain that for each $B\subseteq\{2,\dots,m\}$, we have \begin{equation} \label{invertedrelationcharacters} H_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B;q,t)=\frac{1}{\|H_m\|}\sum_{y \in H_m}H^y_c(\mathcal{M}^+_C;q,t)\,\overline{\chi_B}(y) \end{equation} The equation (\ref{invertedrelationcharacters}) allows to express the cohomology of the $\PGl_2$-character stack $\mathcal{M}_{\mathcal{C}}$ with coefficient in the local system $\mathcal{F}_B$ just in terms of the cohomology of $\mathcal{M}^+_C$ and its $H_m$-action. \bigskip For any $m_1,m_2,r \in \N$, denote by $C_{m_1,m_2,r}$ the coefficient of $y^rx^{m_1+m_2-r}$ in the product $(x-y)^{m_1}(x+y)^{m_2}$, and define $$ \mathbb{A}_r(z,w):=\begin{cases}\frac{(w^2+1)^{k-r}(1-w^2)^r}{(z^2-w^2)(1-w^4)}+\frac{(1-z^2)^r(z^2+1)^{k-r}}{(z^4-1)(z^2-w^2)}&\text{ if }0<r\leq m,\\ &\\ \frac{(w^2+1)^k}{(z^2-w^2)(1-w^4)}+\frac{(z^2+1)^k}{(z^4-1)(z^2-w^2)}-\frac{2^{k-1}}{(z^2-1)(1-w^2)}&\text{ if }r=0. \end{cases} $$ \bigskip \begin{remark}Notice that $\mathbb{A}_0(z,w)=\mathbb{H}_{\bm \mu}(z,w)$ for ${\bm \mu}=((1^2),\dots,(1^2))$ (see Proposition \ref{H}) and for $r>0$, $\mathbb{A}_r(z,w)=\mathbb{H}_{\bm \mu}^y(z,w)$ with $y\in H_m$ such that $r(y)=r$ (see Proposition \ref{H-twisted}). \label{remA}\end{remark} We have the following results concerning the cohomology of the local systems $\mathcal{F}_B$. \begin{teorema} \label{Epolynomialgenericdegenerate} For any $B \subseteq \{2,\dots,m\}$, we have \begin{align} E(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B;q):&=H_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_ B;q,-1)\\&=\frac{q^{k-3}}{2^{m-1}}\sum_{\substack{r=0\\r \text{ even}}}^{m}C_{\|B\|+1,m-\|B\|-1,r}\,\,\mathbb{A}_r\left(\frac{1}{\sqrt{q}},\sqrt{q}\right). \end{align} \end{teorema} \begin{proof} From Formula (\ref{invertedrelationcharacters}) and Theorem \ref{Ey} we have \begin{align} E(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B;q)&=\frac{1}{\|H_m\|}\sum_{y \in H_m}E^y(\mathcal{M}^+_C;q)\,\overline{\chi_B}(y)\\ &=\frac{q^{d_{\bm \mu}/2}}{\|H_m\|}\sum_{y\in H_m}\mathbb{H}^y_{{\bm \mu}}\left(\frac{1}{\sqrt{q}},\sqrt{q}\right) \,\overline{\chi_B}(y) \end{align} First of all notice that $d_{\bm\mu}=2k-6$ when ${\bm\mu}=((1^2),\dots,(1^2))$ and from Proposition \ref{H-twisted} we see that $\mathbb{H}^y_{\bm\mu}(z,w)=\mathbb{A}_{r(y)}(z,w)$ depends only on the number $r(y)$ of $i=1,\dots,m$ such that $y_i=-1$ (notice that $r(y)$ must be even to have $y\in H_m$). Fix an even number $r$ and $y \in H_m$ such that $r(y)=r$. The value $\overline{\chi_B}(y)$ depends on the parity of $\#\{i\in B \cup \{1\}\,\|\, y_i=-1\}$, it equals $1$ if this number is even and it equals $-1$ is this number is odd. For any even $r \in \{0,\dots,m\}$, we thus have \begin{align} \sum_{\substack{y \in H_m\\ r(y)=r}}\overline{\chi_B}(y)\,\mathbb{H}^y_{\bm \mu}(z,w)&=\mathbb{A}_r(z,w)\sum_{\substack{y \in H_m\\ r(y)=r}}(-1)^{\#\{i\in B \cup \{1\}\,\|\, y_i=-1\}}\\ &=\mathbb{A}_r(z,w)\sum_{\substack{A \subseteq \{1,\dots,m\}\\\|A\|=r}}(-1)^{\|A \cap (B \cup \{1\})\|} \\ &=C_{\|B\|+1,m-\|B\|-1,r}\, \mathbb{A}_r(z,w). \end{align} \end{proof} The following conjecture is a consequence of Conjecture \ref{conj-twisted} and Formula (\ref{invertedrelationcharacters}). \begin{conjecture} \label{conjmhslocalsystems} For any $B \subseteq \{2,\dots,m\}$, we have \begin{equation} H_c(\mathcal{M}_\mathcal{C},\mathcal{F}_B;q,t)=\frac{(qt^2)^{k-3}}{2^{m-1}}\sum_{\substack{r=0\\r \text{ even}}}^{m}C_{\|B\|+1,m-\|B\|-1,r}\,\,\mathbb{A}_r\left(\frac{1}{\sqrt{q}},-t\sqrt{q}\right). \end{equation} In particular, the Poincar\'e polynomial of the pure part is given by \begin{equation} PH_c(\mathcal{M}_\mathcal{C},\mathcal{F}_B;q)=\frac{(qt^2)^{k-3}}{2^{m-1}}\sum_{\substack{r=0\\r \text{ even}}}^{m}C_{\|B\|+1,m-\|B\|-1,r}\,\,\mathbb{A}_r\left(0,\sqrt{q}\right). \label{pure-conj}\end{equation} \end{conjecture} \section{Review on Deligne-Lusztig theory and Lusztig's character-sheaves theory for ${\rm GL}_2$ and ${\rm SL}_2$} In this section we review the construction of the complex character table of $\Gl_2(\F_q)$, $\Sl_2(\F_q)$ as well as the construction of the character-sheaves by Lusztig \cite{Lusztig}\cite{CS2}\cite{mars}. \bigskip We assume that $q$ is odd. \bigskip On $G=\Gl_n(\overline{\F}_q),\,\Sl_n(\overline{\F}_q)$ we will only consider the standard Frobenius $F:G\rightarrow G$ that raises matrix coefficients to their $q$-th power so that the finite group $G^F$ of points fixed by $F$ is exactly matrices with coefficients in the finite field $\F_q$. \bigskip As we will work with $\ell$-adic sheaves, we need to choose a prime $\ell$ which does not divide $q$. \subsection{Deligne-Lusztig theory for $\mathrm{GL}_2$ and $\mathrm{SL}_2$}\label{table-SL} Consider a connected reductive algebraic group $G$ over $\overline{\F}_q$ with a geometric Frobenius $F:G\rightarrow G$, an $F$-stable maximal torus $T$ of $G$, a Borel subgroup $B$ containing $T$ and we denote by $U$ the unipotent radical of $B$. We have $B=T\ltimes U$. For $G=\Gl_n$ or $\Sl_n$, an example of such a pair would be the upper triangular matrices for $B$ and the diagonal matrices for $T$ in which case both $T$ and $B$ are $F$-stable if $F$ is the standard Frobenius, but we will need also $F$-stable maximal tori which are not contained in $F$-stable Borel subgroups. \bigskip Denote by $$ L:G\rightarrow G,\hspace{.5cm}g\mapsto g^{-1}F(g) $$ the Lang map (it is surjective by a theorem of Steinberg). \bigskip The affine algebraic variety $L^{-1}(U)$ is equipped with an action of $G^F$ by left multiplication and with an action of $T^F$ by right multiplication (because $U$ is a normal subgroup of $B$). These actions induce actions on the compactly supported $\ell$-adic cohomology groups $H_c^i(L^{-1}(U),\overline{\Q}_{\ell})$. For a linear character $\theta$ of $T^F$ we define the Deligne-Lusztig virtual character $R_T^G(\theta)$ of $G^F$ $$ R_T^G(\theta)(g)=\sum_ i(-1)^i\tr\left(g,\, H_c^i(L^{-1}(U),\overline{\Q}_\ell)\otimes_{\overline{\Q}_\ell[T^F]}\theta\right) $$ for all $g\in G^F$. These virtual characters were defined by Deligne and Lusztig \cite{DL} (see also \cite{DM}). These are very important in the classification of the irreducible characters of $G^F$ (analogue of Jordan decomposition of irreducible characters by Lusztig). When $G=\Gl_n$, we can express all irreducible characters of $G^F$ as a linear combination of these Deligne-Lusztig characters (this is not the case for $\Sl_2$ as we will see below). \bigskip When the maximal torus $T$ is contained in some $F$-stable Borel subgroup $B$ (we say that $T$ is \emph{split}), then $R_T^G(\theta)$ has the following more concrete description (see \cite[below Definition 11.1]{DM}): Consider the canonical projection $\pi:B^F\rightarrow T^F$, then $$ R_T^G(\theta)={\rm Ind}_{B^F}^{G^F}(\pi^(\theta)) $$ where ${\rm Ind}_{B^F}^{G^F}$ is the classical induction in representations of finite groups. If moreover $\theta$ is the trivial character then $R_T^G(\theta)$ is the character afforded by the $G^F$-module $\overline{\Q}_\ell[G^F/B^F]$ where $G^F$ acts by left multiplication on $G^F/B^F$. \bigskip We have the following result. \begin{prop}\cite[Proposition 13.22]{DM} Let $T$ be an $F$-stable maximal torus of $\Gl_n$ and denote by $T'$ the maximal torus $T\cap\Sl_n$ of $\Sl_n$. Let $\theta$ be a linear character of $T^F$, then $$ R_{T'}^{\Sl_n}(\theta\|_{T'{^F}})=R_T^{\Gl_n}(\theta)\|_{\Sl_n^F}. $$ \end{prop} When $G=\Gl_2$, we have two $G^F$-conjugacy classes of $F$-stable maximal tori whose representatives are chosen as follows $$ T_1=\left.\left\{\left(\begin{array}{cc}a&0\\0&b\end{array}\right)\right\|\, a,b\in\overline{\F}_ q^\right\},\hspace{0.5cm}T_\sigma= \left.\Biggl\{\dfrac{1}{\alpha^q-\alpha}\begin{pmatrix} a\alpha^q-b\alpha &-a+b\\ (a-b)\alpha^{q+1} &-a\alpha+b\alpha^q \end{pmatrix} \ \right\| \ a,b \in \overline{\F}_q^ \Biggr\} . $$ where $\alpha$ is a fixed element of $\F_{q^2}$ which is not in $\F_ q$. Here $\sigma$ denotes the non-trivial element of the Weyl group $W=S_2$ of $\Gl_2$ with respect to $T_1$. Then $T_1^F\simeq \F_q^\times\F_q^$ and $$T_\sigma^F=\Biggl\{\dfrac{1}{\alpha^q-\alpha}\begin{pmatrix} a\alpha^q-b\alpha &-a+b\\ (a-b)\alpha^{q+1} &-a\alpha+b\alpha^q \end{pmatrix} \ \| \ a \in \F_{q^2}^* \text{ and } b=a^q \Biggr\}\simeq\F_{q^2}^. $$ Notice that the diagram \begin{equation} \xymatrix{T_1\ar[rr]^{\sigma F:=\sigma\circ F}\ar[d]^g&& T_1\ar[d]^g\\ T_\sigma\ar[rr]^F&&T_\sigma} \label{conj1}\end{equation} where the vertical arrows are the conjugation $t\mapsto gtg^{-1}$ by $g=\left(\begin{array}{cc}1&1\\\alpha&\alpha^q\end{array}\right)$, commutes since $g^{-1}F(g)=\sigma$. A matrix in $T_\sigma^F$ is thus geometrically conjugate to a diagonal matrix with eigenvalues of the form $x$ and $x^q$ with $x\in\F_{q^2}^$. \bigskip When $G=\Sl_2$, we have also two $G^F$-conjugacy classes of $F$-stable maximal tori whose representatives can be chosen as $T'_w:=T_w\cap\Sl_n$ with $w=1,\sigma$. \bigskip The following table gives the values of Deligne-Lusztig characters $R_{T_w}^G(\theta)$ in the case of $G=\Gl_2$ (and by restriction we get also the values in the $G=\Sl_2$ case), see \cite[\S 15.9]{DM} \begin{scriptsize} \begin{equation} \label{tableGL} \begin{array}{\|c\|c\|c\|c\|c\|} \hline &&&&\\ &\left(\begin{array}{cc}a&0\\0&a\end{array}\right)&\left(\begin{array}{cc}a&0\\0&b\end{array}\right)& \left(\begin{array}{cc}x&0\\0& x^q\end{array}\right)&\left(\begin{array}{cc}a&1\\0&a\end{array}\right)\\ &a\in\F_q^&a\neq b\in\F_q^&x\neq x^q\in\F_{q^2}^&a\in\F_q^\\ \hline &&&&\\ R_{T_1}^{\Gl_2}(\alpha,\beta)&(q+1)\alpha(a)\beta(a)&\alpha(a)\beta(b)+\alpha(b)\beta(a)&0&\alpha(a)\beta(a) \\ \alpha,\beta\in\widehat{\F_q^}&&&&\\ \hline &&&&\\ R_{T_\sigma}^{\Gl_2}(\omega)&(1-q)\omega(a)&0&\omega(x)+\omega(x^q)&\omega(a)\\ \omega\in\widehat{\F_{q^2}^}&&&&\\ \hline \end{array} \end{equation} \end{scriptsize} \bigskip The character tables of $\Gl_2(\F_q)$ and $\Sl_2(\mathbb{F}_q)$ from Deligne-Lusztig theory can be found for instance in \cite[15.9]{DM}. \bigskip {\bf List of the irreducible characters of $\Gl_2(\F_q)$ :} \bigskip (1) If $\alpha\neq\beta\in\widehat{\F_q^}$, then the Deligne-Lusztig characters $R_{T_1}^{\Gl_2}(\alpha,\beta)$ are irreducible (from Lusztig-Jordan decomposition they correspond to conjugacy classes of the second column). (2) If $\omega\neq\omega^q\in\widehat{\F_{q^2}^}$, then $-R_{T_\sigma}^{\Gl_2}(\omega)$ is an irreducible character (it corresponds to conjugacy classes of the third column). (3) The irreducible characters of the form $\alpha\circ{\rm det}$ and $(\alpha\circ{\rm det})\otimes{\rm St}$ where $\alpha\in\widehat{\F_q^}$ and where ${\rm St}$ (called the Steinberg character) is the non-trivial irreducible constituent of the character $R_{T_1}^{\Gl_2}({\rm Id})$. They decomposes into Deligne-Lusztig characters as $$ \alpha\circ{\rm det}=\frac{1}{2}\left(R_{T_1}^{\Gl_2}(\alpha\circ{\rm det})+R_{T_\sigma}^{\Gl_2}(\alpha\circ{\rm det})\right)$$ $$ (\alpha\circ{\rm det})\otimes{\rm St}=\frac{1}{2}\left(R_{T_1}^{\Gl_2}(\alpha\circ{\rm det})-R_{T_\sigma}^{\Gl_2}(\alpha\circ{\rm det})\right)$$ \bigskip {\bf List of the irreducible characters of $\Sl_2(\F_q)$ :} \bigskip First of all notice that $$ T'_1{^F}\simeq \F_q^,\hspace{1cm}T'_\sigma{^F}\simeq \mu_{q+1}\subset\F_{q^2}^. $$ (1) If $\alpha\in\widehat{\F_q^}$ satisfies $\alpha^2\neq 1$, then $R_{T'_1}^{\Sl_2}(\alpha)$ is an irreducible character of $\Sl_2^F$. (2) If $\omega\in\widehat{\mu_{q+1}}$ satisfies $\omega^2\neq 1$, then $-R_{T'_\sigma}^{\Sl_2}(\omega)$ is irreducible. (3) We have the unipotent characters ${\rm Id}$ and ${\rm St}$ which are the restrictions of the unipotent characters of $\Gl_2(\F_q)$. (4) If $\alpha_o$ is the nontrivial square of $1$ in $\widehat{\F_q^}$, then $R_{T'_1}^{\Sl_2}(\alpha_o)$ splits as a sum of two irreducible characters $\chi_{\alpha_o}^+$ and $\chi_{\alpha_o}^-$ of same degree $\frac{q+1}{2}$. (5) If $\omega_o$ is the non-trivial square of $1$ in $\widehat{\mu_{q+1}}$, then $-R_{T_\sigma'}^{\Sl_2}(\omega_o)$ splits as a sum of two irreducible characters $\chi_{\omega_o}^+$ and $\chi_{\omega_o}^-$ both of same degree $\frac{q-1}{2}$. \bigskip All irreducible characters of $\Sl_2(\F_q)$ in cases (1), (2) and (3) arise as the restriction of irreducible characters of $\Gl_2(\F_q)$ and so can be written as a linear combination of Deligne-Lusztig characters of $\Sl_2(\F_q)$. The irreducible characters $\chi_{\alpha_o}^{\pm}$ and $\chi_{\omega_o}^\pm$ are not restrictions of irreducible characters of $\Gl_2$ and can not be written as a linear combination of Deligne-Lusztig characters of $\Sl_2$. \bigskip Recall that we have an exact sequence $$ \xymatrix{1\ar[r]&\mu_{q+1}\ar[r]&\F_{q^2}^\ar[r]^N&\F_q^\ar[r]&1} $$ where $N:\F_{q^2}^\rightarrow\F_q^, x\mapsto x^{q+1}$ is the norm map. \bigskip This dualizes into an exact sequence $$ \xymatrix{1\ar[r]&\widehat{\F_q^}\ar[r]^{N^}&\widehat{\F_{q^2}^}\ar[r]^-{\rm res}&\widehat{\mu_{q+1}}\ar[r]&1} $$ where ${\rm res}$ is the restriction. The embedding $N^$ identifies $\widehat{\F_q^}$ with the linear characters $\omega$ of $\F_{q^2}^$ such that $\omega^q=\omega$. Therefore, any $\omega\in {\rm res}^{-1}(\omega_o)$ satisfies $\omega^q\neq \omega$. \bigskip We thus have the following lemma. \begin{lemma} The (non-irreducible) characters $R_{T'_1}^{\Sl_2}(\alpha_o)$ and $-R_{T'_\sigma}^{\Sl_2}(\omega_o)$ are the restrictions respectively of the irreducible characters $R_{T_1}^{\Gl_2}(1,\alpha_o)$ and $-R_{T_\sigma}^{\Gl_2}(\omega)$ (for $\omega\in{\rm res}^{-1}(\omega_o)$). \label{restrict}\end{lemma} We will see in the next sections that all the irreducible characters of $\Gl_2$ and those of $\Sl_2$ in cases (1), (2) and (3) have a geometric nature which is not the case of the irreducible characters of $\Sl_2$ in case (4). More precisely Lusztig has defined certain simple perverse sheaves (called \emph{character-sheaves}) on $\Sl_2(\overline{\F}_q)$ whose characteristic functions coincide with the irreducible characters of $\Sl_2(\F_q)$ in cases (1), (2) and (3) but differs from the irreducible characters in case (4) and (5). \subsection{Characteristic functions of orbital complexes} In what follows, $X$ is an algebraic variety over $\overline{\F}_q$ equipped with a geometric Frobenius $F:X\rightarrow X$ (or equivalently, an $\F_q$-scheme $X_o$ such that $X=X_o\times_{\F_q}\overline{\F}_q$), for more details see for instance \cite[Chapter 4]{DM}. Let $D^b_c(X)$ be the category of $\overline{\Q}_{\ell}$-complexes with constructible cohomology. An $F$-equivariant structure (or Weil structure) on $\mathcal{F}\in D^b_c(X)$ is an isomorphism $$\phi:F^(\mathcal{F}) \to \mathcal{F} .$$ \begin{remark}If $\mathcal{F}$ is the pullback of a complex $\mathcal{F}_o$ on $X_o$, then it admits a canonical $F$-equivariant structure, see for instance \cite[Chapter 1]{Weil}. \end{remark} \bigskip We then say that $(\mathcal{F},\phi)$ is an $F$-\emph{equivariant complex} on $X$. Given two $F$-equivariant complexes $(\mathcal{F},\phi)$ and $(\mathcal{F}',\phi')$, the Frobenius $F$ acts on ${\rm Hom}(\mathcal{F},\mathcal{F}')$ as $$ f\mapsto \phi'\circ F^(f)\circ\phi^{-1}. $$ We denote by $D_c^b(X;F)$ the category of $F$-equivariant complexes on $X$ with ${\rm Hom}(\mathcal{F},\mathcal{F'})^F$ as set of morphisms $(\mathcal{F},\phi)\rightarrow(\mathcal{F}',\phi')$. The characteristic function of $(\mathcal{F},\phi)\in D_c^b(X;F)$ is the function $\X_{\mathcal{F},\phi}:X^F \to \overline{\Q}_{\ell}$ defined as $$\X_{\mathcal{F},\phi}(x)\coloneqq \sum_{i \in \Z}(-1)^i \tr(\phi_x^i: \mathcal{H}^i_x(\mathcal{F}) \to \mathcal{H}^i_x(\mathcal{F})) .$$ The function $\X_{\mathcal{F},\phi}$ does depend on the choice of the isomorphism $\phi$. However, in all the cases of relevance for this article, we can make a canonical choice of the isomorphism $\phi$ and we will often drop it from the notation. In particular, if $X$ is an algebraic group, we will always assume that $\phi_e=Id$. \bigskip Assume that $G$ is a connected linear algebraic group over $\overline{\F}_q$ equipped with a geometric Frobenius $F:G\rightarrow G$. For an $F$-stable conjugacy class $C$ of $G$ together with an $F$-stable $G$-equivariant irreducible local system $\mathcal{E}$ on $C$, we denote by $IC^\bullet_{\overline{C},\mathcal{E}}$ the intersection cohomology complex on the Zariski closure $\overline{C}$ of $C$ with coefficients in $\mathcal{E}$. We also fix an $F$-equivariant structure $\phi:F^(\mathcal{E})\simeq\mathcal{E}$ and we denote again by $\phi$ the induced $F$-equivariant structure on $IC^\bullet_{\overline{C},\mathcal{E}}$ (as $IC^\bullet_{\overline{C},\mathcal{E}}$ is irreducible, any two $F$-equivariant structure differ by a scalar). \begin{prop} \cite[Proposition 4.4.13]{Let} The set $\{{\bf X}_{IC^\bullet_{\overline{C},\mathcal{E}},\phi}\}$, where $(C,\mathcal{E})$ runs over the pairs as above, forms a basis of the space $\mathcal{C}(G^F)$ of class functions $G^F/G^F\rightarrow\overline{\Q}_\ell$. \end{prop} The above basis is a geometric counterpart of the basis of characteristic functions of conjugacy classes of $G^F$. When $G$ is reductive, the characteristic functions of the $F$-stable character-sheaves will be the geometric counter-part of the basis of irreducible characters of $G^F$. \subsection{Character-sheaves on tori} Let $T$ be a torus defined over $\F_q$, with geometric Frobenius $F:T \to T$. A \textit{Kummer local system} $\mathcal{E}$ is a $\overline{\Q}_{\ell}$-local system on $T$ such that $\mathcal{E}^{\otimes m}\simeq\overline{\Q}_\ell$ for some $m \in \N$ such that $(m,q)=1$. Notice that in particular every Kummer local system is of rank $1$ and thus simple. For any $F$-stable Kummer local system $\mathcal{E}$ on $T$, the characteristic function $\X_{\mathcal{E}}$ is a linear character of the finite group $T^F$ and any linear character of $T^F$ is obtained in this way, i.e. \begin{prop}\cite[Proposition 2.3.1]{mars} \label{bijectionkummer} The map $\mathcal{E} \to \X_{\mathcal{E}}$ is an isomorphism between the group of $F$-stable isomorphism classes of Kummer local systems on $T$ and the group $\widehat{T^F}$ of linear characters of $T^F$. \end{prop} The Kummer local systems are the character-sheaves for $T$. \begin{esempio} \label{examplekummer} Consider $T=\mathbb{G}_m$ with the Frobenius $F(x)=x^q$ for $x \in \mathbb{G}_m$. In this case, we have $T^F=\F_q^$. Consider a linear character $\alpha:\F_q^ \to \C^$ and let $n$ be the order of $\alpha$. In particular, $n$ divides $q-1$. Fix a surjection $q_n:\F_q^ \to \Z/n\Z$ (by sending a generator $\zeta$ of the cyclic group $\F_q^$ to its subgroup of order $n$ generated by $\zeta^{\frac{q-1}{n}}$). Since $\alpha^n=1$, there exists a linear character $\mu:\Z/n\Z \to \C^$ such that $\mu \circ q_n=\alpha$. Consider now the $\Z/n\Z$-Galois cover $f_n:\mathbb{G}_m \to \mathbb{G}_m$ given by $f_n(z)=z^n$. We have a splitting $$(f_n)_(\overline{\Q}_{\ell})=\bigoplus_{\xi \in \widehat{\Z/n\Z}} \mathcal{E}_{\xi} .$$ Since $f_n$ commutes with $F$, the local systems $\mathcal{E}_{\xi}$ are defined over $\F_q$ and have thus a canonical $F$-equivariant structure, and we have $\X_{\mathcal{E}_{\mu}}=\alpha$. \end{esempio} \subsection{Grothendieck-Springer resolution and geometric induction} \label{sectionspringerresolution} For more details and references on this section, see \cite[\S 6.1, 6.2]{laumonletellier} and also \cite[\S 2.4, 2.7]{laumonletellier2}. \bigskip Consider a connected reductive algebraic group $G$ over $\overline{\F}_q$, a maximal torus $T \subseteq G$ and a Borel subgroup $B \supseteq T$. Let $\pi:B \to T$ be the canonical projection and $W=W_G(T)$ be the Weyl group of $G$ with respect to $T$. We denote by $\widetilde{G}$ the variety $$\widetilde{G}\coloneqq \{(xB,g)\in (G/B) \times G \ \| \ x^{-1}gx \in B\} .$$ We denote by $p:\widetilde{G} \to G$ and $q:\widetilde{G} \to T$ the maps given by $p((xB,g))=g$ and $q((xB,g))=\pi(x^{-1}gx)$. For any $\mathcal{F} \in D^b_c(T)$, we define the geometric induction $\mathcal{I}_T^G(\mathcal{F}) \in D^b_c(G)$ as $$\mathcal{I}_T^G(\mathcal{F})\coloneqq p_!q^(\mathcal{F})[{\rm dim}\,G-{\rm dim}\, T] . $$ By \cite{BY}, the functor $\mathcal{I}_T^G$ preserves perverse sheaves and induces thus a functor $\mathcal{I}_T^G:\mathcal{M}(T)\rightarrow\mathcal{M}(G)$ between categories of perverse sheaves. \bigskip Consider the morphism of correspondences \begin{equation} \xymatrix{&&\ar[lld]_-q\widetilde{G}\ar[rrd]^-p\ar[d]^{(q,p)}&&\\ T&&S:=T\times_{T/\!/W}G\ar[rr]^{{\rm pr}_2}\ar[ll]_{{\rm pr}_1}&&G} \label{cor}\end{equation} The morphism $(q,p)$ is small and so $(q,p)_!\overline{\Q}_\ell$ is the intersection cohomology complex $IC^\bullet_{S,\overline{\Q}_\ell}$ with coefficients in the constant sheaf $\overline{\Q}_\ell$. From the projection formula we deduce that $$ \mathcal{I}_T^G(\mathcal{F})={\rm pr}_{2\,!}\left({\rm pr}_1^(\mathcal{F})\otimes IC^\bullet_{S,\overline{\Q}_\ell}\right). $$ Notice also that ${\rm pr}_1$ and ${\rm pr}_2$ are $W$-equivariant for the natural action of $W$ on $T$ and the trivial action of $W$ on $G$. Therefore, if $\mathcal{F}$ is equipped with a $W$-equivariant structure, namely if we are given a collection $\theta=(\theta_w)_{w\in W}$ of isomorphisms $$ \theta_w:w^(\mathcal{F})\rightarrow\mathcal{F} $$ such that $\theta_{ww'}=\theta_w\circ w^(\theta_{w'})$ and $\theta_1={\rm Id}$, then we get an action of $W$ on $\mathcal{I}_T^G(\mathcal{F})$, i.e. we have a group homomorphism $$ \theta^G:W\rightarrow{\rm Aut}(\mathcal{I}_T^G(\mathcal{F})). $$ \begin{remark}Notice that $W$ does not act on $\widetilde{G}$ so, to construct $\theta^G$, we need the factorization through $S$ (Stein factorization) together with the fact that $IC^\bullet_{S,\overline{\Q}_\ell}$ is naturally $W$-equivariant. \end{remark} Therefore if $\mathcal{F}$ is equipped with a $W$-equivariant structure $\theta$, then we have a decomposition \begin{equation} \mathcal{I}_T^G(\mathcal{F})=\bigoplus_{\chi\in\widehat{W}}\mathcal{I}_T^G(\mathcal{F})_\chi \label{W-decomp}\end{equation} where $\mathcal{I}_T^G(\mathcal{F})_\chi\rightarrow\mathcal{I}_T^G(\mathcal{F})$ is the kernel of the idempotent $1-e_\chi\in{\rm End}\left(\mathcal{I}_T^G(\mathcal{F})\right)$ with \begin{align} e_\chi&=\frac{\chi(1)}{\|W\|}\sum_{w\in W}\overline{\chi(w)}\,\theta^G(w)\\&=\frac{\chi(1)}{\|W\|}\sum_{w\in W}\chi(w)\,\theta^G(w) \end{align} (the last equality follows from the fact that the irreducible characters of Weyl groups have integer values). \bigskip Now we assume that $T$ is $F$-stable and that $F$ acts trivially on $W$ (this is will be our case since we will consider $\Gl_2$, $\Sl_2$ and $\PGl_2$). We do not assume that $B$ is $F$-stable. \bigskip The morphisms ${\rm pr}_1$ and ${\rm pr}_2$ are $F$-equivariant as well as the complex $IC^\bullet_{S,\overline{\Q}}$. Therefore the functor $\mathcal{I}_T^G$ induces functors $D_c^b(T;F)\rightarrow D_c^b(G;F)$ and $\mathcal{M}(T;F)\rightarrow\mathcal{M}(G;F)$ between $F$-equivariant objects. \bigskip Consider $\mathcal{F}\in D_c^b(T)$ equipped with both a $W$-equivariant structure $\theta$ and an $F$-equivariant structure $\phi:F^(\mathcal{F})\simeq \mathcal{F}$ such that the two are compatible, namely the following diagram commutes for all $w\in W$, see \cite[Diagram (6.1)]{laumonletellier}: $$ \xymatrix{w^F^(\mathcal{F})\ar[rr]^{F^(\theta_w)}\ar[d]_{w^(\phi)}&&F^(\mathcal{F})\ar[d]^{\phi}\\ w^(\mathcal{F})\ar[rr]^{\theta_w}&&\mathcal{F}}. $$ Then we have the following compatibility $$ \xymatrix{F^\left(\mathcal{I}_T^G(\mathcal{F})\right)\ar[rr]^{F^(\theta^G(w)))}\ar[d]_{\phi^G}&&F^\left(\mathcal{I}_T^G(\mathcal{F})\right)\ar[d]^{\phi^G}\\ \mathcal{I}_T^G(\mathcal{F})\ar[rr]^{\theta^G(w)}&&\mathcal{I}_T^G(\mathcal{F})} $$ For any $\chi\in\widehat{W}$, we see from this compatibility condition (see \cite[Remark 6.1.3]{laumonletellier} for more details) that the canonical $F$-equivariant structure $\phi^G:F^\left(\mathcal{I}_T^G(\mathcal{F})\right)\simeq\mathcal{I}_T^G(\mathcal{F})$ restricts to an $F$-equivariant structure $\phi^G_\chi:F^(\mathcal{I}_T^G(\mathcal{F})_\chi)\simeq\mathcal{I}_T^G(\mathcal{F})_\chi$ and $$ {\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\phi^G}=\sum_{\chi\in\widehat{W}}{\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\phi^G_\chi}. $$ We have also a formula for ${\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\theta^G(w)\circ\phi^G}$ for any $w\in W$ which simplifies when the group $W$ is commutative (which will be our case), i.e. \begin{prop}Assume that $G=\Gl_2,\Sl_2$ or $\PGl_2$. Then $${\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\theta^G(w)\circ\phi^G}=\sum_{\chi\in\widehat{W}}\chi(w)\,{\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\phi^G_\chi} $$ for all $w\in W$. \label{Lu}\end{prop} \begin{proof} $\theta^G$ induces an action of $W$ on $\mathcal{I}_T^G(\mathcal{F})_\chi$ for all $\chi\in\widehat{W}$ which we also denote by $\theta^G$. Then $${\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\theta^G(w)\circ\phi^G}=\sum_{\chi\in\widehat{W}}{\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\theta^G(w)\circ\phi^G_\chi} $$ We thus need to see that \begin{equation} {\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\theta^G(w)\circ \phi^G_\chi}=\chi(w)\,{\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\phi^G_\chi}. \label{w}\end{equation} But this follows from the fact that for a representation $$ \rho:H\rightarrow \Gl(V) $$ of a commutative group $H$ in a finite-dimensional complex vector space $V$, then $H$ acts on $V_\chi={\rm Ker}(1-e_\chi)$, with $e_\chi=\frac{1}{\|H\|}\sum_{h\in H}\overline{\chi(h)}\rho(h)$, by the character $\chi$. \end{proof} \begin{remark}Note that Formula (\ref{w}) for $w=1$ is obviously false if $\chi$ is of degree $>1$. Hence the proof works for commutative $W$ only. For general $W$ we need to decompose further the complexes $\mathcal{I}_T^G(\mathcal{F})_\chi$ which may not be indecomposable. \end{remark} From the orthogonality relation on irreducible characters of $W$ we deduce. \begin{prop}Under the assumption of Proposition \ref{Lu} we have $$ {\bf X}_{\mathcal{I}_T^G(\mathcal{F})_\chi,\phi^G_\chi}=\frac{1}{\|W\|}\sum_{w\in W}\chi(w)\,{\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\theta^G(w)\circ\phi^G}. $$ \label{inv}\end{prop} \subsection{Character-sheaves on $\mathrm{SL}_2$}\label{CS-SL} Let us start with some generalities: $G$ is an arbitrary connected reductive algebraic group with Frobenius $F$ and $T$ is a maximal torus of $G$. \begin{teorema}[Lusztig] Let $\mathcal{L}$ be a Kummer local system (character-sheaf up to a shift) on $T$, then $\mathcal{I}_T^G(\mathcal{L}[{\rm dim}\, T])$ is a semisimple perverse sheaf on $G$. \end{teorema} \begin{proof}See for instance \cite[\S 4]{shoji}. \end{proof} \bigskip If $\mathcal{L}$ is a Kummer local system on $T$, then the irreducible constituents of $\mathcal{I}_T^G(\mathcal{L}[{\rm dim}\, T])$ are examples of character-sheaves on $G$. When $G=\Gl_n$, we get all character-sheaves in this way. We will see below that this is not true for $\Sl_2$. \bigskip Assume that $T$ is $F$-stable. Since the morphism ${\rm pr}_2$ of Diagram (\ref{cor}) is $G$-equivariant, for any $\mathcal{F}\in D_c^b(T)$, the complex $\mathcal{I}_T^G(\mathcal{F})$ is $G$-equivariant and so $\mathcal{I}_T^G:D_c^b(T;F)\rightarrow D_c^b(G;F)$ induces a $\overline{\Q}_\ell$-linear map $$ I_T^G:\mathcal{C}(T^F)\rightarrow\mathcal{C}(G^F),\hspace{.5cm} I_T^G({\bf X}_{\mathcal{F},\phi})={\bf X}_{\mathcal{I}_T^G(\mathcal{F}),\phi^G}. $$ We have the following particular case of the main result of \cite{LUG}. \begin{teorema}[Lusztig] The geometric induction coincides with the Deligne-Lusztig induction, i.e. we have $$ I_T^G=R_T^G. $$ \label{GreenLu}\end{teorema} \bigskip Thanks to the above theorem, the characteristic functions of the $F$-stable character-sheaves are closely related to the irreducible characters of $G^F$. When $G=\Gl_n$, the characteristic functions of the $F$-stable character-sheaves are exactly (up to a sign) the irreducible characters of $\Gl_n(\F_q)$. This is not true in general (as we will see in details with $\Sl_2$). Notice that there is in fact no reason that the decomposition of $\mathcal{I}_T^G(\mathcal{L})$ as a direct sum of simple objects in the category of perverse sheaves would agree with the decomposition of the character $R_T^G({\bf X}_\mathcal{L})$ as a direct sum of irreducible characters. \bigskip We assume now that $G=\Sl_2$ with $W=S_2=\{1,\sigma\}$. The $F$-stable maximal torus $T$ is either the maximal torus of diagonal matrices $T_1$ or the twisted $F$-stable maximal torus $T_\sigma$ (see \S \ref{table-SL}). Recall that $\sigma\in W$ acts by $x\mapsto x^{-1}$ on $T\simeq \Gl_1$. \bigskip Fix a an $F$-stable Kummer local system $\mathcal{L}$ on $T$. \begin{remark}Notice that if $\mathcal{L}$ is $W$-equivariant on $T=T_1$, then $\mathcal{L}$ will be also $\sigma F$-stable. Since the pairs $(T_1,\sigma F)$ is $G$-conjugate to $(T_\sigma, F)$ (see Diagram (\ref{conj1})), we get an $F$-stable Kummer local system $\mathcal{L}_\sigma$ on $T_\sigma$ and we have $$ \mathcal{I}_T^G(\mathcal{L}[1])\simeq\mathcal{I}_{T_\sigma}^G(\mathcal{L}_\sigma[1]) $$ as perverse sheaves (not as $F$-equivariant perverse sheaves). To avoid redundancy in the list of character-sheaves we will thus consider only the torus $T_1$ in this situation. \label{Wequiv}\end{remark} \bigskip We therefore have the following cases : \bigskip (1) $T=T_1$ or $T_\sigma$ and $\mathcal{L}^{\otimes 2}\not\simeq \overline{\Q}_\ell$, then $\sigma^(\mathcal{L})\not\simeq\mathcal{L}$. The perverse sheaf $\mathcal{I}_T^G(\mathcal{L}[{\rm dim}\, T])$ is irreducible and so is a character-sheaf. (2) $T=T_1$ and $\mathcal{L}=\overline{\Q}_\ell$, then $\mathcal{L}$ is naturally $W$-equivariant and the decomposition (\ref{W-decomp}) reads $$ \mathcal{I}_T^G(\overline{\Q}_\ell[1])\simeq\overline{\Q}_\ell[3]\oplus{\rm St}[3] $$ where ${\rm St}[3]$ is the character-sheaf corresponding to the sign character of $W$. (3) $T=T_1$ and $\mathcal{L}^{\otimes 2}\simeq\overline{\Q}_\ell$ with $\mathcal{L}\not\simeq\overline{\Q}_\ell$, then, by Example \ref{examplekummer}, $\mathcal{L}$ is the non-trivial constituent $\mathcal{A}_o$ of $$ (f_2)_(\overline{\Q}_\ell)=\overline{\Q}_\ell\oplus\mathcal{A}_o $$ where $f_2:T_1\rightarrow T_1$, $z\mapsto z^2$. The Kummer local system $\mathcal{A}_o$ corresponds to the character $\alpha_o$ of $\F_q^\simeq T_1^F$ (which take the value $1$ at squares and $-1$ at non-squares). Since $f_2$ is $W$-equivariant, the local system $\mathcal{A}_o$ must be $W$-equivariant and by (\ref{W-decomp}) we have $$ \mathcal{I}_T^G(\mathcal{A}_o[1])=\mathcal{X}_1[3]\oplus \mathcal{X}_{\epsilon}[3] $$ where $X_\epsilon$ is the constituent corresponding to the sign character of $W$. \bigskip Notice that we do not get all character-sheaves here, we are missing the two \emph{cuspidals} which are the two non-trivial $G$-equivariant irreducible local systems on the conjugacy classes of $$ \left(\begin{array}{cc}1&1\\0&1\end{array}\right),\hspace{1cm}\text{and}\hspace{1cm}\left(\begin{array}{cc}-1&1\\0&-1\end{array}\right) $$ In the first two cases (1) and (2), the characteristic functions of the character-sheaves coincide with the irreducible "characters" $R_T^G({\bf X}_{\mathcal{L}})$, ${\rm Id}$ and ${\rm St}$ of $\Sl_2(\F_q)$ (here by "character" we mean up to a sign, since $R_T^G(\theta)$ is the negative of a character when $T$ is the twisted torus). This is not true in the last case as we now see. \bigskip Denote by $X_1$ and $X_\epsilon$ the characteristic functions of $\mathcal{X}_1$ and $\mathcal{X}_\epsilon$. \begin{teorema} \begin{align} X_1&=\frac{1}{2}\left(R_{T_1}^G(\alpha_o)+R_{T_\sigma}^G(\omega_o)\right)\\ X_\epsilon&=\frac{1}{2}\left(R_{T_1}^G(\alpha_o)-R_{T_\sigma}^G(\omega_o)\right) \end{align} \label{unif}\end{teorema} \begin{proof}Proposition \ref{inv} says that \begin{align} X_1&=\frac{1}{2}\left({\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o),\phi^G}+{\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o), \theta^G(\sigma)\circ\phi^G}\right)\\ X_\epsilon&=\frac{1}{2}\left({\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o),\phi^G}-{\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o), \theta^G(\sigma)\circ\phi^G}\right) \end{align} By Theorem \ref{GreenLu} applied to $T=T_1$ we have $$ {\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o),\phi^G}=R_{T_1}^G(\alpha_o). $$ As in Remark \ref{Wequiv}, the $F$-equivariant local system $\mathcal{A}_o$ (affording the character $\alpha_o$ on $T_1^F$) being naturally $W$-equivariant (with $W$-equivariant structure compatible with $F$-equivariant structure) is transported to an $F$-equivariant local system $\mathcal{A}_\sigma$ on $T_\sigma$ which corresponds to the non-trivial square $\omega_o$ in $\widehat{\mu_{q+1}}$ of the identity character. Therefore by Theorem \ref{GreenLu} applied to $T=T_\sigma$ we have \begin{align} {\bf X}_{\mathcal{I}_{T_1}^G(\mathcal{A}_o),\theta^G(\sigma)\circ\phi^G}&={\bf X}_{\mathcal{I}_{T_\sigma}^G(\mathcal{A}_\sigma)}\\ &=R_{T_\sigma}^G(\omega_o) \end{align} (the first equality being formal). \end{proof} Now notice that \begin{lemma} $$\omega_o(-1)=-\alpha_o(-1).$$ \label{alphaomega}\end{lemma} \begin{proof} This follows from the fact that $-1$ is a square in $\F_q^$ if and only if it is a non-square in $\mu_{q+1}\subset\F_{q^2}^$. \end{proof} From Theorem \ref{unif}, Lemma \ref{restrict} and the Table (\ref{tableGL}) we deduce the following table for the values of $X_1$ and $X_\epsilon$: \begin{scriptsize} \begin{equation} \label{tableSL} \begin{array}{\|c\|c\|c\|c\|c\|c\|c\|} \hline &&&&&&\\ &\left(\begin{array}{cc}1&0\\0&1\end{array}\right)&\left(\begin{array}{cc}-1&0\\0&-1\end{array}\right)&\left(\begin{array}{cc}a&0\\0&a^{-1}\end{array}\right)& \left(\begin{array}{cc}x&0\\0& x^q\end{array}\right)&\left(\begin{array}{cc}1&1\\0&1\end{array}\right)&\left(\begin{array}{cc}-1&1\\0&-1\end{array}\right)\\ &&&a\neq a^{-1}\in\F_q^\times&x^{q+1}=1, x\neq x^q&&\\ \hline &&&&&&\\ X_1&1&q\alpha_o(-1)&\alpha_o(a)&0&1&0 \\ &&&&&&\\ \hline &&&&&&\\ X_\epsilon&q&\alpha_o(-1)&0&\omega_o(x)&1&\alpha_o(-1)\\ &&&&&&\\ \hline \end{array} \end{equation} \end{scriptsize} We saw in \S \ref{table-SL}, that the irreducible characters $\chi_{\alpha_o}^\pm$ and $\chi_{\omega_o}^\pm$ have degrees $\frac{q+1}{2}$ and $\frac{q-1}{2}$ and so $X_1$ and $X_\epsilon$ can not be irreducible characters of $\Sl_2(\F_q)$. \section{Main results} \subsection{Statements}\label{statement} Let $K$ be an algebraically closed field of characteristic $\neq 2$ ($K=\mathbb{C}$ or $\overline{\F}_q$) and $\mathcal{C}=(\mathcal{C}_1,\dots,\mathcal{C}_k)$ be a $k$-tuple of regular semisimple conjugacy classes of $\PGl_2(K)$. When $K=\overline{\F}_q$, we assume that $\mathcal{C}$ respects the conditions explained at the beginning of \cref{cohomology}. We denote by $\mathcal{M}_C(K)$ the corresponding character stack. \bigskip {\bf The non-degenerate case} \bigskip We assume that all conjugacy classes $\mathcal{C}_i$ are non-degenerate (i.e. that their stabilisers are connected). \bigskip We fix a \emph{generic} $k$-tuple $R'=(R'_1,\dots,R'_k)$ of irreducible characters of $\Sl_2(\F_q)$ of the form $$ R'=\left(R_{T'_1}^{\Sl_2}(\gamma_1),\dots,R_{T'_1}^{\Sl_2}(\gamma_k)\right) $$ (see \S \ref{generic-char} for the definition of generic $R'$). \begin{teorema}We have (1) \begin{equation} E(\mathcal{M}_{\mathcal{C}}(\C);q)=E(\mathcal{M}_\mathcal{C}(\overline{\F}_q);q)=\left\langle1_{(\mathcal{C}_1)^F}\cdots1_{(\mathcal{C}_k)^F},1_{\{1\}}\right\rangle_{\PGl_2(\F_q)}.\label{Eseries-ndegen} \end{equation} (2) \begin{equation} \left\langle R'_1\otimes\cdots\otimes R'_k,1\right\rangle_{\Sl_2(\F_q)}=2\, \mathbb{A}_0(0,\sqrt{q}). \label{pure-part-ndegen}\end{equation} where $\mathbb{A}_r(z,w)$ is defined in \S\ref{cohomology}. \end{teorema} \bigskip We will see that these polynomials do not depend on the choice of the eigenvalues as long as this choice is generic and the conjugacy classes regular non-degenerate. \bigskip The proof of the first assertion is completely similar to the analogous statement for $\Gl_n(\C)$-character varieties (see \cite{HA}). Namely we first notice that the RHS counts the number of points of $\mathcal{M}_\mathcal{C}$ over $\F_q$ and we conclude by Theorem \ref{Katz}. \bigskip The second assertion will be a particular case of Theorem \ref{theoR'}(2) with $k_2=0$. \bigskip It follows from Conjecture \ref{conjEnondegenerate} and Remark \ref{remA} that \bigskip \begin{center} \begin{tikzcd} \label{diag3} & & & H_c(\mathcal{M}_\mathcal{C}(\C);q,t) \arrow[d,"t \mapsto -1"] \arrow[rd, "\text{ pure part}"]\\ & & & \langle 1_{(\mathcal{C}_1)^F}\cdots1_{(\mathcal{C}_k)^F},1_1 \rangle_{\PGl_2(\F_q)} & q^{k-3} \langle R'_1\otimes \cdots\otimes R'_k,1 \rangle_{\Sl_2(\F_q)} \end{tikzcd} \end{center} \bigskip {\bf The degenerate case} \bigskip We fix a non-negative integer $1\leq m<k$ and we assume that $\mathcal{C}_1=\dots=\mathcal{C}_m=\mathcal{C}_{-1}$ and that the $\mathcal{C}_i$'s are non-degenerate for $i>m$. \bigskip Recall that $\mathcal{L}_\epsilon$ denotes the non-trivial irreducible $\PGl_2(K)$-equivariant local system on $\mathcal{C}_{-1}$. \bigskip If $K=\overline{\F}_q$, it is equipped with a natural $F$-equivariant structure and we denote by $$ Y_\epsilon:\PGl_2(\F_q)\rightarrow\overline{\Q}_\ell $$ the characteristic function of $\mathcal{L}_\epsilon$. It is supported on $(\mathcal{C}_{-1})^F$. \bigskip \begin{remark} The set $(\mathcal{C}_{-1})^F$ is not a single conjugacy class of $\PGl_2(\F_q)$. Since the stabiliser of an element of $\mathcal{C}_{-1}$ has two connected components, it decomposes as $$ (\mathcal{C}_{-1})^F=\mathcal{O}_{e}\sqcup\mathcal{O}_{\sigma}, $$ the disjoint union of two $\PGl_2(\F_q)$-conjugacy classes. Here $\mathcal{O}_{e}$ is the $\PGl_2(\F_q)$-conjugacy class of the image $D_e=p(g_{-1})$ of the matrix $g_{-1}=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$ in $\PGl_2$. For the other one we fix $x\in\F_{q^2}^$ such that $x^q=-x$. The matrix $$\left(\begin{array}{cc}x&0\\0&-x\end{array}\right) $$ is $\sigma F$-stable and so is $\Gl_2(\overline{\F}_q)$-conjugate to an $F$-stable matrix $g_{\sigma}$ in $T_\sigma$. The image $D_\sigma=p(g_{\sigma})$ in $\PGl_2$ of this latter matrix is $F$-stable and is geometrically conjugate to $D_e$ (but not rationally as $x\notin\F_q$). The second conjugacy class $\mathcal{O}_{\sigma}$ is the $\PGl_2(\F_q)$-conjugacy class of $D_\sigma$. Then $$ Y_\epsilon=1_{\mathcal{O}_e}-1_{\mathcal{O}_\sigma}, $$ i.e. it takes the value $1$ at $\mathcal{O}_e$ and $-1$ at $\mathcal{O}_\sigma$. \label{rmdecomp}\end{remark} \bigskip Fix a subset $B\subset\{2,\dots,m\}$ (possibly empty) and let $({\bf Y}_1,\dots,{\bf Y}_k)$ be the $k$-tuple of functions ${\bf Y}_i:\PGl_2(\F_q)\rightarrow\overline{\Q}_\ell$ given by $$ {\bf Y}_i=\begin{cases}Y_\epsilon&\text{ if }i\in \{1\}\cup B,\\ 1_{(\mathcal{C}_i)^F}&\text{ otherwise.}\end{cases} $$ where $1_{(\mathcal{C}_i)^F}$ is the characteristic function of $(\mathcal{C}_i)^F$ that takes the value $1$ on $(\mathcal{C}_i)^F$ and $0$ elsewhere. \bigskip \begin{teorema} We have \label{theoremEulerspeciaz} $$ E(\mathcal{M}_{\mathcal{C}}(\C),\mathcal{F}_B;q)=E(\mathcal{M}_\mathcal{C}(\overline{\F}_q),\mathcal{F}_B;q)=\left\langle {\bf Y}_1\cdots{\bf Y}_k,1_{\{1\}}\right\rangle_{\PGl_2(\F_q)}$$ \end{teorema} \bigskip Consider a \emph{generic} $k$-tuple $({\bold X}_1,\dots, {\bf X}_k)$ of characteristic functions of character-sheaves on $\Sl_2$ (for the definition of genericity, see in \S \ref{proofB}) such that $$ {\bf X}_i=\begin{cases}X_\epsilon&\text { if }i\in \{1\}\cup B,\\ X_1&\text{ if }i\in \{2,\dots,m\}\backslash B,\\ R_{T'_2}^{\Sl_2}(\gamma_i)&\text{ otherwise.} \end{cases} $$ \begin{teorema}We have \begin{equation} \left\langle {\bf X}_1 \cdots {\bf X}_k,1\right\rangle_{\Sl_2}=\frac{1}{2^{m-1}}\sum_{\substack{r=0\\r \text{ even}}}^{m}C_{\|B\|+1,m-\|B\|-1,r}\,\,\mathbb{A}_r\left(0,\sqrt{q}\right). \label{B'}\end{equation} \label{ThB}\end{teorema} \bigskip We deduce from the above theorems together with the conjectural formula (\ref{pure-conj}) the following conjectural picture : \begin{center} \begin{tikzcd} \label{diag1} & & & H_c(\mathcal{M}_\mathcal{C}(\C),\mathcal{F}_B;q,t) \arrow[d,"t \mapsto -1"] \arrow[rd, "\text{ pure part}"]\\ & & & \langle {\bf Y}_1\cdots{\bf Y}_k,1_1 \rangle_{\PGl_2(\F_q)} & q^{k-3} \langle {\bf X}_1\cdots{\bf X}_k,1 \rangle_{\Sl_2(\F_q)} \end{tikzcd} \end{center} We will start with the proof of Theorem \ref{ThB}. Notice that the computation of the LHS of (\ref{B'}) will reduce to some computations in $\Gl_2$ thanks to the Frobenius reciprocity formula $$ \left\langle {\rm Res}^{\Gl_2(\F_q)}_{\Sl_2(\F_q)}(f), 1\right\rangle_{\Sl_2}=\left\langle f,{\rm Ind}_{\Sl_2(\F_q)}^{\Gl_2(\F_q)}(1)\right\rangle_{\Gl_2} $$ for any class function $f$ on $\Gl_2(\F_q)$. \subsection{Some result for $\mathrm{GL}_2$} Consider a $k$-tuple $R=(R_1,\dots,R_k)$ of Deligne-Lusztig characters of $\Gl_2(\F_q)$ (possible not irreducible) of the form $$ R=\left(R_{T_1}^{\Gl_2}(\alpha_1^1,\alpha^1_2),\dots,R_{T_1}^{\Gl_2}(\alpha^{k_1}_1,\alpha^{k_1}_2),-R_{T_\sigma}^{\Gl_2}(\omega_1),\dots,-R_{T_\sigma}^{\Gl_2}(\omega_{k_2})\right) $$ with $k_1>0$ (but we allow $k_2$ to be zero). \bigskip We have the following definition \cite[Definition 6.8.6]{letellier2}. \begin{definizione} We say that $R$ is \emph{generic} if \bigskip (1) $$\prod_{i=1}^{k_1}(\alpha^i_1\alpha^i_2)\prod_{j=1}^{k_2}(\omega_j\|_{\F_q^})=1.$$ (2) If $k_2=0$, we further ask that for any $w_1,\dots,w_k\in S_2$, we have $$ \prod_{i=1}^k\alpha^i_{w_i(1)}\neq 1. $$ \label{def-generic}\end{definizione} \begin{teorema}\cite[Theorem 6.10.1]{letellier2} If $R$ is generic then $$ \left\langle R_1\otimes\cdots\otimes R_k,1\right\rangle_{\Gl_2}=(-1)^{k_2}\mathbb{A}_{k_2}(0,\sqrt{q}). $$ \label{JEMS}\end{teorema} \bigskip \begin{remark} It follows from the theorem that $$ \left\langle R_1\otimes\cdots\otimes R_k,1\right\rangle_{\Gl_2} $$ does not depend on the choice of the characters $(\alpha_1^i,\alpha_2^i)$ and $\omega^j$ as long as the choice is generic. \label{independ}\end{remark} \bigskip \begin{remark} If the condition (1) in Definition \ref{def-generic} is not satisfied, then $$ \left\langle R_1\otimes\cdots\otimes R_k,1\right\rangle_{\Gl_2}=0. $$ This is analoguous to the condition (\ref{det}) for the character stack $\mathcal{M}_{\mathcal{C}}$ to be non-empty. \label{cond1}\end{remark} \subsection{From $\mathrm{GL}_2$ to $\mathrm{SL}_2$} Consider a $k$-tuple $R'=(R'_1,\dots,R'_k)$ of Deligne-Lusztig characters of $\Sl_2(\F_q)$ (possibly not irreducible) of the form $$ R'=\left(R_{T'_1}^{\Sl_2}(\gamma_1),\dots,R_{T'_1}^{\Sl_2}(\gamma_{k_1}),-R_{T'_\sigma}^{\Sl_2}(\eta_1),\dots,-R_{T'_\sigma}^{\Sl_2}(\eta_{k_2})\right). $$ Each $R'_i$ is the restriction of a Deligne-Lusztig character $R_i$ of $\Gl_2(\F_q)$ of the form $$ R_i=\begin{cases}R_{T_1}^{\Gl_2}(1,\gamma_i)&\text{ if } 1\leq i\leq k_1,\\-R_{T_\sigma}^{\Gl_2}(\omega_i)&\text{ if }k_1+1\leq i\leq k_2. \end{cases} $$ By the Frobenius reciprocity formula we have $$ \left\langle R'_1\otimes\cdots\otimes R'_k\right\rangle_{\Sl_2}=\left\langle R_1\otimes\cdots\otimes R_k,{\rm Ind}_{\Sl_2}^{\Gl_2}(1)\right\rangle_{\Gl_2}. $$ Since $$ {\rm Ind}_{\Sl_2}^{\Gl_2}(1)=\sum_{\delta\in\widehat{\F_q^}}\delta\circ\det $$ we have $$ \left\langle R'_1\otimes\cdots\otimes R'_k\right\rangle_{\Sl_2}=\sum_{\delta\in\widehat{\F_q^}}\left\langle R_1\otimes\cdots\otimes R_k\otimes(\delta^{-1}\circ\det),1\right\rangle_{\Gl_2} $$ By Remark \ref{cond1}, the multiplicity $$ \left\langle R_1\otimes\cdots\otimes R_k\otimes(\delta^{-1}\circ\det),1\right\rangle_{\Gl_2} $$ vanishes unless $$ \gamma_1\cdots\gamma_{k_1}(\omega_1\|_{\F_q^})\cdots(\omega_{k_2}\|_{\F_q^})=\delta^2. $$ We need the following lemma. \begin{lemma}Given $\gamma\in\widehat{\F_q^}$, the equation $$ \delta^2=\gamma $$ has a solution if and only if $\gamma(-1)=1$. In this case the two solutions differ by the character $\alpha_o$. \end{lemma} \begin{proof} The group homomorphism $$ \phi:\widehat{\F_q^}\rightarrow\{-1,1\},\hspace{1cm}\gamma\mapsto \gamma(-1) $$ is surjective and we see that the square elements of $\widehat{\F_q^}$ live in the kernel of $\varphi$. The subgroup ${\rm Ker}(\varphi)$ is thus of index $2$ in $\widehat{\F_q^}$ and it contains the subgroup of square elements of $\widehat{\F_q^}$ which is also of index $2$ hence the result. \end{proof} We thus have the following result. \begin{prop} (1) If $\left(\gamma_1\cdots\gamma_{k_1}(\omega_1\|_{\F_q^})\cdots(\omega_{k_2}\|_{\F_q^})\right)(-1)=-1$, then $$ \left\langle R'_1\otimes\cdots\otimes R'_k,1\right\rangle_{\Sl_2}=0. $$ (2) If $\left(\gamma_1\cdots\gamma_{k_1}(\omega_1\|_{\F_q^})\cdots(\omega_{k_2}\|_{\F_q^})\right)(-1)=1$, then we choose a square root $$ \lambda_R=\sqrt{\gamma_1\cdots\gamma_{k_1}(\omega_1\|_{\F_q^})\cdots(\omega_{k_2}\|_{\F_q^})}$$and we have \bigskip $\left\langle R'_1\otimes\cdots\otimes R'_k,1\right\rangle_{\Sl_2}$ $$=\left\langle R_1\otimes\cdots\otimes R_k\otimes(\lambda_R^{-1}\circ\det),1\right\rangle_{\Gl_2}+\left\langle R_1\otimes\cdots\otimes R_k\otimes(\lambda_R^{-1}\alpha_o\circ\det),1\right\rangle_{\Gl_2}.$$ \label{GL-SL}\end{prop} \subsection{Generic $k$-tuples of semisimple characters of $\mathrm{SL}_2(\F_q)$}\label{generic-char} Let $R'=(R'_1,\dots,R'_k)$ be as in the previous section. \begin{definizione}The $k$-tuple is said to be \emph{generic} if we have : (1) $$\prod_{i=1}^{k_1}\gamma_i(-1)\prod_{j=1}^{k_2}\eta_i(-1)=1.$$ (2) If $k_2=0$, we further ask that for all $w_1,\dots,w_k\in W=S_2=\{1,\sigma\}$, $$ {^{w_1}}\gamma_1\cdots{^{w_k}}\gamma_k\neq 1 $$ where ${^\sigma}\gamma$ is the character $\gamma^{-1}$. \label{gen-SL}\end{definizione} \begin{prop} The $k$-tuple $R'$ is generic if and only if the two $(k+1)$-tuples of irreducible characters of $\Gl_2(\F_q)$ $$R^+=(R_1,\dots,R_k, \lambda_R^{-1}\circ\det),\hspace{.5cm}\text{ and }\hspace{.5cm} R^-=(R_1,\dots,R_k, \lambda_R^{-1}\alpha_o\circ\det) $$ are both generic. \label{gen}\end{prop} \begin{remark} If $T$ is an $F$-stable maximal torus of $\Gl_2$, $\theta$ a character of $T^F$ and $\lambda$ a linear character of $\F_q^$, then $$ R_T^{\Gl_2}(\theta)\otimes (\lambda\circ\det)=R_T^{\Gl_2}(\theta\otimes(\lambda\circ\det)).$$ The $(k+1)$-tuple $R^+$ is generic if and only if the $k$-tuple $(R^\lambda_1,R_2,\dots,R_k)$, where $$ R^\lambda_1=R_{T_1}^{\Gl_2}\left((1,\gamma_1)\otimes(\lambda^{-1}_R\circ\det)\right),$$ is generic. \label{DL-gen}\end{remark} \begin{proof} From our choice of $\lambda_R$, we have that $$\lambda_R^{-2}\prod_{i=1}^{k_1}\gamma_i \prod_{j=1}^{k_2}(\omega_j\|_{\F_q^})=(\lambda_R\alpha_o)^{-2}\prod_{i=1}^{k_1}\gamma_i \prod_{j=1}^{k_2}(\omega_j\|_{\F_q^})=1 .$$ We see thus that the condition $(1)$ of Definition \ref{def-generic} is verified for the $(k+1)$-tuples $R^+,R^-$. Assume now that $k_2=0$ and consider $w_1,\dots,w_k \in W$. Let $I\subseteq \{1,\dots,k\}$ be the subset $$I \coloneqq \{i \in \{1,\dots,k\}\ \| \ w_i=\sigma\} .$$ Condition $(2)$ of Definition \ref{def-generic} for the $(k+1)$-tuples $R^+,R^-$ and the elements $w_1,\dots,w_k$ is thus \begin{equation} \label{inequalities-genericity} \lambda_R^{-1}\prod_{i \in I}\gamma_i \neq 1 \ \ \text{ and } \ \ \lambda_R^{-1}\alpha_o\prod_{i \in I}\gamma_i \neq 1. \end{equation} Notice that (\ref{inequalities-genericity}) above holds if and only if \begin{equation} \label{inequalities-genericity1} \lambda_R^{-2}\prod_{i \in I}\gamma^2_i \neq 1. \end{equation} We have now $$\lambda_R^{-2}\prod_{i \in I}\gamma^2_i=\prod_{i \in I}\gamma_i \prod_{j \in I^{c}}\gamma_j^{-1}=({^{w_1}}\gamma_1\cdots{^{w_k}}\gamma_k)^{-1} ,$$ i.e. (\ref{inequalities-genericity}) is equivalent to the Condition $(2)$ of Definition \ref{gen-SL} for the $k$-tuple $R^{'}$ and the elements $w_1,\dots,w_k$. \end{proof} \begin{teorema}If $R'$ is generic then (1) $$\left\langle R_1\otimes\cdots\otimes R_k\otimes(\lambda_R^{-1}\circ\det),1\right\rangle_{\Gl_2}=\left\langle R_1\otimes\cdots\otimes R_k\otimes(\lambda_R^{-1}\alpha_o\circ\det),1\right\rangle_{\Gl_2}$$ (2) $$ \left\langle R'_1\otimes\cdots\otimes R'_k,1\right\rangle_{\Sl_2}=2(-1)^{k_2}\mathbb{A}_{k_2}(0,\sqrt{q}). $$ \label{theoR'}\end{teorema} \begin{proof}The first assertion is a consequence of Proposition \ref{gen}, Remark \ref{DL-gen} and Remark \ref{independ}. The second assertion is a consequence of the assertion (1) together with Proposition \ref{GL-SL} and Theorem \ref{JEMS}. \end{proof} \subsection{Proof of Theorem \ref{ThB}}\label{proofB} We use the notation of \S \ref{statement}. We assume that the $k$-tuple $({\bf X}_1,\dots,{\bf X}_k)$ is \emph{generic}, meaning that the $k$-tuple $$ \left(\underbrace{R_{T'_1}^{\Sl_2}(\alpha_o),\dots,R_{T'_1}^{\Sl_2}(\alpha_o)}_m,{\bf X}_{m+1},\dots,{\bf X}_k\right)$$ of Deligne-Lusztig characters of $\Sl_2(\F_q)$ is generic. \bigskip Put $m_1=\|B\|+1$ and $m_2=m-m_1$. We have $$ \left\langle {\bf X}_1\otimes\cdots\otimes{\bf X}_k,1\right\rangle_{\Sl_2}=\left\langle (X_\epsilon)^{\otimes m_1}\otimes (X_1)^{\otimes m_2}\otimes\bigotimes_{j=m+1}^kR_{T'_1}^{\Sl_2}(\gamma_j),1\right\rangle_{\Sl_2} $$ From Theorem \ref{unif}, we have \bigskip $\left\langle {\bf X}_1\otimes\cdots\otimes{\bf X}_k,1\right\rangle_{\Sl_2}$ \begin{align}&=\frac{1}{2^m}\left\langle (R_{T_1}^{\Sl_2}(\alpha_o)-R_{T'_\sigma}^{\Sl_2}(\omega_o))^{\otimes m_1}\otimes (R_{T_1}^{\Sl_2}(\alpha_o)+R_{T'_\sigma}^{\Sl_2}(\omega_o))^{\otimes m_2}\otimes\bigotimes_{j=m+1}^k R_{T'_1}^{\Sl_2}(\gamma_j),1\right\rangle_{\Sl_2}\\ &=\frac{1}{2^m}\sum_{r=0}^m C_{\|B\|+1,m-\|B\|-1,r}\left\langle R_{T'_\sigma}^{\Sl_2}(\omega_o)^{\otimes r}\otimes R_{T'_1}^{\Sl_2}(\alpha_o)^{\otimes m-r}\otimes\bigotimes_{j=m+1}^kR_{T'_1}^{\Sl_2}(\gamma_j),1\right\rangle_{\Sl_2} \end{align} By the genericity assumption we have $$ \alpha_o(-1)^m(\gamma_{m+1}\cdots\gamma_k)(-1)=1. $$ Since $\omega_o(-1)=-\alpha_o(-1)$ by Lemma \ref{alphaomega} we deduce that \begin{equation} \omega_o(-1)^r\alpha_o(-1)^{m-r}(\gamma_{m+1}\cdots\gamma_k)(-1)=(-1)^r. \label{eq-r}\end{equation} By Proposition \ref{GL-SL}(1) we see that $$ \left\langle R_{T'_\sigma}^{\Sl_2}(\omega_o)^{\otimes r}\otimes R_{T'_1}^{\Sl_2}(\alpha_o)^{\otimes m-r}\otimes\bigotimes_{j=m+1}^kR_{T'_1}^{\Sl_2}(\gamma_j),1\right\rangle_{\Sl_2} $$ vanishes unless $r$ is even. Therefore \bigskip $\left\langle {\bf X}_1\otimes\cdots\otimes{\bf X}_k,1\right\rangle_{\Sl_2}$ $$ =\frac{1}{2^m}\sum_{\substack{r=0\\r \text{ even}}}^m C_{\|B\|+1,m-\|B\|-1,r}\left\langle R_{T'_\sigma}^{\Sl_2}(\omega_o)^{\otimes r}\otimes R_{T'_1}^{\Sl_2}(\alpha_o)^{\otimes m-r}\otimes\bigotimes_{j=m+1}^kR_{T'_1}^{\Sl_2}(\gamma_j),1\right\rangle_{\Sl_2} $$ Now notice that the $k$-tuple $$ \left(\underbrace{R_{T'_\sigma}^{\Sl_2}(\omega_o),\dots,R_{T'_\sigma}^{\Sl_2}(\omega_o)}_r,\underbrace{R_{T'_1}^{\Sl_2}(\alpha_o),\dots,R_{T'_1}^{\Sl_2}(\alpha_o)}_{m-r},R_{T'_1}^{\Sl_2}(\gamma_{m+1}),\dots,R_{T'_1}^{\Sl_2}(\gamma_k)\right) $$ is generic for all even integer $r$ ($0\leq r\leq m$). Indeed, if $r=0$, then this is by assumption, if $0<r$ the condition (1) of Definition \ref{gen-SL} is satisfied by Formula (\ref{eq-r}) since $r$ is even (the condition (2) also since $k_2=r>0$). \bigskip We deduce Theorem \ref{ThB} from Theorem \ref{theoR'}(2). \subsection{The Euler specialization} \label{eulerspecializationproof} In this section, we prove Theorem \ref{theoremEulerspeciaz}. We give two proofs : a formal one and a computational one. The second proof is "dual" to the proof of Theorem \ref{ThB} given in \cref{proofB}. \subsubsection{A formal proof} Recall that $\mathcal{C}$ is a generic $k$-tuple of conjugacy classes of $\PGl_2(\overline{\F}_q)$ such that $\mathcal{C}_1=\cdots=\mathcal{C}_m$ are degenerate and $\mathcal{C}_j$ is non-degenerate for $j >m$. Fix $C=(C_1,\dots,C_k)$ a corresponding $k$-tuple of $F$-stable conjugacy classes of $\Gl_2(\overline{\F}_q)$ with the assumptions made at the beginning of \cref{cohomology}. \vspace{2 pt} Notice that, for any $(x_1,\dots,x_k) \in X_{\mathcal{C}}(\F_q)$, we have that \begin{equation} {\bf X_{\mathcal{L}_B}}(x_1,\dots,x_k)={\bf Y}_1(x_1) \cdots {\bf Y}_k(x_k). \end{equation} We thus have \begin{equation} \left\langle {\bf Y}_1 \ast \cdots \ast {\bf Y}_k,1_{\{1\}} \right\rangle=\dfrac{1}{\|\PGl_2(\F_q)\|}\sum_{(x_1,\dots,x_k) \in X_{\mathcal{C}}(\F_q)} {\bf Y}_1(x_1) \cdots {\bf Y}_k(x_k)= \end{equation} \begin{equation} \label{computationconvolutionfq} =\dfrac{1}{\|\PGl_2(\F_q)\|}\sum_{(x_1,\dots,x_k) \in X_{\mathcal{C}}(\F_q)}{\bf X_{\mathcal{L}_B}}(x_1,\dots,x_k)=\sum_{z \in \mathcal{M}_{\mathcal{C}(\F_q)}}{\bf X}_{\mathcal{F}_B}(z). \end{equation} From the last equality of formula (\ref{computationconvolutionfq}) and Grothendieck's trace formula, we deduce that $$\left\langle {\bf Y}_1 \ast \cdots \ast {\bf Y}_k,1_{\{1\}} \right\rangle=\sum_{k} (-1)^k\tr\left(F\,\|\,H^k_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B)\right)$$ Notice that, since the action of $H_m$ and $F$ on $H^_c(\mathcal{M}_{\mathcal{C}})$ commute, the decompositon (\ref{decompositioncohomologyirred}) and the inversion formula in the character ring of $H_m$ imply that $$ \tr\left(F\,\|\,H^k_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B)\right)=\frac{1}{\|H_m\|}\sum_{y \in H_m}\tr\left(yF\,\|\,H^k_c(\mathcal{M}^+_C)\right)\,\overline{\chi_B}(y) $$ for any $k$ and thus \begin{equation} \sum_{k} (-1)^k \tr\left(F\,\|\,H^k_c(\mathcal{M}_{\mathcal{C}},\mathcal{F}_B)\right)=\dfrac {1}{2^{m-1}}\sum_{y \in H_m}\left(\sum_k (-1)^k \tr\left(yF\,\|\,H^k_c(\mathcal{M}_C^+)\right)\right)\overline{\chi_B}(y) \end{equation} Theorem \ref{theoremEulerspeciaz} is now a consequence of Theorem \ref{theoremtwistedE} together with Grothendieck's trace formula and Formula (\ref{invertedrelationcharacters}). \subsubsection{A computational proof} \bigskip We give here another proof of Theorem \ref{theoremEulerspeciaz}, by direct computations of convolution products of class functions of $\PGl_2(\F_q)$. \vspace{2 pt} We start by the following more general remark. Consider $k$ semisimple conjugacy classes of $\PGl_2(\F_q)$ denoted by $\mathcal{O}_1,\dots,\mathcal{O}_k$ and corresponding conjugacy classes $O_1,\dots,O_k$ of $\Gl_2(\F_q)$ such that $\pi(O_j)=\mathcal{O}_j$. We assume that $\mathcal{O}_1,\dots,\mathcal{O}_m$ are degenerate (i.e. $\mathcal{O}_i=\mathcal{O}_{e}$ or $\mathcal{O}_i=\mathcal{O}_{\sigma}$ for $i=1,\dots,m$) and $\mathcal{O}_{m+1},\dots,\mathcal{O}_k$ are non-degenerate. If $\mathcal{O}_j=\mathcal{O}_e$, we take as $O_j$ the conjugacy class of $g_{-1}$ and if $\mathcal{O}_j=\mathcal{O}_{\sigma}$ we take as $O_j$ the conjugacy class of the matrix $g_{\sigma}$. We have the following equality \begin{equation} \label{convolutionfinitegroups} \langle 1_{\mathcal{O}_1} \ast \cdots \ast 1_{\mathcal{O}_k},1_{\{1\}} \rangle_{\PGl_2(\F_q)}=\dfrac{(q-1)}{2^m}\sum_{z \in \F_q^}\langle 1_{O_1} \ast \cdots \ast 1_{O_k},1_{\{z\}} \rangle_{\Gl_2(\F_q)}. \end{equation} Indeed, the LHS of Formula (\ref{convolutionfinitegroups}) is $$\dfrac{\Big\|\{(x_1,\dots,x_k) \in \mathcal{O}_1 \times \cdots \times \mathcal{O}_k \ \| \ x_1\cdots x_k=1\}\Big\|}{\|\PGl_2(\F_q)\|} $$ and the RHS of Formula (\ref{convolutionfinitegroups}) is $$\dfrac{\Bigg\|\displaystyle\bigsqcup_{z \in \F_q^} \{(y_1,\dots,y_k) \in O_1 \times \cdots \times O_k \ \| \ y_1 \cdots y_k=z\} \Bigg\|}{2^{m}\|\PGl_2(\F_q)\|} .$$ A similar argument to the one used in Proposition \ref{propdescriptiongeometry} in the geometric context, shows now that the projection map $$\displaystyle\bigsqcup_{z \in \F_q^} \{(y_1,\dots,y_k) \in O_1 \times \cdots \times O_k \ \| \ y_1 \cdots y_k=z\} \to \{(x_1,\dots,x_k) \in \mathcal{O}_1 \times \cdots \times \mathcal{O}_k \ \| \ x_1\cdots x_k=1\} $$ $$(y_1,\dots,y_k) \to (p(y_1),\dots,p(y_k)) $$ is a quotient map for the free $(\Z/2\Z)^m$-action obtained by multiplication on the first $m$ coordinates. Notice that $\langle 1_{O_1} \ast \cdots \ast 1_{O_k},1_{\{z\}} \rangle_{\Gl_2(\F_q)}=0$ if $\det(O_1)\cdots \det(O_k) \neq z^2$. In particular, we deduce that \begin{equation} \label{oddr} \det(O_1) \cdots \det(O_k) \not \in (\F_q^)^2 \Rightarrow \langle 1_{\mathcal{O}_1} \ast \cdots \ast 1_{\mathcal{O}_k},1_{\{1\}} \rangle_{\PGl_2(\F_q)}=0 \end{equation} where $(\F_q^)^2$ is the subgroup of squares and otherwise, if $$ \det(O_1)\cdots \det(O_k)=\lambda_O^2 $$ with $\lambda_O \in \F_q^$, then \begin{equation} \label{eqconvolutionGL2} \langle 1_{\mathcal{O}_1} \ast \cdots \ast 1_{\mathcal{O}_k},1_{\{1\}} \rangle_{\PGl_2(\F_q)}=\dfrac{(q-1)\left(\langle 1_{O_1} \ast \cdots \ast 1_{O_k},1_{\{\lambda_O\}}\rangle_{\Gl_2(\F_q)} +\langle 1_{O_1} \ast \cdots \ast 1_{O_k},1_{\{-\lambda_O\}} \rangle_{\Gl_2(\F_q)}\right)}{2^{m-1}} \end{equation} \bigskip We apply Formula (\ref{eqconvolutionGL2}) in the following way. Recall that by hypothesis we are assuming that $$\det(C_1)\cdots \det(C_k)=(-1)^m \det(C_{m+1})\cdots \det(C_{k}) \in (\F_q^)^2 .$$ \vspace{4 pt} Fix a subset $B \subseteq \{2,\dots,m\}$ and the corresponding class functions $\bf{Y}_1,\dots,{\bf{Y}}_k$ as in \cref{statement}. Put $m_1=\|B\|+1$ and $m_2=m-m_1$. Since the convolution product is commutative, we have \begin{equation} \left\langle {\bf Y}_1 \ast \cdots \ast {\bf Y}_k,1_{\{1\}} \right\rangle=\left\langle Y_{\epsilon}^{\ast m_1} \ast Y_{1}^{\ast m_2} \ast 1_{\mathcal{C}_{m+1}^F} \ast \cdots \ast 1_{\mathcal{C}_{k}^F},1_{\{1\}} \right\rangle= \end{equation} \begin{equation} \left\langle (1_{\mathcal{O}_e}-1_{\mathcal{O}_{\sigma}})^{\ast m_1} \ast (1_{\mathcal{O}_e}+1_{\mathcal{O}_{\sigma}})^{\ast m_2} \ast 1_{\mathcal{C}_{m+1}^F} \ast \cdots \ast 1_{\mathcal{C}_{k}^F},1_{\{1\}} \right\rangle= \end{equation} \begin{equation} \sum_{r=0}^{m}C_{\|B\|+1,m-\|B\|-1,r}\left \langle 1_{\mathcal{O}_{\sigma}}^{\ast r} \ast 1_{\mathcal{O}_e}^{\ast m-r}\ast 1_{\mathcal{C}_{m+1}^F} \ast \cdots \ast 1_{\mathcal{C}_{k}^F},1_{\{1\}}\right\rangle. \end{equation} For each $r$, we have that $$\det(O_{\sigma})^r \det(O_{e})^{m-r}\det(C_{m+1})\cdots \det(C_{k})=x^{2r}(-1)^m \det(C_{m+1})\cdots \det(C_{k}) .$$ Since $x^2 \in \F_q^* \setminus (\F_q^)^2$, from (\ref{oddr}), we have $$ \left \langle 1_{\mathcal{O}_{\sigma}}^{\ast r} \ast 1_{\mathcal{O}_e}^{\ast m-r}\ast 1_{\mathcal{C}_{m+1}^F} \ast \cdots \ast 1_{\mathcal{C}_{k}^F},1_{\{1\}}\right\rangle=0 ,$$ unless $r$ is even. Given the genericity of $\mathcal{C}$, Formula (\ref{eqconvolutionGL2}) and an argument similar to the one used at the end of the proof of Theorem \ref{Ey} show that, for each $r$ even, we have $$\left \langle 1_{\mathcal{O}_{\sigma}}^{\ast r} \ast 1_{\mathcal{O}_e}^{\ast m-r}\ast 1_{\mathcal{C}_{m+1}^F} \ast \cdots \ast 1_{\mathcal{C}_{k}^F},1_{\{1\}}\right\rangle=\dfrac{1}{2^{m-1}}\mathbb{A}_r\left(\frac{1}{\sqrt{q}},\sqrt{q}\right) .$$ Theorem \ref{theoremEulerspeciaz} is now a consequence of Theorem \ref{Epolynomialgenericdegenerate}. \begin{thebibliography}{} {\small \bibitem{ballandras}{\sc Ballandras, M.}: Intersection cohomology of character varieties for punctured Riemann surfaces, {\em J. \'Ec. polytech. Math.} {\bf 10} (2023), 141--198. \bibitem{BY}{\sc Bezrukavnikov, R. {\rm and }Yon Din, A.}: On parabolic restriction of perverse sheaves, {\em Publ. Res. Inst. Math. 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Orbites et faisceaux pervers, Astérisque} {\bf 173-174} (1989), 111-198. \bibitem{MellitD}{\sc Mellit, A.}: Integrality of Hausel-Letellier-Villegas kernels, \emph{Duke Math. J.} {\bf 167} (2018), 3171--3205. \bibitem{Mellit}{\sc Mellit, A.}: Poincar\'e polynomials of character varieties, Macdonald polynomials and affine Springer fibers, \emph{Ann. of Math. (2)} 1{\bf 92} (2020), no. 1, 165--228. \bibitem{scognamiglio1}{\sc Scognamiglio, T.}: A generalization of Kac polynomials and tensor product of representations of $\Gl_n(\F_q)$, {\em Transformation Groups} (2024). \bibitem{scognamiglio2}{\sc Scognamiglio, T.}: Cohomology of non-generic character stacks, {\em J. \'Ec. polytech. Math.} {\bf 11} (2024), 1287--1371. \bibitem{schiffmann}{\sc Schiffmann, O.}: Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, {\em Ann. of Math. (2)} {\bf 183} (2016), no. 1, 297--362. \bibitem{shoji}{\sc Shoji, T.}: Geometry of orbits and Springer correspondence, {\em Ast\'erisque} {\bf 168} (1988), 61-140. } \end{thebibliography} \end{document} \vspace{4 pt} \begin{oss} In the cases of interest to this article, i.e. when $G=\Gl_n,\Sl_n$, the unipotent characters of $G^F$ have the following description. Denote by $\mathcal{P}_n$ the set of partitions of $n$. Recall that $\mathcal{P}_n$ is in bijection with the set of irreducible characters $\widehat{S_n}$. We denote by $\chi^{\lambda} \in \widehat{S_n}$ the character associated to $\lambda \in \mathcal{P}_n$. In our bijection, $(n)$ is associated to the trivial character. \vspace{4 pt} For $G=\Gl_n,\Sl_n$, put $T=T_n,T_n'$ respectively, where $T_n,T'n$ are the corresponding $F$-stable maximal tori of diagonal matrices. In both cases, we have that $W_G(T)=S_n$ and the corresponding Frobenius action on $S_n$ is trivial. Denote now by $R_{\lambda}$ the following class function of $G^F$: $$R_{\lambda}=\dfrac{1}{n!}\sum_{\sigma \in S_n}R^G_{T_{\sigma}}(1)\chi^{\lambda}(\sigma) .$$ For $G=\Gl_n,\Sl_n$, each $R_{\lambda}$ is an irreducible character and, more precisely, the $R_{\lambda}$s are the unipotent irreducible characters of $G^F$. Moreover, we have a decomposition: $$R^G_T(1)=\sum_{\lambda \in \mathcal{P}_n}R_{\lambda}\chi^{\lambda}(e) ,$$ i.e. the unipotent characters of $G^F$ all appear in the decomposition of $R^G_T(1)$. \end{oss} \begin{oss} In the bijection (\ref{bijectionKummer}), the trivial character of $\widehat{T^F}$ is sent to the trivial local system $\overline{\Q}_{\ell}$. We have in particular, from eq.(\ref{charactersheavesbuilding}), an equality \begin{equation} R^G_T(1)=\chi_{r_!\psi^\overline{\Q_{\ell}}}. \end{equation} Consider now the case of $G=\Gl_n,\Sl_n$ and $T$ the torus $T_n,T'_n$ respectively. Since from Lemma \ref{springermaplemma} the map $r$ is a $S_n$-Galois covering over $G_{rs}$ and $r$ is small, we have a decomposition $$\chi_{r_!\psi^\overline{\Q_{\ell}}}=\bigoplus_{\lambda \in \mathcal{P}_n}\mathcal{R}_{\lambda}^{\oplus \chi^{\lambda}(e)} ,$$ where each $\mathcal{R}_{\lambda}$ is a (shifted) simple perverse sheaf. In this case, we have an equality \begin{equation} R_{\lambda}=\chi_{\mathcal{R}_{\lambda}}, \end{equation} i.e. the unipotent characters coincide with the characteristic functions of the character sheaves appearing in the decomposition of the trivial local system. This is not true in general for other reductive groups. \end{oss} \subsection{Preliminaries: $\mathrm{PGL}_2(\mathbb{F}_q)$} In this paragraph, we describe some geometrical properties of projective linear groups over $\F_q$. In what follows, $\PGl_2$ is $\PGl_2(\overline{\F_q})$ and $\Gl_2=\Gl_2(\overline{\F_q})$ and we denote by $F:\PGl_2 \to \PGl_2$, $F:\Gl_2 \to \Gl_2$ the respective canonical Frobenius morphisms, raising each entry of a matrix to the $q$-th power. Denote by $\widetilde{M_{-1}}$ the analogous element of $\PGl_2(\F_q)$ and by $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}$ the corresponding conjugacy class in $\PGl_2$ and similarly for $M_{-1}$ and $\mathcal{O}_{M_{-1}}$. In the following, it will be clear whether the base field is $\C$ or $\F_q$ from the context. The conjugacy class $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}$ is $F$-stable. However, unlike the case of $\Gl_2$ it is not true that $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}$ is the conjugacy class of $\widetilde{M_{-1}}$ in $\PGl_2(\F_q)$. More precisely, we have a decomposition $$\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^F=\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{+,F} \bigsqcup \widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{-,F} ,$$ where $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{+,F}$ is the conjugacy class of $\widetilde{M_{-1}}$ in $\PGl_2(\F_q)$ and $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{-,F}$ is the conjugacy class of $$p_2\left(\begin{pmatrix}0 &-1\\ \eta^{q+1} &0 \end{pmatrix} \right),$$ where $\eta \in \F_{q^2}^\setminus \F_q^$ and $F(\eta)=-\eta$. \vspace{2 pt} As in the complex case, the morphism $p_2$ restricts to a $2$-covering $$p_2:\mathcal{O}_{M_{-1}} \to \widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}$$ and therefore we have a splitting of $\ell$-adic sheaves over $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}$ \begin{equation} \label{splittingFq} p_2(\overline{\Q}_{\ell}) \cong \overline{\Q}_{\ell} \oplus \mathcal{L}_{\F_q} .\end{equation} Taking the characteristic functions of the sheaves appearing in the RHS of eq.(\ref{splittingFq}), we get two functions $$\phi^+,\phi^-:\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^F \to \overline{\Q}_{\ell} .$$ Extending the two functions by $0$ outside $\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^F$ and fixing an isomorphism $\overline{\Q}_{\ell} \cong \C$, we obtain two functions $$\phi^+,\phi^-:\PGl_2(\F_q) \to \C .$$ Notice that $\phi^+,\phi^-$ are class functions since the decomposition of eq.(\ref{splittingFq}) is $\PGl_2$-equivariant. It is not difficult to verify that we have the following equalities in $\mathcal{C}(\PGl_2(\F_q))$ \begin{equation} \phi^+=1_{\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{+,F}}+1_{\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{-,F}} \end{equation} \begin{equation} \phi^-=1_{\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{+,F}}-1_{\widetilde{\mathcal{O}}_{\widetilde{M_{-1}}}^{-,F}} \end{equation} \section{Correspondence conjugacy class and local system and character sheaves} Consider the following more general situation. We have $(G,F)$ and $(G^,F^)$ two dual reductive groups over $\F_q$. Fix a split $F$-stable maximal torus $T \subseteq G$ and a corresponding dual split $F^$-stable maximal torus $T^* \subseteq G^$. Let $W=W_{G}(T)=W_{G^}(T^)$. Notice that $F$ acts trivially on $W$. Recall that we have the following correspondence \begin{equation}(T^F)^{\vee} \longleftrightarrow (T^)^{F^} ,\end{equation} see for example \cite[Proposition 11.1.14]{DM}. Recall moreover that we have another bijection \begin{equation} (T^F)^{\vee} \longleftrightarrow \{F-\text{stable Kummer local systems on } T \text{ up to isom. }\}. \end{equation} \vspace{2 pt} Fix now an $F^$-stable regular semisimple conjugacy class $C^* \subseteq G^$ and a representative $s^ \in (T^* \cap C^)(\F_q)$. Put $$W_{s^}=\{w \in W \ \| \ w \cdot s^=s^\} .$$ Since $s^$ is regular semisimple, we have that $$W_{s^}=C_{G^}(s^)/C^{\circ}_{G^}(s^) .$$ Recall that we have the following bijection \begin{equation} (C_{G^}(s^)/C^{\circ}_{G^}(s^))^{\vee}=(W_{s^})^{\vee} \longleftrightarrow \{\text{ Irreducible and } G^-\text{equivariant local systems on } C^\}. \end{equation} \vspace{2 pt} The element $s^$ determines a character $s:T^F \to \C^$ and therefore an $F$-stable Kummer local system $\mathcal{L}$. Put $$W_{\mathcal{L}}=\{w \in W \ \| \ w^\mathcal{L} \cong \mathcal{L}\} .$$ We have that $$W_{s^}=W_{\mathcal{L}} .$$ Assume that $m$ is the smallest integer such that $\mathcal{L}^{\otimes m} \cong \overline{\Q}_{\ell}$. Denote by $\theta_m:T \to T$ the covering given by $\theta_m(t)=t^m$ and put $T[m]\coloneqq \Ker(\theta_m)$, i.e. $T[m]$ is given by the element whose order divides $m$. We have a decomposition \begin{equation} (\theta_m)_!\overline{\Q}_{\ell} \cong \bigoplus_{\phi:T[m] \to \C^} \mathcal{L}_{\phi} \end{equation} analogous to that of eq.(\ref{decompositionsplitting}) and there exists $\phi:T[m] \to \C^$ such that $\mathcal{L}_{\phi}=\mathcal{L}$. The character $\phi$ is $W_{\mathcal{L}}$ invariant (for the conjugation action). \vspace{2 pt} Fix an $F$-stable Borel $B \supseteq T$ and consider $\widetilde{G}=\{(x,gB) \in G \times G/B \ \| \ g^{-1}xg \in B\}$ and the usual maps $p:\widetilde{G} \to G$ and $q:\widetilde{G} \to T$. Consider the perverse sheaf $$\mathcal{F}=p_!q^\mathcal{L} .$$ Let $G_{reg} \subseteq G$ be the subset of semisimple regular elements. Recall that we can identify $$G_{reg}=G/T \times_W T_{reg}$$ and $$p^{-1}(G_{reg})=G/T \times T_{reg}$$ in such a way that $p(gT,t)=[(gT,t)]$. Since $p$ is a $W$-covering over $G_{reg}$, we have that $\mathcal{F}\|_{G_{reg}}$ is local system and $$p_!q^\mathcal{L}=\IC(\mathcal{F}_{G_{reg}}) .$$ Let now $p_m\coloneqq p(id \times \theta_m):G/T \times T_{reg} \to G_{reg}$. The map $p_m$ is a Galois covering with group $W \rtimes T[m]$ and we have a decomposition $$(p_m)_!(\overline{\Q}_{\ell}) \cong \bigoplus_{\rho \in (W \rtimes T[m])^{\vee}} \mathcal{M}_{\rho}^{\oplus d_{\rho}}$$ where $d_{\rho}=\dim(\rho)$. Moreover, if $\displaystyle \Ind_{T[m]}^{W \rtimes T[m]}(\phi)=\bigoplus_{\rho \in (W \rtimes T[m])^{\vee}}\rho^{\oplus m_{\rho}}$, we have that $$\mathcal{F}_{G_{reg}} \cong \mathcal{M}_{\rho}^{\oplus m_{\rho}} .$$ Notice that $$\Ind_{T[m]}^{W \rtimes T[m]}(\phi)=\Ind_{W_{\mathcal{L}} \rtimes T[m]}^{W \rtimes T[m]}(\Ind_{T[m]}^{W_{\mathcal{L}} \rtimes T[m]}(\phi)) .$$ Since the character $\phi$ is $W_{\mathcal{L}}$-invariant, we can extend it to a character $\phi :W_{\mathcal{L}} \rtimes T[m] \to \C^$. We have therefore that $$\Ind_{T[m]}^{W_{\mathcal{L}} \rtimes T[m]}(\phi)=\C[W_{\mathcal{L}}] \otimes \phi$$ as $W_{\mathcal{L}} \rtimes T[m]$-modules. Moreover, for any $\chi \in W_{\mathcal{L}}^{\vee}$, the character $\Ind_{W_{\mathcal{L}} \rtimes T[m]}^{W \rtimes T[m]}(\chi \otimes \phi)$ is an irreducible character of $W \rtimes T[m]$. Let $d_{\chi}=\deg(\chi)$. In particular, we have a decomposition $$\Ind_{T[m]}^{W \rtimes T[m]}(\phi)=\bigoplus_{\chi \in W_{\mathcal{L}}^{\vee}} \Ind_{W_{\mathcal{L}} \rtimes T[m]}^{W \rtimes T[m]}(\chi \otimes \phi)^{\oplus d_{\chi}}$$ and therefore $$\mathcal{F}_{G_{reg}}=\bigoplus_{\chi \in W_{\mathcal{L}}^{\vee}} \mathcal{M}_{\Ind_{W_{\mathcal{L}} \rtimes T[m]}^{W \rtimes T[m]}(\chi \otimes \phi)}^{\oplus d_{\chi}} .$$ In particular, the irreducible perverse sheaves (i.e. the character sheaves) appearing in the decomposition of $\mathcal{F}$ are indexed by the irreducible local systems appearing in the decomposition of $\mathcal{F}_{G_{reg}}$, i.e. are indexed by $W_{\mathcal{L}}^{\vee}=\mathcal{W}_{s^}^{\vee}$. ------------------------------------ \subsection{Pure cohomology of $\mathrm{PGL}_2(\C)$-character stacks and Langlands duality over finite fields} In this section, we explay how Theorem \ref{multiplicitiescharactersheaves} generalize the results of \cref{purecohomologyparagraph} through a sort of Langlands duality argument. We start by the following more general considerations. \vspace{2 pt} Consider $(G,F)$ and $(G^,F^)$ two dual split reductive groups over $\F_q$. For a definition, see for instance \cite[Definition 11.1.10]{DM}. Fix a split $F$-stable maximal torus $T \subseteq G$ and a corresponding dual split $F^$-stable maximal torus $T^* \subseteq G^$. Recall the following result, see for example \cite[Proposition 11.1.14]{DM}. \begin{prop} \label{bijectionelementscharacters1} We have an isomorphism of groups \begin{equation} \label{bijectionelementscharacters} \widehat{T^F} \longleftrightarrow (T^)^{F^} .\end{equation} \end{prop} \vspace{2 pt} In what follows, we will consider the case where $(G,F)=\Sl_2$ and $T=T_2'$ the torus of diagonal matrices and $(G^,F^)=\PGl_2$ and $T^=\mathcal{T}$, the torus obtained as the projection of the torus of diagonal matrices. Fix a generator $\zeta_{q-1}$ of $\F_q^$. The bijection (\ref{bijectionelementscharacters}) can be explicitly written down as follows $$\Phi:\Hom(\F_q^,\C^) \to \mathcal{T}(\F_q)$$ \begin{equation} \ \ \ \ \ \ \ \ \ \alpha \longrightarrow p_2\left(\begin{pmatrix} 1 &0\\ 0 &\alpha(\zeta_{q-1}) \end{pmatrix}\right) \end{equation} In particular, we have $$\Phi(\alpha_0)=p_2\left(\begin{pmatrix} 1 &0\\ 0 &-1 \end{pmatrix}\right) .$$ This correspondence has the following geometric interpretation. For each $\alpha \in \Hom(\F_q^,\C^)$, pick the $F$-stable Kummer local system $\mathcal{A}$ on $T_2'$ such that $\chi_{\mathcal{A}}=\alpha.$ Let now $s=\Phi(\alpha)$ and denote by $\mathcal{G}_s$ the skyscraper sheaf $(i_s)_(\overline{\Q}_{\ell})$, where $i_s:s \to \mathcal{T}$ is the closed embedding. Through Proposition \ref{bijectionkummer} and bijection (\ref{bijectionelementscharacters}), we are thus associating to each Kummer local system $\mathcal{A}$ on $T_2'$ a corresponding skyscraper sheaf $\mathcal{G}_s$ on $\mathcal{T}$. We now compare their respective geometric inductions $\mathcal{I}^{\Sl_2}_{T_2'}(\mathcal{A})$ and $\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_s)$. Notice that $\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_s)=r_!\widetilde{\mathcal{G}}_{s}$, where $\widetilde{\mathcal{G}}_{s}=(i_{\psi^{-1}(s)})_\overline{\Q}_{\ell}$ and $i_{\psi^{-1}(s)}:\psi^{-1}(s) \to \widetilde{\PGl_2}$ is the closed embedding. We assume that $\mathcal{A} \neq \overline{\Q}_{\ell}$, i.e. that $s$ is a regular element. In this case, we have that $r(\psi^{-1}(s))=\mathcal{C}_s$, the $\PGl_2$-conjugacy class of $s$ and through the identifications of Lemma \ref{lemmaspringerresolution}, we have $\psi^{-1}(s)=\PGl_2/\mathcal{T}$. We have thus two possibilities: \begin{itemize} \item $\mathcal{A} \neq \mathcal{A}_0$. In this case, on the one side, the (shifted) perverse sheaf $\mathcal{I}^{\Sl_2}_{T_2'}(\mathcal{A})$ is irreducible and, taking characteristic function, $\chi_{\mathcal{I}^{\Sl_2}_{T_2'}(\mathcal{A})}=R^{\Sl_2}_{T_{\epsilon'}}(\alpha)$. On the other side, since $s \neq p(g_{-1})$, we have that $r:\PGl_2/\mathcal{T} \to \mathcal{C}_{s} $ is an isomorphism, i.e. $\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_s)$ is $(i_{\mathcal{C}_{s}})_{}(\overline{\Q}_{\ell})$. In particular, we have $\chi_{\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_s)}=1_{\mathcal{C}_{s}(\F_q)}$, i.e. the characteristic function of the conjugacy class $\mathcal{C}_{s}(\F_q)$. \item $\mathcal{A}=\mathcal{A}_0$. In this case, on the one side, as explained in \cref{sectioncharactersheaves}, we have $\mathcal{I}_{T_2'}^{\Sl_2}(\mathcal{A}_0)=X \oplus Y$. On the other side, we have that $r:\PGl_2/\mathcal{T} \to \mathcal{C}_{-1}$ is the analog over $\F_q$ of the $\Z/2\Z$-covering introduced in \cref{premilinariesconjclassesC} over $\C$ for the degenerate class $\mathcal{C}_{-1}$. Similarly to the complex setting, we have therefore a splitting $\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_{p(g_{-1})})=\overline{\Q}_{\ell} \oplus \mathcal{L}_{\F_q}$, where $\mathcal{L}_{\F_q}$ is associated to the character $\sgn$. In particular, $\chi_{\mathcal{I}^{\PGl_2}_{\mathcal{T}}(\mathcal{G}_{p(g_{-1})})}=1_{\mathcal{C}_{-1}}+\chi_{\mathcal{L}_{\F_q}}$. Notice that, since $\mathcal{L}_{\F_q}$ is $\PGl_2$-equivariant, $\chi_{\mathcal{L}_{\F_q}}$ is a class function. \end{itemize} The bijection of Proposition \ref{bijectionelementscharacters} can therefore be extended to the following correspondence between characters of $\Sl_2(\F_q)$ and local systems on conjugacy classes of $\PGl_2(\overline{\F_q})$ (and their characteristic functions): $$\alpha \neq 1,\alpha_0 ,\ \ R^{\Sl_2}_{T'_2}(\alpha) \rightsquigarrow \text{local system } \overline{\Q}_{\ell} \text{ over } \mathcal{C}_s \ \ (1_{\mathcal{C}_s}) $$ $$\chi_Y \rightsquigarrow \text{ local system } \overline{\Q}_{\ell} \text{ over the degenerate conjugacy class } \mathcal{C}_{-1} \ \ (1_{\mathcal{C}_{-1}})$$ $$\chi_X \rightsquigarrow \text{ local system } \mathcal{L}_{\F_q}\text{ over the degenerate conjugacy class } \mathcal{C}_{-1} \ \ (\chi_{\mathcal{L}_{\F_q}}) .$$ \vspace{4 pt} In this framework, Conjecture \ref{conjmhslocalsystems} and Theorem \ref{multiplicitiescharactersheaves} have the following interpretation in terms of the duality here explained. \subsubsection{Preliminaries representations} We denote by $\alpha_0$ be the non-trivial character $$\alpha_0:\F_q^* \to \{\pm 1 \} .$$ We have the following. \begin{prop} Given $\gamma \in \Hom(\F_q^,\C^)$, there exists a solution to the equation \begin{equation} \delta^2=\gamma \end{equation} if and only if $\gamma(-1)=1$. In this case, the solutions are $\{\delta,\alpha_0 \delta\}$. \end{prop} \subsubsection{Character varieties} We propose thus the following definition of genericity for $k$-tuples of conjugacy classes of $\PGl_2(\C)$. \begin{definizione} Consider a $k$-tuple $\widetilde{\mathcal{C}}$ with $\widetilde{C}_i=\widetilde{C}_{x_i}$ and $x_1,\dots,x_k \in \C^\setminus\{ 1\}$. We say that $\widetilde{C}$ is generic if and only if $\mathcal{C}(1),\mathcal{C}(-1)$ are generic in the usual sense. \end{definizione} \begin{oss} Notice that if there exists $j$ such that $x_j=-1$, the $k$-tuple $\mathcal{C}(1)$ is generic if and only if $\mathcal{C}(-1)$ is generic. \end{oss} \begin{oss} Notice that if $\mathcal{C}(1),\mathcal{C}(-1)$ are generic, we expect $M_{\mathcal{C}}$ and $ M_{\mathcal{C}(-1)}$ to have the same cohomology. If $x_1,\dots,x_k \neq -1$ we should thus have \begin{equation} \label{genericity1} H_c(M_{\widetilde{\mathcal{C}}},q,t)=2H_c(M_{\mathcal{C}},q,t). \end{equation} \end{oss} \begin{oss} Let $\sigma_{\widetilde{\mathcal{C}}} \in (\C^)^I$ be the element $$\sigma_{\widetilde{\mathcal{C}}}\coloneqq(x_1\cdots x_k,x_1^{-2},\dots,x_k^{-2}) .$$ It is not difficult to check that $\mathcal{C}(1),\mathcal{C}(-1)$ are generic if and only if $\sigma_{\widetilde{\mathcal{C}}}$ is generic with respect to $\alpha$, i.e if and only if $$\mathcal{H}^{}_{\sigma_{\widetilde{\mathcal{C}}},\alpha}=\{\delta \in (\N^I)^ \ \| \ \sigma_{\widetilde{\mathcal{C}}}^{\delta}=1 \text{ and } \delta \leq \alpha \}=\{\alpha\} .$$ \end{oss} \vspace{6 pt} \subsubsection{Representations} Consider a $k$-tuple $$\widetilde{\Ch}=(R^{\Sl_2}_{T'}(\gamma_1),\dots,R^{\Sl_2}_{T'}(\gamma_k)) $$ and a $k$-tuple $$\Ch=(R^{\Gl_2}_{T}((\alpha_1,\beta_1),\dots,R^{\Gl_2}_{T}((\alpha_k,\beta_k)) $$ such that, for any $i$, we have $R^{\Sl_2}_{T_2'}(\gamma_i)=\Res^{T_2}_{T_2'}R^{\Gl_2}_{T_2}((\alpha_i,\beta_i))$. For any such $k$-tuple $\Ch$ $=(R^{\Gl_2}_{T}((\alpha_1,\beta_1)),\dots,R^{\Gl_2}_{T}((\alpha_k,\beta_k)))$ put $$\lambda_{\Ch}=\prod_{i=1}^k \alpha_i \beta_i .$$ \vspace{2 pt} From eq.(\ref{restriction1}) and Frobenius reciprocity, we deduce that, for any $\widetilde{\Ch}=(R^{\Sl_2}_{T'}(\gamma_1),\dots,R^{\Sl_2}_{T'}(\gamma_k))$ we have \begin{equation} \langle \bigotimes_{i=1}^k R^{\Sl_2}_{T'}(\gamma_i),1 \rangle= \langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)), \bigoplus_{\delta \in \Hom(\F_q^,\C^)} \delta \circ \det \rangle. \end{equation} Let $\Ch=(R^{\Gl_2}_{T}(\gamma_1,1),\dots,R^{\Gl_2}_{T}(\gamma_k,1))$. Notice that, for any $\delta \in \Hom(\F_q^,\C^)$, we have that $\langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)), \delta \circ \det \rangle \neq 0$ if and only if $$\delta^2=\lambda_{\Ch} .$$ We deduce therefore that $$ \langle \bigotimes_{i=1}^k R^{\Sl_2}_{T'}(\gamma_i),1 \rangle \neq 0$$ if and only if $\gamma_1 \cdots \gamma_k$ is a square, i.e if and only if $\gamma_1\cdot \gamma_k(-1)=1$. In this case, if we pick $\delta$ such that $\delta^2=\gamma_1 \cdots \gamma_k$, we have \begin{equation} \langle \bigotimes_{i=1}^k R^{\Sl_2}_{T'}(\gamma_i),1 \rangle=\langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)), \delta \circ \det \rangle+ \langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)), \alpha_0\delta \circ \det \rangle. \end{equation} We propose the following Definition of genericity for a $k$-tuple $\widetilde{\Ch}$ as above. \begin{definizione} The $k$-tuple $\widetilde{\Ch}$ is generic if and only if $(\Ch,\delta^{-1} \circ \det)$ and $(\Ch,\alpha_0 \delta^{-1} \circ \det)$ are generic in the usual sense. \end{definizione} \begin{oss} Let $\sigma_{\widetilde{\Ch}} \in \Hom(\F_q^,\C^)^I$ be the element $$\sigma_{\widetilde{\Ch}}=(\gamma_1,\dots,\gamma_k,\gamma_1^{-2},\dots,\gamma_k^{-2}) .$$ It is not difficult to check that $(\Ch,\delta^{-1} \circ \det)$ and $(\Ch,\alpha_0 \delta^{-1} \circ \det)$ are generic if and only if $\sigma_{\widetilde{\Ch}}$ is generic with respect to $\alpha$, i.e if and only if $$\mathcal{H}^_{\sigma_{\widetilde{\Ch}},\alpha}=\{\alpha\} .$$ \end{oss} \begin{oss} If $(\Ch,\delta^{-1} \circ \det)$ and $(\Ch,\alpha_0 \delta^{-1} \circ \det)$ are both generic, the associated multiplicities are equal and we find therefore $$\langle \bigotimes_{i=1}^k R^{\Sl_2}_{T'}(\gamma_i),1 \rangle=2 \langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)) \otimes \delta^{-1} \circ \det, 1 \rangle .$$ Consider generic $k$-tuples $\mathcal{C},\widetilde{\mathcal{C}}$ as above, with $\widetilde{\mathcal{C}}_i=\widetilde{\mathcal{C}}_{x_i}$ and $x_1,\dots,x_k \neq -1$. Conjecturally, we have that $$\langle \bigotimes_{i=1}^k R^{\Gl_2}_{T}((\gamma_i,1)) \otimes \delta^{-1} \circ \det, 1 \rangle=PH_c(M_{\mathcal{C}},q)$$ adn thus we should have $$\langle \bigotimes_{i=1}^k R^{\Sl_2}_{T'}(\gamma_i),1 \rangle=2PH_c(M_{\mathcal{C}},q)=PH_c(M_{\widetilde{\mathcal{C}}},q) .$$ \end{oss} \subsection{Character sheaves} Recall that we have a decomposition $$R^{\Sl_2}_{T'}(\alpha_0)=\chi^{+}_{\alpha_0}+\chi^{-}_{\alpha_0}$$ where $\chi^{+}_{\alpha_0},\chi^{-}_{\alpha_0}$ are irreducible. There are $4$ character sheaves which are not irreducible characters for $\Sl_2(\F_q)$. \begin{proof} From the first point of Definition \ref{definitiongenericitycharacterSln}, we have that $\gamma_1\cdots \gamma_k(-1)=1$ and therefore from eq.(\ref{eqmult5}), we have that \begin{equation} \langle R^{\Sl_2}_{T'_2}(\gamma_1) \otimes \cdots \otimes R^{\Sl_2}_{T'_2}(\gamma_k),1 \rangle= \end{equation} \begin{equation} \langle R^{\Gl_2}_{T_2}((\gamma_1,1))\otimes \cdots \otimes R^{\Gl_2}_{T_2}((\gamma_k,1))\otimes \lambda_{\Ch}^{-1}\circ \det,1 \rangle+\langle R^{\Gl_2}_{T_2}((\gamma_1,1))\otimes \cdots \otimes R^{\Gl_2}_{T_2}((\gamma_k,1)) \otimes \lambda_{\Ch}^{-1}\alpha_0\circ \det,1 \rangle. \end{equation} From Lemma \ref{lemmagenericitysl2}, we have that the $(k+1)$-tuples $ R^{\Gl_2}_{T_2}((\gamma_1,1)),\dots, R^{\Gl_2}_{T_2}((\gamma_k,1)),\lambda_{\Ch}^{-1}\circ \det)$ and $ R^{\Gl_2}_{T_2}((\gamma_1,1)),\dots, R^{\Gl_2}_{T_2}((\gamma_k,1)),\lambda_{\Ch}^{-1}\alpha_0\circ \det)$ are both generic. From \cite[Theorem 6.1.1]{HA}, we deduce that we have $$\langle R^{\Gl_2}_{T_2}((\gamma_1,1))\otimes \cdots \otimes R^{\Gl_2}_{T_2}((\gamma_k,1))\otimes \lambda_{\Ch}^{-1}\circ \det,1 \rangle=\langle R^{\Gl_2}_{T_2}((\gamma_1,1))\otimes \cdots \otimes R^{\Gl_2}_{T_2}((\gamma_k,1)) \otimes \lambda_{\Ch}^{-1}\alpha_0\circ \det,1 \rangle=$$ $$\mathbb{H}_{\mathbb{\mu}}(0,\sqrt{q}) .$$ We deduce therefore eq.(\ref{theoremmultiplicitiesformula}) \end{proof} Multiplicities for generic $k$-tuples have a well-understood combinatorial description in terms of the generating series $\Omega(z,w)$ introduced before. More precisely, in \cite{letellier2} to any $k$-tuple $\Ch$, it is associated a function $s_{\omega_{\Ch}} \in \Lambda(\mathbf{x}_1,\dots,\mathbf{x}_k)$. We have the following Theorem, see \cite[Theorem 6.10.1]{letellier2} \begin{teorema} \label{theoremgenericmultiplicitiesgln} If $\Ch$ is generic, we have \begin{equation} \label{genericmultiplicitiesglnformula} \langle \Ch_1 \otimes \cdots \otimes \Ch_k,1 \rangle=\mathbb{H}_{\omega_{\Ch}}(0,\sqrt{q}) \end{equation} where $\mathbb{H}_{\omega_{\Ch}}(z,w):=(z^2-1)(1-w^2)\left\langle {\rm Log}\,\Omega(z,w),s_{\omega_{\Ch}}\right\rangle$ \end{teorema} Fix now an $F^$-stable regular semisimple conjugacy class $C^* \subseteq G^$ and a representative $s^ \in (T^* \cap C^)(\F_q)$. Put $$W_{s^}=\{w \in W \ \| \ w \cdot s^=s^\} .$$ Since $s^$ is regular semisimple, we have that $$W_{s^}=C_{G^}(s^)/C^{\circ}_{G^}(s^) .$$ Recall that we have the following bijection \begin{equation} (C_{G^}(s^)/C^{\circ}_{G^}(s^))^{\vee}=(W_{s^})^{\vee} \longleftrightarrow \{\text{ Irreducible and } G^-\text{equivariant local systems on } C^\}. \end{equation} \vspace{2 pt} From the bijections \ref{bijectionelementscharacters},\ref{bijectionKummer}, the element $s^$ determines a character $s:T^F \to \C^$ and therefore an $F$-stable Kummer local system $\mathcal{E}$. Notice that we have $$W_{s^}=W_{\mathcal{E}} .$$ From the arguments given at the end of \cref{sectioncharactersheaves}, we have therefore a bijection \begin{equation} \label{bijectionlocalsystems} \{\text{ Irreducible and } G^-\text{equivariant local systems on } C^\} \Leftrightarrow \{\text{Irreducible components of } r_!\psi^\mathcal{E}\} .\end{equation} Apply this to the case where $(G,F)=\Sl_2$ and $(G^,F^)=\PGl_2$, as in Example \ref{exampleLanglandsduality}. Put $C^=\mathcal{C}_{-1,\F_q}$ and $s^*=M_{-1}$, where we denote by $\mathcal{C}_{-1,\F_q}$ the conjugacy class of $M_{-1}$ in $\PGl_2$. As remarked in the Exmaple \ref{exampleLanglandsduality}, in this case $\mathcal{E}=\mathcal{A}_0$. The same reasoning applied in \cref{premilinariesconjclassesC}, in that case over $\C$, translates to the case of finite fields and we have therefore $$\{\text{ Irreducible and } \PGl_2-\text{equivariant local systems on } \mathcal{C}_{-1}\} =\{\overline{\Q}_{\ell},\mathcal{L}_{\F_q}\} ,$$ where $\mathcal{L}_{\F_q}$ is the analogous of $\mathcal{L}$ over $\F_q$. The bijection of eq.(\ref{bijectionlocalsystems}) becomes therefore the bijection $$\{\overline{\Q}_{\ell},\mathcal{L}_{\F_q}\} \Leftrightarrow \{X,Y\} $$ $$\mathcal{L}_{\F_q} \to X .$$
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\newcommand{\Tsref}[1]{Tables~\ref{#1}} \newcommand{\sref}[1]{Sec.~\ref{#1}} \newcommand{\Sref}[1]{Section~\ref{#1}} \newcommand{\ssref}[1]{Secs.~\ref{#1}} \newcommand{\Ssref}[1]{Sections~\ref{#1}} \newcommand{\fref}[1]{Fig.~\ref{#1}} \newcommand{\Fref}[1]{Figure~\ref{#1}} \newcommand{\fsref}[1]{Figs.~\ref{#1}} \newcommand{\Fsref}[1]{Figures~\ref{#1}} \newcommand{\aref}[1]{Appendix~\ref{#1}} \newcommand{\asref}[1]{Appendices~\ref{#1}} \newcommand{\Aref}[1]{Appendix~\ref{#1}} \newcommand{\Asref}[1]{Appendices~\ref{#1}} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} \setcounter{theorem}{0} \setcounter{lemma}{0} \setcounter{section}{0} \def\<{\langle} \def\>{\rangle} \newcommand{\rcite}[1]{Ref.~\cite{#1}} \newcommand{\rscite}[1]{Refs.~\cite{#1}} \newcommand{\equad}{\,\hphantom{=}\,} \begin{document} \title{Optimal estimation of three parallel spins with genuine and restricted collective measurements} \author{Changhao Yi} \affiliation{State Key Laboratory of Surface Physics, Department of Physics, and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China} \affiliation{Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China} \affiliation{Shanghai Research Center for Quantum Sciences, Shanghai 201315, China} \author{Kai Zhou} \affiliation{CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China} \affiliation{CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China} \author{Zhibo Hou} \affiliation{CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China} \affiliation{CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China} \affiliation{Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, People's Republic of China} \author{Guo-Yong Xiang} \affiliation{CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China} \affiliation{CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China} \affiliation{Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, People's Republic of China} \author{Huangjun~Zhu} \email{[email protected]} \affiliation{State Key Laboratory of Surface Physics, Department of Physics, and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China} \affiliation{Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China} \affiliation{Shanghai Research Center for Quantum Sciences, Shanghai 201315, China} \begin{abstract} Collective measurements on identical and independent quantum systems are advantageous in information extraction than individual measurements. However, little is known about the distinction between restricted collective measurements and genuine collective measurements in the multipartite setting. In this work we establish a rigorous performance gap based on a simple and old estimation problem, the estimation of a random spin state given three parallel spins. Notably, we derive an analytical formula for the maximum estimation fidelity of biseparable measurements and clarify its fidelity gap from genuine collective measurements. Moreover, we show that this estimation fidelity can be achieved by two- and one-copy measurements assisted by one-way communication in one direction, but not the other way. Our work reveals a rich landscape of multipartite nonclassicality in quantum measurements instead of quantum states and is expected to trigger further studies. \end{abstract} \date{\today} \maketitle \section{Introduction} Quantum measurements are the key for extracting information from quantum systems \cite{NielC10book} and play crucial roles in various tasks in quantum information processing, such as quantum state estimation, quantum metrology, quantum communication, and quantum computation. When two or more quantum systems are available, collective measurements on all quantum systems together may extract more information than individual measurements \cite{PereW91,MassP95,GisiP99,Mass00,BagaBGM06S,Zhu12the,ZhuH18U}, even if there is no entanglement or correlation among these quantum systems. This intriguing phenomenon is a manifestation of nonclassicality in quantum measurements rather than quantum states. Moreover, collective measurements are quite useful in many practical applications, including quantum state estimation \cite{MassP95,BagaBGM06S,HaahHJW16,ODonW16,Zhu12the,ZhuH18U}, direction estimation \cite{GisiP99,Mass00}, multi-parameter estimation \cite{VidrDGJ14,LuW21,ChenCY22}, shadow estimation \cite{Aaro18,GriePS24,LiuLYZ24,LiYZZ24}, quantum state discrimination \cite{PereW91,HiggDBP11,MartHSS21}, quantum learning \cite{HuanKP21,ChenCHL22,AharCQ22,GrewIKL24}, entanglement detection and distillation \cite{Horo03,LindMP98,DehaVDV03}, and nonlocality distillation \cite{EftaWC23}. The power of collective measurements have been demonstrated in a number of experiments \cite{HouTSZ18,TangHSZ20,ZhouYYH23,ConlVMP23,ConlELA23,TianYHX24}. Although collective measurements are advantageous for many applications, their realization in experiments is quite challenging, especially for multi-copy collective measurements. Actually, almost all experiments in this direction are restricted to two-copy collective measurements, and a genuine three-copy collective measurement was realized only very recently \cite{ZhouYYH23}. In view of this situation, it is natural to ask whether there is a fundamental gap between restricted collective measurements on limited copies of quantum states and genuine collective measurements, which represent the ultimate limit. This problem is of interest to both foundational studies and practical applications. Unfortunately, little is known about this problem, although the counterpart for quantum states has been well studied \cite{HoroHHH09,BrunCPS13}. Conceptually, the very basic definitions remain to be clarified. Technically, it is substantially more difficult to analyze the performances of restricted collective measurements. In this work we start to explore the rich territory of multipartite nonclassicality in quantum measurements by virtue of a simple and old estimation problem, the estimation of a random spin state given three parallel spins \cite{MassP95,DerkBE98,LatoPT98,Haya98,BrusM99,GisiP99,Mass00,AcinLP00,BagaBM02,HayaHH05}. To set the stage, we first introduce rigorous definitions of biseparable measurements (which encompass all restricted collective measurements) and genuine collective measurements. Then, we derive an analytical formula for the maximum estimation fidelity of biseparable measurements, which clearly demonstrates a fidelity gap from genuine collective measurements. In addition, we determine the maximum estimation fidelity based on one- and two-copy collective measurements assisted by one-way communication. Surprisingly, such strategies can reach the maximum estimation fidelity of biseparable measurements if the communication direction is chosen properly. By contrast, the maximum estimation fidelity achievable is strictly smaller if the communication direction is reversed. Our work reveals a strict hierarchy of multi-copy collective measurements and a plethora of nonclassical phenomena rooted in quantum measurements, which deserve further studies. The rest of this paper is organized as follows. We begin with the formal definitions positive operator-valued measures (POVMs), biseparable POVMs and genuine collective POVMs in \sref{sec:definition}. In \sref{sec:state_estimation}, we review an old estimation problem and the concept of estimation fidelity. In \sref{sec:CollectiveOpt} we review an optimal POVM for estimating three parallel spins and show that it is genuinely collective, although all its POVM elements are biseparable. In \sref{sec:RestrictedOpt}, we determine the maximum estimation fidelities of biseparable measurements, 2+1 adaptive measurements, and 1+2 adaptive measurements, respectively, and construct optimal estimation strategies explicitly. \Sref{sec:Summary} summarizes this paper. \section{Separable and collective measurements}\label{sec:definition} \subsection{Quantum states and quantum measurements} Let $\caH$ be a given finite-dimensional Hilbert space and $\caL(\caH)$ the space of linear operators on $\caH$. Quantum states on $\caH$ are represented by positive (semidefinite) operators of trace 1. Quantum measurements on $\caH$ can be described by POVMs when post-measurement quantum states are irrelevant \cite{NielC10book}. Mathematically, a POVM is composed of a set of positive operators that sum up to the identity operator, which is denoted by $I$ henceforth. If we perform the POVM $\scrA=\{A_j\}_j$ on the quantum state $\rho$, then the probability of obtaining outcome $j$ is $\tr(\rho A_j)$ according to the Born rule. Given two POVMs $\scrA=\{A_j\}_j$ and $\scrB=\{B_k\}_k$ on $\caH$, $\scrA$ is a \textit{coarse graining} of $\scrB$ if it can be realized by performing $\scrB$ followed by data processing \cite{MartM90,Zhu22}. In other words, the POVM elements $A_j$ of $\scrA$ can be expressed as follows, \begin{equation} A_j = \sum_{B_k\in\scrB}\Lambda_{jk}B_k \quad \forall A_j\in\scrA, \end{equation} where $\Lambda$ is a stochastic matrix satisfying $\Lambda_{jk}\geq 0$ and $\sum_j \Lambda_{jk} = 1$. A convex combination of $\scrA$ and $\scrB$ is the disjoint union of $\{wA_j\}_j$ and $\{(1-w)B_k\}_k$ with $0\leq w\leq 1$. Convex combinations of three or more POVMs can be defined in a similar way. \subsection{Separable and collective measurements} Next, we turn to quantum states and POVMs on a bipartite system shared by Alice and Bob, where the total Hilbert space is a tensor product of the form $\caH_\rmT=\caH_\rmA\otimes \caH_\rmB$. A quantum state $\rho$ on $\caH_\rmT$ is a product state if it is a tensor product of two states on $\caH_\rmA$ and $\caH_\rmB$, respectively. The state $\rho$ is separable if it can be expressed as a convex sum of product states; otherwise, it is entangled. Note that a pure state on $\caH_\rmT$ is separable if and only if (iff) it is a product state. A positive operator on $\caH_\rmT$ is separable if it is proportional to a separable state. A POVM on $\caH_\rmT$ is separable if every POVM element is separable. Let $\scrA=\{A_j\}_j$ and $\scrB=\{B_k\}_k$ be two POVMs on $\caH_\rmA$ and $\caH_\rmB$, respectively. The tensor product of $\scrA$ and $\scrB$ is defined as $\scrA\otimes \scrB:=\{A_j\otimes B_k\}_{j,k}$. Such product POVMs are prominent examples of separable POVMs, but there are more interesting examples. A POVM is $\rmA \rightarrow \rmB$ one-way adaptive if it has the form $\{A'_j\otimes B_{jk}'\}_{j,k}$, where $\scrA'=\{A_j'\}_j$ is a POVM on $\caH_\rmA$ and $\scrB_j'=\{B_{jk}'\}_k$ for each $j$ is a POVM on $\caH_\rmB$. Such a POVM can be realized by first performing the POVM $\scrA'$ on $\caH_\rmA$ and then performing the POVM $\scrB_j'$ on $\caH_\rmB$ if the first measurement yields outcome $j$. \subsection{Biseparable and genuine collective measurements} Next, we turn to a tripartite quantum system, which represents the simplest setting that can manifest multivariate quantum correlations. Although multipartite quantum correlations in quantum states have been studied by numerous researchers, the counterparts in quantum measurements remain uncharted. Now, the total Hilbert space is a tensor product of the form $\caH_\rmT=\caH_\rmA\otimes \caH_\rmB\otimes \caH_\rmC$, which is shared by Alice, Bob, and Charlie. A quantum state on $\caH_\rmT$ is biseparable with respect to the bipartition $\rmA\rmB\|\rmC$ if $\rho$ is separable when $\rmA\rmB$ is regarded as whole. Biseparable states with respect to the bipartition $\rmA\rmC\|\rmB$ or $\rmB\rmC\|\rmA$ can be defined in a similar way. The quantum state $\rho$ is biseparable if it can be expressed as a convex sum of three states $\rho_{\rmA\rmB\|\rmC}$, $\rho_{\rmA\rmC\|\rmB}$, and $\rho_{\rmB\rmC\|\rmA}$, which are biseparable with respect to three bipartitions, respectively. Otherwise, the state $\rho$ has genuine multipartite entanglement. By contrast, a POVM $\scrM=\{M_j\}_j$ on $\caH_\rmT$ is biseparable with respect to the bipartition $\rmA\rmB\|\rmC$ if every POVM element $M_j$ is biseparable with respect to the bipartition, which means $\scrM$ is separable when $\rmA\rmB$ is regarded as whole. The POVM is 2+1 adaptive (with respect to the bipartition $\rmA\rmB\|\rmC$) if it can be realized by one-way communication from $\rmA\rmB$ to $\rmC$; it is 1+2 adaptive if it can be realized by one-way communication from $\rmC$ to $\rmA\rmB$; see \fref{fig:POVMRC} for an illustration. Generalization to other bipartitions is immediate. The POVM $\scrM$ is biseparable if it can be expressed as a coarse graining of a convex combination of three POVMs $\scrM_{\rmA\rmB\|\rmC}$, $\scrM_{\rmA\rmC\|\rmB}$, and $\scrM_{\rmB\rmC\|\rmA}$, which are biseparable with respect to three bipartitions, respectively. Otherwise, the POVM $\scrM$ is genuinely collective. \begin{figure}[tbp] \centering \includegraphics[width = 0.45\textwidth]{biseparable.pdf} \caption{Three types of biseparable measurements on three qubits: two-way adaptive (a), 2+1 adaptive (b), and 1+2 adaptive (c). Note that (a) contains (b) and (c) as special cases.} \label{fig:POVMRC} \end{figure} \section{Optimal quantum state estimation} \label{sec:state_estimation} \subsection{A simple estimation problem and estimation fidelity} Here we reexamine an old estimation problem: A quantum device produces $N$ copies of a Haar random pure state $\rho=\|\psi\>\<\psi\|$ on $\caH$, where $\caH$ has dimension~$d$, and our task is to estimate the identity of $\rho$ based on quantum measurements \cite{MassP95,DerkBE98,LatoPT98,Haya98,BrusM99,GisiP99,Mass00,AcinLP00,BagaBM02,HayaHH05}. The performance of an estimation protocol is quantified by the average fidelity. Suppose we perform a POVM $\scrM=\{M_j\}_j$ on $\rho^{\otimes N}$, then the probability of obtaining outcome $j$ reads $p_j = \tr(M_j \rho^{\otimes N})$. If we choose $\hat{\rho}_j$ as the estimator associated with outcome $j$, then the average estimation fidelity achieved by this protocol reads \begin{equation} \overline{F} = \sum_j \int_{\text{Haar}}d\psi \tr\bigl[(\|\psi\>\<\psi\|)^{\otimes N}M_j\bigr]\<\psi\|\hat{\rho}_j\|\psi\>, \end{equation} where the integral means taking the average over the ensemble of Haar random pure states. Let $\Sym_N(\caH)$ be the symmetric subspace in $\caH^{\otimes N}$ and $P_N$ the projector onto $\Sym_N(\caH)$. Define \begin{align} \caQ(M_j) &:= (N+1)!\tr_{1,2,\ldots, N}[P_{N+1}(M_j\otimes I)], \label{eq:Qmap}\\ F(\scrM)&:= \sum_j\frac{\\|\caQ(M_j)\\|}{d(d+1)\cdots (d+N)},\label{eq:EstimationFidDef} \end{align} where $\\|\cdot\\|$ is the spectral norm. Then $\overline{F}\leq F(\scrM)$, and the inequality is saturated if each $\hat{\rho}_j$ is supported in the eigenspace of $\caQ(M_j)$ with the maximum eigenvalue by \rcite{Zhu22}. In view of this fact, $F(\scrM)$ is called the estimation fidelity of $\scrM$ henceforth. The definition of the estimation fidelity is still applicable when $\scrM$ is an incomplete POVM, which means $\sum_j M_j\leq I^{\otimes N}$. If there is no restriction on the POVMs that can be performed, then the maximum estimation fidelity is $(N+1)/(N+d)$, and optimal POVMs can be constructed from $t$-designs with $t=N$ \cite{MassP95,BrusM99,HayaHH05,Zhu22}. \subsection{Properties of the map $\caQ$ and estimation fidelity} The basic properties of the map $\caQ$ and estimation fidelity $F(\scrM)$ are clarified in \rcite{Zhu22}. Here we introduce some additional results that are relevant to the following discussion. Note that the argument of $\caQ$ is not restricted to POVM elements and is not necessarily Hermitian. In analogy to $P_N$, let $P_N^\rmA$ be the projector onto the antisymmetric subspace in $\caH^{\otimes N}$. Let $\caS_N$ be the symmetric group of the $N$ parties associated with the $N$ copies of $\rho$. For each $\sigma\in \caS_N$, let $\bbW_\sigma$ be the unitary operator on $\caH^{\otimes N}$ tied to the permutation $\sigma$. Then \begin{align} P_N = \frac{1}{N!}\sum_{\sigma\in \caS_N}\bbW_\sigma,\; P^\rmA_N = \frac{1}{N!}\sum_{\sigma\in \caS_N}\sgn(\sigma)\bbW_\sigma, \end{align} where $\sgn(\sigma)=1$ when $\sigma$ is an even permutation and $\sgn(\sigma)=-1$ when $\sigma$ is an odd permutation. The following lemma is a simple corollary of the definition of $\caQ$ in \eref{eq:Qmap}. \begin{lemma}\label{lem:Q} Suppose $M\in \caL(\caH^{\otimes N})$; then \begin{gather} \label{eq:B1} \caQ(\bbW_\sigma M \bbW_\tau) = \caQ(M) \quad \forall \sigma,\tau \in \caS_N,\\ \label{eq:B2} \caQ[(P_{N-1}\otimes I)M(P_{N-1}\otimes I)] = \caQ(M). \end{gather} If in addition $N\geq 3$, then \begin{equation} \label{eq:B3} \caQ\bigl[\bigl(P_{N-1}^\rmA\otimes I\bigr)M\bigl(P_{N-1}^\rmA\otimes I\bigr)\bigr] =0. \end{equation} \end{lemma} Given any POVM $\scrM=\{M_j\}_j$ on $\caH^{\otimes N}$, define \begin{equation}\label{eq:POVMsymAsym} \begin{aligned} \scrM_\rmS&:=\{(P_{N-1}\otimes I) M_j (P_{N-1}\otimes I)\}_j,\\ \scrM_\rmA&:=\bigl\{\bigl(P_{N-1}^\rmA \otimes I\bigr) M_j \bigl(P_{N-1}^\rmA \otimes I\bigr)\bigr\}_j. \end{aligned} \end{equation} Here $\scrM_\rmS$ ($\scrM_\rmA$) is called the symmetric (antisymmetric) part of $\scrM$. The following lemma is a simple corollary of \lref{lem:Q}. \begin{lemma}\label{lem:POVMsym} Suppose $\scrM$ is a POVM on $\caH^{\otimes N}$. Then \begin{align} F(\scrM_\rmS)=F(\scrM),\quad F(\scrM_\rmA)=0. \label{eq:POVMsym} \end{align} \end{lemma} \section{\label{sec:CollectiveOpt}Optimal estimation of three parallel spins with genuine collective measurements} From now on we turn to the estimation of three parallel spins, which means $d=2$, $N=3$, and $\caH$ is a single-qubit Hilbert space. In the following discussion, we label the three copies of $\caH$ by $\rmA$, $\rmB$, and $\rmC$, respectively. To benchmark the performances of restricted collective measurements, we first reexamine an optimal estimation strategy when there is no restriction on the POVMs that can be performed. In this case, the maximum estimation fidelity is $4/5$, \cite{MassP95,LatoPT98}. Consider the three Pauli operators \begin{align} X=\begin{pmatrix} 0 &1\\ 1 &0 \end{pmatrix}, \;\; Y=\begin{pmatrix} 0 &-\rmi\\ \rmi &0 \end{pmatrix}, \;\; Z=\begin{pmatrix} 1 &0\\ 0 &-1 \end{pmatrix}, \end{align} and let $\|\psi_j\>$ for $j=1,2,...,6$ be the six eigenstates, which form a regular octahedron when represented on the Bloch sphere. Then an optimal POVM $\scrM_\col$ can be constructed from the following seven POVM elements \cite{LatoPT98}, \begin{equation}\label{eq:POVMcol} \begin{aligned} E_j &:= \frac{2}{3}\|\psi_j\>\<\psi_j\|^{\otimes 3},\quad j = 1,2,\ldots,6,\\ E_7 &:= I^{\otimes 3} - \sum_{j=1}^6 E_j = I^{\otimes 3} - P_3, \end{aligned} \end{equation} where $P_3$ is the projector onto $\Sym_3(\caH)$. Although this POVM was constructed more than 20 years ago, its intriguing properties have not been fully appreciated. Let $\Pi=P_2^\rmA\otimes I$ and let $\bbW$ be the unitary operator on $\caH^{\otimes 3}$ that is associated with a cyclic permutation. Then $\bbW^2=\bbW^\dag$ and the POVM element $E_7$ can be expressed as follows, \begin{align} E_7=\frac{2}{3}\bigl(\Pi +\bbW \Pi\bbW^\dag + \bbW^\dag \Pi\bbW \bigr), \end{align} which means $E_7$ is biseparable. So all POVM elements in the optimal POVM $\scrM_\col$ are biseparable. However, the POVM $\scrM_\col$ itself is not biseparable as shown in the following proposition and proved in \aref{app:POVMgcolProof}. \begin{proposition}\label{pro:POVMgcol} The POVM $\scrM_\col$ defined in \eref{eq:POVMcol} is genuinely collective. \end{proposition} Collective measurements are in general not easy to realize. If we can only perform local measurements on individual copies, then the maximum estimation fidelity is $(3 +\sqrt{3}\lsp)/6$, and the maximum can be attained when Alice, Bob, and Charlie perform Pauli $X$, $Y$, and $Z$ measurements, respectively. \section{\label{sec:RestrictedOpt}Optimal estimation of three parallel spins with restricted collective measurements} Although the maximum estimation fidelity of general collective measurements has been well known for a long time, the performances of restricted collective measurements are still poorly understood. To fill this gap, here we shall determine the maximum estimation fidelities of biseparable measurements, 2+1 adaptive measurements, and 1+2 adaptive measurements, respectively, and construct optimal estimation strategies explicitly. It turns out 2+1 adaptive measurements can achieve the same estimation fidelity as biseparable measurements. By symmetry, the maximum estimation fidelity of biseparable POVMs with respect to the bipartition $\rmA\rmB\|\rmC$ is identical to the counterpart with respect to the bipartition $\rmA\rmC\|\rmB$ and the counterpart with respect to the bipartition $\rmB\rmC\|\rmA$. Moreover, this maximum estimation fidelity is also the maximum estimation fidelity of general biseparable POVMs, given that coarse graining cannot increase the estimation fidelity \cite{Zhu22}. In addition, it suffices to consider rank-1 POVMs to determine the maximum estimation fidelity. Therefore, we can focus on rank-1 POVMs that are biseparable with respect to the bipartition $\rmA\rmB\|\rmC$ in the following discussion. Furthermore, the maximum estimation fidelity of biseparable POVMs on $\caH^{\otimes 2}\otimes \caH$ is identical to the maximum estimation fidelity of separable POVMs on $\Sym_2(\caH)\otimes \caH$ thanks to \lref{lem:POVMsym}. If $\scrM$ is an optimal biseparable POVM on $\caH^{\otimes 2}\otimes \caH$, then $\scrM_\rmS$ defined in \eref{eq:POVMsymAsym} is an optimal separable POVM on $\Sym_2(\caH)\otimes \caH$. Conversely, if $\scrM$ is an optimal separable POVM on $\Sym_2(\caH)\otimes \caH$, then an optimal biseparable POVM on $\caH^{\otimes 2}\otimes \caH$ can be constructed as follows, \begin{align} \scrM\cup \bigl\{P_2^\rmA\otimes I\bigr\}. \end{align} Similar remarks apply to 2+1 adaptive POVMs and 1+2 adaptive POVMs. \subsection{Biseparable measurements} According to the previous discussion, to determine the maximum estimation fidelity of biseparable POVMs on $\caH^{\otimes 3}$, it suffices to consider separable rank-1 POVMs on $\Sym_2(\caH)\otimes \caH$. Suppose $\scrM=\{M_j\}_j$ is such a POVM, then $M_j$ have the form \begin{align}\label{eq:povm} M_j=w_j\|\Psi_j\>\<\Psi_j\|,\quad \|\Psi_j\>= \|\Phi_j\>\otimes\|\varphi_j\>, \end{align} where $\|\Phi_j\>\in \Sym_2(\caH)$ and $\|\varphi_j\>\in \caH$. In addition, \begin{equation}\label{eq:POVMbs} \begin{gathered} w_j > 0, \quad \sum_j w_j =6,\\ \sum_j w_j\|\Phi_j\>\<\Phi_j\|\otimes\|\varphi_j\>\<\varphi_j\| = P_2\otimes I, \end{gathered} \end{equation} where $P_2$ is the projector onto $\Sym_2(\caH)$. In conjunction with \eref{eq:EstimationFidDef}, the maximum estimation fidelity of biseparable POVMs can be expressed as follows, \begin{gather} F_\bs = \max_{\scrM} \frac{1}{120}\sum_j w_j \\|\caQ(\|\Phi_j\>\<\Phi_j\|\otimes\|\varphi_j\>\<\varphi_j\|)\\|,\label{eq:BisepOpt} \end{gather} where the maximization is subjected to the constraints in \eref{eq:POVMbs}. Based on \eqsref{eq:POVMbs}{eq:BisepOpt} we can derive an analytical formula for the maximum estimation fidelity, as shown in \thref{thm:bisep} below and proved in \aref{app:bisepProof}. Note that biseparable measurements can achieve a higher estimation fidelity than local measurements, but there is a fundamental gap from genuine collective measurements, as summarized in \tref{tab:summary}. \begin{theorem} \label{thm:bisep} In the estimation of three parallel spins, the maximum estimation fidelity of biseparable POVMs reads \begin{equation} F_\bs=\frac{1}{2} + \frac{\sqrt{22}}{16}. \end{equation} \end{theorem} \begin{table}[htbp] \centering \caption{\label{tab:summary}Maximum estimation fidelities of four different types of measurements.} \renewcommand\arraystretch{2} \begin{tabular}{ c \| c } \hline \hline Measurement & Maximum estimation fidelity \\ \hline \hline Local & $\frac{1}{2} + \frac{\sqrt{3}}{6} \approx 0.78868$ \cite{BagaMM05}\\ \hline 1+2 adaptive & $\frac{1}{2} + \frac{11 + \sqrt{41}}{60} \approx 0.79005$ \\ \hline \makecell{Biseparable\\ (or 2+1 adaptive)} & $\frac{1}{2} + \frac{\sqrt{22}}{16} \approx 0.79315$ \\ \hline Genuine collective & $\frac{4}{5}$ \cite{MassP95}\\ \hline \hline \end{tabular} \end{table} \subsection{2+1 adaptive measurements} Here we show that 2+1 adaptive measurements can achieve the same estimation fidelity as biseparable measurements. \begin{theorem}\label{thm:2+1} In the estimation of three parallel spins, the maximum estimation fidelity of 2+1 adaptive POVMs is \begin{equation}\label{eq:2+1} F_{2\rightarrow 1}= \frac{1}{2} + \frac{\sqrt{22}}{16}. \end{equation} \end{theorem} Since 2+1 adaptive POVMs are biseparable, to prove \thref{thm:2+1}, it suffices to construct a 2+1 adaptive POVM that can attain the estimation fidelity in \eref{eq:2+1}. A direct proof can be found in \aref{app:2+1}. Our construction is based on a subgroup of the single-qubit Clifford group $\Cl_1$. Recall that the Pauli group is generated by the three Pauli operators $X,Y,Z$. The Clifford group $\Cl_1$ is the normalizer of the Pauli group. Up to overall phase factors, it is generated by the Hadamard gate $H$ and phase gate~$S$, where \begin{align}\label{eq:HS} H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 &1\\ 1 &-1 \end{pmatrix} , \quad S=\begin{pmatrix} 1 &0\\ 0 &\rmi \end{pmatrix}. \end{align} Let $V=HS$ and let $\caG$ be the group generated by $X$ and $V$, then $\caG$ is a subgroup of $\Cl_1$ that contains the Pauli group. Let \begin{align} \overline{\caG}:=\bigl\{I, V, V^2\bigr\}\times \{I, X, Y, Z\}. \end{align} Then $\overline{\caG}$ can be identified as the quotient group of $\caG$ after identifying operators that are proportional to each other. Define \begin{equation} \begin{aligned} \|\tPhi\> &:= \frac{\sqrt{8+3\sqrt{7}}}{4} \|00\> + \frac{\sqrt{8 - 3\sqrt{7}}}{4}\|11\>,\\ \|\tPsi\> &:=\|\tPhi\>\otimes \|+\>, \end{aligned} \end{equation} where $\|+\>=(\|0\>+\|1\>)/\sqrt{2}$ is the eigenstate of $X$ with eigenvalue 1. Then an optimal biseparable POVM can be constructed as follows, \begin{align}\label{eq:OptPOVMbs2+1} \scrM_\bs:=\Bigl\{\frac{1}{2}U^{\otimes 3}\|\tPsi\>\<\tPsi\|U^{\dag\otimes 3}\,\Big\|\, U\in \overline{\caG} \Bigr\}\cup \bigl\{P_2^\rmA\otimes I\bigr\}. \end{align} Moreover, this optimal POVM can be realized by a 2+1 adaptive strategy. To be specific, let \begin{equation} \begin{aligned} \!\!\scrM_2&:= \Bigl\{\frac{1}{2}U^{\otimes 2}\|\tPhi\>\<\tPhi\|U^{\dag\otimes 2}\Big\| U\in \overline{\caG}_2\Bigr\}\cup\bigl\{P_2^\rmA\bigr\},\\ \overline{\caG}_2&:=\bigl\{I, V, V^2\bigr\}\times\{I,X\}; \end{aligned} \end{equation} then $\scrM_2$ is a POVM on $\caH^{\otimes 2}$. Now, $\scrM_\bs$ can be realized as follows: Alice and Bob first perform the POVM $\scrM_2$ and send the outcome to Charlie; if they obtain outcome $U^{\otimes 2}\|\tPhi\>\<\tPhi\|U^{\dag\otimes 2}/2$ (note that the outcome $P_2^\rmA$ can never occur), then Charlie performs the projective measurement on the eigenbasis of $UXU^\dag$. \subsection{1+2 adaptive measurements} In this section, we determine the maximum estimation fidelity of 1+2 adaptive measurements and devise an optimal strategy. \begin{theorem} \label{thm:1+2} In the estimation of three parallel spins, the maximum estimation fidelity of 1+2 adaptive POVMs is \begin{equation}\label{eq:1+2} F_{1\rightarrow 2}= \frac{1}{2} + \frac{11 + \sqrt{41}}{60}. \end{equation} \end{theorem} \Thref{thm:1+2} is proved in \aref{app:1+2}. Here we construct an optimal POVM that can attain the maximum estimation fidelity in \eref{eq:1+2}. Let \begin{equation}\label{eq:pSPhiW} \begin{aligned} p & := \frac{47 - 3\sqrt{41}}{216},\quad \|\bbS\> :=\frac{\|01\> + \|10\>}{\sqrt{2}}, \\ \|\Upsilon\> &:= \sqrt{\frac{1 - 3p}{3-3p}}\,\|00\> + \sqrt{\frac{1}{3-3p}}\left(\|\bbS\> + \|11\>\right),\\ W_j &:=(S\otimes S)^j,\quad j=0,1,2,3, \end{aligned} \end{equation} where $S$ is the phase gate defined in \eref{eq:HS}. Then we can construct two POVMs $\scrK_0, \scrK_1$ on $\caH^{\otimes 2}$ as follows, \begin{equation}\label{eq:scrE01} \begin{aligned} K_j&:=\frac{3-3p}{4}W_j\|\Upsilon\>\<\Upsilon\|W_j^\dag,\quad j=0,1,2,3,\\ K_4 &:=3p \|00\>\< 00\|,\quad K_5:=P_2^\rmA,\\ \scrK_0 & := \{K_j\}_{j=0}^5,\quad \scrK_1:=\bigl\{X^{\otimes 2}K_jX^{\otimes 2}\bigr\}_{j=0}^5. \end{aligned} \end{equation} On this basis, we can construct an optimal 1+2 adaptive POVM, \begin{align}\label{eq:OptPOVM1+2} \scrM_{1\rightarrow 2} :=\{\scrK_0\otimes \|0\>\<0\|\}\cup \{\scrK_1\otimes \|1\>\<1\|\}. \end{align} This POVM can be realized as follows: Charlie first performs the $Z$-basis measurement on his qubit and sends the measurement outcome to Alice and Bob. If the outcome is 0, then Alice and Bob perform the POVM $\scrK_0$ on their qubits; if the outcome is 1, then they perform the POVM $\scrK_1$ instead. \section{Summary} \label{sec:Summary} In this work we introduced rigorous definitions of biseparable measurements and genuine collective measurements, thereby setting the stage for exploring the rich territory of multipartite nonclassicality in quantum measurements instead of quantum states. By virtue of a simple estimation problem, we established a rigorous fidelity gap between biseparable measurements and collective measurements. Moreover, we showed that the maximum estimation fidelity of biseparable measurements can be attained by 2+1 adaptive measurements, but not by 1+2 adaptive measurements. Optimal estimation protocols in all these settings are constructed explicitly. Our work shows that quantum measurements in the multipartite setting may exhibit a rich hierarchy of nonclassical phenomena, which offer exciting opportunities for future studies. \section{Acknowledgments} The work at Fudan University is supported by the National Natural Science Foundation of China (Grant No.~92165109), National Key Research and Development Program of China (Grant No.~2022YFA1404204), and Shanghai Municipal Science and Technology Major Project (Grant No.~2019SHZDZX01). The work at the University of Science and Technology of China is supported by the Innovation Program for Quantum Science and Technology (Grant No.~2023ZD0301400), the National Natural Science Foundation of China (Grants Nos. 62222512, 12104439, and 12134014), the Anhui Provincial Natural Science Foundation (Grant No.~2208085J03), and USTC Research Funds of the Double First-Class Initiative (Grant Nos.~YD2030002007 and YD2030002011). \onecolumngrid \bigskip \appendix \section{Appendix} In this Appendix we prove the key results on optimal estimation of three parallel spins presented in the main text, namely, \pref{pro:POVMgcol} and \thsref{thm:bisep}-\ref{thm:1+2}. Throughout this appendix, we assume that $\caH$ is a single-qubit Hilbert space. \section{\label{app:POVMgcolProof}Proof of \pref{pro:POVMgcol}} \begin{proof}[Proof of \pref{pro:POVMgcol}] Suppose on the contrary that $\scrM_\col$ is biseparable. Then $\scrM_\col$ can be expressed as a coarse graining of three POVMs $\scrM_{\rmA\rmB\|\rmC}$, $\scrM_{\rmA\rmC\|\rmB}$, and $\scrM_{\rmB\rmC\|\rmA}$, which are biseparable with respect to three bipartitions, respectively. Without loss of generality, we can assume that the three POVMs are all rank-1. Note that every POVM element in $\scrM_\col$ is supported either in $\Sym_3(\caH)$ or in its orthogonal complement $\Sym_3(\caH)^\perp$, so every POVM element in $\scrM_{\rmA\rmB\|\rmC}$ is also supported either in $\Sym_3(\caH)$ or in $\Sym_3(\caH)^\perp$. Let $\scrM_{\rmA\rmB\|\rmC}'$ be the subset of POVM elements in $\scrM_{\rmA\rmB\|\rmC}$ that are supported in $\Sym_3(\caH)^\perp$; then $\scrM_{\rmA\rmB\|\rmC}'$ forms a POVM on $\Sym_3(\caH)^\perp$, which means \begin{align}\label{eq:POVMgcolProof} \sum_{M\in \scrM_{\rmA\rmB\|\rmC}'}M=I^{\otimes 3}-P_3. \end{align} On the other hand, by assumption each POVM element in $\scrM_{\rmA\rmB\|\rmC}'$ has the form $w_j\|\Phi_j\>\<\Phi_j\|\otimes \|\varphi_j\>\<\varphi_j\|$, where $w_j\geq 0$, $\|\Phi_j\>\in \caH^{\otimes 2}$, and $\|\varphi_j\>\in\caH$. According to \lref{lem:P3orthogonal} below, we have $\|\Phi_j\>\<\Phi_j\|=P_2^\rmA$, so every POVM element in $\scrM_{\rmA\rmB\|\rmC}'$ is supported in $\Sym_2(\caH)^\perp\otimes \caH$, which has dimension 2. Consequently, $\sum_{M\in \scrM_{\rmA\rmB\|\rmC}'}M$ has rank at most 2, which contradicts \eref{eq:POVMgcolProof}. This contradiction shows that $\scrM_\col$ cannot be biseparable and is thus genuinely collective. \end{proof} Next, we prove an auxiliary lemma employed in the proof of \pref{pro:POVMgcol}. \begin{lemma}\label{lem:P3orthogonal} Suppose $\caH$ has dimension 2, $\|\Phi\>\in \caH^{\otimes 2}$, and $\|\varphi\>\in\caH$. Then $\tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)]=0$ iff $\|\Phi\>\<\Phi\|=P_2^\rmA$. \end{lemma} \begin{proof} Let $\bbW_{(12)}$ be the swap operator acting on the first two parties. If $\|\Phi\>\<\Phi\|=P_2^\rmA$, then \begin{equation} \tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)]=\tr\bigl[P_3\bbW_{(12)}(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)\bigr]=-\tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)], \end{equation} which implies that $\tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)]=0$. Next, suppose $\tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)]=0$. Let $\rmU(\caH)$ be the group of all unitary operators on $\caH$. Choose $U\in \rmU(\caH)$ such that $U\|\varphi\>=\|0\>$ and let $\|\Phi'\>=U^{\otimes 2}\|\Phi\>$. Then \begin{align} 0&=\tr[P_3(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)]=\tr\bigl[P_3U^{\otimes 3}(\|\Phi\>\<\Phi\|\otimes \|\varphi\>\<\varphi\|)U^{\dag\otimes 3}\bigr]=\tr[P_3(\|\Phi'\>\<\Phi'\|\otimes \|0\>\<0\|)]\nonumber\\ &=\tr(\|\Phi'\>\<\Phi'\| R), \end{align} where \begin{align} R=\tr_\rmC[P_3(I\otimes I\otimes \|0\>\<0\|)]=\|00\>\<00\|+\frac{2\|\bbS\>\<\bbS\|}{3}+\frac{\|11\>\<11\|}{3}, \end{align} with $\|\bbS\>=(\|01\> + \|10\>)/\sqrt{2}$ as defined in \eref{eq:pSPhiW}. Note that $R$ has full support in $\Sym_2(\caH)$, so $\|\Phi'\>$ is necessarily supported in the orthogonal complement of $\Sym_2(\caH)$, which is spanned by $(\|01\>-\|10\>)/\sqrt{2}$. Therefore, $\|\Phi\>\<\Phi\|=\|\Phi'\>\<\Phi'\|=P_2^\rmA$, which completes the proof of \lref{lem:P3orthogonal}. \end{proof} \section{Proof of \thref{thm:bisep}} \label{app:bisepProof} \subsection{Auxiliary results} Suppose $a,b,c,x,y,\phi$ are real numbers. Define \begin{align} \eta(a,b,c,\phi)&:= 15 a^2 + 10b^2 + 5c^2 + \sqrt{8b^2\|3a + 2c\rme^{\rmi\phi}\|^2 + (9a^2 + 2b^2 - c^2)^2}, \label{eq:etaabcphiDef}\\ \eta(a,b,c)&:= 15 a^2 + 10b^2 + 5c^2 + \sqrt{8b^2(3\|a\| + 2\|c\|)^2 + (9a^2 + 2b^2 - c^2)^2},\\ f(x,y)& := \sqrt{4(1-x)(3\sqrt{x + y} + 2\sqrt{x - y}\lsp)^2 + (2x + 5y + 2)^2}. \label{eq:fxy} \end{align} Given any state $\|\Psi \>$ in $\caH^{\otimes 3}$, define \begin{align} \caI_\Psi := \int_{\mathrm{Haar}}\rmd U U^{\otimes 3} \|\Psi\>\<\Psi\| U^{\dag\otimes 3}, \label{eq:IpsiDef} \end{align} where the integration is taken over the normalized Haar measure on the unitary group $\rmU(\caH)$. Alternatively, the integration can be replaced by summation over a unitary 3-design. Recall that a set of unitaries $\caU = \{U_k\}_k$ is a unitary $t$-design \cite{GrosAE07} if the following equation holds \begin{equation} \frac{1}{\|\caU\|}\sum_{U_k\in\caU}U_k^{\otimes t}O U_k^{\dag\otimes t} = \int_{\mathrm{Haar}} dU U^{\otimes t}O U^{\dag\otimes t}\quad \forall O\in \caL\bigl(\caH^{\otimes t}\bigr). \end{equation} \begin{lemma}\label{lem:q_abc} Suppose $\|\Phi \> = a\|00\> + b\|\bbS\> + c\rme^{\rmi\phi}\|11\>\in \caH^{\otimes 2}$ and $\|\Psi\>=\|\Phi\>\otimes \|0\>\in \caH^{\otimes 3}$, where $a,b,c,\phi$ are real numbers and $\|\bbS\>=(\|01\>+\|10\>)/\sqrt{2}$. Then \begin{gather} 6\tr(P_3 \|\Psi\>\<\Psi\|) = 6a^2+4b^2+2c^2, \label{eq:P3Psi0}\\ \caI_\Psi = \frac{3a^2+2b^2+c^2}{12}P_3+\frac{b^2+2c^2}{6}(P_2\otimes I-P_3), \label{eq:IPsi} \\ \\|\caQ(\|\Psi\>\<\Psi\|)\\| = \eta(a,b,c,\phi)\le \eta(a,b,c)=15 a^2 + 10b^2 + 5c^2 + f\bigl(a^2+c^2, a^2-c^2\bigr), \label{eq:QPsinorm2} \end{gather} and the last inequality is saturated when $\phi=0$. \end{lemma} When $a^2 = c^2$, \eref{eq:IPsi} implies that \begin{equation}\label{eq:a2c2} \caI_\Psi = \frac{1}{6}P_2\otimes I. \end{equation} In this case, if $\{U_k\}_{k=1}^m$ is a unitary 3-design, then $\bigl\{6 U_k\|\Psi\>\<\Psi\|U_k^\dag/m \bigr\}_{k=1}^m$ is a POVM on $\Sym_2(\caH)\otimes \caH$. This conclusion will be useful to constructing optimal biseparable POVMs. \begin{proof}[Proof of \lref{lem:q_abc}] \Eref{eq:P3Psi0} can be verified by straightforward calculation. By Schur-Weyl duality, $\caI_\Psi$ is a linear combination of $\bbW_\sigma$ with $\sigma\in \caS_3$. In addition $\caI_\Psi=\bbW_{(12)}\caI_\Psi= \caI_\Psi\bbW_{(12)}$, where $(1 2)$ denotes the transposition of the first two parties. So $\caI_\Psi$ can only be a linear combination of $P_3$ and $P_2\otimes I$. Now \eref{eq:IPsi} is a simple corollary of \eref{eq:P3Psi0}. Next, straightforward calculation yields \begin{align} \caQ(\|\Psi\>\<\Psi\|) &= \bigl(15a^2 + 10b^2 +5c^2\bigr)I + 2\sqrt{2}\lsp b[3a + 2c \cos (\phi)] X + 4\sqrt{2}\lsp b c \sin(\phi) Y + \bigl(9a^2 + 2b^2 - c^2\bigr)Z, \end{align} which implies \eref{eq:QPsinorm2} given the definitions in Eqs.~\eqref{eq:etaabcphiDef}-\eqref{eq:fxy}. \end{proof} \begin{lemma}\label{lem:fxyUB} Suppose $0 \le x \le 1$ and $-x \le y \le x$. Then the function $f(x,y)$ defined in \eref{eq:fxy} satisfies \begin{equation}\label{eq:fxyUB} f(x,y) \le \frac{5}{2}\sqrt{\frac{11}{2}} + 8\sqrt{\frac{2}{11}}\,y; \end{equation} and the inequality is saturated iff $x=9/16$ and $y=0$. \end{lemma} \begin{proof} Due to continuity, it suffices to prove \eref{eq:fxyUB} when $0<x<1$ and $-x<y<x$. Direct calculation yields \begin{gather} \frac{5}{2}\sqrt{\frac{11}{2}} + 8\sqrt{\frac{2}{11}}\,y\geq \frac{5}{2}\sqrt{\frac{11}{2}} - 8\sqrt{\frac{2}{11}}>0,\quad \left(\frac{5}{2}\sqrt{\frac{11}{2}} + 8\sqrt{\frac{2}{11}}\,y\right)^2 - f^2(x,y) = r(x,z), \end{gather} where $z := y^2<x^2$ and \begin{equation} r(x,z): = \frac{243}{8} + 48x^2 - \frac{147z}{11} + 48(x-1)\sqrt{x^2 - z} - 60x. \end{equation} The partial derivative of $r(x,z)$ over $z$ reads \begin{equation} \frac{\partial r}{\partial z} = \frac{24(1-x)}{\sqrt{x^2 - z}} - \frac{147}{11}. \end{equation} If $0<x< 88/137$, then this derivative is always positive. Therefore, \begin{equation}\label{eq:fxyUBproof} r(x,z) \ge r(x,0) = \frac{3}{8}(9-16x)^2\geq 0, \end{equation} which implies \eref{eq:fxyUB}. If instead $88/137\leq x< 1$, then the partial derivative $\partial r/\partial z$ has a unique zero, denoted by $z_0$ henceforth. In addition, $z_0$ satisfies the equation \begin{equation} \sqrt{x^2 - z_0} = \frac{264}{147}(1-x), \end{equation} which means \begin{align} z_0=\frac{-5343x^2+15488x-7744}{2401}. \end{align} Therefore, \begin{equation} r(x,z) \ge r(x,z_0)=-\frac{3(12168x^2-37664x+18293)}{4312}\geq \frac{91875}{150152}>0, \end{equation} which implies \eref{eq:fxyUB}. Here the last inequality follows from the assumption $88/137\leq x< 1$. If the inequality in \eref{eq:fxyUB} is saturated, then $0<x< 88/137$ and the two inequalities in \eref{eq:fxyUBproof} are saturated simultaneously, which means $x=9/16$ and $y=z=0$, in which case the inequality in \eref{eq:fxyUB} is indeed saturated. \end{proof} \subsection{Proof of \thref{thm:bisep}} To determine the maximum estimation fidelity of biseparable POVMs on $\caH^{\otimes 3}$, it suffices to consider separable rank-1 POVMs on $\Sym_2(\caH)\otimes \caH$. More specifically, it suffices to solve the optimization problem in \eref{eq:BisepOpt} subject to the constraints in \eref{eq:POVMbs}. Note that $\|\Psi_j\> =\|\Phi_j\>\otimes \|\varphi_j\>$ can be expressed as follows, \begin{align} \|\Psi_j\>=U_j^{\otimes 3} \bigl[\bigl(a_j\|00\> + b_j\|\bbS\> + c_j\rme^{\rmi \phi_j}\|11\>\bigr)\otimes \|0\>\bigr], \end{align} where $U_j \in \rmU(\caH)$, $a_j, b_j,c_j\geq0$, $a_j^2+b_j^2+c_j^2=1$, and $0\leq\phi_j<2\pi$. By virtue of \lref{lem:q_abc} we can deduce that \begin{align} 4=\sum_j w_j\tr\left(P_3 \|\Psi_j\>\<\Psi_j\|\right)&=\sum_j \frac{w_j\bigl(3a_j^2+2b_j^2+c_j^2\bigr)}{3}=\sum_j\frac{w_j\bigl(2+a_j^2-c_j^2\bigr)}{3}=4+\sum_j \frac{w_j\bigl(a_j^2-c_j^2\bigr)}{3}, \end{align} which means $\sum_j w_j a_j^2=\sum_j w_j c_j^2$. By virtue of \eref{eq:EstimationFidDef} and \lref{lem:q_abc} we can further deduce that \begin{align} F(\scrM)&=\frac{1}{120}\\|\caQ(w_j\|\Psi_j\>\<\Psi_j\|)\\|= \frac{1}{120}\sum_j w_j \eta(a_j,b_j,c_j,\phi_j)\leq \frac{1}{120}\sum_j w_j \eta(a_j,b_j,c_j) \nonumber\\ &=\frac{1}{20}\sum_{j} p_j[10 + 5y_j + f(x_j,y_j)]=\frac{1}{2}+\frac{1}{20}\sum_{j} p_jf(x_j,y_j), \label{eq:bsProof1} \end{align} where \begin{align} p_j :=\frac{w_j}{6},\quad x_j := a_j^2+c_j^2,\quad y_j := a_j^2-c_j^2, \quad \sum_j p_j=1,\quad \sum_j p_jy_j=0. \end{align} Now, in conjunction with \lref{lem:fxyUB}, we can conclude that \begin{align} F(\scrM)&\leq \frac{1}{2}+\frac{1}{20}\sum_{j} p_jf(x_j,y_j) \le \frac{1}{2}+\frac{1}{20}\sum_j p_j \left(\frac{5}{2}\sqrt{\frac{11}{2}} + 8\sqrt{\frac{2}{11}}\, y_j\right) = \frac{1}{2} + \frac{\sqrt{22}}{16}. \label{eq:bsProof2} \end{align} If the upper bound for $F(\scrM)$ in \eref{eq:bsProof2} is saturated, then the two inequalities are saturated simultaneously. The saturation of the second inequality in \eref{eq:bsProof2} means $x_j=9/16$ and $y_j=0$, that is, $a_j=c_j=3\sqrt{2}/8$ and $b_j =\sqrt{7}/4$, for all $j$ by \lref{lem:fxyUB}. The saturation of the first inequality in \eref{eq:bsProof2}, which means the saturation of the inequality in \eref{eq:bsProof1}, further implies that $\phi_j=0$ for all $j$ according to the definitions in Eqs.~\eqref{eq:etaabcphiDef}-\eqref{eq:fxy} (cf. \lref{lem:q_abc}). Conversely, if $a_j=c_j=3\sqrt{2}/8$, $b_j =\sqrt{7}/4$, and $\phi_j=0$ for all $j$, then $F(\scrM)$ can attain the upper bound in \eref{eq:bsProof2}. Suppose $\{U_j\}_{j=1}^m$ is a unitary 3-design (say the Clifford group \cite{Zhu17,Webb16}), $w_j=6/m$, and $\|\Psi_j\>$ have the form \begin{equation} \|\Psi_j\> = U_j^{\otimes 3}\left[\left(\frac{3\sqrt{2}}{8}\|00\> + \frac{\sqrt{7}}{4}\|\bbS\> + \frac{3\sqrt{2}}{8}\|11\>\right)\otimes \|0\>\right],\quad j=1,2,\ldots,m; \end{equation} then $\scrM=\{w_j \|\Psi_j\>\<\Psi_j\|\}_{j=1}^m$ is an optimal separable POVM on $\Sym_2(\caH)\otimes \caH$ that saturates the upper bound in \eref{eq:bsProof2}. Accordingly, $\scrM\cup \{P_2^\rmA\otimes I\} $ is an optimal biseparable POVM on $\caH^{\otimes 3}$. This observation completes the proof of \thref{thm:bisep}. \section{Direct proof of \thref{thm:2+1}} \label{app:2+1} \subsection{Auxiliary results} \begin{lemma}\label{lem:Sym2CanonicalForm} Suppose $\|\Phi\>\in \Sym_2(\caH)$. Then there exists $U\in \rmU(\caH)$ and $\xi \in [0,\pi/2]$ such that \begin{equation}\label{eq:Sym2CanonicalForm} U^{\otimes 2}\|\Phi\> =\cos\frac{\xi}{2}\|00\> + \sin\frac{\xi}{2}\|11\>. \end{equation} \end{lemma} \begin{proof} By assumption $\|\Phi\>$ can be expressed as follows, \begin{equation} \|\Phi\> = W^{\dag\otimes 2}\bigl(a\|00\> + b\|\bbS\> + c \rme^{\rmi\chi}\|11\>\bigr),\quad a,b,c \ge 0,\ a^2 + b^2 + c^2=1, \ \chi \in [0,2\pi), \end{equation} where $W\in \rmU(\caH)$. Consider a unitary operator of the form \begin{equation} U_1(\theta,\phi) = \left(\begin{array}{cc} \cos\theta & \sin\theta \rme^{\rmi\phi} \\ -\sin\theta \rme^{-\rmi\phi} & \cos\theta \end{array}\right). \end{equation} Hence, we have \begin{equation} [U_1(\theta,\phi)W]^{\otimes 2}\|\Phi\> = u(\theta,\phi)\|00\> + v(\theta,\phi)\|\bbS\> + w(\theta,\phi)\|11\>,\\ \end{equation} where \begin{equation} \begin{aligned} u(\theta,\phi) &= a\cos^2\theta + \frac{1}{\sqrt{2}}b\rme^{\rmi\phi} \sin 2\theta + c\rme^{2\rmi(\phi + \chi)}\sin^2\theta,\\ v(\theta,\phi) &= b\cos(2\theta) + \frac{1}{\sqrt{2}}\left[c\rme^{\rmi(\phi + \chi)} - a\rme^{-\rmi\phi}\right]\sin(2\theta)\\ &=b \cos(2\theta) + \frac{1}{\sqrt{2}}\left[c \cos(\phi + \chi) - a \cos(\phi)\right] \sin(2\theta) + \frac{\rmi}{\sqrt{2}}\left[c \sin(\phi + \chi) + a \sin(\phi)\right] \sin(2\theta),\\ w(\theta,\phi) &= c\rme^{\rmi\chi}\cos^2\theta - \frac{1}{\sqrt{2}}b\rme^{-\rmi\phi}\sin(2\theta) + a\rme^{-\rmi2\phi}\sin^2\theta. \end{aligned} \end{equation} Let $\phi_0$ be a solution of the equation \begin{equation} c \sin(\phi_0 + \chi) + a \sin(\phi_0) = 0, \end{equation} and let $\theta_0$ be a solution of the equation \begin{equation} b \cos(2\theta_0) + \frac{1}{\sqrt{2}}\left[c \cos(\phi_0 + \chi) - a \cos\phi_0\right] \sin(2\theta_0) = 0. \end{equation} Then $v(\theta_0,\phi_0)=0$ and \begin{equation} [U_1(\theta_0,\phi_0)W]^{\otimes 2}\|\Phi\> = u(\theta_0,\phi_0) \|00\> + w(\theta_0,\phi_0) \|11\>. \end{equation} Now it is easy to find a diagonal unitary operator $U_2$ (with respect to the computational basis) such that $U_2^{\otimes 2}[u(\theta_0,\phi_0) \|00\> + w(\theta_0,\phi_0) \|11\>]=\cos(\xi/2)\|00\> + \sin(\xi/2)\|11\>$ with $\xi \in [0,\pi/2]$. Let $U=U_2 U_1(\theta_0,\phi_0) W$, then \eref{eq:Sym2CanonicalForm} holds, which completes the proof of \lref{lem:Sym2CanonicalForm}. \end{proof} \begin{lemma}\label{lem:proofofthm2+1} Suppose $\|\Phi\>\in \Sym_2(\caH)$, and $\scrM = \{M_j\}_j$ is a POVM on $\caH$. Then \begin{equation}\label{eq:PsiPOVM2+1} \sum_j \\|\caQ(\|\Phi\>\<\Phi\|\otimes M_j)\\| \le 20 + \frac{5\sqrt{22}}{2}, \end{equation} and the upper bound is saturated when \begin{equation}\label{eq:PsiPOVM2+1Saturate} \|\Phi\>= \cos\frac{\xi_0}{2}\|00\> + \sin\frac{\xi_0}{2}\|11\>,\quad \xi_0 := \arcsin\left(1/8\right),\quad \scrM = \{\|+\>\< +\|, \|-\>\< -\|\}. \end{equation} \end{lemma} \begin{proof} Thanks to \lref{lem:Sym2CanonicalForm}, we can assume that $\|\Phi\>$ has the form $\|\Phi\> = \cos(\xi/2)\|00\> + \sin(\xi/2)\|11\>$ with $\xi \in [0,\pi/2]$ without loss of generality. According to \rcite{BagaMM05}, we can further assume that $\scrM$ is a rank-1 projective measurement that has the form $\scrM = \{\|\varphi_+\>\<\varphi_+\|,\|\varphi_-\>\<\varphi_-\|\}$, where \begin{equation} \|\varphi_+\> = \cos\frac{\theta}{2}\|0\> + \sin\frac{\theta}{2} \rme^{\rmi \phi}\|1\>,\quad \|\varphi_-\> = \sin\frac{\theta}{2}\|0\> - \cos\frac{\theta}{2} \rme^{\rmi \phi}\|1\>,\quad \theta\in[0,\pi],\; \phi\in[0,2\pi). \end{equation} Then \begin{align} \sum_j \\|\caQ(\|\Phi\>\<\Phi\|\otimes M_j)\\| & = \\|\caQ(\|\Phi\>\<\Phi\|\otimes\|\varphi_+\>\<\varphi_+\|)\\| + \\|\caQ(\|\Phi\>\<\Phi\|\otimes\|\varphi_-\>\<\varphi_-\|)\\| \nonumber\\ &= 20 + \sqrt{\sin^2\theta(9 + \sin^2\xi + 6\sin\xi \cos 2\phi) + (5\cos\xi + 4\cos\theta)^2}\nonumber \\ &\equad + \sqrt{\sin^2\theta(9 + \sin^2\xi + 6\sin\xi \cos 2\phi) + (5\cos\xi - 4\cos\theta)^2}\nonumber \\ &\le 20+q(\xi,\theta), \label{eq:PsiPOVM2+1Proof} \end{align} where \begin{equation} q(\xi,\theta) := \sqrt{h_+}+\sqrt{h_-},\quad h_\pm := \sin^2\theta(3 + \sin\xi)^2 + (5\cos\xi \pm 4\cos\theta)^2, \end{equation} and the inequality above is saturated when $\phi = 0$. To prove \eref{eq:PsiPOVM2+1}, it suffices to prove the following inequality \begin{align}\label{eq:qxithetaUB} q(\xi,\theta)\leq\frac{5\sqrt{22}}{2}. \end{align} If $\theta=0$ or $\theta=\pi$, then \begin{align} q(\xi,\theta) &= \|5 \cos \xi + 4\| + \|5 \cos \xi - 4\| \le 10<\frac{5\sqrt{22}}{2}, \label{eq:qxithetaUB1} \end{align} which confirms \eref{eq:qxithetaUB}. Next, suppose $0<\theta<\pi$, then $\sin\theta>0$ and $h_\pm>0$. To determine the extremal points of $q(\xi,\theta)$, we can evaluate the partial derivative of $q(\xi,\theta)$ over $\theta$, with the result \begin{gather} \frac{\partial q(\xi,\theta)}{\partial \theta} = \sin\theta\left(-\frac{g_+}{\sqrt{h_+}} + \frac{g_-}{\sqrt{h_-}}\right) = \frac{\sin\theta \bigl(g_- \sqrt{h_+} - g_+ \sqrt{h_-}\lsp\bigr)}{\sqrt{h_+ h_-}},\\ g_\pm := 20\cos\xi \mp \cos\theta(\sin^2\xi + 6\sin\xi - 7) . \end{gather} In addition, \begin{equation} g^2_-h_+ - g^2_+ h_- = 480\cos\theta \left(\cos\frac{\xi}{2} - \sin\frac{\xi}{2}\right)^3\left(\cos\frac{\xi}{2} + \sin\frac{\xi}{2}\right)(3+\sin\xi)^2 (3+4\sin\xi). \end{equation} If $\partial q(\xi,\theta)/\partial \theta = 0$, then $g^2_-h_+ - g^2_+ h_-=0$, which means $\cos\theta = 0$ or $\cos(\xi/2)=\sin(\xi/2)$, that is, $\theta=\pi/2$ or $\xi = \pi/2$, given that $0<\theta<\pi$ and $0\leq \xi\leq \pi/2$ by assumption. In the latter case, we have \begin{equation} q(\xi,\theta)= q(\pi/2, \theta) = 8 \quad \forall \theta. \label{eq:qxithetaUB2} \end{equation} In the former case, we have \begin{align}\label{eq:qxithetaUB3} q(\xi,\theta) = q(\xi,\pi/2) &= 2\sqrt{(3+\sin\xi)^2 + 25\cos^2\xi} \le \frac{5\sqrt{22}}{2}, \end{align} where the inequality is saturated when $\xi =\xi_0= \arcsin(1/8)$. In conjunction with \eqsref{eq:qxithetaUB1}{eq:qxithetaUB2}, this observation completes the proof of \eref{eq:qxithetaUB}. Now, \eref{eq:PsiPOVM2+1} is a simple corollary of \eqsref{eq:PsiPOVM2+1Proof}{eq:qxithetaUB}. If $\|\Phi\>$ and $\scrM$ have the form in \eref{eq:PsiPOVM2+1Saturate}, then the inequalities in \eqsref{eq:PsiPOVM2+1Proof}{eq:qxithetaUB3} are saturated, so the upper bound in \eref{eq:PsiPOVM2+1} is saturated accordingly, which can also be verified by straightforward calculation. \end{proof} \subsection{Direct proof of \thref{thm:2+1}} Thanks to \lref{lem:POVMsym}, to determine the maximum estimation fidelity of 2+1 adaptive POVMs on $\caH^{\otimes 3}$, it suffices to consider 2+1 adaptive rank-1 POVMs on $\Sym_2(\caH)\otimes \caH$. A general 2+1 adaptive rank-1 POVM $\scrM$ on $\Sym_2(\caH)\otimes \caH$ can be expressed as follows, \begin{align} \scrM=\bigcup_j w_j\|\Phi_j\>\<\Phi_j\|\otimes \scrM_j, \end{align} where $\{w_j\|\Phi_j\>\<\Phi_j\|\}_j$ forms a POVM on $\Sym_2(\caH)$, which means $\|\Phi_j\>\in \Sym_2(\caH)$, $w_j\geq 0$, and $\sum_j w_j=3$; in addition, each $\scrM_j$ is a POVM on $\caH$. By virtue of \eref{eq:EstimationFidDef} and \lref{lem:proofofthm2+1} we can deduce that \begin{align} F(\scrM) &= \frac{1}{120}\sum_{j}\sum_{M\in \scrM_j} \\|\caQ(w_j \|\Phi_j\>\<\Phi_j\|\otimes M)\\| \le \frac{1}{120}\left(20 + \frac{5}{2}\sqrt{22}\right)\sum_{j}w_j = \frac{1}{2} + \frac{\sqrt{22}}{16}. \end{align} An optimal POVM that saturates the upper bound is presented in \eref{eq:OptPOVMbs2+1} in the main text, which completes the proof of \thref{thm:2+1}. \section{Proof of \thref{thm:1+2}} \label{app:1+2} \subsection{Auxiliary results} As a complement to \lref{lem:fxyUB}, here we first derive another tight linear upper bound for the function $f(x,y)$ defined in \eref{eq:fxy}. Let \begin{equation}\label{eq:p0gamma} \begin{gathered} p := \frac{47 - 3\sqrt{41}}{216},\quad x_0 := \frac{\frac{2}{3} - p}{1 - p}=\frac{2003 + 27 \sqrt{41}}{3524},\quad y_0 := \frac{-p}{1 - p}=\frac{-1039 + 81 \sqrt{41}}{3524},\\ \alpha := f_x(x_0,y_0) = \frac{5}{2} - \frac{43}{2\sqrt{41}},\;\; \beta := f_y(x_0,y_0) = \frac{9}{2} - \frac{13}{2\sqrt{41}},\;\; \gamma := f(x_0, y_0) - \alpha x_0 - \beta y_0 = 2 + \frac{28}{\sqrt{41}}. \end{gathered} \end{equation} Here $p$ is reproduced from \eref{eq:pSPhiW}, $f_x=\partial f/\partial x$, and $f_y=\partial f/\partial y$. Note that $\alpha$ and $y_0$ are negative, while the other four numbers are positive. \begin{lemma}\label{lem:fxyABC} Suppose $0 \le x \le 1$ and $-x \le y \le x$. Then the function $f(x,y)$ defined in \eref{eq:fxy} satisfies \begin{equation}\label{eq:fxyABC} f(x,y) \le \alpha x + \beta y + \gamma, \end{equation} where $\alpha,\beta,\gamma$ are defined in \eref{eq:p0gamma}, and the inequality is saturated iff $x = y = 1$ or $x = x_0, y = y_0$. \end{lemma} \begin{proof} Note that $\alpha x + \beta y + \gamma\geq \gamma+\alpha-\beta=13/\sqrt{41}>0$. Define the difference function \begin{align} \Delta(x,y) &:= (\alpha x + \beta y + \gamma)^2 - f(x,y)^2\nonumber\\ &\;= -4 - 108 x + 96 x^2 - 40 y - 25 y^2 + 48 (-1 + x)\bigl(-x + \sqrt{x^2 - y^2}\lsp\bigr)\nonumber\\ &\;\equad + \left[ \left(\frac{5}{2} - \frac{43}{2\sqrt{41}}\right)x + \left(\frac{9}{2} - \frac{13}{2\sqrt{41}}\right)y + 2 + \frac{28}{\sqrt{41}}\right]^2. \end{align} Then to prove \eref{eq:fxyABC}, it suffices to prove the inequality $\Delta(x,y) \ge 0$ for $0 \le x \le 1$ and $-x \le y \le x$. When $x=0$, which means $y=0$, it is straightforward to verify that $\Delta(x,y)>0$ and $f(x,y)<\alpha x + \beta y + \gamma$. By assumption $y$ can be expressed as $y = x\cos \zeta$ with $\zeta\in [0,\pi]$, and $\zeta$ is uniquely determined by $x$ and $y$ when $x\neq 0$. Accordingly, $\Delta(x,y)$ can be expressed as \begin{gather} \Delta(x,y) =\Delta(x,x\cos \zeta)= g_2(\zeta) x^2-2g_1(\zeta)x + \gamma^2 - 4, \end{gather} where \begin{equation}\label{eq:g12} g_2(\zeta): =48 + (\alpha + \beta\cos \zeta)^2 - 25\cos^2 \zeta + 48\sin \zeta ,\quad g_1(\zeta):=30 - \alpha\gamma - (\beta\gamma - 20)\cos \zeta + 24\sin \zeta. \end{equation} Note that \begin{equation}\label{eq:g12LB} g_2(\zeta) \ge 23,\quad g_1(\zeta)\geq g_1(0)=50-\alpha\gamma - \beta\gamma>33, \end{equation} given that $\beta\gamma-20>0$. Let $x^(\zeta) : =g_1(\zeta)/g_2(\zeta)$; then $\Delta(x,x\cos \zeta)\geq \Delta(x^(\zeta),x^(\zeta)\cos \zeta)$, and the inequality is saturated iff $x= x^(\zeta)\leq 1$. Let \begin{equation}\label{eq:ellzeta} \ell(\zeta) :=g_2(\zeta)-g_1(\zeta)= 18 + \alpha^2 + \alpha \gamma - (20 - \beta\gamma - 2\alpha\beta)\cos \zeta -\bigl(25 - \beta^2\bigr)\cos^2 \zeta + 24\sin \zeta; \end{equation} then $x^(\zeta)\leq 1$ iff $\ell(\zeta) \geq 0$. If $\pi/2\leq \zeta\leq \pi$, then \begin{align} \ell(\zeta)\geq 18 + \alpha^2 + \alpha \gamma-\bigl(25 - \beta^2\bigr)=\frac{907-139\sqrt{41}}{41}>0,\quad x^(\zeta)<1, \end{align} given that $(20 - \beta\gamma - 2\alpha\beta)>0$ and $25 - \beta^2>0$. If instead $0\leq \zeta\leq \pi/2$, then $\ell(\zeta)$ is monotonically increasing in $\zeta$. Meanwhile, $\ell(0)<0$ and $\ell(\pi/2)>0$. Therefore, $\ell(\zeta)$ has a unique zero for $\zeta\in [0,\pi]$. Let $\zeta_$ be this unique zero; numerically, we have $\zeta_ \approx 0.12988$ and $\cos \zeta_* \approx 0.99158$. Then $x^(\zeta_)=1$, $x^(\zeta)>1$ for $\zeta\in [0,\zeta_)$, and $x^(\zeta)<1$ for $\zeta\in (\zeta_, \pi]$. Define \begin{equation} \Delta^(\zeta) := \begin{cases} \Delta(1,\cos \zeta) & \zeta \in [0,\zeta_),\\ \Delta(x^(\zeta),x^(\zeta)\cos \zeta) & \zeta \in [\zeta_,\pi]; \end{cases} \end{equation} then $\Delta(x,x\cos \zeta) \ge \Delta^(\zeta)$, and the inequality is saturated iff \begin{align}\label{eq:xzetaMin} x=\begin{cases} 1 & \zeta \in [0,\zeta_),\\ x^(\zeta) & \zeta \in [\zeta_,\pi]. \end{cases} \end{align} To prove \eref{eq:fxyABC}, it suffices to prove the inequality $\Delta^(\zeta) \ge 0$ for $\zeta\in [0,\pi]$. If $\zeta \in [0,\zeta_)$, then \begin{align} \Delta^(\zeta)=\Delta(1,\cos \zeta)=\frac{1}{41}\bigl[433 + 117 \sqrt{41} + \bigl(305 + 117 \sqrt{41}\lsp\bigr) \cos\zeta\bigr]\sin^2\frac{\zeta}{2}\geq 0. \label{eq:DeltazetaLB1} \end{align} If instead $\zeta \in [\zeta_,\pi]$, then \begin{align} \Delta^(\zeta)=\Delta(x^(\zeta),x^(\zeta)\cos \zeta)=\frac{(\gamma^2-4)g_2(\zeta) -g_1(\zeta)^2}{g_2(\zeta)}=\frac{h(\zeta)}{41g_2(\zeta)}, \end{align} where \begin{align} h(\zeta) &:= c_0 \cos (2\zeta) + c_1 \sin (2\zeta) + c_2 \cos \zeta + c_3 \sin \zeta + c_4\nonumber\\ &\;=2c_0\cos^2\zeta+ c_2 \cos \zeta+c_4-c_0+(2c_1\cos\zeta+c_3)\sin\zeta, \label{eq:hzeta} \end{align} with \begin{equation} \begin{gathered} c_0 = -4197 + 977 \sqrt{41},\quad c_1 = 24 \bigl(-633 + 113 \sqrt{41}\lsp\bigr),\quad c_2 = 4\bigl(-14667 + 2191 \sqrt{41}\lsp\bigr),\\ c_3 = 48 \bigl(-843 + 139 \sqrt{41}\lsp\bigr),\quad c_4 = -53775 + 8403 \sqrt{41}. \end{gathered} \end{equation} Note that $c_0,c_1, c_3,c_4>0$ and $c_2<0$; in addition, $g_2(\zeta)>0$ by \eref{eq:g12LB}. To prove the inequality $\Delta^(\zeta)\geq 0$ for $\zeta \in [\zeta_,\pi]$, it suffices to prove the inequality $h(\zeta)\geq 0$. Let $u=\cos\zeta$ and define \begin{equation} h_2(u) := \bigl(2c_0 u^2 + c_2 u + c_4 - c_0\bigr)^2 - \left(2c_1 u + c_3\right)^2\bigl(1-u^2\bigr). \end{equation} Then $h_2(u)=0$ whenever $h(\zeta)=0$ according to \eref{eq:hzeta}. Calculation shows that $h_2(u)$ has the following three distinct zeros: \begin{equation}\label{eq:u0pm} u_0 := \frac{-308 + 27\sqrt{41}}{565},\quad u_\pm:=\frac{37529139\sqrt{41}-239145719\pm 576\sqrt{-35743460158+5587351798\sqrt{41}}}{294550033-45301173\sqrt{41}}, \end{equation} where $u_-<u_0<u_+$ and the zero $u_0$ has multiplicity 2. By contrast, $h(\zeta)$ has two distinct zeros, namely, $\zeta_0:=\arccos u_0\approx 1.81228$ and $\zeta_+:=\arccos u_+\approx 0.07235$; note that $\arccos u_-$ is not a zero of $h(\zeta)$. In addition, $0<\zeta_+<\zeta_$ and $\zeta_<\zeta_0<\pi$, so $\zeta_0$ is the unique zero of $h(\zeta)$ within the interval $[\zeta_,\pi]$. Straightforward calculation shows that $h(\zeta_), h(\pi)>0$ as illustrated in \fref{fig:hzeta}, which implies that $h(\zeta),\Delta^(\zeta) \geq 0$ for $\zeta\in [\zeta_,\pi]$ by continuity. In conjunction with \eref{eq:DeltazetaLB1} we can deduce that $\Delta(x,x\cos \zeta)\geq \Delta^(\zeta)\geq 0$ for $\zeta\in [0,\pi]$, which implies \eref{eq:fxyABC}; in addition, $\Delta^(\zeta)$ has only two zeros in this interval, namely, 0 and $\zeta_0$. If $x = y = 1$ or if $x = x_0$ and $ y = y_0$, then the inequality in \eref{eq:fxyABC} is saturated by straightforward calculation. Conversely, if the inequality in \eref{eq:fxyABC} is saturated, then $\Delta^(\zeta)=\Delta(x,x\cos \zeta)=\Delta(x,y)=0$, which means $\zeta=0$ or $\zeta=\zeta_0$. According to \eref{eq:xzetaMin}, if $\zeta=0$, then $y=x=1$; if instead $\zeta=\zeta_0$, then \begin{align} x=x^(\zeta_0)=x_0, \quad y=x^(\zeta_0)\cos\zeta_0=y_0. \end{align} This observation completes the proof of \lref{lem:fxyABC}. \end{proof} \begin{figure} \centering \includegraphics[width = 9cm]{hz.pdf} \caption{\label{fig:hzeta}A plot of the function $h(\zeta)$ defined in \eref{eq:hzeta} for $\zeta\in [0,\pi]$. Here $\zeta_+=\arccos u_+$ and $\zeta_0=\arccos u_0$ are the two zeros of $h(\zeta)$ [see \eref{eq:u0pm}], while $\zeta_$ is the unique zero of the function $\ell(\zeta)$ defined in \eref{eq:ellzeta}.} \end{figure} \begin{lemma}\label{lem:canonical1+2} Suppose $\scrM = \{w_j \|\Psi_j\>\<\Psi_j\|\}_j$ is a 1+2 adaptive POVM on $\Sym_2(\caH)\otimes\caH$, where $\|\Psi_j\>$ have the form \begin{equation} \|\Psi_j\> = U_j^{\otimes 3}\bigl[\bigl(a_j\|00\> + b_j\|\bbS\> + c_j \rme^{\rmi \phi_j}\|11\>\bigr)\otimes\|0\>\bigr], \end{equation} with $U_j\in \rmU(\caH), a_j, b_j, c_j \ge 0, a_j^2 + b_j^2 + c_j^2 = 1 $, and $\phi_j\in[0,2\pi)$. Then \begin{equation} \sum_j w_j a_j^2 = \sum_j w_j b_j^2 = \sum_j w_j c_j^2 = 2,\quad \sum_j w_j a_j b_j = \sum_j a_j c_j \rme^{\rmi\phi_j} = \sum_j b_j c_j \rme^{\rmi\phi_j} = 0. \label{eq:constraint2} \end{equation} \end{lemma} \begin{proof} Let $\|\Phi_j\>=a_j\|00\> + b_j\|\bbS\> + c_j \rme^{\rmi \phi_j}\|11\>$, then $\{(w_j/2) \|\Phi_j\>\<\Phi_j\|\}_j$ forms a POVM on $\Sym_2(\caH)$. Therefore, \begin{align} \sum_j w_j\|\Phi_j\>\<\Phi_j\|=2P_2 =2\|00\>\<00\|+2\|\bbS\>\<\bbS\|+2\|11\>\<11\|, \end{align} which implies \eref{eq:constraint2} and completes the proof of \lref{lem:canonical1+2}. \end{proof} \subsection{Proof of \thref{thm:1+2}} Thanks to \lref{lem:POVMsym}, to determine the maximum estimation fidelity of 1+2 adaptive POVMs on $\caH^{\otimes 3}$, it suffices to consider 1+2 adaptive rank-1 POVMs on $\Sym_2(\caH)\otimes \caH$. Suppose $\scrM = \bigl\{M_j\}_j$ is an arbitrary 1+2 rank-1 POVM on $\Sym_2(\caH)\otimes\caH$. Then $M_j$ can be expressed as follows, $M_j=w_j U_j^{\otimes 3}\|\Psi_j\>\<\Psi_j\|U_j^{\dag\otimes 3}$, where $w_j> 0$, $\sum_j w_j=6$, $U_j\in \rmU(\caH)$, and $\|\Psi_j\>$ have the form \begin{equation} \|\Psi_j\> = \bigl(a_j\|00\> + b_j\|\bbS\> + c_j \rme^{\rmi \phi_j}\|11\>\bigr)\otimes\|0\> \end{equation} with $a_j,b_j,c_j\ge 0, a_j^2 + b_j^2 + c_j^2 = 1$, and $\phi_j\in[0,2\pi)$. By virtue of \eref{eq:EstimationFidDef} and \lsref{lem:q_abc}, \ref{lem:canonical1+2} we can deduce that \begin{align} F(\scrM)&=\frac{1}{120}\sum_j w_j \caQ(\|\Psi_j\>\<\Psi_j\|)=\frac{1}{120}\sum_j w_j \eta(a_j,b_j,c_j,\phi_j) \le \frac{1}{120}\sum_j w_j\bigl[15 a_j^2+10b_j^2+5c_j^2 +f(x_j,y_j)\bigr]\nonumber\\ &= \frac{1}{2}+\frac{1}{20}\sum_{j} p_jf(x_j,y_j), \label{eq:1+2sat} \end{align} where the relevant parameters satisfy the following constraints (see \lref{lem:canonical1+2}): \begin{equation} p_j := \frac{w_j}{6},\quad x_j := a_j^2 + c_j^2,\quad y_j := a_j^2 - c_j^2,\quad \sum_{j} p_j = 1, \quad \sum_{j} p_j y_j = 0,\quad \sum_j p_j x_j = \frac{2}{3}. \label{eq:opt3} \end{equation} Note that the inequality in \eref{eq:1+2sat} is saturated when $\phi_j=0$ for all $j$. Next, in conjunction with \eref{eq:p0gamma} and \lref{lem:fxyABC} we can deduce that \begin{align}\label{eq:proofthe4} F(\scrM) \le \frac{1}{2}+\frac{1}{20}\sum_j p_j \left(\alpha x_j + \beta y_j + \gamma\right) = \frac{1}{2} + \frac{1}{30}\alpha + \frac{1}{20}\gamma = \frac{1}{2} + \frac{11 + \sqrt{41}}{60}. \end{align} The saturation of the above inequality means either $(x_j,y_j)=(x_0,y_0)$ or $(x_j,y_j)=(1,1)$ for each $j$, where $x_0,y_0$ are defined in \eref{eq:p0gamma}. In conjunction with \eref{eq:opt3} we can deduce that \begin{align} \sum_{j\,\|\,(x_j,y_j)=(1,1)} p_j=p, \end{align} where $p$ is defined in \eref{eq:pSPhiW} and is reproduced in \eref{eq:p0gamma}. Moreover, the upper bound in \eref{eq:proofthe4} is saturated when $\scrM$ has the form \begin{equation} \scrM=\{\scrK_0'\otimes \|0\>\<0\|\}\cup \{\scrK_1'\otimes \|1\>\<1\|\},\quad \scrK_0' := \{K_j\}_{j=0}^4,\quad \scrK_1':=\bigl\{X^{\otimes 2}K_jX^{\otimes 2}\bigr\}_{j=0}^4, \end{equation} where $K_j$ for $j=0,1,2,3,4$ are defined in \eref{eq:scrE01}. Note that $\scrK_0'$ and $\scrK_1'$ are POVMs on $\Sym_2(\caH)$. Accordingly, we can construct an optimal 1+2 adaptive POVM on $\caH^{\otimes 3}$ as shown in \eref{eq:OptPOVM1+2} in the main text. This observation completes the proof of \thref{thm:1+2}. \bibliography{all_references} \end{document}
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\definecolor{transitive}{rgb}{0.0, 0.8, 0.6} \definecolor{non-transitive}{rgb}{0.9, 0.17, 0.31} \tikzstyle{start} = [rectangle, rounded corners, minimum width = 2cm, minimum height = 1cm, draw = blue, fill = white, text centered, very thick] \tikzstyle{stop} = [rectangle, rounded corners, minimum width = 2cm, minimum height = 1cm, draw = blue, fill = white, text width = 2cm, text centered, very thick] \tikzstyle{stop-t} = [rectangle, rounded corners, minimum width = 2cm, minimum height = 1cm, draw = transitive, fill = white, text width = 2cm, text centered, very thick] \tikzstyle{stop-nt} = [rectangle, rounded corners, minimum width = 2cm, minimum height = 1cm, draw = non-transitive, fill = white, text width = 2cm, text centered, very thick] \tikzstyle{stage} = [rectangle, minimum width = 1.5cm, minimum height = 1cm, draw = blue, fill = white, text centered, very thick] \tikzstyle{stage-NP} = [rectangle, minimum width = 2.1cm, minimum height = 1cm, draw = white, fill = white, text centered, very thick] \tikzstyle{tag} = [rectangle, draw = white, fill = white, text centered] \tikzstyle{decision} = [diamond, minimum width = 1.6cm, minimum height = 1cm, draw = black, fill = white, text width = 1.9cm, text centered, thick] \tikzstyle{input} = [trapezium, trapezium left angle = 70, trapezium right angle = 110, draw = cyan, fill = white, text width = 2cm, text centered] \tikzstyle{arrow} = [thick, ->, >=stealth, gray] \title[Galois groups of low dimensional Abelian varieties over finite fields]{Galois groups of low dimensional Abelian\\ varieties over finite fields} \date{December 4, 2024} \author{Santiago Arango-Piñeros} \address{Department of Mathematics, Emory University, Atlanta, GA 30322, USA} \email{[email protected]} \urladdr{\url{https://sarangop1728.github.io/}} \author{Sam Frengley} \address{School of Mathematics, University of Bristol, Bristol, BS8 1UG, UK} \email{[email protected]} \urladdr{\url{https://samfrengley.github.io/}} \author{Sameera Vemulapalli} \address{Department of Mathematics, Harvard University} \email{[email protected]} \urladdr{\url{https://web.math.princeton.edu/~sameerav/}} \allowdisplaybreaks \begin{document} \begin{abstract} We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the \emph{weighted permutation representation} which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds. \end{abstract} \maketitle \vspace{-5mm} \setcounter{tocdepth}{1} \tableofcontents \vspace{-5mm} \section{Introduction} \label{sec:intro} The purpose of this article is to study the surprisingly subtle interactions between three isogeny invariants of abelian varieties over finite fields. We analyze which triples of invariants may occur and what restrictions they impose on the abelian variety in question. The interaction of these invariants is discussed in a letter from Serre to Ribet \cite[pp.~6]{Serre89} and has gained renewed interest following the publication of the database of abelian varieties over finite fields in the LMFDB \cite{DupuyKedlayaRoeVincent22}. The availability of this data has demonstrated that the interaction between these invariants is more intricate than initially thought \cite{DupuyKedlayaRoeVincent21}, prompting the development of more refined invariants to better understand these relationships \cite{DupuyKedlayaZureick-Brown22}. This article makes progress towards that goal. Even though this subject is interesting in its own right, it has applications to the Tate conjecture \cite{Zarhin15, Zarhin22}, monodromy groups of abelian varieties over number fields \cite{Zywina22}, Frobenius distributions \cite{AhmadiShparlinsky10, APBS2023}, and prime number races and Chebyshev biases in the context of function fields \cite{keliher2024}. By the Honda--Tate theorem \cite{Tate1966, Honda1968, Tate1971}, the isogeny class of an abelian variety $A$ is determined by its Frobenius polynomial, which is the characteristic polynomial of the Frobenius endomorphism acting on the $\ell$-adic Tate module of $A$ (where $\ell$ is a prime number which is not equal to the characteristic of the base field $\Fq$). All of our invariants are derived from the Frobenius polynomial; the first invariant is the \cdef{Newton polygon} of the Frobenius polynomial, which determines the $p$-adic valuations of the roots, the second invariant is the \cdef{angle rank} (see \Cref{defn:angle-rank}), which measures the nontrivial multiplicative relations between the roots of the Frobenius polynomial, and finally, we have the \cdef{Galois group} of the Frobenius polynomial as our third invariant. We classify triples of invariants that occur for abelian varieties of dimension $\leq 3$; the dimension $3$ case already exhibits some subtleties that should be expected in general. In \cite{DupuyKedlayaZureick-Brown22}, the authors noticed that the Galois group, Newton polygon, and angle rank are not independent. The Galois group acts on the $p$-adic valuations of the roots (we visualize each root as a ball of radius proportional to its $p$-adic valuation) and this \cdef{weighted permutation representation} (see \Cref{def:perm-rep}) determines the Galois group, Newton polygon, and angle rank. However, this does \emph{not} imply that the Galois group and Newton polygon determine the angle rank. For example, the isogeny classes of abelian threefolds over $\FF_2$ with LMFDB \cite{lmfdb} labels \avlink{3.2.ac_a_d} and \avlink{3.2.a_a_ad} have the same Newton polygon and Galois group, but different angle ranks. This example illustrates the need to consider more than just the isomorphism class of the Galois group. \subsection{Statement of main results} The authors of \cite{DupuyKedlayaZureick-Brown22} defined the \emph{Newton hyperplane representation} of a geometrically simple abelian variety to encode this information. The central contribution of this paper is to reinterpret the Newton hyperplane representation in terms of a \emph{weighted permutation representation} and prove additional constraints upon it (see \Cref{sec:divisor-map}). We leverage these results to classify weighted permutation representations for elliptic curves (\Cref{lemma:main-thm-ecs}), abelian surfaces (\Cref{thm:main-thm-surfaces}) and simple abelian threefolds (\Cref{thm:main-thm-3folds}). The flowcharts in Figures~\ref{fig:flowchart2-simple}--\ref{fig:flowchart3} distill information from the tables in our main theorems. The purpose of these flowcharts is to serve as a ``user's guide'' to the tables in \Cref{thm:main-thm-surfaces} and \Cref{thm:main-thm-3folds}; in particular, if one has in hand an abelian surface or threefold, then the flowchart rules out certain Galois groups. \begin{theorem} \label{thm:main-surfaces} If $A$ is an abelian surface, then the possible isomorphism classes of the Galois group $G_A$ are determined by \Cref{fig:flowchart2-simple} in the simple case, and by \Cref{fig:flowchart2-non-simple} in the nonsimple case. Moreover, every possibility occurs. \end{theorem} \begin{theorem} \label{thm:main-simple-threefolds} If $A$ is a simple abelian threefold, then the possible isomorphism classes of the Galois group $G_A$ are determined by \Cref{fig:flowchart3}. Moreover, every possibility occurs. \end{theorem} \begin{figure}[p] \centering \resizebox{0.8\textwidth}{!}{ \begin{tikzpicture}[node distance = 2.7cm] \node (start) [start] {Simple abelian surface over $\mathbf{F}_q$}; \node (B) [stage-NP, below of = start, yshift=-1cm] {\includegraphics[scale=0.45]{images/ao_2.png}}; \node (B-tag) [tag, above of=B, yshift=-1cm, xshift=-1cm] {(B)}; \node (A) [stage-NP, left of = B, xshift = -3cm] {\includegraphics[scale=0.45]{images/o_2.png}}; \node (A-tag) [tag, above of=A, yshift=-1cm] {(A)}; \node (C) [stage-NP, right of = B, xshift = 3cm] {\includegraphics[scale=0.45]{images/ss_2.png}}; \node (C-tag) [tag, above of=C, yshift=-1cm] {(C)}; \node (C-delta0) [stage, below of=C,yshift=-1cm] {$\delta_A = 0$}; \node (A-delta2) [stage, below of=A,xshift=-1.5cm,yshift=-1cm] {$\delta_A = 2$}; \node (A-delta1) [stage, below of=A,xshift=1.5cm,yshift=-1cm] {$\delta_A = 1$}; \node (B-delta2) [stage, right of=A-delta1,xshift=1.6cm] {$\delta_A = 2$}; \node (W4) [stop, below of=B, yshift=-4cm, xshift = -6cm] {$C_2\wr S_2$}; \node (C4) [stop, below of=B, yshift=-4cm, xshift=-2cm] {$C_4$}; \node (V4) [stop, below of=B, yshift=-4cm, xshift = 2cm] {$V_4$}; \node (C2) [stop, below of=B, yshift=-4cm, xshift = 6cm] {$C_2$}; \draw[arrow] (start) -- (A); \draw[arrow] (start) -- (B); \draw[arrow] (start) -- (C); \draw[arrow] (B) -- (B-delta2); \draw[arrow] (C) -- (C-delta0); \draw[arrow] (A) -- (A-delta2); \draw[arrow] (A) -- (A-delta1); \draw[arrow] (B-delta2) -- (W4); \draw[arrow] (A-delta2) -- (W4); \draw[arrow] (A-delta2) -- (C4); \draw[arrow] (A-delta1) -- (V4); \draw[arrow] (C-delta0) -- (V4); \draw[arrow] (C-delta0) -- (C2); \end{tikzpicture} } \caption{Possible isomorphism classes of Galois groups of simple abelian surfaces in terms of their Newton polygon, and angle rank $\delta_A$.} \label{fig:flowchart2-simple} \end{figure} \begin{figure}[p] \centering \resizebox{0.8\textwidth}{!}{ \begin{tikzpicture}[node distance = 2.7cm] \node (start) [start] {Nonsimple abelian surface $E_1\times E_2$ over $\mathbf{F}_q$}; \node (B) [stage-NP, below of = start, yshift=-1cm] {\includegraphics[scale=0.45]{images/ao_2.png}}; \node (B-tag) [tag, above of=B, yshift=-1cm, xshift=-1cm] {(B)}; \node (A) [stage-NP, left of = B, xshift = -3cm] {\includegraphics[scale=0.45]{images/o_2.png}}; \node (A-tag) [tag, above of=A, yshift=-1cm] {(A)}; \node (C) [stage-NP, right of = B, xshift = 3cm] {\includegraphics[scale=0.45]{images/ss_2.png}}; \node (C-tag) [tag, above of=C, yshift=-1cm] {(C)}; \node (B-delta1) [stage, below of=B,yshift=-1cm] {$\delta_A = 1$}; \node (C-delta0) [stage, below of=C,yshift=-1cm] {$\delta_A = 0$}; \node (A-delta2) [stage, below of=A,xshift=-1.5cm,yshift=-1cm] {$\delta_A = 2$}; \node (A-delta1) [stage, below of=A,xshift=1.5cm,yshift=-1cm] {$\delta_A = 1$}; \node (V4) [stop, below of=B, yshift=-4cm, xshift = -4cm] {$V_4$}; \node (C2) [stop, below of=B, yshift=-4cm] {$C_2$}; \node (C1) [stop, below of=B, yshift=-4cm, xshift = 4cm] {$C_1$}; \draw[arrow] (start) -- (A); \draw[arrow] (start) -- (B); \draw[arrow] (start) -- (C); \draw[arrow] (B) -- (B-delta1); \draw[arrow] (C) -- (C-delta0); \draw[arrow] (A) -- (A-delta2); \draw[arrow] (A) -- (A-delta1); \draw[arrow] (A-delta2) -- (V4); \draw[arrow] (A-delta1) -- (C2); \draw[arrow] (B-delta1) -- (V4); \draw[arrow] (B-delta1) -- (C2); \draw[arrow] (C-delta0) -- (V4); \draw[arrow] (C-delta0) -- (C2); \draw[arrow] (C-delta0) -- (C1); \end{tikzpicture} } \caption{Possible isomorphism classes of Galois groups of simple abelian surfaces in terms of their Newton polygon, and angle rank $\delta_A$.} \label{fig:flowchart2-non-simple} \end{figure} \begin{figure} \centering \resizebox{\textwidth}{!}{ \begin{tikzpicture}[node distance = 2.7cm] \node (start) [start] {Simple abelian threefold over $\mathbf{F}_q$}; \node (C) [stage-NP, below of = start, yshift=-1cm] {\includegraphics[scale=0.36]{images/k3type_3.png}}; \node (C-tag) [tag, above of=C, yshift=-1.3cm] {(C)}; \node (B) [stage-NP, left of = C, xshift = -1.5cm] {\includegraphics[scale=0.36]{images/ao_3.png}}; \node (B-tag) [tag, above of=B, yshift=-1.3cm] {(B)}; \node (A) [stage-NP, left of = B, xshift = -1.5cm] {\includegraphics[scale=0.36]{images/o_3.png}}; \node (A-tag) [tag, above of=A, yshift=-1.3cm] {(A)}; \node (D) [stage-NP, right of = C, xshift = 1.5cm] {\includegraphics[scale=0.36]{images/prk0_3.png}}; \node (D-tag) [tag, above of=D, yshift=-1.3cm] {(D)}; \node (E) [stage-NP, right of = D, xshift = 1.5cm] {\includegraphics[scale=0.36]{images/ss_3.png}}; \node (E-tag) [tag, above of=E, yshift=-1.3cm] {(E)}; \node (A-delta3) [stage, below of = A, xshift = -1cm,yshift=-1cm]{$\delta_A = 3$}; \node (A-delta1) [stage, below of = A, xshift=1cm,yshift=-1cm]{$\delta_A = 1$}; \node (B-delta3) [stage, below of = B, xshift = -1cm,yshift=-1cm]{$\delta_A = 3$}; \node (B-delta2) [stage, below of = B, xshift = 1cm,yshift=-1cm]{$\delta_A = 2$}; \node (C-delta3) [stage, right of = B-delta2, xshift=0.5cm]{$\delta_A = 3$}; \node (D-delta3) [stage, below of = D, xshift = -1cm,yshift=-1cm]{$\delta_A = 3$}; \node (D-delta1) [stage, below of = D, xshift = 1cm,yshift=-1cm]{$\delta_A = 1$}; \node (E-delta0) [stage, right of = D-delta1]{$\delta_A = 0$}; \node (W6) [stop, below of = C, yshift = -6cm, xshift = -6cm]{$C_2\wr S_3$}; \node (C2wrC3) [stop, below of = C, yshift = -6cm, xshift = -2cm]{$C_2\wr C_3$}; \node (D6) [stop, below of = C, yshift = -6cm, xshift = 2cm]{$D_6$}; \node (C6) [stop, below of = C, yshift = -6cm, xshift = 6cm]{$C_6$}; \draw[arrow] (start) -- (A-tag); \draw[arrow] (start) -- (B-tag); \draw[arrow] (start) -- (C-tag); \draw[arrow] (start) -- (D-tag); \draw[arrow] (start) -- (E-tag); \draw[arrow] (E) -- (E-delta0); \draw[arrow] (A) -- (A-delta3); \draw[arrow] (A) -- (A-delta1); \draw[arrow] (B) -- (B-delta3); \draw[arrow] (B) -- (B-delta2); \draw[arrow] (C) -- (C-delta3); \draw[arrow] (D) -- (D-delta3); \draw[arrow] (D) -- (D-delta1); \draw[arrow] (A-delta3) .. controls + (down:2cm) .. (W6); \draw[arrow] (A-delta3) .. controls + (down:2cm) .. (C2wrC3); \draw[arrow] (A-delta3) .. controls + (down:2cm) .. (D6); \draw[arrow] (A-delta3) .. controls + (down:2cm) .. (C6); \draw[arrow] (A-delta1) .. controls + (down:1cm) .. (D6); \draw[arrow] (A-delta1) .. controls + (down:1cm) .. (C6); \draw[arrow] (B-delta3) -- (W6); \draw[arrow] (B-delta3) -- (C2wrC3); \draw[arrow] (B-delta2) -- (C6); \draw[arrow] (C-delta3) -- (W6); \draw[arrow] (C-delta3) -- (C2wrC3); \draw[arrow] (C-delta3) -- (D6); \draw[arrow] (D-delta3) -- (W6); \draw[arrow] (D-delta3) -- (C2wrC3); \draw[arrow] (D-delta1) -- (D6); \draw[arrow] (D-delta1) -- (C6); \draw[arrow] (E-delta0) -- (C6); \end{tikzpicture} } \caption{Possible isomorphism classes of Galois groups of simple abelian threefolds, in terms of their Newton polygon and angle rank $\delta_A$.} \label{fig:flowchart3} \end{figure} \subsection{Outline} In \Cref{sec:background} we introduce background and notation used throughout this paper. A reader familiar with abelian varieties may choose to skip directly to \Cref{sec:weighted-perm-rep}, where we introduce a key tool in our paper, the weighted permutation representation. In \Cref{sec:warmup} we warm up by classifying weighted permutation representations of elliptic curves. In \Cref{sec:surfaces} and \Cref{sec:threefolds} we classify weighted permutation representations of abelian surfaces and simple abelian threefolds respectively. We finish by listing further inverse Galois questions in \Cref{sec:questions}. The code associated to this article is written in \texttt{Magma}~\cite{Magma} and is publicly available from the \GitHub repository \cite{OurElectronic}. \subsection{Acknowledgements} We thank Raymond van Bommel, Deewang Bhamidipati, Soumya Sankar, John Voight, and David Zureick-Brown for helpful conversations about this project. We also thank Everett Howe for several useful correspondences. SF was supported by the Woolf Fisher and Cambridge Trusts through a Woolf Fisher scholarship and by C{\'e}line Maistret's Royal Society Dorothy Hodgkin Fellowship. SV was supported by the National Science Foundation under grant number DMS2303211. \section{Background and notation}\label{sec:background} \subsection{Honda--Tate theory}\label{sec:HT-theory} Let $A/\FF_q$ be an abelian variety. A celebrated theorem of Honda and Tate classifies isogeny classes of abelian varieties over finite fields. Let $g \colonequals \dim(A) > 0$ and recall that $P_A(T) \in \ZZ[T]$ is the characteristic polynomial of Frobenius. The roots of $P_A(T)$ have absolute value $\sqrt{q}$ in all their complex embeddings; an algebraic integer with this property is called \cdef{$q$-Weil number}. A \cdef{$q$-Weil polynomial} is a monic integral polynomial whose roots are all $q$-Weil numbers. Fix an algebraic closure $\Qbar$ of $\QQ$ inside of $\CC$, and an embedding $\Qbar \hookrightarrow \Qbar_p$. We write $\nu$ for the $p$-adic valuation on $\Qbar_p$, normalized so that $\nu(q) = 1$. The statement below is the one presented in \cite[Theorem 4.2.12]{Poonen06}. \begin{theorem}[Honda--Tate Theorem] \label{thm:HT} \noindent \begin{enumerate} \item \label{enum:HT1} If $A$ is a simple abelian variety, then $P_A(T) = h_A(T)^e$ for some irreducible polynomial $h_A(T) \in \mathbb{Z}[T]$ and some $e \geq 1$. \item \label{enum:HT2} There is a bijection between isogeny classes of simple abelian varieties over $\mathbb{F}_q$ and conjugacy classes of $q$-Weil numbers. \item \label{enum:HT3} Given $\Gal(\Qbar/\QQ)$-conjugacy class of $q$-Weil numbers, let $h_A(T)$ be the minimal polynomial of any element of this conjugacy class. Then there exists a unique integer $e_A \colonequals e \geq 1$ such that $h_A(T)^e = P_A(T)$ for some simple abelian variety $A$ over $\Fq$. Moreover, $e$ is the smallest positive integer such that: \begin{enumerate} \item $h_A(0)^e > 0$, and \item For each monic $\QQ_p$-irreducible factor $g(T) \in \QQ_p[T]$ of $h_A(T)$, the valuation $\nu(g(0)^e)$ is in $\ZZ$. \end{enumerate} \end{enumerate} \end{theorem} The polynomial $h_A(T)$ is the \cdef{minimal polynomial of the $q$-Frobenius endomorphism of $A$}. When $P_A(T)$ is totally complex (i.e., has no real roots), the degree of $h_A(T)$ is equal to $2d$ for some positive integer $d$. In general, the isogeny factorization of $A$ yields a factorization of the Frobenius polynomial. In particular, by the Honda--Tate theorem, two abelian varieties $A$ and $B$ are isogenous if and only if $P_A(T)$ is equal to $P_B(T)$. This observation is sufficient to understand the classification of our three isogeny invariants in the case of elliptic curves (see \Cref{sec:warmup}). We now proceed to give more background that will be useful for higher dimensional abelian varieties. We will use the following as a running example throughout the article to consolidate our definitions and notations. This example appears in \cite[Example 6.1]{Shioda81} and \cite[Example 1.7]{Zywina22} (see also \cite[Example 4.2.9]{Gallese2024}). \begin{example}[Shioda's example] \label{example:Shioda} Let $q = p = 19$, and let $A$ be the Jacobian of the hyperelliptic curve $C$ with affine equation $y^2 = x^9 - 1$, defined over the field $\FF_{19}$. The curve $C$ has genus $g = 4$ and therefore $A$ is an abelian fourfold. By calculating $\#C(\FF_{19^r})$ for $r=1,2,3,4$, we are able to estimate the zeta function of $C$ to enough precision to recover the Frobenius polynomial $P_A(T)$. It is given by: \begin{equation} P_A(T) = T^8 + 8T^7 + 28T^6 + 8T^5 - 170T^4 + 152T^3 + 10108T^2 + 54872T + 130321. \end{equation} This polynomial factors as $P_A(T) = P_E(T)P_B(T)$, where $E$ is the elliptic curve $y^2 = x^3-1$ in the isogeny class \avlink{1.19.i} with Frobenius polynomial $P_E(T) = T^2 + 8T + 19$, and $B$ is an abelian threefold in the isogeny class \avlink{3.19.a_j_acm} with Frobenius polynomial $P_B(T) = T^6 + 9T^4 - 64T^3 + 171T^2 + 6859$. By the Honda--Tate theorem $A$ is isogenous to the product $E \times B$. \end{example} \subsection{Newton polygons of Frobenius polynomials} Let $A$ be a $g$-dimensional abelian variety over $\Fq$. At this moment, we do not assume that $A$ is simple. \begin{defn}[$q$-Newton polygon] \label{def:NP} The \cdef{$q$-Newton polygon} of $A$ is the $\nu$-adic Newton polygon of $P_A(T)$. More precisely, if $P_A(T) = \sum_{j = 0}^{2g} a_{2g-j}T^j$, then the $q$-Newton polygon of $A$ is the lower convex hull of the finite set \begin{equation} \brk{(j,\nu(a_j)) : 0 \leq j \leq 2g,\mand a_j \neq 0} \subset \RR^2. \end{equation} \end{defn} \begin{example} Continuing with \Cref{example:Shioda}, we note that both $E$ and $B$ are ordinary varieties. Since $A \sim E \times B$, the Newton polygon of $A$ is also ordinary, and it is obtained by concatenating those of $E$ and $B$. Alternatively, one can notice that $-170$ (the middle coefficient of $P_A(T)$) is not divisible by $p=19$. \end{example} \subsection{Angle rank of Frobenius polynomials} We recall the following definition from \cite{DupuyKedlayaRoeVincent22,DupuyKedlayaZureick-Brown22,APBS2023}. \begin{defn} \label{defn:angle-rank} Consider the multiplicative subgroup $U_A \subset \Qbar^\times$ generated by the normalized Frobenius eigenvalues $u \coloneqq \alpha/\sqrt{q}$ where $\alpha$ ranges over the roots of $h_A(T)$. The \cdef{angle rank of $A$} is denoted $\delta_A$ and is defined to be dimension of $U_A \otimes \QQ$. \end{defn} \subsection{Galois groups of Frobenius polynomials} Given an abelian variety $A$, denote by $R_A$ the \cdef{set of roots}, without multiplicity, of the Frobenius polynomial $P_A(T)$. \begin{defn}[Galois group] \label{def:gal-grp} The \cdef{Galois group} of $A$ is the Galois group of the minimal polynomial of Frobenius $h_A(T)$. Equivalently, $G_A$ is the largest quotient of $\Gal(\Qbar/\QQ)$ over which the permutation action on $R_A$ factors. \end{defn} \begin{defn} \label{def:K} In the case that $A$ is simple, we will denote by $K_A$ the center of the endomorphism algebra $\End(A)\otimes\QQ$. We have that $K_A \cong \QQ[T]/(h_A(T))$ is a number field. \end{defn} For ``most'' abelian varieties the polynomial $P_A(T)$ is totally complex (i.e., has no real roots), and $K_A$ is a \cdef{complex multiplication} (CM) number field. See \cite{Dodson1984} for some background on Galois groups of CM number fields, and \cite[Section 2.2]{DupuyKedlayaZureick-Brown22} for a discussion on Galois groups of $q$-Weil polynomials. \begin{example} \label{example:part-2} Continuing with \Cref{example:Shioda}, we have that the splitting field of $P_A(T)$ is the degree $6$ field $\QQ(\zeta_9)$, where $\zeta_9$ is a primitive $9^\text{th}$-root of unity. This already implies that the $8$ roots of $P_A(T)$ are algebraically dependent. Recalling that $A \sim E \times B$, observe that $K_E = \QQ(\sqrt{-3})$, which is contained in $K_B = \QQ(\zeta_9)$. Note that $G_A$ is a permutation group acting on the $8$ element set $R_A$. But as abstract groups, $G_A \cong \Gal(\QQ(\zeta_9)/\QQ) \cong C_6$. Denote by $R_E = \brk{\eta, 19\eta^{-1}}$ and $R_B = \brk{\alpha,\beta,\gamma,19\alpha^{-1},19\beta^{-1}, 19\gamma^{-1}}$ the sets of roots of $P_E(T)$ and $P_B(T)$ respectively. For an appropriate such labelling, we have \begin{equation} \label{eq:mult-rel-example} \eta = \dfrac{\alpha\cdot\beta\cdot\gamma}{19}. \end{equation} In \Cref{example:angle-rank} that the $\QQ$-vector space $U_A\otimes \QQ$ has dimension 3, i.e., $\delta_A = 3$. \end{example} Instead of viewing $G_A$ as a permutation subgroup of $S_n$ for $n = \# R_A$, it will be convenient to use a ``CM specific'' permutation representation. \subsection{The group of signed permutations}\label{sec:W2d} We now describe the abstract group which Galois groups of totally complex $q$-Weil numbers are naturally contained in. First note that when $h_A(T)$ is totally complex of degree $2d$ and the roots of $h_A(T)$ come in complex conjugate pairs. Moreover, the action of the Galois group $G_A$ respects this partition. Let $X_{2d}$ be the set consisting of the symbols $1, \bar{1}, \dots, d, \bar{d}$. Define $W_{2d}$ to be the subgroup of $\Sym(X_{2d})$ which preserves the partition \begin{equation} X_{2d} = \brk{1,\bar{1}}\sqcup \cdots \sqcup \brk{d,\bar{d}}. \end{equation} Upon a choice of labelling of roots, the Galois group of $h_A(T)$ may naturally be embedded in $W_{2d}$. We refer to the element $\iota \colonequals (1\bar{1})\ldots (d\bar{d})$ as \cdef{complex conjugation}. \subsubsection{Subgroup labelling} \label{sec:subgroup-labelling} We now briefly describe our naming conventions for subgroups of $W_{2d}$. A group $H$ is denoted $\texttt{G.d.t.letter.k}$ if it is isomorphic to $G$, contained in $W_{2d}$, and acts transitively on $X_{2d}$. The $W_{2d}$ conjugacy class is indexed by $\texttt{letter}$, and the groups in that conjugacy class are indexed by the tiebreaker $\texttt{k}$ (a positive integer). The use of $\texttt{nt}$ instead of $\texttt{t}$ indicates that $H$ acts intransitively on $X_{2d}$. For example, the group $\wl{C2.4.nt.c.1}$ refers to a group which is isomorphic to $C_2$, contained in $W_4$, and is intransitive. The label $\texttt{c}$ refers to the conjugacy class in $W_4$ and the index $\texttt{1}$ means that it is the first listed in its conjugacy class. \begin{remark} In the code associated to this article \cite{OurElectronic} we provide functions which compute and label the transitive subgroups of $W_{2d}$ which contain complex conjugation in the file \href{https://github.com/sarangop1728/Galois-Frob-Polys/blob/main/src/W2d-subgroups.m}{\texttt{src/W2d-subgroups.m}}. Our labelling convention is well defined and essentially follows lexicographic ordering of the subgroups of the symmetric group $S_{2d} \supset W_{2d}$ (as described in \cite{HulpkeLinton:TotalOrdering}). See the file \href{https://github.com/sarangop1728/Galois-Frob-Polys/blob/main/src/subgroup-labelling.m}{\texttt{src/subgroup-labelling.m}}. \end{remark} \section{The weighted permutation representation} \label{sec:weighted-perm-rep} In this section we introduce a key tool in this paper -- the notion of a weighted permutation representation. This construction is heavily inspired by the definition of \emph{Newton hyperplane arrangement} of Dupuy, Kedlaya, and Zureick-Brown \cite{DupuyKedlayaZureick-Brown22}. \subsection{The weighted permutation representation} \label{sec:perm-rep} We now describe the weighted permutation representation associated to an abelian variety, which is the central isogeny invariant in this article. It determines the Galois group, angle rank, and Newton polygon. The main purpose of our paper is to determine which weighted permutation representations occur from low dimensional abelian varieties. The weighted permutation representation is a reinterpretation of the \emph{Newton hyperplane representation} discussed in \cite{DupuyKedlayaZureick-Brown22}. \begin{defn}[Weighted permutation representations] \label{def:weighted-perm-rep} Given a finite group $G$, a \cdef{weighted permutation representation of $G$} is a pair $(w,\rho)$, where $w \colon X_{2d} \rightarrow \QQ_{\geq 0}$ is a map of sets and $\rho \colon G \xhookrightarrow{} W_{2d}$ is an inclusion of groups. We say that a pair of weighted permutation representations $(w, \rho)$ and $(w, \rho')$ of $G$ are \cdef{$w$-conjugate} if they are conjugate by an element of $\Stab(w)$, i.e., if there exists an element $\sigma \in W_{2d}$ such that $w = w \circ \sigma$ and $\rho' = \sigma^{-1} \rho(g) \sigma$ for all $g \in G$. \end{defn} \begin{defn}[Roots] \label{def:roots} Recall that for an abelian variety $A$ we write $R_A$ for the set of roots of $h_A(T)$ without multiplicity. We define $\rts_A$ to be the \cdef{multiset} of roots of $h_A(T)$, that is: \begin{enumerate} \item When $h_A(T)$ is totally complex, $\rts_A = R_A$. \item When $h_A(T)$ has a real root $\alpha$, it is counted with multiplicity two, and its duplicate is denoted by $\oalpha$. \end{enumerate} We define $d_A = \#\rts_A / 2$. \end{defn} \begin{defn}[Indexing and weighting] \label{def:indexing} An \cdef{indexing of roots} of $h_A(T)$ is a bijection $\indx\colon X_{2d} \to \rts_A$ which satisfies the following conditions: \begin{enumerate} \item $\indx$ respects complex conjugation, i.e., $\indx(\overline{k}) = \overline{\indx(k)}$ for each $1 \leq k \leq d$, and \item the indices climb the Newton polygon, i.e., \[ \nu(\alpha_i) \leq \nu(\alpha_j) \leq \nu(\oalpha_j) \leq \nu(\oalpha_i) \] for each pair of indices $1 \leq i \leq j \leq d$. \end{enumerate} \end{defn} Any indexing of the roots naturally gives a \cdef{weighting} $w_A \colon X_{2d} \rightarrow \QQ_{\geq 0}$ given by $k \mapsto \nu(\alpha_k)$ and $\bar{k} \mapsto \nu(\oalpha_k)$ for $k \in \brk{1,\dots,d}$. We omit the choice of indexing from the notation, since every choice of indexing yields the same weighting $w_A$. Note that the weighting $w_A$ associated to an abelian variety $A$ is uniquely determined by the $q$-Newton polygon of $A$. \begin{defn}[Weighted permutation representation, totally complex case] \label{def:perm-rep} Suppose $h_A(T)$ is totally complex. Given an indexing $\mathcal{I}$, we obtain representation \begin{equation} \rho_{\mathcal{I}} \colon G_A \xhookrightarrow{} \Sym(X_{2d}) \cong S_{2d} \end{equation} whose image lies in $W_{2d}$. The pair $(w_{A}, \rho_{\mathcal{I}})$ is the \cdef{weighted permutation representation} associated to $A$ with respect to the indexing $\mathcal{I}$. \end{defn} The definition above \emph{canonically} defines the weighted permutation representation associated to $A$ up to $w_A$-conjugacy when $P_A(T)$ is totally complex. The following definition gives a notion of weighted permutation representation when $P_A(T)$ is totally real; although the definition does not appear canonical, we will show later that it is in fact unique up to isomorphism. The simple isogeny classes of abelian varieties with real eigenvalues are highly constrained. \begin{lemma}[{\cite[pp.~528]{Waterhouse1969}}] \label{lem:real-eigenvalues} Let $A$ be a simple abelian variety whose Frobenius eigenvalues are real, then: \begin{enumerate}[label=(\arabic)] \item \label{enum:real-ECs} If $q$ is a square, $A$ is a supersingular elliptic curve, and either $h_A(T) = (T \pm \sqrt{q})$ for some choice of sign, or \item \label{enum:complex-ECs} If $q$ is not a square, $A$ is a supersingular abelian surface, and $h_A(T) = (T^2-q)$. \end{enumerate} Moreover, in case~\ref{enum:real-ECs} we have $\rts_A = \brk{\sqrt{q}, \overline{\sqrt{q}}}$ or $\brk{-\sqrt{q}, -\overline{\sqrt{q}}}$, and in case~\ref{enum:complex-ECs} we have $\rts_A = \brk{\sqrt{q}, \overline{\sqrt{q}}, -\sqrt{q}, -\overline{\sqrt{q}}}$. \end{lemma} \begin{defn}[Weighted permutation representation, totally real case] \label{def:perm-rep-real} Suppose that $A$ has only real Frobenius eigenvalues. \begin{enumerate} \item[(1a)] If $q$ is a square and $A$ has one eigenvalue, then $G_A$ is trivial. Define $\rho_{\mathcal{I}} \colon G_A \to W_2$ to be the trivial map for all indexings. \item[(1b)] If $q$ is a square and $A$ has two eigenvalues, then $G_A$ is trivial. Define $\rho_{\mathcal{I}} \colon G_A \to W_4$ to be the trivial map for all indexings. \item[(2)] If $q$ is not a square, then $G_A \cong C_2$. Define $\rho_{\mathcal{I}} \colon G_A \to W_4$ to be the homomorphism sending the nontrivial element on $G_A$ to $(12)(\bar{1}\bar{2})$ for all indexings. \end{enumerate} The pair $(w_A, \rho_\mathcal{I})$ is the \cdef{weighted permutation representation} associated to $A$ with respect to the indexing $\mathcal{I}$. \end{defn} We now define the weighted permutation representation associated to an abelian variety $A$ for which $h_A(T)$ need not be totally real or totally complex; it is essentially the direct sum of its totally real and totally complex parts. In this case $A$ is isogenous to a product $B \times C$ where $h_B(T)$ is totally real and $h_C(T)$ is totally complex. Let $(w_B,\rho_B)$ and $(w_C,\rho_C)$ be the weighted permutation representations corresponding to the factors $B$ and $C$ respectively. Let $(w_B \oplus w_C, \rho_B \oplus \rho_C)$ be the direct sum of these weighted permutation representations; conjugate by $W_{2d}$ so that the weight function is nondecreasing, i.e., so that \[ w(i) \leq w(j) \leq w(\bar{j}) \leq w(\bar{i}) \] for $i \leq j$. We define the weighted permutation representation associated to $A$ to be the resulting weighted permutation representation. \Cref{lemma:well-def-conj-class} follows by construction. \begin{lemma} \label{lemma:well-def-conj-class} The weighted permutation representation associated to an abelian variety $A$ is well defined up to $w_A$-conjugacy (irrespective of indexing). In particular, its image in $W_{2d}$ is a subgroup $\cclass{A}$ which is well-defined up to $w_A$-conjugacy. \end{lemma} \subsection{The angle rank}\label{sec:newton-rep} It turns out that the angle rank can be computed from the weighted permutation representation of an abelian variety in the following way. \begin{defn} Given a weighted permutation representation $(w,\rho \colon G \xhookrightarrow{} W_{2d})$, define the \cdef{angle rank of $(w,\rho)$} to be $\rank(M) - 1$ where $M$ is the $(d \times \|G\|)$-matrix whose $i^\text{th}$-column has entries $w(\sigma(i))$ where $\sigma$ ranges over elements of $G$. \end{defn} The angle rank of $(w,\rho)$ is equal to the rank of the $(d \times \|G\|)$-matrix whose $i^\text{th}$-column has entries $w(\sigma(i)) - w(\sigma(\bar{i}))$, which is referred to as the Newton hyperplane matrix \cite[Remark 3.4]{DupuyKedlayaZureick-Brown22}. We show in \Cref{prop:newton-matrix-rank} that the angle rank of an abelian variety is equal to the angle rank of its weighted permutation. \begin{example}\label{example:angle-rank} Continuing with \Cref{example:part-2}, fix a prime $\mfp$ of the splitting field $K = \QQ(\zeta_9)$ above $p = 19$, and let $\nu$ be the extension of the $19$-adic valuation of $\QQ$ extended to $K$. An indexing of the roots $R_A$ according to \Cref{def:indexing} is for example \begin{equation} \alpha_1 = \alpha, \quad \alpha_2 = \beta, \quad \alpha_3 = \eta, \quad \alpha_4 = \gamma, \end{equation} if we ensure that $\nu(\alpha) = \nu(\beta) = \nu(\gamma) = \nu(\eta) = 0$. With this indexing, the weighted permutation representation $\rho\colon \Gal(\QQ(\zeta_9)/\QQ) \to W_8$ has image $H = \langle h \rangle$, where $h = (1\bar{2}\bar{4}\bar{1}24)(3\bar{3})$. With respect to this indexing, the multiplicative relation in \Cref{eq:mult-rel-example} becomes \begin{equation} \alpha_3 = \dfrac{\alpha_1\alpha_2\oalpha_4}{19} =\dfrac{\alpha_1\alpha_2}{\alpha_4}. \end{equation} One can find this multiplicative relation by considering the Newton hyperplane matrix of this weighted permutation representation, and computing its kernel. By direct calculation we see that the rank of the Newton hyperplane matrix (which is equal to $\delta_A$) is three. \end{example} \subsection{The divisor map} \label{sec:divisor-map} Many of the proofs in this paper make use of the same key idea, which we elucidate here. Given a simple totally complex $g$-dimensional abelian variety $A$ with Galois group $G_A$, we may construct the following. Let $L$ be the Galois closure of the field $K$, i.e., the field generated by the roots of $P_A(T)$. Let $\cP_L$ be the set of primes in $L$ above $p$. Let $\vecsp{\QQ}{R_A}$ denote the $\QQ$-vector space of dimension $2d$ whose basis consists of the formal symbols $[\alpha]$ where $\alpha$ ranges over the set of Frobenius eigenvalues. Let $\Div_{\QQ}(\OO_L)$ be the free $\QQ$-module supported on the prime ideals of $\OO_L$ and let $\div_A \colon \OO_L \to \Div_{\QQ}(\OO_L)$ be the map sending each element $x \in \OO_L$ to $ \sum_{\mfp} a_{\mfp} \mfp$ where $(x) = \prod_\mfp \mfp^{a_{\mfp}}$ is the prime factorization of the ideal generated by $x$. By abuse of notation we write \[ \div_A \colon \vecsp{\QQ}{R_A} \rightarrow \vecsp{\QQ}{\cP_L} \] for the $\QQ$-linear map given by linearly extending $\div_A$ on the roots $\alpha \in R_A$. Note that $\vecsp{\QQ}{R_A}$ naturally inherits the structure of a $G_A$-module from the action of $G_A$ on $R_A$, and similarly $\vecsp{\QQ}{\cP_L}$ has an action of $G_A$ inherited from the action of $G_A$ on $\cP_L$. The map $\div_A$ is $G_A$-equivariant. Note that $\div_A$ determines the weighted permutation representation, and thus also determines the angle rank, Newton polygon, and Galois group. It is natural to ask which $G_A$-module homomorphisms are permissible as the divisor map of a simple abelian variety. We list some necessary conditions which follow immediately from the construction. \begin{proposition} \label{prop:div-properties} Let $A$ be a simple abelian variety with no real Frobenius eigenvalues. Then: \begin{enumerate}[label=(\arabic)] \item $\div_A([\alpha] + [\overline{\alpha}]) = \div_A(q)$ for all $\alpha \in R_A$; \item the $G_A$-action on $R_A$ is transitive; \item the $G_A$-action on $\cP_L$ is transitive; and \item \label{enum:div-property-4} if the Newton polygon has a segment of length $m$ which contains exactly two lattice points, then $\# \cP_L \mid \frac{1}{m}\# G_A$. \end{enumerate} \end{proposition} We now quickly use the divisor map to show the following lemma. \begin{lemma} \label{prop:newton-matrix-rank} The angle rank $\delta_A$ of an abelian variety is equal to the angle rank of its weighted permutation representation. Moreover, $\delta_A = \rank(\div_A) - 1$. \end{lemma} \begin{proof} Consider the subspace $V_p = \Span(\div_A(p)) \subset \vecsp{\QQ}{\cP_L}$. We first claim that the angle rank of an abelian variety is precisely the rank of the linear map $\vecsp{\QQ}{R_A} \rightarrow \vecsp{\QQ}{\cP_L}/ V_p$ induced by $\div_A$. Clearly a multiplicative relation between the Frobenius eigenvalues gives rise to an element in the kernel of this map. Conversely, given an element $\sum_{i = 1}^{2g} k_i\alpha_i$ in the kernel, we have \[ \bigg (\prod_i\alpha_i^{k_i}\bigg ) = (q)^{\sum k_i /2}, \] so there exists $u \in \OO_L^{\times}$ such that \[ \prod_iq^{-k_i/2}\alpha_i^{k_i} = u. \] Because $u$ has length $1$ in all complex embeddings, $u$ is a root of unity. Now observe that the rank of $\vecsp{\QQ}{R_A} \rightarrow \vecsp{\QQ}{\cP_L} / V_p$ is precisely one less than the rank of $\div_A \colon \vecsp{\QQ}{R_A} \rightarrow \vecsp{\QQ}{\cP_L}$. After removing duplicate rows and columns, the matrix representing the latter is precisely the matrix given in the definition of the angle rank of a weighted permutation representation. \end{proof} Observe that every such map $\vecsp{\QQ}{X_{2d}} \rightarrow \QQ^{\ell}$ satisfying the properties above gives rise to a weighted permutation representation. To prove that a certain weighted representation cannot occur, it suffices to show that there does not exist a lift $\QQ^{2d} \rightarrow \QQ^{\ell}$ satisfying the conditions above. \section{Warm-up: elliptic curves}\label{sec:elliptic-curves} \label{sec:warmup} We begin with the case when $A$ is an elliptic curve. In accordance with the labelling convention described in \Cref{sec:W2d} we write $\texttt{W2.2.t.a.1}$ and $\texttt{C1.2.t.a.1}$ for the subgroups of order $2$ and $1$ of $W_2$. The following \namecref{lemma:main-thm-ecs} is immediate. \begin{proposition} \label{lemma:main-thm-ecs} Let $A$ be an elliptic curve. Then the image, $\cclass{A}$, of the weighted permutation representation associated to $A$ is equal to $W_2$ except when $q$ is a square and $P_A(T) = (T \pm \sqrt{q})^2$, in which case $A$ is supersingular and $\cclass{A} = \{ \operatorname{id} \}$. More precisely, the possible combinations of Galois group, Newton polygon, and angle rank that occur are displayed in Tables~\ref{tab:EC-A}~and~\ref{tab:EC-B}. \end{proposition} \begin{table}[H] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Example(s) \\ \hline $\texttt{W2.2.t.a.1}$ & $1$ & Yes & \avlink{1.2.ab} \\ \hline $\texttt{C1.2.nt.a.1}$ & $0$ & No & \\ \hline \end{tabular} \caption{The $w_A$-conjugacy classes of subgroups $G \subset W_2$ which occur as the image of the weighted permutation representation associated to an ordinary elliptic curve.} \label{tab:EC-A} \end{table} \begin{table}[H] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Example(s) \\ \hline $\texttt{W2.2.t.a.1}$ & $0$ & Yes & \avlink{1.2.ac} \\ \hline $\texttt{C1.2.nt.a.1}$ & $0$ & Yes & \avlink{1.4.ae} \\ \hline \end{tabular} \caption{The $w_A$-conjugacy classes of subgroups $G \subset W_2$ which occur as the image of the weighted permutation representation associated to a supersingular elliptic curve.} \label{tab:EC-B} \end{table} \FloatBarrier \section{Abelian surfaces}\label{sec:surfaces} In this section we prove the following theorem. \begin{theorem} \label{thm:main-thm-surfaces} Let $A$ be an abelian surface. \begin{itemize} \item \thmemph{Permutation representations in $W_4$:} If $A$ has permutation representation contained in $W_4$, then the possible images $\cclass{A}$ of the weighted permutation representation associated to $A$ are given in Tables~\ref{tab:dim2-A}~and~\ref{tab:dim2-A-ns} when $A$ is ordinary, in Tables~\ref{tab:dim2-B}~and~\ref{tab:dim2-B-ns} when $A$ is almost ordinary, and in Tables~\ref{tab:dim2-C}~and~\ref{tab:dim2-C-ns} when $A$ is supersingular. The permutation representation determines whether $A$ is geometrically simple; this information can be seen in the tables as well. \item \thmemph{Permutation representations in $W_2$ when $A$ is simple:} If $A$ has a permutation representation contained in $W_2$ and is simple, then $A$ is supersingular and hence the angle rank is $0$. Both trivial Galois group and Galois group $C_2$ occur. $A$ is not geometrically simple. \item \thmemph{Permutation representations in $W_2$ when $A$ is not simple:} Now suppose $A$ has a permutation representation contained in $W_2$ and is not simple. Then $A$ is isogenous (over $\Fq$) to $E^2$ for some elliptic curve $E$ then the weighted permutation representation associated to $A$ takes values in $W_2$, and its image is equal to that associated to $E$ (and classified in \Cref{lemma:main-thm-ecs}). \end{itemize} \end{theorem} The rest of the section is dedicated to proving \Cref{thm:main-thm-surfaces}. The third point follows from the definition and the section on elliptic curves. The second point follows from the classification given in \cite{Xing1994}; see also \cite[Theorem~2.9~(SS2)]{MaisnerNart02}. In the remainder of this section we assume $A$ has permutation representation contained in $W_4$. \subsection{Permutation representations of subgroups of \texorpdfstring{$W_4$}{W4}} The group $W_4$ is isomorphic to $D_4$, the dihedral group of order $8$. The following lemma classifies the $w$-conjugacy classes of subgroups of $W_4$. \begin{lemma} \label{lemma:w-conj-w4} There are exactly $3$ transitive and $7$ intransitive subgroups of $W_4$ and each is recorded in \Cref{tab:W4-and-W6-subgroups}. Moreover, the only distinct subgroups which are $w_A$-conjugate are: \begin{enumerate} \item $\wl{C2.4.nt.b.1}$ and $\wl{C2.4.nt.b.2}$ when $A$ is either ordinary or supersingular, and \item $\wl{C2.4.nt.c.1}$ and $\wl{C2.4.nt.c.2}$ when $A$ is supersingular. \end{enumerate} Each $w_A$-conjugacy class gives rise to an isomorphism class of weighted permutation representation, and the angle ranks of these $w_A$-conjugacy classes $\cclass{} \subset W_4$ are recorded in Tables~\ref{tab:dim2-A}--\ref{tab:dim2-C-ns}. \end{lemma} \begin{proof} The first claim then follows by a direct calculation. The angle ranks are computed using our implementation of \Cref{prop:newton-matrix-rank} in the file \href{https://github.com/sarangop1728/Galois-Frob-Polys/blob/main/src/weighted-perm-rep.m}{\texttt{src/weighted-perm-rep.m}} of our \GitHub repository \cite{OurElectronic}. \end{proof} To prove \Cref{thm:main-thm-surfaces} it suffices to show that: \begin{enumerate} \item the cases we claim do not occur, actually do not occur; and \item in the cases that do occur, the weighted permutation representation determines whether the abelian surface is geometrically simple. \end{enumerate} In the remaining cases, we provide an example in Tables~\ref{tab:dim2-A}--\ref{tab:dim2-C-ns}, which realizes the given weighted permutation representation. We remark that the angle ranks displayed in the LMFDB are numerical approximations, but we verify these examples explicitly via a slower (but deterministic) algorithm; see the file \href{https://github.com/sarangop1728/Galois-Frob-Polys/blob/main/tables/verify-angle-rank.m}{\texttt{tables/verify-angle-rank.m}} in our \GitHub repository \cite{OurElectronic}. \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-surfaces}}{Theorem ??} in the simple case} For a simple abelian surface, the permutation representation acts transitively except when $A$ is supersingular with real Frobenius eigenvalues. \subsubsection{The ordinary case} In this case, every possible transitive weighted permutation representation occurs. Therefore, to prove the theorem in this case, it suffices to show that a simple ordinary abelian surface is not geometrically simple if and only if its weighted permutation representation is \wl{V4.4.t.a.1}. \begin{proposition} \label{lemma:simple-splits-power-of-EC} A simple abelian variety has a unique simple factor over every finite extension of the base field. In particular, if $A$ is simple of prime dimension and it splits over a finite extension of $\FF_q$, then it does so as the power of an elliptic curve. \end{proposition} \begin{proof} This follows immediately from \cite[Proposition 1.2.6.1]{chai-conrad-oort}. \end{proof} \begin{coro} \label{coro:not-geom-simple=>angle-rank-1} If $A$ is a simple abelian variety of prime dimension which is not geometrically simple and not supersingular, then $A$ is ordinary and has angle rank $1$. \end{coro} \begin{proof} Since $A$ is simple but not geometrically simple by \Cref{lemma:simple-splits-power-of-EC} there exists an extension $\FF_{q^k}$ of $\FF_q$ over which $A$ becomes isogenous to $E^g$ for some ordinary elliptic curve $E/\FF_{q^k}$ (in particular $A$ is ordinary). Angle rank is invariant under base change, so it follows that $\delta_A = 1$. \end{proof} \begin{lemma} \label{lemma:abs-simp-ord=>max-angle-rank} An ordinary geometrically simple abelian surface $A$ has angle rank $2$. \end{lemma} \begin{proof} We prove the contrapositive. Assume that $\delta_A < 2$. This implies that there exists a multiplicative relation among the normalized Frobenius eigenvalues $u_1^{r_1}u_2^{r_2} = 1$. Note that $r_1r_2 = 0$ implies that some $u_i$ is a root of unity, which would imply that $A$ is not ordinary. Thus, we have that both $r_1$ and $r_2$ are nonzero integers. Let $\alpha_1$ and $\alpha_2$ be the two Frobenius eigenvalues with $\nu(\alpha_1) = \nu(\alpha_2) = 0$. From the multiplicative relation we deduce that $r_1 = -r_2$, so that $(u_1/u_2)^{r_1} = 1$ and $\alpha_2 = \zeta_k\alpha_1$ where $\zeta_k$ is a $k^{\text{th}}$ root of unity. The Frobenius eigenvalues of the base change of $A$ to $\FF_{q^k}$ are precisely the $k$-th powers of the Frobenius eigenvalues of $A$. Because $\alpha_1^k = \alpha_2^k$, the Honda--Tate theorem implies that the base change of $A$ to $\FF_{q^k}$ is isogenous to the square of an elliptic curve, so $A$ is not geometrically simple. \end{proof} \begin{lemma} Let $A$ be a simple ordinary abelian surface. Then, exactly one of the following conditions holds. \begin{enumerate}[label=(\arabic)] \item \label{enum:o-dim2-gs} $A$ is geometrically simple and $G_A \cong C_4$ or $W_4$. \item \label{enum:o-dim2-ngs} $A$ is not geometrically simple and $G_A \cong V_4$. \end{enumerate} \end{lemma} \begin{proof} By \Cref{lemma:abs-simp-ord=>max-angle-rank} if $A$ is geometrically simple then $A$ has angle rank 2, and by \Cref{coro:not-geom-simple=>angle-rank-1} if $A$ is not geometrically simple the it has angle rank $1$. The claim follows from the angle ranks computed in \Cref{lemma:w-conj-w4} (and recorded in \Cref{tab:dim2-A}). \end{proof} \subsubsection{The almost ordinary case} Every simple almost ordinary abelian surface is geometrically simple by \Cref{coro:not-geom-simple=>angle-rank-1}, so the following lemma completes the proof. \begin{lemma} \label{lemma:ao-dim2} A simple almost ordinary abelian surface has Galois group $W_4$. \end{lemma} \begin{proof} Suppose for the sake of contradiction that $A$ is an almost ordinary abelian variety with Galois group $C_4$ or $V_4$. Let $\div_A$ be the corresponding divisor map as discussed in \Cref{prop:div-properties}. By point \ref{enum:div-property-4}, we have $\# \cP_L \mid 2$, where $\cP_L$ is the set of primes above $p$ in $L$. From \Cref{prop:newton-matrix-rank}, we have $\delta_A = \rank(\div_A) -1 \leq \#\cP_L - 1$, which contradicts \Cref{tab:dim2-B}. \end{proof} \begin{remark} One can also see \Cref{lemma:ao-dim2} as follows: by the theory of Newton polygons and the Honda--Tate theorem, $P_A(T)$ has an irreducible linear and quadratic factor over $\QQ_p$, so the quartic extension $K/\QQ$ is not Galois. \end{remark} \subsubsection{The supersingular case} It follows from the Honda--Tate theorem that every supersingular abelian variety is geometrically isogenous to the power of an elliptic curve. The proof is concluded with the following well known result which shows that no supersingular abelian variety of dimension $g > 1$ can have Galois group $W_{2g}$. \begin{lemma} \label{lemma:big-G} Suppose $A$ has Galois group $G_A \cong W_{2d}$. Then either: \begin{enumerate} \item $A$ is a power of a supersingular elliptic curve $E$ with $P_E(T) \neq (T\pm\sqrt{q})^2$, or \item $A$ has angle rank $\delta_A = d$. \end{enumerate} \end{lemma} \begin{proof} Suppose that $A$ is not supersingular. Then, every normalized Frobenius eigenvalue $u = \alpha/\sqrt{q}$ satisfies $\nu(u) \neq 0$. Fix an indexing $\mathcal{I}\colon X_{2d} \to R_A$ and consider the vector $\mathbf{v} \colonequals (v_1, \dots, v_d)$ with nonzero entries $v_j \colonequals \nu(u_j)$. Acting on $\mathbf{v}$ by the transpositions $(k\bar{k})\in W_{2d}$, we see that the Newton hyperplane matrix contains the $(d \times d)$-minor \begin{equation} \begin{bmatrix} v_1 & v_1 & \cdots & v_1 \\ v_2 & -v_2 & \cdots & v_2 \\ \vdots & \vdots & \ddots & \vdots \\ v_d & v_d & \cdots & -v_d \end{bmatrix}, \end{equation} which is visibly similar to the diagonal matrix $\mathrm{diag}(v_1,\dots,v_d)$. \end{proof} \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-surfaces}}{Theorem ??} in the nonsimple case} If $A$ is isogenous to the square of an elliptic curve, then the claim follows immediately from \Cref{lemma:main-thm-ecs}. Therefore suppose that $A$ is isogenous over $\Fq$ to a product $E_1 \times E_2$ of non-isogenous elliptic curves. Note that when $A$ is supersingular, there is nothing to show so it suffices to consider the ordinary and almost ordinary cases. Let $\alpha_i$ and $\oalpha_i$ be the Frobenius eigenvalues of $E_i$ for each $i = 1, 2$. \subsubsection{Nonsimple ordinary abelian surfaces} In this case, both $E_1$ and $E_2$ are ordinary elliptic curves and the Frobenius eigenvalues $\alpha_i,\oalpha_i$ are not real, and therefore $\cclass{A}$ contains complex conjugation. It is now easy to see that the Galois group is $\wl{V4.4.nt.a.1}$ if $\QQ(\alpha_1) \neq \QQ(\alpha_2)$ and $\wl{C2.4.nt.a.1}$ otherwise. \begin{remark} In fact, using Lemma 5.3 of \cite{KrajicekScanlon00}, it is possible to show that in this case, the two elliptic curves are geometrically isogeneous if and only if the Galois group is $\wl{C2.4.nt.a.1}$, but we will not need this. \end{remark} \subsubsection{Nonsimple almost ordinary abelian surfaces} In this case we may assume without loss of generality that $E_1$ is ordinary and $E_2$ is supersingular. First note that $\cclass{A}$ cannot be the $w_A$-conjugate to $\wl{C2.4.nt.b.1}$ or $\wl{C1.4.nt.a.1}$ since in this case the Frobenius eigenvalues of $E_1$ are fixed by $G_A$, which cannot occur. \begin{lemma} Let $E_1$ and $E_2$ be elliptic curves over $\FF_q$. Suppose that $E_1$ is ordinary and $E_2$ is supersingular. Then, the corresponding number fields satisfy $K_1 \cap K_2 = \QQ$. \end{lemma} \begin{proof} If $K_2 = \QQ$ there is nothing to show. Suppose that this is not the case, and assume in search of a contradiction that $K_1 = K_2 = K$. Then $\alpha_1 = u \alpha_2$ for $u \in K$ of absolute value $1$. Note that $u$ can't be a root of unity, since that would imply that $\alpha_1$ is a supersingular $q$-Weil number. Since $\alpha_2$ is a supersingular $q$-Weil number, some power of $u$ is an \emph{algebraic integer}. This implies that $u$ is also an algebraic integer which has length $1$ in all complex embeddings, and thus $u \in \OO_K^\times$. But $K$ is quadratic imaginary, implying that $u$ is a root of unity, a contradiction. \end{proof} \FloatBarrier \section{Abelian threefolds}\label{sec:threefolds} \begin{theorem} \label{thm:main-thm-3folds} Let $A$ be a simple abelian threefold. \begin{itemize} \item \thmemph{Permutation representations in $W_6$}. If $A$ has permutation representation contained in $W_6$, then the possible images of the weighted permutation representation associated to a simple abelian threefold is given in Tables~\ref{tab:threefoldA}--\ref{tab:threefoldE}. Each table corresponds to one Newton polygon. The permutation representation determines whether $A$ is geometrically simple or not, and this information can be found in the tables. \item \thmemph{Permutation representations in $W_2$}. If the permutation representation of $A$ is not contained in $W_6$, then it is contained in $W_2$. In this case $e_A = 1$, where $e_A$ is as in the statement of the Honda Tate--theorem. Such abelian varieties have Galois group $C_2$, angle rank $1$, and Newton polygon type $(D)$. They are geometrically simple. \end{itemize} \end{theorem} The second point follows from \cite{Xing1994}; see \cite[Theorem 6.1.1, Lemma 6.1.2]{APBS2023} for more detail. In the remainder of this section we assume all abelian threefolds in question are simple and have permutation representation contained in $W_6$. We first classify the weighted permutation representations that occur, and then in \Cref{subsec:geom-simple} classify whether weighted permutation representations that occur are geometrically simple or not. The rest of this section is dedicated to proving \Cref{thm:main-thm-3folds}. \subsection{Signed permutations on three elements}\label{sec:signed-permutations-3} The following lemma follows by a direct calculation in \texttt{Magma}. See the file \href{https://github.com/sarangop1728/Galois-Frob-Polys/blob/main/src/W2d-subgroups.m}{\texttt{src/W2d-subgroups.m}} in our \GitHub repository \cite{OurElectronic}. \begin{lemma} \label{prop:trans-subs} There are exactly $10$ transitive subgroups in $W_6$ which contain the complex conjugation element $\iota \in W_6$, and each is listed in \Cref{tab:W4-and-W6-subgroups}. These $10$ subgroups are contained in exactly $4$ $W_6$-conjugacy classes, namely those of: \begin{enumerate} \item $W_6$, \item $C_2 \wr C_3$ (transitive label \texttt{6T6}) generated by $(123)(\bar{1}\bar{2}\bar{3})$, $(1\bar{1})$, $(2\bar{2}),$ and $(3\bar{3})$, \item $D_6$ generated by $(123)(\bar{1}\bar{2}\bar{3})$ and $(12)(\bar{1}\bar{2})$, \item $C_6$ generated by $(123\bar{1}\bar{2}\bar{3})$. \end{enumerate} Moreover, for each possible Newton polygon of an abelian threefold $A$, the $w_A$-conjugacy classes of transitive subgroups of $W_6$ are recorded in Tables~\ref{tab:threefoldA}--\ref{tab:threefoldE}. \end{lemma} Similarly to the case of surfaces, to prove \Cref{thm:main-thm-3folds} it suffices to show that: \begin{enumerate} \item the cases we claim do not occur, actually do not occur; and \item in the cases that do occur, the weighted permutation representation determines whether the abelian surface is geometrically simple (this is in \Cref{subsec:geom-simple}). \end{enumerate} In the remaining cases, we provide an example in Tables~\ref{tab:threefoldA}--\ref{tab:threefoldE} which realizes the given weighted permutation representation. We now prove $(1)$. \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-3folds}}{Theorem ??} in the ordinary case}\label{sec:o-3} In this case, the table shows that every possible weighted permutation representation occurs, so there is nothing to prove. \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-3folds}}{Theorem ??} in the almost ordinary case}\label{sec:ao-3} We first classify weighted permutation representations that occur, and then give proofs of geometric simplicity. The following two lemmas complete the classification of weighted permutation representations in the case of simple almost ordinary abelian threefolds. \begin{lemma} \label{lem:ao-not-C6} An almost ordinary abelian threefold cannot have Galois group $C_6$. \end{lemma} \begin{proof} Suppose for the sake of contradiction that $A$ is an almost ordinary abelian threefold with Galois group $G_A \cong C_6$. Note that all weighted permutation representations of $C_6$ are conjugate in the almost ordinary case, so it suffices to show that the image of the weighted permutation representation is not conjugate to $\wl{C6.6.t.a.2}$, which is generated by $\sigma = (123\bar{1}\bar{2}\bar{3})$. We now show that this weighted permutation representation doesn't lift to a divisor map with the properties listed in \Cref{prop:div-properties}. By \Cref{prop:div-properties}\ref{enum:div-property-4}, we have $\# \cP_L \mid 3$ where $\cP_L$ is the set of primes of $K = L$ dividing $p$. But then the action of $G_A$ on $\cP_L$ factors through $C_3$ and in particular the sequence \[ (\nu(\alpha_1), \nu(\sigma \alpha_1), ..., \nu(\sigma^5 \alpha_1)) = \left(0, 0, \tfrac{1}{2}, 1, 1, \tfrac{1}{2} \right) \] should be $3$-periodic, a contradiction. \end{proof} \begin{lemma} \label{lem:aoC6} An almost ordinary abelian threefold with Galois group $D_6$ has angle rank $2$. \end{lemma} \begin{proof} From \Cref{tab:threefoldB}, it suffices to show that the weighted permutation representation \wl{D6.6.t.a.1}, which has angle rank $3$, does not occur. Suppose for the sake of contradiction that $A$ is an almost ordinary abelian threefold with weighted permutation representation \wl{D6.6.t.a.1}. We exhibit a nontrivial multiplicative relation between the Frobenius eigenvalues (a contradiction to the angle rank being maximal) -- in particular we show that $\alpha_1\oalpha_3/q$ is a root of unity. \newcounter{claimcount}\setcounter{claimcount}{1} \claim[\theclaimcount]{$\#\cP_L = 6$.} \addtocounter{claimcount}{1} By \Cref{prop:div-properties}\ref{enum:div-property-4}, we have $\#\cP_L \mid 6$. The order $6$ cyclic subgroup of \wl{D6.6.t.a.1} is generated by $\sigma = (1 \bar{2} 3 \bar{1} 2 \bar{3})$. As in the proof of \Cref{lem:aoC6}, because the sequence \[ (\nu(\alpha_1), \nu(\sigma \alpha_1), ..., \nu(\sigma^5 \alpha_1 )) = \left( 0, 1, \tfrac{1}{2}, 1, 0, \tfrac{1}{2} \right) \] is not periodic, we must have $\#\cP_L = 6$. \claim[\theclaimcount]{$\alpha_1 \oalpha_3/q$ is a root of unity.} Let $\mfp_1$ be the prime of $\OO_L$ corresponding to the valuation $\nu$ and let \[ (\mfp_1,\op_2,\mfp_3,\op_1,\mfp_2,\op_3) = (\mfp_1, \sigma(\mfp_1),\dots,\sigma^5(\mfp_1)). \] Because $D_6$ has a unique transitive permutation representation on a $6$ element set, the action of $D_6$ on $\cP_L$ is exactly the rigid symmetries of the hexagon as depicted in \Cref{fig:hexagon}. Because \[ (\nu(\alpha_1), \nu(\sigma \alpha_1), ..., \nu(\sigma^5 \alpha_1)) = \left( 0, 1, \tfrac{1}{2}, 1, 0, \tfrac{1}{2} \right), \] we have \[ \div_A(\alpha_1) = en \big( [\op_3] + \tfrac{1}{2}[\mfp_2] + [\op_1] + \tfrac{1}{2} [\op_2] \big) \] where $q = p^n$ and $e$ is the ramification index of $p$ in $L$. Now consider the element $\tau = (1\bar{3})(2\bar{2})(\bar{1}3) \in \wl{D6.6.t.a.1}$ depicted in \Cref{fig:hexagon}. Note that $\tau$ is \emph{not} complex conjugation and in fact $\tau(\alpha_1) = \oalpha_3$. The action of $D_6$ on $\mathcal{P}_L$ allows us to compute $\div_A(\oalpha_3)$, and it is: \[ \div_A(\oalpha_3) = \tau(\div_A(\alpha_1)) = en \left( [\mfp_1] + \tfrac{1}{2}[\oq_2] + [\mfp_3] + \tfrac{1}{2}[\mfp_2] \right). \] We have $(\oalpha_3\alpha_1)\OO_L = q\OO_L$, so $\oalpha_3\alpha_1 = qu$ for some unit $u \in \OO_L^\times$. Because $\alpha_1$ and $\oalpha_3$ are both $q$-Weil numbers, the unit $u=\alpha_1\oalpha_3/q$ has absolute value $1$ over all complex places, thus it is a root of unity. \end{proof} \begin{figure}[ht] \centering \scalebox{0.99}{ \begin{tikzpicture} \draw[thick] (30:1) node[anchor=south west] {$\op_1$} -- (90:1) node[anchor=south] {$\mfp_3$} -- (150:1) node[anchor=south east] {$\op_2$} -- (210:1) node[anchor=north east] {$\mfp_1$} -- (270:1) node[anchor=north] {$\op_3$} -- (330:1) node[anchor=north west] {$\mfp_2$} -- cycle; \draw[dotted, thick, blue] (60:1.5) -- (240:1.5); \end{tikzpicture} } \caption{The action of $G_A$ on $\cP_L$. The permutation $\tau \in G_A$ is reflection with respect to the dotted line.} \label{fig:hexagon} \end{figure} \FloatBarrier \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-3folds}}{Theorem ??} in the Newton polygon (C) case} The following lemma completes the proof. \begin{lemma} An abelian threefold with Newton polygon (C) cannot have Galois group $G_A \cong C_6$. \end{lemma} \begin{proof} Suppose for the sake of contradiction that $G_A \cong C_6$. It suffices to show that the image of the weighted permutation representation is not conjugate to $\wl{C6.6.t.a.2}$, which is generated by $\sigma = (123\bar{1}\bar{2}\bar{3})$. By \Cref{prop:div-properties}\ref{enum:div-property-4}, we have $\# \cP_L \mid 3$. But then the action of $G_A$ on $\cP_L$ factors through $C_3$ and in particular the sequence \[ (\nu(\alpha_1), \nu(\sigma \alpha_1), ..., \nu(\sigma^5 \alpha_1)) = \left(0, \tfrac{1}{2}, \tfrac{1}{2}, 1, \tfrac{1}{2}, \tfrac{1}{2} \right) \] should be $3$-periodic, a contradiction. \end{proof} \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-3folds}}{Theorem ??} in the Newton polygon (D) case}\label{sec:non-ss-prank0-3} The following lemmas complete the proof. \begin{lemma} A type (D) abelian threefold cannot have weighted permutation representation whose image is $w_A$-conjugate to either $\wl{C6.6.t.a.2}$ or $\wl{D6.6.t.a.4}$. \end{lemma} \begin{proof} Suppose for the sake of contradiction that the image of the weighted permutation representation is $w_A$-conjugate to either \wl{C6.6.t.a.2} or \wl{D6.6.t.a.4}. By \Cref{prop:div-properties}\ref{enum:div-property-4}, we have $\# \cP_L \mid 4$ where $\cP_L$ is the set of primes of $L$ dividing $p$. But then the action of the order $6$ element $\sigma = (123\bar{1}\bar{2}\bar{3}) \in G_A$ on $\cP_L$ factors through $C_2$. But the sequence \[ (\nu(\alpha_1), \nu(\sigma\alpha_1), ..., \nu(\sigma^5\alpha_1)) = \left( \tfrac{1}{3}, \tfrac{1}{3}, \tfrac{1}{3}, \tfrac{2}{3}, \tfrac{2}{3}, \tfrac{2}{3} \right) \] is not $2$-periodic, a contradiction. \end{proof} \subsection{Proof of \texorpdfstring{\Cref{thm:main-thm-3folds}}{Theorem ??} in the supersingular case}\label{sec:ss-3} \begin{lemma} The sextic field generated by the Frobenius eigenvalues of a simple supersingular abelian threefold must be $\QQ(\zeta_7)$ or $\QQ(\zeta_9)$, both of which have Galois group $C_6$. \end{lemma} \begin{proof} We follow the argument in \cite[Proposition~2.1]{NartRitzenthaler08} (note that the characteristic of the base field in \textit{loc. cit.} is $2$). Supersingular abelian varieties have angle rank $0$, so the sextic field $K$ (as defined in \Cref{def:K}) is of the form $K \cong \QQ(\zeta_m \sqrt{q})$ where $\zeta_m$ is a primitive $m^\text{th}$-root of unity. Choose $m$ to be the smallest integer such that $\zeta_m\sqrt{q}$ generates $K$. If $q$ is a square, then the only cyclotomic fields of degree $6$ are $\QQ(\zeta_7)$ or $\QQ(\zeta_9)$, so we are done. Suppose now that $q$ is not a square. If $m$ is odd then $K$ contains the field $\QQ(\zeta_m)$. Thus $[\QQ(\zeta_m) : \QQ] \leq 6$ and $m = 3,7,9$. If $m = 3$ then $[K : \QQ] \leq [\QQ(\zeta_3,\sqrt{q}) : \QQ] = 4$. If $m = 7,9$, then $6 = [K : \QQ] \geq [\QQ(\zeta_m) : \QQ] = 6$, so $K = \QQ(\zeta_m)$. Now suppose $m = 2n$ is even. Since $K$ contains $\QQ(\zeta_m^2) = \QQ(\zeta_{n})$ and in particular we have $n = 3,4,6,7,9,14$. If $n = 7,9,14$ then $6 = [K : \QQ] \geq [\QQ(\zeta_{n}) : \QQ] = 6$, so $K = \QQ(\zeta_{n})$. If $n = 3$, then $[K : \QQ] \leq [\QQ(\zeta_6, \sqrt{q}) : \QQ] = 4$, contradiction. If $n = 6$, then $K = \QQ(\zeta_{12} \sqrt{q})$. However, this is a quadratic extension of the quadratic field $\QQ(\zeta_6)$, so $[K : \QQ] = 4$, which is a contradiction. \end{proof} \subsection{When are simple abelian threefolds geometrically simple?} \label{subsec:geom-simple} By the Honda--Tate theorem every supersingular abelian variety is not geometrically simple. Moreover, by \Cref{coro:not-geom-simple=>angle-rank-1} a simple ordinary threefold which is neither supersingular nor geometrically simple must be ordinary. Suppose that $A$ is a simple ordinary abelian threefold. In this case all possible weighted permutation representations occur, and it suffices to show that if $A$ is geometrically simple if it has angle rank $3$ and not geometrically simple if it has angle rank $1$. \begin{lemma} \label{lem:geom-simple<=>max-angle-rank} Let $A$ be a simple ordinary abelian threefold. Then the angle rank of $A$ is $3$ if $A$ is geometrically simple and $1$ otherwise. \end{lemma} \begin{proof} If $A$ is geometrically simple, then $\delta_A = 3$ by \cite[Lemma~6.2.2]{APBS2023}. If $A$ is not geometrically simple then $A$ has angle rank $1$ by \Cref{coro:not-geom-simple=>angle-rank-1}. \end{proof} \FloatBarrier \section{Inverse Galois Problems}\label{sec:questions} We state a slight refinement of a conjecture Dupuy, Kedlaya, Roe, and Vincent \cite[Conjecture 2.7]{DupuyKedlayaRoeVincent21}. \begin{conjecture}[Refined inverse Galois problem for abelian varieties] \label{conj:IGP} Fix a prime number $p$ and let $G \subset W_{2d}$ be a transitive subgroup containing the complex conjugation element. Then: \begin{enumerate} \item there exists an integer $r \geq 1$ and a simple abelian variety $A/\FF_{p^r}$ of dimension $d$ such that $G$ is $w_A$-conjugate to the image of the weighted permutation representation associated to $A$, and \item the abelian variety $A$ may be taken to be ordinary. \end{enumerate} In particular, $G$ is isomorphic to the Galois group of some abelian variety $A$. \end{conjecture} We note that \Cref{conj:IGP} is a very strong statement. In particular it implies the inverse Galois problem holds even when we are restricted to totally real fields. \begin{proposition} Assume \Cref{conj:IGP}, then the inverse Galois problem holds for totally real fields. More precisely, let $d \geq 1$ and let $G \subset S_d$ be a transitive subgroup, then $G$ is the Galois group of a polynomial $P^+(T)$ of degree $d$ over $\QQ$ and moreover the splitting field of $P^+(T)$ may be taken to be totally real. \end{proposition} \begin{proof} Let $G \subset S_d$ be a transitive subgroup, and consider the group $\widetilde{G} = C_2 \wr G$ equipped with its natural embedding $\widetilde{G} \hookrightarrow W_{2d}$. But $\widetilde{G}$ is a transitive subgroup of $W_{2d}$ which contains the complex conjugation element, so by assumption it occurs as the Galois group of a $q$-Weil polynomial $P_A(T)$. Let $P_A^+(T)$ be the \cdef{trace polynomial} of $P_A(T)$ defined by the equation \begin{equation} P_A^+(T) = \prod_{\alpha}(T - (\alpha + \oalpha)), \end{equation} where $\alpha$ ranges over the roots of $P_A(T)$. It follows by construction that $\Gal(P_A^+(T))$ is isomorphic to $G$ and that every root of $P_A^+(T)$ is real. \end{proof} \appendix\section{Tables} In \Cref{tab:W4-and-W6-subgroups} we record our labelling convention for subgroups of $W_4$ and $W_6$. In particular, we list every subgroup of $W_4$ and every transitive subgroup of $W_6$ containing complex conjugation. \subsection{Abelian surfaces} Tables~\ref{tab:dim2-A}--\ref{tab:dim2-C-ns} record the possible subgroups $G \subset W_4$ which may occur as the ($w_A$-conjugacy class of the) images of weighted permutation representations associated to abelian surfaces $A$. For each case which does occur, we provide an example from the LMFDB~\cite{lmfdb}. The tables are separated by Newton polygon (according to the conventions in \Cref{fig:flowchart2-simple}) and by whether $A$ is simple. \subsubsection{Simple abelian surfaces} These cases are treated in Tables~\ref{tab:dim2-A}--\ref{tab:dim2-C}. We do not list intransitive subgroups which do not occur as the image of the weighted permutation representation associated to a simple abelian surface -- note that an intransitive subgroup can only occur for the supersingular abelian surface in \Cref{lem:real-eigenvalues}\ref{enum:complex-ECs}. In each case we record whether every isogeny class of abelian surfaces with the recorded image of the weighted permutation representation is geometrically simple. \subsubsection{Non-simple abelian surfaces} These cases are treated in Tables~\ref{tab:dim2-A-ns}--\ref{tab:dim2-C-ns}. Since transitive subgroups of $W_4$ cannot occur as the image of the weighted permutation representation associated to an abelian surface, we do not record them. \subsection{Abelian threefolds} Tables~\ref{tab:threefoldA}--\ref{tab:threefoldE} record the possible subgroups $G \subset W_6$ which may occur as the ($w_A$-conjugacy class of the) images of weighted permutation representations associated to simple abelian threefolds $A$. For each case which does occur, we provide an example from the LMFDB~\cite{lmfdb}. The tables are separated by Newton polygon (according to the conventions in \Cref{fig:flowchart3}). \FloatBarrier { \vfill \begin{table}[ht] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.15} \centering \begin{tabular}{\|c\|C{4.5cm}\|\|c\|C{4.5cm}\|} \hline \rowcolor{headercolor} Label of $G$ & Generators of $G$ & Label of $G$ & Generators of $G$ \\ \hline \mkwl{W4.4.t.a.1} & $W_4$ & \mkwl{W6.6.t.a.1} & $W_6$ \\ \hline \mkwl{V4.4.t.a.1} & $\iota, (12)(\bar{1}\bar{2})$ & \mkwl{6T6.6.t.a.1} & $(123)(\bar{1}\bar{2}\bar{3})$, $(1\bar{1})$, $(2\bar{2})$, $(3\bar{3})$ \\ \hline \mkwl{C4.4.t.a.1} & $(12\bar{1}\bar{2})$ & \mkwl{D6.6.t.a.1} & $(1\bar{2}3\bar{1}2\bar{3}), (23)(\bar{2}\bar{3})$ \\ \hline \mkwl{V4.4.nt.a.1} & $(1\bar{1}), (2\bar{2})$ & \mkwl{D6.6.t.a.2} & $(12\bar{3}\bar{1}\bar{2}3)$, $(23)(\bar{2}\bar{3})$ \\ \hline \mkwl{C2.4.nt.a.1} & $\iota$ & \mkwl{D6.6.t.a.3} & $(1\bar{2}\bar{3} \bar{1}23)$, $(2\bar{3})(\bar{2}3)$ \\ \hline \mkwl{C2.4.nt.b.1} & $(1\bar{1})$ & \mkwl{D6.6.t.a.4} & $(123\bar{1}\bar{2}\bar{3})$, $(2\bar{3})(\bar{2}3)$ \\ \hline \mkwl{C2.4.nt.b.2} & $(2\bar{2})$ & \mkwl{C6.6.t.a.1} & $(1\bar{2}3\bar{1}2\bar{3})$ \\ \hline \mkwl{C2.4.nt.c.1} & $(12)(\bar{1}\bar{2})$ & \mkwl{C6.6.t.a.2} & $(123\bar{1}\bar{2}\bar{3})$ \\ \hline \mkwl{C2.4.nt.c.2} & $(1\bar{2})(\bar{1}2)$ & \mkwl{C6.6.t.a.3} & $(12\bar{3}\bar{1}\bar{2}3)$ \\ \hline \mkwl{C1.4.nt.a.1} & $\operatorname{id}$ &\mkwl{C6.6.t.a.4} & $(1\bar{2}\bar{3}\bar{1}23)$ \\ \hline \end{tabular} \caption{Labels for subgroups of $W_4$ (left) and subgroups of $W_6$ (right).} \label{tab:W4-and-W6-subgroups} \end{table} \begin{table}[ht] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example(s) \\ \hline \wl{W4.4.t.a.1} & $2$ & Yes & Yes & \avlink{2.2.ac_d} \\ \hline \wl{V4.4.t.a.1} & $1$ & Yes & No & \avlink{2.2.ad_f} \\ \hline \wl{C4.4.t.a.1} & $2$ & Yes & Yes & \avlink{2.3.ad_f} \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to a simple ordinary abelian surface (Newton polygon (A) in \Cref{fig:flowchart2-simple}).} \label{tab:dim2-A} \end{table} \vfill } \begin{table}[ht] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example(s) \\ \hline \wl{W4.4.t.a.1} & $2$ & Yes & Yes & \avlink{2.2.ab_a} \\ \hline \wl{V4.4.t.a.1} & $2$ & No & & \\ \hline \wl{C4.4.t.a.1} & $2$ & No & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple almost ordinary abelian surfaces (Newton polygon (B) in \Cref{fig:flowchart2-simple}).} \label{tab:dim2-B} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example(s) \\ \hline \wl{W4.4.t.a.1} & $2$ & No & & \\ \hline \wl{V4.4.t.a.1} & $0$ & Yes & No & \avlink{2.2.ac_c} \\ \hline \wl{C4.4.t.a.1} & $0$ & Yes & No & \avlink{2.4.ac_e} \\ \hline \wl{C2.4.nt.c.1} & \multirow{2}{}{$0$} & \multirow{2}{}{Yes} & \multirow{2}{}{No} & \multirow{2}{}{\avlink{2.2.a_ae}} \\ \wl{C2.4.nt.c.2} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple supersingular abelian surfaces (Newton polygon (C) in \Cref{fig:flowchart2-simple}).} \label{tab:dim2-C} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Example(s) \\ \hline \wl{V4.4.nt.a.1} & $1$ & Yes & \avlink{2.3.ad_i} \\ \hline \wl{C2.4.nt.a.1} & $1$ & Yes & \avlink{2.2.a_d} \\ \hline \wl{C2.4.nt.b.1} & \multirow{2}{}{$1$} & \multirow{2}{}{No} & \\ \wl{C2.4.nt.b.2} & & & \\ \hline \wl{C2.4.nt.c.1} & $1$ & No & \\ \hline \wl{C2.4.nt.c.2} & $1$ & No & \\ \hline \wl{C1.4.nt.a.1} & $0$ & No & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to non-simple ordinary abelian surfaces (Newton polygon (A) in \Cref{fig:flowchart2-non-simple}).} \label{tab:dim2-A-ns} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Example(s) \\ \hline \wl{V4.4.nt.a.1} & $1$ & Yes & \avlink{2.2.ad_g} \\ \hline \wl{C2.4.nt.a.1} & $1$ & No & \\ \hline \wl{C2.4.nt.b.1} & $1$ & No & \\ \hline \wl{C2.4.nt.b.2} & $1$ & Yes & \avlink{2.4.ah_u} \\ \hline \wl{C2.4.nt.c.1} & $1$ & No & \\ \hline \wl{C2.4.nt.c.2} & $1$ & No & \\ \hline \wl{C1.4.nt.a.1} & $0$ & No & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to a non-simple almost ordinary abelian surfaces (Newton polygon (B) in \Cref{fig:flowchart2-non-simple}).} \label{tab:dim2-B-ns} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Example(s) \\ \hline \wl{V4.4.nt.a.1} & $0$ & Yes & \avlink{2.2.ac_e} \\ \hline \wl{C2.4.nt.a.1} & $0$ & Yes & \avlink{2.2.a_a} \\ \hline \wl{C2.4.nt.b.1} & \multirow{2}{}{$0$} & \multirow{2}{}{Yes} & \multirow{2}{}{\avlink{2.4.ag_q}} \\ \wl{C2.4.nt.b.2} & & & \\ \hline \wl{C2.4.nt.c.1} & \multirow{2}{}{$0$} & \multirow{2}{}{No} & \\ \wl{C2.4.nt.c.2} & & & \\ \hline \wl{C1.4.nt.a.1} & $0$ & Yes & \avlink{2.4.a_ai} \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to non-simple supersingular abelian surfaces (Newton polygon (C) in \Cref{fig:flowchart2-non-simple}).} \label{tab:dim2-C-ns} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example(s) \\ \hline \wl{W6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.2.ad_f_ah} \\ \hline \wl{6T6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.2.ad_g_aj} \\ \hline \wl{D6.6.t.a.1} & $1$ & Yes & No & \avlink{3.2.a_a_ad} \\ \hline \wl{D6.6.t.a.2} & \multirow{3}{}{$3$} & \multirow{3}{}{Yes} & \multirow{3}{}{Yes} & \multirow{3}{}{\avlink{3.2.ac_a_d}} \\ \wl{D6.6.t.a.3} & & & & \\ \wl{D6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.1} & $1$ & Yes & No & \avlink{3.2.ae_j_ap} \\ \hline \wl{C6.6.t.a.2} & \multirow{3}{}{$3$} & \multirow{3}{}{Yes} & \multirow{3}{}{Yes} & \multirow{3}{}{\avlink{3.7.ak_bw_afv}} \\ \wl{C6.6.t.a.3} & & & & \\ \wl{C6.6.t.a.4} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple ordinary abelian threefolds (Newton polygon (A) in \Cref{fig:flowchart3}).} \label{tab:threefoldA} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w_A$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example \\\hline \wl{W6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.2.ab_ab_c} \\ \hline \wl{6T6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.4.ac_ab_g} \\ \hline \wl{D6.6.t.a.1} & \multirow{2}{}{$3$} & \multirow{2}{}{No} & & \\ \wl{D6.6.t.a.3} & & & & \\ \hline \wl{D6.6.t.a.2} & \multirow{2}{}{$2$} & \multirow{2}{}{Yes} & \multirow{2}{}{Yes} & \multirow{2}{}{\avlink{3.2.ac_b_a}} \\ \wl{D6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.1} & \multirow{2}{}{$3$} & \multirow{2}{}{No} & & \\ \wl{C6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.2} & \multirow{2}{}{$2$} & \multirow{2}{}{No} & & \\ \wl{C6.6.t.a.3} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple almost ordinary abelian threefold (Newton polygon (B) in \Cref{fig:flowchart3}).} \label{tab:threefoldB} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example \\ \hline \wl{W6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.2.ab_a_a} \\ \hline \wl{6T6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.4.ab_c_a} \\ \hline \wl{D6.6.t.a.1} & \multirow{4}{}{$3$} & \multirow{4}{}{Yes} & \multirow{4}{}{Yes} & \multirow{4}{}{\avlink{3.4.ab_a_ae}} \\ \wl{D6.6.t.a.2} & & & & \\ \wl{D6.6.t.a.3} & & & & \\ \wl{D6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.1} & \multirow{4}{}{$3$} & \multirow{4}{}{No} & & \\ \wl{C6.6.t.a.2} & & & & \\ \wl{C6.6.t.a.3} & & & & \\ \wl{C6.6.t.a.4} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to a simple abelian threefolds with Newton polygon (C) in \Cref{fig:flowchart3}.} \label{tab:threefoldC} \end{table} \begin{table}[p] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example \\ \hline \wl{W6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.2.ac_c_ac} \\ \hline \wl{6T6.6.t.a.1} & $3$ & Yes & Yes & \avlink{3.3.ad_j_ap} \\ \hline \wl{D6.6.t.a.1} & $1$ & Yes & Yes & \avlink{3.2.a_a_ac} \\ \hline \wl{D6.6.t.a.2} & \multirow{3}{}{$3$} & \multirow{3}{}{No} & & \\ \wl{D6.6.t.a.3} & & & & \\ \wl{D6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.1} & $1$ & Yes & Yes & \avlink{3.7.a_a_abj} \\ \hline \wl{C6.6.t.a.2} & \multirow{3}{}{$3$} & \multirow{3}{}{No} & & \\ \wl{C6.6.t.a.3} & & & & \\ \wl{C6.6.t.a.4} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple abelian threefolds with Newton polygon (D) in \Cref{fig:flowchart3}.} \label{tab:threefoldD} \end{table} \FloatBarrier \begin{table}[H] \setlength{\arrayrulewidth}{0.3mm} \setlength{\tabcolsep}{5pt} \renewcommand{\arraystretch}{1.2} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \rowcolor{headercolor} $w$-conjugacy class & Angle rank & Occurs & Geometrically simple & Example \\ \hline \wl{W6.6.t.a.1} & $0$ & No & & \\ \hline \wl{6T6.6.t.a.1} & $0$ & No & & \\ \hline \wl{D6.6.t.a.1} & \multirow{4}{}{$0$} & \multirow{4}{}{No} & & \\ \wl{D6.6.t.a.2} & & & & \\ \wl{D6.6.t.a.3} & & & & \\ \wl{D6.6.t.a.4} & & & & \\ \hline \wl{C6.6.t.a.1} & \multirow{4}{}{$0$} & \multirow{4}{}{Yes} & \multirow{4}{}{No} & \multirow{4}{}{\avlink{3.3.a_a_aj}} \\ \wl{C6.6.t.a.2} & & & & \\ \wl{C6.6.t.a.3} & & & & \\ \wl{C6.6.t.a.4} & & & & \\ \hline \end{tabular} \caption{The images of the weighted permutation representations associated to simple almost ordinary abelian threefolds (Newton polygon (E) in \Cref{fig:flowchart3}).} \label{tab:threefoldE} \end{table} \FloatBarrier \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} MR } \providecommand{\MRhref}[2]{ \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{DKRV21b} \bibitem[ABS24]{APBS2023} Santiago {Arango-Pi{\~n}eros}, Deewang {Bhamidipati}, and Soumya {Sankar}, \emph{Frobenius {D}istributions of {L}ow {D}imensional {A}belian {V}arieties {O}ver {F}inite {F}ields}, Int. 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2412.03500v2	http://arxiv.org/abs/2412.03500v2	$(σ, τ)$-Derivations of Number Rings with Coding Theory Applications	\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{epstopdf}\usepackage{hyperref} \hypersetup{ colorlinks = true, urlcolor = blue, linkcolor = blue, citecolor = red } \usepackage[caption=false]{subfig} \usepackage{amsmath} \usepackage{csquotes} \usepackage{tikz} \usepackage{upgreek} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsfonts} \numberwithin{equation}{section} \usepackage{booktabs,caption} \usepackage[flushleft]{threeparttable} \usepackage{stmaryrd} \usepackage{wasysym} \usepackage{makecell} \usepackage{float} \usepackage{orcidlink} \usepackage{longtable} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{notation}[theorem]{Notation} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \usepackage{nccmath} \usepackage{mathtools} \usepackage[reftex]{theoremref} \usepackage{tikz-cd} \usepackage[left=1in, right=1in, top=1in, bottom=1in]{geometry} \setcounter{MaxMatrixCols}{20} \begin{document} \title{$(\sigma, \tau)$-Derivations of Number Rings with Coding Theory Applications} \author{Praveen Manju and Rajendra Kumar Sharma} \date{} \maketitle \begin{center} \noindent{\small Department of Mathematics, \\Indian Institute of Technology Delhi, \\ Hauz Khas, New Delhi-110016, India$^{1}$} \end{center} \footnotetext[1]{{\em E-mail addresses:} \url{[email protected]}(Corresponding Author: Praveen Manju), \url{[email protected]}(Rajendra Kumar Sharma).} \medskip \begin{abstract} In this article, we study $(\sigma, \tau)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-algebras. We begin by characterizing all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic number field. Then we characterize all $(\sigma, \tau)$-derivations of the ring of algebraic integers $\mathbb{Z}[\zeta]$ of a $p^{\text{th}}$-cyclotomic number field $\mathbb{Q}(\zeta)$ ($p$ odd rational prime and $\zeta$ a primitive $p^{\text{th}}$-root of unity). We also conjecture (using SAGE and MATLAB) an \enquote{if and only if} condition for a $(\sigma, \tau)$-derivation $D$ on $\mathbb{Z}[\zeta]$ to be inner. We further characterize all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the bi-quadratic number ring $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$ ($m$, $n$ distinct square-free rational integers). In each of the above cases, we also determine the rank and an explicit basis of the derivation algebra consisting of all $(\sigma, \tau)$-derivations of the number ring. Finally, we give the applications of the above work in coding theory by giving the notion of a Hom-IDD code. \end{abstract} \textbf{Keywords:} $(\sigma, \tau)$-derivation, inner $(\sigma, \tau)$-derivation, number ring, ring of algebraic integers, quadratic, cyclotomic, bi-quadratic \textbf{Mathematics Subject Classification (2010):} Primary: 13N15, 11R04; Secondary: 11C20, 11R11, 11R16, 11R18. \section{Introduction}\label{section 1} Let us denote a commutative ring with unity by $R$, an algebra over $R$ by $\mathcal{A}$, and two different non-zero $R$-algebra endomorphisms of $\mathcal{A}$ by $\sigma$, $\tau$. An $R$-linear map $d:\mathcal{A} \rightarrow \mathcal{A}$ that satisfies $d(\alpha \beta) = d(\alpha) \beta + \alpha d(\beta)$ for every $\alpha, \beta \in \mathcal{A}$, is called a derivation on $\mathcal{A}$. It is called inner if $d(\alpha) = \beta \alpha - \alpha \beta$ for all $\alpha \in \mathcal{A}$, for some $\beta \in \mathcal{A}$. An $R$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ that satisfies $D(\alpha \beta) = D(\alpha) \tau(\beta) + \sigma(\alpha) D(\beta)$ for all $\alpha, \beta \in \mathcal{A}$, is called a $(\sigma, \tau)$-derivation on $\mathcal{A}$. Again, it is called inner if $D(\alpha) = \beta \tau(\alpha) - \sigma(\alpha) \beta$ for every $\alpha \in \mathcal{A}$, for some $\beta \in \mathcal{A}$. $\mathcal{D}_{(\sigma, \tau)}(\mathcal{A})$ denotes the set of all $(\sigma, \tau)$-derivations on $\mathcal{A}$. Defining componentwise sum and module action, $\mathcal{D}_{(\sigma, \tau)}(\mathcal{A})$ becomes an $R$- as well as $\mathcal{A}$-module. If $1$ is the unity in $\mathcal{A}$, $\sigma$, $\tau$ are unital, that is, $\sigma(1) = \tau(1) = 1$ and $D$ is a $(\sigma, \tau)$-derivation of $\mathcal{A}$, then $D(1) = 0$. We shall need some definitions and results from algebraic number theory for which we refer the reader to \cite{Marcus2018}, \cite{IanStewart2002}. Still, we summarize a few, as follows: A complex number is an algebraic number if it is a zero of some non-constant polynomial over $\mathbb{Q}$. A number field is a finite field extension of $\mathbb{Q}$ that contains algebraic numbers. A number ring is any subring of a number field. An algebraic integer is a complex number that is a zero of some monic polynomial over $\mathbb{Z}$. The ring of algebraic integers of a number field $K$, denoted by $O_{K}$, is defined as $O_{K} = K \cap \mathbb{B}$, where $\mathbb{B}$ is the ring of all algebraic integers. Thus $O_{K}$ contains precisely all those algebraic integers which belong to $K$. A subring of $O_{K}$ which is also a $\mathbb{Z}$-module of rank $[K: \mathbb{Q}]$ is called an order of $K$. A set of algebraic numbers $\{\alpha_{1}, \alpha_{2}, ..., \alpha_{r}\}$ is called an integral basis of $K$ (or $O_{K}$) if each element in $O_{K}$ is uniquely expressible as a $\mathbb{Z}$-linear combination of $\alpha_{1}, \alpha_{2}, ..., \alpha_{r}$. Every integral basis is a $\mathbb{Q}$-basis. An integral basis having form $\{1, \theta, ..., \theta^{n-1}\}$ for some $\theta \in O_{K}$, is called a power basis of $K$. A number field $K$ that has a power basis is called monogenic. A number field $K$ that has degree $2$ over $\mathbb{Q}$ is called quadratic. A number field that has the form $K = \mathbb{Q}(\zeta)$, where $\zeta = e^{2 \pi i / m}$ ($m \geq 1$) is a primitive complex $m^{\text{th}}$ root of unity, is called an $m^{\text{th}}$ cyclotomic field. We state below some results that we may need later. \begin{theorem}[{\cite[Theorem 2.2]{IanStewart2002}}]\th\label{theorem 1.1} Let $K$ be a number field. Then $K = \mathbb{Q}(\theta)$ for some algebraic number $\theta$, or in other words, number fields are finite simple extensions of $\mathbb{Q}$. In fact, $K = \mathbb{Q}(\theta)$ for some algebraic integer $\theta$. \end{theorem} \begin{theorem}[{\cite[Theorem 2.16]{IanStewart2002}}]\th\label{theorem 1.2} A number field $K$ always has an integral basis. Further, the additive group of $O_{K}$ is free abelian having rank equal to the degree of $K$. \end{theorem} \begin{theorem}[{\cite[Proposition 3.1]{IanStewart2002}}]\th\label{theorem 1.3} Every quadratic field is of the form $\mathbb{Q}(\sqrt{d})$ for some square-free rational integer $d$. \end{theorem} \begin{theorem}[{\cite[Theorem 3.2]{IanStewart2002}}]\th\label{theorem 1.4} Let $K = \mathbb{Q}(\sqrt{d})$ be a quadratic field. Then \begin{itemize} \item[(i)] $O_{K} = \mathbb{Z}[\sqrt{d}]$ when $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, \item[(ii)] $O_{K} = \mathbb{Z}[\frac{1 + \sqrt{d}}{2}]$ when $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. \end{itemize} \end{theorem} \begin{theorem}[{\cite[Theorem 3.3]{IanStewart2002}}]\th\label{theorem 1.5} Let $K = \mathbb{Q}(\sqrt{d})$ be a quadratic field. Then $\mathcal{B}$ is an integral basis of $K$, where \begin{itemize} \item[(a)] $\mathcal{B} = \{1, \sqrt{d}\}$ when $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. \item[(b)] $\mathcal{B} = \{1, \frac{1 + \sqrt{d}}{2}\}$ when $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. \end{itemize} \end{theorem} \begin{theorem}[{\cite[Lemma 3.4]{IanStewart2002}}]\th\label{theorem 1.6} If $p$ is an odd prime and $\zeta$ is a primitive $p^{\text{th}}$ root of unity, then the minimal polynomial of $\zeta$ over $\mathbb{Q}$ is $\phi_{p}(x) = x^{p-1} + x^{p-2} + ... + 1$. Hence $[\mathbb{Q}(\zeta):Q] = p-1$. \end{theorem} \begin{theorem}[{\cite[Theorem 3.5]{IanStewart2002}}]\th\label{theorem 1.7} Let $p$ be an odd prime and $K = \mathbb{Q}(\zeta)$ be a $p^{\text{th}}$ cyclotomic field. Then $O_{K} = \mathbb{Z}[\zeta]$. Hence, $\{1, \zeta, \zeta^{2}, ..., \zeta^{p-2}\}$ is an integral basis of $K = \mathbb{Q}(\zeta)$. \end{theorem} \begin{theorem}[{\cite[Chapter 2]{Marcus2018}}]\th\label{theorem 1.8} Let $m \in \mathbb{N}$ and $K = \mathbb{Q}(\zeta)$ be an $m^{\text{th}}$ cyclotomic field, then $O_{K} = \mathbb{Z}[\zeta]$. \end{theorem} Derivations have been studied in various fields like algebra, functional analysis, Fourier analysis, measure theory, operator theory, harmonic analysis, differential equations, differential geometry, etc. They play an essential role in many important branches of mathematics and physics. The notion of derivation in rings has been applied to and studied in various algebras to produce their theory of derivations. For example, BCI-algebras \cite{Muhiuddin2012}, von Neumann algebras \cite{Brear1992}, incline algebras which have many applications \cite{Kim2014}, MV-algebras \cite{KamaliArdakani2013}, \cite{Mustafa2013}, Banach algebras \cite{Raza2016}, lattices that have a very crucial role in various fields like information theory: information recovery, management of information access and cryptanalysis \cite{Chaudhry2011}. Differentiable manifolds, operator algebras, $\mathbb{C}^{}$-algebras, and representation theory of Lie groups are continued to be studied using derivations \cite{Klimek2021}. Derivations on rings and algebras help in the study of their structure. For a historical account and more applications of derivations, we refer the reader to \cite{MohammadAshraf2006}, \cite{Atteya2019}, \cite{Haetinger2011}. The idea of a $(s_{1}, s_{2})$-derivation was introduced by Jacobson \cite{Jacobson1964}. These derivations were later on commonly called as $(\sigma, \tau)$ or $(\theta, \phi)$-derivation. These derivations have been highly studied in prime and semiprime rings and have been principally used in solving functional equations \cite{Brear1992}. Derivations, and especially $(\sigma, \tau)$-derivations of rings have various applications in coding theory \cite{Boucher2014}, \cite{Creedon2019}. For more applications of $(\sigma, \tau)$-derivations, we refer the reader to \cite{AleksandrAlekseev2020}. There is no literature available on twisted derivations of number rings except in \cite{Chaudhuri} where the author studies $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic number field. In this article, we mainly study $(\sigma, \tau)$-derivations of number rings. We partition the article into three sections. In Section \ref{section 2}, we first obtain results on $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of a commutative unital algebra $\mathcal{A}$ over a commutative ring $R$ with unity $1$. We find a necessary condition for an $R$-linear map $D$ on $\mathcal{A}$ to be a $(\sigma, \tau)$-derivation and give counterexamples showing that the condition is not sufficient. This motivates us to study these derivations in certain number rings where this condition is both necessary as well as sufficient. We also determine a necessary and sufficient condition for a $(\sigma, \tau)$-derivation on $\mathcal{A}$ to be inner. In Section \ref{section 3}, we apply our results obtained in Section \ref{section 2} to study $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-algebras. Section \ref{section 3} is further subdivided into three subsections. In Subsection \ref{subsection 3.1}, we study $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic number field. We obtain necessary and sufficient conditions for a $\mathbb{Z}$-linear map $D$ on the rings of algebraic integers of a quadratic field to be a $(\sigma, \tau)$-derivation. We also classify the inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic field. In Subsection \ref{subsection 3.2}, we study $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a cyclotomic number field. We obtain necessary and sufficient conditions for a $\mathbb{Z}$-linear map $D$ on the rings of algebraic integers $\mathbb{Z}[\zeta]$ of a $p^{\text{th}}$ cyclotomic field $\mathbb{Q}(\zeta)$ to be a $(\sigma, \tau)$-derivation. We prove that $\mathcal{D}_{(\sigma, \tau)}(\mathbb{Z}[\zeta])$ is a $\mathbb{Z}$-module having rank $p-1$. Further, in this process, we also propose two conjectures giving a necessary and sufficient condition on a $(\sigma, \tau)$-derivation of $\mathbb{Z}[\zeta]$ to be inner. This is done with the help of SAGE and MATLAB. In Subsection \ref{subsection 3.3}, a beautiful application of \th\ref{lemma 2.1} is given in determining all $(\sigma, \tau)$-derivations of the number ring $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$ for every pair $(\sigma, \tau)$ of distinct ring endomorphisms $\sigma$ and $\tau$ of $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$. As a result, we establish that $D_{(\sigma, \tau)}(\mathbb{Z}[\sqrt{m}, \sqrt{n}])$ is a $\mathbb{Z}$-module having rank $4$. Further, we obtain a necessary and sufficient condition for a $(\sigma, \tau)$-derivation of $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$ to be inner. We discuss the applications of our work in coding theory in Section \ref{section 4} by giving the notion of a Hom-IDD code. In Section \ref{section 5}, we conclude our findings. \section{Some useful results on \texorpdfstring{$(\sigma, \tau)$}{Lg}-derivations of commutative algebras}\label{section 2} $\mathcal{A}$, in this section, denotes a commutative algebra with unity $1$ over a commutative ring $R$ having unity $1$ and $\sigma$, $\tau$ are two different non-zero unital $R$-algebra endomorphisms of $\mathcal{A}$. First, we have an important lemma below. \begin{lemma}\th\label{lemma 2.1} Let $\mathcal{A}$ be of finite rank $n$ as an $R$-module and let $\{\alpha_{1}, \alpha_{2}, ..., \alpha_{n}\}$ be an $R$-basis of $\mathcal{A}$. Then an $R$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ is a $(\sigma, \tau)$-derivation if and only if $$D(\alpha_{i} \alpha_{j}) = D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j})$$ for every $i, j \in \{1, 2, ..., n\}$. \end{lemma} \begin{proof} Let $D:\mathcal{A} \rightarrow \mathcal{A}$ be an $R$-linear map, and $x, y \in \mathcal{A}$. Then $x = \sum_{i=1}^{n} a_{i} \alpha_{i}$ and $y = \sum_{j=1}^{n} b_{j} \alpha_{j}$ for some $a_{i}, b_{j} \in R$ ($i, j \in \{1, 2, ..., n\}$). Observe that $$xy = \left(\sum_{i=1}^{n} a_{i} \alpha_{i}\right) \left(\sum_{j=1}^{n} b_{j} \alpha_{j} \right) = \sum_{i, j} (a_{i}b_{j}) (\alpha_{i} \alpha_{j}),$$ and \begin{eqnarray} D(xy) & = & \sum_{i,j=1}^{n} (a_{i}b_{j}) D(\alpha_{i} \alpha_{j}) = \sum_{i,j=1}^{n} (a_{i}b_{j}) (D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j})) \\ & = & D \left(\sum_{i=1}^{n} a_{i}\alpha_{i}\right) \tau \left(\sum_{j=1}^{n} b_{j} \alpha_{j}\right) + \sigma \left(\sum_{i=1}^{n} a_{i}\alpha_{i}\right) D \left(\sum_{j=1}^{n} b_{j} \alpha_{j}\right) \\ & = & D(x) \tau(y) + \sigma(x) D(y). \end{eqnarray} Hence $D$ is a $(\sigma, \tau)$-derivation. The converse is straightforward. \end{proof} For a non-negative integer $k$, define $S_{k}$ as the set containing all ordered pairs $(i,j)$ for which $i+j = k$. Note that the sets $S_{k} \text{'s}$ are pairwise disjoint. Therefore, $\|\cup_{i=0}^{m} S_{i}\| = \sum_{i=0}^{m} \|S_{i}\| = 1 + 2 + 3 + ... + m + (m+1) = \frac{(m+1)(m+2)}{2}$ for every non-negative integer $m$. \begin{lemma}\th\label{lemma 2.2} For a $(\sigma, \tau)$-derivation $D$ of $\mathcal{A}$, $$D(\alpha^{k}) = \left(\sum_{(i,j) \in S_{k-1}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha),$$ for every $\alpha \in \mathcal{A}$ and for every $k \in \mathbb{N}$. \end{lemma} \begin{proof} We use induction. For $k = 1$, the equality trivially holds. Let the result hold for $k=n$, that is, $D(\alpha^{n}) = \left(\sum_{(i,j) \in S_{n-1}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha)$. Then \begin{eqnarray} D(\alpha^{n+1}) = D(\alpha^{n} \alpha) & = & D(\alpha^{n}) \tau(\alpha) + \sigma(\alpha^{n}) D(\alpha) \\ & = & \left(\sum_{i+j = n-1} \sigma(\alpha^{i}) \tau(\alpha^{j+1}) + \sigma(\alpha^{n}) \tau(\alpha^{0})\right) D(\alpha) \\ & = & \left(\sum_{i+j = n} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha) = \left(\sum_{(i,j) \in S_{n}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha). \end{eqnarray} Induction is complete and so is proof. \end{proof} The theorem is now straightforward. \begin{theorem}\th\label{theorem 2.3} Let $\mathcal{A}$ be of finite rank $n$ and suppose that $\mathcal{A}$ has an $R$-basis of the form $\{1, \alpha, \alpha^{2}, ..., \alpha^{n-1}\}$ for some $\alpha \in \mathcal{A}$. If an $R$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ is a $(\sigma, \tau)$-derivation, then \begin{equation}D(\alpha^{k}) = \left(\sum_{(i,j) \in S_{k-1}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha)\end{equation} for all $k \in \{1, 2, ..., n-1\}$. \end{theorem} The converse of the \th\ref{theorem 2.3} is not true. Some examples are given below.\vspace{10pt} \noindent \textbf{Example 1.} Let $R[X]$ be the polynomial ring over a commutative unital ring $R$. Let $f(X) \in R[X]$ be monic with degree $n$ and $\mathcal{A} = \frac{R[X]}{\langle f(X) \rangle}$. Then $\mathcal{A}$ is a commutative $R$-algebra with unity $\bar{1} = 1 + f(X)$. Suppose $\alpha = X + f(X)$. Then $\{1, \alpha, \alpha^{2}, ..., \alpha^{n-1}\}$ is an $R$-basis of the $R$-module $\mathcal{A}$. We denote the zero element $0 + \langle f(X) \rangle$ of $\mathcal{A}$ by $\bar{0}$. In particular, take $R = \mathbb{Z}$ and $f(X) = X^{4} - 1$ so that $\{\bar{1}, \alpha, \alpha^{2}, \alpha^{3}\}$ is a $\mathbb{Z}$-basis of $\mathcal{A} = \frac{R[X]}{\langle f(X) \rangle}$. Let $\sigma$ and $\tau$ be $\mathbb{Z}$-algebra endomorphisms of $\mathcal{A}$ given by $\sigma(\alpha) = \alpha$ and $\tau(\alpha) = \alpha^{2}$. Then $\sigma(\bar{1}) = \bar{1}$ and $\tau(\bar{1}) = \bar{1}$. Define $D:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(\bar{1}) = \bar{0}$ and $$D(\alpha^{r}) = \left(\sum_{(i,j) \in S_{r-1}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha), \hspace{0.1cm} \forall \hspace{0.1cm} r \in \{1, 2, 3\}.$$ $\alpha^{4} = \bar{1}$ so that $D(\alpha^{4}) = \bar{0}$. Also, \begin{eqnarray} \left(\sum_{(i,j) \in S_{3}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha) & = & \left(\sigma(\alpha^{3}) + \sigma(\alpha^{2}) \tau(\alpha) + \sigma(\alpha) \tau(\alpha^{2}) + \tau(\alpha^{3})\right)D(\alpha) \\ & = & \left(\alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6}\right)D(\alpha) = \left(\bar{1} + \alpha + \alpha^{2} + \alpha^{3}\right)D(\alpha). \end{eqnarray} Now $\bar{1} + \alpha + \alpha^{2} + \alpha^{3} \neq \bar{0}$ since $\{\bar{1}, \alpha, \alpha^{2}, \alpha^{3}\}$ is a basis of $\mathcal{A}$. Also, $\mathcal{A}$ is an integral domain, so if $D(\alpha) \neq 0$, then the above expression cannot be zero. Further, observe that $D(\alpha^{4}) \neq \left(\sum_{(i,j) \in S_{3}} \sigma(\alpha^{i}) \tau(\alpha^{j})\right)D(\alpha)$. Hence by \th\ref{lemma 2.2}, $D$ cannot be a $(\sigma, \tau)$-derivation of $\mathcal{A}$. \vspace{10pt} \noindent \textbf{Example 2.} Consider the set $$\mathcal{A} = \{aI + bA + cA^{2} \mid a, b, c \in \mathbb{Z}\},$$ where $A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$. Then $\mathcal{A}$ is a commutative subalgebra of the $\mathbb{Z}$-algebra $M_{3}(\mathbb{Z})$ with unity $I$ and $\{I, A, A^{2}\}$ is a $\mathbb{Z}$-basis of $\mathcal{A}$. Let $\sigma$ and $\tau$ be $\mathbb{Z}$-algebra endomorphisms of $\mathcal{A}$ given by $\sigma(aI + bA + cA^{2}) = aI + bA + cA^{2}$ and $\tau(aI + bA + cA^{2}) = aI + bA^{2} + cA$ for all $a, b, c \in \mathbb{Z}$. Then $\sigma(I) = \tau(I) = I$. Define $D:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(I) = 0$ and such that $$D(A^{r}) = \left(\sum_{(i,j) \in S_{r-1}} \sigma(A^{i}) \tau(A^{j})\right)D(A), \hspace{0.8cm} \text{for} \hspace{0.2cm} r = 1, 2.$$ Then $D(A^{3}) = D(I) = 0$. Also, \begin{eqnarray}\left(\sum_{(i,j) \in S_{2}} \sigma(A^{i}) \tau(A^{j})\right)D(A) & = & \left(\sigma(A^{2}) + \sigma(A) \tau(A) + \tau(A^{2})\right)D(A) = \left(I + A + A^{2}\right)D(A).\end{eqnarray} Again, $I + A + A^{2} \neq 0$, so if $D(A) \in \mathcal{A}$ is such that $\left(I + A + A^{2}\right)D(A) \neq 0$, for example, $D(A) = A$ or $A^{2}$, then $D(A^{3}) \neq \left(\sum_{(i,j) \in S_{2}} \sigma(A^{i}) \tau(A^{j})\right)D(A)$. Hence, $D$ is not a $(\sigma, \tau)$-derivation. \vspace{10pt} \noindent \textbf{Example 3.} Let $n \in \mathbb{N}$ and $A \in M_{n \times n}(\mathbb{Z})$ be an idempotent matrix. Then the subset $$\mathcal{A} = \{aI + bA \mid a, b \in \mathbb{Z}\}$$ of $M_{n \times n}(\mathbb{Z})$ is a commutative $\mathbb{Z}$-algebra with unity $I$ and $\{I, A\}$ is a basis of $\mathcal{A}$. Let $\sigma$ and $\tau$ be $\mathbb{Z}$-algebra endomorphisms of $\mathcal{A}$ given by $\sigma(aI + bA) = (a+b)I$ and $\tau(aI + bA) = aI + bA$ for all $a, b \in \mathbb{Z}$. Define $D:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(I) = 0$ and such that $$D(A^{r}) = \left(\sum_{(i,j) \in S_{r-1}} \sigma(A^{i}) \tau(A^{j})\right)D(A), \hspace{0.8cm} \text{for} \hspace{0.2cm} r = 1.$$ $$\left(\sum_{(i,j) \in S_{1}} \sigma(A^{i}) \tau(A^{j})\right)D(A) = \left(\sigma(A) + \tau(A)\right)D(A) = \left(I + A\right)D(A).$$ $D(A) \neq \left(I + A\right)D(A)$ provided $A D(A) \neq 0$. Hence, $D$ is not a $(\sigma, \tau)$-derivation of $\mathcal{A}$ provided $D(A)$ is chosen in $M_{n \times n}(\mathbb{Z})$ such that $A D(A) \neq 0$.\vspace{10pt} \noindent \textbf{Example 4.} Let $n \in \mathbb{N}$ and $A \in M_{n \times n}(\mathbb{Z})$ be a nilpotent matrix, that is, $A^{k} = 0$ for some least positive integer $k$. Then the subset $$\mathcal{A} = \{\sum_{i=0}^{k-1} a_{i} A^{i} \mid a_{i} \in \mathbb{Z}, \hspace{0.1cm} \forall \hspace{0.1cm} i \in \{0, 1, ..., k-1\}\}$$ of $M_{n \times n}(\mathbb{Z})$ is a commutative $\mathbb{Z}$-algebra with unity $I$ and $\{A^{0}=I, A, ..., A^{k-1}\}$ forms a basis of the $\mathbb{Z}$-module $\mathcal{A}$. Let $\sigma$ and $\tau$ be $\mathbb{Z}$-algebra endomorphisms of $\mathcal{A}$ given by $\sigma(\sum_{i=0}^{k-1} a_{i} A^{i}) = \sum_{i=0}^{k-1} a_{i} A^{i}$ and $\tau(\sum_{i=0}^{k-1} a_{i} A^{i}) = \sum_{i=0}^{k-1} a_{i} B^{i}$ for all $a_{i} \in \mathbb{Z}$ $(0 \leq i \leq k-1),$ where $B = m A$ for some $m \in \mathbb{N}$ and $m \neq 1$. Define $D:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(I) = 0$ and such that $$D(A^{r}) = \left(\sum_{(i,j) \in S_{r-1}} \sigma(A^{i}) \tau(A^{j})\right)D(A), \hspace{0.8cm} \text{for} \hspace{0.2cm} r \in \{1, ..., k-1\}.$$ Then $D(A^{k}) = 0$. Also, \begin{eqnarray}\left(\sum_{(i,j) \in S_{k-1}} \sigma(A^{i}) \tau(A^{j})\right)D(A) = \left(\sum_{i+j=k-1} A^{i} B^{j}\right)D(A) & = & \left(\sum_{j=0}^{k-1} m^{j} A^{k-1} \right)D(A) \\ & = & \left(\frac{1-m^{k}}{1-m}\right)A^{k-1}D(A).\end{eqnarray} $0 \neq \left(\frac{1-m^{k}}{1-m}\right)A^{k-1}D(A)$ provided $A^{k-1} D(A) \neq 0$. Hence, $D$ cannot be a $(\sigma, \tau)$-derivation of $\mathcal{A}$ provided $D(A)$ is chosen in $M_{n \times n}(\mathbb{Z})$ such that $A^{k-1} D(A) \neq 0$.\vspace{10pt} If $R$ is a ring and $G$ is a group, then the group ring of $G$ over $R$ is defined as the set $$RG = \{\sum_{g \in G} a_{g} g \mid a_{g} \in R, \forall g \in G \hspace{0.2cm} \text{and} \hspace{0.2cm} \|\text{supp}(\alpha)\| < \infty \},$$ where for $\alpha = \sum_{g \in G} a_{g} g$, $\text{supp}(\alpha)$ is the support of $\alpha$ that consists of elements from $G$ that appear in the expression of $\alpha$. $RG$ is a ring with respect to the componentwise addition and multiplication defined respectively by: For $\alpha = \sum_{g \in G} a_{g} g$, $\beta = \sum_{g \in G} b_{g} g$ in $RG$, $$(\sum_{g \in G} a_{g} g ) + (\sum_{g \in G} b_{g} g) = \sum_{g \in G}(a_{g} + b_{g}) g \hspace{0.2cm} \text{and} \hspace{0.2cm} \alpha \beta = \sum_{g, h \in G} a_{g} b_{h} gh.$$ If the ring $R$ is commutative having unity $1$ and the group $G$ is abelian having identity $e$, then $RG$ becomes a commutative unital algebra over $R$ with unity $1 = 1e$. If $\sigma$ and $\tau$ are algebra endomorphisms of $RG$ which are $R$-linear extensions of group homomorphisms of $G$ and $D:RG \rightarrow RG$ is a $(\sigma, \tau)$-derivation, then $$D(e) = D(ee) = D(e) \tau(e) + D(e) \sigma(e) = D(e) + D(e)$$ so that $D(e) = 0$.\vspace{10pt} \noindent \textbf{Example 5.} Let $C_{n} = \langle g \mid g^{n} = 1\rangle$ be a cyclic group having order $n \geqslant 2$ and consider the group ring $\mathbb{Z}C_{n}$. Let $\sigma$ and $\tau$ be $\mathbb{Z}$-algebra endomorphisms of $\mathbb{Z}G$ which are $\mathbb{Z}$-linear extensions of group homomorphisms of $G$, say, $\sigma(g) = 1$ and $\tau(g) = g$. Let $D:\mathbb{Z}C_{n} \rightarrow \mathbb{Z}C_{n}$ be a $\mathbb{Z}$-linear map with $D(1) = 0$ and $$D(g^{r}) = \left(\sum_{(i,j) \in S_{r-1}} \sigma(g^{i}) \tau(g^{j})\right)D(g), \hspace{0.1cm} \forall \hspace{0.1cm} r \in \{1, ..., n-1\}.$$ So $D(g^{n}) = D(1) = 0$. Also, \begin{equation} \begin{aligned} \left(\sum_{(i,j) \in S_{n-1}} \sigma(g^{i}) \tau(g^{j})\right)D(g) & = (\sigma(g^{n-1}) + \sigma(g^{n-2})\tau(g) + \sigma(g^{n-3})\tau(g^{2}) + ... \\ &\quad + \sigma(g^{2}) \tau(g^{n-3}) + \sigma(g) \tau(g^{n-2}) + \tau(g^{n-1}))D(g) \\ & = \left(e + g + ... + g^{n-1}\right)D(g) \end{aligned} \end{equation} Note that $e + g + ... + g^{n-1} \neq 0$ since $e, g, ..., g^{n-1}$ are linearly independent being basis elements of $\mathbb{Z}C_{n}$. So if $D(g) \in \mathbb{Z}C_{n}$ is such that $\left(e + g + ... + g^{n-1}\right)D(g) \neq 0$, then $D(g^{n}) \neq \left(\sum_{(i,j) \in S_{n-1}} \sigma(g^{i}) \tau(g^{j})\right)D(g)$. Hence, $D$ cannot be a $(\sigma, \tau)$-derivation of $\mathbb{Z}C_{n}$. \begin{lemma}\th\label{lemma 2.4} Let $\mathcal{A}$ be of finite rank $n$ and $\{\alpha_{1}, \alpha_{2}, ..., \alpha_{n}\}$ be an $R$-basis of $\mathcal{A}$. Let $D:\mathcal{A} \rightarrow \mathcal{A}$ be a $(\sigma, \tau)$-derivation. Suppose that there exists some $\beta \in \mathcal{A}$ such that $D(\alpha_{i}) = \beta (\tau - \sigma)(\alpha_{i})$ for all $i \in \{1, 2, ..., n\}$. Then $D(\alpha) = \beta (\tau - \sigma)(\alpha)$ for all $\alpha \in \mathcal{A}$. \end{lemma} \begin{proof} For $\alpha = \sum_{i=1}^{n} a_{i} \alpha_{i} \in \mathcal{A}$, \begin{eqnarray}D(\alpha) = \sum_{i=1}^{n} a_{i} D(\alpha_{i}) = \sum_{i=1}^{n} a_{i} \beta (\tau - \sigma)(\alpha_{i}) & = & \beta \left(\sum_{i=1}^{n} a_{i} \tau(\alpha_{i}) - \sum_{i=1}^{n} a_{i} \sigma(\alpha_{i})\right) \\ & = & \beta \left(\tau\left(\sum_{i=1}^{n} a_{i} \alpha_{i}\right) - \sigma\left(\sum_{i=1}^{n} a_{i} \alpha_{i}\right)\right) \\ & = & \beta (\tau - \sigma)(\alpha)\end{eqnarray} Since $\alpha \in \mathcal{A}$ is arbitrary, therefore, $D$ is inner. \end{proof} The theorem now follows immediately. \begin{theorem}\th\label{theorem 2.5} Let $\mathcal{A}$ be of finite rank $n$ and $\{\alpha_{1}, \alpha_{2}, ..., \alpha_{n}\}$ be an $R$-basis of $\mathcal{A}$. Then a $(\sigma, \tau)$-derivation $D:\mathcal{A} \rightarrow \mathcal{A}$ is inner if and only if there exists some $\beta \in \mathcal{A}$ such that $D(\alpha_{i}) = \beta (\tau - \sigma)(\alpha_{i})$ for all $i \in \{1, 2, ..., n\}$. \end{theorem} \begin{lemma}\th\label{lemma 2.6} Let $D: \mathcal{A} \rightarrow \mathcal{A}$ be a $(\sigma, \tau)$-derivation and $\alpha \in \mathcal{A}$. If there exists some $\beta \in \mathcal{A}$ such that $D(\alpha) = \beta (\tau - \sigma)(\alpha)$, then $$D(\alpha^{n}) = \beta (\tau - \sigma)(\alpha^{n})$$ for all $n \in \mathbb{N}$. \end{lemma} \begin{proof} We again use induction on $n$. If $n=1$, the result holds trivially. Now suppose that $D(\alpha^{k}) = \beta (\tau - \sigma)(\alpha^{k})$. Then \begin{eqnarray} D(\alpha^{k+1}) & = & D(\alpha^{k}) \tau(\alpha) + \sigma(\alpha^{k}) D(\alpha) = \beta (\tau - \sigma)(\alpha^{k}) \tau(\alpha) + \sigma(\alpha^{k}) \beta (\tau - \sigma)(\alpha) \\ & = & \beta \left(\tau(\alpha^{k+1}) - \sigma(\alpha^{k}) \tau(\alpha) + \sigma(\alpha^{k}) \tau(\alpha) - \sigma(\alpha^{k+1})\right) \\ & = & \beta (\tau - \sigma)(\alpha^{k+1}) \end{eqnarray} Induction is complete and so is proof. \end{proof} \begin{lemma}\th\label{lemma 2.7} Let $\mathcal{A}$ be of finite rank $n$ and suppose that $\mathcal{A}$ has an $R$-basis of the form $\{1, \alpha, \alpha^{2}, ..., \alpha^{n-1}\}$ for some $\alpha \in \mathcal{A}$. Let $D: \mathcal{A} \rightarrow \mathcal{A}$ be a $(\sigma, \tau)$-derivation. If there exists some $\beta \in \mathcal{A}$ such that $D(\alpha) = \beta (\tau - \sigma)(\alpha)$, then $D$ is inner. \end{lemma} \begin{proof} Since $D(1) = 0$ and $\sigma(1) = \tau(1) = 1$, so $D(1) = \beta (\tau - \sigma)(1)$. Further since $D(\alpha) = \beta (\tau - \sigma)(\alpha)$, therefore, by \th\ref{lemma 2.6}, $$D(\alpha^{k}) = \beta (\tau - \sigma)(\alpha^{k})$$ for all $k \in \mathbb{N}$. So $D(\alpha^{k}) = \beta (\tau - \sigma)(\alpha^{k})$ for every $k \in \{0, 1, ..., n-1\}$. Now we use \th\ref{lemma 2.4}. \end{proof} The theorem below follows immediately. \begin{theorem}\th\label{theorem 2.8} Let $\mathcal{A}$ be of finite rank $n$ and suppose that $\mathcal{A}$ has an $R$-basis of the form $\{1, \alpha, \alpha^{2}, ..., \alpha^{n-1}\}$ for some $\alpha \in \mathcal{A}$. Then a $(\sigma, \tau)$-derivation $D:\mathcal{A} \rightarrow \mathcal{A}$ is inner if and only if there exists some $\beta \in \mathcal{A}$ such that $D(\alpha) = \beta (\tau - \sigma)(\alpha)$. \end{theorem} \section{\texorpdfstring{$(\sigma, \tau)$}{Lg}-derivations of number rings}\label{section 3} We know by \th\ref{theorem 1.1} that if $K$ is a number field, then $K = \mathbb{Q}(\theta)$ for some algebraic integer $\theta$. \th\ref{theorem 2.3} gives the lemma below: \begin{lemma}\th\label{lemma 3.1} Let $K = \mathbb{Q}(\theta)$ ($\theta$ an algebraic integer) be a number field of degree $n$ such that $\{1, \theta, ..., \theta^{n-1}\}$ is a power basis of $K$. Let $\sigma$ and $\tau$ be two different non-zero $\mathbb{Z}$-algebra endomorphisms of $O_{K}$. If a $\mathbb{Z}$-linear map $D:O_{K} \rightarrow O_{K}$ is a $(\sigma, \tau)$-derivation, then $$D(\theta^{k}) = \left(\sum_{(i,j) \in S_{k-1}} \sigma(\theta^{i}) \tau(\theta^{j})\right)D(\theta)$$ for all $k \in \{1, 2, ..., n-1\}$. \end{lemma} \subsection{\texorpdfstring{$(\sigma, \tau)$}{Lg}-derivations of quadratic number rings}\label{subsection 3.1} By \th\ref{theorem 1.3}, if $K$ is a quadratic field, then $K = \mathbb{Q}(\sqrt{d})$ for some square-free rational integer $d$. In this section, we take $K = \mathbb{Q}(\sqrt{d})$ and $\sigma$, $\tau$ as two different non-zero ring endomorphisms of $O_{K}$. Note that any ring endomorphism of the ring $O_{K}$ is a unital $\mathbb{Z}$-algebra endomorphism of the unital $\mathbb{Z}$-algebra $O_{K}$. As an application of \th\ref{lemma 2.1}, we present a different proof of a result of \cite{Chaudhuri} stated as follows. \begin{theorem}[{\cite[Theorem 4.2]{Chaudhuri}}] \th\label{theorem 3.2} Every $\mathbb{Z}$-linear map $D:O_{K} \rightarrow O_{K}$ with $D(1)=0$ is a $(\sigma, \tau)$-derivation of $O_{K}$. \end{theorem} \begin{proof} Only two possibilities arise: $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$ and $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. Since $(\sqrt{d})^{2} = d$, so if $\theta$ is a non-zero ring endomorphism of $O_{K}$, then $(\theta(\sqrt{d}))^{2} = d$ or that $\theta(\sqrt{d}) = \pm \sqrt{d}$. When $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, then from \th\ref{theorem 1.4}, $O_{K} = \mathbb{Z}[\sqrt{d}]$ and from \th\ref{theorem 1.5}, $\{1, \sqrt{d}\}$ is a $\mathbb{Z}$-basis of $O_{K}$. In this case, exactly two possibilities arise. For every $a, b \in \mathbb{Z}$, \begin{itemize} \item[(a)] $\sigma(a + b \sqrt{d}) = a + b \sqrt{d}$; $\tau(a + b \sqrt{d}) = a - b \sqrt{d}$, \item[(b)] $\sigma(a + b \sqrt{d}) = a - b \sqrt{d}$; $\tau(a + b \sqrt{d}) = a + b \sqrt{d}.$ \end{itemize} When $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, then by \th\ref{theorem 1.4}, $O_{K} = \mathbb{Z}[\frac{1+\sqrt{d}}{2}]$ and by \th\ref{theorem 1.5}, $\{1,\frac{1+\sqrt{d}}{2}\}$ is a $\mathbb{Z}$-basis of $O_{K}$. Thus in this case, too, exactly two possibilities arise. For all $a, b \in \mathbb{Z}$, \begin{itemize} \item[(a)] $\sigma(a + b (\frac{1+\sqrt{d}}{2})) = a + b (\frac{1+\sqrt{d}}{2})$; $\tau(a + b (\frac{1+\sqrt{d}}{2})) = a + b (\frac{1-\sqrt{d}}{2})$, \item[(b)] $\sigma(a + b (\frac{1+\sqrt{d})}{2})) = a + b (\frac{1-\sqrt{d}}{2})$; $\tau(a + b (\frac{1+\sqrt{d}}{2})) = a + b (\frac{1+\sqrt{d}}{2})$. \end{itemize} Now the proof trivially follows from \th\ref{lemma 2.1}. \end{proof} Further, in the same paper \cite{Chaudhuri}, the author has proved the theorem stated below. \begin{theorem}[{\cite[Theorem 4.2]{Chaudhuri}}] \th\label{theorem 3.3} Let $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. Let $D:O_{K} \rightarrow O_{K}$ be a $(\sigma, \tau)$-derivation and $D(\sqrt{d}) = c_{0} + c_{1} \sqrt{d}$ for some $c_{0}, c_{1} \in \mathbb{Z}$. If $2 d$ divides $c_{0}$ and $c_{1}$ is even, then $D$ is inner. \end{theorem} We present a different proof of this theorem using \th\ref{theorem 2.8} and also prove that the conditions in the above theorem are necessary. \begin{theorem}\th\label{theorem 3.4} Let $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. Let $D:O_{K} \rightarrow O_{K}$ be a $(\sigma, \tau)$-derivation and $D(\sqrt{d}) = c_{0} + c_{1} \sqrt{d}$ for some $c_{0}, c_{1} \in \mathbb{Z}$. Then $D$ is inner if and only if $c_{0}$ is divisible by $2 d$ and $c_{1}$ is even. In particular, $O_{K}$ has non-trivial outer $(\sigma, \tau)$-derivations. \end{theorem} \begin{proof} Since $d \not\equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, so $\{1, \sqrt{d}\}$ is a basis of the $\mathbb{Z}$-module $O_{K} = \mathbb{\mathbb{Z}}[\sqrt{d}]$. $O_{K} = \mathbb{\mathbb{Z}}[\sqrt{d}]$ has precisely two different non-zero ring endomorphisms: $\phi_{1}(a + b \sqrt{d}) \\ = a + b \sqrt{d}$ and $\phi_{2}(a + b \sqrt{d}) = a - b \sqrt{d}$ ($a, b \in \mathbb{Z}$). Therefore, $(\sigma, \tau) = (\phi_{1}, \phi_{2})$ or $(\phi_{2}, \phi_{1})$. We prove the result for $(\sigma, \tau) = (\phi_{2}, \phi_{1})$ since the other follows similarly. First, let $D$ be inner. Then there exists some $\beta = b_{0} + b_{1} \sqrt{d}$ for some $b_{0}, b_{1} \in \mathbb{Z}$ such that $D(\sqrt{d}) = \beta (\tau - \sigma)(\sqrt{d})$. Therefore, \begin{eqnarray} c_{0} + c_{1} \sqrt{d} = D(\sqrt{d}) & = & \beta (\tau - \sigma)(\sqrt{d}) \\ & = & (b_{0} + b_{1} \sqrt{d}) (\tau(\sqrt{d}) - \sigma(\sqrt{d})) \\ & = & 2d b_{1} + 2b_{0} \sqrt{d} \end{eqnarray} Since $\{1, \sqrt{d}\}$ is a basis of the $\mathbb{Z}$-module $O_{K} = \mathbb{\mathbb{Z}}[\sqrt{d}]$, therefore, $$2d b_{1} = c_{0} \hspace{0.2cm} \text{and} \hspace{0.2cm} 2 b_{0} = c_{1}.$$ Obviously, $b_{0} = \frac{c_{1}}{2}$ and $b_{1} = \frac{c_{0}}{2d}$. Since $b_{0}, b_{1} \in \mathbb{Z}$, therefore, $c_{1}$ is divisible by $2$ and $c_{0}$ is divisible by $2d$. Conversely, let $c_{0}$ be divisible by $2d$ and $c_{1}$ be even. Define $\beta = \left(\frac{c_{0}}{2d}\right) + \left(\frac{c_{1}}{2} \right) \sqrt{d}$. Then it can be verified that $\beta (\tau - \sigma)(\sqrt{d}) = D(\sqrt{d})$ as $D(\sqrt{d}) = c_{0} + c_{1} \sqrt{d}$. Now the result follows immediately by \th\ref{theorem 2.8}. \end{proof} \begin{theorem} Let $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$. Let $D:O_{K} \rightarrow O_{K}$ be a $(\sigma, \tau)$-derivation and $D(\sqrt{d}) = c_{0} + c_{1} (\frac{1+\sqrt{d}}{2})$ for some $c_{0}, c_{1} \in \mathbb{Z}$. Then $D$ is inner if and only if $d$ divides $-c_{0} + c_{1} \left(\frac{d-1}{2}\right)$ and $2 c_{0} + c_{1}$. In particular, the following conditions hold: \begin{enumerate} \item[(i)] $O_{K}$ has non-trivial outer $(\sigma, \tau)$-derivations. \item[(ii)] $D$ is inner if $d$ divides both $c_{0}$ and $c_{1}$. \end{enumerate} \end{theorem} \begin{proof} Since $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, so $\{1, \frac{1+\sqrt{d}}{2}\}$ is a basis of the $\mathbb{Z}$-module $O_{K} = \mathbb{\mathbb{Z}}[\frac{1+\sqrt{d}}{2}]$. $O_{K}$ has precisely two different non-zero ring endomorphisms, namely, $\phi_{1}(a + b (\frac{1+\sqrt{d}}{2})) = a + b (\frac{1+\sqrt{d}}{2})$ and $\phi_{2}(a + b (\frac{1+\sqrt{d}}{2})) = a + b (\frac{1-\sqrt{d}}{2})$ ($a, b \in \mathbb{Z}$). Therefore, $(\sigma, \tau) = (\phi_{1}, \phi_{2})$ or $(\phi_{2}, \phi_{1})$. We prove the result for $(\sigma, \tau) = (\phi_{2}, \phi_{1})$ since the other follows similarly. First, let $D$ be inner. Then there exists some $\beta = b_{0} + b_{1} (\frac{1+\sqrt{d}}{2})$ for some $b_{0}, b_{1} \in \mathbb{Z}$ such that $D(\frac{1+\sqrt{d}}{2}) = \beta (\tau - \sigma)(\frac{1+\sqrt{d}}{2})$. Therefore, \begin{eqnarray} c_{0} + c_{1} \left(\frac{1+\sqrt{d}}{2}\right) & = & D\left(\frac{1+\sqrt{d}}{2}\right) = \beta \left(\tau - \sigma \right)\left(\frac{1+\sqrt{d}}{2}\right) \\ & = & \left(b_{0} + b_{1} \left(\frac{1+\sqrt{d}}{2}\right)\right) \left(\tau \left(\frac{1+\sqrt{d}}{2} \right) - \sigma \left(\frac{1+\sqrt{d}}{2}\right) \right) \\ & = & \left(b_{0} + b_{1} \left(\frac{1+\sqrt{d}}{2} \right) \right) \left( \left(\frac{1+\sqrt{d}}{2}\right) - \left(\frac{1-\sqrt{d}}{2}\right) \right) \\ & = & \left(-b_{0} + b_{1} \left( \frac{d-1}{2} \right)\right) + (2b_{0} + b_{1}) \left(\frac{1+\sqrt{d}}{2}\right) \end{eqnarray} Since $\{1, \frac{1+ \sqrt{d}}{2}\}$ is a basis of $O_{K} = \mathbb{\mathbb{Z}}[\frac{1+ \sqrt{d}}{2}]$, therefore, $$-b_{0} + b_{1} \left( \frac{d-1}{2} \right) = c_{0} \hspace{0.2cm} \text{and} \hspace{0.2cm} 2b_{0} + b_{1} = c_{1}.$$ Note that since $d \equiv 1 \hspace{0.1cm} (\text{mod} \hspace{0.1cm} 4)$, so $\frac{d-1}{2} \in \mathbb{Z}$. Solving, we get, $$b_{0} = \frac{1}{d} \left(- c_{0} + c_{1} \left(\frac{d-1}{2} \right)\right) \hspace{0.1cm} \text{and} \hspace{0.1cm} b_{1} = \frac{1}{d} \left(2 c_{0} + c_{1}\right).$$ Since $b_{0}, b_{1} \in \mathbb{Z}$, therefore, $d$ divides $- c_{0} + c_{1} \left(\frac{d-1}{2} \right)$ and $2 c_{0} + c_{1}$. Conversely, let $d$ divide $- c_{0} + c_{1} \left(\frac{d-1}{2} \right)$ and $2 c_{0}+c_{1}$. Define $$\beta = \left(\frac{1}{d} \left(- c_{0} + c_{1} \left(\frac{d-1}{2} \right)\right)\right) + \left(\frac{1}{d} \left(2 c_{0} + c_{1}\right)\right)\left(\frac{1+\sqrt{d}}{2}\right).$$ Then it can be verified that $\beta \left(\tau - \sigma \right)\left(\frac{1+\sqrt{d}}{2}\right) = D\left(\frac{1+\sqrt{d}}{2}\right)$ as $D\left(\frac{1+\sqrt{d}}{2}\right) = c_{0} + c_{1} \left(\frac{1+\sqrt{d}}{2}\right)$. The result now follows immediately by \th\ref{theorem 2.8}. \end{proof} \subsection{\texorpdfstring{$(\sigma, \tau)$}{Lg}-derivations of cyclotomic number rings}\label{subsection 3.2} In this section, $K = \mathbb{Q}(\zeta)$ denotes a $p^{\text{th}}$ cyclotomic field, where $p$ is an odd rational prime. By \th\ref{theorem 1.7}, the ring of algebraic integers of $K$ is $O_{K} = \mathbb{Z}[\zeta]$ and $\{1, \zeta, \zeta^{2}, ..., \zeta^{p-2}\}$ is an integral basis of $K$. Further, $\sigma$ and $\tau$ denote any two different non-zero ring endomorphisms of $O_{K} = \mathbb{Z}[\zeta]$. \begin{lemma}\th\label{lemma 3.6} Let for each $k \in \{0, 1, ..., p-2\}$, $S_{k}$'s be sets as defined in Section \ref{section 2}. Then $$\sum_{(i,j) \in \cup_{k=0}^{p-2} S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = 0.$$ \end{lemma} \begin{proof} All the non-zero ring endomorphisms $\phi_{i}$ of $O_{K}$ are given by $\phi_{i}(\zeta) = \zeta^{i},$ where $i \in \{1, 2, ..., p-1\}$. Since $\sigma$ and $\tau$ are non-zero and different, therefore, $$\sigma(\zeta) = \zeta^{u} \hspace{0.2cm} \text{and} \hspace{0.2cm} \tau(\zeta) = \zeta^{u+v}$$ for some $u, v \in \{1, 2, ..., p-1\}$ such that $u+v \neq p$. Now, $$\sum_{(i,j) \in \cup_{k=0}^{p-2} S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = \sum_{k=0}^{p-2} \sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}).$$ For any $k \in \{0, 1, ..., p-2\}$, $$\sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = \sum_{(i,j) \in S_{k}} \zeta^{ku} \zeta^{jv}.$$ There are three observations for each $k \in \{0, 1, ..., p-2\}$: \begin{enumerate} \item[(a)] The sum $\sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j})$ contains $k+1$ terms and all these $k+1$ terms are different from each other since $u+v \neq p$. \item[(b)] The sum $\sum_{(i,j) \in \cup_{k=0}^{p-2} S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j})$ contains exactly $\frac{p(p-1)}{2}$ terms. \item[(c)] Since the prime $p$ is odd, therefore, for each $k \in \{1, 2, ..., p-1\}$, $\zeta^{k}$ is a primitive $p^{\text{th}}$ root of unity. \end{enumerate} These together imply that each term from the set $\{1, \zeta, \zeta^{2}, ..., \zeta^{p-1}\}$ is being repeated $\frac{p-1}{2}$ times in the sum $\sum_{(i,j) \in \cup_{k=0}^{p-2} S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j})$. Hence, $$\sum_{(i,j) \in \cup_{m=0}^{p-2} S_{m}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = \frac{p-1}{2} \left(1 + \zeta + \zeta^{2} + ... + \zeta^{p-1}\right) = 0,$$ since $1 + \zeta + \zeta^{2} + ... + \zeta^{p-1} = 0$ by \th\ref{theorem 1.6}. This proves the required result. \end{proof} \begin{lemma}\th\label{lemma 3.7} Let $D:O_{K} \rightarrow O_{K}$ be a $\mathbb{Z}$-linear map with $D(1) = 0$ and \begin{equation}\label{eq 3.1}D(\zeta^{k}) = \left(\sum_{(i,j) \in S_{k-1}} \sigma(\zeta^{i}) \tau(\zeta^{j})\right)D(\zeta)\end{equation} for all $k \in \{1, 2, ..., p-2\}$. Then $D$ is a $(\sigma, \tau)$-derivation. \end{lemma} \begin{proof} According to \th\ref{lemma 2.1}, the result can be concluded by establishing that \begin{equation}\label{eq 3.2}D(\zeta^{i+j}) = D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j})\end{equation} for all $i, j \in \{0, 1, ..., p-2\}$. The relations (\ref{eq 3.2}) hold trivially when atleast one of $i$ or $j$ is $0$, using the fact that $\sigma(1) = \tau(1) = 1$ and $D(1) = 0$. So now let $i, j \in \{1, 2, ..., p-2\}$. Using (\ref{eq 3.1}), we get: \begin{eqnarray} D(\zeta^{i}) \tau(\zeta^{j}) = \left(\sum_{(s,t) \in S_{i-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) \tau(\zeta^{j}) & = & \left(\sum_{(s,t) \in S_{i-1}} \sigma(\zeta^{s}) \tau(\zeta^{t+j})\right)D(\zeta) \\ & = & \left(\sum_{s=0}^{i-1} \sigma(\zeta^{s}) \tau(\zeta^{i-1-s+j})\right)D(\zeta)\end{eqnarray} and \begin{eqnarray} \sigma(\zeta^{i}) D(\zeta^{j}) = \sigma(\zeta^{i})\left(\sum_{(s,t) \in S_{j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) & = & \left(\sum_{(s,t) \in S_{j-1}} \sigma(\zeta^{i+s}) \tau(\zeta^{t})\right)D(\zeta) \\ & = & \left(\sum_{s=0}^{j-1} \sigma(\zeta^{i+s}) \tau(\zeta^{j-1-s})\right)D(\zeta) \\ & = & \left(\sum_{s=i}^{i+j-1} \sigma(\zeta^{s}) \tau(\zeta^{j-1+i-s})\right)D(\zeta). \end{eqnarray} Therefore, \begin{equation}\label{eq 3.3} D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) = \left(\sum_{(s,t) \in S_{i+j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta). \end{equation} Since $i, j \in \{1, 2,..., p-2\}$, so $i+j \in \{2, 3 ..., 2(p-2)\}$. We partition the proof into the following four cases.\vspace{10pt} Case 1: $i+j \leq p-2$. Since $i+j \leqslant p-2$, so by (\ref{eq 3.1}), $D(\zeta^{i+j}) = \left(\sum_{(s,t) \in S_{i+j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta)$. Then by (\ref{eq 3.3}), $D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) = D(\zeta^{i+j})$. Therefore, in this case, relations (\ref{eq 3.2}) hold.\vspace{10pt} Case 2: $i+j = p-1$. By \th\ref{lemma 3.6}, $\sum_{(i,j) \in \cup_{k=0}^{p-2} S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = 0$. $\Rightarrow \sum_{k=0}^{p-2} \sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = 0$. $\Rightarrow \sum_{(i,j) \in S_{p-2}} \sigma(\zeta^{i}) \tau(\zeta^{j}) = - \left(\sum_{k=0}^{p-3} \sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j})\right)$. $\Rightarrow \left(\sum_{(i,j) \in S_{p-2}} \sigma(\zeta^{i}) \tau(\zeta^{j})\right)D(\zeta) = - \left(\sum_{k=0}^{p-3} \left(\sum_{(i,j) \in S_{k}} \sigma(\zeta^{i}) \tau(\zeta^{j})\right)D(\zeta)\right)$. Therefore, using (\ref{eq 3.1}) and (\ref{eq 3.3}), $$D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) = -\left(\sum_{k=0}^{p-3} D(\zeta^{k+1})\right) = -D(1 + \zeta + \zeta^{2} + ... + \zeta^{p-2}) = D(\zeta^{p-1}).$$ So (\ref{eq 3.2}) holds in this case too.\vspace{10pt} Case 3: $i+j = p$. Since the ring endomorphisms $\sigma$ and $\tau$ are non-zero and different, therefore, $\sigma(\zeta) = \zeta^{u} \hspace{0.2cm} \text{and} \hspace{0.2cm} \tau(\zeta) = \zeta^{u+v}$ for some $u, v \in \{1, 2, ..., p-1\}$ such that $u+v \neq p$. Now by (\ref{eq 3.3}), \begin{equation} \begin{aligned} D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) & = \left(\sum_{(s,t) \in S_{i+j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) = \left(\sum_{(s,t) \in S_{p-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) \\ & = \left(\sum_{s=0}^{p-1} \sigma(\zeta^{s}) \tau(\zeta^{p-1-s})\right)D(\zeta) = \zeta^{(p-1)u} \left(\sum_{s=0}^{p-1} \zeta^{(p-1-s)v})\right)D(\zeta) \\ & = \zeta^{(p-1)u} \left(1 + \zeta^{v} + \zeta^{2v} + ... + \zeta^{(p-1)v}\right)D(\zeta) = 0, \end{aligned} \end{equation} as $\zeta^{v}$ satisfies the cyclotomic polynomial. Therefore, $D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) = D(\zeta^{p})$ as $\zeta^{p} = 1$ and $D(1) = 0$. So (\ref{eq 3.2}) holds in this case as well.\vspace{10pt} Case 4: $p < i+j \leq 2(p-2)$. Then $i+j = m + p$ for some $m \in \{1, 2, ..., p-2\}$. By (\ref{eq 3.1}), \begin{eqnarray}D(\zeta^{i+j}) = D(\zeta^{m}) = \left(\sum_{(s,t) \in S_{m-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) & = & \left(\sum_{t=0}^{m-1} \sigma(\zeta^{m-1-t}) \tau(\zeta^{t})\right)D(\zeta) \\ & = & \zeta^{(m-1)u} \left(\sum_{t=0}^{m-1} \zeta^{tv}\right)D(\zeta).\end{eqnarray} Further, by (\ref{eq 3.3}), $D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j}) = \left(\sum_{(s,t) \in S_{i+j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta)$ \begin{equation} \begin{aligned} & = \left(\sum_{(s,t) \in S_{m+p-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta) = \left(\sum_{t=0}^{m+p-1} \sigma(\zeta^{m+p-1-t}) \tau(\zeta^{t})\right)D(\zeta) \\ & = \zeta^{(m+p-1)u} \left(\sum_{t=0}^{m+p-1} \zeta^{tv}\right)D(\zeta) = \zeta^{(m+p-1)u} \left(\sum_{t=0}^{m-1} \zeta^{tv} + \sum_{t=m}^{m+p-1} \zeta^{tv}\right)D(\zeta) \\ & = \zeta^{(m-1)u} \left(\sum_{t=0}^{m-1} \zeta^{tv} + \zeta^{mv} \left(\sum_{t=0}^{p-1} \zeta^{tv}\right)\right)D(\zeta) = \zeta^{(m-1)u} \left(\sum_{t=0}^{m-1} \zeta^{tv}\right)D(\zeta) \end{aligned} \end{equation} since $\sum_{t=0}^{p-1} \zeta^{tv} = 0$ as $v \in \{1, 2, ..., p-1\}.$ Therefore, $D(\zeta^{i+j}) = D(\zeta^{i}) \tau(\zeta^{j}) + \sigma(\zeta^{i}) D(\zeta^{j})$. This gives again the relations (\ref{eq 3.2}). We conclude that the relations (\ref{eq 3.2}) hold for all $i, j \in \{0, 1, ..., p-2\}$. The result now can be concluded from \th\ref{lemma 2.1}. \end{proof} As a consequence of \th\ref{lemma 3.1} and \th\ref{lemma 3.7}, we have the main theorem. \begin{theorem}\th\label{theorem 3.8} Let $D:O_{K} \rightarrow O_{K}$ be a $\mathbb{Z}$-linear map with $D(1) = 0$. Then $D$ is a $(\sigma, \tau)$-derivation if and only if the following relations hold for all $k \in \{1, 2, ..., p-2\}$: $$D(\zeta^{k}) = \left(\sum_{(i,j) \in S_{k-1}} \sigma(\zeta^{i}) \tau(\zeta^{j})\right)D(\zeta).$$ \end{theorem} \begin{corollary}\th\label{corollary 3.9} The $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(O_{K})$ is finitely generated of rank $p-1$. \end{corollary} \begin{proof} For every $i \in \{0, 1, ..., p-2\}$, define $D_{i}:O_{K} \rightarrow O_{K}$ as a $\mathbb{Z}$-linear map with $D(1) = 0$ and $$D_{i}(\zeta) = \zeta^{i}.$$ More precisely, for each $i \in \{0, 1, ..., p-2\}$, let $D_{i}:O_{K} \rightarrow O_{K}$ be a map defined by $$D_{i}\left(\sum_{j=0}^{p-2} a_{j} \zeta^{j}\right) = \sum_{j=1}^{p-2} a_{j} D_{i}(\zeta^{j}),$$ where for each $j \in \{1, 2, ..., p-2\}$, $D_{i}(\zeta^{j})$ is defined as $$D_{i}(\zeta^{j}) = \left(\sum_{(s,t) \in S_{j-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D_{i}(\zeta).$$ Then by \th\ref{theorem 3.8}, for each $i \in \{0, 1, ..., p-2\}$, $D_{i}:O_{K} \rightarrow O_{K}$ is a $(\sigma, \tau)$-derivation. Further, it can be easily verified that $\{D_{0}, D_{1}, ..., D_{p-2}\}$ forms a linearly independent subset of the $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(O_{K})$ that generates it. Hence the result is proved. \end{proof} We observe from the preceding corollary that the rank of the $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(O_{K})$ is equal to $p-1$, the degree of $K = \mathbb{Q}(\zeta)$ and which, by \th\ref{theorem 1.2}, is also equal to the rank of the free abelian (additive) group $O_{K} = \mathbb{Z}[\zeta]$. Some immediate corollaries can be stated below. \begin{corollary}\th\label{corollary 3.10} Let $K = Q(\zeta)$ be a $3^{\text{th}}$ cyclotomic field. Then any $\mathbb{Z}$-linear map $D:O_{K} \rightarrow O_{K}$ with $D(1)=0$ is a $(\sigma, \tau)$-derivation. \end{corollary} \begin{corollary}\th\label{corollary 3.11} There always exists a non-zero $(\sigma, \tau)$-derivation of $O_{K} = \mathbb{Z}[\zeta]$. \end{corollary} We propose the following conjecture. \begin{conjecture}\th\label{conjecture 3.12} Suppose $\beta = \sum_{i=0}^{p-2} b_{i} \zeta^{i} \in O_{K}$ and $\beta (\tau - \sigma) (\zeta) = \sum_{i=0}^{p-2} \left( \sum_{j=0}^{p-2} a_{ij} b_{j} \right) \zeta^{i}$. Then $A = [a_{ij}]$ is a $(p-1) \times (p-1)$ matrix with determinant $p$. \end{conjecture} We have verified the conjecture for all odd primes less than $100$ using SAGE and MATLAB. Calculations for primes $3, 5, 7, 11, 13$ were done on SAGE (explicit matrix $A$ and $\text{det}(A)$ were found for all possible values of the pair $(\sigma(\zeta), \tau(\zeta))$ (see the appendix)). We then developed a code on MATLAB and calculations for the remaining primes were done by running that MATLAB code. For larger primes, MATLAB took a long time to provide the output after running the code. Nevertheless, we believe the conjecture to be true for all odd primes. As a consequence, we propose another conjecture which too will hold once the above conjecture is proved. Let $\beta = \sum_{i=0}^{p-2} b_{i} \zeta^{i} \in K$, $D$ be a $(\sigma, \tau)$-derivation of $O_{K}$ and $D(\zeta) = \sum_{i=0}^{p-2} c_{i} \zeta^{i} \in O_{K}$. Put $X^{T} = (b_{0} ~ b_{1} ~ ... ~ b_{p-2})$ and $C^{T} = (c_{0} ~ c_{1} ~ ... ~ c_{p-2})$. We denote by $\mathbb{Z}^{p-1}$ the set of all $(p-1) \times 1$ column matrices with entries from $\mathbb{Z}$. Also, $Adj(A)$ denotes the adjoint of the matrix $A$. Note that since $\sigma \neq \tau$, therefore, $D(\zeta) = \beta(\tau - \sigma)(\zeta)$ always has a solution $\beta$ in $K$. Furthermore, $D(\zeta) = \beta(\tau - \sigma)(\zeta)$ has a solution $\beta$ in $O_{K}$ if and only if $AX = C$ has a solution in $\mathbb{Z}^{p-1}$. \begin{conjecture}\th\label{conjecture 3.13} With the above notations, $D$ is inner if and only if $\frac{1}{p}(Adj(A)C) \in \mathbb{Z}^{p-1}$. In particular, the following conditions hold: \begin{enumerate} \item[(i)] $O_{K}$ has non-trivial outer $(\sigma, \tau)$-derivations. \item[(ii)] If $p$ divides $c_{i}$ for each $i \in \{0, 1, ..., p-2\}$, then $D$ is inner. \end{enumerate} \end{conjecture} Given below are some interesting examples. \begin{example} Let $p \geq 5$. Then not every $\mathbb{Z}$-linear map $D:\mathbb{Z}[\zeta] \longrightarrow \mathbb{Z}[\zeta]$ with $D(1)=0$ is a $(\sigma, \tau)$-derivation. In fact, there exists a non-zero $\mathbb{Z}$-linear map $D:\mathbb{Z}[\zeta] \longrightarrow \mathbb{Z}[\zeta]$ with $D(1)=0$ which is not a $(\sigma, \tau)$-derivation for every pair $(\sigma, \tau)$ ($\sigma \neq 0, \tau \neq 0, \sigma \neq \tau$). As an example, define $D:\mathbb{Z}[\zeta] \longrightarrow \mathbb{Z}[\zeta]$ by $D(\sum_{i=0}^{p-2} a_{i} \zeta^{i}) = a_{1} D(\zeta),$ where $D(\zeta) \in O_{K} \setminus \{0\}$, say, $D(\zeta) = \zeta$. Then $D$ is a non-zero well-defined $\mathbb{Z}$-linear map with $D(1) = 0.$ But in view of \th\ref{theorem 3.8}, $D$ is not a $(\sigma, \tau)$-derivation, since $$D(\zeta^{2}) \neq \left(\sum_{(i,j) \in S_{1}} \sigma(\zeta^{i}) \tau(\zeta^{j}) \right) D(\zeta).$$ \end{example} \begin{example} Let $p = 5$. Let $\sigma$ and $\tau$ be given by $\sigma(\zeta) = \zeta$ and $\tau(\zeta) = \zeta^{2}$. (i) Define $D:O_{K} \rightarrow O_{K}$ by $$D(\sum_{i=0}^{3} a_{i} \zeta^{i}) = \sum_{i=1}^{3} a_{i} D(\zeta^{i}), \hspace{0.2cm} \forall \hspace{0.2cm} a_{i} \in \mathbb{Z} \hspace{0.1cm} (i \in \{0, 1, 2, 3\}),$$ where for each $i \in \{1, 2, 3\}$, $D(\zeta^{i})$ is defined as $$D(\zeta^{i}) = \left(\sum_{(s,t) \in S_{i-1}} \sigma(\zeta^{s}) \tau(\zeta^{t})\right)D(\zeta).$$ Note that $D$ is a $\mathbb{Z}$-linear map with $D(1) = 0$. Take $D(\zeta) \in O_{K} \setminus \{0\}$. Then by \th\ref{theorem 3.8}, $D$ is a non-zero $(\sigma, \tau)$-derivation. But as shown below, $D$ is not inner if in particular, we take $D(\zeta) = \zeta$. Thus in a cyclotomic field $K$, it is not necessary that every $(\sigma, \tau)$-derivation of its ring of algebraic integers $O_{K}$ is inner. (ii) If possible, suppose that the $(\sigma, \tau)$-derivation $D:O_{K} \rightarrow O_{K}$ is inner. Then there exists some $\beta \in O_{K}$ such that $D(\alpha) = \beta(\sigma - \tau)(\alpha), \hspace{0.1cm} \forall \hspace{0.1cm} \alpha \in O_{K}$. Since $\beta \in O_{K}$, so $\beta = \sum_{i=0}^{3} b_{i} \zeta^{i}$ for some unique $b_{i} \in \mathbb{Z}$ for all $i \in \{0, 1, 2, 3\}$. \begin{eqnarray} \beta(\sigma - \tau)(\zeta) & = & \beta (\sigma(\zeta) - \tau(\zeta)) \\ & = & (b_{0} + b_{1} \zeta + b_{2} \zeta^{2} + b_{3} \zeta^{3})(\zeta - \zeta^{2}) \\ & = & (b_{2} - 2b_{3}) + (b_{0} + b_{2} - b_{3}) \zeta + (-b_{0} + b_{1} + b_{2} - b_{3}) \zeta^{2} + (-b_{1} + 2b_{2} - b_{3})\zeta^{3} \end{eqnarray} Now $\beta(\sigma - \tau)(\zeta) = D(\zeta)$ with $D(\zeta) = \zeta$ gives $$(b_{2} - 2b_{3}) + (b_{0} + b_{2} - b_{3}) \zeta + (-b_{0} + b_{1} + b_{2} - b_{3}) \zeta^{2} + (-b_{1} + 2b_{2} - b_{3})\zeta^{3} = 0 + 1 \zeta + 0 \zeta^{2} + 0 \zeta^{3}.$$ $\Rightarrow b_{2} - 2b_{3} = 0$; $b_{0} + b_{2} - b_{3} = 1$; $-b_{0} + b_{1} + b_{2} - b_{3} = 0$; $-b_{1} + 2b_{2} - b_{3} = 0$. \noindent This system of linear equations can be written as $AX = C$, where $$A = \begin{pmatrix} 0 & 0 & 1 & -2 \\ 1 & 0 & 1 & -1 \\ -1 & 1 & 1 & -1 \\ 0 & -1 & 2 & -1 \end{pmatrix}, ~ X^{T} = \begin{pmatrix} b_{0} & b_{1} & b_{2} & b_{3} \end{pmatrix} ~ \text{and} ~ C^{T} = \begin{pmatrix} 0 & 1 & 0 & 0 \end{pmatrix}.$$ This can have a (unique) integral solution if and only if $A$ is a unimodular matrix, that is, the determinant of $A$ is either $1$ or $-1$. But the determinant of $A$ is $5$. Therefore, there is no integer solution. Hence a contradiction. Therefore, the given $(\sigma, \tau)$-derivation $D:O_{K} \rightarrow O_{K}$ is not inner for $D(\zeta) = \zeta$. \end{example} \subsection{\texorpdfstring{$(\sigma, \tau)$}{Lg}-derivations of bi-quadratic number rings}\label{subsection 3.3} Here we present an elegant application of \th\ref{lemma 2.1} in classifying all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the bi-quadratic number ring $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$. Throughout this subsection, $K$ denotes the quartic number field $K = \mathbb{Q}(\sqrt{m}, \sqrt{n})$, where $m$ and $n$ are distinct square-free rational integers, $\mathcal{A}$ the number ring $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ with $\mathbb{Z}$-basis $\{1, \sqrt{m}, \sqrt{n}, \sqrt{mn}\}$, and $\sigma$ and $\tau$ any two non-zero ring endomorphisms of $\mathcal{A}$. \begin{lemma}\th\label{lemma 3.16} If $D:\mathcal{A} \rightarrow \mathcal{A}$ is a non-zero $(\sigma, \tau)$-derivation of $\mathcal{A}$, then precisely one case amongst (i), (ii) or (iii) holds. \begin{enumerate} \item[(i)] $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$, $D(\sqrt{mn}) \neq 0$ and $D(\sqrt{mn}) = \pm \sqrt{m} D(\sqrt{n})$. Further, in this case, the possible values of the pair $(\sigma, \tau)$ are $(\phi_{1}, \phi_{2}), (\phi_{2}, \phi_{1}), (\phi_{3}, \phi_{4}), (\phi_{4}, \phi_{3})$. In fact, $D(\sqrt{mn}) = \sqrt{m} D(\sqrt{n})$ if $(\sigma, \tau) = (\phi_{1}, \phi_{2})$ or $(\phi_{2}, \phi_{1})$ and $D(\sqrt{mn}) \\ = - \sqrt{m} D(\sqrt{n})$ if $(\sigma, \tau) = (\phi_{3}, \phi_{4})$ or $(\phi_{4}, \phi_{3})$.\vspace{0.2cm} \item[(ii)] $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$, $D(\sqrt{mn}) \neq 0$ and $D(\sqrt{mn}) = \pm \sqrt{n} D(\sqrt{m})$. Further, in this case, the possible values of the pair $(\sigma, \tau)$ are $(\phi_{1}, \phi_{3}), (\phi_{2}, \phi_{4}), (\phi_{3}, \phi_{1}), (\phi_{4}, \phi_{2})$. In fact, $D(\sqrt{mn}) = \sqrt{n} D(\sqrt{m})$ if $(\sigma, \tau) = (\phi_{1}, \phi_{3})$ or $(\phi_{3}, \phi_{1})$ and $D(\sqrt{mn}) \\ = - \sqrt{n} D(\sqrt{m})$ if $(\sigma, \tau) = (\phi_{2}, \phi_{4})$ or $(\phi_{4}, \phi_{2})$.\vspace{0.2cm} \item[(iii)] $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) \neq 0$, $D(\sqrt{mn}) = 0$ and $\sqrt{m} D(\sqrt{n}) = \pm \sqrt{n} D(\sqrt{m})$. Further, in this case, the possible values of the pair $(\sigma, \tau)$ are $(\phi_{1}, \phi_{4}), (\phi_{2}, \phi_{3}), (\phi_{4}, \phi_{1}), (\phi_{3}, \phi_{2})$. In fact, $\sqrt{m} D(\sqrt{n}) = \sqrt{n} D(\sqrt{m})$ if $(\sigma, \tau) = (\phi_{1}, \phi_{4})$ or $(\phi_{4}, \phi_{1})$ and $\sqrt{m} D(\sqrt{n}) \\ = - \sqrt{n} D(\sqrt{m})$ if $(\sigma, \tau) = (\phi_{2}, \phi_{3})$ or $(\phi_{3}, \phi_{2})$. \end{enumerate} Here $\phi_{1}, \phi_{2}, \phi_{3}, \phi_{4}$ are the non-zero ring endomorphisms of $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ given by \begin{enumerate} \item[(a)] $\phi_{1}(\sqrt{m}) = \sqrt{m}$, $\phi_{1}(\sqrt{n}) = \sqrt{n}$. \item[(b)] $\phi_{2}(\sqrt{m}) = \sqrt{m}$, $\phi_{2}(\sqrt{n}) = - \sqrt{n}$. \item[(c)] $\phi_{3}(\sqrt{m}) = - \sqrt{m}$, $\phi_{3}(\sqrt{n}) = \sqrt{n}$. \item[(d)] $\phi_{4}(\sqrt{m}) = - \sqrt{m}$, $\phi_{4}(\sqrt{n}) = - \sqrt{n}$. \end{enumerate} \end{lemma} \begin{proof} If $\phi$ is a non-zero ring endomorphisms of $\mathcal{A}$, then $(\phi(\sqrt{m}))^{2} = m \hspace{0.2cm} \text{and} \hspace{0.2cm} (\phi(\sqrt{n}))^{2} = n$. So $\phi(\sqrt{m}) = \pm \sqrt{m}$ and $\phi(\sqrt{n}) = \pm \sqrt{n}.$ Therefore, $\mathcal{A}$ has precisely the four non-zero ring endomorphisms $\phi_{1}, \phi_{2}, \phi_{3}, \phi_{4}$ given in the table below. \begin{table}[H] \caption{} \centering \begin{tabular}{\|c\|\|c\|c\|c\|c\|} \hline & $1$ & $\sqrt{m}$ & $\sqrt{n}$ & $\sqrt{mn}$ \\ \hline \hline $\phi_{1}$ & $1$ & $\sqrt{m}$ & $\sqrt{n}$ & $\sqrt{mn}$ \\ \hline $\phi_{2}$ & $1$ & $\sqrt{m}$ & $- \sqrt{n}$ & $- \sqrt{mn}$ \\ \hline $\phi_{3}$ & $1$ & $- \sqrt{m}$ & $\sqrt{n}$ & $- \sqrt{mn}$ \\ \hline $\phi_{4}$ & $1$ & $- \sqrt{m}$ & $- \sqrt{n}$ & $\sqrt{mn}$ \\ \hline \end{tabular} \label{table 1} \end{table} Put $\alpha_{1} = 1$, $\alpha_{2} = \sqrt{m}$, $\alpha_{3} = \sqrt{n}$ and $\alpha_{4} = \sqrt{mn}$. Then by \th\ref{lemma 2.1}, $D(\alpha_{i} \alpha_{j}) = D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j}), \hspace{0.1cm} \forall \hspace{0.1cm} i, j \in \{1, 2, 3, 4\}$. Also, $D$ is $\mathbb{Z}$-linear. So we get the following table \ref{table 2}. \begin{table}[H] \caption{} \centering \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline $(i,j)$ & $\alpha_{i}$ & $\alpha_{j}$ & $D(\alpha_{i} \alpha_{j})$ & $D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j})$ \\ \hline \hline $(1, 1)$ & $1$ & $1$ & $0$ & $0$ \\ \hline $(1, 2)$ & $1$ & $\sqrt{m}$ & $D(\sqrt{m})$ & $D(\sqrt{m})$ \\ \hline $(1, 3)$ & $1$ & $\sqrt{n}$ & $D(\sqrt{n})$ & $D(\sqrt{n})$ \\ \hline $(1, 4)$ & $1$ & $\sqrt{mn}$ & $D(\sqrt{mn})$ & $D(\sqrt{mn})$ \\ \hline $(2, 1)$ & $\sqrt{m}$ & $1$ & $D(\sqrt{m})$ & $D(\sqrt{m})$ \\ \hline $(2, 2)$ & $\sqrt{m}$ & $\sqrt{m}$ & $0$ & $(\sigma(\sqrt{m}) + \tau(\sqrt{m}))D(\sqrt{m})$ \\ \hline $(2, 3)$ & $\sqrt{m}$ & $\sqrt{n}$ & $D(\sqrt{mn})$ & $D(\sqrt{m}) \tau(\sqrt{n}) + \sigma(\sqrt{m}) D(\sqrt{n})$ \\ \hline $(2, 4)$ & $\sqrt{m}$ & $\sqrt{mn}$ & $mD(\sqrt{n})$ & $D(\sqrt{m}) \tau(\sqrt{mn}) + \sigma(\sqrt{m}) D(\sqrt{mn})$ \\ \hline $(3, 1)$ & $\sqrt{n}$ & $1$ & $D(\sqrt{n})$ & $D(\sqrt{n})$ \\ \hline $(3, 2)$ & $\sqrt{n}$ & $\sqrt{m}$ & $D(\sqrt{mn})$ & $D(\sqrt{n}) \tau(\sqrt{m}) + \sigma(\sqrt{n}) D(\sqrt{m})$ \\ \hline $(3, 3)$ & $\sqrt{n}$ & $\sqrt{n}$ & $0$ & $(\sigma(\sqrt{n}) + \tau(\sqrt{n}))D(\sqrt{n})$ \\ \hline $(3, 4)$ & $\sqrt{n}$ & $\sqrt{mn}$ & $n D(\sqrt{m})$ & $D(\sqrt{n}) \tau(\sqrt{mn}) + \sigma(\sqrt{n}) D(\sqrt{mn})$ \\ \hline $(4, 1)$ & $\sqrt{mn}$ & $1$ & $D(\sqrt{mn})$ & $D(\sqrt{mn})$ \\ \hline $(4, 2)$ & $\sqrt{mn}$ & $\sqrt{m}$ & $mD(\sqrt{n})$ & $D(\sqrt{mn}) \tau(\sqrt{m}) + \sigma(\sqrt{mn}) D(\sqrt{m})$ \\ \hline $(4, 3)$ & $\sqrt{mn}$ & $\sqrt{n}$ & $nD(\sqrt{m})$ & $D(\sqrt{mn}) \tau(\sqrt{n}) + \sigma(\sqrt{mn}) D(\sqrt{n})$ \\ \hline $(4, 4)$ & $\sqrt{mn}$ & $\sqrt{mn}$ & $0$ & $(\sigma(\sqrt{mn}) + \tau(\sqrt{mn}))D(\sqrt{mn})$ \\ \hline \end{tabular} \label{table 2} \end{table} From Table \ref{table 2} containing the values of $D(\alpha_{i} \alpha_{j})$ and $D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j})$ ($i, j \in \{1, 2, 3, 4\}$), we see that the relation $D(\alpha_{i} \alpha_{j}) = D(\alpha_{i}) \tau(\alpha_{j}) + \sigma(\alpha_{i}) D(\alpha_{j})$ is being satisfied by the pairs $$(i,j) = (1,1), (1,2), (1,3), (1,4), (2,1), (3,1), (4,1).$$ Then using the fact that the $\mathbb{Z}$-algebra $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ is commutative, we get that $D$ will be a $(\sigma, \tau)$-derivation if and only if it satisfies the following nine equations simultaneously. \begin{equation} \label{eq 3.4} (\sigma(\sqrt{m}) + \tau(\sqrt{m}))D(\sqrt{m}) = 0 \end{equation} \begin{equation} \label{eq 3.5} D(\sqrt{m}) \tau(\sqrt{n}) + \sigma(\sqrt{m}) D(\sqrt{n}) = D(\sqrt{mn}) \end{equation} \begin{equation} \label{eq 3.6} D(\sqrt{m}) \tau(\sqrt{mn}) + \sigma(\sqrt{m}) D(\sqrt{mn}) = m D(\sqrt{n}) \end{equation} \begin{equation} \label{eq 3.7} (\sigma(\sqrt{n}) + \tau(\sqrt{n}))D(\sqrt{n}) = 0 \end{equation} \begin{equation} \label{eq 3.8} D(\sqrt{n}) \tau(\sqrt{mn}) + \sigma(\sqrt{n}) D(\sqrt{mn}) = n D(\sqrt{m}) \end{equation} \begin{equation} \label{eq 3.9} (\sigma(\sqrt{mn}) + \tau(\sqrt{mn}))D(\sqrt{mn}) = 0 \end{equation} \begin{equation} \label{eq 3.10} D(\sqrt{n}) \tau(\sqrt{m}) + \sigma(\sqrt{n}) D(\sqrt{m}) = D(\sqrt{mn}) \end{equation} \begin{equation} \label{eq 3.11} D(\sqrt{mn}) \tau(\sqrt{m}) + \sigma(\sqrt{mn}) D(\sqrt{m}) = m D(\sqrt{n}) \end{equation} \begin{equation} \label{eq 3.12} D(\sqrt{mn}) \tau(\sqrt{n}) + \sigma(\sqrt{mn}) D(\sqrt{n}) = n D(\sqrt{m}) \end{equation} Since $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ is an integral domain, so in view of equations (\ref{eq 3.4}), (\ref{eq 3.7}) and (\ref{eq 3.9}), there are four possible cases.\vspace{0.2cm} Case 1: All of $D(\sqrt{m})$, $D(\sqrt{n})$ and $D(\sqrt{mn})$ are non-zero. Since $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ is an integral domain, therefore, from (\ref{eq 3.4}), (\ref{eq 3.7}) and (\ref{eq 3.9}), $$\tau(\sqrt{m}) = - \sigma(\sqrt{m}), \hspace{0.2cm} \tau(\sqrt{n}) = - \sigma(\sqrt{n}) \hspace{0.2cm} \text{and} \hspace{0.2cm} \tau(\sqrt{mn}) = - \sigma(\sqrt{mn}).$$ But from Table \ref{table 1}, we observe that there is no such pair $(\sigma, \tau)$ which satisfies the above three equations simultaneously. Therefore, this case is not possible.\vspace{0.2cm} Case 2: Exactly one of $D(\sqrt{m})$, $D(\sqrt{n})$ and $D(\sqrt{mn})$ is non-zero. Then there are three possible subcases. Subcase 2.1: $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$, $D(\sqrt{mn}) \neq 0$. By equations (\ref{eq 3.7}) and (\ref{eq 3.9}) respectively, $\tau(\sqrt{n}) = - \sigma(\sqrt{n})$ and $\tau(\sqrt{mn}) = - \sigma(\sqrt{mn})$. Also, by (\ref{eq 3.5}) and (\ref{eq 3.10}), $\tau(\sqrt{m}) = \sigma(\sqrt{m})$. Therefore, in view of Table \ref{table 1}, the possible values of the pair $(\sigma, \tau)$ are $$(\phi_{1}, \phi_{2}), \hspace{0.2cm} (\phi_{2}, \phi_{1}), \hspace{0.2cm} (\phi_{3}, \phi_{4}), \hspace{0.2cm} (\phi_{4}, \phi_{3}).$$ Further, from (\ref{eq 3.5}), $D(\sqrt{mn}) = \sigma(\sqrt{m}) D(\sqrt{n})$ and $\sigma(\sqrt{m}) = \pm \sqrt{m}$, therefore, $D$ must satisfy $D(\sqrt{mn}) = \pm \sqrt{m} D(\sqrt{n})$. More precisely, we have the following table \ref{table 3}: \begin{table}[H] \caption{} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline $(\sigma, \tau)$ & $D(\sqrt{mn}) = \pm \sqrt{m} D(\sqrt{n})?$ \\ \hline \hline $(\phi_{1}, \phi_{2})$ & $D(\sqrt{mn}) = \sqrt{m} D(\sqrt{n})$ \\ \hline $(\phi_{2}, \phi_{1})$ & $D(\sqrt{mn}) = \sqrt{m} D(\sqrt{n})$ \\ \hline $(\phi_{3}, \phi_{4})$ & $D(\sqrt{mn}) = - \sqrt{m} D(\sqrt{n})$ \\ \hline $(\phi_{4}, \phi_{3})$ & $D(\sqrt{mn}) = - \sqrt{m} D(\sqrt{n})$ \\ \hline \end{tabular} \label{table 3} \end{table} Subcase 2.2: $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$, $D(\sqrt{mn}) \neq 0$. By equations (\ref{eq 3.4}) and (\ref{eq 3.9}) respectively, $\tau(\sqrt{m}) = - \sigma(\sqrt{m})$ and $\tau(\sqrt{mn}) = - \sigma(\sqrt{mn})$. Also, by (\ref{eq 3.5}) and (\ref{eq 3.10}), $\tau(\sqrt{n}) = \sigma(\sqrt{n})$. Therefore, in view of Table \ref{table 1}, the possible values of the pair $(\sigma, \tau)$ are $$(\phi_{1}, \phi_{3}), \hspace{0.2cm} (\phi_{2}, \phi_{4}), \hspace{0.2cm} (\phi_{3}, \phi_{1}), \hspace{0.2cm} (\phi_{4}, \phi_{2}).$$ Further from (\ref{eq 3.10}), $D(\sqrt{mn}) = \sigma(\sqrt{n}) D(\sqrt{m})$ and $\sigma(\sqrt{n}) = \pm \sqrt{n}$, therefore, $D$ must satisfy $D(\sqrt{mn}) = \pm \sqrt{n} D(\sqrt{m})$. More precisely, we have the following table \ref{table 4}: \begin{table}[H] \caption{} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline $(\sigma, \tau)$ & $D(\sqrt{mn}) = \pm \sqrt{n} D(\sqrt{m})?$ \\ \hline \hline $(\phi_{1}, \phi_{3})$ & $D(\sqrt{mn}) = \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{2}, \phi_{4})$ & $D(\sqrt{mn}) = - \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{3}, \phi_{1})$ & $D(\sqrt{mn}) = \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{4}, \phi_{2})$ & $D(\sqrt{mn}) = - \sqrt{n} D(\sqrt{m})$ \\ \hline \end{tabular} \label{table 4} \end{table} Subcase 2.3: $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) \neq 0$, $D(\sqrt{mn}) = 0$. By equations (\ref{eq 3.4}) and (\ref{eq 3.7}) respectively, $\tau(\sqrt{m}) = - \sigma(\sqrt{m})$ and $\tau(\sqrt{n}) = - \sigma(\sqrt{n})$. Also, by equations (\ref{eq 3.6}) and (\ref{eq 3.11}), $\tau(\sqrt{mn}) = \sigma(\sqrt{mn})$. Therefore, in view of Table \ref{table 1}, the possible values of the pair $(\sigma, \tau)$ are $$(\phi_{1}, \phi_{4}), \hspace{0.2cm} (\phi_{2}, \phi_{3}), \hspace{0.2cm} (\phi_{4}, \phi_{1}), \hspace{0.2cm} (\phi_{3}, \phi_{2}).$$ Further from equation (\ref{eq 3.11}), $m D(\sqrt{n}) = \sigma(\sqrt{mn}) D(\sqrt{m})$ and $\sigma(\sqrt{mn}) = \pm \sqrt{mn}$, therefore, $D$ must satisfy $m D(\sqrt{n}) = \pm \sqrt{mn} D(\sqrt{m})$ so that $\sqrt{m} D(\sqrt{n}) = \pm \sqrt{n} D(\sqrt{m})$. More precisely, we have the following table \ref{table 5}: \begin{table}[H] \caption{} \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline $(\sigma, \tau)$ & $\sqrt{m} D(\sqrt{n}) = \sqrt{n} D(\sqrt{m})?$ \\ \hline \hline $(\phi_{1}, \phi_{4})$ & $\sqrt{m} D(\sqrt{n}) = \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{2}, \phi_{3})$ & $\sqrt{m} D(\sqrt{n}) = - \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{3}, \phi_{2})$ & $\sqrt{m} D(\sqrt{n}) = - \sqrt{n} D(\sqrt{m})$ \\ \hline $(\phi_{4}, \phi_{1})$ & $\sqrt{m} D(\sqrt{n}) = \sqrt{n} D(\sqrt{m})$ \\ \hline \end{tabular} \label{table 5} \end{table} Case 3: Exactly two of $D(\sqrt{m})$, $D(\sqrt{n})$ or $D(\sqrt{mn})$ are $0$. Then there are three possible subcases. Subcase 3.1: $D(\sqrt{m}) = 0$, $D(\sqrt{n}) = 0$, $D(\sqrt{mn}) \neq 0$. Subcase 3.2: $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$, $D(\sqrt{mn}) = 0$. Subcase 3.3: $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$, $D(\sqrt{mn}) = 0$. Then in each of the three possible subcases, $D$ will become a zero derivation using equations (\ref{eq 3.4}) to (\ref{eq 3.12}) and the facts that $\{1, \sqrt{m}, \sqrt{n}, \sqrt{mn}\}$ is a $\mathbb{Z}$-basis of the $\mathbb{Z}$-module $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ and $D:\mathcal{A} \rightarrow \mathcal{A}$ is $\mathbb{Z}$-linear with $D(1)=0$. But this gives a contradiction because $D$ is non-zero. Therefore, this case is not possible.\vspace{0.2cm} Case 4: All $D(\sqrt{p})$, $D(\sqrt{q})$ and $D(\sqrt{pq})$ are zero. The proof is similar to Case 3. \end{proof} The proof of the following lemma is trivial using \th\ref{lemma 2.1}. \begin{lemma}\th\label{lemma 3.17} Let $D:\mathcal{A} \rightarrow \mathcal{A}$ be any $\mathbb{Z}$-linear map with $D(1)=0$. Then $D$ is a $(\sigma, \tau)$-derivation in the following cases. \begin{enumerate} \item[(i)] $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) \neq 0$. \begin{itemize} \item[(a)] If $D(\sqrt{mn}) = \sqrt{m} D(\sqrt{n})$ and $(\sigma, \tau) = (\phi_{1}, \phi_{2})$ or $(\phi_{2}, \phi_{1})$. \item[(b)] If $D(\sqrt{mn}) = - \sqrt{m} D(\sqrt{n})$ and $(\sigma, \tau) = (\phi_{3}, \phi_{4})$ or $(\phi_{4}, \phi_{3})$. \end{itemize} \item[(ii)] $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$ and $D(\sqrt{mn}) \neq 0$. \begin{itemize} \item[(a)] If $D(\sqrt{mn}) = \sqrt{n} D(\sqrt{m})$ and $(\sigma, \tau) = (\phi_{1}, \phi_{3})$ or $(\phi_{3}, \phi_{1})$. \item[(b)] If $D(\sqrt{mn}) = - \sqrt{n} D(\sqrt{m})$ and $(\sigma, \tau) = (\phi_{2}, \phi_{4})$ or $(\phi_{4}, \phi_{2})$. \end{itemize} \item[(iii)] $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) = 0$. \begin{itemize} \item[(a)] If $\sqrt{m} D(\sqrt{n}) = \sqrt{n} D(\sqrt{m})$ and $(\sigma, \tau) = (\phi_{1}, \phi_{4})$ or $(\phi_{4}, \phi_{1})$. \item[(b)] If $\sqrt{m} D(\sqrt{n}) = - \sqrt{n} D(\sqrt{m})$ and $(\sigma, \tau) = (\phi_{2}, \phi_{3})$ or $(\phi_{3}, \phi_{2})$. \end{itemize} \end{enumerate} \end{lemma} \begin{theorem}\th\label{theorem 3.18} A non-zero $\mathbb{Z}$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ with $D(1)=0$ is a $(\sigma, \tau)$-derivation if and only if exactly one of the cases (i), (ii) or (iii) of \th\ref{lemma 3.16} holds. \end{theorem} \begin{corollary}\th\label{corollary 3.19} The $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(\mathcal{A})$ is finitely generated of rank $4$. \end{corollary} \begin{proof} $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ has exactly four distinct ring endomorphisms as given in Table \ref{table 1}. In view of \th\ref{theorem 3.18}, there are three possible cases. Case 1: $(\sigma, \tau) \in \{(\phi_{1}, \phi_{2}), (\phi_{2}, \phi_{1}), (\phi_{3}, \phi_{4}), (\phi_{4}, \phi_{3})\}$. For each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(1) = 0$ such that $$D_{1}(\sqrt{n}) = 1, \hspace{0.2cm} D_{2}(\sqrt{n}) = \sqrt{m}, \hspace{0.2cm} D_{3}(\sqrt{n}) = \sqrt{n}, \hspace{0.2cm} D_{4}(\sqrt{n}) = \sqrt{mn}.$$ More precisely, for each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ by $$D_{i}(a_{1} + a_{2} \sqrt{m} + a_{3} \sqrt{n} + a_{4} \sqrt{mn}) = \begin{cases} (a_{3} + a_{4} \sqrt{m}) D_{i}(\sqrt{n}) & \text{if $(\sigma, \tau) = (\phi_{1}, \phi_{2}) \hspace{0.2cm} \text{or} \hspace{0.2cm} (\phi_{2}, \phi_{1})$} \\ (a_{3} - a_{4} \sqrt{m}) D_{i}(\sqrt{n}) & \text{if $(\sigma, \tau) = (\phi_{3}, \phi_{4}) \hspace{0.2cm} \text{or} \hspace{0.2cm} (\phi_{4}, \phi_{3})$} \end{cases}$$ where for each $i \in \{1, 2, 3, 4\}$, $D_{i}(\sqrt{n})$ is defined as above.\vspace{10pt} Case 2: $(\sigma, \tau) \in \{(\phi_{1}, \phi_{3}), (\phi_{3}, \phi_{1}), (\phi_{2}, \phi_{4}), (\phi_{4}, \phi_{2})\}$. For each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(1) = 0$ such that $$D_{1}(\sqrt{m}) = 1, \hspace{0.2cm} D_{2}(\sqrt{m}) = \sqrt{m}, \hspace{0.2cm} D_{3}(\sqrt{m}) = \sqrt{n}, \hspace{0.2cm} D_{4}(\sqrt{m}) = \sqrt{mn}.$$ More precisely, for each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ by $$D_{i}(a_{1} + a_{2} \sqrt{m} + a_{3} \sqrt{n} + a_{4} \sqrt{mn}) = \begin{cases} (a_{2} + a_{4} \sqrt{n}) D_{i}(\sqrt{m}) & \text{if $(\sigma, \tau) = (\phi_{1}, \phi_{3}) \hspace{0.2cm} \text{or} \hspace{0.2cm} (\phi_{3}, \phi_{1})$} \\ (a_{2} - a_{4} \sqrt{n}) D_{i}(\sqrt{m}) & \text{if $(\sigma, \tau) = (\phi_{2}, \phi_{4}) \hspace{0.2cm} \text{or} \hspace{0.2cm} (\phi_{4}, \phi_{2})$} \end{cases}$$ where for each $i \in \{1, 2, 3, 4\}$, $D_{i}(\sqrt{m})$ is defined as above.\vspace{10pt} Case 3: $(\sigma, \tau) \in \{(\phi_{1}, \phi_{4}), (\phi_{4}, \phi_{1}), (\phi_{2}, \phi_{3}), (\phi_{3}, \phi_{2})\}$. Suppose $\text{gcd}(m, n) = k$. Then $m = kr$ and $n = ks$ for some $r, s \in \mathbb{Z}$. For each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ as a $\mathbb{Z}$-linear map with $D(1) = 0$ such that $$D_{1}(\sqrt{m}) = m, \hspace{0.2cm} D_{2}(\sqrt{m}) = \sqrt{m}, \hspace{0.2cm} D_{3}(\sqrt{m}) = r \sqrt{n}, \hspace{0.2cm} D_{4}(\sqrt{m}) = \sqrt{mn}$$ so that $$D_{1}(\sqrt{n}) = \sqrt{mn}, \hspace{0.2cm} D_{2}(\sqrt{n}) = \sqrt{n}, \hspace{0.2cm} D_{3}(\sqrt{n}) = s \sqrt{m}, \hspace{0.2cm} D_{4}(\sqrt{n}) = n.$$ More precisely, for each $i \in \{1, 2, 3, 4\}$, define $D_{i}:\mathcal{A} \rightarrow \mathcal{A}$ by $$D_{i}(a_{1} + a_{2} \sqrt{m} + a_{3} \sqrt{n} + a_{4} \sqrt{mn}) = a_{2} D_{i}(\sqrt{m}) + a_{3} D_{i}(\sqrt{n})$$ where for each $i \in \{1, 2, 3, 4\}$, $D_{i}(\sqrt{m})$ and $D_{i}(\sqrt{n})$ are defined as above. In all three cases, it can be easily verified that $\{D_{1}, D_{2}, D_{3}, D_{4}\}$ is a linearly independent subset of the $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(\mathcal{A})$ that generates it. \end{proof} We note from the preceding corollary that the rank of the $\mathbb{Z}$-module $\mathcal{D}_{(\sigma, \tau)}(\mathcal{A})$ is equal to $4$, the degree of $K = \mathbb{Q}(\sqrt{m}, \sqrt{n})$ and which, by \th\ref{theorem 1.2}, is also equal to the rank of the free abelian group $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ as a $\mathbb{Z}$-module. \begin{corollary}\th\label{corollary 3.20} There always exists a non-zero $(\sigma, \tau)$-derivation of $\mathcal{A}$. \end{corollary} \begin{corollary}\th\label{corollary 3.21} There always exists a non-zero $\mathbb{Z}$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ with $D(1)=0$ that is not a $(\sigma, \tau)$-derivation. In fact, there exists a non-zero $\mathbb{Z}$-linear map $D:\mathcal{A} \rightarrow \mathcal{A}$ with $D(1)=0$ which is not a $(\sigma, \tau)$-derivation for every pair $(\sigma, \tau)$ of any two different non-zero ring endomorphisms of $\mathcal{A}$. \end{corollary} \begin{theorem}\th\label{theorem 3.22} Let $D:\mathcal{A} \rightarrow \mathcal{A}$ be a non-zero $(\sigma, \tau)$-derivation. \begin{enumerate} \item[(i)] If $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) \neq 0$, then $D$ is inner if and only if $\frac{D(\sqrt{n})}{2 \sqrt{n}} \in \mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$. In particular, $\mathcal{A}$ has non-trivial outer $(\sigma, \tau)$-derivations, where $(\sigma, \tau) \in \{(\phi_{1}, \phi_{2}), (\phi_{2}, \phi_{1}), (\phi_{3}, \phi_{4}), (\phi_{4}, \phi_{3})\}$. \item[(ii)] If $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$ and $D(\sqrt{mn}) \neq 0$, then $D$ is inner if and only if $\frac{D(\sqrt{}m)}{2 \sqrt{m}} \in \mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$. In particular, $\mathcal{A}$ has non-trivial outer $(\sigma, \tau)$-derivations, where $(\sigma, \tau) \in \{(\phi_{1}, \phi_{3}), (\phi_{2}, \phi_{4}), (\phi_{3}, \phi_{1}), (\phi_{4}, \phi_{2})\}$. \item[(iii)] If $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) = 0$, then $D$ is inner if and only if either $\frac{D(\sqrt{}m)}{2 \sqrt{m}} \in \mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ or $\frac{D(\sqrt{n})}{2 \sqrt{n}} \in \mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$. In particular, $\mathcal{A}$ has non-trivial outer $(\sigma, \tau)$-derivations, where $(\sigma, \tau) \in \{(\phi_{1}, \phi_{4}), (\phi_{2}, \phi_{3}), (\phi_{4}, \phi_{1}), (\phi_{3}, \phi_{2})\}$. \end{enumerate} \end{theorem} \begin{proof} $\mathcal{A} = \mathbb{Z}[\sqrt{m}, \sqrt{n}]$ has exactly four distinct ring endomorphisms as given in Table \ref{table 1}. In view of \th\ref{theorem 3.18}, there are three possible cases. Case 1: $D(\sqrt{m}) = 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) \neq 0$. In this case, $(\sigma, \tau) \in \{(\phi_{1}, \phi_{2}), (\phi_{2}, \phi_{1}), (\phi_{3}, \phi_{4}), (\phi_{4}, \phi_{3})\}$. In view of Subcase 2.1 in the proof of \th\ref{lemma 3.16}, in this case, $$\tau(\sqrt{m}) = \sigma(\sqrt{m}), \hspace{0.2cm} \tau(\sqrt{n}) = - \sigma(\sqrt{n}) \hspace{0.2cm} \text{and} \hspace{0.2cm} \tau(\sqrt{mn}) = - \sigma(\sqrt{mn}).$$ Note that $(\tau - \sigma)(1) = 0$, $(\tau - \sigma)(\sqrt{m}) = 0$, $(\tau - \sigma)(\sqrt{n}) = 2 \tau(\sqrt{n})$ and $(\tau - \sigma)(\sqrt{mn}) = 2 \tau(\sqrt{mn}) = 2 \tau(\sqrt{m}) \tau(\sqrt{n})$. By \th\ref{theorem 2.5}, $D$ is inner if and only if there is a $\beta \in \mathcal{A}$ with $$D(1) = \beta (\tau - \sigma)(1), \hspace{0.2cm} D(\sqrt{m}) = \beta (\tau - \sigma)(\sqrt{m}),$$ $$D(\sqrt{n}) = \beta (\tau - \sigma)(\sqrt{n}), \hspace{0.2cm} D(\sqrt{mn}) = \beta (\tau - \sigma)(\sqrt{mn}).$$ Note that $D(1) = \alpha (\tau - \sigma)(1)$ and $D(\sqrt{m}) = \alpha (\tau - \sigma)(\sqrt{m})$ hold for all $\alpha \in \mathcal{A}$. So $D$ will be inner if and only if $$D(\sqrt{n}) = \beta (\tau - \sigma)(\sqrt{n}) \hspace{0.2cm} \text{and} \hspace{0.2cm} D(\sqrt{mn}) = \beta (\tau - \sigma)(\sqrt{mn}),$$ that is, if and only if $$D(\sqrt{n}) = 2 \beta \tau (\sqrt{n}) \hspace{0.2cm} \text{and} \hspace{0.2cm} D(\sqrt{mn}) = 2 \beta \tau (\sqrt{m}) \tau (\sqrt{n}).$$ \noindent Further since in this case, $D(\sqrt{mn}) = \begin{cases} \sqrt{m} D(\sqrt{n}) \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\phi_{2}$} \\ - \sqrt{m} D(\sqrt{n}) \hspace{0.2cm} \text{if $\tau = \phi_{3}$ or $\phi_{4}$} \end{cases}$ and \\ $2 \beta \tau(\sqrt{m}) \tau(\sqrt{n}) = \begin{cases} 2 \beta \sqrt{mn} \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\tau = \phi_{4}$} \\ - 2 \beta \sqrt{mn} \hspace{0.2cm} \text{if $\tau = \phi_{2}$ or $\tau = \phi_{3}$} \end{cases}$ therefore, \\ $D(\sqrt{mn}) = 2 \beta \tau (\sqrt{m}) \tau (\sqrt{n})$ if and only if $D(\sqrt{n}) = \begin{cases} 2 \beta \sqrt{n} \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\phi_{3}$} \\ - 2 \beta \sqrt{n} \hspace{0.2cm} \text{if $\tau = \phi_{2}$ or $\phi_{4}$} \end{cases}.$ From the above discussion, it can be easily concluded that $D$ is inner if and only if $$D(\sqrt{n}) = \begin{cases} 2 \beta \sqrt{n} \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\phi_{3}$} \\ - 2 \beta \sqrt{n} \hspace{0.2cm} \text{if $\tau = \phi_{2}$ or $\phi_{4}$} \end{cases}.$$ Therefore, $D$ is inner if and only if there is a $\beta \in \mathcal{A}$ with $$\beta = \begin{cases} \frac{D(\sqrt{n})}{2 \sqrt{n}} \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\phi_{3}$} \\ - \frac{D(\sqrt{n})}{2 \sqrt{n}} \hspace{0.2cm} \text{if $\tau = \phi_{2}$ or $\phi_{4}$} \end{cases}.$$ Therefore, in this case, $D$ is inner if and only if $\frac{D(\sqrt{n})}{2 \sqrt{n}} \in \mathcal{A}$.\vspace{10pt} Case 2: $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) = 0$ and $D(\sqrt{mn}) \neq 0$. As in Case 1, it can be shown that $D$ is inner if and only if there is a $\beta \in \mathcal{A}$ with $$\beta = \begin{cases} \frac{D(\sqrt{m})}{2 \sqrt{m}} \hspace{0.2cm} \text{if $\tau = \phi_{1}$ or $\phi_{2}$} \\ - \frac{D(\sqrt{m})}{2 \sqrt{m}} \hspace{0.2cm} \text{if $\tau = \phi_{3}$ or $\phi_{4}$} \end{cases}.$$ Therefore, in this case, $D$ is inner if and only if $\frac{D(\sqrt{m})}{2 \sqrt{m}} \in \mathcal{A}$.\vspace{10pt} Case 3: $D(\sqrt{m}) \neq 0$, $D(\sqrt{n}) \neq 0$ and $D(\sqrt{mn}) = 0$. On similar lines, it can be shown that in this case, $D$ is inner if and only if either $\frac{D(\sqrt{m})}{2 \sqrt{m}} \in \mathcal{A}$ or $\frac{D(\sqrt{n})}{2 \sqrt{n}} \in \mathcal{A}$ (as $\frac{D(\sqrt{m})}{2 \sqrt{m}} = \frac{D(\sqrt{n})}{2 \sqrt{n}}$). Hence the result. \end{proof} The corollary is immediate. \begin{corollary}\th\label{corollary 3.23} There always exists a non-zero $(\sigma, \tau)$-inner derivation of $\mathcal{A}$. \end{corollary} \section{Coding Theory Applications}\label{section 4} In this section, we give the applications of the work of previous sections in coding theory. We give the notion of a Hom-IDD code over finite fields. \subsection{IDD Code in $O_{K}$}\label{subsection 4.1} Let $K = \mathbb{Q}(\theta)$ ($\theta$ an algebraic integer) be a number field of degree $n$ over $\mathbb{Q}$, which is a normal extension of $\mathbb{Q}$ and $\mathcal{B} = \{\theta_{1} = \theta, \theta_{2}, ..., \theta_{n}\}$ be an ordered integral basis of $O_{K}$. Let $\sigma, \tau:O_{K} \rightarrow O_{K}$ be two different $\mathbb{Z}$-algebra endomorphisms of $O_{K}$. Let $D$ be a $(\sigma, \tau)$-derivation of $O_{K}$. The range $\text{Im}(D)$ of $D$ is a $\mathbb{Z}$-submodule of the $\mathbb{Z}$-module $O_{K}$ with the generating set $D(\mathcal{B}) = \{D(\theta_{i}) \| 1 \leq i \leq n\}$. It generates an $(n,r)$-code in $O_{K}$, where $r = \text{rank}(Im(D))$, the rank of the $\mathbb{Z}$-module $\text{Im}(D)$. Suppose $S = \{D(\theta_{i_{1}}),D(\theta_{i_{2}}),...,D(\theta_{i_{s}})\}$ is a $\mathbb{Z}$-linearly independent subset of $\text{Im}(D)$. Then $S$ generates a submodule of $\text{Im}(D)$ with rank $s$. Thus, $S$ generates an $(n,s)$-code. \begin{definition} Let $T = \{\theta_{k_{1}}, \theta_{k_{2}}, ..., \theta_{k_{t}}\}$ ($\{k_{1}, k_{2}, ..., k_{t}\} \subseteq \{1, 2, ..., n\}$) be a subset of the integral basis $\{\theta_{1}, \theta_{2}, ..., \theta_{n}\}$ such that $D(T)$ is $\mathbb{Z}$-linearly independent subset of $O_{K}$ (or $\text{Im}(D)$). Then the submodule $W = \langle D(T) \rangle$ generates an $(n,t)$-code, and is called as Image of Derivation-Derived Code, or simply an IDD Code. \end{definition} \subsection{Equivalent IDD Code in $\mathbb{Z}^{n}$}\label{subsection 4.2} In this subsection, we construct a code in $\mathbb{Z}^{n}$ equivalent to an IDD code. Suppose $\Delta:\mathbb{Z}^{n} \rightarrow O_{K}$ be the mapping given by $$\Delta(a_{1}, a_{2}, ..., a_{n}) = \sum_{i=1}^{n} a_{i} \theta_{i}.$$ \noindent For each $i \in \{1, 2, ..., n\}$, suppose $D(\theta_{i}) = \sum_{j=1}^{n} a_{ij}\theta_{j}$. Then $\Delta^{-1}(D(\theta_{i})) = (a_{i1}, a_{i2}, ..., a_{in})$. Construct an $n \times n$ matrix $B$ with the rows $\Delta^{-1}(D(\theta_{1})), \Delta^{-1}(D(\theta_{2})), ..., \Delta^{-1}(D(\theta_{n}))$. More precisely, $$B = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & ... & a_{nn} \\ \end{pmatrix}.$$ \begin{theorem} Any rows $i_{1}, i_{2}, ..., i_{t}$ of $B$ are $\mathbb{Z}$-linearly independent if and only if the set $\{D(\theta_{i_{1}}), D(\theta_{i_{2}}), ..., D(\theta_{i_{t}})\}$ is $\mathbb{Z}$-linearly independent. \end{theorem} Below, we give the steps of construction: \begin{enumerate} \item[(i)] Suppose $W$ is an Image of Derivations-Derived (IDD) code. Then $W = \langle S \rangle$, for some $\mathbb{Z}$-linearly independent subset $S$ of $D(\mathcal{B})$, say, $S = \{D(\theta_{k_{1}}), D(\theta_{k_{2}}), ..., D(\theta_{k_{s}})\}$.\vspace{6pt} \item[(ii)] Pick $\underline{w} \in \mathbb{Z}^{s}$ and $\underline{w} = (\lambda_{1}, \lambda_{2}, ..., \lambda_{s})$.\vspace{6pt} \item[(iii)] Write $\underline{w}$ as $\underline{x}$ in $\mathbb{Z}^{n}$ with $\lambda_{j}$ in position $k_{j}$ and zero elsewhere.\vspace{6pt} \item[(iv)] $\underline{x}$ can be mapped to an element $O_{K}$ by $x = \Delta(\underline{x}) = \sum_{j=1}^{s}\lambda_{j}\theta_{k_{j}}$ and a codeword $$D(x) = D(\Delta(\underline{x})) = D(\sum_{j=1}^{s}\lambda_{j}\theta_{k_{j}}) = \sum_{j=1}^{s}\lambda_{j}D(\theta_{k_{j}})$$ equated with a codeword in $\mathcal{E}$ given by $$\Delta^{-1}(D(x)) = \underline{x}B.$$ \item[(v)] Thus, we obtain an $(n,s)$ code $$\mathcal{E} = \{\underline{x}B \mid \underline{x} ~ \text{extends} ~ \underline{w} \in \mathbb{Z}^{s} ~ \text{to} ~ \mathbb{Z}^{n}\}.$$ \end{enumerate} \subsection{Hom-IDD Code in $(\mathbb{Z}_{q})^{n}$}\label{subsection 4.3} For a positive integer $q$ and for an equivalent IDD code $C$ in $\mathbb{Z}^{n}$, we construct a Hom-IDD code $\Omega(C)$ in $(\mathbb{Z}_{q})^{n}$. \begin{lemma}\label{lemma 4.3} Let $q$ be a positive integer. Then the map $\Omega:\mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$ defined by $$\Omega(\lambda_{1}, \lambda_{2}, ..., \lambda_{n}) = (\overline{\lambda_{1}}, \overline{\lambda_{2}}, ..., \overline{\lambda_{n}}),$$ where $\overline{\lambda_{i}} = \lambda_{i}$ (mod $q$) for each $i \in \{1, 2, ..., n\}$ is a $\mathbb{Z}$-module homomorphism. \end{lemma} \begin{proof} Both $\mathbb{Z}^{n}$ and $(\mathbb{Z}_{q})^{n}$ are abelian groups with respect to the usual operations of componentwise addition. And since every abelian group is a $\mathbb{Z}$-module with respect to the usual operation of scalar multiplication. More precisely, if $M$ is an additive abelian group, then $M$ is a $\mathbb{Z}$-module with respect to the scalar multiplication operation defined by: $\mathbb{Z} \times M \rightarrow M$, where $$(\lambda, m) \mapsto \lambda m,$$ where $$\lambda m = \begin{cases} \underbrace{m + m + ... + m}_{\lambda ~ \text{times}} & \text{if} ~ \lambda > 0 \\ \underbrace{(-m) + (-m) + ... + (-m)}_{-\lambda ~ \text{times}} & \text{if} ~ \lambda < 0 \\ 0 & \text{if} ~ \lambda = 0 \end{cases}.$$ First note that $\Omega$ is indeed a well-defined map from $\mathbb{Z}^{n}$ to $(\mathbb{Z}_{q})^{n}$. Let $x = (\lambda_{1}, \lambda_{2}, ..., \lambda_{n}), y = (\mu_{1}, \mu_{2}, ..., \mu_{n}) \in \mathbb{Z}^{n}$ and $\lambda \in \mathbb{Z}$. Then \begin{align} \Omega(x +_{1} y) & = \Omega(\lambda_{1} + \mu_{1}, \lambda_{2} + \mu_{2}, ..., \lambda_{n} + \mu_{n}) \\ & = (\overline{\lambda_{1} + \mu_{1}}, \overline{\lambda_{2} + \mu_{2}}, ..., \overline{\lambda_{n} + \mu_{n}}), \end{align} where for each $i \in \{1, 2, ..., n\}$, $\overline{\lambda_{i} + \mu_{i}} = (\lambda_{i} + \mu_{i}) (\text{mod} ~ q)$. Now, for $i \in \{1, 2, ..., n\}$, we have that \begin{align} \overline{\lambda_{i} + \mu_{i}} & = (\lambda_{i} + \mu_{i}) (\text{mod} ~ q) \\ & = (\lambda_{i} (\text{mod} ~ q) + \mu_{i} (\text{mod} ~ n))(\text{mod} ~ q) \\ & = (\overline{\lambda_{i}} + \overline{\mu_{i}})(\text{mod} ~ q) \\ & = \overline{\lambda_{i}} \oplus_{q} \overline{\mu_{i}} \end{align} So from above, we get that \begin{align} \Omega(x +_{1} y) & = (\overline{\lambda_{1} + \mu_{1}}, \overline{\lambda_{2} + \mu_{2}}, ..., \overline{\lambda_{n} + \mu_{n}}) \\ & = (\overline{\lambda_{1}} \oplus_{q} \overline{\mu_{1}}, \overline{\lambda_{2}} \oplus_{q} \overline{\mu_{2}}, ..., \overline{\lambda_{n}} \oplus_{q} \overline{\mu_{n}}) \\ & = (\overline{\lambda_{1}}, \overline{\lambda_{2}}, ..., \overline{\lambda_{n}}) +_{2} (\overline{\mu_{1}}, \overline{\mu_{2}}, ..., \overline{\mu_{n}}) \\ & = \Omega(x) +_{2} \Omega(y) \end{align} Now, first suppose that $\lambda > 0$. \begin{align} \lambda \odot_{1} x & = \underbrace{x +_{1} x +_{1} ... +_{1} x}_{\lambda ~ \text{times}} \\ & = (\underbrace{\lambda_{1} + \lambda_{1} + ... + \lambda_{1}}_{\lambda ~ \text{times}}, \underbrace{\lambda_{2} + \lambda_{2} + ... + \lambda_{2}}_{\lambda ~ \text{times}}, ..., \underbrace{\lambda_{n} + \lambda_{n} + ... + \lambda_{n}}_{\lambda ~ \text{times}}) \\ & = (\lambda \lambda_{1}, \lambda \lambda_{2}, ..., \lambda \lambda_{n}) \end{align} Now, \begin{align} \Omega(\lambda \odot_{1} x) & = \Omega(\lambda \lambda_{1}, \lambda \lambda_{2}, ..., \lambda \lambda_{n}) \\ & = (\overline{\lambda \lambda_{1}}, \overline{\lambda \lambda_{2}}, ..., \overline{\lambda \lambda_{n}}) \end{align} Now, for each $i \in \{1, 2, ..., n\}$, \begin{align} \overline{\lambda \lambda_{i}} & = (\lambda \lambda_{i})(\text{mod} ~ q) \\ & = (((\lambda (\text{mod} ~ q)) (\lambda_{i} (\text{mod} ~ q))) (\text{mod} ~ q) \\ & = (\overline{\lambda} ~ \overline{\lambda_{i}}) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q). \end{align} \begin{align} \lambda \odot_{2} \Omega(x) & = \underbrace{\Omega(x) +_{2} \Omega(x) +_{2} ... +_{2} \Omega(x)}_{\lambda ~ \text{times}} \\ & = (\underbrace{\overline{\lambda_{1}} \oplus_{q} \overline{\lambda_{1}} \oplus_{q} ... \oplus_{q} \overline{\lambda_{1}}}_{\lambda ~ \text{times}}, \underbrace{\overline{\lambda_{2}} \oplus_{q} \overline{\lambda_{2}} \oplus_{q} ... \oplus_{q} \overline{\lambda_{2}}}_{\lambda ~ \text{times}}, ..., \underbrace{\overline{\lambda_{n}} \oplus_{q} \overline{\lambda_{n}} \oplus_{q} ... \oplus_{q} \overline{\lambda_{n}}}_{\lambda ~ \text{times}}) \end{align} Now, let $i \in \{1, 2, ..., n\}$. Then we get that \begin{align} \underbrace{\overline{\lambda_{i}} \oplus_{q} \overline{\lambda_{i}} \oplus_{q} ... \oplus_{q} \overline{\lambda_{i}}}_{\lambda ~ \text{times}} & = (\underbrace{\overline{\lambda_{i}} + \overline{\lambda_{i}} + ... + \overline{\lambda_{i}}}_{\lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = (\underbrace{\lambda_{i} + \lambda_{i} + ... + \lambda_{i}}_{\lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q) \end{align} So we have proved that $\Omega(\lambda \odot_{1} x) = \lambda \odot_{2} \Omega(x)$ for $\lambda > 0$. The above equation holds trivially for $\lambda = 0$. Now, suppose $\lambda < 0$. Then \begin{align} \lambda \odot_{1} x & = \underbrace{(-x) +_{1} (-x) +_{1} ... +_{1} (-x)}_{- \lambda ~ \text{times}} \\ & = (\underbrace{(-\lambda_{1}) + (-\lambda_{1}) + ... + (-\lambda_{1})}_{- \lambda ~ \text{times}}, \underbrace{(- \lambda_{2}) + (- \lambda_{2}) + ... + (-\lambda_{2})}_{- \lambda ~ \text{times}}, ..., \\ &\quad \underbrace{(- \lambda_{n}) + (- \lambda_{n}) + ... + (- \lambda_{n})}_{- \lambda ~ \text{times}}) \\ & = ((- \lambda) (- \lambda_{1}), (- \lambda) (- \lambda_{2}), ..., (- \lambda) (- \lambda_{n})) \\ & = (\lambda \lambda_{1}, \lambda \lambda_{2}, ..., \lambda \lambda_{n}) \end{align} Then as shown above, we will have $$\Omega(\lambda \odot_{1} x) = (\overline{\lambda \lambda_{1}}, \overline{\lambda \lambda_{2}}, ..., \overline{\lambda \lambda_{n}}),$$ where for each $i \in \{1, 2, ..., n\}$, $\overline{\lambda \lambda_{i}} = (\lambda \lambda_{i}) (\text{mod} ~ q)$. Using the fact that $\Omega$ is a group homomorphisms from the additive group $\mathbb{Z}^{n}$ to the additive group $(\mathbb{Z}_{q})^{n}$, we get that \begin{align} \lambda \odot_{2} \Omega(x) & = \underbrace{(- \Omega(x)) +_{2} (- \Omega(x)) +_{2} ... +_{2} (- \Omega(x))}_{- \lambda ~ \text{times}} \\ & = \underbrace{\Omega(-x) +_{2} \Omega(-x) +_{2} ... +_{2} \Omega(-x)}_{- \lambda ~ \text{times}} \\ & = (\underbrace{\overline{- \lambda_{1}} \oplus_{q} \overline{- \lambda_{1}} \oplus_{q} ... \oplus_{q} \overline{- \lambda_{1}}}_{- \lambda ~ \text{times}}, \underbrace{\overline{- \lambda_{2}} \oplus_{q} \overline{- \lambda_{2}} \oplus_{q} ... \oplus_{q} \overline{- \lambda_{2}}}_{- \lambda ~ \text{times}}, ..., \\ & \quad \underbrace{\overline{- \lambda_{n}} \oplus_{q} \overline{- \lambda_{n}} \oplus_{q} ... \oplus_{q} \overline{- \lambda_{n}}}_{- \lambda ~ \text{times}}) \end{align} Now, for $i \in \{1, 2, ..., n\}$, \begin{align} \underbrace{\overline{- \lambda_{i}} \oplus_{q} \overline{- \lambda_{i}} \oplus_{q} ... \oplus_{q} \overline{- \lambda_{i}}}_{- \lambda ~ \text{times}} & = (\underbrace{(\overline{- \lambda_{i}}) + (\overline{- \lambda_{i}}) + ... + (\overline{- \lambda_{i}})}_{- \lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = (\underbrace{(- \lambda_{i}) + (- \lambda_{i}) + ... + (- \lambda_{i})}_{- \lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = ((-\lambda) (- \lambda_{i})) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q) \end{align} So we have proved that $\Omega(\lambda \odot_{1} x) = \lambda \odot_{2} \Omega(x)$ for $\lambda < 0$. In all possible cases, we have proved that $\Omega(\lambda \odot_{1} x) = \lambda \odot_{2} \Omega(x)$ for all $\lambda \in \mathbb{Z}$. Therefore, $\Omega$ is a $\mathbb{Z}$-module homomorphism from the $\mathbb{Z}$-module $\mathbb{Z}^{n}$ to the $\mathbb{Z}$-module $(\mathbb{Z}_{q})^{n}$. Hence proved. \end{proof} \begin{lemma}\label{lemma 4.4} \begin{enumerate} \item[(i)] If $W$ is a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$, then $\Omega(W)$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. \item[(ii)] If $V$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$, then $\Omega^{-1}(V) = \{w \in \mathbb{Z}^{n} \mid \Omega(w) \in V\}$ is a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$. \end{enumerate} \end{lemma} \begin{proof} \textbf{(i)} Let $W$ be a $\mathbb{Z}$-submodule of the $\mathbb{Z}$-module $\mathbb{Z}^{n}$. Since $\Omega: \mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$, therefore, $\Omega(W) \subseteq (\mathbb{Z}_{q})^{n}$. Also, $\Omega(W)$ is non-empty as $W$ being a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$ is non-empty. Let $x, y \in \Omega(W)$ and $\lambda \in \mathbb{Z}$. Then $x = \Omega(w)$ and $y = \Omega(w')$ for some $w, w' \in W$. $x + y = \Omega(w) + \Omega(w') = \Omega(w + w')$ as $\Omega: \mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$ is a $\mathbb{Z}$-module homomorphism and $W \subseteq \mathbb{Z}^{n}$. As $w, w' \in W$ and $W$ is a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$, therefore, $w + w' \in W$. This implies that $x+y = \Omega(w + w') \in \Omega(W)$. Further, since $\Omega: \mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$ is a $\mathbb{Z}$-module homomorphism, therefore, $\lambda x = \lambda \Omega(w) = \Omega(\lambda w)$. Since $\lambda \in \mathbb{Z}$, $w \in W$ and $W$ is a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$, therefore, $\lambda w \in W$. This implies that $\lambda x = \Omega(\lambda w) \in \Omega(W)$. Therefore, $\Omega(W)$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. Hence proved.\vspace{16pt} \textbf{(ii)} Let $V$ be a $\mathbb{Z}$-submodule of the $\mathbb{Z}$-module $(\mathbb{Z}_{q})^{n}$. Clearly, $\Omega^{-1}(V)$ is a subset of $\mathbb{Z}^{n}$. Also, $\Omega^{-1}(V)$ is non-empty as $V$ being a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$ is non-empty (note that $0 \in V$ and $\Omega(0) = 0$). Let $w, w' \in \Omega^{-1}(V)$ and $\lambda \in \mathbb{Z}$. Then $w, w' \in \mathbb{Z}^{n}$ such that $\Omega(w), \Omega(w') \in V$. But then the fact that $V$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$ implies that $\Omega(w) + \Omega(w') \in V$. Since by Lemma \ref{lemma 4.3}, $\Omega : \mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$ is a $\mathbb{Z}$-module homomorphism and $w, w' \in \mathbb{Z}^{n}$, so $\Omega(w + w') = \Omega(w) + \Omega(w')$. Therefore, from above, we get that $\Omega(w + w') \in V$. $\Rightarrow w + w' \in \Omega^{-1}(V)$. Also, since $\lambda \in \mathbb{Z}$, $\Omega(w) \in V$ and $V$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$, therefore, $\lambda \Omega(w) \in V$. Since by Lemma \ref{lemma 4.3}, $\Omega : \mathbb{Z}^{n} \rightarrow (\mathbb{Z}_{q})^{n}$ is a $\mathbb{Z}$-module homomorphism, $w \in \mathbb{Z}^{n}$ and $\lambda \in \mathbb{Z}$, therefore, $\Omega(\lambda w) = \lambda \Omega(w)$. Therefore, from above, we get that $\Omega(\lambda w) \in V$. $\Rightarrow \lambda w \in \Omega^{-1}(V)$. Therefore, $\Omega^{-1}(V)$ is a $\mathbb{Z}$-submodule of $\mathbb{Z}^{n}$. Hence proved. \end{proof} \begin{lemma}\label{lemma 4.5} A subset $W$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$ if and only if $W$ is a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$. \end{lemma} \begin{proof} Since $(\mathbb{Z}_{q})^{n}$ is a commutative unital ring with respect to the usual operations of componentwise addition module $q$ and multiplication module $q$, therefore, $(\mathbb{Z}_{q})^{n}$ is also a $\mathbb{Z}_{q}$-module with respect to the usual operation of componentwise scalar multiplication module $q$. Let $\odot_{2}$ denote the scalar multiplication of the $\mathbb{Z}$-module $(\mathbb{Z}_{q})^{n}$ and $\odot$ denote the scalar multiplication of the $\mathbb{Z}_{q}$-module $(\mathbb{Z}_{q})^{n}$. First, let $W$ be a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. We show that $W$ is a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$. $W$ being a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$ is a non-empty subset of $(\mathbb{Z}_{q})^{n}$. Let $w, w' \in W$ and $\lambda \in \mathbb{Z}_{q}$. Suppose $w = (\lambda_{1}, \lambda_{2}, ..., \lambda_{n})$ for some $\lambda_{i} \in \mathbb{Z}_{q}$ ($1 \leq i \leq n$). Since $w, w' \in W$ and $W$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$, therefore, $w+w' \in W$. $\lambda \in \mathbb{Z}_{q}$ implies that $\lambda \in \mathbb{Z}$. Now, \begin{align} \lambda \odot_{2} w & = \underbrace{w +_{2} w +_{2} ... +_{2} w}_{\lambda ~ \text{times}} \\ & = (\underbrace{\lambda_{1} \oplus_{q} \lambda_{1} \oplus_{q} ... \oplus_{q} \lambda_{1}}_{\lambda ~ \text{times}}, \underbrace{\lambda_{2} \oplus_{q} \lambda_{2} \oplus_{q} ... \oplus_{q} \lambda_{2}}_{\lambda ~ \text{times}}, ..., \underbrace{\lambda_{n} \oplus_{q} \lambda_{n} \oplus_{q} ... \oplus_{q} \lambda_{n}}_{\lambda ~ \text{times}}) \end{align} Also, \begin{align} \lambda \odot w & = (\lambda \otimes_{q} \lambda_{1}, \lambda \otimes_{q} \lambda_{2}, ..., \lambda \otimes_{q} \lambda_{n}) \\ \end{align} \textbf{Claim: For each $i \in \{1, 2, ..., n\}$, $\underbrace{\lambda_{i} \oplus_{q} \lambda_{i} \oplus_{q} ... \oplus_{q} \lambda_{i}}_{\lambda ~ \text{times}} = \lambda \otimes_{q} \lambda_{1}$.} So let $i \in \{1, 2, ..., n\}$. Then we get \begin{align} \underbrace{\lambda_{i} \oplus_{q} \lambda_{i} \oplus_{q} ... \oplus_{q} \lambda_{i}}_{\lambda ~ \text{times}} & = (\underbrace{\lambda_{i} + \lambda_{i} + ... + \lambda_{i}}_{\lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q) \\ & = \lambda \otimes_{q} \lambda_{i} \end{align} So we have proved that $\lambda \odot_{2} w = \lambda \odot w$. Since $W$ is a $\mathbb{Z}$-submodule, therefore, $\lambda \odot_{2} w \in W$. Further, since $\lambda \odot_{2} w = \lambda \odot w$, therefore, $\lambda \odot w \in W$. Since $w, w' \in W$ and $\lambda \in \mathbb{Z}_{q}$ are arbitrary, therefore, $w+w', \lambda \odot w \in W$ for all $w, w' \in W$ and $\lambda \in \mathbb{Z}_{q}$. Therefore, $W$ is also a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$. Conversely, let $W$ be a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$. We show that $W$ is also a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. Being a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$, $W$ is a non-empty subset of $(\mathbb{Z}_{q})^{n}$. Let $w, w' \in W$ and $\lambda \in \mathbb{Z}$. Since $W$ is a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$, therefore, $w + w' \in W$. Since $w \in W \subseteq (\mathbb{Z}_{q})^{n}$, so $w = (\lambda_{1}, \lambda_{2}, ..., \lambda_{n})$ for some $\lambda_{i} \in \mathbb{Z}_{q}$ ($1 \leq i \leq n$). Now, if $\lambda > 0$, then \begin{align} \lambda \odot_{2} w & = \underbrace{w +_{2} w +_{2} ... +_{2} w}_{\lambda ~ \text{times}} \\ & = (\underbrace{\lambda_{1} \oplus_{q} \lambda_{1} \oplus_{q} ... \oplus_{q} \lambda_{1}}_{\lambda ~ \text{times}}, \underbrace{\lambda_{2} \oplus_{q} \lambda_{2} \oplus_{q} ... \oplus_{q} \lambda_{2}}_{\lambda ~ \text{times}}, ..., \underbrace{\lambda_{n} \oplus_{q} \lambda_{n} \oplus_{q} ... \oplus_{q} \lambda_{n}}_{\lambda ~ \text{times}}). \end{align} $\overline{\lambda} = \lambda (\text{mod} ~ q)$. Then \begin{align} \overline{\lambda} \odot w & = (\overline{\lambda} \otimes_{q} \lambda_{1}, \overline{\lambda} \otimes_{q} \lambda_{2}, ..., \overline{\lambda} \otimes_{q} \lambda_{n}). \end{align} \textbf{Claim: For each $i \in \{1, 2, ..., n\}$, $\underbrace{\lambda_{i} \oplus_{q} \lambda_{i} \oplus_{q} ... \oplus_{q} \lambda_{i}}_{\lambda ~ \text{times}} = \overline{\lambda} \otimes_{q} \lambda_{i}$.} We get \begin{align} \underbrace{\lambda_{i} \oplus_{q} \lambda_{i} \oplus_{q} ... \oplus_{q} \lambda_{i}}_{\lambda ~ \text{times}} & = (\underbrace{\lambda_{i} + \lambda_{i} + ... + \lambda_{i}}_{\lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q) \\ & = (\overline{\lambda} \overline{\lambda_{i}}) (\text{mod} ~ q) \\ & = \overline{\lambda} \otimes_{q} \lambda_{i} \end{align} because $\overline{\lambda_{i}} = \lambda_{i}$ as $\lambda_{i} \in \mathbb{Z}_{q}$. So we have proved that for $\lambda > 0$, $\lambda \odot_{2} w = \overline{\lambda} \odot w$. For $\lambda = 0$, the result holds trivially. If $\lambda < 0$, then \begin{align} \lambda \odot_{2} w & = \underbrace{(-w) +_{2} (-w) +_{2} ... +_{2} (-w)}_{- \lambda ~ \text{times}} \\ & = (\underbrace{(- \lambda_{1}) \oplus_{q} (- \lambda_{1}) \oplus_{q} ... \oplus_{q} (- \lambda_{1})}_{- \lambda ~ \text{times}}, \underbrace{(- \lambda_{2}) \oplus_{q} (- \lambda_{2}) \oplus_{q} ... \oplus_{q} (- \lambda_{2})}_{- \lambda ~ \text{times}}, ..., \\ &\quad \underbrace{(- \lambda_{n}) \oplus_{q} (- \lambda_{n}) \oplus_{q} ... \oplus_{q} (- \lambda_{n})}_{- \lambda ~ \text{times}}) \end{align} \textbf{Claim: For each $i \in \{1, 2, ..., n\}$, $\underbrace{(- \lambda_{i}) \oplus_{q} (- \lambda_{i}) \oplus_{q} ... \oplus_{q} (- \lambda_{i})}_{- \lambda ~ \text{times}} = \overline{\lambda} \otimes_{q} \lambda_{i}$.} We get \begin{align} \underbrace{(- \lambda_{i}) \oplus_{q} (- \lambda_{i}) \oplus_{q} ... \oplus_{q} (- \lambda_{i})}_{- \lambda ~ \text{times}} & = (\underbrace{(- \lambda_{i}) + (- \lambda_{i}) + ... + (- \lambda_{i})}_{- \lambda ~ \text{times}}) (\text{mod} ~ q) \\ & = ((- \lambda) (- \lambda_{i})) (\text{mod} ~ q) \\ & = (\lambda \lambda_{i}) (\text{mod} ~ q) \\ & = (\overline{\lambda} \overline{\lambda_{i}}) (\text{mod} ~ q) \\ & = \overline{\lambda} \otimes_{q} \lambda_{i} \end{align*} because $\overline{\lambda_{i}} = \lambda_{i}$ as $\lambda_{i} \in \mathbb{Z}_{q}$. So we have proved that for $\lambda < 0$, $\lambda \odot_{2} w = \overline{\lambda} \odot w$. Therefore, for any $\lambda$, $\lambda \odot_{2} w = \overline{\lambda} \odot w$. But since $\overline{\lambda} \in \mathbb{Z}_{q}$, $w \in W$ and $W$ is a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$, therefore, $\overline{\lambda} \odot w \in W$. And then the fact that $\lambda \odot_{2} w = \overline{\lambda} \odot w$ implies that $\lambda \odot_{2} w \in W$. Therefore, $W$ is also a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. \end{proof} Suppose $W$ is an IDD code and $\mathcal{E}$ is its corresponding equivalent IDD code in $\mathbb{Z}^{n}$. Then $\mathcal{E}$ is a $\mathbb{Z}$-submodule of the $\mathbb{Z}$-module $(\mathbb{Z})^{n}$. By Lemma \ref{lemma 4.4} (i), $\Omega(\mathcal{E})$ is a $\mathbb{Z}$-submodule of $(\mathbb{Z}_{q})^{n}$. Then by Lemma \ref{lemma 4.5}, $\Omega(\mathcal{E})$ is a $\mathbb{Z}_{q}$-submodule of $(\mathbb{Z}_{q})^{n}$. Hence, $\Omega(\mathcal{E})$ is a code over $\mathbb{Z}_{q}$ and in $(\mathbb{Z}_{q})^{n}$. So we have the following definition. \begin{definition} Let $q$ be a positive integer. With the notations of Subsection \ref{subsection 4.1}, let $W$ be an IDD code and $\mathcal{E}$ be its equivalent IDD code in $\mathbb{Z}^{n}$. Then the $\mathbb{Z}_{q}$-submodule $\Omega(\mathcal{E})$ of $(\mathbb{Z}_{q})^{n}$ is called Hom-IDD code in $(\mathbb{Z}_{q})^{n}$. \end{definition} \subsection{Example}\label{subsection 4.4} \begin{example} Let $K = \mathbb{Q}(\zeta)$, $\zeta$ primitive $p^{\text{th}}$-root of unity, where $p=17$. Let $\sigma(\zeta) = \zeta$ and $\tau(\zeta) = \zeta^{3}$. Then by Theorem \ref{theorem 3.8}, the map $D:O_{K} \rightarrow O_{K}$ given by $$D(\zeta) = 1 + \zeta + \zeta^{2} + \zeta^{3} + \zeta^{5} + \zeta^{7} + \zeta^{8} + \zeta^{11}$$ is a $(\sigma, \tau)$-derivation of $O_{K} = \mathbb{Z}[\zeta]$. Below, we have obtained (binary) Hom-IDD codes of length $n=16$ over the finite field $\mathbb{Z}_{2}$ with various parameters, using the construction discussed above. \begin{center} \begin{tabular}{\|c\|c\|c\|c\|} \hline \textbf{Basis $S_{i}$} & \thead{\textbf{Code Description:}\\ $[n,k,d]$} & \textbf{Code Properties} & \thead{\textbf{Dual Code} \\ \textbf{Description:} \\ $[n,k,d]$} \\ \hline \hline $S_{1}$ & $[16,8,4]$ & LCD & $[16,8,4]$ \\ \hline $S_{2}$ & $[16,8,4]$ & non-LCD & $[16,8,4]$ \\ \hline $S_{3}$ & $[16,7,4]$ & LCD & $[16,9,3]$ \\ \hline $S_{4}$ & $[16,9,3]$ & non-LCD & $[16,7,4]$ \\ \hline $S_{5}$ & $[16,7,4]$ & non-LCD & $[16,9,3]$ \\ \hline $S_{6}$ & $[16,6,4]$ & non-LCD & $[16,10,3]$ \\ \hline $S_{7}$ & $[16,5,5]$ & non-LCD & $[16,11,2]$ \\ \hline $S_{8}$ & $[16,4,7]$ & non-LCD & $[16,12,2]$ \\ \hline $S_{9}$ & $[16,5,5]$ & LCD & $[16,11,2]$ \\ \hline $S_{10}$ & $[16,8,3]$ & non-LCD & $[16,8,4]$ \\ \hline $S_{11}$ & $[16,9,3]$ & LCD & $[16,7,4]$ \\ \hline $S_{12}$ & $[16,8,4]$ & LCD & $[16,8,3]$ \\ \hline $S_{13}$ & $[16,6,5]$ & LCD & $[16,10,3]$ \\ \hline \end{tabular} \end{center} \begin{center} \begin{tabular}{\|c\|} \hline \textbf{Basis $S_{i}$} \\ \hline \hline $S_{1} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{4}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{10}), D(\zeta^{13})\}$ \\ \hline $S_{2} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{4}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{9}), D(\zeta^{12})\}$ \\ \hline $S_{3} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{9}), D(\zeta^{12})\}$ \\ \hline $S_{4} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{9}), D(\zeta^{10}), D(\zeta^{12}), D(\zeta^{14})\}$ \\ \hline $S_{5} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{9}), D(\zeta^{12}), D(\zeta^{14})\}$ \\ \hline $S_{6} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{9}), D(\zeta^{12})\}$ \\ \hline $S_{7} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{7})\}$ \\ \hline $S_{8} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{6}), D(\zeta^{7})\}$ \\ \hline $S_{9} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{15})\}$ \\ \hline $S_{10} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{6}), D(\zeta^{7}), D(\zeta^{10}), D(\zeta^{13}), D(\zeta^{14}), D(\zeta^{15})\}$ \\ \hline $S_{11} = \{D(\zeta), D(\zeta^{2}), D(\zeta^{4}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{10}), D(\zeta^{13}), D(\zeta^{14}), D(\zeta^{15})\}$ \\ \hline $S_{12} = \{D(\zeta), D(\zeta^{4}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{10}), D(\zeta^{13}), D(\zeta^{14}), D(\zeta^{15})\}$ \\ \hline $S_{13} = \{D(\zeta), D(\zeta^{4}), D(\zeta^{5}), D(\zeta^{6}), D(\zeta^{10}), D(\zeta^{13})\}$ \\ \hline \end{tabular} \end{center} \end{example} \section{Conclusion}\label{section 5} In this article, we studied the $(\sigma, \tau)$-derivations of number rings and their applications to coding theory. First, in Section \ref{section 2}, we obtained some useful results for the forthcoming sections. In Section \ref{section 3}, we studied $(\sigma, \tau)$-derivations of the rings of algebraic integers of quadratic and cyclotomic number fields, and that of the bi-quadratic number ring. In quadratic case, we reformulated and reinvented the results formed in \cite{Chaudhuri}. We obtained a characterization in Subsection \ref{subsection 3.2} for a $\mathbb{Z}$-linear map vanishing on unity to be a $(\sigma, \tau)$-derivation of the ring of algebraic integers of a $p^{\text{th}}$-cyclotomic number field. We also conjectured a criterion under which a $(\sigma, \tau)$-derivation of the ring of algebraic integers of a $p^{\text{th}}$ cyclotomic number field is inner. In Subsection \ref{subsection 3.3}, we explicitly obtained all $(\sigma, \tau)$-derivations of a bi-quadratic number ring. We also obtained a criterion under which a $(\sigma, \tau)$-derivation of a bi-quadratic number ring is inner. Finally, in Section \ref{section 4}, we discussed the applications of our work in coding theory by giving the notion of Hom-IDD codes. \vspace{20pt} \noindent \textbf{Declaration of Interest Statement}\vspace{8pt} \noindent The authors report there are no competing interests to declare. \begin{thebibliography}{10} \bibitem{AleksandrAlekseev2020} Aleksandr Alekseev, Andronick Arutyunov, and Sergei Silvestrov. \newblock On $(\sigma, \tau)$-derivations of group algebra as category characters. \newblock {\em arXiv preprint arXiv:2008.00390}, 2020. \bibitem{Atteya2019} Mehsin~Jabel Atteya. \newblock New types of permuting n-derivations with their applications on associative rings. \newblock {\em Symmetry}, 12(1):46, 2019. \bibitem{Brear1992} M.~Bresar. \newblock On the composition of $(\alpha, \beta)$-derivations of rings and applications to von {N}eumann algebras. \newblock {\em Acta Sct. 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\newcommand{\mbfd}{\mathbf{d}} \newcommand{\mbfx}{\mathbf{x}} \newcommand{\mbfy}{\mathbf{y}} \newcommand{\Euc}{\mathrm{euc}} \newcommand{\orig}{\mathbf{0}} \newcommand{\Div}{\operatorname{div}} \newcommand{\spine}{\operatorname{spine}} \newcommand{\odist}{\overline{\dist}} \DeclareMathOperator{\vol}{Vol} \DeclareMathOperator{\area}{Area} \DeclareMathOperator{\Index}{index} \DeclareMathOperator{\genus}{genus} \DeclareMathOperator{\Clos}{Clos} \DeclareMathOperator{\spt}{spt} \DeclareMathOperator{\Graph}{Graph} \DeclareMathOperator{\rad}{radius} \DeclareMathOperator{\ang}{angle} \DeclareMathOperator{\loc}{loc} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\id}{\text{Id}} \DeclareMathOperator{\Ent}{\text{Ent}} \DeclareMathOperator{\Sing}{\text{Sing}} \DeclareMathOperator{\rot}{\text{rot}} \newenvironment{pf}{\paragraph{Proof:}}{\hfill$\square$ \newline} \linespread{1.1} \begin{document} \title[Fate of flow]{On the long-time limit of the mean curvature flow in closed manifolds} \begin{abstract} In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed $3$--manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument then we construct piecewise almost regular flows which either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in $3$--manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods. \end{abstract} \author{Alexander Mramor and Ao Sun} \address{Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark 2300 \newline \newline \indent Department of Mathematics, Lehigh University, Chandler--Ullmann Hall, Bethlehem, PA 18015 \newline} \email{[email protected], [email protected]} \maketitle \section{Introduction} The mean curvature flow, as the gradient flow of the area functional, is a natural tool to find minimal surfaces -- in particular area minimizing and stable ones. The $1$-dimensional mean curvature flow in surfaces, known as the curve shortening flow, has been proven to be a very powerful tool for constructing geodesics. An especially important accomplishment was by Grayson \cite{Grayson89_CSF}, who used the curve shortening flow to settle Lusternik and Schnirelman's proposal to construct $3$ distinct closed embedded geodesics in $(S^2,g)$. A central obstruction to using $n$--dimensional mean curvature flow in $(n+1)$--dimensional ambient manifold to construct minimal surfaces is the more complicated nature of singularity formation in dimensions $n \geq 2$, and weak notions of the flow are necessary. There are a number of reasonable definitions of weak flows to handle this issue; these are discussed in section \ref{prelim}. Very recently, Bamler and Kleiner \cite{BamlerKleiner23_Multiplicity1} introduced the notion of almost regular flow, which are well-behaved Brakke flows whose so--called local scale functions satisfy a certain integrability condition. They proved that in $\R^3$ all generic flows (i.e. flows which only develop mean convex singularities) and hence, by an approximation argument, inner and outermost flows emanating from an embedded closed surface are almost regular. In this framework, they then proved the multiplicity one conjecture of Ilmanen \cite{Ilmanen95_Sing2D}. In this article we use the notions of generic and almost regular flows in general $3$--manifolds and first show the following: \begin{thm}\label{longterm}\label{smoothfate} Let $M$ be a $2$-sided properly embedded surface in a closed Riemannian $3$--manifold $(N,g)$, and let $M_t$ be an almost regular flow from $M$. Then either $M_t$ goes extinct in finite time or there exists a sequence $t_i \to \infty$ for which $M_{t_i}$ converges to $m_1\Sigma_1+\cdots+m_k\Sigma_k$ in the sense of varifolds, where each $\Sigma_j$ is a smoothly embedded minimal surface and each $m_j$ is a positive integer. Moreover, all the $\Sigma_j$'s are disjoint. \end{thm} We point out that some of the $\Sigma_j$ may be nonorientable, even if the initial data is $2$--sided: consider for instance a situation where the boundary of the tubular neighborhood of an embedded projective plane $P$ (say in the 3 manifold $N = \R P^3$), homeomorphic to $S^2$, flows back to $P$ as $t \to \infty$. This can also be interpreted as saying that the topology of the flow may actually "increase" in the limit; we discuss more precisely to what extent the topology of the $\Sigma_i$ can be controlled in terms of the initial data $M$ in Proposition \ref{limit_topology} below. We note that apriori the limit in Theorem \ref{longterm} may not be unique; if the ambient manifold is analytic and the convergence is with multiplicity one we may appeal to the \L{}ojasiewicz-Simon inequality to see this is the case, but we assume/show neither in the statement above. However, we are able to show the limit stable minimal surface is unique whenever it is strictly stable. Recall below that a stable minimal surface is strictly stable if the first eigenvalue of the linearized operator is positive, and as pointed out for example in \cite{Urbano13_one-sided} this notion continues to make sense even for $1$-sided minimal surfaces: \begin{thm}\label{longterm_uniqueness} In the setting of Theorem \ref{longterm}, if the limit stable minimal surfaces are strictly stable, then any varifold converging limit of $M_{t}$ converges to this limit with the same multiplicities as $t\to\infty$. \end{thm} See Theorem \ref{longterm_uniqueness1} below. We would like to point out that the uniqueness of limit in geometric measure theory, in general, is quite hard when the multiplicity of convergence is greater than one. Next, we further study what the possible limit minimal surfaces can be. Because the flow is the gradient of the area functional one expects to typically find area minimizing or at least stable minimal surfaces using it but nevertheless, in general, the long-time limit of mean curvature flow can be unstable. For curve shortening flow, a well-known example is the flow $\gamma_t$ of a closed embedded curve $\gamma \subset (S^2, g_{round})$ which bisects the sphere. It will converge, as $t \to \infty$, smoothly to an equator of the sphere; curves $\gamma$ which do not bisect the sphere instead flow to round points. Most curves $\gamma$ do not bisect the sphere and we can always perturb one that does to one which does not, of course, so most curve shortening flows cannot converge to an unstable geodesic. This observation motivates us to consider to which extent we can show the flow will generically avoid unstable minimal surfaces. The main tool we use to pursue this is the avoidance principle. In $\R^{n+1}$, if two mean curvature flows of closed hypersurfaces are disjoint at time $0$, then they will remain disjoint for all the future time. Moreover, the distance between these two flows is increasing. This basic property motivates the definition of weak set flows by Ilmanen \cite{Ilmanen92_LSF} and White \cite{White95_WSF_Top}. The avoidance principle in a manifold is more subtle, because in general two disjoint mean curvature flows can actually get closer -- consider for instance the flow of small graphs over strictly area minimizing minimal surfaces. In a manifold equipped with a positive Ricci curvature metric, the avoidance principle is very strong though and lets us easily show the following: \begin{thm}\label{Rc_thm} In a Ricci positive manifold $(N^n,g)$, the mean curvature flow of a generic embedded hypersurface $M$ will go extinct in finite time. \end{thm} This can be viewed as a generalization of the generic behavior of the curve shortening flows in $S^2$. As a corollary this yields another proof of the classical fact that $H_{2}(N,\R)$ for $3$--manifolds $N$ of positive Ricci curvature, see Corollary \ref{vanish}. In the case $(N,g)$ is not Ricci positive, there could be locally minimizing surfaces and we cannot expect the flows starting from generic embedded hypersurfaces to go extinct in finite time. However, we can use the avoidance principle to show that after perturbations, the flow can avoid those unstable limits. Our next theorem, inspired by work of Colding and Minicozzi \cite{ColdingMinicozzi12_generic}, says that in closed $3$--manifolds one may construct, from any embedded initial data, piecewise generic mean curvature flows which avoid unstable minimal surfaces as limits: \begin{thm} \label{pw_thm} Let $M^2$ be a $2$--sided properly embedded surface of a closed compact Riemannian $3$--manifold $(N^3,g)$. Then there exists a piecewise almost regular flow emanating from it which either goes extinct in finite time, or there exist $t_i \to \infty$, such that $M_{t_i}$ converges possibly with multiplicity to a (potentially disconnected) stable minimal surface. \end{thm} When the ambient metric is bumpy, which is a generic condition by work of White \cite{White91_Bumpy, White17_Bumpy}, we observe that by Theorem \ref{longterm_uniqueness} the limit in the statement above is unique. Note that in light of \cite{ChenSun24_Multi2inManifold} there are examples where the flow asymptotically converges to a stable limiting surface with multiplicity greater than one so higher multiplicity convergence in the above may actually occur, and in the case the limit is strictly stable seems to be robust under further perturbations. We will give the definition of piecewise almost regular flow in the section below but roughly speaking it is a concatenation of almost regular flows where by--hand isotopies are performed at the jumps; the number of perturbations by isotopy taken will be finite, and the perturbations can be taken as small as we wish. In that sense the statement above is a statement on generic flows following the convention of Colding and Minicozzi \cite{ColdingMinicozzi12_generic}. Of course, it is certainly desirable to only require perturbing the initial data. The obstruction to doing so in the statement above essentially boils down to the possibility of convergence to an unstable minimal surface with multiplicity; in analogy to singularity analysis and the resolution of the multiplicity one conjecture by Bamler and Kleiner \cite{BamlerKleiner23_Multiplicity1} one might expect this to not occur; see the analogous situation in Almgren-Pitts minmax theory \cite{Zhou19_multi1}. We recall the following conjecture in \cite{ChenSun24_Multi2inManifold}: \begin{conj}[Conjecture 1.3 in \cite{ChenSun24_Multi2inManifold}]\label{conj_multi1} Suppose the long-time limit of an almost regular mean curvature flow in $(N^n,g)$ is an unstable minimal surface, then the convergence must be multiplicity $1$. \end{conj} Despite this, we point out that as applications we get flow proofs of the following existence theorems for stable minimal surfaces. In the proofs of these we will often specifically use piecewise generic flows, for instance because the topological change through singularities is easy to understand. We also recall that obtaining stability of the minimal surfaces is helpful because these often have better controlled topology and geometry, for instance in the context of positive scalar curvature. These corollaries are not new and there have been long standing proofs of them using geometric measure theory or harmonic map theory, see \cite{FedererFleming60_Current, SchoenYau79_ExistenceIncompressibleMinSurf, SacksUhlenbeck81_minimal2sphere, MeeksYau80_Top3d,MeeksSimonYau82_MinSurf, FreedmanHassScott83_LeastArea}, among others. On the other hand, the construction of these minimal surfaces via flows, along with a number of fairly explicit perturbations, is quite amenable to numerical approximation with the caveat that one has found appropriate initial data to start with. In the following statements $(N,g)$ is a compact orientable Riemannian $3$--manifold, where in this case oriented surfaces are $2$--sided and vice versa: \begin{cor}\label{existence1} Suppose $\pi_2(N)$ is nontrivial. Then $N$ contains an embedded stable minimal sphere or projective plane. \end{cor} Note that in the statement above we invoke the sphere theorem to find embedded initial data for which to start the flow and so the above does not constitute a new proof of it. As we pointed out already compared to other techniques a pitfall of using the mean curvature flow to find minimal surfaces is that it is best to start with well-prepared (i.e. embedded) initial data. In some cases the flow can produce stable minimal surfaces of nontrivial topology, the most basic of which is the following. \begin{cor}\label{existence2} Suppose $N$ has trivial second homotopy group and contains a homologically nontrivial embedded torus. Then $N$ contains an embedded stable minimal surface homeomorphic either to a torus or a Klein bottle. \end{cor} Recall that hyperbolic manifolds are aspherical, so the space of such $3$--manifolds with $\pi_2$ trivial is quite large and is even generic amongst $3$--manifolds from a number of perspectives, for example see \cite{DunfieldTHurston06_Random3Mfd, Maher10_RandomHeegaard}. For surfaces of more general topology, we can show the following result which is very much in the spirit of the existence result of Freedman, Hass, and Scott \cite{FreedmanHassScott83_LeastArea} -- in fact it implies the result above as we discuss in section \ref{applications}. The surfaces below are known as algebraically incompressible surfaces, and are incompressible in the more geometric sense via Dehn's lemma: \begin{cor}\label{existence3} Suppose $\iota: \Sigma \to N$ is an orientable properly embedded surface of $N$ of genus $g \geq 1$ for which $\iota_{} \pi_1(\Sigma) \to \pi_1(N)$ is injective, and that $N$ has trivial second homotopy group. Then either $M = \iota(\Sigma)$ is homotopic to an orientable stable minimal surface or a double cover of a stable minimally embedded connect sum of $g+1$ projective planes, where $g$ is the genus of $\Sigma$. \end{cor} By double cover of an embedded surface $P \subset N$ here we mean specifically a surface which is the boundary of a tubular neighborhood of $P$ in $N$. It is a fact (see for instance Lemma 3.6 in \cite{Hatcher07_Notes3Mfd}) that every class of $H_2(N,\Z)$ can be represented by an orientable properly embedded surface $M$. Recalling the Thurston norm $\|\|\sigma\|\|_{T}$ of $\sigma\in H_2(N,\Z)$ is the smallest attained genus of embedded surfaces homologous to $\sigma$, we see any such surface $M$ realizing the norm must be algebraically incompressible. If $\|\|\sigma\|\|_{T} \geq 1$ we can apply the Corollary above then, and clearly this must be the case if we suppose that $\pi_2(N)$ is trivial. We can record the discussion above as follows: \begin{cor}\label{existence0} Suppose $\sigma\in H_2(N, \Z)$, with $\pi_2(N)$ trivial. Then either there exists an embedded oriented stable minimal surface $\Sigma$ in the class of $\sigma$ that realizes the Thurston norm of $\sigma$ or there is an element realizing the Thurston norm which is the double cover of a embedded nonoriented stable minimal surface. \end{cor} Finally, we note that Theorem \ref{longterm} lets us substitute the tightening process in minmax for mean curvature flow, at least in some cases. A major issue in applying the flow to a sweepout is that the flow of a sweepout may not remain a sweepout, but this can be compensated for if the flow of the elements of it will be nonfattening. In particular we show the following: \begin{cor}\label{existence5} Suppose $N$ is diffeomorphic to $S^3$. Then $N$ contains a minimal embedded $S^2$. \end{cor} Originally this result was proved by Simon and Smith using minmax theory, see also \cite{ColdingDeLellis03_minmax}. As to the actual best result known we point out that recently Haslhofer and Ketover in \cite{HaslhoferKetover19_Min2Sphere} showed there are at least two minimal spheres in $S^3$ with a bumpy metric, and very recently Zhichao Wang and Xin Zhou in \cite{WangZhou23_4Sphere} proved that there are at least $4$ embedded minimal spheres in $S^3$ equipped with either a bumpy or Ricci positive metric. We expect that if Conjecture \ref{conj_multi1} is true then the mean curvature flow can also be used to recover their results. We emphasize that in the above we are not applying Theorem \ref{pw_thm} and the minimal surface above is potentially unstable, as must be the case when for instance $N$ is Ricci positive. We conclude the introduction with a remark on the compactness assumptions made throughout. When the ambient manifold is not closed, but merely complete with uniform bounded curvature and injectivity radius lower bound, Theorems \ref{smoothfate} and \ref{pw_thm} remain valid, if we include the possibility that the flow can escape from any compact set. For example if $N$ is $S^2\times\R$ with an infinite long trumpet shape (i.e. a cusp) in one direction the flow of an appropriate $S^2$ section of it will escape to spatial infinity/clear out, neither terminating or converging to a minimal surface. It seems to be a more complicated manner yet to generalize our results to the case $M$ is allowed to be noncompact (say properly embedded in a noncompact $N$), one very basic reason being that the area can be infinite. As before the flow may clear out in the manner of the example above but it seems that in some cases if the flow doesn't clear out or go extinct one could show it must flow to a minimal surface. For a basic example, a well known consequence of Ecker and Huisken \cite{EckerHuisken89_EntireGraph} says that uniformly Lipschitz graphs of bounded height in $\R^n$ must converge to flat, hence minimal, planes. \subsection{Acknowledgments} A.M. thanks Alec Payne and Felix Schulze for their interest and helpful comments. In the course of preparing the article he was supported by CPH-GEOTOP-DNRF151 from the Danish National Research Foundation and CF21-0680 from the Carlsberg Foundation (via GeoTop and N.M. M{\o}ller respectively) and is grateful for their assistance. A.S. is supported by the AMS-Simons travel grant. He is grateful to the Copenhagen Centre for Geometry \& Topology (GeoTop) at the University of Copenhagen, especially N.M. M{\o}ller, for the invitation to visit in Spring 2024 and participate in the meeting "Masterclass: Recent Progress on Singularity Analysis and Applications of the Mean Curvature Flow". Part of the work was initiated during the visit. \section{Background and preliminary lemmas}\label{prelim} Let $N^{n+1}$ be a Riemannian manifold and $X: M \to N^{n+1}$ be an embedding of a manifold $M^n$ realizing it as a $2$--sided smooth closed hypersurface of $N$, whose image by abuse of notation we also refer to as $M$. Then the mean curvature flow $M_t$ of $M$ is given by (the image of) $X: M \times [0,T) \to N^{n+1}$ satisfying the following: \begin{equation}\label{MCF_equation} \frac{dX}{dt} = \vec{H}, \text{ } X(M, 0) = X(M) \end{equation} An immediate consequence of the first variation formula and the evolution equation above is that the mean curvature flow is the gradient flow of the area functional, in the sense that $\frac{d}{dt}\text{Area}(M_t) = - \int_{M_t} \|\vec H\|^2 d\mu_t$, where $\mu_t$ is the area measure; from this one sees the mean curvature is well adapted to finding minimal surfaces, which are critical points of the area functional. The first obstruction to the construction of minimal surfaces using mean curvature flow is singularity formation: as a highly nonlinear system, mean curvature flow can develop finite time singularities; then it is even unclear how to continue the flow after the singular time. To overcome this issue, there are three major weak flow approaches: \begin{itemize} \item The Brakke flow \item The weak set flow and level set flow \item The surgery flow \end{itemize} The \textbf{Brakke flow}, introduced by Brakke in his thesis \cite{Brakke78} is a geometric measure-theoretic definition in terms of (typically, integral) varifolds that satisfy the eponymous Brakke inequality. Here we present a definition essentially due to Ilmanen \cite{Ilmanen94_EllipReg}, and the expository is from \cite{BamlerKleiner23_Multiplicity1}. An $n$-dimensional Brakke flow defined on the time-interval $I\subset\R$ in a $(n+1)$-dimensional manifold $(N,g)$ is a family of Radon measures $(\mu_t)_{t\in I}$ such that: \begin{itemize} \item For a.e. $t\in I$ the measure is integer $\cH^n$-rectifiable and the associated varifold has locally bounded first variation with variational vector $\vec H$ in $L^1$. \item For any compact set $K\subset N$, and any $[t_1,t_2]\subset I$, \[ \int_{t_1}^{t_2}\int_K \|\vec H\|^2 d\mu_t dt<+\infty, \] \item For any $[t_1,t_2]\subset I$ and $u\in C_c^1(N\times[t_1,t_2])$, \begin{equation}\label{eq:BrakkeF} \left.\int_N u(\cdot,t) d\mu_t \right\|_{t=t_1}^{t=t_2} \leq \int_{t_1}^{t_2}\int_N (\pr_t u+\nabla u\cdot \vec H-u\|\vec H\|^2)d\mu_t dt. \end{equation} \end{itemize} One can quickly see that the Radon measures associated to a smooth/classical mean curvature flow, that is a family of smooth manifolds satisfying \eqref{MCF_equation}, is a Brakke flow. A Brakke flow suffers from a serious problem: as \eqref{eq:BrakkeF} is only an inequality, the whole connected components of (the support of) a Brakke flow are allowed to disappear instantaneously! In the same monograph, Ilmanen \cite{Ilmanen94_EllipReg} (see also White \cite{White09_CurrentsVarifolds}) gives a construction via elliptic regularization of a "well-behaved" Brakke flow out of smooth initial data for which these issues don't occur. Such a Brakke flow is known to be: \begin{itemize} \item unit-regular: if the density of the Brakke flow at a spacetime point is $1$, then it is a smooth mean curvature flow in a spacetime neighborhood. \item cyclic mod $2$: suppose the unique associated rectifiable mod-$2$ flat chain of $\mu_t$ is $V(t)$, then $\pr[V(t)]=0$ for a.e. $t\in I$. \end{itemize} Even amongst such well-behaved Brakke flows there happens to be an issue of uniqueness, which brings us to the level set flow which originated from the viscosity method \cite{EvansSpruck91, ChenGigaGoto91_LSF}. Later Ilmanen \cite{Ilmanen92_LSF} and White \cite{White95_WSF_Top} introduced a purely set-theoretic definition using the avoidance principle. The classical avoidance principle says that two disjoint hypersurfaces in a manifold $N$ will stay disjoint, at least as long as one of them is compact. The \textbf{weak set flow} is defined to be a closed subset $\cM$ of $N\times\R$ such that if a smooth mean curvature flow does not intersect $\cM\cap\{t\}$, then it is disjoint from $\cM\cap \{t'\}$ for all $t'>t$. White \cite{White24_Avoidance} proved the following avoidance principle for weak set flows. \begin{thm}\label{ambient_avoidance} Suppose that $N$ is a complete, connected Riemannian manifold with positive injectivity radius such that $\|\nabla^k \text{Riem} \|$ is bounded for each nonnegative integer $k$. Let $\Lambda$ be a lower bound for the Ricci curvature of $N$, and suppose that $t \in [0, \infty) \to X(t), Y(t)$ are weak set flows in $N$. Then \begin{equation}\label{dist} e^{-\Lambda t} d(X(t), Y(t)) \end{equation} is an increasing function of $t$. \end{thm} In particular, note that for $\Lambda > 0$ two flows don't merely stay disjoint but actively "repel" exponentially quickly -- this will play an important role in the proof of Theorem \ref{Rc_thm} below. Indeed the support of a Brakke flow is a weak set flow, as shown in Ilmanen's monograph. Although weak set flows may not be unique starting from a given hypersurface, there is a canonical weak set flow associated to any initial data called the \textbf{level set flow}: for a given set $A$ is as the envelope of all families of sets initially agreeing with $A$ that satisfy the avoidance principle. As an aside we point out that in fact the level set flow can even be defined this way for closed initial data that is not necessarily a regular hypersurface. While the existence and uniqueness are guaranteed, the level set flow of initial data may develop interior even if it is smooth, as first noted by Ilmanen and White \cite{White02ICM} (and recently appeared in \cite{IlmanenWhite24_Fattening}, see also related works \cite{AngenentIlmanenVelazquez17_fattening, LeeZhao24_MCFconical, Ketover24_Fattening}) -- this fattening can be seen as an indication of nonuniqueness for level set flows. To study the level set flow even in the presence of fattening Hershkovits and White \cite{HershkovitsWhite20_Nonfattening} introduced the notions of innermost/outermost flows. Given a closed hypersurface $\Sigma \subset \R^3$ that is the boundary of a domain $\Omega$, we can define: \begin{itemize} \item The innermost flow $\cM^-$ is the boundary of the level set flow in $N\times\R$ generated by $\Omega\times\{0\}$; \item The outermost flow $\cM^+$ is the boundary of the level set flow in $N\times\R$ generated by $(N\backslash\Omega)\times\{0\}$. \end{itemize} Hershkovits and White noticed that if the level set flow $\cM$ starting from $\Sigma$ fattens, then $\cM^-$ and $\cM^+$ are different. It is expected that if the level set flow fattens, the innermost/outermost flows are natural candidates to serve as the canonical representatives of the weak set flow. It is not immediate to define inner and outermost flows in general $3$--manifolds from a $2$--sided surface $M$ because such a surface need not be separating, as a cross section of $S^2 \times S^1$ illustrates. In the sequel we will define such flows as the following: for a choice of normal vector $\nu$ consider the domains $\cM^-$ and $\cM^+$ bounded by $M$ and $M \pm \epsilon \nu$ respectively, for some small $\epsilon > 0$, where $M \pm \epsilon \nu$ are surfaces that are generated by pushing $M$ in $\epsilon\nu$ direction by the exponential map. Then we define the inner and outermost flows to be the boundary component of the level set flows of $\cM^-$, $\cM^+$ respectively emanating from $M$. The third weak flow is the \textbf{mean curvature flow with surgery}, where one preemptively cuts out regions developing high curvature, classifies them topologically, and then continues the flow in a piecewise manner, see \cite{Head13_MCF2convex, Lauer13_MCFSurgery, BrendleHuisken16_MCFSurgeryR3, HaslhoferKleiner17_MCFsurgery, Daniels-Holgate22_SurgeryApprox}. Locally, the high curvature regions will be modeled on round spheres or cylinders in coordinate patches which will bound solid balls/cylinders respectively. Regions where high curvature transitions into low curvature (i.e. "neckpinches") the surface are modeled on cylinders and at these points one "cuts" the neck at an appropriate spot and places in caps/discs of controlled geometry, in particular so that the remaining low curvature regions have curvature upper bounded by a controlled constant (which is amongst the analytical difficulties in establishing the flow with surgery). We note that the high curvature regions after cap placement will be homeomorphic to either $S^n$ or $S^{n-1} \times S^1$, which makes the flow with surgery a useful tool in topology. This particular consequence won't be so relevant in this work but the well controlled, concretely understandable change of topology across surgeries will be helpful in the sequel. \subsection{Almost regular flow and generic approximation} While these notions of weak flows seem quite different, in $3$--manifolds there is a relatively complete picture to relate all of them. In \cite{BamlerKleiner23_Multiplicity1}, Bamler and Kleiner introduced a new notion of weak flow called \textbf{almost regular flow} which are well behaved Brakke flows in the sense described above along with an extra integrability condition on what they call scale functions, which measure the size of neighborhoods in which the flow is smooth. We refer the readers to \cite{BamlerKleiner23_Multiplicity1} for the definition and properties of almost regular flows. Here we only state properties of almost regular flows that we reference: \begin{enumerate} \item Generic flows, i.e. ones which only encounter mean convex singularities (see discussion and references below) are almost regular. \item The innermost/outermost flows that start from a closed smooth embedded surface can be approximated by generic flows as a consequence are almost regular flows for $t\geq 0$. In particular, if the level set flow from a closed smooth embedded surface is nonfattening then it is almost regular. \item An almost regular flow is regular for a.e. $t\in I$. \item (Lemma 2.7 in \cite{BamlerKleiner23_Multiplicity1}) If $\cM$ is almost regular, then for any test function $u\in C_c^1(N\times[t_1,t_2])$ where $[t_1,t_2]\subset \text{Int} I$, $t\to\int_N u(\cdot,t)d\mu_t$ is continuous and the equality holds in \eqref{eq:BrakkeF}. \end{enumerate} While the almost regular flows have many nice properties, they still possibly develop complicated singularities which obscures the possible topological change of the flow through them. On the other hand, as we have described before, the mean curvature flow with surgery has a clear description of the topological change of the flow. Therefore, we would like to use mean curvature flow with surgery to approximate the almost regular flows we use in the sequel, particularly in showing (some of) the applications mentioned in the introduction. In order to use mean curvature flow with surgery to approximate the almost regular flows we use in practice, we first use a well behaved (unit regular, cyclic) Brakke flow with only cylindrical and spherical singularities to approximate them. Such flows we call generic. It has been conjectured by Huisken that mean curvature flow with generic initial data can only have cylindrical and spherical singularities, and recently there has been various progress on this conjecture, \cite{ColdingMinicozzi12_generic, CCMS20_GenericMCF, chodoshchoischulze2023mean, SunXue2021_initial_closed, SunXue2021_initial_conical}. Using the result of Chodosh, Choi, Mantoulidis and Schulze \cite{CCMS20_GenericMCF} and Chodosh, Choi, and Schulze \cite{chodoshchoischulze2023mean}, Bamler and Kleiner \cite{BamlerKleiner23_Multiplicity1} proved that a mean curvature flow starting from a generic closed embedded surface in a three-manifold can only have cylindrical and spherical singularities. Furthermore, Daniels-Holgate \cite{Daniels-Holgate22_SurgeryApprox} proved that the mean curvature flows with surgeries can approximate a mean curvature flow with only cylindrical and spherical singularities, where the approximation is in the Hausdorff distance. In addition, this approximation is smooth away from the singular set. Combining all the ingredients above gives the following approximation result in $\R^3$: \begin{prop}\label{goodflow_ambient} The inner and outermost flows starting from a closed embedded surface in $\R^3$ can be approximated by mean curvature flows with surgeries in Hausdorff distance, and the approximation is smooth away from the singular set. \end{prop} Note that in the classical case i.e. $N = \R^{3}$ properly embedded hypersurfaces are automatically orientable. If $N$ is more generally simply connected it is not hard to see this true by an intersection number argument, but we point out it does not suffice if $N$ is merely orientable with as the example $\R P^2 \subset \R P^3$ shows. With this in mind, we claim that the following holds true in general closed $3$--manifolds: \begin{prop} Let $M$ be a closed, properly embedded $2$--sided surface $M$ in a closed $3$--manifold $N$. Then the inner and outermost flows of $M$, as defined above, can be approximated by mean curvature flows with surgeries in Hausdorff distance over any fixed time interval and the approximation is smooth away from the singular set. \end{prop} \begin{proof} The main point is to establish the existence of generic mean curvature flow ala \cite{chodoshchoischulze2023mean} and its subsequent approximation by surgery flows, then the statement about inner and outermost flows follows exactly as in the Euclidean case by an approximation argument as discussed in section 7 of \cite{BamlerKleiner23_Multiplicity1}. As is well known, in a blowup limit singularities will still be modeled on solitons in $\R^3$ so we mostly just supply a sketch of this argument. Noting that because $M$ is 2-sided one may meaningfully discuss one-sided perturbations of $M$ and the point made in the previous sentence one readily sees from \cite{CCMS20_GenericMCF, chodoshchoischulze2023mean} that if one encounters a nongeneric singularity of multiplicity one that the initial data can be perturbed to avoid the singularity at that spacetime point. Because $M$ is a compact surface it has finite genus and so by genus monotonicity (see Section 11 of \cite{CCMS20_GenericMCF}) and the classification of genus $0$ shrinkers by Brendle \cite{Brendle16_genus0}, only finitely many (arbitrarily small) perturbations of the initial data are required to produce a generic flow up at least to the first time a high multiplicity tangent flow develops. Now all these generic flows, as shown in section 7 of \cite{BamlerKleiner23_Multiplicity1}, are almost regular flows; note this uses the resolution of the mean convex neighborhood conjecture by Choi, Haslhofer, and Hershkovits \cite{ChoiHaslhoferHershkovits18_MeanConvNeighb} which the authors note carries over in the curved ambient setting. Because under rescaling as in singularity analysis the ambient metric converges smoothly to the Euclidean metric, one can see by contradiction that the resolution of the multiplicity one conjecture applies in the curved ambient setting because all the relevant geometric quantities can be arranged to evolve as they do in the flat case up to arbitrarily small errors -- their argument at a high level is by showing in the case of high multiplicity convergence in a tangent flow that a certain quantity satisfies contradictory bounds so this suffices for our needs. As a result we get that for generic choice of data there will be a flow along which only mean convex singularities may develop for all time; by the resolution of the mean convex neighborhood conjecture \cite{ChoiHaslhoferHershkovits18_MeanConvNeighb} and the barrier argument of Hershkovits and White \cite{HershkovitsWhite20_Nonfattening, HershkovitsWhite23_Avoid}, it is unique among almost regular flows starting from $M$. The approximation theorem of Daniels-Holgate also applies in this setting; most importantly besides the mean convex neighborhood conjecture the required local estimates for surgery also hold in the general ambient setting as shown by Haslhofer and Ketover \cite{HaslhoferKetover19_Min2Sphere}. \end{proof} We next present some properties of such almost regular flows, where $M$ is as usual a properly embedded $2$--sided surface in a given closed $3$--manifold $N$. \begin{lem}\label{lem:PreserveOriented} Suppose that $M_t$ is a generic or inner/outermost flow out of $M$. Then $M_t$ will also be 2--sided for all regular times $t$. \end{lem} \begin{proof} It's obvious if $M_t$ is smooth then it will remain 2--sided, because the flow is an isotopy. In the case that $M_t$ has singularities, we first observe that it is easy to see for the approximating smooth flows $S_t$ to $M_t$ produced in \cite{Daniels-Holgate22_SurgeryApprox} that $S_t$ will remain orientable if $M$ is. The convergence of the approximating surgery flows is smooth during regular times, giving the claim. \end{proof} An important subtlety to remember though, as mentioned in the introduction, is that orientability is not necessarily inherited by the limit we discuss in the next section. In the next statement we write $M\sim \sigma$ to indicate that $M$ is homologous to $\sigma$. \begin{lem}\label{lem:PreserveHomology} Suppose that $M_t$ is a generic or inner/outermost flow out of $M$, with $M\sim \sigma$ where $\sigma\in H_2(N,\Z)$. Then $M_t \sim \sigma$ for all regular times $t\geq 0$. \end{lem} \begin{proof} Again using that the approximation by surgery flows is smooth during regular times, one only needs to note that surgery does not change the $2$nd homology class of a regular surface. \end{proof} \begin{lem}\label{lem:GenusNonincreasing} Suppose that $M_t$ is a generic or inner/outermost flow out of $M$. Then the genus of $M_t$, considered at smooth times, is nonincreasing along the flow. \end{lem} \begin{proof} As in the previous proofs it suffices to note this is the case for approximating surgery flows. Now in the surgery components are either deleted or cylindrical regions are cut and replaced with caps and both of these actions do not increase the genus, giving the claim. \end{proof} We show in the next section that an almost regular flow will either go extinct or flow to a potentially unstable smooth minimal surface, possibly with multiplicity. If the minimal surface is unstable we will afterwards wish to perturb the flow to ensure it doesn't flow to it, but as discussed in the introduction for technical reasons it seems difficult to carry this out by just perturbing the initial data if the multiplicity is high (i.e. greater or equal than $2$). With this in mind we will consider the following piecewise almost regular flows in the sequel: \begin{defn}\label{pwflow_def} Let $M$ be a properly embedded, $2$--sided surface of a $3$--manifold $N$. Then a piecewise almost regular flow $M_t$ emanating from $M$ defined on $[0, \infty)$ is given by an increasing (possibly finite) sequence of times $t_i \to \infty$ for which \begin{enumerate} \item $M_t$ on $[0, t_1]$ is an almost regular flow out of $M$ satisfying the conclusions of Proposition \ref{goodflow_ambient} above and so that $M_{t_1}$ is a smooth surface. \item Denoting by $M_t^1$ on $[0, t_1)$ the flow from above, we let $\tilde{M}^1_{t_1}$ be a surface that is isotopic to $M^1_{t_1}$, and we define $M^2_t$ out of this perturbed surface $\tilde{M}^1_{t_1}$ as above. \item Similarly, we define $M^i_t$ for $i \geq 2$ inductively. \end{enumerate} \end{defn} We end this section with the following technical tool to ensure the limit of the flow we take is smooth: \subsection{Ilmanen's regularity theorem} Because the mean curvature flow is the gradient flow of area functional, the long-time limit is expected to be a critical point to the area functional. From geometric measure theory, the limit is known to be a stationary varifold, and without any other assumptions, the limit can have singularities. In \cite{Ilmanen95_Sing2D}, Ilmanen used Simon's sheeting theorem \cite{Simon94_Willmore} to show that the tangent flow of a first-time singularity of a mean curvature flow of closed embedded surfaces in $\R^3$ must support on a smooth embedded shrinker, possibly with multiplicity. Ilmanen's argument is very general and can be adapted to study the long-time limit of mean curvature flow. For the sake of clarity we precisely state the version that we will use in this paper: \begin{thm}[Ilmanen's regularity for limit surfaces]\label{thm:IlReg} Suppose $\{M_i\}_{i\in\Z_+}$ is a sequence of closed embedded surfaces in a closed manifold $(N^3,g)$, satisfies the following assumptions: \begin{enumerate} \item $\area(M_i)$ and $\text{genus}(M_i)$ are uniformly bounded from above; \item $\int_{M_i}\|\vec H\|^2 d\mu_{M_i}\to 0$ as $i\to\infty$; \item (Area growth bound) there exists $r_0>0$ such that there exists a constant $C$ such that for any $x\in N$ and $r\in(0,r_0)$, $\area(M_i\cap B_r(x))\leq Cr^2$. Here $B_r(x)$ can either be the intrinsic ball in $(N,g)$ or the extrinsic ball for a fixed isometric embedding $(N,g)\to (\R^m,g_{\text{Euc}})$. \end{enumerate} Then there exists a subsequence of $\{M_i\}_{i\in\Z_+}$, still denoted by $\{M_i\}$, that converges to an integral varifold $V$ in the sense of varifold, such that $V$ is supported on a finitely many disjoint closed embedded minimal surface $\Sigma_1,\cdots,\Sigma_k$, with multiplicity $m_1,\cdots,m_k\geq 1$. Moreover, for any $j=1,2,\cdots,k$, there exists finitely many points $p_{j,1},\cdots,p_{j,\ell_j}$ inside $\Sigma_j$, such that for any $r>0$ that is sufficiently small, for sufficiently large $i$, $M_i\backslash \bigcup_{l=1}^{\ell_j} B_{r}(p_{j,l})$ is the union of $m_k$ number of disjoint connected components, and each component is isotopic to and converges in the varifold sense with either multiplicity $1$ or $2$ to $\Sigma_j \backslash \bigcup_{l=1}^{\ell_j} B_{r}(p_{j,l})$ depending on if the limit is two sided or not. \end{thm} \section{Long-time behavior of almost regular flows: the proof of Theorem \ref{longterm}} In this section we show that in the long term an almost regular flow either goes extinct in finite time or limits to a collection of smooth, but possibly unstable, minimal surfaces. Note that almost regular flows are not necessarily generic, although in the further applications we will often suppose our flow is -- for posterities sake, we work in this more general class as much as possible. We finish with some properties of the minimal surfaces we find should the limit be nonempty. We start with the following total curvature bound: \begin{lem}\label{tabounds} Let $(N^3,g)$ be a closed manifold and $(M_t)_{t\in[0,\infty)}$ is an almost regular flow. For each $\epsilon > 0$, there exists a $C = C(\epsilon, M_0,g)$ such that $\int_{M_t} \|A\|^2\leq C$ for $t\in[0,\infty)$ away from a set of measure $\epsilon$. \end{lem} \begin{proof} Because $M_t$ is regular for a.e. time $t\geq 0$, we can apply the Gauss-Codazzi equations to get $R_N = K + 2 \Ric_N(\nu,\nu) + \|A\|^2 - H^2$, where $R_N$ and $\Ric_N$ are the scalar and Ricci curvatures on $N$ respectively. With this in hand, by Gauss--Bonnet we have for any $t > 0$ that is a regular time: \begin{equation} \int_{M_t} \|A\|^2 = \int_{M_t} H^2 - 4 \pi \chi(M_t) + C\text{Vol}(M_t) \end{equation} Where the constant $C$, independent of $t$, depends on the curvature of $N$. By the property of almost regular flow \cite{BamlerKleiner23_Multiplicity1}, the genus of $M_t$ is bounded by a uniform constant $G$, and hence $\chi(M_t)$ is also bounded by a constant. Of course $\text{Vol}(M_t)$ is bounded in terms of $\text{Vol}(M)$. Since the time derivative of the area is given by $- \int_{M_t} H^2$ (for weak flows this is a consequence of Brakke's inequality) we see measure of the set of times for which $\int H^2$ is greater than a constant $C_1$ is bounded by $C_1/\text{area}(M_0)$. These two observations combined give what we want. \end{proof} Similarly, we have the following: \begin{lem}\label{decay} $\liminf\limits_{t\to\infty}\int_{M_{t}} H^2 = 0$. \end{lem} \begin{proof} Supposing not, then there exists some $\epsilon > 0$ for which $\int_{M_t} H^2 > \epsilon$ for all sufficiently large $t$. This contradicts that $M$ is compact and so of finite area and that $\frac{d}{dt} \text{Area}(M_t) \leq -\int_{M_t} H^2$ by the Brakke inequality. \end{proof} With Lemma \ref{decay} in hand, Allard's compactness theorem implies the following convergence theorem: \begin{cor}\label{lem:VariConv} There exists a sequence of time $t_i\to\infty$ such that $M_{t_i}$ converges to an integral stationary varifold $V$. \end{cor} Of course, from general theory $\supp V$ is not necessarily a regular minimal surface. However, for surfaces in $3$--manifolds, we have more powerful tools to pick a convergence subsequence whose limit is regular, using Ilmanen's regularity theory \cite{Ilmanen95_Sing2D}. We adopt his argument here; Lemma \ref{tabounds} gives us the total curvature bounds we require in the application of Simon's sheeting theorem, but there is a central feature of mean curvature flows in $\R^3$ that does not hold for mean curvature flows in a $3$--manifold which requires us to take some extra care. Namely Huisken's monotonicity formula shows that a mean curvature flow $M_t$ of surfaces in $\R^3$ has a uniform area growth bound $\area(M_t\cap B_R(x))\leq D\pi R^2$ for all $R>0$, $x\in\R^3$ and $t>0$, where $D$ only depends on $M_0$; see Proposition 3.3 in \cite{sun-generic-multi-1}. This area growth bound is crucial in applying Simon's sheeting theorem in \cite{Ilmanen95_Sing2D}; in particular see Theorem 10 and Corollary 11 of \cite{Ilmanen95_Sing2D}. When $M_t$ is a mean curvature flow in a closed manifold $(N,g)$, by isometric embedding $(N,g)$ into $\R^m$, $M_t$ is a mean curvature flow with additional forces $\beta$, where $\beta$ is a tangent vector field on $N$, depending on the second fundamental form of $(N,g)$ in $\R^m$. Because of the forcing term, Huisken's monotonicity formula can not directly provide uniform area growth bound for all future time $t$, but luckily a modified version of it can for short times which will suffice for our purposes. To start, let us fix an isometric embedding of $(N,g)$ into $\R^m$; recalling that we suppose $N$ is compact and denote by $L < \infty$ an upper diameter bound for $N$ under this embedded. Then it happens that the forcing term $\beta$, $\beta(x)\in T_xN$, only depends on the second fundamental form of the isometric embedding of $(N,g)$. Hence $\\|\beta\\|_{L^\infty}\leq C$ for some fixed constant $C$. Considering \begin{equation} \rho_{x_0,t_0}(x,t)=(4\pi(t_0-t))^{-1}e^{-\frac{\|x-x_0\|^2}{4(t_0-t)}} \end{equation} which is the backward heat kernel in $2$-dimensions, the monotonicity formula of mean curvature flow with additional forces \cite{White97_Stratif, AM_CMgenericambient} shows that for $0\leq t_1\leq t_2<t_0$, \begin{equation}\label{eq:MonoForce} \int_{M_{t_2}}\rho_{x_0,t_0} (x,t_2) dx \leq e^{C(t_2-t_1)}\int_{M_{t_1}}\rho_{x_0,t_0}(x,t_1)dx. \end{equation} We remark that the monotonicity formula also works for Brakke flows with additional forces, so this inequality holds even $M_t$ is not necessarily smooth for $t\in[t_1,t_2]$. The following lemma shows that, while we may not get a uniform area growth bound for all $t>0$, we can show that for sufficiently large $t$, $M_t$ has a uniform area growth bound. \begin{lem}\label{lem:AreaGrowth} There exists $D>0$ and $\mathbf{t}>0$ such that for $t>\mathbf{t}$, and any $R\in(0,L]$, $\tau\in(1,2)$ and $x_0\in \R^m$ that $\area(B_R(x_0)\cap M_{t_i+\tau})\leq D\pi R^2$ where $L$ is the diameter bound of $N$ as above. \end{lem} \begin{proof} Because $\int_{M_t} \rho_{x_0,t_0}(x,t_0-R^2)\geq (4\pi R^2)^{-1}e^{-\frac{1}{4}}\area(M_t\cap B_R(x_0))$, we have \[ \area(M_t\cap B_R(x_0))\leq 4\pi R^2\int_{M_t} \rho_{x_0,t_0}(x,t_0-R^2)dx. \] By \eqref{eq:MonoForce}, it suffices to show that exists a fixed constant $\mathbf{t}>0$ such that \[ \int_{M_{t-\mathbf{t}}}\rho_{x_0,t+R^2}(x,t-\mathbf{t})dx \] has a uniform upper bound for all $t>\mathbf{t}$, $x_0\in N$ and $R\in(0,L]$. We observe that \[ \begin{split} \int_{M_{t-\mathbf{t}}}\rho_{x_0,t+R^2}(x_0,t-\mathbf{t}) =& \int_{M_{t-\mathbf{t}}} (4\pi(R^2+\mathbf{t}))^{-1} e^{-\frac{\|x-x_0\|^2}{4(R^2+\mathbf{t})}}dx \\ \leq & (4\pi(R^2+\mathbf{t}))^{-1}\area(M_{t-\mathbf{t}}) \leq (4\pi\mathbf{t})^{-1}\area(M_0). \end{split} \] Therefore, we can choose $\mathbf{t}=1$ to get a uniform upper bound, which yields the uniform area growth bound for $R\in(0,L]$. \end{proof} We are now ready to show the first result discussed in the introduction: \begin{proof}[proof of Theorem \ref{smoothfate}] If the area of $M_t$ is sufficiently small the clearing out lemma \cite[12.2 and 12.5]{Ilmanen94_EllipReg} shows that $M_t$ goes extinct (i.e. is supported on a set of zero measure) a short time later, so we may suppose its area is uniformly bounded below for all finite times. Considering a sequence $t_i \to \infty$ as in Lemma \ref{decay} above, by Allard compactness we may extract a converging subsequence which will be a stationary varifold $V$. By the almost all time regularity of the almost regular flows, we may suppose this set of times is regular as well. We also have the area growth estimate of $M_{t_i}$ from Lemma \ref{lem:AreaGrowth}. With this in hand we may employ Simon's sheeting theorem ala Ilmanen \cite{Ilmanen95_Sing2D} along a sequence of smooth times $t_i \to \infty$ of $M_{t_i}$ to say that $V$ supports on a finite collection of smooth, embedded minimal surfaces $\{\Sigma_1,\cdots,\Sigma_k\}$ and the convergence is $m_j$--sheeted away from a finite number of ``bridge points'' towards $\Sigma_j$. By the regularity of the limit, and the embeddedness of $M_{t_i}$, all the $\Sigma_j$'s are disjoint via the maximum principle. \end{proof} Higher multiplicity convergence can indeed occur, see \cite{ChenSun24_Multi2inManifold}, and in the case one of the $\Sigma_i$ is strictly stable should be a robust phenomenon. Also, it is easy to see the limiting set could be disconnected. Take for instance the initial data given by two disconnected area-minimizing closed embedded (say, strictly stable) minimal surfaces connected by a very thin neck. Then shortly after the flow is started the neck will quickly pinch and retract to each minimizing surface, and the limit of the flow will be the union of these two disconnected minimal surfaces. It is interesting to ask to what extent the limit is unique. If the convergence has multiplicity $1$, and the manifold is analytic, then by the classic \L{}ojasiewicz-Simon inequality, together with the Brakke/White regularity theory, $M_t$ indeed convergence to $\Sigma$ as $t\to\infty$, not only subsequentially. Without this machinery even in the most well-known case, the curve shortening flow on surfaces, this question is not completely understood. In fact in \cite{Grayson89_CSF} Grayson conjectured that there exists an example of curve shortening flow whose limit geodesics are not unique. It is not so hard though to see the limit will be unique if the limiting minimal surfaces are strictly stable, which we show next. To start the following lemma shows that if we carefully choose $t_i$, $M_{t_i}$ will stay in a neighborhood of the limit surfaces. In other words, those necks described in the example above indeed must disappear in a very short period of time. \begin{lem}\label{lem:LimNbhd} In the context of Corollary \ref{smoothfate} given a sequence $s_i \to \infty$ along which the flow converges to a collection of minimal surfaces $\Sigma_j$ for any $\delta>0$ we can choose another sequence of $t_i\to\infty$ such that, when $t_i$ is sufficiently large, $M_{t_i}$ must lie inside the $\delta$-tubular neighborhood of $\cup_{j=1}^k \Sigma_j$. \end{lem} \begin{proof} Denote by $\cN_{\delta}(\Sigma)$ the (open) $\delta$-tubular neighborhood of a closed embedded surface $\Sigma$ in $N$, and consider a sequence of $s_i\to\infty$ as in Corollary \ref{smoothfate} such that $M_{s_i}$ converges to $m_1\Sigma_1+\cdots+m_k\Sigma_k$ in the sense of varifolds. For any $x_0\in N\backslash(\bigcup_{j=1}^k\cN_{\delta}(\Sigma_j))$, $\area(M_{s_i}\cap B_{\delta}(x_0))\to 0$ because $B_{\delta}(x_0)\cap(\bigcup_{j=1}^k\Sigma)=\emptyset$. In particular, this shows that for any $\eps>0$, $x'\in B_{\delta/2}(x_0)$, and $\eta\in(1/2,3/4)$, the Gaussian weight \[ \int_{M_{s_i}}\rho_{x',s_i+(\eta\delta)^2}(x,s_i)dx<\eps, \text{when $s_i$ is sufficiently large.} \] Then by \eqref{eq:MonoForce} again, we see that when $\eps$ is chosen sufficiently small, the Gaussian density of $x'\in B_{\delta/2}(x_0)$ of $M_t$ for $t\in(s_i+\delta/2,s_i+3\delta/4)$ is strictly less than $1$, hence must be $0$. In other words, for $t\in(s_i+\delta/2,s_i+3\delta/4)$, $M_t$ is disjoint from $B_{\delta/2}(x_0)$ for sufficiently large $s_i$. Because $N\backslash \bigcup_{j=1}^k\cN_{\delta}(\Sigma_j)$ is compact, we can cover it by finitely many $B_{\delta/2}(x_0)$'s. This allows us to pick a sequence of $t_i\to\infty$ such that $M_{t_i}\subset \bigcup_{j=1}^k\cN_{\delta}(\Sigma_j)$ that have bounded total curvature and total mean curvature tending to zero. The argument of Corollary \ref{smoothfate} may then be rerun on this sequence to extract a subsequence which converges to a union of smoothly embedded minimal surfaces $\Sigma_j'$. Using these minimal surfaces the conclusion follows by the triangle inequality. \end{proof} We remark that in the above proof conceivably $\Sigma_i \neq \Sigma_i'$, and in particular the further chosen subsequence $M_{t_j}$ may not converge to the same collection of closed embedded minimal surfaces $m_1\Sigma_1+\cdots+m_k\Sigma_k$. But their limit must be very close to them, in the sense that the limit lies in a tubular neighborhood of these minimal surfaces. Now, recall that a stable minimal surface $\Sigma$ is strictly stable if the linearized operator $L_\Sigma$ has only positive eigenvalues. Using a barrier argument and the lemma above then we observe we may show that if the limit stable minimal surfaces are strictly stable, then the limit is unique, even if the convergence is possibly in high multiplicity and in a merely smooth (but not analytic) background manifold $N$: \begin{thm}\label{longterm_uniqueness1} Suppose $M_t$ is an almost regular flow such that there exists $t_i\to\infty$ such that $M_{t_i} \to m_1\Sigma_1+\cdots+m_k\Sigma_k$ as in Theorem \ref{smoothfate}, and each $\Sigma_j$ is strictly stable. Then the limit is unique: namely for any sequence of $s_i\to\infty$ so that $M_{s_i}$ converges as varifolds the limit must be $m_1\Sigma_1+\cdots+m_k\Sigma_k$. \end{thm} \begin{proof} For any $\delta>0$, we can apply Lemma \ref{lem:LimNbhd} to claim that there is another sequence, which we still label $t_i$, such that $M_{t_i}$ lies inside the union of $\delta$-tubular neighborhood of the $\Sigma_j$'s when $t_i$ is sufficiently large. To proceed, we use the neighborhood foliation structure of strictly stable minimal surfaces: because each $\Sigma_j$ is strictly stable, if it is $2$-sided, then there exists a tubular neighborhood $\cN^j$ of $\Sigma_j$ such that $\cN^j$ is foliated by surfaces $\Sigma^t$ for $t\in(-s,s)$, such that when $t\in(-s,0)\cup(0,s)$, $\Sigma^t$ has positive (inward pointing) mean curvature. For $1$-sided $\Sigma_j$, there is a similar structure of the neighborhood. For a proof of this fact, see \cite[Lemma A.1]{StevensSun24_LargeArea} for $2$-sided case, and \cite[Section 6]{StevensSun24_LargeArea} for $1$-sided case. Using this fact, we can find a tubular neighborhood $\cN^j$ of $\Sigma_j$ such that $\pr\cN^j$ has strictly positive mean curvature. In particular, if at the beginning we choose $\delta>0$ sufficiently small so that the $\delta$-tubular neighborhood is contained in $\cN^j$, then $M_t$ will lie within $\bigcup_{j=1}^k\cN^j$ for $t\geq t_i$. Hence we see any limit of $M_t$ has to be supported in $\bigcup_{j=1}^k\cN^j$. If $\delta > 0$ is sufficiently small we may apply Brakke regularity to see the flows of boundary components of $\bigcup_{j=1}^k\cN^j$ are smooth for all $t > 0$ and asymptote back to $\bigcup \Sigma_j$, so by the comparison principle any converging limit of $M_t$ has to be supported on $\bigcup_{j=1}^k\Sigma_j$. The question of convergence in varifold topology to this entire set is more subtle though, because in the above we may only apply Ilmanen's analysis to special sequence of times $s_i \to \infty$. To deal with this, first we point out that for any sequence $s_i \to \infty$ the argument from the proof of Theorem \ref{smoothfate} does apply we get the limit must be supported precisely on $\bigcup_{j=1}^k\Sigma_j$. Moreover by the monotonicity of area along the mean curvature flow the multiplicities may not vary either, so that for any sequence $t_i \to \infty$ for which $M_{t_i}$ converges "well" as in the proof of Theorem \ref{smoothfate}, $M_{t_i}\to m_1\Sigma_1+\cdots+m_k\Sigma_k$ as $t_i\to\infty$. Now, for a given $1 \leq j \leq k$ choose a small $\epsilon > 0$ let $u_{j, \epsilon, D}$ be a smooth function which is supported on the $\epsilon$ tubular neighborhood of a domain $D \subset \Sigma_j$. Because almost regular flows are smooth for almost all times and continuous in the weak topology in time (see Lemma 2.7 in \cite{BamlerKleiner23_Multiplicity1}) Lemmas \ref{tabounds}, \ref{decay} gives us that for any $\delta > 0$ we can approximate any sequence $s_i \to \infty$ by another sequence $s_i' \to \infty$ so that $\int_{M_{s_i}} u_{j, \epsilon, D} - \int_{M_{s_i'}} u_{j, \epsilon, D} < \delta$ and that Ilmanen's analysis applies to the sequence $s_i'$. Because from the above any converging limit of $M_t$ must be contained in $\bigcup \Sigma_j$ if the sequence of $M_{s_i}$ converges as varifolds its support must be contained in $\bigcup_{j=1}^k\Sigma_j$, and so by ranging over $D, j$ and letting $\epsilon, \delta \to 0$ we thus get that if the sequence $M_{s_i}$ converges as varifolds the limit must be precisely $m_1\Sigma_1+\cdots+m_k\Sigma_k$ as claimed. \end{proof} Indeed if one of the minimal surfaces $\Sigma_i$ is merely stable but not strictly stable, the above contracting tubular neighborhood may not exist. An example of such a minimal surface can be found in \cite[Appendix B]{StevensSun24_LargeArea}. To wrap up this section we summarize some topological properties of the limit minimal surfaces in the case $M_t$ can be approximated by generic flows: \begin{prop}\label{limit_topology} Suppose that $M_t$ is an inner or outermost flow of $M$ or is a generic flow whose long term limit is nonempty, and let $m_1\Sigma_1+\cdots+m_k\Sigma_k$ be limiting minimal surfaces as in Theorem \ref{smoothfate}, and that $N$ is oriented implying 2--sidedness is equivalent to orientability. Then the following hold: \begin{enumerate} \item $m_1\Sigma_1+\cdots+m_k\Sigma_k$ in Corollary \ref{smoothfate} is homologous to $M$ in $H_2(N,\Z_2)$. \item If $\Sigma_j$ is $1$--sided, then $m_j$ is even. \item When all the $\Sigma_j$ are orientable $\sum\limits_{j=1}^k m_j\cdot \text{genus}(\Sigma_j)\leq \text{genus}(M)$. More generally \begin{equation} \sum\limits_{j, \text{ }\Sigma_j \text{ 2--sided}} m_i \cdot\text{genus}(\Sigma_j) + \sum\limits_{j, \text{ }\Sigma_j \text{ 1--sided}} \frac{m_j}{2} \cdot (\text{genus}(\Sigma_j) - 1)\leq \text{genus}(M) \end{equation} \end{enumerate} \end{prop} \begin{proof} By Lemma \ref{lem:PreserveHomology}, item (1) holds because this homology group is closed under limits. Now for the other parts we first note that because the convergence to the $\Sigma_j$ is multisheeted away from a discrete set of points, each sheet in the convergence corresponds to a connected component of the boundary of the tubular neighborhood of the limit (or, in other words, a connected component of its normal bundle). If $\Sigma_j$ is $1$-sided there is only one connected component, but about a point locally there are two components giving item (2). For item (3) by genus of nonoriented $\Sigma_j$ we refer to the demigenus and, since this may be somewhat less familiar, first we consider the case all the $\Sigma_j$ are orientable. In this case because all of the limit surfaces $\Sigma_j$ are orientable each sheet over $\Sigma_j$ in the convergence is homeomorphic to $\Sigma_j$ away from a finite set of points following the discussion above; then the claim follows easily from Lemma \ref{lem:GenusNonincreasing}. In case some are nonorientable we note by the classification of surfaces they must be homeomorphic to a connected sum of projective planes, say $k_j$ of them. The demigenus of such a connect sum is $k_j$ and the double cover of such a connect sum, corresponding to a single sheet in the convergence, happens to have genus $k_j - 1$. Because these numbers agree we obtain the formula in item (3). \end{proof} \section{The long--time limit is generically stable or empty: the proofs of Theorems \ref{Rc_thm} and \ref{pw_thm}} From the previous section we see that if the limit of an almost regular flow is nonempty it converges to a collection of smooth minimal surfaces, possibly with multiplicity. We also obtain some rough topological information about the limit. Even so it is still often desirable to specifically produce stable minimal surfaces because they carry more structure; for instance as a basic but important example we recall that stable surfaces in PSC $3$--manifolds must be diffeomorphic to $S^2$ or $\R P ^2$ but just in $S^3$ there are minimal surfaces of arbitrarily large genus. In this section, we show that with appropriately timed small isotopies, we can construct piecewise flows whose long-time limits are either empty or stable minimal surfaces. We consider first the Ricci positive case. As promised in the introduction, arguing in this case turns out to be simple because in this case the avoidance principle says two disjoint flows actually "repel" In fact we don't even need the regularity statements from the previous statement to proceed and indeed our argument works in all dimensions. \begin{proof}[proof of Theorem \ref{Rc_thm}] Where $M$ and $N$ are as in the statement of the theorem, suppose that the flow $M_t$ of $M$ doesn't go extinct in finite time. Since the flow is gradient of the area functional, we may pick a sequence of times $t_i \to \infty$ for which $\int_{M_{t_i}} H^2 \to 0$. After potentially passing to a subsequence the sequence $M_{t_i}$ of surfaces limits to a varifold $\Sigma$ which by the choice of $t_i$ must be stationary. Now consider a one sided perturbation $M'$ of $M$, for which say $d(M, M') > \delta$ for some $\delta > 0$. Since $N$ is compact and Ricci positive, there exists some $\Lambda > 0$ for which the Ricci curvature of $N$ is bounded below by $\Lambda$. Hence by the avoidance principle, Theorem \ref{ambient_avoidance} above, we have that the distance between $M'_t$ and $\Sigma$ is bounded below by $\delta e^{\Lambda t}$ for all times; because $N$ is compact so has bounded diameter for $t$ sufficiently large this gives that the level set flow $M'_t$ of $M'$ must be empty. \end{proof} We point out that as a consequence of the above, that in an orientable $3$--manifold every $2$--cycle can be represented by a properly embedded $2$--sided surface (see, for instance, Lemma 3.6 in \cite{Hatcher07_Notes3Mfd}), and Proposition \ref{limit_topology} we have another proof of the following classical fact: \begin{cor}\label{vanish} $H_{2}(N, \R)$ of a compact orientable $3$--manifold $N$ which admits a metric of positive Ricci curvature vanishes. \end{cor} Indeed, by Hodge theory and the Bochner formula $H^1(N)$ vanishes so the claim follows by Poincare duality. Now in the case $N$ doesn't have Ricci positive curvature the avoidance principle doesn't give as strong information and indeed it is easy to concoct examples where two disjoint flows may converge to each other as $t \to \infty$. For instance let $N$ be a compact manifold for which $H_2(N,\Z)$ is nontrivial (so cannot have positive Ricci curvature by the above). By the direct method in a nontrivial class of $H_2(N,\Z)$ there is an area minimizing surface $\Sigma$, and by perturbing the metric to be bumpy we may even suppose it is strictly area minimizing. By considering disjoint nearby surfaces (say graphical over $\Sigma$) their flows must converge to $\Sigma$ in the long term. With this in mind we now move on to Theorem \ref{pw_thm}. Recall the definition of piecewise generic flow, Definition \ref{pwflow_def}, and note that after potentially perturbing $M$ be a small amount without loss of generality we may suppose we are in the setting of Corollary \ref{smoothfate}. If $M_t$ goes extinct or all the minimal surfaces $\Sigma_j$ it subconverges to, defined as in the previous section, are stable minimal surfaces, then we are done. Otherwise, we apply isotopes to $M_{t}$ at a smooth time far along the flow which ensures the perturbation must not converge to the same surfaces after restarting the flow: \begin{lem}\label{perturb} Let $M$ be a $2$--sided properly embedded surface in a closed compact Riemannian $3$--manifold $(N^3,g)$, and suppose $M_t$ is an almost regular flow out of $M$ which can be approximated by generic flows (if not generic itself, the inner or outermost flow from $M$). Suppose $M_{t_i}$ converges to $m_1\Sigma_1+\cdots+m_k\Sigma_k$ as varifolds, and one of $\Sigma_j$ is unstable. Then for sufficiently large $t_i$ we may isotope $M_{t_i}$ so that any almost regular flow starting from the perturbed surface can not enter the tubular neighborhood of that unstable $\Sigma_j$. \end{lem} \begin{proof} Suppose $\Sigma_1$ is an unstable minimal surface, and in the following, we simply call it $\Sigma$. We first consider the situation that $\Sigma$ is unstable and $2$-sided. Then we can find a unit normal vector field $\nu$ over it, and we may assume $\varphi$ is the first eigenfunction of the linearized operator $L_\Sigma$. Let us use $(x,s)\in\Sigma\times(-\tau,\tau)$ to parametrize a tubular neighborhood $\cN_{\tau}$ of $\Sigma$ by $(x,s)\to\exp_x(s\varphi(x)\nu(x))$. We first define an isotope $\Psi_i$ of the ambient manifold as follows: we define the vector field $\pr_s$ in $\cN_{\tau/2}$ using the coordinate $(x,s)$, then we smoothly extend it to $0$ outside $\cN_{\tau}$ to get a smooth vector field $W$. Then we define $\{\Psi(w)\}_{w\in\R}$ to be the isotopy generated by the vector field $W$. Let us choose a very small $\delta>0$, such that $\cN_\delta(\Sigma)$ is completely contained in the domain given by the parametrization $(x,s)\to\exp_x(s\varphi(x)\nu(x))$ for $s\in(-\tau/10,\tau/10)$, and $\Psi(\tau/2)\cN_\delta(\Sigma)$ will completely leave $\cN_\delta(\Sigma)$. Following Lemma \ref{lem:LimNbhd}, we may assume $M_{t_i}\subset \bigcup_{j=1}^k\cN_{\delta}(\Sigma_j)$ for sufficiently large $t_i$. Then $\tilde M_{t_i}:=\Psi(\tau/2)M_{t_i}$ must be completely disjoint from $\cN_\delta(\Sigma)$, and this is a $2$--sided flow from which we may safely restart the flow from. Recalling that the support of Brakke flows are weak set flows, it's easy to see that any Brakke (and hence almost regular) flow starting from such $M' := \tilde M_{t_i}$ will never enter $\cN_{\delta}(\Sigma)$ again by the avoidance principle. To see this, note that the domain generated by the parametrization $(x,s)\to\exp_x(s\varphi(x)\nu(x))$ for $s\in(-\tau/10,\tau/10)$ has compact, smooth and outwardly mean convex boundary, with two components, which we denote by $\Sigma_{\pm \tau/10}$. By the strict mean convexity and the compactness there exists $\delta, \eta > 0$ so that the smooth mean curvature flow of the boundary exists on $[0, \delta)$ and the distance between $\Sigma_{\pm \tau/10}$ and $(\Sigma'_{\pm \tau/10})_\delta$ is bounded below by $\eta$. The avoidance principle says $(\Sigma_{\pm \tau/10})_t$ and $M'_t$ are disjoint on $[0, \delta)$, and because $\eta > 0$ we can iterate this argument over the intervals $[\delta, 2\delta)$ et. cetera to get the claim. Finally, we briefly describe the situation that $\Sigma$ is unstable and $1$--sided. The argument is similar but now the tubular neighborhood of $\Sigma$ is only parametrized by $\overline{\Sigma}\times(0,\tau)\bigcup \Sigma$, where $\overline{\Sigma}$ is the oriented double cover of $\Sigma$. In this case, we can still construct the isotopy to push the flow outside this tubular neighborhood, and verbatim arguments validate all the previous arguments. \end{proof} With this in hand, we are now ready to prove the main result of this section: \begin{proof}[proof of Theorem \ref{pw_thm}] We may clearly restart the flow after the perturbation in Lemma \ref{perturb}, but apriori the set of times for which we perturb may accumulate to a finite time; we claim that only finitely many perturbations are required though to ensure that afterwards none of the $\Sigma_j$ could be unstable. Suppose $\Sigma$ is an unstable minimal surface that the flow was perturbed to avoid by the lemma; then if the flow $M_t$ starting from this perturbation converges to another unstable minimal surface it must converge to one which is disjoint from $\Sigma$ and hence disjoint from the component of the flow which was just perturbed -- but by the avoidance principle argument above the flow of this connected component may never converge to any such unstable minimal surface. Now because we can take the parameter $\delta$ in the proof above as small as we wish we may suppose the area of the piecewise mean curvature flow is bounded by, say, twice the area of $M$ after the $k$-th perturbation/iteration of Lemma \ref{perturb} for any $k \in \mathbb{N}$. In particular by the clearing out lemma only finitely many components of $M_t$ at a given smooth time might go on to flow to a minimal surface, with the rest must quickly being exhausted; in fact by this we may crudely uniformly upper bound (independent of time) the number of such components. Applying the reasoning above to each such component then gives what we want.\end{proof} \section{Applications to existence questions for minimal surfaces: proofs of corollaries \ref{existence1}, \ref{existence2}, \ref{existence3}, and \ref{existence5}}\label{applications} In this section (aside from the last application, where we use Theorem \ref{longterm}), we consider various circumstances where we can show that the piecewise almost regular flows from the last section will be nonempty as $t \to \infty$ and say something about the topology of the limit. The first is on the existence of stable minimal spheres when $\pi_2(N) \neq 0$. \begin{proof}[proof of Corollary \ref{existence1}] First we note that by the sphere theorem in $3$--manifold topology \cite{Papakyriakopoulos57_Dehn} we can find, using $N$ is orientable, a homotopically nontrivial embedded $2$--sphere $M$. Of course $M$ is itself orientable, and we may use it as initial data in Theorem \ref{pw_thm}. By considering the approximating surgery flows $M_t$ during smooth times will consist of a collection (possibly multiple, if there are nontrivial neckpinches) of embedded $2$--spheres. Now, either the flow from the conclusion will terminate in finite time or converge to a stable minimal surface $\Sigma$ which, by the nature of the convergence discussed in Theorem \ref{smoothfate} and the proof of Proposition \ref{limit_topology}, must be a collection of embedded stable minimal spheres or projective planes because each component for large (smooth) times is covered $k \geq 1$ times by $M_t $ for large $t$, away from mentioned bridge points and each component of this is an embedded $2$--sphere. Picking any of these components then gives the statement. So, it suffices to rule out the flow going extinct in finite time. Supposing this is actually the case, considering the approximating surgery flow one can show that $M$ is isotopic to a "marble" tree as in \cite{BuzanoHaslhoferHershkovits21_Moduli} or similarly \cite{Mramor18_Finiteness}, taking some care because in our setting the flow may not be mean convex even if the high curvature regions are. Because in the mean curvature flow with surgery for large parameters the high curvature regions all bound handlebodies one can show such marble trees bound $3$--balls in $N$ giving that $M$ is in fact homotopically trivial, leading to a contradiction.\end{proof} We remind the reader the next corollary is on the existence of stable minimal tori in the aspherical case. Corollary \ref{existence2} in fact follows from Corollary \ref{existence3}: it isn't hard to see by Dehn's lemma and the trivial $\pi_2$ assumption that if the torus $T$ isn't algebraically incompressible then it bounds a 3 cycle. That being said we give the proof below for the sake of exposition, because it is somewhat simpler than the more general statement. \begin{proof}[proof of Corollary \ref{existence2}] Denote by $M$ the torus in the statement. Because the torus is orientable and $N$ is assumed oriented we have that $M$ is $2$--sided, so that we may apply Theorem \ref{pw_thm} to it. First we argue that the flow must not go extinct. Supposing it does, we note that approximating surgery flows go extinct as well. Considering one of these and denoting it by $S_t$, we note that high curvature regions which are discarded certainly bound $3$--cycles, homeomorphic to either $3$--balls or solid tori $\simeq S^1 \times D^2$, since different time slices along a smooth mean curvature flow are homologous this gives that $M$ is homologous to $S_t$ for all $t > 0$ and hence bounds a $3$--cycle, contradicting that $M$ is assumed to be homologically nontrivial. Next, we note that for any finite smooth time $t$, $M_t$ cannot be given by a union of embedded spheres $P_1, \ldots, P_k$. To see this, because $\pi_2(N)$ is trivial each of the $P_i$ are nullhomotopic and hence are the boundary of a $3$--cycle, so since $M_t$ is homologous to $M$ this gives that $M$ is homologically trivial giving a contradiction. In particular, a connected component of $M_t$ at a smooth time must be homeomorphic to a torus. Now, because the flow doesn't go extinct as before we may produce a sequence of times $t_i \to \infty$ for which $M_{t_i}$ converge to an embedded stable minimal surface $\Sigma$, and we claim one of the connected components of $\Sigma$ is a torus or a Klein bottle (which is the nonorientable surface covered by the torus) -- note this doesn't follow immediately from the case that a component of $M_t$ is a torus. For example, one can take a standardly embedded torus in $\R^3$, tighten the "donut hole" and then push the sides of the inner cylinder along the outer sides of the donut so in this fashion covering the sphere twice, away from bridge points. Of course if one of the components is a torus or Klein bottle we are done, so it suffices to rule out the case none of them are. By Proposition \ref{limit_topology} in this case all the components of $\Sigma$ are spheres or projective planes. Suppose first all the components are spheres: then by assumption they are nullhomotopic and so each of these spheres then could be homotoped to be within coordinate patches homeomorphic to open balls. As these patches are homeomorphic to an open ball in $\R^3$ all surfaces contained within are nullhomologous. Using that homotopic surfaces are homologous as before this implies $M$ itself is nullhomologous, giving a contradiction. If some of the components are projective planes we can argue similarly, because the boundary of their tubular neighborhoods are diffeomorphic to spheres. \end{proof} As the reader recalls from the introduction, the next statement concerns the existence of stable minimal surfaces in the presence of algebraically incompressible initial data, which is to say the inclusion map on fundamental groups injects; by Dehn's lemma, algebraically incompressible surfaces are incompressible in the standard sense i.e. that there are no compressing discs in $N$ along $M$. This topological assumption is arguably most naturally suited to how the topology of a flow may change across singularities, at least generic ones. This is because the effect of flowing through a neckpinch is essentially to glue a thickened disc along the flow of the corresponding domain, and so if the surface is incompressible only spheres can be pinched off. \begin{proof}[proof of Corollary \ref{existence3}] Supposing that $M = \iota(\Sigma)$ is as in the statement, we consider as before the approximating surgery flow $S_t$ to $M_t$. We first claim that at any smooth time a given connected component of it is either homeomorphic to either a sphere or $M$. Supposing not and let $T$ be the smallest time, necessarily right after a surgery, that this isn't the case -- for a fixed choice of approximating surgery flow this time is well defined. Then it's easy to see that the cross section curve $\gamma$ at one of the necks where surgery was performed on $S_T^{-}$ (the presurgery surface) is homotopically nontrivial in $M_T$. The surgery shows that $\gamma$ bounds a disc in $N$ though, so that $S_T^{-}$ is not algebraically incompressible. Since $T$ was the least such time this implies that $M$ itself is not algebraically incompressible, which gives a contradiction. Although it is the case that the flow preserves the topology of $M$ in this sense it could be the case that there is strict genus drop in the limit as $t \to \infty$. In the following we focus only on the connected component $\tilde{M_t}$ of $M_t$ homeomorphic to $M$, and denote its corresponding limiting stable minimal surface $\Sigma$; the reason this suffices for us because, by the $\pi_2$ triviality assumption and that the other components pinched off the flow will be spheres, at any given time $M_t$ will be homotopic to just this component. Now, if the convergence is with multiplicity one then it will be the case $\Sigma$ is orientable and isotopic to $M$. Supposing this is not the case, then from the proof of Theorem \ref{pw_thm} we have that either $\Sigma$ is nonorientable and $M_t$ converges to $\Sigma$ with multiplicity 2, or $M_t \to \Sigma$ with some multiplicity $m$ and there is at least one bridge point $p$, within which for some large time $s$ and some small $r > 0$ we have $M_s \cap B(p,r)$ has less than $m$ components -- our task is to rule out this latter case. Taking $r$ slightly larger if necessary $M \cap (B(p,2r) \setminus B(p, r))$ is comprised of $m$ disjoint annuli, where $m$ is the multiplicity of convergence; after a slight isotopy we may suppose these annuli $A_i$ lay in parallel planes and so may order them from "top" to "bottom" in the obvious sense, with $A_1$ being the top annulus. Without loss of generality the top annulus at the point $p$ is nontrivial in the sense it doesn't border a disc in $M_s \cap B(p,r)$; if not below we can consider the next lowest disc and so on. Considering one of the boundary components $\gamma$ of $A_1$, we may clearly find an embedded disc $D$ in $N$ disjoint from $M_ s$ with boundary $\gamma$. If $\gamma$ is nonseparating or splits $M_s$ into two components of genus greater than zero then $D$ will be a compressing disc because $\gamma$ will be homotopically nontrivial in $M$, which gives a contradiction. If the number of bridge points connected to the top sheet (i.e. where $B(p_i, r_i)$ are the bridge points and neighborhoods, the connected component of $M \setminus \cup_i B(p_i, r_i)$ containing $A_1$) is greater than one then $\gamma$ is nonseparating, so that we are done; let us suppose that $p$ is the only bridge point where the top sheet connects to the lower sheets then. If $\Sigma$ is not an embedded sphere or projective plane then $\gamma$ would disconnect $M$ into two components of nonzero genus, completing the argument in this case. If $\Sigma$ is a sphere or projective plane a bit more arguing is required. As before first suppose that $\Sigma$ is a sphere. Then because $N$ has trivial $\pi_2$, we may homotope $\Sigma$ and hence $M_s$ to lay in a small coordinate chart; of course because $\R^3$ is contractible $M_s$ in this case cannot be incompressible. If it is a projective plane again the boundary of its tubular neighborhood is a sphere, and we may proceed as in the first case. \end{proof} Leaving the world of stable minimal surfaces we now consider the use of our first convergence statement, Theorem \ref{longterm}, to essentially replace the tightening process in minmax. Recalling that in the setting of Corollary \ref{existence5} setting $N \simeq S^3$ we may consider the standard one parameter sweepout $\Sigma_s$ of it by $2$--spheres, where there are two singular values of the sweepout that are single points corresponding to the north and south poles under a fixed choice of diffeomorphism from $N$ to $S^3$. Perhaps the most natural thing to do then is to consider the flow of the sweepout under the flow, by which we mean the family of the flows $(\Sigma_s)_t$ of the individual leaves $\Sigma_s$. Recalling that the width of any sweepout of $S^3$ by spheres is positive by the isoperimetric inequality, we see if we could show that if the flow will remain a sweepout, then Theorem \ref{longterm} says that the flow of at least one leaf will not go extinct, and because $S^3$ is simply connected and embedded hypersurfaces in such spaces are orientable Proposition \ref{limit_topology} gives the limit will be an embedded minimal sphere, as usual potentially with multiplicity. There are some issues with the approach though, unfortunately. The biggest issue in general appears to be that the level set flow of some of the leaves may fatten, but in our particular setting this may be ruled out as we discuss below. There is also an issue with preserving the typical regularity assumptions one make in the definition of sweepout. Recall that a sweepout of $N$ (here, $1$--parameter) is a family of surfaces $\Sigma_s$, $s \in [0,1]$, which for finitely many $s$ is singular in a finite set of points and smooth elsewhere and whose union is all of $N$. Now, we know that for a fixed leaf $\Sigma_s$ the flow $(\Sigma_s)_t$ will be smooth for almost all times $t$, but because we wish to consider the flow of the family, parameterized over the whole unit interval (which is uncountable so Baire category theorem cannot apply), it doesn't seem clear if we can show the flow of the whole family will be smooth for almost all or even many times. The current state of the art also doesn't rule out the size of the singular set of a flow being greater than a finite union of points, either. \begin{proof}[proof of Corollary \ref{existence5}] With respect to the discussion above one may be able to proceed by generalizing the results of minmax theory to more general sweepouts including those "generated" by the flow but instead we will take a somewhat more hands on approach. Denote by $E(s)$ the extinction time of the flow of $\Sigma_s$; if this never occurs of course we set $E(s) = \infty$ so it is a function into the extended reals. We next claim that $E(s)$ is sequentially continuous. To see this, we first note that because each of the $\Sigma_s$ are spheres their singularities under the flow must be mean convex and so each of their flows is nonfattening -- implying for instance that the almost regular flows $(\Sigma_{s_i})_t $ are well defined. Fixing an $s_0 \in [0,1]$, nonfattening also implies that for a sequence $s_i \to s_0$, after potentially passing to a subsequence by Brakke compactness, $\lim\limits_{s_j \to s_0} (\Sigma_{s_i})_t \to (\Sigma_{s_0})_t$ as Brakke flows because $\lim\limits_{s_j \to s_0} (\Sigma_{s_i})_t $ is a weak flow associated to $\Sigma_{s_0}$. With this in mind suppose then that there was a sequence $s_i \to s_0$ and $\epsilon > 0$ for which $E(s_0) < A - \epsilon$, where $A = \lim\limits_{i \to \infty} E(s_i)$. At time $T = E(s_0)$, by Colding and Minicozzi \cite{ColdingMinicozzi16_SingularSet} $(\Sigma_{s_0})_T$ is the union of a finite number of compact embedded Lipschitz curves corresponding to cylindrical singularities along with a countable collection of points corresponding to spherical singularities. This gives that for $i$ sufficiently large we make take the mass of $(\Sigma_{s_i})_T$ to be as small as we like, so in particular the clearing out lemma says these flows must go extinct before $t = A - \epsilon/2$, giving a contradiction to show that $E(s)$ is upper semicontinuous. Similarly $E(s)$ is lower semicontinuous, giving our claim of sequential continuity. Because $E(s)$ is sequentially continuous, by the compactness of the unit interval $\sup\limits_{s \in [0,1]} E(s) = T$ is obtained for some time $s_0$, recycling notation. Now as above we have $(\Sigma_{s_0})_T$ must consist of the union of a finite number of compact embedded Lipschitz curves corresponding to cylindrical singularities along with a countable collection of points corresponding to spherical singularities. Picking some smooth time $t^$ right before $T$, the resolution of the mean convex neighborhood conjecture gives that $(\Sigma_{s_0})_{t_}$ is mean convex and after rescaling is locally modeled on bowl solitons, round cylinders, or round spheres. To see why this is useful, first note that $s_0 \in (0,1)$: this is because for $s$ very near $0$ or $1$ that $\Sigma_s$ are approximately small concentric round spheres contained in a coordinate chart so that $\lim\limits_{s \to 0, 1} E(s) = 0$; on the other hand for a fixed smooth $\Sigma_s$ its smooth flow exists for a short but strictly positive time. Secondly, the flows $(\Sigma_{s})_t$ are pairwise disjoint in $s$ because $\Sigma_s$ are and almost regular flows are weak set flows. In particular, for $s - s_0$ sufficiently small $s$ Brakke regularity says that $(\Sigma_s)_{t^}$ is mean convex and a graph over $(\Sigma_{s_0})_{t_}$ laying on one side or the other of it, and we can find $s$ so that the flow at time $t^$ is on either side. Hence we see that for $s$ either slightly greater or smaller than $s_0$ $(\Sigma_{s})_T$ must not be singular, or in other words that their flow lasts at least slightly after time $T$ giving a contradiction to the definition of $T$. \end{proof} \bibliographystyle{alpha} \bibliography{GMT} \end{document} The following corollary directly results from the fact that the varifold convergence can only lead to genus dropping. \begin{cor} Suppose $(N,g)$ is a closed $3$-manifold with bumpy metric. Then for any homology class $\sigma\in H_2(N,\Z)$, there exists a finitely many disjoint closed embedded stable minimal surfaces $\Sigma_1,\cdots,\Sigma_k$ such that $[\Sigma_1]+\cdots+[\Sigma_k]=\sigma$. Moreover, if $\sigma$ can be represented by a closed embedded surface with genus $g$, then $\sum_{j=1}^k \text{genus}(\Sigma_i)\leq g$. \end{cor} In particular, if we run the piecewise flow starting from a closed embedded surface in the homology class with the least genus, the limit surface must have the same number of genus, counting multiplicity. \begin{cor}\label{cor:StableGenus} Suppose $(N,g)$ is a closed $3$-manifold with bumpy metric. Then for any homology class $\sigma\in H_2(N,\Z)$ there exists a finitely many disjoint closed embedded stable minimal surfaces $\Sigma_1,\cdots,\Sigma_k$ such that \[ \sum_{j=1}^k m_i\cdot\text{genus}(\Sigma_i) = \min_{\Sigma\in \sigma} \text{genus}(\Sigma). \] \end{cor} \begin{rem} Corollary \ref{cor:StableGenus} seems to be a feature that can only be detected by flows. In fact, the minimizing surface in a homology class may have genus strictly larger than the minimizing genus representative in the homology class. Consider a manifold is topologically $S^2\times S^1$. We equip it with a $SO(2)$-invariant metric where the $SO(2)$ acting on the $S^2$ factor. Then after taking the quotient, the manifold becomes $[-1,1]\times S^1$, and any $SO(2)$-invariant surface can be represented as a curve in $[-1,1]\times S^1$, and the area of the surface equals a weighted length of the curve. Now we equip $[-1,-1/2]\times S^1$ and $[1/2,1]\times S^1$ with the standard round metrics on $S^2$ product with $S^1$, and we equip the rest part with a metric such that a closed loop $\ell$ has the minimizing weighted length and the weighted length is strictly less than the weighted length of $[-1,-1/2]\times \{\}$ plus $[1/2,1]\times \{\}$. Consider the homology class $2[S^2\times\{\}]$. The minimizing surface $\Sigma$ in this homology class must be $SO(2)$-invariant as well. Otherwise, one can apply $SO(2)$ action to $\Sigma$ to get another surface $\Sigma'$, which is also minimizing, but $\Sigma'$ intersects $\Sigma$. If the rotation is small enough, $\Sigma'$ would be a graph over $\Sigma$, and we can take the smaller area pieces of $\Sigma$ and $\Sigma'$ to get another surface $\Sigma''$ with possibly smaller area, but in the same homology class. If $\Sigma''$ has area strictly smaller than $\Sigma$, then we get a contradiction; if $\Sigma''$ has area equal to $\Sigma$, then it is also an area minimizer in the homology class. However, it is not regular as $\Sigma$ and $\Sigma'$ intersect, contradicting the fact that area-minimizing surfaces are regular in $ 3$ dimensions. Therefore, because $\ell$ is the weighted length minimizing curve in the quotient space, $\ell\times S^1$ is a minimizing surface in the homology class $2[S^2\times\{\}]$. It has genus $1$. However, $2[S^2\times\{\}]$ can be realized by two disconnected spheres, which have genus $0$. \end{rem} Perturbation argument in version 1 As discussed before, lemma \ref{lbound} shows that for a given genus $g$ the area bound $m$ is strictly positive. Now suppose $M \subset N$ is as in the statement of the theorem. By an arbitrarily small perturbation as discussed in the preliminaries and used in the section above we may suppose that $M_t$ develops only mean convex singularities. As we showed in the section above, in particular by lemmas \ref{tabounds}, \ref{decay} and corollary \ref{smoothfate} we know that either the weak flow $M_t$ will go extinct in finite time or there is a sequence of times $t_i \to \infty$ for which $M_{t_i}$ converges to a smooth orientable minimal surface $\Sigma$. Because the area of $M$ is less than $2m$ the convergence is with multiplicity one, so for $t_i$ sufficiently large by Brakke regularity \cite{W1} $M_{t_i}$ can be written as the graph of a smooth function $u$ (depending on $i$, but this will be supressed) over $\Sigma$. Of course, if $\Sigma$ is stable then we are finished. Now, imaginably between the times $t_i$ the flow $M_t$ might not be graphical over $\Sigma$ but its a consequence by Brakke regularity that for small forward times from $t_i$ that it is and we write $u(x,t)$ for the corresponding function; since $\Sigma$ is minimal and the convergence is smooth we also see for $i$ large enough $M_t$ can be guaranteed to be written graphically over $\Sigma$ for as long a time as we wish. Because the graph of $u$ moves by the mean curvature flow, it turns out that $u$ solves the following PDE: \begin{equation} \frac{du}{dt} = L_\Sigma u + E(u) \end{equation} where $L_\Sigma = \Delta + \|A\|^2 + \Ric(\nu,\nu)$ is the Jacobi operator of $\Sigma$, and $E(u)$ is an error term which is $O(\|\|u\|\|_{C^2(\Sigma)})$, proceeding as in section 2.2 of \cite{CHL}. Of course $L_\Sigma$ is a self adjoint operator; denote by $\lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots \leq \lambda_I < 0 < \lambda_{I + 1} \leq \cdots$ the eigenvalues of $L_\Sigma$ where $I$ is the index of $\Sigma$. Denoting the corresponding eigenfunctions by $f_i(x)$, the functions $e^{-\lambda_i t}f_i(x)$ each solve the linear evolution equation $\frac{du}{dt} = L_\Sigma u$. With this in mind consider the fourier series representation $u(x,t_i + 1) = \sum_{i = 1}^\infty c_i f_i$. The solution $\tilde{u}$ to the linearized flow with this as initial data is given by $\tilde{u}(x,t) = \sum_{i = 1}^\infty c_i e^{-\lambda_i t} f_i$. This well approximates the actual solution $u$ to the flow: \begin{lem} For any $s > 1$, there exists $\epsilon_0, C > 0$ so that when $\|\|u(\cdot, T + 1)\|\|_{C^{1}}$ is sufficiently small \begin{equation} \|\| u( \cdot, T + 1 + t) - \widetilde{u}( \cdot, t)\|\|_{C^{2, \alpha}} \leq C \|\|u\|\|_{C^{1}}^{1 + \epsilon_0} \end{equation} on the interval $[1,s]$. \end{lem} Because $M_{t_i}$ converges to $\Sigma$ note $\|\|u(\cdot ,t_i )\|\|_{C^1}$ and hence, since $\Sigma$ is minimal, $\|\|u(\cdot ,t_i + 1)\|\|_{C^1} \to 0$ as $t_i \to \infty$. Note in particular this says that in the above $c_i = 0$ for $1 \leq i \leq I$ by the lemma. Recalling that the first eigenfunction of $L_\Sigma$ has a sign, the idea is that a sufficiently small one sided perturbation $M'$ of $M$ will also flow to a graph over $\Sigma$ of a function $w$ at a fixed time $t_i$, but in this case the Fourier expansion of $w(x,t_i + 1)$ will have $c_1 \neq 0$. If the flow is smooth up to time $t_i $ that $M'_{t_i}$ will be a graph over $\Sigma$ is a consequence of smooth dependence on initial conditions but its also true in the presence of singularities by the nonfattening of $M_t$. Now we claim that $M'_t$ cannot converge to $\Sigma$: suppose on the contrary it did. Then we can apply the lemma above with $s$ as large as we wish (using $i$ sufficiently large) but $c_1 e^{-\lambda_1 t}$ grows exponentially quickly. Because $f_1$ is the first eigenfunction of $L_\Sigma$ which corresponds to the second derivative of area, the area of $M_t'$ must be strictly less than the area of $\Sigma$ by similar arguing -- in particular very small perturbations of M' will never flow back to $\Sigma$. By the compactness of minimal surfaces in a bumpy 3 manifold of bounded genus and area and the Lojasiewicz-Simon inequality \cite{LS} there are only finitely many possible areas for minimal surfaces under these conditions so we can repeat the argument above finitely many times to reach the conclusion. Pilgrim process Because there is no Huisken's monotonicity formula in a general closed manifold, this uniform area growth bound is not necessarily true. On the other hand, we can still use Ilmanen's argument, after applying further isotopes, as follows. First, suppose the varifold limit of $M_{t_i}$ is $k\Sigma$, a priori $\Sigma$ is not necessarily a smoothly embedded minimal surface, but it is a stationary varifold. In particular, for any $p\in \supp\Sigma$ and sufficiently small $r>0$, when $t_i$ is sufficiently large, $\area(M_{t_i}\cap B_r(p))\leq (k+1)\theta(\Sigma,p)\pi r^2$. Here $\theta(\Sigma,p)$ is the density of $\Sigma$ at $p$. Next, we show that $\Sigma$ has finite density, namely there exists $m\in\Z_+$ such that $\theta(\Sigma,p)\leq m$ for all $p\in\Sigma$. In fact, by monotonicity formula in Riemannian manifold (\textcolor{violet}{see \href{https://web.math.princeton.edu/~rcabral/pdfs/minimalsurfaces.pdf}{this note, Theorem 7.11}. Is there a better reference?}), there exists $r_0>0$, $A>0$ only depending on $(N,g)$ such that for any $p\in \Sigma$ and $r<r_0$, $\|B_r(p)\cap \Sigma\|\geq e^{-Ar^2}\theta(\Sigma,p)$. But $\area(B_r\cap M_{t_i})\to k\|B_r(p)\cap \Sigma\|$ as $t_i\to\infty$, so we must have an uniform upper bound on $\theta(\Sigma,p)$. Suppose $\theta(\Sigma,p)\leq m$ for all $p\in\Sigma$. Then there exists $\eps_0$ depending on $N,m$ and $k$ such that if $\int_{M_{t_i}\cap B_r(p)}\|A\|^2<\eps<\eps_0$, $M_{t_i}\cap B_{r/2}(p)$ would be a disjoint union of topological disks $D_1,\cdots,D_\ell$, and away from a collection of closed disks $P_1,\cdots,P_N$ (Simon called them ``pimples''), $D_i$'s are graphs over some plane, and the gradient of the graphs are controlled by $\eps^{1/2}$. The sum of the diameters of $P_i$'s are bounded by $C\eps^{1/2}r$. Now we isotope $M_{t_i}$ to ``pilgrim'' the pimples, so that they will only contribute to area by $\pi(C\eps^{1/2}r)^2$. This procedure (and the terminology ``pilgrim'') has been discussed in \cite{??? Almgren-Simon}. In fact, this can be done by finding little round loops on the graph regions of $D_1,\cdots, D_\ell$ ($\ell$ is uniformly bounded by $M_0,m$) with total length bounded by $C'\cdot C\eps^{1/2}r$ where $C'$ only relies on the geometry of $(N,g)$, so that the pimples lying inside each loop. These round loops can be chosen to be the intersection of geodesic balls in $(N,g)$ with $M_{t_i}$. Then we can use isotopes to shrink the pimple to a minimal disk whose boundary is the loop. It is clear that after this process, each pimple will have only area $C'\pi (r')^2$ if the geodesic ball has radius $r'$. Let us call $\tilde{M}_{t_i}$ which is the surface $M_{t_i}$ after this isotopes procedure. Now we can repeat the proof of Ilmanen as we have the desired area growth estimate for each disk $D_1,\cdots, D_\ell$ in the above setting. Then after passing to limit, $\tilde{M}_{t_i}$ converges to an embedded minimal surface $\tilde \Sigma$ except at finitely many points where curvature concentrates. Again by the removability of isolated singularities of minimal surfaces, we can show that $\tilde \Sigma$ is a smoothly embedded minimal surface.
2412.03536v1	http://arxiv.org/abs/2412.03536v1	On unique continuation in measure for fractional heat equations	\documentclass[11pt]{amsart} \usepackage{amssymb,amsmath,epsfig,mathrsfs, enumerate, xparse, mathtools} \usepackage[pagewise]{lineno} \usepackage[nodisplayskipstretch]{setspace} \setstretch{1.5} \usepackage{graphicx}\usepackage[normalem]{ulem} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhead[RO,LE]{\small\thepage} \fancyhead[LO]{\small \emph{\nouppercase{\rightmark}}} \fancyhead[RE]{\small \emph{\nouppercase{\rightmark}}} \fancyfoot[L,R,C]{} \renewcommand{\headrulewidth}{1pt} \usepackage[margin=2.5cm]{geometry} \usepackage[pagebackref,colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref} \hypersetup{ colorlinks = true, urlcolor = blue, linkcolor = blue, citecolor = red , bookmarksopen=true } \theoremstyle{plain} \newtheorem{thrm}{Theorem}[section] \newtheorem{lemma}[thrm]{Lemma} \newtheorem{prop}[thrm]{Proposition} \newtheorem{cor}[thrm]{Corollary} \newtheorem{rmrk}[thrm]{Remark} \newtheorem{dfn}[thrm]{Definition} \newtheorem{prob}[thrm]{Problem} \newtheorem{hyp}{HYPOTHESIS} \setlength{\textheight}{8.7in} \allowdisplaybreaks \begin{document} \newcommand{\sn}{\mathbb{S}^{n-1}} \newcommand{\SL}{\mathcal L^{1,p}( D)} \newcommand{\Lp}{L^p( Dega)} \newcommand{\py}{ \partial_z^a} \newcommand{\La}{\mathscr{L}_a} \newcommand{\CO}{C^\infty_0( \Omega)} \newcommand{\Rn}{\mathbb R^n} \newcommand{\Rm}{\mathbb R^m} \newcommand{\R}{\mathbb R} \newcommand{\Om}{\Omega} \newcommand{\Hn}{\mathbb H^n} \newcommand{\aB}{\alpha B} \newcommand{\eps}{\ve} \newcommand{\BVX}{BV_X(\Omega)} \newcommand{\p}{\partial} \newcommand{\IO}{\int_\Omega} \newcommand{\bG}{\boldsymbol{G}} \newcommand{\bg}{\mathfrak g} \newcommand{\bz}{\mathfrak z} \newcommand{\bv}{\mathfrak v} \newcommand{\Bux}{\mbox{Box}} \newcommand{\e}{\ve} \newcommand{\X}{\mathcal X} \newcommand{\Y}{\mathcal Y} \newcommand{\W}{\mathcal W} \newcommand{\la}{\lambda} \newcommand{\vf}{\varphi} \newcommand{\rhh}{\|\nabla_H \rho\|} \newcommand{\Ba}{\mathcal{B}_\beta} \newcommand{\Za}{Z_\beta} \newcommand{\ra}{\rho_\beta} \newcommand{\n}{\nabla} \newcommand{\vt}{\vartheta} \newcommand{\its}{\int_{\{y=0\}}} \numberwithin{equation}{section} \newcommand{\RN} {\mathbb{R}^N} \newcommand{\Sob}{S^{1,p}(\Omega)} \newcommand{\Dxk}{\frac{\partial}{\partial x_k}} \newcommand{\Co}{C^\infty_0(\Omega)} \newcommand{\Je}{J_\ve} \newcommand{\beq}{\begin{equation}} \newcommand{\bea}[1]{\begin{array}{#1} } \newcommand{\eeq}{ \end{equation}} \newcommand{\ea}{ \end{array}} \newcommand{\eh}{\ve h} \newcommand{\Dxi}{\frac{\partial}{\partial x_{i}}} \newcommand{\Dyi}{\frac{\partial}{\partial y_{i}}} \newcommand{\Dt}{\frac{\partial}{\partial t}} \newcommand{\aBa}{(\alpha+1)B} \newcommand{\GF}{\psi^{1+\frac{1}{2\alpha}}} \newcommand{\GS}{\psi^{\frac12}} \newcommand{\HFF}{\frac{\psi}{\rho}} \newcommand{\HSS}{\frac{\psi}{\rho}} \newcommand{\HFS}{\rho\psi^{\frac12-\frac{1}{2\alpha}}} \newcommand{\HSF}{\frac{\psi^{\frac32+\frac{1}{2\alpha}}}{\rho}} \newcommand{\AF}{\rho} \newcommand{\AR}{\rho{\psi}^{\frac{1}{2}+\frac{1}{2\alpha}}} \newcommand{\PF}{\alpha\frac{\psi}{\|x\|}} \newcommand{\PS}{\alpha\frac{\psi}{\rho}} \newcommand{\ds}{\displaystyle} \newcommand{\Zt}{{\mathcal Z}^{t}} \newcommand{\XPSI}{2\alpha\psi \begin{pmatrix} \frac{x}{\left< x \right>^2}\\ 0 \end{pmatrix} - 2\alpha\frac{{\psi}^2}{\rho^2}\begin{pmatrix} x \\ (\alpha +1)\|x\|^{-\alpha}y \end{pmatrix}} \newcommand{\Z}{ \begin{pmatrix} x \\ (\alpha + 1)\|x\|^{-\alpha}y \end{pmatrix} } \newcommand{\ZZ}{ \begin{pmatrix} xx^{t} & (\alpha + 1)\|x\|^{-\alpha}x y^{t}\\ (\alpha + 1)\|x\|^{-\alpha}x^{t} y & (\alpha + 1)^2 \|x\|^{-2\alpha}yy^{t}\end{pmatrix}} \newcommand{\norm}[1]{\lVert#1 \rVert} \newcommand{\ve}{\varepsilon} \newcommand{\D}{\operatorname{div}} \newcommand{\G}{\mathscr{G}} \newcommand{\sa}{\langle} \newcommand{\da}{\rangle} \title[ucp in measure etc]{ On unique continuation in measure for fractional heat equations} \author{Agnid Banerjee} \address{School of Mathematical and Statistical Sciences\\ Arizona State University}\email[Agnid Banerjee]{[email protected]} \author{Nicola Garofalo} \address{School of Mathematical and Statistical Sciences\\ Arizona State University}\email[Nicola Garofalo]{[email protected]} \keywords{} \subjclass{35A02, 35B60, 35K05} \maketitle \begin{abstract} We prove a theorem of unique continuation in measure for nonlocal equations of the type $(\partial_t - \Delta)^s u= V(x,t) u$, for $0<s <1$. Our main result, Theorem \ref{main}, establishes a delicate nonlocal counterpart of the unique continuation in measure for the local case $s=1$. \end{abstract} \section{Introduction and statement of main result} The problem of unique continuation occupies a central position in the analysis of partial differential equations. A fundamental question in the subject is whether the trivial solution is the only one that can vanish to infinite order at one point. When this is the case, one says that the strong unique continuation property holds. In the study of observability inequalities and/or null-controllability of parabolic evolutions over measurable sets, a different type of unique continuation becomes relevant: can a nontrivial solution vanish on a subset of positive measure? If this cannot happen, one says that the relevant differential operator has the unique continuation property in measure, see for instance \cite{EMZ} and the references therein. The primary objective of this paper is to establish a theorem of unique continuation in measure for solutions to the nonlocal parabolic equation \begin{equation}\label{e0} H^s u(x,t) + V(x,t) u(x,t) = 0,\ \ \ \ \ \ \ \ 0<s<1, \end{equation} in the space-time cylinder $B_1 \times (-1, 0]\subset \Rn_x\times \R_t$. In \eqref{e0} we have denoted by $H^s = (\p_t - \Delta_x)^s$ the fractional power of the heat operator $H = \p_t - \Delta_x$ in $\R^{n+1} = \Rn_x \times \R_t$, and on the potential $V$ suitable assumptions will be specified in \eqref{vassump} below. Throughout this paper, for a function $f:\R^{n+1}\to \mathbb C$, we indicate with \[ \hat f(\xi,\sigma) = \int_{\R^{n+1}} e^{-2\pi i (\sa\xi,x\da + \sigma t)} f(x,t) dx dt \] its Fourier transform. Then the action of $H^s$ on a function $f\in \mathscr S(\R^{n+1})$ is defined by the formula \begin{equation}\label{sHft} \widehat{H^s f}(\xi,\sigma) = (4\pi^2 \|\xi\|^2 + 2\pi i \sigma)^s\ \hat f(\xi,\sigma), \end{equation} with the understanding that we have chosen the principal branch of the complex function $z\to z^s$. As it is well-known, the nonlocal operator $H^s$ is alternatively given by the formula \[ H^{s} f(x,t) = - \frac{s}{\Gamma(1-s)} \int_0^\infty \frac{1}{\tau^{1+s}} \big(P^H_\tau f(x,t) - f(x,t)\big) d\tau, \] where we have let \[ P^H_\tau f(x,t) = (4\pi \tau)^{-\frac n2} \int_{\Rn} e^{-\frac{\|x-y\|^2}{4\tau}} f(y,t-\tau) dy. \] The domain of the operator $H^s$ will be denoted by \[ \operatorname{Dom}(H^s) = \{u\in L^2(\Rn \times \R)\mid H^s u \in L^2(\Rn \times \R)\}. \] We say that a function $u\in \operatorname{Dom}(H^s)$ solves \eqref{e0} in an open set $\Om\subset \R^{n+1}$ if the equation is satisfied for a.e. point $(x,t)\in \Om$. For a measurable set $E \subset \Rn$, we will indicate by $\|E\|$ its $n$-dimensional Lebesgue measure. The ball in $\Rn$ centered at the origin and having radius $r>0$ will be denoted by $B_r$. Our main result is the following. \begin{thrm}\label{main} Let $u \in \operatorname{Dom}(H^s)$ solve \eqref{e0} in the cylinder $B_1 \times (-1, 0]$. Assume that $u$ vanishes in $F \times (-1, 0]$, for some measurable set $F \subset B_1$ with $\|F\|>0$. Then $u \equiv 0$ in $\Rn \times (-1,0]$. \end{thrm} The proof of Theorem \ref{main} will be presented in Section \ref{s:m}. We mention that the main new difficulty with this result is the treatment of the nonlocal operator $H^s$. In the local case, in fact, by an application of the Poincar\'e and energy inequalities, it is easy to show that at any Lebesgue point of its zero set, a solution of the relevant differential operator must vanish to infinite order. This reduces the proof to having the strong unique continuation property for the local operator in question, see for instance \cite{DG, EMZ, Ru2, BZ}. In the framework of the present paper, a corresponding space-like strong unique continuation result has been recently established in \cite{ABDG}, see Theorem \ref{main1} below. However, since the energy inequality for fractional equations has a nonlocal character, a straightforward adaptation of the above mentioned local arguments is all but obvious. This aspect was already mentioned in the introduction of \cite{FF}, where the authors derived the time-independent counterpart of Theorem \ref{main} by analysing the local asymptotic of solutions to the corresponding Caffarelli-Silvestre extension problem for $(-\Delta)^s$. In this note, we present a different approach which, in fact, does reduce the property of unique continuation in measure to that of strong unique continuation. The novel aspects of our proof consist in exploiting in a subtle way some fundamental properties of the solution of the extension problem associated with \eqref{e0}. One of them is Theorem \ref{extmain}, which we use to establish the crucial compactness Lemma \ref{L:pos}. Another important ingredient is the conditional doubling property in Theorem \ref{doub}, which we use in multiple ways in the proof of Theorem \ref{main}. Combining these tools with the trace interpolation inequality in Lemma \ref{L:inter1} (and the Poincar\'e inequality in Lemma \ref{po}), we are able to prove that, if $u$ in Theorem \ref{main} does not vanish identically in the cylinder $B_1 \times (-1, 0]$, then at any Lebesgue point $x_0$ of the set $F$, the following inequality holds for any $\ve>0$ and for all $0<r<r_\ve$ \begin{equation} \int_{Q_r(x_0, t_0)} u^2 dx dt \leq C \ve^{2/n} \int_{Q_{2r}(x_0, t_0)} u^2 dx dt, \end{equation} see \eqref{e5} below. This critical information proves that $u$ vanishes to infinite order at $(x_0,t_0)$. We can thus invoke the nonlocal space-like strong unique continuation property in Theorem \ref{main1} to finally infer that it must be $u(\cdot, t_0) \equiv 0$ in $\Rn$. This contradicts the initial assumption that $u\not\equiv 0$ in $B_1 \times (-1, 0]$, thus allowing us to spread the zero set of $u$. The organisation of the paper is as follows. In section \ref{s:n} we introduce some notation and gather the above mentioned results that are used in the proof of Theorem \ref{main} in section \ref{s:m}. In closing, we mention that for the existing literature on strong unique continuation for $(-\Delta)^s$ and its parabolic counterpart $(\partial_t - \Delta)^s$, the reader should see \cite{FF, Ru1, BG, FPS, BGh, AT}. \section{Notations and Preliminaries}\label{s:n} In this section we introduce the relevant notation and gather some auxiliary results that will be useful in the rest of the paper. Generic points in $\Rn \times \R$ will be denoted by $(x_0, t_0), (x,t)$, etc. For an open set $\Omega\subset \Rn_x\times \R_t$, we indicate with $C_0^{\infty}(\Omega)$ the set of compactly supported, smooth functions in $\Om$. We also denote by $H^{\alpha}(\Omega)$ the non-isotropic parabolic H\"older space, see \cite[p. 46]{Li}. Given the nonlocal operator \eqref{sHft}, the parabolic Sobolev space of fractional order $2s$ is \begin{align}\label{dom} \mathscr H^{2s} & = \operatorname{Dom}(H^s) = \{f\in \mathscr S'(\R^{n+1})\mid f, H^s f \in L^2(\R^{n+1})\} \\ & = \{f\in L^2(\R^{n+1})\mid (\xi,\sigma) \to (4\pi^2 \|\xi\|^2 + 2\pi i \sigma)^s \hat f(\xi,\sigma)\in L^2(\R^{n+1})\}. \notag \end{align} Hereafter in this paper, we assume that $u\in \mathscr H^{2s}$ solves the equation \eqref{e0}, where on the potential $V$ we make the hypothesis that for some $K>0$ one has \begin{equation}\label{vassump} \|\|V\|\|_{C^1(B_1 \times (-1, 0])} \leq K, \ \text{if}\ s \in [1/2, 1),\ \ \ \ \ \|\|V\|\|_{C^2(B_1 \times (-1, 0])} \leq K,\ \text{for}\ s \in (0,1/2). \end{equation} Under such assumptions on $u$ and $V$, various basic results hold. In order to state them we next recall the counterpart for the parabolic nonlocal operator $H^s$ of the Caffarelli-Silvestre extension problem in \cite{CS}. When $s=1/2$ such problem was originally introduced and solved by F. Jones in \cite{Jr}, and later independently and more extensively developed for any $s\in (0,1)$ by N\"ystrom and Sande \cite{NS} and Stinga and Torrea in \cite{ST}. Since we need to consider the half-space $\R^{n+1}_{(x,t)} \times \R^+_z$, it will be convenient to combine the ``extension" variable $z>0$ with $x\in \Rn$, and indicate the generic point in the thick space $\Rn_x\times\R_z$ with the letter $X=(x,z)$. Whenever convenient, we will indicate with the short notation $U(X,t)$ the value at the point $((x,t),z)$ of a function $U:\R^{n+1}_{(x,t)} \times \R^+_z\to \R$. This should not cause any confusion in the reader's mind. For $x_0\in \Rn$ and $r>0$ we let $B_r(x_0) = \{x\in \Rn\mid \|x-x_0\|<r\}$, and denote the upper half-ball by $\mathbb B_r^+(x_0,0)=\{X = (x,z) \in \R^n \times \R^{+}\mid \|x-x_0\|^2 + z^2 < r^2\}$. The parabolic cylinder in the thin space $Q_r(x_0, t_0) = B_r(x_0) \times [t_0, t_0 + r^2)$. We also will need the upper half-cylinder in thick space $\mathbb Q_r^+((x_0,t_0),0)=\mathbb B_r^+(x_0,0) \times (t_0,t_0+r^2]$. When the center $x_0$ of $B_r(x_0)$ is not explicitly indicated, then we are taking $x_0 = 0$. Similar agreement for the thick half-balls $\mathbb B_r^+(x_0,0)$. \begin{rmrk}\label{R:balls} Since in this paper we make extensive use of the work \cite{ABDG}, it is important that we alert the reader that what we are presently indicating with $\mathbb B_r^+(x_0,0)$ and $\mathbb Q_r^+((x_0,t_0),0)$ were respectively denoted by $\mathbb B_r(x_0,0)$ and $\mathbb Q_r((x_0,t_0),0)$ on p. 6 in \cite{ABDG}. The reader should keep this in mind when comparing the definition of the quantity $\theta$ in \eqref{theta} with the one in (3.3) in that paper. \end{rmrk} For notational ease, $\nabla U$ and $\operatorname{div} U$ will respectively refer to the operators $\nabla_X U$ and $ \operatorname{div}_X U$. The partial derivative in $t$ will be denoted by $\p_t U$, and also at times by $U_t$. The partial derivative $\partial_{x_i} U$ will be denoted by $U_i$. At times, the partial derivative $\partial_{z} U$ will be denoted by $U_{n+1}$. Given a number $a\in (-1,1)$ and a $u:\R^n_x\times \R_t\to \R$, we seek a function $U:\R^n_x\times\R_t\times \R_z^+\to \R$ that satisfies the Dirichlet problem \begin{equation}\label{la} \begin{cases} \La U \overset{def}{=} \partial_t (z^a U) - \operatorname{div}(z^a \nabla U) = 0, \\ U((x,t),0) = u(x,t),\ \ \ \ \ \ \ \ \ \ \ (x,t)\in \R^{n+1}. \end{cases} \end{equation} Denote by $\py$ the weighted normal derivative \begin{equation}\label{nder} \py U((x,t),0)\overset{def}{=} \underset{z \to 0^+}{\lim} z^a \partial_z U((x,t),z). \end{equation} Then, the most basic property of the Dirichlet problem \eqref{la} is that if $s = \frac{1-a}2\in (0,1)$, then one has in $L^2(\R^{n+1})$ \begin{equation}\label{np} 2^{-a}\frac{\Gamma(\frac{1-a}{2})}{\Gamma(\frac{1+a}{2})} \py U((x,t),0)= - H^s u(x,t), \end{equation} where $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt$ is Euler gamma function. In \cite[Corollary 4.6]{BG} we proved that if $u\in \mathscr H^{2s}$, then the function $U$ in \eqref{la} is a weak solution of \begin{equation}\label{wk} \begin{cases} \La U=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \R^{n+1}\times \R^+_z, \\ U((x,t),0)= u(x,t)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for}\ (x,t)\in \R^{n+1}, \\ \py U((x,t),0)= 2^{a} \frac{\Gamma(\frac{1+a}{2})}{\Gamma(\frac{1-a}{2})} V(x,t) u(x,t)\ \ \ \ \text{for}\ (x,t)\in B_1 \times (-1,0]. \end{cases} \end{equation} Note that the third equation in \eqref{wk} is justified by \eqref{e0} and \eqref{np}. For notational purposes, it will be convenient henceforth to work with the following backward version of problem \eqref{wk} \begin{equation}\label{exprob} \begin{cases} z^a \partial_t U + \operatorname{div}(z^a \nabla U)=0\ \ \ \ \ \ \ \ \ \ \ \ \text{in} \ \R^{n+1} \times \R^+_z, \\ U((x,t),0)= u(x,t) \\ \py U((x,t),0)= V(x,t) u(x,t)\ \ \ \ \ \ \ \ \ \text{in}\ B_4 \times [0,16). \end{cases} \end{equation} We note that the former can be transformed into the latter by changing $t \to -t$ and a parabolic rescaling $U_{r_0}(X,t)= U(r_0X, r_0^2t)$ for small enough $r_0$. We also emphasise that, to simplify the notation in \eqref{exprob}, we have incorporated in the potential $V$ the normalising constant $2^{a} \frac{\Gamma(\frac{1+a}{2})}{\Gamma(\frac{1-a}{2})}$ in \eqref{wk}. In the next section we will need the following regularity result established in \cite[Section 5]{BG}. \begin{lemma}\label{reg1} Let $U$ be a weak solution of \eqref{exprob}. Then there exists $\alpha>0$ such that one has up to the thin set $\{z=0\}$ \[ U_i,\ U_t,\ z^a U_z\ \in\ H^{\alpha}(\mathbb B_\frac{1}{2}^+ \times [0, 1/4)),\ \ \ \ i=1,2,..,n. \] Moreover, the relevant H\"older norms are bounded by $\int_{\mathbb B_1^+ \times (0, 1)} U^2 z^a dX dt$. Furthermore, we also have \begin{equation}\label{est} \int_{\mathbb B_{1/2}^+ \times [0, 1/4)} ( \|\nabla_x U\|^2 + \|\nabla \nabla_x U\|^2 + U_t^2 ) z^a dX dt \leq C \int_{\mathbb B_1^+ \times (0, 1)} U^2 z^a dX dt. \end{equation} \end{lemma} In the proof of Theorem \ref{main} we will need the following basic conditional doubling property established in \cite[Theorem 3.5]{ABDG}. With $U$ as in \eqref{exprob}, define \begin{equation}\label{theta} \theta \overset{def}{=}\frac{\int_{\mathbb Q_4^+} U(X,t)^2 z^adXdt }{\int_{\mathbb B_1^+} U(X,0)^2 z^adX}. \end{equation} Notice that in \eqref{theta} the notation $U(X,t)$ means $U((x,t),z)$. Similarly, $U(X,0)$ indicates $U((x,0),z)$. Concerning the definition of $\theta$, the reader should also keep Remark \ref{R:balls} in mind. \begin{thrm}\label{doub} Let $U$ be a solution of \eqref{exprob}. There exists $N>2$, depending on $n$, $a$ and the $C^1$-norm of $V$, such that $N\log(N\theta) \ge 1$, and for which: \begin{itemize} \item[(i)] For $r \leq 1/2,$ we have $$\int_{\mathbb B_{2r}^+}U^2(X,0)z^adX \leq (N \theta)^N\int_{\mathbb B_{r}^+}U^2(X,0)z^adX.$$ \end{itemize} Moreover, for $r \leq 1/\sqrt{N \operatorname{log}(N \theta)}$ the following inequality holds: \begin{itemize} \item[(ii)]$$\int_{\mathbb Q_{2r}^+} U^2 z^adXdt \leq \operatorname{exp}(N \operatorname{log}(N \theta) \operatorname{log}(N \operatorname{log}(N \theta)))\int_{\mathbb Q_r^+}U^2z^adXdt.$$ \end{itemize} \end{thrm} We say that a function $u(x,t)$ vanishes to infinite order at $(0,0)$ if for all $k>0$ one has for $r \to 0$ \begin{equation}\label{vp} \int_{B_r \times (-r^2, 0]} u(x,t)^2 dx dt = O(r^k). \end{equation} \begin{rmrk}\label{R:norms} It is worth mentioning here that in the following strong unique continuation results from \cite{ABDG}, Theorems \ref{extmain} and \ref{main1}, which will be needed in the proof of Theorem \ref{main}, the notion of vanishing to infinite order used the $L^\infty$ norm, instead of the $L^2$ norm as in \eqref{vp}. Since the proofs hold unchanged with $L^2$ norms, and since such norms will play an important role in the blowup analysis, we will use \eqref{vp}. \end{rmrk} Theorem \ref{doub}, combined with an appropriate blow-up analysis, was used in \cite{ABDG} to derive the following strong unique continuation property which represents the central result of that work. \begin{thrm}\label{extmain} Let $U$ be a solution of \eqref{exprob} in $\mathbb Q_{2}^+$ with $V$ satisfying the assumption in \eqref{vassump}. If the function $u:Q_2 \to \mathbb R$ defined by $u(x,t) \overset{def}= U((x,t), 0)$ vanishes to infinite order at $(0,0)$, then $U((x,0),z) \equiv 0$ in $\mathbb B_{2}^+$. \end{thrm} \begin{proof} Theorem \ref{extmain} was not explicitly stated in \cite{ABDG}, but its proof is embedded in that of \cite[Theor. 1.1, p. 35-38]{ABDG}. We thus refer the reader to that source. \end{proof} The next result is the just quoted Theorem 1.1 from \cite{ABDG}. Since this result will be used in the proof of Theorem \ref{main}, we need to make a comment here. \begin{thrm}\label{main1} Let $u \in \operatorname{Dom}(H^s)$ solve \eqref{e0} in $B_1 \times (-1, 0]$, and assume that $u$ vanishes to infinite order at $(0,0)$. Then it must be $u(\cdot, 0) \equiv 0$ in $\Rn$. \end{thrm} In Section \ref{s:m} we will also need the following trace interpolation estimate in \cite[Lemma 2.4]{ArB1}, inspired to a related time-independent result in \cite{RS}. \begin{lemma}\label{interpolation} Let $s \in (0,1)$. There exists a constant $C(n,s)>0$ such that, for any $0<\eta <1$ and $f \in C^2_0(\R^n \times \R_+)$, the following holds \begin{align}\label{inte} \int_{\Rn} \|\nabla_x f\|^2 dx \le C \eta^{2s} \int_{\R^n \times \R_+} ( \|\nabla_x f\|^2 + \|\nabla \nabla_x f\|^2) z^a dX dt + C\eta^{-2} \|\|f\|\|^2_{L^2(\R^n)}. \end{align} \end{lemma} Using Lemmas \ref{reg1} and \ref{interpolation}, by arguing as in \cite[(5.14)-(5.16) on p.195]{ArB1}, we obtain the following estimate. \begin{lemma}\label{L:inter1} Let $U$ be as in \eqref{exprob}. For any $0< \eta <1$, there exists $C>0$, depending on $n, a, K$, such that the following holds \begin{equation} \int_{Q_1} \|\nabla_x u\|^2 dxdt \leq C\eta^{2s} \int_{\mathbb Q_4^+} U^2 z^a dXdt + C\eta^{-2} \int_{Q_2} u^2 dxdt. \end{equation} \end{lemma} Finally, we also need the following well-known form of Poincar\'e inequality, see e.g. \cite[Lemma 3.4, p. 54]{LU}. \begin{lemma}\label{po} Let $v \in W^{1,2}(B_1)$ and denote by $F= \{x\in B_1\mid v(x)=0\}$. If $E\subset B_1$ is any measurable set, one has for some $C=C(n)>0$, \begin{equation}\label{po1} \left(\int_{E} v^2\right)^{1/2} \leq C\ \frac{\|E\|^{1/n}}{\|F\|} \left(\int_{B_1} \|\nabla_x v\|^2\right)^{1/2}. \end{equation} \end{lemma} \vskip 0.3in \section{Proof of Theorem \ref{main}}\label{s:m} We begin by proving a quantitative result that allows to concentrate near the thin space the $L^2$ norm in the thick space of a solution to \eqref{exprob}. In the statement of the next lemma we denote by $\tilde V(x,t)$ a function satisfying the hypothesis \eqref{vassump}. \begin{lemma}\label{L:pos} Let $W$ be a solution to \begin{equation}\label{ex1} \begin{cases} z^a \partial_t W + \operatorname{div}(z^a \nabla W)=0\ \ \ \ \ \ \ \ \ \ \ \ \text{in} \ \mathbb Q_4^+ \\ \py W((x,t),0)= \tilde V(x,t) W((x,t),0)\ \ \ \ \ \ \ \ \ \text{in}\ B_4 \times [0,16), \end{cases} \end{equation} such that for some $C_1>1$ one has \begin{equation}\label{ret} \int_{\mathbb Q_{2}^+} W^2 z^a dXdt \leq C_1. \end{equation} Assume furthermore that \begin{equation}\label{norm} \int_{\mathbb Q_1^+} W^2 z^a dXdt =1. \end{equation} Then there exists $C_0>0$, depending on $n, a, K$ and $C_1$, such that \begin{equation}\label{pos1} \int_{Q_1} W((x,t),0)^2 dxdt \geq C_0. \end{equation} \end{lemma} \begin{proof} We argue by contradiction, and assume that for every $k\in \mathbb N$ there exists $\tilde V_k$ satisfying \eqref{vassump}, and a solution to \eqref{ex1}, $W_k$, which verifies \eqref{ret} and \eqref{norm}, and such that \begin{equation}\label{sm} \int_{Q_1} W_k((x,t),0)^2dxdt \leq \frac{1}{k}. \end{equation} By the theorem of Ascoli-Arzel\`a, we can extract a subsequence $\tilde V_k \to \tilde V_\infty$, uniformly in $\overline{\mathbb Q_1^+}$. Clearly, the limit function $\tilde V_\infty$ will satisfy \eqref{vassump}. Because of \eqref{ret} and the regularity estimate in Lemma \ref{reg1}, it follows that, up to a subsequence, $W_k \to W_\infty$. Furthermore, $W_\infty$ is a weak solution to \begin{equation}\label{Winfty} \begin{cases} z^a \partial_t W_{\infty} + \operatorname{div}(z^a \nabla W_{\infty})=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in} \ \mathbb Q_2^+, \\ \py W_\infty((x,t),0)= \tilde V_\infty (x,t) W_{\infty} ((x,t),0)\ \ \ \ \ \ \ \ \ \text{in}\ B_2 \times [0,4). \end{cases} \end{equation} Because of uniform convergence and \eqref{sm}, it follows that $w(x,t) = W_{\infty}((x,t),0)=0$ for all $(x,t)$ in $Q_1$. This implies, in particular, that $w(x,t)$ vanishes to infinite order at \emph{every point} $(x_1,t_1)\in Q_1$. Invoking Theorem \ref{extmain} we thus infer that $W_\infty((x,0),z) \equiv 0$ in $\mathbb B_{1}^+$. Consider now any $t_1\in [0,1)$ and denote $\tilde W_\infty((x,t),z) = W_\infty((x,t+t_1),z)$, and $\tilde w(x,t) = \tilde W_\infty((x,t),0)$. From what we have observed above, $\tilde w(x,t)$ vanishes to infinite order at $(0,0)$. Since the extension equation in \eqref{ex1} is translation-invariant in $t$, again by Theorem \ref{extmain} we conclude that $W_\infty((x,t_1),z) = \tilde W_\infty((x,0),z) \equiv 0$ in $\mathbb B_{1}^+$. By the arbitrariness of $t_1\in (0,1)$, we finally conclude that $W_\infty \equiv 0$ in $\mathbb Q_1^+$. On the other hand, again using uniform convergence and \eqref{norm}, we can also assert that the following holds \[ \int_{\mathbb Q_1^+} W_{\infty}^2 z^a dXdt =1, \] thus reaching a contradiction. This proves the lemma. \end{proof} We now turn to the \begin{proof}[Proof of Theorem \ref{main}] In keeping with the presentation in Section \ref{s:n}, we will work with the adjoint nonlocal equation $(\p_t + \Delta)^s u + V(x,t) u = 0$, and the corresponding extension problem \eqref{exprob}. Without loss of generality, we assume that $F \subset B_1$ and that the adjoint version of \eqref{e0} holds in $B_4 \times [0,16)$. We will show that $u \equiv 0$ in $B_1 \times [0,16)$. Using Theorem \ref{main1}, we can then spread the zero set and reach the desired conclusion. We argue by contradiction and suppose that there exists $t_0 \in [0,16)$ such that \begin{equation}\label{assume} u(\cdot,t_0) \not \equiv 0,\ \ \ \ \text{in}\ B_1. \end{equation} Since for the corresponding extension function $U$ in \eqref{exprob} we have $U((x,t_0),0) = u(x,t_0)$, by continuity we must also have $U((x,t_0),z)) \not \equiv 0$ for $(x,z)\in \mathbb B^+_1$, and therefore in particular \[ \int_{\mathbb B^+_1} U((x,t_0),z))^2 z^a dX > 0. \] We claim that this implies that for every $x_0\in B_1$ one has \begin{equation}\label{pos} \int_{\mathbb B_1^+(x_0, 0)} U((x, t_0),z)^2 z^a dX>0. \end{equation} To see \eqref{pos}, consider the function $\tilde U((x,t),z) = U((x,t+t_0),z)$, which also solves \eqref{exprob}. For such function we have \[ \int_{\mathbb B_1^+} \tilde U(X,0)^2 z^adX = \int_{\mathbb B^+_1} U((x,t_0),z))^2 z^a dX > 0, \] and therefore the corresponding $\theta$ in \eqref{theta} is well-defined. We can thus apply the doubling condition (i) in Theorem \ref{doub}, obtaining for any $r>0$ (sufficiently small) \begin{equation}\label{i} \int_{\mathbb B_r^+} \tilde U(X,0)^2 z^a dX > 0. \end{equation} Given now any $x_0\in B_1$, the triangle inequality gives $\mathbb B_{1-\|x_0\|}^+ \subset \mathbb B^+_1(x_0,0)$. Applying \eqref{i} with $r = 1-\|x_0\|$, we conclude \[ \int_{\mathbb B_1^+(x_0, 0)} U((x, t_0),z)^2 z^a dX \ge \int_{\mathbb B_r^+} U((x, t_0),z)^2 z^a dX = \int_{\mathbb B_r^+} \tilde U(X,0)^2 z^a dX >0, \] which proves \eqref{pos}. Consider now the function $\bar{U}((x,t),z)= U((x+x_0,t+t_0),z)$, which is also a solution of \eqref{exprob}. Since by \eqref{pos} we have \[ \int_{\mathbb B_1^+} \bar{U}(X,0)^2 z^adX = \int_{\mathbb B_1^+(x_0, 0)} U((x, t_0),z)^2 z^a dX > 0, \] the corresponding \eqref{theta} for $\bar U$ is also well-defined, and from the doubling inequality (ii) in Theorem \ref{doub} we can assert that for some $C_1>1, r_1>0$, the following holds for all $r \leq r_1$ \begin{equation}\label{doub1} \int_{\mathbb Q_{2r}^+((x_0, 0, t_0))} U^2 z^a dX dt \leq C_1 \int_{\mathbb Q_r^+ ((x_0, 0, t_0))} U^2 z^a dX dt. \end{equation} Let now $x_0 \in B_1$ be a Lebesgue point of $F$, i.e. \begin{equation} \lim_{r \to 0} \frac{\|F \cap B_r(x_0)\|}{\|B_r(x_0)\|}=1. \end{equation*} This implies that given any $\ve>0$, there exists $r_\ve>0$ (which without loss of generality we can assume $< \frac{r_1}{100}$) such that for all $r< r_\ve$, one has $\|F \cap B_r(x_0)\|> (1-\ve)\|B_r(x_0)\|$. This implies for $r<r_\ve$ \begin{equation}\label{l1} \|B_r(x_0)\setminus F\| < \ve \|B_r(x_0)\|. \end{equation} From \eqref{l1} and the rescaled version of the Poincar\'e inequality in Lemma \ref{po} it follows for all $r < r_\ve$ \begin{equation}\label{e1} \int_{Q_r(x_0, t_0)} u^2 dx dt\leq Cr^2 \ve^{2/n} \int_{Q_r(x_0, t_0)}\|\nabla_x u\|^2 dx dt. \end{equation} Since a change of scale with $r<1$ for the potential $V(x,t)$ decreases the bound $K$ on its $C^{k}$-norm ($k=1$ or $2$) in \eqref{vassump}, from the rescaled version of the interpolation estimate in Lemma \ref{L:inter1} it follows \begin{equation}\label{res1} \int_{Q_r(x_0, t_0)}\|\nabla_x u\|^2 dx dt \leq \frac{C\eta^{2s}}{r^{3+a}}\int_{\mathbb Q_{4r}^+((x_0, 0, t_0))} U^2 z^a dX dt + \frac{C \eta^{-2}}{r^2} \int_{Q_{2r}(x_0, t_0)} u^2 dx dt. \end{equation} Using \eqref{res1} in \eqref{e1}, we obtain for a new $C>0$ \begin{align}\label{e2} \int_{Q_r(x_0, t_0)} u^2 dx dt & \leq C \eta^{-2} \ve^{2/n} \int_{Q_{2r}(x_0, t_0)} u^2 dx dt+ \frac{C\eta^{2s}\ve^{2/n}}{r^{1+a}} \int_{\mathbb Q_{4r}^+((x_0, 0, t_0))} U^2 z^a dX dt\\ \notag & \leq C \eta^{-2} \ve^{2/n} \int_{Q_{2r}(x_0, t_0)} u^2 dxdt+ \frac{C \eta^{2s} \ve^{2/n}}{r^{1+a}} \int_{\mathbb Q_{r}^+((x_0, 0, t_0))} U^2 z^a dX dt, \notag \end{align} where in the second inequality we have used \eqref{doub1} (which we can, since $r < r_{\ve}< \frac{ r_1}{100}$). Consider now the function \[ W((x,t),z)= \frac{U((x_0 + rx, t_0 + r^2 t),rz)}{\bigg(\frac{1}{r^{n+3+a}}\int_{\mathbb Q_{r}^+((x_0, 0, t_0))} U^2 z^a dX dt\bigg)^{1/2}}. \] Then $W$ solves \eqref{ex1} with a potential given by \[ \tilde V(x,t) = r^{2s} V(x_0 + rx, t_0 + r^2 t). \] Furthermore, by \eqref{doub1} and a change of variable it is seen that $W$ satisfies the bounds \begin{equation}\label{norm1} \begin{cases} \int_{\mathbb Q_1^+} W^2 z^a dX dt =1, \\ \int_{\mathbb Q_2^+} W^2 z^a dX dt \leq C_1. \end{cases} \end{equation} Applying Lemma \ref{L:pos}, we reach the conclusion that $W$ satisfies the inequality \eqref{pos1}. By the change of variable $y= x_0+rx$, $\tau = t_0 + r^2 t$, we infer that $U$ verifies the estimate \begin{equation}\label{pos12} \int_{\mathbb Q_r^+((x_0, 0, t_0))} U^2 z^a dX dt \leq C_0^{-1} r^{1+a} \int_{Q_r(x_0, t_0)} u^2 dx dt. \end{equation} We now insert \eqref{pos12} in \eqref{e2}, obtaining for yet another $C>0$ the following basic inequality \begin{equation}\label{e4} \int_{Q_r(x_0, t_0)} u^2 dx dt \leq C \eta^{-2} \ve^{2/n} \int_{Q_{2r}(x_0, t_0)} u^2 dx dt + C \eta^{2s} \ve^{2/n} \int_{Q_r(x_0, t_0)} u^2 dx dt. \end{equation} Since $\ve<1$, by taking $\eta$ sufficiently small (it suffices to choose $C \eta^{2s}\le 1/2$), we can absorb the second term in the right-hand side of \eqref{e4} in the left-hand side, and finally arrive at the following bound \begin{equation}\label{e5} \int_{Q_r(x_0, t_0)} u^2 dx dt \leq C \ve^{2/n} \int_{Q_{2r}(x_0, t_0)} u^2 dx dt, \end{equation} valid for some $C>1$. Finally, we want to show that \eqref{e5} implies that $u$ vanishes to infinite order at $(x_0,t_0)$. We first observe that, if $f:[0,1]\to [0,\infty)$ is an increasing function such that for every $\delta\in (0,1)$, there exists $r_\delta\in (0,1)$ such that $f(r)\le \delta f(2r)$ for $0<r<r_\delta$, then we have for every $0<r<r_\delta$ \begin{equation}\label{f} f(r) \le \delta \left(\frac r{r_\delta}\right)^{\frac{\log \delta}{\log 2}} f(1). \end{equation} To prove \eqref{f}, we fix $m\in \mathbb N\cup\{0\}$ such that $2^{m} < \frac{r_\delta}{r} \le 2^{m+1}$. We thus find \[ f(1) \ge f(r_\delta) \ge f(2^{m} r) \ge \delta^{-1} f(2^{m} r) \ge\ ...\ge \delta^{-m} f(r). \] This gives for every $0<r<r_\delta$ \begin{equation}\label{fast} f(r) \le f(1) \left(\frac{r}{r_\delta}\right)^{\log\left(\frac{1}{\delta}\right)^{\frac{1}{\log 2}}}. \end{equation} If now $k\in \mathbb N$ is arbitrarily chosen, pick $\delta(k)\in (0,1)$ such that \[ \log\left(\frac{1}{\delta}\right)^{\frac{1}{\log 2}} \cong k. \] Corresponding to such choice, in view of \eqref{fast} there exists $r_k = r_{\delta(k)}\in (0,1)$ such that for every $0<r<r_k$ \[ f(r) \cong f(1) \left(\frac{r}{r_\delta}\right)^{k}. \] This shows that $f(r) = O(r^k)$ as $r\to 0^+$. By the arbitrariness of $k$ we infer that $f$ vanishes to infinite order at $r= 0$. Applying these observations to \[ f(r) = \int_{Q_r(x_0, t_0)} u^2 dx dt, \] by taking $\delta = C\ve^{2/n}$ in \eqref{e5}, we reach the conclusion that $u$ vanishes to infinite order at $(x_0,t_0)$. 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2412.03834v2	http://arxiv.org/abs/2412.03834v2	Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function	\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{geometry} \usepackage{amsmath} \usepackage{amsfonts} \usepackage[mathscr]{eucal} \usepackage{amsthm} \usepackage{bookmark} \usepackage{enumerate} \usepackage{extarrows} \usepackage{changes}\usepackage{color} \usepackage{bbm} \usepackage{verbatim} \usepackage{graphicx} \usepackage{courier} \usepackage{helvet} \usepackage{courier} \usepackage{type1cm} \usepackage{multicol} \usepackage[bottom]{footmisc} \geometry{left=3cm,right=3cm,top=3cm,bottom=3cm} \date{} \theoremstyle{plain} \newtheorem{thm}{Main Theorem} \newtheorem{lem}{Main Lemma} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{question}[theorem]{Question} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}[theorem]{Example} \newtheorem{claim}{Claim} \numberwithin{equation}{section} \theoremstyle{plain} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \author{Shilei Fan} \address{Shilei FAN: School of Mathematics and Statistics, and Key Lab NAA--MOE, Central China Normal University, Wuhan 430079, China} \email{[email protected]} \title{ Tiling the field $\Q_p$ of $p$-adic numbers by a function } \date{\today} \def\Card{{\rm Card}} \def\R{{\textbf{R}}} \def\H{\mathcal{H}} \def\fh{\f{1}{2}} \def\p{\partial} \def\dsum{\displaystyle\sum} \def\f{\frac} \def\R{\mathbb R} \def\N{\mathbb N} \def\1{\mathbbm 1} \def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\C{\mathbb C} \def\df{\displaystyle\frac} \def\e{\mathrm{e}} \def\d{\mathrm{d}} \def\i{\mathrm{i}} \def\v{\mathbf{v}} \def\x{\mathbf{x}} \def\w{\mathbf{w}} \def\g{\mathbf{g}} \def\df{\displaystyle\frac} \def\vsp{\vspace{1mm}} \def\ee{\mathbf{e}} \def\u{\mathbf{u}} \def\pt{\df{\partial}{\partial t}} \def\ff{\mathbf{f}} \def\x{\mathbf{x}} \def\sp{\mathrm{ess\,sup}} \def\al{\alpha} \def\be{\beta} \def\la{\lambda} \def\Om{\Omega} \def\va{\varepsilon} \def\ds{\displaystyle} \def\l{\left} \def\r{\right} \def\v{\varphi} \def\n{\nabla} \def\m{\mathfrak{m}} \thanks{ S. L. FAN was partially supported by NSFC (grants No. 12331004 and No. 12231013). } \begin{document} \maketitle \begin{abstract} This study explores the properties of the function which can tile the field $\Q_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\Q_p$ is by translation uniformly locally constancy. As an application, in the field $\Q_p$, we addressed the question posed by H. Leptin and D. M\"uller, providing the necessary and sufficient conditions for a discrete set to correspond to a uniform partition of unity. The study also connects these tiling properties to the Fuglede conjecture, which states that a measurable set is a tile if and only if it is spectral. The paper concludes by characterizing the structure of tiles in \(\mathbb{Q}_p \times \mathbb{Z}/2\mathbb{Z}\), proving that they are spectral sets. \emph{Keywords:} periodic, translation tile, spectral set, $p$-adic field. \end{abstract} \section{Introduction} Consider a locally compact abelian group $G$ equipped with a Haar measure $\mu$. Let $f$ be a function in $L^1(G)$. The function $f$ is said to {\em tile} $G$ at level $w$ by a translation set $T$ if \[ \sum_{t \in T} f(x - t) = w \quad \text{a.e.} \ x\in G. \] Here, the convergence is absolute, the translation set $T$ is required to be locally finite, i.e. for any compact set $K$, the set $T\cap K$ is finite and we allow elements of $T$ to occur with finite multiplicities. We say that $(f, T)$ is a tiling pair at level $w$. If $f = 1_\Omega$ is an indicator function of the measurable set $\Omega$ and $w = 1$, we say that $\Omega$ {\em tiles} the group $G$ by the translation set $T$, and $(\Omega, T)$ is a {\em tiling pair} (at level $1$). The set $\Omega$ is called a {\em tile } of $G$ In the case where \( G = \mathbb{R} \), it was proven in \cite{LW96Invent} that if a bounded measurable set \(\Omega\) with measure-zero boundary tiles the line \(\mathbb{R}\) by a translation set \(T\), then \( T \) is a periodic set, that is, \( T + \tau = T \) for some \( \tau > 0 \). This result was extended in \cite{KL96Duke} (and proved earlier in \cite{LM91}) to tiling \(\mathbb{R}\) by a function \( f \in L^1(\mathbb{R}) \) with compact support. It was shown that if \( (f, T) \) is a tiling pair at some level \( w \) with \( T \) of bounded density and \( f \) is not identically zero, then \( T \) must be a finite union of periodic sets. For more related results and questions, interested readers can refer to \cite{KL21}. For the case \( G = \mathbb{Q}_p \), the field of \( p \)-adic numbers, it was proven in \cite{FFLS} that if \((\Omega, T)\) forms a tiling pair, then \( \Omega \) is a compact open set up to a set of measure zero, which implies \[ 1_\Omega(x) = 1_{\Omega}(x + \tau), \quad \text{a.e.} \ x \in \mathbb{Q}_p \] for some \( \tau \in \mathbb{Q}_p \setminus \{0\} \). This means that \( 1_\Omega(x) \) is a periodic function or the measurable set \( \Omega \) is periodic. Let \( \mathbb{Z}_p \) be the ring of \( p \)-adic integers. A compact open set \( \Omega \) can be expressed as \[ \Omega = \bigsqcup_{c \in C} (c + p^\gamma \mathbb{Z}_p) \] for some finite set \( C \subset \mathbb{Q}_p \) and some integer \( \gamma \in \mathbb{Z} \). Note that \( p^\gamma \mathbb{Z}_p \) is a subgroup of \( \mathbb{Q}_p \) under addition. Hence, a compact open set is a finite union of cosets of some subgroup. Moreover, the structure of the tile \( \Omega \) and the translation set \( T \) are characterized by \( p \)-homogeneous trees in \cite{FFS}. Let $f \in L^{1}(\mathbb{Q}_p)$ be a function. We say that $f$ is \emph{periodic} if \[f(x + \tau) = f(x) \quad \text{a.e. } x \in \mathbb{Q}_p,\] for some $\tau \in \mathbb{Q}_p \setminus \{0\}$. In this note, we show that any function capable of tiling $\mathbb{Q}_p$ is inherently periodic. \begin{theorem}\label{thm-tilingperiodic} Let $f \in L^{1}(\mathbb{Q}_p)$ and $V \subset \mathbb{Z} \setminus \{0\}$ be a finite set of non-zero integers. Suppose that $T$ is a locally finite subset in $\mathbb{Q}_p$, and $v_t \in V$ for $t \in T$ are such that \begin{align} \sum_{t \in T} v_t \cdot f(x-t) = w, \quad \text{a.e. } x \in \mathbb{Q}_p, \end{align} for some $w \in \mathbb{R}$.Then $f$ is uniformly locally constancy, i.e. there exists $n\in \mathbb{Z}$ such that \begin{align}\label{eq-periodic} \forall u\in B(x,p^n), \quad f(x+u)=f(x) \quad \text{a.e. } x\in \Q_p. \end{align} \end{theorem} Remark that Equality \eqref{eq-periodic} implies that $f$ is periodic. Actually, the definition regarding tiling doesn't necessarily require the group to be commutative (abelian). It is proved in \cite{HN79} that in every connected nilpotent group $G$ there exists a discrete subset $T$ and a corresponding non-negative smooth function $\phi$ with compact support in $G$ such that \[\sum_{t\in T} \phi(t x)=1 \quad \text{for all } x\in G,\] i.e., the family $\{\phi(t x): t\in T\}$ forms a partition of unity in $G$, which is called a \emph{uniform partition of unity} in $G$. H. Leptin and D. M\"uller \cite{LM91} proposed two questions: 1. Which locally compact groups do admit uniform partitions of unity? 2. Can one describe the set $T$ which corresponds to a uniform partition of unity?\\ And a positive answer for the first question was obtained by the authors. However, the authors claimed that a comprehensive answer to the second question seems to be beyond the present possibilities. They restricted their considerations to the simplest non-trivial case, namely to the group of reals: $ G = \mathbb{R}$ and provided a complete solution. In the case of $G=\mathbb{Q}_p$, the field of $p$-adic numbers, we have also presented a complete solution. For $x\in \Q_p$ and $r>0$, denote by \[B(x, r)=\{y\in \Q_p: \|x-y\|_p\leq r\}\] the closed ball of radius $r$ centered at $x$. \begin{theorem} \label{thm:card} A discrete set $T \subset \mathbb{Q}_p$ corresponds to a uniform partition of unity if and only if there exists a sufficiently large integer $n$ such that \[ \#(B(x,p^n)\cap T) =\#(B(y,p^n)\cap T) \] for all $x, y \in \mathbb{Q}_p$. \end{theorem} For other motivation, it is illuminating to observe the inherent connection between tiling by a set and tiling by a function, particularly within the framework of the Fuglede conjecture \cite{MR0470754}. The conjecture stats that a measurable set $\Omega \in \R^d$ of unit measure can tile $\R^d$ by translation if and only if there exists a set $\Lambda \in \R^d$ such that $\{e^{2\pi i \lambda x}, \lambda \in \Lambda\}$ forms an orthonormal basis of $L^2(\Omega)$. Here, the set $\Lambda$ is called a spectrum of $\Omega$ and $(\Omega,\Lambda)$ is called a spectral pair. Denote by $f_\Omega$ the so-called power spectrum of $1_\Omega$ \[f_{\Omega}(\xi)=\|\widehat{1_{\Omega}}(\xi)\|^2.\] Then $(\Omega,\Lambda)$ is a spectral pair if and only if $(f_{\Omega},\Lambda)$ form a tiling pair at level $1$. Hence, the Fuglede conjecture takes the following symmetric form \begin{align}\label{equ-fug} \Omega\text{ tiles }\R^d \Longleftrightarrow \|\widehat{1_{\Omega}}\|^2 \text{ tiles } \R^d. \end{align} There are many positive results under different extra assumptions before the work\cite{MR2067470} where Tao gave a counterexample: there exists a spectral subset of ${\R}^{n}$ with $n\ge 5$ which is not a tile. After that, Matolcsi\cite{MR2159781}, Matolcsi and Kolountzakis\cite{MR2264214,MR2237932}, Farkas and Gy \cite{MR2221543}, Farkas, Matolcsi and Mora\cite{MR2267631} gave a series of counterexamples which show that both directions of Fuglede's conjecture fail in ${\R}^{n}\left ( n\ge 3 \right ) $. However, the conjecture is still open in low dimensions $n=1,2$. Generally, for a locally compact abelian group $G$, denote by $\widehat{G} $ its dual group. Consider a Borel measurable subset $\Omega $ in $G$ of unit measure. We say that $\Omega $ is a \textit{spectral set} if there exists a set $\Lambda \subset \widehat{ G} $ which forms an orthonormal basis of the Hilbert space $L^{2} \left ( \Omega \right ) $. Such a set $\Lambda $ is called a \textit{spectrum} of $\Omega $ and $\left ( \Omega ,\Lambda \right ) $ is called a \textit{spectral pair}. The Fuglede conjecture can be generalized to locally compact abelian groups: {\em a Borel measurable set $\Omega \subset G$ of unit measure is a spectral set if and only if it can tile $G$ by translation.} Similarly, the generalized Fuglede conjecture also takes the following symmetric form \begin{align}\label{equ-fugloc} \Omega\text{ tiles }G \Longleftrightarrow \|\widehat{1_{\Omega}}\|^2 \text{ tiles } G. \end{align} In its generality, this generalized conjecture does not hold true. Instead, we should inquire about the groups for which it holds. This question even arises for finite groups. The counterexamples in $\mathbb{R}^d$, $d\geq 3$, are actually constructed based on counterexamples in finite groups. Substantial work has been done for some finite groups \cite{MR3684890, MR4402630, FFS, MR3649367, MR4085122, MR4493731, MR1772427, MR2237932, MR1895739, MR4422438, MR3695475, MR4186120, MR4604197}. For infinite groups, based on \eqref{equ-fugloc}, Fan et al. \cite{FFLS} proved that Fuglede's conjecture holds in the field $\mathbb{Q}_p$ of $p$-adic numbers. In fact, this is the first example of an infinite abelian group where Fuglede's conjecture holds. It is natural to find other infinite locally compact abelian groups where Fuglede's conjecture holds. As an application of Theorem \ref{thm-tilingperiodic}, we can characterize the structure of the tiles in the infinite group $\Q_p\times \Z/2\Z$. \begin{theorem}\label{thm1.3} Assume that $\Omega= \Omega_0\times\{0\} \cup \Omega_1\times\{1\} $ tiles $\Q_p\times \Z/2\Z$ by translation. \begin{enumerate}[{\rm (1)}] \item If $p>2$, then either $\mu(\Omega_0\cap\Omega_1)=0$ and $ \Omega_0\cup \Omega_1$ can tile $\Q_p$ by translation or $\Omega_0$ and $\Omega_1$ can tiles $\Q_p$ with a common translation set $T_0\subset \Q_p$. \item If $p=2$, then we have three cases: \begin{itemize} \item[(i)] $\mu(\Omega_0\cap \Omega_1)=0$ and $\Omega_0\cup \Omega_1$ tiles $\Q_2$ by translation, \item[(ii)]$\Omega_0$ and $\Omega_1$ can tiles $\Q_2$ with a common translation set $T_0\subset \Q_2$ \item[(iii)] both $\Omega_0$ and $\Omega_1$ are compact open (up to a measure zero set) and there exist $j_0\in \{0,\cdots, n-1\}$ such that $\Omega_0$ and \[\widetilde{\Omega}_1 = \{ x + 2^{n-j_0-1}: x \in \Omega_1\}\] are disjoint (up to a measure zero set) and $\Omega_0\cup \widetilde{\Omega}_1$ tiles $\Q_2$ by translation. \end{itemize} \end{enumerate} \end{theorem} As a consequence, we establish that spectrality of tiles in the infinite group $\Q_p\times \Z/2\Z$. \begin{corollary} Tiles in $\Q_p \times \Z/2\Z$ are spectral sets. \end{corollary} \section{Preliminaries} \subsection{The field $\Q_p$ of $p$-adic numbers }\label{p-adicfield} Let's start with a quick review of \(p\)-adic numbers. Consider the field \(\mathbb{Q}\) of rational numbers and a prime \(p \ge 2\). Any nonzero number \(r \in \mathbb{Q}\) can be expressed as \(r = p^v \frac{a}{b}\), where \(v, a, b \in \mathbb{Z}\) and \((p, a) = 1\) and \((p, b) = 1\) (here \((x, y)\) denotes the greatest common divisor of the integers \(x\) and \(y\)). We define \(\|r\|_p = p^{-v_p(r)}\) for \(r \neq 0\) and \(\|0\|_p = 0\). Then \(\|\cdot\|_p\) is a non-Archimedean absolute value, meaning: \begin{itemize} \item[(i)] \(\|r\|_p \ge 0\) and equality only when \(r = 0\), \item[(ii)] \(\|r s\|_p = \|r\|_p \|s\|_p\), \item[(iii)] \(\|r + s\|_p \leq \max\{ \|r\|_p, \|s\|_p\}\). \end{itemize} The field \(\mathbb{Q}_p\) of \(p\)-adic numbers is the completion of \(\mathbb{Q}\) under \(\|\cdot\|_p\). The ring \(\mathbb{Z}_p\) of \(p\)-adic integers is the set of \(p\)-adic numbers with absolute value at most 1. A typical element \(x\) of \(\mathbb{Q}_p\) is written as \begin{equation}\label{HenselExp} x = \sum_{n = v}^\infty a_n p^{n} \quad (v \in \mathbb{Z}, a_n \in \{0,1,\dotsc, p-1\}, \text{ and } a_v \neq 0). \end{equation} Here, \(v_p(x) := v\) is called the \(p\)-{\em valuation} of \(x\). A non-trivial continuous additive character on $\mathbb{Q}_p$ is defined by $$ \chi(x) = e^{2\pi i \{x\}}, $$ where $\{x\} = \sum_{n=v_p(x)}^{-1} a_n p^n$ is the fractional part of $x$ in (\ref{HenselExp}). Remark that \begin{align}\label{one-in-unit-ball} \chi(x) = e^{2\pi i k/p^n}, \quad \text{if } x \in \frac{k}{p^n} + \mathbb{Z}_p \ \ (k, n \in \mathbb{Z}), \end{align} and \begin{align}\label{integral-chi} \int_{p^{-n}\mathbb{Z}_p} \chi(x) \,dx = 0 \ \text{for all } n \geq 1. \end{align} From the character $\xi$, we can obtain all characters $\chi_\xi$ of $\mathbb{Q}_p$ by defining $\chi_\xi(x) = \chi(\xi x)$. The map $\xi \mapsto \chi_{\xi}$ from $\mathbb{Q}_p$ to $\widehat{\mathbb{Q}}_p$ is an isomorphism. We write $\widehat{\mathbb{Q}}_p \simeq \mathbb{Q}_p$ and identify a point $\xi \in \mathbb{Q}_p$ with the point $\chi_\xi \in \widehat{\mathbb{Q}}_p$. For more information on $\mathbb{Q}_p$ and $\widehat{\mathbb{Q}}_p$, the reader is referred to the book \cite{VVZ94}. The following notation will be used in the whole paper. \medskip \\ \noindent {\bf Notation}: \begin{itemize} \item $\mathbb{Z}_p^\times := \mathbb{Z}_p\setminus p\mathbb{Z}_p=\{x\in \mathbb{Q}_p: \|x\|_p=1\}$, the group of units of $\mathbb{Z}_p$. \item $B(0, p^{n}): = p^{-n} \mathbb{Z}_p$, the (closed) ball centered at $0$ of radius $p^n$. \item $B(x, p^{n}): = x + B(0, p^{n})$. \item $ S(x, p^{n}): = B(x, p^{n})\setminus B(x, p^{n-1})$, a ``sphere". \end{itemize} \subsection{Fourier Transform} The Fourier transform of $f\in L^1(\Q_p)$ is defined to be $$\widehat{f}(\xi)=\int_{\Q_p}f(x)\overline{\chi_\xi(x)} dx \quad (\forall \xi\in \widehat{\Q}_p\simeq \Q_p).$$ A complex function $f$ defined on $\Q_p$ is called \textit{uniformly locally constancy} if there exists $n\in \mathbb{Z}$ such that \[f(x+u)=f(x) \quad \forall x\in \Q_p, \forall u \in B(0, p^n).\] The following proposition shows that for an integrable function $f$, having compact support and being uniformly locally constant are dual properties for $f$ and its Fourier transform. \begin{proposition}[\cite{FFLS}]\label{Prop-compactConstant} Let $f\in L^1(\Q_p)$ be a complex-value integrable function. Then $f$ has compact support if and only if $\widehat{f}$ is uniformly locally constancy. \end{proposition} The Fourier transform of a positive finite measure $\mu $ on $\Q_p$ is defined to be \[\widehat{\mu}(\xi)=\int_{\Q_p}\overline{\chi_\xi(x)} d\mu(x) \quad (\forall \xi\in \widehat{\Q}_p\simeq \Q_p).\] Let $\nu $ be a signed measure on ${\Q}_{p}$, with the finite total variation $\left \| \nu \right \| $. The measures $$\nu ^{+} =\frac{1}{2} \left ( \left \| \nu \right \| +\nu \right ) ,\mu ^{-} =\frac{1}{2} \left ( \left \| \nu \right \| -\nu \right )$$ are called the positive and negative variations of $\nu$. The Fourier transform of $\nu $ is defined to be \[\widehat{\nu}(\xi) =\widehat{\nu^{+} }(\xi)-\widehat{\nu^{-} }(\xi)=\int_{{\Q}_{p} }\overline{\chi_{\xi} (x) } d\nu^{+}(\xi) -\int_{{\Q}_{p} }\overline{\chi_{\xi} (x) } d\nu^{-}(x)=\int_{{\Q}_{p} }\overline{\chi_{\xi} (x) } d\nu(x)\] for all $\xi \in \widehat{{\Q}}_{p}\cong {\Q}_{p}$. \subsection{$\mathbb{Z}$-module generated by $p^n$-th roots of unity} \label{Zmodule} Let $m\ge 2$ be an integer and let $\omega_m = e^{2\pi i/m}$, which is a primitive $m$-th root of unity. Denote by $\mathcal{M}_m$ the set of integral points $(a_0, a_1, \dotsc, a_{m-1}) \in \mathbb{Z}^m$ such that $$ \sum_{j=0} ^{m-1} a_j \omega_m^j =0. $$ The set $\mathcal{M}_m$ is clearly a $\mathbb{Z}$-module. When $m=p^n$ is a power of a prime number, the relation between the coefficients is established. \begin{lemma} [{\cite[Theorem 1]{Sch64}}]\label{SchLemma} If $(a_0,a_1,\dotsc, a_{p^n-1})\in \mathcal{M}_{p^n}$, then for any integer $0\le j\le p^{n-1}-1$ we have $a_j=a_{j+sp^{n-1}}$ for all $s=0,1,\dots, p-1$. \end{lemma} A subset $C\subset \Q_p$ is called a {\em $p$-cycle} in $ \Q_{p}$ is a set of $p$ elements such that \[\sum_{c\in C} \chi(c)=0\] which is equivalent to \begin{align}\label{eq-pcycle} C \equiv \left \{ r,r+\frac{1}{p} ,r+\frac{2}{p} ,\dots ,r+\frac{p-1}{p} \right \} \mod \Z_{p} \end{align} for some $r\in \Q_{p} $. Note that \eqref{eq-pcycle} is also equivalent to that $C$ is a subset of $p$ elements such that for distinct $c, c^{\prime}\in C$, \[ \|c-c^{\prime}\|_p=p.\] For a $p$-cycle $C$ in $\Q_p$ and $x\in \Z_p^{\times}$, it is easy to see that $x\cdot C$ is also a $p$-cycle in $\Q_p$, which is to say \[\sum_{c\in C} \chi(x c)=0.\] Let $ E\subset \Q_{p}$ be a finite subset. If $\sum_{c\in E} \chi \left ( c \right ) =0$, then by Lemma \ref{SchLemma}, $E$ can be decomposed into some $p$-cycles. Hence, the property $\sum_{c\in E} \chi(c) = 0$ of the set $E\subset \Q_p$ is invariant under rotations. \begin{lemma}[{\cite[ Lemma 2.6]{FFS}}]\label{lem6} Let $\xi _{0}, \xi _{1},\cdots ,\xi _{m-1}$ be $m$ points in $\Q_{p}$. If\: $\sum_{j=0}^{m-1} \chi \left ( \xi _{j} \right ) =0$, then $p\mid m$ and $$\sum_{j=0}^{m-1} \chi \left ( x\xi _{j} \right ) =0$$for all $x\in \Z_{p}^{\times } $. \end{lemma} Actually, by Lemmas \ref{SchLemma} and \ref{lem6}, we have the following corollary. \begin{corollary}\label{corollary2} Let $\xi _{0}, \xi _{1},\cdots ,\xi _{m-1}$ be $m$ points in $\Q_{p}$. If $\sum_{j=0}^{m-1} \alpha _{j} \chi \left ( \xi _{j} \right ) =0$, where $\alpha _{j}\in \Z $, then $$\sum_{j=0}^{m-1}\alpha _{j} \chi \left ( x\xi _{j} \right ) =0$$for all $x\in \Z_{p}^{\times } $. \end{corollary} \subsection{Zeros of the Fourier Transform of a finite discrete signed measure} Let $E$ be a non-empty finite subset of $\Q_p$ and \[\nu=\sum_{x\in E} \alpha_{x}\delta_x,\quad \alpha_{x}\in \Z\setminus\{0\}\] be a finite measure supported in the set $E$. By definition, we have \[ \widehat{\nu} (\xi)= \sum_{x\in E} \alpha_{x}\overline{\chi(\xi x)}= \sum_{x\in E} \alpha_{x}e^ {2\pi i\{-\xi x\}}.\] By Corollary \ref{corollary2}, the zero set of the Fourier transform of a finite discrete measure $\nu$ is invariant under rotations. \begin{lemma}\label{lem-fourierzero} If $\widehat{\nu}(\xi)=0$, then $\widehat{\nu}(\xi \cdot \zeta)=0$ for all $\zeta\in \Z_{p}^{\times }$. \end{lemma} A necessary condition for a point to be a zero of the Fourier transform of $\nu$ is provided. \begin{lemma}\label{lem-necessary} For $\xi\in \Q_p^{\times}$, if $\widehat{\nu}(\xi)=0$, then for any $x\in E$, there exists a distict point $x^{\prime}\in E$ such that \[\|x-x^{\prime}\|_p\leq \frac{p}{\|\xi\|_p}. \] \end{lemma} \begin{proof} Since $E$ is a finite subset of $\mathbb{Q}_p$, it is bounded. Specifically, let's assume that for each $x \in E$, the $p$-adic absolute value $\|x\|_p$ satisfies $\|x\|_p \leq p^n$ for some integer $n$. For $\xi \in \Q_p^{\times}$, we have $\|\xi\|_p=p^m$ for some $m\in \Z$. By Lemma \ref{lem-fourierzero}, $\widehat{\nu} (\xi)=0$ if and only if $\widehat{\nu} (-1/p^m)=0$, which is equivalent to \begin{align}\label{eq-Fourierzero} \sum_{x\in E} \alpha_{x}\chi_{1/p^m}(x)=0. \end{align} Note that the character $\chi_{1/p^m}$ is uniformly locally constant on the disks with radius $1/p^m$. Consider the map $\chi_{1/p^m}:E \to \{1, e^{2\pi i /p^{m+n}}, \cdots, e^{2\pi i (p^{m+n}-1)/p^{m+n}}\}$. Denote by $E_j=\{x\in E: \chi_{1/p^m}(x)=e^{2\pi i j/p^{m+n}} \}$. Equality \eqref{eq-Fourierzero} is equivalent to \[ \sum_{j=0} ^{p^{m+n}-1} a_j e^{2\pi i j /p^{m+n}} =0, \] where $a_j=\sum_{x\in E_j} \alpha_x$. For $x\in E$, assume that $\chi_{1/p^m}(x)=e^{2\pi i j /p^{m+n}} $ for some $0\leq j\leq p^{m+n}-1$. By Lemma \ref{SchLemma}, $a_j=a_{j+p^{m+n-1} \mod p^{m+n}}$. If $a_j=0$, then there exists $x^{\prime}\in E_j$ such that $x^{\prime}\neq x$. Noting that $\chi_{1/p^m}(x)= \chi_{1/p^m}(y)$ if and only if $\|x-y\|_p\leq 1/p^m$. Hence $\|x-x^{\prime}\|\leq 1/p^m$. On the other hand, if $a_j\neq 0$, then there exists $x^{\prime}\in E$ such that \[\chi_{1/p^m}(x^{\prime})=e^{2\pi i \frac{j+p^{m+n-1}}{p^{m+n}} },\] which implies \[\|x-x^{\prime}\|_p=\frac{1}{p^{m-1}}.\] \end{proof} \subsection{Bruhat-Schwartz distributions in $\Q_p$}\label{subsec2.4} Here, we provide a concise overview of the theory based on the works of \cite{AKS10, Tai75, VVZ94}. Let $\mathcal{E}$ represent the space of uniformly locally constancy functions. The space $\mathcal{D}$ of \textit{Bruhat-Schwartz test functions} is defined as functions that are uniformly locally constancy and have compact support. A test function $f\in \mathcal{D}$ is a finite linear combination of indicator functions of the form $1_{B(x,p^k)}(\cdot)$, where $k\in \mathbb{Z}$ and $x\in \Q_p$. The largest such number $k$ is denoted by $\ell:= \ell(f)$ and referred to as the \textit{parameter of constancy} of $f$. As $f\in \mathcal{D}$ has compact support, the minimal number $\ell':=\ell'(f)$ such that the support of $f$ is contained in $B(0, p^{\ell'})$ exists and is named the \textit{parameter of compactness} of $f$. Certainly, we have $\mathcal{D}\subset \mathcal{E}$. The space $\mathcal{D}$ is endowed with the topology of a topological vector space as follows: a sequence ${\phi_n }\subset \mathcal{D}$ is termed a \textit{null sequence} if there exist fixed integers $l$ and $l^{\prime}$ such that each $\phi_n$ is constancy on every ball of radius $p^l$, is supported by the ball $B(0,p^{l^{\prime}})$, and the sequence $\phi_n$ uniformly converges to zero. A \textit{Bruhat-Schwartz distribution} $f$ on $\Q_p$ is defined as a continuous linear functional on $\mathcal{D}$. The value of $f$ at $\phi \in \mathcal{D}$ is denoted by $\langle f, \phi \rangle$. Notably, linear functionals on $\mathcal{D}$ are automatically continuous, facilitating the straightforward construction of distributions. Let $\mathcal{D}'$ denote the space of Bruhat-Schwartz distributions, equipped with the weak topology induced by $\mathcal{D}$. A locally integrable function $f$ is treated as a distribution: for any $\phi \in \mathcal{D}$, $$ \langle f,\phi\rangle=\int_{\Q_p} f\phi d\m. $$ Let $E$ be a locally finite discrete subset in $\Q_p$, and let $\nu$ be a discrete signed measure supported in $E$ defined as \begin{align}\label{eq-discretemeasure} \nu = \sum_{x\in E} \alpha_{x}\delta_x, \quad \alpha_{x}\in \Z\setminus\{0\}, \end{align} where $\{\alpha_x: x\in E\}$ is a bounded. The discrete measure $\nu$ defined by \eqref{eq-discretemeasure} is also a distribution: for any $\phi \in \mathcal{D}$, $$ \langle \nu,\phi\rangle= \sum_{x\in E} \alpha_x \phi(x). $$ Here for each $\phi$, the sum is finite because $E$ is locally finite and each ball contains at most a finite number of points in $E$. Since the test functions in $\mathcal{D}$ are uniformly locally constancy and have compact support, the following proposition is a direct consequence of Proposition \ref{Prop-compactConstant} or of the fact (see also \cite[Lemma 4]{Fan15}) that \begin{equation}\label{FB} \widehat{1_{B(c, p^k)}}(\xi) = \chi(-c \xi)\cdot p^{k} \cdot 1_{B(0, p^{-k})}(\xi). \end{equation} \begin{proposition}[\cite{Tai75}, Chapter II 3] The Fourier transformation $f \mapsto \widehat{f}$ is a homeomorphism from $\mathcal{D}$ onto $\mathcal{D}$. \end{proposition} Let $f \in \mathcal{D}'$ be a distribution in $\Q_p$. A point $x\in \Q_p$ is called a {\em zero} of $f$ if there exists an integer $n_0$ such that $$ \langle f, 1_{B(y,p^{n})}\rangle=0, \quad \text {for all } y\in B(x,p^{n_0}) \text{ and all integers } n\leq n_0 \text.$$ Denote by $\mathcal{Z}_f$ the set of all zeros of $f$. Remark that $\mathcal{Z}_f$ is the maximal open set $O$ on which $f$ vanishes, i.e. $\langle f, \phi\rangle=0$ for all $\phi \in \mathcal{D}$ such that the support of $\phi$ is contained in $O$. The {\em support} of a distribution $f$ is defined as the complementary set of $\mathcal{Z}_f$ and is denoted by $\operatorname{supp}(f)$. The Fourier transform of a distribution $f \in \mathcal{D}'$ is a new distribution $\widehat{f}\in \mathcal{D}'$ defined by the duality \[ \langle \widehat{f},\phi\rangle=\langle f,\widehat{\phi}\rangle, \quad \forall \phi \in \mathcal{D}.\] The Fourier transform $f\mapsto \widehat{f}$ is a homeomorphism on $\mathcal{D}'$ under the weak topology \cite[Chapter II 3]{Tai75}. \subsection{Convolution and multiplication of distributions} Denote $$ \Delta_k:=1_{B(0,p^k)}, \quad \theta_k:=\widehat{\Delta}_k=p^k \cdot 1_{B(0,p^{-k})}. $$ Let $f,g\in \mathcal{D}'$ be two distributions. We define the {\em convolution} of $f$ and $g$ by $$ \langle fg,\phi \rangle =\lim\limits_{k\to \infty} \langle f(x), \langle g(\cdot),\Delta_k(x)\phi(x+\cdot) \rangle \rangle, $$ if the limit exists for all $\phi \in \mathcal{D}$. {\begin{proposition} [\cite{AKS10}, Proposition 4.7.3 ] If $f\in \mathcal{D}^{\prime}$, then $f\theta_k\in \mathcal{E}$ with the parameter of constancy at least $-k$. \end{proposition} } We define the {\em multiplication} of $f$ and $g$ by $$ \langle f\cdot g,\phi \rangle =\lim\limits_{k\to \infty} \langle g, ( f\theta_k)\phi \rangle, $$ if the limit exists for all $\phi \in \mathcal{D}$. The above definition of convolution is compatible with the usual convolution of two integrable functions and the definition of multiplication is compatible with the usual multiplication of two locally integrable functions. The following proposition shows that both the convolution and the multiplication are commutative when they are well defined and the convolution of two distributions is well defined if and only if the multiplication of their Fourier transforms is well defined. \begin{proposition} [\cite{VVZ94}, Sections 7.1 and 7.5] \label{Conv-Mul} Let $f, g\in \mathcal{D}'$ be two distributions. Then\\ \indent {\rm (1)} \ If $fg$ is well defined, so is $gf$ and $fg=gf$.\\ \indent {\rm (2)} \ If $ f\cdot g$ is well defined, so is $g\cdot f$ and $ f\cdot g= g\cdot f$.\\ \indent {\rm (3)} \ $fg$ is well defined if and only $\widehat{f}\cdot \widehat{g}$ is well defined. In this case, we have $ \widehat{fg}=\widehat{f}\cdot\widehat{g}$ and $ \widehat{f\cdot g}=\widehat{f}\widehat{g}. $ \end{proposition} \section{Proof of Theorem \ref{thm-tilingperiodic}} In this section, we will study the following equation \begin{equation}\label{EQ} f\nu = w \end{equation} where $ f\in L^1(\Q_p)$ with $\int_{\Q_p}fd\m>0$, and $\nu$ is a locally finite signed measure defined by \eqref{eq-discretemeasure} in $\Q_p$. Our discussion is based on the functional equation \[\widehat{f}\cdot\widehat{\nu}=w\delta_0,\] which is implied by $f\nu=w$ (see Proposition \ref{Conv-Mul}). \subsection{Zeros of the Fourier transform of the measure $\nu$} Let $\nu$ be a discrete signed measure defined by \eqref{eq-discretemeasure}. The support of $\nu$ is the discrete set $E$. Denote by $\nu_n$ the restriction of $\nu$ on the ball $B(0,p^n)$,i.e. \[ \nu_n=\sum_{x\in E\cap B(0,p^n) } \alpha_{x}\delta_x. \] Consider $\nu$ as a distribution on $\Q_p$. For simplicity, denote by $E_n=E\cap B(0,p^n) $ the restriction of $E$ on the ball $B(0, p^n)$. The following proposition characterizes the structure of $ \mathcal{Z}_{\widehat{\nu}}$, the set of zeros of the Fourier transform of $\nu$. It is bounded and is a union of spheres centered at $0$. Recall that the support of $\nu$ is a locally finite subset. Define \begin{align}\label{def-nv} n_{\nu}:=\inf_{\lambda \in E } \max_{\substack {\lambda^{\prime}\in E\\ \lambda\neq \lambda^{\prime}} } v_p(\lambda- \lambda^{\prime}). \end{align} Remark that $n_{\nu}$ can be $-\infty$. \begin{proposition}\label{zeroofE} Let $\nu$ be a discrete measure defined by \eqref{eq-discretemeasure} in $\Q_p$.\\ \indent {\rm (1)} If $\xi\in \mathcal{Z}_{\widehat{\nu}}$, then $S(0,\|\xi\|_p)\subset \mathcal{Z}_{\widehat{\nu}}$.\\ \indent {\rm (2)} The set $ \mathcal{Z}_{\widehat{\nu}}$ is contained in $B(0,p^{n_\nu+1})$. \end{proposition} \begin{proof} First remark that by using (\ref{FB}) we get \begin{equation}\label{FofMu} \langle \widehat{\nu}, 1_{B(\xi,p^{-n})} \rangle =\langle \nu, \widehat{1_{B(\xi,p^{-n})}} \rangle =p^{-n}\sum_{x\in E_n} \alpha_x \cdot \overline{\chi(\xi x)}=p^{-n} \widehat{\nu_n}(\xi). \end{equation} This expression will be used several times. \indent {\rm (1)} By definition, $\xi\in \mathcal{Z}_{\widehat{\nu}}$ implies that there exists an integer $n_0$ such that $$\langle \widehat{\nu}, 1_{B(\xi,p^{-n})} \rangle=0 , \quad \forall~ n\geq n_0. $$ By (\ref{FofMu}), this is equivalent to \begin{equation}\label{zeross} \sum_{x\in E_n} \alpha_x \cdot \chi(\xi x)=0 ,\quad \forall~ n\geq n_0. \end{equation} For any $\xi^{\prime}\in S(0,\|\xi\|_p)$, we have $\xi' = u \xi$ for some $u\in \mathbb{Z}_p^\times$. By Corollary \ref{corollary2} and the equality (\ref{zeross}), we obtain $$ \sum_{x\in E_n}\alpha_x \cdot \chi(\xi^{\prime}x )= \sum_{x\in E_n}\alpha_x \cdot \chi(\xi x)=0,\quad \forall~ n\geq n_0. $$ Thus, again by (\ref{FofMu}), $\langle \widehat{\nu}, 1_{B(\xi',p^{-n})} \rangle=0 $ for $ n\geq n_0$. We have thus proved $S(0,\|\xi\|_p)\subset \mathcal{Z}_{\widehat{\nu}}$. \indent {\rm (2)} We distinguish two cases based on the value of $n_{\nu}$ : $n_\nu=-\infty$ or $n_\nu>-\infty$. { \bf Case 1: $n_\nu=-\infty$.} Fix $\xi \in \Q_p\setminus\{0\}$, since $n_\nu=-\infty$, there exists $x_0\in E$ such that $$\forall x \in E \setminus \{x_0\}, \quad \|x-x_0\|_p\geq 1/\|\xi\|_p.$$ We are going to show that $ \mathcal{Z}_{\widehat{\nu}}\subset \{0\}$, which is equivalent to that for each $\xi \in \Q_p\setminus\{0\},$ \[ \langle \widehat{\nu}, 1_{B(\xi,p^{-n})} \rangle \neq 0\] for sufficient large $n$ such that $x_0\in B(0,p^n)$. In fact, if this is not the case, then by (\ref{FofMu}), $\widehat{\nu_n}(\xi)=0.$ Thus by Lemma \ref{lem-necessary}, we can find $x\in E_n$ such that \[ \|x-x_0\|=p/\|\xi\|_p<1/\|\xi\|_p,\] a contradiction. {\bf Case 2: $n_\nu>-\infty$.} Hence, there exists a point $x_0 \in E$ such that \[n_\nu= \max_{\substack {x\in E\\ x\neq x_0} } v_p(x- x_0).\] It follows that $$\forall x \in E \setminus \{x_0\}, \quad \|x-x_0\|_p\geq p^{-n_\nu}.$$ We are going to show that $ \mathcal{Z}_{\widehat{\nu}}\subset B(0,p^{n_\nu+1})$, which is equivalent to that $\xi \not \in \mathcal{Z}_{\widehat{\nu}}$ when $\|\xi\|_p\geq p^{n_\nu+2}$. To this end, we will prove that for all integer $n$ large enough such that $x_0 \in B(0, p^n)$, we have $$\langle \widehat{\nu}, 1_{B(\xi,p^{-n})} \rangle \neq 0.$$ In fact, if this is not the case, then by (\ref{FofMu}), $\widehat{\nu_n}(\xi)=0.$ Thus, by Lemma \ref{lem-necessary}, for the given $x_0$, we can find $x \in E_n$, such that \[\|x-x_0\|_p\leq p/\|\xi\|_p <p^{-n_{\nu}},\] a contradiction. \end{proof} \subsection{Proof of Theorem \ref{thm-tilingperiodic}} For a function $g: \Q_p \rightarrow \mathbb{R}$, denote $$ \mathcal{N}_g :=\{x \in \Q_p: g(x)=0\}. $$ If $g\in C(\Q_p)$ is a continuous function, then $\mathcal{N}_g $ is a closed set and $ \mathcal{Z}_g$ is the set of interior points of $\mathcal{N}_g $. However, the support of $g$ as a continuous function is equal to the support of $g$ as a distribution. \begin{proposition}[{\cite[Proposition 3.2]{FFLS}}]\label{mainlem} Let $g\in C(\Q_p)$ be a continuous function and let $G\in \mathcal{D}'$ be a distribution. Suppose that the product $H=g\cdot G$ is well defined. Then $$\mathcal{Z}_{H} \subset \mathcal {N}_{g} \cup \mathcal{Z}_{G}.$$ \end{proposition} For $f\in L^1(\Q_p)$, it is evident that $\widehat{f}$ is a continuous function. Consequently, we obtain the following immediate consequence. \begin{corollary}\label{cor:3.3} If $f\nu=w$ with $f\in L^1(\mathbb{Q}_p)$, then $\Q_p \setminus \{0\} \subset \mathcal {N}_{\widehat{f}} \cup \mathcal{Z}_{\widehat{\nu}},$ which is equivalent to \begin{align}\label{mainprop} \{\xi\in\Q_p: \widehat{f}(\xi) \neq 0 \} \setminus \{0\} \subset \mathcal{Z}_{\widehat{\nu}}. \end{align} \end{corollary} \begin{proof}[Proof of Theorem \ref{thm-tilingperiodic}] By Proposition \ref{zeroofE}, we have $ \mathcal{Z}_{\widehat{\nu}} \subset B(0,p^{n_v+1})$. According to Corollary \ref{cor:3.3}, ${ \rm supp }(\widehat{f}) \subset B(0,p^{n_v+1})$, which is bounded and, therefore, compact. Following Proposition \ref{Prop-compactConstant}, we conclude that $f$ is uniformly locally constancy. \end{proof} \section{Density of $\nu$ and the proof of Theorem \ref{thm:card}} We say that the discrete measure $\nu$ has a {\em bounded density} if the following limit exists for some $x_0\in \Q_p$, $$ D(\nu):=\lim\limits_{k\to \infty}\frac{ \nu(B(x_0,p^k))}{\m (B(x_0,p^k))}, $$ which is called {\em the density} of the discrete measure $\nu$. Actually, if the limit exists for some $x_0 \in \Q_p$, then it exists for all $x\in \Q_p$ and the limit is independent of $x$. In fact, for any $x_0, x_1\in \Q_p$, when $k$ is large enough such that $\|x_0-x_1\|_p<p^k$, we have $B(x_0, p^k)=B(x_1, p^k)$. The following theorem gets together some properties of the solution $(f, \nu)$ of the equation \eqref{EQ}, which will be proved in this section. \begin{theorem} \label{M1} Let $f\in L^1(\mathbb{Q}_p)$ with $\int_{\mathbb{Q}_p}fd\m>0$, and $\nu$ be a discrete measure defined by \eqref{eq-discretemeasure} with locally finite support in $\Q_p$. Suppose that the equation {\rm (\ref{EQ})} is satisfied by $f$ and $\nu$ for some $w\geq 0$. Then the following statements hold.\\ \indent {\rm (1)} The set $\mathcal{Z}_{\widehat{\nu}}$ is bounded and it is the union of the punctured ball and some spheres with the same center $0$. \\ \indent {\rm (2)}\ The density $D(\nu)$ exists and equals to $1/\int_{\Q_p} f d\m$. Furthermore, there exists an integer $n_f\in \mathbb{Z}$ such that for all integers $n \geq n_f$ we have $$ \forall \xi \in \Q_p, \quad \nu \big(B(\xi, p^{n})\big) = p^{n} D(\nu). $$ \end{theorem} Theorem \ref{M1} (1) and will be proved in $\S 4.1$, the distribution of $\nu$ will be discussed in $\S 4.2$ and the equality $D(\nu)=w/ \int_{\Q_p} f d \m$ will be proved in $\S 4.3$. \subsection{ Structure of $\mathcal{Z}_{\widehat{\nu}}$} Our discussion is based on the functional equation $\widehat{f}\cdot\widehat{\nu}= w \cdot \delta_0$, which is implied by $f\mu_\nu=w$ (see Proposition \ref{Conv-Mul}). Notice that $\widehat{f}(0)=\int_{\Q_p} f d \m>0$ and that $\widehat{f}$ is a continuous function. It follows that there exists a small ball where $\widehat{f}$ is nonvanishing. Let \begin{align} \label{nf} n_{f}:=\min \{n\in\mathbb{Z}: \widehat{f}(x)\neq 0,\text{ if } x\in B(0,p^{-n}) \}. \end{align} \begin{proposition}\label{prop:4.2} Let $f\in L^1(\mathbb{Q}_p)$ with $\int_{\mathbb{Q}_p}fd\m>0$, and $\nu$ be a discrete measure defined by \eqref{eq-discretemeasure} with locally a finite support in $\Q_p$. Then, \begin{equation} \label{eq:4.2} B(0,p^{-n_f})\setminus \{0\}\subset \mathcal{Z}_{\widehat{\nu}}. \end{equation} \end{proposition} \begin{proof} Actually, \eqref{eq:4.2} is an immediately consequence of \eqref{mainprop}. \end{proof} By Proposition \ref{zeroofE} and \ref{prop:4.2}, the set $\mathcal{Z}_{\widehat{\nu}}$ is bounded and it is the union of the punctured ball $B(0,p^{-n_f})$ and some spheres with the same center $0$. \subsection{Distribution of $\nu$} The discrete measure $\nu$ involved in the equation (\ref{EQ}) shares the following uniform distribution property. \begin{proposition}\label{Structure} The measure $\nu$ of the ball $B(\xi, p^{n_f})$ is independent of $\xi \in \mathbb{Q}_p$. Consequently, the measure $\nu$ admits a bounded density $D(\nu)$. Moreover, for all integers $n \geq n_f$, we have \begin{equation} \label{numberE} \forall \xi \in \mathbb{Q}_p, \quad \nu\big(B(\xi, p^{n})\big) = p^n D(\nu). \end{equation} \end{proposition} \begin{proof} For any given $\xi \in \Q_p$, let $k=-v_p(\xi)$. If $k\leq n_f$, then $B(0,p^{n_f})=B(\xi,p^{n_f})$. So obviously $\nu\big(B(0,p^{n_f})\big)= \nu\big(B(\xi,p^{n_f})\big)$. Now we suppose $k> n_f$. Then, consider any $\eta$ satisfying $$B(\eta,p^{-k})\subset B(0,p^{-n_f})\setminus \{0\}.$$ By \eqref{eq:4.2} in Proposition \ref{prop:4.2}, we have $\langle \widehat{\nu}, 1_{B(\eta,p^{-k})} \rangle =0$, which by (\ref{FofMu}), is equivalent to \begin{equation}\label{eq:4.4} \sum_{x\in E_k} \alpha_x \cdot \chi(\eta x)=0, \end{equation} where $E_k= {\rm supp} \nu \cap B(0,p^k).$ Taking $\eta= p^{k-1}$ in \eqref{eq:4.4}, we have $ \sum_{x\in E_k} \alpha_x \cdot{\chi(p^{k-1}x)}=0. $ Observe that $$B(0,p^k)= \bigsqcup_{i=0}^{p-1} B(ip^{-k},p^{k-1})$$ and that the function $\chi(p^{k-1}\cdot)$ is constant on each ball of radius $p^{k-1}$. So we have $$ 0=\sum_{x\in E_k}\alpha_x \cdot {\chi(p^{k-1}x)}=\sum_{i=0}^{p-1} {\chi\Big(\frac{i}{p}\Big)}\nu\big(B(ip^{-k},p^{k-1})\big). $$ Applying Lemma \ref{SchLemma}, we obtain \begin{equation} \label{ud1} \nu \big(B(ip^{-k},p^{k-1})\big) =\nu\big(B(jp^{-k},p^{k-1})\big), \quad \forall \ 0\le i, j \le p-1. \end{equation} Similarly, taking $\eta=p^{k-2}$ in (\ref{eq:4.4}), we have $$ 0=\sum_{0\le i, j \le p-1}{\chi\Big(\frac{i}{p^2}+\frac{j}{p}\Big)} \nu \big(B(\frac{i}{p^{k}}+\frac{j}{p^{k-1}}, p^{k-2})\big). $$ Again, Lemma \ref{SchLemma} implies \[ \nu\big(B(\frac{i}{p^{k}}+\frac{j}{p^{k-1}}, p^{k-2}) \big)=\nu\big(B(\frac{i}{p^{k}}+\frac{m}{p^{k-1}}, p^{k-2})\big), \quad \forall \ 0\le i, j,m \le p-1. \] Since $$\sum_{j=0}^{p-1} \nu \big(B(\frac{i}{p^{k}}+\frac{j}{p^{k-1}}, p^{k-2})\big)= \nu\big(B(\frac{i}{p^k}, p^{k-1})\big),$$ by (\ref{ud1}), we get $$ \nu\big(B(\frac{i}{p^{k}}+\frac{j}{p^{k-1}}, p^{k-2}) \big)=\nu\big(B(\frac{l}{p^{k}}+\frac{m}{p^{k-1}}, p^{k-2})\big) \quad \forall\ 0\le i, j, l,m \le p-1. $$ We continue these arguments for all $\eta= p^{k-1}, \dotsc, p^{n_f}$. By induction, we have \begin{align}\label{equalnumber} \nu\big( B(\xi_1, p^{n_f})\big)=\nu\big( B(\xi_2, p^{n_f})\big), \quad \forall \xi_1,\xi_2 \in D(0,p^k). \end{align} Thus by (\ref{equalnumber}), $$ \nu \big(B(\xi, p^{n_f})\big)=\nu \big( B(0, p^{n_f} \big).$$ The formula (\ref{numberE}) follows immediately because each ball of radius $p^n$ with $ n\ge n_f$ is a disjoint union of $p^{n-n_f}$ balls of radius $p^{n_f}$ so that $$ \nu \big( B(0,p^n)\big) = p^{n-n_f} \cdot \nu \big( B(0,p^{n_f})\big) . $$ \end{proof} \subsection {Equality $D(\nu)=w/ \int_{\Q_p} f d\m$} \begin{proposition}\label{fisrtprop} The density $D(\nu)$ of $\nu$ satisfies $$D(\nu)=\frac{w}{\int_{\Q_p}f(x)dx}. $$ \end{proposition} \begin{proof} By the integrability of $f$, the quantity $$ \epsilon_n := \int_{\Q_p\setminus B(0, p^n)} f(x) dx $$ tends to zero as the integer $n\to \infty$. Integrating the equality (\ref{EQ}) over the ball $B(0, p^n)$, we have \begin{eqnarray}\label{ee} w \cdot \m (B(0,p^n)) =\sum_{\lambda \in E} \int_{B(0,p^n)}\alpha_{\lambda}\cdot f(x-\lambda)dx. \end{eqnarray} Now we split the sum in (\ref{ee}) into two parts, according to $\lambda \in E\cap B(0, p^n)$ or $\lambda \in E\setminus B(0, p^n)$. Denote $I:=\int_{\Q_p} f d\m$. For $\lambda \in B(0, p^n)$, we have \begin{eqnarray} \int_{B(0,p^n)} f(x-\lambda)dx = \int_{B(0,p^n)} f(x)dx = I - \epsilon_n. \end{eqnarray} It follows that \begin{equation}\label{ee1} \sum_{\lambda \in E\cap B(0, p^n)} \int_{B(0,p^n)} \alpha_{\lambda} \cdot f(x-\lambda)dx = \nu\big( B(0, p^n)\big)\cdot (I-\epsilon_n). \end{equation} Notice that \begin{equation}\label{ee2} \int_{B(0,p^n)} f(x-\lambda)dx = \int_{B(-\lambda,p^n)} f(x)dx. \end{equation} For $\lambda \in E\setminus B(0, p^n)$, the ball $B(-\lambda,p^n)$ is contained in $\Q_p\setminus B(0, p^n)$. We partition the support $\Q_p\setminus B(0, p^n)$ into $B_j$'s such that each $B_j$ is a ball of radius $p^n$. Thus the integrals in (\ref{ee2}) for the $\lambda$'s in the same $B_j$ are equal. Let $\lambda_j$ be a representative of $P_j$. Then we have \begin{eqnarray} \sum_{\lambda \in E\setminus B(0,p^n)} \int_{B(0,p^n)} \alpha_{\lambda} \cdot f(x-\lambda)dx = \sum_j \nu(B_j) \cdot \int_{B(-\lambda_j,p^n)} f(x)dx. \nonumber \end{eqnarray} However, by (\ref{numberE}) in Proposition \ref{Structure}, $\nu (B_j) = D(\nu) \cdot \m\big(B(0, p^n)\big)$ if $n \geq n_f$. Thus, for each integer $n\geq n_f$, \begin{align} \sum_{\lambda\in E\setminus B(0,p^n)} \int_{B(0,p^n)}\alpha_{\lambda} \cdot f(x-\lambda)dx & = D(\nu) \cdot \m\big(B(0, p^n)\big) \sum_j \int_{B_j} f(x)dx \nonumber \\ & = D(\nu)\cdot \m\big(B(0, p^n)\big) \int_{\Q_p \setminus B(0, p^n)} f(x) dx \nonumber \\ &= D(\nu)\cdot \m\big(B(0, p^n)\big)\cdot \epsilon_n. \label{ee3} \end{align} Thus from (\ref{ee}), (\ref{ee1}) and (\ref{ee3}), we finally get $$ \left\| w\cdot \m\big(B(0, p^n)\big) - \nu\big( B(0, p^n)\big)\cdot I \right\| \le 2 D(\nu) \cdot \m(B(0, p^n))\cdot \epsilon_n. $$ We conclude by dividing $\m\big(B(0, p^n)\big)$ and then letting $n\to \infty$. \end{proof} \subsection{Proof of Theorem \ref{thm:card} } \begin{proof}[Proof of Theorem \ref{thm:card}] Consider the convolution equation $f\nu=1$, where $f\in L^{1}(\Q_p)$ is a non-negative, $\nu=\sum_{t\in T}\delta_t$ and $T$ is a discrete subset in $\Q_p$. By Proposition \ref{Structure}, for integers $n \geq n_f$, we have \[ \#(B(x,p^n)\cap T) =\#(B(y,p^n)\cap T), \quad \forall x, y \in \mathbb{Q}_p.\] On the other hand, if \[ \#(B(x,p^n)\cap T) = \#(B(y,p^n)\cap T), \quad \forall x, y \in \mathbb{Q}_p.\] for some $n\geq 0$. Take \[f=\frac{1_{B(0,p^n)}}{\#(B(x,p^n)\cap T)}. \] Then $f\nu=1$ with $\nu=\sum_{t\in T}\delta_t$. \end{proof} \section{The structure of tiles on $\mathbbm {\Z}/ p^{n}q\mathbbm {\Z}$ and $\Z/p^n\Z\times\Z/p\Z$} In this section, we primarily utilize the results established in \cite{FFS} and \cite{FKL} to characterize the structure of tiles on $\mathbbm {\Z}/ p^{n}q\mathbbm {\Z}$. The structural properties of tiles in $\mathbb{Z}/p^n\mathbb{Z}$ are characterized by $p$-homogeneity, as demonstrated in \cite{FFS}. \subsection{ $p$-homogeneity of tiles in $\Z/p^n\Z$} Let $nn$ be a positive integer. To any finite sequence $t_0 t_1 \cdots t_{n-1} \in \{0, 1, \dots, p-1\}^{n}$, we associate the integer \[ c = c(t_0 t_1 \cdots t_{n-1}) = \sum_{i=0}^{n-1} t_i p^i \in \{0, 1, \dots, p^{n} - 1\}. \] This establishes a bijection between ${\Z}/p^{n}{\Z}$ and $\{0, 1, \dots, p-1\}^{n}$, which we consider as a finite tree, denoted by ${\mathcal T}^{(n)}$ (see Figure~\ref{fig:1}). The set of vertices of ${\mathcal T}^{(n)}$ is the disjoint union of the sets ${\Z}/p^\gamma{\Z}$, for $0 \le \gamma \le n$. Each vertex, except the root, is identified with a sequence $t_0 t_1 \cdots t_{\gamma-1}$ in ${\Z}/p^\gamma{\Z}$, where $0 \le \gamma \le n$ and $t_i \in \{0, 1, \dots, p-1\}$. The set of edges consists of pairs $(x, y) \in {\Z}/p^\gamma{\Z} \times {\Z}/p^{\gamma+1}{\Z}$ such that $x \equiv y \pmod{p^n}$, where $0 \le \gamma \le n-1$. Each point $c$ of ${\Z}/p^n{\Z}$ is identified with the boundary point $\sum_{i=0}^{n-1} t_i p^i \in \{0, 1, \dots, p^{n} - 1\}$ of the tree. Each subset $C \subset {\Z}/p^n{\Z}$ determines a subtree of ${\mathcal T}^{(n)}$, denoted by ${\mathcal T}_C$, which consists of the paths from the root to the boundary points in $C$. For each $0 \le \gamma \le n$, we denote by \[ C_{\bmod p^\gamma} := \{x \in \{0, 1, \dots, p^\gamma-1\} : \exists y \in C \text{ such that } x \equiv y \pmod{p^\gamma}\} \] the subset of $C$ modulo $p^\gamma$. The set of vertices of ${\mathcal T}_C$ is the disjoint union of the sets $C_{\bmod p^\gamma}$, for $0 \le \gamma \le n$. The set of edges consists of pairs $(x, y) \in C_{\bmod p^\gamma} \times C_{\bmod p^{\gamma+1}}$ such that $x \equiv y \pmod{p^\gamma}$, where $0 \le \gamma \le n-1$. For vertices $u \in C_{\bmod p^{\gamma+1}}$ and $s \in C_{\bmod p^\gamma}$, we call $s$ the \emph{parent} of $u$ or $u$ the \emph{descendant} of $s$ if there exists an edge between $s$ and $u$. Now, we proceed to construct a class of subtrees of ${\mathcal T}^{(n)}$. Let $I$ be a subset of $\{0, 1, \dots, n-1\}$, and let $J$ be its complement. Thus, $I$ and $J$ form a partition of $\{0, 1, \dots, n-1\}$, and either set may be empty. We say a subtree ${\mathcal T}_C$ of ${\mathcal T}^{(n)}$ is of ${\mathcal T}_{I}$-form if its vertices satisfy the following conditions: \begin{enumerate} \item If $i \in I$ and $t_0 t_1\dots t_{i-1} $ is given, then $t_i$ can take any value in $\{0, 1, \dots, p-1\}$. In other words, every vertex in $C_{\bmod p^{i}}$ has $p$ descendants. \item If $i \in J$ and $t_0 t_1 \dots t_{i-1}$ is given, we fix a value in $\{0, 1, \dots, p-1\}$ that $t_i$ must take. That is, $t_i$ takes only one value from $\{0, 1, \dots, p-1\}$, which depends on $t_0 t_1 \dots t_{i-1}$. In other words, every vertex in $C_{\bmod p^{i}}$ has one descendant. \end{enumerate} \begin{figure} \centering \includegraphics[width=0.7\linewidth]{1} \caption{The set ${\Z}/3^4{\Z}=\{0,1,2,\cdots,80\}$ is considered as a tree ${\mathcal T}^{\left(4\right)} $. } \label{fig:1} \end{figure} \begin{figure} \centering \includegraphics[width=0.7\linewidth]{2} \caption{For $p=3$, a ${{\mathcal T}_{I,J }}$-form tree with $n=5$, $I=\{0,2,4\} ,J=\{1,3\}$.} \label{fig:2} \end{figure} Note that such a subtree depends not only on $I$ and $J$ but also on the specific values assigned to $t_i$ for $i \in J$. A ${\mathcal T}_{I}$-form tree is called a finite \emph{$p$-homogeneous tree.} An example of a ${\mathcal T}_{I}$-form tree is shown in Figure~\ref{fig:2}. A set $C \subset {\Z}/p^n{\Z}$ is said to be \emph{$p$-homogeneous} subset of ${\Z}/p^n{\Z}$ with the \emph{branched level set} $I$ if the corresponding tree ${\mathcal T}_C$ is $p$-homogeneous of form ${\mathcal T}_{I}$. If we consider $C \subset \{0, 1, \dots, p^n - 1\}$ as a subset of ${\Z}_p$, then the tree ${\mathcal T}_C$ can be identified with the finite tree determined by the compact open set \[ \Omega = \bigsqcup_{c \in C} (c + p^n{\Z}_p). \] A criterion for a subset $C \subset \Z/p^n\Z$ to be $p$-homogeneous is given in \cite{FFS}. \begin{theorem}[{\cite[Theorem 2.9]{FFS}}] \label{thm5.4} Let $n$ be a positive integer, and let $C \subseteq {\Z}/p^n{\Z}$ be a multiset. Suppose that \begin{enumerate}[{\rm(1)}] \item $\#C \le p^k$ for some integer $k$ with $1 \le k \le n$; \item there exist $k$ integers $1 \le j_1 < j_2 < \dots < j_k \le n$ such that \[ \sum_{c \in C} e^{2\pi i c p^{-j_t}} = 0 \quad \text{for all } 1 \le t \le k. \] \end{enumerate} Then $\#C = p^k$ and $C$ is $p$-homogeneous. Moreover, the tree ${\mathcal T}_C$ is a ${\mathcal T}_{I}$-form tree with $I = \{j_1-1, j_2-1, \dots, j_k-1\}$. \end{theorem} \begin{theorem}[{\cite[Theorem 4.2]{FFS}}] Let $n$ be a positive integer, and let $C \subseteq {\Z}/p^n{\Z}$. Then $C$ tiles ${\Z}/p^n{\Z}$ if and only if $C$ is p-homogeneous. \end{theorem} \subsection{The structure of tiles in $\Z/p^nq\Z$ and $\Z/p^n\Z\times\Z/p\Z$} Write $\Z/p^nq\Z$ as the product form $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$. The geometrical characterization of tiles in $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$ was obtained in \cite{FKL}. \begin{theorem}[\cite{FKL}]\label{prop5.7} Let $A$ be a tile of $\mathbb{Z}/p^n\Z \times \Z/ q\Z$. For $0\leq j \leq q-1$, let $A_j=\{x\in \mathbb{Z}/p^n\Z : (x,j)\in A\}$. \begin{enumerate}[{\rm(1)}] \item If $\#A=p^t$, then $A_j$ are disjoint and $\cup_{j=0}^{q-1}A_j$ tiles $\Z/p^n\Z$ by translation. \item If $\#A=p^tq$, then all $A_j$ can tiles $\mathbb{Z}/p^n\Z $ with a common branched level set. \end{enumerate} \end{theorem} Consider the finite abelian group $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$. Let $ A$ be a tile of $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$. It is evident that $\#A$ divide $p^{n+1}$. Hence, assume that $\#A=p^i$ for some $1 \leq i\leq n$. Define a map $\pi_1$ from the group $ \mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ to the group $\Z/p^n\mathbb{Z}$ by \[ \pi_1(a, b)=a, \quad \hbox{ for } (a, b)\in \mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}. \] Let \[\mathcal{Z}_A=\Big\{g\in \mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}: \widehat {1_ A}(g)=0 \Big\}\] be the set of zeros of the Fourier transform of the function $1_A$. The geometrical characterization of tiles in $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ was established in \cite{FKL}. \begin{theorem}[\cite{FKL}]\label{thmfinite} Assume $A$ is a tile of $\mathbb{Z}/p^n\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ with $\#A=p^t$. Let $$ \mathcal{I}_{A}=\big\{i\in \{0,\cdots \ n-1\}: (p^i, 0)\in \mathcal{Z}_ A \big\}. $$ We distinguish three cases. \begin{enumerate}[{\rm(1)}] \item If $\# \mathcal{I}_{A}=t$, then the set $\pi_1(A)$ is a $p$-{\it homogeneous} in $\Z/p^n\Z$ with $\#\pi_1( A)=p^t.$ \item If $\# \mathcal{I}_{A}=t-1$ and $ (p^j, b)\notin \mathcal{Z}_A$ for each $ j\in \{0,1,\cdots, n\}\setminus \mathcal{I}_{A}$ and $b\in \{1,\cdots,p-1\}$, then the sets \[ A_i=\{x\in \Z/{p^n}\Z: (x, i) \in A\}\] are $p$-homogenous in $\Z/p^n\Z$ with $\#A_i=p^{t-1}$. \item If $\#\mathcal{I}_{A}=t-1$ and $(p^{j}, b) \in \mathcal{Z}_{A}$ for some $j\in \{0,1,\cdots, n\}\setminus \mathcal{I}_{A}$ and $b\in\{1,\cdots,p-1\}$, then the set \[\widetilde{A} = \{ x +b_0 y p^{n-j_0-1}: (x, y) \in A\} \] is $p$-homogeneous in $\Z/{p^n}\Z$ with $\# \widetilde{A}=p^t$, where $j_0$ is the minimal number in $\{0,1,\cdots, n\}\setminus\mathcal{I}_{A}$ such that $(p^{j_0},b_0)\in \mathcal{Z}_A $ for some $ b_0\in \{0, \cdots, p-1\}$. \end{enumerate} \end{theorem} \section{Structure of tiles in $\Q_p\times \Z/2\Z$.} Let $\mu$ be the Haar measure $\Q_p$ such that $\mu(\Z_p)=1$. Equip $\Q_p\times \Z/2\Z$ with the Haar measure $\nu$ with $\nu(\Z_p\times\{0\})=1$. \subsection{Structure of tiles} \begin{proof}[Proof of Theorem \ref{thm1.3}] Let $\left ( \Omega ,T \right ) $ be a tiling pair in $\Q_p\times \Z/2\Z$. Write \[\Omega =\left ( \Omega_{0} \times \left \{ 0 \right \} \right ) \sqcup \left ( \Omega_{1}\times \left \{ 1 \right \} \right ),\] and \[T =\left ( T_{0} \times \left \{ 0 \right \} \right ) \sqcup \left ( T_{1}\times \left \{ 1 \right \} \right ).\] Since $\left ( \Omega ,T \right ) $ be a tiling pair in $\Q_p\times \Z/2\Z$, by definition, we have \begin{equation}\label{equ61} 1_{ \Omega_{0} } \ast \mu_{T_{0} }+1_{ \Omega_{1}}\ast \mu_{T_{1} }=1, \quad \mu-a.e. \end{equation} and \begin{equation}\label{equ62} 1_{ \Omega_{1} } \ast \mu_{T_{0} }+1_{ \Omega_{0}}\ast \mu_{T_{1} }=1, \quad \mu-a.e. \end{equation} where $\mu_T=\sum_{t\in T} \delta_t $ is a discrete measure in $\Q_p$. Next, we will transform the tiling problem on $\mathbb{Q}_p \times \mathbb{Z}/2\mathbb{Z}$ by a set into tiling problem on $\mathbb{Q}_p$ by a function. Adding both sides of Equation~\eqref{equ61} and Equation~\eqref{equ62} separately, we get \begin{equation}\label{equ63} \left ( 1_{ \Omega_{0} } +1_{ \Omega_{1}} \right ) \ast \left ( \mu_{T_{0} } +\mu_{T_{1} } \right ) =2. \end{equation} Subtracting Equation \eqref{equ62} from Equation \eqref{equ61}, we get \begin{equation}\label{equ64} \left ( 1_{ \Omega_{0} } -1_{ \Omega_{1}} \right ) \ast \left ( \mu_{T_{0} } -\mu_{T_{1} } \right ) =0. \end{equation} Hence, by Theorem \ref{thm-tilingperiodic}, we deduce from \eqref{equ63} that the function $ f= 1_{ \Omega_{0} } +1_{ \Omega_{1}}$ is uniformly locally constancy. Then, we have $ \Omega_{0} \cap \Omega_{1} $ and $\left ( \Omega_{0} \setminus \Omega_{1} \right ) \cup \left ( \Omega_{1} \setminus \Omega_{0} \right ) $ are almost compact open. Now, we distinguish two cases: $\left ( 1 \right )$ If $\mu_{T_{0}} -\mu_{T_{1}}=0 $; $\left ( 2 \right )$ If $\mu_{T_{0}} -\mu_{T_{1}}\neq 0 $. $\left ( 1 \right )$ If $\mu_{T_{0}} -\mu_{T_{1}}=0 $, then we deduce from (\ref{equ63}) that $$\left ( 1_{ \Omega_{0} } +1_{ \Omega_{1}} \right ) \ast \mu_{T_{0}}=1.$$ Hence $\left ( \Omega_{0}\cup \Omega_{1} ,T_{0} \right ) $ is a tiling pair in ${\Q}_{p} $ and $ \Omega_{0}\cup \Omega_{1}$ is almost compact open. $\left ( 2 \right )$ If $\mu_{T_{0}} -\mu_{T_{1}}\neq 0 $, then by Theorem \ref{thm-tilingperiodic}, we deduce from \eqref{equ64} that the function $ f= 1_{ \Omega_{0} } -1_{ \Omega_{1}}$ is uniformly locally constancy, which implies that $ \Omega_{0} \setminus \Omega_{1}$ and $ \Omega_{1} \setminus \Omega_{0}$ are almost compact open. Hence, both $ \Omega_0$ and $ \Omega_1$ are compact open. Without loss of generality, assume that $ \Omega_0, \Omega_1$ are compact open set in $\Z_p$. Hence, there exist a positive integer $n$ and $C_0,C_1 \in \Z/p^n \Z$ such that \[ \Omega_0=\bigsqcup_{c \in C_0} c + p^n \mathbb{Z}_p, \quad \Omega_1=\bigsqcup_{c \in C_1} c + p^n \mathbb{Z}_p,\] up to measure zero sets. Note that $ \Omega_0\times\{0\} \cup \Omega_1\times \{1\}$ tiles $\Q_p\times\Z/2\Z$ by translation if and only if $C_0\times\{0\} \cup C_1\times \{1\}$ tiles $\Z/p^n \Z \times\Z/2\Z$ by translation. When $p\geq 3$, by Theorem \ref{prop5.7}, either $C_0\cap C_1=\emptyset$ and $C_0\cup C_1$ tiles $\Z/p^n\Z$ by translation or $C_0, C_1$ can tile $\Z/p^n\Z$ by translation with a common translation set. When $p=2$, by Theorem \ref{thmfinite}, we have three cases: (1) $C_0\cap C_1=\emptyset$ and $C_0\cup C_1$ tiles $\Z/2^n\Z$ by translation, (2) $C_0, C_1$ can tile $\Z/2^n\Z$ by translation with a common translation set, (3) there exist $j_0\in \{0,\cdots, n-1\}$ and such that $C_0$ and $\widetilde{C}_1 = \{ x + 2^{n-j_0-1}: x \in C_1\}$ are disjoint and $C_0\cup \widetilde{C}_1$ tiles $\Z/2^n\Z$ by translation. The combination of the two cases led to our final conclusion. \end{proof} \subsection{Tiles are spectral in $\Q_p\times \Z/2\Z$} It is proved in \cite{FFS,FFLS}that Fuglede conjecture holds in $\Q_p$, spectral sets are compact open and are characterized by their $p$-homogeneity. \begin{theorem}[\cite{FFS}]\label{thm6} Let $\Omega $ be a compact open set in $\Q_{p}$. The following statements are equivalent: \begin{enumerate}[{\rm(1)}] \item $\Omega $ is a spectral set; \item $\Omega $ tiles $\Q_{p}$ by translation; \item $\Omega $ is $p$-homogeneous. \end{enumerate} \end{theorem} Let $E$ be a subset of $\Q_{p}$. Define $$I_{E} =\left \{ i\in \Z:\:\exists \: x,y\in E \: such \:that \: v_{p}\left ( x-y \right )=i \right \} $$ as the set of admissible $p$-orders corresponding to the set $E$. The structure of spectra for spectral sets is also characterized. \begin{theorem}[\cite{FFS} Theorem 5.1]\label{thm5} Let $\Omega \subset \Q_{p} $ be a $p$-homogeneous compact open set with the admissible $p$-order set $I_{\Omega } $. \begin{enumerate}[{\rm(1)}] \item The set $\Lambda $ is a spectrum of $\Omega$ if and only if it is $p$-homogeneous discrete set with admissible $p$-order set $I_{\Lambda }=-\left ( I_{\Omega }+1 \right ) $. \item The set $T$ is a tiling complement of $\Omega$ if and only if it is a $p$-homogeneous discrete set with admissible $p$-order set $I_{T} =\Z \setminus I_{\Omega }$. \end{enumerate} \end{theorem} Let $\Omega\subset \Q_p\times \Z/2\Z$ be a tile. Write $\Omega$ as \[\Omega =\left ( \Omega_{0} \times \left \{ 0 \right \} \right ) \sqcup \left ( \Omega_{1}\times \left \{ 1 \right \} \right ).\] When $p>2$, by Theorem \ref{thm1.3}, there are two mutually exclusive cases: {\rm (1)} $\mu(\Omega_0\cap\Omega_1)=0$ and $ \Omega_0\cup \Omega_1$ can tile $\Q_p$ by translation, {\rm (2)} $\Omega_0$ and $\Omega_1$ can tiles $\Q_p$ with a common translation set $T_0\subset \Q_p$. For the first case, $\Omega_0\cup \Omega_1 $ tiles $\Q_p$ by translation. Assume that $(\Omega_0\cup \Omega_1,\Lambda)$ is a spectral pair. Therefore, $(\Omega, \Lambda \times \left \{ 0 \right \})$ is a spectral pair in $\Q_p\times \Z/2\Z$. For the second case, the set $\Omega_0$ tiles $\Q_p$ by translation. It follows that $\Omega_0$ and $\Omega_1$ are spectral sets in $\Q_p$ with a common spectrum $\Lambda$. Therefore, $(\Omega,\left ( \Lambda \times \left \{ 0 \right \} \right ) \sqcup \left ( \Lambda \times \left \{ 1 \right \} \right ))$ is also a spectral pair. When $p=2$, by Theorem \ref{thm1.3}, there are three cases. For the first two cases, it is similar as $p>2$. For the third case, by Theorem \ref{thm1.3}, $\Omega_0$ and $\Omega_1$ are compact open up to measure zero sets. Without loss of generality, assume that $ \Omega_0, \Omega_1$ are compact open set in $\Z_2$. Hence, there exist a positive integer $n$ and $C_0,C_1 \in \Z/2^n \Z$ such that \[ \Omega_0=\bigsqcup_{c \in C_0} c + 2^n \mathbb{Z}_2, \quad \Omega_1=\bigsqcup_{c \in C_1} c + 2^n \mathbb{Z}_2.\] Hence, $C=C_{0}\times\{0\} \cup C_{0}\times\{1\}$ tiles $\Z/2^n \Z \times \Z/2\Z$ by translation. Let $$ \mathcal{I}_{C}=\big\{i\in \{0,\cdots \ n-1\}: (2^i, 0)\in \mathcal{Z}_ C \big\}, $$ and let $j_0\in \{0,1,\cdots, n-1\}\setminus \mathcal{I}_{C}$ such that $(2^{j_0},1)$ in $\mathcal{Z}_C$ and $(2^i, 1)\notin \mathcal{Z}_C$ for $i\in \mathcal{I}_{C}$ with $i<j_0$. Define $$ \Lambda=\Big\{(\sum \limits_{i\in \mathcal{I}_{C}}s_ip^{i-n}, 0)+s_{j_0}(p^{j_0-n},1): s_i, s_j \in \{0, 1, \cdots, p-1\} \Big\}+(\mathbb{L}_{n}\times\{0\}), $$ where \[\mathbb{L}_n=\{\frac{k}{p^{l}}: l> n, 1\leq k\leq p^{l} \hbox{ and } \gcd(k,p)=1\}.\] Hence, $(\Omega, \Lambda)$ forms a spectral pair of $\Q_2\times \Z/2\Z$. \begin{thebibliography}{10} \bibitem{AKS10} S.~Albeverio, A.~Yu. Khrennikov, and V.~M. 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2412.03806v1	http://arxiv.org/abs/2412.03806v1	Dynamical Persistent Homology via Wasserstein Gradient Flow	\documentclass[twoside,11pt]{article} \usepackage[abbrvbib, preprint]{jmlr2e} \usepackage{jmlr2e} \newcommand{\dataset}{{\cal D}} \newcommand{\fracpartial}[2]{\frac{\partial #1}{\partial #2}} \usepackage{amsmath} \usepackage{amssymb} \usepackage{algorithm} \usepackage[noend]{algpseudocode} \DeclareMathOperator{\argmin}{argmin} \usepackage{lastpage} \usepackage{tikz} \usetikzlibrary{matrix, shapes, arrows.meta, calc, decorations.pathreplacing, positioning, quotes, fit} \tikzset{>=latex} rstpageno{1} \begin{document} \title{\scalebox{1.15}{\large Dynamical Persistent Homology via Wasserstein Gradient Flow}} \author{\name Minghua Wang \email [email protected] \\ \addr Department of Computer Science and Engineering\\ State University of New York at buffalo \AND \name Jinhui Xu \email [email protected]\\ \addr Department of Computer Science and Engineering\\ State University of New York at Buffalo} \editor{My editor} \maketitle \begin{abstract} In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate variations in persistence diagrams back into the data space. This advancement enables direct manipulation of the data, guided by observed changes in persistence diagrams, offering a powerful tool for data analysis and interpretation in the context of topological data analysis. \end{abstract} \begin{keywords} persistent homology, optimal transport, Wasserstein gradient flow \end{keywords} \section{Introduction} \label{sec:intro} Persistent homology \citep{edelsbrunner2000topological} analyzes the multi-scale topological features of data, and a persistence diagram summarizes these features by keeping their birth and death across a filtration. Persistence diagram is also the most common topological feature that applied in topological data analyais (TDA) machine learning whose typical extraction can be presented as: \[ \text{Data} \rightarrow \text{Simplicial Complex} \xrightarrow{\operatorname{Dgm}_p \circ F} \text{Persistence Diagram}. \] Here, we begin with input data, which can be point clouds, graphs, images, or manifolds, and use these to construct abstract simplicial complexes. A filtration, denoted by \( F \), creates a sequence of nested simplicial complexes from the data. With this structure, we use a matrix reduction algorithm \citep{edelsbrunner2022computational} to compute the persistence diagram, \(\text{Dgm}_p\), for the \( p \)-th homology group. The persistence diagram is a collection of points, each representing the birth and death of a topological feature. Traditionally, the pipeline flows in one direction: from data to persistence diagrams. However, recent research highlights the need to reverse this process, allowing for filtration or even data adjustments through manipulation of persistence diagrams \citep{gameiro2016continuation, hofer2020graph, oudot2020inverse, ballester2024expressivity}. This paper focuses on adapting data based on the dynamics of persistence diagrams along Wasserstein gradient flows. This builds on prior work by using Wasserstein gradient flows to guide the process of modifying data to achieve desired topological characteristics. By developing algorithms that translate changes in persistence diagrams back to the data space, this research enables data analysis and interpretation through TDA. \section{Related Work} As mentioned in Introduction~\ref{sec:intro}, persistence diagrams are a powerful tool for summarizing topological information \citep{edelsbrunner2000topological, edelsbrunner2022computational, chazal2018density}. They have found applications in diverse fields including shape analysis \citep{poulenard2018topological}, image analysis \citep{li2014persistence, singh2007topological}, graph classification \citep{li2012effective, rieck2019persistent, carriere2020perslay, wangpersistent}, link prediction \citep{yan2021link, aktas2019persistence}, and various machine learning tasks \citep{chen2021topological, chen2021z, chen2024topogcl, wang2024multiset}. One key aspect of persistence diagrams is their stability with respect to perturbations in the data they are built upon. The most common metric used to compare persistence diagrams is the bottleneck distance, which corresponds to the $\infty$-Wasserstein distance. The stability of persistence diagrams under the bottleneck distance has been well studied \citep{cohen2005stability,edelsbrunner2006persistence,chazal2009proximity}. However, the bottleneck distance can be sensitive to outliers and often provides overly pessimistic bounds, especially in high-dimensional settings. Recent work has shifted focus to the use of $p$-Wasserstein distances, A central challenge in this area has been establishing the stability of persistence diagrams with respect to the $p$-Wasserstein distance\citep{skraba2020wasserstein}. The differentiability of persistence diagrams is another important consideration for optimization problems. Early work relied on explicitly computing the derivatives of persistence diagrams with respect to function values, often requiring combinatorial updates \citep{poulenard2018topological}. More recently, researchers have leveraged tools from real analytic geometry and o-minimal theory to establish the differentiability of persistence-based functions. \citet{carriere2021optimizing} demonstrated that the persistence map can be viewed as a semi-algebraic map between Euclidean spaces, enabling the definition and computation of gradients for a wide range of persistence-based functions. Moreover, \citet{leygonie2022differential, leygonie2022framework} not only provide a theoretical foundation for the application of gradient-based optimization techniques to persistence-based problems, but also establish a framework for understanding how changes in data parameters impact persistence diagrams. Several previous studies have explored methods for manipulating data through modifications to persistence diagrams. \citet{gameiro2016continuation} developed a method for point cloud continuation by using the Newton-Raphson method to move a persistence diagram towards a target diagram, iteratively updating the point cloud to achieve the desired topological features. \citet{poulenard2018topological} presented an approach for optimizing real-valued functions based on topological criteria, using derivatives of persistence diagrams to guide the modification of the function and achieve desired topological properties. \citet{corcoran2020regularization} focused on regularizing the computation of persistence diagram gradients to improve the optimization process in data science applications. These works demonstrate the potential of using persistence diagrams as a tool for manipulating data to achieve desired topological characteristics. The current work aims to build upon these efforts by developing novel algorithms that translate variations in persistence diagrams back into the data space, with a focus on the $p$-Wasserstein distance and gradient-based optimization techniques. \section{Preliminaries} This study's theoretical foundation rests on two key areas: Wasserstein gradient flows (WGFs) and differentiable persistence diagrams. In this Preliminaries section, we first introduce the concept of gradient flows in Hilbert spaces, providing the necessary mathematical background. We then extend this to gradient flows in Wasserstein spaces, a crucial framework for our analysis. Finally, we present the fundamentals of differentiable persistent homology, an essential tool in topological data analysis that connects persistent homology with differentiable optimization techniques. These concepts form the basis for the methodologies and analyses presented in subsequent chapters. \subsection{Gradient Flow in Hilbert Space} Gradient flow in Hilbert space \citep{ambrosio2008gradient} is a fundamental concept in the analysis of variational problems and partial differential equations. It describes the evolution of a function in a Hilbert space under the influence of its gradient, leading to a minimization of the associated energy functional. Let $\mathcal{H}$ be a Hilbert space with inner product $\langle \cdot, \cdot \rangle$ and norm $\\|\cdot\\|$. Consider an energy functional $E: \mathcal{H} \to \mathbb{R}$, which is typically assumed to be Fréchet differentiable. The gradient flow of $E$ is the solution to the differential equation \begin{equation} \left\{\begin{array}{l} \dfrac{d u(t)}{dt} = -\nabla E(u(t))\quad \text{for } t>0,\\[0.5em] u(0) = u_0 \end{array}\right. \label{eq:hilbert_gradient_flow} \end{equation} where $u(t) \in \mathcal{H}$ represents the state of the system at time $t$, and $\nabla E(u)$ denotes the gradient of $E$ at $u$. The gradient $\nabla E(u)$ is defined by the property that for all $v \in \mathcal{H}$, \begin{equation} \left. \frac{d}{d\epsilon} E(u + \epsilon v) \right\|_{\epsilon=0} = \left\langle \nabla E(u), v \right\rangle. \end{equation} This ensures that the direction of the gradient $\nabla E(u)$ is the direction of the steepest ascent of the functional $E$. Consequently, the negative gradient $-\nabla E(u)$ points in the direction of the steepest descent, which is why it appears in Equation~\eqref{eq:hilbert_gradient_flow}. One of the key properties of gradient flow is that it decreases the energy functional over time. Specifically, if $u(t)$ is a solution to Equation~\eqref{eq:hilbert_gradient_flow}, then \begin{equation} \frac{d}{dt} E(u(t)) = \left\langle \nabla E(u(t)), \frac{d u(t)}{dt} \right\rangle = -\\|\nabla E(u(t))\\|^2 \leq 0. \label{eq:hilbert_gradient} \end{equation} This implies that $E(u(t))$ is non-increasing along the curve \(u(t)\). Moreover, since \(\frac{d}{d t} E(u(t))=0 \Leftrightarrow\\|\nabla E(u(t))\\|=0\), if \(E\) has a unique stationary point \(u^\star\), $u(t)$ converges to it as $t \to \infty$ under appropriate conditions. \subsection{Proximal Map in Hilbert Space} From a computational perspective, discretization schemes are essential for approximating gradient flows in Hilbert spaces. The implicit Euler scheme, also known as the minimizing movement scheme or proximal map \citep{bubeck2015convex}, is particularly effective for this purpose. This scheme approximates the continuous-time gradient flow by a sequence of discrete minimization problems: \begin{equation} u_{k+1} = \arg\min_{v \in \mathcal{H}} \left( \frac{1}{2\tau} \\|u_k - v\\|^2 + E(v) \right) \end{equation} where $\tau > 0$ is the time step, $u_k$ represents the approximation at the $k$-th time step, $E$ is the energy functional, and $\mathcal{H}$ is the Hilbert space. This scheme not only provides a practical method for numerical computations but also offers insights into the theoretical properties of gradient flows, such as their connection to proximal operators and convex analysis. \subsection{Gradient Flow in Wasserstein Space} \label{sec:gfw} Gradient flow in Wasserstein space extends the concept of gradient flow from Hilbert spaces to the space of probability measures. This framework is particularly valuable for studying partial differential equations and variational problems involving mass transport. While this subsection provides a brief introduction, readers are encouraged to consult \citet{ambrosio2008gradient} for a more comprehensive treatment. The Wasserstein space is defined as the space of probability measures with finite second moments, equipped with the Wasserstein distance. More formally, let $(X, d)$ be a metric space. The Wasserstein space of order $p$ over $(X, d)$, denoted by $\mathcal{P}_p(X)$, is the set of all probability measures $\mu$ on $X$ with finite $p$-th moment: \begin{equation} \mathcal{P}_p(X) = \left\{ \mu \in \mathcal{P}(X) \mathrel{\mid} \int_X d(x, x_0)^p \, d\mu(x) < \infty \right\} \label{eq:wass_space} \end{equation} where $x_0$ is a fixed reference point in $X$. This definition ensures that the measures in $\mathcal{P}_p(X)$ have well-behaved moments, which is crucial for various analytical properties. The Wasserstein distance of order $p$ between two probability measures $\mu, \nu \in \mathcal{P}_p(X)$ is defined as: \[W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p \, d\pi(x, y) \right)^{\frac{1}{p}}\] where $\Pi(\mu, \nu)$ represents the set of all joint probability measures on $X \times X$ with marginals $\mu$ and $\nu$. This distance metric quantifies the optimal cost of transporting mass from one probability distribution to another, providing a natural way to compare and analyze probability measures in the context of gradient flows. With a series of works by \citet{benamou2000computational, otto2001geometry, ambrosio2008gradient}, the concept of gradient flow in Wasserstein space has been rigorously established. Consider an energy functional $J: \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ defined on a Wasserstein space, where $\mathcal{P}_2(\mathbb{R}^d)$ denotes the space of probability measures with finite second moments as in Equation~\eqref{eq:wass_space}. A curve $(\mu_t)_{t \geq 0}$ of probability measures is called a Wasserstein gradient flow for the functional $J$ if it satisfies the following continuity equation in a weak sense: \begin{equation} \frac{\partial \mu_t}{\partial t} = \nabla \cdot \left( \mu_t \nabla \frac{\delta J}{\delta \mu}(\mu_t) \right). \label{eq:wass_gd} \end{equation} Here, $\frac{\delta J}{\delta \mu}: \mathbb{R}^d \rightarrow \mathbb{R}$ represents the first variation of $J$ at $\mu$, defined by \[ \frac{d}{d\varepsilon} J(\mu + \varepsilon \xi) \bigg\|_{\varepsilon=0} = \int \frac{\delta J}{\delta \mu} d\xi = \left\langle \frac{\delta J}{\delta \mu}, \xi \right\rangle, \quad \text{for all } \xi \text{ with } \int \mathrm{d} \xi=0. \] This first variation can be interpreted as the ``functional derivative" of $J$ with respect to $\mu$, extending the concept of derivatives to measure spaces. Consequently, $\nabla\frac{\delta J}{\delta \mu} (\mu_t): \mathbb{R}^d \rightarrow \mathbb{R}^d$ for $t \geq 0$ represents a time-dependent family of vector fields. \citet{chewi2020svgd} refer to this as the Wasserstein gradient, denoted as \(\nabla_{W_2} (\mu_t):= \nabla\frac{\delta J}{\delta \mu} (\mu_t)\). This term draws an analogy with the standard gradient in Euclidean spaces due to its similar role in describing the flow of probability measures. Additionally, some literature defines the Wasserstein gradient as \(\nabla_{W_2} (\mu_t):= -\nabla \cdot \left( \mu_t \nabla \frac{\delta J}{\delta \mu}(\mu_t)\right)\) to maintain consistency with Equation~\eqref{eq:hilbert_gradient_flow}. \subsection{JKO Scheme} The JKO scheme, named after Jordan, Kinderlehrer, and Otto, is a time-discretization method for approximating the Wasserstein gradient flow \citep{jordan1998variational}, It involves solving a sequence of minimization problems: \[\mu_{k+1} = \arg \min_{\mu} \left( \frac{1}{2\tau} W_2^2(\mu, \mu_k) + J(\mu) \right),\] where \(\tau\) is the time step. The transition from the continuous Wasserstein gradient flow to the discrete JKO scheme can be understood through the concept of minimizing movement which approximates the continuous flow by discrete steps. The convergence of the JKO scheme to the continuous Wasserstein gradient flow as \(\tau \to 0\) is established through \(\Gamma\)-convergence \citep{ambrosio2008gradient}. The functional \(J(\mu)\) often represents the internal energy or potential energy of the system, and its gradient with respect to the Wasserstein metric drives the evolution of the distribution \citep{villani2009optimal,villani2021topics}. The JKO scheme is particularly useful in numerical simulations of diffusion processes and other phenomena described by partial differential equations (PDEs) in the Wasserstein space \citep{santambrogio2015optimal}. Applications of the JKO scheme include image processing, machine learning, and fluid dynamics, where the evolution of distributions is of interest \citep{peyre2019computational}. \subsection{Differentiability on Persistence Diagrams} \citet{nigmetov2024topological} introduces a novel method that directly uses the cycles and chains involved in the persistence computation to define gradients for larger subsets of the data. This contrasts with prior methods that focused solely on individual PD points. The authors also demonstrate that this approach can significantly reduce the number of steps required for optimization. Our work builds upon this idea, allowing it to adapt data based on variations in PDs along Wasserstein gradient flows. \section{Methodology} We introduce our basic settings as follows. Let a function \( f: \mathcal{K} \to \mathbb{R} \), where \( \mathcal{K} \) denotes a simplicial complex, induce a filtration. This function assigns a real value to each simplex in the complex, thereby defining a sequence of subcomplexes \( \mathcal{K}(t) = \{ \sigma \in \mathcal{K} \mid f(\sigma) \leq t \} \) for \( t \in \mathbb{R} \). Let \(\operatorname{dgm}_p(\mathcal{K}, f)\) be the algorithm to compute \(p\)-th persistence homology based on \citet[Algorithm 1]{nigmetov2024topological}. In this study, we don't take essential features into account. To construct the dynamical persistence diagram, we consider two situations: with or without a target persistence diagram. \subsection{Dynamical Persistent Homology via McCann Interpolation} In this subsection, we assume that the target persistence diagram is known \textit{a priori}. Our objective is to develop a learnable process for generating persistence diagrams guided by a dynamical system. To approach this problem, we draw upon two fundamental theorems from optimal transport theory: the Benamou-Brenier Theorem and the McCann Interpolation Theorem. The Benamou-Brenier Theorem provides a dynamic formulation of optimal transport, allowing us to view the transport problem as a fluid flow minimizing a kinetic energy functional. Meanwhile, the McCann Interpolation Theorem offers a way to construct geodesics in the space of probability measures, which is crucial for understanding the geometry of persistence diagrams. By leveraging these theorems, we can formulate a continuous evolution of persistence diagrams that converges to the target diagram while respecting the underlying topological structure of the data. The Benamou-Brenier theorem shows that the optimal transport cost can be expressed as the infimum of the action integral over all possible time-dependent probability measures \(\mu_t\) and velocity fields \(v_t\) that transport \(\mu_0\) to \(\mu_1\). \begin{theorem}[Benamou-Brenier \citep{benamou2000computational}] \label{thm:bb} Let $\mu_0$ and $\mu_1$ be two probability measures on $\mathbb{R}^d$ with finite second moments. The squared Wasserstein distance $W_2^2(\mu_0, \mu_1)$ between $\mu_0$ and $\mu_1$ can be expressed as: \begin{equation} W_2^2(\mu_0, \mu_1) = \inf_{(\mu_t, v_t)} \int_0^1 \int_{\mathbb{R}^d} \\|v_t(x)\\|^2 \, d\mu_t(x) \, dt \label{eq:bb_main} \end{equation} subject to the continuity constraints: \begin{equation} \frac{\partial \mu_t}{\partial t} + \nabla \cdot (\mu_t v_t) = 0. \label{eq:bb_continuity} \end{equation} \end{theorem} On the other hand, McCann's interpolation theorem describes the displacement interpolation between two probability measures in the context of optimal transport, showing that the interpolation is a geodesic in the Wasserstein space. \begin{theorem}[McCann Interpolation \citep{villani2021topics}] \label{thm:mccann} Let $\mu_0$ and $\mu_1$ be two probability measures on $\mathbb{R}^d$ with finite second moments, and let $T$ be the optimal transport map pushing $\mu_0$ forward to $\mu_1$. Define the displacement interpolation $\mu_t$ for $t \in [0,1]$ by: \begin{equation} \mu_t = ((1-t) \operatorname{Id} + tT)\# \mu_0, \label{eq:McCann_interplotation} \end{equation} where $\#$ denotes the push-forward operation, $\operatorname{Id}$ is the identity map and $(1-t) \operatorname{Id} + tT$ is the interpolation map. Then: \begin{enumerate} \item $\mu_t$ is a probability measure for each $t \in [0,1]$, \item The curve $\{\mu_t\}_{t \in [0,1]}$ is a geodesic in the Wasserstein space $(\mathcal{P}_2(\mathbb{R}^d), W_2)$, \item The Wasserstein distance between $\mu_0$ and $\mu_1$ can be expressed as: \begin{equation} W_2^2(\mu_0, \mu_1) = \int_{\mathbb{R}^d} \\|x - T(x)\\|^2 \, d\mu_0(x). \end{equation} \end{enumerate} \end{theorem} Therefore, we have the following remark. \begin{remark}[McCann Interpolation is a WGF~\citep{ambrosio2008gradient}] \label{rem:mccann-bb} Let \(T_t = (1-t)\operatorname{Id} + tT\). Then, \((\mu_t, v_t)\) given by McCann interpolation \begin{align} \mu_t & = T_t\#\mu_0 \\ v_t(x) & = \left\{\begin{array}{ll} T(x_0) - x_0 & \text{if } x=T_t (x_0), \text{where } x_0 \text{ is the initial position to } x.\\ 0 & \text{otherwise} \end{array}\right. \end{align} satisfies Equations~\eqref{eq:bb_main} and \eqref{eq:bb_continuity} in Theorem~\ref{thm:bb}. \end{remark} The McCann interpolation can be viewed as a specific instance of a dynamical process. Based on this insight, we have developed an algorithm that leverages McCann interpolation to guide the optimization within the context of persistence diagrams. \begin{algorithm} \caption{Dynamical Persistent Homology via McCann Interpolation} \label{alg:maccann} \begin{algorithmic}[1] \State \textbf{Input:} $\mathcal K = \{\sigma_i\}_{i=1}^l$ the simplices, $f^{(1)}: \mathcal K \mapsto \mathbb R$ the initial filtration function, $K$ the total number of step for the dynamic process, $Z$ the target persistence diagram, $S$ the number of steps to perform filtration updates. \For{$k \gets 1$ to $K$} \State $t\gets \frac{1}{K-k+1}$ \State $X^{(k)} \gets \operatorname{dgm}_p(\mathcal K, f^{(k)})=\{x_i^{(k)}\}_{i=1}^n$ \Comment{Compute persistence diagram} \State $\pi \gets \operatorname{OptimalTransportPlan}(X^{(k)}, Z)$ \Comment{Use Sinkhorn Algorithm~\citep{cuturi2013sinkhorn}} \State $X_1 \gets \operatorname{TargetDgm}(\pi)$ \Comment{Use Equation~\eqref{eq:barycenter}} \State $Y^{(k)} \gets (1-t) X^{(k)} + t X_1$ \Comment{Use McCann Interpolation} \State $f \gets f^{(k)}$ \For{$s\gets 1$ to $S$} \State $\mathcal L \gets \operatorname{Loss}(\operatorname{dgm}_p(\mathcal K, f), Y^{(k)})$ \State $\nabla f \gets \operatorname{CriticalSetMethod}(\mathcal L, f)$ \Comment{\citep[Algorithm 3]{nigmetov2024topological}} \State $f \gets f - \eta \nabla f$ \EndFor \State $f^{(k+1)} \gets f$ \EndFor \Return $\{X^{(k)}\}_{k=1}^K, \{Y^{(k)}\}_{k=1}^K, \{f^{(k)}\}_{k=1}^K$ \end{algorithmic} \end{algorithm} In the Algorithm~\ref{alg:maccann}, the $\operatorname{OptimalTransportPlan}$ can be efficiently computed using the Sinkhorn algorithm~\citep{cuturi2013sinkhorn}, known for its rapid convergence and computational efficiency. Alternatively, conventional linear programming methods can also be employed. The optimal transportation plan \(\pi\) is represented as an \(n \times m\) matrix, where \(n\) and \(m\) denote the cardinalities of the source persistence diagram \(X^{(k)}\) and the target persistence diagram \(Z\), respectively. Each entry \(\pi_{ij}\) in the matrix indicates the amount of mass to be transported from \(x_i \in X^{(k)}\) to \(z_j \in Z\). The mass associated with each point \(x_i \in X^{(k)}\) and \(z_j \in Z\) is \(1/n\) and \(1/m\), respectively. In cases where \(n\) and \(m\) are not equal, the \(\operatorname{TargetDgm}\) function computes an appropriate target persistence diagram. This is achieved by using the barycenter to determine the target locations, calculated as follows: \begin{equation} x_i = \frac{\sum_{j=1}^{m} \pi_{ij} z_j}{\sum_{j=1}^{m} \pi_{ij}} \quad \text{for all } i. \label{eq:barycenter} \end{equation} The Algorithm~\ref{alg:maccann} demonstrates the use of McCann interpolation to guide optimization on persistence diagrams. The core idea is to obtain the next persistence diagram through McCann interpolation and employ a learnable scheme, specifically the critical set method \citep{nigmetov2024topological}, to determine the subsequent filtration. This approach allows for the construction of the entire dynamical process of the persistence diagram. The results $X^{(k)}, Y^{(k)}, f^{(k)}$ represent the persistence diagram, the next target persistence diagram, and the filtration at step $k$, respectively. We explicitly store $Y^{(k)}$ to maintain the next step's persistence diagram because the learnable scheme cannot guarantee that the persistence diagram is always achievable \citep{nigmetov2024topological}. This is also why we do not compute the full trajectories before learning the filtration. Instead, we compute the McCann interpolation at each step to ensure that the persistence diagram progresses towards the target persistence diagram. In other words, this algorithm optimizes the filtration towards the final target rather than intermediate states. The proposed algorithm is particularly well-suited for applications where a target persistence diagram is explicitly defined. Additionally, in the context of McCann interpolation, our algorithm guarantees that the measure $\mu_t$ evolves along the geodesic in Wasserstein space, provided that the target persistence diagram is achievable at each step. This property ensures that the topological features of the data, as captured by persistence diagrams, change towards the target diagram in the best possible manner. \subsection{Dynamical Persistent Homology via Energy Functional} The previous subsection demonstrated the dynamical persistence diagrams along McCann interpolation trajectories, given a target persistence diagram. In this subsection, we extend our methodology to scenarios without a predefined target diagram. As introduced in Subsection~\ref{sec:gfw}, a dynamical system describes the temporal evolution of a state according to a set of rules or equations. In the context of Wasserstein gradient flow, the probability measure \( \mu_t \) evolves following the gradient flow of an energy functional \( J \) in the Wasserstein space. This evolution follows a path that locally minimizes \( J \) most efficiently with respect to the Wasserstein distance. The choice of functional \(J\) determines the system's ultimate configuration, allowing for diverse outcomes depending on the specific energy landscape. To perform the dynamical persistence diagram through Wasserstein gradient flows, we propose the following Algorithm~\ref{alg:wgf}. \begin{algorithm} \caption{Dynamical Persistent Homology via Energy Functional} \label{alg:wgf} \begin{algorithmic}[1] \State \textbf{Input:} $\mathcal K = \{\sigma_i\}_{i=1}^l$ the simplices, $f^{(1)}: \mathcal K \mapsto \mathbb R$ the initial filtration function, $K$ the total number of step for the dynamic process, $S$ the number of steps to perform filtration updates, $\tau$ the step size in JKO, $J$ the energy functional. \For{$k \gets 1$ to $K$} \State $X^{(k)} \gets \operatorname{dgm}_p(\mathcal{K}, f^{(k)}) = \{x_i^{(k)}\}_{i=1}^n$ \Comment{Compute persistence diagram} \State $\left(y_1^{(k)}, y_2^{(k)}, \ldots, y_n^{(k)}\right) \leftarrow\left(x_1^{(k)}, x_2^{(k)}, \ldots, x_n^{(k)}\right)$ \State $\mu_k \gets \frac{1}{n} \sum_{i=1}^n \delta_{x_i^{(k)}}$ \State $\mu \gets \frac{1}{n} \sum_{i=1}^n \delta_{y_i^{(k)}}$ \Comment{Initialize a new measure} \State $\mu \gets \argmin_{\mu} \frac{1}{2 \tau} W_2^2(\mu_k, \mu)+J(\mu)$ \Comment{Perform JKO} \State $Y^{(k)}\gets \{y_i^{(k)}\}_{i=1}^n$ \Comment{The target at step $k$} \State $f \gets f^{(k)}$ \For{$s\gets 1$ to $S$} \State $\mathcal L \gets \operatorname{Loss}(\operatorname{dgm}_p(\mathcal K, f), Y^{(k)})$ \State $\nabla f \gets \operatorname{CriticalSetMethod}(\mathcal L, f)$ \Comment{\citep[Algorithm 3]{nigmetov2024topological}} \State $f \gets f - \eta \nabla f$ \EndFor \State $f^{(k+1)} \gets f$ \EndFor \Return $\{X^{(k)}\}_{k=1}^K, \{Y^{(k)}\}_{k=1}^K, \{f^{(k)}\}_{k=1}^K$ \end{algorithmic} \end{algorithm} Algorithm~\ref{alg:wgf}, akin to Algorithm~\ref{alg:maccann}, outputs $X^{(k)}$, $Y^{(k)}$, and $f^{(k)}$, which represent the current persistence diagram, the target persistence diagram, and the filtration at step $k$, respectively. The key distinction lies in Algorithm~\ref{alg:wgf}'s utilization of the JKO scheme to compute the target persistence diagram at each iteration. Importantly, we initialize the target persistence diagram with the current persistence diagram, ensuring equal cardinalities. This approach enables us to leverage the energy functional to guide the optimization process without prior knowledge of the final target persistence diagram. \section{Case Illustrations} The most beneficial aspect of the proposed methodology is its ability to capture the dynamic evolution of topological features in complex data. To demonstrate the effectiveness of our approach, inspired by the examples in \citet{carriere2021optimizing}, we present two case illustrations: circle denoising and circle emerging. These examples showcase the power of dynamical persistence diagrams in uncovering the underlying topological structures of noisy data and emerging patterns. In both cases, we use Rips filtration. \subsection{Circle Denoising} Our first illustration concerns the denoising of a circle. The input data consists of a set of points in a 2D plane, sampled from a circle with added Gaussian noise. This noisy data forms a perturbed circle, as depicted in Figure~\ref{fig:noised_circle}. The objective is to remove the noise and highlight the primary 1-dimensional topological feature, namely the circle itself. \begin{figure}[H] \centering \includegraphics[width=0.42\textwidth]{img/noised_circle.png} \caption[Noised circle]{The input data is a set of points in a 2D plane, sampled from a circle with added Gaussian noise.} \label{fig:noised_circle} \end{figure} \subsubsection{Repulsion Loss} In this experiment, we need to prevent the points from clustering together. To achieve this, we introduce an auxiliary loss function designed to enforce point separation. Specifically, given the point set \(\mathcal{K}_0 = \{ \sigma_i \}_{i=1}^n\), where each \(\sigma_i \in \mathbb{R}^2\), the repulsion loss is defined as follows: \[ \text{loss} = \sum_{i=1}^{n} \sum_{j=1, j \neq i}^{n} \frac{1}{\\|\sigma_i - \sigma_j\\|^2 + \epsilon} \] This formula calculates the repulsion loss for the point set \(\mathcal{K}_0\). The parameter \(\epsilon\) is a small positive constant that ensures the denominator does not become zero, thereby avoiding numerical instability. \subsubsection{Circle Denoising via McCann Interpolation on 0th Persistence Diagram} The target persistence diagram represents a 1-dimensional circle, which is the primary topological feature of interest. We apply Algorithm~\ref{alg:maccann} to denoise the 0th persistence diagram, using the target persistence diagram as a reference. More specifically, we set the target of the 0th persistence diagram to $(0, 0.05)$ in the experiment, aiming to give these points a reasonable distance from nearby points. For the 1st persistence diagram, we employ a built-in denoising method. The evolution of the denoising process is illustrated in Figure~\ref{fig:noised_circle_mccann_evolution}, where the denoised circle becomes clearly visible. \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_mccann.png} \caption[Circle denoising via McCann Interpolation]{ The evolution of the noisy circle data towards the target circle persistence diagram using McCann interpolation on the 0th persistence diagram and a denoising algorithm \citep{nigmetov2024topological} on the 1st persistence diagram.} \label{fig:noised_circle_mccann_evolution} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_mccann_0th_dgm.png} \caption[0th PD evolution of circle denoising]{ The evolution of the 0th persistence diagram.} \label{fig:noised_circle_mccann_evolution_0th_dgm} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_mccann_1st_dgm.png} \caption[1th PD evolution of circle denoising]{ The evolution of the 1st persistence diagram.} \label{fig:noised_circle_mccann_evolution_1th_dgm} \end{figure} We present the evolution of the 0th and 1st persistence diagrams in Figure~\ref{fig:noised_circle_mccann_evolution_0th_dgm} and Figure~\ref{fig:noised_circle_mccann_evolution_1th_dgm}, respectively. The denoising process effectively eliminates noise from the data, thereby revealing the underlying circular structure. As shown in the figures, the persistence diagrams progressively converge to the target diagram, illustrating the successful denoising of the circle data. However, we observe a gap at the top of the circle in the results. This occurs because the McCann interpolation is applied solely to the 0th persistence diagram, neglecting the 1st persistence diagram. The built-in denoising process lacks control over the upper left 1st persistence pair, which represents the circle. To address this issue, we apply Algorithm~\ref{alg:wgf} to denoise the 1st persistence diagram and let the persistence pair move towards the left. \subsubsection{Circle Denoising via Energy Functional on 1st Persistence Diagram} To achieve the goal we mentioned above, we apply Algorithm~\ref{alg:wgf} with the energy functional \[J(\mu) = \frac{1}{2} \mathbb{E}_{(x, y) \sim \mu} \left[ \min\left(x^2 + (y - 1.2)^2, \frac{(x - y)^2}{2}\right) \right].\] It makes the points near to the diagonal move towards the diagonal, while points far from the diagonal move towards \((0,1.2)\). The results of this process are illustrated in Figure~\ref{fig:noised_circle_wgf_evolution}, Figure~\ref{fig:noised_circle_wgf_evolution_0th_dgm}, and Figure~\ref{fig:noised_circle_wgf_evolution_1st_dgm}. In JKO, we use the sliced Wasserstein implementation from \citet{bonet2022efficient} for efficiency purposes. As shown, the algorithm not only effectively removes noise from the data but also fills gaps at the top of the circle successfully. \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_all.png} \caption[Circle denoising via McCann Interpolation and Energy Functional]{ The evolution of the noisy circle data towards the target circle persistence diagram using McCann interpolation on the 0th persistence diagram and energy functional on the 1st persistence diagram.} \label{fig:noised_circle_wgf_evolution} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_all_0th_dgm.png} \caption[0th PD evolution of circle denoising]{ The evolution of the 0th persistence diagram.} \label{fig:noised_circle_wgf_evolution_0th_dgm} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_denoise_all_1st_dgm.png} \caption[1st PD evolution of circle denoising]{ The evolution of the 1st persistence diagram.} \label{fig:noised_circle_wgf_evolution_1st_dgm} \end{figure} \subsection{Circle Emerging} Given a set of random 2D points on a plane, we observe that the first persistence diagram is not always empty. In this case, the objective is to enhance the circles represented by the first persistence diagram. In the experiment, we independently and identically sample 250 points from the uniform distribution on the square $[-1,1] \times [-1,1]$. Similar to the previous case, we apply the McCann algorithm with a target of $(0, 0.08)$ on the 0th persistence diagrams. Additionally, we apply Algorithm~\ref{alg:wgf} with the functional \[ J(\mu) = \frac{1}{4} \mathbb{E}_{(x, y) \sim \mu} \left[ (y - (x + 0.15))^2 + x^2 \right] \] to encourage the first persistence pairs above a certain threshold to evolve away from the diagonal line and drift to the left. The results of this process are illustrated in Figure~\ref{fig:emerging_circle_wgf_evolution}. We also present 0th and 1st persistence diagrams in Figure~\ref{fig:emerging_circle_wgf_evolution_0th_dgm} and Figure~\ref{fig:emerging_circle_wgf_evolution_1st_dgm}, respectively. The results demonstrate how the persistence diagrams evolve over time, highlighting the emergence of the circle from the random 2D points. \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_emerging_all.png} \caption[Circle emerging via Wasserstein gradient flow]{ The evolution of the random 2D points towards the target circle persistence diagram using McCann interpolation on the 0th persistence diagram and energy functional on the 1st persistence diagram.} \label{fig:emerging_circle_wgf_evolution} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_emerging_all_0th_dgm.png} \caption[0th PD evolution of circle emerging]{ The evolution of the 0th persistence diagram.} \label{fig:emerging_circle_wgf_evolution_0th_dgm} \end{figure} \begin{figure}[H] \centering \includegraphics[width=.92\textwidth]{img/circle_emerging_all_1st_dgm.png} \caption[1st PD evolution of circle emerging]{ The evolution of the 1st persistence diagram.} \label{fig:emerging_circle_wgf_evolution_1st_dgm} \end{figure} \section{Limitations and Further Works} First, the current learnable persistence diagram scheme cannot be executed on a GPU, which hinders the scalability of the proposed algorithms when applied to large datasets. This limitation restricts the practical application of these methods in real-world scenarios where computational efficiency is crucial. Second, the proposed methods exclusively consider persistence diagrams. It is worth exploring whether other topological features, such as Zigzag persistence or multiparameter persistence, can be integrated into the proposed framework to enhance its versatility. Third, there is a need to establish statistical theories for the proposed methods when utilizing different energy functionals. For instance, when deriving dynamical persistence diagrams from a time series of real-world data, it is essential to determine the statistical properties of these diagrams. Additionally, identifying an appropriate energy functional that governs the generation of dynamical persistence diagrams from the time series is crucial. Furthermore, guidelines for selecting suitable energy functionals for various tasks should be developed to ensure optimal performance. Finally, a more effective approach is required to address the conflicts arising from singleton losses \citep{nigmetov2024topological} in the proposed methods, as these conflicts can significantly impact the accuracy and reliability of the results. \section{Conclusion} In this study, we propose two methods to integrate Wasserstein gradient flow into the learning process for persistence diagrams. First, by leveraging McCann interpolation, we develop an algorithm that optimizes persistence diagrams along the geodesic in the Wasserstein space. This approach ensures that the optimization path respects the intrinsic geometry of the space, leading to more accurate and meaningful results. Second, we introduce an energy functional-based approach that guides the optimization process without requiring a predefined target persistence diagram. This method allows for greater flexibility and adaptability in various applications. We demonstrate the effectiveness of our methods through case studies, including the denoising of a circle and the emergence of a circle from noisy data. These examples illustrate how our techniques can enhance the temporal evolution of topological features, highlighting the potential of dynamical persistence diagrams in topological data analysis. \newpage \section{Acknowledgments} We are grateful for the insightful discussion with Dr. Ziyun Huang at the outset of this paper, which played a crucial role in shaping its development. \newpage \vskip 0.2in \bibliography{ref} \end{document}
2412.03863v2	http://arxiv.org/abs/2412.03863v2	Further analysis on the second frequency of union-closed set families	\documentclass[a4paper,12pt]{article} \usepackage[english]{babel} \newcommand{\arXiv}[1]{\href{http://arxiv.org/abs/#1}{arXiv:#1}} \usepackage[margin=2cm]{geometry} \usepackage{type1cm} \usepackage{titlesec} \usepackage{fancyhdr} \usepackage[dvipsnames]{xcolor} \usepackage{graphicx} \usepackage{titling} \usepackage{tabularx} \usepackage[shortlabels]{enumitem} \usepackage{amsmath, amsthm, amssymb} \usepackage{mathtools} \usepackage{tikz-cd} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usepackage[breakable]{tcolorbox} \usepackage{standalone} \usetikzlibrary{decorations.pathmorphing} \usetikzlibrary{calc, arrows, matrix} \usetikzlibrary{external} \tikzexternalize[prefix=tikz/] \usepackage[unicode=true, pdfborder={0 0 0}, bookmarksdepth=-1]{hyperref} \usepackage[capitalise,nameinlink]{cleveref} \setlength{\headheight}{15pt} \setlength{\droptitle}{-1.5cm} \parindent=24pt \newtheoremstyle{mystyle} {6pt}{15pt} {\it} {} {\bf} {.} {1em} {} \theoremstyle{mystyle} \renewcommand{\proofname}{\bf Proof:\\} \newtheorem{theorem}{Theorem}[section] \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{property}[theorem]{Property} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}{Problem} \newtheorem{question}{Question} \newtheorem{fact}{fact}[subsection] \newtheorem{recall}{Recall} \newtheorem{remark}{Remark} \newtheorem{claim}{Claim} \newtheorem{observation}{Observation} \newtheorem{conjecture}{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\E}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} \newcommand{\6}{\partial} \DeclarePairedDelimiter{\norm}{\lVert}{\rVert} \DeclarePairedDelimiter{\gen}{\langle}{\rangle} \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} \DeclarePairedDelimiter{\innerp}{\langle}{\rangle} \pagestyle{fancy} \lhead{} \chead{} \lfoot{} \cfoot{} \rfoot{\thepage} \renewcommand{\headrulewidth}{0.4pt} \renewcommand{\footrulewidth}{0.4pt} \DeclareMathOperator{\Freq}{Freq} \linespread{1.5} \begin{document} \newcommand{\F}{\mathcal{F}} \newcommand{\fq}{\Freq_{\F}} \title{Further analysis on the second frequency of union-closed set families} \author{Saintan Wu\thanks{ National Taiwan University, [email protected]}} \date{\today} \maketitle \begin{abstract} The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\frac{1}{2^{k-1}+1}$ proportion of the sets in the family. Das and Wu showed that this conjecture is true for $k\geq 3$ and $k=2$ if $\|\mathcal{F}\|$ is outside a particular range. In this companion paper, we analyse further when $\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming. \end{abstract} \section{Introduction} A \textit{union-closed} set family $\mathcal{F}$ is a set family satisfying $A,B\in \mathcal{F}\implies A\cup B\in \mathcal{F}$. Nagel \cite{nagel2023notes} conjectured that: \begin{conjecture}\label{conj:Nagel} For any union-closed set family $\|\mathcal{F}\|$ with ground set $\geq k$, there are $k$ elements, each of them lies in at least $\frac{1}{2^{k-1}+1}\|\mathcal{F}\|$ many sets in $\mathcal{F}$. \end{conjecture} Let $F_k(\mathcal{F})$ denote the ratio of sets in $\mathcal{F}$ that contain the $k$-th most frequent element. So the above conjecture states that $F_k(\mathcal{F})\geq \frac{1}{2^{k-1}+1}$ for such families. Notice that when $k=1$, it conjectures the well-known Union-Closed Sets Conjecture, that every nonempty union-closed family has an element appearing in at least $\frac{\|\mathcal{F}\|}{2}$ sets. \cite{our_main_paper} gives a notion about $k$-good set, which here we will only use $2$-good sets: \begin{definition} Given a union-closed family $\mathcal{F}$, a set $S$ is \textbf{$2$-good} for $\mathcal{F}$ if it does not contain $1$, and for each set $A$ that is not $\varnothing$ nor $\{1\}$, $S\cap A \neq\varnothing$. A $2$-good set is called \textbf{minimal} if its proper subsets are not $2$-good. \end{definition} Then, in the paper, \cite{our_main_paper} proved, via the entropic method for large families and shattering method for small families, that \begin{theorem} \label{thm:main} Let $k \geq 2$, and let $\mathcal{F}$ be a union-closed set family with $\|\mathcal{F}\| = m$ and $\|\cup_{F \in \mathcal{F}} F\| \ge k$. Then the $k$th-most frequent element lies in at least $\frac{m}{2^{k-1} + 1}$ sets in $\mathcal{F}$, with equality only if $\mathcal{F}$ is a near-$k$-cube, provided: \begin{itemize} \item[(i)] $k \ge 3$, or \item[(ii)] $k = 2$, and either $m \le 44$ or $m \ge 114$. \end{itemize} \end{theorem} We still did not know if there is a counterexample for $k=2$ when $\|\mathcal{F}\|\in [45,113]$. We will analyse this case more carefully and narrow the range. Our main result in this paper is the following: \begin{theorem}\label{thm:sidemain} Let $\F$ be union-closed with $\|\cup_{F \in \F} F\| \ge 2$, and that $\F$ is not a near-$2$-cube. Then, if $f_2(\F) \le \frac13$, we must have \begin{enumerate}[(i)] \item $81 \le \|\F\| \le 113$, and \item All minimal $2$-good sets $S$ have size $4$. \end{enumerate} \end{theorem} In the following sections, we will focus on the case when $\F$ has $f_2(\F)\leq \frac13$ and is in the case that \cref{thm:main} didn't cover. That is, when $\|\F\|\in [45,113]$ and any of its minimal $2$-good sets $S$ has size $4$ or $5$. We will further assume that $\varnothing\in \mathcal{F}$, and $1$ is the most abundant element in sets of $\mathcal{F}$. \subsection{Paper structure and notations} In Section 2, we will briefly explain the idea behind the proof. Then, we will prove several lemmas mentioned in the heuristics. After that, two separate cases $\|S\|=4$ and $\|S\|=5$ will be done in \cref{sec:s=4} and \cref{sec:s=5}, respectively. For simplicity, for any set $S$ and element $x$, we use $S+x$ for $S\cup\{x\}$, and $S-x$ for $S\setminus\{x\}$. We will also write, for example, $abd$ for short of $\{a,b,d\}$. \section{Heuristics behind the proof} \subsection{Review of the shattering strategy} Let us first have a brief review of the proof for the case $4\leq \|\mathcal{F}\|\leq 113$ and $k=2$. We considered a minimal $2$-good set $S$ and showed that by the minimality, for any element $y\in S$, there is $F_y\in \mathcal{F}$ that $F_y\cap S=\{y\}$ because if there was no such $F_y$, $S-y$ became a smaller $2$-good set. By the union-closed property, for any $T\subseteq S$, we find that $F_T:= \bigcup_{y\in T} F_y$ is in $\mathcal{F}$, and $F_T\cap S=T$. This shows that $\|S\|\leq \log_2\|\mathcal{F}\|$ because these $2^{\|S\|}$ sets $F_T,T\subseteq S$, are surely different. Lastly, using $F_T$'s to analyse the \textit{incidence} $\sum_{A\in \mathcal{F}} \|A\cap S\|$ of $\mathcal{F}$ over $S$ when $f_2(\mathcal{F})\leq \frac13$, we get that either $s=4$ and $m \geq 45$, or $s=5$ and $m\geq 70.5$. \subsection{Linear programming and two ways of winning} We can make a more careful estimate: For each subset $T$ of $S$, we define $q_T$ as the quantity of sets $A$ in $\mathcal{F}$ that $S\cap A=T$. \begin{itemize} \item By $S$ being $2$-good, there are at most $2$ sets that do not intersect $S$, namely $\varnothing$ and $\{1\}$. Thus \[q_\varnothing\leq 2.\] \item By $F_T\cap S=T$, we obtain \[q_T\geq 1 \qquad\text{for each }T\subseteq S.\] \item As $f_2(\mathcal{F})\leq \frac 13$, we obtain \[\sum_{y\in T}q_T\leq \frac m3 \qquad \text{for each }y\in S.\] \end{itemize} It turns out that these are all linear constraints for $q_T$'s and $m$, so we can solve it by setting the following linear programming $L_0$: \begin{description} \item[Variables] $m,q_T:T\subseteq S$ \item[Minimize] $m$ \item[Subject to] \begin{equation} \begin{array}{r@{\,}ll} q_\varnothing &\leq 2 ,&\\ q_T & \geq 1 ,&T\subseteq S\\ \sum_{T\subseteq S} q_T &= m,&\\ \sum_{y\in T}q_T & \leq \frac m3 ,& y\in S.\\ \end{array} \end{equation} \end{description} By solving this directly, we get same bounds $m\geq 45$ for $\|S\|=4$ and $m\geq 70.5$ for $\|S\|=5$. So what happens when this LP reaches the lower bound? A solution reaching the lower bound of $m$ when $\|S\|=4$ is {\linespread{1} \[q_{T}=\begin{cases} 2,&T=\varnothing;\\ 8,&\|T\|=1;\\ 1,&\|T\|\geq 2. \end{cases}\] } This configuration can cause many dependencies. For example, while there are many pairs of choices of $F_a, F_b$ in the proof, they all result in the unique $F_{ab}:= F_a\cup F_b$! This phenomenon could only happen when each element in $F_{ab}$ is either a common element of $F_a$s or a common element of $F_b$s. In this scenario, elements \textit{outside} $S$ appear in many sets. Hence, we can strengthen $L_0$ by considering another carefully chosen element $x$. This $x$ allows two ways of winning: Either \begin{itemize} \item this $x$ helps us find different sets (by spotting $x\notin F_T, x\in F_T'$) that $F_{T}\cap S=T=F_T'\cap S$ for big $T$. In this way, it gives a greater lower bound for $q_T$. Or, \item $x$ itself lies in many sets. In this way, it gives an extra constraint for $L_0$, as $x$ cannot lie in over $\frac{m}{3}$ sets if $x\neq 1$. \end{itemize} \subsection{The choice of $x$ and three lemmas} We will pick $x\neq 1$ and an $a\in S$ such that there are two sets $F_a,F_a'$ with $F_a\cap (S+x)=a, F_a'\cap (S+x)=ax$. For any choice of $F_y, y\in S$, such that $F_y\cap S=\{y\}$, we associate \[ F_{T}:= \bigcup_{y\in T} F_y, T\subseteq S.\] Now we see how the flexibility of choice of $F_a$ helps us: If we can choose $F_y$ for $y\in S$ such that $x\notin F_T$, then $F_T\cup F_a$ and $F_T\cup F_a'$ are two different sets in $\mathcal{F}$, both of which intersect $S$ at $T+a$, thus $q_{T+a}\geq 2$. We can do much better than this, but we give some definitions before we can precisely state the lemma: \begin{definition}[flexible, covered]\mbox{} For $y\in S$, we say $a$ is $x$-\textit{flexible} if there are two sets $F_y,F_y'$ with $F_y\cap (S+x)=y, F_y'\cap (S+x)=xy$. We say $y$ is \textit{covered} (by $x$) if all choices of $F_y$ contains $x$. Equivalently, $y$ is covered by $x$ iff $S+x-y$ is $2$-good. \end{definition} Let $C$ denote the set of covered elements. \begin{lemma}\label{lemma:smallway} Let $a\in S$ be $x$-flexible. Then $q_T\geq 2$ for each $T\subseteq S$ such that \begin{itemize} \item either $a\in T$ and $T\cap C=\varnothing$, or \item $T$ contains at least $2$ elements in $C$. \end{itemize} \end{lemma} On the other hand, if there is a choice of $\{F_y:y\in S\}$ such that $x\in F_T$, then $x\in F_{T'}$ for any $T\subseteq T'\subseteq S$. we can find many sets in $\mathcal{F}$ that contains $x$ if there are many covered elements. It leads to the second lemma: \begin{lemma}\label{lemma:largeway} Let $a\in S$ be $x$-flexible. and $x$ covers the elements in $C$. Then \[(2^{s}-2^{s-1-\|C\|}) + \sum_{c\in C} q_c - \|C\| \leq \frac{m}{3}.\] \end{lemma} These lemmas handle cases where $\|C\|$ is small and large well. For $\|C\|=1$, it turns out that we can benefit from assuming $S$ has the best incidence among all minimal $2$-good $s$-sets of $\mathcal{F}$. \begin{lemma}\label{lemma:middleway} Let $a\in S$ be $x$-flexible. Suppose that $S$ is a minimal $2$-good set which maximizes the incidence $\sum_{A\in \mathcal{F}} \|A\cap S\|$ among all minimal $2$-good $s$-sets of $\mathcal{F}$. Suppose that $x$ only covers $b\in S$. Then \begin{itemize} \item There are $2^{\|S\|-2}$ sets in $\mathcal{F}$ that intersects $S$ at $\geq 2$ elements, contains $b$ but not $x$. \item There are $2^{\|S\|-2}-1$ sets in $\mathcal{F}$ that intersects $S$ at $\geq 3$ elements, contains $b$ but not $x$. \item There are $2^{\|S\|-3}-1$ sets in $\mathcal{F}$ that intersects $S$ at $\geq 4$ elements, contains $b$ but not $x$. \end{itemize} \end{lemma} In the next section, we will justify the choice of $a,x$ and prove these three lemmas. Remind that these lemmas are all under the assumption $f_2(\mathcal{F})\leq \frac13$. To utilise \cref{lemma:middleway}, we will assume that $S$ maximises the incidence among the minimal $2$-good sets with the corresponding size. We will separate the analysis regarding the size of $C$. We obtain a bound for $m$ by first obtaining extra constraints for $q_T$s using \cref{lemma:smallway,lemma:largeway,lemma:middleway}, then solve the linear program $L_0$ with these extra constraints. Detailed LP calculations are available in the companion Git repository\footnote{\url{https://github.com/haur576/k-Union-Closed-Sets-Conjecture}}. A summary of the bounds for $m$ is shown in \cref{fig:summary}. One can find that this is sufficient to reach a contradiction unless $s=4$ and $\|C\|=0,1$. \begin{figure}[htbp!] \centering \[\begin{array}{c\|cccc} \text{Bound for $m$ if $f_2(\mathcal{F})\leq \frac 13$} & \|C\|=0 &\|C\|=1 & \|C\|=2 & \|C\|\geq 3\\\hline s=4 & \textcolor{red}{m\geq 81} & \textcolor{red}{m\geq 81} & m\geq 114 & \text{infeasible}\\ s=5 & m\geq 118.5 & m\geq 115.5 & m\geq 122 & m\geq 114\\ \text{Using Lemma...} &\text{\labelcref{lemma:smallway}}&\text{\labelcref{lemma:smallway,lemma:middleway}}&\text{\labelcref{lemma:smallway,lemma:largeway}}&\text{\labelcref{lemma:largeway}} \end{array}\] \caption{Summary of bound found in cases} \label{fig:summary} \end{figure} \section{Proofs of the lemmas} First of all, we show that there is always a choice of $a$ and $x$. \begin{lemma} There is $a\in S$ and $x\notin S\cup\{1\}$ such that $a$ is $x$-flexible. \end{lemma} \begin{proof} By the constraints of $L_0$, we find that $\sum_{a\in S}q_a\geq 40$ for $\|S\|=5$ and $q_y\geq 8$ for each $y\in S, \|S\|=4$. Therefore, there is $a\in S$ such that $q_a\geq 8$. So $\{F\in \mathcal{F}\| F\cap S=\{a\}\}$ has size $\geq 8$. We pick any two sets in this family that differ not only at $1$. Then there is $x \neq 1$ such that $x$ lies in one set but not the other. These two are desired $F_a'$ and $F_a$, respectively. \end{proof} Now, we prove three lemmas introduced in the previous section. We prove \cref{lemma:largeway} first: \begin{proof}[proof of \cref{lemma:largeway}] This lemma is by simple counting: For each $y\in S$, we choose $F_y$ such that $x\in F_y$ if $y=a$ or $y\in C$, and $\notin F_y$ otherwise. By the choice and the definition of covering, $x$ lies in \begin{itemize} \item $F_T$, where $T$ contains $a$ or an element in $C$, and \item any $F\in \mathcal{F}$ such that $F\cap S=\{c\}$ for some $c\in C$. \end{itemize} There are $2^{\|S\|}-2^{\|S\|-1-\|C\|}$ sets from the first case, $\sum_{c\in C}q_c$ sets from the second case and $\|C\|$ sets from both. As $x$ cannot lie in more than $\frac{m}{3}$ sets, the result follows from the Inclusion-Exclusion Principle. \end{proof} \begin{proof}[proof of \cref{lemma:smallway}] The first case is direct: By the definition of non-covered elements, we can pick $F_y$ for $y\notin C$ such that $x\notin F_y$. If $a\in T$ and $T\cap C=\varnothing$, consider $F_T=\bigcup_{y\in T} F_y$ and $F_T'=F_a'\cup F_T$. These two are different as $x\notin F_T, x\in F_T'$. For the second case, there is always an $F$ that $F\cap S=T$ with $x\in F$, and we will find another which $x\notin F$. We first show that for any $b,c\in C$, $S+x-b-c$ is \textit{not} $2$-good. Suppose conversely that $S+x-b-c$ is $2$-good. If $\|S\|=4$, then it means that there is a $2$-good set with size $3$, which implies $f_2(\mathcal{F})>\frac 13$. Hence we consider $\|S\|=5$. Notice that now any set $F\in \mathcal{F}$ that $F\cap S=\{b,c\}$ also contains $x$. By the same counting method of proving \cref{lemma:largeway}, $x$ lies in $q_b+q_c+q_{bc}+28-3$ sets. Consider the LP $L_0$ along with $q_b+q_c+q_{bc}+28-3\leq \frac{m}{3}$, we get $m\geq 129$, contradicting to the assumption. For any $2$ elements $b,c\in C$, since $S+x-b-c$ is not $2$-good, there is a set $G_{bc}\neq \varnothing,\{1\}$ that does not intersect $S+x-b-c$. However, since $S+x-b$ and $S+x-c$ are $2$-good, both $G_{bc}\cap (S+x-b)$ and $G_{bc}\cap (S+x-c)$ are nonempty. This implies $G_{bc}\cap (S+x)=bc$. Now for $T$ that contains $2$ or more elements in $C$, we consider $F_T'$ to be the union of \begin{itemize} \item $G_{bc}$ for each pair $b,c$ of elements in $T\cap C$, and \item $F_y$ for $y\in T\setminus C$ with $x\notin F_y$. \end{itemize} Then $F'_T\cap S=T$ and $x\notin F_T$, as desired. \end{proof} Lastly, we prove \cref{lemma:middleway}. \begin{proof}[proof of \cref{lemma:middleway}] We first show that there are $2^{\|S\|-2}$ sets in $\mathcal{F}$ containing $b$ but not $x$. As $x$ only covers $b$, we may choose $F_y$ such that $x\in F_a, x\in F_b$ and $x\notin F_y$ for $y\neq a,b$. Then there are $2^{\|S\|-2}$ resulting $F_T$ in $\mathcal{F}$ that contains $x$ but not $b$, namely those with $a\in T, b\notin T$. Note that $S+x-b$ is a $2$-good set and is minimal. Indeed, for each $y\neq b$ in $S$, $S+x-y$ is not $2$-good, thus $S+x-b-y$ is not $2$-good. (Also, $S+x-b-x=S-b$ is not $2$-good.) As $S$ maximises the incidence among minimal $2$-good $s$-sets, $S+x-b$ has no larger incidence than $S$. Thus, the frequency of $b$ is at least the one of $x$. That is, there are at least $2^{\|S\|-2}$ sets which contains $b$ but not $x$. In the following we will show that in these $2^{\|S\|-2}$ sets which contains $b$ but not $x$, \begin{itemize} \item $2^{\|S\|-2}$ of them intersect $S$ at $\geq 2$ elements, \item $2^{\|S\|-2}-1$ of them intersect $S$ at $\geq 3$, and \item $2^{\|S\|-3}-1$ of them intersect $S$ at $\geq 4$. \end{itemize} Let us call the other $\|S\|-2$ elements $c,d$ and possibly $e$ if $s=5$. for the choice of $\{F_y\}$, we choose $F_y$ such that $x\notin F_y$ for each $y\neq b$. Among these $2^{\|S\|-2}$ sets, we pick one $F$ with minimised intersection size of intersection with $S$. \begin{itemize} \item If $\|F\cap S\|=1$, then $F\cap S=\{b\}$ but $F$ does not contain $x$. It contradicts that $b$ is covered. \item If $\|F\cap S\|=2$, suppose that $F\cap S = ab$ (the case for $bc,bd,be$ are similar). We consider the union \[G_T:= F\cup \bigcup_{y\in T} F_y \in \mathcal{F}\] for any $T$ that does not contain $a$ nor $b$. These $2^{\|S\|-2}$ sets contain $b$ but not $x$. Also, \begin{itemize} \item $2^{\|S\|-2}$ of them intersect $S$ at $\geq 2$ elements, \item $2^{\|S\|-2}-1$ of them intersect $S$ at $\geq 3$ elements, and \item $2^{\|S\|-2}-1-(\|S\|-2) \geq 2^{\|S\|-3}-1$ of them intersect $S$ at $\geq 4$ elements. \end{itemize} Thus, they meet the conditions. \footnote{For the inequality, we used the fact that $2^{\|S\|-2}-1-(\|S\|-2) \geq 2^{\|S\|-3}-1$ when $\|S\|\geq 4$.} \item $\|F\cap S\|=3$: Then all these $2^{\|S\|-2}$ sets intersect $S$ at $\geq 3$ elements, satisfying the first two requirements. Assume that $F\cap S= abc$ (the cases for $abd, bcd, abe, bce, bde$ are similar). We consider the union \[F_T':=F\cup \bigcup_{y\in T} F_y \in \mathcal{F}\] for any $T$ does not contain $a,b,c$. These $2^{\|S\|-3}$ sets contain $b$ but not $x$, and $2^{\|S\|-3}-1$ of them intersect $S$ at $\geq 4$ elements. Thus, they meet the conditions. \item $\|F\cap S\|\geq 4$: Then all $2^{\|S\|-2}$ sets meet $S$ at $\geq 4$ elements, satisfying these three conditions. \end{itemize} \end{proof} \section{$s=4$}\label{sec:s=4} Now we assume that $S=\{a,b,c,d\}$ and $x,ax\in \mathcal{F}_{S+x}$. \subsection{When $\|C\|=0$} From \cref{lemma:smallway} we have \[q_T\geq 2 \text{ for any } a\in T\subseteq S.\] Solving the LP $L_0$ along with this additional constraint gives $m\geq 81$. \subsection{When $\|C\|=1$} Assume that $b$ is covered. From \cref{lemma:smallway} we have \[q_T\geq 2 \text{ for } T =a, ac,ad,acd.\] From \cref{lemma:middleway} we have the following inequalities: \begin{corollary}\mbox{} \begin{equation} \begin{aligned} \sum_{b\in T\subseteq S, \|T\|\geq 2} q_T &\geq 7+4,\\ \sum_{b\in T\subseteq S, \|T\|\geq 3} q_T &\geq 4+3, \text{ and }\\ \sum_{b\in T\subseteq S, \|T\|\geq 4} q_T &\geq 1+1. \end{aligned} \end{equation} \begin{proof} We count $F\in \mathcal{F}$ for those that contain both $b$ and $x$ and those that contain only $b$ but no $x$. By choosing $\{F_y\}$ such that $x\in F_b$ and $x\notin F_y$ for other $y\neq b$, we find that there are $7,4,1$ $F_T$'s with $\|F_T\cap S\|\geq 2,3,4$, respectively, each of which contains both $b$ and $x$. From \cref{lemma:middleway}, there are $4,3,1$ $F$'s with $\|F\cap S\|\geq 2,3,4$, respectively, each of which contains $b$ but not $x$. \end{proof} \end{corollary} Solving $L_0$ with these $4+3$ extra constraints gives $m\geq 81$. \subsection{When $\|C\|=2$} Assume that $b,c$ are covered. From \cref{lemma:smallway} we get \[q_T \geq 2 \text{ for } T=a, ad, bc, bcd, abc, abcd.\] From \cref{lemma:largeway} we get $12 + q_b+q_c \leq \frac{m}{3}$. Solving $L_0$ with these two additional constraints gives $m\geq 114$. \subsection{When $\|C\|\geq 3$} Assume that $b,c,d$ are covered. From \cref{lemma:largeway} we get $12 + q_b+q_c+q_d \leq \frac m3$. The linear program $L_0$ with this constraint is infeasible. \section{$s=5$}\label{sec:s=5} Now we assume that $S=\{a,b,c,d,e\}$ and $x,ax\in \mathcal{F}_{S+x}$. \subsection{When $\|C\|=0$} From \cref{lemma:smallway} we get \[q_T\geq 2 \text{ for any } a\in T\subseteq S.\] Solving the LP $L_0$ along with this additional constraint gives $m\geq 118.5$. \subsection{When $\|C\|=1$} We assume $b\in C$. From \cref{lemma:smallway} we get \[q_T\geq 2 \text{ for any } T\subseteq S \text{ such that }a\in T,b\notin T.\] By \cref{lemma:middleway} we have the following inequalities: \begin{corollary} \begin{equation} \begin{aligned} \sum_{b\in T\subseteq S, \|T\|\geq 2} q_T &\geq 15+8,\\ \sum_{b\in T\subseteq S, \|T\|\geq 3} q_T &\geq 11+7, \text{ and }\\ \sum_{b\in T\subseteq S, \|T\|\geq 4} q_T &\geq 5+3. \end{aligned} \end{equation} \begin{proof} We count $F\in \mathcal{F}$ for those that contain both $b$ and $x$ and those that contain only $b$ but no $x$. By choosing $\{F_y\}$ such that $x\in F_b$ and $x\notin F_y$ for other $y\neq b$, we find that there are $15,11,5$ $F_T$'s with $\|F_T\cap S\|\geq 2,3,4$, respectively, each of which contains both $b$ and $x$. From \cref{lemma:middleway}, there are $8,7,3$ $F$'s with $\|F\cap S\|\geq 2,3,4$, respectively, that contains $b$ but not $x$. \end{proof} \end{corollary} Consider $L_0$ with these $8+3$ extra constraints, and we get $m \geq 115.5$. \subsection{When $\|C\|=2$} We assume $b,c\in C$. From \cref{lemma:smallway} we get \[q_T\geq 2 \text{ for } T=a, ad, ae, ade, bc, bcd, bce, bcde, abc, abcd, abce, abcde.\] From \cref{lemma:largeway} we get $26 + q_b+q_c \leq \frac{m}{3}$. Solving $L_0$ with these two types of additional constraints gives $m\geq 122$. \subsection{When $\|C\|\geq 3$} Assume $b,c,d\in C$ are covered. From \cref{lemma:largeway} we get $30 + q_b+q_c+q_d-3 \leq \frac{m}{3}$ or $30 + q_b+q_c+q_d+q_e-4 \leq \frac{m}{3}$. Either we get $30 + q_b+q_c+q_d-3 \leq \frac{m}{3}$. Solving $L_0$ with these additional constraints gives us $m\geq 114$. In conclusion, the case $s=5$ is completely solved. \section{Concluding remark} In this paper, we further investigated the potential counterexample for \cref{conj:Nagel} and narrowed down the searching range of their sizes from $[45,113]$ to $[81,113]$ and showed that there is some uniformity to the collection of minimal $2$-good sets. To further improve the bound, a possible approach is to pick $x$ more carefully. In our proof, we can see that there are many choices for extra elements. With such many choices, one may possibly push the bound further. Another perspective is to investigate the uniformity of minimal $2$-good sets of the family. To do this, one could answer this question: \begin{question} What can we say about the minimal $2$-good sets if they all have the same size? \end{question} To first ignore the asymmetry induced by the most common element $1$, we consider a definition slightly different from $k$-good sets. \begin{definition} Let $\mathcal{F}$ be a family, a \textbf{cover} $S$ of $\mathcal{F}$ is a set that intersects every set in $\mathcal{F}$. A \textbf{minimal cover} is a cover whose proper subsets are not covers. Let $\mathcal{MC}(\mathcal{F})$ denote the family of minimal covers of $\mathcal{F}$. \end{definition} Equivalently, a cover is a vertex cover when we see $\mathcal{F}$ as the edge set of a hypergraph on $n$ vertices. In this case, we can characterize when $\mathcal{MC}(F)$ is uniform: \begin{theorem} Let $\mathcal{F}$ be a family, not necessarily be union-closed. \begin{enumerate} \item $\mathcal{MC}(\mathcal{F})$ is an antichain. In other words, any two elements in $\mathcal{MC}(\mathcal{F})$ are incomparable. \item If $\mathcal{F}$ is an antichain, then $\mathcal{MC}(\mathcal{MC}(\F))=\F$. \item Subsequently, let $\mathcal{G}$ be the subfamily of minimal elements in $\mathcal{F}$, then $\mathcal{MC}(\F)$ is $k$-uniform if and only if $\mathcal{G}$ is the minimal covers of a $k$-uniform hypergraph. \end{enumerate} \end{theorem} \begin{proof} We only have to prove the second claim. \begin{description} \item[$\F\subseteq \mathcal{MC}(\mathcal{MC}(\F))$:] For $F\in \F$, it is a cover of $\mathcal{MC}(\F)$ since every $C\in \mathcal{MC}(\F)$ intersects $F$. To show that $F\setminus\{f\}$ is not a cover of $\mathcal{MC}(\F)$ for each element $f\in F$, we need to find a minimal cover $C\in \mathcal{MC}(\F)$ contained in $F^{\complement}\cup\{f\}$. Since $\F$ is an antichain, $F^{\complement}\cup\{f\}$ is indeed a cover and therefore such $C$ exists. \item[$\mathcal{MC}(\mathcal{MC}(\F))\subseteq \F$:] For $C$ a minimal cover of $\mathcal{MC}(F)$, we show that it covers a set in $\F$. If this is true, $C$ must in $\mathcal{F}$ as sets in $\mathcal{F}$ are covers of $\mathcal{MC}(\F)$. Were it not, then for each $F\in \F$, there is an element $a_F\in F$ that $C$ does not have. Notice that $\{a_F:F\}$ is a cover of $\F$, it contains a minimal cover of $\F$. It lies in $\mathcal{MC}(\F)$ but does not intersect $C$, giving a contradiction. \end{description} \end{proof} We hope to understand the potential counterexamples for \cref{conj:Nagel} when $k=2$ better by understanding the families whose minimal $2$-good sets are uniform. \bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} MR } \providecommand{\MRhref}[2]{ \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{1} \bibitem{our_main_paper} Shagnik Das and Saintan Wu, \emph{Frequent elements in union-closed set families}, 2024, \arXiv{2412.03862}. \bibitem{nagel2023notes} Nicolas Nagel, \emph{Notes on the union closed sets conjecture}, 2023, \arXiv{2208.03803}. \end{thebibliography} \end{document}
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\newcommand{\move}{{\mathfrak m}} \newcommand{\id}{\mathrm{id}} \newcommand{\qnt}{{\mathfrak q}} \newcommand{\lightcone}{{\mathbf L}} \newcommand{\trace}{\text{trace}} \title[$\SLE_\kappa(\rho)$ processes in the light cone regime on Liouville quantum gravity]{$\SLE_{\kappa}(\rho)$ processes in the light cone regime on\\ Liouville quantum gravity} \author{Konstantinos Kavvadias and Jason Miller} \begin{document} \begin{abstract} We study the relationship between certain $\SLE_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution with parameter $\kappa$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $\rho > -2-\kappa/2$ and in this work we will focus on the light cone regime, meaning that $\kappa \in (0,4)$ and $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$. Such processes are self-intersecting even though ordinary $\SLE_\kappa$ curves are simple for $\kappa \in (0,4)$. We show that such a process drawn on top of an independent $\sqrt{\kappa}$-LQG surface called a weight $(\rho+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $\alpha$-stable L\'evy process with $\alpha = 1-2(\rho+2)/\kappa$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an $\SLE_\kappa(\kappa-4)$ curve on an independent $\sqrt{\kappa}$-LQG surface for $\kappa \in (4/3,2)$. \end{abstract} \date{\today} \maketitle \setcounter{tocdepth}{1} \tableofcontents \parindent 0 pt \setlength{\parskip}{0.20cm plus1mm minus1mm} \section{Introduction} \label{sec:introduction} \subsection{Overview and setting} The purpose of this work is to study the connection between Liouville quantum gravity (LQG) surfaces and the $\SLE_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution ($\SLE_\kappa$) \cite{schramm2000scaling} in which one keeps track of an extra marked point \cite[Section~8.3]{lsw2003restriction}. These processes are defined whenever $\rho > -2-\kappa/2$ and we will focus on the so-called light cone regime \cite{ms2019lightcone} which means that $\kappa \in (0,4)$ and $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$. These processes exhibit rather different behavior from ordinary $\SLE_\kappa$. For example, they are self-intersecting even though $\kappa \in (0,4)$. We will begin by giving a brief overview of the objects which will be relevant for this work. Recall that an LQG surface is a random two-dimensional Riemannian manifold which is formally described by the metric tensor \begin{equation} \label{eqn:lqg_form} e^{\gamma h(z)} (dx^2 + dy^2) \end{equation} where $z=x+iy$, $dx^2 + dy^2$ is the Euclidean metric on a domain $D \subseteq \C$, $h$ is (some form of) the Gaussian free field (GFF) on $D$, and $\gamma \in (0,2]$ is a parameter. Since the GFF $h$ and its variants are random variables which live in the space of distributions rather than in a space of functions, some care is required in order to make sense of~\eqref{eqn:lqg_form}. The volume form $\qmeasure{h}$ associated with~\eqref{eqn:lqg_form} was constructed in \cite{ds2011lqgkpz} as the limit as $\epsilon \to 0$ of \begin{equation} \label{eqn:volume_limit} \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)} dx dy \end{equation} where $dx dy$ denotes Lebesgue measure and $h_\epsilon(z)$ denotes the average of $h$ on the circle $\partial B(z,\epsilon)$. The boundary length measure $\qbmeasure{h}$ on a linear segment $L$ was also constructed in \cite{ds2011lqgkpz} as the limit as $\epsilon \to 0$ of \begin{equation} \label{eqn:boundary_limit} \epsilon^{\gamma^2/4} e^{\gamma h_\epsilon(x)/2} dx \end{equation} where $dx$ denotes Lebesgue measure on $L$ and $h_\epsilon(x)$ denotes the average of $h$ on the semi-circle $D \cap \partial B(x,\epsilon)$. The measures $\qmeasure{h}$, $\qbmeasure{h}$ can also be understood as Gaussian multiplicative chaos measures in the framework developed by Kahane \cite{k1985gmc}. The metric (two-point distance function) associated with~\eqref{eqn:lqg_form} was later constructed in the case $\gamma=\sqrt{8/3}$ in \cite{ms2020qle1,ms2021qle2,ms2021qle3} and in \cite{dddf2020tightness,gm2021metric} in the case that $\gamma \in (0,2)$ where the latter works are based on a regularization procedure analogous to~\eqref{eqn:volume_limit} while the former takes a more indirect approach. LQG surfaces are important in statistical mechanics, string theory, and conformal field theory. The regularization procedure~\eqref{eqn:volume_limit}, \eqref{eqn:boundary_limit} implies that the following is true. Suppose that $h$ is an instance of (some form of) the GFF on $D$, $\varphi \colon \wt{D} \to D$ is a conformal map, and \begin{align} \label{eqn:change_of_coordinates} \wt{h} = h \circ \varphi + Q \log \|\varphi'\| \quad\text{where}\quad Q = \frac{2}{\gamma} + \frac{\gamma}{2}. \end{align} Then $\qmeasure{h}(\varphi(A)) = \qmeasure{\wt{h}}(A)$ for all $A \subseteq \wt{D}$ Borel. We say that two pairs $(D,h)$, $(\wt{D},\wt{h})$ consisting of a planar domain and a field are equivalent as quantum surfaces if they are related as in~\eqref{eqn:change_of_coordinates} and a \emph{quantum surface} is an equivalence class under the equivalence relation defined by~\eqref{eqn:change_of_coordinates}. This definition extends to quantum surfaces which have extra marked points where one requires the conformal map $\varphi$ to take the marked points associated with $\wt{h}$ to those associated with $h$. There has been a substantial amount of work in recent years focused on developing connections between LQG and $\SLE$ and its variants. This started with \cite{she2016zipper} in which it was shown that it is possible to conformally weld together two independent copies $\CW_1,\CW_2$ of a certain type of quantum surface called a (weight $2$) quantum wedge and obtain a quantum wedge $\CW$ (but with weight $4$) where the welding interface $\eta$ is given by an $\SLE_\kappa$ curve which is independent of $\CW$. Roughly, a quantum wedge is an infinite volume surface with two marked points (the ``origin'' and the ``infinity'' point) and two boundary rays which have locally finite quantum length. Here, it is important that the parameters $\kappa$ and $\gamma$ are matched by $\kappa = \gamma^2 \in (0,4)$. A number of other welding results of this type were proved in \cite{dms2014mating} with different types of quantum wedges and depending on the setting one obtains as the welding interface an $\SLE_\kappa(\rho_1;\rho_2)$ process with $\rho_1, \rho_2 > -2$. Recall that the $\SLE_\kappa(\rho)$ processes are variants of $\SLE_\kappa$ in which one keeps track of extra marked points \cite[Section~8]{lsw2003restriction} called force points and the $\rho$'s are the weights, which determine how the force points affect the behavior of the curve. By $\SLE_\kappa(\rho_1;\rho_2)$ we typically mean the case where there is a force point of weight $\rho_1$ (resp.\ $\rho_2$) located at $0_-$ (resp.\ $0_+$). These processes can be boundary intersecting even though ordinary $\SLE_\kappa$ for $\kappa \in (0,4)$ does not intersect the boundary. The threshold $-2$ is important because when $\rho > -2$ an $\SLE_\kappa(\rho)$ process is absolutely continuous with respect to an ordinary $\SLE_\kappa$ process when it is away from the boundary while this is not the case for $\rho \leq -2$. It is also shown in \cite{dms2014mating} that if one welds the two boundary rays of a quantum wedge to each other then one obtains a quantum cone, which is a surface homeomorphic to $\C$, decorated by an independent whole-plane $\SLE_\kappa(\rho)$ process. This ultimately leads to the proof that one can explore and encode an LQG surface by an appropriate space-filling version of $\SLE_{\kappa'}$ for $\kappa' > 4$ (throughout we will use the imaginary geometry \cite{ms2016imag1} convention that $\kappa \in (0,4)$, $\kappa'=16/\kappa > 4$) which can be thought of as the Peano curve which traces the interface between a pair of continuum random trees (CRTs) $\CT_1$, $\CT_2$ drawn in the plane and whose branches are themselves whole-plane $\SLE_\kappa(\rho)$ processes (with $\rho=2-\kappa$) \cite{ms2017imag4}. This yields the so-called ``mating of trees'' representation of LQG, whose importance is that it provides one avenue for making connections between LQG, $\SLE$, and random planar maps ($\RPM$s). The reason for this is that a number of $\RPM$ models can be encoded using so-called tree bijections and the mating of trees representation of LQG can be thought of as a continuous analog of such a bijection. In particular, there have been a number of scaling limit results for $\RPM$s towards $\SLE$ on LQG in which it is shown that the contour functions which encode a pair of discrete trees used to construct a planar map using a tree bijection converge in the scaling limit to the pair of contour functions which encode the continuous trees associated with space-filling $\SLE_{\kappa'}$ on LQG. Such scaling limit results are referred to as being of ``peanosphere'' type and we will discuss this point in more detail later on. As we mentioned earlier, the $\SLE_\kappa(\rho)$ processes with $\rho < -2$ exhibit a rather different character from the $\SLE_\kappa(\rho)$ processes with $\rho > -2$ (we exclude the case $\rho = -2$ because it is in a certain sense degenerate). In particular, they are self-intersecting for $\kappa \in (0,4)$ even though the ordinary $\SLE_\kappa$ processes are not. There are two regimes of $\rho$ values in which these processes are defined: \begin{enumerate}[(i)] \item The loop-making regime: $-2-\kappa/2 < \rho \leq \kappa/2-4$. \item The light cone regime: $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$. \end{enumerate} The basic properties of the $\SLE_\kappa(\rho)$ processes in the loop-making regime were studied in \cite{msw2017clepercolations} and their relationship with LQG in \cite{msw2020simplecle}. The particular case $\rho=\kappa-6$ in this regime is of special significance because it corresponds to the conformal loop ensembles ($\CLE$) \cite{s2009cle,sw2012cle}. This case is thus motivated because it conjecturally corresponds to the scaling limit of $\RPM$ models decorated by a statistical mechanics model such as the Ising model. In the light cone regime, the basic properties were studied in \cite{ms2019lightcone} and the purpose of the present paper is to study the relationship between these processes and LQG. One application of this regime is that it makes it possible to connect bipolar oriented random planar maps with large faces to LQG \cite{km2022bplargefaces} using a ``peanosphere'' type scaling limit as was briefly described above. \subsection{Main results} \label{subsec:main_results} We now turn to state our main results, which are stated in terms of quantum wedges and $\SLE_\kappa(\rho)$ processes in the light cone regime. We will give the precise definition of the former in Section~\ref{subsec:lqg} and the latter in Section~\ref{subsec:slekapparho}. The theorem statements that we will give concern an $\SLE_\kappa(\rho)$ process $\eta$ with $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$ drawn on top of an independent quantum wedge $\CW = (\h,h,0,\infty)$ and that the quantum surfaces which are parameterized by the components of $\h \setminus \eta$ together form a quantum wedge. We emphasize that $\CW$ will be homeomorphic to $\h$ while the quantum wedge corresponding to $\h \setminus \eta$ is not homeomorphic to $\h$. As we will describe in further detail in Section~\ref{subsec:lqg}, the starting point for the construction of this latter surface is a Bessel process $Y$ with dimension in $(0,1)$ and therefore has a well-defined local time $\ell$ for the times that it hits $0$. \begin{theorem} \label{thm:quantum_natural_time_cutting} Fix $\kappa \in (0,4) , \rho \in (\kappa/2 - 4, -2) \cap (-2 -\kappa/2 , -2)$, and suppose that $\CW = (\h,h,0,\infty)$ is a quantum wedge of weight $\rho + 4$. Let $\eta$ be an $\SLE_{\kappa}(\rho)$ process in $\h$ from $0$ to $\infty$ with a single force point of weight $\rho$ located at $0_+$ and assume that $\eta$ is independent of $h$. Then the following hold: \begin{enumerate}[(i)] \item The law of the beaded surface consisting of the components of $\h \setminus \eta$ which are to the right of $\eta$ is that of a quantum wedge of weight $\rho + 2$. \item Let $\qnt_{u}$ be the first capacity time $t$ for $\eta$ that the local time at $0$ of the Bessel process $Y$ which encodes the weight $\rho + 2$ wedge is equal to $u$. For each $u > 0$, we have (as path-decorated quantum surfaces) that $(h,\eta)$ and $(h \circ f_{\qnt_{u}}^{-1} + Q\log\|(f_{\qnt_{u}}^{-1})'\|,f_{\qnt_{u}}(\eta))$ have the same law, where $(f_t)$ is the forward centered Loewner flow corresponding to $\eta$. \end{enumerate} \end{theorem} \begin{definition} \label{def:quantum_natural_time} We call $\qnt_u$ the quantum natural time parameterization of the bubbles which are cut off by $\eta$ from $\infty$. \end{definition} In \cite{dms2014mating}, a similar definition of quantum natural time for $\SLE_{\kappa'}$ processes with $\kappa' \in (4,8)$ is given. In contrast to the case of $\SLE_{\kappa'}$ processes, an $\SLE_\kappa(\rho)$ process as in Theorem~\ref{thm:quantum_natural_time_cutting} parameterized by quantum natural time is not continuous, the reason being that as the process draws the boundary of a large component it does not discover any small components. \begin{figure}[ht!] \begin{center} \includegraphics[scale=0.85]{figures/boundary_length_evolution.pdf} \end{center} \caption{\label{fig:boundary_length_evolution} Shown is an $\SLE_\kappa(\rho)$ process in $\h$ from $0$ to $\infty$ with a single force point located at $0_+$ in the light cone regime and drawn up to a typical quantum natural time $\qnt_u$. The arc which disconnects each component from $\infty$ is drawn in its entirety before $\eta$ ``creeps'' up it disconnects further disks. Its left boundary length $L_u$ is equal to the quantum length of the dark blue arc (counterclockwise from $\eta(\qnt_u)$ to $A_u$) minus the quantum length of the light blue arc ($[A_u,0]$). Similarly, its right boundary length $R_u$ is equal to the quantum length of the dark green arc (clockwise from $\eta(\qnt_u)$ to $B_u$) minus the quantum length of the light green arc ($[0,B_u]$). We prove in Theorem~\ref{thm:boundary_length_evolution} that $(L,R)$ evolves as an $\alpha$-stable L\'evy process where $\alpha$ is determined by $\rho$ as in~\eqref{eqn:alpha_value}.} \end{figure} Theorem~\ref{thm:quantum_natural_time_cutting} allows us to describe the law of the boundary length evolution of $\eta$ when it has the quantum natural time parameterization. More precisely, for each $u > 0$ we let $A_u$ (resp.\ $B_u$) be the leftmost (resp.\ rightmost) point of intersection of $\eta([0,\qnt_u])$ with $\R_-$ (resp.\ $\R_+$). Let also $L_u$ be the difference between the quantum length of the segment of the outer boundary of $\eta([0,\qnt_u])$ connecting $\eta(\qnt_u)$ and $A_u$ with the quantum length of $[A_u,0]$. Similarly, let $R_u$ be the difference between the quantum length of the segment of the outer boundary of $\eta([0,\qnt_u])$ connecting $\eta(\qnt_u)$ and $B_u$ with the quantum length of $[0,B_u]$. Then the following theorem describes the evolution of $(L,R)$. Recall that $(L,R)$ is an $\alpha$-stable L\'evy process if it has stationary and independent increments and for each $c > 0$ we have that $(L_u,R_u)_{u \geq 0}$ and $(c^{-1/\alpha}L_{c u},c^{-1/\alpha}R_{c u})_{u \geq 0}$ have the same law. \begin{theorem} \label{thm:boundary_length_evolution} Suppose that we have the setup described in the statement of Theorem~\ref{thm:quantum_natural_time_cutting} and the process $(L,R)$ is as described just above. Then $(L,R)$ is an $\alpha$-stable L\'evy process with \begin{align}\label{eqn:alpha_value} \alpha = 1 - \frac{2(\rho+2)}{\kappa} \in (1,2). \end{align} Moreover, $L$ (resp.\ $R$) has only upward (resp.\ downward) jumps and the jump times of $L$ coincide with the jump times of $R$. In the case that $\kappa \in (\frac{4}{3},2)$ and $\rho = \kappa - 4$ the sequence of jumps of $(L,-R)$ forms a Poisson point process (p.p.p.) $\Lambda^$ whose law can be sampled from as follows. \begin{itemize} \item Firstly, we sample a p.p.p.\ $\Lambda = \{(s_j,t_j) : j \in \N \}$ on $\R_+ \times \R_+$ with intensity measure given by $c(du \times t^{-4/\kappa}dt)$, where $c > 0$ is a constant depending only on $\kappa$. \item Next, we sample independently a sequence of i.i.d.\ random variables $(u_j)_{j \in \N}$ which are uniform on $[0,1]$ and consider \begin{align} \Lambda^* = \{(t_j u_j, (1-u_j)t_j) : (s_j,t_j) \in \Lambda, j \in \N\}. \end{align} \end{itemize} \end{theorem} Note that by \cite{bertoin1996levy}, the law of a two-dimensional $\alpha$-stable L\'evy process is determined by the law of its jumps. Hence when $\rho = \kappa - 4$, Theorem~\ref{thm:boundary_length_evolution} gives a complete description of the law of the process $(L,R)$. Moreover, combining Theorem~\ref{thm:boundary_length_evolution} with the one-dimensional $\text{KPZ}$ formula for quantum boundary length, we obtain the following formula for the Hausdorff dimension of $\eta \cap \R_+$. \begin{corollary} \label{cor:dim_of_boundary_intersection} Fix $\kappa \in (0,4), \rho \in (\kappa/2-4,-2) \cap (-2 - \kappa/2,-2)$ and let $\eta$ be an $\SLE_{\kappa}(\rho)$ process in $\h$ from $0$ to $\infty$ with its force point located at $0_+$. Then, a.s.\ the Hausdorff dimension of the set $\eta \cap \R_+$ is given by \begin{equation} \label{eqn:dim_of_intersection_with_R_+} -\frac{(2+\rho)(\kappa + 8 + 2\rho)}{2\kappa}. \end{equation} \end{corollary} We note that the analog of Corollary~\ref{cor:dim_of_boundary_intersection} for $\rho > -2$ was given in \cite{mw2017slepaths} and for $-2-\kappa/2 < \rho \leq \kappa/2-4$ follows from the loop-trunk decomposition of such processes given in \cite{msw2017clepercolations} combined with the $\rho > -2$ case given in \cite{mw2017slepaths}. We note that for $\rho = \kappa/2-4$ the value from~\eqref{eqn:dim_of_intersection_with_R_+} is equal to $2-8/\kappa'$ where $\kappa'=16/\kappa$. This, in turn, is equal to the dimension of the intersection of an $\SLE_{\kappa'}$ process in $\h$ with $\partial \h$ determined in \cite{as2011covariant}. This is not a coincidence as it was shown in \cite{ms2019lightcone} that the law of the range of an $\SLE_\kappa(\kappa/2-4)$ process agrees with that of an $\SLE_{\kappa'}(\kappa'/2-4)$ process (but the corresponding curves visit the points in their range in a different order). \subsection{Mating of trees interpretation} \label{subsec:interpretation} We will now describe how Theorems~\ref{thm:quantum_natural_time_cutting} and~\ref{thm:boundary_length_evolution} have the interpretation of giving a mating of trees representation of $\SLE_\kappa(\rho)$ processes in the light cone regime drawn on top of LQG, how it contrasts with the mating of trees representations given in \cite{dms2014mating}, and how this is related to the scaling limits of certain types of $\RPM$s. \begin{figure}[ht!] \begin{center} \includegraphics[scale=0.85]{figures/mot_representation.pdf} \end{center} \caption{\label{fig:mot_representation} Illustration of the ``mating of trees representation'', which describes the topology of an $\SLE_\kappa(\rho)$ process $\eta$ drawn on top of an independent quantum wedge of weight $\rho+4$. {\bf Left:} Points on the graph of $\wh{R}$ are equivalent if they can be connected by a horizontal line which is below the graph of $\wh{R}$ and points on the graph of $3-\wh{L}$ are equivalent if they can be connected by a horizontal line which is above the graph of $3-\wh{L}$. Points on the graphs of $\wh{R}$ and $3-\wh{L}$ are equivalent if they can be connected by a vertical line which does not correspond to a jump time of $(L,R)$. The jump times $(L,R)$ are the yellow regions and in the quotient correspond to doubly marked disks which are cut out by $\eta$. The opening (resp.\ closing) point of such a disk $D$ is given by the projection of the left (resp.\ right) vertical boundary of the corresponding yellow region $Y$ and the left (resp.\ right) boundary of $D$ is given by the projection of the part of $\partial Y$ which is on the graph of $3-\wh{L}$ (resp.\ $\wh{R}$). {\bf Right:} The time $\tau$ is a jump time for $L$, $R$, which corresponds to a time at which $\eta$ disconnects a topological disk from $\infty$ (shown in yellow); $K_{\tau-}$ is the hull cut out by $\eta$ up to just before time $\tau$. The opening point of the disk is $\eta(\tau^-)$ (blue) and the closing point is $\eta(\tau)$ (red). At the time $\tau$, $R$ (resp.\ $L$) makes a downward (resp.\ upward) jump of size equal to the quantum length counterclockwise (resp.\ clockwise) arc of the disk boundary from the opening point to the closing point.} \end{figure} Suppose that $(L,R)$ is an $\alpha$-stable L\'evy process with law as in Theorem~\ref{thm:boundary_length_evolution} so that $L$ (resp.\ $R$) has only upward (resp.\ downward) jumps and that the jump times of $L$ and $R$ coincide. Let $\wt{L}$ (resp.\ $\wt{R}$) be the continuous process obtained by starting with $L$ (resp.\ $R$) and then replacing each jump with a linear segment whose length is equal to square of the jump made by $L-R$ at this time. Since the sum of the squares of the jumps of an $\alpha$-stable L\'evy process with $\alpha \in (1,2)$ on any compact time interval is finite, it follows that $(\wt{L},\wt{R})$ is well-defined. Let $\phi$ be a homeomorphism from $\R$ to $(-1,1)$ which takes $-\infty$ (resp.\ $\infty$) to $-1$ (resp.\ $1$) and fixes $0$. Then the processes $\wh{L}_u = \phi(\wt{L}_{\phi^{-1}(u)})$, $\wh{R}_u = \phi(\wt{R}_{\phi^{-1}(u)})$ are defined on $[0,1)$ and take values in $(-1,1)$. Let $\CI$ be the image under $\phi$ of those intervals where $\wt{L}$, $\wt{R}$ are linear. Suppose that we draw the graph of $\wh{R}$ and the graph of $3-\wh{L}$ in the rectangle $\CR = [0,1] \times [-1,4]$. Let $\CA$ consist of those vertical lines $[(u, \wh{R}_u), (u, 3-\wh{L}_u)]$ where $u \in \CI$. We consider the finest equivalence relation $\sim$ on $\CR$ such that points in $\CR \setminus \CA$ which can be connected by: \begin{itemize} \item a horizontal line which lies below the graph of $\wh{R}$, or \item a horizontal line which lies above the graph of $3-\wh{L}$, or \item a vertical line which connects the graphs of $\wh{R}$ and $3-\wh{L}$ whose $x$-coordinate is not in $\CI$. \end{itemize} Arguing as in \cite{dms2014mating}, it is possible to check using Moore's theorem that the topological space $\CR / \sim$ is homeomorphic to $\h$. Moreover, one can consider the continuous curve $\eta$ which is given by the projection of the curve $u \mapsto 3 - \wh{L}_u$ under the quotient map. Then $\eta$ is homeomorphic to an $\SLE_\kappa(\rho)$ process where $\rho$ is determined by $\alpha$ as in~\eqref{eqn:alpha_value}. If $I$ is a connected component of $\CI$ (i.e., $I = (a,b)$ for some $-1 < a < b < 1$), then the topological disk given by the union of the vertical segments $[(u, \wh{R}_u), (u, 3-\wh{L}_u)]$ for $u \in I$ corresponds to a component which is disconnected from $\infty$ by $\eta$. Recall that a planar map is a graph $G = (V,E)$ together with an embedding into the plane so that no two edges cross, considered up to orientation preserving homeomorphism. A $\RPM$ is a planar map chosen according to some probability measure. Examples of $\RPM$s include planar triangulations and quadrangulations chosen uniformly at random \cite{lg2013bm,m2013bm,lg2019survey,miermontstflour} and weighted by the partition function of various models from statistical mechanics \cite{s2016qginv,kmsw2019bipolar,lsw2017schnyder,gm2021saw,gm2017percolation,gkmw2018active}. One approach to study $\RPM$s is to encode them in terms of random trees and random walks via combinatorial bijections. An important class of such bijections are the so-called \emph{mating of trees bijections} which represent a planar map decorated by a statistical mechanics model as the gluing of a pair of discrete trees. Since discrete trees can be represented in terms of their contour function, such an encoding is equivalent to encoding the map by a two-dimensional walk. The construction of LQG surfaces using matings of trees \cite{dms2014mating} therefore leads to a natural topology on surfaces which is called the \emph{peanosphere topology}. In particular, a surface decorated by a tree can be encoded in terms of a pair of continuous functions $(L,R)$ where $L$ (resp.\ $R$) is given by the contour function of the tree (resp.\ dual tree) on the surface. Then if we have two tree-decorated surfaces with associated pairs of contour functions $(L,R)$ and $(L',R')$, we define the distance between the two surfaces to be the $L^\infty$ distance between $(L,R)$ and $(L',R')$. In other words, the peanosphere topology is the restriction of the $L^\infty$ metric to the space of continuous functions which arise as the contour functions related to tree-decorated surfaces. Since applying a rescaling to a planar map corresponds to applying a rescaling to the discrete pair of trees encoding the map, we say that a sequence of rescaled $\RPM$s converges in distribution to a tree-decorated random quantum surface as the number of vertices (or faces) of the map tends to infinity, if the rescaled random walks encoding the $\RPM$s converge in distribution with respect to the~$L^\infty$ metric to the pair of continuous functions $(L,R)$ encoding the tree-decorated quantum surface. It was shown in \cite{kmsw2019bipolar} that if we pick a bipolar oriented planar map uniformly at random with fixed boundary lengths and whose faces are all triangles then the corresponding rescaled random walk converges in the limit to a correlated two-dimensional Brownian motion. Using the construction related to the peanosphere topology, this result can be interpreted as being a scaling limit result towards the $\SLE$ with parameter $\kappa = 12$ on a $\sqrt{4/3}$-LQG surface. The case of random bipolar oriented planar maps ($\BPRPM$s) with \emph{large faces}, meaning that the law of the face degree is in the domain of attraction of an $\alpha$-stable random variable, are considered in \cite{km2022bplargefaces}. It is shown in \cite[Theorem 1.3]{km2022bplargefaces} that the contour functions for the pair of discrete trees which encode such a $\BPRPM$ converge in the scaling limit to an $\alpha$-stable L\'evy process with the same law as in Theorem~\ref{thm:boundary_length_evolution} in the case that $\rho = \kappa-4$ and $\alpha=4/\kappa-1$. This allows us to interpret $\SLE_\kappa(\kappa-4)$ drawn on top of a weight $\rho+4$ quantum wedge as the scaling limit of $\BPRPM$s with large faces in the peanosphere sense. \subsection{Outline} The remainder of this article is structured as follows. In Section~\ref{sec:preliminaries}, we will collect a number of preliminaries. Next, in Section~\ref{sec:slegff_couplings} we will collect some results on the $\SLE$/GFF coupling. Finally, we will complete the proofs of our main theorems in Section~\ref{sec:main_theorems_proof}. \subsection{Acknowledgements} K.K.'s work was supported by the EPSRC grant EP/L016516/1 for the University of Cambridge CDT (CCA) and by ERC starting grant 804166 (SPRS). J.M.'s work was supported by ERC starting grant 804166. \section{Preliminaries} \label{sec:preliminaries} The purpose of this section is collect a number of preliminaries which will be used in the remainder of this article. In Section~\ref{subsec:bessel}, we will give a brief review of Bessel processes. Next, in Section~\ref{subsec:gff} we will give a review of the GFF and in Section~\ref{subsec:lqg} of LQG. In Section~\ref{subsec:slekapparho} we will review the $\SLE_\kappa(\rho)$ processes and in Section~\ref{subsec:ig} the aspects of imaginary geometry which are relevant for this work. Finally, in Section~\ref{subsec:light_cones} we will review the $\SLE_\kappa(\rho)$ processes in the light cone regime in the imaginary geometry framework. \subsection{Bessel processes} \label{subsec:bessel} In this subsection, we recall a few facts about Bessel processes which will play an important role in this paper. We refer the reader to \cite[Chapter XI]{revuz2013continuous} for a more in depth overview of Bessel processes. Fix $\delta \in \R$ and $x \geq 0$. The squared $\delta$-dimensional Bessel process ($\BESQ^\delta$) starting from $x^2$ is given by the unique strong solution to the SDE \begin{equation}\label{eq:square_bessel_sde} dZ_t = \delta dt + 2\sqrt{Z_t} dB_t,\quad Z_0 = x^2 \end{equation} where $B$ is a standard Brownian motion. If we want to emphasize the starting point of a $\BESQ^\delta$ process we will write $\BESQ_{x^2}^\delta$. Standard results for SDEs imply that there is a unique strong solution to~\eqref{eq:square_bessel_sde} up until the first time that $Z$ hits $0$. When $\delta > 0$, there exists a unique strong solution defined for all times which always remains non-negative. Then for $\delta > 0$, the \emph{Bessel process} of dimension $\delta$ ($\BES^{\delta}$), is the process $X_t = \sqrt{Z_t}$, where $Z$ is the unique strong solution to~\eqref{eq:square_bessel_sde}. If we want to emphasize the starting point of a $\BES^\delta$ process we will write $\BES_x^\delta$. If $\delta \geq 2$, then a.s.\ $X_t > 0$ for all $t > 0$ while if $\delta \in (0,2)$, then $X_t$ a.s.\ assumes the value zero on a non-empty random set with zero Lebesgue measure. Also, for all $\delta > 0$, the process $X_t$ is invariant under Brownian scaling, i.e., for every given constant $c > 0$, the processes $(c^{-1/2}X_{ct})$ and $(X_t)$ have the same law. Moreover, when $\delta > 1$, the process $X_t$ is a semimartingale and a strong solution to the SDE \begin{equation} \label{eq:bessel_sde} d X_t = \frac{\delta-1}{2} \cdot \frac{1}{X_t} dt + d B_t, \quad X_0 = x \end{equation} and when $\delta =1$, the process $X_t$ is equal in distribution to $\|B\|$ and still a semimartingale. However, when $\delta \in (0,1]$, \eqref{eq:bessel_sde} holds up until the first time $t$ such that $X_t = 0$, but it does not hold for larger $t$ since the integral $\int_{0}^{t} X_s^{-1}ds$ is infinite a.s. In order to make sense of a solution to~\eqref{eq:bessel_sde} in integrated form for $\delta \in (0,1)$, we make a principal value correction. More precisely, $X$ satisfies the equation \begin{equation} \label{eq:principal_value_correction_equation} X_t = x + \frac{\delta-1}{2}\text{P.V.}\int_{0}^{t}\frac{1}{X_s} ds + B_t. \end{equation} The principal value correction can be expressed in terms of an integral of the local time processes associated with $X$ (see \cite[Chapter XI]{revuz2013continuous}). Next, we recall the \emph{approximate Bessel processes} for $\delta \in (0,1)$ considered in \cite[Section 6]{s2009cle}. For fixed $\epsilon > 0$, we define an $\epsilon$-$\BES^{\delta}$ process $X_t^{\epsilon}$ to be the Markov process beginning at $x > 0$ that solves~\eqref{eq:bessel_sde} except that each time it hits zero it immediately jumps to $\epsilon$ and continues. Then we have that \begin{equation} \label{eq:approximate_bessel_sde} X_t^{\epsilon} = x + \frac{\delta-1}{2} \int_{0}^{t}\frac{1}{X_s^{\epsilon}}ds + B_t + J_t^{\epsilon} \end{equation} for all times $t \geq 0$, where $J_t^{\epsilon}$ is $\epsilon$ times the number of $\epsilon$-jump discontinuities of $X_t^{\epsilon}$ up to and including time $t$. Then if $t_i$ is the time of the $i$th jump, the following was shown in \cite{s2009cle}: \begin{proposition}\label{prop:epsilon_bessel} As $\epsilon \to 0$, the $X_{t}^{\epsilon}$ converge in law to a $\BES_{x}^{\delta}$ with respect to the $L^{\infty}$ metric on a fixed interval $[0,T]$, with $T>0$. As $\epsilon \to 0$ we have that \begin{enumerate}[(i)] \item $J_{T}^{\epsilon} \to 0$ if $\delta>1$, \item $J_{T}^{\epsilon} \to \infty$ if $0<\delta<1$, and \item\label{it:jtsquared} $J_{T}^{\epsilon^{2}} := \sum_{t_i \leq T} \epsilon^{2} \to 0$ for all $\delta>0$. \end{enumerate} \end{proposition} \subsection{Gaussian free fields} \label{subsec:gff} Let $D \subseteq \C$ be a simply connected domain with harmonically non-trivial boundary and let $H_{0}(D)$ be the Hilbert space closure of $C_{0}^{\infty}(D)$ with respect to the Dirichlet inner product \[ (f,g)_{\nabla} = \frac{1}{2\pi}\int_{D}\nabla f(z) \cdot \nabla g(z) dz.\] Then the zero boundary Gaussian free field (\text{GFF}) $h$ is the random distribution defined by \begin{equation}\label{eq:gff_expression} h = \sum_{n \geq 1}\alpha_n \phi_n \end{equation} where $(\phi_n)_{n \geq 1}$ is an orthonormal basis of $H_{0}(D)$ with respect to $(\cdot,\cdot)_{\nabla}$ and $(\alpha_n)_{n \geq 1}$ is a sequence of i.i.d.\ random variables with distribution $N(0,1)$. Note that a \text{GFF} on $D$ should not be considered as a function but rather as a random variable taking values in the space of distributions. The above construction also applies for other variants of the GFF. In particular, a free boundary \text{GFF} is defined in the same way except that we replace $H_{0}(D)$ by the closure $H(D)$ with respect to $(\cdot,\cdot)_{\nabla}$ of the space of functions $f \in C^{\infty}(D)$ such that $\int_{D}f(z)dz = 0$. Note that in this way, the free boundary \text{GFF} is defined in a space of distributions modulo additive constant. However, we can fix the additive constant for the field by fixing its value when acting on a fixed test function which does not have mean zero. We also note that we can define the free boundary \text{GFF} in the following way. Let $G_{D}^{N}$ be the Green's function with Neumann boundary conditions on $\partial D$ and recall that \begin{equation} \label{eqn:neumann_greens} G_{\h}^{N}(z,w) = -\log \|z-w\| - \log \|z-\overline{w}\|. \end{equation} Then, we can define the free boundary \text{GFF} on $\h$ as the centered Gaussian process with covariance kernel~$G_{\h}^{N}$. We can also define the free boundary \text{GFF} in other simply connected domains by conformally mapping a free boundary \text{GFF} on~$\h$. \subsection{Liouville quantum gravity} \label{subsec:lqg} Recall the definition of the \text{LQG} surfaces given in Section~\ref{sec:introduction}. We can also consider \text{LQG} surfaces with marked points $(D,h,x_1,\dots,x_n)$, $(\wt{D},\wt{h},\wt{x}_1,\dots,\wt{x}_n)$ where we consider them to be equivalent if~\eqref{eqn:change_of_coordinates} holds and $x_j = \varphi(\wt{x}_j)$ for all $j=1,\dots,n$. Next we give the definition of a so-called \emph{thick} quantum wedge, which is a type of quantum surface which is homeomorphic to $\h$ and has two marked points: the \emph{origin} and \emph{infinity}. Compact neighborhoods of the former have finite quantum area while any neighborhood of the latter has infinite quantum area. We recall that a quantum surface is an equivalence class under the equivalence relation defined by~\eqref{eqn:change_of_coordinates}. Thus to specify the law of a quantum surface, we are free to choose which domain we will use to parameterize it. In the case of a quantum wedge, the most convenient choice is the infinite strip $\strip = \R \times (0,\pi)$. \begin{definition} \label{def:thick_quantum_wedge} Fix $\gamma \in (0,2)$, $W > \gamma^2 / 2$, and set $\alpha = \frac{\gamma}{2} + Q - \frac{1}{\gamma}W$. A quantum wedge of weight $W$ is the quantum surface together with two marked boundary points at $-\infty$ and $+\infty$ (when parameterized by $\strip$) described by the distribution $h$ on $\strip$ whose law can be sampled from as follows. Let $X_s$ be the process defined for $s \in \R$ such that \begin{itemize} \item For $s \geq 0$, $X_s = B_{2s} + (Q - \alpha)s$ where $B$ is a standard Brownian motion with $B_0 = 0$ conditioned so that $B_{2u} + (Q - \alpha)u > 0$ for all $u > 0$ and \item For $s < 0$, $X_s = \wh{B}_{-2s} + (Q - \alpha)s$ where $\wh{B}$ is a standard Brownian motion independent of $B$ with $\wh{B}_0 = 0$. \end{itemize} Let $\mathcal{H}_1(\strip)$ be the subspace of $H(\strip)$ which consists of those functions which are constant on vertical lines of the form $u + [0,i\pi]$ for $u \in \R$ and let $\mathcal{H}_2(\strip)$ be the subspace of $H(\strip)$ which consists of those functions which have a common mean on all such vertical lines. Then $h$ is the field with projection onto $\mathcal{H}_1(\strip)$ given by the function whose common value on $u + [0,i\pi]$ is $X_u$ for $u \in \R$ and its projection onto $\mathcal{H}_2(\strip)$ is given by the corresponding projection of a free boundary GFF on $\strip$. We fix the additive constant for the projection of $h$ onto $\mathcal{H}_1(\strip)$ (resp.\ $\mathcal{H}_2(\strip)$) so that its average on $[0,i\pi]$ vanishes. \end{definition} \begin{remark} The above embedding of a quantum wedge into $\strip$ is the so-called \emph{circle average embedding}. Note that we can change the embedding $\strip$ while keeping the marked points fixed by applying~\eqref{eqn:change_of_coordinates} and using a conformal transformation $\phi : \strip \rightarrow \strip$ fixing $-\infty$ and $+\infty$, and this corresponds to translating the field $h$ by a constant $r \in \R$. \end{remark} Throughout this paper, we will use the notation $\langle \cdot \rangle$ in order to denote the quadratic variation. \begin{remark} Suppose that $X$ is as in Definition~\ref{def:thick_quantum_wedge} and let $Z_t = \exp( \gamma X_t/2)$. By It\^o's formula, \begin{align} d\langle Z \rangle_{t} = \frac{\gamma^2}{2} Z_t^2 dt. \end{align} By \cite[Proposition 3.4]{dms2014mating}, we see that if we reparameterize $Z$ by its quadratic variation then it evolves as a $\BES^{\delta}$ with \begin{align} \label{eqn:bessel_dimension_wrt_weight} \delta = 2 + \frac{2(Q-a)}{\gamma} = 1 + \frac{2}{\gamma^2}W. \end{align} This gives another way of sampling $h$ as follows. Start with a $\BES^{\delta}$ process $Z$ with $\delta$ as in~\eqref{eqn:bessel_dimension_wrt_weight}. Then we sample the projection of $h$ onto $\mathcal{H}_1(\strip)$ to be given by reparameterizing $2\gamma^{-1}\log(Z)$ to have quadratic variation $2dt$ and taking the horizontal translation so that it last hits $0$ at time $0$. The projection onto $\mathcal{H}_2(\strip)$ is sampled as in Definition~\ref{def:thick_quantum_wedge} and independently of $Z$. This is how we define the $a = Q$, $W = \gamma^2/2$ quantum wedge. When $\delta \geq 2$, we call the resulting wedge a \emph{thick} quantum wedge. \end{remark} We can also consider quantum surfaces parameterized by a closed set (not necessarily homeomorphic to a disk) such that each component of its interior together with its prime-end boundary is homeomorphic to the closed unit disk, and $h$ is defined as a distribution on each of these components. \begin{definition} \label{def:thin_quantum_wedge} Fix $\gamma \in (0,2)$ and $W \in (-\gamma^2 / 2, \gamma^2 / 2)$. A quantum wedge of weight $W$ on $\strip$ (with marked points at $-\infty$ and $+\infty$) is the random distribution $h$ on $\strip$ that can be sampled as follows. Let $Y$ be a $\BES^{\delta}$ process where $\delta$ is as in~\eqref{eqn:bessel_dimension_wrt_weight}. Let $\mathcal{H}_1(\strip), \mathcal{H}_2(\strip)$ be as before. We sample a countable collection of distribution valued random variables $h_{e}$ on $\strip$ indexed by the excursions~$e$ of~$Y$ from~$0$ and take the projection of $h_{e}$ onto $\mathcal{H}_1(\strip)$ to be given by reparameterizing $2\gamma^{-1}\log(e)$ to have quadratic variation $2dt$ and the projection of $h_{e}$ onto $\mathcal{H}_2(\strip)$ to be sampled according to the law of the corresponding projection of a free boundary GFF on $\strip$ (with additive constant fixed so that its average on $[0,i\pi]$ vanishes) taken to be independent of the corresponding projection of the other excursions from $0$ of $Y$. Note that $\delta \in (0,2)$ and in that case we call the quantum wedge \emph{thin} quantum wedge. \end{definition} In \cite{dms2014mating}, the thin quantum wedges are defined only when $W \in (0,\gamma^2/2)$. The definition given in Definition~\ref{def:thin_quantum_wedge} is the same as in \cite{dms2014mating} except it is for $W \in (-\gamma^2/2,\gamma^2/2)$. The difference between the regime $W \in (-\gamma^2/2,0]$ and $W \in (0,\gamma^2/2)$ is that for the latter the sum of the boundary lengths of the bubbles which make up the wedge is locally finite while in the former this is not the case. \begin{remark} Let $D \subseteq \C$ be a simply connected domain. Then we can consider a quantum wedge on $D$ with distinct marked points $x,y \in \partial D$ by starting with a quantum wedge on $\strip$ and then applying~\eqref{eqn:change_of_coordinates} for some conformal transformation $\phi : \strip \rightarrow D$ such that $\phi(-\infty) = x$ and $\phi(+\infty) = y$. \end{remark} \subsection{$\SLE_{\kappa}(\rho)$ processes with $\kappa > 0$ and $\rho > -2 - \kappa/2$} \label{subsec:slekapparho} We will now review the definition of the $\SLE_\kappa(\rho)$ processes, first focusing on the forward case and then on the reverse case. In order to differentiate between the former and the latter, we will use tildes for quantities associated with the reverse case. \subsubsection{Forward $\SLE_\kappa(\rho)$} Fix $\kappa > 0$, $\rho > -2 - \kappa/2$, and let $\delta$ be such that \begin{equation} \label{eqn:delta_and_rho} \delta = 1 + \frac{2(\rho + 2)}{\kappa}. \end{equation} Let $X$ be a $\BES^{\delta}$ process and let \begin{align} V_t = \frac{2}{\sqrt{\kappa}}\text{P.V.}\int_{0}^{t} \frac{1}{X_s} ds\quad \text{and} \quad W = V - \sqrt{\kappa}X. \end{align} Then an $\SLE_{\kappa}(\rho)$ process is described by the solution to the \text{ODE} \begin{align} \partial_{t}g_t(z) = \frac{2}{g_t(z) - W_t},\quad g_0(z) = z. \end{align} For each $t \geq 0$ we let $\h_t$ be the domain of $g_t$ and $K_t = \h \setminus \h_t$ be the hull at time $t$ associated with the $\SLE_\kappa(\rho)$ process. The location of the force point at time $0$ is given by $V_0$ and at time $t$ is given by $g_t^{-1}(V_t)$. When $\rho > -2$, we have that $\delta > 1$ and so the force point at time $t$ is located at the rightmost intersection of the hull at time $t$ with $\R$. The continuity of the process in that case was shown in \cite{ms2016imag1}. When $\rho \in (-2-\kappa/2,-2)$, the force point does not stay in $\R$ because of the principal value correction and since $\delta \in (0,1)$. The continuity of the process in the case that $\rho \in (-2-\kappa/2,\kappa/2-4]$ and $\kappa \in (2,4)$ was shown in \cite{msw2017clepercolations} while the continuity when $\rho \in (-2-\kappa/2,-2) \cap (\kappa/2-4,-2)$ and $\kappa \in (0,4)$ was shown in \cite{ms2019lightcone}. We can more generally consider the $\SLE_\kappa(\rho)$ processes with multiple force points, which we will denote by $\SLE_\kappa(\ul{\rho})$. More specially, suppose that $\ul{x} = (x_1,\ldots,x_n)$ are distinct points in $\partial \h$ (where we consider $0_-$ and $0_+$ to be distinct) and that $\ul{\rho} = (\rho_1,\ldots,\rho_n) \in \R^n$. Suppose that $B$ is a standard Brownian motion and let $W$ be the solution to the SDE \begin{align} \label{eqn:sle_kappa_rho_sde} dW_t = \sqrt{\kappa} dB_t + \sum_{i=1}^n \frac{\rho_i}{W_t - V_t^i} dt, \quad dV_t^i = \frac{2}{V_t^i-W_t} dt, \quad V_0^i = x_i. \end{align} It is shown in \cite{ms2016imag1} that there is a unique solution~\eqref{eqn:sle_kappa_rho_sde} up until the so-called \emph{continuation threshold}, which is the first time $t$ that $\sum_{i : V_t^i = W_t} \rho_i = -2$ and that the corresponding Loewner flow is generated by a continuous curve up until this time. \subsubsection{Reverse $\SLE_\kappa(\wt{\rho})$} \label{subsubsec:reverse_sle_kappa_rho} Fix $\kappa > 0$, $\wt{\rho} > 2-\kappa/2$, and let \begin{equation} \label{eqn:delta_and_rho_reverse} \wt{\delta} = 1 + \frac{2(\wt{\rho}-2)}{\kappa}. \end{equation} Let \[ \wt{V}_{t} = -\frac{2}{\sqrt{\kappa}}\text{P.V.} \int_{0}^{t}\frac{1}{\wt{X}_{s}}ds\quad\text{and}\quad \wt{W} = \wt{V} - \sqrt{\kappa}\wt{X}.\] A reverse $\SLE_{\kappa}(\wt{\rho})$ (with the force point located at $0^{+}$) centered Loewner flow is defined to be the family of conformal maps ($\wt{f_{t}}$) given by $\wt{g_{t}}(z) - \wt{W_{t}}$ where ($\wt{g_{t}}$) solve the reverse Loewner equation \begin{align} \label{eqn:rev_loewner} \partial_{t}{\wt{g}_{t}}(z) = -\frac{2}{\wt{g}_{t}(z) - \wt{W}_{t}},\quad \wt{g}_0(z) = z. \end{align} We note that~\eqref{eqn:rev_loewner} has a unique solution for every fixed $z$ in $\h$ defined for all times $t$. \subsection{Imaginary geometry} \label{subsec:ig} Fix $\kappa \in (0,4)$, $\kappa'=16/\kappa > 4$, and let \begin{equation} \label{eqn:chi_value} \lambda = \frac{\pi}{\sqrt{\kappa}},\quad \lambda' = \frac{\pi}{\sqrt{\kappa'}}, \quad\text{and}\quad \chi = \frac{2}{\sqrt{\kappa}} - \frac{\sqrt{\kappa}}{2}. \end{equation} It was shown in \cite{she2016zipper,ms2016imag1} (see also \cite{dub2009partition}) that the $\SLE_{\kappa}(\ul{\rho})$ processes with $\kappa \in (0,4)$ and $\rho > -2$ can be considered as the flow lines of the vector field $e^{ih / \chi}$, where $h$ is a GFF with fixed boundary data. Let us first explain how this works in the case of the $\SLE_\kappa$ processes. Suppose that~$h$ is a GFF on~$\h$ with boundary conditions given by $-\lambda$ on $\R_-$ and $\lambda$ on $\R_+$. Then it is shown in \cite{ms2016imag1} that there exists a unique coupling of $h$ with an $\SLE_\kappa$ process $\eta$ in $\h$ from $0$ to $\infty$ so that the following is true. Let $(g_t)$ be the Loewner flow for $\eta$, $W$ its driving function, and let $f_t = g_t - W_t$ be its centered Loewner flow. Then for every stopping time $\tau$ for $\eta$ we have that \[ h \circ f_\tau^{-1} - \chi \arg (f_\tau^{-1})' \stackrel{d}{=} h.\] In this coupling, we moreover have that $\eta$ is determined by $h$. More generally, suppose that $\eta$ is an $\SLE_\kappa(\ul{\rho})$ process where $\ul{\rho} = (\ul{\rho}^L; \ul{\rho}^R)$, $\ul{\rho}^L = (\rho_1^L,\ldots,\rho_\ell^L)$, $\ul{\rho}^R = (\rho_1^R,\ldots,\rho_k^R)$ with force points located at $\ul{x} = (\ul{x}^L;\ul{x}^R)$ with $-\infty = x_{\ell+1}^L < x_\ell^L < \cdots < x_1^L \leq x_0^L = 0_-$ and $0_+ = x_0^R \leq x_1^R < \cdots < x_k^R < x_{k+1}^R = +\infty$. Suppose that $h$ is a GFF on $\h$ with boundary conditions given by \begin{align} -\lambda\left(1+ \sum_{i=1}^j \rho_i^L \right) \quad&\text{in}\quad (x_{i+1}^L,x_i^L] \quad\text{for}\quad 0 \leq i \leq \ell \quad \quad\text{and}\\ \lambda\left( 1+ \sum_{i=1}^j \rho_i^R \right) \quad&\text{in}\quad (x_i^R,x_{i+1}^R] \quad\text{for}\quad 0 \leq i \leq k. \end{align} Let $(f_t)$ be the centered Loewner flow for $\eta$. Then there exists a unique coupling of $\eta$ with $h$ so that for every stopping time $\tau$ for $\eta$ we have that $h \circ f_\tau^{-1} - \chi \arg (f_\tau^{-1})'$ is a GFF on $\h$ with boundary conditions given by \begin{align} -\lambda\left(1+ \sum_{i=1}^j \rho_i^L \right) \quad&\text{in}\quad (f_\tau(x_{i+1}^L),f_\tau(x_i^L)] \quad\text{for}\quad 0 \leq i \leq \ell \quad \quad\text{and}\\ \lambda\left( 1+ \sum_{i=1}^j \rho_i^R \right) \quad&\text{in}\quad (f_\tau(x_i^R),f_\tau(x_{i+1}^R)] \quad\text{for}\quad 0 \leq i \leq k \end{align} where we take the convention that $f_\tau(x_0^L) = 0_-$ and $f_\tau(x_0^R) = 0_+$. In this coupling, we moreover have that $\eta$ is determined by $h$. In both of the above situations, we refer to $\eta$ as the flow line of $h$ from $0$ to $\infty$. We can more generally define the flow line of $h$ from $x$ to $\infty$ by translating $h$. Also, if $\theta \in \R$ then we define the flow line of $h$ with angle $\theta$ to be the flow line of $h + \theta \chi$. The manner in which the flow lines interact was determined in \cite{ms2016imag1}. In particular, suppose that $\eta_{x_i}^{\theta_i}$ for $i=1,2$ are the the flow lines of $h$ starting from $x_1 < x_2$ with angles $\theta_1,\theta_2$. If $\theta_1 > \theta_2$, then $\eta_{x_1}^{\theta_1}$ stays to the left of $\eta_{x_2}^{\theta_2}$. If $\theta_1 = \theta_2$, then $\eta_{x_1}^{\theta_1}$ merges with $\eta_{x_2}^{\theta_2}$ and does not subsequently separate. Finally, if $\theta_1 \in (\theta_2-\pi,\theta_2)$, then $\eta_{x_1}^{\theta_1}$ crosses $\eta_{x_2}^{\theta_2}$ upon intersecting and does not subsequently cross back. The $\SLE_{\kappa'}$ curves are also coupled with the GFF in a similar manner but the interpretation of the coupling is different. Suppose that $h$ is a GFF on $\h$ with boundary conditions given by $\lambda'$ on $\R_-$ and $-\lambda'$ on $\R_+$. Then it is shown in \cite{ms2016imag1} that there exists a unique coupling of $h$ with an $\SLE_{\kappa'}$ process $\eta'$ in $\h$ from $0$ to $\infty$ so that the following is true. Let $(g_t)$ be the Loewner flow for $\eta'$, $W$ its driving function, and let $f_t = g_t - W_t$ be its centered Loewner flow. Then for every stopping time $\tau$ for $\eta'$ we have that \[ h \circ f_\tau^{-1} - \chi \arg (f_\tau^{-1})' \stackrel{d}{=} h.\] In this coupling, we moreover have that $\eta'$ is determined by $h$. More generally, suppose that $\eta$ is an $\SLE_\kappa(\ul{\rho})$ process where $\ul{\rho} = (\ul{\rho}^L; \ul{\rho}^R)$, $\ul{\rho}^L = (\rho_1^L,\ldots,\rho_\ell^L)$, $\ul{\rho}^R = (\rho_1^R,\ldots,\rho_k^R)$ with force points located at $\ul{x} = (\ul{x}^L;\ul{x}^R)$ with $-\infty = x_{\ell+1}^L < x_\ell^L < \cdots < x_1^L \leq x_0^L = 0_-$ and $0_+ = x_0^R \leq x_1^R < \cdots < x_k^R < x_{k+1}^R = +\infty$. Suppose that $h$ is a GFF on $\h$ with boundary conditions given by \begin{align} \lambda' \left(1+ \sum_{i=1}^j \rho_i^L \right) \quad&\text{in}\quad (x_{i+1}^L,x_i^L] \quad\text{for}\quad 0 \leq i \leq \ell \quad \quad\text{and}\\ -\lambda' \left( 1+ \sum_{i=1}^j \rho_i^R \right) \quad&\text{in}\quad (x_i^R,x_{i+1}^R] \quad\text{for}\quad 0 \leq i \leq k. \end{align} Let $(f_t)$ be the centered Loewner flow for $\eta'$. Then there exists a unique coupling of $\eta'$ with $h$ so that for every stopping time $\tau$ for $\eta'$ we have that $h \circ f_\tau^{-1} - \chi \arg (f_\tau^{-1})'$ is a GFF on $\h$ with boundary conditions given by \begin{align} \lambda'\left(1+ \sum_{i=1}^j \rho_i^L \right) \quad&\text{in}\quad (f_\tau(x_{i+1}^L),f_\tau(x_i^L)] \quad\text{for}\quad 0 \leq i \leq \ell \quad \quad\text{and}\\ -\lambda'\left( 1+ \sum_{i=1}^j \rho_i^R \right) \quad&\text{in}\quad (f_\tau(x_i^R),f_\tau(x_{i+1}^R)] \quad\text{for}\quad 0 \leq i \leq k \end{align} where we take the convention that $f_\tau(x_0^L) = 0_-$ and $f_\tau(x_0^R) = 0_+$. In this coupling, we moreover have that $\eta'$ is determined by $h$. In both of the above situations, we refer to $\eta'$ as the counterflow line of $h$ from $0$ to $\infty$. In \cite{ms2016imag1} the manner in which the flow and counterflow lines of the GFF interact is also described. In particular, suppose that $h$ is a GFF on $\h$ and $\eta'$ is the counterflow line of $h$ from $\infty$ to $0$. Then the left (resp.\ right) boundary of $\eta'$ is equal to the flow line of $h$ from $0$ to $\infty$ with angle $\pi/2$ (resp.\ $-\pi/2$). It is also possible to consider flow lines starting from interior points \cite{ms2017imag4} but we will not need to consider this in the present work. \subsection{Light cones} \label{subsec:light_cones} Now, we review some basic properties of \emph{light cones} and their relation with the $\SLE_{\kappa}(\rho)$ processes. We will first review the definition of the light cone. We will focus on the case that it is defined on~$\h$; the definition on another simply connected domain is given by conformal mapping. Suppose that $h$ is a GFF on $\h$ and $x \in \partial \h$ with piecewise constant boundary conditions that change only finitely many times. Fix angles $\theta_1 \leq \theta_2 \leq \theta_1 + \pi$. The $\SLE_{\kappa}$ \emph{light cone} $\lightcone_x(\theta_1,\theta_2)$ of $h$ starting from $x$ with angle range $[\theta_1,\theta_2]$ is given by the closure of the set of points accessible by the flow lines of $h$ starting from $x$ with angles which are either rational and contained in $[\theta_1,\theta_2]$ or equal to $\theta_1$ or $\theta_2$ and which change angles a finite number of times and only at positive rational times. More generally, if $A$ is a segment of $\partial D$, we let $\lightcone_A(\theta_1,\theta_2)$ be the set of points accessible by flow lines of $h$ starting from a countable dense subset of $A$ with angles which are either rational and contained in $[\theta_1,\theta_2]$ or equal to $\theta_1$ or $\theta_2$ which change angles only a finite number of times and only at positive rational times. It was shown in \cite[Theorem 1.2]{ms2019lightcone} that the range of an $\SLE_{\kappa}(\rho)$ process for $\kappa \in (0,4)$, $\rho \in [\kappa/2-4,-2) \cap (-2 - \kappa/2,-2)$ is equal to $\lightcone_{\R_-}(0,\theta)$ when the boundary of $h$ and~$\theta$ are chosen appropriately. More precisely, the following was shown in \cite{ms2019lightcone}: \begin{theorem} \label{thm:ligthcone_and_sle} Fix $\kappa \in (0,4)$, $\rho \in [\kappa/2-4,-2)$ and $\rho > -2 - \kappa/2$, and suppose that $h$ is a \text{GFF} on $\h$ whose boundary data is given by $-\lambda$ on $\R_-$ and $\lambda (1+\rho)$ on $\R_+$. Let $\eta$ be an $\SLE_{\kappa}(\rho)$ process on $\h$ from $0$ to $\infty$ where its force point is located at $0_+$. For each $t \geq 0$, let $K_t$ denote the closure of the complement of the unbounded connected component of $\h \setminus \eta([0,t])$, let $g_t : \h \setminus K_t \to \h$ be the unique conformal transformation with $g_t(z)-z \to 0$ as $z \to \infty$, and let $(W,V)$ be the Loewner driving pair for $\eta$. There exists a unique coupling of $h$ and $\eta$ such that the following is true. For each $\eta$-stopping time $\tau$, the conditional law of \begin{align} h \circ g_{\tau}^{-1} - \chi \arg(g_{\tau}^{-1})' \end{align} given $\eta\|_{[0,\tau]}$ is that of a \text{GFF} on $\h$ with boundary conditions given by \begin{align} h\|_{(-\infty,W_{\tau}]} \equiv -\lambda,\quad h\|_{( W_{\tau},V_{\tau}]} \equiv \lambda,\quad \text{and} \quad h\|_{(V_{\tau},\infty)}\equiv \lambda (1+\rho), \end{align} where $\lambda = \pi / \sqrt{\kappa}$. Moreover, in the coupling $(h,\eta)$, $\eta$ is a.s.\ determined by $h$. Finally, let \begin{equation}\label{eqn:theta_and_rho} \theta = \theta_{\rho} = \pi \!\left ( \frac{\rho + 2}{\kappa / 2 - 2} \right ). \end{equation} Then the range of $\eta$ is a.s.\ equal to $\lightcone_{\R_-}(0,\theta)$. \end{theorem} Theorem~\ref{thm:ligthcone_and_sle} allows to interpret an $\SLE_{\kappa}(\rho)$ process with the above range of values of $\rho$ as an ordered light cone of flow lines of $e^{ih / \chi}$. \section{SLE/GFF couplings} \label{sec:slegff_couplings} The purpose of this section is to prove two results related to couplings of $\SLE$ with the GFF. In Section~\ref{subsec:reverse_coupling} we will focus on the so-called reverse coupling and will establish a version of it which holds for reverse $\SLE_\kappa(\wt{\rho})$ processes with $\wt{\rho} < 2$. In Section~\ref{subsec:law_of_sle_excursion} we use the forward coupling of $\SLE$ with the GFF in order to describe the law of an excursion of an $\SLE_\kappa(\rho)$ process in the light cone regime. \subsection{Reverse coupling with $\wt{\rho} < 2$} \label{subsec:reverse_coupling} In this subsection we will state and prove a version of the reverse coupling of $\SLE_\kappa(\wt{\rho})$ with the GFF when $\wt{\rho} < 2$. In the case of ordinary reverse $\SLE_\kappa$, this was first proved in \cite{she2016zipper} and it was extended to the case of the reverse $\SLE_\kappa(\wt{\rho})$ processes with $\wt{\rho} > 2$ in \cite{dms2014mating}. In both \cite{she2016zipper} and \cite{dms2014mating}, the process which drives the reverse $\SLE$ is a semimartingale which allows one to use the tools of ordinary stochastic calculus. In contrast, when $\wt{\rho} < 2$ the driving process for a reverse $\SLE_\kappa(\wt{\rho})$ process is not a semimartingale and this introduces some extra complications in the version that we will prove. Throughout, we fix $\kappa >0$, $\wt{\rho} \in (2-\kappa/2 ,2)$, and let \[ \gamma=\min\left( \sqrt{\kappa},\frac{4}{\sqrt{\kappa}} \right) \quad\text{and}\quad Q=\frac{2}{\gamma}+\frac{\gamma}{2}.\] (Recall from Section~\ref{subsubsec:reverse_sle_kappa_rho} that reverse $\SLE_\kappa(\wt{\rho})$ is only defined for $\wt{\rho} > 2-\kappa/2$.) Let $G_\h^N$ be the Neumann Green's function on $\h$ (recall~\eqref{eqn:neumann_greens}). Let $(\wt{f}_t)$ be the centered Loewner flow associated with a reverse $\SLE_\kappa(\wt{\rho})$ process and let \begin{align} \wt{G}_{t}(y,z) = G_\h^N(\wt{f}_{t}(y),\wt{f}_{t}(z)),\quad \wh{\Fh}_{0}(z) = \frac{2}{\sqrt{\kappa}}\log\|z\|,\quad \text{and}\quad \wh{\Fh}_{t}(z) = \wh{\Fh}_{0}(\wt{f}_{t}(z)) + Q\log\|\wt{f}_{t}'(z)\|. \end{align} The following is the main theorem of this subsection and serves to extend \cite[Theorem~1.2]{she2016zipper} and \cite[Theorem~5.1]{dms2014mating}. \begin{theorem} \label{thm:reverse_coupling} Let $\kappa$, $\wt{\rho}$ be as above, let $\wt{W}$ be the driving function associated with a reverse $\SLE_{\kappa}(\wt{\rho})$ with a single force point located at $0_+$ of weight $\wt{\rho}$, and let $(\wt{f}_{t})$ be the centered reverse Loewner evolution driven by $\wt{W}$. For each $t \geq 0$, let \begin{align} \wt{\Fh}_{t}(z) = \wh{\Fh}_{t}(z) + \frac{\wt{\rho}}{2\sqrt{\kappa}}\wt{G}_{t}(0_+,z) \end{align} and let $h$ be an instance of the free boundary $\text{GFF}$ on $\h$ independent of $\wt{W}$. Suppose that $\wt{\tau}$ is an a.s.\ finite stopping time for the filtration generated by the Brownian motion which drives $\wt{W}$. Then \begin{align} \wt{\Fh}_{0} + h \stackrel{d}{=} \wt{\Fh}_{\wt{\tau}} + h \circ \wt{f}_{\wt{\tau}} \end{align} where the left and right sides are viewed as distributions modulo additive constant. \end{theorem} Recall from Section~\ref{subsubsec:reverse_sle_kappa_rho} that $\wt{f}_t(0_+)/\sqrt{\kappa}$ is a $\BES^{\wt{\delta}}$ process with \[ \wt{\delta} = 1+ \frac{2(\wt{\rho}-2)}{\kappa}.\] The proofs of \cite[Theorem~1.2]{she2016zipper} and \cite[Theorem~5.1]{dms2014mating} make use of stochastic calculus and require that $\int_0^t \wt{f}_s(0_+)^{-1}ds$ is finite for all $t\geq 0$ a.s. However, when $\wt{\rho} \in (2-\kappa/2, 2)$ we have that $\wt{\delta} \in (0,1)$ so that this integral is infinite a.s.\ and this fact is the key difference in the proof of Theorem~\ref{thm:reverse_coupling} in comparison to that given in \cite{she2016zipper,dms2014mating}. In order to overcome this difficulty, we are going to use Proposition~\ref{prop:epsilon_bessel} to approximate $\wt{f}_t(0_+)/\sqrt{\kappa}$ using an $\epsilon$-$\BES^{\wt{\delta}}$ process and then take a limit as $\epsilon \to 0$. Moreover, we will also make use of the following generalization of It\^o's formula for semimartingales with discontinuities. \begin{theorem} \label{thm:general_Ito} Suppose that $Z_{t} = X_{t} + iY_{t}$ is a semimartingale (not necessarily continuous) with $X_{t} = \re(Z_{t})$ and $Y_{t} = \im(Z_{t})$ and $Z_{t} \in U$ for all $t$ for $U \subseteq \C$ open. Then if $f$ is an analytic function on $U$, it holds that \begin{align} f(Z_{t}) - f(Z_{0}) &= \int_{0}^{t}f'(Z_{s})dZ_{s} + \frac{1}{2}\int_{0}^{t}f''(Z_{s^{-}})d\langle Z^{c}\rangle_{s}\\ &+\sum_{0 \leq s \leq t}\left( f(Z_{s}) - f(Z_{s^{-}}) - f'(Z_{s^{-}})\Delta Z_{s}\right) \end{align} \end{theorem} Let us now perform some initial calculations which will be used in the proof of Theorem~\ref{thm:reverse_coupling}. Let~$B$ be the Brownian motion which drives $\wt{X}$ and let $\wt{X}^{\epsilon}$ be the solution of~\eqref{eq:approximate_bessel_sde}. Let also $\wt{J}_t^\epsilon$ be as in~\eqref{eq:approximate_bessel_sde} and $\wt{J}_t^{\epsilon^2}$ be as in part~\eqref{it:jtsquared} of Proposition~\ref{prop:epsilon_bessel} (so that $\wt{J}_t^{\epsilon^2}$ is $\epsilon^2$ times the number of $\epsilon$-jump discontinuities of $X_t^\epsilon$ up to and including time $t$). We consider the processes \begin{align} \wt{Y}^\epsilon = \frac{2}{\wt{\delta} - 1}(\wt{X}^{\epsilon} - B) \quad\text{and}\quad \wt{W}^{\epsilon} = -\frac{2}{\sqrt{\kappa}}\wt{Y}^{\epsilon} - \sqrt{\kappa}\wt{X}^{\epsilon} \end{align} By Proposition~\ref{prop:epsilon_bessel}, we can couple $\wt{X}^{\epsilon}$ and $\wt{X}$ such that $\wt{X}^{\epsilon}$ converges to $\wt{X}$ as $\epsilon \to 0$ uniformly on compact intervals a.s. Fix $z \in \h$ and let \begin{align} \wt{Z}_{t}^{\epsilon} = \wt{g}_{t}(z) + \frac{2}{\sqrt{\kappa}}\wt{Y}_{t}^{\epsilon} + \sqrt{\kappa}\wt{X}_{t}^{\epsilon}. \end{align} Note that $\wt{Z}^{\epsilon}$ converges to $\wt{f}_{t}(z)$ as $\epsilon \to 0$ uniformly on compact intervals a.s. Since \[ \langle \wt{Z}^{\epsilon}\rangle_{t} = \kappa t + \left(\sqrt{\kappa} + \frac{4}{\sqrt{\kappa}(\wt{\delta} - 1)}\right)^2 \wt{J}_{t}^{\epsilon^{2}},\] by applying Theorem~\ref{thm:general_Ito} we obtain that \begin{align} \frac{2}{\sqrt{\kappa}}\left( \log \wt{Z}_{t}^{\epsilon} - \log\wt{Z}_{0}^{\epsilon}\right) = \wt{\alpha}_{t}^{\epsilon} + \frac{2}{\sqrt{\kappa}}\int_{0}^{t}\frac{1}{\wt{Z}_{s^{-}}^{\epsilon}}d \wt{Z}_{s}^{\epsilon} - \sqrt{\kappa}\int_{0}^{t}\frac{1}{(\wt{Z}_{s^{-}}^{\epsilon})^{2}}ds \end{align} where \begin{align} \wt{\alpha}_{t}^{\epsilon} = \frac{2}{\sqrt{\kappa}}\sum_{0 \leq s \leq t}\left( \log \wt{Z}_{s}^{\epsilon} - \log \wt{Z}_{s^-}^{\epsilon} - \frac{\Delta \wt{Z}_{s}^{\epsilon}}{\wt{Z}_{s^{-}}^{\epsilon}}\right). \end{align} Similarly we have that \[ \langle \wt{A}^{\epsilon}\rangle_{t} = \frac{16 \wt{J}_{t}^{\epsilon^{2}}}{\kappa (\wt{\delta} -1)^{2}} \quad\text{where}\quad \wt{A}_{t}^{\epsilon} = \wt{g}_t(z) + \frac{2}{\sqrt{\kappa}} \wt{Y}_{t}^{\epsilon}\] and hence by applying Theorem~\ref{thm:general_Ito} again we obtain that \begin{align} \frac{\wt{\rho}}{\sqrt{\kappa}}\left( \log \wt{A}_{t}^{\epsilon} - \log\wt{A}_{0}^{\epsilon}\right) = \wt{\beta}_{t}^{\epsilon} + \frac{\wt{\rho}}{\sqrt{\kappa}}\int_{0}^{t}\frac{1}{\wt{A}_{s^-}^{\epsilon}}d\wt{A}_{s}^{\epsilon} \end{align} where \begin{align} \wt{\beta}_{t}^{\epsilon} = \frac{\wt{\rho}}{\sqrt{\kappa}}\sum_{0 \leq s \leq t}\left(\log \wt{A}_{s}^{\epsilon} - \log \wt{A}_{s^-}^{\epsilon} - \frac{\Delta \wt{A}_{s}^{\epsilon}}{\wt{A}_{s^-}^{\epsilon}}\right). \end{align} Let $(t_i)$ be as in~\eqref{eq:approximate_bessel_sde}. Since $\wt{X}_{t_{i}^-}^{\epsilon} = 0$ we have that $\wt{A}_{t_{i}^{-}}^{\epsilon} = \wt{Z}_{t_{i}^-}^{\epsilon}$ and so \begin{align} &\frac{2}{\sqrt{\kappa}}\left( \log \wt{Z}_{t}^{\epsilon} - \log \wt{Z}_{0}^{\epsilon} \right) - \frac{\wt{\rho}}{\sqrt{\kappa}} \left( \log\wt{A}_{t}^{\epsilon} - \log \wt{A}_{0}^{\epsilon} \right) \nonumber\\ =&(\wt{\alpha}_{t}^{\epsilon} - \wt{\beta}_{t}^{\epsilon}) +\int_{0}^{t}\frac{2}{\wt{Z}_{s^-}^{\epsilon}}dB_{s} - \frac{2}{\sqrt{\kappa}}\int_{0}^{t}\frac{2}{\wt{Z}_{s^-}^{\epsilon}\wt{f}_{s}(z)}ds\nonumber\\ &-\sqrt{\kappa}\int_{0}^{t}\frac{1}{(\wt{Z}_{s^-}^{\epsilon})^{2}}ds + \frac{2\wt{\rho}}{\sqrt{\kappa}}\int_{0}^{t}\frac{\wt{Z}_{s^-}^{\epsilon} - \wt{f}_{s}(z)}{\wt{A}_{s^-}^{\epsilon}\wt{f}_{s}(z) \wt{Z}_{s^-}^{\epsilon}}ds.\label{eqn:basic_equation} \end{align} Fix $T > 0$. It is easy to see that \[ \log \wt{Z}_{t_i}^{\epsilon} - \log \wt{Z}_{t_{i}^-}^{\epsilon} - \frac{\Delta \wt{Z}_{t_i}^{\epsilon}}{\wt{Z}_{t_{i}^-}^{\epsilon}} = O((\Delta \wt{Z}_{t_i}^{\epsilon})^2) \quad\text{for all}\quad t_i \leq T,\] where the implicit constant is random but independent of $\epsilon$. Since \[ \Delta \wt{Z}_{t_i}^{\epsilon} = \left( \sqrt{\kappa} + \frac{4}{\sqrt{\kappa}(\wt{\delta} - 1)} \right) \epsilon\] we obtain that $\wt{\alpha}_{T}^{\epsilon} = O(\wt{J}_{T}^{\epsilon^2})$, where the implicit constant is random and independent of $\epsilon$. Therefore by applying Proposition~\ref{prop:epsilon_bessel} we obtain that $\wt{\alpha}_{t}^{\epsilon}$ converges to zero as $\epsilon \to 0$ uniformly on compact intervals a.s. A similar analysis holds for $\wt{\beta}_{t}^{\epsilon}$. Moreover, it is easy to see that \[ \int_{0}^{t}\frac{1}{\wt{Z}_{s^-}^{\epsilon}}dB_{s} \to \int_{0}^{t}\frac{1}{\wt{f}_{s}(z)}dB_{s} \quad\text{in}\quad L^{2} \quad\text{as}\quad \epsilon \to 0.\] In particular, the convergence holds in probability. Therefore the above observations imply that the right hand side of~\eqref{eqn:basic_equation} converges in probability as $\epsilon \to 0$ to \begin{align} \label{eq:RHS_convergence} \int_{0}^{t} \frac{2}{\wt{f}_{s}(z)}dB_{s} - Q\int_{0}^{t} \frac{2}{(\wt{f}_{s}(z))^2}ds \end{align} while the left hand side of~\eqref{eqn:basic_equation} converges a.s.\ as $\epsilon \to 0$ to \begin{align} \label{eq:LHS_convergence} \frac{2}{\sqrt{\kappa}}\left(\log \wt{f}_{t}(z) - \log z\right) - \frac{\wt{\rho}}{\sqrt{\kappa}}\left(\log(\wt{f}_{t}(z) - \wt{f}_{t}(0^{+})) - \log z \right) \end{align} Hence the quantities in~\eqref{eq:RHS_convergence} and~\eqref{eq:LHS_convergence} are equal for all $t$ a.s. Next, we follow the same strategy as in \cite[Section~5.1]{dms2014mating} in order to prove Theorem~\ref{thm:reverse_coupling}. We will restate the lemmas used in order to make clear how the equality of the expressions in~\eqref{eq:RHS_convergence} and~\eqref{eq:LHS_convergence} is used. \begin{lemma}[$\text{\cite[Lemma~5.2]{dms2014mating}}$] \label{lem:helping_lemma} For each $z \in \h$, we have that \begin{align} d\wt{\Fh}_{t}(z) = \re\frac{2}{\wt{f}_{t}(z)}dB_{t} \end{align} For each $y,z \in \h$ distinct, we have that \begin{align} d\wt{G}_{t}(y,z) = -\re\left(\frac{2}{\wt{f}_{t}(y)}\right)\re\left(\frac{2}{\wt{f}_{t}(z)}\right)dt \end{align} \end{lemma} \begin{proof} By~\eqref{eqn:rev_loewner} we obtain that \begin{align} d\wt{f}_{t}'(z) = \frac{2\wt{f}_{t}'(z)}{\wt{f}_{t}^{2}(z)}dt \end{align} and thus \begin{align} \log \wt{f}_{t}'(z) = \int_{0}^{t}\frac{2}{(\wt{f}_{t}(z))^{2}}ds. \end{align} The first claim then follows by taking real parts in~\eqref{eq:RHS_convergence} and~\eqref{eq:LHS_convergence} and then adding $Q\log\|\wt{f}_{t}'(z)\|$ to both of them. \newline As for the second claim, we apply the same argument as in \cite[Lemma~5.2]{dms2014mating}. \end{proof} \begin{lemma}[$\text{\cite[Lemma~5.3]{dms2014mating}}$]\label{lem:square_integrable_martingale} For each $ t \geq 0$, $\wt{\Fh}_{t} + h \circ\wt{f}_{t}$ a.s.\ takes values in the space of modulo additive constant distributions. Suppose that $\phi \in C_{0}^{\infty}(\h)$ with $\int_{\h}\phi(z)dz = 0$. Then both $(\wt{\Fh}_{t} + h \circ \wt{f}_{t},\phi)$ and $(\wt{\Fh}_{t},\phi)$ are a.s.\ continuous in $t$ and the latter is a square-integrable martingale. \end{lemma} \begin{proof} It follows from the proof of \cite[Lemma~5.3]{dms2014mating}. \end{proof} For each $\phi \in C_{0}^{\infty}(\h)$ with $\int_{\h}\phi (z)dz = 0$ and $t \geq 0$ we let \begin{align} \wt{E}_{t}(\phi) = \int_{\h}\int_{\h} \phi(y)\wt{G}_{t}(y,z)\phi(z)dydz \end{align} be the conditional variance of $(h \circ\wt{f}_{t},\phi)$ given $\wt{f}_{t}$. \begin{lemma}[$\text{\cite[Lemma~5.4]{dms2014mating}}$] \label{lem:variation_formula} For each $\phi \in C_{0}^{\infty}(\h)$ with $\int_{\h} \phi (z)dz = 0$ we have that \begin{align} d\langle (\wt{\Fh}_{\cdot},\phi)\rangle_t = -d\wt{E}_{t}(\phi) \end{align} \end{lemma} \begin{proof} It follows from the proof of \cite[Lemma~5.4]{dms2014mating}. \end{proof} We are now ready to prove Theorem~\ref{thm:reverse_coupling}. \begin{proof} It follows by combining Lemmas~\ref{lem:helping_lemma}, \ref{lem:square_integrable_martingale}, and~\ref{lem:variation_formula} as in the proof of \cite[Theorem~5.1]{dms2014mating}. \end{proof} \subsection{Law of an $\SLE_{\kappa}(\rho)$ excursion}\label{subsec:law_of_sle_excursion} In this section, we will give the conditional law of the excursion of an $\SLE_{\kappa}(\rho)$ process in $\h$ with $\kappa \in (0,4)$ and $\rho \in (\kappa/2-4,-2) \cap (-2 -\kappa/2,-2)$ with the force point at $0_+$ which separates a given boundary point $x > 0$ from $\infty$ given the realization of the path before the start of the excursion. \begin{theorem} \label{thm:law_of_sle_excursion} Fix $\kappa \in (0,4)$, $\rho \in (\kappa/2 - 4, -2)$, $\rho > -\kappa/2 - 2$. Let $\eta$ be an $\SLE_{\kappa}(\rho)$ process in $\h$ from $0$ to $\infty$ with the force point located at $0_+$. Fix $x > 0$ and let $T_x$ be the first time that~$\eta$ separates~$x$ from $\infty$. Let $S_x = \sup\{ t < T_x : W_t = V_t\}$ be the most recent time before $T_x$ that~$\eta$ hits its force point. Then the conditional law of $\eta\|_{[S_x,T_{x}]}$ given $\eta\|_{[0,S_x]}$ and $\eta(T_x)$ is that of an $\SLE_{\kappa}(\rho+2; \kappa-4-\rho)$ process from $\eta(S_x)$ to $\eta(T_x)$ in $\h_{S_x}$ with the weight $\rho+2$ force point located at $\infty$ and the weight $\kappa-4-\rho$ force point located at $(\eta(S_x))^+$. \end{theorem} In order to give the proof of Theorem~\ref{thm:law_of_sle_excursion}, we first need to collect the following lemma. \begin{lemma} \label{lem:sle_separates_points} Fix $\kappa \in (0,4)$, $\rho \in (\kappa/2 - 4, -2)$, $\rho > -\kappa/2 - 2$. Let $\eta$ be an $\SLE_{\kappa}(\rho)$ process in~$\h$ from~$0$ to~$\infty$ with the force point located at $0_+$. Then for each fixed $x>0$, we have that $\eta$ does not hit $x$ and separates $x$ from $\infty$ a.s. \end{lemma} \begin{proof} \noindent{\it Overview.} The proof will be carried out in three steps. First of all, by the scale invariance of the law of $\eta$, it suffices to prove the claim of the lemma when $x=1$. In Step $1$, we show that $\eta$ hits $(1,\infty)$ with positive probability. In Step 2, we use the conformal Markov property of $\SLE_\kappa(\rho)$ in order to show that $\eta$ in fact hits $(1,\infty)$ a.s. Finally, in Step 3 we prove that $\eta$ a.s.\ does not hit $1$. \noindent{\it Step 1. $\eta$ hits $(1,\infty)$ with positive probability.} Let $(V,W)$ be the driving pair encoding $\eta$ as in Section~\ref{subsec:slekapparho} and let $\tau$ be the first time that $\eta$ hits $(1,\infty)$. Let also $(g_t)$ be the family of conformal maps solving the Loewner equation used to define $\eta$. Since $\eta$ is continuous, there exists $\epsilon_1>0$ so that $\p[\eta([0,\epsilon_1]) \subseteq B(0,\frac{1}{2})] > 0$. Moreover, since $\kappa^{-1/2} (V-W)$ is a $\BES^{\delta}$ process starting from $0$ with $\delta$ as in~\eqref{eqn:delta_and_rho}, we obtain that $V_{\epsilon_1} \neq W_{\epsilon_1}$ a.s.\ and so there exist $p,\epsilon_2 \in (0,1)$ such that $\p[A] \geq p$, where $A$ is the event that $\eta([0,\epsilon_1]) \subseteq B(0,\frac{1}{2})$ and $\epsilon_2 \leq V_{\epsilon_1} - W_{\epsilon_1} \leq \epsilon_2^{-1}$. The Markov property of $\eta$ implies that if $\mathcal{F}_t = \sigma(\eta\|_{[0,t]})$, then conditioned on $\mathcal{F}_{\epsilon_1}$ and restricted to $A$, the curve $\wt{\eta}(t) = g_{\epsilon_1}(\eta(\epsilon_1 + t)) - W_{\epsilon_1}$ has the law of an $\SLE_{\kappa}(\rho)$ process in $\h$ from $0$ to $\infty$ with the force point located at $V_{\epsilon_1} - W_{\epsilon_1}$. Let $\gamma$ be the concatenation of the segments $[0,i]$, $[i,i+\epsilon_2^{-1}]$ and $[i+\epsilon_2^{-1},\epsilon_2^{-1}]$. By \cite[Lemma~2.5]{mw2017slepaths}, we obtain that there exists a constant $q \in (0,1)$ such that conditionally on $\mathcal{F}_{\epsilon_1}$ and on the event $A$, we have that with probability at least $q$, the curve $\wt{\eta}$ hits $(\epsilon_2^{-1}-\epsilon_2,\epsilon_2^{-1}+\epsilon_2)$ before exiting the $\epsilon_2 / 2$-neighborhood of $\gamma$. The claim then follows possibly by taking $\epsilon_1$ to be smaller since standard estimates for conformal maps imply that $\|g_{\epsilon_1}(z)-z\|\leq 6 \epsilon_1$ for all $z \in \h_{\epsilon_1}$, where~$\h_{\epsilon_1}$ is the unbounded connected component of~$\h \setminus \eta([0,\epsilon_1])$. \noindent{\it Step 2. $\eta$ hits $(1,\infty)$ a.s.} By Step 1, we have that $\eta$ hits $(1,\infty)$ with positive probability. Let $p$ be the probability that $\eta$ hits $(1,\infty)$ and assume that $p \in (0,1)$. Fix $\epsilon \in (0,p)$. Then there exists $T > 0$ so that the probability that $\eta\|_{[0,T]}$ hits $(1,\infty)$ is at least $p-\epsilon$. Let $\tau = \inf\{t \geq T : \eta(t) \in \R_+\}$. On the event that $\eta\|_{[0,\tau]}$ has not hit $(1,\infty)$, we let $\wt{\eta} = (g_\tau(\eta) - W_\tau)/(g_\tau(1) - W_\tau)$. Then we have that~$\wt{\eta}$ hits $(1,\infty)$ with probability $p$. Overall, this implies that the probability that $\eta$ hits $(1,\infty)$ is at least $p - \epsilon + (1-p+\epsilon)p$. By taking $\epsilon > 0$ sufficiently small we see that $p - \epsilon + (1-p+\epsilon)p > p$, which is a contradiction. Therefore $p=1$. \noindent{\it Step 3. $\eta$ does not hit $1$ a.s.} Let $p$ be the probability that $\eta$ hits $1$. By the argument given in Step~1, we have that $p < 1$. Assume that $p > 0$ and fix $q \in (0,1)$ so that $p < q < 1$. Fix $\delta > 0$ sufficiently small so that the probability that $\eta$ hits $B(1,\delta)$ is at most $q$. Let $\tau = \inf\{t \geq 0 : \eta(t) \in B(1,\delta)\}$. On the event that $\tau < \infty$, let $\phi_\tau = (g_\tau - W_\tau)/(V_\tau - W_\tau)$ so that $\wt{\eta} = \phi_\tau(\eta)$ is an $\SLE_\kappa(\rho)$ process in $\h$ from $0$ to $\infty$ with its force point located at $1$. Let $\wt{\tau} = \inf\{t \geq 0 : \wt{\eta}(t) \in (1,\infty)\}$. Let $p_1(\delta)$ be the probability that $\wt{\eta}(\wt{\tau}) \neq \phi_{\tau}(1)$ conditionally on the event that $\tau < \infty$. Then, \cite[Lemma~2.5]{mw2017slepaths} implies that there exists $p_2 > 0$ independent of $\delta$ so that $p_1(\delta) \geq p_2$. Let $p_{1,L}(\delta)$ be the probability that $\wt{\eta}(\wt{\tau}) < \phi_\tau(1)$ and $p_{1,R}(\delta)$ be the probability that $\wt{\eta}(\wt{\tau}) > \phi_\tau(1)$, conditionally on the event that $\tau<\infty$, so that $p_1(\delta) = p_{1,L}(\delta) + p_{1,R}(\delta)$. On the event that $\wt{\eta}(\wt{\tau}) < \phi_\tau(1)$, by the conformal Markov property of $\SLE_\kappa(\rho)$ the probability that it subsequently hits $\phi_\tau(1)$ is $p$. Altogether, we have that \[ p \leq q ( (1-p_1(\delta)) + p_{1,L}(\delta) p) \leq q( (1-p_1(\delta)) + p_1(\delta) p) \leq q(1-p_2(1-p)).\] Since $ 1-p_2(1-p)< 1$, this is a contradiction unless $p = 0$ since by taking $\delta > 0$ small we can assume that $p < q < 1$ is arbitrarily close to $p$. \end{proof} \begin{figure}[ht!] \begin{center} \includegraphics[scale=0.85]{figures/sle_excursion_strip.pdf} \end{center} \caption{\label{fig:sle_excursion_strip} Illustration of the setup to prove Theorem~\ref{thm:law_of_sle_excursion}. Shown in orange is the $\SLE_\kappa(\rho)$ process $\eta$, which we view as the light cone path from $0$ to $i\pi$ coupled with the GFF on $\strip$ with the indicated boundary data. The counterflow lines $\eta_{\mathrm{L}}'$ and $\eta_{\mathrm{R}}'$ are shown in red and blue, respectively, and their common outer boundary gives the completion of the excursion that $\eta$ is drawing at the time $\sigma$ (shown in green).} \end{figure} \begin{proof}[Proof of Theorem~\ref{thm:law_of_sle_excursion}] \noindent{\it Step 1. Setup and outline of the proof.} First we note that Lemma~\ref{lem:sle_separates_points} implies that $T_x < \infty$ a.s. We are going to apply arguments which are similar to those presented in the proofs of \cite[Theorem~5.6]{dms2014mating} and \cite[Proposition~5.12]{ms2016imag2}. We will in particular make use of the coupling of $\eta$ with the $\text{GFF}$ light cone introduced in \cite{ms2019lightcone} (see Section~\ref{subsec:light_cones} for a review of this). First, we apply a conformal transformation $\h \to \strip$ taking $0$ to $0$ and $\infty$ to $i \pi$. Then we can view $\eta$ as the light cone exploration path from $0$ to $i \pi$ associated with a $\text{GFF}$ $h$ on $\strip$ and $h$ has boundary values given by $-\lambda$ on $\R_-$, $\lambda (1+\rho)$ on $\R_+$, $-\lambda - \pi \chi$ on $\R_- + i \pi$, and $\lambda (1+\rho) + \pi \chi$ on $\R_+ + i\pi$. (The reason for making this change of coordinates is that it will be easier to read off the law of different objects associated with the GFF on $\strip$ versus $\h$.) Let $\eta_{\text{L}}'$ be the counterflow line of $h+ \pi \chi/2$ starting from $i \pi$ and targeted at $0$. As usual, we let $\kappa'=16/\kappa$. Then we note that $\eta_{\text{L}}'$ has the law of an $\SLE_{\kappa}(\rho_{\text{L}}' ; \rho_{\text{R}}')$ process where $\rho_{\text{L}}' = \kappa'(1+\rho/4)-4$ and $\rho_{\text{R}}' = \kappa'/2-2$ where the force points are located at $(i\pi)_-$ and $(i\pi)_+$. Indeed, the reason for this is that $\rho_{{\mathrm L}}'$, $\rho_{{\mathrm R}}'$ are determined by the equations \[ \lambda'(1+\rho_{{\mathrm L}}') = \lambda(1+\rho)+ \frac{3\pi\chi}{2} \quad\text{and}\quad -\lambda'(1+\rho_{{\mathrm R}}') = -\lambda -\frac{\pi \chi}{2}.\] Hence, a.s.\ $\eta_{\text{L}}'$ does not intersect either $\R_-$ or $\R_- + i \pi$ since $\kappa'/2-2$ is the critical value below which such processes can hit the boundary. The proof is then divided into three subsequent steps. In Step~2, we will describe the interaction between $\eta_{\text{L}}'$ and the pockets formed by $\eta$. In Step~3, we are going to prove that the joint law of $\eta\|_{[S_x,T_x]}$ together with certain flow lines satisfies the same resampling property as stated in \cite[Proposition~5.12]{ms2016imag2}. Finally, in Step~4, we complete the proof by observing that if $\eta\|_{[S_x,T_x]}$ had the law of an $\SLE_{\kappa}(\rho+2 ; \kappa-4-\rho)$ process, then it would satisfy the same resampling property. \cite[Proposition~5.10]{ms2016imag2} implies that the conditional law of $\eta\|_{[S_x,T_x]}$ is uniquely determined by this resampling property and so this completes the proof of the lemma. \noindent{\it Step 2. Interaction between $\eta_{\text{L}}'$ and the pockets of $\eta$.} We now describe the interaction between $\eta_{\text{L}}'$ and the pockets formed by $\eta$ as described in \cite{ms2019lightcone}. Let $P$ be a pocket of formed by $\eta$ with $u$ (resp.\ $v$) its opening (resp.\ closing) point. Let $\eta_1(P)$ be the $0$-angle flow line part of $\partial P$ and $\eta_2(P)$ be the $\theta_{\rho}$-angle flow line part of $\partial P$ with $\theta_{\rho}$ as in~\eqref{eqn:theta_and_rho}. For $z \in \strip$, we let $\eta_z'$ be the counterflow line of $h+\pi \chi/2$ starting from $i \pi$ and targeted at $z$. Then $\eta_{\text{L}}'$ and $\eta_z'$ agree up until they separate $z$ from $0$ a.s., and the right side of $\eta_z'$ stopped upon hitting $z$ is equal to the flow line of $h$ starting from $z$ with angle $0$. It in particular follows that $\eta_{\text{L}}'$ hits the pocket of $\eta$ containing $z$ provided the flow line of $h$ starting from $z$ exits $\strip$ in $\R_+$ or $\R_+ + i \pi$. Assume that we are on this event. Combining with the flow line interaction rules \cite[Theorem~1.7]{ms2017imag4} with \cite[Section~3]{ms2019lightcone}, we obtain that $\eta_{\text{L}}'$ interacts with $P$ in the following way. \begin{enumerate}[(i)] \item $\eta_{\text{L}}'$ enters the interior of $P$ at $u$ after filling the left side of $\eta_1(P)$ in reverse chronological order, where we view $\eta_1(P)$ as an oriented path from $u$ to $v$. \item Upon intersecting $P$, $\eta_{\text{L}}'$ visits some points on the right side of $\eta_1(P)$ as it travels from $u$ to $v$. It does not touch $\eta_2(P)$ up until hitting $v$. \item Upon hitting $v$, it visits all the points of $\eta_2(P)$ in reverse chronological order and, while doing so, $\eta_{\text{L}}'$ makes excursions both into and out of $P$. \end{enumerate} The above interaction rules imply that $\eta_{\text{L}}'$ contains the pockets $P$ of $\eta$ such that the the flow line corresponding to $\eta_1(P)$ exits $\strip$ in $\R_+$ or $\R_+ + i \pi$ a.s. In particular, if $\eta$ makes an excursion away from where it hits its force point and the excursion terminates in $\R_+$ or $\R_+ + i \pi$, then $\eta_{\text{L}}'$ has to hit it at its tip and then fill the path in reverse chronological order a.s. Hence, the excursion that $\eta$ completes at time $T_x$ is contained in $\eta_{\text{L}}'$ a.s.\ and corresponds to an excursion that $\eta_{\text{L}}'$ makes away from $\R_+ \cup (\R_+ + i \pi)$. \noindent{\it Step 3. Resampling argument.} Suppose that $\sigma$ is a stopping time for $\eta$. We assume that we are working on the event that $\eta$ does not hit its force point at time $\sigma$ and that $S_x \leq \sigma \leq T_x$. We then let~$\eta_{\text{R}}'$ be the counterflow line of the restriction of $h-\frac{\pi \chi}{2}$ to the connected component $D$ of $\strip \setminus \eta([0,\sigma])$ whose boundary contains $i\pi$, which starts from the point corresponding to where~$\eta$ has most recently before time $\sigma$ hit its force point, and it is targeted at $\eta(\sigma)$. Then $\eta_{\text{R}}'$ has the law of an $\SLE_{\kappa}(\rho_{1,\text{L}}';\rho_{1,\text{R}}',\rho_{2,\text{R}}')$ process in $D$ with $\rho_{1,\text{L}}' = \kappa'/2-2$, $\rho_{1,\text{R}}' = -(\kappa'/4)(2+\rho)$, and $\rho_{1,\text{R}}'+\rho_{2,\text{R}}'=\kappa'-4$, and the force points are located immediately to the left and right of its starting point and at $i\pi$. Indeed, the reason for this is that $\rho_{1,\text{L}}'$, $\rho_{1,\text{R}}'$, and $\rho_{2,\text{R}}'$ are determined by the equations \[ \lambda'(1+\rho_{1,\text{L}}') = \lambda + \frac{\pi\chi}{2}, \quad -\lambda'(1+\rho_{1,{\mathrm R}}') = \lambda(1+\rho) + \frac{\pi\chi}{2} ,\quad\text{and}\quad -\lambda'(1+\rho_{1,{\mathrm R}}'+\rho_{2,{\mathrm R}}') = -\lambda- \frac{3\pi \chi}{2}.\] Let $\ol{T}$ be a stopping time for the filtration generated by $\eta\|_{[0,\sigma]}$ and the time-reversal of $\eta\|_{[\sigma,T_x]}$. Note that the segment of $\eta$ from $\eta(\sigma)$ to $\eta(T_x)$ in the clockwise order has boundary conditions $\lambda'+\chi \cdot \text{winding}$ (resp. $-\lambda'+\chi \cdot \text{winding}$) on its right (resp. left) side. Also, the flow line interaction rules imply that the right outer boundary of $\eta_{\text{R}}'$ is given by the flow line of $h\|_D$, starting from $\eta(\sigma)$. Conditionally on $\eta\|_{[0,\sigma]}$, we let $T_q'$ be the first time that $\eta_q'$ hits $\eta(\ol{T})$ for $q \in \{L,R\}$. Then, $\eta([\ol{T},T_x])$ is equal to the common part of the outer boundaries of $\eta_{\text{L}}'([0,T_L'])$ and $\eta_{\text{R}}'([0,T_R'])$. Let $\eta_{\text{L}}$ (resp.\ $\eta_{\text{R}}$) be the right (resp.\ left) outer boundary of $\eta_{\text{L}}'([0,T_{\text{L}}'])$ (resp.\ $\eta_{\text{R}}'([0,T_{\text{R}}'])$). Then, arguing as in the proof of \cite[Lemma~5.13]{ms2016imag2} gives that the conditional law of $\eta\|_{[\sigma,\ol{T}]}$ given $\eta\|_{[0,\sigma]}$, $\eta_{\text{L}}$, $\eta_{\text{R}}$, and $\eta\|_{[\ol{T},T_x]}$, is that of an $\SLE_{\kappa}(\kappa/2-2;\kappa/2-2)$ process in $D$ starting from $\eta(\sigma)$ and targeted at $\eta(\ol{T})$ with the force points located at $i \pi$ and at the left of the starting point of $\eta_{\text{R}}'$. \noindent{\it Step 4. Conclusion of the proof.} Now we are going to identify the law of $\eta\|_{[S_x,T_x]}$ given $\eta\|_{[0,S_x]}$ and $\eta(T_x)$. Conditionally on $\eta\|_{[0,T_{x}]}$, we sample an $\SLE_{\kappa}(\rho+2 ; \kappa-4-\rho)$ process $\wt{\eta}$ in $D$ from~$\eta(\sigma)$ to~$\eta(T_x)$ where the left (resp.\ right) force point is located at $i \pi$ (resp.\ $(\eta(S_x))^+$). Let~$\wt{T}$ be a stopping time for $\wt{\eta}$ and $\wt{T}'$ be a reverse stopping time for $\wt{\eta}$ given $\wt{\eta}\|_{[0,\wt{T}]}$ and assume that we are working on the positive probability event that $\wt{\eta}\|_{[0,\wt{T}]}$ and $\wt{\eta}\|_{[\wt{T}',\infty)}$ are disjoint. Let $A = \eta([0,\sigma]) \cup \wt{\eta}$ and let $\wt{h}$ be $\text{GFF}$ on $\strip \setminus A$ with the same boundary conditions as $h$ on $\partial{(\strip \setminus A)} \setminus \wt{\eta}$ and with boundary conditions along $\wt{\eta}$ as if $\wt{\eta}$ were the flow line of $\wt{h}$ starting from~$\eta(\sigma)$ and targeted at~$\eta(T_x)$. Let $\wt{\eta}_{L}$ (resp.\ $\wt{\eta}_{R}$) be the flow line of $\wt{h}$ starting from $\wt{\eta}(\wt{T}')$ with angle $\pi$ (resp.\ $-\pi$) and targeted at $i\pi$ (resp. the point corresponding to where $\eta$ has most recently before time $\sigma$ hit its force point). Then by construction we have that the conditional law of $\wt{\eta}_{L}$ and $\wt{\eta}_{R}$ given $A$ is the same as the conditional law of $\eta_{L}$ and $\eta_{R}$ given $\eta\|_{[0,T_{u}]}$ in the analogous set up for $\eta$. Moreover, the proof of \cite[Proposition~5.12]{ms2016imag2} implies that the conditional law of $\wt{\eta}\|_{[\wt{T},\wt{T}']}$ given $\wt{\eta}_{L}, \wt{\eta}_{R}, \eta\|_{[0,\sigma]}, \wt{\eta}\|_{[0,\wt{T}]}$ and $\wt{\eta}\|_{[\wt{T}',\infty)}$ is that of an $\SLE_{\kappa}(\kappa/2 - 2; \kappa/2 - 2)$ process in the connected component of $\strip \setminus (\eta([0,\sigma]) \cup \wt{\eta}_{L} \cup \wt{\eta}_{R} \cup \wt{\eta}([0,\wt{T}]))$ with $\wt{\eta}(\wt{T})$ on its boundary from $\wt{\eta}(\wt{T})$ to $\wt{\eta}(\wt{T}')$. Since the stopping time $\sigma$ was arbitrary, the proof will be complete if we manage to prove that this resampling property determines the law of the triple $(\eta, \eta_{L}, \eta_{R})$ conditional on $\eta\|_{[0,T]}$ and $\eta\|_{[T',T_x]}$ where $T,T'$ are the corresponding stopping times for $\eta$. To see this, let $T_{1}, T_{2}$ be two stopping times for $\eta_{L},\eta_{R}$ respectively. Then the law of the triple $(\eta_{1}, \eta_{L}$, $\eta_{R})$ with $\eta_{1}$ = $\eta\|_{[T,T']}$ conditional on $\eta\|_{[0,T]}$, $\eta\|_{[T',T_x]}$ , $\eta_{L}\|_{[0,T_{1}]}$ and $\eta_{R}\|_{[0,T_{2}]}$ is the same as that of ordinary flow lines of a $\text{GFF}$ conditioned on the positive probability event that $\eta_{L}$ and $\eta_{R}$ do not intersect since this conditioned law satisfies the same resampling properties. In particular, the marginal law of $(\eta_{L},\eta_{R})$ satisfies the hypothesis of \cite[Proposition 5.10]{ms2016imag2} with $\theta_{2} = -\pi$ and $\theta_{1} = \pi$, and so its marginal law is uniquely determined. Since we know the conditional law of $\eta_{1}$ given $(\eta_{L},\eta_{R})$, the law of the triple $(\eta_{1}, \eta_{L}, \eta_{R})$ is uniquely determined and this completes the proof. \end{proof} \section{Poissonian structure} \label{sec:main_theorems_proof} The purpose of this section is to prove Theorems~\ref{thm:quantum_natural_time_cutting} and~\ref{thm:boundary_length_evolution} as well as Corollary~\ref{cor:dim_of_boundary_intersection}. Fix $\kappa \in (0,4)$ and $\rho \in (\kappa/2-4,-2) \cap (-2-\kappa/2,-2)$, and set $\gamma = \sqrt{\kappa} \in (0,2)$. The main step in establishing all of these results is to describe the Poissonian structure of the quantum surfaces which are cut out of a quantum wedge of weight $\rho+4$ by an independent $\SLE_{\kappa}(\rho)$ process. Our strategy will be similar to the one followed in \cite[Section~6]{dms2014mating} for the $\rho \in (-2,\kappa/2-2)$ case. However, as discussed in Section~\ref{subsec:reverse_coupling}, there will be some key differences in the proofs since $\SLE_{\kappa}(\rho)$ processes are not simple for the range of $\rho$ that we consider. As in \cite{dms2014mating}, associated with the quantum surfaces which are cut out by the $\SLE_{\kappa}(\rho)$ process is a local time which comes from the encoding of such surfaces using Bessel processes (see \cite[Section~4]{dms2014mating}). By \cite[Proposition~19.12]{dms2014mating}, we have that this local time should be considered as counting the number of small bubbles which have been cut out by the $\SLE_{\kappa}(\rho)$ process. More precisely, it is the limit as $\epsilon \to 0$ of the number of excised bubbles of quantum mass in $[\epsilon,2\epsilon]$, times an appropriate power of $\epsilon$. If we consider the right-continuous inverse of this local time, then we obtain a different time-parameterization for the $\SLE_{\kappa}(\rho)$ process which is intrinsic to the bubbles that it cuts out viewed as quantum surfaces. Following the terminology of \cite{dms2014mating}, we refer to this notion of time as ``quantum natural time''. We will show that when the $\SLE_{\kappa}(\rho)$ process is drawn on top of a certain independent quantum wedge and it is parameterized by quantum natural time, then the law of the surfaces is invariant under zipping according to a fixed amount of quantum natural time, and hence completing the proof of Theorem~\ref{thm:quantum_natural_time_cutting}. Moreover, by combining the previous results with standard properties of stable processes, we will identify the law of the process associated with the evolution of the changes in boundary lengths, and hence complete the proof of Theorem~\ref{thm:boundary_length_evolution}. Finally, we will give the proof of Corollary~\ref{cor:dim_of_boundary_intersection} as a consequence of Theorem~\ref{thm:boundary_length_evolution}. As before, we let $\strip = \R \times (0,\pi)$ be the the strip. We also let $\strip_+ = \R_+ \times (0,\pi)$ be the positive part of the strip and $\strip_- = \R_- \times (0,\pi)$ be the negative part of the strip. \subsection{Strategy} \label{subsec:strategy} We will give an outline of the strategy used to prove Theorem~\ref{thm:quantum_natural_time_cutting}, as it will take quite a bit of work to complete. Let $h$ be a free boundary GFF on~$\h$ with a log singularity at $0$ and which is coupled with a reverse $\SLE_{\kappa}(\wt{\rho})$ process with $\wt{\rho} = \rho + 4$ as in Theorem~\ref{thm:reverse_coupling}. More precisely, we have that $h = \wh{h} - \frac{\rho + 2}{\gamma}\log \|\cdot\|$, where $\wh{h}$ is a free boundary $\text{GFF}$ on $\h$. Next, we let $\eta$ be an $\SLE_{\kappa}(\rho)$ process in $\h$ from $0$ to $\infty$ with a single boundary force point at $0_+$ and which is independent of $h$. We will fix the additive constant for $h$ so that the average of $h \circ f_{T_u}^{-1} + Q \log \|(f_{T_u}^{-1})'\|$ on $\h \cap \partial \D$ is equal to $0$, where $T_u$ is the right-continuous inverse of the local time of $V-W$ at $0$ with $(V,W)$ being the driving pair for~$\eta$, and $(f_t)$ the centered forward Loewner evolution. We will also take $u>0$ to be large but fixed. \begin{enumerate} \item[Step 1.] We will describe the Poissonian structure of the quantum surfaces which are cut out by $\eta\|_{[0,T_u]}$ from $\infty$ ordered reverse chronologically in comparison to how they are first visited by $\eta\|_{[0,T_u]}$. We will in particular show they have the same structure as that of the bubbles arising from a weight $\rho + 2$ quantum wedge. This step will be carried out in Section~\ref{subsec:lightcone_on_free_boundary_gff} just below (see Theorem~\ref{thm:wedge_cutting}). \item[Step 2.] The weight $\rho+2$ quantum wedge structure of these bubbles leads to a quantum measure which is supported on the union of $\eta([0,T_u]) \cap \partial \h$ and the set of self-intersection points of $\eta$. This measure corresponds to the local time of the Bessel process encoding the weight $\rho+2$ wedge. Using this, we will see that the bubbles cut out by $\eta\|_{[0,T_u]}$ (at least near the typical point) ordered according to when they are first visited by $\eta$ have the structure of a weight $\rho+2$ wedge. This will be described in Section~\ref{subsec:natural_time} below. \item[Step 3.] Combining Steps 1 and 2, we will complete the proof of the first part of Theorem~\ref{thm:quantum_natural_time_cutting} by identifying the law of the bubbles which are cut out by $\eta$ when drawn on top of an independent weight $\rho + 4$ quantum wedge. The weight $\rho + 4$ appears because of the combination of the initial singularity at the origin given by $-(2+\rho)/\gamma \log \|\cdot\|$ and an additional $-\gamma (2-d) \log \|\cdot-w\|$ singularity with $d = 1 + (2/\gamma^2)(\rho + 2)$ which arises from observing the field at a quantum typical point $w$ (using the measure from Step 2). This will be described in Section~\ref{subsec:natural_time}. \item[Step 4.] As for the second part of Theorem~\ref{thm:quantum_natural_time_cutting}, it will follow from the fact that if we shift the bubbles near the typical point when viewed as quantum surfaces by a fixed amount of local time, then their Poissonian structure remains the same. This will be described in Section~\ref{subsec:natural_time}. \end{enumerate} Once we have proved Theorem~\ref{thm:quantum_natural_time_cutting}, we will use the scaling properties to determine the boundary length evolution and complete the proof of Theorem~\ref{thm:boundary_length_evolution} in Section~\ref{subsec:boundary_length_evolution}. We will also give the proof of Corollary~\ref{cor:dim_of_boundary_intersection} here. \subsection{Poissonian structure of light cone path on a free boundary GFF} \label{subsec:lightcone_on_free_boundary_gff} We will now carry out Step 1 as described in Section~\ref{subsec:strategy}. That is, we will determine the law of the quantum surfaces which are cut out by an $\SLE_{\kappa}(\rho)$ process for $\kappa \in (0,4)$ and $\rho \in (\kappa/2 - 4, - 2) \cap (- 2 - \kappa/2, - 2)$ drawn on top of a free boundary $\text{GFF}$ with an appropriate log singularity and with the additive constant fixed in a certain way. \subsubsection{Statement} \begin{theorem} \label{thm:wedge_cutting} Fix $\kappa \in (0,4)$, $\rho \in (\kappa/2 - 4 , - 2) \cap ( - 2 - \kappa/2 , - 2)$, $\wt{\rho} = \rho + 4$ and let $\gamma = \sqrt{\kappa}$. Let $\wh{h}$ be a free boundary $\text{GFF}$ on $\h$ and let \begin{align} h = \wh{h} + \frac{(2 - \wt{\rho})}{\gamma}\log\|\cdot \| = \wh{h} - \frac{2 + \rho}{\gamma}\log\| \cdot\| \end{align} Let $\eta$ be an $\SLE_{\kappa}(\rho)$ process starting from $0$ with a single boundary force point of weight $\rho$ located at $0_+$ and assume that $\eta$ is independent of $h$. Let $(f_{t})$ be the centered Loewner flow for $\eta$, $(W , V)$ be its driving process, $L_{t}$ be the local time of $V - W$ at $0$, and let $T_{u} = \inf \{s >0 : L_{s} > u\}$ be the right-continuous inverse of $L_{t}$. Fix $u > 0$ large and assume that the additive constant for $h$ is fixed such that the average of $h \circ f_{T_{u}}^{-1} + Q\log\|(f_{T{u}}^{-1})'\|$ on $\h \cap \partial \D$ is equal to $0$. Then the law of the quantum surfaces parameterized by the bounded components of $\h \setminus \eta([0,T_{u}])$ ordered in reverse from when they are first visited by $\eta$, is equal to those of a weight $\rho + 2$ wedge up to a time $s_{u}$ which tends to $\infty$ in probability as $u \to \infty$. \end{theorem} Note that in our setup there are some subtleties in the proof of Theorem~\ref{thm:wedge_cutting} since the force point of an $\SLE_{\kappa}(\rho)$ process for the range of the values of $\rho$ that we consider might not lie on the boundary of the domain. However the main idea of the proof of \cite[Theorem~6.1]{dms2014mating} still works in our case. \subsubsection{Setup} \label{sec:setup} First, we recall from~\eqref{eqn:bessel_dimension_wrt_weight} that the dimension of the Bessel process $Y$ associated with a wedge of weight $\rho + 2$ is given by: \begin{align} \label{eq:bessel_dimension} \delta = \delta (\kappa , \rho) = 1 + \frac{2\rho + 4}{\kappa} = 1 + \frac{2\rho + 4}{\gamma^{2}} \end{align} Note that the value of $\delta(\kappa , \rho)$ varies in $(0,1)$ as $\rho$ varies in $(-\kappa/2 -2 , -2)$. Let $(g_{t})$ be the family of conformal maps solving the Loewner equation driven by $W$ and for all $t \geq 0$ let $P_{t}$ be the point of last intersection of $\eta$ either with itself or $\partial \h$ by time $t$. Then we know that $X =\kappa^{-\frac{1}{2}}(V - W)$ has the law of a Bessel process starting from zero of dimension $\delta(\kappa,\rho)$ where $V_{t} = g_{t}(P_{t}^+)$. Let $(T_{u})$ be the family of the right-continuous inverses for the local time of $X$ at $0$. We also consider $\wt{\eta}^0$ to be an $\SLE_{\kappa}(\rho)$ in $\h$ from $0$ to $\infty$ which is independent of $W$ and $h$ a free boundary $\text{GFF}$ on $\h$ independent of $(W , \wt{\eta}^0)$. Fix $u > 0$ and note that $T_{u} \to \infty$ a.s.\ as $u \to \infty$. We consider the process $\wt{X} = (\wt{X}_{t})$ with $\wt{X}_{t} = X_{T_{u} -t}$ for $0 \leq t \leq T_{u}$ and $\wt{X}_{t} = X_{t}$ for $t \geq T_{u}$, and with corresponding family of local times $(\wt{L}_{t}^x)$. We also consider the process $\wt{V} = (\wt{V}_{t})$ with $\wt{V}_{t} = V_{T_{u} - t} - V_{T_{u}}$ for $0 \leq t \leq T_{u}$ and $\wt{V}_{t} = - V_{t}$ for $t \geq T_{u}$. Note that the time-reversal symmetries of Bessel processes (see \cite[Proposition~3.1]{dms2014mating}) imply that $X$ and $\wt{X}$ have the same law. It is also easy to see that \[ \wt{V}_{t} = -\frac{2}{\sqrt{\kappa}}\text{P.V.} \int_{0}^{t}\frac{1}{\wt{X}_{s}}ds.\] Thus $\wt{W}$ has the law of a reverse Loewner driving function, where $\wt{W}_t = W_{T_u - t} - W_{T_u}$ for $0 \leq t \leq T_u$ and $\wt{W}_t = -V_t - \sqrt{\kappa}X_t$ for $ t \geq T_u$. Let $(\wt{g}_t)$ be the family of conformal maps solving the reverse Loewner equation driven by $\wt{W}$ with $\wt{g}_0(z) = z$ and $\wt{f}_t(z) = \wt{g}_t(z)-\wt{W}_t$. Let also $\wt{\eta}^{T_{u}}$ be the random curve such that $\wt{\eta}^{T_{u}}\|_{[0,T_{u}]}$ has driving function $W^{T_{u}}$ given by $t \mapsto \wt{W}_{T_{u} - t} - \wt{W}_{T_{u}}$ and $\wt{\eta}^{T_{u}}(s) = \wt{f}_{T_u}(\wt{\eta}^0(s - T_{u}))$ for $s \geq T_{u}$. For $0 \leq s \leq t$ we set $\wt{f}_{s,t} = \wt{f}_{t} \circ (\wt{f}_{s})^{-1}$ and for $0 \leq t \leq T_{u}$ we consider the continuous curve $\wt{\eta}^t$ defined by $\wt{\eta}^t(s) = (\wt{f}_{t,T_{u}})^{-1}(\wt{\eta}^{T_{u}}(T_{u} - t + s))$ for $0 \leq s \leq t$ and $\wt{\eta}^t(s) = \wt{f}_{t}(\wt{\eta}^0(s - t))$ for $s \geq t$. It is easy to check that $W_{s}^t = \wt{W}_{t - s} - \wt{W}_{t}$ and $g_{s}^t(z) = \wt{g}_{t - s}((\wt{f}_{t})^{-1}(z)) - \wt{W}_{t}$ where $W^t$ is the driving function of $\wt{\eta}^t\|_{[0,t]}$ and $(g_{s}^t)$ the conformal maps solving the Loewner equation driven by $W^t$. For all $0 \leq s \leq t \leq T_{u}$, let $P_{s}^t$ be the last point of intersection of $\wt{\eta}^t$ before time $s$ either with itself or with $\partial \h$, and set $V_{s}^t = g_{s}^t(P_{s}^t)$. It is again easy to see that $V_{s}^t = V_{T_{u} - t + s}^{T_{u}} - W_{T_{u} - t}^{T_{u}}$ by applying certain uniqueness arguments for the Loewner equation. Also, the above imply that $\wt{U}_{t} = f_{T_{u} - t}^{T_{u}}(\wt{\eta}^{T_{u}}(s))$ where $s = \sup\{0 \leq r \leq T_{u} - t : \wt{\eta}^{T_{u}}(r) \in \partial \h \cup \wt{\eta}^{T_{u}}([0,r))\}$ and $\wt{U}_{t} = \sqrt{\kappa}\wt{X}_{t}$. By construction, $\wt{\eta}^{T_{u}}$ has the law of an $\SLE_{\kappa}(\rho)$ and so if $P$ is a pocket drawn by $\wt{\eta}^{T_{u}}$ and $S_{1}(P)$ is the left side of $P$ with zero angle boundary conditions, then every point on the right side of $S_{1}(P)$ is hit exactly once. Moreover, since the set of times during which $\wt{\eta}^{T_{u}}$ draws $S_{1}(Q)$ for some pocket $Q$, is dense in $\R_+$, we obtain that $\wt{\eta}^{T_{u}}((T_{u} - t,\infty))$ does not intersect the right side of $\wt{\eta}^{T_{u}}([s,T_{u} - t])$. This shows that $\wt{U}_{t}$ lies on the boundary of the connected component of $\h \setminus f_{T_{u} - t}^{T_{u}}(\wt{\eta}^{T_{u}}([T_{u} - t,\infty)))$ whose boundary contains $0_+$. Also, $\wt{\eta}^t([0,t]) = f_{T_{u} - t}^{T_{u}}(\wt{\eta}^{T_{u}}([T_{u} - t,T_{u}]))$ and $\wt{\eta}^t([t,\infty)) = \wt{f}_{t}(\wt{\eta}^0([0,\infty))) = f_{T_{u} - t}^{T_{u}}(\wt{\eta}^{T_{u}}([T_{u},\infty)))$ and hence $\wt{\eta}^t([0,\infty)) = f_{T_{u} - t}^{T_{u}}(\Tilde{\eta}^{T_{u}}([T_{u}-t,\infty)))$. Therefore $\wt{U}_{t}$ lies on the connected component of $\h \setminus \wt{\eta}^t([0,\infty))$ whose boundary contains the origin. We also observe that $\wt{\eta}^t([t,t + r]) = \wt{f}_{t}(\wt{\eta}^0([0,r]))$ for all $0 \leq t \leq T_{u}$ and for all $r > 0$, and that the hull of $\wt{\eta}^t([0,t])$ is equal to the hull of $\wt{f}_{t}(f_{T_{u}}^{T_{u}}(\wt{\eta}^{T_{u}}([0,T_{u}])))$. But $f_{T_{u}}^{T_{u}}(\wt{\eta}^{T_{u}}([0,T_{u}])) \subseteq \partial \h$ and so the hull of $\wt{\eta}^t([0,t + r])$ is equal to the hull of $\wt{f}_{t}(\partial \h \cup \wt{\eta}^0([0,r]))$. Now set \begin{align} \wt{h}^{T_{u}} = h + \frac{2}{\gamma}\log\|\cdot\| - \frac{\wt{\rho}}{\gamma}\log\|\cdot- \wt{U}_{T_u}\| = h + \frac{2}{\gamma}\log\|\cdot\| - \frac{\wt{\rho}}{\gamma}\log\|\cdot\| \end{align} By Theorem~\ref{thm:reverse_coupling}, we obtain that the random distributions \begin{align}\label{eq:law_equivalence} &h + \frac{2 - \wt{\rho}}{\gamma}\log\|\cdot\|, \nonumber \\ & h \circ \wt{f}_{T_{u}} + \frac{2-\wt{\rho}}{\gamma}\log\|\wt{f}_{T_{u}}\| +Q\log\|(\wt{f}_{T_{u}})'\|, \quad\text{and} \nonumber \\ & h \circ \wt{f}_{t} + \frac{2}{\gamma}\log\|\wt{f}_{t}\| - \frac{\wt{\rho}}{\gamma}\log\|\wt{f}_{t} - \wt{U}_{t}\| +Q\log\|(\wt{f}_{t})'\| \end{align} have the same law for all $0 \leq t \leq T$. Set \begin{align} \wt{h}^t = \wt{h}^{T_{u}} \circ \wt{f}_{t,T_{u}} + Q\log\|(\wt{f}_{t,T_{u}})'\| \end{align} Then, the equality in law in~\eqref{eq:law_equivalence} implies that $\wt{h}^t$ and $h + \frac{2}{\gamma}\log\|z\| - \frac{\wt{\rho}}{\gamma}\log\|z - \wt{U}_{t}\|$ have the same law for all $0 \leq t \leq T_u$. The above observations show that for every fixed $u \in (0,\infty)$, we can find a random family $(\wt{h}^t,\wt{\eta}^t)_{0\leq t \leq T_u}$, where $\wt{\eta}^t$ is a random continuous curve in $\h$ from $0$ to $\infty$ and $\wt{h}^t$ is a random distribution on $\h$ such that the following hold; \begin{enumerate}[(i)] \item $(\wt{\eta}^t)_{0 \leq t \leq T_u}$ is independent of $\wt{h}^0$ and $\wt{\eta}^0$ has the law of an $\SLE_{\kappa}(\rho)$. \item $(W_{T_u - t}^{T_u} - W_{T_u}^{T_u})_{0 \leq t \leq T_u}$ has the law of a reverse Loewner driving flow restricted to $[0,T_u]$ with dimension $\delta(\kappa,\rho)$ and $(\wt{U}_{t}^t)_{0 \leq t \leq T_u}$ has the law of a Bessel process with dimension $\delta(\kappa,\rho)$ and multiplied by $\sqrt{\kappa}$, restricted to $[0,T_u]$, where $\wt{U}_{t}^{T_u} = V_{T_u - t}^{T_u} - W_{T_u - t}^{T_u} + V_{T_u}^{T_u} - W_{T_u}^{T_u}$. Here, $W^{T_u} \|_{[0,T_u]}$ is the Loewner driving function of $\wt{\eta}^{T_u} \|_{[0,T_u]}$. \item If $(\wt{f}_{t}^{T_u})_{0 \leq t \leq T_u}$ is the reverse centered Loewner flow associated with $(W_{T_u - t}^{T_u} - W_{T_u}^{T_u})_{0 \leq t \leq T_u}$, then for all $0 \leq t \leq T_u$ and $r >0 $, the hull of $\Tilde{\eta}^{t}([0,t + r])$ is equal to the hull of $\wt{f}_{t}(\partial \h \cup \wt{\eta}^0([0,r]))$. \item $\wt{h}^s = \wt{h}^t \circ \wt{f}_{s,t}^{T_u} + Q\log\|(\wt{f}_{s,t}^{T_u})'\|$ and $\wt{h}^t$ can be expressed as the sum of a free boundary $\text{GFF}$ on $\h$ plus $\frac{2}{\gamma}\log\|z\| - \frac{\wt{\rho}}{\gamma}\log\|z - \wt{U}_{t}\| $ for all $0 \leq s \leq t \leq T_u$. \item $\wt{U}_{t}^{T_u}$ lies on the boundary of the component of $\h \setminus \wt{\eta}^t([0,\infty))$ whose boundary contains 0 for all $t \in [0,T_u]$. \end{enumerate} The following lemma gives us a way to identify the laws of the fields which arise when performing the zipping and the unzipping operations. It is the analogue of \cite[Lemma~6.2]{dms2014mating}. \begin{lemma}[$\text{\cite[Lemma~6.2]{dms2014mating}}$] \label{lem:extension_lemma} There exists a random family $(\wt{h}^t,\wt{\eta}^t)_{t \geq 0}$ where $\wt{h}^t$ is a distribution on $\h$ and $\wt{\eta}^t$ is a continuous curve on $\overline{\h}$ from $0$ to $\infty$ such that the following hold: \begin{enumerate}[(i)] \item $(\wt{\eta}^t)_{t \geq 0}$ is independent of $\wt{h}^0$ and $\wt{\eta}^0$ has the law of an $\SLE_{\kappa}(\rho)$. \item $\wt{W} = (\wt{W}_{t})_{t \geq 0}$ with $\wt{W}_{t} = -W_{t}^t$ has the law of a reverse Loewner driving function with dimension $\delta$ and let $(\wt{f}_{t})_{t \geq 0}$ be the corresponding centered reverse Loewner flow. Also, $\wt{U} = (\wt{U}_{t})_{t \geq 0}$ with $\wt{U}_{t} = \wt{U}_{t}^t$ has the law of a Bessel process with dimension $\delta$ multiplied by $\sqrt{\kappa}$. \item The hull of $\wt{\eta}^t([0,t + r])$ is equal to the hull of $\wt{f}_{t}(\partial \h \cup \wt{\eta}^0([0,r]))$ for all $t \geq 0$ and $r > 0$. \item $\wt{U}_{t}$ lies on the boundary of the component of $\h \setminus \wt{\eta}^t([0,\infty))$ whose boundary contains the origin. \item $\wt{h}^s = \wt{h}^t \circ \wt{f}_{s,t} + Q\log\|(\wt{f}_{s,t})'\|$ and $\wt{h}^t$ can be expressed as the sum of a free boundary $\text{GFF }$ on $\h$ plus $\frac{2}{\gamma}\log\|z\| - \frac{\wt{\rho}}{\gamma}\log\|z - \wt{U}_{t}\|$ for all $0 \leq s \leq t$. \item $\wt{W} = (\wt{W}_{t})_{t \geq 0}$ is the reverse Loewner driving function corresponding to $\wt{U}$. \end{enumerate} \end{lemma} \begin{proof} Noting that Theorem~\ref{thm:reverse_coupling} is an analogue of \cite[Theorem~5.1]{dms2014mating}, the proof follows from the same argument used to prove \cite[Lemma~6.2]{dms2014mating}. \end{proof} We will assume throughout that $(\wt{h}^t)$ and $(\wt{\eta}^t)$ are as in the statement of Lemma~\ref{lem:extension_lemma}. We fix the additive constant for $\wt{h}^0$, hence $\wt{h}^t$ for all $t \geq 0$, by taking its average on $\h \cap \partial \D$ to be equal to $0$. When shifting the above collection of pairs of fields and curves by a fixed amount of local time, it will be important to identify the law of the resulting pair as seen by the description of our strategy for the proof of Theorem~\ref{thm:quantum_natural_time_cutting}. It turns out that the laws of the pairs are invariant when zipping by a fixed amount of local time. However, we note that for general $t \geq 0$, the curve $\wt{\eta}^t$ does not necessarily have the law of a forward $\SLE_{\kappa}(\rho)$ process. \begin{lemma}[$\text{\cite[Lemma~6.3]{dms2014mating}}$] \label{lem:equality_in_law} Suppose that we have the setup as described above. Let $\wt{L}_{t}$ denote the local time of $\wt{U}_{t}$ at $0$ and, for each $u > 0$, we let $\wt{T}_{u}$ be the right-continuous inverse of $\wt{L}_{t}$. For each $u > 0$, we have that $\wt{\eta}^{\wt{T}_{u}}$ has the law of a forward $\SLE_{\kappa}(\rho)$ process. In particular, the laws of the fields $(\wt{h}^{\wt{T}_{u}},\wt{\eta}^{\wt{T}_{u}})$ and $ (\wt{h}^0 , \wt{\eta}^0)$ are the same, where $\wt{h}^{\wt{T}_{u}}$ is viewed as a distribution modulo additive constant. \end{lemma} \begin{proof} It follows from the proof of \cite[Lemma~6.3]{dms2014mating}. \end{proof} We are now going to define the objects which are used in the proof of Theorem~\ref{thm:wedge_cutting}. We will follow the same terminology as in \cite[Section~6.2]{dms2014mating}. For each $t \geq 0$ such that $\wt{U}_{t} \neq 0$, we let $\wt{B}_{t}$ denote the component of $\h \setminus \wt{\eta}^t([0,\infty))$ with $\wt{U}_{t}$ on its boundary. If $\wt{U}_{t} = 0$, we take $\wt{B}_{t} = \emptyset$. $\wt{B}_{t}$ is a Jordan domain as its boundary consists of part of the curve $\wt{\eta}^t$ and an interval in $\partial \h$. For $t\geq 0$ such that $\wt{B}_{t} \neq \emptyset$, we let \begin{enumerate}[(i)] \item $\wt{\zeta}_{t}$ be the closing point of $\partial{\wt{B}_{t}}$ (the opening point is $0$). \item $\wt{\varphi}_{t}: \wt{B}_{t}\rightarrow \strip$ be the unique conformal transformation which takes $\wt{U}_{t}$ to $+\infty$, $0$ to $0$, and $\wt{\zeta}_{t}$ to $-\infty$. \item $\wh{h}^t = \wt{h}^t \circ \wt{\varphi}_{t}^{-1} + Q\log\|(\wt{\varphi}_{t}^{-1})'\|$. \item $\wh{X}^t$ be the projection of $\wh{h}^t$ onto $\CH_{1}(\strip)$ \end{enumerate} We note that in the setup of Lemma~\ref{lem:extension_lemma}, the transformation from $(\wt{h}^s,\wt{\eta}^s)$ to $(\wt{h}^t,\wt{\eta}^t)$ for $t>s$ corresponds to zipping up for $t-s$ units of capacity time and the transformation from $(\wt{h}^s,\wt{\eta}^s)$ to $(\wt{h}^t,\wt{\eta}^t)$ for $t<s$ corresponds to unzipping for $s-t$ units of capacity time. The reason that we consider the quantum surfaces parameterized by the domains $\wt{B}_t$ is the following. When we perform the forward Loewner flow at times when the $\SLE_{\kappa}(\rho)$ process completes an entire excursion either away from $\partial \h$ or away from itself, we have that a region with finite and positive quantum area parameterized by some $\wt{B}_t$ is cut out from $\infty$. If we apply the reverse Loewner flow, then a region with positive quantum area parameterized by some $\wt{B}_t$ is added all at once. Moreover, we have that every quantum surface parameterized by a bubble lying to the right of the $\SLE_{\kappa}(\rho)$ process $\eta$ will correspond to a newly zipped in quantum surface parameterized by some $\wt{B}_t$ (modulo coordinate change). Furthermore, the latter is determined by the values of the field in a very small region and the conditional law of the field given its values in that region is very close to the unconditioned law. The latter fact will lead to the quantum surfaces which are cut out by $\eta$ to be independent. \subsubsection{Filtration and stopping times} First, we start by constructing the filtration and stopping times that we are going to consider. The definitions and notation are similar to those in \cite[Section~6.2.3]{dms2014mating}. Suppose that we have the setup of Lemma~\ref{lem:extension_lemma}. When performing the zipping and the unzipping operations, we need to keep track of the path and the field as well. However, as we mentioned at the end of Section~\ref{sec:setup}, we will only need to know the values of the field outside of the region that is currently zipped in. Also, it will be convenient to use $\strip$ as the underlying domain. We also note that the dimension of the Bessel process encoding the bubbles which are zipped in is determined by the form of the projection of the field which describes such a bubble onto $\mathcal{H}_1(\strip)$ with fixed additive constant and it does not depend on its projection onto $\mathcal{H}_2(\strip)$. The above observations motivate to consider the following filtration. Consider the process $\wt{V} = (\wt{V}_{s})_{s \geq 0}$ with $\wt{V}_{s} = \wt{U}_{s} + \wt{W}_{s}$ and for $t \geq 0$ we let $\wt{\CF_{t}}$ be the $\sigma$-algebra with respect to which the following are measurable: \begin{enumerate}[(i)] \item Both $\eta = \wt{\eta}^0$ and the driving pair $(\wt{W}_{s},\wt{V}_{s})$ of the reverse Loewner flow for $s \leq t$. \item The field $\wt{h}^t$ restricted to $\h \setminus \wt{B}_{t}$ (i.e., $\wt{h}^t$ restricted to any component of $\h \setminus \wt{\eta}^t$ other than the one currently been generated). \item The restriction of $\wh{h}^t$ to $\strip_-$. \end{enumerate} Note that $\wt{\eta}^t([0,t]) = f_{T_u - t}^{T_u}(\wt{\eta}^{T_u}([T_u - t , T_u]))$ for $t < T_u$ and so $\wt{\eta}^t([0,t])$ is determined by $W_{s+T_u -t}^{T_u}-W_{T_u - t}^{T_u}, s \in [0,t]$. But $\wt{\eta}^{T_u}$ hits either itself or $\R_+$ at time $T_u$ and so $\wt{B}_t$ is a connected component of $\h \setminus \wt{\eta}^t([0,t])$. Also, $W_{s+T_u - t}^{T_u} - W_{T_u}^{T_u} = \wt{W}_{t-s}-\wt{W}_t$ for all $s \in [0,t]$ and so $\wt{B}_t$ is determined by $(\wt{W}_s,\wt{V}_s), 0 \leq s \leq t$. It follows that $\wt{\zeta}_t,\wt{\varphi}_t$, and $\wh{X}^t\|_{\R_-}$ are determined by $\wt{\CF}_t$. Moreover, by arguing as in \cite[Section~6.2.3]{dms2014mating}, we obtain that $(\wt{\CF}_{t})$ is a filtration. Next we fix $t_{1},t_{2} \in (0,\infty)$ such that $\wt{U}_{t_{1}} = \wt{U}_{t_{2}} = 0$ and $\wt{U}_{r} > 0 $ for all $r \in (t_{1},t_{2})$. Fix $t_{0} \in (t_{1},t_{2})$. Then the quantum length of $\partial{\wt{B}_{t}}$ with respect to $\wt{h}^t$ remains the same for all $t \in (t_{1},t_{2})$, since $\wt{h}^t\|_{\wt{B}_{t}}$ and $\wt{h}^s\|_{\wt{B}_{s}}$ remain equivalent as surfaces for all $t,s \in (t_{1},t_{2}) $. Also the same holds for the quantum length of $[\wt{U}_{t},\wt{\zeta}_{t}]$ with respect to $\wt{h}^t$ for $t \in (t_{1},t_{2})$. This shows that the quantum length of $\partial{\wt{B}_{t}} \setminus \partial{\h}$ tends to $0$ as $t \to t_{1}$ and it tends to the quantum length of the clockwise part of $\partial{\wt{B}_{t_{0}}}$ from $\wt{U}_{t_{0}}$ to $\wt{\zeta}_{t_{0}}$ as $t \to t_{2}$. Hence if $\alpha(t)$ is such that $\wh{h}^t$ comes by translating $\wh{h}^{t_{0}}$ horizontally by $\alpha(t)$ units, then $\alpha(t) \to -\infty$ as $t \to t_{1}$ and $\alpha(t) \to +\infty$ as $t \to t_{2}$, and $\alpha$ is continuous in $t$. Therefore if $\sup_{u \in \R}(\wh{X}^{t_{0}}_{u}) \geq r$, then there exists $t \in (t_{1},t_{2})$ such that $\inf\{u \in \R : \wh{X}^{t}_{u} = r\} =0$. Now we are ready to define the stopping times that we are going to consider. We note that our goal is to analyze the structure of the bubbles which are zipped in when viewed as quantum surfaces modulo coordinate change. It will be convenient for the proof of Theorem~\ref{thm:quantum_natural_time_cutting} to analyze the structure of a given bubble which has only been partially zipped in. Also, we will first analyze the collection of ``large bubbles'' in the sense of quantum area and then obtain an asymptotic coupling of these bubbles with the bubbles cut out from the $\SLE_{\kappa}(\rho)$ process as their quantum areas tend to zero. Following again \cite[Section~6.2.3]{dms2014mating}, we define the stopping times as follows. Fix $\epsilon > 0, \overline{\epsilon} > 0$ and $r \in \R$. We define stopping times inductively as follows. We let $\wt{\tau_{1}}^{\epsilon,\overline{\epsilon},r}$ be the first time $t \geq 0$ such that the following hold simultaneously: \begin{enumerate}[(i)] \item The quantum length of $\partial{\wt{B}_{t}} \setminus \partial{\h}$ is at least $\epsilon$ with respect to the $\gamma$-$\text{LQG}$ length measure induced by $\wt{h}^t$, \item $\inf\{u \in \R : \wh{X}^{t}_{u} = r\} =0$, and \item The $\gamma$-$\text{LQG}$ mass of $\strip_-$ associated with the field $\wh{h}^t$ is at least $\overline{\epsilon}$. \end{enumerate} We note that the time $\wt{\tau}_{1}^{\epsilon,\overline{\epsilon},r}$ is a.s.\ finite since our previous observations imply that if $\sup_{u \in \R}\wh{X}^{t}_{u}$ is at least $r$, there will be a time $s$ such that $\wh{X}^s$ first hits $r$ at $u = 0$. We then let $\wt{\sigma}_{1}^{\epsilon,\overline{\epsilon},r}$ be the first time $t$ after time $\wt{\tau}_{1}^{\epsilon,\overline{\epsilon},r}$ that $\wt{U}_{t} = 0$. Given that $\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r},\wt{\sigma}_{j}^{\epsilon,\overline{\epsilon},r}$ for $1 \leq j \leq k$ have been defined for some $k \in \N$, we let $\wt{\tau}_{k + 1}^{\epsilon,\overline{\epsilon},r}$ be defined in exactly the same way as $\wt{\tau}_{1}^{\epsilon,\overline{\epsilon},r}$ except with $t \geq \wt{\sigma}_{k}^{\epsilon,\overline{\epsilon},r}$. We then let $\wt{\sigma}_{k}^{\epsilon,\overline{\epsilon},r}$ be the first time $t$ after time $\wt{\tau}_{k + 1}^{\epsilon,\overline{\epsilon},r}$ that $\wt{U}_{t} = 0$. For each $j$, we let $\wh{X}^{j,\epsilon,\overline{\epsilon},r}$ be given by $\wh{X}^{\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r}}$ and $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ be given by $\wh{h}^{\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r}}$. We emphasize that the process $ \wh{X}_{u}^{j,\epsilon,\overline{\epsilon},r}$ hits $r$ for the first time at time $0$. As we mentioned earlier, the law of the sequences $(\wh{X}^{j,\epsilon,\overline{\epsilon},r})$ and $(\wh{X}^{j,\overline{\epsilon},r}) = (\wh{X}^{j,0,\overline{\epsilon},r})$ will determine the dimension of the Bessel process encoding the quantum surfaces parameterized by the bubbles which are zipped in. In particular, we have that $(\wh{X}^{j,\epsilon,\overline{\epsilon},r})$ and $(\wh{X}^{j,\overline{\epsilon},r})$ are both given by independent Brownian motions with a common downward drift. Following \cite[Section~4]{dms2014mating}, we have that each such Brownian motion corresponds to an excursion of the Bessel process away from $0$. We record the result about the form of the drift in the following proposition. \begin{proposition}[$\text{\cite[Proposition~6.4]{dms2014mating}}$]\label{prop:form_of_drift} Let \begin{align} \alpha = \frac{\rho + 4}{\gamma} - Q = \frac{\rho + 2}{\gamma} - \frac{\gamma}{2}. \end{align} Let $\wh{B}_{2t}^{j,\epsilon,\wt{\epsilon},r} = \alpha t - \wh{X}_{t}^{j,\epsilon,\wt{\epsilon},r}$ for $t \geq 0$. Given $\wt{\CF}_{\tau_{j}^{r,\epsilon,\overline{\epsilon}}}$, the process $\wh{B}^{j,\epsilon,\overline{\epsilon},r}$ is a standard Brownian motion with $\wh{B}_{0}^{j,\epsilon,\overline{\epsilon},r} = -r$. \end{proposition} \begin{proof} It follows from the proof of \cite[Proposition~6.4]{dms2014mating}. \end{proof} \subsubsection{Proof of Theorem~\ref{thm:wedge_cutting}} Proposition~\ref{prop:form_of_drift} implies that the projections of the fields $(\wh{h}_j^{\epsilon,\overline{\epsilon},r})_{j \in \N}$ onto $\mathcal{H}_1(\strip)$ and restricted to $\strip_+$ are i.i.d. and they are related to the excursions of a Bessel process from $0$. However, in order to prove Theorem~\ref{thm:wedge_cutting}, we will need to know the dependency between the projections of the fields onto $\mathcal{H}_2(\strip)$ and restricted to $\strip_+$. Note that the Markov property of the $\text{GFF}$ implies that conditionally on $\wt{\CF}_{\wt{\tau}_j^{\epsilon,\overline{\epsilon},r}}$, the field $\wh{h}_j^{\epsilon,\overline{\epsilon},r}\|_{\strip_+}$ can be expressed as the sum of a $\text{GFF}$ in $\strip_+$ with Dirichlet boundary conditions on $[0,\pi i]$ and free boundary conditions on $\partial \strip_+ \setminus [0,\pi i]$. Hence, it follows that the way the field $\wh{h}_j^{\epsilon,\overline{\epsilon},r}\|_{\strip_+}$ is correlated with the fields $\wh{h}_1^{\epsilon,\overline{\epsilon},r},\cdots,\wh{h}_{j-1}^{\epsilon,\overline{\epsilon},r}$ is encoded by the function which is harmonic in $\strip_+$ with boundary conditions given by the values of $\wh{h}_j^{\epsilon,\overline{\epsilon},r}$ on $[0,\pi i]$ and Neumann boundary conditions on $\partial \strip_+ \setminus [0,\pi i]$. Next we follow the notation and the strategy of \cite[Section~6.2.5]{dms2014mating}. First, we note that the quantum length of $(-\infty,0]$ with respect to $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ is at least $\epsilon > 0$ by the definition of $\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r}$, and so there exists a unique $\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r} \in (-\infty,0]$ such that the quantum length of $(-\infty,\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r}]$ with respect to $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ is equal to $\epsilon > 0$. Also, a.s.\ there exist unique $t_{j,1}^{\epsilon,\overline{\epsilon},r},t_{j,2}^{\epsilon,\overline{\epsilon},r} > 0$ such that $\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r} \in (t_{j,1}^{\epsilon,\overline{\epsilon},r},t_{j,2}^{\epsilon,\overline{\epsilon},r})$ and $\wt{U}_{t_{j,1}^{\epsilon,\overline{\epsilon},r}} = \wt{U}_{t_{j,2}^{\epsilon,\overline{\epsilon},r}} = 0$. We let $t_{j}^{\epsilon,\overline{\epsilon},r}$ be the first time $t$ in $(t_{j,1}^{\epsilon,\overline{\epsilon},r},\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r}]$ such that the quantum length of $\partial{\wt{B}_{t}} \setminus \partial{\h}$ with respect to $\wt{h}^t$ is equal to $\epsilon > 0$. Equivalently, $t_{j}^{\epsilon,\overline{\epsilon},r}$ is the first time $t$ in $(t_{j,1}^{\epsilon,\overline{\epsilon},r},\wt{\tau}_{j}^{\epsilon,\overline{\epsilon},r}]$ such that the quantum length of $(-\infty,0]$ with respect to $\wh{h}^t$ is equal to $\epsilon > 0$. Note that $\wh{h}^{t_{j}^{\epsilon,\overline{\epsilon},r}}$ can be derived by translating $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ horizontally by $\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r}$ units. Let $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}$ be the function which is harmonic in $[\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r},+\infty) \times [0,i\pi]$ with Neumann boundary conditions on the horizontal parts of the strip boundary and at $+\infty$, and Dirichlet boundary conditions on $\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r} + [0,i\pi]$ with boundary values given by those of $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$. Then $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}$ determines the harmonic part of $\wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ on $[\wh{\omega}_{j}^{\epsilon,\overline{\epsilon},r},+\infty) \times [0,i\pi]$. In particular, $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}$ determines the harmonic part of $\wh{h}_{j}^{t_{j}^{\epsilon,\overline{\epsilon},r}}$ on $\strip_+$. By applying the Markov property to $\wh{h}_{j}^{t_{j}^{\epsilon,\overline{\epsilon},r}}$, we obtain that conditional on $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}$, the restriction of $\wh{h}_{j}^{t_{j}^{\epsilon,\overline{\epsilon},r}}$ to $\strip_-$ has the law of a zero boundary $\text{GFF}$ on $\strip_-$ plus the harmonic extension to $\strip_-$ of the function whose boundary values coincide with those of $\wh{h}_{j}^{t_{j}^{\epsilon,\overline{\epsilon},r}}$. Since the zero boundary part is independent of of $\wh{h}_{1}^{\epsilon,\overline{\epsilon},r},\cdots,\wh{h}_{j - 1}^{\epsilon,\overline{\epsilon},r}$ and the harmonic part is determined by $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}$, we obtain that conditional on $\wh{\psi}_{j}^{\epsilon,\overline{\epsilon},r}, \wh{h}_{j}^{\epsilon,\overline{\epsilon},r}$ is independent of $\wh{h}_{1}^{\epsilon,\overline{\epsilon},r},\cdots,\wh{h}_{j -1 }^{\epsilon,\overline{\epsilon},r}$. As we explained above, if we show that the functions $\wh{\psi}_j^{\epsilon,\overline{\epsilon},r}$ become trivial as $\epsilon \to 0$, then this will imply that the bubbles which are successively zipped in are independent quantum surfaces. This is the purpose of the next lemma. \begin{lemma}[$\text{\cite[Lemma~6.6]{dms2014mating}}$] Fix $\overline{\epsilon} > 0, j \in \N$ and $r \in \R$. There exists a sequence $(\epsilon_{k})$ of positive numbers decreasing to $0$ such that $\wh{\psi}_{j}^{\epsilon_{k},\overline{\epsilon},r}$ a.s.\ converges to the $0$ function on $\strip_+$ as $k \to +\infty$ with respect to the topology of local uniform convergence modulo a global additive constant. \end{lemma} \begin{proof} The proof follows essentially from the same argument used in the proof of \cite[Lemma~6.6]{dms2014mating}. Nevertheless, we are going to emphasize the parts of the proof where Lemma~\ref{lem:sle_separates_points} and Theorem~\ref{thm:law_of_sle_excursion} are used. Fix $u > 0$ large. For $x > 0$ rational, we let $\CV_{x}$ be the connected component of $\h \setminus \wt{\eta}^{\wt{T}_{u}}$ with $x$ on its boundary. Note that $\wt{\eta}^{\wt{T}_{u}}$ has the law of an $\SLE_{\kappa}(\rho)$ process and so by Lemma~\ref{lem:sle_separates_points} we obtain that a.s.\ such connected component exists for all $x \in \Q_+$. We fix $\epsilon >0$ and we assume that we are working on the event that the quantum length of $\partial{\CV}_x \setminus \partial{\h}$ with respect to $\wt{h}^{\wt{T}_{u}}$ is at least $\epsilon > 0$. Note that the probability of this event tends to 1 as $\epsilon \to 0$ for $x \in \Q_+$ fixed. We let $\xi$(resp.\ $\zeta$) be the first (resp.\ last) time that $\wt{\eta}^{\wt{T}_{u}}$ hits a point on $\partial{\CV_x}$ and let $\varphi^{\epsilon} : \CV_x \rightarrow \strip$ be a conformal transformation with $\varphi^{\epsilon}(\wt{\eta}^{\wt{T}_{u}}(\xi)) = +\infty$ and $\varphi^{\epsilon}(\wt{\eta}^{\wt{T}_{u}}(\zeta)) = -\infty$. Note that $\varphi^{\epsilon}$ is determined up to a horizontal translation, so we can determine $\varphi^{\epsilon}$ by requiring that the $\gamma$-$\text{LQG}$ boundary measure of $(-\infty,0]$ associated with $\wt{h}^{\wt{T}_{u}} \circ (\varphi^{\epsilon})^{-1} + Q\log\|((\varphi^{\epsilon})^{-1})'\|$ is equal to $\epsilon$. Let $\psi^{\epsilon}$ be the function which is harmonic in $\strip_+$ with Neumann boundary conditions on $\partial{\strip_+} \setminus [0,i\pi]$ and Dirichlet boundary conditions on $[0,i\pi]$ with boundary values given by the values of $\wt{h}^{\wt{T}_{u}} \circ (\varphi^{\epsilon})^{-1} + Q\log\|((\varphi^{\epsilon})^{-1})'\|$ on $[0,i\pi]$. Then, arguing as in the proof of \cite[Lemma~6.6]{dms2014mating}, we obtain that if we show that as $\epsilon \to 0$, the law of $\psi^{\epsilon}$ converges weakly with respect to the topology of local uniform convergence in $\strip_+$ modulo a global additive constant to a function which is harmonic in $\strip_+$, then we have that $\wh{\psi}_j^{\epsilon,\overline{\epsilon},r}$ will converge to a constant. Therefore, it suffices to show the convergence of $\psi^{\epsilon}$ as $\epsilon \to 0$. We are going to deduce this from Theorem~\ref{thm:law_of_sle_excursion} and the independence of $\wt{\eta}^{\wt{T}_{u}}$ and $\wt{h}^{\wt{T}_{u}}$ (viewed modulo additive constant). For each $\delta > 0$ we let $\eta^{\delta}(t) = \delta^{-1} (\wt{\eta}^{\wt{T}_{u}}(\zeta - t) - \wt{\eta}^{\wt{T}_{u}}(\zeta))$. Then Theorem~\ref{thm:law_of_sle_excursion} combined with the time-reversal symmetries of $\SLE_{\kappa}(\rho_1 ; \rho_2)$ processes (\cite[Theorem 1.1]{ms2016imag2}) imply that the law of $\eta^{\delta}$ converges as $\delta \to 0$ to the law of a continuous curve in $\overline{\h}$. We note that $(\wt{h}^{\wt{T}_{u}},\wt{\eta}^{\wt{T}_{u}})$ and $(\wt{h}^0,\wt{\eta}^0)$ have the same law, where we view the distributions modulo additive constants. Let $h^{\delta}$ be the field which arises by precomposing $\wt{h}^{\wt{T}_{u}}$ with $z \mapsto \delta (z + \wt{\eta}^{\wt{T}_{u}}(\zeta))$. Then $h^{\delta}$ (modulo additive constants) converges as $\delta \to 0$ to a free boundary $\text{GFF}$ on $\h$ by the scale and translation invariance of free boundary $\text{GFF}$s and the independence of $\wt{h}^{\wt{T}_{u}}$ (modulo additive constant) and $\wt{\eta}^{\wt{T}_{u}}$. Therefore $(\eta^{\delta},h^{\delta})$ converges in law as $\delta \to 0$ to the law of a pair consisting of a continuous curve in $\overline{\h}$ and an independent free boundary $\text{GFF}$ on $\h$. For each $t \in (0,\zeta - \xi)$, we let $\wt{\varphi}_t^{\epsilon}$ be the conformal transformation mapping $\h \setminus \wt{\eta}^{\wt{T}_u}([\zeta-t,\zeta])$ onto $\strip$ with $\wt{\eta}^{\wt{T}_u}(\zeta)$ mapped to $-\infty$ and $\wt{\eta}^{\wt{T}_u}(\zeta-t)$ mapped to $+\infty$, and the horizontal translation fixed so that the $\gamma-\text{LQG}$ boundary length of $(-\infty,0]$ with respect to $\wt{h}^{\wt{T}_u} \circ (\wt{\varphi}_t^{\epsilon})^{-1} + Q \log \|((\wt{\varphi}_t^{\epsilon})^{-1})'\|$ is equal to $\epsilon$. Then, arguing as in the proof of \cite[Lemma~6.6]{dms2014mating}, we obtain that as $t,\epsilon \to 0$ with $\epsilon \to 0$ at a sufficiently fast rate relative to the rate at which $t \to 0$, the law of the function which is harmonic in $\strip_+$ with Neumann boundary conditions on $\partial \strip_+ \setminus [0,\pi i]$ and with Dirichlet boundary conditions on $[0,\pi i]$ given by the values of $\wt{h}^{\wt{T}_u} \circ (\wt{\varphi}_t^{\epsilon})^{-1} + Q \log \|((\wt{\varphi}_t^{\epsilon})^{-1})'\|$ on $[0,\pi i]$ converges weakly with respect to the topology of local uniform convergence modulo global additive constant. Finally, arguing as in the last paragraph of the proof of \cite[Lemma~6.6]{dms2014mating} gives that the total variation distance between the fields $\wt{h}^{\wt{T}_u} \circ (\wt{\varphi}_t^{\epsilon})^{-1} + Q \log \|((\wt{\varphi}_t^{\epsilon})^{-1})'\|$ and $\wt{h}^{\wt{T}_u} \circ (\varphi^{\epsilon})^{-1} + Q \log \|((\varphi^{\epsilon})^{-1})'\|$ tends to $0$ as $t \to 0$ and $\epsilon$ tends to $0$ much faster than $t$. This proves the convergence of $\varphi^{\epsilon}$ and hence the convergence of $\psi^{\epsilon}$. This completes the proof of the lemma. \end{proof} Now we are ready to prove Theorem~\ref{thm:wedge_cutting}. \begin{proof}[Proof of Theorem~\ref{thm:wedge_cutting}] We will be brief since the proof is essentially the same as that of \cite[Theorem~6.1]{dms2014mating}. Fix $u>0$ large and let $Y$ be a Bessel process of dimension $\delta(\kappa,\rho)$ as in~\eqref{eq:bessel_dimension}. As in the proof of \cite[Theorem~6.1]{dms2014mating}, we can use the previous results to obtain an asymptotic coupling where the excursions of $Y$ away from $0$ up until some time $s_u$ each correspond to a connected component of $\h \setminus \wt{\eta}^{\wt{T}_u}$ which is to the left of $X_u$ and this correspondence is bijective, where $X_u$ is the last intersection point of $\wt{\eta}^{\wt{T}_u}$ with $\R_+$ before time $\wt{T}_u$. Note also that $Y$ encodes these bubbles as quantum surfaces from right to left. Moreover, by the construction of the coupling, we have that $\wt{U}_t = V_{T_u-t}^{T_u}-W_{T_u-t}^{T_u}$ for all $t \in [0,T_u]$, where $T_u$ is the right continuous inverse of the local time at $0$ of $V^{T_u} - W^{T_u}$ at $u$. Hence $T_u = \wt{T}_u$ a.s.\ and $\wt{h}^{\wt{T}_u}$ is independent of $\wt{\eta}^{\wt{T}_u}$ with the law of the restriction of $\wt{h}^{\wt{T}_u}$ to the bubbles of $\h \setminus \wt{\eta}^{\wt{T}_u}([0,\wt{T}_u])$ ordered from right to left to be determined by the law of $Y$. This completes the proof of the theorem. \end{proof} \subsection{Zipping according to quantum natural time} \label{subsec:natural_time} Suppose that we have the setup of Theorem~\ref{thm:wedge_cutting}. Then Theorem~\ref{thm:wedge_cutting} implies that if $u>0$ is sufficiently large but fixed, we have that the quantum surfaces which are cut out by $\eta\|_{[0,T_u]}$ from $\infty$ and they are to the right of $\eta$ have the same Poissonian structure as those arising from a weight $\rho + 2$ quantum wedge. Let $Y$ be the Bessel process encoding the law of the above bubbles when viewed as quantum surfaces modulo coordinate change. Recall that $Y$ induces a quantum measure $m$ on the union of $\eta([0,T_u]) \cap \R_+$ with the set of self-intersection points of $\eta\|_{[0,T_u]}$. In Lemma~\ref{lem:main_lemma}, we will identify the conditional law of the pair $(h,\eta)$ when we sample $w$ from $m$ and condition on it, and then show that when we zoom in near $w$ as in \cite[Proposition~4.7]{dms2014mating}, we have that the beaded quantum surface which consists of the connected components of $\h \setminus \eta$ lying to the right of $w$ converges to that of a quantum wedge of weight $\rho + 2$. However, when we condition on $w$, we add to the field an extra log singularity at $w$ and the law of the curve is weighted by a certain Radon-Nikodym derivative. This leads to the weight $\rho + 4$ that we eventually obtain when we zoom in near $w$. Combining, we obtain part (i) of Theorem~\ref{thm:quantum_natural_time_cutting}. Finally, part (ii) of Theorem~\ref{thm:quantum_natural_time_cutting} will follow by observing that the total variation distance between the laws of the pairs of the fields and curves tends to $0$ as $u \to \infty$ when we zoom in near $w$ and when we zoom in near the point obtained by shifting $w$ according to a fixed amount of quantum measure. As we mentioned in the previous paragraph, we will have to deal with fields perturbed by adding certain smooth functions. Therefore, it will be useful to know how the quantum measure described above changes when we add such a function. This is the content of the next lemma. \begin{lemma}[$\text{\cite[Lemma~6.19]{dms2014mating}}$] \label{lem:smooth_perturbation} Assume that we have the same setup as in Theorem~\ref{thm:wedge_cutting}(with the additive constant for $h$ fixed in the same way) and let $L$ be the local time at $0$ associated with the Bessel process $Y$ which encodes the weight $\rho + 2$ wedge corresponding to the bounded components of $\h \setminus \eta([0,T_{u}])$. For each smooth function $\phi$, let $Y^{\phi}$ be the corresponding process with $h$ replaced by $h + \phi$ (which we assume to be parameterized according to its quadratic variation). Let $\overline{\nu}$ (resp.\ $\overline{\nu}^{\phi}$) denote the empirical measure of the excursions made by $Y$ (resp.\ $Y^{\phi}$) from $0$ which correspond to the bubbles cut off $\infty$ by $\eta([T_{r},T_{u}])$. Let $d$ be the Bessel dimension of $Y$. We a.s.\ have (off a common set of measure $0$) for all $r \in [0,u]$ and all such smooth functions $\phi$ that the limit $m^{\phi}([r,u])$ of $\frac{\overline{\nu}^{\phi}(\CE(\epsilon))}{\nu_{d}^{\BES}(\CE(\epsilon))}$ as $\epsilon \to 0$ exists and (with $m = m^0$) \begin{align} m^{\phi}([r,u]) = \int_{r}^{u}\exp{ \left(\frac{\gamma(2 - d)}{2}\phi(\eta(T_{v}))\right)dm(v)}. \end{align} \end{lemma} \begin{proof} It follows from the proof of \cite[Lemma~6.19]{dms2014mating}. \end{proof} We note that Lemma~\ref{lem:smooth_perturbation} implies that the law of $Y^{\phi}$ is mutually absolutely continuous with respect to the law of $Y$ and moreover the local time of $Y^{\phi}$ at $0$ is defined for all smooth $\phi$ simultaneously. Now, we are ready to state and prove Lemma~\ref{lem:main_lemma}. \begin{lemma}[$\text{\cite[Lemma~6.21]{dms2014mating}}$] \label{lem:main_lemma} Assume that we have the setup of Theorem~\ref{thm:wedge_cutting} (with the additive constant for $h$ fixed in the same way) and recall that \begin{align} \label{eq:law_of_the _field} h = \wh{h} - \frac{2 + \rho}{\gamma}\log\|\cdot\| \end{align} where $\wh{h}$ is a free boundary $\text{GFF}$ on $\h$. For each $0 \leq q < r \leq u$, we let $m_{q,r}$ be the restriction of $m$ as in Lemma~\ref{lem:smooth_perturbation} to subsets of $[q,r]$. Consider the law on $(w,h,\eta)$ triples given by $\CZ^{-1}_{q,r}dm_{q,r}dhd\eta$ where $dh$ denotes the law as in~\eqref{eq:law_of_the _field} and $\CZ_{q,r}^{-1}$ is a normalization constant. (Note that $m_{q,r}$ depends on $h$ and $\eta$.) \begin{enumerate}[(i)] \item Given $w$ and $\eta$, the conditional law of $h$ is equal to the law of $$\wh{h} - \frac{2 + \rho}{\gamma}\log\|z\| + \frac{\gamma(2 - d)}{2}G(z,\eta(T_w)) + \psi \circ f_{T_{u}}$$ where $\wh{h}$ is a free boundary $\text{GFF}$ on $\h$, $d$ is the dimension of the Bessel process $Y$, $\psi$ is a function which is harmonic outside of $\h \cap \partial{\D}$, and the additive constant is fixed in the same manner as for $h$. \item Given $w$ and $\eta\|_{[0,T_{w}]}$, the conditional law of $\eta\|_{[T_{w},\infty)}$ is that of an $\SLE_{\kappa}(\rho)$ process in the unbounded component of $\h \setminus \eta([0,T_{w}])$ from $\eta(T_{w})$ to $\infty$ with a single boundary force point of weight $\rho$ located at $(\eta(T_{w}))^{+}$ weighted by the Radon-Nikodym derivative \begin{align} \CZ^{-1}\exp\!\left(\frac{\gamma^{2}(2-d)^2}{8}\left(\int \int G(f_{T_u}^{-1}(x),f_{T_u}^{-1}(y))dxdy - 2 \int G(\eta(T_w),f_{T_u}^{-1}(x))dx\right)\right) \end{align} where $G$ denotes the Neumann Green's function on $\h$ and $\CZ^{-1}$ is a normalizing constant and the integrals are all over $\partial \D \cap \partial \h$. \item If one zooms in near $\eta(T_{w})$ as in part (ii) of \cite[Proposition 4.7]{dms2014mating}, then the law of the beaded surface which consists of the components of $\h \setminus \eta$ which are to the right of $\eta\|_{[T_{w},\infty)}$ converges to that of a quantum wedge of weight $\rho + 2$. \end{enumerate} \end{lemma} \begin{proof} The proofs of parts (i) and (ii) follow from the proofs of parts (i) and (ii) respectively of \cite[Lemma~6.21]{dms2014mating}. We note that the harmonic function $\psi \circ f_{T_u}$ obtained in part (i) can be defined on $\h$ since the Lebesgue measure of $\eta \cap \h$ is equal to $0$ a.s. The latter follows since the law of $\eta$ on the intervals that it is not colliding with its force point is mutually absolutely continuous with respect to the law of an $\SLE_{\kappa}$ restricted to the corresponding intervals. So the above harmonic function extends to a function defined on $\h$. Now we turn to the proof of part (iii). First of all, we recall that the beaded surface consisting of the bubbles parameterized by the components of $\h \setminus \eta([0,T_r])$ from right to left is given by that of a quantum wedge $\CW$ of weight $\rho + 2$, up to a given amount of local time $M$, which is a random time. Without loss of generality, we can assume that $\CW$ is a bi-infinite wedge (which is just a concatenation of two independent weight $\rho + 2$-wedges). More precisely, let $\CW_1,\CW_2$ be two independent weight $\rho + 2$-wedges with corresponding encoding Bessel processes $(Y_{t}^1)_{t \geq 0}$ and $(Y_{t}^2)_{t \geq 0}$ respectively. Then, their concatenation is the beaded surface $\CW$ encoded by the process $Y = (Y_t)_{t \in \R}$ defined by $Y_t = Y_{t}^1$ for $t \geq 0$ and $Y_t = Y_{-t}^2$ for $t \leq 0$. For $\epsilon \in (0,1)$ we consider the conformal transformation $\varphi_{\epsilon} \colon \h \to \h$ with $\varphi_{\epsilon}(z) = z/\epsilon$ and the $\text{GFF}$ on $\h$, \begin{align} h^{\epsilon,C} = h \circ f_{T_w}^{-1} \circ \varphi_{\epsilon}^{-1} + Q\log \|(f_{T_w}^{-1})' \circ \varphi_{\epsilon}^{-1}\| + Q\log (\epsilon) + \frac{C}{\gamma} \end{align} and set $\epsilon(C) = \sup\{\epsilon \in [0,1] : h_{1}^{\epsilon,C}(0) = 0\}$ and $h^{C} = h^{\epsilon(C),C}$. Here, $h_{1}^{\epsilon,C}(0)$ is the average of the field $h^{\epsilon,C}$ on $\h \cap \partial \D$. We want to show that the beaded surface with respect to $h^{C}$ which consists of the components of $\h \setminus \varphi_{\epsilon}(\eta^w)$ which are to the right of $\varphi_{\epsilon}(\eta^w)$ converges to that of a quantum wedge of weight $\rho + 2$ as $C \to \infty$, where $\eta^w(t) = f_{T_w}(\eta(t+T_w))$ for $t \geq 0$. Next, we fix a large number $N$ and we assume that $X \in [0,N]$ is sampled uniformly at random. If we take $\CW$ and recenter it after shifting by $X$ units of local time, i.e., by considering the wedge encoded by the process $Y^T = (Y_{t}^T)_{t \in \R}$ with $Y_{t}^{T} = Y_{T - t}$ for $t \in \R$ and $T = T_X$, then the bubbles of $\CW$ going from left to right have the law of the bubbles in a weight $\rho + 2$-wedge because the law of the bubbles is invariant under the time-reversal and recentering (simply because the Poisson law has this property). Therefore the law of the beaded surface with respect to $h^C$ which consists of the bubbles of $\h \setminus \varphi_{\epsilon}(\eta^w)$ lying on $B(0,\epsilon(C)^{-1})$ can be sampled as follows. Let $\wt{\CW}$ be a bi-infinite wedge of weight as above coupled with $\eta$ such that the bubbles to the right of $\eta$ (starting from $\eta(T_r)$ and going from right to left) agree with the bubbles of the wedge encoded by $(\wt Y_t)_{t \geq 0}$, where $\wt Y $ is the encoding process for $\wt{\CW}$, and up until $M$ units of local time. We fix $N \in \N$ large and we pick $X \in [0,N]$ uniformly at random. Then we condition on the event that $X \in [0,M]$ and we consider the beaded surface which is part of $\wt{\CW}$ and it is parameterized by the bubbles which are images under $f_{T_X}^{-1} \circ \varphi_{\epsilon}^{-1}$ of the bubbles of $\h \setminus \varphi_{\epsilon}(\eta^X)$ lying on the right of $\varphi_{\epsilon}(\eta^X)$. Then by letting $N \to \infty$, we have that the law of the above surface converges to the law of the surface parameterized by the bubbles of $\h \setminus \varphi_{\epsilon}(\eta^X)$ lying to the right of $\varphi_{\epsilon}(\eta^X)$. Fix $K>0$ and let $\CF_{C,N}$ be the $\sigma$-algebra generated by the bubbles of the previous paragraph. Then the law of these bubbles when conditioning on $\CF_{C,N}$ has Radon-Nikodym derivative with respect to the law without conditioning equal to \begin{align} \frac{\p[ X \in [0,M]\,\|\,\CF_{C,N}]}{\p[ X \in [0,M] ]}. \end{align} Suppose that we have shown that the $\sigma$-algebra $\CF_{C,N}$ becomes trivial when $C \to \infty$, for every fixed $N \in \N$. Then, by the backwards martingale convergence theorem, we have that the Radon-Nikodym derivative converges to $1$ as $C \to \infty$ a.s. Also, the unconditional law of the bubbles is the same as if we replaced $h^C$ by a wedge of weight $\rho + 2$. By letting $N \to \infty$ we obtain the claim of part (iii). Hence, in order to complete the proof of part (iii), we need to show that $\cap_{C>0} \CF_{C,N}$ is trivial for all $N \in \N$. Let $\CF_{C,N}^1$ (resp.\ $\CF_{C,N}^2$) be the $\sigma$-algebra generated by the restriction of $h_X = h \circ f_{T_X}^{-1} + Q \log \|(f_{T_X}^{-1})'\|$ to $\h \cap B(0,K \epsilon(C))$ (resp.\ the bubbles cut off $\infty$ by $\eta^X$ which lie on $B(0,K \epsilon(C))$). Set $\CG_{N,C} = \sigma(\CF_{C,N}^1,\CF_{C,N}^2)$ and $\CG_{N} = \cap_{C>0}\CG_{N,C}$. Then it suffices to show that $\CG_N$ is trivial for every fixed $K$. For the latter, it suffices to prove that the law of the pair $((h_X,\phi),\eta^X)$ given $\CG_{C,N}$ converges as $C \to \infty$ to the unconditional law of $((h_X, \phi),\eta^X)$, for every fixed $\phi \in C_{0}^{\infty}(\h)$ with $\int_{\h}\phi(z)dz = 0$. To show this, first we note that the conditional law of the pair consisting of $h_X$ and $\eta^X$ given $X$ is equal to that of an $\SLE_{\kappa}(\rho)$ process $\eta$ in $\h$ with a single force point at $0_+$ weighted by the Radon-Nikodym derivative of part (ii) and $h$ can be expressed as $\wh{h} + \left(\frac{\rho +2 -\gamma^2}{\gamma}\right) \log \|\cdot\| + \psi \circ f_{T_{N-X}}$, where $\wh{h}$ is a free boundary $\text{GFF}$ and the additive constant is taken so that the average of the field after applying the coordinate change with $f_{T_{N-X}}$ on $\h \cap \partial \D$ is equal to $0$. This form of the conditional law follows by combining part (ii) with the proof of \cite[Theorem~6.16]{dms2014mating}. Set $\wh{\psi} = \left( \frac{\rho + 2 - \gamma^2}{\gamma}\right) \log \|\cdot\| + \psi \circ f_{T_{N-X}}$ and note that conditional on $X$, $\wh{\psi}$ is determined by $\eta^X$. Note also that the Markov property of the $\text{GFF}$ implies that $\wh{h}$ can be decomposed as $\wh{h} = \wh{h}_C +\wh{ \mathcal{H}}_C$, where $\wh{h}_C$ is a $\text{GFF}$ on $\h \setminus B(0,K \epsilon(C))$ with free boundary conditions on $\h \cap \partial B(0,K\epsilon(C))$ and $\wh{\mathcal{H}}_C$ is a harmonic function on $\h$ with boundary conditions given by those of $\wh{h}$ on $\h \cap \partial B(0,K \epsilon(C))$ and free boundary conditions on $\partial \h \setminus B(0,K\epsilon(C))$. Hence the conditional law of $(h_X,\phi)$ given $X$ and $\eta^X$ is that of a Gaussian random variable with mean $(\wh{\mathcal{H}}_C,\phi)$ and variance $\int_{\h}\int_{\h} \phi(y)G_C(y,z)\phi(z)dydz$, where $G_C$ is the Green's function on $\h \setminus B(0,K \epsilon(C))$ with Dirichlet (resp.\ Neumann) boundary conditions on $\h \cap \partial B(0,K \epsilon(C))$ (resp.\ $\partial \h \setminus B(0,K \epsilon(C)))$. Note that \begin{equation}\label{eq:green_convergence} \int_{\h}\int_{\h} \phi(y) G_C(y,z) \phi(z) dydz \to \int_{\h}\int_{\h} \phi(y) G(y,z) \phi(z) dydz \,\,\,\text{as}\,\,\,C \to \infty, \end{equation} since $G_C \to G$ as $C \to \infty$ locally uniformly. Also, given $X$, $\eta^X$ and $\CG_{C,N}$, the random variable $(\wh{\mathcal{H}}_C,\phi)$ is a Gaussian with mean zero and covariance given by $\int_{\h}\int_{\h} \phi(y) \cov(\wh{\mathcal{H}}_C(y),\wh{\mathcal{H}}_C(z)) \phi(z) dydz$. Then, using the explicit form of the Poisson kernel in $\h \setminus B(0,\epsilon)$, we have that the latter covariance tends to $0$ as $C \to \infty$. Therefore, we obtain that the conditional law of $(h_X,\phi)$ given $(X,\eta^X)$ and $\mathcal{G}_{C,N}$ converges to the conditional law of $(h_X,\phi)$ given $(X,\eta^X)$ as $C \to \infty$. Moreover, $X$ is independent of $\mathcal{G}_{C,N}$ and the conditional law of $\eta^X$ given $X$ and $\mathcal{G}_{C,N}$ converges to the conditional law of $\eta^X$ given $X$ as $C \to \infty$. The claim then follows since $(h_X,\phi)$ and $\eta^X$ are independent given $X$. This completes the proof of part (iii). \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:quantum_natural_time_cutting}.] Now that we have proved Theorem~\ref{thm:wedge_cutting} and Lemma~\ref{lem:main_lemma}, the proof follows the same argument as that of the proof of \cite[Theorem~6.16]{dms2014mating}. \end{proof} \subsection{Quantum boundary length evolution} \label{subsec:boundary_length_evolution} Suppose that we have the setup of Theorem~\ref{thm:quantum_natural_time_cutting}. In this subsection we are going to describe the law of the boundary length evolution of $\eta$ when it has the quantum natural time parameterization as in Theorem~\ref{thm:quantum_natural_time_cutting}, and hence proving Theorem~\ref{thm:boundary_length_evolution}. As an intermediate step in the proof of Theorem~\ref{thm:boundary_length_evolution}, we are going to describe the Poissonian structure of the quantum lengths of the beads of a surface corresponding to a quantum wedge of weight $\rho + 2$ with $\kappa \in (0,4)$ and $\rho \in (\kappa/2-4,-2) \cap (-2-\kappa/2,-2)$. Finally, we are going to prove Corollary~\ref{cor:dim_of_boundary_intersection} as a consequence of Theorem~\ref{thm:boundary_length_evolution}. We start by analyzing the Poissonian structure of the quantum lengths of the beads of a weight $\rho + 2$ quantum wedge. Our strategy will be similar to that used to prove \cite[Proposition~4.18]{dms2014mating}. In particular, we prove the following. \begin{theorem} \label{thm:law_of_quantum_lengths} Fix $\kappa$ and $\rho$ as above. Let $\CW$ be a quantum wedge of weight $\rho + 2$ and let $h$ be the corresponding field. Let $Y$ be the Bessel process encoding $\CW$ and $(e_{j})_{j \in \N}$ be its excursions away from zero. Let $(h_{j})_{j \in \N}$ be the fields encoding the surfaces determined by the above excursions. Let also $t_{j}$ be the boundary quantum length associated with $h_{j}$ for all $j \in \N$. Then $(t_{j})$ has the law of a $\text{p.p.p.}$ with intensity measure given by $c( du \times t^{\alpha}dt)$ where $\alpha = -2 + \frac{2(\rho + 2)}{\kappa}$ and $c > 0$ is a constant depending only on $\alpha$ and $\rho$. \end{theorem} \begin{proof} Note that we can sample $Y$ by first sampling a $\text{p.p.p.}$ $\Lambda^$ according to $du \times \nu_{\delta}^(dt)$ with $\nu_{\delta}^(dt) = c_{\delta}^* t^{\delta - 3}dt$ for some constant $c_{\delta}^>0$ depending only on $\delta$ and then sampling $Y$ by associating with each $(u,e) \in \Lambda^$ a $\BES^{\delta}$ excursion away from zero whose maximum value is given by $e^$. Here $\delta$ satisfies~\eqref{eq:bessel_dimension}. Note that for fixed $e^$ the law of the latter can be sampled by joining back to back two independent Bessel processes with dimension $\wt{\delta} = 4 - \delta$ starting from $0$ and up until the first time they hit $e^$ (see \cite[Remark~3.7]{dms2014mating}). Let $\mu_{\delta}^{x}$ be the law on such paths for $x = e^* \in (0,+\infty)$. Suppose that $X = (X_{t})_{0 \leq t \leq T}$ is sampled from $\mu_{\delta}^c$, where $T$ is the length of the excursion. Since Bessel processes satisfy Brownian scaling, we obtain that the process $\wt{X} = (c^{-1}X_{c^{2}t})_{0 \leq t \leq \wt{T}}$ with $\wt{T} = c^{-2}T$ is sampled from $\mu_{\delta}^{1}$. For $\epsilon \in (0,1)$ we set $T_{\epsilon} = \inf \{t \geq 0 : X_{t} \geq \epsilon \} $, $\wt{T}_{\epsilon} = \inf \{t \geq 0 : \wt{X}_{t} \geq \epsilon \}$ and \begin{align} \sigma^{\epsilon}(t) = \inf \left\{s \geq 0 : \frac{4}{\gamma^{2}}\int_{0}^{s}\frac{1}{(X^{\epsilon}_{r})^{2}}dr \geq 2t \right \} \\ \wt{\sigma}^{\epsilon}(t) = \inf \left\{s \geq 0 : \frac{4}{\gamma^{2}}\int_{0}^{s}\frac{1}{(\wt{X}^{\epsilon}_{r})^{2}}dr \geq 2t \right \}, \end{align} where $X^{\epsilon}_{t} = X_{T_{\epsilon} + t}$ for $0 \leq t \leq T^{\epsilon}$ and $\wt{X}^{\epsilon}_{t} = \wt{X}_{\wt{T}_{\epsilon} + t }$ for $0 \leq t \leq \wt{T}^{\epsilon}$, and $T^{\epsilon} = T - T_{\epsilon}$, $\wt{T}^{\epsilon} = \wt{T} - \wt{T}_{\epsilon}$. It follows from \cite[Proposition~3.4]{dms2014mating} that $Z^{\epsilon}$ (resp.\ $\wt{Z}^{\epsilon}$) evolves as a Brownian motion starting from $\frac{2}{\gamma}\log(\epsilon)$ and run twice the speed with drift $\frac{(2-\delta)}{2}\gamma$ up until the first time it hits $\frac{2}{\gamma}\log(c)$ (resp.\ $0$) and then it evolves independently as a Brownian motion starting from $\frac{2}{\gamma}\log(c)$ (resp.\ $0$) and run twice the speed with drift $\frac{ (\delta - 2)}{2}\gamma$, where $Z^{\epsilon} = \frac{2}{\gamma}\log(X_{\sigma^{\epsilon}(t)}^{\epsilon})$ and $\wt{Z}_{t}^{\epsilon} = \frac{2}{\gamma}\log(\wt{X}_{\wt{\sigma}^{\epsilon}(t)}^{\epsilon})$ for $t \geq 0$. Also, we have that $\wt{T}_{\epsilon} = c^{-2}T_{\epsilon c}$ and hence $\wt{X}_{t}^{\epsilon} = c^{-1} X_{c^{2}t}^{\epsilon c}$ and $\wt{\sigma}^{\epsilon}(t) = c^{-2}\sigma^{\epsilon c}(t)$ for all $t \geq 0$. Moreover if $\sigma^{\epsilon} = \inf \{t \geq 0 : Z_{t}^{\epsilon} \geq \frac{2}{\gamma} \log(c) \}$ (resp.\ $\wt{\sigma}^{\epsilon} = \inf\{t \geq 0 : \wt{Z}_{t}^{\epsilon} \geq 0 \}$), then $Z_{\sigma^{\epsilon}+ t}^{\epsilon}$ for $t \geq - \sigma^{\epsilon}$ (resp.\ $\wt{Z}_{\wt{\sigma}^{\epsilon} + t}$ for $t \geq - \wt{\sigma}^{\epsilon}$) converges weakly as $\epsilon \to 0$ with respect to the local uniform topology to a process $Z$ (resp.\ $\wt{Z}$) indexed by $\R$. Note also that $\wt{Z}_t = Z_t - \frac{2}{\gamma}\log(c)$ for all $t \in \R$. The above observations imply that the total quantum length of a surface associated with a given $(u,e^)$ is given by $e^U_{e}$ where the $U_{e}$ are i.i.d. random variables indexed by the excursions of $Y$ away from $0$. Also the law of $U_{e}$ is given by the total quantum length of the random surface $\wt{h}$ sampled as follows: \begin{enumerate}[(i)] \item Let $X$ be a sample from $\mu_{\delta}^{1}$. The projection of $\wt{h}$ onto $\CH_{1}(\strip)$ is given by parameterizing $\frac{2}{\gamma}\log(X)$ to have quadratic variation $2dt$. \item The projection of $\wt{h}$ onto $\CH_{2}(\strip)$ is given by taking an independent sample of the law of the corresponding projection onto $\CH_{2}(\strip)$ of a free boundary $\text{GFF}$ on $\strip$. \end{enumerate} Note that $ \alpha = \delta - 3$, where $\alpha$ is as in the statement of the theorem, and if $U$ is sampled from the law of $U_{e}$ then $\E[U^{-\alpha - 1}] = c < \infty$ since $-\alpha -1 = 1 - \frac{2(\rho + 2)}{\kappa} \in (0,\frac{4}{\kappa})$ and so \cite[Lemma~4.20]{dms2014mating} applies. Therefore $\{(u,e^U_{e}) : (u,e^) \in \Lambda^* \}$ is a $\text{p.p.p.}$ with intensity measure given by $c(du \times t^{\alpha}dt)$ by applying \cite[Lemma~4.19]{dms2014mating}. We also observe that the above $\text{p.p.p.}$ corresponds to the sequence of jumps of a positive stable subordinator with parameter $-\alpha -1 \in (1,2)$. This completes the proof. \end{proof} Now suppose that we have the same setup as in Theorem~\ref{thm:quantum_natural_time_cutting}. We note that the quantum natural time $(\qnt_{u})_{u \geq 0}$ can be expressed as follows. Let $(x_{j},t_{j})_{j \in \N}$ be the $\text{p.p.p.}$ encoding the local time at $0$ $L$ of $Y$ and let $(e_{j})_{j \in \N}$ be the corresponding excursions of $Y$ away from zero. For all $t \geq 0$ we define $A(t)$ to be the set of $j \in \N$ such that the bubble corresponding to $e_{j}$ has been drawn completely by $\eta$ by capacity time $t$ and $x(t) = \sum_{j \in A(t)}t_{j}$. Then we set \begin{align} \qnt_{u} = \inf \{t \geq 0 : L_{x(t)} > u \} \end{align} for all $u \geq 0$. Now we are ready to prove Theorem~\ref{thm:boundary_length_evolution}. Recall the definitions of $A_u, B_u, X_u$ and $Z_u$ in Section~\ref{sec:introduction}. \begin{proof}[Proof of Theorem~\ref{thm:boundary_length_evolution}.] First, we prove that $(X_u,Z_u)$ is an $\alpha$-stable L\'evy process. Note that if we run the process for $u$ units of quantum natural time, the unexplored region (quantum surface parameterized by the unbounded component) is again a weight $\rho + 4$ wedge and the curve is an $\SLE_{\kappa}(\rho)$ (Theorem~\ref{thm:quantum_natural_time_cutting}). Hence the increments of $(X,Z)$ are independent and so it is a Levy process. For all $\epsilon > 0$, let $\CE(\epsilon)$ be the set of excursions of $Y$ with time length at least $\epsilon$. For all $t \geq 0$ let $\eta_{t}(\CE(\epsilon))$ be the number of excursions completed by $Y$ by time $t$ which lie in $\CE(\epsilon)$. Let $\nu_{\delta}$ be the It\^o excursion measure of a $\BES^{\delta}$ process. It follows from \cite[Lemma~6.18]{dms2014mating} that there exists a constant $C_{\delta} > 0$ depending only on $\delta$ such that $\nu_{\delta}(\CE(\epsilon)) = C_{\delta}\epsilon^{\frac{\delta}{2} - 1}$ for all $\epsilon > 0$. We also have from \cite[Proposition~19.12]{kallenberg1997foundations} that $L_{t} = \lim_{\epsilon \to 0}\frac{\eta_{t}(\CE(\epsilon))}{\nu_{\delta}(\CE(\epsilon))}$ for all $t \geq 0$ a.s. Fix $C \in \R$ and let $\wt{Y}$ be the $\BES^{\delta}$ process encoding $h + C$. Then it holds that $\wt{Y}_{t} = e^{\frac{C\gamma}{2}}Y_{e^{-\gamma C}t}$ for all $t \geq 0$. Let $\wt{L},\wt{q},\wt{\eta}_{t}$ be the corresponding quantities for $\wt{Y}$. Then $\wt{\eta}_{t}(\CE(\epsilon)) = \eta_{e^{-\gamma C}t}(\CE(\epsilon e^{-\gamma C}))$ and \begin{align} \frac{\nu_{\delta}(\CE(\epsilon))}{\nu_{\delta}(\CE(\epsilon e^{-\gamma C}))} = \frac{C_{\delta}\epsilon^{\frac{\delta}{2}-1}}{C_{\delta}\epsilon^{\frac{\delta}{2} - 1}e^{\gamma C (1 - \frac{\delta}{2})}} = e^{\gamma C (\frac{\delta}{2} - 1)} \end{align} and hence \begin{align} \frac{\wt{\eta}_{t}(\CE(\epsilon))}{\nu_{\delta}(\CE(\epsilon))} = \frac{\eta_{e^{-\gamma C}t}(\CE(\epsilon e^{-\gamma C}))}{\nu_{\delta}(\CE(\epsilon e^{-\gamma C}))}e^{\gamma C(1 - \frac{\delta}{2})} \end{align} converges to $e^{\gamma C(1 - \frac{\delta}{2})}L_t$ as $\epsilon \to 0$ for all $t \geq 0$ a.s. Thus $\wt{L}_{t} = e^{\gamma C(1 - \frac{\delta}{2})}L_t$ for all $t \geq 0$ a.s. Moreover $\wt{x}(t) = \sum_{j \in A(t)}\wt{t}_{j} = e^{\gamma C}\sum_{j \in A(t)}t_{j}$ and so $\wt{L}_{\Tilde{x}(t)} = e^{\gamma C(1 - \frac{\delta}{2})}L_{e^{-\gamma C}\wt{x}(t)} = e^{\gamma C(1 - \frac{\delta}{2})}L_{x(t)}$ for all $t \geq 0$ a.s. Therefore we have that $\wt{q}_{u} = \qnt_{e^{\gamma C(\frac{\delta}{2} - 1)}u}$ for all $u \geq 0$ a.s. Let $(\wt{X},\wt{Z})$ be the process describing the change in the boundary length relative to time zero corresponding to the field $h + C$. Since the quantum length scales by $e^{\frac{\gamma C}{2}}$ when we add $C$ to the field, we obtain that \begin{align} (\wt{X}_{u},\wt{Z}_{u}) = \left(e^{\frac{\gamma C}{2}}X_{e^{\gamma C(\frac{\delta}{2} - 1)}u},e^{\frac{\gamma C}{2}}Z_{e^{\gamma C(\frac{\delta}{2} - 1)}u}\right) \end{align} for all $u \geq 0$. Set $\mu = e^{\gamma C(\frac{\delta}{2} - 1)}$. Then $\mu^{-\frac{1}{\alpha}} = e^{\frac{\gamma C}{2}}$ and so $(\wt{X}_{u},\wt{Z}_{u}) = (\mu^{-\frac{1}{\alpha}}X_{\mu u},\mu^{-\frac{1}{\alpha}}Z_{\mu u})$ for all $u \geq 0$. The claim of the theorem then follows since $(h,\eta)$ and $(h + C,\eta)$ have the same law for all $C \in \R$ (\cite[Proposition~4.8]{dms2014mating}) and $(\wt{X},\wt{Z})$ is determined by $(h + C,\eta)$. We note that $(X,Z)$ makes a jump at time $u$ if and only if the right continuous inverse $T$ of $L$ makes a jump at time $u$. Note also that there is a correspondence between the discontinuities of $T$ and the excursions $(e_{j})_{j \in \N}$ made by $Y$ away from zero. Let $(h_{j})_{j \in \N}$ be the corresponding sequence of quantum surfaces. We also observe that when $(X,Z)$ makes a jump, both of $X$ and $Z$ make a jump with positive and negative sign respectively. The jump sizes of $X$ and $-Z$ are equal to the quantum lengths of $\R$ and $\R \times \{\pi\}$ respectively under the corresponding surface $(\strip,h_{j},-\infty,+\infty)$, where the opening (resp.\ closing) point of the pocket made by $\eta$ which corresponds to the excursion $e_j$ is mapped to $-\infty$ (resp.\ $+\infty$). We are now ready to describe the law of the boundary length evolution in the case where $\rho = \kappa - 4$ and $\kappa \in (\frac{4}{3},2) $. Since $\rho = \kappa - 4$, we have that the corresponding surfaces have the law of i.i.d. quantum disks. By the resampling property characterizing the two marked points of a unit boundary length quantum disk (\cite[Proposition~A.8]{dms2014mating}) and the Poissonian structure of the sequence of their quantum boundary lengths (Theorem~\ref{thm:law_of_quantum_lengths}), we obtain that the sequence of jumps of $(X,-Z)$ can be sampled as follows: \begin{enumerate}[(i)] \item Firstly, we sample a $\text{p.p.p.}$ $\Lambda = \{(s_{j},t_{j})\}_{j \in \N}$ on $\R_+ \times \R_+$ with intensity measure $c(du \times t^{\alpha}dt)$ where $\alpha$ and $c$ are as in Theorem~\ref{thm:law_of_quantum_lengths}. In our case, we have that $\alpha = -\frac{4}{\kappa}$. \item Then we sample independently a sequence of i.i.d. random variables with the uniform law on $[0,1]$, $\{U_{j}\}_{j \in \N}$ and we consider \begin{align} \Lambda^ = \{(t_{j}u_{j} , (1 - u_{j})t_{j}) : (s_{j},t_{j}) \in \Lambda \}_{j \in \N} \end{align} \end{enumerate} Note that $\wt{\Lambda} = \{(s_{j},t_{j},u_{j})\}_{j \in \N}$ is a $\text{p.p.p.}$ on $\R_+ \times \R_+ \times [0,1]$ with intensity measure given by $\nu = c(du \times t^{\alpha}dt) \times \mu$, where $\mu$ is the law of a uniform random variable on $[0,1]$. Consider the function $F \colon \R_+ \times \R_+ \times [0,1] \to \R_+ \times \R_+$ with $F(s,t,u) = (tu,t(1 - u))$. Then $\Lambda^$ is a $\text{p.p.p.}$ on $ \R_+ \times \R_+$ with intensity measure given by $\nu_{}F$. Set $\wt{\nu}(dx,dy) = c(x + y)^{\alpha - 1}dxdy$ to be defined on $\R_+ \times \R_+$. It is easy then to check that $\nu_{}F([x_{1},x_{2}] \times [y_{1},y_{2}]) = \wt{\nu}([x_{1},x_{2}] \times [y_{1},y_{2}])$ for all $0 < x_{1} < x_{2}$ and $0 < y_{1} < y_{2}$. Therefore we have that $\nu_{*}F = \wt{\nu}$ and so we have described the law of $(X,-Z)$ since it is determined by the law of the jumps. This completes the proof of the theorem. \end{proof} Now that we have completed the proof of Theorem~\ref{thm:boundary_length_evolution}, we are ready to prove Corollary~\ref{cor:dim_of_boundary_intersection}. \begin{proof}[Proof of Corollary~\ref{cor:dim_of_boundary_intersection}.] Suppose that we have the setup of Theorems~\ref{thm:quantum_natural_time_cutting} and~\ref{thm:boundary_length_evolution} where the $\SLE_{\kappa}(\rho)$ process $\eta$ is drawn on top of an independent quantum wedge $\CW = (\h,h,0,\infty)$ of weight $\rho + 4$ and we parameterize $\eta$ by quantum natural time with respect to $\CW$. Let $(X,Z)$ be the pair encoding the change of the quantum lengths of the left and right outer boundaries as in Theorem~\ref{thm:boundary_length_evolution}. Then Theorem~\ref{thm:boundary_length_evolution} implies that $(X,-Z)$ has the law of an $\alpha$-stable L{\'e}vy process where $\alpha$ satisfies~\eqref{eqn:alpha_value}. We parameterize $\R_+$ according to quantum length with respect to $h$ and set $I_t = \inf_{s \in [0,t]} Z_s$, $S_t = \sup_{s \in [0,t]}(-Z_s)$ for all $t \geq 0$. Let $L = (L_t)$ be the local time of $Z-I$ at $0$ which is the same as the local time of $S+Z$ at $0$. Let also $L^{-1}$ be the right-continuous inverse of $L$. Then, it follows from \cite[Chapter~VIII]{bertoin1996levy} that $L^{-1}$ is a stable subordinator with index $1 - \frac{1}{\alpha}$. Note that the ladder height process is defined by $H_t = S_{L_t^{-1}}$ for all $t \geq 0$ (see \cite[Chapter~VI]{bertoin1996levy}). Moreover \cite[Lemma~1, Chapter~VIII]{bertoin1996levy} implies that $H$ is a L{\'e}vy stable process with index $\alpha-1 \in (0,1)$ and hence it is an $(\alpha-1)$-stable subordinator. We note that $Z$ achieves a record minimum at time $t$, i.e., $Z_t = I_t$, if and only if $x = \eta(t) \in \R_+$ and then $x = -Z_t$. It follows that $\{\nu_h([0,x]) : x \in \eta \cap \R_+\} = \overline{\{H_t : t \geq 0\}}$. We also note that $H$ has Laplace exponent $\Phi$ given by $\Phi(\lambda) = \lambda^{\alpha-1}$ for all $\lambda > 0$. Therefore, combining with \cite[Theorem~15, Chapter~III]{bertoin1996levy}, we obtain that \begin{equation}\label{eqn:hausdorff_dim_equality} \text{dim}_{\mathcal{H}}(\{H_t : t \geq 0\}) = \text{dim}_{\mathcal{H}}(\{\nu_h([0,x]) : x \in \eta \cap \R_+\}) = \alpha-1 = -\frac{2(\rho+2)}{\kappa}\,\,\text{a.s.} \end{equation} where $\text{dim}_{\mathcal{H}}$ denotes Hausdorff dimension. We note that it follows from \cite[Theorem~4.1]{rhodes2008kpz} that if $\wt{h}$ is a free boundary $\text{GFF}$ on $\h$ and $K \subseteq [0,\infty)$ is a deterministic set, then if $\wh{K} = \{\nu_{\wt{h}}([0,x]) : x \in K\}$, it holds that \begin{equation}\label{eqn:kpz_formula_equation} \text{dim}_{\mathcal{H}}(K) = \left(1+\frac{\gamma^2}{4}\right) \text{dim}_{\mathcal{H}}(\wh{K}) -\frac{\gamma^2}{4}\text{dim}_{\mathcal{H}}(\wh{K})^2\,\,\text{a.s.} \end{equation} Note that if we consider $\CW$ with the circle average embedding, then the law of the restriction of $h$ to any subdomain of $\h$ which is bounded away from $0,\infty$ and $\h \cap \partial \D$ is mutually absolutely continuous with respect to the law of the corresponding restriction of a free boundary $\text{GFF}$ on $\h$ normalized to have average zero on $\h \cap \partial \D$. Since $\eta$ is independent of $h$, we obtain that~\eqref{eqn:kpz_formula_equation} holds a.s.\ when $K$ is replaced by $\eta \cap \R_+$ and $\wt{h}$ is replaced by $h$. Then, \eqref{eqn:dim_of_intersection_with_R_+} follows by combining with~\eqref{eqn:hausdorff_dim_equality}. This completes the proof of the corollary. \end{proof} \bibliographystyle{abbrv} \bibliography{biblio} \end{document}
2412.04148v1	http://arxiv.org/abs/2412.04148v1	Recursively Extended Permutation Codes under Chebyshev Distance	\documentclass[journal,draft,onecolumn]{IEEEtran} \usepackage[dvipdfmx]{graphicx,xcolor} \usepackage[fleqn]{amsmath} \usepackage[percent]{overpic} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{newtxtext} \usepackage[varg]{newtxmath} \usepackage{here} \usepackage{bbm} \usepackage{mathtools} \mathtoolsset{showonlyrefs=true} \let\labelindent\relax \usepackage[inline, shortlabels]{enumitem} \usepackage{tablefootnote} \usepackage{mathdots} \usepackage{algorithmic} \usepackage{amssymb} \usepackage{algorithm} \usepackage{xspace} \input{notation} \def\stkeqa{\stackrel{\rm{(a)}}{=}} \def\stkeqb{\stackrel{\rm{(b)}}{=}} \def\stkeqc{\stackrel{\rm{(c)}}{=}} \def\stkeqd{\stackrel{\rm{(d)}}{=}} \def\preqa{\stackrel{\rm{(a)}}{=}} \def\preqb{\stackrel{\rm{(b)}}{=}} \def\preqc{\stackrel{\rm{(c)}}{=}} \def\preqd{\stackrel{\rm{(d)}}{=}} \def\preqe{\stackrel{\rm{(e)}}{=}} \def\eqa{\stackrel{\rm{(a)}}{=}} \def\eqb{\stackrel{\rm{(b)}}{=}} \def\eqc{\stackrel{\rm{(c)}}{=}} \def\eqd{\stackrel{\rm{(d)}}{=}} \def\eqe{\stackrel{\rm{(e)}}{=}} \def\Klove{Kl\o ve } \def\proofname{\bf {Proof}} \def\vector#1{\mbox{\boldmath $#1$}} \theoremstyle{definition}\newtheorem{teiri}{Theorem} \newtheorem{giron}{Discussion} \newtheorem{lem}{Lemma} \newtheorem{example}{Example} \newtheorem{remark}{Remark} \newtheorem{pp}{Proposition} \newtheorem{df}{Definition} \newcommand{\dmin}{d_{\mathrm{min}}} \newcommand{\dinf}{d_{\infty}} \def\Prob{P} \ifCLASSINFOpdf \else \hyphenation{op-tical net-works semi-conduc-tor} \begin{document} \title{Recursively Extended Permutation Codes under Chebyshev Distance} \author{Tomoya~Hirobe,~\IEEEmembership{Non-Member,~IEEE,} and~Kenta~Kasai,~\IEEEmembership{Member,~IEEE}\thanks{T. Hirobe was with Department of Information and Communications Engineering, School of Engineering, Tokyo, 152-8550 Japan.} \thanks{K. Kasai was with Department of Information and Communications Engineering, School of Engineering, Tokyo, 152-8550 Japan.} } \markboth{Journal of \LaTeX\ Class Files,~Vol.~14, No.~8, August~2015}{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for IEEE Journals} \maketitle \begin{abstract} This paper investigates the construction and analysis of permutation codes under the Chebyshev distance. The direct product group permutation (DPGP) codes, introduced independently by \Klove et al. and Tamo et al., represent the best-known permutation codes in terms of both size and minimum distance. These codes possess algebraic structures that facilitate efficient encoding and decoding algorithms. In particular, this study focuses on recursively extended permutation (REP) codes, which were also introduced by \Klove et al. We examine the properties of REP codes and prove that, in terms of size and minimum distance, the optimal REP code is equivalent to the DPGP codes. Furthermore, we present efficient encoding and decoding algorithms for REP codes. \end{abstract} \begin{IEEEkeywords} permutation codes, Chebyshev distance, $\ell_\infty$ distance, recursively extended permutation codes \end{IEEEkeywords} \IEEEpeerreviewmaketitle \input{body} \ifCLASSOPTIONcaptionsoff \newpage \bibliographystyle{IEEEtran} \bibliography{IEEEabrv,../../../literature/00kasai} \end{document} \newcommand\scalemath[2]{\scalebox{#1}{\mbox{\ensuremath{\displaystyle #2}}}} \def\qed{\hfill $\Box$} \def\QED{\hfill $\Box$} \usepackage{bbm} \usepackage{mathrsfs} \makeatletter \newcommand\RedeclareMathOperator{ \@ifstar{\def\rmo@s{m}\rmo@redeclare}{\def\rmo@s{o}\rmo@redeclare}} \newcommand\rmo@redeclare[2]{ \begingroup \escapechar\m@ne\xdef\@gtempa{{\string#1}}\endgroup \expandafter\@ifundefined\@gtempa {\@latex@error{\noexpand#1undefined}\@ehc} \relax \expandafter\rmo@declmathop\rmo@s{#1}{#2}} \newcommand\rmo@declmathop[3]{ \DeclareRobustCommand{#2}{\qopname\newmcodes@#1{#3}}} 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\def\bigfloor#1{\big\lfloor #1\big\rfloor} \def\floor#1{\lfloor #1\rfloor} \def\Bigceil#1{\Big\lceil #1\Big\rceil} \def\bigceil#1{\big\lceil #1\big\rceil} \def\ceil#1{\lceil #1\rceil} \def\af{\mathfrak{A}} \def\Af{\mathfrak{A}} \def\Bf{\mathfrak{B}} \def\Cf{\mathfrak{C}} \def\Df{\mathfrak{D}} \def\Ef{\mathfrak{E}} \def\Ff{\mathfrak{F}} \def\Gf{\mathfrak{G}} \def\Hf{\mathfrak{H}} \def\Jf{\mathfrak{J}} \def\Kf{\mathfrak{K}} \def\Lf{\mathfrak{L}} \def\Mf{\mathfrak{M}} \def\Nf{\mathfrak{N}} \def\Of{\mathfrak{O}} \def\Pf{\mathfrak{P}} \def\Qf{\mathfrak{Q}} \def\Rf{\mathfrak{R}} \def\Sf{\mathfrak{S}} \def\Tf{\mathfrak{T}} \def\Uf{\mathfrak{U}} \def\Vf{\mathfrak{V}} \def\Wf{\mathfrak{W}} \def\Xf{\mathfrak{X}} \def\Yf{\mathfrak{Y}} \def\Zf{\mathfrak{Z}} \def\Ab{\mathbb{A}} \def\Bb{\mathbb{B}} \def\Cb{\mathbb{C}} \def\Db{\mathbb{D}} \def\Eb{\mathbb{E}} \def\Fb{\mathbb{F}} \def\Gb{\mathbb{G}} \def\Hb{\mathbb{H}} \def\Ib{\mathbb{I}} \def\Jb{\mathbb{J}} \def\Kb{\mathbb{K}} \def\Lb{\mathbb{L}} 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\def\sigmaH{\hat{\sigma}} \def\tauH{\hat{\tau}} \def\upsilonH{\hat{\upsilon}} \def\phiH{\hat{\phi}} \def\chiH{\hat{\chi}} \def\psiH{\hat{\psi}} \def\omegaH{\hat{\omega}} \def\GammaH{\hat{\Gamma}} \def\DeltaH{\hat{\Delta}} \def\ThetaH{\hat{\Theta}} \def\LambdaH{\hat{\Lambda}} \def\XiH{\hat{\Xi}} \def\PiH{\hat{\Pi}} \def\SigmaH{\hat{\Sigma}} \def\UpsilonH{\hat{\Upsilon}} \def\PhiH{\hat{\Phi}} \def\PsiH{\hat{\Psi}} \def\OmegaH{\hat{\Omega}} \def\varepsilonH{\hat{\varepsilon}} \def\varthetaH{\hat{\vartheta}} \def\varpiH{\hat{\varpi}} \def\varrhoH{\hat{\varrho}} \def\varsigmaH{\hat{\varsigma}} \def\varphiH{\hat{varphi}} \def\0U{\underline{0}} \def\1U{\underline{1}} \def\aU{\underline{a}} \def\bU{\underline{b}} \def\cU{\underline{c}} \def\dU{\underline{d}} \def\eU{\underline{e}} \def\fU{\underline{f}} \def\gU{\underline{g}} \def\hU{\underline{h}} \def\iU{\underline{i}} \def\jU{\underline{j}} \def\kU{\underline{k}} \def\lU{\underline{l}} \def\mU{\underline{m}} \def\nU{\underline{n}} \def\oU{\underline{o}} \def\pU{\underline{p}} \def\qU{\underline{q}} \def\rU{\underline{r}} \def\sU{\underline{s}} \def\tU{\underline{t}} \def\uU{\underline{u}} \def\vU{\underline{v}} \def\wU{\underline{w}} \def\xU{\underline{x}} \def\yU{\underline{y}} \def\zU{\underline{z}} \def\aV{\vec{a}} \def\bV{\vec{b}} \def\cV{\vec{c}} \def\dV{\vec{d}} \def\eV{\vec{e}} \def\fV{\vec{f}} \def\gV{\vec{g}} \def\hV{\vec{h}} \def\iV{\vec{i}} \def\jV{\vec{j}} \def\kV{\vec{k}} \def\lV{\vec{l}} \def\mV{\vec{m}} \def\nV{\vec{n}} \def\oV{\vec{o}} \def\pV{\vec{p}} \def\qV{\vec{q}} \def\rV{\vec{r}} \def\sV{\vec{s}} \def\tV{\vec{t}} \def\uV{\vec{u}} \def\vV{\vec{v}} \def\wV{\vec{w}} \def\xV{\vec{x}} \def\yV{\vec{y}} \def\zV{\vec{z}} \def\AV{\vec{A}} \def\BV{\vec{B}} \def\CV{\vec{C}} \def\DV{\vec{D}} \def\EV{\vec{E}} \def\FV{\vec{F}} \def\GV{\vec{G}} \def\HV{\vec{H}} \def\IV{\vec{I}} \def\JV{\vec{J}} \def\KV{\vec{K}} \def\LV{\vec{L}} \def\MV{\vec{M}} \def\NV{\vec{N}} \def\OV{\vec{O}} \def\PV{\vec{P}} \def\QV{\vec{Q}} \def\RV{\vec{R}} \def\SV{\vec{S}} \def\TV{\vec{T}} \def\UV{\vec{U}} \def\VV{\vec{V}} \def\WV{\vec{W}} \def\XV{\vec{X}} \def\YV{\vec{Y}} \def\ZV{\vec{Z}} \def\aBF{\mathbf{a}} \def\bBF{\mathbf{b}} \def\cBF{\mathbf{c}} \def\dBF{\mathbf{d}} \def\eBF{\mathbf{e}} \def\fBF{\mathbf{f}} \def\gBF{\mathbf{g}} \def\hBF{\mathbf{h}} \def\iBF{\mathbf{i}} \def\jBF{\mathbf{j}} \def\kBF{\mathbf{k}} \def\lBF{\mathbf{l}} \def\mBF{\mathbf{m}} \def\nBF{\mathbf{n}} \def\oBF{\mathbf{o}} \def\pBF{\mathbf{p}} \def\qBF{\mathbf{q}} \def\rBF{\mathbf{r}} \def\sBF{\mathbf{s}} \def\tBF{\mathbf{t}} \def\uBF{\mathbf{u}} \def\vBF{\mathbf{v}} \def\wBF{\mathbf{w}} \def\xBF{\mathbf{x}} \def\yBF{\mathbf{y}} \def\zBF{\mathbf{z}} \def\ABF{\mathbf{A}} \def\BBF{\mathbf{B}} \def\CBF{\mathbf{C}} \def\DBF{\mathbf{D}} \def\EBF{\mathbf{E}} \def\FBF{\mathbf{F}} \def\GBF{\mathbf{G}} \def\HBF{\mathbf{H}} \def\IBF{\mathbf{I}} \def\JBF{\mathbf{J}} \def\KBF{\mathbf{K}} \def\LBF{\mathbf{L}} \def\MBF{\mathbf{M}} \def\NBF{\mathbf{N}} \def\OBF{\mathbf{O}} \def\PBF{\mathbf{P}} \def\QBF{\mathbf{Q}} \def\RBF{\mathbf{R}} \def\SBF{\mathbf{S}} \def\TBF{\mathbf{T}} \def\UBF{\mathbf{U}} \def\VBF{\mathbf{V}} \def\WBF{\mathbf{W}} \def\XBF{\mathbf{X}} \def\YBF{\mathbf{Y}} \def\ZBF{\mathbf{Z}} \def\aT{\tilde{a}} \def\bT{\tilde{b}} \def\cT{\tilde{c}} \def\dT{\tilde{d}} \def\eT{\tilde{e}} \def\fT{\tilde{f}} \def\gT{\tilde{g}} \def\hT{\tilde{h}} \def\iT{\tilde{i}} \def\jT{\tilde{j}} \def\kT{\tilde{k}} \def\lT{\tilde{l}} \def\mT{\tilde{m}} \def\nT{\tilde{n}} \def\oT{\tilde{o}} \def\pT{\tilde{p}} \def\qT{\tilde{q}} \def\rT{\tilde{r}} \def\sT{\tilde{s}} \def\tT{\tilde{t}} \def\uT{\tilde{u}} \def\vT{\tilde{v}} \def\wT{\tilde{w}} \def\xT{\tilde{x}} \def\yT{\tilde{y}} \def\zT{\tilde{z}} \def\AT{\tilde{A}} \def\BT{\tilde{B}} \def\CT{\tilde{C}} \def\DT{\tilde{D}} \def\ET{\tilde{E}} \def\FT{\tilde{F}} \def\GT{\tilde{G}} \def\HT{\tilde{H}} \def\IT{\tilde{I}} \def\JT{\tilde{J}} \def\KT{\tilde{K}} \def\LT{\tilde{L}} \def\MT{\tilde{M}} \def\NT{\tilde{N}} \def\OT{\tilde{O}} \def\PT{\tilde{P}} \def\QT{\tilde{Q}} \def\RT{\tilde{R}} \def\ST{\tilde{S}} \def\TT{\tilde{T}} \def\UT{\tilde{U}} \def\VT{\tilde{V}} \def\WT{\tilde{W}} \def\XT{\tilde{X}} \def\YT{\tilde{Y}} \def\ZT{\tilde{Z}} \def\alphaT{\tilde{\alpha}} \def\betaT{\tilde{\beta}} \def\gammaT{\tilde{\gamma}} \def\deltaT{\tilde{\delta}} \def\epsilonT{\tilde{\epsilon}} \def\zetaT{\tilde{\zeta}} \def\etaT{\tilde{\eta}} \def\thetaT{\tilde{\theta}} \def\iotaT{\tilde{\iota}} \def\kappaT{\tilde{\kappa}} \def\lambdaT{\tilde{\lambda}} \def\muT{\tilde{\mu}} \def\nuT{\tilde{\nu}} \def\xiT{\tilde{\xi}} \def\piT{\tilde{\pi}} \def\rhoT{\tilde{\rho}} \def\sigmaT{\tilde{\sigma}} \def\tauT{\tilde{\tau}} \def\upsilonT{\tilde{\upsilon}} \def\phiT{\tilde{\phi}} \def\chiT{\tilde{\chi}} \def\psiT{\tilde{\psi}} \def\omegaT{\tilde{\omega}} \def\GammaT{\tilde{\Gamma}} \def\DeltaT{\tilde{\Delta}} \def\ThetaT{\tilde{\Theta}} \def\LambdaT{\tilde{\Lambda}} \def\XiT{\tilde{\Xi}} \def\PiT{\tilde{\Pi}} \def\SigmaT{\tilde{\Sigma}} \def\UpsilonT{\tilde{\Upsilon}} \def\PhiT{\tilde{\Phi}} \def\PsiT{\tilde{\Psi}} \def\OmegaT{\tilde{\Omega}} \def\varepsilonT{\tilde{\varepsilon}} \def\varthetaT{\tilde{\vartheta}} \def\varpiT{\tilde{\varpi}} \def\varrhoT{\tilde{\varrho}} \def\varsigmaT{\tilde{\varsigma}} \def\varphiT{\tilde{varphi}} \def\JOc{\overline{{\mathcal{J}}}} \def\aO{\overline{a}} \def\bO{\overline{b}} \def\cO{\overline{c}} \def\dO{\overline{d}} \def\eO{\overline{e}} \def\fO{\overline{f}} \def\gO{\overline{g}} \def\hO{\overline{h}} \def\iO{\overline{i}} \def\jO{\overline{j}} \def\kO{\overline{k}} \def\lO{\overline{l}} \def\mO{\overline{m}} \def\nO{\overline{n}} \def\oO{\overline{o}} \def\pO{\overline{p}} \def\qO{\overline{q}} \def\rO{\overline{r}} \def\sO{\overline{s}} \def\tO{\overline{t}} \def\uO{\overline{u}} \def\vO{\overline{v}} \def\wO{\overline{w}} \def\xO{\overline{x}} \def\yO{\overline{y}} \def\zO{\overline{z}} \def\AO{\overline{A}} \def\BO{\overline{B}} \def\CO{\overline{C}} \def\DO{\overline{D}} \def\EO{\overline{E}} \def\FO{\overline{F}} \def\GO{\overline{G}} \def\HO{\overline{H}} \def\IO{\overline{I}} \def\JO{\overline{J}} \def\KO{\overline{K}} \def\LO{\overline{L}} \def\MO{\overline{M}} \def\NO{\overline{N}} \def\OO{\overline{O}} \def\PO{\overline{P}} \def\QO{\overline{Q}} \def\RO{\overline{R}} \def\SO{\overline{S}} \def\UO{\overline{U}} \def\VO{\overline{V}} \def\WO{\overline{W}} \def\XO{\overline{X}} \def\YO{\overline{Y}} \def\ZO{\overline{Z}} \def\IMIFU{\red{\ovalbox{意味不明}}} \def\alphaO{\overline{\alpha}} \def\thetaO{\overline{\theta}} \def\tauO{\overline{\tau}} \def\betaO{\overline{\beta}} \def\varthetaO{\overline{\vartheta}} \def\piO{\overline{\pi}} \def\upsilonO{\overline{\upsilon}} \def\gammaO{\overline{\gamma}} \def\varpiO{\overline{\varpi}} \def\phiO{\overline{\phi}} \def\deltaO{\overline{\delta}} \def\kappaO{\overline{\kappa}} \def\rhoO{\overline{\rho}} \def\varphiO{\overline{\varphi}} \def\epsilonO{\overline{\epsilon}} \def\lambdaO{\overline{\lambda}} \def\varrhoO{\overline{\varrho}} \def\chiO{\overline{\chi}} \def\varepsilonO{\overline{\varepsilon}} \def\muO{\overline{\mu}} \def\sigmaO{\overline{\sigma}} \def\psiO{\overline{\psi}} \def\zetaO{\overline{\zeta}} \def\nuO{\overline{\nu}} \def\varsigmaO{\overline{\varsigma}} \def\omegaO{\overline{\omega}} \def\etaO{\overline{\eta}} \def\xiO{\overline{\xi}} \def\GammaO{\overline{\Gamma}} \def\LambdaO{\overline{\Lambda}} \def\SigmaO{\overline{\Sigma}} \def\PsiO{\overline{\Psi}} \def\DeltaO{\overline{\Delta}} \def\UpsilonO{\overline{\Upsilon}} \def\OmegaO{\overline{\Omega}} \def\ThetaO{\overline{\Theta}} \def\PiO{\overline{\Pi}} \def\PhiO{\overline{\Phi}} \def\RhoO{\overline{\Rho}} \def\alphaU{\underline{\alpha}} \def\iotaU{\underline{\iota}} \def\thetaU{\underline{\theta}} \def\tauU{\underline{\tau}} \def\betaU{\underline{\beta}} \def\varthetaU{\underline{\vartheta}} \def\piU{\underline{\pi}} \def\upsilonU{\underline{\upsilon}} \def\gammaU{\underline{\gamma}} \def\gammaU{\underline{\gamma}} \def\varpiU{\underline{\varpi}} \def\phiU{\underline{\phi}} \def\deltaU{\underline{\delta}} \def\kappaU{\underline{\kappa}} \def\rhoU{\underline{\rho}} \def\varphiU{\underline{\varphi}} \def\epsilonU{\underline{\epsilon}} \def\lambdaU{\underline{\lambda}} \def\varrhoU{\underline{\varrho}} \def\chiU{\underline{\chi}} \def\varepsilonU{\underline{\varepsilon}} \def\muU{\underline{\mu}} \def\sigmaU{\underline{\sigma}} \def\psiU{\underline{\psi}} \def\zetaU{\underline{\zeta}} \def\nuU{\underline{\nu}} \def\varsigmaU{\underline{\varsigma}} \def\omegaU{\underline{\omega}} \def\etaU{\underline{\eta}} \def\xiU{\underline{\xi}} \def\GammaU{\underline{\Gamma}} \def\LambdaU{\underline{\Lambda}} \def\SigmaU{\underline{\Sigma}} \def\PsiU{\underline{\Psi}} \def\DeltaU{\underline{\Delta}} \def\UpsilonU{\underline{\Upsilon}} \def\OmegaU{\underline{\Omega}} \def\ThetaU{\underline{\Theta}} \def\PiU{\underline{\Pi}} \def\PhiU{\underline{\Phi}} \def\RhoU{\underline{\Rho}} \def\AU{\underline{A}} \def\BU{\underline{B}} \def\CU{\underline{C}} \def\DU{\underline{D}} \def\EU{\underline{E}} \def\FU{\underline{F}} \def\GU{\underline{G}} \def\HU{\underline{H}} \def\IU{\underline{I}} \def\JU{\underline{J}} \def\KU{\underline{K}} \def\LU{\underline{L}} \def\MU{\underline{M}} \def\NU{\underline{N}} \def\OU{\underline{O}} \def\PU{\underline{P}} \def\QU{\underline{Q}} \def\RU{\underline{R}} \def\SU{\underline{S}} \def\TU{\underline{T}} \def\UU{\underline{U}} \def\VU{\underline{V}} \def\WU{\underline{W}} \def\XU{\underline{X}} \def\YU{\underline{Y}} \def\ZU{\underline{Z}} \def\St{\mathtt{S}} \def\ct{\mathtt{c}} \def\vt{\mathtt{v}} \def\ub{\underbrace} \def\ob{\overbrace} \def\tIMPLIES{\mbox{ implies }} \def\tOF{\mbox{ of }} \def\tST{\mbox{ s.t. }} \def\tTRUE{\mbox{true}} \def\tFOR{\mbox{ for }} \def\tMINIMIZE{\mbox{minimize }} \def\tWITH{\mbox{ with }} \def\tFORALL{\mbox{ for all }} \def\tFORANY{\mbox{ for any }} \def\tFORSOME{\mbox{ for some }} \def\tAND{\mbox{ and }} \def\tOR{\mbox{ or }} \def\tIF{\mbox{ if }} \def\tIFF{\mbox{ iff }} \def\tST{\mbox{ subject to }} \def\tSUCHTHAT{\mbox{ such that }} \def\tTHEN{\mbox{ then }} \def\tOUTPUT{\mbox{ output }} \def\ttrue{\rm{true}} \def\OF{\mbox{ of }} \def\ST{\mbox{ s.t. }} \def\MINIMIZE{\mbox{minimize }} \def\WITH{\mbox{ with }} \def\FORANY{\mbox{ for any }} \def\FORSOME{\mbox{ for some }} \def\IFF{\mbox{ iff }} \def\ST{\mbox{ subject to }} \def\SUCHTHAT{\mbox{ such that }} \def\THEN{\mbox{ then }} \def\OUTPUT{\mbox{ output }} \def\true{\rm{true}} \def\pack{\mathrm{pack}} \def\cov{\mathrm{cov}} \def\eff{\mathrm{eff}} \def\LLR{\mathrm{LLR}} \def\RHS{\mathrm{RHS}} \def\LHS{\mathrm{LHS}} \def\Rho{P} \def\v{\mathtt{v}} \def\c{\mathtt{c}} \def\Id{\mathrm{Id}} \def\Aut{\mathrm{Aut}} \def\End{\mathrm{End}} \def\lcm{\mathrm{lcm}} \def\coef{\mathrm{coef}} \def\ord{\mathrm{ord}} \def\Int{\mathrm{Int}} \def\Re{\mathrm{Re}\,} \RedeclareMathOperator{\Im}{Im} \DeclareMathOperator{\Ker}{Ker} \def\cl{\mathrm{Cl}} \def\ann{\mathrm{ann}} \def\dom{\mathrm{Dom}} \def\de{\mathrm{de}} \def\ad{\mathrm{ad}} \def\rank{\mathrm{rank}} \def\diag{\mathrm{diag}} \def\floor#1{\lfloor #1\rfloor} \def\op#1{\|\|#1\|\|_\mathrm{op}} \def\y{\underline{y}} \def\s{\underline{s}} \def\x{\underline{x}} \def\ES{\mathcal{S}} \def\Y{\mathcal{Y}} \def\d{\mathrm{d}} \def\H{\mathcal{H}} \def\O{\mathcal{O}} \def\B{\mathcal{B}} \def\N{\mathbb{N}} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\Z{\mathbb{Z}} \def\GF{\mathbb{F}} \def\E{\mathbb{E}} \def\C{\mathbb{C}} \def\vx{\vec{x}} \def\vh{\vec{h}} \def\vy{\vec{y}} \def\vz{\vec{z}} \def\vs{\vec{s}} \def\dl{\mathrm{l}} \def\dr{\mathrm{r}} \def\dg{\mathrm{g}} \def\cw{\mathrm{cw}} \def\ss{\boxast} \def\cs{\oast} \def\G{\mathcal{G}} \def\A{\mathcal{A}} \newcommand{\error}{\mathrm{error}} \def\thepage{\arabic{page}} \newcommand{\T}{\mathsf{T}} \newcommand{\I}{\mathbbm{1}} \section{Introduction} In this paper, we explore the subject of {\itshape permutation codes}, which are subsets of all permutations of a fixed length \(n\). The concept of permutation codes originated in the 1960s \cite{1445610}. Vinck et al. later applied permutation codes to power-line communication and \(m\)-ary frequency shift keying (FSK) modulation systems \cite{Vinck2000,866429}, renewing interest in permutation codes \cite{BLAKE19791,1302307,Swart2007}. In \(m\)-ary FSK systems, individual frequencies are assigned to time slots to represent permutation symbols. The use of time and frequency diversity helps reduce the impact of various types of noise, such as background noise, impulse noise, and persistent frequency interference commonly seen in power-line communication systems. For multilevel flash memory applications, the \(\ell_\infty\) norm, known as the Chebyshev distance, is effective for managing issues related to recharging and error correction. Among the distance metrics employed for permutation codes, Chebyshev distance has been thoroughly examined, covering aspects like the Gilbert–Varshamov bound and ball-packing bound \cite{5466536,7342968,7908949}, efficient encoding and decoding algorithms \cite{5466546,5466536}, and systematic code construction methods \cite{8648459,6937135}. \Klove et al. \cite[Sec.~III.A]{5466546} and Tamo et al. \cite[\black{Construction 1}]{5466536} independently introduced a construction of permutation codes based on the Chebyshev distance. In \cite{5466536}, the coordinates are partitioned into \(\mathbb{Z}/d\mathbb{Z}\), and the construction is viewed as a direct product of sub-groups over the symmetric group \(\mathcal{S}_n\), with \(d\) symmetric groups acting as constituent groups. Based on this framework, these codes are termed direct product group permutation (DPGP) codes in this paper. Efficient algebraic encoding and decoding algorithms for DPGP codes have been proposed \cite{5466536,5466546}. DPGP codes demonstrate strong asymptotic normalized minimum distance for permutation codes. As far as the authors are aware, DPGP codes provide the largest code size for a given code length and minimum distance \cite[Fig.~1]{5466536}, except for codes derived using the methods from the Gilbert–Varshamov (GV) bound proof \cite[Thm.~26]{5466536} and short-length codes obtained through greedy algorithms \cite[Sec.~IV.B]{5466546}. DPGP codes form the foundation for various extended code constructions and are thus of significant importance. For example, \cite[Construction 2]{5466536} extends DPGP codes, while \cite{8648459} employs right coset codes of \((n,M,d)\) DPGP codes in \(\mathcal{S}_n\) to construct an alternative structured permutation code distinct from the one proposed in \cite{7435295}. \Klove et al. introduced code extension methods in \cite[Sec.~III.C]{5466546}, referred to here as recursively extended codes (REP). When a code is extended, its size increases by a factor of \( q \), with \( q \) distinct leading elements. For the case \( q=2 \), a simple encoding and decoding method was designed \cite[Sec.~III.C]{5466546}. Because the factor graph connecting the input and output of this encoder forms a tree, MAP decoding becomes feasible using this graph. Kawasumi and Kasai enhanced decoding performance by concatenating this code with LDPC codes \cite{19950201,KawasumiKasai2023}. However, for the general case with \( q>2 \), no specific encoding and decoding scheme has been proposed. The rest of this paper is organized as follows. Section II introduces the necessary notation and fundamental concepts related to the construction of general permutation codes and DPGP codes. Section III describes the properties of extended codes and provides several lemmas that will be used in the proofs in subsequent sections. Section IV discusses REP code properties and presents key theorems regarding optimal REP codes. Section V covers encoding algorithms for REP codes, including both natural and recursive methods, and introduces decoding methods for optimal REP codes. \section{Notation and Preliminaries} For a positive integer $n$, we define $[n]$ as the set $\{0, 1, \ldots, n-1\}$. We denote the set $\{x_0, \ldots, x_{n-1}\}$ by $\{x_j\}_{j=0}^{n-1}$, or simply by $\{x_j\}$ when the context makes the range of $j$ clear. We denote the array $(x_0, \ldots, x_{n-1})$ by $x_0^{n-1}$. Let \( \Sc_n \) be the symmetric group on \([n]\). More precisely, let \( \Sc_{[n]} \), or simply \( \Sc_n \), denote the set of permutations over \([n]\), which can be defined as the set of bijective functions \( f: [n] \to [n] \). To represent a permutation \( f \in \Sc_n \) as an array, we use \( \fU = \left[ f(0), \ldots, f(n-1) \right]\). Let us represent arrays with an underlined variable such as $\xU$. We write the \( j \)-th element of the array \(\xU\) as an array of square brackets: \( x_j \): \(\xU = [x_0, x_1, \ldots, x_{n-1}] \). A subset \( C \subset \Sc_n \) of the symmetric group \( \Sc_n \) is called a {\itshape permutation code} of length \( n \), or simply a code of length \( n \). The elements of \( C \) are called {\itshape codewords}. Let \( C \) be a code of length \( n \) with \( C \subset \Sc_n \), and let \( \cU \) and \( \cU' \) be two codewords in \( C \). The Chebyshev distance between \( \cU \) and \( \cU' \) is defined as \( d_\infty(\cU, \cU') = \max_{j \in [n]} \|c_j - c_j'\|. \) The minimum distance between different codewords in \( C \) is referred to as the minimum distance of \( C \) and is denoted by \( d_\infty(C) \): \( d_\infty(C) := \min_{\cU, \cU' \in C : \cU \neq \cU'} d_\infty(\cU, \cU'). \) For a code \( C \) containing only one codeword, the minimum distance is defined as infinity. We call a code $\Cc\subset\Sc_n$ and $(n,M,d)$ code if $\Cc$ is of length $n$, of size $M$ and of minimum distance at least $d$. \subsection{Direct Product Group Permutation Codes}\label{212515_27Aug24} In this section, we review a simple permutation code independently discovered by \Klove et al. \cite[Explicit Construction]{5466546} and Tamo et al. \cite[\black{Construction 1}]{5466536}. In this paper, we will refer to the codes as {\it direct product group permutation} (DPGP) codes based on the properties of the fact described below \cite{5466536}. The DPGP code \(G\) of length \(n\) and minimum distance \(d\) is defined as a set of permutations \((\pi_0, \ldots, \pi_{n-1}) \in \Sc_n\) that satisfy the following condition: \( \pi_i \equiv i \pmod{d} \text{ for all } i \in [n]. \) Let \(A_i\) be the set of integers in \([n]\) congruent to \(i\) modulo \(d\). For all \(i \in [d]\), we define \(A_i\) as follows: \( A_i = (d\mathbb{Z} + i) \cap [n] = \{j \in [n] \mid j \equiv i \ (\bmod\ d)\}. \) Then, we can express \(G\) as the direct product of symmetric groups on \(A_i\): \( G = \Sc_{A_0} \times \Sc_{A_1} \times \cdots \times \Sc_{A_{d-1}}. \) The size of \(A_i\) is \(\left\lfloor \frac{n}{d} \right\rfloor \) when \( i \geq (n \bmod\ d)\), and \(\left\lceil \frac{n}{d} \right\rceil \) when \( i < (n \bmod\ d)\). Consequently, the size of the code \(\|G\| = \|A_0\| \cdots \|A_{d-1}\|\) can be expressed as $ \|G\|=\left(\left\lceil \frac{n}{d} \right\rceil !\right)^{n \bmod\ d} \left(\left\lfloor \frac{n}{d} \right\rfloor !\right)^{d - (n \bmod\ d)}. $ This expression simplifies to \(\|G\| = \left(\left(\frac{n}{d}!\right)^d\right)\) when \(d\) divides \(n\). We offer an alternative expression for $\|G\|$. The size of $G$ can be represented as the product of $n$ factors, as shown below: $\|G\|=\prod_{j=0}^{n-1} {\left( \left\lfloor j/d \right\rfloor + 1 \right)}$ Now, let us proceed with proving this. First, express $n$ in terms of the quotient $q$ and remainder $r$ when divided by $d$, i.e., $n = qd + r$. The product $\prod_{j=0}^{n-1} \left( \left\lfloor \frac{j}{d} \right\rfloor + 1 \right)$ can be rewritten as follows: \begin{align} &\prod_{j=0}^{qd-1} {\left( \left\lfloor j/d \right\rfloor + 1 \right)}\times\prod_{j=qd}^{qd+r-1} {\left( \left\lfloor j/d \right\rfloor + 1 \right)}. \\&= \prod_{p=0}^{q-1} \prod_{s=0}^{d-1} \left( \left\lfloor \frac{pd+s}{d} \right\rfloor + 1 \right) \times \prod_{s=0}^{r-1} \left( \left\lfloor \frac{qd+s}{d} \right\rfloor + 1 \right) \\&= \Bigl(\prod_{p=0}^{q-1}{\left( p + 1 \right)^d}\Bigr)\times(q+1)^r \\ &= (\overbrace{1\cdots 1}^{d \text{ times}})(\overbrace{2\cdots 2}^{d \text{ times}})\cdots(\overbrace{q\cdots q}^{d \text{ times}})\times(q+1)^r \\&= (q!)^d\times(q+1)^r \\&\stkeqa \left( \left\lceil n/d \right\rceil ! \right)^{r}\left( \left\lfloor n/d \right\rfloor ! \right)^{d - r}=\|G\|. \end{align} The validity of (a) becomes evident upon considering that $ \left\lceil j/d\right\rceil$ is $\left\lfloor n/d\right\rfloor $ if $r=0$ and is $\left\lfloor n/d\right\rfloor+1$, otherwise. \section{Code Extension} In Section \cite[III.~C]{5466546}, \Klove et al. introduced the concept of code extension. In this section, we provide a comprehensive overview of these codes, followed by a discussion of their encoding methods in the subsequent section. The properties of code extension detailed here are either directly derived from or previously established in \cite{5466546}. While the original work in presents several valuable insights regarding code extension, its presentation is somewhat fragmented, making it challenging to cite relevant points clearly. Therefore, the goal of this section is to systematically consolidate the key findings on code extension. By organizing the material in a more cohesive manner, we aim to clarify the relationships and properties associated with code extension, enabling a more straightforward understanding and application of these ideas in further research. \subsection{Definition} The concept of an \textit{extension of a permutation} was introduced in \cite[Section III.C]{5466546}\footnote{Note that the definition provided in \cite[Section III.C]{5466546} contains a minor error, using a strict inequality, specifically defining $\phi_s(x) := x + \mathbbm{1}[x > s]$. This formulation fails to yield a valid permutation for the subsequently defined $\piU^s$.}. Let $\piU = (\pi_0, \dots, \pi_{n-1}) \in \Sc_n$ be a permutation of length $n \geq 1$. The \textit{extended permutation} of $\piU$ with a \textit{head} $s \in [n+1]$ is defined as a permutation of length $n+1$: \begin{align} \piU^s := \bigl[s, \pi_0^s, \pi_1^s, \dots, \pi_{n-1}^s\bigr], \label{001517_13Apr23} \end{align} where $x^s := \phi_s(x) := x + \mathbbm{1}[x \geq s]$. Here, the indicator function $\mathbbm{1}[P]$ equals 1 if the proposition \( P \) is true, and 0 otherwise. Next, we introduce the extension of permutation codes. For $\Cc \subset \Sc_n$ and a set \( S \subset [n+1] \), which we refer to as the \textit{head set}, the \textit{extended code} with head set \( S \) is defined by \[ \Cc^S := \{\piU^s \in \Sc_{n+1} \mid s \in S, \piU \in \Cc\}. \] Since \( \Cc^S \) is empty if \( S \) is empty, we assume throughout this paper, unless otherwise noted, that the head set is non-empty. This code is the set of permutations obtained by extending each codeword \( \piU \in \Cc \) with a head \( s \in S \). To facilitate a more concise definition, we introduce a {formal codeword of length zero}, denoted as \( \varepsilonU \), which satisfies the condition: \[ \varepsilonU^0 = [0]. \] For a subset \( S \subset [n+1] \), we define the \textit{minimum distance} of \( S \) as the smallest difference between distinct elements, formally given by: \[ \dmin(S) \defeq \min \bigl\{ \|s - s'\| : s, s' \in S, s \neq s' \bigr\}. \] For sets containing only a single distinct element, the minimum distance is defined to be \( \infty \). \begin{example} The extended codeword of $[0123]$ with head $2$ is $[0123]^2=[20134]$. For $\Cc=\{[0123]\}$ and $S=\{0,2,4\}$, we have $\Cc^S=\{[01234],\allowbreak [20134],[40123]\}. $ \end{example} \subsection{Some Properties on Extensions} In this section, we derive several useful properties related to extensions for $n\ge 1$. \begin{lem}\label{144109_3Jun24} For $\pi,\sigma\in [n]$ and $s\in [n+1]$, \begin{enumerate}[label=\roman), series = tobecont, itemjoin = \quad] \item \label{145251_3Jun24} $\pi < \sigma$ implies $\pi^s < \sigma^s$. \item \label{145256_3Jun24} $\pi \le \sigma$ implies $\pi^s \le \sigma^s$. \end{enumerate} \end{lem} \begin{proof}\label{144130_3Jun24} \ref{145251_3Jun24}. In the case where $s \le \pi < \sigma$: $\pi^s = \pi + 1 < \sigma + 1 = \sigma^s$. In the case where $\pi < \sigma < s$: $\pi^s = \pi < \sigma = \sigma^s$. In the case where $\pi < s \le \sigma$: $\pi^s = \pi < \sigma + 1 = \sigma^s$. From \ref{145251_3Jun24} and the fact that $\pi^s = \sigma^s$ when $\pi = \sigma$, \ref{145256_3Jun24} is evident. \end{proof} The following theorem gives a lower bound on $\|S\|$ in terms of $\dmin(S)$. \begin{teiri}\label{230931_30Oct23} For any subset $S \subset [n+1]$ such that $\dmin(S) \ge d$, the following inequality holds: $d(\|S\|-1) \le n$. From this, it follows that: $\|S\| \le \left\lfloor \frac{n}{d} + 1 \right\rfloor$. Conversely, by setting $S = \{0, d, 2d, \ldots, (\|S\|-1)d\} \subset [n+1]$, we achieve $\|S\| = \left\lfloor \frac{n}{d} + 1 \right\rfloor$ and $\dmin(S) = d$. \end{teiri} \begin{proof}\label{001824_29Oct23} Consider the set of integers with the following inclusion: \[ \{s_1\} \cup \bigcup_{i=1}^{\|S\|-1} (s_i, s_{i+1}] \subset [n+1] \] Each constituent set on the left-hand side is disjoint. Considering the sizes of both sides, we have: \[ 1 + \sum_{i=1}^{\|S\|-1} \|(s_i, s_{i+1}]\| \le n + 1 \] Moreover, since $d \le \|s_i - s_{i+1}\| = \|(s_i, s_{i+1}]\|$, it follows that: \[ 1 + d(\|S\|-1) \le 1 + \sum_{i=1}^{\|S\|-1} \|(s_i, s_{i+1}]\| \le n + 1 \] This concludes the proof. \end{proof} \begin{example}\label{141422_10Jan24} For $n=5$, $S=\{0,3,5\}$, we have $\dmin(S)=2$, $\|S\|=3$, $\floor{(n+1)/\dmin(S)+1}=\floor{5/2+1}=3$. For $n=6$, $S=\{0,3,6\}$, we have $\dmin(S)=2$, $\|S\|=3$, $\floor{(n+1)/\dmin(S)+1}=\floor{6/3+1}=3$. \end{example} \begin{lem}\label{010525_27Nov23} For a permutation $\piU \in \Sc_n$ and $s, t \in [n+1]$, we have: \begin{align} \dinf(\piU^s, \piU^t) = \|s - t\|. \end{align} \end{lem} \begin{proof}\label{012433_27Nov23} The result is clear when $s = t$, as both sides are zero. Now, consider the case when $s \neq t$. We have $\dinf(\piU^s,\piU^t)=\max_{j\in [n+1]}\|(\piU^s)_j-(\piU^t)_j\|=\max\{\|s-t\|,\|\pi_j^s-\pi_j^t\| \tFOR j\in [n]\}=\|s-t\|$. \end{proof} \begin{lem}\label{lem:phi_kakudai} For \( n > 0 \), \( s \in [n+1] \), and \( \pi, \sigma \in [n] \), we have: \begin{align} \left\| \pi^s - \sigma^s \right\| = \left\| \pi - \sigma \right\| + \mathbbm{1}[s \in (\pi, \sigma]]. \label{142940_10Apr23} \end{align} \end{lem} \begin{proof} The following equation provides the proof for the claim. $ \|\pi^s-\sigma^s\| = \left\|\phi_s\left(\pi\right)-\phi_s\left(\sigma\right)\right\| =\left\|\left(\pi-\sigma\right)+\left(\mathbbm{1}\left\{\pi \geq s\right\}-\mathbbm{1}\left\{\sigma \geq s\right\}\right)\right\| = \|\pi-\sigma\|+\mathbbm{1}[\sigma< s\le \pi \tOR \pi< s\le \sigma] = \|\pi-\sigma\|+\mathbbm{1}[s\in (\pi, \sigma]]. $ \end{proof} \begin{lem}\label{224431_12Nov23} Let $\Cc$ be a code of length $n$. For distinct codewords $\piU, \sigmaU \in \Cc$ and $s \in [n+1]$, it holds that $\dinf(\piU, \sigmaU) \le d_\infty\bigl(\piU^s, \sigmaU^s\bigr) \le \dinf(\piU, \sigmaU) + 1. $ \end{lem} \begin{proof} The 0-th element of both $\piU^s$ and $\sigmaU^s$ is $s$. From Lem.~\eqref{001517_13Apr23}, we have $d_\infty\bigl(\piU^s, \sigmaU^s\bigr) = \max_{j \in [n+1]}\|\pi_j^s - \sigma_j^s\|$. From Lem.~\ref{lem:phi_kakudai}, it follows that $\|\pi_j - \sigma_j\| \le \|\pi_j^s - \sigma_j^s\| \le \|\pi_j - \sigma_j\| + 1$. The equality in the second inequality holds if and only if $s \in (\pi_j^s, \sigma_j^s]$. Taking the maximum over all $j \in [n]$, we derive the assertion of the lemma. \end{proof} \begin{lem}[$\phi$ is expansive w.r.t. its second argument]\label{143827_10May23} For a permutation code $\Cc$ of length $n$ and a subset $S \subset [n+1]$, for arbitrary $\piU, \sigmaU \in \Cc$ and $s, t \in S$, the following inequality holds true: $d_\infty\bigl(\piU^s, \sigmaU^t\bigr) \ge \|s - t\|.$ Equality holds when $\piU = \sigmaU$. \end{lem} \begin{proof} The claim is evident from the following inequality: \begin{align} d_\infty\left(\piU^s, \sigmaU^t\right) = \max_{j \in [n+1]} \left\| (\piU^s)_j - (\sigmaU^t)_j \right\| \geq \left\| (\piU^s)_0 - (\sigmaU^t)_0 \right\| = \left\| s - t \right\|. \end{align} From Lem.~\ref{010525_27Nov23}, it is clear that equality holds when $\piU = \sigmaU$. \end{proof} \begin{lem}\label{152042_1Oct23} $\phi:(\Sc_n\times [n+1])\to \Sc_{n+1}$ is a one-to-one mapping. \end{lem} \begin{proof} It is sufficient to show $(\piU,s) \neq (\sigmaU,t)$ implies $\piU^s \neq \sigmaU^t$. First, let's consider the case when $s \neq t$. From Lem.~\ref{143827_10May23}, $s\neq t$ implies $\piU^s \neq \sigmaU^t$. Next, let's consider the case when $\piU \neq \sigmaU$ and $s = t$. There exists $i \in [n]$ such that $\pi_i \neq \sigma_i$. According to Lem.~\ref{lem:phi_kakudai}, we have $\|\phi_s(\pi_i) - \phi_s(\sigma_i)\| \geq \|\pi_i - \sigma_i\|$, which in turn implies $\piU^s \neq \sigmaU^t$. \end{proof} From these lemmas, the following theorem is immediately derived. \begin{teiri}\label{205810_6Jul23} For a code $\Cc$ of length $n$ and a subset $S \subset [n+1]$, we have: $\|\Cc^S\| = \|\Cc\| \times \|S\|.$ \end{teiri} \subsection{Lower Bounds on Minimum Distance Through Extension} In this section, we provide several lower bounds on minimum distance through extension. \begin{teiri}\label{153755_6Jan24} For any permutation code $C\subset \Sc_{n}$ and any head set $S \subset [n+1]$, \begin{align} \dmin\left(\Cc^S\right) \geq \min \left(\dmin(S), \dmin(\Cc)\right) \end{align} \end{teiri} \begin{proof}\label{153804_6Jan24} First, consider the case where $\|S\|=1$, for which $\dmin(S)=\infty$. Let $S=\{s\}$. Any distinct pair of codewords from $\Cc^S$ can be expressed as $(\piU^s, \sigmaU^s)$, with $\piU$ and $\sigmaU$ being distinct elements of $\Cc$. We then have $ \dinf\left(\piU^s, \sigmaU^s\right) \stkgea \dinf\left(\piU, \sigmaU\right) \ge \dmin(\Cc)$, which leads to the inequality $ \dmin(\Cc) = \min \left\{\dmin(S), \dmin(\Cc)\right\}$. The result follows from Lemma~\ref{224431_12Nov23} as used in (a). Now, consider the case where $\|S\| \ge 2$. For any distinct codewords $\piU^s \neq \sigma^t \in \Cc^S$, we aim to show that $\dinf\left(\piU^s, \sigma^t\right) \geq \min \left(\dmin(S), \dmin(\Cc)\right)$. We examine the following two cases: \begin{itemize} \item If $s \neq t$: From Lemma~\ref{143827_10May23}, we know that $\dinf\left(\piU^s, \sigmaU^t\right) \geq \|s-t\| \geq \dmin(S)$. \item If $\piU \neq \sigmaU$ and $s = t$: According to Lemma~\ref{224431_12Nov23}, for distinct $\piU$ and $\sigmaU$ in $\Cc$, we have $\dinf\left(\piU^s, \sigmaU^s\right) \geq \dinf\left(\piU, \sigmaU\right) \geq \dmin(\Cc)$. In either case, it follows that $\dinf\left(\piU^s, \sigmaU^t\right) \geq \min \left\{\dmin(S), \dmin(\Cc)\right\}$. \end{itemize} \end{proof} The following theorem provides sufficient conditions on $\Cc$ and $S$ to construct an extended code $\Cc^S$ while ensuring the minimum distance remains at least $d$. \begin{teiri}[{\cite[Theorem 4]{5466546}}]\label{163227_14May23} For a code $\Cc$ of length $n$ and a subset $S \subset [n+1]$, the following holds: $\dmin(S) \ge d$ and $\dmin(\Cc) \ge d$ implies $\dmin(\Cc^S) \ge d$. \end{teiri} \begin{proof}\label{161444_6Jan24} The assumption is equivalent to $\min(\dmin(S), \dmin(\Cc)) \geq d$. By applying Theorem~\ref{153755_6Jan24}, we conclude that $\dmin(\Cc^S) \geq d$. \end{proof} \subsection{Upper Bounds on Minimum Distance Through Extension} The following two theorems give upper bound of the minimum distance of the extended code. \begin{teiri}[Upper bound on $\dmin(\Cc^S)$]\label{003746_1Oct23} Let $\Cc$ be a code of length $n$ and $S \subset [n+1]$ be a head set. \begin{align} \dmin(\Cc^S) \le \dmin(S). \end{align} \end{teiri} \begin{proof} If $\|S\|=1$, the claim of the theorem would be $d_\infty(\Cc^S) \le \infty$, which renders the claim meaningless. Therefore, we consider the case where $\|S\| \ge 2$. It suffices to show that there exists a pair of codeword $\Cc^S$, whose distance is $\dmin(S)$. Select $s, t \in S$ such that $\|s-t\| = \dmin(S)$. For any $\piU \in \Cc$, by Lem.~\ref{010525_27Nov23}, we have $d_\infty\bigl(\piU^s, \piU^t\bigr) = \|s-t\| = \dmin(S)$. \end{proof} \begin{teiri}\label{161110_1Oct23} Let $\Cc$ be a code of length $n$ and $S \subset [n+1]$ be a head set. Then, it holds that $d_{\infty}(\Cc^S) \le \dmin(\Cc)+1$. \end{teiri} \begin{proof} When $\|C\| = 1$, $\dmin(\Cc) = \infty$, so the claim is true. Consider the case where $\|C\| \ge 2$. It is sufficient to show that there exists a pair of codewords in $\Cc^S$ whose distance is less than or equal to $\dmin(\Cc) + 1$. Select distinct $\piU, \sigmaU \in \Cc$ such that $d_\infty(\piU, \sigmaU) = \dmin(\Cc)$. From Lemma~\ref{152042_1Oct23}, we observe that for any $s \in S$, $\piU^s$ and $\sigmaU^s$ are distinct codewords in $\Cc^S$. Hence, it follows that $\dinf\bigl(\piU^s, \sigmaU^s\bigr) \stackrel{\ref{224431_12Nov23}}{\leq} \dinf(\piU, \sigmaU) + 1 = \dmin(\Cc) + 1$, where the inequality is derived using Lemma~\ref{224431_12Nov23}. \end{proof} For $\Cc \subset \Sc_n$ and $S \subset [n+1]$, consider the extension $\Cc \to \Cc^S$. When $\|S\| = 1$, the size remains unchanged after the extension, i.e., $\|\Cc\| = \|\Cc^S\|$, as stated in Theorem~\ref{205810_6Jul23}. Such an extension is referred to as {\itshape size-preserving}. In cases where $\|\Cc\| < \|\Cc^S\|$, the extension is called {\itshape size-increasing}. If $\dmin(\Cc) < \dmin(\Cc^S)$, we describe the extension as distance-increasing. We probide an example of an extension that is both size-preserving and distance-increasing. \begin{example}\label{134606_2Oct23} Let $\Cc=\{0123,3012\}$ and $S=\{1\}$. Then, $\Cc^S=\{10234,14023\}$, with $\dmin(S)=\infty$, $\dmin(\Cc)=3$ and $\dmin(\Cc^S)=4$. This a size-preserving and distance-increasing extension. For $\Cc=\{0123,1032\}$ and $S=\{1,3\}$, we have $\Cc^S=\{10234,30124,12043,31042\}$, $\dmin(S)=2$,$\dmin(C)=1$ and $ \dmin(\Cc^S)=2$. This a size-increasing and distance-increasing extension. \end{example} \subsection{Codeword Pairs, Interval Sets, and Maximum Intervals} In this subsection, we derive the lemmas on extensions that are used in the proof of the theorem in Section \ref{164043_17Sep24}. For integers $x, y \ge 0$, we define {\itshape interval} between $y$ and $x$ and denoted it by $(y, x]$ as follows: $(y, x]$ is defined as $\{a \in \Zb \mid y < a \le x\}$ if $y < x$, as $\{a \in \Zb \mid x < a \le y\}$ if $x < y$, and as an empty set if $x = y$. The length of interval $I=(x,y]$ is defined as $\|x-y\|$ and denoted by $\|I\|$. For an interval $I=(x,y]\subset [n]$ and $s\in [n+1]$, we define $I^s:=\{a^s:a\in I, s\in S\}=(x^s,y^s]$. \begin{lem}\label{054652_14Jul24} For an interval $I\subset [n+1]$, it holds that $\|I^s\|=\|I\|+\I[s\in I]$. \end{lem} \begin{proof} Let $I=(x,y]\subset[n+1]$. The claim is obvious from the following: $\|I^s\|=\|(x^s,y^s]\|=\|x^s-y^s\|\stkeqa\|x-y\|+\I[s\in (x,y]]=\|I\|+\I[s\in I]$. In (a), we used \ref{lem:phi_kakudai}. \end{proof} In this section, we define interval sets and maximum intervals for codeword pairs and provide sufficient conditions for increasing the distance when the codeword pairs are extended, using the maximum intervals of the codeword pairs. \begin{lem}\label{153103_22May24} If intervals $I$ and $J$ are disjoint, then $I^s$ and $J^s$ are disjoint. \end{lem} \begin{proof} Without loss of generality, we can write $I = (\pi_1, \sigma_1]$ and $J = (\pi_2, \sigma_2]$ using $\pi_1 < \sigma_1 \le \pi_2 < \sigma_2$. From Lem.~\ref{144109_3Jun24}, we have $\pi_1^s < \sigma_1^s \le \pi_2^s < \sigma_2^s$, so $I^s$ and $J^s$ are disjoint. \end{proof} \begin{lem}\label{195413_3Jun24} If intervals $I$ and $J$ satisfy $I \subset J$, then $I^s \subset J^s$. \end{lem} \begin{proof} Without loss of generality, we can write $I = (\pi_1, \sigma_1]$ and $J = (\pi_2, \sigma_2]$ using $\pi_1 \le \pi_2 < \sigma_2 \le \sigma_1$. From Lem.~\ref{144109_3Jun24}, we have $\pi_1^s \le \pi_2^s < \sigma_2^s \le \sigma_1^s$, so $I^s \subset J^s$ holds. \end{proof} For a pair of permutations $\piU, \sigmaU$ of length $n$, we define the following: \begin{enumerate} \item The set of intervals $(\pi_j, \sigma_j]$ for $j=0, \ldots, n-1$ of non-zero length is called the {\it interval set} between $\piU$ and $\sigmaU$, or simply the interval set, and is denoted by $(\piU, \sigmaU]$. To be precise, $(\piU, \sigmaU] \defeq \{(\pi_j, \sigma_j] \mid \pi_j \neq \sigma_j, j=0, \ldots, n-1\}.$ \item For a pair of codewords $\piU, \sigmaU$, if an interval $(\pi_j, \sigma_j] \in (\piU, \sigmaU]$ contains all other intervals $(\pi_i, \sigma_i] \in (\piU, \sigmaU]$, i.e., $(\pi_j, \sigma_j] \supset (\pi_i, \sigma_i]$, then $(\pi_j, \sigma_j]$ is called the maximum interval of the pair $\piU, \sigmaU$. From the definition, we see that if a maximum interval exists for $\piU, \sigmaU$, it is unique. \end{enumerate} From the definition, the following holds: $(\piU^s, \sigmaU^s] = \{(\pi^s, \sigma^s] \mid (\pi, \sigma] \in (\piU, \sigmaU]\}$. Furthermore, the maximum length of the intervals in the interval set $(\piU, \sigmaU]$ is equal to the distance between $\piU$ and $\sigmaU$: $ \displaystyle \dinf(\piU, \sigmaU) = \max_{J \in (\piU, \sigmaU]} \ell(J).$ For a pair of permutations $\Psf:=(\piU, \sigmaU)$ in $\Sc_n$ and a head $s\in[n+1]$, we denote a pair of permutations $(\piU^s, \sigmaU^s)$ in $\Sc_{n+1}$ by $\Psf^s$. \begin{lem}\label{060127_4Mar24} For a pair of permutations $\Psf:=(\piU, \sigmaU)$ in $\Sc_n$ of length $n$ that has a maximum interval $I$, the following holds: \begin{enumerate}[label=\roman), series = tobecont, itemjoin = \quad] \item\label{142848_5Mar24} The permutation pair $\Psf^s$ has maximum interval $I^s$. \item $\dinf(\piU, \sigmaU) = \|I\|$ \item\label{153314_30Dec23} $\dinf(\piU^s, \sigmaU^s) = \dinf(\piU, \sigmaU) + \I[s \in I]$ \item$\|I^s\| = \|I\| + \I[s \in I]$ \end{enumerate} \end{lem} \begin{proof}\label{162409_30Dec23} First, we prove \ref{142848_5Mar24}. Since $I$ is maximum, we have $J \subset I$ for any interval $J \in (\piU, \sigmaU]$. From this and Lem.~\ref{195413_3Jun24}, it follows that $J^s \subset I^s$. Therefore, $I^s$ is maximum in the permutation pair $\Psf^s$. The fact that $I$ is the unique interval of length $\dinf(\piU, \sigmaU)$ contained in $(\piU^s, \sigmaU^s)$, together with Lem.~\ref{054652_14Jul24}, makes \ref{153314_30Dec23} evident. \end{proof} \begin{lem}\label{000804_8Dec23} Let $\piU$ be a codeword of length $n$, and let $s<t$ for $s,t \in [n+1]$. Then the following holds: \begin{enumerate}[label=\roman), series = tobecont, itemjoin = \quad] \item \label{133646_30Dec23} $ (\piU^s,\piU^t]=\{(s,t],(s,s+1],\ldots,(t-1,t]\}$ \item $\#(\piU^s,\piU^t]=\|s-t\|+1$ \item The codeword pair $(\piU^s,\piU^t)$ has the largest interval $(s,t]$. \end{enumerate} \end{lem} \begin{proof} We will prove \ref{133646_30Dec23}. Without loss of generality, we can assume that $\piU$ is the identity permutation $\iotaU=[0,1,\ldots,n-1]$. From the definition of extension \eqref{001517_13Apr23}, we have the following: \begin{align} \begin{array}{llll} \iotaU^s=[\black{s},&0,1,\ldots,s-1,\black{s+1},&\black{s+2,\ldots,t}, &t+1,\ldots,n], \\ \iotaU^t=[\black{t},&0,1,\ldots,s-1,\black{s}, &\black{s+1,\ldots,t-1},&t+1,\ldots,n]. \end{array} \end{align} The remaining claims follow from this. \end{proof} \section{Recursively Extended Permutation Codes}\label{164043_17Sep24} In the previous section, we investigated the changes in minimum distance and size resulting from a single code extension. In this section, we investigate permutation codes that repeatedly extended. For each $j=0,\ldots,n-1$, let $S^{(j)}$ be a non-empty subset of $[j+1]$. The construction method for the permutation code $\Cc^{(n)}$ of length $n$ is as follows: First, we define $\Cc^{(0)} := \{\varepsilonU\}$. Next, for $j=1, \ldots, n$, we recursively construct $\Cc^{(j)}$ from $\Cc^{(j-1)}$ using the equation: $\Cc^{(j)} = \phi(\Cc^{(j-1)}; S^{(j-1)}).$ We refer to $\Cc^{(n)}$ constructed in this manner as a {\it recursively extended permutation} (REP) code generated by $\{S^{(j)}\}_{j=0}^{n-1}$. We denote it by $\Cc^{(n)} = \<\{S^{(0)},\ldots,S^{(n-1)}\}\>$. From Thm.~\ref{205810_6Jul23}, we obtain the following: $\|\Cc^{(n)}\| = \prod_{j=0}^{n-1} \|S^{(j)}\|.$ \begin{example} In \cite[III. D]{5466546}, a construction of $(n,q^{n-(q-1)d}, d)$ REP code with head sets $S^{(j)}\subset [j+1]$ for $j\in [n]$ is proposed as follows. For integers $n,d,q$ with $q \ge 2$ and $(q-1)d< n$, set $S^{(j)} = \{0\}$ for $0 \le j < (q-1)d$. Set $S^{(j)} = \{\lfloor {j}/{(q-1)}\rfloor x: x = 0, \ldots, q-2\} \cup \{j\}$ for $(q-1)d \le j \le n-1$. We can interpret such $S^{(j)}$ as the positioning of $q$ points within $[j+1]$, ensuring a minimum spacing of $d$ between each point. We observe that $\|S^{(j)}\|$ is 1 and $\dmin(S^{(j)}) = \infty$ for $0 \le j < (q-1)d$ and $\|S^{(j)}\| = q$ and $\dmin(S^{(j)}) \ge d$ for $(q-1)d \le j \le n-1$. The size of the code is given by $\|\Cc^{(n)}\| = \prod_{j=0}^{n-1}\|S^{(j)}\| = q^{n-(q-1)d}$. Since $\Cc^{(0)} = \{\epsilonU\}$, it follows that $\dmin(\Cc^{(0)}) = \infty$. By repeatedly applying Thm.~\ref{163227_14May23}, it holds that $\dmin(\Cc^{(n)}) \ge d$. \end{example} As seen in the example above, from Thm.~\ref{163227_14May23}, if $\dmin (S^{(j)}) \ge d$ for $j=0,1,\ldots,n-1$, then $\dmin (\Cc^{(n)}) \ge d$. The converse is not true. To achieve $\dmin (\Cc^{(n)}) \ge d$, it is not necessary that $\dmin (S^{(j)}) \ge d$ for $j=0,1,\ldots,n-1$. For instance, consider $S^{(0)} = {0,1}$ and $S^{(1)} = {1}$, where $\dmin(S^{(0)}) = 1$. Then, we have, $\Cc^{(0)} = \{0\}, \ \Cc^{(1)} = \{01, 10\}, \ \Cc^{(2)} = \{102, 120\}, $ and thus $\dmin (\Cc^{(2)}) = 2$. \subsection{The necessary number of size-preserving extensions for increasing minimum distance} A code with a minimum distance of at least $d$ and a length of $n$ is referred to an $[n, d]$ code. In this subsection, we identify the $[n, d]$ code with the largest possible size. From the results of the previous section, it is clear that the minimum distance can increase with extensions. It is difficult to derive a tight upper bound on the size of an $[n, d]$ code from the conventional bounds derived in previous section. We need to evaluate the number of size-preserving extensions required to increase the minimum distance through extensions. Let $\Cc_0$ be a permutation code of some code length. In this subsection, we investigate the number of size-preserving extensions, denoted as $c_1(\Cc_0;d)$, required to increase the minimum distance of extended code of $\Cc_0$ to $d$.We provide both lower and upper bounds on $c_1(\Cc_0;d)$. These bounds will be used in the proof for the optimal REP codes in the next subsection. The code $\Cc_0$ is extended with head set $S_j$ as $\Cc_{j+1} = \Cc_j^{S_j}$ for $j \ge 0$. In this context, we denote the minimum number of size-preserving extension needed for $\Cc_k$ to achieve a minimum distance of $d$ as $c_1^{(k)}(\Cc_0; S_0, \ldots, S_{k-1})$. Formally, this can be written as follows: \begin{align} &c_1(\Cc_0; d) \defeq \min_{k \ge 0} c_1^{(k)}(\Cc_0; d) \label{151909_28Apr24} \\ &c_1^{(k)}(\Cc_0; d) \defeq \min_{S_0, \ldots, S_{k-1} : \dmin(\Cc_k) \ge d} \# \{0 \le l \le k-1 : \|S_l\| = 1\} \end{align} The following lemma provides an upper bound for $c_1(\Cc^{(n)};d)$. \begin{lem}[Upper bound on $c_1$]\label{165523_23May24} Let $n > d \ge 1$. For any REP code $\Cc^{(n)}$ such that $\dmin(\Cc^{(n)}) \ge d$, for any $1 \le k \le n$, the following holds: \begin{align} c_1(\Cc^{(k)};d) \le n - k \label{031411_8May24} \end{align} \end{lem} \begin{proof} First, since $\dmin(\Cc^{(n)}) \ge d$, we have $c_1(\Cc^{(n)};d) = 0$. Next, we show that $c_1(\Cc^{(n-1)};d) \le 1$. From the assumption: $S^{(n)} = (\Cc^{(n-1)})^{S^{(n-1)}}$, we have $c_1(\Cc^{(n-1)};d) = 1$ if $\|S^{(n-1)}\| = 1$, and $c_1(\Cc^{(n-1)};d) = 0$, otherwise. Continuing this process, we obtain \eqref{031411_8May24}. \end{proof} In \eqref{151909_28Apr24}, we defined $c_1(\Cc;d)$ for a code $\Cc \subset \Sc_n$. Below, with a slight abuse of notation, we define $c_1(S;d)$ for a head set $S \subset [n+1]$. First, for $S$ with $\|S\| = 1$, we define $c_1(S;d) = 0$. Next, for $S$ with $\|S\| \ge 2$, let us write $S = \{s_1, s_2, \ldots\}$ with $s_1 < s_2 < \cdots$. We define $c_1(S;d)$ as the minimum number of increments required to extend the length of each interval $(s_i, s_{i+1}]$ of length less than $d$ to length $d$. More precisely, it is defined as follows: \begin{align} c_1(S;d) \defeq \sum_{j:\|s_j-s_{j+1}\|<d} (d - \|s_j-s_{j+1}\|) \label{151901_28Apr24} \end{align} This gives a lower bound for $c_1(\Cc^{S_0};d)$ in Thm.~\ref{231412_20May24}. The following lemma generalizes Thm.~\ref{230931_30Oct23}, which provides an upper bound for $\|S\|$. By setting $c = 0$, it reduces to Thm.~\ref{230931_30Oct23}. \begin{lem}\label{234753_24Apr24} For $S \subset [n]$, suppose $c \ge c_1(S;d)$. Then, the following holds: \begin{align} \|S\| \le \frac{n-1+c}{d} + 1\label{132307_12Jul24} \end{align} \end{lem} \begin{proof} Let $J := \{1, \ldots, \|S\|-1\}$. Define $\JU := \{j \in J : \|s_j-s_{j+1}\| < d\}$ and $\JO := \{j \in J : \|s_j-s_{j+1}\| \ge d\}$. We have $\|\JU\| + \|\JO\| = \|S\|-1$. The following holds: \begin{align} n &\stkgea 1 + \sum_{j \in J} \|s_j-s_{j+1}\| \\&= 1 + \sum_{j \in \JU} \|s_j-s_{j+1}\| + \sum_{j \in \JO} \|s_j-s_{j+1}\| \\&\stkgeb 1 + \|\JU\|d - c + \|\JO\|d \\&= 1 - c + (\|S\|-1)d \end{align} In (a), we used the union bound for the inclusion $[n] \supset \{s_1\} \cup \bigcup_{j \in J} (s_j, s_{j+1}]$, we obtain: In (b), we used the assumption: $c \ge c_1(S;d) = \sum_{j \in \JU} (d - \|s_j-s_{j+1}\|)$ and the fact that $\sum_{j \in \JO} \|s_j-s_{j+1}\| \ge \|\JO\|d$. This inequality immediately gives \eqref{132307_12Jul24}. \end{proof} \begin{teiri}\label{153614_27May24} For a code $\Cc_0 \subset \Sc_n$ and a head set $S \subset [n+1]$, let $\Cc_1 = \Cc_0^{S_0}$. For $d \ge 1$, the following holds: \begin{enumerate}[label=\roman), series = tobecont, itemjoin = \quad] \item \label{161656_27May24} $c_1(\Cc_0;d) \le c_1(\Cc_1;d) + 1$ \item \label{161706_27May24} $c_1(\Cc_0;d) = c_1(\Cc_1;d) + 1$ implies $\|S_0\| = 1$. \end{enumerate} \end{teiri} \begin{proof} \ref{161656_27May24}. Suppose $c_1(\Cc_0;d) > c_1(\Cc_0^{S_0};d) + 1$ and derive a contradiction. Then, there exist $k>0$ and $k-1$ head sets $S_i \subset [n+1+i]$ $(i = 1,2 \ldots, k-1)$ of which at most $c_1(\Cc_0;d) - 2$ head sets are of size one, that satisfy $\dmin(\Cc_k = \Cc_1^{S_1 \cdots S_{k-1}}) \ge d$. This implies $\dmin(\Cc_k = \Cc_0^{S_0 \cdots S_{k-1}}) \ge d$ which contradicts the minimality of $c_1(\Cc_0;d)$. \ref{161706_27May24}. Suppose $\|S_0\| \neq 1$ and derive a contradiction. There exist head sets $S_i \subset [n+1+i]$ $(i = 1,2 \ldots, k-1)$ of which $c_1(\Cc_1;d)$ head sets are of size one, that satisfy $\dmin(\Cc_k = \Cc_1^{S_1 \cdots S_{k-1}}) \ge d$. From the fact that $\dmin(\Cc_k = \Cc_0^{S_0 \cdots S_{k-1}}) \ge d$ and the assumption $\|S_0\| \neq 1$, we see that this contradicts the minimality of $c_1(\Cc_0;d)$. \end{proof} For a codeword pair $(\piU, \sigmaU) =: \Psf$, we denote $(\piU^s, \sigmaU^s)$ by $\Psf^s$. We can rewrite Lem.~\ref{060127_4Mar24} as $ \dinf(\Psf^s) = \dinf(\Psf) + \I[s \in J]$. From this, when $(\piU, \sigmaU)$ has maximum interval $I$, it holds that $ \|I^s\| = \|I\| + \I[s \in I]$. \begin{lem}\label{152448_18May24} For a code $\Cc$ of length $n$, suppose there are $k$ codeword pairs $\Psf_1, \Psf_2, \ldots, \Psf_k$ each having maximum intervals $I_1, I_2, \ldots, I_k$ that are mutually disjoint. For $S \subset [n+1]$, there exist $k$ codeword pairs $\Qsf_1, \ldots, \Qsf_k$ in the extended code $\Cc^S$ each having mutually disjoint maximum intervals $J_1, J_2, \ldots, J_k$ satisfying the following: when $\|S\| = 1$, $\|J_1\| \le \|I_1\| + 1$, and $\|J_i\| \le \|I_i\|$ for $i \neq 1$ and when $\|S\| \ge 2$, $\|J_i\| \le \|I_i\|$ for $1 \le i \le k$. \end{lem} \begin{proof}\label{225701_20May24} First, we prove the case with $\|S\| = 1$. Let $S = \{s\}$ and $\Qsf_i := \Psf_i^s$ for $1 \le i \le k$. From Lem.~\ref{060127_4Mar24}~\ref{142848_5Mar24} and Lem.~\ref{153103_22May24}, each $\Qsf_i$ has a mutually disjoint maximum interval $J_i := I_i^s$ satisfying $\|J_i\| = \|I_i\| + \I[s \in I_i]$. Since $\{I_i\}$ are mutually disjoint, $s$ can be contained in at most one of the intervals $I_i$. Next, we prove the case with $\|S\| \ge 2$. Let $s$ and $t$ be distinct elements in $S$. Since $I_1, \ldots, I_k$ are disjoint, it suffices to consider the following three cases without loss of generality:\\ i) If $s$ and $t$ are contained in the same interval: let $s, t \in I_1$. For $i \neq 1$, since $I_1$ and $I_i$ are disjoint, $s, t \notin I_i$. For any $\piU \in \Cc$, let $\Qsf_1 = (\piU^s, \piU^t)$. From Lem.~\ref{000804_8Dec23}, $\Qsf_1$ has the maximum interval $J_1 = (s, t] \subsetneq I_1$. This inequality implies $\|J_1\| < \|I_1\|$. For $i \neq 1$, let $\Qsf_i = \Psf_i^s$. From Lem.~\ref{153103_22May24} and the fact that $s \notin I_i$, the pairs $\{\Qsf_i\}_{i \neq 1}$ are mutually disjoint. From Lem.~\ref{060127_4Mar24}~\ref{142848_5Mar24}, each $\Qsf_i$ has the maximum interval $J_i = I_i^s = I_i$, hence we have $\|J_i\|=\|I_i\|$. Thus, the intervals $\{J_i\}$ are disjoint.\\ ii) If $s$ is not contained in any interval: There is no $i$ such that $s \in I_i$. For $1 \le i \le k$, let $\Qsf_i = \Psf_i^s$. By the same argument as above, $\Qsf_i$ has the maximum interval $J_i$ with $\|J_i\| = \|I_i\|$ and these intervals are disjoint.\\ iii) If $s$ and $t$ are contained in different intervals: let $s \in I_1$ and $t \in I_2$. Let $\Qsf_1 = \Psf_2^s$, $\Qsf_2 = \Psf_1^t$, and $\Qsf_i = \Psf_i^s$ for $i \ge 2$. By the same argument as above, $\{\Qsf_i\}$ are mutually disjoint, each of which has the maximum interval $J_i$ with $\|J_i\| = \|I_i\|$. \end{proof} \begin{teiri}\label{231412_20May24} For a code $\Cc_0 \subset \Sc_n$ and a head set $S_0 \subset [n+1]$, let $\Cc_1 := \Cc_0^{S_0}$. Then, $c_1(\Cc_1;d) \ge c_1(S_0;d)$ holds. \end{teiri} \begin{proof}\label{231611_20May24} For head sets $S_j \subset [j+1]$ for $j = 1, 1, 2, \ldots$, define $\Cc_{j+1} := \Cc_j^{S_j}$. It is sufficient to show that there are at least $c_1(S_0; d)$ head set of size one among $S_1, \ldots, S_{m-1}$ for any $m\ge 1$ and $S_0,\ldots,S_{m-1}$ such that $\dmin(\Cc_m) \ge d$. Let the elements of $S_0$ be $s_1 < \cdots < s_{k+1}$. Denote $k := \|S_0\| - 1$. Choose some $\piU \in \Cc_0$ and denote $k$ codeword pairs $(\piU^{s_i}, \piU^{s_{i+1}})$ in $\Cc_1$ by $\Psf_i^0$. Each codeword pair $\Psf_i^0$ has a maximum interval $I_i^0 := (s_i, s_{i+1}]$, and these intervals are mutually disjoint. According to Lem.~\ref{152448_18May24}, there exist $k$ corresponding codeword pairs in $\Cc_1$, each with a mutually disjoint maximum interval. Let these pairs be denoted as $\{\Psf_i^1\}$. Continue this procedure for $\Cc_{i+1}$ for $i = 1, \ldots, m-1$. Consequently, there will be $k$ corresponding codeword pairs in $\Cc_m$, each with a mutually disjoint maximum interval, denoted as $\{\Psf_i^m\}_{i=1}^k$. Since $\dmin(\Cc_m) \ge d$, the length of the intervals for the codeword pairs $\{\Psf_i^m\}_{i=1}^k$ must be at least $d$. From Lem.~\ref{152448_18May24}, it follows that during each extension, at most one corresponding interval increases in length, and the increase is by at most one. Therefore, to increment the size of one of these $k$ disjoint intervals during the $j$-th extension by $S_j$, we need $\|S_j\| = 1$. By definition, $c_1(S_0; d)$ represents the total number of increments needed to increase the length of each interval $(s_i, s_{i+1}]$ from less than $d$ to $d$. Hence, the number of $j$ such that $\|S_j\| = 1$ is at least $c_1(S_0; d)$, which completes the proof. \end{proof} \subsection{Optimal REP codes} \label{161538_26Aug24} In this subsection, we prove the following for any $n > d \geq 1$: 1) An upper bound on the size of an $[n,d]$ REP code. 2) There exists an $[n,d]$ REP code whose size achieves the upper bound. 3) The upper bound matches the size of an $[n,d]$ DPGP code. Some readers might conclude from these results that the REP code and DPGP code share the same structure. However, as far as the authors have investigated, no such structure has been found. \begin{teiri}\label{165623_23May24} Let $\Cc^{(n)}$ be an $[n,d]$ REP code. Then it holds that $ \|\Cc^{(n)}\|\le \prod_{j=0}^{n-1}\floor{j/d+1}$. \end{teiri} \begin{proof} To simplify notation, we write $c^{(k)} := c_1(\Cc^{(k)}; d)$ for $0\le k <n$. We denote the sets of non-decreasing and decreasing points in the sequence $\{c^{(k)}\}$ by $K$ and $K^c$, respectively. Formally, $K \defeq \{0 \le k <n : c^{(k)} \le c^{(k+1)}\}$, $K^c \defeq \{0 \le k <n : c^{(k)} > c^{(k+1)}\}$. For $k \in K^c$, from Thm.~\ref{153614_27May24}, we have $c^{(k)} = c^{(k+1)} + 1$ and $\|S^{(k)}\| = 1$. Therefore, the following holds: $\|\Cc^{(n)}\|=\prod_{k=0}^{n-1} \|S^{(k)}\| = \prod_{k \in K} \|S^{(k)}\|. $ Furthermore, we can express it as follows: \begin{align} \prod_{k \in K} \|S^{(k)}\| &\stklea \prod_{k \in K} \left\lfloor \frac{k + c^{(k+1)}}{d} + 1 \right\rfloor \label{020938_27Apr24} \\ &\stkeqb \prod_{i=1}^{\|K\|} \left\lfloor \frac{k_i + c^{(k_i+1)}}{d} + 1 \right\rfloor \end{align} In (a), we used the fact that from Thm.~\ref{231412_20May24}, $c^{(k+1)} \ge c_1(S^{(k)}; d)$, and from Lem.~\ref{234753_24Apr24}, $\|S^{(k)}\| \le \left\lfloor \frac{k + c^{(k+1)}}{d} + 1 \right\rfloor$. In (b), we wrote the elements of $K$ in ascending order as $k_1 < k_2 < \cdots < k_{\|K\|}$. For $\|K\| = 1$, from Lem.~\ref{165523_23May24}, we have $k_1 + c^{(k_1+1)} \le n-1$, thus proving the theorem. Let us consider the case $\|K\| \ge 2$. The following holds: \begin{align} \prod_{i=1}^{\|K\|} \left\lfloor \frac{k_i + c^{(k_i+1)}}{d} + 1 \right\rfloor &\stklec \prod_{i=1}^{\|K\|} \left\lfloor \frac{n-1-(\|K\|-i)}{d} + 1 \right\rfloor \end{align} In (c), we used Lem.~\ref{021210_24Jul24}. The proof completes by considering $\prod_{i=1}^{\|K\|} \left\lfloor \frac{n-1-(\|K\|-i)}{d} + 1 \right\rfloor = \prod_{j=n-\|K\|}^{n-1} \left\lfloor \frac{j}{d} + 1 \right\rfloor \le \prod_{j=0}^{n-1} \left\lfloor \frac{j}{d} + 1 \right\rfloor. $ \end{proof} \begin{lem}\label{021210_24Jul24} Denote $m:=\|K\|$. For $i=1,\ldots,m-1$, it holds that \begin{align} k_i + c^{(k_i+1)} \le n-1 -(m-i) .\label{014925_24Jul24} \end{align} \end{lem} \begin{proof} First, we prove that for $i=1, \ldots, m-1$ \begin{align} k_i + c^{(k_i+1)} < k_{i+1} + c^{(k_{i+1}+1)}. \label{172920_9May24} \end{align} Note that we have $c^{(k_{i+1})} \le c^{(k_{i+1}+1)}$ since $k_{i+1} \in K$. It is sufficient to consider the following two cases. i) The case $k_i$ and $k_{i+1}$ are consecutive, $ k_i + 1=k_{i+1}$, hence $c^{(k_i+1)} = c^{(k_{i+1})}$: Therefore it holds that $k_i + c^{(k_i+1)} \le k_{i+1} + c^{(k_{i+1}+1)} - 1$. ii) The case $k_i$ and $k_{i+1}$ are not consecutive, $k_i+1< k_{i+1}$: For $k_i+1 \le k \le k_{i+1}-1$, it holds $k \in K^c$, then we obtain $c^{(k)} = c^{(k+1)} + 1$ from Thm.~\ref{153614_27May24}. This implies $c^{(k_i+1)} - c^{(k_{i+1})} = k_{i+1} - k_i - 1$. Thus, $k_i + c^{(k_i+1)} \le k_{i+1} + c^{(k_{i+1}+1)} - 1$. From Lem.~\ref{165523_23May24}, we have $k_{m} + c^{(k_{m}+1)} \le n-1$. Applying \eqref{172920_9May24} for $i=m-1$, we get $k_{m-1} + c^{(k_{m-1}+1)} \le k_{m} + c^{(k_{m}+1)} - 1 \le n-2$. Repeating this, we obtain \eqref{014925_24Jul24}. \end{proof} \begin{teiri}\label{223851_4Jun24} For $n > d \ge 1$, there exists an $[n,d]$ REP code $\Cc^{(n)}$ of size $\|\Cc^{(n)}\| = \prod_{j=0}^{n-1}(\lfloor j/d\rfloor+1)$. \end{teiri} \begin{proof} We construct a REP code $\Cc^{(n)}$ by choosing $S^{(j)} \subset [j+1]$ such that $\|S^{(j)}\| = \lfloor j/d \rfloor + 1, \quad S^{(j)} := \{0, d, 2d, \ldots, (\|S^{(j)}\|-1)d\} \quad \text{for} \; j = 0, \ldots, n-1$. We see that $\dmin(S^{(j)}) = \infty$ for $0 \le j < d$, and $\dmin(S^{(j)}) = d$ for $d \le j < n$. From Thm.~\ref{230931_30Oct23}, we understand that such $S^{(j)}$ are the largest possible sets that satisfy $\dmin(S^{(j)}) \ge d$. The subsequent result is obtained by applying Thm.~\ref{205810_6Jul23} and Thm.~\ref{163227_14May23} repeatedly: $\|\Cc^{(n)}\| = \prod_{j=0}^{n-1} \|S^{(j)}\| = \prod_{j=0}^{n-1} (\lfloor j/d \rfloor + 1)$, $\dmin(\Cc^{(j)}) = \infty$ for $0 \le j \le d$ and $\dmin(\Cc^{(j)}) \ge d$ for $d < j \le n$. \end{proof} Recall Sec.~\ref{212515_27Aug24}. The size of $[n,d]$ optimal code size is the same as the size of $[n,d]$ DPGP codes whose size is $\prod_{j=0}^{n-1}(\lfloor j/d\rfloor+1)$. \section{Encoding and Decoding Algorithms} In this section, we present several encoding algorithms of REP code $\Cc^{(n)}=\<S^0,\ldots,S^{n-1}\>$. We consider $(s^{(0)},\ldots,s^{(n-1)})\in S^{(0)}\times\cdots\times S^{(n-1)}$ as input to the encoder\footnote{ The size of the code $\Cc^{(n)}$ constructed by $S^{(0)},\ldots,S^{(n-1)}$ is given by $\prod_{j=0}^{n-1}\|S^{(j)}\|$, as we recall. We represent the message array $\underline{x} = \left(x_0, \ldots, x_{n-1}\right)$, where each $x_j$ is independently chosen from $[\|S^{(j)}\|]$. We denote the $x_j$-th smallest element in $S^{(j)}$ as $s^{(j)}$. Since, $(x_0, \ldots, x_{n-1})$ and $(s_0, \ldots, s_{n-1})$ correspond one-to-one in this mapping for given $S^{(0)},\ldots,S^{(n-1)}$, we can consider $\sU$ as input. }. \subsection{Natural Encoding Algorithm} \label{182618_25May23} The codewords of $\Cc^{(j+1)}$ are generated by extending the codewords of the $j$-th code $\Cc^{(j)},$ using each element of $S^{(j)}$. By considering the freedom in the selection of each element in $S^{(j)}$ as message, the following {\itshape natural encoding algorithm} is derived. Recall that $\Cc^{(j)}=\phi(\Cc^{(j-1)};S^{(j-1)})$ is defined recursively. Thus, the codeword $\piU^{(j)}$ of $\Cc^{(j)}$ can be expressed as $\piU^{(j)}=\phi(\piU^{(j-1)};s^{(j-1)})$ with $\piU^{(j-1)}\in \Cc^{(j-1)}$ and $s^{(j-1)}\in S^{(j-1)}$. From this observation, it is evident that all codewords of $\Cc^{(n)}$ are exhaustively generated by the naturally defined encoding algorithm. We use $s_j \in S^{(j)}$ for $j \in [n]$ as input to the encoder. Equivalently, we can use $x_j \in [\|S^{(j)}\|]$ for $j \in [n]$ as the input, where $s_j$ is the $x_j$-th smallest element in $S^{(j)}$. This yields $\piU^{(n)}$ as a codeword of $\Cc^{(n)}$. We denote this encoder, with some abuse of notation, as $\piU^{(n)} := \Cc^{(n)}(\sU)$. \input{natural_encode_figure} \begin{figure}[!t] \begin{algorithm}[H] \caption{Natural Encoding Algorithm of $\Cc^{(n)}$} \begin{algorithmic}[1]\label{natural_encode} \renewcommand{\algorithmicrequire}{\textbf{Input:}} \renewcommand{\algorithmicensure}{\textbf{Output:}} \REQUIRE $ (s^{(0)} , \dotsc, s^{(n-1)})\in S^{(0)}\times\cdots\times S^{(n-1)}$\\ \ENSURE $(\pi_0^{(n)} , \dotsc, \pi_{n-1}^{(n)})\in \Cc^{(n)}$ \FOR {$j:= 1$ to $n$} \STATE $\pi^{(j)} _0 := s^{(j-1)}$ \ENDFOR \FOR {$k := 1$ to $n-1$} \FOR {$j:= k$ to $n$}\label{141220_6Aug23} \STATE $\pi_k^{(j)}:=\phi\left(\pi_{k-1}^{(j-1)} ; s^{(j-1)}\right)$\label{141312_6Aug23} \ENDFOR\label{141314_6Aug23} \ENDFOR \end{algorithmic} \end{algorithm} \end{figure} The formal component-wise description of this encoder is given in Alg.~\ref{natural_encode}. In Fig.~\ref{125759_3Aug23}, we depict the dependencies of each variable that appears in this algorithm for the case of $n=8$. Although natural encoding algorithms are simple, it requires computational complexity of $O(n^2)$. \subsection{Sequential Encoding Algorithm} \label{182606_25May23} For a given encoding algorithm $\xU \mapsto \piU^{(n)}$ for the recursively extended code $\Cc^{(n)}$, the algorithm is said to be sequential if the following condition is met: for each $j\in [n]$ the algorithm determines the $j$-th output $\pi_j^{(n)}$ based on the input $x_j$ and some state variables. The computational order can be rearranged to make natural encoding algorithms sequential. Specifically, the components depicted in Fig.~\ref{125759_3Aug23}, originally calculated from bottom to top, can alternatively be computed from left to right, thereby rendering the algorithm sequential. Despite these modifications, the computational complexity remains $O(n^2)$. In this subsection, we propose an efficient sequential encoding algorithm with computational cost $O(n\log n)$ So far, we have considered $\phi_s(\cdot)$ as a map $\Sc_n\to \Sc_n$ or a map $[n]\to[n+1]$, for head $s\in [n+1]$ We now extend the domain of $\phi_s(\cdot)$ to permutations on $[n]$ without duplicate elements, as follows. For a set $A \subset [n-1]$, define $\phi_s(A) \defeq \{\phi_s(a) \mid a \in A\}\cup\{s\}$ We denote $\min_{r\text{ th}}(A)$ or $\min(A; r)$ denote the $r$-th smallest element in the array $A$, where the smallest element is denoted as $\min(A; 0)$. \begin{lem}\label{181528_2Aug23} Let $\piU^{(n)}\in \Sc_{n}$ and $s,r\in [n]$. Then, the following holds: \begin{align} \phi_s(\min_{r\text{ th}} (\piU^{(n)}))=\min_{r\text{ th}}\bigl(\phi_s' (\piU^{(n)})\bigr), \label{223340_2Aug23} \end{align} where we define $\phi_s'(A) \defeq \{\phi_s(a) \mid a \in A\}$. \end{lem} \begin{proof} Let the elements of \( \piU^{(n)} \) be enumerated in ascending order as \( \sigma_0 < \cdots < \sigma_{n-1} \). Then the LHS of \eqref{223340_2Aug23} is \( \phi_s(\sigma_r)\). Recalling that \( \phi_s(\sigma_r) = \sigma_r + \mathbbm{1}[\sigma_r \ge s] \), it is evident that \( \phi_s(\cdot) \) preserves the order: $\phi_s(x)<\phi_s(y)$ if $x<y$. Since $\phi_s' (\piU^{(n)})= \{\phi_s(\sigma_0),\ldots,\phi_s(\sigma_{n-1})\}$, enumerating the elements of \( \phi_s' (\piU^{(n)}) \) in ascending order yields: $ \phi_s(\sigma_0) < \cdots < \phi_s(\sigma_{n-1})$. Consequently, the RHS of \eqref{223340_2Aug23} is \( \phi_s(\sigma_r) \). \end{proof} Thus far, we have represented a permutation $\fU := [f_0, \ldots, f_{n-1}] \in \Sc_n$ as an array. However, in the following lemma, we will also interpret it as a set of elements for simplicity. To simplify notation, for a set $X \subset [n]$, let $\overline{X}^{[n]} := [n] \setminus X$. \begin{lem}\label{224537_2Aug23} For any $A \subset [n-1]$ and $s\in [n-1]$, it holds that $\phi_s'(\overline{A}^{[n-1]}) = \overline{\phi_s(A)}^{[n]}.$ \end{lem} \begin{proof} For disjoint sets $X$ and $Y$, we write $X \oplus Y$ instead of $X \cup Y$. Since $\phi'_s(\cdot)$ is a bijection from $[n-1]$ to $[n] \setminus \{s\}$, we can partition $[n]$ as follows: $[n] = \phi'_s([n-1]) \oplus \{s\} = \phi'_s(A \oplus \overline{A}^{[n-1]}) = \phi'_s(\overline{A}^{[n-1]}) \oplus \phi'_s(A) \oplus \{s\}. $ From this, the claim immediately follows: $\overline{\phi_s(A)}^{[n]} = \overline{\phi'_s(A) \oplus \{s\}}^{[n]} = \phi'_s(\overline{A}^{[n-1]}).$ \end{proof} Consider Alg.~\ref{214627_21Aug24} for message $\sU$. The following theorem shows that this algorithm functions as the encoder for the code $\Cc^{(n)}$. Specifically, it confirms that the output is identical to that of the natural encoding algorithm. \begin{teiri}\label{153741_1Aug23} Let $\pi_j^{(n)}$ and $\piT_j^{(n)}$ denote the outputs of Alg.~\ref{natural_encode} and Alg.~\ref{214627_21Aug24}, respectively. Then, it holds that $\piT^{(n)}_j = \pi^{(n)}_j$ for any $n > 1$ and $0 \le j < n$. \end{teiri} \begin{proof} For any $n>1$ and $j = 0$, we have $\pi^{(n)}_0 = s^{(n-1)}$, and from Alg.~\ref{214627_21Aug24}, we have $\piT^{(n)}_0 = s_{n-1}$. Therefore, $\piT^{(n)}_j = \pi^{(n)}_j$ for any $n>1$ and $0\le j<n$ holds. We use induction for $j$: we assume that $\piT^{(n-1)}_{j-1} = \pi^{(n-1)}_{j-1}$ for any $n>0$ and derive that $\piT^{(n)}_{j} = \pi^{(n)}_{j}$ for any $n>0$. We have: \begin{align} \phi_{s_{n-1}}(\piU_{[j-1]}^{(n-1)}) &= [s_{n-1}, \phi_{s_{n-1}}(\pi_0^{(n-1)}), \ldots, \phi_{s_{n-1}}(\pi_{j-1}^{(n-1)})] \\ &= [s_{n-1}, \pi_1^{(n)}, \ldots, \pi_j^{(n)}] =: \piU_{[j]}^{(n)} \label{225827_2Aug23}, \end{align} where we denote $\piU_{[j]}^{(n)}$ the array consisting of the first $j$ elements of $\piU^{(n)}$. Since $(n-1)-1-(j-1) = \jO$, we have $\pi_{j-1}^{(n-1)} = \piT_{j-1}^{(n-1)} = \min_{s_{\jO} \text{th}}([n-1] \setminus \tilde{\piU}_{[j-1]}^{(n-1)})$. Applying Lem.~\ref{181528_2Aug23}, we get $\pi_j^{(n)} = \phi_{s_{n-1}}(\pi_{j-1}^{(n-1)}) = \min_{s_{\jO} \text{th}} \Bigl( \phi_{s_{n-1}}'\bigl([n-1] \setminus \tilde{\piU}_{[j-1]}^{(n-1)} \bigr) \Bigr)$. Furthermore, from Lem.~\ref{224537_2Aug23} and \eqref{225827_2Aug23}, we have $\phi'_{s_{n-1}}\bigl([n-1] \setminus \tilde{\piU}_{[j-1]}^{(n-1)} \bigr) = [n] \setminus \phi_{s_{n-1}}(\tilde{\piU}_{[j-1]}^{(n-1)}) = [n] \setminus \tilde{\piU}_{[j]}^{(n)}$. Summarizing the above, we get $\pi_j^{(n)} = \min_{s_{\jO} \text{th}}\bigl([n] \setminus \tilde{\piU}_{[j]}^{(n)} \bigr) = \piT_j^{(n)}$. \end{proof} In Alg.~\ref{214627_21Aug24}, for each index $j$, the process of selecting the $s_j$-th smallest element from the set $[n] \setminus {\pi_0^{(n)}, \ldots, \pi_{j-1}^{(n)}}$ can be performed efficiently. This selection process requires at most $O(\log n)$ steps when implemented with an appropriate search or sorting method. Therefore, the overall computational complexity of the encoding algorithm is $O(n \log n)$. \begin{figure}[!t] \label{002543_29Jun24} \begin{algorithm}[H] \caption{Sequential Encoder of $\Cc^{(n)}=\<S^{(0)},\ldots,S^{(n-1)}\>$} \begin{algorithmic}[1]\label{214627_21Aug24} \renewcommand{\algorithmicrequire}{\textbf{Input:}} \renewcommand{\algorithmicensure}{\textbf{Output:}} \REQUIRE $ (s^{(0)} , \dotsc, s^{(n-1)})\in S^{(0)}\times\cdots\times S^{(n-1)}$\\ \ENSURE $(\piT^{(n)}_0,\ldots,\piT^{(n)}_{n-1})\in \Cc^{(n)}$ \FOR {$j = 0$ to $n-1$} \STATE $\piT^{(n)}_j:=\displaystyle\min_{s_{\jO}\text{ th}}([n]\setminus \{\piT_0^{(n)},\ldots,\piT_{j-1}^{(n)}\})$\label{173310_26Aug24} \ENDFOR \RETURN $(\piT^{(n)}_0,\ldots,\piT^{(n)}_{n-1})$ \end{algorithmic} \end{algorithm} \end{figure} \subsection{Decoding Algorithm of Optimal REP Codes}\label{143813_28Aug23} In Sec.~\ref{161538_26Aug24}, we showed that REP codes $\Cc^{(n)} = \<S^{(0)}, \ldots, S^{(n-1)}\>$ satisfying $\dmin(S^{(j)}) \ge d$ are optimal among $[n, d]$ codes. Let $\sU$ and $\piU$ denote the message and the corresponding codeword. Let $\rhoU$ and $\hat{\sU}$ denote the corresponding received word and the estimated message of the decoder. We propose a decoding algorithm for such codes as described in Alg.~\ref{160732_26Aug24}. The function $\psi_i(\cdot; \cdot)$ defined in line \ref{173558_26Aug24} of this algorithm mirrors the computation described in line \ref{173310_26Aug24} of Alg.~\ref{214627_21Aug24}. It is important to note that $\hat{s}_{\iO} \in S^{(\iO)}$ is chosen so that $\psi_i(\hat{s}_{\iO})$ is closest to $\rho_i$: $\|\psi_i(\sH_{\iO})-\rho_{i}\|\ge \|\psi_i(s_{\iO})-\rho_{i}\|$ for any $s_{\iO} \in S^{(\iO)}$. We will show that the decoder can successfully correct any errors as long as $\dinf(\piU, \rhoU) < d/2$, where $\piU$ and $\rhoU$ are the transmitted codeword and the received word, respectively. \begin{figure}[!t] \begin{algorithm}[H] \caption{Sequential Decoder of $\Cc^{(n)}=\<S^{(0)},\ldots,S^{(n-1)}\>$} \begin{algorithmic}[1]\label{160732_26Aug24} \renewcommand{\algorithmicrequire}{\textbf{Input:}} \renewcommand{\algorithmicensure}{\textbf{Output:}} \REQUIRE Received array $(\rho^{(n)}_0,\ldots,\rho^{(n)}_{n-1})$ \ENSURE Estimated message $(\sH_0\in S^{(0)},\ldots,\sH_{n-1}\in S^{(n-1)})$ \FOR {$i = 0$ to $n-1$} \STATE Let $\displaystyle\psi_i(s;\underline{\piH}_{[i]}^{(n)})\defeq\min_{s \text{ th}}[n]\setminus\{\piH_0^{(n)},\ldots,\piH_{i-1}^{(n)}\}$.\label{173558_26Aug24} \STATE $\displaystyle\sH_{\overline{i}}:=\argmin_{s\in S^{(\overline{i})}}\|\rho_i^{(n)}-\psi_i(s;\underline{\piH}_{[i]}^{(n)}))\|$ \STATE $\piH^{(n)}_i:=\psi_i(\sH_{\overline{i}};\underline{\piH}_{[i]}^{(n)}))$ \ENDFOR \RETURN $(\sH^{(0)},\ldots,\sH^{(n-1)})$ \end{algorithmic} \end{algorithm} \end{figure} \begin{teiri}\label{221904_14Jun24} Consider the setting of optimal REP code and decoder described above. Assume that $\|\pi_i - \rho_i\| < d/2$ for all $i \in [n]$. Then, it follows that $\hat{\sU} = \sU$. \end{teiri} \begin{proof} Let $s_i$ and $\sH_i$ denote the $i$-th message and its estimate, respectively. We will prove that $\sH_{\overline{i}} = s_{\overline{i}}$ for all $i\in [n]$ by induction. It clear that $\sH_{\overline{0}} = s_{\overline{0}}$. Now, assume that the decoder has correctly estimated up to step $i-1$, specifically: $\sH_{\overline{0}} = s_{\overline{0}}, \ldots, \sH_{\overline{i-1}} = s_{\overline{i-1}}$. We will now derive $\sH_{\overline{i}} = s_{\overline{i}}$. By Alg.~\ref{214627_21Aug24}, we have $\piH_{\overline{0}} = \pi_{\overline{0}}, \ldots, \piH_{\overline{i-1}} = \pi_{\overline{i-1}}$, then, $\pi_{i}=\psi_i(s_{\iO};\hat{\piU}_{[i]}^{(n)}))$. Now, assume for contradiction that $\sH_{\overline{i}} \neq s_{\overline{i}}$. We will derive a contradiction from this assumption. Recall that $\hat{s}_{\iO} \in S^{(\iO)}$ is chosen such that $\psi_i(\hat{s}_{\iO};\underline{\piH}_{[i]}^{(n)})$ is the closest head in $S^{(i)}$ to $\rho_i$. Hence, we have $\|\psi_i(\sH_{\iO};\underline{\piH}_{[i]}^{(n)}) - \rho_i\| \leq \|\psi_i(s_{\iO};\underline{\piH}_{[i]}^{(n)}) - \rho_i\| = \|\pi_i - \rho_i\|$. Since from the premise $\|\pi_i - \rho_i\| < d/2$, we obtain: $\|\psi_i(\sH_{\iO};\underline{\piH}_{[i]}^{(n)}) - \psi_i(s_{\iO};\underline{\piH}_{[i]}^{(n)})\| = \|\psi_i(\sH_{\iO};\underline{\piH}_{[i]}^{(n)}) - \pi_i\| \leq \|\psi_i(\sH_{\iO};\underline{\piH}_{[i]}^{(n)}) - \rho_i\| + \|\pi_i - \rho_i\| \leq 2\|\pi_i - \rho_i\| < d. $ On the other hand, from the premise $\dmin(S^{(i)}) \ge d$ and $\sH_{\iO}, s_{\iO} \in S^{(i)}$, we have $\|\sH_{\iO} - s_{\iO}\| \geq d$, and from the definition of $\psi(\cdot;\cdot)$, we have $\|\psi_i(\sH_{\iO};\underline{\piH}_{[i]}^{(n)}) - \psi_i(s_{\iO};\underline{\piH}_{[i]}^{(n)})\| \geq d.$ \end{proof} Since $\dinf(\piU, \rhoU) < d/2$ implies $\|\pi_i - \rho_i\| < d/2$ for all $i \in [n]$, the condition in Thm.~\ref{221904_14Jun24} can be replaced with $\dinf(\piU, \rhoU) < d/2$. This shows that the performance of this decoder is equivalent to or better than that of the bounded distance decoder. \section{Conclusions} In this paper, we investigate recursively extended permutation (REP) codes. We prove that the optimal REP code matches DPGP codes in terms of size and minimum distance. Moreover, we developed efficient encoding and decoding algorithms for REP codes. \begin{figure}\label{125759_3Aug23} \begin{center} {\scriptsize \begin{tabular}{ @{\hspace{0mm}}l\|\|@{\hspace{2mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{\hspace{0mm}} } $\piU^{(8)}$&$s_{(7)}=$&$\pi_0^{(8)}$ && $\pi_1^{(8)}$ && $\pi_2^{(8)}$ && $\pi_3^{(8)}$ && $\pi_4^{(8)}$ && $\pi_5^{(8)}$ && $\pi_6^{(8)}$ && $\pi_7^{(8)}$ \\ $\uparrow_7$&&& $\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&&$\nearrow\hspace{-3mm}{}^7$&& \\ $\piU^{(7)}$&$s_{(6)}=$&$\pi_0^{(7)}$ && $\pi_1^{(7)}$ && $\pi_2^{(7)}$ && $\pi_3^{(7)}$ && $\pi_4^{(7)}$ && $\pi_5^{(7)}$ && $\pi_6^{(7)}$ && \\ $\uparrow_6$&&& $\nearrow\hspace{-3mm}{}^6$&&$\nearrow\hspace{-3mm}{}^6$&&$\nearrow\hspace{-3mm}{}^6$&&$\nearrow\hspace{-3mm}{}^6$&&$\nearrow\hspace{-3mm}{}^6$&&$\nearrow\hspace{-3mm}{}^6$&& \\ $\piU^{(6)}$&$s_{(5)}=$&$\pi_0^{(6)}$ && $\pi_1^{(6)}$ && $\pi_2^{(6)}$ && $\pi_3^{(6)}$ && $\pi_4^{(6)}$ && $\pi_5^{(6)}$ && \\ $\uparrow_5$&&& $\nearrow\hspace{-3mm}{}^5$&&$\nearrow\hspace{-3mm}{}^5$&&$\nearrow\hspace{-3mm}{}^5$&&$\nearrow\hspace{-3mm}{}^5$&&$\nearrow\hspace{-3mm}{}^5$&& \\ $\piU^{(5)}$&$s_{(4)}=$&$\pi_0^{(5)}$ && $\pi_1^{(5)}$ && $\pi_2^{(5)}$ && $\pi_3^{(5)}$ && $\pi_4^{(5)}$ && \\ $\uparrow_4$&&& $\nearrow\hspace{-3mm}{}^4$&&$\nearrow\hspace{-3mm}{}^4$&&$\nearrow\hspace{-3mm}{}^4$&&$\nearrow\hspace{-3mm}{}^4$&& \\ $\piU^{(4)}$&$s_{(3)}=$&$\pi_0^{(4)}$ && $\pi_1^{(4)}$ && $\pi_2^{(4)}$ && $\pi_3^{(4)}$ && \\ $\uparrow_3$&&& $\nearrow\hspace{-3mm}{}^3$&&$\nearrow\hspace{-3mm}{}^3$&&$\nearrow\hspace{-3mm}{}^3$&& \\ $\piU^{(3)}$&$s_{(2)}=$&$\pi_0^{(3)}$ && $\pi_1^{(3)}$ && $\pi_2^{(3)}$ && \\ $\uparrow_2$&&& $\nearrow\hspace{-3mm}{}^2$&&$\nearrow\hspace{-3mm}{}^2$&& \\ $\piU^{(2)}$&$s_{(1)}=$&$\pi_0^{(2)}$ && $\pi_1^{(2)}$ && \\ $\uparrow_1$&&& $\nearrow\hspace{-3mm}{}^1$&& \\ $\piU^{(1)}$&$s_{(0)}=$&$\pi_0^{(1)}$ && \end{tabular} } \end{center} \caption{Dependency of variables in the natural encoding algorithm for the case of $n=8$. We write $a \xrightarrow{j} b$ when $b = \phi(a; s^{(j)})$ holds.} \end{figure}
2412.04143v1	http://arxiv.org/abs/2412.04143v1	Pin Classes I: Growth Rates and Bounds	\documentclass[11pt,a4paper]{article} \frenchspacing \setlength{\parskip}{9pt plus 3pt minus 1pt} \setlength{\parindent}{0pt} \usepackage[margin=3cm]{geometry} \usepackage{amsmath} \usepackage{amssymb,latexsym} \usepackage{mathabx} \usepackage{amsthm} \usepackage[shortlabels]{enumitem} \usepackage{array} \usepackage{multirow} \usepackage[hang,flushmargin,bottom]{footmisc} \usepackage[margin=0.35in,format=hang,font=small,labelfont=bf]{caption} \usepackage[normalem]{ulem} \usepackage{needspace} \usepackage{graphicx,tikz} \usepackage[colorlinks=true,urlcolor=blue,linkcolor=blue,citecolor=blue]{hyperref} \usepackage{palatino} \usepackage{eulervm} \usetikzlibrary{patterns} \input{tikzperm} \setlength{\textwidth}{6.3in} \setlength{\textheight}{8.7in} \setlength{\topmargin}{0pt} \setlength{\headsep}{0pt} \setlength{\headheight}{0pt} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} 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\newcommand{\gridded}{\sharp} \newcommand{\sfl}{\mathsf{l}} \newcommand{\sfr}{\mathsf{r}} \newcommand{\sfu}{\mathsf{u}} \newcommand{\sfd}{\mathsf{d}} \newcommand{\tikzcircle}[2][black,fill=black]{\tikz[baseline=-0.5ex]\draw[#1,radius=#2] (0,0) circle ;} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\rulebreak}{\vspace{24pt}\hrule\vspace{15pt}} \newcommand{\note}[1]{ \marginpar{\raggedright{\footnotesize\it #1}} } \newcommand{\noterb}[1]{\note{\color{red!33!black}\textbf{RB:}~#1}} \newcommand{\commentrb}[1]{{\color{red!33!black}[\emph{\textbf{RB:}~#1}]}} \tikzstyle{vertex}=[circle, draw, fill=black, inner sep=0pt, minimum width=4pt] \newcommand{\smallgrid}{\setplotptradius{2pt}\draw (-.5,0)--++(1,0) (0,-.5)--++(0,1);} \newcommand{\sept}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (.25,-.25) {};}} \newcommand{\swpt}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (-.25,-.25) {};}} \newcommand{\nept}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (.25,.25) {};}} \newcommand{\nwpt}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (-.25,.25) {};}} \newcommand{\netwo}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (.15,.35) {};\node[permpt] at (.35,.15) {};}} \newcommand{\nwtwo}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (-.15,.35) {};\node[permpt] at (-.35,.15) {};}} \newcommand{\swtwo}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (-.15,-.35) {};\node[permpt] at (-.35,-.15) {};}} \newcommand{\nthree}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (.15,.35) {};\node[permpt] at (.35,.15) {};\node[permpt] at (-.25,.25) {};}} \newcommand{\wthree}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (-.15,-.35) {};\node[permpt] at (-.35,-.15) {};\node[permpt] at (-.25,.25) {};}} \newcommand{\nwfive}{\tikz[scale=.5,baseline=-3pt]{\smallgrid\node[permpt] at (.15,.35) {};\node[permpt] at (.35,.15) {};\node[permpt] at (-.15,-.35) {};\node[permpt] at (-.35,-.15) {};\node[permpt] at (-.25,.25) {};}} \newcommand{\plottwocellperm}[2][black]{ \foreach \i [count=\nn] in {#2} {\global\let\n\nn}; \foreach \j [count=\i] in {#2} { \ifnum0=\j { \draw (\i,.5) -- ++(0,\n-1); } \else { \node[permpt,fill=#1,draw=#1] (\j) at (\i,\j) {}; }; } \newcommand{\twocell}[1]{\tikz[scale=.4,baseline=-3pt]{\setplotptradius{2pt} \foreach \i [count=\nn] in {#1} {\global\let\n\nn}; \begin{scope}[yscale=1/(\n-1),xscale=1/3] \foreach \j [count=\i] in {#1} { \ifnum0=\j { \draw (\i,.5-\n/2) -- ++(0,\n-1); } \else { \node[permpt] (\j) at (\i,\j-\n/2) {}; }; \end{scope} }} \title{Pin Classes I: Growth Rates and Bounds} \author{Ben Jarvis\footnote{[email protected]}\\ \small School of Mathematics and Statistics\\ \small The Open University, UK } \begin{document} \maketitle \begin{abstract} Pin sequences play an important role in the structural study of permutation classes. In this paper, we study the permutation classes that comprise all the finite subpermutations contained in an infinite pin sequence. We prove that these permutation classes have proper growth rates and establish a procedure for calculating these growth rates. \end{abstract} \section{Introduction} Since their introduction by Brignall, Huczynska and Vatter~\cite{brignall:decomposing-sim:}, pin sequences have proven to be objects of considerable interest in the study of permutation classes, especially in connection with simple permutations and infinite antichains. Bassino, Bouvel and Rossin~\cite{bassino:enumeration-of-:} showed that the class of all pin sequences has a rational generating function and established its growth rate, and recently, Brignall and Vatter~\cite{bv:wqo-uncountable:} used pin sequences to construct uncountably many well-quasi-ordered permutation classes with distinct enumeration sequences. Given an (infinite) pin sequence, the \emph{pin class} is formed from all finite permutations that are contained in the sequence. In this paper we take up the study of pin classes in a more systematic manner than has previously been attempted. Our main result is that pin classes have \emph{proper} growth rates, not merely upper growth rates as guaranteed by the Marcus-Tardos Theorem~\cite{marcus:excluded-permut:}. We also describe a procedure for finding the growth rate of a given pin class in terms of the recurrent subfactors of the defining pin sequence. Pin classes are most naturally situated in the context of \textbf{centred permutations}, which we define in Section \ref{sec:2}. We shall see in Section \ref{sec:3} that this change of perspective allows us to state a particuarly nice structure theorem for \emph{recurrent} pin classes in terms of the $\boxplus$-sum (an operation specific to centred permutations). This gives us an extraordinarily large family of permutation classes whose generating functions we can find by solving a simple combinatorial problem concerning subfactors of the defining pin sequence. In Section \ref{sec:4} we extend this theory to pin classes defined by non-recurrent pin sequences: in this case we can no longer find generating functions but we can still prove the existence of growth rates and find these by considering a centred class called the $\boxplus$-interior of a pin class. \section{Centred Permutation Classes} \label{sec:2} We refer the reader to Bevan~\cite{bevan2015defs} for background information and basic definitions concerning permutations and permutation classes. \subsection{Centred Permutations} Our main focus in this paper will be a particular method for converting a binary sequence into a permutation class; the resulting permutation classes will be known as \emph{pin classes}. It shall transpire that pin classes are in fact most naturally understood not as classes of permutations in the ordinary sense, but as \emph{centred permutations}. A centred permutation is essentially a permutation with an extra point, designated as the \emph{origin}, which does not contribute to the length of the permutation. A formal definition follows: \begin{defn}[Centred Permutations] A centred permutation of length $n$ is a permutation of length $n+1$ in which one point is designated as the origin, which does not contribute to the length of the centred permutation. \end{defn} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (-2,-1) {}; \node[permpt] (2) at (-1,2) {}; \node[permpt] (3) at (1,1) {}; \begin{scope}[shift={(12,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (-1,-1) {}; \node[permpt] (2) at (1,2) {}; \node[permpt] (3) at (2,1) {}; \end{scope} \end{tikzpicture} \end{center} \caption{Two centred permutations of length $3$. Note that a centred permutation of length $3$ consists of $4$ points (no two of which share either an $x$- or $y$-coordinate) in the plane, of which one is designated as the origin, which does not contribute to the length. Here we denote the origin with an empty circle, whereas the `true' points are represented by solid circles.} \label{fig:centred0} \end{figure} We usually distinguish between centred and uncentred permutations by placing a circle in the superscript: so $\pi$ is a regular permutation and $\pi^{\circ}$ is a centred permutation. (Similarly $\mathcal{C}^{\circ}$ is a class of centred permutations, whereas $\mathcal{C}$ is a class of uncentred permutations.) \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node at (1,1) {1}; \node at (-1,1) {2}; \node at (-1,-1) {3}; \node at (1,-1) {4}; \draw[thick] (0,-2) -- ++ (0,4); \draw[thick] (-2,0) -- ++ (4,0); \end{tikzpicture} \end{center} \caption{The standard quadrant numbering: the origin point of a centred permutation splits the plane into four quadrants, numbered anticlockwise.} \label{fig:quadnumbering} \end{figure} Every centred permutation $\sigma^{\circ}$ of length $n$ is naturally associated with an \emph{underlying} permutation $\sigma$ of length $n$, obtained by removing the origin point from $\sigma^{\circ}$. We also have the \emph{filled-in} permutation $\sigma^{\tikzcircle{1.5pt}}$, the permutation of length $n+1$ obtained by replacing the origin of $\sigma^{\circ}$ with a true point. Though we usually study centred permutation classes in order to understand their associated underlying permutation classes, it is the latter of these notions which provides us with the most convenient notation for a centred permutation: \begin{defn}[One-Line Notation for Centred Permutations] Suppose that $\sigma^{\circ}$ is a centred permutation of length $n$, with corresponding filled-in permutation $\sigma^{\tikzcircle{1.5pt}}$ of length $n+1$. Suppose that the $k$th entry of $\sigma^{\tikzcircle{1.5pt}}$ is the origin of $\sigma^{\circ}$. Then we can write $\sigma^{\circ}$ in one-line notation as follows: \[ \sigma^{\circ} = \sigma^{\tikzcircle{1.5pt}}(1)\sigma^{\tikzcircle{1.5pt}}(2)\dots\underline{\sigma^{\tikzcircle{1.5pt}}(k)}\dots\sigma^{\tikzcircle{1.5pt}}(n)\sigma^{\tikzcircle{1.5pt}}(n+1) \] In other words, this is the one-line notation for $\sigma^{\tikzcircle{1.5pt}}$ with the centre point underlined. See Fig. \ref{fig:centred1} for an example. \end{defn} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,3) {}; \node[permpt] (1) at (1,4) {}; \node[permpt] (2) at (2,2) {}; \node[permpt] (3) at (3,6) {}; \node[permpt] (4) at (5,5) {}; \node[permpt] (5) at (6,1) {}; \draw[thick] (4,0.5) -- ++ (0,6); \draw[thick] (0.5,3) -- ++ (6,0); \end{tikzpicture} \end{center} \caption{The centred permutation $426\underline{3}51$.} \label{fig:centred1} \end{figure} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (-2,-1) {}; \node[permpt] (2) at (-1,2) {}; \node[permpt] (3) at (1,1) {}; \draw[thick] (0,-3) -- ++ (0,6); \draw[thick] (-3,0) -- ++ (6,0); \node at (6,0) {$\neq$}; \begin{scope}[shift={(12,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (-1,-1) {}; \node[permpt] (2) at (1,2) {}; \node[permpt] (3) at (2,1) {}; \draw[thick] (0,-3) -- ++ (0,6); \draw[thick] (-3,0) -- ++ (6,0); \end{scope} \end{tikzpicture} \end{center} \caption{The centred permutations $14\underline{2}3$ and $1\underline{2}43$ are both of length $3$ (the origin point doesn't count) and both have the same underlying (uncentred) permutation $132$, but they are distinct as centred permutations as the origins are in a different place.} \label{fig:centred2} \end{figure} Analogously with the uncentred case we define the notion of centred permutation containment: \begin{defn}[Centred Permutation Containment] Suppose that $\sigma^{\circ}$ is a centred permutation of length $n$ and $\pi^{\circ}$ is a centred permutation of length $m \leq n$. We say that $\pi^{\circ}$ is \emph{contained} in $\sigma^{\circ}$, denoted $\pi^{\circ} \leq \sigma^{\circ}$, if $\pi^{\tikzcircle{1.5pt}}$ can be embedded in $\sigma^{\tikzcircle{1.5pt}}$ in such a way that the origins match up. Explicitly, if \[ \pi^{\circ} = \pi^{\tikzcircle{1.5pt}}(1)\pi^{\tikzcircle{1.5pt}}(2)\dots\underline{\pi^{\tikzcircle{1.5pt}}(k)}\dots\pi^{\tikzcircle{1.5pt}}(m)\pi^{\tikzcircle{1.5pt}}(m+1) \] and \[ \sigma^{\circ} = \sigma^{\tikzcircle{1.5pt}}(1)\sigma^{\tikzcircle{1.5pt}}(2)\dots\underline{\sigma^{\tikzcircle{1.5pt}}(l)}\dots\sigma^{\tikzcircle{1.5pt}}(n)\sigma^{\tikzcircle{1.5pt}}(n+1) \] then $\pi^{\circ} \leq \sigma^{\circ}$ if and only if there is an ascending sequence of $m+1$ indices \begin{equation} 1 \leq i_{1} < i_{2} < \dots < i_{m+1} \leq n + 1 \end{equation} such that the sequence $\sigma^{\tikzcircle{1.5pt}}(i_{1})\sigma^{\tikzcircle{1.5pt}}(i_{2})\dots\sigma^{\tikzcircle{1.5pt}}(i_{m+1})$ is order-isomorphic to $\pi^{\tikzcircle{1.5pt}}$ \emph{and} $i_{k} = l$. \end{defn} As with regular permutation containment, centred permutation containment is much easier to understand graphically - see Fig. \ref{fig:centcont} for an example. This is one of the reasons that we generally prefer to draw centered permutations out rather than work with one-line notation. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.5] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (3,2) {}; \node[permpt] (1) at (1,1) {}; \node[permpt] (2) at (2,4) {}; \node[permpt] (3) at (4,3) {}; \draw[thick] (0,2) -- ++ (5,0); \draw[thick] (3,0) -- ++ (0,5); \node at (7,2) {$\leq$}; \begin{scope}[shift={(9,-2)}] \node[circle, red, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,4) {}; \node[permpt] (1) at (1,7) {}; \node[permpt,red] (2) at (2,1) {}; \node[permpt,red] (3) at (3,8) {}; \node[permpt] (4) at (4,3) {}; \node[permpt,red] (5) at (6,5) {}; \node[permpt] (6) at (7,2) {}; \node[permpt] (7) at (8,6) {}; \draw[thick] (0,4) -- ++ (9,0); \draw[thick] (5,0) -- ++ (0,9); \end{scope} \end{tikzpicture} \end{center} \caption{The centred permutation $14\underline{2}3$ is contained in $7183\underline{4}526$; the relevant embedding is shown by the points marked in red in the diagram of $7183\underline{4}526$. Note that the embedding must make the origins match up.} \label{fig:centcont} \end{figure} Centred permutation containment, of course, immediately allows us to define the centred analogue of a permutation class: \begin{defn}[Centred Permutation Class] A \textbf{centred permutation class} is a non-empty set $\mathcal{C}^{\circ}$ of centred permutations which is downward-closed under centred permutation containment. We call a centred permutation class \textbf{proper} if the set of all underlying (uncentred) permutations of centred permutations in $\mathcal{C}^{\circ}$ does not contain every (uncentred) permutation. \end{defn} Note that $\sigma^{\circ} \leq \pi^{\circ}$ implies that $\sigma \leq \pi$. This means that, for every centred permutation class $\mathcal{C}^{\circ}$, the set $\mathcal{C}$ consisting of the underlying permutations of $\mathcal{C}^{\circ}$ is itself an uncentred permutation class. We call $\mathcal{C}$ the \textbf{underlying} permutation class of $\mathcal{C}^{\circ}$. We wish to use centred permutation classes as a means to study regular permutation classes: there are many interesting permutation classes that are most succinctly described as the underlying class of some given centred permutation class. In particular, we are interested in the growth rates of (centred and uncentred) permutation classes, so we recall the following: \begin{defn}[Growth Rates] Let $(C_{n})$ be a sequence of non-negative integers. The \textbf{upper} and \textbf{lower growth rates} of $(C_{n})$ are defined, respectively, to be \[ \overline{gr}((C_n)) = \limsup_{n\to\infty}\sqrt[n]{C_n}, \quad\text{ and }\quad \underline{gr}((C_n))=\liminf_{n\to\infty}\sqrt[n]{C_n}, \] when these quantities exist. If both quantities exist and $\overline{gr}((C_n))=\underline{gr}((C_n))$, then the sequence has a \textbf{(proper) growth rate}, typically denoted $\gr(C_{n})$. If $\mathcal{C}$ is a (centred or uncentred) permutation class, we define $\mathcal{C}_{n}$ to be the set of permutations of length $n$ in $\mathcal{C}$ and write $C_{n} = \|\mathcal{C}_{n}\|$. We call $C_{n}$ the \textbf{enumeration sequence} of $\mathcal{C}_{n}$ and say that $\mathcal{C}_{n}$ has (upper, lower, proper) growth rate $\rho$ if $\rho$ is the (upper, lower, proper) growth rate of the enumeration sequence. We write $gr(\mathcal{C})$ for $gr(C_{n})$, and analogously for upper and lower growth rates. \end{defn} We plan to use centred permutation classes to study the growth rates of their underlying (uncentred) permutation classes, so the following is crucial to note: \begin{prop} \label{eqgrs} Suppose that $\mathcal{C}^{\circ}$ is a proper centred permutation class whose underlying permutation class is $\mathcal{C}$. Then: \begin{enumerate} \item The upper growth rate of $\mathcal{C}^{\circ}$ exists and is equal to the upper growth rate of $\mathcal{C}$. \item The lower growth rate of $\mathcal{C}^{\circ}$ exists and is equal to the lower growth rate of $\mathcal{C}$. \item The proper growth rate $gr(\mathcal{C}^{\circ})$ exists if and only if $gr(\mathcal{C})$ exists; if so then $gr(\mathcal{C}^{\circ}) = gr(\mathcal{C})$. \end{enumerate} \end{prop} \begin{proof} As $\mathcal{C}^{\circ}$ is a proper centred permutation class, $\mathcal{C}$ is, by definition, a proper permutation class. Hence by the Marcus-Tardos Theorem~\cite{marcus:excluded-permut:}, $\mathcal{C}$ has an upper growth rate. On noting that a centred permutation can be identified with a $2$-by-$2$ gridded permutation, the fact that $\mathcal{C}^{\circ}$ has the same upper and lower growth rates follows immediately from~{Vatter~\cite[Proposition 2.1]{vatter:small-permutati:}}. (Informally, there are $(n+1)^2$ ways of placing a centre in a permutation of length $n$; given that this factor is polynomial it cannot affect the (exponential) asymptotics of the class.) \end{proof} We also note the following immediate consequence of~{Vatter~\cite[Proposition 2.1]{vatter:small-permutati:}}: \begin{lemma} \label{eqgrs2} Let $\mathcal{C}^{\circ}$ be a proper centred permutation class and let $\mathcal{C}^{\circ+n}$ denote the class consisting of centred permutations in $\mathcal{C}^{\circ}$ with at most $n$ points added anywhere. Then $\overline{gr}(\mathcal{C}^{\circ+n})$ exists and is equal to $\overline{gr}(\mathcal{C}^{\circ})$. Similarly, $\underline{gr}(\mathcal{C}^{\circ+n}) = \underline{gr}(\mathcal{C}^{\circ})$. \end{lemma} We shall also require a centred analogue of the notion of an interval of a permutation: \begin{defn}[$\circ$-intervals of a Centred Permutation] A \textbf{centred-interval} (also written \textbf{$\circ$-interval}) of a centred permutation $\pi^{\circ}$ is an interval containing the origin. We call a centred interval non-trivial if it contains at least one point in addition to the origin. A non-trivial centred interval is \textbf{minimal} if it contains no smaller non-trivial centred interval. \end{defn} \begin{lemma}[Minimal non-trivial $\circ$-intervals in a centred permutation] \label{minint} Let $\pi^{\circ}$ be a centred permutation. Then $\pi^{\circ}$ has either: \begin{itemize} \item one minimal non-trivial $\circ$-interval $\mathcal{I}$; \emph{or} \item two minimal non-trivial $\circ$-intervals $\mathcal{I}_{1}$,$\mathcal{I}_{2}$. In this case, each $\mathcal{I}_{i}$ will contain points in one quadrant only, and from opposite quadrants to each other. \end{itemize} \end{lemma} \begin{proof} Suppose that $\pi^{\circ}$ has two distinct minimal non-trivial $\circ$-intervals $\mathcal{I}_{1}$,$\mathcal{I}_{2}$. Then: \begin{itemize} \item as the intersection of two $\circ$-intervals is a $\circ$-interval, $\mathcal{I}_{1} \cap \mathcal{I}_{2}$ is a $\circ$-interval. \item Hence $\mathcal{I} = \mathcal{I}_{1} \cap \mathcal{I}_{2}$ is a $\circ$-interval contained in both $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$; as these were assumed to be distinct we can deduce that $\mathcal{I}$ is \emph{strictly} contained in at least one of these. Hence, by the assumed minimality of $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$, $\mathcal{I} = \mathcal{I}_{1} \cap \mathcal{I}_{2} = \{\circ\}$ \item Hence $\mathcal{I}_{1}, \mathcal{I}_{2}$ are non-trivial $\circ$-intervals with trivial overlap. Suppose that $\mathcal{I}_{1}, \mathcal{I}_{2}$ both have at least one point in the upper half-plane: say $p_{1} \in \mathcal{I}_{1},p_{2} \in \mathcal{I}_{2}$. Note that $p_{1} \notin \mathcal{I}_{2},p_{2} \notin \mathcal{I}_{1}$ (by $\mathcal{I}_{1} \cap \mathcal{I}_{2} = \{\circ\}$) and that one of $p_{1},p_{2}$ must be lower than the other - without loss of generality, say that $p_{1}$ is lower than $p_{2}$. But then $p_{1}$ is vertically between the origin and $p_{2}$ and hence slices the rectangle $\mathcal{I}_{2}$, contradicting the assumption that $\mathcal{I}_{2}$ is a $\circ$-interval. Hence our supposition is impossible, and $\mathcal{I}_{1}, \mathcal{I}_{2}$ cannot both occupy the upper half-plane. See Fig. \ref{fig:minintervals} for an illustration of this argument. \item By precisely the same logic $\mathcal{I}_{1}, \mathcal{I}_{2}$ cannot both occupy any half-plane, and thus the only possibility remaining is that $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are both one-quadrant intervals occupying opposite quadrants, as required. Note that this argument immediately precludes the existence of a third minimal non-trivial $\circ$-interval $\mathcal{I}_{3}$ as this would have to be a one-quadrant interval opposite to both $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$. \end{itemize} \end{proof} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt,label={\tiny $p_{1}$}] (1) at (-2,1) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (1,2) {}; \node[empty,label={\tiny $\pi_{2}^{\circ}$}] (3) at (2.5,0) {}; \draw[thick] (0,-1) -- ++ (0,4); \draw[thick] (-3,0) -- ++ (6,0); \draw[dotted] (-0.5,-0.5) rectangle (1.5,2.5); \end{tikzpicture} \end{center} \caption{The 'interval' $\pi_{2}^{\circ}$ is intersected by the point $p_{1}$ - and hence is not an interval at all.} \label{fig:minintervals} \end{figure} \subsection{The $\boxplus$-decomposition} The following operation on centred permutations will be fundamental in describing the structure of the permutation classes we work with in the next section: \begin{defn}[The Box Sum] Given two centred permutations $\pi^{\circ}$ and $\sigma^{\circ}$ their \textbf{$\boxplus$-sum} (read: \textbf{box-sum}), written $\pi^{\circ} \boxplus \sigma^{\circ}$, is obtained by inflating the origin of $\sigma^{\circ}$ with a copy of $\pi^{\circ}$ - see Fig. \ref{fig:boxsumdef} for an illustration. \end{defn} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \begin{scope}[shift={(0,4)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,3) {}; \node[permpt] (1) at (5,5) {}; \node[permpt] (2) at (2,4) {}; \node[permpt] (3) at (3,1) {}; \node[permpt] (4) at (1,2) {}; \draw[thin] (4,0.5) -- ++ (0,6); \draw[thin] (0.5,3) -- ++ (6,0); \node at (9,3) {$\boxplus$}; \end{scope} \begin{scope}[shift={(11,2)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,5) {}; \node[permpt] (1) at (3,3) {}; \node[permpt] (2) at (1,4) {}; \node[permpt] (3) at (2,1) {}; \node[permpt] (4) at (5,2) {}; \draw[thin] (4,0.5) -- ++ (0,8); \draw[thin] (0.5,5) -- ++ (7,0); \node at (10,5) {=}; \end{scope} \begin{scope}[shift={(23,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (7,7) {}; \node[permpt] (1) at (8,9) {}; \node[permpt] (2) at (5,8) {}; \node[permpt] (3) at (6,5) {}; \node[permpt] (4) at (4,6) {}; \node[permpt] (1) at (3,3) {}; \node[permpt] (2) at (1,4) {}; \node[permpt] (3) at (2,1) {}; \node[permpt] (4) at (9,2) {}; \draw[thin] (7,0.5) -- ++ (0,12); \draw[thin] (0.5,7) -- ++ (12,0); \draw[dotted] (8.5,0.5) -- ++ (0,12); \draw[dotted] (3.5,0.5) -- ++ (0,12); \draw[dotted] (0.5,9.5) -- ++ (12,0); \draw[dotted] (0.5,4.5) -- ++ (12,0); \end{scope} \end{tikzpicture} \end{center} \caption{The $\boxplus$-sum of the centred permutations $\pi^{\circ} = 241\underline{3}5$ and $\sigma^{\circ} = 413\underline{5}2$ is $413685\underline{7}92$; the origin of $\sigma^{\circ}$ is simply replaced ('inflated') by a copy of $\pi^{\circ}$.} \label{fig:boxsumdef} \end{figure} Fig. \ref{fig:boxsumdef} suggests a close relationship between the $\boxplus$-sum and $\circ$-intervals. Note that if $\mathcal{I}$ is a $\circ$-interval of a centred permutation $\pi^{\circ}$ then $\mathcal{I}$ contains the origin, and so the set of points contained in $\mathcal{I}$ forms a centred permutation $\sigma^{\circ}$. We say that $\mathcal{I}$ \textbf{encloses} the centred permutation $\sigma^{\circ}$, and we immediately observe the following: \begin{obs}[The $\boxplus$-sum and centred intervals] \label{intsum} Let $\pi^{\circ},\sigma^{\circ}$ be (centred) permutations, with $\sigma^{\circ}$ having strictly smaller length than $\pi^{\circ}$. Then: \begin{enumerate} \item $\pi^{\circ} = \sigma^{\circ} \boxplus \tau^{\circ}$ for some centred permutation $\tau^{\circ}$ if and only if $\pi^{\circ}$ contains a (necessarily unique) $\circ$-interval $\mathcal{I}_{\sigma}$ enclosing $\sigma^{\circ}$. \item If $\pi^{\circ} = \sigma^{\circ} \boxplus \tau^{\circ}$ then $\tau^{\circ}$ is the permutation obtained from $\pi^{\circ}$ by deleting all non-origin points contained in the (unique) $\circ$-interval $\mathcal{I}_{\sigma}$. \end{enumerate} \end{obs} We note the following immediate consequence of this result: \begin{cor}[Left-cancellation] Suppose $\sigma^{\circ},\tau^{\circ}_{1},\tau^{\circ}_{2}$ are centred permutations such that \[ \sigma^{\circ} \boxplus \tau^{\circ}_{1} = \sigma^{\circ} \boxplus \tau^{\circ}_{2}. \] Then $\tau^{\circ}_{1} = \tau^{\circ}_{2}$. \end{cor} \begin{proof} Let $\pi^{\circ} = \sigma^{\circ} \boxplus \tau^{\circ}_{1} = \sigma^{\circ} \boxplus \tau^{\circ}_{2}$. Then $\pi^{\circ}$ contains a unique $\circ$-interval $\mathcal{I}_{\sigma}$ enclosing $\sigma^{\circ}$ by Observation \ref{intsum}. Deleting all (non-origin) points in $\mathcal{I}_{\sigma}$ from $\pi^{\circ}$ will result in $\tau^{\circ}_{1}$, but by the same token will also result in $\tau^{\circ}_{2}$; hence $\tau^{\circ}_{1} = \tau^{\circ}_{2}$. \end{proof} Right-cancellation is also easy to prove, but will not be required here. \begin{defn}[$\boxplus$-decomposables and -indecomposables] We say that a centred permutation $\sigma^{\circ}$ is $\boxplus$-\textbf{decomposable} if it can be written as $\pi_{1}^{\circ} \boxplus \pi_{2}^{\circ}$ for two smaller centred permutations $\pi_{1}^{\circ}$ and $\pi_{2}^{\circ}$. Conversely, $\sigma^{\circ}$ is $\boxplus$-\textbf{indecomposable} if it cannot be split up like this. \end{defn} It will be important to note for later that $\sigma^{\circ}$ is $\boxplus$-decomposable if and only if there is a proper, non-trivial $\circ$-interval containing the origin; this is an immediate consequence of Observation \ref{intsum}. For example, the inner rectangle in Fig. \ref{fig:boxsumdef} is an interval because there are no points in the 'shadow' of the rectangle in either the $x$- or $y$-direction. Any $\boxplus$-decomposable centred permutation can be decomposed as a $\boxplus$-sum of $\boxplus$-indecomposables. This decomposition is not necessarily unique, as certain elements commute, as in Fig. \ref{fig:commpair}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,2) {}; \node[permpt] (2) at (3,4) {}; \node[permpt] (3) at (4,3) {}; \draw[thin] (2,-1) -- ++ (0,6); \draw[thin] (-1,2) -- ++ (6,0); \node at (7,2) {$\boxplus$}; \begin{scope}[shift={(10,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,2) {}; \node[permpt] (1) at (1,1) {}; \draw[thin] (2,-1) -- ++ (0,6); \draw[thin] (-1,2) -- ++ (6,0); \node at (7,2) {=}; \end{scope} \begin{scope}[shift={(20,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,2) {}; \node[permpt] (1) at (1,1) {}; \node[permpt] (2) at (3,4) {}; \node[permpt] (3) at (4,3) {}; \draw[thin] (2,-1) -- ++ (0,6); \draw[thin] (-1,2) -- ++ (6,0); \node at (7,2) {=}; \end{scope} \begin{scope}[shift={(30,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,2) {}; \node[permpt] (1) at (1,1) {}; \draw[thin] (2,-1) -- ++ (0,6); \draw[thin] (-1,2) -- ++ (6,0); \node at (7,2) {$\boxplus$}; \end{scope} \begin{scope}[shift={(40,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,2) {}; \node[permpt] (2) at (3,4) {}; \node[permpt] (3) at (4,3) {}; \draw[thin] (2,-1) -- ++ (0,6); \draw[thin] (-1,2) -- ++ (6,0); \end{scope} \end{tikzpicture} \end{center} \caption{The two $\boxplus$-indecomposable centred permutations $\underline{1}32$ and $1\underline{2}$ commute under the $\boxplus$-sum, due to being one-quadrant permutations from opposite quadrants.} \label{fig:commpair} \end{figure} More generally, we have: \begin{lemma}[Commuting elements under the $\boxplus$-sum] \label{commuters}\ \\ Suppose $\pi^{\circ}$ and $\sigma^{\circ}$ are $\boxplus$-indecomposable centred permutations. Then \[ \pi^{\circ} \boxplus \sigma^{\circ} = \sigma^{\circ} \boxplus \pi^{\circ} \] if and only if either $\pi^{\circ} = \sigma^{\circ}$ or $\pi^{\circ}$ and $\sigma^{\circ}$ are one-quadrant permutations from opposite quadrants. \end{lemma} \begin{proof} One direction here is obvious: if either $\pi^{\circ} = \sigma^{\circ}$ or $\pi^{\circ}$ and $\sigma^{\circ}$ are one-quadrant permutations from opposite quadrants then clearly $\pi^{\circ}$ and $\sigma^{\circ}$ commute under the $\boxplus$-sum. For the converse, we assume that the centred permutations $\pi^{\circ}$ and $\sigma^{\circ}$ commute and aim to deduce that they must be of one of these two forms: Consider the centred permutation $\tau^{\circ} = \pi^{\circ} \boxplus \sigma^{\circ} = \sigma^{\circ} \boxplus \pi^{\circ}$. This can be generated in two ways: either by inflating the origin of $\sigma^{\circ}$ by $\pi^{\circ}$ or vice versa. Thus there is a $\circ$-interval $\mathcal{I}_{1}$ of $\tau^{\circ}$ enclosing the permutation $\pi^{\circ}$ and another $\circ$-interval $\mathcal{I}_{2}$ of $\tau^{\circ}$ enclosing the permutation $\sigma^{\circ}$. As $\pi^{\circ}, \sigma^{\circ}$ are both $\boxplus$-indecomposable, $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are both minimal non-trivial $\circ$-intervals. Hence, by Lemma \ref{minint}, either $\mathcal{I}_{1} = \mathcal{I}_{2}$, in which case $\pi^{\circ} = \sigma^{\circ}$, or $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ each contain points in only one quadrant, and from opposite quadrants to each other, in which case $\pi^{\circ},\sigma^{\circ}$ are one-quadrant permutations from opposite quadrants, as required.\end{proof} We shall henceforth call a pair $\{\pi^{\circ},\sigma^{\circ}\}$ of one-quadrant centred permutations from opposite quadrants a \textbf{commuting pair}. In fact, commutativity of these commuting pairs is the \emph{only} way that uniqueness of the $\boxplus$-decomposition can fail, as is demonstrated by the following result: \begin{thm} \label{uniqueness} Let $\pi^{\circ}$ be a centred permutation. Then there exist $\boxplus$-indecomposables $\sigma_{1}^{\circ}, \sigma_{2}^{\circ}, \dots, \sigma_{n}^{\circ}$ such that \[ \pi^{\circ} = \sigma_{1}^{\circ} \boxplus \sigma_{2}^{\circ} \boxplus \dots \boxplus \sigma_{n}^{\circ}. \] Furthermore, this decomposition is unique up to commutativity of adjacent commuting pairs. In other words, if \[ \pi^{\circ} = \tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{k}^{\circ}. \] where $\tau_{1}^{\circ}, \tau_{2}^{\circ}, \dots, \tau_{k}^{\circ}$ are $\boxplus$-indecomposables, then: \begin{enumerate} \item k = n; \item $[\sigma_{1}^{\circ}, \sigma_{2}^{\circ}, \dots, \sigma_{n}^{\circ}] = [\tau_{1}^{\circ}, \tau_{2}^{\circ}, \dots, \tau_{k}^{\circ}]$ as multisets; \item the $n$-tuple $(\sigma_{1}^{\circ}, \sigma_{2}^{\circ}, \dots, \sigma_{n}^{\circ})$ can be transformed into $(\tau_{1}^{\circ}, \tau_{2}^{\circ}, \dots, \tau_{k}^{\circ})$ by a sequence of transpositions of adjacent commuting pairs. \end{enumerate} \end{thm} \begin{proof} On noting that centred permutations are equivalent to $2$-by-$2$-gridded permutations, this follows from~{Bevan, Brignall and Ru\v{s}kuc~\cite[Lemma 3.4]{bbr:unicyclicgrids:}}. We outline a proof here: Existence is clear: just take minimal non-trivial $\circ$-intervals inductively. For uniqueness the key is to prove that if $\sigma_{1}^{\circ} \neq \tau_{1}^{\circ}$ then there is some $l \in \{2,3, \dots, k\}$ such that $\sigma_{1}^{\circ} = \tau_{l}^{\circ}$ and $\sigma_{1}^{\circ} = \tau_{l}^{\circ}$ commutes with $\tau_{j}^{\circ}$ for $j = 1,2, \dots, l-1$. All three claims follow inductively from this. To prove this, note that, as \[ \pi^{\circ} = \tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{k}^{\circ} \] there is, by Lemma \ref{intsum} an increasing sequence of $\circ$-intervals $\mathcal{J}_{1}, \mathcal{J}_{2}, \dots, \mathcal{J}_{k}$ where $\mathcal{J}_{j}$ encloses the centred permutation $\tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{j}^{\circ}$. Also by Lemma \ref{intsum} there is a unique $\circ$-interval $\mathcal{I}$ enclosing $\sigma_{1}^{\circ}$. Note that $\mathcal{J}_{1} \cap \mathcal{I} = \{\circ\}$ (as these are distinct and $\mathcal{I}$ is minimal due to $\boxplus$-indecomposability of $\sigma_{1}^{\circ}$), and so by Lemma \ref{minint} these are both one-quadrant intervals from opposite quadrants - wlog say that $\sigma_{1}^{\circ}$ is entirely contained in quadrant $1$ and $\tau_{1}^{\circ}$ is in quadrant $3$. Now, as $\mathcal{I}$ is minimal $\mathcal{J}_{j} \cap \mathcal{I}$ is either $\{\circ\}$ or $\mathcal{I}$ for all $j$. Further $\mathcal{J}_{k} \cap \mathcal{I} = \mathcal{I}$. As $\mathcal{J}_{j}$ is an increasing sequence of intervals, this implies that there is some $l$ such that $\mathcal{J}_{l} \cap \mathcal{I} = \mathcal{I}$ but $\mathcal{J}_{j} \cap \mathcal{I} = \{\circ\}$ for all $j \leq l-1$. As $\mathcal{J}_{l-1} \cap \mathcal{I} = \{\circ\}$, Lemma \ref{minint} implies that $\tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{l-1}^{\circ}$ is entirely contained in quadrant $3$, so $\sigma_{1}^{\circ}$ commutes with all of $\tau_{j}^{\circ}$ for $j = 1,2, \dots, l-1$. And as $\mathcal{J}_{l} \cap \mathcal{I} = \mathcal{I}$, $\tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{l}^{\circ}$ contains a $\circ$-interval enclosing $\sigma_{1}^{\circ}$. But $\tau_{1}^{\circ} \boxplus \tau_{2}^{\circ} \boxplus \dots \boxplus \tau_{l}^{\circ}$ can be thought of as '$\tau_{l}^{\circ}$ plus some points in the third quadrant', whereas $\sigma_{1}^{\circ}$ is entirely contained in the first quadrant. Hence the interval enclosing $\sigma_{1}^{\circ}$ is in fact entirely contained in $\tau_{l}^{\circ}$, hence $\sigma_{1}^{\circ} \leq \tau_{l}^{\circ}$. But by $\boxplus$-indecomposability of $\tau_{l}^{\circ}$ we conclude that $\sigma_{1}^{\circ} = \tau_{l}^{\circ}$.\end{proof} \subsection{The Generating Function Specification} We say that a centred permutation class $\mathcal{C}^{\circ}$ is \textbf{$\boxplus$-closed} if, for any $\pi^{\circ},\sigma^{\circ} \in \mathcal{C}^{\circ}$, $\pi^{\circ} \boxplus \sigma^{\circ} \in \mathcal{C}^{\circ}$. For any centred permutation class $\mathcal{C}^{\circ}$ we define its \textbf{$\boxplus$-closure}, denoted $\boxplus\mathcal{C}^{\circ}$, to be the smallest $\boxplus$-closed class containing $\mathcal{C}^{\circ}$. Equivalently, $\boxplus\mathcal{C}^{\circ}$ is the set of all centred permutations of the form \[ \sigma^{\circ}_{1} \boxplus \sigma^{\circ}_{2} \boxplus \dots \boxplus \sigma^{\circ}_{n} \] where all $\sigma^{\circ}_{i}$ are $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$. Of course, a centred permutation class is $\boxplus$-closed if and only if it is equal to its own $\boxplus$-closure. We aim in this section to describe a general method for determining the generating function of a $\boxplus$-closed centred permutation class from the enumeration sequence of its $\boxplus$-indecomposables. Suppose we have a $\boxplus$-closed centred permutation class $\mathcal{C}^{\circ}$. Then every element of $\mathcal{C}^{\circ}$ is of the form \[ \pi^{\circ} = \sigma_{1}^{\circ} \boxplus \sigma_{2}^{\circ} \boxplus \dots \boxplus \sigma_{n}^{\circ} \] where each $\sigma_{i}^{\circ}$ is a $\boxplus$-indecomposable in $\mathcal{C}^{\circ}$. Suppose we have the generating function $g(z)$ of the (non-empty) $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$ (crucially, for the class of permutation classes that will be the main focus of this paper, this generating function will be relatively easy to find). \emph{If this decomposition were unique} we could then immediately obtain the generating function of the entire class by applying the \textbf{sequent operator}: \[ \begin{split} f(z) = Seq(f(z)) & = 1 + g(z) + g(z)^{2} + g(z)^{3} + \dots \\ & = \frac{1}{1 - g(z)} \end{split} \] Of course, as was demonstrated in the previous section, this decomposition is \emph{not} in general unique: the issue is that pairs of $\boxplus$-indecomposables from opposite quadrants commute, and so elements of $\mathcal{C}^{\circ}$ cannot be uniquely identified with $n$-tuples of $\boxplus$-indecomposables from the class. As it happens, however, we can easily amend the sequent operator to provide a specification of the generating function of $\mathcal{C}^{\circ}$ in terms of $g(z)$, though, unsurprisingly, we shall also have to use the generating functions $g_{i}(z)$ of the $\boxplus$-indecomposables of $\mathcal{C}^{\circ}$ contained entirely in the $i$th quadrant, for each $i \in \{1,2,3,4\}$, as these are the elements that introduce commutativity into the $\boxplus$-decomposition: \begin{thm}[Generating Function Specification for a $\boxplus$-closed Class]\label{genfuncspec} Let $\mathcal{C}^{\circ}$ be a centred permutation class, with $\boxplus$-closure $\boxplus\mathcal{C}^{\circ}$. Let \[ G(z) = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \] where $g(z)$ is the generating function of the (non-empty) $\boxplus$-indecomposables of $\mathcal{C}^{\circ}$; and $g_{i}(z)$ is the generating function of the (non-empty) one-quadrant $\boxplus$-indecomposables of $\mathcal{C}^{\circ}$ contained entirely in the $i$th quadrant. Then \[ f(z) = Seq(G(z)) = \frac{1}{1 - G(z)} \] is the generating function of $\boxplus\mathcal{C}^{\circ}$. If $\mathcal{C}^{\circ}$ is itself $\boxplus$-closed then $f(z)$ is in fact the generating function of $\mathcal{C}^{\circ}$. \end{thm} \begin{proof} This follows almost immediately from much more general result of Cartier and Foata~\cite{cartier:problemes-combi:} on trace monoids; see also~\cite[Note V.10]{flajolet:analytic-combin:}. \end{proof} \subsection{The Exponential Growth Theorem} We now have a method for determining the generating function of a $\boxplus$-closed centred permutation class. This may seem initially to be of limited interest, as we are mostly interested in the underlying (uncentred) classes, and these will of course have different generating functions to their centred counterparts. We recall, however, that if $\mathcal{C}^{\circ}$ is a centred permutation class with underlying class $\mathcal{C}$ then $\mathcal{C}^{\circ}$ and $\mathcal{C}$ have the same (upper) growth rate by Proposition \ref{eqgrs}, and by the following consequence of Pringsheim's Theorem we can calculate this from the generating function of $\mathcal{C}^{\circ}$: \begin{thm}[Exponential Growth Theorem, {Flajolet and Sedgewick~\cite{flajolet:analytic-combin:}}]\label{EGT} Suppose that $f(z)$ is the generating function of a proper centred permutation class $\mathcal{C}^{\circ}$. Then the upper growth rate of $\mathcal{C}^{\circ}$ is equal to the reciprocal of the radius of convergence of $f(z)$. \end{thm} \begin{proof} This is an immediate consequence of~\cite[Theorem IV.7]{flajolet:analytic-combin:}, on noting that $f(z)$ certainly has non-negative coefficients. \end{proof} Suppose now that $\mathcal{C}^{\circ}$ is a proper centred permutation class with amended $G$-sequence $G(z)$. Then the generating function of $\boxplus\left(\mathcal{C}^{\circ}\right)$ is given by \[ f(z) = \frac{1}{1 - G(z)} \] By Theorem \ref{EGT} the growth rate of $\boxplus\mathcal{C}^{\circ}$ is then equal to the reciprocal of the radius of convergence of $f(z)$. At first glace it looks like this radius of convergence should be equal to the modulus of the smallest solution of $G(z)=1$ (and we will see that this is true for the classes that we are interested in in this paper) but we need to be careful about $G(z)$ itself possibly having singularities before any solutions of $G(z)=1$. For the time-being we will deal with this issue on a case-by-case basis. \subsection{Examples} Theorems \ref{genfuncspec} and \ref{EGT} are remarkably useful in tandem, suggesting as they do a powerful method for constructing a large number of permutation classes whose growth rates we can easily calculate. For example, the $\boxplus$-closure of a given finite set of $\boxplus$-indecomposables is now easy to calculate, as demonstrated by the following examples: \begin{example}[$\boxplus$-closure of $41\underline{3}52$] $\pi^{\circ} = 41\underline{3}52$ is a $\boxplus$-indecomposable centred permutation. We shall determine the growth rate of its $\boxplus$-closure $\mathcal{C}^{\circ} = \boxplus(\pi^{\circ})$, that is, the smallest $\boxplus$-closed class containing $\pi^{\circ}$. Equivalently, this is the downward closure of the set of centred permutations of the form $\pi^{\circ} \boxplus \pi^{\circ} \boxplus \dots \boxplus \pi^{\circ}$. As $\mathcal{C}^{\circ}$ is (by definition) $\boxplus$-closed we need only enumerate the $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$ in each quadrant. First, the only $\boxplus$-indecomposables contained in $\pi^{\circ}$ (and hence in $\mathcal{C}^{\circ}$) are $\pi^{\circ}$ itself and the four single point centred permutations. Hence the generating function of the $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$ is $g(z) = 4z + z^4$. Of course, as the only single-quadrant $\boxplus$-indecomposables in this list are the four single point permutations, the generating functions $g_{i}(z)$ of the single-quadrant $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$ are given by $g_{1}(z) = g_{2}(z) = g_{3}(z) = g_{4}(z) = z$. Hence the generating function of $\mathcal{C}^{\circ}$ is given by: \[ \begin{split} f(z) & = \frac{1}{1 - [(4z + z^4) - 2z^2]} \\ & = \frac{1}{1 - 4z + 2z^2 - z^4} \end{split} \] And so by Theorem \ref{EGT} we can deduce that the growth rate of $gr(\mathcal{C}^{\circ})$ is the reciprocal of the modulus of the smallest root of the denominator, which is $\approx 3.44372$. \end{example} \begin{example}[The $\mathcal{X}$-class] Next, we turn to a class introduced by Waton~\cite{waton:on-permutation-:} in the context of picture classes: the $\mathcal{X}$-class, consisting of all permutations which can be drawn on an $X$. Waton showed that this class has generating function \[ \frac{z(1 - 2z)}{1 - 4z + 2z^2} \] and hence growth rate $2 + \sqrt{2} \approx 3.41421$ We shall show that this growth rate (but not the generating function) can be derived quickly by instead considering the \emph{centred} permutation class $\mathcal{X}^{\circ}$, defined to be the $\boxplus$-closure of the four single point centred permutations: $\mathcal{X}^{\circ} = \boxplus\{\nept,\nwpt,\swpt,\sept\}$. Clearly $\mathcal{X}^{\circ}$ has $\mathcal{X}$ as its underlying (uncentred) class: we can think of this as simply adding an origin point at the centre of the $X$ diagram used to define $\mathcal{X}$. But then for $\mathcal{X}^{\circ}$ we clearly have $g(z) = 4z$ and $g_{1}(z) = g_{2}(z) = g_{3}(z) = g_{4}(z) = z$; hence $\mathcal{X}^{\circ}$ has generating function: \[ \frac{1}{1 - 4z + 2z^2} \] Note that this has the same denominator as the uncentred class $\mathcal{X}$, demonstrating, as expected, that these two classes have the same growth rate. \end{example} \section{Pin Sequences and Pin Classes} \label{sec:3} \subsection{Pin Sequences} We begin with a definition, following Brignall, Ru\v{s}kuc and Vatter~\cite{brignall:simple-permutat:b}: \begin{defn} A \emph{pin sequence} is a word (finite or infinite) over the language \begin{equation} \mathcal{L}_{P} = \{1,2,3,4\}(\{\text{l},\text{r}\}\{\text{u},\text{d}\})^{}\cup \{1,2,3,4\}(\{\text{u},\text{d}\}\{\text{l},\text{r}\})^{} \end{equation} \end{defn} We tend to refer to a finite pin sequence as a \emph{pin word} and reserve the term 'pin sequence' for the infinite case. For example, $w=3ldrdrdlurdl$ is a pin word and $4(urul)^{}$ is a pin sequence (we use the notation $(f)^{}$ to denote the factor $f$ recurring indefinitely, so $4(urul)^{} = 4urulurulurulurul\dots$). We think of the letters $u,d,l,r$ as representing up, down, left, right, respectively. We then say that letters $u$ and $d$ have \textbf{vertical alignment} and letters $l$ and $r$ have \textbf{horizontal alignment}. We can then informally descrive a pin sequence as consisting of an inital numeral from $\{1,2,3,4\}$ followed by a sequence of letters from $\{u,l,d,r\}$ which must alternate between horizontal and vertical alignment. Note that this implies that there are $4$ pin words of length $1$ and $2^{n+2}$ pin words of every length $n \geq 2$. We shall also have cause to refer to the language \begin{equation} \mathcal{L}^{}_{P} = (\{\text{l},\text{r}\}\{\text{u},\text{d}\})^{}\cup (\{\text{u},\text{d}\}\{\text{l},\text{r}\})^{} \end{equation} Note that a pin word consists of an initial numeral between $1$ and $4$ followed by a (possibly empty) word over $\mathcal{L}^{}_{P}$. A finite pin word $w$ of length $n$ can be converted into a centred permutation $\pi_{w}^{\circ}$ of length $n$ by the following procedure (see Fig. \ref{fig:pinpermexample} for an illustration of this process): \begin{defn}[The $\pi$-map and pin permutations] Let $n \in \mathbb{N}$; furthermore, let $\Pi_{n}$ denote the set of pin words of length $n$, and $S_{n}^{\circ}$ denote the set of centred permutations of length $n$. The $\pi$-\emph{map} is the map \[ \pi^{\circ}: \Pi_{n} \rightarrow S_{n}^{\circ} \] defined by the following procedure: given $w \in \Pi_{n}$: \begin{enumerate} \item Place the initial origin point $p_{0}$; use this point to split the plane into four numbered quadrants, as in Fig. \ref{fig:quadnumbering}. \item Place the first point $p_{1}$ in the quadrant specified by the initial numeral of the pin word $w$. \item Once the first $k-1$ points have been placed, place the point $p_{k}$ either up, down, left or right (depending on the letter $u$, $d$, $l$, or $r$ appearing in the $k$-th place in the pin sequence) of the bounding rectangle of all previous points $\{p_{0},p_{1}, \dots , p_{k-1} \}$ at the end of a 'pin' which separates the last point $p_{k-1}$ from all previous points. \item Once all $n$ points have been placed, read off the centred permutation $\pi^{\circ}_{w}$ given by the points $\{p_{0}, p_{1}, \dots , p_{n} \}$, with $p_{0}$ as the origin. \end{enumerate} We refer to $\pi^{\circ}_{w}$ as the (centred) pin permutation associated with $w$; similarly, we write $\pi_{w}$ for the underlying (uncentred) permutation of $\pi^{\circ}_{w}$ and refer to this as the (uncentred) pin permutation associated with $w$. We refer to a centred permutation $\sigma^{\circ}$ as a (centred) \textbf{pin permutation} if there is some pin word $w$ such that $\sigma^{\circ} \leq \pi_{w}$. Similarly, an uncentred permutation $\sigma$ is an (uncentred) pin permutation if it is the underlying permutation of a centred pin permutation. Finally, we refer to a centred permutation $\sigma^{\circ}$ as a \textbf{contiguous} pin permutation is there is some pin word $w$ such that $\sigma^{\circ} = \pi_{w}$. \end{defn} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.5] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (6,4) {}; \node[permpt,label={\tiny $p_{1}$}] (1) at (5,6) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (3,5) {}; \draw[thin] (2) -- ++ (2.5,0); \node[permpt,label={\tiny $p_{3}$}] (3) at (4,8) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{4}$}] (4) at (8,7) {}; \draw[thin] (4) -- ++ (-4.5,0); \node[permpt,label={\tiny $p_{5}$}] (5) at (7,2) {}; \draw[thin] (5) -- ++ (0,5.5); \node[permpt,label={\tiny $p_{6}$}] (6) at (1,3) {}; \draw[thin] (6) -- ++ (6.5,0); \node[permpt,label={\tiny $p_{7}$}] (7) at (2,1) {}; \draw[thin] (7) -- ++ (0,2.5); \draw[thick] (6,0) -- ++ (0,9); \draw[thick] (0,4) -- ++ (9,0); \end{tikzpicture} \end{center} \caption{The (centred) pin permutation $31586\underline{4}27$ (and its underlying uncentred permutation $3147526$), constructed from the pin sequence $2lurdld$.} \label{fig:pinpermexample} \end{figure} \begin{obs} \label{slicing} By definition the point $p_{k}$ slices the bounding rectangle $rec(p_{0},p_{1}, \dots , p_{k-1})$ of all previous points, including the origin. In fact, the definition implies that $p_{k}$ is the \textbf{only} point after $p_{k-1}$ that slices this rectangle; this means that if we remove $p_{k}$ from the corresponding centred permutation $rec(p_{0},p_{1}, \dots , p_{k-1})$ becomes a $\circ$-interval, and hence the resulting centred permutation is $\boxplus$-decomposable. This will be important in our later structure theorem for a pin class. \end{obs} We denote the class consisting of \emph{all} centred pin permutations $\mathcal{P}^{\circ}$, and its uncentred counterpart by $\mathcal{P}$. We call this the \textbf{complete pin class}. Bassino, Bouvel and Rossin~\cite{bassino:enumeration-of-:} proved that $\mathcal{P}$ has rational generating function and growth rate $\omega_{\infty} \approx 5.24112$. We shall later find the generating function of $\mathcal{P}^{\circ}$, which of course has the same growth rate as $\mathcal{P}$ by Proposition \ref{eqgrs}. We are primarily interested in this paper in certain subclasses of $\mathcal{P}$, known as \textbf{pin classes}. These are defined somewhat analogously to pin permutations: just as we can convert a (finite) pin word into a permutation, we can also convert an (infinite) pin sequence into a pin class: \begin{defn}[Pin Classes] Suppose that $w$ is an infinite pin sequence. Let $\pi^{\circ}_{n}$ be the pin permutation obtained by the above process from the inital contiguous subsequence of $w$ of length $n$ (which is itself a finite pin word), and consider the set $\Pi = \{ \pi^{\circ}_{1}, \pi^{\circ}_{2}, \pi^{\circ}_{3}, \dots \}$. The downward closure of $\Pi$ under the pattern containment order $\leq$ forms a centred permutation class $\mathcal{C}^{\circ}_{w}$, known as the centred \emph{pin class} of $w$. We similarly define the uncentred pin class of $w$, $\mathcal{C}_{w}$, to be the underlying class of $\mathcal{C}^{\circ}_{w}$. \end{defn} Informally, we can think of this as using the infinite pin sequence $w$ to draw an 'infinite diagram' by the same process as in the finite case; the pin class $\C_{w}$ is then the class of all permutations that can be found somewhere within this infinite diagram (by picking a finite collection of points and 'forgetting' everything else). \begin{example} Taking $w=1(ldrdluru)^{}$ we generate the pin diagram shown in Fig. \ref{fig:fountain}. We could of course extend this diagram indefinitely. The class $\mathcal{C}_{w}$ then consists of all permutations that can be found somewhere in this diagram (with $\mathcal{C}^{\circ}_{w}$ being the class of all \emph{centred} permutations that can be found). \end{example} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (9,9) {}; \node[permpt,label={\tiny $p_{1}$}] (1) at (10,11) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (7,10) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt,label={\tiny $p_{3}$}] (3) at (8,7) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt,label={\tiny $p_{4}$}] (4) at (12,8) {}; \draw[thin] (4) -- ++ (-4.5,0); \node[permpt,label={\tiny $p_{5}$}] (5) at (11,5) {}; \draw[thin] (5) -- ++ (0,3.5); \node[permpt,label={\tiny $p_{6}$}] (6) at (5,6) {}; \draw[thin] (6) -- ++ (6.5,0); \node[permpt,label={\tiny $p_{7}$}] (7) at (6,13) {}; \draw[thin] (7) -- ++ (0,-7.5); \node[permpt,label={\tiny $p_{8}$}] (8) at (14,12) {}; \draw[thin] (8) -- ++ (-8.5,0); \node[permpt,label={\tiny $p_{9}$}] (9) at (13,15) {}; \draw[thin] (9) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{10}$}] (10) at (3,14) {}; \draw[thin] (10) -- ++ (10.5,0); \node[permpt,label={\tiny $p_{11}$}] (11) at (4,3) {}; \draw[thin] (11) -- ++ (0,11.5); \node[permpt,label={\tiny $p_{12}$}] (12) at (16,4) {}; \draw[thin] (12) -- ++ (-12.5,0); \node[permpt,label={\tiny $p_{13}$}] (13) at (15,1) {}; \draw[thin] (13) -- ++ (0,3.5); \node[permpt,label={\tiny $p_{14}$}] (14) at (1,2) {}; \draw[thin] (14) -- ++ (14.5,0); \node[permpt,label={\tiny $p_{15}$}] (15) at (2,17) {}; \draw[thin] (15) -- ++ (0,-15.5); \node[permpt,label={\tiny $p_{16}$}] (16) at (17,16) {}; \draw[thin] (16) -- ++ (-15.5,0); \draw[thick] (9,0) -- ++ (0,18); \draw[thick] (0,9) -- ++ (18,0); \end{tikzpicture} \end{center} \caption{The pin class $\mathcal{C}^{\circ}_{w}$ generated by the pin sequence $w = 1(ldrdluru)^{}$.} \label{fig:fountain} \end{figure} \subsubsection{Oscillations} One particular collection of pin permutations is already well-known and will play a key role in our later theory: \begin{defn}[Oscillations] Let $w$ be a (finite) pin word that stays in its initial quadrant (for example, $1ururur$, $1rururur$, $3ldl$ and $4drdrd$). We call the permutation $\pi^{\circ}_{w}$ generated by $w$ an \textbf{oscillation} (sometimes a \textbf{one-quadrant oscillation} for emphasis). \end{defn} Oscillations have natural `staircase' structures - see Fig. \ref{fig:oscexamples} for examples. Note that there are precisely two oscillations of each length $n\geq3$ in each quadrant (eg. those generated by $1ururu$ and $1rurur$ of length $6$ in the first quadrant), but only one of length $2$ (as $1u$ and $1r$ in fact generate the same (centred) permutation \netwo - see Fig. \ref{fig:2osccollision} for an illustration). \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.3] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (2,1) {}; \node[permpt] (2) at (1,3) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (4,2) {}; \draw[thin] (3) -- ++ (-3.5,0); \node[permpt] (4) at (3,5) {}; \draw[thin] (4) -- ++ (0,-3.5); \node[permpt] (5) at (6,4) {}; \draw[thin] (5) -- ++ (-3.5,0); \node[permpt] (6) at (5,6) {}; \draw[thin] (6) -- ++ (0,-2.5); \draw[thick] (-1,0) -- ++ (8,0); \draw[thick] (0,-1) -- ++ (0,8); \node at (0,-2) {$1ururu$}; \node at (0,-4) {$\underline{1}426375$}; \begin{scope}[shift={(12,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (1,2) {}; \node[permpt] (2) at (3,1) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt] (3) at (2,4) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (5,3) {}; \draw[thin] (4) -- ++ (-3.5,0); \node[permpt] (5) at (4,6) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt] (6) at (6,5) {}; \draw[thin] (6) -- ++ (-2.5,0); \draw[thick] (-1,0) -- ++ (8,0); \draw[thick] (0,-1) -- ++ (0,8); \node at (0,-2) {$1rurur$}; \node at (0,-4) {$\underline{1}352746$}; \end{scope} \begin{scope}[shift={(29,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (-1,2) {}; \node[permpt] (2) at (-3,1) {}; \draw[thin] (2) -- ++ (2.5,0); \node[permpt] (3) at (-2,4) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (-5,3) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (-4,5) {}; \draw[thin] (5) -- ++ (0,-2.5); \draw[thick] (1,0) -- ++ (-7,0); \draw[thick] (0,-1) -- ++ (0,8); \node at (0,-2) {$2lulu$}; \node at (0,-4) {$46253\underline{1}$}; \end{scope} \end{tikzpicture} \end{center} \caption{Three oscillations and the pin words that generate them. Note the two distinct oscillations of length $6$ in the first quadrant. In general, there are two distinct oscillations of each length $n \geq 2$ in each quadrant.} \label{fig:oscexamples} \end{figure} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.3] \begin{scope}[shift={(10,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (1,2) {}; \node[permpt] (2) at (2,1) {}; \draw[thin] (2) -- ++ (-1.5,0); \node[empty] (-1) at (0,4) {}; \draw[thick] (-3,0) -- ++ (6,0); \draw[thick] (0,-3) -- ++ (0,6); \node[] (4) at (0,-5) {$1r$}; \end{scope} \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (2,1) {}; \node[permpt] (2) at (1,2) {}; \draw[thin] (2) -- ++ (0,-1.5); \draw[thick] (-3,0) -- ++ (6,0); \draw[thick] (0,-3) -- ++ (0,6); \node[] (3) at (5,0) {=}; \node[] (4) at (0,-5) {$1u$}; \end{tikzpicture} \end{center} \caption{Whilst there are two distinct oscillations in each quadrant for all lengths $n \geq 3$, there is only one at length $2$. This is because the pin words $1u$ and $1r$ in fact generate the \emph{same} centred permutation $\underline{1}32$. This is an example of a \textbf{collision} of pin factors, a phenomenon that we will study in detail later.} \label{fig:2osccollision} \end{figure} We can also consider the pin class $\mathcal{O}^{\circ}$ of \textbf{increasing oscillations}, defined to be $\mathcal{C}^{\circ}_{w}$ for the pin word $1(ru)^{}$ (see Fig. \ref{fig:oscclass}). This class has been shown to have growth rate $\kappa \approx 2.20557$, a fact which we shall be able to prove later. We will also be able to prove that $\kappa$ is in fact the \emph{smallest} possible growth rate of a pin class, and is only achievable by pin classes which are `essentially' $\mathcal{O}^{\circ}$. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (1,2) {}; \node[permpt] (2) at (3,1) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt] (3) at (2,4) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (5,3) {}; \draw[thin] (4) -- ++ (-3.5,0); \node[permpt] (5) at (4,6) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt] (6) at (7,5) {}; \draw[thin] (6) -- ++ (-3.5,0); \node[permpt] (7) at (6,8) {}; \draw[thin] (7) -- ++ (0,-3.5); \node[permpt] (8) at (9,7) {}; \draw[thin] (8) -- ++ (-3.5,0); \node[permpt] (9) at (8,10) {}; \draw[thin] (9) -- ++ (0,-3.5); \node[permpt] (10) at (11,9) {}; \draw[thin] (10) -- ++ (-3.5,0); tellipsis {1a} {1b}; tellipsis {6a} {6b} ; \draw[thick] (0,0) -- ++ (10,0); \draw[thick] (0,0) -- ++ (0,12); \end{tikzpicture} \end{center} \caption{The class $\mathcal{O}^{\circ}$ of \textbf{increasing oscillations}, generated by the pin sequence $1(ru)^{}$.} \label{fig:oscclass} \end{figure} Oscillations have been studied extensively in the literature. In fact, to a significant extent pin permutations are worth studying because they generalise the oscillations and end up being interesting for much the same reasons. For example, Bevan~\cite{bevan:intervals} demonstrated that an infinite antichain contained in the two-point extension $\mathcal{O}^{+2}$ can be used to construct permutation classes (all of which contain long oscillations) at \emph{every} growth rate $\lambda \geq 2.35526$. This construction in fact generalises to pin classes in general, allowing us to construct genuinely distinct antichains at every real number which is the growth rate of a pin class. We thus have established a method for converting a pin sequence into a (centred or uncentred) permutation class. This allows us to describe an extraordinarily large number of classes which would be awkward to describe by other means (it would, for example, take a lot of work to describe the class in Fig. \ref{fig:fountain} in terms of its basis). There are two main reasons that this construction will prove useful. First, controlling the initial pin sequence allows us to control various properties of the permutation class produced (for example, the aforementioned connection with simple permutations allows us to control the number of simples in a given pin class). Second, and most importantly, we have a process for determining the growth rates of the pin permutation classes produced: this will depend on an in-built structure possessed by pin permutation classes, the \textbf{$\boxplus$-decomposition}. \subsection{Pin Factors} We will need to have a notion of a 'subsequence' of a pin sequence, one that should hopefully preserve pattern containment in both the centred and uncentred case: \begin{defn}[Pin Factor] Suppose that $w$ is a (finite or infinite) pin word. A \textbf{pin factor} of $w$ is either: \begin{itemize} \item an initial contiguous subsequence of $w$; or: \item a non-initial contiguous subsequence of $w$ in which the first letter is replaced by the quadrant in which the corresponding point appears in $\pi_{w}^{\circ}$. \end{itemize} \end{defn} Note that a pin factor of a given pin word is by definition itself a pin word. We write $w_{i,j}$ for the pin factor obtained by taking the contiguous subsequence of $w$ between the $i$th and $j$th place. \begin{example} Consider the pin word $w = 2ruldlurdru$. Then $w_{1,2} = 2r$, $w_{1,4} = 2rul$, $w_{1,9} = 2ruldlurd$ and $w$ itself are all pin factors of $w$ as they are all (finite) initial subsequences. If instead we take out the contiguous subsequence from the fifth to ninth terms we obtain the word $dlurd$. This is a factor in the word sense, but it is not a pin factor as it does not begin with a numeral (and so is not a pin word). However, if we draw out the (centred) permutation $\pi_{w}^{\circ}$ generated by $w$ (see Fig. \ref{fig:pinfactorexample}) we can see that the fifth point placed is in the $3$rd quadrant, so we replace the initial $d$ in $dlurd$ with a $3$ to obtain the pin word $3lurd$, which is now a pin factor of $w$; specifically $w_{5,9} = 3lurd$. For an internal pin word we can always work our what this inital numeral should be by looking at the previous letter: for example, if we wish to work out $w_{9,11}$ we first take out the subword $dru$ (between the $9$th and $11$th places); to work out which numeral we replace the initial $d$ with, note that it was preceded by an $r$ and the $d$ in any $rd$ clearly must be placed in the $4$th quadrant. Hence $w_{9,11} = 4ru$. This suggests a correspondence between pin factors of length $n$ and $\mathcal{L}^{}$-subwords of length $n+1$, stated below in Lemma \ref{pfsubwcorr}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (6,5) {}; \node[permpt] (1) at (5,7) {}; \node[permpt] (2) at (8,6) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[permpt] (3) at (7,9) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (3,8) {}; \draw[thin] (4) -- ++ (4.5,0); \node[permpt,red] (5) at (4,3) {}; \draw[thin] (5) -- ++ (0,5.5); \node[permpt,red] (6) at (1,4) {}; \draw[thin] (6) -- ++ (3.5,0); \node[permpt,red] (7) at (2,11) {}; \draw[thin] (7) -- ++ (0,-7.5); \node[permpt,red] (8) at (10,10) {}; \draw[thin] (8) -- ++ (-8.5,0); \node[permpt,red] (9) at (9,1) {}; \draw[thin] (9) -- ++ (0,9.5); \node[permpt] (10) at (12,2) {}; \draw[thin] (10) -- ++ (-3.5,0); \node[permpt] (11) at (11,12) {}; \draw[thin] (11) -- ++ (0,-10.5); \draw[thick] (6,0.5) -- ++ (0,12); \draw[thick] (0.5,5) -- ++ (12,0); \draw[dotted] (3.5,2.5) rectangle (4.5,3.5); \end{tikzpicture} \end{center} \caption{The construction of the (centred) permutation $\pi_{w}^{\circ}$ from the pin word $w = 2ruldlurdru$. Note that the fifth-placed point, $p_{5}$, corresponding to the first $d$ in the word, is in the $3$rd quadrant, so when we extract the word factor $dlurd$ from $w$ we replace the inital $d$ with a $3$ to obtain the pin factor $\tilde{w} = 3lurd$. Note that the (centred) permutation $\sigma_{\tilde{w}}^{\circ}$ (highlighted in red) is a subpermutation of $\pi_{w}^{\circ}$.} \label{fig:pinfactorexample} \end{figure} \end{example} As Fig. \ref{fig:pinfactorexample} suggests, this notion preserves pattern containment: \begin{obs} \label{pinfactorcontainment} Suppose $w_{1},w_{2}$ are (finite) pin words and that $w_{1}$ is a pin factor of $w_{2}$. Then: \[ \pi^{\circ}_{w_{1}} \leq \pi^{\circ}_{w_{2}} \] \end{obs} It's also worth noting that, while pin factors of $w$ of length $n$ cannot in general be identified with subwords of $w$ of length $n$ (for example, the subword $ulur$ could instantiate either $1lur$ or $2lur$ depending on the context) they can \emph{almost} be identified with subwords of length $n+1$ (with the first letter determining the quadrant numeral): the only issue being initial pin factors, which may not reappear. In practice this will be only a minor inconvenience (which may be completely ignored in the recurrent case) as initial pin factors cannot affect the growth rate of a pin class, at worst only affecting the enumeration sequence. We now have the terminology to define an important category of pin sequences, whose associated pin classes (we shall see) have a structure which will be particularly amenable to the enumeration methods developed in Section 1: \begin{defn}[Recurrent Pin Sequences] We refer to an infinite pin sequence $w$ as \textbf{recurrent} if every pin factor that occurs in $w$ occurs infinitely often; similarly, $w$ is \textbf{eventually recurrent} if every pin factor that occurs after a certain point occurs infinitely often. We will often refer to the pin class $\mathcal{C}^{\circ}_{w}$ generated by a recurrent pin sequence as a \textbf{recurrent pin class}. \end{defn} Note that this definition is subtly different from recurrence in the usual (subword) sense, due to complications introduced by initial subsequences (which begin with a numeral): these initial subwords can never occur as subwords again in $w$ by the definition of a pin sequence, but (in order for $w$ to be recurrent) they \emph{must} appear infinitely often a pin factors. This means it can often be difficult to tell whether a pin sequence is recurrent or not, even in the periodic case, until it is drawn out: for example, $2(ul)^{}$ is recurrent, but $1(ul)^{}$ is not, as $w_{1,2}=1u$ never reoccurs as a pin factor. We wish to ignore, as far as possible, this subtle and often irritating distinction between subwords and pin factors. First, we call a pin factor of $w$ \emph{fully internal} if it occurs in $w$ starting after the second place; that is, fully internal pin factors are those of the form $w_{i,j}$ where $3 \leq i < j$. Note that fully internal pin factors can be found somewhere in $w$ with a preceeding letter (as opposed to numeral). This letter is enough to reconstruct the initial numeral of the pin factor, thus leading to the following: \begin{lemma} \label{pfsubwcorr} Let $w$ be a pin sequence. Then the number of fully internal pin factors of $w$ of length $n \geq 2$ is equal to the number of $\mathcal{L}^$-subwords of $w$ of length $n+1$. \end{lemma} Of course, the fully internal restriction can be removed if we further assume that $w$ is recurrent, as in this case any pin factor beginning in the first or second place can also be found later on; hence all pin factors are fully internal, giving us the following simplification: \begin{cor}\label{recpfsubwcorr} Let $w$ be a recurrent pin sequence. Then the number of pin factors of $w$ of length $n$ is equal to the number of $\mathcal{L}^$-subwords of $w$ of length $n+1$. \end{cor} \subsubsection{Left-truncations} We shall also require an infinite analogue of a pin factor of $w$, the (infinite) pin \emph{sequence} obtained by starting $w$ at a later point: \begin{defn}[Left-truncation of a pin sequence] Let $w$ be a pin sequence and $n \in \mathbb{N}$. The $n$th \textbf{left-truncation} of $w$, $w_{\geq n}$, is the pin sequence obtained from $w$ by replacing the symbol in the $n$th place with the number of the quadrant in which $p_{n}$ is placed, and then removing all of the previous symbols. \end{defn} Informally, $w_{\geq n}$ is `$w$ but started in the $n$th place'. For example, if $w = 3rurdlurur$ then $w_{\geq 4} = 1dlurur$ and $w_{\geq 7} = 2rur$. Again, we can always work out the numeral to replace the letter in the $n$th place with by looking at the previous symbol. Crucially, left-truncating a pin sequence does not affect the asymptotics of its pin class: \begin{lemma}[Finite Prefix Lemma] \label{fplem} Let $w$ be a pin sequence and $n \in \mathbb{N}$. Then $\overline{gr}(\mathcal{C}^{\circ}_{w}) = \overline{gr}(\mathcal{C}^{\circ}_{w_{\geq n}})$ and $\underline{gr}(\mathcal{C}^{\circ}_{w}) = \underline{gr}(\mathcal{C}^{\circ}_{w_{\geq n}})$. \end{lemma} \begin{proof} Recall that we write $\mathcal{C}^{\circ+k}$ for the $k$-point extenstion of $\mathcal{C}^{\circ}$ (that is, the class of centred permutations from $\mathcal{C}^{\circ}$ with at most $k$ extra points added anywhere). Note that the (infinite) pin diagram generated by $w$ is simply that of $w_{\geq n}$ with $n-1$ extra points, hence: \[ \mathcal{C}^{\circ}_{w_{\geq n}} \subseteq \mathcal{C}^{\circ}_{w} \subseteq \mathcal{C}^{\circ+(n-1)}_{w_{\geq n}} \] and by Lemma \ref{eqgrs2} the classes on the left and right have the same (upper and lower) growth rates. \end{proof} We call this the \textbf{Finite Prefix Lemma}, as it tells us that any finite prefix of a pin sequence cannot affect the growth rate of the associated pin class, a fact that we shall use often. \subsection{The Pin Decomposition} The key idea we will use to enumerate pin classes comes from the following observation: when we take a centred permutation generated by a pin word and remove any interior point, we decompose the resulting permutation as the $\boxplus$-sum of two smaller pin permutations. Fig. \ref{fig:boxdecomposition} illustrates this process. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.3] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (7,6) {}; \node[permpt] (1) at (8,8) {}; \node[permpt] (2) at (5,7) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (6,4) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (3,5) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (4,2) {}; \draw[thin] (5) -- ++ (0,3.5); \node[permpt] (6) at (10,3) {}; \draw[thin] (6) -- ++ (-6.5,0); \node[permpt] (7) at (9,10) {}; \draw[thin] (7) -- ++ (0,-7.5); \node[permpt] (8) at (12,9) {}; \draw[thin] (8) -- ++ (-3.5,0); \node[permpt] (9) at (11,12) {}; \draw[thin] (9) -- ++ (0,-3.5); \node[permpt] (10) at (1,11) {}; \draw[thin] (10) -- ++ (10.5,0); \node[permpt] (11) at (2,1) {}; \draw[thin] (11) -- ++ (0,10.5); \draw[thick] (7,0.5) -- ++ (0,12); \draw[thick] (0.5,6) -- ++ (12,0); \draw[dotted] (9.5,2.5) rectangle (10.5,3.5); \node at (15,6) {$\rightarrow$}; \begin{scope}[shift={(17,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (7,6) {}; \node[permpt] (1) at (8,8) {}; \node[permpt] (2) at (5,7) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (6,4) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (3,5) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (4,2) {}; \draw[thin] (5) -- ++ (0,3.5); \node[permpt] (7) at (9,10) {}; \node[permpt] (8) at (12,9) {}; \draw[thin] (8) -- ++ (-3.5,0); \node[permpt] (9) at (11,12) {}; \draw[thin] (9) -- ++ (0,-3.5); \node[permpt] (10) at (1,11) {}; \draw[thin] (10) -- ++ (10.5,0); \node[permpt] (11) at (2,1) {}; \draw[thin] (11) -- ++ (0,10.5); \draw[thick] (7,0.5) -- ++ (0,12); \draw[thick] (0.5,6) -- ++ (12,0); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (0.5,1.5) rectangle (2.5,8.5); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (8.5,1.5) rectangle (11.5,8.5); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (2.5,8.5) rectangle (8.5,12.5); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (2.5,0.5) rectangle (8.5,1.5); \node at (15,6) {=}; \end{scope} \begin{scope}[shift={(34,2)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,4) {}; \node[permpt] (1) at (6,6) {}; \node[permpt] (2) at (3,5) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (4,2) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (1,3) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (2,1) {}; \draw[thin] (5) -- ++ (0,2.5); \draw[thick] (5,0.5) -- ++ (0,6); \draw[thick] (0.5,4) -- ++ (6,0); \node at (9,4) {$\boxplus$}; \end{scope} \begin{scope}[shift={(45,2)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (3,2) {}; \node[permpt] (1) at (4,4) {}; \node[permpt] (2) at (6,3) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt] (3) at (5,6) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (1,5) {}; \draw[thin] (4) -- ++ (4.5,0); \node[permpt] (5) at (2,1) {}; \draw[thin] (5) -- ++ (0,4.5); \draw[thick] (3,0.5) -- ++ (0,6); \draw[thick] (0.5,2) -- ++ (6,0); \end{scope} \end{tikzpicture} \end{center} \caption{When we remove the point $p_{6}$ from the pin permutation generated by $1ldldruruld$ we remove the only point that slices the bounding rectangle of the first five points; we can thus contract this rectangle down to a single point and in doing so express the resulting centred permutation as the box sum of $\pi^{\circ}_{1ldld}$ and $\pi^{\circ}_{1ruld}$.} \label{fig:boxdecomposition} \end{figure} We formalise this as follows: \begin{lemma}[Removing an interior point from a pin permutation]\ \\ Suppose that $w$ is a pin word of length $n$, and that $p_{1}, p_{2}, \dots, p_{n}$ are the corresponding points of the centred pin permutation $\pi_{w}^{\circ}$. Let $k \in \{2,3,\dots,n-1\}$. Then: \[ \pi_{w}^{\circ} - \{p_{k}\} = \pi_{w_{1,k-1}}^{\circ} \boxplus \pi_{w_{k+1,n}}^{\circ} \] \end{lemma} \begin{proof} We note that by Observation \ref{slicing}, $p_{k}$ is the \emph{only} point after $p_{k-1}$ that slices the bounding rectangle of $p_{0}, p_{1}, \dots, p_{k-1}$. Hence when $p_{k}$ is removed, $rec(p_{0}, p_{1}, \dots, p_{k-1})$ becomes a $\circ$-interval $\mathcal{I}$ in the resulting permuation, enclosing the permutation $\pi^{\circ}_{w_{1,k-1}}$. Hence, by Observation \ref{intsum}: \[ \pi_{w}^{\circ} - \{p_{k}\} = \pi_{w_{1,k-1}}^{\circ} \boxplus \tau^{\circ} \] for some centred permuation $\tau^{\circ}$. That $\tau^{\circ}$ is $\pi_{w_{k+1,n}}^{\circ}$ follows from Observation \ref{pinfactorcontainment}, as this corresponds to the remaining points. \end{proof} This process equips pin classes with an in-built structure theorem: \begin{thm}[The Pin Decomposition] \label{pindec}\ \\ Suppose that $w$ is an infinite pin sequence and that $\C_{w}^{\circ}$ is the pin class it generates. Then \[ \sigma^{\circ} \in \C_{w}^{\circ} \ \text{iff} \ \sigma^{\circ} = \pi_{w_{1}}^{\circ} \boxplus \pi_{w_{2}}^{\circ} \boxplus \dots \boxplus \pi_{w_{k}}^{\circ} \] where $w_{1}, w_{2}, \dots w_{k}$ is a sequence of pin factors of $w$ that occur \textbf{in that order, in non-overlapping instances and separated from each other by at least one letter} in $w$. \end{thm} \begin{proof} This is almost immediate from the process derived above: $\sigma^{\circ} \in \C_{w}^{\circ}$ must be contained in some centred permutation generated by an inital subsequence $w^{}$ of $w$, so we can obtain $\sigma^{\circ}$ by deleting letters from $w^{}$; but every time we do we split the resulting permutation into the $\boxplus$-sum of the pin factors directly before and after. Repeatedly doing this gives the required result. \end{proof} We note one immediate consequence of this theorem: \begin{cor}[$\boxplus$-indecomposables in a pin class] \label{pinfactorsareindecs}\ \\ Let $w$ be a pin sequence and $\C_{w}^{\circ}$ the associated pin class. Suppose $\pi^{\circ} \in \C_{w}^{\circ}$ is a $\boxplus$-indecomposable pin permutation. Then $\pi^{\circ} = \pi_{\tilde{w}}^{\circ}$ for some pin factor $\tilde{w}$ of $w$. \end{cor} The structure theorem \ref{pindec} is often awkward to apply due to the conditions on the pin factors $w_{i}$; it becomes much easier however, if we assume that $w$ is a \emph{recurrent} pin sequence - that is, every pin factor of $w$ occurs infinitely often. The theorem then becomes: \begin{thm}[The Pin Decomposition - Recurrent Case]\ \\ Suppose that $w$ is an recurrent infinite pin sequence and that $\C_{w}^{\circ}$ is the pin class it generates. Then: \[ \sigma^{\circ} \in \C_{w}^{\circ} \ \text{iff} \ \sigma^{\circ} = \pi_{w_{1}}^{\circ} \boxplus \pi_{w_{2}}^{\circ} \boxplus \dots \pi_{w_{k}}^{\circ} \] where $w_{1}, w_{2}, \dots w_{k}$ is a sequence of pin factors of $w$, and $\pi_{w_{i}}^{\circ}$ is the (centred) permutation generated from $w_{i}$. \end{thm} This has the following crucial corollary: \begin{cor}[Recurrent pin classes are $\boxplus$-closed]\label{recbox}\ \\ Suppose that $w$ is a recurrent pin sequence. Then the pin class $\mathcal{C}^{\circ}_{w}$ is $\boxplus$-closed. \end{cor} Corollary \ref{recbox} tells us that to enumerate a \emph{recurrent} pin class $\C_{w}^{\circ}$ it will suffice to enumerate its $\boxplus$-indecomposables and then apply the generating function specification given in Theorem \ref{genfuncspec}. It is thus important for us to know how to enumerate the $\boxplus$-indecomposables in a pin class; Corollary \ref{pinfactorsareindecs} suggests a way of doing this that we will study in a later section. \subsection{Recurrent Pin Classes have Growth Rates} We know that pin classes all have \emph{upper} growth rates by Proposition \ref{eqgrs}. In this section we show that \emph{recurrent} pin classes in fact have \emph{proper} growth rates. In fact, as our eventual aim is to show that \emph{all} pin classes have proper growth rates we will have to be slightly more general here and aim to prove that all $\boxplus$-closed subclasses of $\mathcal{P}^{\circ}$ (the class of all pin permutations) have proper growth rates. Our strategy in proving this will be to follow Arratia~\cite{arratia:on-the-stanley-:} who proved that $\oplus$-closed permutation classes have proper growth rates. Arratia's strategy was to note that, in a $\oplus$-closed class $\mathcal{C}$, the map \[ \left(\sigma, \tau\right) \mapsto \sigma \oplus \tau \] is an injection from $\mathcal{C}_{m} \times \mathcal{C}_{n}$ to $\mathcal{C}_{m+n}$, and so the enumeration sequence of $\mathcal{C}$ satisfies the \textbf{supermultiplicative inequality:} \[ C_{m+n} \geq C_{m}C_{n} \] Finally, Arratia applies the supermultiplicative form of Fekete's Lemma to deduce that $gr(C_{n})$ exists. We aim to apply this method to $\boxplus$-closed centred permutation classes, but we have a problem: as we have seen, the $\boxplus$-decomposition is certainly \emph{not} unique and so the map \begin{equation}\label{pairinjection} \begin{split} \mathcal{C}^{\circ}_{m} \times \mathcal{C}^{\circ}_{n} & \rightarrow \mathcal{C}^{\circ}_{m+n} \\ \left(\sigma^{\circ}, \tau^{\circ}\right) & \mapsto \sigma^{\circ} \boxplus \tau^{\circ} \end{split} \end{equation} is not necessarily an injection and we will not, in general, be able to prove the supermultiplicative inequality for a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$. We note, however, that even the weaker `supermultiplicative-like' inequality $C_{m+n} \geq kC_{m}C_{n}$ (for some constant $k<1$) would be sufficient to deduce the existence $gr(C_{n})$, on applying Fekete's Lemma to $D_{n}=kC_{n}$ instead. Unfortunately, the enumeration sequence of a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$ does not in general satisfy even this weaker form of supermultiplicativity: for example, if we take $\mathcal{C}^{\circ}$ to be the $\boxplus$-closure of $\nept$ and $\swpt$ we can easily show that $\lim_{n\rightarrow\infty}\|\mathcal{C}^{\circ}_{2n}\|/\|\mathcal{C}^{\circ}_{n}\|^{2} = 0$. The problem in this example is that there is \emph{too much commutativity} due to the class only containing points in the first and third quadrants, which are opposite to each other. As it turns out, this is really the only thing that can go wrong in proving a supermultiplicative-like identity: if we have any points from a pair of adjacent quadrants then we already have enough pairs that don't commute to conclude that $C_{m+n} \geq kC_{m}C_{n}$ for some constant $k$. We thus exclude the bad case in which we have only points from opposite quadrants via the following definition: \begin{defn}[Adjacency Condition] Let $\mu^{\circ}_{i}$ denote the centred permutation consisting of a single point in the $i$th quadrant (so $\mu^{\circ}_{1} = \nept$, etc.). We say that a centred permutation class $\mathcal{C}^{\circ}$ satisfies the \textbf{adjacency condition} if it contains either precisely one of $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}, \mu^{\circ}_{4}$ or it contains a pair $\mu^{\circ}_{i}, \mu^{\circ}_{j}$ from adjacent quadrants. \end{defn} Assuming this condition we may now prove a weak form of supermultiplicativity for the enumeration sequence of a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$: \begin{prop}\label{supermultgen} Let $\mathcal{C}^{\circ}$ be a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$ which satisfies the adjacency condition. Let $\mathcal{C}^{\circ}_{n}$ denote the set of centred permutations of length $n$ in $\mathcal{C}^{\circ}$ and write $C_{n} = \|\mathcal{C}^{\circ}_{n}\|$. Then, for all $m,n \in \mathbb{N}$: \[ C_{m+n} \geq C_{m-1}C_{n-1} \] \end{prop} \begin{proof} Note first that certainly $\mu_{i} \in \mathcal{C}^{\circ}$ for some $i$: now, if $\sigma^{\circ} \in \mathcal{C}^{\circ}_{n-1}$ then $\mu_{i} \boxplus \sigma^{\circ} \in \mathcal{C}^{\circ}_{n}$, and by left-cancellation the implied map from $\mathcal{C}^{\circ}_{n-1}$ to $\mathcal{C}^{\circ}_{n}$ is injective. Hence: \[ C_{n} \geq C_{n-1} \] We now split into cases based on how many of the one-point permutations $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}, \mu^{\circ}_{4}$ are contained in $\mathcal{C}^{\circ}$: \begin{itemize} \item \textbf{Case 1:} Suppose $\mathcal{C}^{\circ}$ contains either one or two of $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}, \mu^{\circ}_{4}$. Then, \emph{by the Adjacency Condition}, $\mathcal{C}^{\circ}$ is contained entirely within one half-plane. Hence there is no commutativity in $\mathcal{C}^{\circ}$ and so (by Theorem \ref{uniqueness}) the map \[ \begin{split} \mathcal{C}^{\circ}_{m} \times \mathcal{C}^{\circ}_{n} & \rightarrow \mathcal{C}^{\circ}_{m+n} \\ \left(\sigma^{\circ}, \tau^{\circ}\right) & \mapsto \sigma^{\circ} \boxplus \tau^{\circ} \end{split} \] is an injection, hence \[ \begin{split} C_{m+n} & \geq C_{m}C_{n} \\ & \geq C_{m-1}C_{n-1} \end{split} \] as required. \item \textbf{Case 2:} Suppose $\mathcal{C}^{\circ}$ contains precisely three of $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}, \mu^{\circ}_{4}$. Without loss of generality, assume that these are $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}$. Note that $\mu^{\circ}_{2}$ commutes with nothing in $\mathcal{C}^{\circ}$. Thus \[ \begin{split} \mathcal{C}^{\circ}_{m} \times \mathcal{C}^{\circ}_{n-1} & \rightarrow \mathcal{C}^{\circ}_{m+n} \\ \left(\sigma^{\circ}, \tau^{\circ}\right) & \mapsto \sigma^{\circ} \boxplus \mu^{\circ}_{2} \boxplus \tau^{\circ} \end{split} \] is an injection. Hence: \[ \begin{split} C_{m+n} & \geq C_{m}C_{n-1} \\ & \geq C_{m-1}C_{n-1} \end{split} \] as required. \item \textbf{Case 3:} Suppose $\mathcal{C}^{\circ}$ contains all four of $\mu^{\circ}_{1}, \mu^{\circ}_{2}, \mu^{\circ}_{3}, \mu^{\circ}_{4}$. We claim that \[ \begin{split} \mathcal{C}^{\circ}_{m-1} \times \mathcal{C}^{\circ}_{n-1} & \rightarrow \mathcal{C}^{\circ}_{m+n} \\ \left(\sigma^{\circ}, \tau^{\circ}\right) & \mapsto \sigma^{\circ} \boxplus \mu^{\circ}_{2} \boxplus \mu^{\circ}_{3} \boxplus \tau^{\circ} \end{split} \] is an injection (as nothing can commute with both of the interior permutations). Hence: \[ C_{m+n} \geq C_{m-1}C_{n-1} \] as required. \end{itemize} \end{proof} This, as it stands, is slightly weaker than a supermultiplicativity result (which should relate $C_{m+n}$ with $C_{m}C_{n}$, not $C_{m-1}C_{n-1}$), but we could easily convert it into such if we could obtain a bound on the ratio between consecutive terms in the enumeration sequence of a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$. To find such a bound we notice that all pin permutations of length $n+1$ can be obtained as `extensions' of pin permutations of length $n$, motivating the following definition: \begin{defn}[Pin Representations] Let $\sigma^{\circ}$ be a pin permutation. We call a $k$-tuple of pin words $\left(w_{1}, w_{2}, \dots, w_{k}\right)$ a \textbf{pin representation of $\sigma^{\circ}$} if \[ \sigma^{\circ} = \pi^{\circ}_{w_{1}} \boxplus \pi^{\circ}_{w_{2}} \boxplus \dots \boxplus \pi^{\circ}_{w_{k}} \] where each $w_{i}$ is a pin word. \end{defn} Clearly, every pin permutation $\sigma^{\circ}$ has a pin representation (as every $\boxplus$-indecomposable is of the form $\pi^{\circ}_{\tilde{w}}$; but note that we do not require that each $\pi^{\circ}_{w_{i}}$ is $\boxplus$-indecomposable). Note, however, that pin representations are in general highly non-unique (it can, for example, easily be seen that $\left(1, 3, 1ul\right)$, $\left(3, 1, 1ul\right)$ and $\left(1, 1uld\right)$ are all representations of the same centred permutation, namely $51\underline{2}364$). We can now formalise the notion of one pin permutation being an extension of another: \begin{defn}[One-point Extensions] Let $\sigma^{\circ}, \widetilde{\sigma}^{\circ} \in \mathcal{P}^{\circ}$ with $\|\widetilde{\sigma}^{\circ}\| = \|\sigma^{\circ}\| + 1$. We say that $\widetilde{\sigma}^{\circ}$ is a \textbf{one-point extension of $\sigma^{\circ}$} if there is some pin representation $\left(w_{1}, w_{2}, \dots, w_{k}\right)$ of $\sigma^{\circ}$ for which there exists \textbf{either:} \begin{itemize} \item some $L \in \{u,d,l,r\}$ such that the concatenation $w_{k}L$ is a valid pin word and \[ \left(w_{1}, w_{2}, \dots, w_{k}L\right) \] is a pin representation of $\widetilde{\sigma}^{\circ}$; \textbf{or:} \item some $Q \in \{1,2,3,4\}$ such that \[ \left(w_{1}, w_{2}, \dots, w_{k}, Q\right) \] is a pin representation of $\widetilde{\sigma}^{\circ}$. \end{itemize} \end{defn} (Informally, $\widetilde{\sigma}^{\circ}$ is a one-point extension of $\sigma^{\circ}$ if there is a pin representation of $\sigma^{\circ}$ which can be extended to a pin representation of $\widetilde{\sigma}^{\circ}$ by the appendment of one final symbol, either as an extension of the final pin word in the representation, or as an additional pin word which consists of a single quadrant numeral.) We note the following crucial facts about one-point extensions: \begin{lemma}\label{onepointfacts} Suppose $\mathcal{C}^{\circ}$ is a subclass of the complete pin class $\mathcal{P}^{\circ}$. Then: \begin{enumerate} \item Every $\widetilde{\sigma}^{\circ} \in \mathcal{C}^{\circ}$ of length $n+1$ is a one-point extension of some $\sigma^{\circ} \in \mathcal{C}^{\circ}$ of length $n$. \item Let $\sigma^{\circ} \in \mathcal{C}^{\circ}$. Then $\sigma^{\circ}$ has at most $12$ one-point extensions in $\mathcal{C}^{\circ}$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item Let $\widetilde{\sigma}^{\circ} \in \mathcal{C}^{\circ}$ of length $n+1$ and take a pin representation $\left(w_{1}, w_{2}, \dots, w_{k}\right)$ of $\widetilde{\sigma}^{\circ}$. Simply by deleting the final symbol of $w_{k}$ (which may be all of $w_{k}$ if $w_{k}$ is simply a quadrant numeral) we obtain a pin representation of a permutation $\sigma^{\circ}$ which must also be in $\mathcal{C}^{\circ}$ (by Observation \ref{pinfactorcontainment} and closure of $\mathcal{C}^{\circ}$ under the containment order $\leq$) and which has $\widetilde{\sigma}^{\circ}$ as a one-point extension. \item Let $\sigma^{\circ} \in \mathcal{C}^{\circ}$ and suppose that $\widetilde{\sigma}^{\circ}$ is a one-point extension of $\sigma^{\circ}$ (which may or may not be in $\mathcal{C}^{\circ}$). We claim that \emph{regardless of the pin representation chosen for} $\sigma^{\circ}$, $\widetilde{\sigma}^{\circ}$ must be obtained by adding a point in one of the $12$ positions indicated in Fig. \ref{fig:extensionpossibilities}: if we append a numeral $Q$ to the pin representation then $\widetilde{\sigma}^{\circ} = \sigma^{\circ} \boxplus \mu^{\circ}_{Q}$ and we have added a point in one of the extreme corners; if, on the other hand, we append a letter $L$ to the pin representation then we add a point which is the most extreme in the direction indicated by $L$ and the second-most extreme in another (perpendicular) direction, determined by the final symbol of the pin representation of $\sigma^{\circ}$. \end{enumerate} \end{proof} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (5,3) {}; \draw[thin] (1) -- ++ (-2.5,0); \node[permpt] (2) at (3,5) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (-3,5) {}; \draw[thin] (3) -- ++ (0,-2.5); \node[permpt] (4) at (-5,3) {}; \draw[thin] (4) -- ++ (2.5,0); \node[permpt] (5) at (-5,-3) {}; \draw[thin] (5) -- ++ (2.5,0); \node[permpt] (6) at (-3,-5) {}; \draw[thin] (6) -- ++ (0,2.5); \node[permpt] (7) at (3,-5) {}; \draw[thin] (7) -- ++ (0,2.5); \node[permpt] (8) at (5,-3) {}; \draw[thin] (8) -- ++ (-2.5,0); \node[permpt] (9) at (5,5) {}; \node[permpt] (10) at (-5,5) {}; \node[permpt] (11) at (-5,-5) {}; \node[permpt] (11) at (5,-5) {}; \draw[thick] (-7,0) -- ++ (14,0); \draw[thick] (0,-7) -- ++ (0,14); \draw[thick] (-4,-4) rectangle (4,4); \end{tikzpicture} \end{center} \caption{If $\sigma^{\circ}$ is contained in the square then \emph{any} one-point extension of $\sigma^{\circ}$ in $\mathcal{C}^{\circ}$ is formed by adding a point in one of the $12$ positions shown.} \label{fig:extensionpossibilities} \end{figure} Lemma \ref{onepointfacts} immediately implies the following: \begin{prop} \label{upratiobound} Let $\mathcal{C}^{\circ}$ be a subclass of the complete pin class $\mathcal{P}^{\circ}$. Let $\mathcal{C}^{\circ}_{n}$ denote the set of centred permutations of length $n$ in $\mathcal{C}^{\circ}$ and write $C_{n} = \|\mathcal{C}^{\circ}_{n}\|$. Then, for all $n \in \mathbb{N}$: \[ C_{n} \leq 12C_{n-1} \] \end{prop} Finally, we combine Propositions \ref{supermultgen} and \ref{upratiobound} to deduce a supermultiplicativity-like identity for $\boxplus$-closed subclasses of $\mathcal{P}^{\circ}$ satisfying the adjacency condition: \begin{cor}\label{supermultbound} Let $\mathcal{C}^{\circ}$ be a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$ which satisfies the adjacency condition and let $C_{n} = \|\mathcal{C}^{\circ}_{n}\|$. Then, for all $m,n \in \mathbb{N}$: \[ C_{m+n} \geq \frac{1}{144}C_{m}C_{n} \] \end{cor} This is finally enough to prove our main result: \begin{thm} \label{grexist} Suppose that $\mathcal{C}^{\circ}$ is a $\boxplus$-closed subclass of $\mathcal{P}^{\circ}$ which satisfies the adjacency condition. Then $\mathcal{C}^{\circ}$ has a proper growth rate. \end{thm} \begin{proof} Let $C_{n}$ be the enumeration sequence of $\mathcal{C}^{\circ}$ and set $D_{n} = \frac{1}{144}C_{n}$. Then, by Corollary \ref{supermultbound}, \[ D_{m+n} \geq D_{m}D_{n}, \] so we may apply Fekete's Lemma to deduce that $\lim_{n\rightarrow\infty}\sqrt[n]{D_{n}}$ exists and is equal to $\sup_{n\in\mathbb{N}}\sqrt[n]{D_{n}}$, which is finite by Marcus-Tardos. Hence: \[ \begin{split} \lim_{n\rightarrow\infty}\sqrt[n]{C_{n}} & = \lim_{n\rightarrow\infty}(\sqrt[n]{144})(\sqrt[n]{D_{n}}) \\ & = \lim_{n\rightarrow\infty}\sqrt[n]{D_{n}} \\ & < \infty, \end{split} \] which is to say that $gr(\mathcal{C}^{\circ})$ exists. \end{proof} As any recurrent pin class is $\boxplus$-closed and satisfies the adjacency condition, we immediately obtain: \begin{cor}[Recurrent pin classes have proper growth rates] \label{recgrs} Let $w$ be a recurrent pin word. Then the pin class $\mathcal{C}^{\circ}_{w}$ has a proper growth rate. \end{cor} Of course, Corollary \ref{recgrs} (along with Proposition \ref{eqgrs}) immediately implies that \emph{uncentred} pin classes generated by recurrent pin sequences also have growth rates, which will equal that of their centred counterparts: $gr(\mathcal{C}_{w}) = gr(\mathcal{C}^{\circ}_{w})$. \subsection{Classification of Collisions and $\boxplus$-decomposables} We now know that any recurrent pin class $\mathcal{C}^{\circ}_{w}$ has a proper growth rate. In order to find the growth rate of a given recurrent pin class in practice we shall have to enumerate the $\boxplus$-indecomposables contained in $\mathcal{C}^{\circ}_{w}$: if we can do this then Theorem (\ref{genfuncspec}) will allow us to write down the generating function of the class, from which we can deduce the growth rate via the Exponential Growth Theorem. Accordingly, in this section we take up the problem of enumerating the $\boxplus$-indecomposables in a pin class. We begin by noting that Corollary \ref{pinfactorsareindecs} immediately implies the following property of the restriction of the $\pi$-map to pin factors of length $n$: \begin{prop}[Induced $\pi$-map contains all $\boxplus$-indecomposables in its image] Let $w$ be a pin sequence and $\mathcal{C}^{\circ}_{w}$ the associated (centred) pin class. Let $P_{n}$ denote the set of pin factors of $w$ of length $n$ and $\mathcal{B}^{\circ}_{n}$ denote the set of $\boxplus$-indecomposables in $\mathcal{C}^{\circ}_{w}$ of length $n$. Then the induced map \[ \pi_{n}: P_{n} \rightarrow \mathcal{C}^{\circ}_{w} \] defined by \[ \pi_{n}: \tilde{w} \mapsto \pi^{\circ}_{\tilde{w}} \] contains the set $\mathcal{B}^{\circ}_{n}$ in its image. \end{prop} The study of this induced map will be vital for further progress as it suggests a natural method for counting the $\boxplus$-indecomposables in a given pin class $\mathcal{C}^{\circ}_{w}$: if (for some $n$) we had the following two properties: \begin{itemize} \item the image $\pi_{n}(P_{n})$ is equal to $\mathcal{B}^{\circ}_{n}$ (that is, $\pi_{n}(\tilde{w})$ is $\boxplus$-indecomposable for all pin factors $\tilde{w}$ of $w$); \item $\pi_{n}$ is injective \end{itemize} then $\pi_{n}$ would be a bijection between $P_{n}$ and $\mathcal{B}^{\circ}_{n}$, thus reducing the problem of enumerating the $\boxplus$-indecomposables of $\mathcal{C}^{\circ}_{w}$ to the (much simpler) combinatorial problem of counting the pin factors of $w$. We do in fact have some intuitive reasons for suspecting that these two properties should hold, at least for sufficently large $n$: the definition of the $\pi$-map (specifically, the placing of each new point in order to intersect the bounding rectangle of all previous points) seems almost designed to ensure that $\pi_{\tilde{w}}$ can have no proper non-trivial $\circ$-interval, and pin permutations have so much internal structure that it seems unlikely that two distinct pin sequences could conspire to generate the exact same centred permutation. Alas, both turn out not to be true for any $n$: for every length $n \geq 2$ there are pin sequences which generate $\boxplus$-decomposable permutations, as well as pairs of pin sequences which generate the same (centred) permutation. Fortunately, as our intuition suggests, both of these 'bad behaviours' are somewhat pathological and easily classified, so with some minor adjustments we will be able to make our strategy for counting $\boxplus$-indecomposables in a pin class work. We begin by introducing some terminology for these pathological behaviours. First, those pin words whose image under the induced $\pi$-map is not contained in $\mathcal{B}^{\circ}_{n}$: \begin{defn}[$\boxplus$-decomposable pin words] We call a (finite) pin word $w$ a \textbf{$\boxplus$-decomposable pin word} if the centred permution it generates, $\pi^{\circ}_{w}$, is $\boxplus$-decomposable. \end{defn} And those pin words which make injectivity of the induced $\pi$-map fail: \begin{defn}[Collisions] We call a set $\{w_{1}, w_{2}, \dots , w_{k}\}$ of pin words which generate the same centred permutation $\pi^{\circ}$ a \textbf{collision} (sometimes a $k$-\textbf{collision} when we wish to emphasise the size of the set) of pin words. \end{defn} It is fairly easy to find all $\boxplus$-decomposables and collisions of length $n\leq5$ through an exhaustive search, especially on applying symmetries of the square. This search reveals families of $\boxplus$-decomposables and collisions which generalise to all longer lengths. We can then, with considerable effort, prove that \emph{all} examples of lengths greater than $6$ belong to these families, giving us a full classification of these confounding behaviours. This is the content of the following two theorems: the statements will be crucial and will be quoted in full here; the proofs are technical and long and will be relegated to the appendix. \newpage \begin{thm}[Classification of $\boxplus$-decomposables and Collisions] \label{classification} The following two tables form a complete list of $\boxplus$-decomposable pin words and collisions: \end{thm} \begin{tabular}{@{}\|>{\centering}m{2em}\|>{\centering}m{4em}\|>{\centering\arraybackslash}m{10em}\|>{\centering}m{5em}\|>{\centering\arraybackslash}m{6.5em}\|>{\centering\arraybackslash}m{6.5em}\|} \hline \multicolumn{6}{\|c\|}{}\\[0.5pt] \multicolumn{6}{\|c\|}{\textbf{List of $\boxplus$-decomposable pin words:}}\\[6pt] \hline \multicolumn{2}{\|c\|}{\textbf{Length}}&\textbf{Representative}&\textbf{Total number}&\multicolumn{2}{\|c\|}{\textbf{Full List}}\\ \hline \multicolumn{2}{\|c\|}{$2$} & \begin{tikzpicture}[scale=0.25] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (2,1) {}; \node[permpt] (1) at (3,3) {}; \node[permpt] (2) at (1,2) {}; \draw[thin] (2) -- ++ (2.5,0); \node[permpt,white] (3) at (2,5) {}; \draw[thick] (0,1) -- ++ (4,0); \draw[thick] (2,0) -- ++ (0,4); \node[] (4) at (2,-1) {$1l$}; \draw[dashed] (0.5,0.5) rectangle (2.5,2.5); \end{tikzpicture} & 8 & \multicolumn{2}{\|c\|}{$1l, 1d, 2r, 2d, 3u, 3r, 4l, 4u$} \\ \hline \multicolumn{2}{\|c\|}{$3$} & \begin{tikzpicture}[scale=0.25] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (3,2) {}; \node[permpt] (1) at (4,4) {}; \node[permpt] (2) at (1,3) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (2,1) {}; \draw[thin] (3) -- ++ (0,2.5); \node[permpt,white] (4) at (3,6) {}; \draw[thick] (0,2) -- ++ (5,0); \draw[thick] (3,0) -- ++ (0,5); \node[] (4) at (3,-1) {$1ld$}; \draw[dashed] (1.5,0.5) rectangle (3.5,2.5); \end{tikzpicture} & 8 & \multicolumn{2}{\|c\|}{$1ld, 1dl, 2dr, 2rd, 3ru, 3ur, 4ul, 4lu$} \\ \hline \multirow{2}{}{$n\geq4$} & \textbf{Type 1:} & \begin{tikzpicture}[scale=0.25] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (3,2) {}; \node[permpt] (1) at (6,6) {}; \draw[thin] (1) -- ++ (0,-1.5); \node[permpt] (2) at (8,5) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt] (3) at (7,8) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (1,7) {}; \draw[thin] (4) -- ++ (6.5,0); \node[permpt] (5) at (2,1) {}; \draw[thin] (5) -- ++ (0,6.5); \begin{scope}[shift={(0.5,0.5)}] \node[circle,fill,inner sep=0.5pt] (6) at (3.5,2.5) {}; \node[circle,fill,inner sep=0.5pt] (7) at (4,3) {}; \node[circle,fill,inner sep=0.5pt] (8) at (4.5,3.5) {}; \node[circle,fill,inner sep=0.5pt] (8) at (5,4) {}; \end{scope} \node[empty] (-1) at (3,10) {}; \draw[thick] (0,2) -- ++ (9,0); \draw[thick] (3,0) -- ++ (0,9); \node[] (4) at (4,-1.5) {$1(ur)^{k}uld$ \text{or} $1(ru)^{k}ld$}; \draw[dashed] (1.5,0.5) rectangle (3.5,2.5); \end{tikzpicture} & 8 & \textbf{$n \geq 4$ even:} \[ \begin{split} 1(ur)^{k}uld, \\ 1(ru)^{k}rdl, \\ 2(ul)^{k}urd, \\ 2(lu)^{k}ldr, \\ 3(dl)^{k}dru, \\ 3(ld)^{k}lur, \\ 4(dr)^{k}dlu, \\ 4(rd)^{k}rul \end{split} \] & \textbf{$n \geq 5$ odd:} \[ \begin{split} 1(ru)^{k}ld, \\ 1(ur)^{k}dl, \\ 2(lu)^{k}rd, \\ 2(ul)^{k}dr, \\ 3(ld)^{k}ru, \\ 3(dl)^{k}ur, \\ 4(rd)^{k}lu, \\ 4(dr)^{k}ul \end{split} \] \\ \cline{2-6} & \textbf{Type 2:} & \begin{tikzpicture}[scale=0.25] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,4) {}; \node[permpt] (1) at (6,6) {}; \node[permpt] (2) at (3,5) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (4,2) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (1,3) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (2,1) {}; \draw[thin] (5) -- ++ (0,2.5); \begin{scope}[shift={(-4,-2.25)}] \node[circle,fill,inner sep=0.5pt] (6) at (3.5,2.5) {}; \node[circle,fill,inner sep=0.5pt] (7) at (4,3) {}; \node[circle,fill,inner sep=0.5pt] (8) at (4.5,3.5) {}; \node[circle,fill,inner sep=0.5pt] (8) at (5,4) {}; \end{scope} \node[empty] (-1) at (5,8) {}; \draw[thick] (-2,4) -- ++ (9,0); \draw[thick] (5,-2) -- ++ (0,9); \node[] (4) at (3,-3.5) {$1l(dl)^{k}$ \text{or} $1l(dl)^{k}d$}; \draw[dashed] (-1,-0.25) rectangle (5.5,5.5); \end{tikzpicture} & 8 & \textbf{$n \geq 4$ even:} \[ \begin{split} 1l(dl)^{k}, \\ 1d(ld)^{k}, \\ 2r(dr)^{k}, \\ 2d(rd)^{k}, \\ 3r(ur)^{k}, \\ 3u(ru)^{k}, \\ 4l(ul)^{k}, \\ 4u(lu)^{k} \end{split} \] & \textbf{$n \geq 5$ odd:} \[ \begin{split} 1(ld)^{k}, \\ 1(dl)^{k}, \\ 2(rd)^{k}, \\ 2(dr)^{k}, \\ 3(ru)^{k}, \\ 3(ur)^{k}, \\ 4(lu)^{k}, \\ 4(ul)^{k} \\ \end{split} \] \\ \hline \end{tabular} \newpage { \centering \begin{tabular}{@{}\|>{\centering}m{1.5em}\|>{\centering}m{5.5em}\|>{\centering\arraybackslash}m{12em}\|>{\centering}m{4em}\|>{\centering\arraybackslash}m{17.5em}\|} \hline \multicolumn{5}{\|c\|}{}\\[0.4pt] \multicolumn{5}{\|c\|}{\textbf{List of collisions of pin factors:}}\\[6pt] \hline \multicolumn{2}{\|c\|}{\textbf{Length}}&\textbf{Representative}&\textbf{Total number}&\textbf{Full List}\\ \hline \multicolumn{2}{\|c\|}{$2$} & \begin{tikzpicture}[scale=0.2] \begin{scope}[shift={(10,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (1,2) {}; \node[permpt] (2) at (2,1) {}; \draw[thin] (2) -- ++ (-1.5,0); \node[empty] (-1) at (0,4) {}; \draw[thick] (-3,0) -- ++ (6,0); \draw[thick] (0,-3) -- ++ (0,6); \node[] (4) at (0,-5) {$1r$}; \end{scope} \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (0,0) {}; \node[permpt] (1) at (2,1) {}; \node[permpt] (2) at (1,2) {}; \draw[thin] (2) -- ++ (0,-1.5); \draw[thick] (-3,0) -- ++ (6,0); \draw[thick] (0,-3) -- ++ (0,6); \node[] (3) at (5,0) {=}; \node[] (4) at (0,-5) {$1u$}; \end{tikzpicture} & 4 \ \text{pairs} & \[ \{1u, 1r\}, \{2l, 2u\}, \{3d, 3l\}, \{4r, 4d\} \] \\ \hline \multicolumn{2}{\|c\|}{$3$} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (1,0) {}; \node[permpt] (1) at (3,1) {}; \node[permpt] (2) at (2,3) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (0,2) {}; \draw[thin] (3) -- ++ (2.5,0); \draw[thick] (-2,0) -- ++ (6,0); \draw[thick] (1,-3) -- ++ (0,7); \node[] (3) at (6,0) {=}; \node[] (4) at (1,-4) {$1ul$}; \begin{scope}[shift={(10,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (1,0) {}; \node[permpt] (1) at (0,2) {}; \node[permpt] (2) at (3,1) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[permpt] (3) at (2,3) {}; \draw[thin] (3) -- ++ (0,-2.5); \node[empty] (-1) at (1,5) {}; \draw[thick] (-2,0) -- ++ (6,0); \draw[thick] (1,-3) -- ++ (0,7); \node[] (4) at (1,-4) {$2ru$}; \end{scope} \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1ul, 2ru\}, \{1rd, 4ur\}, \{2ur, 1lu\}, \{2ld, 3ul\},\\ \{3lu, 2dl\}, \{3dr, 4ld\}, \{4ru, 1dr\}, \{4dl, 3rd\} \end{split} \] \\ \hline \multicolumn{2}{\|c\|}{$4$} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (3,3) {}; \node[permpt] (1) at (4,5) {}; \draw[thin] (1) -- ++ (0,-3.5); \node[permpt] (2) at (1,4) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (2,1) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (5,2) {}; \draw[thin] (4) -- ++ (-3.5,0); \node[empty] (4) at (3,7) {}; \draw[thick] (0,3) -- ++ (6,0); \draw[thick] (3,0) -- ++ (0,6); \node[] (4) at (3,-1) {$1ldr = 2dru = 3rul = 4uld$}; \end{tikzpicture} & 2 \ \text{quad.s} & \[ \begin{split} \{1ldr, 2dru, 3rul, 4uld\}, \\ \{1dlu, 2rdl, 3urd, 4lur\} \end{split} \] \\ \hline \multirow{2}{}{$5$} & \textbf{Pathological:} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,3) {}; \node[permpt] (1) at (6,4) {}; \node[permpt] (2) at (5,6) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (2,5) {}; \draw[thin] (3) -- ++ (3.5,0); \node[permpt] (4) at (3,1) {}; \draw[thin] (4) -- ++ (0,4.5); \node[permpt] (5) at (1,2) {}; \draw[thin] (5) -- ++ (2.5,0); \node[empty] (-1) at (4,8) {}; \draw[thick] (0,3) -- ++ (8,0); \draw[thick] (4,0) -- ++ (0,7); \node[] (3) at (10,3) {=}; \node[] (4) at (4,-1) {$1uldl$}; \begin{scope}[shift={(12,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,3) {}; \node[permpt] (1) at (6,4) {}; \draw[thin] (1) -- ++ (-4.5,0); \node[permpt] (2) at (5,6) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (2,5) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (3,1) {}; \node[permpt] (5) at (1,2) {}; \draw[thin] (5) -- ++ (2.5,0); \draw[thick] (0,3) -- ++ (8,0); \draw[thick] (4,0) -- ++ (0,7); \node[] (4) at (4,-1) {$3luru$}; \end{scope} \end{tikzpicture} & 4 \ \text{pairs} & \[ \begin{split} \{1uldl, 3luru\}, \{1rdld, 3drur\}, \\ \{2urdr, 4rulu\}, \{2ldrd, 4dlul\} \end{split} \] \\ \cline{2-5} & \textbf{Regular:} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,3) {}; \node[permpt] (1) at (6,5) {}; \node[permpt] (2) at (3,4) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (4,1) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (1,2) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (2,6) {}; \draw[thin] (5) -- ++ (0,-4.5); \draw[thick] (0,3) -- ++ (8,0); \draw[thick] (5,0) -- ++ (0,7); \node[] (3) at (10,3) {=}; \node[] (4) at (5,-1) {$1ldlu$}; \begin{scope}[shift={(12,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,3) {}; \node[permpt] (1) at (6,5) {}; \draw[thin] (1) -- ++ (-4.5,0); \node[permpt] (2) at (3,4) {}; \node[permpt] (3) at (4,1) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (1,2) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (2,6) {}; \draw[thin] (5) -- ++ (0,-4.5); \draw[thick] (0,3) -- ++ (8,0); \draw[thick] (5,0) -- ++ (0,7); \node[] (4) at (5,-1) {$2dlur$}; \end{scope} \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1ldlu, 2dlur\}, \{1dldr, 4ldru\}, \\ \{2rdru, 1drul\}, \{2drdl, 3rdlu\}, \\ \{3urul, 2ruld\}, \{3rurd, 4urdl\}, \\ \{4luld, 3uldr\}, \{4ulur, 1lurd\} \end{split} \] \\ \hline \multicolumn{2}{\|c\|}{$n \geq 6$ \ \textbf{even:}} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,8) {}; \node[permpt] (1) at (9,10) {}; \node[permpt] (2) at (6,9) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (7,6) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (4,7) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (5,5) {}; \draw[thin] (5) -- ++ (0,2.5); \node[permpt] (6) at (1,3) {}; \draw[thin] (6) -- ++ (1.5,0); \node[permpt] (7) at (2,1) {}; \draw[thin] (7) -- ++ (0,2.5); \node[permpt] (8) at (11,2) {}; \draw[thin] (8) -- ++ (-9.5,0); \node[permpt] (9) at (10,11) {}; \draw[thin,dashed] (9) -- ++ (0,-9.5); \node[empty] (-1) at (8,13) {}; \node[circle,fill,inner sep=0.5pt] (10) at (4.5,5.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4,5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (3.5,4.5) {}; \node[circle,fill,inner sep=0.5pt] (13) at (3,4) {}; \node[circle,fill,inner sep=0.5pt] (14) at (2.5,3.5) {}; \draw[thick] (8,0) -- ++ (0,12); \draw[thick] (0,8) -- ++ (12,0); \draw[dotted] (9.5,10.5) rectangle (10.5,11.5); \node[] (15) at (6.5,-1) {$1(ld)^{k}r=2(dl)^{k}dru$}; \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1(ld)^{k}r, 2(dl)^{k}dru\}, \{1(dl)^{k}u, 4(ld)^{k}lur\}, \\ \{2(dr)^{k}u, 3(rd)^{k}rul\}, \{2(rd)^{k}l, 1(dr)^{k}dlu\}, \\ \{3(ru)^{k}l, 4(ur)^{k}uld\}, \{3(ur)^{k}d, 2(ru)^{k}rdl\}, \\ \{4(ul)^{k}d, 1(lu)^{k}ldr\}, \{4(lu)^{k}r, 3(ul)^{k}urd\} \end{split} \] \\ \hline \multicolumn{2}{\|c\|}{$n \geq 7$ \ \textbf{odd:}} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,6) {}; \node[permpt] (1) at (9,8) {}; \node[permpt] (2) at (6,7) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (7,4) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (5,5) {}; \draw[thin] (4) -- ++ (2.5,0); \node[permpt] (5) at (3,1) {}; \draw[thin] (5) -- ++ (0,1.5); \node[permpt] (6) at (1,2) {}; \draw[thin] (6) -- ++ (2.5,0); \node[permpt] (7) at (2,10) {}; \draw[thin] (7) -- ++ (0,-8.5); \node[permpt] (8) at (10,9) {}; \draw[thin,dashed] (8) -- ++ (-8.5,0); \node[empty] (-1) at (8,12) {}; \node[circle,fill,inner sep=0.5pt] (8) at (5.5,4.5) {}; \node[circle,fill,inner sep=0.5pt] (9) at (5,4) {}; \node[circle,fill,inner sep=0.5pt] (10) at (4.5,3.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4,3) {}; \node[circle,fill,inner sep=0.5pt] (12) at (3.5,2.5) {}; \draw[thick] (8,0) -- ++ (0,11); \draw[thick] (0,6) -- ++ (11,0); \draw[dotted] (9.5,8.5) rectangle (10.5,9.5); \node[] (13) at (6,-1) {$1(ld)^{k}lu=2(dl)^{k}ur$}; \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1(ld)^{k}lu, 2(dl)^{k}ur\}, \{1(dl)^{k}dr, 4(ld)^{k}ru\}, \\ \{2(dr)^{k}dl, 3(rd)^{k}lu\}, \{2(rd)^{k}ru, 1(dr)^{k}ul\}, \\ \{3(ru)^{k}rd, 4(ur)^{k}dl\}, \{3(ur)^{k}ul, 2(ru)^{k}ld\}, \\ \{4(ul)^{k}ur, 1(lu)^{k}rd\}, \{4(lu)^{k}ld, 3(ul)^{k}dr\} \end{split} \] \\ \hline \end{tabular}\par } Note: in the previous table we write $(f)^{k}$ to denote the factor $f$ repeating $k$ times for some $k \geq 0$. \begin{obs} We emphasise some noteworthy features of these lists: \begin{enumerate} \item There is no overlap between the two lists: no pin word which generates a $\boxplus$-decomposable is also involved in a collision. \item All collisions are pairs except for the two quadruples at length $4$. \item There are two distinct families of collisions at length $5$: the \textbf{regular} length $5$ collisions, which generalise to a family of collisions for all $n \geq 6$; and the \textbf{pathological} length $5$ collisions, which are a family unique to length $5$. \item All even collisions of length $n \geq 4$ require all four quadrants, whereas the odd collisions of length $n \geq 5$ require only three. \end{enumerate} \end{obs} \subsection{Enumeration of Recurrent Pin Classes} We now have a general strategy for obtaining the generating function of the $\boxplus$-closure of a pin class: \begin{procedure}[$\boxplus\left(\mathcal{C}^{\circ}_{w}\right)$ Enumeration Procedure]\label{recproc} Suppose that $w$ is a pin word. We can find the generating function of the centred pin class $\boxplus\left(\mathcal{C}^{\circ}_{w}\right)$ by the following procedure: \begin{enumerate} \item Find the generating function $h(z)$ of the pin factors of $w$; \item Enumerate the pin factors of $w$ that generate $\boxplus$-decomposable pin permutations, as well as colliding sets of pin factors of $w$, by comparison with the lists in Theorem \ref{classification}. Subtract these from $h(z)$ to obtain $g(z)$, the generating function of $\boxplus$-indecomposables in $\mathcal{C}^{\circ}_{w}$; \item Amend for commutativity: find the generating function $g_{i}(z)$ of the one-quadrant $\boxplus$-indecomposables of $\mathcal{C}^{\circ}_{w}$ in the $i$th quadrant, for each $i \in \{1,2,3,4\}$. (In other words: enumerate the one-quadrant oscillations.) Set: \[ G(z) = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z), \] this is the amended $G$-sequence of $\mathcal{C}^{\circ}_{w}$; \item Then the generating function of $\boxplus\left(\mathcal{C}^{\circ}_{w}\right)$ is given by \[ f(z) = \frac{1}{1 - G(z)} \] and the growth rate of $\boxplus\left(\mathcal{C}^{\circ}_{w}\right)$ is the reciprocal of the radius of convergence of $f(z)$. \end{enumerate} Of course, if $w$ is recurrent then $\mathcal{C}^{\circ}_{w}$ is $\boxplus$-closed and so $f(z)$ is in fact the generating function of $\mathcal{C}^{\circ}_{w}$. \end{procedure} We illustrate this procedure with a sequence of examples, beginning with the class of increasing oscillations $\mathcal{O}^{\circ}$, already defined in Fig. \ref{fig:oscclass}: \begin{example}[The Increasing Oscillations] The growth rate of $\mathcal{O}^{\circ}$ is $\kappa \approx 2.20557$, defined to be the reciprocal of the smallest positive real root of \[ 1 - 2z - z^3 = 0 \] \end{example} \begin{proof} The defining pin sequence of $\mathcal{O}^{\circ}$ is $w=1(ru)^{}$, which has two pin factors of every length $\geq 2$ (corresponding to the two distinct starting positions within the period) and one pin factor (namely $1$ itself) of length $1$, so \[ h(z) = z + 2z^2 + 2z^3 + 2z^4 + 2z^5 \dots \] By comparison with the lists in Theorem \ref{classification}, there are no $\boxplus$-decomposables amongst the pin factors of $w$ and precisely one colliding pair, namely $\{1u,1r\}$, at length $2$. We thus subtract $1$ at length $2$ to obtain the generating function of the $\boxplus$-indecomposables in $\mathcal{O}^{\circ}$: \[ \begin{split} g(z) & = z + z^2 + 2z^3 + 2z^4 + 2z^5 + \dots \\ & = \frac{z+z^3}{1-z} \end{split} \] Noting that $g_{2}(z) = g_{3}(z) = g_{4}(z) = 0$ we see that $G(z)=g(z)$ and the generating function of $\mathcal{O}^{\circ}$ is \[ \begin{split} f(z) & = \frac{1}{1 - g(z)} \\ & = \frac{1-z}{1 - 2z - z^3} \end{split} \] and we may now deduce the growth rate by the Exponential Growth Theorem \ref{EGT}. \end{proof} We shall see later that $\kappa$ is in fact the smallest possible growth rate of a pin class. For now we give three more classes as examples: \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \draw[very thick] (9,2) -- ++ (0,16); \draw[very thick] (0,2) -- ++ (17,0); \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (9,2) {}; \node[permpt] (3) at (7,3) {}; \node[permpt] (4) at (8,6) {}; \draw[thin] (4) -- ++ (0,-3.5); \node[permpt] (5) at (13,5) {}; \draw[thin] (5) -- ++ (-5.5,0); \node[permpt] (6) at (12,8) {}; \draw[thin] (6) -- ++ (0,-3.5); \node[permpt] (7) at (5,7) {}; \draw[thin] (7) -- ++ (7.5,0); \node[permpt] (8) at (6,10) {}; \draw[thin] (8) -- ++ (0,-3.5); \node[permpt] (9) at (15,9) {}; \draw[thin] (9) -- ++ (-9.5,0); \node[permpt] (10) at (14,12) {}; \draw[thin] (10) -- ++ (0,-3.5); \node[permpt] (11) at (3,11) {}; \draw[thin] (11) -- ++ (11.5,0); \node[permpt] (12) at (4,14) {}; \draw[thin] (12) -- ++ (0,-3.5); \node[permpt] (13) at (17,13) {}; \draw[thin] (13) -- ++ (-13.5,0); \node[permpt] (14) at (16,16) {}; \draw[thin] (14) -- ++ (0,-3.5); \node[permpt] (15) at (1,15) {}; \draw[thin] (15) -- ++ (15.5,0); \node[permpt] (16) at (2,17) {}; \draw[thin] (16) -- ++ (0,-2.5); \end{tikzpicture} \end{center} \caption{The pin class $\mathcal{V}^{\circ}$, defined by the pin sequence $w = 2(urul)^{}$} \label{fig:2pinclassV} \end{figure} \begin{example}[Pin Class $\mathcal{V}$] The pin class $\mathcal{V}$ is the underlying (uncentred) permutation class of the centred class $\mathcal{V}^{\circ} = \mathcal{C}^{\circ}_{w}$, where $w = 2(urul)^{}$ - see Fig. \ref{fig:2pinclassV}. Writing this pin sequence out with letters subscripted by the corresponding quadrant number \[ w = 2u_{2}r_{1}u_{1}l_{2}u_{2}r_{1}u_{1}l_{2}u_{2}r_{1}u_{1}l_{2}u_{2}r_{1}u_{1}l_{2} \dots \] we note immediately that $w$ has $2$ pin factors of length $1$ and $4$ of every greater length. By comparison with our list of $\boxplus$-decomposables and collisions, we note that we have to remove $2$ from this count for lengths $2$ and $3$ (due to the $\boxplus$-decomposables $1l$ and $2r$ at length $2$ and the collisions $\{1ul,2ru\}$ and $\{2ru,1lu\}$ at length $3$) to obtain the generating function of the $\boxplus$-indecomposables of $\mathcal{V}^{\circ}$: \[ \begin{split} g(z) & = 2z + 2z^2 + 2z^3 + 4z^4 + 4z^5 + 4z^6 + 4z^7 + \dots \\ & = \frac{2z + 2z^4}{1-z} \end{split} \] As we are in the upper half-plane, $g_{3}(z)=g_{4}(z)=0$ so the amended $G$-sequence is just $g(z)$ itself, and the generating function of $\mathcal{V}^{\circ}$ is given by \[ \begin{split} f(z) & = \frac{1}{1 - g(z)} \\ & = \frac{1-z}{1 - 3z - 2z^4} \end{split} \] We deduce that $gr(\mathcal{V})$, which we shall call $\nu$, is the reciprocal of the smallest real root of \[ 1 - 3z - 2z^4 = 0; \] explicitly: \[ \nu = gr(\mathcal{V}) = gr(\mathcal{V}^{\circ}) \approx 3.06918 \] \end{example} Next, a class in three quadrants which will force us to confront commutativity: \begin{example}[The class $\mathcal{Y}$] Let $w$ be the pin sequence $1(uldlur)^{}$. We refer to $\mathcal{C}_{w}$, the associated pin class, as $\mathcal{Y}$ - see Fig. (\ref{fig:Y}). Then the growth rate of $\mathcal{Y}$ is $\gamma$, the reciprocal of the smallest positive real root of \begin{equation} 1 - 4z + 2z^2 + z^3 - z^4 - 2z^5 - 3z^6 = 0. \end{equation} Explicitly, $\gamma \approx 3.36637$. \end{example} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (9,5) {}; \node[permpt] (1) at (11,6) {}; \node[permpt] (2) at (10,8) {}; \draw[thin] (2) -- ++ (0,-2.5); \node[permpt] (3) at (7,7) {}; \draw[thin] (3) -- ++ (3.5,0); \node[permpt] (4) at (8,3) {}; \draw[thin] (4) -- ++ (0,4.5); \node[permpt] (5) at (5,4) {}; \draw[thin] (5) -- ++ (3.5,0); \node[permpt] (6) at (6,10) {}; \draw[thin] (6) -- ++ (0,-6.5); \node[permpt] (7) at (13,9) {}; \draw[thin] (7) -- ++ (-7.5,0); \node[permpt] (8) at (12,12) {}; \draw[thin] (8) -- ++ (0,-3.5); \node[permpt] (9) at (3,11) {}; \draw[thin] (9) -- ++ (9.5,0); \node[permpt] (10) at (4,1) {}; \draw[thin] (10) -- ++ (0,10.5); \node[permpt] (11) at (1,2) {}; \draw[thin] (11) -- ++ (3.5,0); \node[permpt] (12) at (2,13) {}; \draw[thin] (12) -- ++ (0,-11.5); \draw[thick] (9,0) -- ++ (0,14); \draw[thick] (0,5) -- ++ (14,0); \end{tikzpicture} \end{center} \caption{The pin class $\mathcal{Y}^{\circ}$, generated by the pin sequence $w = 1(uldlur)^{}$.} \label{fig:Y} \end{figure} \begin{proof} We shall find the generating function of the corresponding centred class $\mathcal{Y}^{\circ} = \mathcal{C}^{\circ}_{w}$. We note that $w$, being periodic, is recurrent and we can thus use Procedure \ref{recproc}. First, we enumerate the pin factors of $w=1(uldlur)^{}$; this will be easier if we first write out $w$ with letters subscripted by the corresponding quadrant number: \[ w = 1u_{1}l_{2}d_{3}l_{3}u_{2}r_{1}u_{1}l_{2}d_{3}l_{3}u_{2}r_{1}u_{1}l_{2}d_{3}l_{3}u_{2}r_{1}u_{1}l_{2}d_{3}l_{3}u_{2}r_{1}\dots \] Clearly there are three pin factors of length $1$, and for $n \geq 2$ the six distinct starting points in the period give six pin factors of length $n$. Hence the generating function of the pin factors of $w$ is \[ h(z) = 3z + 6z^2 + 6z^3 + 6z^4 + 6z^5 + 6z^6 + 6z^7 + \dots \] We now list the collisions and $\boxplus$-decomposables found amongst the pin factors of $w$ by comparison with the lists given in Theorem \ref{classification}: \[ \begin{split} \textbf{Collisions:} \ & \{1ul, 2ru\}, \{2dl, 3lu\} \ (\text{length} \ 3)\\ & \{1uldl, 3luru\}, \{1ldlu, 2dlur\}, \{3urul, 2ruld\} \ (\text{length} \ 5) \\ \textbf{$\boxplus$-dec.s:} \ & 1l, 2d, 3u, 2r \ (\text{length} \ 2) \\ & 1ld, 3ur \ (\text{length} \ 3) \\ & 1uld, 1ldl, 3lur, 3uru \ (\text{length} \ 4) \end{split} \] Hence we subtract these from the generating function of pin factors to obtain the generating function of $\boxplus$-indecomposables of $\mathcal{Y}^{\circ}$: \[ \begin{split} g(z) & = h(z) - (2z^3 + 3z^5) - (4z^2 + 2z^3 + 4z^4) \\ & = 3z + 2z^2 + 2z^3 + 2z^4 + 3z^5 + 6z^6 + 6z^7 + 6z^8 + 6z^9 + \dots \\ \end{split} \] Next we must enumerate the one-quadrant $\boxplus$-indecomposables in each quadrant; this is easily-done by looking at the diagram: $g_{1}(z) = g_{3}(z) = z + z^2$, $g_{2}(z)=z$ and $g_{4}(z)=0$. Hence we can calculate the amended $G$-sequence: \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & = 3z + z^2 + z^4 + 3z^5 + 6z^6 + 6z^7 + 6z^8 + 6z^9 + \dots \\ & = \frac{3z - 2z^2 - z^3 + z^4 + 2z^5 + 3z^6}{1-z} \end{split} \] FInally, by the Generating Function Specification \ref{genfuncspec}, the generating function of $\mathcal{Y}^{\circ}$ is given by: \[ \begin{split} f(z) & = \frac{1}{1 - G(z)} \\ & = \frac{1-z}{1-4z+2z^{2}+z^{3}-z^{4}-2z^{5}-3z^{6}} \end{split} \] This is a rational function, whose singularities are precisely the roots of \[ 1-4z+2z^{2}+z^{3}-z^{4}-2z^{5}-3z^{6} = 0 \] By computation, the smallest root of this equation is positive and real and has reciprocal $\gamma \approx 3.36637$, which by the Exponential Growth Theorem is therefore the growth rate of $\mathcal{Y}$.\end{proof} And finally, a four-quadrant pin class that has in fact been studied before in the literature: the Widdershins Spiral, $\mathcal{W}^{\circ}$, generated by the pin sequence $w = 1(ldru)^{}$ (see Fig. \ref{fig:widdershins}). This class was first introduced by Murphy~\cite{murphy:restricted-perm:}, who gave the generating function of the uncentred class by an explicit enumeration. Using our specification we can now give an alternative derivation of its growth rate: \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (7,7) {}; \node[permpt] (1) at (8,9) {}; \node[permpt] (2) at (5,8) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (6,5) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (10,6) {}; \draw[thin] (4) -- ++ (-4.5,0); \node[permpt] (5) at (9,11) {}; \draw[thin] (5) -- ++ (0,-5.5); \node[permpt] (6) at (3,10) {}; \draw[thin] (6) -- ++ (6.5,0); \node[permpt] (7) at (4,3) {}; \draw[thin] (7) -- ++ (0,7.5); \node[permpt] (8) at (12,4) {}; \draw[thin] (8) -- ++ (-8.5,0); \node[permpt] (9) at (11,13) {}; \draw[thin] (9) -- ++ (0,-9.5); \node[permpt] (10) at (1,12) {}; \draw[thin] (10) -- ++ (10.5,0); \node[permpt] (11) at (2,1) {}; \draw[thin] (11) -- ++ (0,11.5); \node[permpt] (12) at (14,2) {}; \draw[thin] (12) -- ++ (-12.5,0); \node[permpt] (13) at (13,14) {}; \draw[thin] (13) -- ++ (0,-12.5); \draw[thick] (0,7) -- ++ (15,0); \draw[thick] (7,0) -- ++ (0,15); \end{tikzpicture} \end{center} \caption{The Widdershins Spiral: this is the smallest pin class in four quadrants.} \label{fig:widdershins} \end{figure} \begin{example}[The Widdershins Spiral] \ \\ Let $\mathcal{W} = \mathcal{C}_{w}$ be the pin class generated by the pin sequence $w = 1(ldru)^{}$. Then the generating function of the corresponding centred class $\mathcal{W}^{\circ}$ is given by: \begin{equation} f(z) = \frac{1-z}{1 - 5z + 6z^2 - 2z^3 - z^4 - 3z^5} \end{equation} Hence $\omega_{0} = gr(\mathcal{W}^{\circ}) \approx 3.48806$. \end{example} \begin{proof} The defining pin sequence $w$ is recurrent so we can enumerate $\mathcal{W}^{\circ}$ using our standard procedure. We first determine the generating function of pin factors of $w$: as $w$ has period $4$ and all four pin factors are already distinct at length $1$ we have: \[ h(z) = 4z + 4z^2 + 4z^3 + 4z^4 + 4z^5 + 4z^6 + \dots \] We must amend this for collisions and $\boxplus$-decomposables to obtain the generating function $g(z)$ of $\boxplus$-indecomposables in $\mathcal{W}^{\circ}$. By comparision with the lists in Theorem \ref{classification} we see that the only collision amongst the pin factors of $w$ is the colliding quadruple $\{1ldr, 2dru, 3rul, 4uld\}$ and that there are precisely four $\boxplus$-decomposables of length $2$ ($1l$, $2d$, $3r$ and $4u$), four $\boxplus$-decomposables of length $3$ ($1ld$, $2dr$, $3ru$ and $4ul$), and none of length $n \geq 4$. Hence the generating function of $\boxplus$-indecomposables in $\mathcal{W}^{\circ}$ is given by: \[ \begin{split} g(z) & = h(z) - 3z^4 - (4z^2 + 4z^3) \\ & = 4z + z^4 + 4z^5 + 4z^6 + 4z^7 + 4z^8 + \dots \\ & = \frac{4z - 4z^2 + z^4 + 3z^5}{1-z} \end{split} \] Next we note that there is precisely $1$ one-quadrant $\boxplus$-indecomposable of length $1$ in each quadrant and none of any greater length: \[ g_{1}(z) = g_{2}(z) = g_{3}(z) = g_{4}(z) = z \] Hence the amended $G$-sequence of $\mathcal{W}^{\circ}$ is given by: \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & = \frac{4z - 4z^2 + z^4 + 3z^5}{1-z} - 2z^2 \\ & = \frac{4z - 6z^2 + 2z^3 + z^4 + 3z^5}{1-z} \end{split} \] And so, by the Generating Function Specification \ref{genfuncspec}, we can derive the generating function of $\mathcal{W}^{\circ}$: \[ \begin{split} f(z) & = \frac{1}{1 - G(z)} \\ & = \frac{1-z}{1 - 5z + 6z^2 - 2z^3 - z^4 - 3z^5} \end{split} \] as required. Finally, we can calculate the growth rate $\omega_{0}$ of $\mathcal{W}^{\circ}$ as the reciprocal of the smallest positive real root of the denominator: $\omega_{0} \approx 3.48806$. \end{proof} \section{Non-recurrent Pin Classes} \label{sec:4} We have now established a general method for determining the growth rate and generating function of a pin class $\mathcal{C}_{w}$ defined by a \emph{recurrent} pin sequence $w$ over the language $\mathcal{L}$. This also allows us to determine the growth rate of an \emph{eventually} recurrent pin class, by the Finite Prefix Lemma \ref{fplem}. This leaves the non-eventually-recurrent case: these classes are not $\boxplus$-closed, so the methods outlined in the previous section will not enable us to determine their generating functions (in fact, the author does not know the generating function of \emph{any} not-eventually-recurrent pin class). Somewhat surprisingly, however, we \emph{can} at least determine the \emph{growth rate} of a general non-recurrent pin class $\mathcal{C}^{\circ}_{w}$. The key idea here is simple: the classes that we know how to enumerate are the $\boxplus$-closed classes, so we enumerate $\mathcal{C}^{\circ}_{w}$ by taking better and better $\boxplus$-closed approximations. The limiting behaviour of these approximations will give us the growth rate, but \emph{not} the generating function, of $\mathcal{C}^{\circ}_{w}$. We begin by noting that we already know how to bound $\mathcal{C}^{\circ}_{w}$ by a $\boxplus$-closed class from above: Procedure \ref{recproc} gives us the generating function of the $\boxplus$-closure of $\mathcal{C}^{\circ}_{w}$; in the non-recurrent case this is not equal to $\mathcal{C}^{\circ}_{w}$ but will function as an upper bound as $\mathcal{C}^{\circ}_{w} \subseteq \boxplus\mathcal{C}^{\circ}_{w}$. We also note that, like pin classes, $\boxplus$-closures of pin classes have proper growth rates: \begin{lemma} Let $w$ be a pin sequence. Then $\boxplus\mathcal{C}^{\circ}_{w}$ has a proper growth rate. \end{lemma} \begin{proof} This is an application of Theorem \ref{grexist}: $\boxplus\mathcal{C}^{\circ}_{w}$ is a $\boxplus$-closed class contained in the complete pin class $\mathcal{P}^{\circ}$ and satisfies that adjacency condition by the same reasoning as $\boxplus\mathcal{C}^{\circ}_{w}$.\end{proof} Hence, for any pin sequence $w$, $gr(\mathcal{C}^{\circ}_{w}) \leq gr(\boxplus\mathcal{C}^{\circ}_{w})$. \subsection{The $\boxplus$-interior of a Pin Class} We now wish to determine a \emph{lower} bound on the growth rate of a pin class, which we again do by comparison with a $\boxplus$-closed class whose generating function we can find. We thus consider the `largest $\boxplus$-closed subclass contained in $\mathcal{C}^{\circ}_{w}$', which we refer to as the $\boxplus$-\emph{interior} of $\mathcal{C}^{\circ}_{w}$. In order to define this rigorously, recall that if $w$ is an infinite pin word over the language $\mathcal{L}$, we use the notation $w_{i,j}$ to denote the pin factor of $w$ taken between the $i$th and $j$th places. Now we can define: \begin{defn}[Recurrent Pin Factors] Let $w$ be a pin sequence and $\widetilde{w}$ a (finite) pin word. We say that $\widetilde{w}$ is a \textbf{recurrent pin factor} of $w$ if for all $n \in \mathbb{N}$ there exist $j \geq i \geq n$ such that $w_{i,j} = \widetilde{w}$. We call a $\boxplus$-indecomposable permutation $\pi^{\circ}_{\tilde{w}}$ generated by a recurrent pin factor $\tilde{w}$ of $w$ a \textbf{recurrent} $\boxplus$-indecomposable in $\mathcal{C}^{\circ}_{w}$. \end{defn} That is, a recurrent pin factor of $w$ is a pin factor that occurs in $w$ infinitely-often. We may now define: \begin{defn}[The $\boxplus$-interior of a pin class] \ \\ Let $w$ be a pin sequence with pin class $\mathcal{C}^{\circ}_{w}$. We define the $\boxplus$-\textbf{interior}, $\mathcal{C}^{\boxplus}_{w}$, of $\mathcal{C}^{\circ}_{w}$ to be the $\boxplus$-closure of the set of pin permutations of the form $\pi^{\circ}_{\tilde{w}}$ where $\tilde{w}$ is a \emph{recurrent} pin factor of $w$. \end{defn} We then have: \begin{obs}[Basic properties of the $\boxplus$-interior] For any pin sequence $w$ over $\mathcal{L}$: \begin{enumerate} \item $\mathcal{C}^{\boxplus}_{w}$ is a (non-empty) $\boxplus$-closed subclass of $\mathcal{C}^{\circ}_{w}$; \item $\mathcal{C}^{\boxplus}_{w}$ is the union of all $\boxplus$-closed subclasses of $\mathcal{C}^{\circ}_{w}$; \item $\mathcal{C}^{\boxplus}_{w} = \mathcal{C}^{\circ}_{w}$ if and only if $\mathcal{C}^{\circ}_{w}$ is $\boxplus$-closed; \item $\sigma^{\circ} \in \mathcal{C}^{\boxplus}_{w}$ if and only if for all $i \in \mathbb{N}$ there exists $j \geq i$ such that $\sigma^{\circ} \leq \pi^{\circ}_{w_{i,j}}$. \item $\mathcal{C}^{\boxplus}_{w}$ is the $\boxplus$-closure of the set of recurrent $\boxplus$-indecomposables in $\mathcal{C}^{\circ}_{w}$. \end{enumerate} \end{obs} Informally, the $\boxplus$-interior of $\mathcal{C}^{\circ}_{w}$ is the set of all centred permutations that can be found in the pin diagram of $w$ in infinitely many (non-overlapping) instances. As the $\boxplus$-interior of a pin class is $\boxplus$-closed, we can determine its generating function in terms of its $G$-sequence, which we can obtain from $w$ as follows: \begin{lemma} Suppose that $g(z)$ is the generating function of the \textbf{recurrent} $\boxplus$-indecomposables in $\mathcal{C}^{\circ}_{w}$ and $g_{1}(z), g_{2}(z), g_{3}(z), g_{4}(z)$ are the generating functions of the \textbf{recurrent} one-quadrant $\boxplus$-indecomposables in quadrants $1,2,3,4$, respectively. Then \[ G(z) = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \] is the amended $G$-sequence for $\mathcal{C}^{\boxplus}_{w}$, and \[ f(z) = \frac{1}{1 - G(z)} \] is the generating function of $\mathcal{C}^{\boxplus}_{w}$. \end{lemma} \begin{proof} Immediate from the Generating Function Specification on noting that $\mathcal{C}^{\circ}_{w}$ is a $\boxplus$-closed class and the recurrent $\boxplus$-indecomposables of $\mathcal{C}^{\circ}_{w}$ are its $\boxplus$-indecomposables. \end{proof} As with the $\boxplus$-closure we briefly note the following: \begin{lemma} Let $w$ be a pin sequence. Then $\mathcal{C}^{\boxplus}_{w}$ has a proper growth rate. \end{lemma} \begin{proof} If $\mathcal{C}^{\boxplus}_{w}$ contains \nept and \swpt then $w$ contains $1$ and $3$ as recurrent pin factors, which is to say that $w$ visits quadrants $1$ and $3$ infinitely-often. Every time $w$ moves from quadrant $1$ to quadrant $3$ it must pass through either quadrant $2$ or $4$, and so must also visit at least one of these quadrants infinitely-often; hence $\mathcal{C}^{\boxplus}_{w}$ also contains either \nwpt or \sept and so $w$ satisfies the adjacency condition. The same reasoning works if $\mathcal{C}^{\boxplus}_{w}$ contains \nwpt or \sept, and so we conclude that $\mathcal{C}^{\boxplus}_{w}$ satisfies the adjacency condition. As $\mathcal{C}^{\boxplus}_{w}$ is $\boxplus$-closed we can now apply Theorem \ref{grexist} to deduce that $\mathcal{C}^{\boxplus}_{w}$ has a proper growth rate. \end{proof} Recall that $\kappa \approx 2.20557$ is the growth rate of the class $\mathcal{O}^{\circ}$ of increasing oscillations. We shall require the following elementary bound: \begin{lemma} Let $w$ be a pin sequence. Then $gr(\mathcal{C}^{\boxplus}_{w}) \geq \kappa$. \end{lemma} \begin{proof} \begin{itemize} \item If $w$ visits only one quadrant recurrently, which by symmetry we may take to be the first quadrant, then $w=\widetilde{w}(ur)^{}$, where $\widetilde{w}$ is some finite prefix. Hence $\mathcal{C}^{\boxplus}_{w}$ is just $\mathcal{O}^{\circ}$ with a finite prefix, and so has growth rate $\kappa$. \item Suppose $w$ visits precisely two (necessarily adjacent) quadrants recurrently - by symmetry we may take these to be quadrants $1$ and $2$. Then after some finite prefix, $w$ stays in the upper half-plane, and moves between quadrants $1$ and $2$ infinitely-often. Hence $w$ contains $1l$ and $2r$ as recurrent pin factors, and these must extend to recurrent pin factors $1lu$ and $2ru$ (as $w$ stays in the upper half-plane from this point). Hence $\pi^{\circ}_{1lu}$ and $\pi^{\circ}_{2ru}$ are contained in $\mathcal{C}^{\boxplus}_{w}$ and we note that \nwtwo and \netwo are contained in these two pin permutations, respectively. Hence $\mathcal{C}^{\boxplus}_{w}$ contains the $\boxplus$-closure $\boxplus\{\nwtwo,\netwo\}$, which by calculation has growth rate $\approx 2.73205 > \kappa$. \item Suppose $w$ visits three or four quadrants recurrently - by symmetry assume these include quadrants $1$, $2$ and $3$. Then $\mathcal{C}^{\boxplus}_{w}$ contains $\boxplus\{\nept,\nwpt,\swpt\}$, which by calculation has growth rate $\approx 2.61803 > \kappa$. \end{itemize}\end{proof} We note that this now implies the following as a corollary: \begin{cor} Let $w$ be a pin sequence. Then: \begin{itemize} \item $\overline{gr}(\mathcal{C}_{w}) \geq \kappa$ \item $\overline{gr}(\mathcal{C}_{w}) = \kappa$ if and only if $w = \widetilde{w}(ur)^{}$ or some symmetry of this form, where $\widetilde{w}$ is a finite prefix. \end{itemize} \end{cor} \subsection{$G$-sequence properties} Let $w$ be a pin sequence, and let $\mathcal{C}^{\circ}$ be either the pin class generated by $w$ or its $\boxplus$-interior. We now know how to obtain the amended $G$-sequence $G(z)$ of $\mathcal{C}^{\circ}$, by enumerating either the pin factors or recurrent pin factors of $w$. Then the generating function of $\boxplus\left(\mathcal{C}^{\circ}\right)$ (note that this is $\mathcal{C}^{\circ}$ itself in the $\boxplus$-interior case) is given by: \[ f(z) = \frac{1}{1 - G(z)} \] We know that the growth rate $\rho$ of $\boxplus\left(\mathcal{C}^{\circ}\right)$ is equal to the reciprocal of the radius of convergence of $f(z)$. We should like to be able to deduce from this that $G(\rho^{-1}) = 1$, as this will give us a strategy for studying growth rates of pin classes in terms of analytic properties of $G(z)$, but this will require a slightly more thorough study of the properties of $G(z)$: \begin{prop}[$G$-sequence of pin classes and $\boxplus$-interiors] \label{Gproperties} Let $w$ be a pin sequence and suppose that $G(z)= \sum_{n=1}^{\infty}a_{n}z^{n}$ ($a_{n} \in \mathbb{Z}$) is the amended $G$-sequence of a class $\mathcal{C}^{\circ}$, which is \textbf{either} the pin class generated by $w$ \textbf{or} its $\boxplus$-interior. Then: \begin{enumerate} \item $a_{1} \in \left\{1, 2, 3, 4\right\}$. \item For all $n \geq 2$: \[ -8n \leq a_{n} < 2^{n+2} \] \item $G(z)$ converges to a smooth function on the interval $\left[0,\frac{1}{2}\right)$, with $G(0) = 0$. \item $G(z) = 1$ has a solution in $\left[0,\frac{1}{\kappa}\right]$. \item Let $\alpha$ be the smallest positive real solution of $G(z)=1$. The growth rate of $\boxplus\mathcal{C}^{\circ}$ is equal to $\alpha^{-1}$. \end{enumerate} \end{prop} \begin{proof} Recall that \begin{equation} \label{amend} G(z) = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \end{equation} where $g(z)$ counts the $\boxplus$-indecomposables in $\mathcal{C}^{\circ}$ and $g_{i}(z)$ counts the one-quadrant $\boxplus$-indecomposables contained in the $i$th quadrant. \begin{enumerate} \item This is clear from \ref{amend}: the $z$-term of $G(z)$ must agree with the $z$-term of $g(z)$ (as the products $g_{1}(z)g_{3}(z)$ and $g_{2}(z)g_{4}(z)$ have no term of degree lower than $z^2$), and this simply counts the number of quadrants that $\mathcal{C}^{\circ}$ visits (either at all or recurrently), which is clearly either $1$, $2$, $3$ or $4$. \item Let $\preceq$ denote coefficient-wise ordering on formal power series, so $p(z) \preceq q(z)$ means that the $z^n$-coefficient of $p(z)$ is less than or equal to the $z^n$-coefficient of $q(z)$ for all $n \in \mathbb{N}$. Then, from \ref{amend} (and using the fact that the coefficients of $g(z),g_{i}(z)$ are all non-negative): \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & \preceq g(z) \\ & \preceq 4z + \sum_{n=2}^{\infty}2^{n+2}z^n \end{split} \] where the final inequality is simply the number of pin words of length $n$ over $\mathcal{L}$. On the other hand, using the fact that the generating function of \emph{all} one-quadrant $boxplus$-indecomposable pin permutations in the $i$th quadrant is \[ g_{i}(z) = z + z^2 + 2z^3 + 2z^4 + 2z^5 + \dots = \frac{z + z^3}{1-z}, \] we obtain: \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & \succeq - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & \succeq -2\frac{(z+z^3)^{2}}{(1-z)^{2}} \\ & = -2z^2 - 4z^3 - 10z^4 - 16z^5 - 24z^6 - 32z^7 - 40z^8 - 48z^9 - \dots \\ & \succeq \sum_{n=1}^{\infty}-8nz^n \end{split} \] Combining these two inequalities gives the desired result. \item This is from the Ratio Test, on noting that the coefficients of $G(z)$ have magnitudes bounded by powers of $2$. \item We know that the all the classes we are dealing with have growth rates greater than or equal to $\kappa$. Hence the generating function of $\boxplus\mathcal{C}^{\circ}$, namely \[ f(z) = \frac{1}{1 - G(z)} , \] has radius of convergence (in $\mathbb{C}$) $R \leq \frac{1}{\kappa} < \frac{1}{2}$. As $f(z)$ has non-negative coefficients (regardless of whether $G(z)$ does), we can apply Pringsheim's Theorem~\cite[Theorem IV.7]{flajolet:analytic-combin:} to deduce that $f(z)$ has a singularity at $z = R$. Singularites of $f(z)$ correspond either to singularities of $G(z)$ or solutions of $G(z)=1$. But $G(z)$ converges on $\left[0,\frac{1}{2}\right)$ which includes $R$, so $G(R)=1$. \item We saw that $R$ solves $G(z)=1$ in the proof of \textit{4.}; there cannot be a smaller solution to this equation as that would give a singularity with magnitude smaller than $R$. Now apply the Exponential Growth Theorem \ref{EGT}. \end{enumerate} \end{proof} We require one more property of $G$-sequences in the $\boxplus$-interior case only: \begin{lemma} \label{G+} Let $G(z)$ be the amended $G$-sequence of the $\boxplus$-interior $\mathcal{C}^{\boxplus}_{w}$ of some pin class $\mathcal{C}^{\circ}_{w}$. Let $\alpha$ be the smallest positive real root of the equation $G(z)=1$, guaranteed to exist by Proposition \ref{Gproperties}. Then $G(z)$ is positive on the interval $(0,\alpha)$. \end{lemma} \begin{proof} We note that this property is obvious if $G(z)$ has non-negative coefficients, which immediately deals with the case in which $w$ visits only two of the quadrants (either recurrently or otherwise). We next suppose that $\mathcal{C}^{\circ}$ has points in precisely three quadrants, which, without loss of generality, we take to be quadrants $1$, $2$ and $3$. Then $\mathcal{C}^{\circ}$ must contain the permutations \nept, \nwpt, \swpt, \netwo and \swtwo (as the defining pin sequence $w$ must turn around in the first and third quadrants infinitely-often), and so: \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) \\ & \succeq (3z + 2z^2 + 2z^3) - \frac{(z+z)^2}{(1-z)^2} \\ & = \frac{3z - 5z^2 + z^3 - 4z^4 + 2z^5 - z^6}{(1-z)^2} \end{split} \] Hence: \[ G(z) = \frac{3z - 5z^2 + z^3 - 4z^4 + 2z^5 - z^6}{(1-z)^2} + F(z) \] where $F(z)$ has non-negative coefficients. By computation the function given here is positive on the desired range, and $F(z)$ is certainly positive here due to non-negativity of its coefficients. Hence $G(z)$ is positive on $(0,\alpha)$. Finally, if $G(z)$ visits all four quadrants then: \[ \begin{split} G(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z)\\ & \succeq 4z - 2\frac{(z+z)^2}{(1-z)^2} \\ & = \frac{4z - 10z^2 + 4z^3 - 4z^4 - 2z^6}{(1-z)^2} \end{split} \] and as in the three quadrant case, this `worst case scenario' function is nevertheless still positive in the desired range. \end{proof} \subsection{$\mathcal{C}^{\circ}_{w}$ has same growth rate as $\mathcal{C}^{\boxplus}_{w}$} The moral of the preceeding sections is that we can sandwich \emph{any} pin class (recurrent or otherwise) between two $\boxplus$-closed (centred) permutation classes that we know how to enumerate. If the pin class happens to be recurrent then these containments are in fact equalities and we have enumerated the pin class. If the pin class is \emph{not} recurrent then we at least have bounds on the growth rate: \begin{equation} \label{conts} gr(\mathcal{C}^{\boxplus}_{w}) \leq gr(\mathcal{C}^{\circ}_{w}) \leq gr(\boxplus\mathcal{C}^{\circ}_{w}) \end{equation} The aim of this section is to prove the remarkable result that the left-hand inequality in (\ref{conts}) is in fact an equality; that is, the growth rate of a pin class is always equal to that of its $\boxplus$-interior. This will enable us to conclude that \emph{any} pin class (recurrent or otherwise) has a proper growth rate, and will enable us to determine this growth rate providing that we can enumerate the recurrent pin factors of the defining pin sequence $w$. The core idea in the proof is relatively simple to understand: if $w$ is a pin sequence we know by the Finite Prefix Theorem that $dfg$ and $cvgbh$ have the same (upper) growth rate and we note that if we take $n$ sufficiently large then the pin factors of $dfg$ are precisely ... \begin{prop} \label{limprop} Let $\mathcal{C}^{\circ}_{w}$ be a pin class generated by a pin sequence $w$. Let $\mathcal{C}^{\boxplus}_{w}$ be the $\boxplus$-interior of $\mathcal{C}^{\circ}_{w}$ and write $w_{\geq n}$ for the left-truncation of $w$ starting in the $n$th position. Then: \[ \lim_{n\rightarrow\infty}gr\left(\boxplus\mathcal{C}^{\circ}_{w_{\geq n}}\right) = gr(\mathcal{C}^{\boxplus}_{w}) \] \end{prop} \begin{proof} Let $g(z)$ be the generating function of the $\boxplus$-indecomposables of $\mathcal{C}^{\boxplus}_{w}$, and $g_{i}(z)$ be the generating function of one-quadrant $\boxplus$-indecomposables of $\mathcal{C}^{\boxplus}_{w}$ in the $i$th quadrant. Then \[ G(z) = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \] is the amended $G$-sequence of $\mathcal{C}^{\boxplus}_{w}$, and $\alpha = gr(\mathcal{C}^{\boxplus}_{w})^{-1}$ is the smallest positive real root of the equation $G(z) = 1$. Now, let $t \in \mathbb{N}$, and consider the set $\mathcal{W}^{}_{t}$ of all \textbf{non}-recurrent pin factors of $w$ of length $\leq t$. This is a finite set and each element of it occurs as a pin factor of $w$ only a finite number of times, so there is some $n(t) \in \mathbb{N}$ such that no element of $\mathcal{W}^{}_{k}$ is contained as a pin factor of $w_{\geq n(t)}$ (for concreteness, we may take $n(t)$ to be the \emph{smallest} positive integer with this property). But of course \emph{all} of the recurrent pin factors of $w$ of length $\leq t$ occur as pin factors of $w_{\geq n(t)}$, so we deduce that the pin factors of $w_{\geq n(t)}$ of length $ \leq t$ are precisely the recurrent pin factors of $w$ of length $\leq t$. Hence if we write $g_{t}(z)$ for the generating function of the $\boxplus$-indecomposables of $\mathcal{C}^{\circ}_{w_{\geq n(t)}}$, and $g_{t, i}(z)$ for generating function of the one-quadrant $\boxplus$-indecomposables of $\mathcal{C}^{\circ}_{w_{\geq n}}$ in the $i$th quadrant, then $g_{t}(z), g_{t,1}(z), g_{t,2}(z), g_{t,3}(z), g_{t,4}(z)$ will agree with $g(z), g_{1}(z), g_{2}(z), g_{3}(z), g_{4}(z)$, respectively, up to and including the $z^{t}$-term. Hence the amended $G$-sequence of $\mathcal{C}^{\circ}_{w_{\geq n(t)}}$, namely \[ G_{t}(z) = g_{t}(z) - g_{t,1}(z)g_{t,3}(z) - g_{t,2}(z)g_{t,4}(z) , \] agrees with $G(z)$ up to and including the $z^{t}$-term. Fix a $t$: we now consider some basic analytic facts about the functions $G(z)$ and $G_{t}(z)$. First, note that by Proposition \ref{Gproperties}, $G(z)$ and $G_{t}(z)$ are smooth functions defined on the interval $\left[0,\frac{1}{\kappa}\right]$. Further, $G(z) = 1$ and $G_{t}(z) = 1$ have solutions in this interval; we call the smallest solution to these equations in this interval $\alpha$ and $\alpha_{t}$, respectively. Then $\mathcal{C}^{\boxplus}_{w}$ and $\boxplus\mathcal{C}^{\circ}_{w_{\geq n(t)}}$ have growth rates $\rho = \alpha^{-1}$ and $\rho_{t} = \alpha^{-1}_{t}$, respectively. Note that we have: \[ \rho = gr(\mathcal{C}^{\boxplus}_{w}) \leq gr(\mathcal{C}^{\circ}_{w}) = gr(\mathcal{C}^{\circ}_{w_{\geq n(t)}}) \leq gr(\boxplus\mathcal{C}^{\circ}_{w_{\geq n(t)}}) = \rho_{t} \] where the two inequalities follow from containment of the corresponding classes, and the middle equality follows from the Finite Prefix Lemma \ref{fplem}. Note that this implies that $\alpha_{t} \leq \alpha$. Now, write $G_{t}(z) = \sum_{n=1}^{\infty}a_{n}z^n$ and $G(z) = \sum_{n=1}^{\infty}b_{n}z^n$: then each $a_{n}, b_{n}$ is an integer (not necessarily positive) with magnitude bounded by $2^{n+2}$ and $a_{n} = b_{n}$ for all $n \leq t$. We combine these facts to deduce a bound on the difference between these two functions: \[ \begin{split} \left\|G_{t}(z) - G(z)\right\| & = \left\|\sum_{n=t+1}^{\infty}(a_{n} - b_{n})z^{n}\right\| \\ & \leq \sum_{n=t+1}^{\infty}\left\|a_{n} - b_{n}\right\|z^n \\ & \leq \sum_{n=t+1}^{\infty}(2^{n+2} + 8n)z^{n} \\ & \leq \sum_{n=t+1}^{\infty}2^{n+3}z^{n} \\ & = \frac{8(2z)^{t+1}}{1-2z} \end{split} \] and so, for $z \in \left[0,\frac{1}{\kappa}\right]$: \[ \begin{split} \left\|G_{t}(z) - G(z)\right\| & \leq \sup_{z \in \left[0,\frac{1}{\kappa}\right]}\left\{\frac{8(2z)^{t+1}}{1-2z}\right\} \\ & = \frac{8(\frac{2}{\kappa})^{t+1}}{1 - \frac{2}{\kappa}} \\ & = \frac{16}{\kappa - 2} \cdot \left(\frac{2}{\kappa}\right)^{t} \end{split} \] Note, crucially, that (as $\kappa > 2$) this expression approaches $0$ as $t \rightarrow \infty$. We claim that this fact implies that $\alpha_{t} \rightarrow \alpha$: Let $\epsilon > 0$ and consider $G(z)$ on the interval $\left[0, \alpha - \epsilon\right]$. As a continuous function on a closed interval, $G(z)$ achieves a maximum value $M$ on $\left[0, \alpha - \epsilon\right]$. Further, as $G(z)$ is positive on $(0,\alpha]$ (by Lemma \ref{G+}) and $G(z)=1$ does not have a root in $\left[0, \alpha - \epsilon\right]$ (as $\alpha$ is, by definition, the \emph{smallest} positive real root of $G(z)=1$), $M$ must be a positive number smaller than $1$. Now, choose a $K \in \mathbb{N}$ such that: \[ \frac{16}{\kappa - 2} \cdot \left(\frac{2}{\kappa}\right)^{K} < 1 - M \] and take any $t \geq K$. Then, for $z \in \left[0, \alpha - \epsilon\right]$: \[ \begin{split} \left\|G_{t}(z)\right\| & = \left\|(G_{t}(z) - G(z)) + G(z)\right\| \\ & \leq \left\|G_{t}(z) - G(z)\right\| + \left\|G(z)\right\| \\ & \leq \left\|G_{t}(z) - G(z)\right\| + G(z) \\ & \leq \frac{16}{\kappa - 2} \cdot \left(\frac{2}{\kappa}\right)^{t} + M \\ & < (1 - M) + M \\ & = 1 \end{split} \] Hence, in particular, $G_{t}(z)=1$ does not have a root in $\left[0, \alpha - \epsilon\right]$. But $G_{t}(z)=1$ certainly does have a root, namely $\alpha_{t}$, which is smaller than $\alpha$. Hence $\alpha_{t} \in \left(\alpha - \epsilon, \alpha\right]$. We have thus proved that for every $\epsilon > 0$ there exists $K \in \mathbb{N}$ such that $\alpha_{t} \in \left(\alpha - \epsilon, \alpha\right]$ for all $t \geq K$; hence $\alpha_{t} \rightarrow \alpha$ as $t \rightarrow \infty$. Hence $gr(\boxplus\mathcal{C}^{\circ}_{\geq n(t)}) \rightarrow gr(\mathcal{C}^{\boxplus}_{w})$ as $t\rightarrow\infty$, as required.\end{proof} We may now finally deduce the main result of this paper: \begin{thm} Let $w$ be a pin sequence. Then the associated pin class $\mathcal{C}^{\circ}_{w}$ (along with its uncentred counterpart $\mathcal{C}_{w}$) has a proper growth rate which is equal to that of its $\boxplus$-interior, $\mathcal{C}^{\boxplus}_{w}$ \end{thm} \begin{proof} Suppose $w$ is a pin sequence. Then $\mathcal{C}^{\boxplus}_{w} \subseteq \mathcal{C}^{\circ}_{w}$ and so \begin{equation}\label{one} gr(\mathcal{C}^{\boxplus}_{w}) = \underline{gr}(\mathcal{C}^{\boxplus}_{w}) \leq \underline{gr}(\mathcal{C}^{\circ}_{w}) \end{equation} where the left-hand equality is due to the fact that $\boxplus$-interiors have (proper) growth rates. Now, letting $n \in \mathbb{N}$ we have \begin{equation}\label{two} \overline{gr}(\mathcal{C}^{\circ}_{w}) = \overline{gr}(\mathcal{C}^{\circ}_{w_{\geq n}}) \end{equation} by the Finite Prefix Lemma \ref{fplem}, and by containment we also have \begin{equation} \label{three} \overline{gr}(\mathcal{C}^{\circ}_{w_{\geq n}}) \leq \overline{gr}(\boxplus\mathcal{C}^{\circ}_{w_{\geq n}}) = gr(\boxplus\mathcal{C}^{\circ}_{w_{\geq n}}) \end{equation} where the right-hand equality follows from existence of proper growth rates of $\boxplus$-closures of pin classes. Combining equations (\ref{one}), (\ref{two}) and (\ref{three}) yields: \[ gr(\mathcal{C}^{\boxplus}_{w}) \leq \underline{gr}(\mathcal{C}^{\circ}_{w}) \leq \overline{gr}(\mathcal{C}^{\circ}_{w}) \leq gr(\boxplus\mathcal{C}^{\circ}_{w_{\geq n}}) \] and taking limits (using Proposition \ref{limprop}) as $n \rightarrow \infty$ forces \[ \underline{gr}(\mathcal{C}^{\circ}_{w}) = \overline{gr}(\mathcal{C}^{\circ}_{w}) \] as required.\end{proof} \subsection{The Complete Class $\mathcal{P}^{\circ}_{c}$} We illustrate the use of this theory by explicitly enumerating the (centred) complete pin class $\mathcal{P}^{\circ}_{c}$. Recall that this is the class of \emph{all} pin permutations. It may not be immediately obvious, but $\mathcal{P}^{\circ}_{c}$ is itself a pin class, and is in fact the \emph{only} pin class that achieves its growth rate: \begin{prop}[Complete Pin Class] \ \begin{enumerate} \item There is a pin sequence $w_{c}$ which contains every finite pin word as a (recurrent) pin factor. \item For any such pin sequence $w_{c}$, $\mathcal{C}^{\circ}_{w} = \mathcal{P}^{\circ}_{c}$. \item The complete pin class has growth rate $\omega_{\infty} \approx 5.24112$, where $\omega_{\infty}$ is defined to be the reciprocal of the smallest positive real root of the equation \[ 1 - 8z + 19z^2 - 26z^3 + 14z^4 - 12z^5 - 8z^6 + 20z^7 - 8z^8 = 0 \] \end{enumerate} \end{prop} \begin{proof} \begin{enumerate} \item Note that $\mathcal{L}^{}$ is countable, we can thus order all $\mathcal{L}^{}$-words and concatenate them (after any initial numeral) in this order, possibly placing a letter in between consecutive words to ensure the alignments alternate. This resulting pin sequence $w_{c}$ will contain all $\mathcal{L}^{}$-words as subword factors, and will hence contain all possible finite pin words as pin factors. \item This is clear by the definition of a pin permutation. \item First, we enumerate the set of pin factors of $w_{c}$, which is to say the set of all finite pin words. Clearly, there are $4$ pin words of length $1$. For length $n \geq 2$, there are $4$ choices for the inital numeral, $4$ choices for the letter in second place, and then $2$ choices for every subsequent letter as the alignments must now alternate. Hence there are $2^{n+2}$ pin words of length $n \geq 2$, and the generating function of the set of all finite pin words is given by: \[ h(z) = 4z + \sum_{n=2}^{\infty}2^{n+2}z^n \] This will be an overcount of the $\boxplus$-indecomposables in $\mathcal{P}^{\circ}$ due to collisions and $\boxplus$-decomposable pin words, but we have already classified all of these in Theorem \ref{classification}. The generating function of the overcount due to collisions is \[ h_{\text{Col}}(z) = 4z^2 + 8z^3 + 6z^4 + 12z^5 + 8z^6 + 8z^7 + 8z^8 + 8z^9 + \dots \] (Note: this is the generating function of the \emph{overcount} due to collisions. At all $n \neq 4$ it is simply equal to the \emph{number} of collisions, as these are all pairs so we need only take away one from the count for each collision. But at $n=4$ there are two colliding \emph{quadruples} so the overcount is $6$.) And the generating function of the $\boxplus$-decomposable pin words is \[ h_{\boxplus\text{-Dec.}}(z) = 8z^2 + 8z^3 + 16z^4 + 16z^5 + 16z^6 + 16z^7 + 16z^8 + 16z^9 + \dots \] Subtracting these from $h(z)$ gives us the generating function of the $\boxplus$-indecomposable pin permutations: \[ \begin{split} g(z) & = h(z) - h_{\text{Col}}(z) - h_{\boxplus\text{-Dec.}}(z) \\ & = 4z + 4z^2 + 16z^3 + 42z^4 + 100z^5 + \sum_{n=6}^{\infty}(2^{n + 2} - 24)z^n \\ & = \frac{4z - 8z^2 + 12z^3 + 2z^4 + 6z^5 + 16z^6 - 8z^7}{(1-z)(1-2z)} \end{split} \] Next, we shall require the generating functions $g_{i}(z)$ ($i \in \{1,2,3,4\}$) of the $\boxplus$-indecomposable pin permutations entirely contained in the $i$th quadrant. Clearly, these are all equal to the generating function of the oscillations; namely: \[ \begin{split} g_{1}(z) = g_{2}(z) = g_{3}(z) = g_{4}(z) & = z + z^2 + 2z^3 + 2z^4 + 2z^5 + 2z^6 + \dots \\ & = \frac{z + z^3}{1-z} \end{split} \] Hence we obtain the amended $G$-sequence for the complete class of pin permutations: \[ \begin{split} G_{\infty}(z) & = g(z) - g_{1}(z)g_{3}(z) - g_{2}(z)g_{4}(z) \\ & = \frac{4z - 14z^2 + 24z^3 - 14z^4 + 12z^5 + 8z^6 - 20z^7 + 8z^8}{(1-z)^{2}(1-2z)} \end{split} \] Thus, we can now apply the Generating Function Specification to obtain the generating function for the class of all (centred) pin permutations: \[ \begin{split} f(z) & = \frac{1}{1 - G_{\infty}(z)} \\ & = \frac{(1-z)^{2}(1-2z)}{1 - 8z + 19z^2 - 26z^3 + 14z^4 - 12z^5 - 8z^6 + 20z^7 - 8z^8} \end{split} \] By the Exponential Growth Rate formula, and the fact that $f(z)$ has positive coefficients, the growth rate of $\mathcal{P}^{\circ}$, and hence also the uncentred class $\mathcal{P}$, is the reciprocal of the smallest positive real root of the denominator, as required. \end{enumerate} \end{proof} \section{Concluding Remarks} \label{sec:5} We have proved that all pin classes have growth rates and established a procedure to determine them. We also have bounds on the possible growth rate of a pin class: for any pin sequence $w$, \[ \kappa \leq gr(\mathcal{C}_{w}) \leq \omega_{\infty} \] where $\kappa \approx 2.20557$ is the reciprocal of the smallest positive real root of \[ 1 - 2z - z^3 = 0 \] and $\omega_{\infty}$ the smallest positive real root of \[ 1 - 8z + 19z^2 - 26z^3 + 14z^4 - 12z^5 - 8z^6 + 20z^7 - 8z^8 = 0 \] A natural further question is what happens within these bounds: what are the possible growth rates of pin classes? We can also ask about bounds on growth rates of pin classes subject to certain characteristics: the number of quadrants visited (recurrently) by a pin class and the length of the longest oscillation contained in $\mathcal{C}^{\circ}_{w}$ are natural characteristics to consider. We can in fact state some answers in the former case already: the pin classes $\mathcal{V}$ and $\mathcal{Y}$ are in fact the smallest pin classes which visit two and three quadrants recurrently, respectively. It is also relatively easy to deduce \emph{upper} bounds on the growth rates of pin classes in two and three quadrants by considering the \textbf{complete pin classes} in these bounded quadrants: for example, the complete class in two quadrants, $\mathcal{V}_{c} = \mathcal{V}_{w_{c}}$ is the pin class generated by a pin sequence $w_{c}$ that contains \emph{all} pin words in quadrants $1$ and $2$ as pin factors. Without too much difficulty (though we omit the proof here) we can calculate the growth rate of $\mathcal{V}_{c}$ to be $\nu_{c} \approx 3.51205$, where $\nu_{c}$ is the reciprocal of the smallest positive real root of \[ 1 - 2z - 4z^2 - 2z^3 - 8z^4 - 4z^5 = 0 \] In the sequel~\cite{brignall-jarvis:pin-classes-ii} we take up the question of what happens within this interval $[\nu,\nu_{c}]$: we move towards a classification of the growth rates of two-quadrant pin classes and observe some interesting structures in this set of growth rates. We show, for example, that there is a point $\nu_{\mathcal{L}} \approx 3.28277$ at which there are uncountably many distinct pin classes and that $\nu_{\mathcal{L}}$ is in fact an accumulation point in the set of pin class growth rates from both above and below. This has potential consequences for the study of well-quasi-ordered permutation classes because $\mathcal{V}$-classes (that is, two-quadrant pin classes) can be used to generate infinite antichains. Potential further directions for study include: \begin{itemize} \item A systematic study of pin class growth rates in three and four quadrants; \item The possibility of conjecturing a classification of `small' antichains (perhaps taking $\nu_{\mathcal{L}}$ as a cut-off) using $\mathcal{V}$-classes; \item The question of whether we can explicitly determine the generating function (not merely the growth rate) or \emph{any} not-eventually-recurrent pin class, such as the Liouville $\mathcal{V}$, introduced in the sequel. \end{itemize} \appendix \section*{Appendix} \renewcommand{\thesubsection}{(\Alph{subsection})} In this appendix we prove Theorem \ref{classification}, that the lists of collisions and $\boxplus$-decomposables given are in fact complete. We begin with collisions: \subsection{Collisions Proof} We can verify that the list of collisions given in the table is complete for lengths $n \geq 5$ by an exhaustive search (which can be done fairly quickly on applying symmetries). We thus aim to prove that the table is complete at lengths $n \geq 6$. We repeat the relevant section of the table for reference: { \centering \begin{tabular}{@{}\|>{\centering}m{1.5em}\|>{\centering}m{5.5em}\|>{\centering\arraybackslash}m{12em}\|>{\centering}m{4em}\|>{\centering\arraybackslash}m{17.5em}\|} \hline \multicolumn{5}{\|c\|}{}\\[0.4pt] \multicolumn{5}{\|c\|}{\textbf{List of collisions of pin factors:}}\\[6pt] \hline \multicolumn{2}{\|c\|}{\textbf{Length}}&\textbf{Representative}&\textbf{Total number}&\textbf{Full List}\\ \hline \multicolumn{2}{\|c\|}{$n \geq 6$ \ \textbf{even:}} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,8) {}; \node[permpt] (1) at (9,10) {}; \node[permpt] (2) at (6,9) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (7,6) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (4,7) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (5,5) {}; \draw[thin] (5) -- ++ (0,2.5); \node[permpt] (6) at (1,3) {}; \draw[thin] (6) -- ++ (1.5,0); \node[permpt] (7) at (2,1) {}; \draw[thin] (7) -- ++ (0,2.5); \node[permpt] (8) at (11,2) {}; \draw[thin] (8) -- ++ (-9.5,0); \node[permpt] (9) at (10,11) {}; \draw[thin,dashed] (9) -- ++ (0,-9.5); \node[empty] (-1) at (8,13) {}; \node[circle,fill,inner sep=0.5pt] (10) at (4.5,5.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4,5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (3.5,4.5) {}; \node[circle,fill,inner sep=0.5pt] (13) at (3,4) {}; \node[circle,fill,inner sep=0.5pt] (14) at (2.5,3.5) {}; \draw[thick] (8,0) -- ++ (0,12); \draw[thick] (0,8) -- ++ (12,0); \draw[dotted] (9.5,10.5) rectangle (10.5,11.5); \node[] (15) at (6.5,-1) {$1(ld)^{k}r=2(dl)^{k}dru$}; \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1(ld)^{k}r, 2(dl)^{k}dru\}, \{1(dl)^{k}u, 4(ld)^{k}lur\}, \\ \{2(dr)^{k}u, 3(rd)^{k}rul\}, \{2(rd)^{k}l, 1(dr)^{k}dlu\}, \\ \{3(ru)^{k}l, 4(ur)^{k}uld\}, \{3(ur)^{k}d, 2(ru)^{k}rdl\}, \\ \{4(ul)^{k}d, 1(lu)^{k}ldr\}, \{4(lu)^{k}r, 3(ul)^{k}urd\} \end{split} \] \\ \hline \multicolumn{2}{\|c\|}{$n \geq 7$ \ \textbf{odd:}} & \begin{tikzpicture}[scale=0.2] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,6) {}; \node[permpt] (1) at (9,8) {}; \node[permpt] (2) at (6,7) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (7,4) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (5,5) {}; \draw[thin] (4) -- ++ (2.5,0); \node[permpt] (5) at (3,1) {}; \draw[thin] (5) -- ++ (0,1.5); \node[permpt] (6) at (1,2) {}; \draw[thin] (6) -- ++ (2.5,0); \node[permpt] (7) at (2,10) {}; \draw[thin] (7) -- ++ (0,-8.5); \node[permpt] (8) at (10,9) {}; \draw[thin,dashed] (8) -- ++ (-8.5,0); \node[empty] (-1) at (8,12) {}; \node[circle,fill,inner sep=0.5pt] (8) at (5.5,4.5) {}; \node[circle,fill,inner sep=0.5pt] (9) at (5,4) {}; \node[circle,fill,inner sep=0.5pt] (10) at (4.5,3.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4,3) {}; \node[circle,fill,inner sep=0.5pt] (12) at (3.5,2.5) {}; \draw[thick] (8,0) -- ++ (0,11); \draw[thick] (0,6) -- ++ (11,0); \draw[dotted] (9.5,8.5) rectangle (10.5,9.5); \node[] (13) at (6,-1) {$1(ld)^{k}lu=2(dl)^{k}ur$}; \end{tikzpicture} & 8 \ \text{pairs} & \small \[ \begin{split} \{1(ld)^{k}lu, 2(dl)^{k}ur\}, \{1(dl)^{k}dr, 4(ld)^{k}ru\}, \\ \{2(dr)^{k}dl, 3(rd)^{k}lu\}, \{2(rd)^{k}ru, 1(dr)^{k}ul\}, \\ \{3(ru)^{k}rd, 4(ur)^{k}dl\}, \{3(ur)^{k}ul, 2(ru)^{k}ld\}, \\ \{4(ul)^{k}ur, 1(lu)^{k}rd\}, \{4(lu)^{k}ld, 3(ul)^{k}dr\} \end{split} \] \\ \hline \end{tabular}\par } In order to prove this we shall first make the observation that in each of the pairs listed the two pin words end in different letters. We shall call a collision a \emph{minimal collision} if all pin words in the tuple differ in their final letter. We shall first prove that the list above is a complete list of minimal collisions, and then deduce from this that there are no non-minimal collisions. \begin{thm}[Classification of Minimal Collisions] Any minimal collision of length $n \geq 6$ is one of the colliding pairs listed in the table above. \end{thm} \begin{proof} In order to prove this we shall first apply symmetries to this list so that one pin word of each pair ends in $dr$: \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,10) {}; \node[permpt] (1) at (10,11) {}; \node[permpt] (2) at (9,8) {}; \draw[thin] (2) -- ++ (0,3.5); \node[permpt] (3) at (6,9) {}; \draw[thin] (3) -- ++ (3.5,0); \node[permpt] (4) at (7,6) {}; \draw[thin] (4) -- ++ (0,3.5); \node[permpt] (5) at (5,7) {}; \draw[thin] (5) -- ++ (2.5,0); \node[permpt] (6) at (3,3) {}; \draw[thin] (6) -- ++ (0,1.5); \node[permpt] (7) at (1,4) {}; \draw[thin] (7) -- ++ (2.5,0); \node[permpt] (8) at (2,1) {}; \draw[thin] (8) -- ++ (0,3.5); \node[permpt] (9) at (12,2) {}; \draw[thin] (9) -- ++ (-10.5,0); \node[permpt] (10) at (11,12) {}; \draw[thin,dashed] (10) -- ++ (0,-10.5); \node[circle,fill,inner sep=0.5pt] (11) at (5.5,6.5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (5,6) {}; \node[circle,fill,inner sep=0.5pt] (13) at (4.5,5.5) {}; \node[circle,fill,inner sep=0.5pt] (14) at (4,5) {}; \node[circle,fill,inner sep=0.5pt] (15) at (3.5,4.5) {}; \draw[thick] (8,0) -- ++ (0,13); \draw[thick] (0,10) -- ++ (13,0); \draw[dotted] (10.5,11.5) rectangle (11.5,12.5); \node[] (16) at (6.5,-1) {$1dldl \dots dldr$}; \node[] (17) at (6.65,-2.5) {$=4ldl \dots dldru$}; \begin{scope}[shift={(18,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (10,5) {}; \node[permpt] (1) at (11,3) {}; \node[permpt] (2) at (8,4) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (9,7) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (6,6) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (7,8) {}; \draw[thin] (5) -- ++ (0,-2.5); \node[permpt] (6) at (3,10) {}; \draw[thin] (6) -- ++ (1.5,0); \node[permpt] (7) at (4,12) {}; \draw[thin] (7) -- ++ (0,-2.5); \node[permpt] (8) at (1,11) {}; \draw[thin] (8) -- ++ (3.5,0); \node[permpt] (9) at (2,1) {}; \draw[thin] (9) -- ++ (0,10.5); \node[permpt] (10) at (12,2) {}; \draw[thin,dashed] (10) -- ++ (-10.5,0); \node[circle,fill,inner sep=0.5pt] (11) at (6.5,7.5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (6,8) {}; \node[circle,fill,inner sep=0.5pt] (13) at (5.5,8.5) {}; \node[circle,fill,inner sep=0.5pt] (14) at (5,9) {}; \node[circle,fill,inner sep=0.5pt] (15) at (4.5,9.5) {}; \draw[thick] (10,0) -- ++ (0,13); \draw[thick] (0,5) -- ++ (13,0); \draw[dotted] (11.5,1.5) rectangle (12.5,2.5); \node[] (16) at (6.5,-1) {$4lulu \dots luld$}; \node[] (17) at (6.5,-2.5) {$=3ulu \dots luldr$}; \end{scope} \end{tikzpicture} \end{center} \caption{Odd collisions ending in $dr$ of length $\geq 6$} \label{fig:oddcollisionsdr} \end{figure} \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,8) {}; \node[permpt] (1) at (9,10) {}; \node[permpt] (2) at (6,9) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (7,6) {}; \draw[thin] (3) -- ++ (0,3.5); \node[permpt] (4) at (4,7) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (5,5) {}; \draw[thin] (5) -- ++ (0,2.5); \node[permpt] (6) at (1,3) {}; \draw[thin] (6) -- ++ (1.5,0); \node[permpt] (7) at (2,1) {}; \draw[thin] (7) -- ++ (0,2.5); \node[permpt] (8) at (11,2) {}; \draw[thin] (8) -- ++ (-9.5,0); \node[permpt] (9) at (10,11) {}; \draw[thin,dashed] (9) -- ++ (0,-9.5); \node[circle,fill,inner sep=0.5pt] (10) at (4.5,5.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4,5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (3.5,4.5) {}; \node[circle,fill,inner sep=0.5pt] (13) at (3,4) {}; \node[circle,fill,inner sep=0.5pt] (14) at (2.5,3.5) {}; \draw[thick] (8,0) -- ++ (0,12); \draw[thick] (0,8) -- ++ (12,0); \draw[dotted] (9.5,10.5) rectangle (10.5,11.5); \node[] (15) at (6.5,-1) {$1ldld \dots ldr$}; \node[] (16) at (6.5,-2.5) {$=2dld \dots ldru$}; \begin{scope}[shift={(18,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (8,4) {}; \node[permpt] (1) at (10,3) {}; \node[permpt] (2) at (9,6) {}; \draw[thin] (2) -- ++ (0,-3.5); \node[permpt] (3) at (6,5) {}; \draw[thin] (3) -- ++ (3.5,0); \node[permpt] (4) at (7,8) {}; \draw[thin] (4) -- ++ (0,-3.5); \node[permpt] (5) at (5,7) {}; \draw[thin] (5) -- ++ (2.5,0); \node[permpt] (6) at (3,11) {}; \draw[thin] (6) -- ++ (0,-1.5); \node[permpt] (7) at (1,10) {}; \draw[thin] (7) -- ++ (2.5,0); \node[permpt] (8) at (2,1) {}; \draw[thin] (8) -- ++ (0,9.5); \node[permpt] (9) at (11,2) {}; \draw[thin,dashed] (9) -- ++ (-9.5,0); \node[circle,fill,inner sep=0.5pt] (10) at (5.5,7.5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (5,8) {}; \node[circle,fill,inner sep=0.5pt] (12) at (4.5,8.5) {}; \node[circle,fill,inner sep=0.5pt] (13) at (4,9) {}; \node[circle,fill,inner sep=0.5pt] (14) at (3.5,9.5) {}; \draw[thick] (8,0) -- ++ (0,12); \draw[thick] (0,4) -- ++ (12,0); \draw[dotted] (10.5,1.5) rectangle (11.5,2.5); \node[] (15) at (6.5,-1) {$4ulul \dots uld$}; \node[] (16) at (6.5,-2.5) {$=1lul \dots uldr$}; \end{scope} \end{tikzpicture} \end{center} \caption{Even collisions ending in $dr$ of length $\geq 6$} \label{fig:evencollisionsdr} \end{figure} We shall now proceed as follows: suppose we have a minimal collision of pin words $w_{1}$ and $w_{2}$ of length $\geq 6$ and that $w_{1}$ ends in $dr$. Then the permutation $\pi^{\circ}$ is of the form given in Fig. \ref{i}. We shall use this to deduce facts about $w_{2}$ in order to show that this collision is in fact one of the pairs listed in Fig.s \ref{fig:oddcollisionsdr} and \ref{fig:evencollisionsdr}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[] (0) at (2,9) {$w_{1} = \_\_\_\_dr$}; \node[permpt] (1) at (2,0) {}; \draw[thin] (1) -- ++ (0,7); \node[permpt] (2) at (6,1) {}; \draw[thin] (2) -- ++ (-5.5,0); \node[] (3) at (2,-1) {\tiny{(A)}}; \node[] (4) at (6,0) {\tiny{(B)}}; \draw (0.5,2.5) rectangle (3.5,5.5); \end{tikzpicture} \end{center} \caption{The permutation $\pi^{\circ}$ generated by the pin word $w_{1}=\_\_\_\_dr$ has this form. Note that $(A)$ is in the lower half-plane and $(B)$ is in the fourth quadrant. All points other than $(A)$ and $(B)$ (of which there are at least $4$ by the assumption that the length of $\pi^{\circ}$ is $\geq 6$) are in the box, with precisely one point on one side of the pin attached to $(A)$ and all other points on the other.} \label{i} \end{figure} As in Fig \ref{i}, we shall call the lowest and second-lowest points of $\pi^{\circ}$ $(A)$ and $(B)$, respectively. These are generated by the final two letters $dr$ of $w_{1}$. We shall split into four cases based on which position the letter of $w_{2}$ which generates $(A)$ is in: the final letter, the inital numeral, or an internal letter. \subsubsection{Case $1$: $(A)$ is the final point} We begin by deducing various facts about the pin word $w_{2}$, using the fact that the permutation $\pi$ looks like Fig. \ref{ii} (with all points other than $(A)$ and $(B)$ in the box), as well as the assumption that the final letter of $w_{2}$ corresponds to the point $(A)$: \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[] (0) at (2,9) {$w_{2} = 4\_\_\_\_d$}; \node[permpt] (1) at (2,0) {}; \draw[thin,dotted] (1) -- ++ (0,7); \node[permpt] (2) at (6,1) {}; \node[] (3) at (2,-1) {\tiny{(A)}}; \node[] (4) at (6,0) {\tiny{(B)}}; \draw (0.5,2.5) rectangle (3.5,5.5); \end{tikzpicture} \end{center} \caption{The fact that the permutation $\pi^{\circ}$ can be generated by the pin word $w_{1}=\_\_\_\_dr$ means that it has this form. Note that $(A)$ is in the lower half-plane and $(B)$ is in the fourth quadrant. All points other than $(A)$ and $(B)$ (of which there are at least $4$ by the assumption that the length of $\pi^{\circ}$ is $\geq 6$) are in the box, with precisely one point on one side of the dotted line attached to $(A)$ and all other points on the other.} \label{ii} \end{figure} \begin{itemize} \item By assumption, the final letter of $w_{2}$ corresponds to the point $(A)$. The point corresponding to the final letter of a pin sequence must be the most extreme point in the direction indicated by that letter. Fig. \ref{ii} shows that $(A)$ is the downmost point but not the most extreme point in any other direction (clearly, $(B)$ is further up and to the right, and the fact that there must be at least one point in the box on each side of the dotted line implies that there is a point further to the left). Hence the final letter of $w_{2}$ \emph{must} be a $d$. \item Consider the pin word $w_{2}^{-}$, formed by removing the final letter of $w_{2}$. This must correspond to a permutation of the shape given in Fig. \ref{iii} (basically Fig. \ref{ii} without the point $(A)$). Note that the point $(B)$ does not separate the bounding rectangle of all other points in the permutation. By the definition of a pin permutation, this can only happen if $(B)$ was the first point placed: so $(B)$ corresponds to the initial numeral of $w_{2}$. But as $(B)$ also corresponds to the final $r$ in the pin word $w_{1} = \_\_\_\_ dr$ it must be in the fourth quadrant. Hence $w_{2}$ must begin with the numeral $4$. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[permpt] (2) at (6,1) {}; \node[] (4) at (6,0) {\tiny{(B)}}; \draw (0.5,2.5) rectangle (3.5,5.5); \end{tikzpicture} \end{center} \caption{The permutation generated by $w_{2}^{-}$ has this form.} \label{iii} \end{figure} \item Hence $w_{2} = 4 \_\_\_\_ d$, with the $4$ corresponding to $(B)$ and the final $d$ corresponding to $(A)$. Suppose that the blank space in the middle contained an $r$. Then this would correspond to a point to the right of all points placed before. But as $(B)$ was the first point placed, this would imply the existence of a point to the right of $(B)$. Fig. \ref{ii} shows that no such point exists, and so $w_{2}$ contains no $r$. \item Similarly, suppose that $w_{2}$ contained another $d$, in addition to the final one. Then this would correspond to a point below $(B)$. But the only point below $(B)$ is $(A)$, already accounted for by the final $d$. Hence $w_{2}$ contains no $d$ apart from its final letter. \item Combining these facts, we see that $w_{2} = 4 \_\_\_\_ d$, with the letters in the blank space alternating between $u$ and $l$. Hence $w_{2} = 4(ul)^{\geq2}d$ or $w_{2} = 4l(ul)^{\geq2}d$, depending of whether the length of $\pi^{\circ}$ is even or odd, respectively. We thus now know what the permutation $\pi$ looks like, as shown in Fig. \ref{iv}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (10,2) {}; \node[permpt] (1) at (12,1) {}; \draw[thin,dotted] (1) -- ++ (-10.5,0); \node[permpt] (2) at (11,4) {}; \draw[thin] (2) -- ++ (0,-3.5); \node[permpt] (3) at (8,3) {}; \draw[thin] (3) -- ++ (3.5,0); \node[permpt] (4) at (9,6) {}; \draw[thin] (4) -- ++ (0,-3.5); \node[permpt] (5) at (6,5) {}; \draw[thin] (5) -- ++ (3.5,0); \node[permpt] (6) at (7,7) {}; \draw[thin] (6) -- ++ (0,-2.5); \node[permpt] (7) at (3,8) {}; \draw[thin] (7) -- ++ (1.5,0); \node[permpt] (8) at (4,10) {}; \draw[thin] (8) -- ++ (0,-2.5); \node[permpt] (9) at (1,9) {}; \draw[thin] (9) -- ++ (3.5,0); \node[permpt] (10) at (2,0) {}; \draw[thin] (10) -- ++ (0,9.5); \node[circle,fill,inner sep=0.5pt] (11) at (5,8) {}; \node[circle,fill,inner sep=0.5pt] (12) at (5.5,7.5) {}; \node[circle,fill,inner sep=0.5pt] (13) at (6,7) {}; \node[circle,fill,inner sep=0.5pt] (14) at (6.5,6.5) {}; \node[] (15) at (2,-1) {\tiny{(A)}}; \node[] (16) at (12,0) {\tiny{(B)}}; \draw[thick] (10,-1) -- ++ (0,12); \draw[thick] (0,2) -- ++ (13,0); \begin{scope}[shift={(18,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (10,3) {}; \node[permpt] (1) at (11,1) {}; \draw[thin,dotted] (1) -- ++ (-9.5,0); \node[permpt] (2) at (8,2) {}; \draw[thin] (2) -- ++ (3.5,0); \node[permpt] (3) at (9,5) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (6,4) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt] (5) at (7,7) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt] (6) at (5,6) {}; \draw[thin] (6) -- ++ (2.5,0); \node[permpt] (7) at (3,9) {}; \draw[thin] (7) -- ++ (0,-1.5); \node[permpt] (8) at (1,8) {}; \draw[thin] (8) -- ++ (2.5,0); \node[permpt] (9) at (2,0) {}; \draw[thin] (9) -- ++ (0,9.5); \node[circle,fill,inner sep=0.5pt] (10) at (4,8) {}; \node[circle,fill,inner sep=0.5pt] (11) at (4.5,7.5) {}; \node[circle,fill,inner sep=0.5pt] (12) at (5,7) {}; \node[circle,fill,inner sep=0.5pt] (13) at (5.5,6.5) {}; \node[] (14) at (2,-1) {\tiny{(A)}}; \node[] (15) at (11,0) {\tiny{(B)}}; \draw[thick] (10,-1) -- ++ (0,11); \draw[thick] (0,3) -- ++ (12,0); \end{scope} \end{tikzpicture} \end{center} \caption{Even and odd cases for the permutation $\pi^{\circ}$, respectively} \label{iv} \end{figure} \item We now return to $w_{1} = \_\_\_\_dr$; as we now know what the permutation $\pi$ looks like, we can deduce that the blank space here also does not contain a $d$ or $r$: first, if the blank space contained a $d$ then the points corresponding to both this and the next letter would be in the lower half-plane. But Fig. \ref{iv} shows that there is at most one point in the lower half-plane in addition to $(A)$ and $(B)$ (which are already accounted for by the final two letters). Similarly, if the blank space contained an $r$ then the points corresponding to this and the next letter (neither of which can be the point $(B)$ as this is accounted for by the final $r$) would be in the right half-plane. But Fig. \ref{iv} shows that there is at most one point in the right half-plane other than $(B)$. Hence $w_{1} = \_\_\_\_dr$ contains no $d$ or $r$ apart from the final two letters. This is now enough to deduce all of $w_{1}$: $w_{1} = 1(lu)^{\geq1}ldr$ in the even case, and $w_{1} = 3(ul)^{\geq2}dr$ in the odd case. \item This means that there is only one (potential) collision of each length $n \geq 6$ in Case $1$: $w_{1} = 1(lu)^{\geq1}ldr$ paired with $w_{2} = 4(ul)^{\geq2}d$ in the even case, and $w_{1} = 3(ul)^{\geq2}dr$ paired with $w_{2} = 4l(ul)^{\geq2}d$ in the odd case. These are the collisions on the right hand sides of Fig.s \ref{fig:oddcollisionsdr} and \ref{fig:evencollisionsdr}. \end{itemize} \subsubsection{Case $2$: $(A)$ is the first point} We now deal with the case in which $(A)$ is the first point placed according to the pin word $w_{2}$, corresponding to the initial numeral. We will again use the shape of the permutation $\pi$ shown in Fig. \ref{i} to deduce the form of $w_{2}$. \begin{itemize} \item First note that, as $(A)$ corresponds to the letter $d$ in $w_{1}$, $(A)$ must be in the $3$rd or $4$th quadrant. Hence $w_{2} = \{3/4\}\_\_\_\_$ \item Next, note that, as $(B)$ is not the first point placed, it must correspond to a letter. If this letter were $d$ or $l$ then $(B)$ would be below or to the left of $(A)$ (as $(A)$ has already been placed), which it is not (see Fig. \ref{i}). If the letter were $u$, then $(B)$ would be the upmost of all points placed so far, and as Fig. \ref{i} shows that all other points of $\pi$ are above $(B)$, this would mean that $(B)$ would have to be the second point placed, so $w_{2}$ begins with either $3u$ or $4u$: in the first case, both $(A)$ and $(B)$ would be in quadrant $3$ and in the second case $(B)$ would be to the left of $(A)$ - Fig. \ref{i} shows that neither of these is true, so $(B)$ cannot correspond to a $u$ in $w_{2}$. Hence, by elimination, $(B)$ corresponds to an $r$ in $w_{2}$. \item Hence $(B)$ is at the end of a right-pin separating the previously placed point from the bounding rectangle of all other previously placed points and the origin. As the origin is above $(B)$ (as we know $(B)$ is in quadrant $4$), the previously placed point must be the only point below $(B)$, namely $(A)$. Hence $(B)$ corresponds to the first letter of $w_{2}$ after the numeral. \item Hence $w_{2}=\{3/4\}r\_\_\_\_$. We can now easily deduce that the blank space here contains no $r$ or $d$: if it contained an $r$ then this would correspond to a point to the right of $(B)$ (as this has already been placed), and it if contained a $d$ then this would correspond to a point below $(A)$; but Fig. \ref{i} clearly shows that no point of either type exists. \item Hence $w_{2} = \{3/4\}rulul \dots$, and so the permutation $\pi^{\circ}$ looks like one of the permutations in Fig. \ref{v}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (5,3) {}; \node[permpt] (1) at (4,1) {}; \node[permpt] (2) at (7,2) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[permpt] (3) at (6,5) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (2,4) {}; \draw[thin] (4) -- ++ (4.5,0); \node[permpt] (5) at (3,7) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt] (6) at (1,6) {}; \draw[thin] (6) -- ++ (2.5,0); \node[circle,fill,inner sep=0.5pt] (7) at (1.5,7) {}; \node[circle,fill,inner sep=0.5pt] (8) at (1,7.5) {}; \node[circle,fill,inner sep=0.5pt] (9) at (0.5,8) {}; \node[circle,fill,inner sep=0.5pt] (10) at (0,8.5) {}; \node[] (11) at (4,0) {\tiny{(A)}}; \node[] (12) at (7,1) {\tiny{(B)}}; \node[] (13) at (7,5) {\tiny{(C)}}; \draw[thick] (5,0) -- ++ (0,8); \draw[thick] (0,3) -- ++ (8,0); \begin{scope}[shift={(13,0)}] \node[circle, draw, fill=none, inner sep=0pt, minimum width=\plotptradius] (0) at (4,3) {}; \node[permpt] (1) at (5,1) {}; \node[permpt] (2) at (7,2) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt] (3) at (6,5) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt] (4) at (2,4) {}; \draw[thin] (4) -- ++ (4.5,0); \node[permpt] (5) at (3,7) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt] (6) at (1,6) {}; \draw[thin] (6) -- ++ (2.5,0); \node[circle,fill,inner sep=0.5pt] (7) at (1.5,7) {}; \node[circle,fill,inner sep=0.5pt] (8) at (1,7.5) {}; \node[circle,fill,inner sep=0.5pt] (9) at (0.5,8) {}; \node[circle,fill,inner sep=0.5pt] (10) at (0,8.5) {}; \node[] (11) at (5,0) {\tiny{(A)}}; \node[] (12) at (7,1) {\tiny{(B)}}; \node[] (13) at (7,5) {\tiny{(C)}}; \draw[thick] (4,0) -- ++ (0,8); \draw[thick] (0,3) -- ++ (8,0); \end{scope} \end{tikzpicture} \end{center} \caption{The two possible cases for the permutation $\pi^{\circ}$, as generated by the pin word $w_{2} = \{3/4\}rulul \dots$. As $n \geq 6$ all the points shown must exist; any further points are a continuation of the oscillation in the second quadrant.} \label{v} \end{figure} \item We now return to $w_{1} = \_\_\_\_dr$ which also generates $\pi^{\circ}$, with the final two letters corresponding to $(A)$ and $(B)$. Note first that all the points in the upper half-plane of Fig. \ref{v} have been placed before $(A)$ in $w_{1}$, and in either case there are at least three points to the left of $(A)$ which have been placed already: this means that the previous letter to the $d$ corresponding to $(A)$ must have been an $r$, and this must correspond to the point marked $(C)$ (the only point other than $(B)$ in the right half-plane). Thus $w_{1} = \_\_\_\_rdr$, with the final three letters corresponding to $(C)$, $(A)$, $(B)$, respectively. Note that the blank space here cannot contain a $d$, as $(A)$ and $(B)$ (already accounted for by the final two letters) are the only points in the lower half-plane. Thus $w_{1} = \_\_\_\_urdr$. But this implies that $(C)$ is the second-upmost point, despite Fig. \ref{v} showing that $(C)$ has at least two points above it. This contradiction implies that there are in fact no collisions in Case $2$. \end{itemize} \subsubsection{Case $3$: $(A)$ is an internal point} We now deal with the case where $(A)$ is generated an internal letter of the pin word $w_{2}$ (ie., neither the initial numeral nor the final letter): \begin{itemize} \item First, we determine which letter represents $(A)$ in $w_{2}$. By looking at Fig. \ref{i}, we see that $(A)$ is in the lower half-plane and is indeed the lowest point in the entire permutation. If $(A)$ were represented by a $u$ in $w_{2}$ it would be in the upper half-plane, so we can exclude this possibility. If $(A)$ were represented by an $l$ or $r$, on the other hand, it would have to follow either the numeral $3$ or $4$ or the letter $d$ (as otherwise $(A)$ would be in the upper half-plane); but in this case $(A)$ would be above the previously-placed point, contradicting the fact that it is the lowest point in the entire permutation. Hence $(A)$ must be represented by the letter $d$ in $w_{2}$. \item Next, note that the $d$ representing $(A)$ must in fact be the \emph{final} $d$ in $w_{2}$: if there were a $d$ after $(A)$ it would place a point below $(A)$, but $(A)$ is the lowest point in the entire permutation. As $(A)$ is internal to $w_{2}$ there is a letter immediately after the $d$ corresponding to $(A)$ in $w_{2}$, either an $l$ or an $r$. The point corresponding to this next letter, which we shall call $(B')$, will be the second-lowest point \emph{so far} when it is placed. But since all points placed after $(B')$ (if there are any) will be in the upper half-plane (as there is no further $d$ in the word), $(B')$ is in fact the second-lowest point in the \emph{entire} permutation. Hence $(B')$ must actually be $(B)$, which must correspond to an $r$ in $w_{2}$ as it is to the right of $(A)$. \item We now know that the points $(A)$ and $(B)$ are generated by a consecutive $dr$ in $w_{2}$, and that there is no letter $d$ in $w_{2}$ after this $dr$ appears; by the same token there is also no further $r$ after this $dr$, as this would generate a point to the right of $(B)$. Hence, after the $dr$ in $w_{2}$ corresponding to $(A),(B)$, the pin word $w_{2}$ alternates between $u$ and $l$ for any remaining points. We now ask how many points are remaining in $w_{2}$ after this final $dr$. \item If there were \emph{no} points after the $dr$ in $w_{2}$ then $w_{1}$ and $w_{2}$ would both end in $dr$, so this would not be a pair of \emph{minimal} collisions. Hence we can assume that there is at least one further letter after the final $dr$, which must be a $u$. \item Now suppose that there were at least two letters after the final $dr$; then $w_{2}$ would have the form $\dots drul \dots$, with any further letters at the end alternating between $u$ and $l$. Thus $\pi^{\circ}$ would look like the permutation shown on the right-hand side of Fig. \ref{vi}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[] (0) at (2,9) {$w_{1} = \dots dr$}; \node[permpt] (1) at (2,0) {}; \draw[thin] (1) -- ++ (0,5.5); \node[permpt] (2) at (5,1) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[] (3) at (2,-1) {\tiny{(A)}}; \node[] (4) at (5,0) {\tiny{(B)}}; \draw (0.5,2) rectangle (3.5,5); \begin{scope}[shift={(13,2)}] \node[] (0) at (1,7) {$w_{2} = \dots drul \dots$}; \node[permpt] (1) at (1,-2) {}; \draw[thin] (1) -- ++ (0,4.5); \node[permpt] (2) at (4,-1) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[permpt] (3) at (3,4) {}; \draw[thin] (3) -- ++ (0,-5.5); \node[permpt] (4) at (-2,3) {}; \draw[thin] (4) -- ++ (5.5,0); \node[] (6) at (1,-3) {\tiny{(A)}}; \node[] (7) at (4,-2) {\tiny{(B)}}; \node[] (8) at (3,5) {\tiny{(C)}}; \node[] (9) at (-2,2) {\tiny{(D)}}; \node[circle,fill,inner sep=0.5pt] (8) at (-1.5,4) {}; \node[circle,fill,inner sep=0.5pt] (9) at (-2,4.5) {}; \node[circle,fill,inner sep=0.5pt] (10) at (-2.5,5) {}; \node[circle,fill,inner sep=0.5pt] (11) at (-3,5.5) {}; \draw (0,0) rectangle (2,2); \end{scope} \end{tikzpicture} \end{center} \caption{The permutation $\pi^{\circ}$ is (under the assumption that there are at least two letters after the $r$ that generates $(B)$ in $w_{2}$) generated by both the words $w_{1} = \dots dr$ and $w_{2} = \dots drul \dots$, so has both of these forms. The points marked $(A)$ and $(B)$ must match up.} \label{vi} \end{figure} \item Hence the permutation $\pi^{\circ}$ is simultaneously of both forms represented in Fig. \ref{vi}. On closer inspection, however, these forms contradict each other, which we can see by counting the points (other than $(B)$) to the left and right of $(A)$: looking first at the diagram generated by $w_{1}$, the box here is non-empty (as the length of $\pi^{\circ}$ is $\geq 6$), and the pin attached to $(A)$ must separate the previously-placed point from all others (including the origin). Hence, excluding $(B)$, there is precisely one point on one side of $(A)$ and all other points (including the origin) are on the other side. Looking at the diagram generated by $w_{2}$, however, we see that the box here is also non-empty (as $(A)$ is internal there is at least one non-origin point preceeding it), with the previous point on one side and all other points (including the origin) on the other. Including the points marked $(C)$ and $(D)$, we now see that there are at least two points (including the origin but excluding $(B)$) on either side of $(A)$, thus contradicting what the diagram generated by $w_{1}$ told us. We conclude that there cannot in fact be more than one point after the final $dr$ in $w_{2}$. \item Thus there is in fact precisely one point after the final $dr$, corresponding to $(A),(B)$ in $w_{2}$, and so $w_{2} = \dots dru$. Hence our permutation $\pi^{\circ}$ is of both the forms shown in Fig. \ref{vii}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.35] \node[] (0) at (2,9) {$w_{1} = \dots dr$}; \node[permpt] (1) at (2,0) {}; \draw[thin] (1) -- ++ (0,5.5); \node[permpt] (2) at (5,1) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[] (3) at (2,-1) {\tiny{(A)}}; \node[] (4) at (5,0) {\tiny{(B)}}; \draw (0.5,2) rectangle (3.5,5); \begin{scope}[shift={(13,2)}] \node[] (0) at (1,7) {$w_{2} = \dots dru$}; \node[permpt] (1) at (1,-2) {}; \draw[thin] (1) -- ++ (0,4.5); \node[permpt] (2) at (4,-1) {}; \draw[thin] (2) -- ++ (-3.5,0); \node[permpt] (3) at (3,4) {}; \draw[thin] (3) -- ++ (0,-5.5); \node[] (6) at (1,-3) {\tiny{(A)}}; \node[] (7) at (4,-2) {\tiny{(B)}}; \node[] (8) at (3,5) {\tiny{(C)}}; \draw (0,0) rectangle (2,2); \end{scope} \end{tikzpicture} \end{center} \caption{The permutation $\pi^{\circ}$ is generated by both the words $w_{1} = \dots dr$ and $w_{2} = \dots dru$, so has both of these forms. The points marked $(A)$ and $(B)$ must match up.} \label{vii} \end{figure} \item But now consider the pin sequence $w_{1}^{-1}$, obtained by removing the final letter from $w_{1}$. This must generate a pin permutation corresponding to the permutations shown in Fig. \ref{vii} with the point $(B)$ removed; in particular, it contains the point $(C)$ which (given the absence of $(B)$) will be both the highest and rightmost point in the corresponding pin permutation $\pi^{\circ}_{w_{1}^{-1}}$. This can only happen if $(C)$ was the \emph{first} point placed in $\pi^{\circ}_{w_{1}^{-1}}$ (otherwise is separates the previously placed point from all others in one direction, making it only the second-most point in that direction); as $(C)$ is clearly in the first quadrant, this means that $w_{1}$ starts with a $1$. Hence $w_{1} = 1 \dots dr$. Further, given that there is no point above $(C)$ in $\pi^{\circ}$ and the only point to the right is $(B)$, accounted for by the final $r$, this means that there is no $u$ in $w_{1}$ and no $r$ except for the final letter. Hence $w_{1}$ is either $1l(dl)^{\geq 1}dr$ or $1(dl)^{\geq 2}dr$, depending on whether the length of $\pi^{\circ}$ is even or odd. Thus $\pi^{\circ}$ is one of the permutations on the left-hand sides of Fig.s \ref{fig:oddcollisionsdr} and \ref{fig:evencollisionsdr}, with $w_{1}$ being the upper word in the pair. Noting that $w_{2} = \dots dru$ cannot contain an $r$ or $u$ apart from the final two letters (all points on the upper and right half-planes are already accounted for) allows us to conclude that $w_{2}$ must be the lower word in the pair. \end{itemize} \end{proof} We can now quickly deduce that \emph{all} collisions are in fact minimal, and thus the list given above is complete: \begin{lemma} Every colliding tuple $\{w_{1},w_{2}, \dots, w_{k}\}$ of pin words is minimal (that is to say, $k \leq 4$ and each pair $w_{i},w_{j}$ differs in the final letter). \end{lemma} \begin{proof} Note that every collision of length $n \leq 5$ (which we have exhaustively listed) is minimal. Now suppose that $\{w_{1},w_{2}\}$ is a non-minimal colliding pair of some length $n \geq 6$. By applying symmetries, we can assume that both $w_{1}$ and $w_{2}$ ends in a $d$. But then this $d$ generates the lowest point in the shared generated pin permutation $\pi^{\circ}$ - in particular, it represents the same point in each pin word. Thus $w_{1}^{-1}$ and $w_{2}^{-1}$ (the pin words obtained by removing the final $d$ from each word in the pair) must also form a collision, indeed a collision of shorter length. Given that $w_{1}$ and $w_{2}$ are not the same word, we can continue this process until we arrive at a minimal collision $w_{1}^{-k},w_{2}^{-k}$. But we have already classified all minimal collisions, and can note that all minimal collisions are of different type: if one of the pair ends in $d$ or $u$ then the other must end in $l$ or $r$, and therefore cannot be extended to a collision of longer length, forming a contradiction. \end{proof} Thus all collisions are minimal and the list given in the table is complete. \subsection{Classification of $\boxplus$-decomposables} We now aim to prove that the list of $\boxplus$-decomposables given in Theorem \ref{classification} is complete: \begin{proof} First, we list the $\boxplus$-decomposables of lengths $2$ and $3$: these are $1l$ and $1ld$ along with their eight symmetries, as can be confirmed by an exhaustive list. We will now construct a $\boxplus$-decomposable pin permutation $\pi^{\circ}$ of length $n \geq 4$, generated by the pin word $w$, and show that it must be one of those listed in the statement of the theorem. We proceed by noting that every $\boxplus$-decomposable pin permutation can be drawn on the diagram in Fig. \ref{fig:boxdec1}, with no points in the shaded region, and at least one point in each of the inner and outer regions. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-4,-11) -- ++ (0,22); \draw[thin] (4,-11) -- ++ (0,22); \draw[thin] (-11,-4) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,4) rectangle (4,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-4) rectangle (-4,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,-11) rectangle (4,-4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (4,-4) rectangle (11,4); \end{tikzpicture} \end{center} \caption{The general shape of a $\boxplus$-decomposable permutation. The shaded region must be empty, whilst both the inner and outer unshaded regions must be non-empty.} \label{fig:boxdec1} \end{figure} By applying symmetries if necessary we can assume that the first point of $\pi^{\circ}$ placed was in the first quadrant. In fact, this point must be placed in the outer region of the first quadrant, as it is impossible to get out of the inner region once a point has been placed in there: if the point $p_{n}$ has been placed in the inner region then the point $p_{n+1}$ will be closer to the origin in the direction specified by the letter that placed $p_{n}$, and hence must also be in the inner region. Hence our permutation $\pi^{\circ}$ starts out like that in Fig. \ref{fig:boxdec2}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-4,-11) -- ++ (0,22); \draw[thin] (4,-11) -- ++ (0,22); \draw[thin] (-11,-4) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \node[permpt,label={\tiny $p_{1}$}] (1) at (5,6) {}; \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,4) rectangle (4,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-4) rectangle (-4,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,-11) rectangle (4,-4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (4,-4) rectangle (11,4); \end{tikzpicture} \end{center} \caption{First point of $\pi^{\circ}$ is placed in the outer region of the first quadrant.} \label{fig:boxdec2} \end{figure} Next, note that it is impossible to get into the inner region of the first quadrant now: every point placed in the first quadrant from now on will be either to the right of or above $p_{1}$. Hence in order to construct a $\boxplus$-decomposable, we must at some point move to another quadrant. Again, by symmetry, we can assume that the next quadrant visited by the pin permutation $\pi^{\circ}$ is the second quadrant, and we now split into cases based on whether this first point in the second quadrant is in the inner or outer region: \subsubsection{Case $1$: First point in the second quadrant is in the outer region} In this case, we may (or may not) begin by oscillating in the first quadrant for any length: if there is an oscillation of length $\geq 2$ this will ensure that the first point placed in the second quadrant will be in the outer region (it will be the second-highest point placed so far, and so will be above $p_{1}$, the first point placed, which is in the outer region); if not, we shall simply make this assumption for now. We thus have a permutation that looks like that in Fig. \ref{fig:boxdec3}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-4,-11) -- ++ (0,22); \draw[thin] (4,-11) -- ++ (0,22); \draw[thin] (-11,-4) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \node[permpt,label={\tiny $p_{1}$}] (1) at (5,6) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (7,5) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt,label={\tiny $p_{3}$}] (3) at (6,8) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{4}$}] (4) at (9,7) {}; \draw[thin] (4) -- ++ (-3.5,0); \node[permpt,label={\tiny $p_{5}$}] (5) at (8,10) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{6}$}] (6) at (-6,9) {}; \draw[thin] (6) -- ++ (14.5,0); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,4) rectangle (4,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-4) rectangle (-4,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,-11) rectangle (4,-4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (4,-4) rectangle (11,4); \draw[pattern=crosshatch,pattern color=red!80,draw=none] (0,-4) rectangle (4,4); \draw[pattern=crosshatch,pattern color=red!80,draw=none] (-4,0) rectangle (0,4); \end{tikzpicture} \end{center} \caption{The first point of $\pi^{\circ}$ outside the first quadrant can be assumed to be in the second quadrant by symmetry.} \label{fig:boxdec3} \end{figure} We note that the inner regions of the first, second and fourth quadrants (shaded in red in Fig. \ref{fig:boxdec3}) are now inaccesible: there are now at least two points in the outer region of the upper half-plane, and because any new point in the upper half-plane must be either the highest or second-highest placed so far, it is now impossible to place any points below those two, hence the inner region of the upper half-plane will remain empty. A similar argument shows that the inner region of the right-halt plane is now also inaccesible: if we had oscillated in the first quadrant initially then there are now at least two points to the right of the inner region in the right half-plane, and the argument goes through exactly as before. If we went directly to the second quadrant with our second point then there would be only one point to the right of the inner region so far, but in order to place anything in the right half-plane again we would need to take a right step, which would create a second, thus rendering the inner region of the right half-plane again inaccesible. In any case, the only part of the inner region which is now accesible is in the third quadrant, so we must be aiming to end up there. If, after placing our first point in the second quadrant, we took either an upward step or a downward step into the outer region of the third quadrant, then the whole of the inner region would be rendered inaccesbile, by the same argument as in the previous paragraph. Hence we must now take a downstep into the inner region of the third quadrant, as in Fig. \ref{fig:boxdec4}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-4,-11) -- ++ (0,22); \draw[thin] (4,-11) -- ++ (0,22); \draw[thin] (-11,-4) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \node[permpt,label={\tiny $p_{1}$}] (1) at (5,6) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (7,5) {}; \draw[thin] (2) -- ++ (-2.5,0); \node[permpt,label={\tiny $p_{3}$}] (3) at (6,8) {}; \draw[thin] (3) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{4}$}] (4) at (9,7) {}; \draw[thin] (4) -- ++ (-3.5,0); \node[permpt,label={\tiny $p_{5}$}] (5) at (8,10) {}; \draw[thin] (5) -- ++ (0,-3.5); \node[permpt,label={\tiny $p_{6}$}] (6) at (-6,9) {}; \draw[thin] (6) -- ++ (14.5,0); \node[permpt,label={\tiny $p_{7}$}] (7) at (-2,-2) {}; \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,4) rectangle (4,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-4) rectangle (-4,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-4,-11) rectangle (4,-4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (4,-4) rectangle (11,4); \draw[pattern=crosshatch,pattern color=red!80,draw=none] (0,-4) rectangle (4,4); \draw[pattern=crosshatch,pattern color=red!80,draw=none] (-4,0) rectangle (0,4); \draw [thin] plot [smooth] coordinates {(-5,9.5) (-5,7) (-4,6) (-2,4) (-2,-2)}; \end{tikzpicture} \end{center} \caption{This is $\boxplus$-indecomposable, but will not be if any right or up step is taken.} \label{fig:boxdec4} \end{figure} Now note that we cannot now place any further points: either a left or right step here will place a point into the shaded region. Hence the only $\boxplus$-decomposable pin permutations in Case $1$ are those shown in Fig. \ref{fig:boxdec4}, which is to say those generated by a pin word of the form $1 \dots ld$, where the only letters in the ellipsis are $u$ and $r$. \subsubsection{Case $2$: First point in the second quadrant is in the inner region} In this case, it is clear that we have to move into the second quadrant immediately after placing the first point in quadrant $1$ (otherwise the first point in quadrant $2$ will be above $p_{1}$ and hence not in the inner region). Hence we start off as in Fig. \ref{fig:boxdec5}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-7,-11) -- ++ (0,22); \draw[thin] (5,-11) -- ++ (0,22); \draw[thin] (-11,-7) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \node[permpt,label={\tiny $p_{1}$}] (1) at (7,6) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (-2,2) {}; \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-7,4) rectangle (5,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-7) rectangle (-7,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-7,-11) rectangle (5,-7); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (5,-7) rectangle (11,4); \draw [thin] plot [smooth] coordinates {(7.5,5) (5,5) (3,4) (0,2) (-2,2)}; \end{tikzpicture} \end{center} \caption{This is $\boxplus$-indecomposable, but will not be if any further point is added.} \label{fig:boxdec5} \end{figure} Note that we can now no longer take an up- or right-step: doing so would place a point above or to the right of $p_{1}$ and therefore within the shaded region. Conversely, if we only take down- and left-steps we can stay in the inner region indefinitely, as shown in Fig. \ref{fig:boxdec6}. \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.25] \draw[very thick] (0,-11) -- ++ (0,22); \draw[very thick] (-11,0) -- ++ (22,0); \draw[thin] (-7,-11) -- ++ (0,22); \draw[thin] (5,-11) -- ++ (0,22); \draw[thin] (-11,-7) -- ++ (22,0); \draw[thin] (-11,4) -- ++ (22,0); \node[permpt,label={\tiny $p_{1}$}] (1) at (7,6) {}; \node[permpt,label={\tiny $p_{2}$}] (2) at (-2,2) {}; \node[permpt,label={\tiny $p_{3}$}] (3) at (-1,-3) {}; \draw[thin] (3) -- ++ (0,5.5); \node[permpt,label={\tiny $p_{4}$}] (4) at (-4,-2) {}; \draw[thin] (4) -- ++ (3.5,0); \node[permpt,label={\tiny $p_{5}$}] (5) at (-3,-5) {}; \draw[thin] (5) -- ++ (0,3.5); \node[permpt,label={\tiny $p_{6}$}] (6) at (-6,-4) {}; \draw[thin] (6) -- ++ (3.5,0); \node[permpt,label={\tiny $p_{7}$}] (6) at (-5,-6) {}; \draw[thin] (6) -- ++ (0,2.5); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-7,4) rectangle (5,11); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-11,-7) rectangle (-7,4); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (-7,-11) rectangle (5,-7); \draw[pattern=crosshatch,pattern color=black!80,draw=none] (5,-7) rectangle (11,4); \draw [thin] plot [smooth] coordinates {(7.5,5) (5,5) (3,4) (0,2) (-2,2)}; \end{tikzpicture} \end{center} \caption{The general form of a $\boxplus$-decomposable pin permutation in Case $2$} \label{fig:boxdec6} \end{figure} Hence we see that we have obtained another family of $\boxplus$-decomposable pin permutations, those generated by the pin words $1ldldl \dots$, and that these are exhaustive in Case $2$. Combining these two cases, we can now see that we have exhaustively obtained all possible $\boxplus$-decomposable pin permutations that begin in the first quadrant and first visit the second. By applying all eight symmetries we thus obtain all $\boxplus$-decomposable pin permutations and can see that these are precisely those described in the statement of the Theorem \ref{classification}. \end{proof} \def\cprime{$'$} \begin{thebibliography}{10} \bibitem{arratia:on-the-stanley-:} Richard Arratia. \newblock On the {S}tanley-{W}ilf conjecture for the number of permutations avoiding a given pattern. \newblock {\em Electron. J. Combin.}, 6:Note 1, 4 pp., 1999. \bibitem{bassino:enumeration-of-:} Fr{\'e}d{\'e}rique Bassino, Mathilde Bouvel, and Dominique Rossin. \newblock Enumeration of pin-permutations. \newblock {\em Electron. J. Combin.}, 18(1):Paper 57, 39, 2011. \bibitem{bbr:unicyclicgrids:} D.~Bevan, R.~Brignall, and N.~Ru\v{s}kuc. \newblock On cycles in monotone grid classes of permutations. \newblock Submitted. \bibitem{bevan2015defs} David Bevan. \newblock Permutation patterns: basic definitions and notation. \newblock arXiv:1506.06673, 2015. \bibitem{bevan:intervals} David Bevan. \newblock Intervals of permutation class growth rates. \newblock {\em Combinatorica}, 38(2):279--303, Apr 2018. \bibitem{brignall-jarvis:pin-classes-ii} R.~Brignall and B.~Jarvis. \newblock Pin classes {II}: {S}mall pin classes. \newblock Submitted. \bibitem{bv:wqo-uncountable:} R.~Brignall and V.~Vatter. \newblock Uncountably many enumerations of well-quasi-ordered permutation classes. \newblock Submitted. \bibitem{brignall:decomposing-sim:} Robert Brignall, Sophie Huczynska, and Vincent Vatter. \newblock Decomposing simple permutations, with enumerative consequences. \newblock {\em Combinatorica}, 28:385--400, 2008. \bibitem{brignall:simple-permutat:b} Robert Brignall, Nik Ru\v{s}kuc, and Vincent Vatter. \newblock Simple permutations: decidability and unavoidable substructures. \newblock {\em Theoret. Comput. Sci.}, 391(1--2):150--163, 2008. \bibitem{cartier:problemes-combi:} P.~Cartier and D.~Foata. \newblock {\em Probl\`emes combinatoires de commutation et r\'earrangements}. \newblock Lecture Notes in Mathematics, No. 85. Springer-Verlag, Berlin, 1969. \bibitem{flajolet:analytic-combin:} Philippe Flajolet and Robert Sedgewick. \newblock {\em Analytic combinatorics}. \newblock Cambridge University Press, Cambridge, 2009. \bibitem{marcus:excluded-permut:} Adam Marcus and G{\'a}bor Tardos. \newblock Excluded permutation matrices and the {S}tanley-{W}ilf conjecture. \newblock {\em J. Combin. Theory Ser. A}, 107(1):153--160, 2004. \bibitem{murphy:restricted-perm:} Maximillian~M. Murphy. \newblock {\em Restricted permutations, antichains, atomic classes, and stack sorting}. \newblock PhD thesis, Univ. of St Andrews, 2002. \bibitem{vatter:small-permutati:} Vincent Vatter. \newblock Small permutation classes. \newblock {\em Proc. Lond. Math. Soc. (3)}, 103:879--921, 2011. \bibitem{waton:on-permutation-:} Steve Waton. \newblock {\em On Permutation Classes Defined by Token Passing Networks, Gridding Matrices and Pictures: Three Flavours of Involvement}. \newblock PhD thesis, Univ. of St Andrews, 2007. \end{thebibliography} \end{document} \usepackage{ifthen} \usetikzlibrary{calc} \usetikzlibrary{decorations.pathreplacing,decorations.markings} \newcommand{\plotptradius}{4pt} \newcommand{\setplotptradius}[1]{\renewcommand{\plotptradius}{#1}} \tikzset{permpt/.style={circle, draw, fill=black, inner sep=0pt, minimum width=\plotptradius}} \tikzset{empty/.style={draw=none, fill=none}} \newcommand\absdot[2]{ \node[permpt] at #1 {}; } \newcommand\absdothollow[2]{ \node[fill=white] at #1 {}; \node[draw=none,fill=none] at #1 [below] {$#2$}; } \newcommand{\plotperm}[2][black]{ \foreach \j [count=\i] in {#2} { \ifnum0=\j {} \else { \node[permpt,fill=#1,draw=#1] (\j) at (\i,\j) {}; }; } \newcommand{\plotpermbox}[4]{ \draw [darkgray, very thick, line cap=round, fill=white] ({#1-0.5}, {#2-0.5}) rectangle ({#3+0.5}, {#4+0.5}); } \newcommand{\plotpermborder}[1]{ \foreach \i [count=\nn] in {#1} {\global\let\n\nn}; \plotpermbox{1}{1}{\n}{\n}; \plotperm{#1}; } \newcommand{\plotpermbordergrid}[1]{ \foreach \i [count=\nn] in {#1} {\global\let\n\nn}; \plotpermbox{1}{1}{\n}{\n}; \draw[step=1cm,gray!50,very thin] (1,1) grid (\n,\n); \plotperm{#1}; } \newcommand{\plotgrid}[1]{ \draw[step=1cm,gray!50,very thin] (0.5,0.5) grid (#1+0.5,#1+0.5); } \newcommand{\plotpermgrid}[1]{ \foreach \i [count=\nn] in {#1} {\global\let\n\nn}; \plotgrid{\n}; \plotperm{#1}; } \newcommand{\plotpinsequence}[1]{ \absdot{(0,0)}{}; \edef\n{0} \edef\s{0} \edef\e{0} \edef\w{0} \edef\x{0} \edef\y{0} \foreach \pin [remember=\pin as \oldpin (initially 1), count=\i] in {#1} { \ifthenelse{\pin=1 \OR \pin=2}{ \ifthenelse{\oldpin=3}{ \xdef\x{\number\numexpr\e-1} }{ \xdef\x{\number\numexpr\w+1} } \ifnum\i=1 \pgfmathparse{\e+1} \xdef\e{\pgfmathresult} }{ \ifthenelse{\oldpin=1}{ \xdef\y{\number\numexpr\n-1} }{ \xdef\y{\number\numexpr\s+1} } \ifnum\i=1 \pgfmathparse{\s-1} \xdef\s{\pgfmathresult} } \ifnum\pin=1 \pgfmathparse{\n+2} \xdef\n{\pgfmathresult} \absdot{(\x,\n)}{}; \ifnum\i>1 \draw (\x,\n) -- (\x,\y-0.5); \else \draw[gray,very thick] (-0.5,-0.5) rectangle (\x+0.5,\n+0.5); \ifnum\pin=2 \pgfmathparse{\s-2} \xdef\s{\pgfmathresult} \absdot{(\x,\s)}{}; \ifnum\i>1 \draw (\x,\s) -- (\x,\y+0.5); \else \draw[gray,very thick] (-0.5,0.5) rectangle (\x+0.5,\s-0.5); \ifnum\pin=3 \pgfmathparse{\e+2} \xdef\e{\pgfmathresult} \absdot{(\e,\y)}{}; \ifnum\i>1 \draw (\e,\y) -- (\x-0.5,\y); \else \draw[gray,very thick] (-0.5,+0.5) rectangle (\e+0.5,\y-0.5); \ifnum\pin=4 \pgfmathparse{\w-2} \xdef\w{\pgfmathresult} \absdot{(\w,\y)}{}; \ifnum\i>1 \draw (\w,\y) -- (\x+0.5,\y); \else \draw[gray,very thick] (0.5,0.5) rectangle (\w-0.5,\y-0.5); }; } \tikzset{ on each segment/.style={ decorate, decoration={ show path construction, moveto code={}, lineto code={ \path [#1] (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast); }, curveto code={ \path [#1] (\tikzinputsegmentfirst) .. controls (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb) .. 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\newcommand{\matt}[9]{\left[\begin{array}{ccc} #1 & #2& #3 \\ #4 & #5& #6 \\ #7 & #8& #9 \end{array}\right]} \newcommand{\xitilde}{\tilde{\xi}} \mathchardef\emptyset="001F \newcommand{\MM}[1]{{\textcolor[rgb]{0,0,1}{[#1]}}} \newcommand{\RC}[1]{{\textcolor{ForestGreen}{[#1]}}} \newcommand{\GF}[1]{{\textcolor[rgb]{0,0.6,0.8}{[#1]}}} \renewcommand{\theenumi}{(\roman{enumi})} \renewcommand{\labelenumi}{\theenumi} \renewcommand{\arraystretch}{1.2} \numberwithin{equation}{section} \newtheorem{defin}{Definition}[section] \newtheorem{remark}[defin]{Remark} \newtheorem{theorem}[defin]{Theorem} \newtheorem{lemma}[defin]{Lemma} \newtheorem{proposition}[defin]{Proposition} \newtheorem{cor}[defin]{Corollary} \newtheorem{example}[defin]{Example} \title{Geometrically constrained walls in three dimensions} \author[R.~Cristoferi]{Riccardo Cristoferi} \address[R.~Cristoferi]{Department of Mathematics - IMAPP, Radboud University, Nijmegen, The Netherlands} \email{[email protected]} \author[G.~Fissore]{Gabriele Fissore} \address[G.~Fissore]{Department of Mathematics - IMAPP, Radboud University, Nijmegen, The Netherlands} \email{[email protected]} \author[M.~Morandotti]{Marco Morandotti} \address[M.~Morandotti]{Dipartimento di Scienze Matematiche ``G.~L.~Lagrange'', Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy} \email{[email protected]} \date{\today} \subjclass[2020]{} \keywords{} \usepackage{subfiles} \makeindex \raggedbottom \begin{document} \begin{abstract} We study geometrically constrained magnetic walls in a three dimensional geometry where two bulks are connected by a thin neck. Without imposing any symmetry assumption on the domain, we investigate the scaling of the energy as the size of the neck vanishes. We identify five significant scaling regimes, for all of which we characterize the energy scaling; in some cases, we are also able to identify the asymptotic behavior of the domain wall. Finally, we notice the emergence of sub-regimes that are not present int previous works due to restrictive symmetry assumptions. \end{abstract} \maketitle \section{Introduction} \label{sec:intro} \emph{Magnetic domain walls} are regions in which the magnetisation of a material changes from one value to another one. In the presence of extreme geometries, such as that of a dumbbell-shaped domain, the magnetic wall is more likely to be found in or around the neck; in this and similar geometry-driven situations, one usually speaks of \emph{geometrically constrained walls}, to stress the fact that the domain shape plays a pivotal role in the localisation of the transition region of the magnetisation, for instance, when prescribing it in the bulky parts of the dumbbell. The study of the behaviour of the magnetisation in a dumbbell-shaped domain is relevant in micro- and nano-electronics application, where the neck of the dumbbell models magnetic point contacts. If one imposes two different values of the magnetisation, one in each of the two macroscopic components, a transition is expected in the vicinity of the neck, as initially observed by Bruno in \cite{Bru99}: if the neck is small enough, so that the geometry of the material varies drastically when passing from one bulk to the other, it can play a crucial role in determining the location of the magnetic wall, by influencing the mere minimisation of the magnetic energy. Three scenarios are to be considered: the transition may happen either completely inside the neck, or partly inside and partly outside of the neck, or completely outside of the neck. The model features a sufficiently smooth potential which is minimal at the imposed values of the magnetisation in the bulk parts of the dumbbell, and a gradient term panalising transition; the two are competing as soon as the values of the magnetisation in the bulks are not the same. To model the geometries that are of interest in the applications, we will consider an infinitesimally small neck, whose size is determined by three parameters $\eps, \delta, \eta>0$: \begin{equation}\label{eq_neck_ep} N_{\eps}\coloneqq\{(x,y,z)\in\R{3}: \|x\|\leq \eps,\|y\|<\delta,\|z\|<\eta\}, \end{equation} with the understanding that all three of them vanish when $\eps\to0$, that is $\delta=\delta(\eps)\to0$ and $\eta=\eta(\eps)\to0$, as $\eps\to0$. The full domain is described by \begin{equation}\label{eq_domain_eps} \Omega_{\eps} \coloneqq \Omega_{\eps}^{\ell} \cup N_{\eps} \cup \Omega_{\eps}^{\rr}, \end{equation} where $\Omega_{\eps}^{\ell}=\Omega^{\ell}-(\eps,0,0)$ and $\Omega_{\eps}^{\rr}=\Omega^{\rr}+(\eps,0,0)$, for certain sets $\Omega^{\ell}\subset\{x<0\}$ and $\Omega^{\rr}\subset\{x>0\}$ such that $0\in\partial\Omega^{\ell}\cap\partial\Omega^{\rr}$. This geometry makes the $x$ direction the preferred one, whereas the $y$- and $z$-direction can be interchanged upon a change of coordinates; this motivates the fact that we will use, throughout the work, the subscript $\eps$ alone as an indication of the smallness of the neck. We are interested in minimising the non-convex energy $F_\eps\colon H^1(\Omega_\eps)\to[0,+\infty)$ defined by \begin{equation}\label{004} F_\eps(u) \coloneqq \frac{1}{2} \int_{\Omega_\eps} \abs{\nabla u(\textbf{x})}^2\,\de \textbf{x} + \int_{\Omega_\eps} W(u(\textbf{x}))\,\de \textbf{x}, \end{equation} where $W\colon \R{} \to [0,\infty)$ is a $C^1$ function such that $W^{-1}(0)=\{\alpha,\beta\}$ for some $\alpha<\beta$, and $\de \textbf{x} = \de x\de y \de z$. In \eqref{004}, the function $u$ represents a suitable quantity related to the magnetisation field defined on $\Omega_\eps$ and the potential $W$ favours the values $u(\textbf{x})=\alpha$ and $u(\textbf{x})=\beta$ for the magnetisation, corresponding to those to be imposed in the bulks. Here, the competition between the potential and the gradient terms is significantly influenced by the geometry of $\Omega_\eps$\,. \subsection{Related literature} The body of literature on this problem counts many physical contributions stemming from Bruno's work \cite{Bru99} and a few mathematical items, which we are going to briefly review to give a context to our novel results. In \cite{Bru99}, Bruno considers the special geometry of a thin ($0<h\ll1$) three-dimensional wall $\Omega=S\times(-h,h)$, where $S$ is a planar region with a dumbbell shape whose neck is located at the origin and the bulks are in $\{x<0\}$ and $\{x>0\}$. He assumes that the preferred directions of the magnetisation vector are $\mathbf{m}=(0,0,\pm1)$ and makes the Ansatz that it varies only in the $y$-$z$ plane, as a function of the $x$-coordinate, namely $$\mathbf{m}=(0,\cos\theta(x),\sin\theta(x)).$$ The energy that Bruno minimises is the one usually describing Bloch walls and turns out to be a functional of $\theta$ alone, with two emerging length scales when imposing that $\mathbf{m}\approx(0,0,\pm1)$ in $\{\pm x>0\}$: one driven by the shape $S$ of the domain, the other one dictated by the physical parameters entering the expression of the energy. Despite Bruno's insightful intuition, the special form of the magnetisation has some limitations, some of which were removed (for instance, by allowing $\mathbf{m}$ to vary also in the $x$-$z$ plane and considering fully three dimensional geometries) in \cite{MolOsiPon02}. Among the mathematical literature related to this topic, we mention \cite{ArrCar04, ArrCarLC06, ArrCarLC09a, ArrCarLC09b, CDLLMur08, HalVeg84, Jim84, Jim88, Jim04} as far as the PDE aspect is concerned, and \cite{CabFreMorPer01, CDLLMur04, CDLLMur04b, RubSchSte04} for variational approaches. Finally, we discuss two more recent contributions in detail, since the results we obtain are similar to them. In the work by Kohn \& Slastikov \cite{KohSla}, the problem is studied in the full three-dimensional setting, with the assumption that the geometry be axisymmetric: the dumbbell $\Omega_{\epsilon}$ is a rotation body around the $x$ axis, so that the shape parameters of the neck are essentially its length $\eps$ and its radius $\delta$. By taking advantage of a scale-invariant Poincar\'{e} inequality for Sobolev functions and by reducing the problem to a one-dimensional variational one, the authors proved the existence of three possible regimes, according to the value of the limit $\lim_{\eps\to0} \delta/\eps=\lambda\in[0,+\infty]$ and singled out a \emph{thin neck} regime ($\lambda=0$), a \emph{normal neck} regime ($\lambda\in(0,+\infty)$), and a \emph{thick neck} regime ($\lambda=+\infty$). In the first case the transition happens entirely inside the neck and is an affine function of the $x$-coordinate, in the second case the transition happens across the neck, partially inside and partially outside, depending on the value of~$\lambda$, whereas in the third case the transition happens entirely outside of the neck. These behaviours are found by studying the energy of particular competitors (essentially, an affine transition inside the neck and a harmonic transition in a spherical shell just outside of the neck) and then rescaling the minimiser in the vicinity of the neck. \smallskip In the works by Morini \& Slastikov \cite{MorSla12, MorSla15} the same problem was addressed in the case of magnetic thin film, that is when the domain has the shape of a dumbbell, but it is two-dimensional, that is, in Bruno's setting in the limit as $h\to0$. Mathematically speaking, the endeavor is more difficult on two accounts: the scale-invariant Poincar\'{e} inequality is not available in dimension two, and the problem loses its variational character. Methods that are typical from the study of PDE's were employed to construct suitable barriers to estimate the solutions. Moreover, due to the slow decay of the logarithm (the fundamental solution to Laplace's equation in two dimensions), sub-regimes became available in additions to the thin, regular, and thick neck regimes already analyzed by Kohn \& Slastikov: the sub-critical, critical, and super-critical thin neck regimes were found according to the value of the limit $\lim_{\eps\to0} (\delta\|\ln\delta\|)/\eps\in[0,+\infty]$, displaying a richer variety. In the case of sub-regimes, the rescaling of the minimisers to study their asymptotic behaviour is not trivial; nonetheless, the authors managed to characterise the profiles as the unique solutions to certain PDE's where the boundary conditions track the expected asymptotic behaviour. Both in Kohn \& Slastikov's and in Morini \& Slastikov's papers the technique involves two steps: (i) rescaling the neck to size of order $1$ to estimate the energy of the minimiser inside it (this is achieved by a simple change of variables) and (ii) rescaling the whole domain $\Omega_\eps$ to $\Omega_\infty$ in a way that either a variational problem or a PDE can be studied in $\Omega_\infty$, which brings to the attention that the boundary $\partial\Omega_\infty$ must be a set in which boundary conditions can be prescribed. \subsection{Novelty of the present contribution and main results} In this paper, we study the full three-dimensional case of the problem with no symmetry assumption on the shape of the neck: it will be a parallelepiped as in \eqref{eq_neck_ep} with all three dimensions independent from one another and all vanishing to zero as $\eps\to0$. When considering the mutual rate of convergence to zero of the three parameters $\eps,\delta,\eta$, we single out five regimes that do not emerge in the analysis of Kohn \& Slastikov, and for each of them we study the energy scaling. We notice that in our three-dimensional setting the scale-invariant Poincar\'{e} inequality is not always available. Therefore, only for those regimes for which it is available, we also investigate the behavior of local minimizers and the associated rescaled problem, which possesses a variational structure. As the roles of $\delta$ and $\eta$ can be interchanged, and noticing that the case $\delta\sim\eta$ corresponds, up to homomorphism, to having a cylindrical neck as in \cite{KohSla}, we always consider $\eta\ll\delta$ and we find are the following regimes: \begin{enumerate} \item[$(i)$] \textit{Super thin}: $\varepsilon\gg\delta\gg\eta$, \item[$(ii)$] \textit{Flat thin}: $\eps\approx\delta\gg\eta$, \item[$(iii)$] \textit{Window thick}: $\delta \gg\eta\gg \eps$, \item[$(iv)$] \textit{Narrow thick}: $\delta\gg\eps\approx\eta$, \item[$(v)$] \textit{Letter-box thick}: $\delta\gg\eps\gg\eta$. \end{enumerate} Due to the geometries that some of these regimes describe, in particular when $\eta\ll\delta$ so that the cross section of the junction of the neck with the bulks is a rectangle with a very large aspect ratio, we find an ellipsoidal competitor carrying less energy than the spherical one proposed in the previous works. As it depends on $\ln(\eta/\delta)$, the energy scaling turns out to exhibit sub-regimes in some of these cases, as described in Section~\ref{sec:heuristics}. The main achievements of the paper are the following. \begin{itemize} \item[(A1)] For all of the above mentioned regimes, we identify where the transition happens. More precisely, we find $\lambda_\varepsilon$, with $\lambda_\eps\to+\infty$ as $\eps\to 0$ such that \[ \lim_{\eps\to0} \lambda_\eps F_\eps(u_\eps) = e_0\in (0,+\infty), \] \[ \lim_{\eps\to 0} \lambda_\eps F_\eps(u_\eps,N_\eps) = e_0^N\in [0,e_0]. \] Their interpretation is the following: $e_0$ is the asymptotic energy in the whole domain, and $e_0^N$ that in the neck. Therefore, if $e_0^N=e_0$, we say that the transition happens entirely inside the neck, if $e_0^N=0$, the transition happens entirely outside of the neck, while if $0<e_0^N<e_0$, the transition happens both inside and outside of the neck. We prove that the transition happens entirely inside the neck in the regimes (i) and (ii) (see Theorem ), entirely outside of the regimes (iii) and (iv), and partly inside and partly outside in (v). Moreover, regime (v) is the richest, because it features three sub-regimes, due to the energetic convenience of the ellipsoidal competitor. \item[(A2)] In the regimes (i) and (ii) we fully characterize the asymptotic behavior of the domain wall: it is a harmonic function in the rescaled domain, and the transition from $\alpha$ to $\beta$ is affine inside the neck. \end{itemize} The plan of the paper is the following. In Section \ref{sec:results} we introduce the standing assumptions, present the heuristics for (A1), and introduce the competitors and compute the energy of each of them. In Section \ref{sec:analysis} tackle the regimes one by one, and provide the proofs of the main results. The study of the asymptotic behavior of the domain wall in the regimes (iii), (iv), and (v) will be discussed in a forthcoming paper. \section{Mathematical statement of the problem}\label{sec:results} We study a mathematical model to characterise magnetic domain walls in a three-dimensional dumbbell-shaped domain. The geometry of the domain can be presented as \begin{equation}\label{001} \Omega_\eps = \Omega_\eps^\ell \cup N_\eps \cup \Omega_\eps^\rr, \end{equation} where \begin{equation}\label{002} \Omega_\eps^\ell = \Omega^\ell + (-\eps,0,0) \qquad \text{and} \qquad \Omega_\eps^\rr = \Omega^\rr + (\eps,0,0), \end{equation} \[ \Omega^\ell\subset \{x<0\}, \qquad \Omega^\rr\subset \{x>0\} \] are two given bounded, connected, open subsets of $\R{3}$, and where \begin{equation}\label{003} N_\eps = [-\eps,\eps]\times(-\delta,\delta)\times(-\eta,\eta) \end{equation} is the \emph{neck} connecting the two bulks. We assume that $\Omega_\eps$ is a Lipschitz and that \begin{itemize}\compresslist \item the origin $(0,0,0)$ belongs to $\partial\Omega^\ell \cap \partial\Omega^\rr$; \item $\Omega^\ell \subset \{x<0\}$ and $\Omega^\rr \subset \{x>0\}$; \item there exists $r_0>0$ such that $(\partial\Omega^\ell) \cap B_{r_0}(0,0,0)$ and $(\partial\Omega^\rr) \cap B_{r_0}(0,0,0)$ are contained in the plane $\{x=0\}$, i.e., the bulks are flat and vertical near conjunction with the neck. \end{itemize} We start by giving the definition of isolated local minimizer for the functional $F_\eps$ introduced in \eqref{004}. \begin{defin}[isolated local minimizer for~$F_\eps$] Define $u_{0,\eps}:\Omega_\varepsilon\to\R{}$ as \begin{align} u_{0,\eps}(\bfx)\coloneqq\begin{cases} \alpha\quad \text{ if } \bfx\in\Omega^\ell_\eps,\\[5pt] \displaystyle 0\quad\text{ if } \bfx\in N_\eps,\\[5pt] \beta\quad \text{ if } \bfx\in\Omega^\rr_\eps. \end{cases} \end{align} we have that $u$ is an isolated local minimizer for $F_\eps$ if \begin{align}\label{eq:min_pb} F_\eps(u)\leq F_\eps(v), \end{align} for every $v\in B_\eps$, where \begin{equation}\label{Beps} B_\eps\coloneqq\{u\in H^1(\O_\eps):\|\|u-u_{0,\eps}\|\|_{L^2(\O_\eps)}<d,\, \\|u\\|_{L^\infty(\Omega_\eps)}<\infty \}, \end{equation} with $d<\min\{\|\O^\ell\|^{1/2},\|\O^\rr\|^{1/2}\}$. \end{defin} \subsection{Heuristics}\label{sec:heuristics} In this section, we show how to heuristically guess where the main part of the energy concentrates, just by looking at the asymptotic relationships between the three geometric parameters $\eps,\delta,\eta$. First of all, we note that, given the privileged role of the parameter~$\eps$, it is trivial to see that the roles of~$\delta$ and~$\eta$ can be interchanged upon switching the coordinate axes~$y$ and~$z$. The regimes investigated in \cite{KohSla} corresponds to the cases where $\delta\sim\eta$. Therefore, here we limit ourselves to consider the other regimes: The regimes we are going to study are: \begin{enumerate} \item[$(i)$] \textit{Super thin}: $\varepsilon\gg\delta\gg\eta$, \item[$(ii)$] \textit{Flat thin}: $\eps\approx\delta\gg\eta$, \item[$(iii)$] \textit{Window thick}: $\delta \gg\eta\gg \eps$, \item[$(iv)$] \textit{Narrow thick}: $\delta\gg\eps\approx\eta$, \item[$(v)$] \textit{Letter-box thick}: $\delta\gg\eps\gg\eta$. \end{enumerate} We now want to guess where the transition will happen: completely inside, completely outside, or in both regions. To understand this, we reason as follows. First of all, we expect the main part of the energy to be the Dirichlet integral. Therefore, we consider two harmonic functions that play the role of competitors for the minimization problem \eqref{eq:min_pb}: one where the transition from $\alpha$ to $\beta$ happens inside the neck, and the other where it happens only outside (and it is constant inside the neck). We then compare their energies (whose computations will be carried out in Section \ref{sec:competitors}) to get a guess of where the transition will happen. The first harmonic function will be referred to as the \emph{affine competitor}, and has energy of order \[ \text{Energy of the affine competitor } = \frac{\delta\eta}{\varepsilon}. \] The second harmonic function will be referred to as the \emph{elliptical competitor}, and has energy of order \[ \text{Energy of the elliptical competitor } = \frac{\|\ln(\eta/\delta)\|}{\delta}. \] When one of the two energies is dominant with respect to the other, it is clear where we expect the transition to happen. In the case they are of the same order, we guess that the transition is both inside and outside of the neck. This will be later confirmed by rigorous analysis. The comparison of the energies of the two harmonic competitors leads to the following heuristics: \begin{enumerate} \item[$(i)$] \textit{Super thin neck}: In this case we have \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\delta}{\eps}\Big(\frac{\eta}{\delta} \Big\|\ln\Big(\frac{\eta}{\delta}\Big)\Big\|\Big)\to 0 \end{align} as $\eps\to0$. We then expect the transition to happen inside of $N_\eps$. \item[$(ii)$] \textit{Flat thin neck}: In that case we obtain \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta}{\delta}\Big\|\ln\Big(\frac{\eta}{\delta}\Big)\Big\|\to 0, \end{align} thus the transition is occurring inside of $N_\eps$. \item[$(iii)$]\textit{Window thick neck}: The comparison of the energies of the harmonic competitors gives \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta}{\eps} \Big\|\ln\Big(\frac{\eta}\delta\Big)\Big\|\to+\infty, \end{align} as $\varepsilon\to0$. The transition is expected to happen entirely outside of the neck \item[$(iv)$] \textit{Narrow thick neck}: In this case we have \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\Big\|\ln\Big(\frac{\eta}\delta\Big)\Big\|\to+\infty, \end{align} as $\eps\to0$. Therefore, we expect the transition to happen outside. \item[$(v)$] \textit{Letter-box neck}: In this case we have the presence of sub-regimes. Indeed, the comparison of the orders of the energies of the harmonic competitors, gives \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta}{\eps}\Big\|\ln\Big(\frac{\eta}{\delta}\Big)\Big\|\approx\frac{\eta\|\ln\eta\|}{\eps}, \end{align} as $\varepsilon\to0$, whose asymptotic is not clear. Therefore, we have to consider the following sub-regimes: \begin{enumerate} \item[$(a)$] \textit{Sub-critical letter-box neck}, when \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta\|\ln\eta\|}{\eps}\to 0, \end{align} as $\eps\to0$. In such a case, the transition happens outside of the neck. \item[$(b)$] \textit{Critical letter-box neck}, when \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta\|\ln\eta\|}{\eps}\to \ell \in(0,\infty), \end{align} as $\eps\to 0$. In this case the transition happens both inside and outside of the neck. \item[$(c)$] \textit{Super critical letter-box neck}, when \begin{align} \frac{\delta\eta}{\eps} \frac{\|\ln(\eta/\delta)\|}{\delta}=\frac{\eta\|\ln\eta\|}{\eps}\to +\infty, \end{align} as $\eps\to0$. Here the transition is expected to happen inside of the neck. \end{enumerate} \end{enumerate} \subsection{Competitors}\label{sec:competitors} The goal of this section is to build two harmonic competitors and to compute the order of their energies. For clarity, we present the affine and elliptic competitors separately. However, at the end of the section, they are mixed together in a more general way. \subsubsection{Affine competitor} Here we build the affine competitor inside the neck and we compute its energy. Let $A,B\in\R{}$, and define the affine function $\xi_\varepsilon \colon \R{3}\to\R{}$ as \begin{align}\label{affinecompetitor} \xi_\varepsilon(x,y,z)\coloneqq\begin{cases} A\quad &\Omega^\ell_\eps\,,\\[5pt] \displaystyle\frac{B-A}{2\eps}x+\frac{B+A}{2}\quad &N_\eps\,,\\[5pt] B\quad& \Omega^\rr_\eps\,. \end{cases} \end{align} Then, we have that \begin{align}\label{AffineBound} \int_{N_\varepsilon} \|\nabla \xi_\varepsilon\|^2 \de \mathbf{x} =\frac{\delta\eta}{\eps}(B-A)^2. \end{align} \subsubsection{Elliptic competitor} In \cite{KohSla}, the authors built an harmonic competitor by imposing boundary condition on \emph{half-spheres} centered at the edges of the neck. The choice of the spherical geometry was dictated by the fact that the authors required $\delta=\eta$. In our case, the geometry will be that of an ellipsoid, driven by the fact that one of the parameters $\delta$ and $\eta$ is larger than the other. In order to define our competitor, we first need to introduce the so called \emph{prolate spheroidal coordinates}. Consider the change of coordinates $(x,y,z) = \Psi(\mu, \nu, \varphi)$, given by \begin{equation}\label{EllipsHConstr} \begin{cases} x = a \sinh \mu \sin \nu \cos \varphi, \\ y = a \cosh \mu \cos \nu, \\ z = a \sinh \mu \sin \nu \sin \varphi, \\ \end{cases} \end{equation} where $(0, \pm a, 0)$ are the coordinates of the foci, $\varphi\in [0,2\pi]$ is the polar angle, $\nu\in[0,\pi]$ is the azimuthal angle and $a,\mu>0$. \\ Define the ellipsoid \begin{equation}\label{ellipsoid} E_M \coloneqq \{\Psi(\mu, \nu, \varphi): \mu < M \}. \end{equation} Moreover, we need consider the left and the right halves of the set $E_M$ translated at the edges of the neck. Namely, we consider the open sets \begin{equation} E_{M,\eps}^\ell \coloneqq \{\Psi(\mu, \nu, \varphi): \mu < 2M, \, \varphi \in (\pi/2, 3\pi/2) \} - (\eps, 0, 0) \subset \Omega_\eps^\ell \end{equation} and \begin{equation} E_{M,\eps}^\rr \coloneqq \{\Psi(\mu, \nu, \varphi): \mu < 2M, \, \varphi \in (-\pi/2, \pi/2) \} + (\eps, 0, 0) \subset \Omega_\eps^\rr. \end{equation} We are now in position to define our competitor $\xi_\eps:\R{3}\to\R{}$ as \begin{equation}\label{testfootball} \xi_\eps(x,y,z)\coloneqq \begin{cases} \alpha & \text{in $\Omega_\eps^\ell\setminus E_{M,\eps}^\ell$\,,} \\[5pt] \displaystyle \frac{\alpha+\beta}{2}-h(x+\eps,y,z) & \text{in $E_{M,\eps}^\ell$\,,} \\[5pt] \displaystyle \frac{\alpha+\beta}{2} & \text{in $N_\eps$\,,} \\[5pt] \displaystyle \frac{\alpha+\beta}{2}+h_\eps(x-\eps,y,z) & \text{in $E_{M,\eps}^\rr$\,,} \\[5pt] \beta & \text{in $\Omega_\eps^\rr\setminus E_{M,\eps}^\rr$\,,} \end{cases} \end{equation} where $h \colon E_{2M} \backslash E_{2m}\to\R{}$ is the solution of the problem \begin{equation} \begin{cases} \Delta h=0 & \text{in } E_{2M} \backslash E_{2m}\,, \\[5pt] \displaystyle h = \frac{\beta-\alpha}{2} & \text{on } \partial E_{2M}\,, \\[5pt] h = 0 & \text{on } \partial E_{2m}\,. \end{cases} \end{equation} Now, our goal is to find explicitly the function $h$ and to estimate, asymptotically, the Dirichlet energy. We look for the solution in the form $h = h(\mu)$. Then the Laplacian in prolate spheroidal coordinates is given by \begin{equation} \Delta h = \frac{1}{a^2 (\sinh^2 \mu + \sin^2 \nu)} \left(h_{\mu \mu} + (\coth \mu) h_\mu \right) = 0 \end{equation} or equivalently \begin{equation} (\sinh \mu h_\mu)_\mu = 0. \end{equation} It follows that \begin{equation} h_\mu = \frac{c}{\sinh \mu}, \end{equation} and thus \begin{equation}\label{hmu} h(\mu) = c \ln \|k \tanh(\mu/2)\|. \end{equation} Enforcing the boundary conditions \begin{equation} h(2M) = \frac{\beta-\alpha}{2}\quad\text{and}\quad h(2m) = 0 \end{equation} yields \begin{equation} k = \coth m \qquad\text{and}\qquad c = \frac{\beta-\alpha}{2\ln\Big(\displaystyle\frac{\tanh M}{\tanh m}\Big)}. \end{equation} Hence, we can write \begin{equation} h(\mu) = \frac{(\beta-\alpha)}{2\ln\Big(\displaystyle\frac{\tanh M}{\tanh m}\Big)}\ln\Big(\displaystyle\frac{\tanh \mu/2}{\tanh m}\Big). \end{equation} We are now in position to compute the Dirichlet energy of the function $\xi_\varepsilon$. Indeed, by using \eqref{hmu}, we obtain \begin{equation}\label{hUB3} \begin{split} \frac{1}{2} \int_{E_{2M}\setminus E_{2m}} \|\nabla \xi_\eps\|^2 \, \de \mathbf{x} =&\, \frac{1}{2} \int_{E_{2M}\setminus E_{2m}} \|\nabla h\|^2 \, \de \mathbf{x} = c^2 a \int_0^\pi \int_0^{2\pi} \int_{2m}^{2M} \frac{\sin \nu}{\sinh \mu} \,\de\nu \de \varphi \de \mu \\ =&\, \frac{\pi a(\beta-\alpha)^2}{\ln \left( \displaystyle\frac{\tanh M}{\tanh m} \right)}. \end{split} \end{equation} In the above expression, there are still two choices we can make: that of the parameter $a$, and of the parameter $m$. We now want to choose $a$ and $m$ in such a way that \begin{equation} \label{fit} (N_\eps \cap \{{x = \pm \eps}\})^\circ \subset \overline{E_{2m} \cap \{x = 0 \}}\pm(\eps,0,0). \end{equation} In order to have \eqref{fit} in force, we note that \eqref{EllipsHConstr} implies that, for all $(x,y,z) \in \partial E_{2m}$ such that $x =0$, it holds \begin{equation}\label{amconditionellipsoid} \frac{y^2}{a^2 \cosh^2 m}+\frac{z^2}{a^2 \sinh^2 m} = 1. \end{equation} Therefore, choosing $a$ and $m$ to satisfy \[ a \sinh m = \sqrt{2} \eta,\quad\quad\quad a \cosh m = \sqrt{2} \delta \] guarantees the validity of \eqref{fit}. We now want to get an asymptotic estimate of \eqref{hUB3}, taking in account the fact that all the regimes in this paper consider the case in which $\eta \ll \delta$. By definition $a^2=\delta^2-\eta^2$, and then $a\approx \delta$ as $\eps\to0$. Moreover \begin{equation} \tanh m = \frac{\eta}{\delta}. \end{equation} Observe that in our regiemes, when $\delta \gg \eta$, then $m \ll 1$ and \begin{equation} \ln \left( \frac{\tanh M}{\tanh m} \right) = \ln \tanh M - \ln \tanh m \approx -\ln \tanh m = -\frac{1}{2}\ln \frac{\eta}{\delta}, \end{equation} for $\eps$ small enough. This, together with \eqref{hUB3} implies \begin{equation}\label{footballenergy} \frac{1}{2} \int_{E_{2M}\backslash E_{2m}} \|\nabla \xi_\eps\|^2 \, \de x \de y \de z \approx \frac{\pi\delta(\beta-\alpha)^2}{\|\ln \frac{\eta}{\delta}\|}, \end{equation} for $\eps$ small enough. \\ Finally, we note that the elliptic competitor just built gives a better upper bound on the energy of the minimizer $u_\varepsilon$ than the one that could be obtained in \cite{KohSla}, with a spherical harmonic function. Indeed, in the spherical harmoic case, they obtained an upper estimate with a term of order $\delta$. Therefore, we can notice that \[ \frac{\delta}{\|\ln(\eta/\delta)\|}\ll\delta, \] as $\eps\to0$. Thus, we obtained a competitor whose order of energy is asymptotically lower than the previous one. This is particularly relevant since such a competitor follows the geometry of our problem, in which the shape of the neck presents the $y$ coordinate way smaller than the $z$ coordinate, ruled by $\delta$ and $\eta$ respectively. \subsubsection{Mixed competitor} The idea now, is to mix the affine competitor in the neck, together with the ellipsoidal just built. The purpose of this new competitor, is to describe whenever the transition happens simultaneously inside and outside of the neck. Recalling \eqref{testfootball} and \eqref{EllipsHConstr}, we define the function $\xi_\varepsilon \colon \R{3}\to\R{}$ as \begin{equation}\label{mixedcompetitor} \xi_\eps(x,y,z)\coloneqq \begin{cases} \alpha & \text{in $\Omega_\eps^\ell\setminus E_{M,\eps}^\ell$,} \\[5pt] h_\eps(x+\eps,y,z) & \text{in $E_{M,\eps}^\ell$,} \\[5pt] \displaystyle\frac{B-A}{2\eps}x+\frac{B+A}{2} & \text{in $N_\eps$,} \\[5pt] g_\eps(x-\eps,y,z) & \text{in $E_{M,\eps}^\rr$,} \\[5pt] \beta & \text{in $\Omega_\eps^\rr\setminus E_{M,\eps}^\rr$}, \end{cases} \end{equation} where $h$ is chosen be the solution to \begin{equation} \begin{cases} \Delta w=0 & \text{in } E_{2M}^\ell \backslash E_{2m}^\ell, \\[5pt] w = \alpha & \text{on } \partial E_{2M}^\ell, \\[5pt] w = A & \text{on } \partial E_{2m}^\ell. \end{cases} \end{equation} and $g$ solves \begin{equation} \begin{cases} \Delta w=0 & \text{in } E_{2M}^r \backslash E_{2m}^r, \\[5pt] w= \beta & \text{on } \partial E_{2M}^\rr, \\[5pt] w = B & \text{on } \partial E_{2m}^\rr. \end{cases} \end{equation} We now want to estimate, asymptotically, its energy. Using the same argument used to obtain \eqref{hmu}, we can write the explicit solution of the problems above as \begin{equation} h(\mu) = c^\ell \ln \|k^\ell \tanh(\mu/2)\|,\quad\quad\quad\text{and}\quad g(\mu) = c^\rr \ln \|k^\rr \tanh(\mu/2)\|. \end{equation} We can explicitly obtain $c^\ell,k^\ell,c^\rr, k^\rr\in\R{}$ by imposing the boundary conditions and we get \begin{align} c^\ell=\frac{\alpha-A}{\ln\Big(\displaystyle\frac{\tanh M}{\tanh m}\Big)},\quad c^\rr=\frac{\beta-B}{\ln\Big(\displaystyle\frac{\tanh M}{\tanh m}\Big)} \end{align} and \begin{align} k^\ell=\frac{\exp\Big(\displaystyle\ln\abs{\frac{\tanh M}{\tanh m}}\frac{A}{\alpha-A}\Big)}{\tanh M}\,\quad k^\rr=\frac{\exp\Big(\displaystyle\ln\abs{\frac{\tanh M}{\tanh m}}\frac{B}{\beta-B}\Big)}{\tanh M}. \end{align} Arguing like in \eqref{hUB3}, we get \begin{align}\label{BoundA} \frac{1}{2}\int_{E_{2M}^\ell\setminus E_{2m}^\ell}\|\nabla h_\eps\|^2=\frac{\pi a(A-\alpha)^2}{\|\ln(\delta/\eta)\|} \end{align} and \begin{align}\label{BoundB} \frac{1}{2}\int_{E_{2M}^\rr\setminus E_{2m}^\rr}\|\nabla g_\eps\|^2\ \de \textbf{x}= \frac{\pi a(B-\beta)^2}{\|\ln(\delta/\eta)\|}, \end{align} as $\eps\to0$. \\ Therefore, from \eqref{AffineBound}, \eqref{BoundA} and \eqref{BoundB}, we obtain \begin{align} F(\xi_\eps)= \frac{\pi a}{\|\ln(\delta/\eta)\|}\big[(A-\alpha)^2+(B-\beta)^2\big]+(B-A)^2\frac{\delta \eta}{\eps}. \end{align} If we notice that since $\eta\ll\delta$m we have $a \approx \delta$, we can write, for $\eps$ small enough \begin{align}\label{lowerbound} F(\xi_\eps)\approx \frac{\pi \delta}{\|\ln(\delta/\eta)\|}\big[(A-\alpha)^2+(B-\beta)^2\big]+(B-A)^2\frac{\delta \eta}{\eps}. \end{align} if we optimize \eqref{lowerbound} for $A$ and $B$ we get \begin{align}\label{optimalAandB} A=\frac{\displaystyle\frac{\pi\alpha}{\abs{\ln(\eta/\delta)}}+\frac{\eta}{\eps}(\alpha+\beta)}{\displaystyle\frac{\pi}{\abs{\ln(\eta/\delta)}}+\frac{2\eta}{\eps}}\quad\text{and}\quad B=\frac{\displaystyle\frac{\pi\beta}{\abs{\ln(\eta/\delta)}}+\frac{\eta}{\eps}(\alpha+\beta)}{\displaystyle\frac{\pi}{\abs{\ln(\eta/\delta)}}+\frac{2\eta}{\eps}}. \end{align} The choice of $A$ and $B$ in \eqref{optimalAandB} will be crucial in the various regimes when we will need to infer the boundary conditions of the rescaled profile at the edge of the neck. In conclusion, from \eqref{lowerbound}, if $u_\eps$ is a local minimizer, we then have \begin{align}\label{upperboundueps} F(u_\eps)\leq\frac{\pi\delta}{\|\ln(\delta/\eta)\|}\big[(A-\alpha)^2+(B-\beta)^2\big]+\frac{\delta \eta}{\eps}(B-A)^2. \end{align} Notice that the right-hand side of \eqref{upperboundueps} makes clear the energetic contribution of the competitor inside and outside of the neck, with their respective order of the energy. \section{Analysis of the problem in the several regimes} \label{sec:analysis} In this section we carry out the rigorous analysis of the asymptotic behavior of the solution, obtaining information on its energy and its behavior inside and close to the neck. \subsection{Super-thin neck} In this regime the parameters are ordered as $\eps \gg \delta \gg \eta$. Namely, we have \begin{align} \lim_{\eps\to0} \frac{\delta}{\eps}=0\quad\text{and}\quad\lim_{\eps\to0}\frac{\eta}{\delta}=0. \end{align} According to the heuristics in Section \ref{sec:heuristics}, we we expect the transition to happen entirely inside the neck. Thus, in order to understand the transition inside the neck, we rescale it to make it the set $N\coloneqq [-1,1]^3$. If $u_\eps$ is a local minimizer of the functional \eqref{004}, the only rescaling that works is \begin{align} v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\delta y, \eta z). \end{align} If we rescale in this way, the limiting domain becomes $$\Omega_\infty = \Omega^\ell_\infty \cup N \cup \Omega^\rr_\infty,$$ where $\Omega^\ell_\infty = \{x<-1\}$ and $\Omega^\rr_\infty = \{x>1\}$. \begin{theorem}\label{Theoremsuperthin} Assume $\eps \gg \delta \gg \eta$. Then, \begin{align} \lim_{\eps\to0} \frac{\eps}{\delta\eta}F_\eps(u_\eps)=\lim_{\eps\to 0} \frac{\eps}{\delta\eta}F_\eps(u_\eps,N_\eps)= (\beta-\alpha)^2, \end{align} which means that the main contribution of the energy is given by the transition inside the neck. Moreover, for $\varepsilon>0$ let $v_\varepsilon \colon [-1,1]^3\to\R{}$ be defined as \[ v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\delta y, \eta z). \] Then, $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x)$, where $v\in H^1(-1,1)$ is the unique minimizer of the variational problem \begin{align} \min\bigg\{ \frac{1}{2} \int_{-1}^1 \|v'\|^2\ \de x:\ v\in H^1(-1,1),\ v(-1)=\alpha,\ v(1)=\beta\bigg\}. \end{align} In particular, $v(x)=\dfrac{\beta-\alpha}{2}x+\dfrac{\alpha+\beta}{2}$. \end{theorem} \begin{proof} \emph{Step 1: limit of $v_\varepsilon$.} We claim that \[ \sup_{\varepsilon>0} \\|v_\varepsilon\\|_{H^1(N)} < \infty. \] First of all, using the fact that $\eps \gg \delta \gg \eta$, we get that \begin{align}\label{eq:est_grad_v_eps} \\|\nabla v_\varepsilon\\|^2_{L^2(N)} &=\frac{\varepsilon}{2\delta\eta} \int_N \bigg((\partial_x u_\eps)^2 + \frac{\delta^2}{\eps^2}(\partial_y u_\eps)^2 + \frac{\eta^2}{\eps^2}(\partial_z u_\eps)^2\bigg)\,\de\textbf{x} \leq\frac{\varepsilon}{\delta\eta} F_\varepsilon(u_\varepsilon) \leq C <\infty, \end{align} where the first inequality follows by using a change of variable, while the second one from \eqref{AffineBound} with $A=\alpha$ and $B=\beta$. Moreover, from \[ \int_{N} \|v_\eps\|^2\,\de \mathbf{x}=\frac{1}{\eps\delta\eta}\int_{N_\eps} \|u_\eps\|^2\,\de\mathbf{x}\leq \frac{\big(\sup_{\mathbf{x}\in N_\eps}\|u_\eps\|^2\big) \|N_\eps\|}{\eps\delta\eta}=C. \] Thus, we get that, up to a subsequence, $v_\varepsilon$ converges weakly in $H^1(N)$ to a function $\hat{v}\in H^1(N)$. The independence of the subsequence will follow from Step $2$, where we show that the limit is the \emph{unique} solution to a variational problem. \emph{Step 2: limiting problem} We now want to characterize the function $v$ as the unique solution to a variational problem. We do this in two steps: first, we identify a functional that will be minimized, and then we identify the boundary conditions.\\ We have that \begin{align}\label{eq:est_energy_stn} \liminf_{\eps\to0}\frac{\eps}{\delta\eta} F(u_\eps)&\geq \liminf_{\eps\to0}\frac{\eps}{2\delta\eta}\Big(\int_{N_\eps}\|\nabla u_\eps\|^2\de \textbf{x} +\int_{N_\eps} W(u_\eps)\ \de\bfx \Big)\nonumber \\[5pt] &=\liminf_{\eps\to0}\frac{1}{2}\int_{N} \bigg((\partial_x v_\eps)^2 + \frac{\eps^2}{\delta^2}(\partial_y v_\eps)^2 + \frac{\eps^2}{\eta^2}(\partial_z v_\eps)^2\bigg)\,\de \textbf{x} +\eps^2\int_N W(v_\eps)\, \de\bfx \nonumber \\[5pt] &\geq\liminf_{\eps\to0}\frac{1}{2}\int_N \|\nabla v_\eps\|^2 \de \textbf{x} \geq \frac{1}{2}\int_N \|\nabla v\|^2\, \de \textbf{x} \end{align} Notice that, from the bound \eqref{eq:est_grad_v_eps} and the fact that $\eps/\eta\to\infty$ and $\eps/\delta\to\infty$, we necessarily have that $v$ does not depend on $y$ and $z$. Namely, $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x)$, where $v\in H^1(-1,1)$. Therefore, from \eqref{eq:est_energy_stn}, we can write \begin{align}\label{eq:est_energy_stnpt2} \frac{1}{2}\int_N \|\nabla v\|^2 \de \textbf{x} = & 2\int_{-1}^1 (\hat{v}')^2\ \de x \geq2\min\Big\{ \int_{-1}^1 (w')^2\ \de x:\ w\in H^1(-1,1),\ w(\pm1)=\hat{v}(\pm1)\Big\} \nonumber \\[5pt] &=(\hat{v}(-1)-\hat{v}(1)\big)^2, \end{align} where last step follows by an explicit minimization. Now we claim that $\hat{v}(-1)=\alpha$ and $\hat{v}(1)=\beta$. We prove the former, since the latter follows by using a similar argument. The idea (introduced in \cite{KohSla}) is to use the scale-invariant Poincaré inequality \begin{align}\label{poincare} \Big(\int_{\Omega_\eps^\ell}\|u_\eps-\bar{u}_\eps\|^6\ \de \textbf{x}\Big)^{\frac{1}{6}}\leq C \Big(\int_{\Omega_\eps^\ell}\|\nabla u_\eps\|^2\ \de \textbf{x}\Big)^\frac{1}{2}, \end{align} where $C>0$ and $\bar{u}_\eps$ is the average on $\Omega_\eps^\ell$ of $u_\eps$. Using a change of variable, we estimate (by neglecting the potential term as in \eqref{eq:est_energy_stnpt2}) the right-hand side as \begin{align} \eps\delta\eta\int_{\Omega_\eps^\ell}\|v_\eps-\bar{v}_\eps\|^6\ \de \textbf{x} \leq C \Big(\frac{\delta\eta}{\eps} \int_{\Omega_\eps^\ell}(\partial_x v_\eps)^2 +\frac{\eps^2}{\delta^2}(\partial_y v_\eps)^2 +\frac{\eps^2}{\eta^2}(\partial_z v_\eps)^2\ \de \textbf{x} \Big)^3 = C \Big(\frac{\delta\eta}{\eps}\Bigr)^3 \left(\frac{\eps}{\delta\eta}F_\eps(u_\eps)\right)^3, \end{align} Thus, \begin{align}\label{poincareestimate} \int_{\Omega_\eps^\ell}\|v_\eps-\bar{v}_\eps\|^6\ \de \textbf{x}\leq C \Big(\frac{\delta\eta}{\eps^2}\Big)^3\left(\frac{\eps}{\delta\eta}F_\eps(u_\eps)\right)^3\to0, \end{align} as $\varepsilon\to0$. Indeed, by assumption $\delta\eta/\eps^2\to0$ as $\eps\to0$ and, thank to \eqref{eq:est_grad_v_eps}, we see that the term $\frac{\eps}{\delta\eta}F_\eps(u_\eps)$ is uniformly bounded in $\varepsilon$ . Therefore, $v_\varepsilon\to \bar{v}_\eps$ in $L^6(\Omega_\eps^\ell)$. We now claim that $\bar{v}_\eps=\alpha$. Indeed, by definition of local minimizer we have that \[ \|\|u_\eps-\alpha\|\|_{L^1(\Omega_\eps^\ell)}\to0 \] as $\eps\to0$, giving that also the average of $u_\eps$ on $\Omega_\eps^\ell$ tends to $\alpha$ in $L^1$. Therefore, in the rescaled profile $v_\eps$ we also get that $\bar{v}_\eps\to \alpha$ as $\eps\to 0$. This gives that $\hat{v}(-1)=\alpha$. The reasoning for $\O^\rr_\eps$ is analogous, so we get that $\hat{v}(1)=\beta$. \emph{Step 3: asymptotic behavior of the energy.} From \eqref{eq:est_energy_stnpt2} we can conclude that \begin{align}\label{liminfsuperthin} \liminf_{\eps\to0}\frac{\eps}{\delta\eta} F(u_\eps)\geq (\beta-\alpha)^2. \end{align} On the other hand, if $w_\eps$ is the affine competitor in \eqref{affinecompetitor}, we have that \begin{align}\label{limsupsupethin} \limsup_{\eps\to 0}\frac{\eps}{\delta\eta} F(u_\eps)\leq\limsup_{\eps\to 0} \frac{\eps}{\delta\eta}F(w_\eps)=(\beta-\alpha)^2. \end{align} Finally, from \eqref{upperboundueps}, \eqref{liminfsuperthin}, and \eqref{limsupsupethin}, we get \begin{align} \lim_{\eps\to0} \frac{\eps}{\delta \eta} F(u_\eps)=\lim_{\eps\to 0} \frac{\eps}{\delta \eta} F(u_\eps,N_\eps)=(\beta-\alpha)^2. \end{align} This concludes the proof. \end{proof} \subsection{Flat-thin neck} In this regime the parameters are ordered as $\eps \approx\delta \gg \eta$. Namely, we have \begin{align} \lim_{\eps\to0} \frac{\delta}{\eps}=m\in(0,+\infty)\quad\text{and}\quad \lim_{\eps\to 0}\frac{\eta}{\eps}=0. \end{align} This case is similar to the super-thin regime and techniques used are similar. Therefore, we only highlight the main differences with theorem \ref{Theoremsuperthin}. Since we expect the transition to happen entirely inside the neck, we would like to use a rescaling for which the neck $N_\eps$ transforms in $N\coloneqq [-1,1]^3$. If $u_\eps$ is a local minimizer of the functional \eqref{004}, the only rescaling that works is \begin{align} v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\eps y, \eta z). \end{align} If we rescale in this way, the limiting domain becomes $$\Omega_\infty = \Omega^\ell_\infty \cup N \cup \Omega^\rr_\infty,$$ where $\Omega^\ell_\infty = \{x<-1\}$ and $\Omega^\rr_\infty = \{x>1\}$. \begin{theorem} Assume $\eps \approx \delta \gg \eta$. Then, \begin{align} \lim_{\eps\to0} \frac{1}{\eta}F_\eps(u_\eps)=\lim_{\eps\to 0} \frac{1}{\eta}F_\eps(u_\eps,N_\eps)= (\beta-\alpha)^2, \end{align} which means that the main contribution of the energy is given by the transition inside the neck. Moreover, for $\varepsilon>0$ let $v_\varepsilon:[-1,1]^3\to\R{}$ be defined as \[ v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\eps y, \eta z). \] Then, $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x)$, where $v\in H^1(-1,1)$ is the unique minimizer of the variational problem \begin{align} \min\bigg\{ \frac{1}{2} \int_{-1}^1 \|v'\|^2\ \de x:\ v\in H^1(-1,1),\ v(-1)=\alpha,\ v(1)=\beta\bigg\}. \end{align} In particular, $v(x)=\dfrac{\beta-\alpha}{2}x+\dfrac{\alpha+\beta}{2}$. \end{theorem} \begin{proof} In the same way as in Theorem \ref{Theoremsuperthin}, we can prove that \begin{align}\label{boundFflatthin} \\|\nabla v_\eps\\|^2_{L^2(N)}\leq \frac{1}{\eta}F_\eps (u_\eps)\leq C<\infty \end{align} and $\|\|v_\eps\|\|_{L^2(N)}<C$ uniformly in $\varepsilon$. Therefore, by compactness there is $v\in H^1(N)$ such that, up to a subsequence, $v_\eps\wto v$ in $H^1(N)$. The independence of the subsequence will follow from the fact that the limit is the \emph{unique} solution to a variational problem. \\ Thus, we can write \begin{align}\label{liminfflatthin} \nonumber \liminf_{\eps\to0}\frac{1}{\eta} F(u_\eps)&\geq \frac{1}{2} \int_{\O_\infty} \bigg((\partial_x v_\eps)^2 + (\partial_y v_\eps)^2 + \frac{\eps^2}{\eta^2}(\partial_z v_\eps)^2\bigg)\,\de\mathbf{x}+\eps^2\eta\int_{\O_\infty} W(v_\eps)\,\de\mathbf{x}\\[5pt] \nonumber&\geq\int_{-1}^1 (\partial_x\hat{v}')^2+(\partial_y \hat{v})^2\ \de x\de y\\[5pt] \nonumber&\geq\min\bigg\{\int_{[-1,1]^2} (\partial_x w)^2+(\partial_y w)^2\ \de x\de y:\ w\in H^1\big([-1,1]^2\big),\\[5pt] & \phantom{\geq\min \bigg\{\;} w(\pm1,y)=\hat{v}(\pm1,y),\ \forall y\in[-1,1]\bigg\}, \end{align} Where in the previous to last step we used \eqref{boundFflatthin} and the fact that $\eps/\eta\to\infty$. Then we necessarily have that $v$ does not depend on $z$. Namely, $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x,y)$, where $v\in H^1([-1,1]^2)$. \\ Now, would like to show that the boundary conditions $\hat{v}(\pm1,y)$ are independent from $y$. This is done by acting similarly as in \eqref{poincare} and \eqref{poincareestimate}. Indeed by using the scale-invariant Poincarè inequality, we get\\ \begin{align}\label{poincareestimate2} \int_{\Omega_\eps^\ell}\|v_\eps-\bar{v}_\eps\|^6\ \de \textbf{x}\leq C \Big(\frac{\eta}{\eps}\Big)^2\Big(\frac{1}{\eta}F_\eps(v_\eps)\Big)^3. \end{align} From \eqref{boundFflatthin} and $\eta/\eps\to0$ as $\eps\to0$ we can conclude that $v=\alpha$ on $\Omega^\ell_\infty$ and $v=\beta$ on $\Omega^r_\infty$ independently on $y$. Therefore, from \eqref{liminfflatthin} we get \begin{align} \liminf_{\eps\to0}\frac{1}{\eta} F(u_\eps,N_\eps) &\geq\min\bigg\{ \int_{[-1,1]^2} (\partial_x w)^2+(\partial_y w)^2\ \de x\de y:\ w\in H^1\big([-1,1]^2\big),\\[5pt] &\phantom{\geq\min\bigg\{} w(-1,y)=\alpha,\ w(1,y)=\beta,\ \forall y\in[-1,1]\bigg\}\\[5pt] &\geq\min\bigg\{ \int_{[-1,1]^2} (\partial_x w)^2\ \de x\de y:\ w\in H^1\big([-1,1]^2\big),\\[5pt] &\phantom{\geq\min\bigg\{} w(-1,y)=\alpha,\ w(1,y)=\beta,\ \forall y\in[-1,1]\bigg\}\\[5pt] &=(\beta-\alpha)^2 \end{align} On the other hand, if $w_\eps$ is the affine competitor in \eqref{affinecompetitor}, we have that \begin{align}\label{limsupsupethin} \limsup_{\eps\to 0}\frac{1}{\eta} F(u_\eps)\leq\limsup_{\eps\to 0} \frac{1}{\eta}F(w_\eps)\leq(\beta-\alpha)^2. \end{align} Finally, by putting all together and considering \eqref{upperboundueps}, we get \begin{align} \lim_{\eps\to0} \frac{1}{ \eta} F(u_\eps)=\lim_{\eps\to 0} \frac{1}{ \eta} F(u_\eps,N_\eps)=(\beta-\alpha)^2. \end{align} This concludes the proof. \end{proof} \subsection{Window thick and narrow thick regimes} Here we consider the scaling of the energy in two regimes: the window thick, where \begin{align} \lim_{\eps\to0}\frac{\eta}{\delta}=0\quad\text{and}\quad\lim_{\eps\to0} \frac{\eta}{\eps}=+\infty, \end{align} and the narrow thick, where \begin{align} \lim_{\eps\to0}\frac{\eta}{\delta}=0\quad\text{and}\quad\lim_{\eps\to0} \frac{\eta}{\eps}=l\in(0,\infty), \end{align} Since the ellipsoidal competitor outside of the neck provides an energy whose order is lower than the energy of the affine competitor in the neck, we expect the transition happening outside of the neck. Therefore, among all the possible rescalings, we choose to preserve a \textit{window} in the $yz$-plane. More generally, we would like to get a non-trivial domain that connects the two bulks. This is possible if we rescale by $(\delta,\delta,\eta)$ in the window thick regime and by $(\eta,\delta,\eta)$ in the narrow thick one. \begin{theorem}\label{Theoremwindowthick} In the window thick or in the narrow thick regimes, it holds that \begin{equation} \lim_{\eps\to0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps)=\lim_{\eps\to 0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps,\O_\eps\setminus N_\eps)= \pi(\beta-\alpha)^2. \end{equation} which translates into the fact that the transition happens entirely outside of the neck. \end{theorem} \begin{proof} \emph{Step 1. Window thick regime.} We can find an energy bound by using the competitor built in \eqref{testfootball}. This, together with \eqref{footballenergy}, we get that \begin{align} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps)\leq \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(\xi_\eps)\leq \pi(\beta-\alpha)^2+C_\eps, \end{align} for some $C_\eps\geq0$ which is going to $0$ as $\eps\to 0$. We choose the rescaling \begin{align} \hat{u}_\eps(x,y,z)\coloneqq u_\eps(\delta x,\delta y ,\eta z). \end{align} In such a way the limiting domain is \begin{align} \Omega_\infty\coloneqq \{x<0\}\cup\Big[\{0\}\times [-1,1]\times [-1,1]\Big]\cup \{x>0\}. \end{align} First, we notice that since $u_\eps$ is a critical point of the energy $F_\eps$, we have that $u_\eps$ satisfies the Euler-Lagrange equation \begin{align} \int_{\Omega_\eps}\nabla u_\eps \cdot \nabla\varphi\ \de\textbf{x}-\int_{\Omega_\eps} W'(u_\eps)\varphi\ \de\textbf{x}=0. \end{align} For every $\varphi\in H^1(\Omega_\eps)$ Therefore, we get \begin{align}\label{nablatozero} \int_{\Omega_\eps}\abs{\nabla u_\eps}^2\ \de\textbf{x}=\int_{\Omega_\eps} W'(u_\eps)u_\eps\ \de \textbf{x}\to 0, \end{align} since $u_\eps$ is uniformly bounded, and $\chi_{\Omega_\eps} W'(u_\eps)\to 0$ in $L^p(\Omega)$, for every $p\geq 1$, as $\eps\to0$. Therefore, if we set $\hat{u}_\eps(x,y,z)\coloneqq u_\eps(\delta x,\delta y ,\eta z)$, we have that $\hat{u}_\eps$ satisfies weakly \begin{align} \begin{cases} \Delta \hat{u}_\eps=\delta^2\eta W'(\hat{u}_\eps)\quad&\text{in}\ \Omega_\infty\\[5pt] \displaystyle\frac{\partial \hat{u}_\eps}{\partial \nu}=0\quad&\text{on}\ \partial\Omega_\infty. \end{cases} \end{align} We would like to apply \cite[Theorem 6.2]{MorSla12}. Since $\delta^2\eta \chi_{\Omega_\eps}W'(\hat{u}_\eps)\to0$ as $\eps\to0$ in $L^p(\Omega)$, for every $p\geq 1$. Therefore, up to a subsequence, there is $u^\ast$, weak solution of the problem \begin{align} \begin{cases} \Delta u^\ast=0\quad&\text{in}\ \Omega_\infty\\[5pt] \displaystyle\frac{\partial u^\ast}{\partial \nu}=0\quad&\text{on}\ \partial\Omega_\infty, \end{cases} \end{align} for which $\hat{u}_\eps \to u^\ast$ in $W^{2,p}_{\text{loc}}(\Omega_\infty)$, as $\eps\to0$. It turns out that $u^\ast$ is locally constant. By the Sobolev embedding, we can also state that such a convergence is locally uniform. Consider the prolate ellipsoidal coordinates $\Psi$ defined in \eqref{EllipsHConstr}. Let $M>0$, that will be chosen later, but large enough such that \begin{align}\label{chiuderelafinestra} \Big[\{0\}\times [-1,1]\times [-1,1]\Big]\subset E_M\cap\{x=0\} \end{align} Consider the ellipsoid \begin{align} E_M\coloneqq\{\Psi(\mu,\nu,\varphi):\mu<2M\}, \end{align} and the sphere centered at the origin $B_M\coloneqq B_M(0,0,0)$ such that $E_M\subset B_M$. By moving into prolate elliptical coordinates, from the local uniform convergence of $\hat{u}_\eps$ we have \begin{align} \hat{u}_\eps\big(\Psi(\mu,\nu,\varphi)\big)\to m\quad\text{locally uniformly in}\ B_M\cap \Omega_\infty. \end{align} In particular, as $\partial B_M\cap \Omega_\infty$ is compact, we get \begin{align} \hat{u}_\eps\big(\Psi(\mu,\nu,\varphi)\big)\to m\quad\text{ uniformly in}\ \partial B_M\cap \Omega_\infty. \end{align} From the above fact, we infer that for every $\eps>0$ there is $\gamma>0$ such that \begin{align}\label{firstboundarycondition} m-\gamma < u_\eps\big(\Psi(\mu,\nu,\varphi)\big)< m+\gamma\quad\text{on } \partial B_{M_{\delta\eta}}, \end{align} where $B_{M_{\delta\eta}}$ is the rescaled sphere obtained from $B_M$. The convergence \eqref{firstboundarycondition} is also true on $\partial E_{_{M_{\delta\eta}}}\subset B_{M_{\delta\eta}}$, where $E_{_{M_{\delta\eta}}}$ is the rescaled ellipsoid obtained from $E_M$. \\ Consider $r_0>0$ for which $\partial \Omega_\infty\cap E_{r_0}$is a vertical ellipses. Let $0<\rho<r_0$ and define \begin{align} E_{\eps,\rho}^\ell &\coloneqq E_\rho\cap \{x<-\eps\}\\[5pt] E_{\eps,\rho}^r&\coloneqq E_\rho\cap \{x>\eps\}. \end{align} By hypothesis and \eqref{nablatozero}, we have that \begin{align} \gamma_\eps^\ell&\coloneqq \|\|u_\eps-\alpha\|\|_{L^\infty(E_{\eps,\rho}^\ell )}\to0,\\[5pt] \gamma_\eps^r&\coloneqq \|\|u_\eps-\beta\|\|_{L^\infty( E_{\eps,\rho}^r )}\to0, \end{align} as $\eps\to0$. From the above fact we infer that, \begin{align}\label{secondboundarycondition} \nonumber\alpha-\gamma_\eps^\ell<&u_\eps<\alpha+\gamma_\eps^\ell\quad\text{on }\partial E_{\eps,\rho}^\ell, \\[5pt] \beta-\gamma_\eps^\rr<&u_\eps<\beta+\gamma_\eps^\rr\quad\text{on }\partial E_{\eps,\rho}^\rr. \end{align} Notice that $E_{M_{\delta\eta}}$ and $E_\rho$ are two concentric ellipsoids. Therefore, from \eqref{firstboundarycondition} and \eqref{secondboundarycondition} we can write \begin{align} \frac{1}{2}\int_{E_{\eps,\rho}^\ell \setminus E_{M_{\delta\eta}}}&\abs{\nabla u}^2\de\textbf{x}\\[5pt] \geq\min\Big\{ \frac{1}{2}\int_{E_{\eps,\rho}^\ell \setminus E_{M_{\delta\eta}}}&\abs{\nabla u_\eps}^2\de\textbf{x}: u=m+\gamma\text{ on } \partial E_{M_{\delta\eta}},\ u=\alpha-\gamma_\eps^\ell\text{ on }\partial E_{\eps,\rho}^\ell \Big\}. \end{align} By doing similar computations as for \eqref{hUB3}, we get the unique minimizer, in ellipsoidal coordinates, given by \begin{align} u(\mu)=c \ln\abs{k \tanh(\mu/2)}, \end{align} where \begin{align} c=\frac{m+\gamma-\alpha+\gamma_\eps^\ell}{\ln\Big(\displaystyle\frac{\tanh(\rho)}{\tanh(M_{\delta\eta})}\Big)}. \end{align} By computing its Dirichlet energy, we get \begin{align}\label{loweboundwithgamma} \frac{1}{2a}\ln\Big(\displaystyle\frac{\tanh(\rho)}{\tanh(M_{\delta\eta})}\Big)\int_{E_{\eps,\rho}^\ell \setminus E_{M_{\delta\eta}} }&\abs{\nabla u_\eps}^2\de\textbf{x}\geq \frac{\pi}{2} (m+\gamma-\alpha+\gamma_\eps^\ell)^2. \end{align} Now, we would like to undestand the asymptotic behaviour of the coefficient in front of the left-hand side of \eqref{loweboundwithgamma}. Notice that on $ \partial E_{M_{\delta\eta}}\cap \{x =0\}$ we have \begin{equation}\label{amconditionellipsoid} \frac{y^2}{a^2 \cosh^2 M_{\delta\eta}}+\frac{z^2}{a^2 \sinh^2 M_{\delta\eta}} = 1. \end{equation} We can choose $M_{\delta\eta}$ to satisfy $a \sinh M_{\delta\eta} = \sqrt{2} \eta$ and $a \cosh M_{\delta\eta} = \sqrt{2} \delta$. In such a way, we have that \eqref{chiuderelafinestra} holds. Therefore, \begin{equation} \tanh M_{\delta\eta} = \frac{\eta}{\delta}. \end{equation} In particular, $a^2=\delta^2-\eta^2\approx \delta^2$, therefore we can choose $a\approx4\delta$ as the eccentricity of the change of the coordinates. As consequence, we get the following asymptotic estimate \begin{align}\label{lowerboundapprox} \frac{1}{8a}\ln\Big(\displaystyle\frac{\tanh(\rho)}{\tanh(M_{\delta\eta})}\Big)=\frac{1}{8a}\big( \ln \tanh \rho - \ln \tanh M_{\delta\eta}\big) \approx\frac{\abs{\ln(\delta/\eta)}}{8\delta}. \end{align} Therefore, from \eqref{loweboundwithgamma}, \eqref{lowerboundapprox}, since $\gamma_\eps^\ell\to 0$ and by arbitrariness of $\gamma$, we get \begin{align} \liminf_{\eps\to0}\frac{\abs{\ln(\delta/\eta)}}{2\delta}\int_{E_{\eps,\rho}^\ell \setminus E_{M_{\delta\eta}} }&\abs{\nabla u_\eps}^2\de\textbf{x}\geq 2\pi (m-\alpha)^2. \end{align} By applying the same idea on $E_{\eps,\rho}^r \setminus E_{M_{\delta\eta}}$ we get \begin{align} \liminf_{\eps\to0}\frac{\abs{\ln(\delta/\eta)}}{2\delta}\int_{E_\rho \setminus E_{M_{\delta\eta}} }\abs{\nabla u_\eps}^2\de\textbf{x}&\geq 2\pi\big[ (m-\alpha)^2+ (m-\beta)^2\big]\\[5pt] &\geq\pi(\beta-\alpha)^2\\[5pt] &\geq\limsup_{\eps\to0}\frac{\abs{\ln(\delta/\eta)}}{2\delta}\int_{E_\rho \setminus E_{M_{\delta\eta}} }\abs{\nabla u_\eps}^2\de\textbf{x}, \end{align} where we optimized for $m$ and we used the energy of the competitor built in \eqref{testfootball}. In conclusion we have \begin{align}\label{malphabeta2} m=\frac{\alpha+\beta}{2} \end{align} and \begin{align} \lim_{\eps\to 0} \frac{\abs{\ln(\delta/\eta)}}{\delta}F_\eps(u_\eps)=\pi(\beta-\alpha)^2. \end{align} From \eqref{upperboundueps}, we can conclude that \begin{align} \lim_{\eps\to 0} F_\eps(u_\eps,\Omega_\eps)=\lim_{\eps\to 0}(u_\eps,\Omega_\eps\setminus N_\eps). \end{align} From \eqref{malphabeta2}, we can infer also that \begin{align} u_\eps(0,0,0)\to\frac{\alpha+\beta}{2}, \end{align} as $\eps\to0$. \emph{Step 2. Narrow thick regime.} The proof corresponds to the previous one, with no differences, as we use the same rescaling \begin{align} \hat{u}_\eps(x,y,z)\coloneqq u_\eps(\delta x,\delta y ,\eta z). \end{align} This concludes the proof. \end{proof} \subsection{Letter-box regime} In this regime the parameters are ordered as $\delta \gg \eps \gg \eta$. Namely, we have \begin{align} \lim_{\eps\to0} \frac{\delta}{\eps}=+\infty\quad\text{and}\quad\lim_{\eps\to0}\frac{\eta}{\delta}=0, \end{align} In this regime, the order of the energy inside and outside of the neck provides two order that are asymptotically the same, namely \begin{align}\label{limitell} \lim_{\eps\to 0}\frac{\eps}{\eta}\abs{\log(\eta/\delta)}=\ell\in(0,\infty). \end{align} In this setting, we expect the transition happening inside and outside of the neck simultaneously. \subsubsection{Critical Letter-box regime} This sub-regime, corresponds to the case in which the limit $\ell$ in \eqref{limitell} is finite and strictly positive. \\ In order to understand the transition inside the neck, we rescale it to make it the set $N\coloneqq [-1,1]^3$. If $u_\eps$ is a local minimizer of the functional \eqref{004}, the only rescaling that works is \begin{align} v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\delta y, \eta z). \end{align} If we rescale in this way, the limiting domain becomes $$\Omega_\infty = \Omega^\ell_\infty \cup N \cup \Omega^r_\infty,$$ where $\Omega^\ell_\infty = \{x<-1\}$ and $\Omega^r_\infty = \{x>1\}$. \begin{theorem}\label{Theoremcriticalletterbox} In the Critical Letter-box regime, it holds that \begin{align} \lim_{\eps\to 0} \frac{\eps}{\delta\eta}F_\eps(u_\eps,N_\eps)= \frac{\pi^2(\beta-\alpha)^2}{(\pi+2\ell)^2}, \end{align} and that \begin{equation} \lim_{\eps\to 0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps,\O_\eps\setminus N_\eps)= 2\pi\frac{\ell^2(\beta-\alpha)^2}{(\pi+2\ell)^2}. \end{equation} \end{theorem} \begin{proof} \emph{Step $1$. Energy estimate inside the neck.} This part of the proof follows the strategy of Theorem \ref{Theoremsuperthin}. However, we remark that the scale-invariant Poincaré inequality in this regimes does not provide us with any information.\\ Consider the rescaling $v_\varepsilon:[-1,1]^3\to\R{}$ be defined as \[ v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\delta y, \eta z). \] It is possible to prove that $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x)$, where $v\in H^1(-1,1)$. \\ We can write \begin{align}\label{letterbox:lbneck} \nonumber\liminf_{\eps\to0}\frac{\abs{\ln(\eta/\delta)}}{\delta} F(u_\eps)&\geq \liminf_{\eps\to0}2\frac{\abs{\ln(\eta/\delta)}}{\delta}\frac{\delta\eta}{\eps}\int_{-1}^1 (\partial_x v_\eps)^2\ \de x\\[5pt] &\geq 2\ell\big(\hat{v}(-1)+\hat{v}(1)\big)^2. \end{align} \emph{Step $2$. Energy estimate in the bulk.} In order to get both boundary conditions at the edge of the neck, we need to perform two rescalings. The choice is made as \begin{align} \hat{u}^\pm_\eps(x,y,z)\coloneqq u_\eps(\eta x\pm\eps,\delta y ,\eta z). \end{align} In such a way the limiting domains are \begin{align} \Omega_\infty^\ell&\coloneqq \{x<0\}\cup \{-1<y<1,\ -1<z<1,\ x>0\}.\\[5pt] \Omega_\infty^\rr&\coloneqq \{x>0\}\cup \{-1<y<1,\ -1<z<1,\ x<0\}. \end{align}By using the same strategy of Theorem \ref{Theoremwindowthick}, we can prove that \begin{align}\label{letterbox:lbbulk} \liminf_{\eps\to0}\frac{\abs{\ln(\delta/\eta)}}{2\delta}\int_{(\O_\eps^\ell\cup\O_\eps^\rr)\setminus N_\eps }\abs{\nabla u_\eps}^2\de\textbf{x}&\geq 2\pi\big[ (\hat{v}(-1)-\alpha)^2+ (\hat{v}(1)-\beta)^2\big] \end{align} \emph{Step 3. Limit of the energy.} By putting together \eqref{letterbox:lbneck} and \eqref{letterbox:lbbulk}, we obtain \begin{align}\label{letterboxtooptimize} \nonumber \liminf_{\eps\to0}\frac{\abs{\ln(\delta/\eta)}}{2\delta}\int_{\O_\eps }&\abs{\nabla u_\eps}^2\de\textbf{x}\\[5pt] &\geq2\pi\big[ (\hat{v}(-1)-\alpha)^2+ (\hat{v}(1)-\beta)^2\big]+2\ell\big(\hat{v}(-1)+\hat{v}(1)\big)^2. \end{align} Now, if we optimize \eqref{letterboxtooptimize} for $\hat{v}(-1)$ and $\hat{v}(1)$, we obtain \begin{align} v(-1)=\frac{\pi\alpha+(\alpha+\beta)\ell}{\pi+2\ell}=\quad\text{and}\quad v(1)=\frac{\pi\beta+(\alpha+\beta)\ell}{\pi+2\ell}. \end{align} In conclusion, from \eqref{upperboundueps}, we have \begin{align} \lim_{\eps\to 0} \frac{\eps}{\delta\eta}F_\eps(u_\eps,N_\eps)= \frac{\pi^2(\beta-\alpha)^2}{(\pi+2\ell)^2}, \end{align} and \begin{equation} \lim_{\eps\to 0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps,\O_\eps\setminus N_\eps)= 2\pi\frac{\ell^2(\beta-\alpha)^2}{(\pi+2\ell)^2}. \end{equation} \end{proof} \begin{remark} From \eqref{letterbox:lbneck}, we can deduce that $\hat{v}$, in the neck, is the unique solution of the variational problem \begin{align} \min\bigg\{ \frac{1}{2} \int_{-1}^1 \|v'\|^2 \de x:\ v\in H^1(-1,1),\ v(-1)=\frac{\pi\alpha+(\alpha+\beta)\ell}{\pi+2\ell},\\[5pt] v(1)=\frac{\pi\beta+(\alpha+\beta)\ell}{\pi+2\ell}\bigg\}, \end{align} whose solution can be explicitly written. \end{remark} The other two critical regimes, obtained when $\ell=0$ or $\ell=\infty$, can easily be obtained by Theorem \ref{Theoremcriticalletterbox}. We list them for completeness, but the proof is omitted. \subsubsection{Super-critical Letter-box regime} In this sub-regime, we have \begin{align} \lim_{\eps\to 0}\frac{\eps}{\eta}\abs{\log(\eta/\delta)}=+\infty. \end{align} In this case, we recover the same result as in Theorem \ref{Theoremwindowthick}. \begin{theorem} In the super-critical letter-box regime, it holds \begin{equation} \lim_{\eps\to0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps)=\lim_{\eps\to 0} \frac{\|\ln(\eta/\delta)\|}{\delta}F_\eps(u_\eps,\O_\eps\setminus N_\eps)= \pi(\beta-\alpha)^2. \end{equation} which translates into the fact that the transition happens entirely outside of the neck. \end{theorem} \subsubsection{Sub-critical Letter-box regime} In this sub-regime, we have \begin{align} \lim_{\eps\to 0}\frac{\eps}{\eta}\abs{\log(\eta/\delta)}=0. \end{align} In this case, we recover the same result as in Theorem \ref{Theoremsuperthin}. \begin{theorem} In the super-critical letter-box regime, it holds \begin{align} \lim_{\eps\to0} \frac{\eps}{\delta\eta}F_\eps(u_\eps)=\lim_{\eps\to 0} \frac{\eps}{\delta\eta}F_\eps(u_\eps,N_\eps)= (\beta-\alpha)^2, \end{align} which means that the main contribution of the energy is given by the transition inside the neck. Moreover, for $\varepsilon>0$ let $v_\varepsilon:[-1,1]^3\to\R{}$ be defined as \[ v_\eps(x,y,z)\coloneqq u_\eps(\eps x,\delta y, \eta z). \] Then, $v_\eps$ converges to a function $\hat{v}\in H^1(N)$, of the form $\hat{v}(x,y,z)=v(x)$, where $v\in H^1(-1,1)$ is the unique minimizer of the variational problem \begin{align} \min\bigg\{ \frac{1}{2} \int_{-1}^1 \|v'\|^2\ \de x:\ v\in H^1(-1,1),\ v(-1)=\alpha,\ v(1)=\beta\bigg\}. \end{align} In particular, $v(x)=\dfrac{\beta-\alpha}{2}x+\dfrac{\alpha+\beta}{2}$. \end{theorem} \noindent\textbf{Acknowledgements} This research was carried out at the departments of Mathematics of Politecnico di Torino and Radboud University, whose hospitality is gratefully acknowledged. MM and RC are members of GNAMPA (INdAM). This study was carried out within the Geometric-Analytic Methods for PDEs and Applications project (2022SLTHCE), funded by European Union -- Next Generation EU within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Universit\`{a} e della Ricerca). This manuscript reflects only the authors’ views and opinions and the Ministry cannot be considered responsible for them. \bibliographystyle{siam} \def\url#1{} \bibliography{bibliography} \end{document}
2412.04195v1	http://arxiv.org/abs/2412.04195v1	Partial Betti splittings with applications to binomial edge ideals	\documentclass[12pt,twoside]{amsart} \usepackage[english]{babel} \usepackage{amsfonts,amssymb,amsthm,amsmath,mathtools,accents,latexsym} \usepackage[a4paper,top=3cm,bottom=3cm,left=2.5cm,right=2.5cm,marginparwidth=1.75cm]{geometry} \setlength{\parskip}{3pt} \usepackage{xcolor} \usepackage{graphicx,comment,mathtools} \usepackage[colorlinks=true, allcolors=blue]{hyperref} \usepackage{cleveref} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{claim}[theorem]{Claim} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{construction}[theorem]{Construction} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{acknowledgement}{Acknowledgement} \newtheorem{notation}[theorem]{Notation} \newtheorem{question}[theorem]{Question} \newcommand{\avj}[1]{\textcolor{purple}{\sffamily ((AVJ: #1))}} \usepackage{tikz} \newcommand\circled[1]{\tikz[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=2pt] (char) {#1};}} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan, citecolor=red } \urlstyle{same} \DeclareMathOperator{\tor}{Tor} \DeclareMathOperator{\In}{in} \DeclareMathOperator{\pd}{pd} \DeclareMathOperator{\reg}{reg} \DeclareMathOperator{\comp}{comp} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\mdeg}{mdeg} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{Im} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\len}{len} \DeclareMathOperator{\Mon}{Mon} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\iv}{iv} \newcommand{\B}{\mathcal{B}} \title{Partial Betti splittings with applications to binomial edge ideals} \date{\today } \author[A.V. Jayanthan]{A.V. Jayanthan} \address[A.V. Jayanthan] {Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India - 600036} \email{[email protected] } \author[A. Sivakumar]{Aniketh Sivakumar} \address[A. Sivakumar] {Department of Mathematics, Tulane University, New Oreans, LA, 70118} \email{[email protected]} \author[A. Van Tuyl]{Adam Van Tuyl} \address[A. Van Tuyl] {Department of Mathematics and Statistics\\ McMaster University, Hamilton, ON, L8S 4L8} \email{[email protected]} \keywords{partial Betti splittings, graded Betti numbers, binomial edge ideals, trees} \subjclass[2020]{13D02, 13F65, 05E40} \begin{document} \begin{abstract} We introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, H\`a, and Van Tuyl. Given a homogeneous ideal $I$ and two ideals $J$ and $K$ such that $I = J+K$, a partial Betti splitting of $I$ relates {\it some} of the graded Betti of $I$ with those of $J, K$, and $J\cap K$. As an application, we focus on the partial Betti splittings of binomial edge ideals. Using this new technique, we generalize results of Saeedi Madani and Kiani related to binomial edge ideals with cut edges, we describe a partial Betti splitting for all binomial edge ideals, and we compute the total second Betti number of binomial edge ideals of trees. \end{abstract} \maketitle \section{Introduction} Given a homogeneous ideal $I$ of a polynomial ring $R = k[x_1,\ldots,x_n]$ over an arbitrary field $k$, one is often interested in the numbers $\beta_{i,j}(I)$, the graded Betti numbers of $I$, that are encoded into the graded minimal free resolution of $I$. In some situations, we can compute these numbers by ``splitting'' the ideal $I$ into smaller ideals and use the graded Betti numbers of these new ideals to find those of the ideal $I$. More formally, suppose $\mathfrak{G}(L)$ denotes a set of minimal generators of a homogeneous ideal $L$. Given a homogeneous ideal $I$, we can ``split'' this ideal as $I = J+K$ where $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. The ideals $I, J, K$ and $J \cap K$ are then related by the short exact sequence $$0 \longrightarrow J\cap K \longrightarrow J \oplus K \longrightarrow J+K = I \longrightarrow 0.$$ The mapping cone construction then implies that the graded Betti numbers of $I$ satisfy \begin{equation}\label{bettisplit} \beta_{i,j}(I) \leq \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J \cap K) ~~\mbox{for all $i,j \geq 0$}. \end{equation} Francisco, H\`a, and Van Tuyl \cite{francisco_splittings_2008} defined $I = J+K$ to be a {\it Betti splitting} if the above inequality is an equality for all $i,j \geq 0$. Betti splittings of monomial ideals first appeared in work of Eliahou and Kervaire \cite{EK1990}, Fatabbi \cite{fatabbi2001}, and Valla \cite{Valla2005}. In fact, these prototypical results provided the inspiration for Francisco, H\`a, and Van Tuyl's introduction of Betti splittings in \cite{francisco_splittings_2008}. Their paper also provided conditions on when one can find Betti splittings of edge ideals, a monomial ideal associated to a graph (see \cite{francisco_splittings_2008} for more details). Betti splittings have proven to be a useful tool, having been used to study: the graded Betti numbers of weighted edge ideals \cite{kara2022}, the classification of Stanley-Reisner ideals of vertex decomposable ideals \cite{moradi2016}, the linearity defect of an ideal \cite{hop2016}, the depth function \cite{ficarra2023}, componentwise linearity \cite{bolognini2016}, and the Betti numbers of toric ideals \cite{FAVACCHIO2021409,gimenez2024}. In general, an ideal $I$ may not have any Betti splitting. However, it is possible that \Cref{bettisplit} may hold for {\it some} $i,j \geq 0$. In order to quantify this behaviour, we introduce a new concept called a {\it partial Betti splitting} of an ideal $I$. Specifically, if $I = J+K$ with $\mathfrak{G}(I)$ equal to the disjoint union $\mathfrak{G}(J) \cup \mathfrak{G}(K)$, then $I = J+K$ is an {\it $(r,s)$-Betti splitting} if \[\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1, j}(J\cap K )\text{\hspace{3mm} for all $(i,j)$ with $i\geq r$ or $j\geq i+s$}.\] Using the language of Betti tables, if $I = J+K$ is an $(r,s)$-Betti splitting, then all the Betti numbers in the $r$-th column and beyond or the $s$-th row and beyond of the Betti table of $I$ satisfy \Cref{bettisplit}. The Betti splittings of \cite{francisco_splittings_2008} will now called {\it complete Betti splittings}. The goal of this paper is two-fold. First, we wish to develop the properties of partial Betti splittings, extending the results of \cite{francisco_splittings_2008}. Note that \cite{francisco_splittings_2008} focused on Betti splittings of monomial ideals; however, as we show, almost all the same arguments work for any homogeneous ideal $I$ of $R = k[x_1,\ldots,x_n]$ when $R$ is graded by a monoid $M$. Among our results, we develop necessary conditions for an $(r,s)$-Betti splitting: \begin{theorem}[\Cref{parcon2}] Let $I$, $J$ and $K$ be homogeneous ideals of $R$ with respect to the standard $\mathbb{N}$-grading such that $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Suppose that there are integers $r$ and $s$ such that for all $i \geq r$ or $j \geq i+s$, $\beta_{i-1,j}(J \cap K) > 0$ implies that $\beta_{i-1,j}(J) = 0$ and $\beta_{i-1,j}(K) = 0$. Then $I = J + K$ is an $(r,s)$-Betti splitting. \end{theorem} Second, we wish to illustrate (partial) Betti splittings by considering splittings of binomial edge ideals. If $G = (V(G,E(G))$ is a graph on the vertex set $V = [n] :=\{1,\ldots,n\}$ and edge set $E$, the {\it binomial edge ideal of $G$} is the binomial ideal $J_G = \langle x_iy_j - x_jy_i ~\|~ \{i,j\} \in E \rangle$ in the polynomial ring $R = k[x_1,\ldots,x_n,y_1,\ldots,y_n]$. Binomial edge ideals, which were first introduced in \cite{herzog_binomial_2010,Ohtani2011}, have connections to algebraic statistics, among other areas. The past decade has seen a flurry of new results about the homological invariants (e.g., Betti numbers, regularity, projective dimension) for this family of ideals (see \cite{ZZ13}, \cite{SZ14}, \cite{deAlba_Hoang_18}, \cite{herzog_extremal_2018}, \cite{KS20}, \cite{jayanthan_almost_2021} for a partial list on the Betti numbers of binomial edge ideals). Interestingly, Betti splittings of binomial edge ideals have not received any attention, providing additional motivation to study this family of ideals. In order to split $J_G$, we wish to partition the generating set $\mathfrak{G}(J_G)$ in such a way that the resulting ideals generated by each partition, say $J$ and $K$, are the binomial edge ideals of some subgraphs of $G$, that is, splittings of the form $J_G = J_{G_1}+J_{G_2}$ where $G_1$ and $G_2$ are subgraphs. We focus on two natural candidates. The first way is to fix an edge $e = \{i,j\} \in E(G)$ and consider the splitting $$J_G = J_{G\setminus e} + \langle x_iy_j- x_jy_i \rangle.$$ where $G\setminus e$ denotes the graph $G$ with the edge $e$ removed. The second way is to fix a vertex $s \in V(G)$ and consider the set $F \subseteq E(G)$ of all edges that contain the vertex $s$. We can then split $J_G$ as follows $$J_G = \langle x_sy_j-x_jy_s ~\|~ \{s,j\} \in F \rangle + \langle x_ky_j-x_jy_k ~\|~ \{k,l\} \in E(G) \setminus F \rangle.$$ We call such a partition an $s$-partition of $G$. Note that the first ideal is the binomial edge ideal of a star graph, while the second ideal is the binomial edge ideal of the graph $G \setminus \{s\}$, the graph with the vertex $s$ removed. These splittings are reminiscent of the edge splitting of edge ideals and the $x_i$-splittings of monomial ideals introduced in \cite{francisco_splittings_2008}. In general, neither of these splitting will give us a complete Betti splitting. This is not too surprising since the edge ideal analogues are not always complete Betti splittings. So it is natural to ask when we have a partial or complete Betti splitting using either division of $J_G$. Among our results in Section 4, we give a sufficient condition on an edge $e$ of $G$ so that the first partition gives a complete Betti splitting. In the statement below, an edge is a cut-edge if $G \setminus e$ has more connected components than $G$, and a vertex is free if it belongs to a unique maximal clique, a subset of vertices of $G$ such that all the vertices are all adjacent to each other. \begin{theorem}[\Cref{singlefreevertex}]\label{them2} Let $e = \{u,v\} \in E(G)$ be a cut-edge where $v$ is a free vertex in $G\setminus e$. Then $J_G = J_{G\setminus e}+\langle x_uy_v-x_vy_u\rangle$ is a complete Betti splitting. \end{theorem} \noindent Theorem \ref{them2} generalizes previous work of Saeedi Madani and Kiani \cite{kiani_regularity_2013-1}, and it allows us to give new proofs for their results about the Betti numbers, regularity, and projective dimension for some classes of binomial edge ideals (see \Cref{freecutedge}). In the case of $s$-partitions, we again do not always have a complete Betti splitting. However, we can derive a result about the partial Betti splittings for all graphs. \begin{theorem}[\Cref{maintheo2}] Let $J_G$ be the binomial edge ideal of a graph $G$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$. Let $c(s)$ be the size of the largest clique that contains $s$. Then $$ \beta_{i,j}(J_G) = \beta_{i,j}(J_{G_1})+\beta_{i,j}(J_{G_2})+\beta_{i-1, j}(J_{G_1}\cap J_{G_2})~~~ \mbox{for all $(i,j)$ with $i\geq c(s)$ or $j\geq i+4$.} $$ In other words, $J_G = J_{G_1}+J_{G_2}$ is a $(c(s), 4)$-Betti splitting. \end{theorem} \noindent Note that if $G$ is a triangle-free graph, then for every vertex $i \in V(G)$ we have $c(i) \leq 2$. We can use the above result to construct a complete Betti splitting for the binomial edge ideals of all triangle-free graphs (see Corollary \ref{trianglefree}). In the final section, we use the complete Betti splitting of \Cref{them2} to explore the (total) graded Betti numbers of binomial edge ideals of trees. In particular, we give formulas for the first and second total Betti numbers for the binomial edge ideal of any tree. Our result extends work of Jayanthan, Kumar, and Sarkar \cite{jayanthan_almost_2021} which computed the first total Betti numbers for these ideals. Our paper is structured as follows. In Section 2 we recall the relevant background. In Section 3 we introduce the notion of a partial Betti splitting and describe some of their basic properties. In Section 4, we consider splittings of $J_G$ using a single edge of $G$, while in Section 5, we consider a splitting of $J_G$ by partitioning the generators on whether or not they contain $x_s$ or $y_s$ for a fixed vertex $s$. In our final section we determine the second total Betti number of binomial edge ideals of trees. \section{Preliminaries} In this section we recall the relevant background on Betti numbers, graph theory, and binomial edge ideals that is required for later results. \subsection{Homological algebra} Throughout this paper $k$ will denote an arbitrary field. Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring over $k$. We will use various gradings of $R$. Recall that if $M$ is a monoid (a set with an addition operation and additive identity), we say a ring $S$ is {\it $M$-graded} if we can write $S = \bigoplus_{j \in M} S_j$, where each $S_j$ is an additive group and $S_{j_1}S_{j_2} \subseteq S_{j_1+j_2}$ for all $j_1,j_2 \in M$. We will primarily use three gradings of $R$ in this paper: (1) $R$ has an $\mathbb{N}$-grading by setting $\deg(x_i) = 1$ for all $i$; (2) $R$ has an $\mathbb{N}^n$-grading by setting $\deg(x_i) = e_i$ for all $i$, where $e_i$ is the standard basis element of $\mathbb{N}^n$; and (3) $R$ has an $\mathbb{N}^2$-grading by setting the degree of some of the $x_i$'s to $(1,0)$, and the degrees of the rest of the $x_i$'s to $(0,1)$. Given an $M$-graded ring $R$, an element $f \in R$ is {\it homogeneous} if $f \in R_j$ for some $j \in M$. We say the {\it degree} of $f$ is $j$ and write $\deg(f) = j$. An ideal $I \subseteq R$ is {\it homogeneous} if it is generated by homogeneous elements. We write $I_j$ to denote all the homogeneous elements of degree $j\in M$ in $I$. We let $\mathfrak{G}(I)$ denote a minimal set of homogeneous generators of $I$. While the choice of elements of $\mathfrak{G}(I)$ may not be unique, the number of generators of a particular degree is an invariant of the ideal. If $I$ is a homogeneous ideal, then the Tor modules ${\rm Tor}_i(k,I)$ are also $M$-graded for all $i \geq 0$. The {\it $(i,j)$-th graded Betti number of $I$} is then defined to be $$\beta_{i,j}(I) := \dim_k {\rm Tor}_i(k,I)_j ~~\mbox{for $i \in \mathbb{N}$ and $j \in M$.}$$ We use the convention that $\beta_{i,j}(I) = 0$ if $i <0$. We are sometimes interested in the (multi)-graded Betti numbers of the quotient $R/I$; we make use of the identity $\beta_{i,j}(R/I) = \beta_{i-1,j}(I)$ for all $i \geq 1$ and $j \in M$. The graded Betti number $\beta_{i,j}(I)$ is also equal to the number of syzygies of degree $j$ in the $i$-th syzygy module of $I$. For further details, see the book of Peeva \cite{P2011}. When $R$ has the standard $\mathbb{N}$-grading, we are also interested in the following two invariants: the {\it (Castelnuovo-Mumford) regularity of $I$}, which is defined as $${\rm reg}(I) = \max\{ j-i ~\|~ \beta_{i,i+j}(I) \neq 0\},$$ and the {\it projective dimension of $I$}, which is defined as $${\rm pd}(I) = \max\{i ~\|~ \beta_{i,j}(I) \neq 0\}.$$ These invariants measure the ``size'' of the minimal graded free resolution of $I$. \subsection{Graph theory} Throughout this paper, we use $G = (V(G),E(G))$ to represent a finite simple graph where $V(G)$ denotes the vertices and $E(G)$ denotes the edges. Most of our graphs will have the vertex set $[n] = \{1,\dots ,n\}$. A {\it subgraph} of $G$ is a graph $H$ such that $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$. An \textit{induced subgraph} on $S\subset V(G)$, denoted by $G[S]$, is a the subgraph with vertex set $S$ and for all $u,v\in S$, if $\{u,v\}\in E(G)$, then $ \{u,v\}\in E(G[S])$. The {\it complement} of a graph, denoted $G^c$, is a graph with $V(G^c) = V(G)$ and $E(G^c) = \{\{u,v\}\mid \{u,v\}\notin E(G)\}$. From a given graph $G = (V(G),E(G))$, if $e \in E(G)$, then we denote by $G\setminus e$ the subgraph of $G$ on the same vertex set, but edge set $E(G\setminus e) = E(G) \setminus \{e\}$. Given any $i \in V(G)$, we let $N_G(i) = \{j ~\|~ \{i,j\} \in E(G)\}$ denote the set of {\it neighbours} of the vertex $i$. The {\it degree} of a vertex $i$ is then $\deg_G i = \|N_G(i)\|$. In the context where there is a fixed underlying graph, we omit the subscript $G$ and write this as $\deg i$. The {\it closed neighbourhood of $i$} is the set $N_G[i] =N_G(i) \cup \{i\}$. If $G = (V(G),E(G))$ is a graph and $e =\{i,j\} \not\in E(G)$, we let $G_e$ denote the graph on $V(G)$, but with edge set $$E(G_e) = E(G) \cup \{\{k,l\} ~\|~ k,l \in N_G(i)~~\mbox{or}~~k,l \in N_G(j) \}.$$ So, $G$ is a subgraph $G_e$. We will require a number of special families of graphs. The \textit{$n$-cycle}, denoted $C_n$, is the graph with vertex set $[n]$ with $n \geq 3$ and edge set $\{\{i,i+1\} ~\|~ i =1,\ldots,n-1\} \cup \{\{1,n\}\}.$ A \textit{chordal graph} $G$ is a graph where all the induced subgraphs of $G$ that are cycles are 3-cycles, that is, there are no induced $n$-cycles with $n\geq 4$. A \textit{triangle-free graph} is a graph $G$ such that $C_3$ is not an induced subgraph of $G$. A \textit{tree} is a graph which has no induced cycles. A particular example of a tree that we will use is the {\it star graph} on $n$ vertices, denoted $S_n$. Specifically, $S_n$ is the graph on the vertex set $[n]$ and edge set $E(S_n) = \{\{1,k\}\mid 1<k\leq n\}$. A \textit{complete graph} is a graph $G$ where $\{u,v\}\in E(G)$ for all $u,v\in V(G)$. If $G$ is a complete graph on $[n]$, we denote it by $K_n$. A \textit{clique} in a graph $G$ is an induced subgraph $G[S]$ that is a complete graph. A \textit{maximal clique} is a clique that is not contained in any larger clique. A vertex $v$ of $G$ is a \textit{free vertex} if $v$ only belongs to a unique maximal clique in $G$, or equivalently, the induced graph on $N_G(v)$ is a clique. An edge $e = \{u,v\}$ in $G$ is a \textit{cut edge} if its deletion from $G$ yields a graph with more connected components than $G$. Note that a tree is a graph where all of its edges are cut edges. A \textit{free cut edge} is a cut edge $\{u,v\}$ such that both ends, $u$ and $v$, are free vertices in $G \setminus e$. We are also interested in cliques combined with other graphs. A graph $G$ is said to be a \textit{clique-sum} of $G_1$ and $G_2$, denoted by $G = G_1 \cup_{K_r} G_2$, if $V(G_1) \cup V(G_2) = V(G)$, $E(G_1) \cup E(G_2) = E(G)$ and the induced graph on $V(G_1) \cap V(G_2)$ is the clique $K_r$. If $r = 1$, then we write $G = G_1 \cup_v G_2$ for the clique-sum $G_1 \cup _{K_1} G_s$ where $V(K_1) = \{v\}$. A graph $G$ is \textit{decomposable} if there exists subgraphs $G_1$ and $G_2$ such that $G_1\cup_{v}G_2 = G$ and $v$ is a free vertex of $G_1$ and $G_2$. So a decomposable graph is an example of a clique-sum on a $K_1$ where the $K_1$ is a free vertex in both subgraphs. \begin{example} Consider the graph $G$ in \Cref{fig:graph5}, with $V(G) = [7]$ and $$E(G) = \{\{1,2\}, \{2,3\}, \\\{2,4\}, \{4,5\}, \{4,6\}, \{4,7\}, \{6,7\}\}.$$ Here, we can see that $G = T \cup_{\{4\}} K_3$, where $T$ is the tree with $V(T) = \{1,2,3,4,5\}$ and $E(T) = \{\{1,2\}, \{2,3\}, \{2,4\}, \{4,5\}\}$ and $K_3$ is the clique of size $3$, with $V(K_3) = \{4,6,7\}$ and $E(K_3) = \{\{4,6\}, \{4,7\}, \{6,7\}\}$. \begin{figure}[ht] \centering \begin{tikzpicture}[every node/.style={circle, draw, fill=white!60, inner sep=2pt}, node distance=1.5cm] \node (1) at (0, 0) {1}; \node (2) at (1.5, 0) {2}; \node (3) at (3, 0) {3}; \node (4) at (1.5, -1.5) {4}; \node (5) at (0, -1.5) {5}; \node (6) at (0.5, -2.5) {6}; \node (7) at (2.5, -2.5) {7}; \draw (1) -- (2); \draw (2) -- (3); \draw (2) -- (4); \draw (4) -- (5); \draw (4) -- (6); \draw (4) -- (7); \draw (6) -- (7); \end{tikzpicture} \caption{$G = T\cup_{\{4\}}K_3$} \label{fig:graph5} \end{figure} \end{example} \subsection{Binomial edge ideals} Suppose that $G = (V(G),E(G))$ is a finite simple graph with $V(G) = [n]$. The {\it binomial edge ideal} of $G$, denoted $J_G$, is the binomial ideal $$J_G = \langle x_iy_j - x_jy_i ~\|~ \{i,j\} \in E(G) \rangle$$ in the polynomial ring $R = k[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In what follows, we will find it convenient to consider different gradings of $R$; we can grade the polynomial ring $R$ either with the standard grading where $\deg x_i=\deg y_i=1$ for all $i$, with an $\mathbb{N}^n$-multigrading where $\deg x_i=\deg y_i=(0,\dots,1,\dots, 0)$, the $i$-th unit vector for all $i$, or with an $\mathbb{N}^2$-grading where $\deg x_i = (1,0)$ for all $i$ and $\deg y_j = (0,1)$ for all $j$. Note that $J_G$ is a homogeneous ideal with respect to all three gradings. We review some useful facts from the literature about the idea $J_G$. Recall that a standard graded ideal $I$ has {\it linear resolution} if $I$ is generated by homogeneous elements of degree $d$ and $\beta_{i,i+j}(I) = 0$ for all $j \neq d$. \begin{theorem}\label{completebetti} Let $G = K_n$ be a complete graph. Then \begin{enumerate} \item The binomial edge ideal $J_G$ has a linear resolution. \item $\beta_{i,i+2}(J_G) = (i+1)\binom{n}{i+2}$ for $i \geq 0$ and $0$ otherwise. \end{enumerate} \end{theorem} \begin{proof} Statement (1) follows from {\cite[Theorem 2.1]{kiani_binomial_2012}}. Statement (2) follows from a more general fact of Herzog, Kiani, and Saaedi Madani \cite[Corollary 4.3]{herzog_linear_2017} on the Betti numbers that appear in the linear strand of a binomial edge ideals applied to $K_n$. \end{proof} The next result is related to a cut edge in a graph. \begin{lemma}[{\cite[Theorem 3.4]{mohammadi_hilbert_2014}}]\label{lemma 3.8} Let $G$ be a simple graph and let $e = \{i,j\}\notin E(G)$ be a cut edge in $G\cup \{e\}$. Let $f_e = x_iy_j-x_jy_i$. Then $J_G:\langle f_e \rangle = J_{G_e}$. \end{lemma} We will require the next result about the Betti polynomials of binomial edge ideals of decomposable graphs. For an $\mathbb{N}$-graded $R$-module $M$, the {\it Betti polynomial of $M$} is $$B_M(s,t) = \sum_{i,j \geq 0} \beta_{i,j}(M)s^it^j.$$ The following result is due to Herzog and Rinaldo, which generalized an earlier result of of Rinaldo and Rauf \cite{rauf_construction_2014}. \begin{theorem}[{\cite[Proposition 3]{herzog_extremal_2018}}]\label{freevertexbetti} Suppose that $G$ is a decomposable graph with decomposition $G = G_1\cup G_2$. Then \[B_{R/J_G}(s, t) = B_{R/J_{G_1}}(s, t)B_{R/J_{G_2}}(s, t).\] \end{theorem} The graded Betti numbers in the linear strand of $J_G$ (all the Betti numbers of the form $\beta_{i,i+2}(J_G))$ were first calculated by Herzog, Kaini, and Saeedi Madani. In the statement below, $\Delta(G)$ is the clique complex of the graph $G$ and $f_{i+1}(\Delta(G))$ is the number of faces in $\Delta(G)$ of dimension $i+1$. \begin{theorem}[{\cite[Corollary 4.3]{herzog_linear_2017}}]\label{linearbinom} Let $G$ be a finite simple graph with binomial edge ideal $J_G$. Then the Betti numbers in the linear strand of $J_G$ are given by \[\beta_{i,i+2}(J_G) = (i+1)f_{i+1}(\Delta(G)) ~~\mbox{for $i\geq 0$.}\] \end{theorem} \begin{example}\label{runningexample} Let $G$ be the finite simple graph on the vertex set $[7]$ with edge set $$E(G) =\{\{1,2\}, \{1,3\}, \{1,4\}, \{1, 5\}, \{1,7\},\{2, 4\}), \{2,5\}, \{2,7\},\{3,7\},\{4,5\},\{6,7\}\}.$$ This graph is drawn in Figure \ref{fig:runningexamp}. \begin{figure}[ht] \centering \begin{tikzpicture}[every node/.style={circle, draw, fill=white!60, inner sep=2pt}, node distance=1.5cm] \node (1) at (1.5, 0) {1}; \node (2) at (1.5, -1.5) {2}; \node (3) at (3, 0) {3}; \node (4) at (0, -1.5) {4}; \node (5) at (0, 0) {5}; \node (6) at (4.5, 0) {6}; \node (7) at (3, -1.5) {7}; \draw (1) -- (2); \draw (1) -- (3); \draw (1) -- (4); \draw (1) -- (5); \draw (1) -- (7); \draw (2) -- (4); \draw (2) -- (5); \draw (2) -- (7); \draw (3) -- (7); \draw (4) -- (5); \draw (6) -- (7); \end{tikzpicture} \caption{Graph $G$} \label{fig:runningexamp} \end{figure} The binomial edge ideal of $G$ is an ideal of $R=k[x_1,\ldots,x_7,y_1,\ldots,y_7]$ with 11 generators. Specifically, \begin{multline} J_G = \langle x_1y_2-x_2y_1, x_1y_3-x_3y_1, x_1y_4-x_4y_1, x_1y_5-x_5y_1, x_1y_7-x_7y_1, x_2y_4-x_4y_2, \\ x_2y_5-x_5y_2, x_2y_7-x_7y_2, x_3y_7-x_7y_3, x_4y_5-x_5y_4, x_6y_7-x_7x_6 \rangle. \end{multline} \end{example} \section{Partial Betti splittings} In this section, we define the notion of a partial Betti splitting, generalising the concept of a Betti splitting first established by Francisco, H\`a, and Van Tuyl \cite{francisco_splittings_2008}. While a Betti splitting of an ideal $I$ is a ``splitting" of $I$ into two ideals $I = J+K$ such that {\it all} of the (multi)-graded Betti numbers of $I$ can be related to those of $J, K$ and $J \cap K$, in a partial Betti splitting, we only require that some of these relations to hold. Betti splittings of ideals were originally defined just for monomial ideals, since the original motivation of \cite{francisco_splittings_2008} was to extend Eliahou and Kevaire's splitting of monomial ideals \cite{EK1990}. However, a careful examination of the proofs of \cite{francisco_splittings_2008} reveals that some of the main results hold for all (multi)-graded ideals in a polynomial ring $R = k[x_1,\ldots,x_n]$. We develop partial Betti splittings within this more general context. Assuming that $R$ is $M$-graded, let $I,J$, and $K$ be homogeneous ideals with respect to this grading such that $I = J + K$ and $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. We have a natural short exact sequence $$0 \longrightarrow J \cap K \stackrel{\varphi}{\longrightarrow} J \oplus K \stackrel{\psi}{\longrightarrow} I = J+K \longrightarrow 0,$$ where the maps $\varphi(f) = (f,-f)$ and $\psi(g,h) = g+h$ have degree $0$, i.e., they map elements of degree $j \in M$ to elements of degree $j \in M$. The mapping cone resolution applied to this short exact sequence then implies that $$\beta_{i,j}(I) \leq \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J \cap K) ~~\mbox{for all $i \geq 0$ and $j \in M$}.$$ We are then interested in when we have an equality. The following lemma gives such a condition for a specific $i \in \mathbb{N}$ and $j \in M$. The proof is essentially the same as \cite[Proposition 2.1]{francisco_splittings_2008} which considered only monomial ideals, but for completeness, we have included the details here. \begin{lemma}\label{singlesplit} Let $R$ be a $M$-graded ring, and suppose that $I, J$, and $K$ are homogeneous ideals with respect to this grading such that $I = J+K$ and $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Let $$0 \longrightarrow J \cap K \stackrel{\varphi}{\longrightarrow} J \oplus K \stackrel{\psi}{\longrightarrow} I = J+K \longrightarrow 0$$ be the natural short exact sequence. Then, for a fixed integer $i > 0$ and $j \in M$, the following two statements are equivalent: \begin{enumerate} \item $\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K) + \beta_{i-1,j}(J\cap K)$; \item the two maps $$\varphi_i:{\rm Tor}_i(k,J \cap K)_j \rightarrow {\rm Tor}_i(k,J)_j \oplus {\rm Tor}_i(k,K)_j$$ and $$\varphi_{i-1}:{\rm Tor}_{i-1}(k,J \cap K)_j \rightarrow {\rm Tor}_{i-1}(k,J)_j \oplus {\rm Tor}_{i-1}(k,K)_j$$ induced from the long exact sequence of \emph{Tor} using the above short sequence are both the zero map. \end{enumerate} \end{lemma} \begin{proof} Fix an integer $i >0$ and $j \in M$. Using the short exact sequence given in the statement, we can use Tor to create a long exact sequence that satisfies \begin{multline} \cdots \rightarrow {\rm Tor}_i(k,J \cap K)_j \stackrel{\varphi_i}{\rightarrow} {\rm Tor}_i(k,J)_j \oplus {\rm Tor}_i(k,K)_j \rightarrow {\rm Tor}_i(k,I)_j \rightarrow \\ {\rm Tor}_{i-1}(k,J \cap K)_j \stackrel{\varphi_{i-1}}\rightarrow {\rm Tor}_{i-1}(k,J)_j \oplus {\rm Tor}_{i-1}(k,K)_j \rightarrow \cdots . \end{multline} Consequently, we have an exact sequence of vector spaces \begin{multline} 0 \rightarrow {\rm Im}(\varphi_i)_j \rightarrow {\rm Tor}_i(k,J)_j \oplus {\rm Tor}_i(k,K)_j \rightarrow {\rm Tor}_i(k,I)_j \rightarrow \\ {\rm Tor}_{i-1}(k,J \cap K)_j \stackrel{\varphi_{i-1}}\rightarrow A_j \rightarrow 0 \end{multline} where $$A = {\rm Im}(\varphi_{i-1}) \cong {\rm Tor}(k,J \cap K)/{\ker \varphi_{i-1}}.$$ We thus have $$\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K) + \beta_{i-1,j}(J\cap K) - \dim_k ({\rm Im}(\varphi_i))_j - \dim_k ({\rm Im}(\varphi_{i-1}))_j.$$ To prove $(1) \Rightarrow (2)$, note that if both $\varphi_i$ and $\varphi_{i-1}$ are the zero map, we have $\beta_{i,j}(I) = \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J \cap K)$. For $(2) \Rightarrow (1)$, if either of $\varphi_i$ or $\varphi_{i-1}$ is not the zero map, either $\dim_k ({\rm Im}(\varphi_i))_j > 0$ or $\dim_k ({\rm Im}(\varphi_{i-1}))_j> 0$, which forces $\beta_{i,j}(I) < \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J \cap K).$ \end{proof} The following corollary, which is \cite[Proposition 3]{francisco_splittings_2008}, immediately follows. \begin{corollary} Let $R$ be a $M$-graded ring, and suppose that $I, J$, and $K$ are homogeneous ideals with respect to this grading such that $I = J+K$ and $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Let $$0 \longrightarrow J \cap K \stackrel{\varphi}{\longrightarrow} J \oplus K \stackrel{\psi}{\longrightarrow} I = J+K \longrightarrow 0$$ be the natural short exact sequence. Then $\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K) + \beta_{i-1,j}(J\cap K)$ for all integers $i \geq 0$ and $j \in M$, if and only if the maps $$\varphi_i:{\rm Tor}_i(k,J \cap K)_j \rightarrow {\rm Tor}_i(k,J)_j \oplus {\rm Tor}_i(k,K)_j$$ induced from the long exact sequence of {\rm Tor} using the above short exact sequence are the zero map for all integers $i \geq 0$ and $j \in M$. \end{corollary} Applying \Cref{singlesplit} directly implies that we would need to understand the induced maps between {\rm Tor} modules in order to determine if a specific $(i,j)$-th graded Betti number of $I$ can be determined from those of $J$, $K$, and $J\cap K$. However, we can now modify Theorem 2.3 from \cite{francisco_splittings_2008} to obtain a a specific ``splitting'' of $\beta_{i,j}(I)$ from other graded Betti numbers. \begin{theorem}\label{parcon} Let $R$ be a $M$-graded ring, and suppose that $I, J$, and $K$ are homogeneous ideals with respect to this grading such that $I = J+K$ and $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Suppose for a fixed integer $i > 0$ and $j \in M$ we have that: \begin{itemize} \item if $\beta_{i,j}(J\cap K)>0$, then $\beta_{i,j}(J) = 0$ and $\beta_{i,j}(K) = 0$, and \item if $\beta_{i-1,j}(J\cap K)>0$, then $\beta_{i-1,j}(J) = 0$ and $\beta_{i-1,j}(K) = 0.$ \end{itemize} Then we have: \begin{equation} \beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1, j}(J\cap K ). \end{equation} \end{theorem} \begin{proof} Since $I = J+K$, we have the short exact sequence \[0\longrightarrow J\cap K \xlongrightarrow{\varphi} J\oplus K \xlongrightarrow{\psi} J+K = I\longrightarrow 0.\] For all integers $\ell \geq 0$ and $j \in M$, we get the induced maps $$\varphi_\ell:{\rm Tor}_\ell(k,J \cap K)_j \rightarrow {\rm Tor}_\ell(k,J)_j \oplus {\rm Tor}_\ell(k,K)_j$$ from the long exact sequence of {\rm Tor} using the short exact sequence. Let $i > 0$ and $j \in M$ be the fixed $i$ and $j$ as in the statement. There are four cases to consider: (1) $\beta_{i,j}(J \cap K)$ and $\beta_{i-,j}(J \cap K)$ both non-zero, (2) $\beta_{i,j}(J\cap K) = 0$ and $\beta_{i-1,j}(J \cap K) > 0$, (3) $\beta_{i,j}(J\cap K) > 0$ and $\beta_{i-1,j}(J \cap K) = 0$, and (4) both $\beta_{i,j}(J\cap K) = \beta_{i-1,j}(J \cap K) = 0$. In case (1), the maps $\varphi_i$ and $\varphi_{i-1}$ must be the zero map since $0 =\beta_{i,j}(J)$ and $0 = \beta_{i,j}(K)$ imply that ${\rm Tor}_i(k,J)_j \oplus {\rm Tor}_i(k,K)_j = 0$, and similarly, $0 =\beta_{i-1,j}(J)$ and $0 = \beta_{i-1,j}(K)$ imply ${\rm Tor}_{i-i}(k,J)_j \oplus {\rm Tor}_{i-1}(k,K)_j = 0$. The conclusion now follows from \Cref{singlesplit}. For case (2), the map $\varphi_{i-1}$ is the zero map using the same argument as above. On the other hand, $0 = \beta_{i,j}(J \cap K) = \dim_k {\rm Tor}(k, J\cap K)_j$ implies that $\varphi_i$ is the zero map. We now apply \Cref{singlesplit}. Cases (3) and (4) are proved similarly, so we omit the details. \end{proof} We now introduce the notion of a partial Betti splitting, that weakens the conditions of a Betti splitting found in \cite{francisco_splittings_2008}. Note that we assume that $R$ has the standard $\mathbb{N}$-grading. \begin{definition}\label{pardef} Let $I$, $J$ and $K$ be homogeneous ideals of $R$ with respect to the standard $\mathbb{N}$-grading such that $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Then $I= J + K$ is an {\it $(r,s)$-Betti splitting} if \[\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1, j}(J\cap K )\text{\hspace{3mm} for all $(i,j)$ with $i\geq r$ or $j\geq i+s$}.\] If $(r,s) \neq (0,0)$ we call an $(r,s)$-Betti splitting $I=J+K$ a {\it partial Betti splitting}. Otherwise, we say that $I = J+K$ is a {\it complete Betti splitting} if it is a $(0,0)$-Betti splitting, that is, $$\beta_{i,j}(I) = \beta_{i,j}(J) + \beta_{i,,j}(K) + \beta_{i-1,j}(J\cap K) ~~\mbox{for all $i,j \geq 0$}.$$ \end{definition} \begin{remark} A complete Betti splitting is what Francisco, H\`a, and Van Tuyl \cite{francisco_splittings_2008} called a Betti splitting. \end{remark} \begin{remark} We can interpret the above definition with the Betti table of $I$. The {\it Betti table of $I$} is a table whose columns are indexed by the integers $i\geq 0$, and in row $j$ and column $i$, we place $\beta_{i,i+j}(I)$. If $I = J+K$ is an $(r,s)$-Betti splitting, then all the Betti numbers in the Betti table of $I$ in the $r$-th column and beyond or in the $s$-th row and beyond are ``split'', that is, they satisfy $\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1, j}(J\cap K ).$ \end{remark} The following observation will be useful. \begin{lemma} Suppose that $I=J+K$ is an $(r,s)$-Betti splitting of $I$. If $r = 0$ or $1$, then $I=J+K$ is a complete Betti splitting. \end{lemma} \begin{proof} Since $I = J+K$ is an $(r,s)$-Betti splitting, we have $\mathfrak{G}(I) = \mathfrak{G}(J) \cup \mathfrak{G}(K)$. Consequently, we always have $$\beta_{0,j}(I) = \beta_{0,j}(J) + \beta_{0,j}(K) + \beta_{-1,j}(J\cap K) = \beta_{0,j}(J)+\beta_{0,j}(K) ~\mbox{for $i=0$ and all $j \geq 0$.}$$ For any $(r,s)$-Betti splitting with $r =0$ or $1$, the definition implies \[\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1, j}(J\cap K ) ~\mbox{for all $i > 0$ and all $j \geq 0$}.\] So, for any $i,j \geq 0$, we have $\beta_{i,j}(I) = \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J \cap K)$, that is, we have a complete Betti splitting. \end{proof} We can now use Theorem \ref{parcon} to get a condition on $(r,s)$-Betti splittings. \begin{theorem}\label{parcon2} Let $I$, $J$ and $K$ be homogeneous ideals of $R$ with respect to the standard $\mathbb{N}$-grading such that $\mathfrak{G}(I)$ is the disjoint union of $\mathfrak{G}(J)$ and $\mathfrak{G}(K)$. Suppose that there are integers $r$ and $s$ such that for all $i \geq r$ or $j \geq i+s$, $\beta_{i-1,j}(J \cap K) > 0$ implies that $\beta_{i-1,j}(J) = 0$ and $\beta_{i-1,j}(K) = 0$. Then $I = J + K$ is an $(r,s)$-Betti splitting. \end{theorem} \begin{proof} Let $r$ and $s$ be as in the statement, and suppose that $(i,j)$ is fixed integer tuple that satisfies $i \geq r$ or $j \geq i+s$. But then $(i+1,j)$ also satisfies $i+1 \geq r$ or $j \geq i+s$. Consequently, for this fixed $(i,j)$, the hypotheses imply \begin{enumerate} \item[$\bullet$] if $\beta_{i-1,j}(J\cap K) >0$, then $\beta_{i-1,j}(J) = \beta_{i-1,j}(K) = 0$, and \item[$\bullet$] if $\beta_{i,j}(J\cap K) > 0$, then $\beta_{i,j}(J) = \beta_{i,j}(K) = 0$. \end{enumerate} By Theorem \ref{parcon}, this now implies that $$\beta_{i,j}(I) = \beta_{i,j}(J)+\beta_{i,j}(K) + \beta_{i-1,j}(J\cap K)$$ for this fixed pair $(i,j)$. But since this is true for all $(i,j)$ with either $i \geq r$ or $j \geq i+s$, this means $I=J+K$ is an $(r,s)$-Betti splitting. \end{proof} We end this section with consequences for the regularity and projective dimension of $I$ for a partial Betti splitting. The case for a complete Betti splitting was first shown in \cite[Corollary 2.2]{francisco_splittings_2008}. \begin{theorem}\label{regprojbounds} Suppose that $I=J+K$ is an $(r,s)$-Betti splitting of $I$. Set \begin{eqnarray} m &= &\max\{ {\rm reg}(J), {\rm reg}(K), {\rm reg}(J\cap K)-1\}, ~~\mbox{and} \\ p &=& \max\{ {\rm pd}(I), {\rm pd}(J), {\rm pd}(J\cap K)+1\}. \end{eqnarray} Then \begin{enumerate} \item if $m \geq s$, then ${\rm reg}(I) = m$. \item if $p \geq r$, then ${\rm pd}(I) = p$. \end{enumerate} \end{theorem} \begin{proof} By applying the mapping cone construction to the the short exact sequence $$0 \longrightarrow J \cap K \longrightarrow J \oplus K \longrightarrow J+K = I \longrightarrow 0,$$ we always have ${\rm reg}(I) \leq m$ and ${\rm pd}(I) \leq p$. Since $m \geq s$, this means for all $i \geq 0$ $$\beta_{i,i+m}(I)=\beta_{i,i+m}(J)+\beta_{i,i+m}(K) +\beta_{i-1,i+m}(J\cap K)$$ because we have an $(r,s)$-Betti splitting. By our definition of $m$, there is an integer $i$ such that at least one of the three terms on the right hand side must be nonzero. This then forces ${\rm reg}(I) \geq m$, thus completing the proof that ${\rm reg}(I) = m$. Similarly, since $p \geq r$, for all $j \geq 0$ we have $$\beta_{p,j}(I) = \beta_{p,j}(J)+\beta_{p,j}(K) +\beta_{p-1,j}(J\cap K).$$ By our definition of $p$, there is at least one $j$ such that one of the terms on the right hand side is nonzero, thus showing ${\rm pd}(I) \geq p$. Consequently, ${\rm pd}(I) = p$. \end{proof} \begin{example}\label{runningexample2} We illustrate a partial Betti splitting using the binomial edge ideal $J_G$ of \Cref{runningexample}. We ``split'' $J_G$ as $J_G = J + K$ where \begin{eqnarray} J & = & \langle x_1y_2-x_2y_1, x_1y_3-x_3y_1, x_1y_4-x_4y_1, x_1y_5-x_5y_1, x_1y_7-x_7y_1 \rangle ~~\mbox{and}\\ K& = & \langle x_2y_4-x_4y_2, x_2y_5-x_5y_2, x_2y_7-x_7y_2, x_3y_7-x_7y_3, x_4y_5-x_5y_4, x_6y_7-x_7x_6 \rangle. \end{eqnarray} We compute the graded Betti tables use in \emph{Macaulay2} \cite{mtwo}. The graded Betti tables of $J$, $K$ and $J \cap K$ are given below. \footnotesize \begin{verbatim} 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 5 total: 5 20 30 18 4 total: 6 15 20 14 4 total: 15 47 73 62 26 4 2: 5 . . . . 2: 6 2 . . . 2: . . . . . . 3: . 20 30 18 4 3: . 13 8 . . 3: 10 9 2 . . . 4: . . . . . 4: . . 12 14 4 4: 5 26 21 4 . . 5: . . . . . 5: . . . . . 5: . 12 50 58 26 4 Betti Table J Betti Table K Betti Table J intersect K \end{verbatim} \normalsize We compare this to the Betti table of $J_G$: \footnotesize \begin{verbatim} 0 1 2 3 4 5 6 total: 11 44 89 103 70 26 4 2: 11 12 3 . . . . 3: . 32 62 39 8 . . 4: . . 24 64 62 26 4 Betti Table J_G \end{verbatim} \normalsize Then $J_G = J+K$ is {\it not} a complete Betti splitting since $$\beta_{2,4}(J_G) = 3 \neq 0+ 0+ 9 =\beta_{2,4}(J) + \beta_{2,4}(K) + \beta_{1,4}( J\cap K).$$ However, this is an example of a $(4,4)$-Betti splitting since $$\beta_{i,j}(J_G) = \beta_{i,j}(J) + \beta_{i,j}(K) + \beta_{i-1,j}(J\cap K) ~~\mbox{for all $i \geq 4$ and $j \geq i+4$.}$$ \end{example} \section{Betti splittings of binomial edge ideals: cut edge case} In this section and the next, we wish to understand when a binomial edge ideal $J_G$ has a (partial) Betti splitting. A natural candidate to consider is when $G_1$ is a single edge $e = \{u,v\}$ of $G$ and $G_2 = G\setminus e$. More formally, if $f_e = x_uy_v-x_vy_u$ is the binomial associated to $e$, we wish to understand when $$J_G = \langle f_e \rangle + J_{G\setminus e}$$ is either a partial or a complete Betti splitting of $J_G$. As we show in this section, with some extra hypotheses on $e$, this splitting of $J_G$ does indeed give a complete Betti splitting. Since Betti splittings require information about the intersection of the two ideals used in the splitting, the following lemma shall prove useful. \begin{lemma}\label{lemma 2.18} Let $G = (V(G),E(G))$ be a simple graph with $e \in E(G)$. Then, using the standard grading of $R$, we have a graded $R$-module isomorphism $$[J_{G\setminus e} \cap \langle f_e \rangle] \cong [J_{G\setminus e}: \langle f_e \rangle](-2).$$ Furthermore, if $e$ is a cut edge, then $$ \beta_{i,j}(J_{(G\setminus e)}\cap \langle f_e\rangle) = \beta_{i,j-2}(J_{(G\setminus e)_e}) ~\mbox{for all $i \geq 0$}.$$ \end{lemma} \begin{proof} By definition of quotient ideals, we have that $J_{G\setminus e}: \langle f_e \rangle \xrightarrow{\cdot f_e} J_{(G\symbol{92} e)}\cap \langle f_e\rangle$ is an $R$-module isomorphism of degree two. This fact implies the first statement. Now suppose that $e$ is a cut edge. From \Cref{lemma 3.8} we have that $J_{(G\setminus e)_e} = J_{G\setminus e}: \langle f_e \rangle$. Using this fact and the above isomorphisms of modules, we have $$ \tor_i(J_{(G\setminus e)_e},k)_{j-2} = \tor_{i}(J_{G\setminus e}:\langle f_e \rangle, k)_{j-2} \cong \tor_{i}(J_{G\setminus e}\cap \langle f_e\rangle, k)_j. $$ This isomorphism imples that $\beta_{i,j}(J_{(G\setminus e)}\cap \langle f_e\rangle) = \beta_{i,j-2}(J_{(G\setminus e)_e})$ for all $i \geq 0$ for $j \geq 2$. Now, for any $i \geq 0$ and $j=0$, $\beta_{i,0}(J_{(G\setminus e)}\cap \langle f_e\rangle) = \beta_{i,0-2}(J_{(G\setminus e)_e}) =0$. Finally, because $J_{(G\setminus e)_e} = J_{G \setminus e} : \langle f_e \rangle$ is generated by degree two binomials, then $J_{G\setminus e} \cap \langle f_e \rangle$ is generated by degree four elements. Thus $\beta_{i,1}(J_{(G\setminus e)}\cap \langle f_e\rangle) = \beta_{i,1-2}(J_{(G\setminus e)_e}) =0$ for all $i \geq 0$ and $j =1$. \end{proof} With the above lemma, we can study splittings where $e = \{u,v\}$ when $v$ is a pendant vertex, that is, $\deg v = 1$. \begin{theorem}\label{maintheo} Let $e = \{u,v\} \in E(G)$ with $v$ a pendant vertex. Then \begin{enumerate} \item $J_G = J_{G\setminus e}+\langle f_e\rangle$ is a complete Betti splitting, and \item $\beta_{i,j}(J_G) = \beta_{i,j}(J_{G\symbol{92}e}) + \beta_{i-1,j-2}(J_{(G\setminus e)_e})$ for all $i\geq 1$ and $j \geq 0$. \end{enumerate} \end{theorem} \begin{proof} (1). Let $J_G = \langle f_e\rangle+J_{G\setminus e} \subseteq R = k[x_1,\ldots,x_n,y_1,\ldots,y_n]$. We consider the $\mathbb{N}^n$-grading on $R$ given by $\deg x_i = \deg y_i = e_i$, the $i$-th standard basis vector of $\mathbb{N}^n$. Note that $J_G$ is a homogeneous ideal with respect to this grading. Since $\langle f_e\rangle\cap J_{G\setminus e}\subseteq \langle f_e \rangle$, all generators of $\langle f_e\rangle\cap J_{G\setminus e}$ are of the form $rf_e = r(x_uy_v-x_vy_u)$, where $r$ is some polynomial in $R$. Hence, the multidegree of the generators, and thus the multigraded Betti numbers of the ideal $\langle f_e\rangle\cap J_{G\setminus e}$ must occur with multidegrees $\mathbf{a} = (a_1,\ldots,a_n)$ where its $v$-th component $a_v$ is non-zero. Because $v$ is a pendant vertex, $J_{G\setminus e}$ contains no generators having $x_v$ or $y_v$. Thus, $\beta_{i,{\bf a}}(J_{G\symbol{92}e}\cap \langle f_e \rangle )>0$ implies that $\beta_{i,{\bf a}}(J_{G \setminus e}) = 0$ for all $i\in \mathbb{N}$ and all multidegrees ${\bf a} \in \mathbb{N}^n$ as defined above. We have that $\beta_{0,2}(\langle f_e\rangle) = 1$ and $\beta_{i,j}(\langle f_e\rangle) = 0$ for $i\neq 0$ and $j\neq 2$ as $\langle f_e\rangle$ is a principal ideal. Since $J_{G\symbol{92}e}\cap \langle f_e\rangle$ is generated by polynomials of degree three or more, this means that $\beta_{i,j}(J_{G\symbol{92}e}\cap \langle f_e\rangle)>0$ implies $\beta_{i,j}(\langle f_e \rangle) = 0$ for all $i\geq 0$ and degrees $j$. It is clear that since this is true for all degrees $j$, this result also holds for all ${\bf a} \in \mathbb{N}^n$ as well, that is, if $\beta_{i,{\bf a}}(J_{G \setminus e} \cap \langle f_e \rangle) > 0$, then $\beta_{i,{\bf a}}(\langle f_e \rangle) =0$ for all $i \geq 0$ and degrees ${\bf a} \in \mathbb{N}^n$. Therefore \Cref{parcon} implies that $$\beta_{i,{\bf a}}(J_G) = \beta_{i,{\bf a}}(J_{G\setminus e})+ \beta_{i,{\bf a}}(\langle f_e \rangle) + \beta_{i-1,{\bf a}}(J_{G\setminus e} \cap \langle f_e \rangle)$$ for all $i \geq 0$ and ${\bf a} \in \mathbb{N}^n$. Since this true for all multidegrees, we can combine them to obtain the same result with the degrees $j$ in the standard grading. Hence we have: $$\beta_{i,j}(J_G) = \beta_{i,j}(\langle f_e\rangle)+ \beta_{i,j}(J_{G\symbol{92} e}) + \beta_{i-1,j}(J_{G\symbol{92} e}\cap \langle f_e\rangle) ~\text{for all $i,j \geq 0$},$$ that is, $J_G = \langle f_e\rangle+J_{G\setminus e}$ is a complete Betti splitting. An edge with a pendant vertex is a cut edge of $G$. So, to prove (2), we can combine (1) and \Cref{lemma 2.18} to give $$\beta_{i,j}(J_G) = \beta_{i,j}(\langle f_e\rangle)+\beta_{i,j}(J_{G\symbol{92} e}) + \beta_{i-1,j-2}(J_{(G\symbol{92} e)_e})$$ for all integers $i \geq 1$ and $j \geq 0$. On the other hand, $\beta_{i,j}(\langle f_e\rangle) = 0$ for $i\neq 0$ or $j\neq 2$. Hence, $\beta_{i,j}(J_G) = \beta_{i,j}(J_{G\symbol{92}e}) + \beta_{i-1,j-2}(J_{(G\symbol{92}e)_e})$ for all $i\geq 1$ and $j \geq 0$. \end{proof} In \Cref{maintheo}, we have proved that when there is a cut edge $e$ where one end is a pendant vertex, then removing $e$ induces a complete Betti splitting. We can now use this result to derive complete Betti splittings for more general types of edges. \begin{theorem}\label{singlefreevertex} Let $e = \{u,v\} \in E(G)$ be a cut-edge where $v$ is a free vertex in $G\setminus e$. Then \begin{enumerate} \item $J_G = J_{G\setminus e}+\langle f_e\rangle$ is a complete Betti splitting, and \item $\beta_{i,j}(J_G) = \beta_{i,j}(J_{G\symbol{92}e}) + \beta_{i-1,j-2}(J_{(G\setminus e)_e})$ for all $i\geq 1$ and $j \geq 0$. \end{enumerate} \end{theorem} \begin{proof} First note that if we can prove $(2)$, then $(1)$ will follow. To see why, it is immediate that $\beta_{0,j}(J_G) = \beta_{0,j}(J_{G\setminus e}) + \beta_{0,j}(\langle f_e \rangle) +\beta_{-1,j}(J_{G\setminus e} \cap \langle f_e \rangle)$ for all $j \geq 0$. If $i \geq 1$, then \begin{eqnarray} \beta_{i,j}(J_G) &=& \beta_{i,j}(J_{G\symbol{92}e}) + \beta_{i-1,j-2}(J_{(G\setminus e)_e}) \\ & = & \beta_{i,j}(J_{G\setminus e}) + \beta_{i,j}(\langle f_e \rangle) + \beta_{i-1,j}(J_{G \setminus e} \cap \langle f_e \rangle) \end{eqnarray} where we are using \Cref{lemma 2.18} and the fact that $\beta_{i,j}(\langle f_e \rangle) = 0$ for all $i \geq 1$. Now note that to prove to $(2)$, we can pass to quotient rings and prove that $$\beta_{i,j}(R/J_G) = \beta_{i,j}(R/J_{G\setminus e}) + \beta_{i-1,j-2}(R/J_{(G\setminus e)_e} ) ~~\mbox{for all $i \geq 2$ and $j \geq 0$}.$$ Let $G$ be a connected graph with cut-edge $e = \{u,v\}$. Let $G_1$ and $G_2$ be the connected components of $G\setminus e$, and suppose $u\in V(G_1)$ and $v\in V(G_2)$. By our hypotheses, the vertex $v$ is a free vertex in $G_2$. Hence, we can see that $G$ is a decomposable graph, with decomposition $G = (G_1\cup \{e\}) \cup_v G_2$ (since pendant vertices are free vertices and $v$ is a pendant vertex of $e$). By \Cref{freevertexbetti} we have \begin{equation}\label{5.21} \beta_{i,j}(R/J_G) = \sum_{\substack{0 \leq i_1\leq i \\ ~0 \leq j_1\leq j}}\beta_{i_1,j_1}(R/J_{G_1\cup \{e\}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}}). \end{equation} Since $e$ is a cut-edge with a pendant vertex in $G_1 \cup \{e\}$, we can now apply \Cref{maintheo} to $R/J_{G_1 \cup \{e_1\}}$. Thus, \begin{multline}\label{1.2} \sum_{\substack{0 \leq i_1\leq i \\0 \leq j_1\leq j}}\beta_{i_1,j_1}(R/{J_{G_1\cup \{e\}}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}}) = \\ \sum_{\substack{2\leq i_1\leq i \\ 0 \leq j_1\leq j}}(\beta_{i_1,j_1}(R/{J_{G_1}}) + \beta_{i_1-1,j_1-2}(R/{J_{(G_1)_e}}))\beta_{i-i_1,j-j_1}(R/{J_{G_2}}) \\ + (\beta_{1,2}(R/{J_{G_1}})+ 1)\beta_{i-1,j-2}(R/{J_{G_2}}) + \beta_{i,j}(R/{J_{G_2}}). \end{multline} Here, we are using the fact that $\beta_{1,j}(R/J_{G_1 \cup \{e\}}) =0$ if $j \neq 2$, and when $j=2$, $J_{G_1 \cup \{e\}}$ has one more generator than $J_{G_1}$, that is, $\beta_{1,2}(R/J_{G_1 \cup \{e\}}) = \beta_{1,2}(R/J_{G_1})+1$. By expanding out and regrouping, we get \footnotesize \begin{align} \label{1.3} \beta_{i,j}(J_G) =& \sum_{ \substack{1\leq i_1\leq i \\ 0\leq j_1\leq j}}\beta_{i_1,j_1}(R/{J_{G_1}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}}) + \beta_{i,j}(R/{J_{G_2}}) \nonumber\\ & + \sum_{\substack{2\leq i_1\leq i \\ 0 \leq j_1\leq j}}\beta_{i_1-1,j_1-2}(R/{J_{(G_1)_e}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}}) +\beta_{i-1,j-2}(R/{J_{G_2}}) \nonumber\\ =& \sum_{ \substack{0 \leq i_1\leq i \\ 0 \leq j_1\leq j}}\beta_{i_1,j_1}(R/{J_{G_1}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}})+ \sum_{\substack{0 \leq i_1\leq i-1 \\ 0 \leq j_1\leq j-2}}\beta_{i_1,j_1}(R/{J_{(G_1)_e}})\beta_{i-1-i_1,j-2-j_1}(R/{J_{G_2}}). \end{align} \normalsize Since $G_1$ and $G_2$ are graphs on disjoint sets of vertices, $J_{G_1}$ and $J_{G_2}$ are ideals on disjoint sets of variables. Hence, \begin{align}\label{1.4} \sum_{\substack{0\leq i_1\leq i \\ 0\leq j_1\leq j}}\beta_{i_1,j_1}(R/{J_{G_1}})\beta_{i-i_1,j-j_1}(R/{J_{G_2}}) & = \beta_{i,j}(R/{J_{G_1}+J_{G_2}}) \nonumber \\ &=\beta_{i,j}(R/{J_{G_1\cup G_2}}) = \beta_{i,j}(R/{J_{(G\setminus e)}}). \end{align} Similarly, the same is true for $(G_1)_e$ and $G_2$. Note, that since $v$ is already a free vertex of $G_2$, we have $(G\setminus e)_e = (G_1)_e \cup G_2$. Hence, \begin{align}\label{1.5} \sum_{\substack{0 \leq i_1\leq i-1 \\ 0 \leq j_1\leq j-2}}\beta_{i_1,j_1}(R/{J_{(G_1)_e}})\beta_{i-1-i_1,j-2-j_1}(R/{J_{G_2}}) & = \beta_{i-1,j-2}(R/{J_{(G_1)_e}+J_{G_2}}) \nonumber\\ & = \beta_{i-1,j-2}(R/{J_{(G_1)_e\cup G_2}}) \nonumber \\ & = \beta_{i-1,j-2}(R/{J_{(G\setminus e)_e}}). \end{align} Thus, substituting \Cref{1.5} with \Cref{1.4} into \Cref{1.3}, we get the desired conclusion. \end{proof} Because we have a complete Betti splitting, \Cref{regprojbounds} implies the collorary. \begin{corollary}\label{singlevertexcor} With the hypotheses as in \Cref{singlefreevertex}, \begin{eqnarray} {\rm reg}(J_G) &= &\max\{{\rm reg}(J_{G\setminus e}), {\rm reg}((J_{G \setminus e})_e) +1\} ~~\mbox{and} \\ {\rm pd}(J_G) &= &\max\{{\rm pd}(J_{G\setminus e}), {\rm pd}(J_{(G \setminus e)_e}) +1\}. \end{eqnarray} \end{corollary} \begin{proof} Because $J_G = J_{G\setminus e} + \langle f_e \rangle$ is a complete Betti splitting, \Cref{regprojbounds} gives \begin{eqnarray} {\rm reg}(J_G) &= &\max\{{\rm reg}(J_{G\setminus e}), {\rm reg}(\langle f_e \rangle), {\rm reg}(J_{G \setminus e} \cap \langle f_e \rangle) -1\} ~~\mbox{and} \\ {\rm pd}(J_G) &= &\max\{{\rm pd}(J_{G\setminus e}), {\rm pd}(\langle f_e \rangle), {\rm pd}(J_{G \setminus e} \cap \langle f_e \rangle) +1\}. \end{eqnarray} The result now follows since $2 = {\rm reg}(\langle f_e \rangle) \leq {\rm reg}(J_{G \setminus e})$ and $0 = {\rm pd}(\langle f_e \rangle)$ and because \Cref{lemma 2.18} implies ${\rm reg}(J_{G \setminus e} \cap \langle f_e \rangle) = {\rm reg}(J_{(G\setminus e)_e}) +2$ and ${\rm pd}(J_{G \setminus e} \cap \langle f_e \rangle) = {\rm pd}(J_{(G \setminus e)_e})$. \end{proof} Recall that an edge $e = \{u,v\}$ is a free cut-edge of $G$ if both $u$ and $v$ are free vertices of $G \setminus e$. When \Cref{singlefreevertex} is applied to a free cut-edge, we can recover the following results of Saeedi Madani and Kiani \cite{kiani_regularity_2013-1}. \begin{corollary}[{\cite[Proposition 3.4]{kiani_regularity_2013-1}}] \label{freecutedge} Let $e = \{u,v\} \in E(G)$ be a free cut-edge. Then \begin{enumerate} \item $\beta_{i,j}(J_G) = \beta_{i,j}(J_{G\setminus e}) + \beta_{i-1,j-2}(J_{G\setminus e})$, \item \rm pd($J_G$) = pd($J_{G\setminus e}) + 1$, and \item \rm reg($J_G$) = reg($J_{G\setminus e}$) + 1. \end{enumerate} \end{corollary} \begin{proof} When $e$ is a free cut-edge of $G$, then $(G\setminus e)_e = G\setminus e$. The results then follow from \Cref{singlefreevertex} and \Cref{singlevertexcor} by using the equality $J_{(G\setminus e)_e} = J_{G\setminus e}.$ \end{proof} One application of \Cref{maintheo} is finding the Betti numbers of the binomial edge ideals of certain graphs. The corollary below is a new proof of \cite[Proposition 3.8]{jayanthan_almost_2021} for the graded Betti numbers of the binomial edge ideals of any star graph $S_n$. \begin{corollary}\label{star} Let $S_n$ denote the star graph on $n$-vertices. Then we have: \[ \beta_{i}(J_{S_n}) = \beta_{i,i+3}(J_{S_n}) = i\binom{n}{i+2} \text{\hspace{4mm} $i\geq 1$}. \] Furthermore, $\beta_0(J_{S_n}) = \beta_{0,2}(S_n) = n-1$. \end{corollary} \begin{proof} Note that the statement about $0$-th graded Betti numbers just follows from the fact that $S_n$ has $n-1$ edges. Consider the edge $e =\{1,n\}$. Since $S_n\setminus e = S_{n-1} \cup \{n\}$, we have $(S_n\setminus e)_e = K_{n-1} \cup \{n\}$. So from \Cref{maintheo}, we have: \[\beta_{i,j}(J_{S_n}) = \beta_{i,j}(J_{S_{n-1}})+\beta_{k-1,j-2}(J_{K_{n-1}}) ~~\text{ for all $i\geq 1$}.\] We can now use induction to show the above assertion. For $n = 2$, we can see that $S_2$ is just an edge. We know that $\beta_{i,j}(J_{S_2}) = 0$ for all $i\geq 1$. Hence, we can see that it agrees with the above formula as $\binom{2}{r} = 0$ when $r>2$. Now assume the formula holds for $n-1$. We must show that it holds for $n$. From \Cref{completebetti}, we know that $\beta_{i,i+2}(J_{K_{n-1}}) = (i+1)\binom{n-1}{i+2}$ and $\beta_{i,j}(J_{K_{n-1}}) = 0$ if $j\neq i+2$. Hence, using induction and \Cref{maintheo}, we can see that $\beta_{i,j}(J_{S_n}) = \beta_{i,j}(J_{S_{n-1}})+\beta_{i-1,j-2}(J_{K_{n-1}})=0+0$, when $j\neq i+3$. We also have \[\beta_{i,i+3}(J_{S_n}) = \beta_{i,i+3}(J_{S_{n-1}})+\beta_{i-1,i+1}(J_{K_{n-1}}) = i\binom{n-1}{i+2}+i\binom{n-1}{i+1} = i\binom{n}{i+2}.\] This verifies the formula of the statement. \end{proof} \section{Partial Betti splittings of binomial edge ideals: \texorpdfstring{$s$}{s}-partitions} In this section we consider the other natural candidate to study in the context of partial Betti splittings. In this case, we fix a vertex $s \in V)$, and let $G_1$ be the graph with $E(G_1)$ equal to the set of edges of $G$ that contain $s$ (so $G_1$ is isomorphic to a star graph) and $G_2 = G \setminus \{s\}$. We formalize this idea in the next definition. \begin{definition}\label{vpart} For $s\in V(G)$, an {\it $s$-partition} of $J_G$ is the splitting $J_G = J_{G_1}+J_{G_2},$ where $G_1$ is the subgraph of $G$ with $V(G_1) = N_G[s]$ and $E(G_1) = \{\{s,k\}\mid k\in N_G(s)\}$, and $G_2=G\setminus \{s\}$. \end{definition} Note that the graph $G_1$ in an $s$-partition is isomorphic to the star graph $S_{\deg(s)+1}$. We will show that an $s$-partition always gives a partial Betti splitting of $J_G$: \begin{theorem}\label{maintheo2} Let $G$ be a graph on $[n]$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$ for some $s\in [n]$. Let $c(s)$ be the size of the largest clique containing $s$. Then, for all $i, j$ with $i \geq c(s)$ or $j \geq i+4$, \begin{equation} \beta_{i,j}(J_G) = \beta_{i,j}(J_{G_1})+\beta_{i,j}(J_{G_2})+\beta_{i-1, j}(J_{G_1}\cap J_{G_2}). \end{equation} In other words, $J_G = J_{G_1}+J_{G_2}$ is a $(c(s), 4)$-Betti splitting. \end{theorem} Our proof hinges on a careful examination of $J_{G_2} \cap J_{G_2}$, which is carried out below. \begin{lemma}\label{deg3gen} Let $G$ be a graph on $[n]$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$ for some $s\in [n]$. Then the set \[ \mathcal{B} = \{x_sf_{a,b}, y_sf_{a,b}\mid a,b\in N_G(s) \text{ and } \{a,b\}\in E(G)\}.\] is a $k$-basis for $(J_{G_1} \cap J_{G_2})_3$. \end{lemma} \begin{proof} Let $N_G(s) = \{v_1,\dots, v_r\}$. Since $E(G_1) \cap E(G_2) = \emptyset$, the generators of $J_{G_1} \cap J_{G_2}$ are of degree at least $3$. First of all observe that $\B_1 = \{x_if_e, y_if_e\mid e \in E(G_1) \text{ and } i\in \{1, \dots, n\}\}$ and $\B_2 = \{x_if_e, y_if_e\mid e\in E(J_{G_2}) \text{ and } i\in \{1, \dots, n\}\}$ form $k$-bases for the subspaces $(J_{G_1})_3$ and $(J_{G_2})_3$ respectively. Let $P \in (J_{G_1} \cap J_{G_2})_3 = (J_{G_1})_3 \cap (J_{G_2})_3$. Write \begin{equation}\label{eq.P} P = \sum_{g_{i,e}\in \B_1}c_{i,e} g_{i,e}, \end{equation} where $c_{i,e} \in k$. We first claim that the coefficients of $x_if_{a,s}$ and $y_if_{a,s}$ in the linear combination of $P$ are zero if $i \notin \{v_1,\ldots, v_r\}$. We prove this for $x_if_{a,s}$ and the other proof is similar. Let $c$ be the coefficient of $x_if_{a,s}$. Observe that, since $i\notin \{v_1,\dots, v_k\}$, the term $y_sx_ix_a$ in $P$, appears in only one basis element, namely $x_if_{a,s}$. Since $P$ is in $(J_{G_2})_3$ as well, we can write \begin{equation}\label{2.8} P = S+ y_s(c x_ix_a+L) = Q + y_s\left(\sum_{f_e\in \mathfrak{G}(J_{G_2})}c'_e f_e\right), \end{equation} where no terms of $S$ and $Q$ are divisible by $y_s$ and $L$ does not have any monomial terms divisible by $x_ix_a$. Since $y_s$ does not divide any term of $S$ and $Q$, the above equality implies that $c x_ix_a+L = \sum_{f_e\in \mathfrak{G}(J_{G_2})}c'_e f_e$. Now by considering the grading on $R$ given by $\deg x_j = (1,0)$ and $\deg y_j = (0,1)$ for all $j$, we can see that $x_ix_a$ is of degree $(2,0)$ but the degree of each term $f_e$ in $\mathfrak{G}(J_{G_2})$ is $(1,1)$. Hence, for \Cref{2.8} to hold, $c=0$. This completes the proof of the claim. Now consider the case where $i\in \{v_1,\dots, v_k\}$. In this case, it can be seen that the term $y_sx_ix_a$ when written as an element of $(J_{G_1})_3$ appears in the basis elements $x_if_{a,s}$ and $x_af_{i,s}$, and in no other basis element. As before, to make sure that there are no elements of degree $(2,0)$, the coefficients of $x_if_{a,v}$ and $x_af_{i,s}$ in \Cref{eq.P} must be additive inverses of each other. Denote the coefficient of $x_if_{a,s}$ by $c$. Then, $$cx_if_{a,s} - cx_af_{i,s} = cx_s(x_ay_i-x_iy_a) = cx_sf_{a,i}.$$ Similar arguments show that the coefficients of $y_if_{a,s}$ and $y_af_{i,s}$ must be additive inverses of each other, and that the corresponding linear combination in the \Cref{eq.P} appears as $c'y_sf_{a,i}$. Therefore, \Cref{eq.P} becomes \[P = \sum_{a,i\in N_G(s)}c_{i,a} x_sf_{a,i}+c'_{i,a} y_sf_{a,i}.\] Since $P \in (J_{G_2})_3$, it is easily observed that $c_{i,a} = 0$ whenever $\{i,a\} \notin E(G)$. Therefore, $\mathcal{B}$ spans the subspace $(J_{G_1} \cap J_{G_2})_3$. Linear independence is fairly straightforward as $s \neq a, b$ for any $a, b \in N_G(s)$. Hence the assertion of the lemma is proved. \end{proof} \begin{remark}\label{deg4} If $G$ is a triangle-free graph, then there does not exist any $a,b\in N_G(s)$ with $\{a,b\}\in E(G)$ for any $s\in V(G)$. Hence it follows from \Cref{deg3gen} that there are no degree 3 generators of $J_{G_1}\cap J_{G_2}$ for any $s$-partition. Hence, $J_{G_1} \cap J_{G_2}$ is generated by elements of degrees $4$ or higher. \end{remark} Since the generators of $J_{G_1}\cap J_{G_2}$ resemble the generators of a binomial edge ideal, we can calculate its linear strand in terms of the linear strand of some binomial edge ideal. \begin{theorem}\label{thm:Betti-intersection} Let $G$ be a graph on $[n]$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$ for some $s\in [n]$. If $G'$ is the induced subgraph of $G$ on $N_G(s)$, then \[\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = 2\beta_{i,i+2}(J_{G'})+\beta_{i-1,i+1}(J_{G'})\text{\hspace{2mm} for all $i\geq 0$}.\] \end{theorem} \begin{proof} From \Cref{deg3gen}, we have that the minimal degree 3 generators for $J_{G_1}\cap J_{G_2}$ are \[L =\{x_sf_{a,b}, y_sf_{a,b}\mid a,b\in N_G(s) \text{ and } \{a,b\}\in E(G)\}.\] Since, $J_{G_1}\cap J_{G_2}$ is generated in degree 3 or higher, if $I$ is the ideal generated by $L$, then $\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = \beta_{i,i+3}(I)$ for all $i \geq 0$. Now consider the partition $I = I_x+I_y$, where $$ \mathfrak{G}(I_x) = \{x_sf_{a,b}\mid \text{ $\{a,b\}\in E(G')$}\} ~\mbox{and} ~ \mathfrak{G}(I_y) = \{y_sf_{a,b}\mid \text{$\{a,b\}\in E(G')$}\}. $$ We now claim that \[I_x\cap I_y = \langle\{x_sy_sf_{a,b}\mid \text{$\{a,b\}\in E(G')$}\}\rangle.\] It is clear that each $x_sy_sf_{a,b} \in I_x\cap I_y$. For the other inclusion, consider $g\in I_x\cap I_y$. Since $g$ is in both $I_x$ and $I_y$, we can write $g$ as \[g = x_s\left(\sum k_{a,b}f_{a,b}\right) = y_s\left(\sum k'_{a,b}f_{a,b}\right),\] where $k_{a,b}, k'_{a,b} \in R$. Since, none of the $f_{a,b}$'s involve the variables $x_s$ and $y_s$, some terms of $k_{a,b}$ are divisible by $y_s$, for each $\{a,b\}\in E(G')$. Separating out the terms which are divisible by $y_s$, write: \[g = x_s\left(\sum k_{a,b}f_{a,b}\right) = x_s\left(\sum y_sh_{a,b}f_{a,b}+L\right),\] where no term of $L$ is divisible by $y_s$. Since $g$ is divisible by $y_s$, we have that $y_s\|L$. But since no term of $L$ is divisible by $y_s$, this implies that $L=0$. Hence, $$g = x_sy_s\left(\sum h_{a,b}f_{a,b}\right)\in \langle\{x_sy_sf_{a,b}\mid \text{$\{a,b\}\in E(G')$}\}\rangle.$$ It is readily seen that $J_{G'}\xrightarrow{\cdot x_s} I_x$, $J_{G'}\xrightarrow{\cdot y_s} I_y$, and $J_{G'}\xrightarrow{\cdot x_sy_s} I_x\cap I_y$ are isomorphisms of degree 1, 1, and 2 respectively. Now, consider $\mathbb{N}^n$ multigrading on $R$ with $\deg x_i = \deg y_i = e_i$ for all $i=1,\ldots, n$. The above isomorphisms imply that: \[\tor_i(I_x,k)_{\mathbf{a}+e_s}\cong \tor_i(J_{G'},k)_{\mathbf{a}} \cong \tor_i(I_y,k)_{\mathbf{a}+e_s} \] and $$\tor_i(I_x\cap I_y,k)_{\mathbf{a}+2e_s}\cong \tor_i(J_{G'},k)_{\mathbf{a}},$$ where $\mathbf{a} = (a_1,\ldots,a_n) \in \mathbb{N}^n$ with $a_s=0$. Summing up all the multigraded Betti numbers, we get $\beta_{i,j}(I_x) = \beta_{i,j-1}(J_{G'}) = \beta_{i,j}(I_y) $ and $\beta_{i,j}(I_x\cap I_y) = \beta_{i,j-2}(J_{G'})$. Observe that all the non-zero multigraded Betti numbers of $I_x\cap I_y$ occur only on multidegrees $\mathbf{a}+2e_s$ while all Betti numbers of $I_x$ and $I_y$ occur only at $\mathbf{a}+e_s$. Hence, by using \Cref{parcon} and combining all multidegrees, we have $$\beta_{i,j}(I) = \beta_{i,j}(I_x)+\beta_{i,j}(I_y)+\beta_{i-1,j}(I_x\cap I_y) ~~\mbox{for all $i,j \geq 0$}.$$ Therefore, \[\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = \beta_{i,i+3}(I) = \beta_{i,i+2}(J_{G'})+\beta_{i,i+2}(J_{G'})+\beta_{i-1,i+1}(J_{G'})\] for all $i \geq 0$. \end{proof} We can now prove the main result of this section: \begin{proof}[Proof of \Cref{maintheo2}] We first prove that $\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = 0$ for all $i\geq c(s)-1$, since we will require this fact later in the proof. It follows from \Cref{thm:Betti-intersection} that for all $i \geq 0$ \[\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = 2\beta_{i,i+2}(J_{G'})+\beta_{i-1,i+1}(J_{G'}),\] where $G'$ is the induced subgraph of $G$ on $N_G(s)$. From \Cref{linearbinom}, we get $\beta_{i,i+2}(J_{G'}) = (i+1)f_{i+1} (\Delta(G'))$, where $f_k(\Delta(G'))$ is the number of faces of $\Delta(G')$ of dimension $k$. Since the largest clique in $G'$ is of size $c(s)-1$, $\beta_{i,i+2}(J_{G'}) = 0$ for all $i\geq c(s)-2$. Hence $\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) = 0$ for all $i\geq c(s)-1$ by the above formula. Consider the $\mathbb{N}^n$-grading on $R$ given by $\deg x_i = \deg y_i = e_i$, the $i$-th unit vector. Now fix any $i \geq 1$ and let ${\bf a} = (a_1,\ldots,a_n) \in \mathbb{N}^n$ with $\sum_{\ell=1}^n a_\ell \geq i+ 4$. All the generators of $J_{G_1}\cap J_{G_2}$ are of the form $fx_s+gy_s$, so their multigraded Betti numbers occur within multidegrees $\mathbf{a}$ such that its $s$-th component, $a_s$ is non-zero. Since $J_{G_2}$ contains no generators of the form $fx_s+gy_s$, $\beta_{i,{\bf a}}(J_{G_1}\cap J_{G_2})>0$ implies that $\beta_{i,{\bf a}}(J_{G_2}) = 0$ for all $i\in \mathbb{N}$, and similarly, $\beta_{i-1,{\bf a}}(J_{G_1} \cap J_{G_2}) > 0$ implies that $\beta_{i,{\bf a}}(J_{G_2}) = 0$ From \Cref{star}, since $G_1$ is a star graph, \[ \beta_{i}(J_{G_1}) = \beta_{i,i+3}(J_{G_1}) = i\binom{\deg(s)}{i+2} ~\mbox{for all $i\geq 1$}.\] Hence, we can see that for all multidegrees ${\bf a} = (a_1,\dots,a_n)$ with $\sum_{\ell=1}^n a_\ell\geq i+4$, we also have $\beta_{i,{\bf a}}(J_{G_1}\cap J_{G_2})>0$ implies that $\beta_{i,{\bf a}}(J_{G_1})=0$, and $\beta_{i-1,{\bf a}}(J_{G_1}\cap J_{G_2})>0$ implies that $\beta_{i-1,{\bf a}}(J_{G_1})=0$. Therefore, from \Cref{parcon}, we have \[\beta_{i,{\bf a}}(J_G) = \beta_{i,{\bf a}}(J_{G_1})+ \beta_{i,{\bf a}}(J_{G_2})+ \beta_{i-1, {\bf a}}(J_{G_1}\cap J_{G_2}),\] for all $i \geq 0$ and multidegrees ${\bf a}$ with $\sum_{\ell=1}^n a_\ell\geq i+4$. Now fix any $i \geq c(s)$ and ${\bf a} \in \mathbb{N}^n$. As argued above, if $\beta_{i,{\bf a}}(J_{G_1} \cap J_{G_2})>0$, then $\beta_{i,{\bf a}}(J_{G_2}) = 0$ (and a similar statement for $\beta_{i-1,{\bf a}}(J_{G_1} \cap J_{G_2})$). We also know that if $\beta_{i,{\bf a}}(J_{G_1} \cap J_{G_2}) > 0$, with $i \geq c(s)-1$, then $\sum_{\ell=1}^n a_l \geq i+4$ since $J_{G_1} \cap J_{G_2}$ is generated in degree three and $\beta_{i,i+3}(J_{G_1}\cap J_{G_2}) =0$ for all $i \geq c(s)-1$. On the other hand, since ${\rm reg}(J_2) = 3$ by \Cref{star}, we have $\beta_{i,{\bf a}}(J_{G_2}) = 0$ for all $\sum_{\ell=1}^n a_\ell \neq i+3$ if $i \geq 1$. So, we have shown that if $\beta_{i,{\bf a}}(J_{G_1} \cap J_{G_2}) > 0$, then $\beta_{i,{\bf a}}(J_{G_2}) = 0$, and also if $\beta_{i-1,{\bf a}}(J_{G_1} \cap J_{G_2}) > 0$, then $\beta_{i-1,{\bf a}}(J_{G_2}) = 0$. So by using \Cref{parcon}, we have \[\beta_{i,{\bf a}}(J_G) = \beta_{i,{\bf a}}(J_{G_1})+ \beta_{i,{\bf a}}(J_{G_2})+ \beta_{i-1, {\bf a}}(J_{G_1}\cap J_{G_2}),\] for all $i \geq c(s)$ and multidegrees ${\bf a} \in \mathbb{N}^n$. Therefore, by combining these two results we have \[\beta_{i,{\bf a}}(J_G) = \beta_{i,{\bf a}}(J_{G_1})+ \beta_{i,{\bf a}}(J_{G_2})+ \beta_{i-1,{\bf a}}(J_{G_1}\cap J_{G_2}),\] for all $i$ and multidegrees ${\bf a}$ with $i\geq c(s)$ or $\sum_{k=1}^n a_k\geq i+4$. By summing over all multidegrees, we obtain the same result for the standard grading, i.e., $$\beta_{i,j}(J_G) = \beta_{i,j}(J_{G_1})+ \beta_{i,j}(J_{G_2})+ \beta_{i-1, j}(J_{G_1}\cap J_{G_2}),$$ for all $i,j$ with $i\geq c(s)$ or $j\geq i+4$. In other words, we have a $(c(s),4)$-Betti splitting. \end{proof} \begin{example} If $G$ is the graph of \Cref{runningexample}, then we saw in \Cref{runningexample2} that the ideal $J_G$ has a $(4,4)$-Betti splitting. Note that the splitting of \Cref{runningexample2} is an example of an $s$-partition with $s=1$. Furthermore, the largest clique that the vertex $s=1$ belongs to has size four (there is a clique on the vertices $\{1,2,4,5\})$. So, by the previous result $J_G$ will have a $(c(1),4)$-Betti splitting with $c(1)=4$, as shown in this example. \end{example} \begin{corollary}\label{trianglefree} Let $G$ be a graph on $[n]$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$ for some $s\in [n]$. If $G$ is a triangle-free graph, then $J_G = J_{G_1}+J_{G_2}$ is a complete Betti splitting. \end{corollary} \begin{proof} Since $G$ is a triangle-free graph, the largest clique containing $s$ is a $K_2$, i.e., $c(s)=2$. Thus \Cref{maintheo2} implies that $J_G = J_{G_1}+J_{G_2}$ is a $(2,4)$-Betti splitting, that is, $$\beta_{i,j}(J_G) = \beta_{i,j}(J_{G_1})+\beta_{i,j}(J_{G_2})+\beta_{i-1, j}(J_{G_1}\cap J_{G_2} )\text{ for all $i\geq 2$ or $j \geq i +4$.}$$ To complete the proof, we just need to show the above formula also holds for the graded Betti numbers $\beta_{i,j}(J_G)$ with $(i,j) \in \{(0,0),(0,1),(0,2),(0,3),(1,1), (1,2),(1,3),(1,4)\}$. We always have $\beta_{0,j}(J_G) = \beta_{0,j}(J_{G_1})+\beta_{0,j}(J_G) + \beta_{-1,j}(J_{G_1}\cap J_{G_2})$ for all $j \geq 0$. Also, since $J_G, J_{G_1}$ and $J_{G_2}$ are generated in degree $2$ and $J_{G_1} \cap J_{G_2}$ generated in degree four (by \Cref{deg4}), we have $$0 = \beta_{1,j}(J_G) = \beta_{1,j}(J_{G_1})+\beta_{1,j}(J_G) + \beta_{0,j}(J_{G_1}\cap J_{G_2}) = 0 + 0 + 0$$ for $j=1,2$. Finally, because $J_{G_1} \cap J_{G_2}$ is generated in degree four, we have $\beta_{1,3}(J_{G_1}\cap J_{G_2}) = \beta_{1,4}(J_{G_1}\cap J_{G_2}) = 0$. Thus, for $(i,j) = (1,3)$ the conditions of \Cref{parcon} are vacuously satisfied (since $\beta_{1,3}(J_{G_1}\cap J_{G_2}) = \beta_{0,3}(J_{G_1}\cap J_{G_2}) = 0$). For $i=1$ and $j=4$, we have $\beta_{1,4}(J_{G_1}\cap J_{G_2}) = 0$ and when $\beta_{0,4}(J_{G_1} \cap J_{G_2}) > 0$, we have $\beta_{0,4}(J_{G_1}) = \beta_{0,4}(J_{G_2}) =0$ since both $J_{G_1}$ and $J_{G_2}$ are generated in degree 2. So again the conditions of \Cref{parcon} are satisfied. Thus $$ \beta_{1,j}(J_G) = \beta_{1,j}(J_{G_1})+\beta_{1,j}(J_{G_2}) + \beta_{1,j}(J_{G_1}\cap J_{G_2}) = \beta_{1,j}(J_{G_1})+\beta_{1,j}(J_G) $$ for $j=3,4$. \end{proof} \begin{corollary} Let $G$ be a graph on $[n]$ and let $J_G = J_{G_1}+J_{G_2}$ be an $s$-partition of $G$ for some $s\in [n]$. \begin{enumerate} \item If $\pd(J_G)\geq c(s)$, then $\pd(J_G) = \max\{ \pd(J_{G_1}), \pd(J_{G_2}), \pd(J_{G_1}\cap J_{G_2})+1\}.$ \item If $\reg(J_G)\geq 4$, then $\reg(J_G) = \max\{\reg(J_{G_2}), \reg(J_{G_1}\cap J_{G_2})-1\}.$ \end{enumerate} \end{corollary} \begin{proof} Given that $\pd(J_G)\geq c(s)$, we know that there is a partial splitting for all $\beta_{i,j}(J_G)$, for all $i\geq c(s)$. Hence, $\pd(J_G) = \max\{ \pd(J_{G_1}), \pd(J_{G_2}), \pd(J_{G_1}\cap J_{G_2})+1\}$. Similarly, if $\reg(J_G)\geq 4$, we know that there is a partial splitting for all $\beta_{i,j}(J_G)$, for all $i\geq c(s)$. Hence, $\reg(J_G) = \max\{ \reg(J_{G_1}), \reg(J_{G_2}), \reg(J_{G_1}\cap J_{G_2})-1\}$. Since $\reg(J_{G_1}) = 3$, we have $\reg(J_G) = \max\{\reg(J_{G_2}), \reg(J_{G_1}\cap J_{G_2})-1\}$. \end{proof} \section{On the total Betti numbers of binomial edge ideals of trees} In this section, we explore an application of \Cref{maintheo} to find certain Betti numbers of trees. In particular, we obtain a precise expression for the second Betti number of $J_T$ for any tree $T$. Note that $\beta_1(J_T)$ was first computed in \cite[ Theorem 3.1]{jayanthan_almost_2021}. We begin with recalling a simple technical result that we require in our main results. \begin{lemma}\label{pendantexist} Let $T$ be a tree which is not an edge with $v\in V(T)$ and let $S_v = \{u\in N_T(v) ~\|~ \deg u > 1\}$. Then, there exists $a\in V(T)$ with $\deg a>1$ such that $\|S_a\|\leq 1.$ \end{lemma} \begin{proof} See \cite[Proposition 4.1]{JK2005}. \end{proof} To compute the second Betti number of $J_T$, we use \Cref{maintheo} to reduce the computation to graphs with a fewer number of vertices. One of the graphs involved in this process becomes a clique sum of a tree and a complete graph. So, we now compute the first Betti number of this class of graphs. \begin{theorem}\label{T+K_m} Let $G=T \cup_{a} K_m$. If $\|V(G)\| = n$, then \begin{eqnarray} \beta_1(J_G) &= &\binom{n-1}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m)}\binom{\deg_G w}{3}+\binom{\deg_G a-m+1}{3} \\ & &+(n-m-1)\binom{m-1}{2} +(m-1)\binom{\deg_G a -m+1}{2}. \end{eqnarray} \end{theorem} \begin{proof} We prove the assertion by induction on $\|V(T)\|$. If $\|V(T)\| = 1$, then $G$ is a complete graph and $n = m$. Therefore, by \Cref{completebetti} \[\beta_1(J_G) = 2\binom{n}{3} = \binom{n-1}{2}+2\binom{n}{3}-\binom{n-1}{2}.\] Hence the assertion is true. Assume now that the assertion is true if $\|V(T)\| \leq n-m$. Let $G = T \cup_a K_m$. Since $E(T)\neq \emptyset$, it follows from \Cref{pendantexist} that there exists $u\in V(T)$ such that $\deg u\neq 1$ and $\|S_u\|\leq 1$. We now split the remaining proof into two cases. \noindent \textbf{Case 1:} $u\neq a$.\\ Let $e= \{u,v\}$ with $\deg_G v = 1$ and let $G' = G \setminus v$. Then $G' = (T\setminus v) \cup_a K_m$ and $J_{G'} = J_{G\setminus e}$. Note that $\deg_{G'} u = \deg_G u - 1$ and $\deg_{G'} w = \deg_G w$ for all $w \neq u$. From \Cref{maintheo}, we have $\beta_1(J_G) = \beta_1(J_{G\setminus e}) + \beta_{0}(J_{(G\setminus e)_e})$. We now compute the two terms on the right hand side of this equation. It follows by induction that \begin{eqnarray} \beta_1(J_{G\setminus e}) &= &\binom{n-2}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m), w\neq u}\binom{\deg_{G'} w}{3}+\binom{\deg_G u-1}{3}\\ & &+\binom{\deg_G a-m+1}{3}+ (n-m-2)\binom{m-1}{2} + (m-1)\binom{\deg_G a -m+1}{2}. \end{eqnarray} Now, $(G\setminus e)_e$ is obtained by adding $\binom{\deg u-1}{2}$ edges to $E(G\setminus e)$. Since $T$ is a tree and $G=T \cup_a K_m$, we have $E(G) = n-m+\binom{m}{2}$. Hence, $G\setminus e$ has $n-m-1 + \binom{m}{2} = n-2+\binom{m-1}{2}$ edges. This means that: \[\beta_0(J_{(G\setminus e)_e}) =\|E((G\setminus e)_e)\| = n-2 + \binom{m-1}{2} +\binom{\deg_G u-1}{2}.\] Therefore, \begin{eqnarray} \beta_1(J_{G}) &= & \beta_1(J_{G\setminus e}) + \beta_{0}(J_{(G\setminus e)_e}) \\ & = & \binom{n-2}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m), w\neq u}\binom{\deg_G w}{3}+\binom{\deg_G u-1}{3} \\ & &+ \binom{\deg_G a-m+1}{3} + (n-m-2)\binom{m-1}{2} + (m-1)\binom{\deg_G a -m+1}{2}\\ & &+ n-2 + \binom{m-1}{2} +\binom{\deg_G u-1}{2}\\ &= & \binom{n-1}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m)}\binom{\deg_G w}{3}+\binom{\deg_G a-m+1}{3}\\ & &+(n-m-1)\binom{m-1}{2} +(m-1)\binom{\deg_G a -m+1}{2}. \end{eqnarray} Therefore, we obtain our desired formula. \noindent \textbf{Case 2:} $u=a$. \noindent Let $e= \{a,v\}$ with $\deg v = 1$. Then, as before, we apply induction to get \begin{eqnarray} \beta_1(J_{G\setminus e}) &= & \binom{n-2}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m)}\binom{\deg_G w}{3}+ \binom{\deg_G a-m}{3}\\ & &+ (n-m-2)\binom{m-1}{2}+(m-1)\binom{\deg_G a -m}{2}. \end{eqnarray} There are $\binom{\deg_G a-m}{2}+(m-1)\binom{\deg_G a-m}{1}$ new edges in $(G\setminus e)_e$. Thus \[\beta_0(J_{(G\setminus e)_e}) = \|E(G\setminus e)_e\| = n-2+\binom{m-1}{2}+\binom{\deg_G a-m}{2} + (m-1)\binom{\deg_G a-m}{1}.\] Using \Cref{maintheo} and the identity $\binom{n}{r} = \binom{n-1}{r}+\binom{n-1}{r-1}$ appropriately, we get: \begin{eqnarray} \beta_1(J_{G}) & = & \binom{n-2}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m)}\binom{\deg_G w}{3}+ \binom{\deg_G a-m}{3}\\ & &+ (n-m-2)\binom{m-1}{2}+(m-1)\binom{\deg_G a -m}{2}\\ & &+ n-2+\binom{m-1}{2}+\binom{\deg_G a-m}{2} + (m-1)\binom{\deg_G a-m}{1} \\ & = & \binom{n-1}{2}+2\binom{m}{3}+\sum_{w\notin V(K_m)}\binom{\deg_G w}{3}+\binom{\deg_G a-m+1}{3}\\ & & +(n-m-1)\binom{m-1}{2} +(m-1)\binom{\deg_G a -m+1}{2}. \end{eqnarray} Thus, we get the desired formula. This completes the proof. \end{proof} As an immediate consequence, we recover \cite[ Theorem 3.1]{jayanthan_almost_2021}: \begin{corollary} Let $T$ be a tree on $[n]$. Then \[ \beta_1(J_T) = \binom{n-1}{2}+\sum_{w \in V(T)}\binom{\deg_T w}{3}. \] \end{corollary} \begin{proof} If $G = T$, it can be trivially written as $G = T\cup_a K_1$, where $V(K_1) = \{a\}$. Therefore, taking $m=1$ in \Cref{T+K_m} we get the desired formula. \end{proof} We now compute the second Betti number of a tree using \Cref{T+K_m} and \Cref{maintheo}. This Betti number also depends upon the number of induced subgraphs isomorphic to the following caterpillar tree. We first fix the notation for this graph. \begin{definition} Let $P$ be the graph with $V(P)=[6]$ and $E(P) = \{\{1,2\}, \{2,3\},\\ \{3,4\}, \{2,5\}, \{3,6\} \}$. Given a tree $T$, we define $\mathcal{P}(T)$ to be the collection of all subgraphs of $T$ which are isomorphic to $P$, as shown in \Cref{fig:graph6}. Let $P(T) = \|\mathcal{P}(T)\|$. \end{definition} \begin{figure}[ht] \centering \begin{tikzpicture}[every node/.style={circle, draw, fill=white!60, inner sep=1.5pt}, node distance=2cm] \node (1) at (0, 0) {1}; \node (2) at (1, 0) {2}; \node (3) at (2, 0) {3}; \node (4) at (3, 0) {4}; \node (5) at (1, -1) {5}; \node (6) at (2, 1) {6}; \draw (1) -- (2); \draw (2) -- (3); \draw (3) -- (4); \draw (2) -- (5); \draw (3) -- (6); \end{tikzpicture} \caption{The graph $P$} \label{fig:graph6} \end{figure} \begin{example}\label{ex:pt} Consider the graph $G$ of \Cref{fig:example of P} with $V(G) = [7]$ and $$E(G) = \{\{1,2\}, \{2,3\}, \{3,4\}, \{2,5\},\\ \{3,6\}, \{3,7\}\}.$$ For this graph, the collection $\mathcal{P}(G)$ will be the induced subgraphs on the following collections of vertices: $\mathcal{P}(G)=\{\{1,2,3,4,5,6\}, \{1,2,3,5,6,7\}, \{1,2,3,4,5,7\}\}$. Hence, $P(G)=3$. \begin{figure}[ht] \centering \begin{tikzpicture}[every node/.style={circle, draw, fill=white!60, inner sep=1.5pt}, node distance=2cm] \node (1) at (0, 0) {1}; \node (2) at (1, 0) {2}; \node (3) at (2, 0) {3}; \node (4) at (3, 0) {4}; \node (5) at (1, -1) {5}; \node (6) at (2, 1) {6}; \node (7) at (2, -1) {7}; \draw (1) -- (2); \draw (2) -- (3); \draw (3) -- (4); \draw (2) -- (5); \draw (3) -- (6); \draw (3) -- (7); \end{tikzpicture} \caption{The graph $G$} \label{fig:example of P} \end{figure} \end{example} \begin{theorem}\label{betti2tree} Let $T$ be a tree on $[n]$, and let $J_T$ be its binomial edge ideal. Then \[\beta_2(J_T) = \binom{n-1}{3}+ 2\sum_{w \in V(T)}\binom{\deg_T w}{4}+\sum_{w \in V(T)}\binom{\deg_T w}{3}(1+\|E(T\setminus w)\|)+P(T).\] \end{theorem} \begin{proof} We prove the assertion by induction on $n$. If $n=2$, then $T$ is an edge. Since $J_T$ is a principal ideal, we have $\beta_{2}(J_T) = 0$, which agrees with the above formula. Now, assume that $n > 2$ and that the above formula is true for trees with $V(T)\leq n-1$. Let $T$ be a tree with $\|V(T)\|=n$. We know from \Cref{pendantexist} that there exists a vertex $u$ such that $\deg u>1$ and $\|S_u\|\leq 1$. Let $e = \{u,v\}$ be an edge such that $v$ is a pendant vertex. If $S_u = \emptyset$, then $T = K_{1,n-1}$. In this situation, the expression in the theorem statement reduces to $\binom{n-1}{3} + 2\binom{n-1}{4} + \binom{n-1}{3}.$ It is an easy verification that this number matches with the formula we obtained in \Cref{star}. We now assume that $\|S_u\| = 1$. By the choice of $u$, we can see that $(T\setminus e)_e = (T\setminus v)\cup_a K_m \sqcup \{v\}$, where $S_u = \{a\}$ and $m = \deg_T u$. Let $G' = (T\setminus v)\cup_a K_m$. Then $\|V(G')\| = n-1$ and $J_{G'} = J_{(T\setminus e)_e}$. Observe that $\deg_{(T\setminus e)_e} a = \deg_T a + m-2$. Thus, from \Cref{T+K_m}, we get \begin{eqnarray} \beta_1\left(J_{(T\setminus e)_e}\right) &= & \binom{n-2}{2} +2\binom{m}{3} + \sum_{w\notin V(K_m)}\binom{\deg_{(T\setminus e)_e} w}{3} +\binom{\deg_{(T\setminus e)_e} a-m+1}{3}\\ & &+(n-m-2)\binom{m-1}{2} + (m-1)\binom{\deg_{(T\setminus e)_e} a -m+1}{2}\\ &= & \binom{n-2}{2} +2\binom{\deg_T u}{3} + \sum_{w\notin V(K_m)}\binom{\deg_T w}{3} +\binom{\deg_T a-1}{3}\\ & &+(n-\deg_T u-2)\binom{\deg_T u-1}{2} + (\deg_T u-1)\binom{\deg_T a-1}{2}. \end{eqnarray} Let $T' = T\setminus v$. Then $J_{T'} = J_{T\setminus e}$. Note that $\|V(T')\| = n-1,$ $\deg_{T'} u = \deg_T u-1$, and $\deg_{T'}x = \deg x$ for all $x \in V(T) \setminus\{u\}.$ Additionally $\|E(T'\setminus u)\| = \|E(T \setminus u)\|$ and $\|E(T' \setminus w)\| = \|E(T \setminus w) \| -1$ for all $w \neq u$. By the induction hypothesis, \begin{eqnarray} \beta_2(J_{T'}) & = & \binom{n-2}{3} + 2\sum_{w\neq u}\binom{\deg_T w}{4} + 2\binom{\deg_T u-1}{4} \\ & &+\sum_{w\neq u}\binom{\deg_T w}{3}(\|E(T\setminus w)\|)+\binom{\deg_T u-1}{3}(\|E(T \setminus u)\|+1)+P(T'). \end{eqnarray} Thus, it follows from \Cref{maintheo} that \begin{eqnarray} \beta_2(J_{T}) &= & \binom{n-2}{3}+ 2\sum_{w\neq u}\binom{\deg_T w}{4}+ 2\binom{\deg_T u-1}{4} \\ & &+\sum_{w\neq u}\binom{\deg_T w}{3}(\|E(T\setminus w)\|)+\binom{\deg_T u-1}{3}(\|E(T \setminus u)\|+1)+P(T')\\ & &+\binom{n-2}{2}+2\binom{\deg_T u}{3}+\sum_{w\notin V(K_m)}\binom{\deg_T w}{3}+\binom{\deg_T a-1}{3}\\ & &+(n-\deg_T u-2)\binom{\deg_T u-1}{2}+(\deg_T u-1)\binom{\deg_T a-1}{2}. \end{eqnarray} Note that for all $w \in N_{T'}(u) \setminus \{a\}$, $\deg_{T'}(w) = 1$. Thus $\binom{\deg_{T'} w}{3} = 0$ for all $w\in N_{T'}(u) \setminus \{a\}$. Hence, none of the $w$, $w \neq a$, for which $\binom{\deg_T w}{3} \neq 0$ belong to $V(K_m)$ in $(T\setminus e)_e$. Thus we can write \[\sum_{w\neq u}\binom{\deg_T w}{3}(\|E(T\setminus w)\|) + \sum_{w\notin V(K_m)}\binom{\deg_T w}{3} = \sum_{w\neq u}\binom{\deg_T w}{3}(\|E(T\setminus w)\|+1).\] To compare $P(T)$ and $P(T\setminus e)$, observe that the only elements of $\mathcal{P}(T)$ which are not in $\mathcal{P}(T\setminus e)$ are the induced subgraphs which contain the edge $e$. Since $a$ is the only neighbor of $u$ having degree more than one, the total number of such graphs is $(\deg_T u -2)\binom{\deg_T a-1}{2}$. Thus $P(T\setminus e) = P(T) - (\deg_T u -2)\binom{\deg_T a-1}{2}.$ Note also that $\|E(T\setminus u)\| =n-\deg_T u -1$. Incorporating the above observations in the expression for $\beta_2(J_T)$, and using the identity $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$, we get \footnotesize \begin{eqnarray} \beta_2(J_T) &= & \binom{n-1}{3} + 2\sum_{w\neq u}\binom{\deg_T w}{4} + 2\binom{\deg_T u-1}{4}+\sum_{w\neq u,a}\binom{\deg_T w}{3}(\|E(T\setminus w)\|+1) \\ & &+\binom{\deg_T a}{3}(\|E(T\setminus a)\|)+\binom{\deg_T u-1}{3}(\|E(T\setminus u)\|+1)+P(T)+\binom{\deg_T a-1}{2}\\ & &+2\binom{\deg_T u}{3}+\binom{\deg_T a-1}{3}+(\|E(T\setminus u)\|-1)\binom{\deg_T u-1}{2}\\ &= & \binom{n-1}{3}+ 2\sum_{w\neq u}\binom{\deg_T w}{4} + 2\binom{\deg_T u-1}{4} +\sum_{w\neq u,a}\binom{\deg_T w}{3}(\|E(T\setminus w)\|+1)\\ & &+\binom{\deg_T a}{3}(\|E(T\setminus a)\|+1)+\binom{\deg_T u}{3}(\|E(T\setminus u)\|+1)\\ & &+P(T)+2\binom{\deg_T u}{3}-2\binom{\deg_T u-1}{2}\\ &= & \binom{n-1}{3}+ 2\sum_{w\neq u}\binom{\deg_T w}{4} + 2\binom{\deg_T u-1}{4}+\sum_{w}\binom{\deg_T w}{3}(\|E(T\setminus w)\|+1)\\ & &+P(T) +2\binom{\deg_T u-1}{3} \\ &= & \binom{n-1}{3} + 2\sum_{w}\binom{\deg_T w}{4} +\sum_{w}\binom{\deg_T w}{3}(1+\|E(T\setminus w)\|)+P(T). \end{eqnarray} \normalsize We have now completed the proof. \end{proof} It can be seen that \Cref{betti2tree} builds on \cite[Theorem 3.1]{jayanthan_almost_2021}. We conclude our article by computing certain graded Betti numbers of binomial edge ideals of trees. \begin{theorem}\label{thirdrow} Let $T$ be a tree and $J_T$ be its corresponding binomial edge ideal. Then, \[\beta_{k,k+3}(J_T) = \sum_{w\in V(T)}k\binom{\deg_T w+1}{k+2}\text{ for all k $\geq 2$}.\] \end{theorem} \begin{proof} We prove the assertion by induction on $\|V(T)\|$. Let $\|V(T)\|=n=2$. Then $J_T$ is the binomial edge ideal of a single edge. Since this is a principal ideal generated in degree $2$, $\beta_{k,k+3}(J_T)=0$ for all $k\geq 2$, which agrees with the formula. Suppose the assertion is true for all trees with $n-1$ vertices. Let $T$ be a tree with $\|V(T)\| = n$. Using \Cref{pendantexist}, consider $e=\{u,v\} \in E(T)$, where $u$ is such that $\deg u>1$ and $\|S_u\|\leq 1$. Then, using \Cref{maintheo}, we get \[\beta_{k,k+3}(J_T) = \beta_{k,k+3}(J_{T\setminus e})+ \beta_{k-1,k+1}(J_{(T\setminus e)_e}).\] Let $T' = T \setminus v$. Then $J_{T'} = J_{T\setminus e}$, $\deg_{T'} u = \deg_T u - 1$ and $\deg_{T'} w = \deg_T w$ for all $w \in V(T') \setminus u$. Also, $(T\setminus e)_e$ is a clique sum of a tree and a complete graph, with the size of the complete graph equal to $\deg u$. Hence using the inductive hypothesis and \Cref{linearbinom} we get: \begin{align} & \beta_{k,k+3}(J_{T\setminus e}) = \sum_{w\neq u}k\binom{\deg_T w+1}{k+2} + k\binom{\deg_T u}{k+2},~~\mbox{and}\\ & \beta_{k-1,k+1}(J_{(T\setminus e)_e}) = k\binom{\deg_T u}{k+1}. \end{align*} Substituting these values into \Cref{maintheo} we get: \[\beta_{k,k+3}(J_T) = \sum_{w\neq u}k\binom{\deg_T w+1}{k+2} + k\binom{\deg_T u}{k+2}+k\binom{\deg_T u}{k+1} = \sum_{w}k\binom{\deg_T w+1}{k+2}.\] \end{proof} \begin{example} To illustrate some of the results of this section, consider the tree $T$ described in \Cref{ex:pt}. The graded Betti table of $J_T$ is given below: \begin{verbatim} 0 1 2 3 4 5 total: 6 20 41 43 21 4 2: 6 . . . . . 3: . 20 12 3 . . 4: . . 29 40 21 4 \end{verbatim} It follows from \Cref{betti2tree} that for this tree $$\beta_2(J_T) = \binom{6}{3} + 2 \binom{4}{4} + \binom{4}{3}(1+2) + \binom{3}{3}(1+3) + 3 = 41.$$ Additionally, the Betti numbers of the form $\beta_{k,k+3}(J_T)$ with $k \geq 2$ satisfy \Cref{thirdrow} since $\beta_{2,5}(J_T) = 2\left[\binom{4}{4} + \binom{5}{4}\right] = 12$ and $\beta_{3,6}(J_T) = 3 \binom{5}{5} = 3$. \end{example} \noindent {\bf Acknowledgments.} Some of the results in this paper first appeared in the MSc thesis of Sivakumar. Van Tuyl’s research is supported by NSERC Discovery Grant 2024-05299. \bibliographystyle{amsplain} \bibliography{Bibliography} \end{document}
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\def\punct{} \newtheoremstyle{dotless}{}{}{\rm}{}{\bf}{\punct}{.5em}{} \theoremstyle{dotless} \newtheorem{taggedthmx}{} \newenvironment{taggedthm}[1] {\renewcommand\thetaggedthmx{#1}\taggedthmx} {\endtaggedthmx} \newtheorem{taggedassumptionx}{} \newenvironment{taggedassumption}[1] {\renewcommand\thetaggedassumptionx{#1}\taggedassumptionx} {\endtaggedassumptionx} \makeatletter \renewcommand{\theequation}{ \thesection.\arabic{equation}} \@addtoreset{equation}{section} \newcommand{\setword}[2]{ \phantomsection #1\def\@currentlabel{\unexpanded{#1}}\label{#2} } \makeatother \begin{document} \title{\bf Solvability of Coupled Forward-Backward Volterra Integral Equations} \author{ Wenyang Li\thanks{School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China (Email: {\tt [email protected]}). } ~~~ Hanxiao Wang\thanks{Corresponding author. School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China (Email: {\tt [email protected]}). This author is supported in part by NSFC Grant 12201424, Guangdong Basic and Applied Basic Research Foundation 2023A1515012104, and the Science and Technology Program of Shenzhen RCBS20231211090537064.} ~~~ Jiongmin Yong\thanks{Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA (Email: {\tt [email protected]}). This author is supported by NSF Grant DMS-2305475.} } \maketitle \no\bf Abstract. \rm Motivated by the optimality system associated with controlled (forward) Volterra integral equations (FVIEs, for short), the well-posedness of coupled forward-backward Voterra integral equations (FBVIEs, for short) is studied. The main feature of FBVIEs is that the unknown $\{(\cX(t,s),\cY(t,s))\}$ has two arguments. By taking $t$ as a parameter and $s$ as a (time) variable, one can regard FBVIE as a system of ordinary differential equations (ODEs, for short), with infinite-dimensional space values $\{(\cX(\cd,s),\cY(\cd,s));\,s\in[0,T]\}$. To establish the well-posedness of such an FBVIE, a new non-local monotonicity condition is introduced, by which a bridge in infinite-dimensional spaces is constructed. Then by generalizing the method of continuation developed by \cite{Hu-Peng1995,Yong1997,Peng-Wu1999} for differential equations, we have established the well-posedness of FBVIEs. The key is to apply the chain rule to the mapping $t\mapsto\big[\int_\cd^T\lan \cY(s,s),\cX(s,\cd)\ran ds+\lan G(\cX(T,T)),\cX(T,\cd)\ran\big](t)$. \ms \no\bf Keywords. \rm forward-backward Volterra integral equation, infinite-dimensional system of ordinary differential equations, two-point boundary problem, method of continuation. \ms \no\bf AMS subject classifications. \rm 45D05, 45J05, 49K21. \section{Introduction} In a number of problems involving biology systems, finance models, and fractional-order differential dynamics, (forward) Volterra integral equations (FVIEs, for short) and the related optimal control theory have attracted strong attention recently. The main feature is that FVIE can be used to describe some dynamics involving memory. Further, if we consider an optimal control problem of FVIE with a Bolza type cost functional, then by applying the Pontryagin maximum principle, one will get a coupled forward-backward Volterra integral equation FBVIE, for short) (see \cite{Angell1976,Carlson1987,Yong2008,Lin2020}). Let us briefly look at such a problem. Consider the following state equation: \bel{state}X(t)=x(t)+\int_0^tb(t,s,X(s),u(s))ds,\qq t\in[0,T],\ee with the following cost functional \bel{cost}J(u(\cd))=h(X(T))+\int_0^Tg(t,X(t),u(t))dt.\ee In the above, $X(\cd)$ is the state valued in $\mathbb{R}^n$, and $u(\cd)$ is the control valued in $\dbR^m$. For simplicity, we assume that $(X,u)\mapsto( b(t,s,X,u),h(X),g(t,X,u))$ is differentiable, and the control domain is the whole space $\dbR^m$. We assume that the state has no constraint (and therefore, the state space is the whole space $\dbR^n$). Our optimal control problem is to minimize \rf{cost}, subject to \rf{state}. Now, let $(\bar X(\cd),\bar u(\cd))$ be an optimal pair of this optimal control problem. For any admissible control $u(\cd)$, let $X(\cd)$ be the solution to the following variational equation: \begin{align} X(t)&=\int_0^t\(b_x(t,s,\bar X(s),\bar u(s))X(s)+b_u(t,s,\bar X(s),\bar u(s))u(s)\)ds\\ &\equiv\f(t)+\int_0^tA(t,s)X(s)ds,\q t\in[0,T], \end{align} where $$ \f(t)=\int_0^tb_u(t,s,\bar X(s),\bar u(s))u(s)ds,\qq A(t,s)=b_x(t,s,\bar X(s),\bar u(s)),\q 0\les s\les t\les T. $$ We let $\bar Y(\cd)$ be a continuous function satisfying the following adjoint equation: $$\bar Y(t)=\psi(t)+\int_t^TA(s,t)^\top\bar Y(s)ds,\q t\in[0,T],$$ for some undetermined $\psi(\cd)$. Then \begin{align} &\int_0^T\bar Y(t)^\top X(t)dt=\int_0^T\[\bar Y(t)^\top\(\f(t)+\int_0^tA(t,s)X(s)ds\)\]dt\\ &\q=\int_0^T\bar Y(t)^\top\f(t)dt+\int_0^T\(\int_s^TA(t,s)^\top \bar Y(t)dt\)^\top X(s)ds\\ &\q=\int_0^T\bar Y(t)^\top\f(t)dt+\int_0^T\(\bar Y(s)-\psi(s)\)^\top X(s)ds. \end{align} Hence, $$\int_0^T\psi(t)^\top X(t)dt=\int_0^T\bar Y(t)^\top\f(t)dt.$$ On the other hand, by the minimality of the optimal pair $(\bar X(\cd),\bar u(\cd))$, one has \begin{align} 0&\les \lim_{\e\downarrow 0}{J(\bar u(\cd)+\e u(\cd))-J(\bar u(\cd))\over\e}\\ &=h_x(\bar X(T))X(T)+\int_0^T\(g_x(s,\bar X(s),\bar u(s))X(s)+g_u(s,\bar X(s),\bar u(s))u(s)\)ds\\ &=h_x(\bar X(T))\int_0^T\(b_x(T,s,\bar X(s),\bar u(s))X(s)+b_u(T,s,\bar X(s),\bar u(s))u(s)\)ds\\ &\q+\int_0^T\(g_x(s,\bar X(s),\bar u(s))X(s)+g_u(s,\bar X(s),\bar u(s))u(s)\)ds\\ &=\int_0^T\(h_x(\bar X(T))b_x(T,s,\bar X(s),\bar u(s))+g_x(s,\bar X(s),\bar u(s))\)X(s)ds\\ &\q+\int_0^T\(h_x(\bar X(T))b_u(T,s,\bar X(s),\bar u(s))+g_u(s,\bar X(s),\bar u(s))\)u(s)ds. \end{align} Now, by taking $$\psi(s)=\(h_x(\bar X(T))b_x(T,s,\bar X(s),\bar u(s))+g_x(s,\bar X(s),\bar u(s))\)^\top,\q s\in[0,T],$$ the above gives \begin{align} 0&\les\int_0^T\bar Y(t)^\top\int_0^tb_u(t,s,\bar X(s),\bar u(s))u(s)dsdt\\ &\q+\int_0^T\(h_x(\bar X(T))b_u(T,s,\bar X(s),\bar u(s))+g_u(s,\bar X(s),\bar u(s))\)u(s)ds\\ &=\int_0^T\(\int_s^T\bar Y(t)^\top b_u(t,s,\bar X(s),\bar u(s))dt\\ &\q+h_x(\bar X(T))b_u(T,s,\bar X(s),\bar u(s))+g_u(s,\bar X(s),\bar u(s))\)u(s)ds. \end{align} Then we have the following {\it optimality system:} \bel{OS}\left\{\begin{aligned} &\bar X(t)=x(t)+\int_0^tb(t,s,\bar X(s),\bar u(s))ds, \\ &\bar Y(t)=g_x(t,\bar X(t),\bar u(t))^\top+b_x(T,t,\bar X(t),\bar u(t))^\top h_x(\bar X(T))^\top\\ &\qq\q+\int_t^Tb_x (s,t,\bar X(t),\bar u(t))^\top\bar Y(s)ds,\\ & g_u(t,\bar X(t),\bar u(t))^\top+b_u(T,t,\bar X(t),\bar u(t))^\top h_x(\bar X(T))^\top\\ &\qq\qq+\int_t^Tb_u(s,t,\bar X(t),\bar u(t))^\top\bar Y(s)ds=0,\end{aligned}\right.\qq t\in[0,T]. \ee This is a coupled FBVIE. The coupling is given in the third relation, which is called a {\it stationarity condition} or an {\it optimality condition}. The solution of \rf{OS} will provide a candidate for the optimal trajectory of the corresponding control problems. Therefore, the well-posedness of \rf{OS} is important, at least for the optimal control theory of FVIEs. \ms A careful observation of the above shows that in the case that if we are considering a problem with a linear state equation and the cost being quadratic in control, i.e., \bel{LCONVEX} \begin{aligned} & b(t,s,X,u)=A(t,s)X+B(t,s)u, \q 0\les s\les t\les T,\\ & g(t,X,u)=Q(t,X)+\lan R(t)u,u\ran,\q h(X)=M(X),\q t\in[0,T], \end{aligned} \ee then the optimality condition reads \bel{LCONVEX1} 2R(t)\bar u(t)+ B(T,t)^\top M_x(\bar X(T)) +\int_t^TB(s,t)^\top\bar Y(s)ds=0,\q t\in[0,T].\ee By assuming the existence of $R(\cd)^{-1}$, we end up with \bel{OS-1} \left\{\begin{aligned} \bar X(t)= & x(t)+\int_0^t \[A(t,s)\bar X(s)-{1\over 2}B(t,s) R(s)^{-1}\int_s^T B(r,s)^\top\bar Y(r)dr\\ &\qq-{1\over 2} B(t,s)R(s)^{-1}B(T,s)^\top M_x(\bar X(T))\] ds, \\ \bar Y(t)= & Q_x(t,\bar X(t))+A(T,t)^\top M_x(\bar X(T))+\int_t^T A(s,t)^{\top}\bar Y(s)ds,\end{aligned}\right. \q t\in[0,T]. \ee Motivated by the above, we consider the following FBVIE with general coefficients: \bel{FBVIE-main}\left\{\begin{aligned} &X(t)= x(t)+\int_0^t f\Big(t,s,X(s),Y(s),\int_s^T K(s,r) Y(r)dr,X(T)\Big) ds, \\ &Y(t)=h(t,X(t),X(T))+\int_t^T g(t,s,X(s), Y(s))ds,\end{aligned}\right.\q t\in[0,T],\ee where $x(\cd)$ is a given continuous function, and $f(\cd)$, $g(\cd)$, $h(\cd)$, and $K(\cd)$ are suitable mappings. We will focus on studying the well-posedness of FBVIE \rf{FBVIE-main}. \ms Another motivation for studying FBVIEs is the so-called time-inconsistent optimal control problems, in which the optimality system is also a coupled FBVIE; see \cite{Hu-Jin-Zhou2012,Wang-Yong2021,Hamaguchi2021-1}, for example. In addition, for the feature of involving memory, FBVIE has potential applications in biology models \cite{Gopalsamy1980,Kot2001,AIOmari-Gourley2003}, finance models \cite{Comte1998,ElEuch2018,ElEuch2019}, and infinite-dimensional partial differential equations \cite{Viens-Zhang2019,Wang-Yong-Zhang2022,Wang-Yong-Zhou2023,Bondi2023,Bonesini2023}. For example, we can regard the forward equation and the backward equation as an evolution system and a utility, respectively. When the evolution system is affected by the utility, one will get a coupled FBVIE immediately. Such a phenomenon has appeared in many applications. A typical example is the so-called large investor model in finance (see Cvitani\'{c} and Ma \cite{Cvitanic1996}). \ms To recover the semi-group property of \rf{FBVIE-main}, inspired by \cite{Viens-Zhang2019,Wang-Yong-Zhang2022}, we introduce the following auxiliary system of ODEs with the unknown $(\cX(\cd,\cd),\cY(\cd,\cd))$ having two arguments: \bel{FBVIE-main1}\left\{\begin{aligned} &\cX_s(t,s)= f\Big(t,s,\cX(s,s),\cY(s,s), \int_s^TK(s,r)\cY(r,r)dr,\cX(T,T)\Big),\\ & \qq\qq\qq\qq (t,s)\in\D_[0,T],\\ &\cY_s(t,s)= -g(t,s,\cX(s,s),\cY(s,s)), \q (t,s)\in\D^[0,T],\\ &\cX(t,0)=x(t),\q \cY(t,T)=h(t,\cX(t,t),\cX(T,T)),\q t\in[0,T], \end{aligned}\right.\ee where $\D_[0,T]=\{(t,s)\in[0,T]^2\hbox{ with } t\ges s\}$ and $\D^[0,T]=\{(t,s)\in[0,T]^2\hbox{ with } t\les s\}$. Indeed, \rf{FBVIE-main1} provides an equivalent representation of \rf{FBVIE-main} with the relationship: \bel{FBVIE-ODEs} X(t)=\cX(t,t),\q Y(t)=\cY(t,t),\q t\in[0,T]. \ee \ms For the forward-backward structure, \rf{FBVIE-main1} is essentially a Fredholm-type integral equation, and thus one cannot use the contraction mapping theorem unless $T>0$ is small enough. When the coefficients do not depend on $t$, \rf{FBVIE-main} reduces to the following forward-backward ODE (FBDE, for short): \bel{FBDE}\left\{\begin{aligned} &p(t)= x+\int_0^t a(s,p(s),q(s)) ds, \\ &q(t)=\psi(p(T))+\int_t^T b(s,p(s), q(s))ds,\end{aligned}\right.\q t\in[0,T].\ee For FBDEs, which can be regarded as a special case of FBSDEs, two types of methods have been developed to prove solvability. The first one is the so-called {\it four-step method} (also called a {\it decoupling method}), which was initiated by Ma, Protter, and Yong \cite{Ma-Protter-Yong1994}. The decoupling method mainly depends on the theory of partial differential equations (PDEs, for short). Since the PDE associated with \rf{FBVIE-main1} is defined on the path space, which is infinite-dimensional, its solvability might be more difficult than that of \rf{FBVIE-main1} itself; see Wang, Yong, and Zhang \cite{Wang-Yong-Zhang2022}, for example. Thus, it seems too difficult to extend the decoupling method of FBDEs to FBVIE \rf{FBVIE-main1}. \ms The second one is called a {\it method of continuation} (or a {\it monotonicity method}), which was introduced by Hu and Peng \cite{Hu-Peng1995}, and then developed by \cite{Yong1997,Peng-Wu1999}. Some further developments on FBSDEs/FBDEs can be found in \cite{Delarue2002,Yong2010,Ma2015,Yu2022}. The main idea of continuation method is to reach the solvability of FBDEs by starting with a known solvable FBDE. The way is to apply the chain rule together with some sort of monotonicity conditions to get certain a priori estimates. More precisely, in Hu and Peng \cite{Hu-Peng1995}, they introduced the following monotonicity condition: \bel{FBDE-MC}\begin{aligned} &\lan a(s,p_1,q_1)-a(s,p_2,q_2),\, q_1-q_2\ran-\lan b(s,p_1,q_1)-b(s,p_2,q_2),\, p_1-p_2\ran\\ &\q\les-\a\|p_1-p_2\|^2-\b\|q_1-q_2\|^2,\q \forall p_1,p_2,q_1,q_2\in\dbR^n,\\ &\lan \psi(p_1)-\psi(p_2),\, p_1-p_2\ran\ges \g \|p_1-p_2\|^2,\q \forall p_1,p_2\in\dbR^n,\end{aligned}\ee where $\a,\b,\g>0$ are given constants, and then applied the chain rule to the following function: \bel{FBDE-Ito} \lan p(t),q(t)\ran,\q t\in[0,T]. \ee In this paper, we will develop this method to prove the solvability of FBVIE \rf{FBVIE-main}. However, it is by no means easy; some comments and discussions can be found in our previous paper \cite[Subsection 5.1]{Wang-Yong-Zhang2022}. \ms We now make a careful analysis of FBVIE \rf{FBVIE-main}. Recall that \rf{FBVIE-main} is equivalent to \rf{FBVIE-main1}. From now on, we will focus on the auxiliary system \rf{FBVIE-main1} rather than \rf{FBVIE-main}. Since at the point $(t,s)$, \rf{FBVIE-main1} involves the values $\cX(s,s)$ and $\cY(s,s)$ at $(s,s)$, it is a non-local system. Alternatively, we can view $t$ as a parameter, and then \rf{FBVIE-main1} can be regarded as a Hamiltonian system of ODEs with the solution $\{(\cX(\cd,s),\cY(\cd,s)\}_{s\in[0,T]}$, and the two-point boundary condition $\cX(\cd,0)=x(\cd),\,\, \cY(\cd,T)=h(\cd,\cX(\cd,\cd),\cX(T,T))$. Thus, \rf{FBVIE-main1} can be regarded as an infinite-dimensional version of FBDE \rf{FBDE}. \ms Note that the monotonicity condition \rf{FBDE-MC} is defined at a local point $(t,x,y)$, but \rf{FBVIE-main1} is a non-local system (or an infinite-dimensional system). Then one immediately realizes that \rf{FBDE-MC} does not work to the system \rf{FBVIE-main1}, for which one should not apply the chain rule to the function $\lan \cX(t,s),\cY(t,s)\ran$ either. So two questions appear naturally: \ms (i) \emph{How can one introduce a proper non-local monotonicity condition for \rf{FBVIE-main1}?} \ms (ii) \emph{Which function should we apply the chain rule?} \ms Our answer is to impose the following {\it non-local monotonicity condition}: \bel{MC-1} \begin{aligned} &\int_t^T \big\lan \h y(s),\, \h f(s,t)\big\ran ds-\Big\lan \h h(t)+\int_t^T\h g(t,s)ds,\, \h x(t) \Big\ran\\ &+\big\lan G(x_1(T))- G(x_2(T)), \,\h f(T,t)\big\ran \les -\gamma\|\h x(t)\|^2,\q \hb{for some constant }\g>0, \end{aligned}\ee and some smooth function $G(\cd)$ satisfying \bel{MC-3} \begin{aligned} &\lan G(x_1(T))- G(x_2(T)),\, x_1(T)-x_2(T)\ran\ges \| G^{1\over 2}_0 [x_1(T)- x_2(T)]\|^2,\\ & \|G(x_1(T))- G(x_2(T))\|\les K\| G^{1\over 2}_0 [x_1(T)- x_2(T)]\|, \end{aligned} \ee where $G_0\ges 0$ is a positive semi-definite matrix, $K>0$ is a constant, and \bel{MC-2} \begin{aligned} &\h x(t)=x_1(t)-x_2(t),\q \h y(t)=y_1(t)-y_2(t),\\ &\h h(t)=h(t,x_1(t), x_1(T))-h(t,x_2(t),x_2(T)),\\ &\h g(t,s)= g(t,s,x_1(s), y_1(s))- g(t,s,x_2(s), y_2(s)),\\ &\h f(t,s)=f\Big(t,s,x_1(s),y_1(s), \int_s^T K(s,r)y_1(r)dr,x_1(T)\Big)\\ &\qq\qq-f\Big(t,s,x_1(s),y_1(s), \int_s^T K(s,r)y_1(r)dr,x_2(T)\Big), \end{aligned}\ee for any $x_i(\cd),y_i(\cd)\in C([0,T];\dbR^n);\,i=1,2$. Note that \rf{MC-1} is a non-local monotonicity condition (or a monotonicity condition defined on the infinite-dimensional space $C([0,T];\dbR^n)$). Then, with the method of continuation, by applying the chain rule to the following function: \bel{FBVIE-Ito} \int_t^T \lan \cX(s,t),\cY(s,s)\ran ds+\lan G( \cX(T,T)),\cX(T,t)\ran,\q t\in[0,T], \ee we can establish the well-posedness of \rf{FBVIE-main1}. \ms The main difficulty is to find an appropriate monotonicity condition for \rf{FBVIE-main1}. This is achieved based on some new observations for FBDEs. From Subsection \ref{subsec:comparison}, we can see that \rf{MC-1} is essentially an infinite-dimensional version of \rf{FBDE-MC}, and when FBVIE \rf{FBVIE-main1} reduces to an FBDE, it is equivalent to \rf{FBDE-MC}. In Section \ref{sec:application}, we will show the standard condition in LQ optimal control problems for FVIEs is sufficient for \rf{MC-1}. With the non-local monotonicity condition \rf{MC-1}, we still need to find a simple and known solvable FBVIE as the start of applying the method of continuation. Interestingly, we will see that this known solvable FBVIE \rf{linear} is essentially an FBDE. \ms The rest of the paper is organized as follows. In Section \ref{sec:pre}, we introduce some notations and state the main result of our paper. The proof is given in Section \ref{sec:well}. More precisely, we show the idea of finding the non-local monotonicity condition \rf{MC-1} in Subsection \ref{subsec:mc}, compare our non-local monotonicity condition \rf{MC-1} with the local one \rf{FBDE-MC} in Subsection \ref{subsec:comparison}, prove the uniqueness result in Subsection \ref{subsec:unique}, and establish the solvability in Subsection \ref{subsec:existence}. Finally, two simple examples are given in Section \ref{sec:application}. \section{Preliminary and the main result}\label{sec:pre} Let $T>0$ be a given time horizon and denote $$ \D_[0,T]=\{(t,s)\in[0,T]^2\hbox{ with } t\ges s\} \hbox{ and } \D^[0,T]=\{(t,s)\in[0,T]^2\hbox{ with } t\les s\}. $$ We introduce the following spaces of functions: \begin{align} & C([0,T];\dbR^n): \hb{~~the space of $\dbR^n$-valued, continuous functions on $[0,T]$};\\ & L^2([0,T];\dbR^n): \hb{~~the space of $\dbR^n$-valued, measurable, square integrable functions on $[0,T]$}. \end{align} Similarly, we can define the spaces of continuous functions on $\D_[0,T]$ and $\D^[0,T]$ as $C(\D_[0,T];\dbR^n)$ and $C(\D^[0,T];\dbR^n)$, respectively. \ms We impose the following assumptions for FBVIE \rf{FBVIE-main1}. \begin{taggedassumption}{(A1)}\label{ass:A1} The non-local monotonicity condition \rf{MC-1}--\rf{MC-3} holds. \end{taggedassumption} \begin{taggedassumption}{(A2)}\label{ass:A2} The coefficients of FBVIE \rf{FBVIE-main1} are continuous functions. Moreover, there exists a constant $L>0$ such that \begin{align} &\|f(t_1,s,x_1,y_1,y_1',x_1')-f( t_2,s, x_2,y_2,y_2',x_2')\|\\ &\q\les L\big(\|t_1- t_2\|^\a+\|x_1-x_2\|+\|y_1-y_2\|+\|G_0( x_1'-x_2')\|+\|y_1'-y_2'\| \big), \\ &\qq \forall (t_i,s,x_i,y_i,y_i',x_i')\in\D_[0,T]\times(\dbR^n)^4,\q i=1,2,\\ &\|g(t_1,s,x_1,y_1)-g(t_2, s,x_2,y_2)\|+\|h(t_1,x_1,x_1')-h(t_2,x_2,x_2')\|\\ &\q\les L\big(\|t_1- t_2\|^\a +\|x_1- x_2\|+\|y_1- y_2\|+\|G_0( x_1'-x_2')\|\big), \\ &\qq \forall (t_i,s,x_i,y_i,x_i')\in\D^[0,T]\times(\dbR^n)^3,\q i=1,2, \end{align} for some $\a\in(0,1]$, where $G_0\ges 0$ is given by \rf{MC-3}. \end{taggedassumption} The main result of our paper can be stated as follows. \begin{theorem}\label{Thm:well-posedness} Let {\rm \ref{ass:A1}} and {\rm \ref{ass:A2}} hold. Then the FBVIE \rf{FBVIE-main1} admits a unique solution $(\cX(\cd,\cd),\cY(\cd,\cd)) \in C(\D_[0,T];\dbR^n)\times C(\D^[0,T];\dbR^n)$. \end{theorem} \begin{remark} The solvability of FBVIEs in a small time horizon was proved by Hamaguchi \cite{Hamaguchi2021} by the classical fixed point theorem. The case with an arbitrarily given $T>0$ is much more challenging. Inspired by \cite{Wang-Yong-Zhang2022}, the first solvability result of FBVIEs in an arbitrarily long time horizon was obtained by Wang, Yong, and Zhou \cite{Wang-Yong-Zhou2023} for linear systems, by introducing the so-called path-dependent Riccati equation. Essentially, this is a four-step method, and heavily depends on the symmetric structure of the linear system. To the best of our knowledge, \autoref{Thm:well-posedness} is the first result on the well-posedness of FBVIEs with general nonlinear coefficients and an arbitrarily given time horizon. \end{remark} \begin{remark} In \cite{Hamaguchi2021,Wang-Yong-Zhang2022,Wang-Yong-Zhou2023}, the authors considered the stochastic case of FBVIEs, which is a generalization of \rf{FBVIE-main1} with an additional It\^{o} integral. We remark that the monotonicity method introduced in the current paper still works for stochastic FBVIEs. Additionally, we need to handle some technical issues. We hope to focus on introducing the basic idea here, and will report the results of stochastic FBVIEs in future publications. \end{remark} \section{Well-posedness}\label{sec:well} \subsection{Find a monotonicity condition in infinite-dimensional space}\label{subsec:mc} It is known that the monotonicity condition plays a central role in establishing the solvability of FBDEs by the method of continuation; see \cite{Hu-Peng1995,Yong1997,Peng-Wu1999}, for example. However, since FBVIE \rf{FBVIE-main1} is a non-local system, the local monotonicity condition \rf{FBDE-MC} defined at a point cannot be applied here. We need to find an infinite-dimensional version of it, and then construct a bridge which can link two infinite-dimensional spaces. \ms The key is to well understand the intuition of calculating $d[\lan p(t),q (t)\ran]$ in Hu and Peng \cite{Hu-Peng1995}. Our idea is to return to the linear-quadratic optimal control theory. \ms \begin{center} \textbf{An insight from value function} \end{center} Consider the controlled linear ODE: \bel{LQ-ODE1} p(t)=x+\int_0^t\[A(s) p(s)+B(s) u(s)\]ds,\q t \in[0,T], \ee and the cost functional of a quadratic form: \bel{LQ-ODE2} J(u(\cd))=\int_0^T\[\lan Q(t)p(t),p(t)\ran+\lan R(t) u(t),u(t)\ran\] d t+\lan G p(T),p(T)\ran. \ee Then the corresponding Hamiltonian system reads \bel{FBDE-LQ}\left\{\begin{aligned} &p(t)= x+\int_0^t \big[A(s)p(s)-B(s)R(s)^{-1}B(s)^\top q(s)\big] ds, \q t\in[0,T],\\ & q(t)=G p(T)+\int_t^T \big[A(s)^\top q(s) +Q(s)p(s)\big] ds,\q t\in[0,T],\end{aligned}\right.\ee with the optimal control $$\bar u(s)=-R(s)^{-1}B(s)^\top q(s),\q s\in[0,T].$$ From Yong and Zhou \cite{Yong-Zhou1999}, one immediately finds that \rf{FBDE-MC} almost coincides with the so-called standard condition, but more importantly, we realize that $\lan p(t),q(t)\ran$ is indeed the function of optimal values; that is $$\lan p(t),q(t)\ran=\int_t^T \[\lan Q(s)p(s),p(s)\ran+\lan R(s)\bar u(s),\bar u(s)\ran\]ds+\lan Gp(T),p(T)\ran.$$ Thus, essentially, Hu and Peng \cite{Hu-Peng1995} calculated the derivative of the optimal value with respect to the time variable, and we know that the derivative is always non-negative under proper conditions. Based on this observation, our approach is beginning to come out: We should calculate the optimal value of optimal control problems for VIEs (i.e., \rf{FBVIE-Ito}), by which we can find a proper monotonicity condition in infinite-dimensional spaces (i.e., \rf{MC-1}). \subsection{Comparison between the non-local monotonicity condition \rf{MC-1} and Hu and Peng's condition \rf{FBDE-MC} }\label{subsec:comparison} In this subsection, we will show that \rf{MC-1} is essentially an infinite-dimensional version of Hu and Peng's monotonicity condition \rf{FBDE-MC} given in \cite{Hu-Peng1995}. When the system \rf{FBVIE-main} reduces to an FBDE, \rf{MC-1} is almost equivalent to \rf{FBDE-MC}. \ms For simplicity, we only consider the Hamiltonian system associated with the LQ optimal control problem \rf{LQ-ODE1}--\rf{LQ-ODE2}. By taking \rf{LQ-ODE1}--\rf{LQ-ODE2} as an LQ optimal control problem for VIEs, the associated FBVIE reads $$ \left\{\begin{aligned} X(t)= & x+\int_0^t \[A(s) X(s)-B(s) R(s)^{-1}\int_s^TB(s)^\top Y(r)dr\\ &\qq-B(s)R(s)^{-1}B(s)^\top G X(T)\] ds, \\ Y(t)= & Q(t)X(t)+A(t)^\top G X(T)+\int_t^T A(t)^{\top} Y(s)ds,\end{aligned}\right. \q t\in [0,T]. $$ Let $$ p(t)=X(t),\q q(t)=\int_t^T Y(s)ds+GX(T),\q t\in[0,T], $$ then we have $$ \left\{\begin{aligned} &p(t)= x+\int_0^t \[A(s) p(s)-B(s) R(s)^{-1}B(s)^\top q(s) \]ds, \\ &q(t)= G p(T)+\int_t^T \[Q(s)p(s)+A(s)^\top q(s)\]ds,\end{aligned}\right.\qq t\in[0,T], $$ which is an FBDE. By \rf{MC-2} and \rf{FBDE-LQ}, the corresponding functions $\h x(\cd),\h y(\cd)$, $\h f(\cd), \h g(\cd), \h h(\cd)$, and $a(\cd), b(\cd), \psi(\cd)$ can be well-defined. Then \begin{align} &\int_t^T \big\lan \h y(s),\, \h f(s,t)\big\ran ds-\Big\lan \h h(t)+\int_t^T\h g(t,s)ds,\, \h x(t) \Big\ran+\big\lan G\h x(T), \,\h f(T,t)\big\ran \\ &\q=\Big\lan \int_t^T \h y(s)ds+G\h x(T),\, \h f(t) \Big\ran -\Big\lan \h h(t)+\int_t^T\h g(s)ds,\, \h x(t) \Big\ran\\ &\q=\big\lan q_1(t)-q_2(t),\, a(t,p_1(t),q_1(t))-a(t,p_2(t),q_2(t))\big \ran\\ &\qq -\big\lan b(t,p_1(t),q_1(t))-b(t,p_2(t),q_2(t)),\, p_1(t)-p_2(t) \big \ran,\q t\in[0,T].\end{align} Note that the above only depends on the values at a local point $t$. Then from \rf{MC-1}, we have \bel{MC-MC}\begin{aligned} &\big\lan a(t,p_1,q_1)-a(t,p_2,q_2),\,q_1-q_2\big \ran-\big\lan b(t,p_1,q_1)-b(t,p_2,q_2),\, p_1-p_2 \big \ran\\ &\q \les -\g \|p_1-p_2\|^2,\q \forall p_1,p_2,q_1,q_2\in\dbR^n,\\ &\lan \psi(p_1)-\psi(p_2),\, p_1-p_2\ran=G\|p_1-p_2\|^2\ges 0,\q \forall p_1,p_2\in\dbR^n. \end{aligned}\ee The non-local monotonicity condition \rf{MC-1} reduces to a local one. Clearly, the above monotonicity condition is almost equivalent to \rf{FBDE-MC}. Indeed, \rf{MC-MC} is a little bit weaker than \rf{FBDE-MC}, and in \cite{Peng-Wu1999}, Peng and Wu proved the well-posedness of \rf{FBDE} under a condition like this. \subsection{Uniqueness}\label{subsec:unique} \begin{proposition}\label{prop:uniqueness} Let {\rm \ref{ass:A1}} and {\rm \ref{ass:A2}} hold. Then FBVIE \rf{FBVIE-main1} admits at most one solution. \end{proposition} \begin{proof} Suppose that $(\cX_i(\cd,\cd),\cY_i(\cd,\cd))\in C(\D_[0,T];\dbR^n)\times C(\D^[0,T];\dbR^n);i=1,2$ satisfy \rf{FBVIE-main1}. Denote \begin{align} &\h \cX(t,s)=\cX_1(t,s)-\cX_2(t,s),\q \h \cY(t,s)=\cY_1(t,s)-\cY_2(t,s), \nonumber\\ &\h h(t)=h(t,\cX_1(t,t),\cX_1(T,T))-h(t,\cX_2(t,t),\cX_2(T,T)),\nn\\ & \h g(t,s)= g(t,s,\cX_1(s,s),\cY_1(s,s))- g(t,s,\cX_2(s,s),\cY_2(s,s)),\nn\\ &\h f(t,s)=f\Big(t,s,\cX_1(s,s),\cY_1(s,s), \int_s^T K(s,r)\cY_1(r,r)dr,\cX_1(T,T)\Big)\nn\\ &\qq\qq-f\Big(t,s,\cX_2(s,s),\cY_2(s,s), \int_s^T K(s,r)\cY_2(r,r)dr,\cX_2(T,T)\Big).\nn \end{align} Then $$ \left\{\begin{aligned} &{d\h\cX(t,s)\over ds}= \h f(t,s), \q (t,s)\in\D_[0,T],\\ &{d \h\cY(t,s)\over ds}= -\h g(t,s), \q (t,s)\in\D^[0,T],\\ &\h\cX(t,0)=0,\q \h\cY(t,T)=\h h(t),\q t\in[0,T], \end{aligned}\right. $$ which implies that \begin{align} &{d\over dt}\[\int_t^T \lan \h\cY(s,s),\h\cX(s,t)\ran ds+\lan G(\cX_1(T,T)) -G(\cX_2(T,T)),\,\h\cX(T,t)\ran\]\\ &\q=-\lan \h\cY(t,t),\h\cX(t,t)\ran+\int_t^T\lan \h\cY(s,s),\h\cX_t(s, t)\ran ds\\ &\qq+\lan G(\cX_1(T,T)) -G(\cX_2(T,T)),\h\cX_t(T, t)\ran\\ &\q=-\Big\lan\h h(t)+\int_t^T \h g(t,s)ds,\, \h\cX(t,t)\Big\ran+\int_t^T \lan \h\cY(s,s),\,\h f(s,t)\ran ds\\ &\qq+\big\lan G(\cX_1(T,T)) -G(\cX_2(T,T)), \,\h f(T,t)\big\ran\\ &\q\les-\g \|\h\cX(t,t)\|^2,\q t\in[0,T]. \end{align} Thus, \begin{align} &\lan G(\cX_1(T,T)) -G(\cX_2(T,T)),\,\h\cX(T,T)\ran+\g\int_0^T \|\h\cX(t,t)\|^2dt\les0. \end{align} Note that $$ \lan G(\cX_1(T,T)) -G(\cX_2(T,T)),\,\h\cX(T,T)\ran\ges\| G^{1\over 2}_0 [\cX_1(T,T)- \cX_2(T,T)]\|^2\ges0. $$ It turns out that $$ \cX_1(t,t)=\cX_2(t,t),\q t\in[0,T]. $$ Then, both $(\cX_1(\cd,\cd),\cY_1(\cd,\cd))$ and $(\cX_2(\cd,\cd),\cY_2(\cd,\cd))$ satisfy \bel{uniqueness-proof1} \left\{\begin{aligned} &{d \cX_i(t,s)\over ds}= f\Big(t,s,\cX(s,s),\cY_i(s,s), \int_s^T K(s,r)\cY_i(r,r)dr,\cX(T,T)\Big), \\ &\qq\qq\qq (t,s)\in\D_[0,T],\\ &{d \cY_i(t,s)\over ds}= - g(t,s,\cX(s,s),\cY_i(s,s)), \q (t,s)\in\D^[0,T],\\ &\cX_i(t,0)=x(t),\q \cY_i(t,T)=h(t,\cX(t,t),\cX(T,T)),\q t\in[0,T], \end{aligned}\right. \ee where $$ \cX(t,t):=\cX_1(t,t)=\cX_2(t,t),\q t\in[0,T]. $$ Clearly, FBVIE \rf{uniqueness-proof1} is a decoupled one, and then admits a unique solution. Thus, $(\cX_1(\cd,\cd),\cY_1(\cd,\cd))=(\cX_2(\cd,\cd),\cY_2(\cd,\cd))$. \end{proof} \subsection{Existence}\label{subsec:existence} We now give an existence result of FBVIE \rf{FBVIE-main1}. We begin with a simple FBVIE, whose solvability can be obtained by modifying the result in Peng and Wu \cite{Peng-Wu1999}. \begin{lemma}\label{lemm:alpha=0} For any given continuous functions $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, $h_0(\cd)$, and $x(\cd)$, the following linear FBVIE \bel{linear}\left\{\ba &\cX_s(t,s)=- \int_s^T \cY(r,r)dr-G(\cX(T,T))+f_0(t,s),\q (t,s)\in\D_[0,T], \\ &\cY_s(t,s)=-g_0(t,s),\q (t,s)\in\D^[0,T],\\ &\cX(t,0)=x(t),\q \cY(t,T)=\cX(t,t)+h_0(t),\q t\in[0,T],\ea\right.\ee admits a unique solution $(\cX(\cd,\cd),\cY(\cd,\cd)) \in C(\D_[0,T];\dbR^n)\times C(\D^[0,T];\dbR^n)$, where $G(\cd)$ is given by the monotonicity condition \rf{MC-3}. \end{lemma} \begin{proof} One can easily check that \rf{linear} satisfies the non-local monotonicity condition \rf{MC-1}. Thus, by \autoref{prop:uniqueness}, it admits at most one solution. For the existence of a solution to \rf{linear}, we divide the proof into two steps. \ms \emph{Step 1.} We first consider the case with the smooth coefficient $f_0(\cd,\cd)$ and smooth initial value $x(\cd)$. By Peng and Wu \cite{Peng-Wu1999}, we know that the following FBDE admits a unique solution $(X(\cd),Y(\cd))$: \bel{lemm:alpha=0-p1}\left\{\ba &dX(t)=\[-Y(t)+f_0(t,t)+x^{\prime}(t)+\int_0^t \frac{\partial f_0(t,r)}{\partial t}dr\]dt, \q t\in[0,T],\\ &-dY(t)=\[X(t)+h_0(t)+\int_t^T g_0(t,r)dr\]dt,\q t\in[0,T],\\ &X(0)=x(0),\q Y(T)=G(X(T)).\ea\right. \ee Let $\cX(t,t)=X(t)$ and $\cY(t,t)=-{d Y(t)\over dt}$, then from the above we have \begin{align} &\cX(t,t)=x(0)+\int_0^t \[-Y(s)+f_0(s,s)+x^{\prime}(s)+\int_0^s\frac{\partial f_0(s,r)}{\partial s}dr\]ds\\ &\q=x(t)-\int_0^t Y(s)ds+\int_0^t f_0(s,s)ds+\int_0^t\int_r^t\frac{\partial f_0(s,r)}{\partial s}dsdr\\ &\q=x(t)+\int_0^t f_0(t,s)ds-\int_0^t \[ Y(T)-\int _s^T{dY(r)\over dr}dr\] ds\\ &\q=x(t)+\int_0^t \[ f_0(t,s)-G(\cX(T,T))-\int _s^T\cY(r,r)dr\] ds,\q t\in[0,T], \end{align} and $$\cY(t,t)=\cX(t,t)+h_0(t)+\int_t^T g_0(t,r)dr,\q t\in[0,T].$$ Further, we define \begin{align} \cX(t,s)&=x(t)+\int_0^s \[ f_0(t,r)-G(\cX(T,T))-\int _r^T\cY(\t,\t)d\t\] dr,\q (t,s)\in\D_[0,T],\\ \cY(t,s)&=\cX(t,t)+h_0(t)+\int_s^T g_0(t,r)dr,\q (t,s)\in\D^[0,T], \end{align} which, clearly, is a solution to \rf{linear}. \ms \emph{Step 2.} We next consider the case with the coefficient $f_0(\cd,\cd)$ and initial value $x(\cd)$ being continuous. By the standard mollification method, we can find a sequence of smooth functions $\{f_{0,n}(\cd,\cd),x_n(\cd)\}_{n>0}$ to uniformly converge to $f_0(\cd,\cd)$ and $x(\cd)$. By the results obtained in Step 1, we know that the following equation admits a unique solution $(\cX_n(\cd,\cd),\cY_n(\cd,\cd))$ for any given $n>0$: \bel{X-m,n}\left\{\ba &{d \cX_n(t,s)\over ds}=- \int_s^T \cY_n(r,r)dr-G(\cX_n(T,T))+f_{0,n}(t,s),\q (t,s)\in\D_[0,T], \\ &{d\cY_n(t,s)\over ds}=-g_0(t,s),\q (t,s)\in\D^[0,T],\\ &\cX_n(t,0)=x_n(t),\q \cY_n(t,T)=\cX_n(t,t)+h_0(t),\q t\in[0,T].\ea\right. \ee Denote $\cX_{m,n}(\cd,\cd)=\cX_m(\cd,\cd)-\cX_n(\cd,\cd)$, $\cY_{m,n}(\cd,\cd)=\cY_m(\cd,\cd)-\cY_n(\cd,\cd)$, $x_{m,n}(\cd)=x_m(\cd)-x_n(\cd)$, $f_{0,m,n}(\cd,\cd)=f_{0,m}(\cd,\cd)-f_{0,n}(\cd,\cd)$ and $\h G(\cX_{m,n}(T,T))=G(\cX_{m}(T,T))-G(\cX_{n}(T,T))$ for any given $m,n>0$. Then, $$\left\{\ba &{d \cX_{m,n}(t,s)\over ds}=- \int_s^T \cY_{m,n}(r,r)dr-\h G(\cX_{m,n}(T,T))+f_{0,m,n}(t,s), \q (t,s)\in\D_[0,T],\\ &{d \cY_{m,n}(t,s)\over ds}=0,\q (t,s)\in\D^[0,T],\\ &\cX_{m,n}(t,0)=x_{m,n}(t),\q \cY_{m,n}(t,T)=\cX_{m,n}(t,t),\q t\in[0,T].\ea\right.$$ Noting that $\cY_{m,n}(t,t)=\cX_{m,n}(t,t)$ for any $t\in[0,T]$, we get \begin{align} &{d\over dt}\[\int_t^T \lan \cX_{m,n}(s,t),\cY_{m,n} (s,s)\ran ds+\lan \h G(\cX_{m,n}(T,T)),\cX_{m,n}(T,t)\ran\]\\ &\q=- \lan \cX_{m,n}(t,t),\cX_{m,n} (t,t)\ran+\int_t^T\Big \lan \cY_{m,n} (s,s) , \[f_{0,m,n}(s,t)- \int_t^T \cY_{m,n}(r,r)dr\\ &\qq-\h G(\cX_{m,n}(T,T))\]\Big\ran ds+\Big\lan\h G(\cX_{m,n}(T,T)),\[f_{0,m,n}(T,t)- \int_t^T \cY_{m,n}(r,r)dr\\ &\qq-\h G(\cX_{m,n}(T,T))\]\Big\ran\\ &\q=- \lan \cX_{m,n}(t,t),\cX_{m,n} (t,t)\ran-\Big\lan\h G(\cX_{m,n}(T,T))+\int_t^T \cY_{m,n} (s,s)ds,\,\\ &\qq\q \h G(\cX_{m,n}(T,T))+\int_t^T \cY_{m,n} (s,s)ds\Big\ran+\int_t^T\lan \cX_{m,n}(s,s),\,f_{0,m,n}(s,t)\ran ds\\ &\qq+\lan\h G(\cX_{m,n}(T,T)),f_{0,m,n}(T,t)\ran,\q t\in[0,T]. \end{align} Further, noting $\cX_{m,n}(t,0)=x_{m,n}(t)$ for any $t\in[0,T]$, we have \begin{align} &\lan\h G(\cX_{m,n}(T,T)),\cX_{m,n}(T,T)\ran-\int_0^T \lan x_{m,n}(s),\cX_{m,n} (s,s)\ran ds-\lan \h G(\cX_{m,n}(T,T)), x_{m,n}(T)\ran\\ &\q=-\int_0^T\[ \big\|\cX_{m,n}(t,t)\big\|^2+\Big\| \h G(\cX_{m,n}(T,T))+\int_t^T \cY_{m,n} (s,s)ds\Big\|^2\]dt\\ &\qq\, +\int_0^T\int_t^T\lan \cX_{m,n}(s,s),\,f_{0,m,n}(s,t)\ran dsdt +\int_0^T\lan \h G(\cX_{m,n}(T,T)),f_{0,m,n}(T,t)\ran dt. \end{align} By Young's inequality, from \rf{MC-3}, the above implies that \begin{align} &\| G_0^{1\over 2}\cX_{m,n}(T,T)\|^2 +\int_0^T\[ \big\|\cX_{m,n}(t,t)\big\|^2+\Big\| \h G(\cX_{m,n}(T,T))+\int_t^T \cY_{m,n} (s,s)ds\Big\|^2\]dt\\ &\q\les \e \int_0^T \big\|\cX_{m,n}(t,t)\big\|^2dt +\e\| G_0^{1\over 2}\cX_{m,n}(T,T)\|^2 +C_\e \int_0^T \big\|x_{m,n}(t)\big\|^2dt\\ &\qq +C_\e\| x_{m,n}(T)\|^2+C_\e \int_0^T\int_t^T \big\|f_{0,m,n}(s,t)\big\|^2dsdt+C_\e\int_0^T \| f_{0,m,n}(T,t)\|^2dt, \end{align} which yields that \begin{align} &\| G_0^{1\over 2}\cX_{m,n}(T,T)\|^2 +\int_0^T\[ \big\|\cX_{m,n}(t,t)\big\|^2+\Big\| \h G(\cX_{m,n}(T,T))+\int_t^T \cY_{m,n} (s,s)ds\Big\|^2\]dt\\ &\q\les C \[\int_0^T \big\|x_{m,n}(t)\big\|^2dt+\| x_{m,n}(T)\|^2+\int_0^T\int_t^T \big\|f_{0,m,n}(s,t)\big\|^2dsdt\\ &\qq\qq+\int_0^T \| f_{0,m,n}(T,t)\|^2dt\]. \end{align} Then noting that $$\| \h G(\cX_{m,n}(T,T))\|\les K \|G_0^{1\over 2}\cX_{m,n}(T,T)\|,$$ by the standard results of VIEs, we have \begin{align} \sup_{t,s}\|\cX_{m,n}(t,s)\|^2&\les C \[\int_0^T \big\|x_{m,n}(t)\big\|^2dt+\| x_{m,n}(T)\|^2+\sup_t\int_0^t \big\|f_{0,m,n}(t,s)\big\|^2ds\\ &\qq+\int_0^T \| f_{0,m,n}(T,t)\|^2dt+\sup_t\|x_{m,n}(t)\|^2\].\end{align} Thus, $\{\cX_n(\cd,\cd)\}_{n>0}$ is a Cauchy sequence in the space of continuous functions. With the fact that $\cY_{m,n}(t,s)=\cX_{m,n}(t,t)$, we know that $\{\cY_n(\cd,\cd)\}_{n>0}$ is also a Cauchy sequence in the space of continuous functions. Let $(\cX(\cd,\cd),\cY(\cd,\cd))$ be the limit of $\{\cX_n(\cd,\cd),\cY_n(\cd,\cd)\}_{n>0}$ as $n\to\infty$. Taking $n\to\infty$ in \rf{X-m,n}, we know that $(\cX(\cd,\cd),\cY(\cd,\cd))$ is a solution to \rf{linear}. \end{proof} From the proof, we can see that \rf{linear} is essentially an FBDE. Next, we consider the following family of FBVIEs parameterized by $\a \in[0,1]$: \bel{para}\left\{\ba \cX_s(t,s)&=\a f\Big(t,s,\cX(s,s),\cY(s,s), \int_s^T K(s,r)\cY(r,r)dr,\cX(T,T)\Big)\\ &\q+(1-\a)\[- \int_s^T \cY(r,r)dr-G(\cX(T,T))\]+f_0(t,s),\\ & \q\qq (t,s)\in\D_[0,T],\\ \cY_s(t,s)&=-\big[ \a g(t,s,\cX(s,s),\cY(s,s))+g_0(t,s) \big],\q (t,s)\in\D^[0,T],\\ \cX(t,0)&=x(t),\q t\in[0,T],\\ \cY(t,T)&=\a h(t,\cX(t,t),\cX(T,T))+(1-\a)\cX(t,t)+h_0(t),\q t\in[0,T],\ea\right.\ee where $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, and $h_0(\cd)$ are continuous functions. Clearly, when $\a=1$, the existence of the solution of \rf{para} implies that of \rf{FBVIE-main1}. \begin{lemma}\label{lem:con} Let {\rm \ref{ass:A1}} and {\rm \ref{ass:A2}} hold. We assume that, for a given $\a_0\in[0,1)$ and for any $f_0(\cd,\cd),g_0(\cd,\cd),h_0(\cd)$, \rf{para} admits a unique solution. Then there exists a constant $\d_0\in(0,1)$, such that for all $\a \in[\a_0,\a_0+\d_0]$, and for any $f_0(\cd,\cd),g_0(\cd, \cd), h_0(\cd)$, \rf{para} admits a unique solution. \end{lemma} \begin{proof} Note that for each $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, $h_0(\cd)$, and $\a_0\in[0,1)$, \rf{para} admits a unique solution. Then, for each pair $(\f(\cd,\cd),\p(\cd,\cd))$, there exists a unique pair $(\cX(\cd,\cd),\cY(\cd,\cd))$ satisfying the following FBVIE: $$ \left\{\ba \cX_s(t,s)&=\a_0 f\Big(t,s,\cX(s,s),\cY(s,s), \int_s^T K(s,r)\cY(r,r)dr,\cX(T,T)\Big)\\ &\q+(1-\a_0)\[- \int_s^T \cY(r,r)dr-G(\cX(T,T))\]\\ &\q+\d f\Big(t,s,\f(s,s),\p(s,s), \int_s^T K(s,r)\p(r,r)dr,\f(T,T)\Big)\\ &\q+\d \[\int_s^T \phi(r,r)dr+G(\f(T,T))\]+f_0(t,s),\q (t,s)\in\D_[0,T], \\ \cY_s(t,s)&=-\a_0 g(t,s,\cX(s,s),\cY(s,s))-\d g(t,s,\f(s,s),\p(s,s))-g_0(t,s),\\ &\qq\qq\qq (t,s)\in\D^[0,T],\\ \cX(t,0)&=x(t),\q t\in[0,T],\\ \cY(t,T)&=\a_0 h(t,\cX(t,t),\cX(T,T))+(1-\a_0)\cX(t,t)\\ &\q+\d\[h(t,\f(t,t),\f(T,T))-\f(t,t)\]+h_0(t),\q t\in[0,T].\ea\right. $$ \ms \ms We are going to prove that the mapping $\G_{\a_0+\d}[\cd,\cd]$, defined by \begin{align} &\G_{\a_0+\d}[\f(\cd,\cd),\p(\cd,\cd)]=(\cX(\cd,\cd),\cY(\cd,\cd)),\\ &\qq \forall (\f(\cd,\cd),\p(\cd,\cd))\in C(\D_[0, T] ;\dbR^n) \times C(\D^[0, T] ;\dbR^n), \end{align} is a contraction. Let $(\f_i(\cd,\cd),\p_i(\cd,\cd))\in C(\D_[0, T] ;\dbR^n) \times C(\D^[0, T] ;\dbR^n)$, and $$ (\cX_i(\cd,\cd),\cY_i(\cd,\cd))=\G_{\a_0+\d}[\f_i(\cd,\cd),\p_i(\cd,\cd)],\q \hbox{for } i=1,2. $$ Denote $(\h\f(\cd,\cd), \h\p(\cd,\cd))=(\f_1(\cd,\cd)-\f_2(\cd,\cd),\p_1(\cd,\cd)-\p_2(\cd,\cd))$ and $(\h\cX(\cd,\cd), \h\cY(\cd,\cd))=(\cX_1(\cd,\cd)-\cX_2(\cd,\cd),\cY_1(\cd,\cd)-\cY_2(\cd,\cd))$. Similar to \rf{MC-2}, we can denote \begin{align} &\h h(t;\f),\q \h f(t,s;\f,\p),\q\h g(t,s;\f,\p), \q\hbox{and}\\ & \h h(t;\cX),\q \h f(t,s;\cX,\cY),\q\h g(t,s;\cX,\cY), \end{align} with $(x_i(\cd),y_i(\cd))$ replaced by $(\f_i(\cd,\cd),\p_i(\cd,\cd))$ and $(\cX_i(\cd,\cd),\cY_i(\cd,\cd))$, respectively. Next, we denote $\h G(\h\cX(T,T))=G(\cX_1(T,T))-G(\cX_2(T,T))$ and $\h G(\h\f(T,T))=G(\f_1(T,T))-G(\f_2(T,T))$. Then, $$ \left\{\ba {d \h\cX(t,s)\over ds}&=\a_0 \h f(t,s;\cX,\cY)+(1-\a_0)\[- \int_s^T \h\cY(r,r)dr-\h G(\h\cX(T,T))\]\\ &\q+\d \h f(t,s;\f,\p)+\d \[\int_s^T \h\phi(r,r)dr+\h G(\h\f(T,T))\], \q (t,s)\in\D_[0,T],\\ {d\h\cY(t,s)\over ds}&=-\a_0 \h g(t,s;\cX,\cY)-\d \h g(t,s;\f,\p),\q (t,s)\in\D^[0,T],\\ \h\cX(t,0)&=0,\q t\in[0,T],\\ \h\cY(t,T)&=\a_0\h h(t;\cX)+(1-\a_0)\h\cX(t,t)+\d[\h h(t;\f)-\h\f(t,t)],\q t\in[0,T].\ea\right. $$ By the same argument as that employed in the proof of \autoref{prop:uniqueness}, we have \begin{align} &{d\over dt}\[\int_t^T \lan \h\cY(s,s),\h\cX(s,t)\ran ds+\lan\h G(\h\cX(T,T)),\h\cX(T,t)\ran\]\\ &\q=-\Big\lan \a_0\h h(t;\cX)+(1-\a_0)\h\cX(t,t)+\d[\h h(t;\f)-\h\f(t,t)]\\ &\qq\qq\qq+\int_t^T [\a_0 \h g(t,s;\cX,\cY)+\d \h g(t,s;\f,\p)]ds,\, \h\cX(t,t)\Big\ran\\ &\qq+\int_t^T\Big\lan \h\cY(s,s),\,\a_0 \h f(s,t;\cX,\cY)+(1-\a_0)\[- \int_t^T \h\cY(r,r)dr-\h G(\h\cX(T,T))\]\\ &\qq\qq\qq +\d \h f(s,t;\f,\p)+\d \[\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\]\Big\ran ds\\ &\qq+\Big\lan \h G(\h\cX(T,T)),\,\a_0 \h f(T,t;\cX,\cY)+(1-\a_0)\[- \int_t^T \h\cY(r,r)dr-\h G(\h\cX(T,T))\]\\ &\qq\qq\qq +\d \h f(T,t;\f,\p)+\d \[\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\]\Big\ran,\q t\in[0,T]. \end{align} Then from the non-local monotonicity condition \rf{MC-1}, we have \begin{align} &{d\over dt}\[\int_t^T \lan \h\cY(s,s),\h\cX(s,t)\ran ds+\lan\h G(\h\cX(T,T)),\h\cX(T,t)\ran\]\\ &\q\les-\d\Big\lan \h h(t;\f)-\h\f(t,t)+\int_t^T \h g(t,s;\f,\p)ds,\, \h\cX(t,t)\Big\ran\\ &\qq+\d \int_t^T\Big\lan \h\cY(s,s),\, \h f(s,t;\f,\p)+\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\Big\ran ds\\ &\qq+\d\Big\lan\h G(\h\cX(T,T)),\,\h f(T,t;\f,\p)+\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\Big\ran\\ &\qq -\a_0 \g\|\h\cX(t,t)\|^2-(1-\a_0)\[\|\h\cX(t,t)\|^2+\Big\|\int_t^T \h\cY(r,r)dr+\h G(\h\cX(T,T))\Big\|^2\],\q t\in[0,T]. \end{align} Thus, noting $\h\cX(\cd,0)\equiv 0$ and $\g>0$, by \rf{MC-3}, we get \begin{align} &\big\| G_0^{1\over 2}\h\cX(T,T)\big\|^2+\int_0^T\|\h\cX(t,t)\|^2dt\\ &\q\les \d C\int_0^T \bigg\{\Big\|\Big\lan \h h(t;\f)-\h\f(t,t)+\int_t^T \h g(t,s;\f,\p)ds,\, \h\cX(t,t)\Big\ran\Big\|\\ &\qq\qq+\Big\|\int_t^T\Big\lan \h\cY(s,s),\, \h f(s,t;\f,\p)+\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\Big\ran ds\Big\|\\ &\qq\qq+\Big\|\Big\lan \h G(\h \cX(T,T)),\,\h f(T,t;\f,\p)+\int_t^T \h\phi(r,r)dr+\h G(\h\f(T,T))\Big\ran\Big\|\bigg\}dt\\ &\q\les {1\over 2} \|G_0^{1\over2}\h\cX(T,T)\|^2+{1\over 2}\int_0^T\|\h\cX(t,t)\|^2dt+\d C\int_0^T\|\h\cY(t,t)\|^2dt\\ &\qq +\d C\|\h\f(T,T)\|^2+ \d C\int_0^T \bigg\{\| \h h(t;\f)\|^2+\|\h\phi(t,t)\|^2+\|\h f(T,t;\f,\p)\|^2+\|\h\f(t,t)\|^2\\ &\qq\qq+\int_t^T\big[ \| \h g(t,s;\f,\p)\|^2 +\|\h f(s,t;\f,\p)\|^2\big]ds\bigg\}dt, \end{align} where the last inequality is obtained by applying the Young's inequality. Thus, \begin{align} &\big\| G_0^{1\over 2}\h\cX(T,T)\big\|^2+\int_0^T\|\h\cX(t,t)\|^2dt\\ &\q\les \d C\int_0^T\|\h\cY(t,t)\|^2dt +\d C\|\h\f(T,T)\|^2+ \d C\int_0^T \big[\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\big]dt. \end{align} By the standard results of BVIEs, we have \bel{Prop:con-p1} \begin{aligned} &\int_0^T \|\h\cY(t,t)\|^2dt \les\d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]\\ &\qq +C\[\int_0^T\|\h\cX(t,t)\|^2dt+\|G_0\h\cX(T,T)\|^2\]\\ &\q\les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]\\ &\qq +C\[\int_0^T\|\h\cX(t,t)\|^2dt+\|G_0^{1\over 2}\h\cX(T,T)\|^2\]. \end{aligned} \ee Combining the above two estimates together, we have \begin{align} & \|G_0^\frac{1}{2}\h\cX(T,T)\|^2+\int_0^T\|\h\cX(t,t)\|^2dt \\ &\q\les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\int_0^T\|\h\cX(t,t)\|^2dt+\|G_0^{1\over 2}\h\cX(T,T)\|^2+\|\h\f(T,T)\|^2\], \end{align} which implies that \begin{align} & \|G_0^\frac{1}{2}\h\cX(T,T)\|^2+\int_0^T\|\h\cX(t,t)\|^2dt \\ &\q\les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]. \end{align} Substituting the above into \rf{Prop:con-p1}, we get \begin{align} &\|G_0^\frac{1}{2}\h\cX(T,T)\|^2+\int_0^T\(\|\h\cX(t,t)\|^2+\|\h\cY(t,t)\|^2\)dt \\ &\q\les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]. \end{align} Then by the standard estimates of FVIEs, we have \bel{Prop:con-p2} \begin{aligned} &\sup _{t,s} \|\h\cX(t,s)\|^2 \les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]\\ &\qq+C\[ \|G_0^\frac{1}{2}\h\cX(T,T)\|^2+\int_0^T\|\h\cY(t,t)\|^2dt\]\\ &\q\les \d C\[\int_0^T\(\|\h\f(t,t)\|^2+\|\h\p(t,t)\|^2\)dt+\|\h\f(T,T)\|^2\]. \end{aligned} \ee On the other hand, the standard estimate of BVIEs implies that \bel{Prop:con-p3} \begin{aligned} &\sup _{t,s} \|\h\cY(t,s)\|^2 \les \d C\[\int_0^T\|\h\p(t,t)\|^2dt+\sup _{t} \|\h\f(t,t)\|^2\]+C\sup _{t} \|\h\cX(t,t)\|^2\\ &\q\les \d C\[\int_0^T\|\h\p(t,t)\|^2dt+\sup _{t} \|\h\f(t,t)\|^2\], \end{aligned} \ee in which the last equality is due to \rf{Prop:con-p2}. Combining the estimates \rf{Prop:con-p2} and \rf{Prop:con-p3} together, we get \begin{align} \sup_{t,s}\[\|\h\cX(t,s)\|^2+\|\h\cY(t,s)\|^2\] \les \d C\sup_{t,s}\[\|\h\f(t,s)\|^2+\|\h\p(t,s)\|^2\]. \end{align} We now choose $\d_0={1\over 2C}$, which is independent of $\a_0$. Clearly, for each fixed $\d\in [0,\d_0]$, the mapping $\G_{\a_0+\d}[\cd,\cd]$ is a contraction. It turns out that this mapping has a unique fixed point $(\cX^{\a_0+\d}(\cd,\cd), \cY^{\a_0+\d}(\cd,\cd))$, which is the unique solution of \rf{para} for $\a=\a_0+\d$. \end{proof} \ms We are ready to give the proof of \autoref{Thm:well-posedness} now. \begin{proof} From \autoref{lemm:alpha=0}, we see immediately that, when $\a=0$, for any $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, and $h_0(\cd)$, FBVIE \rf{para} admits a unique solution. Then by \autoref{lem:con}, for any $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, and $h_0(\cd)$, we can solve the FBVIE \rf{para} successively for the case $\a \in[0, \d_0],[\d_0, 2 \d_0], \ldots, [(N-1)\d_0, 1]$, with $(N-1)\d_0<1\les N\d_0$. It turns out that, when $\a=1$, for any $f_0(\cd,\cd)$, $g_0(\cd,\cd)$, and $h_0(\cd)$, \rf{para} admits a solution, which implies that the solution of \rf{FBVIE-main1} exists. The uniqueness of solutions to \rf{FBVIE-main1} is an immediate consequence of \autoref{prop:uniqueness}. \end{proof} \section{Examples} \label{sec:application} In this section, we shall give two explicit examples of coupled FBVIEs, whose solvability can be obtained from \autoref{Thm:well-posedness}. \begin{example}[Hamiltonian System Derived From Linear-Convex Optimal Control Problems] Consider the controlled Volterra integral equation: $$ X(t)=x(t)+\int_0^t\[A(t, s) X(s)+B(t, s) u(s)\]ds,\q t \in[0,T], $$ and the cost functional: $$ J( u(\cd))=\int_0^T\[Q(t,X(t))+\lan R(t) u(t),u(t)\ran\] d t+M(X(T)). $$ We assume that all the functions involved above are smooth, and \begin{align} &\|Q(t,x)\|\les L(1+\|x\|^2),\q\|M(x)\|\les L(1+\|x\|^2),\q \forall x\in\dbR^n, \\ & R(t)\ges \d I_m,\q \lan Q_{x}(t,x_1)-Q_{x}(t,x_2),\,x_1-x_2\ran\ges \d \|x_1-x_2\|^2,\\ &\|Q_{x}(t,x_1)-Q_{x}(t,x_2)\|\les L\|x_1-x_2\|,\\ & \lan M_{x}(x_1)-M_{x}(x_2),\,x_1-x_2\ran\ges \|G_0^{1\over 2} (x_1-x_2)\|^2,\\ &\| M_{x}(x_1)-M_{x}(x_2)\|\les L\|G_0^{1\over 2} (x_1-x_2)\|,\q \forall x_1,x_2\in\dbR^n, \end{align} for some $\d>0$, $L>0$, and $G_0\ges 0$. A typical example is that \bel{LQ-Form} Q(t,x)=\lan Q(t)x,x\ran, \q \hbox{with } Q(t)\ges \d I_n>0, \q M(x)=\lan G_0 x,x\ran, \ee under which the problem has a linear-quadratic form. In addition, if the state $X(\cd)$ is one-dimensional, we can also take $$ Q(t,x)={x^4-2\over x^2+1}, $$ by which the problem is out of the linear-quadratic framework. The optimal control problem can be stated as follows: Find a control $\bar u(\cd)\in L^2([0,T];\dbR^m)$ such that $$ J(0, x(\cd);\bar u(\cd))=\inf_{u(\cd)\in L^2([0,T];\dbR^m)} J(0, x(\cd);u(\cd)). $$ \ms Clearly, this is the optimal control problem with the form \rf{LCONVEX} given in Introduction. Then from \rf{LCONVEX1} and \rf{OS-1}, the optimal control $\bar u(\cd)$ can be written as: \begin{align}\label{MP-condition} \bar{u}(s)=-{1\over 2}R(s)^{-1}\int_s^T B(r,s)^\top Y(r)dr-{1\over 2}R(s)^{-1}B(T,s)^\top M_x( X(T)), \quad s \in[0, T], \end{align} with the Hamiltonian system \bel{MP-system}\left\{\begin{aligned} X(t)= & x(t)+\int_0^t \[A(t,s) X(s)-{1\over 2}B(t,s) R(s)^{-1}\int_s^T B(r,s)^\top Y(r)dr\\ &\qq-{1\over 2}B(t,s)R(s)^{-1}B(T,s)^\top M_x(X(T))\] ds, \\ Y(t)= & Q_x(t,X(t))+A(T,t)^\top M_x(X(T))+\int_t^T A(s,t)^{\top} Y(s)ds,\end{aligned}\right. \q t\in[0,T]. \ee Then, \begin{align} &f(t,s,x,y,y^\prime,x^\prime)=A(t,s)x-{1\over 2}B(t,s) R(s)^{-1}y^\prime-{1\over 2}B(t,s)R(s)^{-1}B(T,s)^\top M_x(x^\prime),\\ & K(s,r)=B(r,s)^\top,\q g(t,s,x,y)=A(s,t)^\top y,\q h(t,x,x^\prime)=Q_x(t,x)+A(T,t)^\top M_x(x^\prime). \end{align} For any $x_i(\cd),y_i(\cd)\in C([0,T];\dbR^n),\,i=1,2$, denote $\h x(\cd)$, $\h y(\cd)$, $\h f(\cd,\cd)$, $\h g(\cd,\cd)$, and $\h h(\cd)$ as that in \rf{MC-2}. Moreover, we take $G(\cd)=M_x(\cd)$, and denote $$ \h G(T)=G (x_1(T))-G (x_2(T)),\q \h Q_x(t)=Q_x(t,x_1(t))-Q_x(t,x_2(t)). $$ Clearly, \rf{MC-3} holds. Then, \begin{align} &\int_t^T \big\lan \h y(s),\, \h f(s,t)\big\ran ds-\Big\lan \h h(t)+\int_t^T\h g(t,s)ds,\, \h x(t) \Big\ran+\big\lan\h G (T), \,\h f(T,t)\big\ran \nn\\ &\q=\int_t^T \bigg\lan \h y(s),\, A(s,t)\h x(t)-{1\over 2}B(s,t) R(t)^{-1}\int_t^TB(r,t)^\top \h y(r)dr\nn\\ &\qq\qq-{1\over 2}B(s,t)R(t)^{-1}B(T,t)^\top \h G (T)\bigg\ran ds\nn\\ &\qq-\Big\lan\h Q_x(t) +A(T,t)^\top \h G (T)+\int_t^T A(s,t)^\top\h y(s) ds,\, \h x(t) \Big\ran\nn\\ &\qq+\bigg\lan \h G (T), \,A(T,t)\h x(t)-{1\over 2}B(T,t) R(t)^{-1}\int_t^TB(r,t)^\top \h y(r)dr\nn\\ &\qq\qq -{1\over 2} B(T,t)R(t)^{-1} B(T,t)^\top\h G (T)\bigg\ran\nn\\ &\q =-\big\lan\h Q_x(t),\, \h x(t)\big\ran-{1\over 2}\Big\lan R(t)^{-1}\[B(T,t)^\top \h G (T)\nn\\ &\qq\,+\int_t^TB(r,t)^\top \h y(r)dr\],\, \[B(T,t)^\top \h G (T)+\int_t^TB(r,t)^\top \h y(r)dr\]\Big\ran\nn\\ &\q\les -\d \|\h x(t)\|^2,\q t\in[0,T],\nn \end{align} which implies the non-local monotonicity condition \rf{MC-1} holds. Thus, from \autoref{Thm:well-posedness}, FBVIE \rf{MP-system} admits a unique solution. It turns out that the control process $\bar u(\cd)$, given by \rf{MP-condition}, is the unique optimal control. Moreover, if $Q(\cd,\cd)$ and $M(\cd)$ admit the quadratic form \rf{LQ-Form}, then the value function can be given by $$ \begin{aligned} &\int_0^T \lan \cX(s,0),\cY(s,s)\ran ds+\lan G_0\cX(T,0),\cX(T,T)\ran\\ &\q = \int_0^T \big[\lan Q(s)X(s),X(s)\ran +\lan R(s)\bar u(s),\bar u(s)\ran \big]ds+\lan G_0X(T),X(T)\ran, \end{aligned} $$ where $(\cX(\cd,\cd),\cY(\cd,\cd))$ satisfies the following auxiliary system corresponding to \rf{MP-system}: $$ \left\{\begin{aligned} &\cX_s(t,s)= A(t,s)\cX(s,s)-{1\over 2}B(t,s)R(s)^{-1}\int_s^T B(r,s)^\top\cY(r,r)dr\\ &\qq- B(t,s)R(s)^{-1}B(T,s)^\top G_0\cX(T,T),\q (t,s)\in\D_[0,T], \\ &\cY_s(t,s)= - A(s,t)^{\top}\cY(s,s),\q (t,s)\in\D^[0,T], \\ & \cX(t,0)=x(t),\q \cY(t,T)=2 Q(t)\cX(t,t)+2A(T,t)^\top G_0\cX(T,T),\q t\in[0,T]. \end{aligned}\right. $$ \end{example} The following is an FBVIE with some general nonlinear terms. We will show that under some proper assumptions, it also satisfies the non-local monotonicity condition \rf{MC-1}--\rf{MC-2}. \begin{example}[Nonlinear FBVIEs] Consider the following nonlinear FBVIE: \bel{FBVIE-Nonlinear}\left\{\begin{aligned} X(t)= & x(t)+\int_0^t \[A(t,s) X(s)+B(t,s)a(s,X(s))\\ &\qq-B(t,s) \int_s^T B(r,s)^\top Y(r)dr\] ds, \\ Y(t)= & b(t,X(t))+\phi\Big(t,\int_t^T B(r,t)^\top Y(r)dr\Big)\\ &\qq+\int_t^T\[ A(s,t)^{\top} Y(s)+\psi(t,s,X(t))\] ds,\end{aligned}\right.\q t\in[0,T]. \ee Assume that all the coefficients of the above are continuous functions. Moreover, there exist constants $\l,L_a,L_b,L_\phi,L_\psi>0$ such that \begin{align} &\|a(s,x_1)-a(s,x_2)\|\les L_a \|x_1-x_2\|,\q \|b(s,x_1)-b(s,x_2)\|\les L_b \|x_1-x_2\|,\\ & \lan b(s,x_1)-b(s,x_2) ,x_1-x_2\ran\ges\l \|x_1-x_2\|^2,\\ &\|\psi(t,s,x_1)-\psi(t,s,x_2)\|\les L_\psi \|x_1-x_2\|,\q \forall x_1,x_2\in\dbR^n;\\ &\|\phi(s,y^\prime_1)-\phi(s,y^\prime_2)\|\les L_\phi \|y^\prime_1-y^\prime_2\|,\q \forall y^\prime_1,y^\prime_2\in\dbR^n. \end{align} We now show that under proper conditions, the above satisfies the monotonicity condition \rf{MC-1}. Indeed, \begin{align} &\int_t^T \big\lan \h y(s),\, \h f(s,t)\big\ran ds-\Big\lan \h h(t)+\int_t^T\h g(t,s)ds,\, \h x(t) \Big\ran\\ &\q =\int_t^T \Big\lan \h y(s),\, \[A(s,t)\h x(t)+B(s,t)[a(t,x_1(t))-a(t,x_2(t))]\\ &\qq -B(s,t) \int_t^T B(r,t)^\top \h y (r)dr\]\Big\ran ds-\Big\lan\h x(t),\, \[b(t,x_1(t))-b(t,x_2(t))\\ &\qq+\phi\Big(t,\int_t^T B(r,t)^\top y_1(r)dr\Big)-\phi\Big(t,\int_t^T B(r,t)^\top y_2(r)dr\Big)\\ &\qq +\int_t^T\( A(s,t)^{\top} \h y(s)+\psi(t,s,x_1(t))-\psi(t,s,x_2(t))\) ds\]\Big\ran\\ &\q = \Big\lan\int_t^T B(s,t)^\top\h y(s)ds,\, \big[a(t,x_1(t))-a(t,x_2(t))\big]\Big\ran\\ &\qq -\Big\| \int_t^T B(r,t)^\top \h y (r)dr\Big\|^2-\Big\lan\h x(t),\, \[b(t,x_1(t))-b(t,x_2(t))\\ &\qq+\phi\Big(t,\int_t^T B(r,t)^\top y_1(r)dr\Big)-\phi\Big(t,\int_t^T B(r,t)^\top y_2(r)dr\Big)\]\Big\ran\\ &\qq -\Big\lan\h x(t),\,\int_t^T\big[\psi(t,s,x_1(t))-\psi(t,s,x_2(t))\big] ds\Big\ran\\ &\q \les {1\over 2} \Big\| \int_t^T B(r,t)^\top \h y (r)dr\Big\|^2+ {1\over 2}L_a^2 \|\h x(t)\|^2-\Big\| \int_t^T B(r,t)^\top \h y (r)dr\Big\|^2\\ &\qq -\l\|\h x(t)\|^2+{1\over 2} \Big\| \int_t^T B(r,t)^\top \h y (r)dr\Big\|^2+ {1\over 2}L_\phi^2 \|\h x(t)\|^2+L_\psi T\|\h x(t)\|^2\\ &\q=-\Big(\l-{1\over 2}L_a^2 -{1\over 2}L_\phi^2-L_\psi T\Big) \|\h x(t)\|^2,\q t\in[0,T]. \end{align*} Let $$ \l-{1\over 2}L_a^2 -{1\over 2}L_\phi^2-L_\psi T>0, $$ and take $G(\cd)\equiv0$. Then the non-local monotonicity condition \rf{MC-1}--\rf{MC-2} holds. One can see that for the case with only one nonlinear term $b(\cd)$, that is $a(\cd),\phi(\cd),\psi(\cd)=0$, we just need to assume that $\l>0$. Thus, by \autoref{Thm:well-posedness}, \rf{FBVIE-Nonlinear} admits a unique solution. \end{example} \section{Conclusion} The main contribution of this paper is that we provide a method for finding a non-local monotonicity condition for coupled forward-backward Volterra integral equations, under which the system admits a unique solution. From this procedure, we can see the celebrated method of continuation developed for establishing the solvability of FBSDEs (see \cite{Hu-Peng1995,Yong1997,Peng-Wu1999}) is also an optimal control approach in some sense. The stochastic version of coupled FBVIEs has more potential in applications; for example, it can be applied in the popular rough Heston model, and it can provide a probabilistic interpretation for non-local PDEs. 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2412.04527v1	http://arxiv.org/abs/2412.04527v1	Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle	\documentclass{article} \usepackage[utf8]{inputenc} \usepackage[margin=3cm]{geometry} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amsmath} \usepackage{mathtools} \usepackage{amssymb} \usepackage{multicol} \usepackage{cite} \usepackage{caption} \usepackage{subcaption} \usepackage{graphicx} \usepackage{tcolorbox} \usepackage{dsfont} \usepackage{xcolor} \usepackage{tikz-cd} \usepackage{subfiles} \usepackage{longtable} \usepackage{hyperref} \usepackage{bbm} \newtheorem{theorem}{Theorem} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conj}[theorem]{Conjecture} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{defn}[theorem]{Definition} \newtheorem{propn}[theorem]{Proposition} \newtheorem{propn-non}{Proposition} \newenvironment{sproof}{ \renewcommand{\proofname}{Sketch Proof}\proof}{\endproof} \makeatletter \def\blfootnote{\gdef\@thefnmark{}\@footnotetext} \makeatother \newcommand{\B}[1]{\mathbb{#1}} \newcommand{\C}[1]{\mathcal{#1}} \newcommand{\oop}{\preccurlyeq} \newcommand{\is}{\mathbbm{1}} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\erfc}{erfc} \newcommand{\midL}{\binom{n}{\lfloor n/2\rfloor}} \renewcommand{\arraystretch}{1.5} \linespread{1.25} \usepackage[skip=7.5pt plus1pt, indent=20pt]{parskip} \newcommand{\JB}[1]{\textcolor{blue!70!black}{\textbf{JB:} #1}} \title{Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle} \author{Jacob Mercer\footnote{\texttt{[email protected]}, Department of Statistics, University of Oxford}} \date{} \begin{document} \maketitle \begin{abstract} $N$-Brownian bees is a branching-selection particle system in $\B{R}^d$ in which $N$ particles behave as independent binary branching Brownian motions, and where at each branching event, we remove the particle furthest from the origin. We study a variant in which $d=1$ and particles have an additional drift $\mu\in\B{R}$. We show that there is a critical value, $\mu_c^N$, and three distinct regimes (sub-critical, critical, and super-critical) and we describe the behaviour of the system in each case. In the sub-critical regime, the system is positive Harris recurrent and has an invariant distribution; in the super-critical regime, the system is transient; and in the critical case, after rescaling, the system behaves like a single reflected Brownian motion. We also show that the critical drift $\mu_c^N$ is in fact the speed of the well-studied $N$-BBM process, and give a rigorous proof for the speed of $N$-BBM, which was missing in the literature. \end{abstract} \section{Introduction} \blfootnote{This publication is based on work supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling, and Simulation (EP/S023925/1)} \subfile{sections/intro} \section{Some useful couplings of $N$-BBM and Brownian bees} \subfile{sections/couplings} \section{Association of the $N$-BBM} \subfile{sections/appendixB} \section{The Sub-critical Case: $\|\mu\|<\mu_c^N$} \subfile{sections/subcritical} \section{The Super-critical Case: $\|\mu\|>\mu_c^N$} \subfile{sections/supercritical} \section{The Critical Case: $\|\mu\|=\mu_c^N$} \subfile{sections/critical} \section{Proof of Proposition \ref{pathSpaceTfm} and Proposition \ref{asympSmallTC}} \label{techSec} \subfile{sections/technicals} \newpage \section{Appendix: The Asymptotic Velocity of the $N$-BBM} \subfile{sections/appendixA} \section{Acknowledgements} The author thanks Julien Berestycki for his supervision of the project and heplful feedback on the paper, as well as helpful conversations with Bastien Mallein, and constructive feedback from Matthias Winkel and Brett Kolesnik. \bibliographystyle{plain} \bibliography{bps} \subfile{} \end{document} Recall that we are trying to prove here that when $\|\mu\|=\mu_c^N$, the system satisfies an \textit{invariance principle} $$ \left(m^{-1/2}X^{N,\mu_c^N}_i(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} (\frac{\mu}{\|\mu\|}\beta^{-1/2}\sigma \|B(t)\|)_{t\geq 0} $$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$ for any particle $X^{N,\mu}_i$, where $B(t)$ is a standard Brownian motion. We want to further prove that the radius of the cloud of particles converges to zero in the scaling limit; that is $$m^{-1/2}\left(X^{N,\mu_c^N}_N(mt) - X^{N,\mu_c^N}_1(mt)\right) \overset{d}{=} 0.$$ Therefore, in the scaling limit, the cloud of particles behaves like a single point mass moving about as reflected Brownian motion. We start with an analogous result for the $N$-BBM system which we will then couple to the $N$-Brownian bees. For ease of notation, for the rest of this work we will denote the $N$-BBM, $Z^{N,\mu_c^N}$, with positive critical drift $\mu_c^N$ and killing on the right simply as $Z$ and the $N$-Brownian bees $X^{N,\mu_c^N}$ with positive critical drift $\mu_c^N$ simply as $X$. Notice that we want to prove a statement of convergence in the space $\C{D}([0,\infty),\B{R})$, the space of cadlag functions $[0,\infty)\to \B{R}$. However, in it's standard formulation, Donsker's theorem, and the machinery of the paper \cite{bbb} which we use here, are for convergence in the space $\C{D}([0,1],\B{R})$. Therefore before proceeding to the following proposition, we state some results from \cite{bills} which will be needed here. Let \begin{align} g_m(t)=(m-t)\is_{\{m-1\leq t\leq m\}} + \is_{\{t\leq m-1\}} \end{align} So for any $m>0$, $\psi_m:(X(t))_{t\geq 0} \mapsto (X(t)g_m(t))_{0\leq t\leq m}$ is a map $\C{D}([0,\infty),\B{R})\to \C{D}([0,m],\B{R})$ such that $X(m)g_m(m)=0$. Then \begin{lemma} \label{DinftyConv} (Lemma 3 in Section 16 of \cite{bills}) A necessary and sufficient condition for $(Y_n(t))_{t\geq 0}$ to converge to $(Y(t))_{t\geq 0}$ in $\C{D}([0,\infty),\B{R})$ is that $\psi_m((Y_n(t))_{t\geq 0})$ converges to $\psi_m((Y(t))_{t\geq 0})$ in $\C{D}([0,m],\B{R})$ for every $m>0$ \end{lemma} \begin{propn} \label{critNBBMInvar} Fix some $j$ in $\{1,\ldots,N\}$. Then $$\left(m^{-1/2}Z_j(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma B(t))_{t\geq 0}$$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$, where $B$ is a standard Brownian motion, and $\beta,\sigma$ are constants which we will define in the course of the proof. \end{propn} \begin{proof} The first part of this proof will follow almost exactly as the proof of Proposition 1.4 in \cite{bbb}, which we will give a brief summary of now. Suppose we have a stochastic process $W$ and a sequence of finite stopping times, $\tau_1,\tau_2,\tau_3,\ldots$ satisfying the following properties \begin{enumerate} \item $\tau_{i+1}-\tau_i$ are i.i.d with non-zero and finite mean for $i\geq 1$ \item $W(\tau_{i+1})-W(\tau_i)$ are i.i.d for $i\geq 1$ \item $\sup_{\tau_i \leq t\leq \tau_{i+1}} \|W(t)-W(\tau_i)\|$ are i.i.d for $i\geq 1$ \item $\B{E}\left[\sup_{\tau_i\leq t\leq \tau_{i+1}}\|W(t)-W(\tau_i)\|^2\right]<\infty$ \end{enumerate} then $W$ converges to a Brownian motion under the scaling $m^{-1/2}W(mt)$ in distribution in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$. This is a simple corollary of Proposition 1.4 in \cite{bbb}, who go a step further by extending this result to the give a scaling limit for the barycentre of an $N$ particle system, however for our purposes it will be sufficient to show the four properties above. As in Proposition \ref{finExpNBBM}, we consider the $N$-BBM process at some time $t\in [k,k+1]$, for $k\in \B{N}$ as a function of $Z(k)$ and $N$ i.i.d. BBMs. Therefore for $i=1,\ldots, N$, let $$\C{B}^i_k(t)_{0\leq t\leq 1}=\{B^{i,j}_k(t):j=1,\ldots,\C{N}^i_k(t)\}$$ be as in Proposition \ref{finExpNBBM}, but here with drift $\mu=\mu_c^N$. Let the events $A_k$ and the regeneration times $\tau_i$ also be the same as in Proposition \ref{finExpNBBM}. Now in Proposition \ref{finExpNBBM}, we showed that $\{\tau_{i+1}-\tau_i:i\geq 1\}$ are i.i.d. random variables with finite mean, and that $\{Z(\tau_{i+1})-Z(\tau_i):i\geq 1\}$ are i.i.d. random variables. It is also immediate from construction that $\left\{\sup_{\tau_i\leq t \leq \tau_{i+1}}\|Z_j(t)-Z_j(\tau_i)\|:i\geq 1\right\}$ is an i.i.d family of random variables. Therefore in order to apply the result of Proposition 1.4 in \cite{bbb} it only remains to prove that \begin{align} \label{sqBound} \B{E}\left[\sup_{\tau_i\leq t \leq \tau_{i+1}}\|Z_j(t)-Z_j(\tau_i)\|^2\right]<\infty. \end{align} Note that $\sup_{\tau_i \leq t\leq \tau_{i+1}}\|Z_j(t)-Z_j(\tau_i)\|$ is bounded above by $\|Z_N(\tau_i)-Z_1(\tau_i)\|$ plus $\lceil \tau_{i+1}-\tau_i\rceil$ random variables distributed like $\max_{i=1}^{\C{N}(1)}\sup_{0\leq t\leq 1}\|B^i(t)\|$, where $\C{B}(t)=\{B^i(t), i=1,\ldots,\C{N}(t)\}$ is a branching Brownian motion. Therefore by the Cauchy-Schwarz inequality, $\sup_{\tau_i \leq t\leq \tau_{i+1}}\|Z_j(t)-Z_j(\tau_i)\|^2$ is bounded above by \begin{align}\label{boundforSq} (\lceil \tau_{i+1}-\tau_i \rceil + 1)\left(\|Z_N(\tau_i) -Z_1(\tau_i)\|^2 + \Omega_1^2 + \ldots \Omega_{\lceil \tau_{i+1}-\tau_i \rceil}^2\right), \end{align} where the $\Omega_i$'s are i.i.d random variables distributed like $\max_{i=1}^{\C{N}(1)}\sup_{0\leq t\leq 1}\|B^i(t)\|$. Since the maximum of $\C{N}(1)$ positive quantities is bounded above by their sum, thus by the many-to-one lemma: \begin{align} \B{E}\left[\max_{j=1}^{\C{N}(1)} \sup_{0 \leq t\leq 1}\|B^i(t)\|^2\right] \leq e\B{E}\left[ \sup_{0\leq t\leq 1}\|B(t)\|^2\right]. \end{align} Then by the reflection principle: \begin{align} \B{E}\left[\sup_{0\leq t\leq 1}\|B(t)\|^2 \right]&=\int_0^\infty \B{P}(\sup_{0\leq t\leq 1}\|B(t)\|^2 > x)dx = 2\int_0^\infty \B{P}(B(1)>\sqrt{x})dx = 2\int_0^\infty \erfc(\sqrt{x})dx<\infty \end{align} Therefore since $\B{E}[(\tau_{i+1}-\tau_i)^2]<\infty$ (because $\tau_{i+1}-\tau_i$ is bounded above by a geometric random variable), thus (\ref{boundforSq}) is finite, and hence (\ref{sqBound}) is finite, as required. Therefore our process $Z$ satisfies conditions (1)-(4) of Proposition 1.4 of \cite{bbb}. Now define $\beta=\B{E}[\tau_2-\tau_1]$ and $\sigma^2=Var(Z_j(\tau_2)-Z_j(\tau_1))$. Fix any $T\in (0,\infty)$. Then by combining the Lemmas 5.2, 5.3 and Theorems 5.1, 5.5 of \cite{bbb}, we can conclude that the following invariance principle holds: $$ \left(m^{-1/2}(Z_j(mt)- mt\B{E}[Z_j(\tau_2)-Z_j(\tau_1)])\right)_{0\leq t\leq T} \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma B(t))_{0\leq t\leq T} $$ in the Skorokhod topology on $\C{D}([0,T],\B{R})$, for standard Brownian motion $B$. Whislt \cite{bbb} states the result only for the space $\C{D}([0,1],\B{R})$; all spaces $\C{D}([0,T],\B{R})$ are essentially analogous, in the sense that the choice of finite interval $[0,T]$ is arbitrary. We now show that $\B{E}[Z_j(\tau_2)-Z_j(\tau_1)]=0$. Suppose for contradiction that $\B{E}[Z_j(\tau_2)-Z_j(\tau_1)]=c\neq 0$. Then $\frac{\B{E}[Z_j(\tau_i)]}{\tau_i}\to \frac{c}{\B{E}[\tau_2-\tau_1]}\neq 0$. This is a contradiction, as $Z$ has asymptotic drift $\lim_{t}Z_j(t)/t=\mu_c^N-\mu_c^N=0$. Thus: $$ \left(m^{-1/2}Z_j(mt)\right)_{0\leq t\leq T} \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma B(t))_{0\leq t\leq T}. $$ Since the map $$\psi_T:(m^{-1/2}Z_j(mt))_{0\leq t\leq T} \mapsto (m^{-1/2}Z_j(mt)g_T(t))_{0\leq t\leq T}$$ is continuous (see the proof of Lemma 3 in Section 16 of \cite{bills}), we can apply the mapping theorem (Theorem 2.7 of \cite{bills}), to get that $$ \psi_T\left((m^{-1/2}Z_j(mt))_{t\geq 0}\right) \xrightarrow[m\to\infty]{d} \psi_T\left((\beta^{-1/2}\sigma B(t))_{t\geq 0}\right), $$ and therefore by Lemma \ref{DinftyConv}, since our choice of $T$ was arbitrary, we have $$ \left(m^{-1/2}Z_j(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma B(t))_{t\geq 0}, $$ as required. \end{proof} \begin{corollary} \label{nbbmInvarCor} $$\left(m^{-1/2}Z(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma B(t)\underline{1})_{t\geq 0},$$ in the Skorokhod topology, where $\underline{1}=(1,\ldots,1)\in \B{R}^N.$ \end{corollary} \begin{proof} Let $\tau_1,\tau_2,\tau_3,\ldots$ be the regeneration times from Proposition \ref{critNBBMInvar}. Note that since $\tau_i \to \infty$ as $i\to \infty$ and $Z_N(\tau_i)-Z_1(\tau_i), i\geq 0$ are i.i.d with finite mean, thus we have $\tau_i^{-1/2}\|Z_N(\tau_i)-Z_1(\tau_i)\|\xrightarrow{\B{P}} 0$. And therefore in the scaling limit, the radius of the particle system goes to zero. Then by Slutsky's theorem, we have \begin{align}\left( m^{-1/2}Z(mt)\right)_{t\geq 0} &=\left( m^{-1/2}Z_1(mt)\underline{1}\right)_{t\geq 0}+\left(0, m^{-1/2}(Z_2(mt)-Z_1(mt)),\ldots,m^{-1/2}(Z_N(mt)-Z_1(mt)\right)_{t\geq 0}\\ &\xrightarrow[m\to\infty]{d}\left(\beta^{-1/2}\sigma B(t)\underline{1}\right)_{t\geq 0}+(0,\ldots,0)=\left(\beta^{-1/2}\sigma B(t)\underline{1}\right)_{t\geq 0} \end{align} \end{proof} Given that $Z$, in the scaling limit, converges to a Brownian motion with zero drift, we may naturally expect the properties of the Brownian motions `null-recurrence' to also hold for the $N$-BBM with critical drift, $Z$. Thus in the next Proposition we will show that the expected hitting time of $0$ by $Z_1$, uniformly over all initial configurations with $Z_1(0)=1$, is infinite. This will also be helpful for a later result. \begin{propn} \label{infHitNBBM} Let $\mathfrak{X}^+_{N,1}$ be the set of configurations of $N$ particles with the leftmost particle at $1$. Let $\hat{\tau}:=\inf\{t\geq 0: Z_1(t) < 0\}$ be the first hitting time of $0$ by $Z_1$. Then $\B{E}[\hat{\tau}\|Z(0)=\rho]=\infty$ for all $\rho \in \mathfrak{X}^+_{N,1}$. \end{propn} \begin{proof} To prove this result we will consider the process $Z_1$ as a \textit{regenerative process} (see Example 1.3 of \cite{fpMomentsPRW}). Note that $Z$ is a Markov process, but $Z_1$ is not. However, if we consider $Z_1$ specifically at the regeneration times $\tau_1,\tau_2,\ldots$ defined in Proposition \ref{finExpNBBM}, then the process $(Z_1(\tau_i))_{i\geq 1}$ is a Markov process. Therefore define for $n\geq 1$ $$\xi_n :=Z_1(\tau_{n+1}) - Z_1(\tau_n)\text{ and }\eta_n:=\inf_{\tau_{n}\leq t< \tau_{n+1}}Z_1(t)-Z_1(\tau_{n}),$$ so that $\inf_{t\geq \tau_1}Z_1(t) = \inf_{n\geq 1}\{Z_1(\tau_1) + \xi_1 + \cdots \xi_{n-1} + \eta_n\}$. The strategy of the proof will be to show that the hitting time of the barrier $n^{3/8}$ by the mean-centred random walk $S_n := \xi_1 + \cdots + \xi_n$ has infinite expectation and that $\eta_n \leq n^{3/8}$ for all $n$ with positive probability. Putting these two results together we will then therefore be able to show that the hitting time of zero by the process $(Z_1(\tau_1)+S_{n-1} + \eta_n)_{n\geq 1}$ has infinite expectation. By the properties described in Proposition \ref{finExpNBBM}, we know that $S_n:=\xi_1 + \xi_2 + \cdots + \xi_n$ is a random walk with $\B{E}\xi_i = 0$ and $\B{E}\xi_i^2 < \infty$. Therefore by Theorem 3.2 of \cite{permantlePeres} we have that \begin{align}\label{permantlePeresIneq} I := \inf_{n\geq 1} \sqrt{n}\B{P}(S_k \geq k^{3/8} \text{ for } 1 \leq k \leq n) >0.\end{align} Now let us turn our attention to the $\eta_i$'s. By the properties which we proved in Proposition \ref{critNBBMInvar}, we know that $\{\eta_i : i\geq 1\}$ is an i.i.d family of positive random variables with $\B{E}[\eta_i^2]<\infty$. In fact, our proof easily extends to showing that $\B{E}[\eta_i^4]<\infty$. In particular, note that $\tau_{i+1}-\tau_i$ is bounded by a geometric random variable which has bounded fourth moment, and $\B{E}[\sup_{0\leq t\leq 1} \|B(t)\|^4] = 2\int_0^\infty \erfc(x^{1/4})dx < \infty$. Then just as we bound $\eta_i^2$ by (\ref{boundforSq}), we can use the Cauchy-Schwarz inequality to bound $\eta_i^4$ by $$(\lceil \tau_{i+1} - \tau_i \rceil + 1)^3\left(\|Z_N(\tau_i) - Z_1(\tau_i)\|^4 + \Omega_1^4 + ... + \Omega_{\lceil \tau_{i+1} - \tau_i \rceil}^4 \right) < \infty.$$ Then as $\B{E}[\eta_i^4] < \infty$, we can prove that $\B{P}(\eta_i \geq -i^{3/8} \; \forall i\geq 1)>0 $. In particular, as the $\eta_i$'s are i.i.d negative random variables, then by Markov's inequality \begin{align} -\log\left(\B{P}(\eta_i \geq - i^{3/8} \; \forall i \geq 1)\right) &= -\log \left(\prod_{i\geq 1}\B{P}(\eta_i \geq -i^{3/8})\right) = - \sum_{i\geq 1}\log\left(\B{P}(\eta_i \geq - i^{3/8})\right) \\ &=-\sum_{i\geq 1}\log \left(1-\B{P}(\eta_i \leq -i^{3/8})\right) = -\sum_{i\geq 1}\log\left(1-\B{P}(\eta_i^4 \geq i^{3/2})\right) \\ &\leq -\sum_{i\geq 1}\log (1-i^{-3/2}\B{E}[\eta_1^4]) \leq -\sum_{i\geq 1} \frac{-i^{-3/2}\B{E}[\eta_1^4]}{1-i^{-3/2}\B{E}[\eta_1^4]} = \sum_{i\geq 1} \frac{\B{E}[\eta_1^4]}{i^{3/2} - \B{E}[\eta_1^4]} < \infty \end{align} and therefore $\B{P}(\eta_i \geq -i^{3/8} \; \forall i\geq 1) > 0$. Finally it will remain to show that $$\B{P}(S_j > j^{3/8} \; \forall j\leq n, \eta_j \geq -j^{3/8} \; \forall j\leq n+1) \geq \B{P}(S_j > j^{3/8} \; \forall j\leq n)\B{P}(\eta_j \geq - j^{3/8} \; \forall j\leq n+1).$$ Note that the events $A=\{S_j > j^{3/8} \; \forall j\leq n\}$ and $B=\{\eta_j \geq -j^{3/8} \; \forall j\leq n\}$ are both increasing in the sense that if $X(t)\oop X'(t)$ for all $t\geq 0$ and the event $A$ (resp. $B$) occurs for the process $X$, then certainly the event $A$ (resp. $B$) occurs for the process $X'$. We prove in Lemma \ref{nbbmAssoc} that for any $T>0$ the $N$-BBM process $(Z(t))_{0\leq t\leq T}$ is associated and therefore by the definition of association of random variables, increasing events are positively correlated and therefore \begin{align} \B{P}(S_j > j^{3/8} \; \forall j\leq n, \eta_j \geq -j^{3/8} \; \forall j\leq n+1) &\geq \B{P}(S_j > j^{3/8} \; \forall j\leq n)\B{P}(\eta_j \geq - j^{3/8} \; \forall j\leq n+1) \\ &\geq \frac{I}{\sqrt{n}}\B{P}(\eta_j \geq -j^{3/8}\forall j\geq 1)\end{align} where the second inequality follows from the fact that $\B{P}(\eta_j \geq -j^{3/8} \forall j\leq n)\geq \B{P}(\eta_j \geq j^{3/8} \forall j\geq 1)$ and (\ref{permantlePeresIneq}). Define $\sigma:=\inf\{n>0: S_{n-1} + \eta_n \leq 0\}$ to be the hitting time of $0$ by the perturbed random walk $(S_n + \eta_n)_{n\geq 1}$. Then: \begin{align} \B{E}\sigma &= \sum_{n\geq 1}\B{P}(\sigma \geq n) = \sum_{n\geq 1}\B{P}(S_{j-1} + \eta_j >0 \; \forall j\leq n) \\ & \geq \sum_{n\geq 1}\B{P}(S_j > j^{3/8} \; \forall j\leq n-1, \eta_j \geq -j^{3/8} \;\forall j\leq n) \\ & \geq \sum_{n\geq 1}\frac{I}{\sqrt{n}}\B{P}(\eta_j \geq -j^{3/8}\; \forall j\geq 1) = \infty \end{align} Now suppose that $Z_1(\tau_1) > 0$. Therefore since $\B{E}\sigma = \infty$, $\B{E}[\tau_{i+1}-\tau_i]\in (0,\infty)$, and $\inf_{t\geq \tau_1}Z_1(t) = \inf_{n\geq 1}Z_1(\tau_1)+S_n + \eta_n$, thus it follows that certainly $\B{E}[\inf\{t\geq \tau_1: X_1(t)<0\}]=\infty$. We can therefore conclude that, since for every $\rho\in \mathfrak{X}_{N,1}^+$ we have $Z_1(\tau_1)>0$ with positive probability, we have that $\B{E}[\hat{\tau}\|Z(0)=\rho]=\infty$ for every initial condition $\rho\in \mathfrak{X}^+_{N,1}$ \end{proof} Next we prove that an $N$-Brownian bees system with positive critical drift asymptotically spends proportion 1 of its time with all particles in $[0,\infty)$. \begin{lemma} \label{LRExcursions} Let $X$ be $N$-Brownian bees with critical positive drift. Then $$ t^{-1}\int_0^t \is_{X_1(u)<0}du \xrightarrow[t\to\infty]{a.s.}0$$ \end{lemma} \begin{proof} Let $S:=\inf\{s\geq 0: X_1(s)\leq 0\}$ be the first time that $X_1(s)$ exits $(0,\infty)$. Then observe that $$t^{-1}\int_0^t \is_{X_1(u)<0}du \leq t^{-1}\int_S^{S+t} \is_{X_1(u)<0}du$$ so we can assume without loss of generality that $X_1(0)\leq 0$. Now consider the following sequence of stopping times. Fix $\tau_0=0$ and for $i\geq 1$, recursively define $\sigma_{i}:=\inf\{t\geq \tau_{i-1}:X_1(t)\geq 1\}$ and $\tau_{i}:=\inf\{t\geq \sigma_i:X_1(t)\leq 0\}$. Define $N_t:=\max\{n:\tau_n \leq t\}$ to be the index of the most recent stopping time $\tau_i$. Note that all the time that $X_1$ spends in $(-\infty,0]$ is contained in the intervals $[\tau_{i-1},\sigma_i]$ for $i\geq 1$. So then \begin{equation}\label{leftExcBound}t^{-1}\int_0^t \is_{X_1(u)<0}du \leq t^{-1}\sum_{i=1}^{N_t+1} (\sigma_i - \tau_{i-1}) = \frac{N_t+1}{t} \times \frac{1}{N_t +1}\sum_{i=1}^{N_t+1}(\sigma_i - \tau_{i-1}).\end{equation} Therefore the strategy of this proof will be to show that $\B{E}_\xi[\tau_i - \sigma_i]=\infty$ uniformly for all configurations of $X(\sigma_i)$ and that $\B{E}_\xi[\sigma_i - \tau_{i-1}]<\infty$ uniformly for all configurations of $X(\tau_{i-1})$. Then by renewal theory, the first term of the right hand side of (\ref{leftExcBound}) converges to zero and the second term converges to a finite limit, so that that we have the desired convergence to zero by the algebra of limits. Recall that $\mathfrak{X}^+_{N,1}$ is the set of configurations of $N$ particles with the leftmost particle at 1. Now, before $X_1$ hits 0, the process $X$ has the same behaviour as an $N$-BBM. Therefore by Proposition \ref{infHitNBBM}, $\B{E}[\tau_i - \sigma_i \|X(\sigma_i)=\xi]=\infty$ for all $\xi \in \mathfrak{X}^+_{N,1}$, which is the set of possible configurations of $X(\sigma_i)$. We now consider the renewal process $N'_t$ with renewal times distributed like $\tau_i-\tau_{i-1}\|X(\sigma_i)=\underline{1}$. Since we can couple so that $\{\tau_i - \tau_{i-1} \| X(\sigma_i)=\underline{1}\} \leq \{\tau_i - \tau_{i-1} \|X(\sigma_i)=\xi\}$ for any $\xi \in \mathfrak{X}^+_{N,1}$, almost surely, therefore we can couple $(N_t)_{t\geq 0}$ and $(N'_t)_{t\geq 0}$ so that $N_t \leq N'_t$ by coupling the inter-renewal times of $N_t, N'_t$. Therefore by standard renewal theory (see Theorem 2.4.7 in \cite{durrett}), $(N_t+1)/t \leq (N'_t+1)/t \overset{\text{a.s.}}{\to} 0$. We now prove that $\frac{1}{N_t+1}\sum_{i=1}^{N_t+1}(\sigma_i - \tau_{i-1}) \to L<\infty$, by showing that $\sup_{\xi\in \mathfrak{X}^+_{N,0}}\B{E}_\xi[\sigma_i - \tau_{i-1}]<\infty$. To do this we will consider the times at which the particle closest to the origin is in $[0,\infty)$. For this part of the proof, it will be preferable to think of the $N$-Brownian bees in terms of their intrinsic labelling; that is, the process $V^N(t)$ instead of the functional $X^N(t)=\Theta^N(V^N(t))$. We will denote by $V$ the $N$-Brownian bees $V^N$ with critical positive drift $\mu_c^N$. Fix $i$ and define the stopping times $\tau'_{i,0},\tau'_{i,1},\tau'_{i,2},\tau'_{i,3},\ldots$ inductively by $\tau_{i,0}' = \tau_i$ and for $j\geq 0$, $$\tau'_{i,j+1}:=\inf\{s > \tau'_{i,j}: V_{\ell_0}(s)\in [0,\infty), s-\tau_i\in \B{N}\},$$ where $\ell_0=\ell_0(s)$ is the index (under the intrinsic labeling) of the particle closest to the origin at time $s$. As in the proof of Propositions \ref{finExpNBBM},\ref{critNBBMInvar}, we will consider our process, here the $N$-Brownian bees, as being embedded in $N$ independent branching Brownian motions. Therefore for $\ell=1,\ldots, N$, let $\C{B}^\ell_j(t)_{0\leq t\leq 1}=\{B^{\ell,m}_j(t):m=1,\ldots,\C{N}^\ell_j(t)\}$ be as in Proposition \ref{finExpNBBM}. $(\C{B}_j^\ell(t))_{0\leq t\leq 1}$ will be the BBM attached, over the interval $[\tau'_{i,j},\tau'_{i,j}+1]$, to the particle which is $\ell$\textsuperscript{th} smallest at time $\tau'_{i,j}$. As before, particles will have 2 types (`alive' and `ghost') and over the interval $[\tau'_{i,j},\tau'_{i,j}+1]$ will behave according to the following dynamics: \begin{itemize} \item At time $\tau'_{i,j}$, all particles have type `alive'. \item When an `alive' particle branches, it branches into 2 `alive' particles, and simultaneously the `alive' particle furthest from the origin (which may be one of the two particles involved in the branching event) is changed from type `alive' to type `ghost'. \item When a `ghost' particle branches, it branches into 2 `ghost' particles. \end{itemize} Then the $N$-Brownian bees systems is described by the set of $N$ `alive' particles. Explicitly, for $t\in [\tau_{i,j}',\tau_{i,j}'+1]$, write $$V(t)=\Psi(V(\tau_{i,j}'),\{(\C{B}^\ell_j(t))_{0\leq t\leq 1}:\ell=1,\ldots,N\})$$ to denote that $V(t)$ is a function of $V(\tau_{i,j}')$ and the $N$ specified and independent BBMs. Recall that at time $\tau'_{i,j}$, the particle closest to zero has index $\ell_0(\tau'_{i,j})$. Then we can now consider the event $E_j=E^{(1)}_j\cap E^{(2)}_j\cap E^{(3)}_j\cap E^{(4)}_j$, where \begin{align} E^{(1)}_j &= \{\C{N}_j^{\ell_0(\tau'_{i,j})}(1/2)=\C{N}_j^{\ell_0(\tau'_{i,j})}(1)=N, \C{N}_j^{\ell}(1)=1\text{ for }\ell\neq \ell_0(\tau'_{i,j})\}\\ E^{(2)}_j &= \{\text{At each branching time }T_1,\ldots T_{N-1}\text{ of }(\C{B}^{\ell_0(\tau'_{i,j})}_j(t))_{0\leq t\leq 1}\text{ we have }\\ &\quad \quad \quad \text{sign}(V_\ell(\tau'_{i,j}))B^{\ell, 1}_j(T_m - \tau'_{i,j})>1\text{ for }\ell\neq \ell_0(\tau'_{i,j})\text{ and }m=1,\ldots,N-1\}\\ E^{(3)}_j &= \{0>B^{\ell_0(\tau'_{i,j}),\ell}_j(T_m-\tau'_{i,j})>-1\text{ and }0>B_j^{\ell_0(\tau'_{i,j}),\ell}(1/2)>-1\text{ for }m=1,\ldots,N-1\\ &\quad \quad \quad \text{ and }\ell=1,\ldots,\C{N}^{\ell_0(\tau'_{i,j})}_j(T_m - \tau'_{i,j})\}\\ E^{(4)}_j &= \{B^{\ell_0(\tau'_{i,j}),\ell}_j(1))>B^{\ell_0(\tau'_{i,j}),\ell}_j(1/2)+2\text{ for }\ell=1,\ldots,N\} \end{align} In laymans terms, this is the event that particle $V_{\ell_0(\tau_{i,j}')}(\tau_{i,j}')$ branches $N-1$ times, all in the interval $[\tau'_{i,j},\tau'_{i,j}+1/2]$ whilst no other particle branches (the event $E^{(1)}_j$). During the interval $[\tau'_{i,j},\tau'_{i,j}+1/2]$, the particle $\ell_0(\tau_{i,j}')$ and all of it's descendants stay within $(V_{\ell_0(\tau_{i,j}')}(\tau'_{i,j})-1,V_{\ell_0(\tau_{i,j}')}(\tau'_{i,j}))\subseteq (-1,\infty)$ at branching events (the event $E^{(3)}_j$), whilst all other particles get further from the origin by distance at least $1$ at branching events (the event $E^{(2)}_j$). This means that at each branching time, the `alive' particle furthest from the origin is not among the descendants of particle $V_{\ell_0(\tau_{i,j}')}$, and therefore at time $\tau'_{i,j}+1/2$, the $N$ `alive' particles in the system are exactly the $N$ descendants of $V_{\ell_0(\tau_{i,j}')}$. Subsequently, the event $E^{(4)}_j$ ensures that all of these `alive' particles move at least $2$ units to the right, so that all the `alive' particles are in $[1,\infty)$. As in our construction in Proposition \ref{critNBBMInvar}, the probability of this event occurring is independent on the initial configuration of the system, therefore $\B{P}(E_j)=q_0>0$ uniformly for all $j$. Finally, we consider the time increments $\tau'_{i,j+1}-\tau'_{i,j}$. We want to prove that $$\B{E}[\tau'_{i,j+1}-\tau'_{i,j}]<\infty.$$ To see this, note that if at time $t_0$, we have $V_{\ell_0(t_0)}(t_0)<0$, then whilst $V_{\ell_0(t_0)}(t)<0$, $V_{\ell_0(t_0)}(t)$ is bounded below by a Brownian motion with drift $\mu_c^N$. This is because particles, under the intrinsic labelling, only ever jump closer to the origin. As a Brownian motion with positive drift exits $(-\infty,0]$ in finite expected time, certainly the expected time it takes until $V_{\ell_0(t)}(t)$ exits $(-\infty,0]$ is also finite. Therefore $\B{E}_\xi[\tau'_{i,j+1}-\tau'_{i,j}]<M<\infty$ uniformly for any starting configuration $V(\tau'_{i,j})=\xi\in \mathfrak{X}^+_{N,0}$. Therefore uniformly over initial configurations $X(\tau_{i})=\xi\in \mathfrak{X}^+_{N,0}$, we have $ \B{E}_\xi[\sigma_{i+1} - \tau_{i}] \leq M\B{E}[Geom(q_0)]<\infty$. Therefore by the strong law of large numbers $$\lim_{t\to\infty}\frac{1}{N_t+1}\sum_{i=1}^{N_t+1}(\sigma_{i+1} - \tau_i)\leq M\B{E}[Geom(q_0)] < \infty \quad a.s.,$$ Hence we can conclude that the right hand side of equation (\ref{leftExcBound}) almost surely converges to $0$ as $t\to\infty$, and thus the claim holds. \end{proof} Next we state a technical result about the almost sure continuity (with respect to Brownian motion) of a transformation of path space. In particular, the transformation is one which takes a path $[0,\infty)\to \B{R}$ and transforms it into a path $[0,\infty)\to [0,\infty)$ by removing the excursions in the negative half-line. Specifically, this is the transformation $g$ defined by: $$g:(Z(t))_{t\geq 0}\mapsto \left( Z(\inf\{s\geq 0: \int_0^s \is_{Z(u)\geq 0}du \geq t\})\right)_{t\geq 0}$$ The motivation behind this technical result is that it will allow us to apply the mapping theorem (Theorem 2.7 in \cite{bills}), which states that if $A_n \to A$ in distribution and the transformation $g$ has a discontinuity set $\C{D}_g$ such that $\B{P}(A\in \C{D}_g)=0$, then $g(A_n)\to g(A)$ in distribution. Since our random variables are cadlag paths, we must prove that our transformation is almost surely continuous under the Skorokhod metric on $\C{D}([0,\infty),\B{R})$. The transformation described above is chosen because it transforms a Brownian motion into a reflected Brownian motion; therefore by the mapping theorem and Proposition \ref{critNBBMInvar}, the continuity result will give that: $$g((m^{-1/2}Z_i(mt))_{t\geq 0}) \xrightarrow[m\to\infty]{d} (\beta^{-1/2}\sigma\|B(t)\|)_{t\geq 0}$$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$ for any $i=1,2,\ldots,N$. Before stating the next proposition, we recall the definition of the Skorokhod metric $d_T$ on the space $\C{D}([0,T],\B{R})$ for $T>0$ and the Skorokhod metric $d_\infty$ on $\C{D}([0,\infty),\B{R})$. \begin{defn} Fix $T>0$ and let $\Lambda_T$ be the set of continuous and strictly increasing bijections $[0,T]\to[0,T]$. Then for $X,Y\in \C{D}([0,T],\B{R})$ \begin{align} d_T(X,Y)&:=\inf_{\lambda \in \Lambda_T}\left\{\sup_{t\in [0,T]}\|t-\lambda(t)\|\vee \sup_{t\in [0,T]}\|X(t)-Y(\lambda(T))\|\right\} \end{align} and for $X,Y\in \C{D}([0,\infty),\B{R})$ \begin{align} d_\infty(X,Y)&:=\sum_{T\in\B{N}}^\infty 2^{-T}(1\wedge d_T(\psi_T(X),\psi_T(Y))). \end{align} \end{defn} We can now define precisely our transformation and state the following proposition \begin{propn} \label{pathSpaceTfm} Let $(W(t))_{t\geq 0}$ be a stochastic process whose sample paths are in $\C{D}([0,\infty),\B{R})$, the space of cadlag paths $[0,\infty)\to \B{R}$. Let $g:\C{D}([0,\infty),\B{R})\to\C{D}([0,\infty),\B{R})$ be the map given by: $$ g:(W(t))_{t\geq 0} \mapsto \left(W(\inf\{s\geq 0: \int_0^s \is_{W(u) \geq 0} du \geq t\})\right)_{t\geq 0}. $$ Let $D_g\in \C{D}([0,\infty),\B{R})$ be the discontinuity set of $g$ under the metric $d_\infty$. Then $\B{P}((B_t)_{t\geq 0} \in D_g)=0$, where $B$ is a standard Brownian motion. \end{propn} The final result we need before we can prove Theorem \ref{mainThem}.2 is the following, which essentially states that changing a stochastic process by an asymptotically small time change doesn't change the scaling limit. In particular: \begin{propn} \label{asympSmallTC} Let $(W(t))_{t\geq 0}$ be a stochastic process such that $(m^{-1/2}W(mt))_{t\geq 0}$ converges in distribution as $m\to\infty$ to $(B(t))_{t\geq 0}$ (resp. $(\|B(t)\|)_{t\geq 0}$) in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$, where $B$ is a standard Brownian motion. Suppose that for every $T>0$ we have $$\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|\xrightarrow[m\to\infty]{\B{P}} 0.$$ Then $\left(m^{-1/2}W(mt+\alpha(mt))\right)_{t\geq 0}\xrightarrow[m\to\infty]{d}(B(t))_{t\geq 0}$ (resp. $(\|B(t)\|)_{t\geq 0}$) in the Skorohod topology on $\C{D}([0,\infty),\B{R})$ \end{propn} The proofs of these two results are technical, and so we postpone them until section \ref{techSec}. We are now ready to prove an invariance principle to a reflected Brownian motion, which was the second regime in Theorem \ref{mainThem}. \begin{theorem} Let $X$ be the Brownian bees system with critical drift. Then we have the invariance principle: $$ \left(m^{-1/2}X_1(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} \left(\beta^{-1/2}\sigma\|B(t)\|\right)_{t\geq 0}, $$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$, for constants $\beta,\sigma$ defined in Proposition \ref{critNBBMInvar}. \end{theorem} \begin{proof} The idea of this proof is the following: since the Brownian bees model with critical drift spends asymptotically zero time to the left of the origin, it can be coupled (up to an asymptotically small time shift) to the image by a transformation similar to $g$ of an $N$-BBM process with critical drift. The invariance principle for the $N$-BBM $Z$, and the mapping theorem with the almost surely continuous map $g$ then yield the conclusion. Recall that $Z$ is the $N$-BBM with killing on the right and critical drift of $\mu_c^N$. Let $\tilde{Z}$ be a time-change of $Z$ in which we remove the excursions where $Z_1<0$; that is $$(\tilde{Z}(t))_{t\geq 0}:=\left(Z(\inf\{s\geq 0: \int_0^s \is_{Z_1(u)\geq 0}du \geq t\})\right)_{t\geq 0}.$$ Let $\tilde{Z}_1$ denote the position of the leftmost particle of $\tilde{Z}$ at time $t$ and note that $\tilde{Z}_1=g(Z_1)$. Therefore by the mapping theorem (Theorem 2.7 in \cite{bills}) and Proposition \ref{pathSpaceTfm} \begin{align}\label{ztildeConv}\lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(mt))_{t\geq 0} =g\left(\lim_{m\to\infty}(m^{-1/2}Z_1(mt))_{t\geq 0}\right)=g((\beta^{-1/2}\sigma B(t))_{t\geq 0}) \overset{d}{=} (\|\beta^{-1/2}\sigma B(t)\|)_{t\geq 0}.\end{align} We will now carefully construct regeneration times of $X$, by defining events $C_k$ for $k\in \B{N}$ in which the leftmost particle of $X$ is positive and, in some suitably controlled way, `regenerates' the system by becoming the parent of every particle. As with the proofs of Propositions \ref{finExpNBBM}, \ref{critNBBMInvar} and Lemma \ref{LRExcursions}, we will describe the regeneration event by considering the $N$-Brownian bees system as being embedded in $N$ independent $N$-BBMs. Therefore for $i=1,\ldots,N$, $k=1,2,\ldots$, and $t\geq 0$, let $\C{B}^i_k(t)=\{B^{i,j}_k(t),j=1,\ldots,\C{N}_k^i(t)\}$ be as in Proposition \ref{LRExcursions}, with the same dynamics of `alive' and `ghost' particles describing the behaviour of $X$; that is, defined so that at any time $t\in [k,k+1]$, $X(t)$ is given by the positions of the $N$ `alive' particles in the system and can be explicitly writen as $$X(t)=\Psi(X(k),\{(\C{B}^i_k(t))_{0\leq t\leq 1}:i=1,\ldots,N\}).$$ Again, let the branching Brownian motion $(\C{B}_k^i(t))_{0\leq t\leq 1}$ drive the particle which is the $i$\textsuperscript{th} largest at time $k$. Now we can describe the `regeneration' events, $C_k$, of the process $X$, in terms of the branching Brownian motions and the position of the leftmost particle. Let $C_k:=C_k^{(1)}\cap C_k^{(2)}\cap C_k^{(3)}\cap C_k^{(4)}\cap C_k^{(5)}\cap C_k^{(6)}$, where \begin{align} C^{(1)}_k &= \{X_1(k-1)\geq 1\}, \\ C^{(2)}_k &= \{\C{N}_{k-1}^1(1)=N\}, \\ C^{(3)}_k &= \{\C{N}_{k-1}^j(1)=1\text{ for }j\neq 1\}, \\ C^{(4)}_k &= \{\text{At each branching time $T_1,\ldots,T_{N-1}$ of $(\C{B}^1_{k-1}(t))_{0\leq t\leq 1}$, we have $B^{1,j}_{k-1}(T_\ell)<0$ for }\\ &\quad \quad \quad \text{for $\ell=1,\ldots,N-1$ and $j=1,\ldots, \C{N}^1_{k-1}(T_\ell)=\ell+1$}\}, \\ C^{(5)}_k &= \{B^{i,1}_{k-1}(T_\ell)>0\text{ for }\ell=1,\ldots,N-1\text{ and }i\neq 1\}, \\ C^{(6)}_k &= \{\min_{1\leq i\leq N}\min_{1\leq j\leq \C{N}^i_{k-1}(1)}\inf_{t\in [0,1]}B^{i,j}_{k-1}(t)>-1/2 \}, \end{align} and define the regeneration times of $X$ as $\tau_0^X:=0$ and subsequently $$\tau_{i+1}^X:=\inf\{k\in \B{N}:k>\tau_i^X,\text{ and }C_k\text{ occurs}\}.$$ In layman's terms, the event $C_k$ is an event in which the leftmost particle of the system, whilst to the right of the origin, becomes ancestor to every particle in the system over $1$ time unit. Specifically, we start with the leftmost particle in $[1,\infty)$ at time $k-1$ (the event $C^{(1)}_k$) and ask that it branches $N-1$ times in $[k-1,k]$ (the event $C^{(2)}_k$) whilst no other particle branches (the event $C^{(3)}_k$). Then the events $C^{(4)}_k$ and $C^{(6)}_k$ ensure that at each branching time $T_\ell$ of the BBM $\C{B}^1_k$, all descendants of the particle $X_1(k-1)$ are in the interval $[X_1(k-1)-1,X_1(k-1)]\subseteq [1/2,\infty)$, and that at all times \textit{all} particles remain in $[1/2,\infty)$. The event $C_k^{(5)}$ ensures that particles not descended from $X_1(k-1)$, at the branching times, are to the right of their initial position. Together, these conditions ensure that at each branching time, the particle furthest from the origin is not a descendant of the leftmost particle $X_1(k-1)$, and so at time $k$, conditional on the event $C_k$, the $N$ particles alive in the system are all descendants of particle $X_1(k-1)$. We now make the following observation, which demonstrates why we defined the event $C_{k}$ as we did. The events $C^{(2)}_k,\ldots,C^{(6)}_k$ are all independent of the initial condition $X(k-1)$ and depend only on the BBMs driving the system. Therefore the function $\is_{C_k}$ is a function only of $X_1(k-1)$ and the BBMs $\C{B}^1_{k-1},\ldots,\C{B}^N_{k-1}$. Note that, as we see in the proof of Proposition \ref{LRExcursions}, the expected time between successive events such that $\{X_1(t)\geq 1\}$ is finite, say with mean bounded above by $M$ uniformly for all configurations. Furthermore, at each time $k\in \B{N}$ such that $\{X_1(k-1)\geq 1\}$, the event $C_k$ occurs with a non-zero probability $p_0>0$, uniformly for all configurations $X(k-1)$. The number of occasions that $X_1(k-1)\geq 1$ until $C_k$ occurs is therefore a geometric random variable with mean $p_0^{-1}$, therefore certainly we have $\B{E}_\xi[\tau_{i+1}^X-\tau_i^X]\leq M\B{E}[Geom(p_0)]$ for all possible configurations $X(\tau_i^X)=\xi$. Given the process $(X(t))_{t\geq 0}$ we will now construct the process $Z$ (and by extension $\tilde{Z}$) from $X$ so that $Z$ is indeed an $N$-BBM and is coupled to $X$. \textbf{Case 1:} If $X_1(t)\geq 0$ for all $t\in [\tau^X_0,\tau^X_1]$, then $X$ behaves exactly as an $N$-BBM process staying always above zero. Therefore define $(Z(t))_{\tau_0^X \leq t\leq \tau_1^X}:=(X(t))_{\tau_0^X \leq t\leq \tau_1^X}$, and $\tau^Z_1:=\tau_1^X$. Therefore certainly $X(\tau_1^X)=\tilde{Z}(\tau_1^X)$ and $Z$ behaves like an $N$-BBM on $[\tau_0^X,\tau_1^X]$ \textbf{Case 2:} If $X_1$ hits $0$, say for the first time at $\theta_1 \in [\tau^X_0,\tau^X_1]$, then define $(Z(t))_{\tau_0^X \leq t\leq \theta_1}:=(X(t))_{\tau_0^X \leq t\leq \theta_1}$. As all particles are above the origin in this interval, $Z$ certainly behaves like an $N$-BBM. Then on $[\theta_1,\tau_1^X-1]$, we can couple $X$ to an $N$-BBM process $Z$ as in Proposition \ref{generalCouple}, so that $Z(\theta_1)=X(\theta_1)$ and $Z(t)\oop X(t)$ for $t\in [\theta_1,\tau^X_1-1]$. Note that by definition of $\tau_1^X$, $X_1$ must stay positive in $[\tau_1^X-1,\tau_1^X]$, so certainly $\theta_1 < \tau_1^X -1$. Now we consider the position $X_1(\tau_1^X-1)$, the position at which $X_1$ begins the next regeneration event. By our coupling (Proposition \ref{generalCouple}), we certainly have $Z_1(\tau^X_1-1)\oop X_1(\tau^X_1-1)$. So now we continue to run the process $Z$ as an independent $N$-BBM until the time $\theta_2$ at which $Z_1(\theta_2)=X_1(\tau_1^X-1)$. Finally, for $t\in [\theta_2,\theta_2+1]$, we define $Z(t)=\Psi(Z(\theta_2),\{(B^i_{\theta_2}(t))_{0\leq t\leq 1}:i=1,\ldots,N\})$, and $$\tau_1^Z=\inf\left\{s\geq 0: \int_0^s\is_{Z_1(u)\geq 0}du \geq \theta_2+1\right\}.$$ By the construction of our event $C_{\tau_1^X}$, all particles of $X$ stay above $0$ in $[\tau_1^X-1,\tau_1^X]$, and therefore all particles of $Z$ stay above $0$ in $[\theta_2,\theta_2+1]$. Therefore $(Z(t))_{t\in (\theta_2,\theta_2+1]}$ is exactly $(\tilde{Z}(t))_{t\in (\tau_1^Z-1,\tau_1^Z]}$, and $Z$ behaves exactly as an $N$-BBM process. First we claim that $\B{E}_\xi\|\tau_1^X - \tau_1^Z\|$ is bounded uniformly for all initial conditions $X(\tau_0^X)=\tilde{Z}(\tau_0^Z)=\xi$. As above, we know that $\B{E}_\xi[\tau_{i+1}^X - \tau_i^X]\leq M\B{E}[Geom(p_0)]$. Then the boundedness of $\B{E}_\xi[\tau_1^Z]$ follows because, in case 2, $\tau_1^Z$ is at most $\theta_1$ plus the hitting time of $X_1(\tau_1^X -1)>0$ by $\tilde{Z}_1$, plus $1$. $[0,X_1(\tau_1^X-1)]$ is a compact interval, and therefore the expected hitting time of $X_1(\tau_1^X-1)$ by $\tilde{Z}_1$ is finite. Therefore $\B{E}_\xi\|\tau_1^X - \tau_1^Z\|$ is uniformly bounded for all $\xi$. Next we claim that $X(\tau_1^X)=\tilde{Z}(\tau_1^Z)$. Since the event $C_{\tau_1^X-1}$ is dependent only on the driving BBMs in $[\tau_1^X-1,\tau_1^X]$ and the position $X_1(\tau_1^X-1)=\tilde{Z}(\tau^Z_1-1)$, therefore our coupling ensures that: $$X(\tau^X_1)=\Psi(X(\tau_1^X-1),\{(\C{B}^i_{\tau_1^X-1}(t))_{0\leq t\leq 1}:i=1,\ldots,N\}) = \Psi(\tilde{Z}(\tau_1^Z-1),\{(\C{B}^i_{\tau_1^Z-1}(t))_{0\leq t\leq 1}:i=1,\ldots,N\}) = \tilde{Z}(\tau^Z_1).$$ Continuing in the same way inductively, we can define stopping times $\tau_i^X$ and $\tau_i^Z$ for $i\geq 2$ so that $i\geq 0$ \begin{align}\label{xzTildeCoupling} X(\tau_i^X)=\tilde{Z}(\tau_i^Z) = \tilde{Z}\left(\tau_i^X + \sum_{j=1}^i ((\tau_j^Z-\tau_{j-1}^Z)-(\tau_j^X-\tau_{j-1}^X))\right). \end{align} and that, whenever $X_1$ exits $(0,\infty)$ in $[\tau_{j-1}^X,\tau_j^X]$, we have $\B{E}_\xi\left[\|(\tau_j^Z-\tau_{j-1}^Z)-(\tau_j^X-\tau_{j-1}^X)\|\right]<E_{\text{bound}}<\infty$ uniformly for all initial conditions $X(\tau_{j-1}^X)=\tilde{Z}(\tau_{j-1}^Z)=\xi$. Now let $\tau^X(mt)$ be the smallest regeneration time $\tau_i^X$ after $mt$, and let $I(mt)$ denote its index. Let $$\alpha(mt):=\sum_{i=1}^{I(mt)}\left((\tau_i^Z-\tau_{i-1}^Z) - (\tau_i^X-\tau_{i-1}^X)\right),$$ so that we can write $X(\tau^X(mt))=\tilde{Z}(\tau^X(mt)+\alpha(mt))$. Fix $T\in (0,\infty)$. Using the same ideas as in the proof of Lemma \ref{LRExcursions}, we will now show that $\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|\xrightarrow[m\to\infty]{\B{P}}0.$ Define $N_{mt}:=\sum_{j=1}^{I(mt)}\is_{(\tau_j^Z-\tau_{j-1}^Z)\neq (\tau_j^X-\tau_{j-1}^X)}$. So $N_{mt}$ counts the number of intervals $[\tau_{i-1}^X,\tau_i^X]$ up to $[\tau^X_{I(mt)-1},\tau^X_{I(mt)}]$ in which $X_1$ exits $(0,\infty)$, and hence in which $(\tau^Z_j-\tau^Z_{j-1})$ and $(\tau^X_j-\tau^X_{j-1})$ are not necessarily equal. Note that at each time $\tau_i^X$, we have $X_1(\tau_i^X)>1/2$, so we can bound $N_{mt}$ above by a renewal process whose inter-arrival times are distributed like the hitting time of $0$ started from the initial configuration $(1/2,\ldots,1/2)$. By Proposition \ref{infHitNBBM}, these inter-arrival times have infinite expectation, so by standard renewal theory, $N_{mt}/mt\xrightarrow[t\to\infty]{a.s.} 0$. Combining this with the fact above that $\B{E}_\xi[\|(\tau_j^Z - \tau_{j-1}^Z) - (\tau_j^X-\tau_{j-1}^X)\|]<E_{\text{bound}}$, the strong law of large numbers gives that $$\lim_{t\to\infty}\frac{1}{N_{mt}}\sum_{i=1}^{I(mt)}\left\|(\tau_i^Z-\tau_{i-1}^Z) - (\tau_i^X-\tau_{i-1}^X)\right\|<E_{\text{bound}}\quad a.s.,$$ Therefore putting these convergence results together \begin{align}\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|\leq \frac{N_{mt}}{m} \times \frac{1}{N_{mt}}\sum_{i=1}^{I(mt)}\left\|(\tau_i^Z-\tau_{i-1}^Z) - (\tau_i^X-\tau_{i-1}^X)\right\| \xrightarrow[t\to\infty]{a.s.} 0,\end{align} and thus, since our choice of $T$ was arbitrary, therefore by Proposition \ref{asympSmallTC}, we have \begin{align} \label{removingAlpha} \lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(\tau^X(mt)+\alpha(mt)))_{t\geq 0}=\lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(\tau^X(mt))_{t\geq 0}.\end{align} Finally, we note that $\tau^X(mt)-mt$ is, in the language or renewal theory, an `excess lifetime process' or `residual waiting time' process, so it follows that for any $T\in (0,\infty)$, we have $$\sup_{0\leq t\leq T}\left\|m^{-1}(\tau^X(mt)-mt)\right\|\xrightarrow[m\to\infty]{\B{P}}0$$ (see Example 4.4.8 in \cite{durrett}). Therefore applying Proposition \ref{asympSmallTC} again, we have \begin{align} \lim_{m\to\infty}(m^{-1/2}X_1(mt))_{t\geq 0}=&\lim_{m\to\infty}(m^{-1/2}X_1(\tau^X(mt)))_{t\geq 0}\quad \quad && \text{(by Proposition \ref{asympSmallTC})}\\ =&\lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(\tau^X(mt)+\alpha(mt)))_{t\geq 0}\quad \quad && \text{(by (\ref{xzTildeCoupling}))} \\ =&\lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(\tau^X(mt)))_{t\geq 0}\quad \quad && \text{(by (\ref{removingAlpha}))}\\ =&\lim_{m\to\infty}(m^{-1/2}\tilde{Z}_1(mt))\quad \quad && \text{(by Proposition \ref{asympSmallTC})}\\ \overset{d}{=}&\left(\beta^{-1/2}\sigma \|B(t)\|\right)_{t\geq 0}\quad \quad && \text{(by (\ref{ztildeConv}))} \end{align} \end{proof} We can now give our proof of Theorem \ref{mainThem}.2, that is, that $(X(t))_{t\geq 0}$ satisfies the invariance principle that $$\left(m^{-1/2}X(mt)\right)_{t\geq 0}\xrightarrow[m\to\infty]{d}\left(\beta^{-1/2}\sigma\|B(t)\|\underline{1}\right)_{t\geq 0},$$ where $B$ is a standard Brownian motion and $\underline{1}=(1,\ldots,1)\in \B{R}^N$. \begin{proof} (of Theorem 1.2) Let $\tau_1^X,\tau_2^X,\tau_3^X,\ldots$ be defined as in the above theorem. Note that since $\tau_i^X > i$, thus $\tau_i^X \to \infty$ as $i\to \infty$. Furthermore, the random variables $X_N(\tau^X_i)-X_1(\tau^X_i), i\geq 0$ are i.i.d with finite mean, thus we have $\tau_i^{-1/2}\|X_N(\tau^X_i)-X_1(\tau^X_i)\|\xrightarrow[i\to\infty]{\B{P}} 0$. And therefore in the scaling limit, the radius of the particle system goes to zero. Then by Slutsky's theorem, we have \begin{align}\left( m^{-1/2}X(mt)\right)_{t\geq 0} &=\left( m^{-1/2}X_1(mt)\underline{1}\right)_{t\geq 0}+\left(0, m^{-1/2}(X_2(mt)-X_1(mt)),\ldots,m^{-1/2}(X_N(mt)-X_1(mt))\right)_{t\geq 0}\\ &\xrightarrow[m\to\infty]{d}\left(\beta^{-1/2}\sigma B(t)\underline{1}\right)_{t\geq 0}+(0,\ldots,0)=\left(\beta^{-1/2}\sigma B(t)\underline{1}\right)_{t\geq 0} \end{align} \end{proof} The coupling of Lemma \ref{generalCouple} also immediately gives proof of the transience of the $N$-Brownian bees system. Recall that for Theorem 1.3, we want to prove the following. \begin{theorem} Let $X^{N, \mu}(t)$ be a $N$-Brownian bees process with drift $\|\mu\| > \mu_c^N$ and initial condition $X^{N,\mu}(0)=\nu$. Then $$\lim_{t \rightarrow \infty} \frac{X^{N, \mu}_1(t)}{t} = \lim_{t \rightarrow \infty} \frac{X^{N, \mu}_N(t)}{t} = \left(\|\mu\| - \mu_c^N\right) \text{sign}(\mu) \text{ a.s.}$$ \end{theorem} The proof of this result is due to Flynn and can be found as Theorem 3.0.2 of \cite{flynn}. We give the argument in full for completeness. \begin{proof} (Due to Flynn, Theorem 3.0.2 of \cite{flynn}) Suppose without loss of generality that $\mu < -\mu_c^N$. Let $T$ be the random time defined by $$T:=\inf\left\{s\geq 0: t\geq s \implies \frac{X_N^{N,\mu}(t)}{t}\leq \frac{\mu + \mu_c^N}{2}<0\right\}$$ This is almost surely finite since, by the coupling of Proposition \ref{generalCouple} $$\liminf_{t \rightarrow \infty} \frac{X^{N, \mu}_N(t)}{t} \leq \lim_{t \rightarrow \infty } \frac{Z^{N, \mu}_N(t)}{t} = \mu + \mu_c^N < 0 .$$ Fix $\epsilon>0$. Since $T$ is almost surely finite, we can choose a $K \in \mathbb{R}$ such that $\mathbb{P}(T \geq K) \leq \epsilon$. Now we construct a process $Y^{N, \mu, K}$ such that $(Y^{N, \mu, K}(t))_{0\leq t\leq K}$ is an $N$-Brownian bees process with drift $\mu$ started from $Y^{N,\mu,K}(0) = X^{N, \mu}(0)$ and $(Y^{N, \mu, K}(t))_{t\geq K}$ is an $N$-BBM process with drift $\mu$ started from $Y^{N,\mu,K}(K)$. $$\lim_{t \rightarrow \infty} \frac{Y^{N, \mu, K}_1(t)}{t} = \lim_{t \rightarrow \infty} \frac{Y^{N, \mu, K}_N(t)}{t} = \mu_c^N + \mu$$ Moreover, by construction of $K$, with probability at least $1- \epsilon$, the processes $Y^{N, \mu, K}$ and $X^{N, \mu}$ will be identical - since if all the particles lie on the left of $0$ then the $N$ Brownian bees moves identically to the $N$-BBM, hence $$\mathbb{P} \left( \lim_{t \rightarrow \infty} \frac{X^{N, \mu}_1(t)}{t} = \lim_{t \rightarrow \infty} \frac{Y^{N, \mu, K}_1(t)}{t}\right) \geq 1 - \epsilon$$ and similarly for $X^{N, \mu}_N(t)$, $Y^{N, \mu, K}_N(t)$. So since $\epsilon$ is arbitrary we deduce that: $$\lim_{t \rightarrow \infty} \frac{X^{N, \mu}_1(t)}{t} = \lim_{t \rightarrow \infty} \frac{X^{N, \mu}_N(t)}{t} = \mu + \mu_c^N \quad a.s.$$ \end{proof} As we noted in the introduction the behaviour of the $N$-Brownian bees system without drift is known when the number of particles tends to $\infty$. In particular, it is known that as $N\to\infty$, the empirical distribution of particles converges to the solution of a free boundary problem given by: \begin{align} \label{beesFBP1} \begin{cases} u_t=\frac{1}{2}u_{xx} + u & x\in(-R_t,R_t), t>0, \\ \int_{-R_t}^{R_t} u(s,t)ds=1 & t>0, \\ u(t,x)=0 & x\notin (R_t,R_t), t>0, \end{cases} \end{align} with an initial condition based on the initial distribution of particles. This result is proven by Berestycki et al. in \cite{afree}, \cite{beesBBNP}. We conjecture that for Brownian bees with sub-critical drift $\mu$ such that $\|\mu\|<\mu_c^N$, the empirical distribution converges to the solution of the FBP: \begin{align} \begin{cases} u_t=\frac{1}{2}u_{xx} - \mu u_x + u & x\in(-R_t,R_t), t>0, \\ \int_{-R_t}^{R_t} u(s,t)ds=1 & t>0, \\ u(t,x)=0 & x\notin (R_t,R_t), t>0. \end{cases} \end{align} The methods of proof used by Berestycki et al. rely on the symmetry of the system, and view it essentially as a system of reflected Brownian motions on $[0,\infty)$. However these methods no longer work when drift is introduced, since $\|B_t + \mu t\|$ does not behave like a reflected Brownian motion and in fact is not even Markovian. Therefore this conjecture remains open. Recall Proposition \ref{pathSpaceTfm}: \begin{propn-non} Let $(W(t))_{t\geq 0}$ be a stochastic process whose sample paths are in $\C{D}([0,\infty),\B{R})$, the space of cadlag paths $[0,\infty)\to \B{R}$. Let $g:\C{D}([0,\infty),\B{R})\to\C{D}([0,\infty),\B{R})$ be the map given by: $$ g:(W(t))_{t\geq 0} \mapsto \left(W(\inf\{s\geq 0: \int_0^s \is_{W(u) \geq 0} du \geq t\})\right)_{t\geq 0}. $$ Let $D_g\in \C{D}([0,\infty),\B{R})$ be the discontinuity set of $g$ under the metric $d_\infty$. Then $\B{P}((B_t)_{t\geq 0} \in D_g)=0$, where $B$ is a standard Brownian motion. \end{propn-non} \begin{proof} Let $Y$ be a path in $\C{D}([0,\infty),\B{R})$ such that $\{t:Y(t)=0\}$ is closed and Lebesgue null, and $Y$ is $\alpha$-H\"{o}lder continuous for $\alpha \in (0,1/2)$. Note that Brownian motion satisfies these properties almost surely. Fix $T\in (0,\infty)$ and $\epsilon>0$. We will first show that given $\delta>0$ sufficiently small, $d_\infty(X,Y)<\delta$ implies that $d_T(g(X),g(Y))<\epsilon$. In a slight abuse of notation, we use $d_T(g(X),g(Y))$ to mean the metric $d_T$ on $\C{D}([0,T],\B{R})$ of the paths $g(X)$ and $g(Y)$ \textit{restricted to the interval} $[0,T]$. Let $\tilde{T}$ be the smallest integer time such that $\int_0^{\tilde{T}}\is_{Y(u)\geq 0}du \geq T+1$. Since $d_\infty(X,Y)<\delta$, we have $d_{\tilde{T}}(X,Y)<2^T\delta=:\tilde{\delta}$. So there exists a continuous and strictly increasing bijection $\lambda:[0,\tilde{T}]\to [0,\tilde{T}]$ such that $\sup_{t\in [0,\tilde{T}]}\|t-\lambda(t)\|<\tilde{\delta}$ and $\sup_{t\in [0,\tilde{T}]}\|X(t)-Y(\lambda(t))\|<\tilde{\delta}$. We will now show that by choosing $\tilde{\delta}$ sufficiently small, we can make $\|\int_0^s \is_{X(u)\geq 0}du-\int_0^s \is_{Y(\lambda(u))\geq 0}du\|$ arbitrarily small for all $s\in [0,\tilde{T}]$. Now note that the zero set of $Y$ in $[0,\tilde{T}]$, $S_0:=\{t\in [0,\tilde{T}]:Y(t)=0\}\subseteq [0,\tilde{T}]$ is closed and bounded (and therefore compact) and Lebesgue null. Therefore the zero set of $Y\circ \lambda$, $S^\lambda_0:=\{t\in[0,\tilde{T}]:Y(\lambda(t))=0\}$, which is the image of $S_0$ under the continuous bijection $\lambda$, is also closed. Now let $A:=[0,\tilde{T}]\setminus S^\lambda_0$ be the complement of $S^\lambda_0$, which is thus open, and hence can be written as a countable union of disjoint intervals, $A=\bigcup_{i=1}^\infty A_i$. Now since $\lambda^{-1}$ is a bijection, $\lambda^{-1}(A_i)$ are intervals contained in $[0,\tilde{T}]\setminus S_0$, the complement of $S_0$. And since $\sup_t\|t-\lambda(t)\| < \tilde{\delta}$, we have that $Leb(\lambda^{-1}(A_i))\in (Leb(A_i)-\tilde{\delta},Leb(A_i)+\tilde{\delta})$. Therefore choosing $k$ sufficiently large, $Leb\left(\cup_{i=1}^k \lambda^{-1}(A_i)\right)>\tilde{T}-\epsilon/4$. Then choosing $\tilde{\delta} < \epsilon/8k$, we have that $Leb\left(\cup_{i=1}^k A_i\right)>Leb\left(\cup_{i=1}^k \lambda^{-1}(A_i)\right)-2k\tilde{\delta} > \tilde{T}-\epsilon/2$. Then given an open interval $A_i=(a_i,b_i)$, define the closed interval $I_i=[a_i+\frac{\epsilon}{4k},b_i-\frac{\epsilon}{4k}]$, so that $\bigcup_{i=1}^k I_i$ is a finite set of closed intervals contained in $A$ such that $Leb\left( \bigcup_{i=1}^k I_i\right)>Leb(\bigcup_{i=1}^k A_i)-2k\frac{\epsilon}{4k} > \tilde{T}-\epsilon$. By the compactness of $\bigcup_{i=1}^k I_i$, the infimum $\inf\{\|Y(\lambda(t))\|:t\in \cup_{i=1}^k I_i\}$ is attained in $\bigcup_{i=1}^k I_i$ and thus is $>0$. Therefore choosing $\tilde{\delta} < \inf\{\|Y(\lambda(t))\|:t\in \cup_{i=1}^k I_i\}$, the condition $\sup_{t\leq \tilde{T}}\|X(t)-Y(\lambda(t))\|<\tilde{\delta}$ ensures that on $\bigcup I_i$ we have either $X(t),Y(\lambda(t))> 0$ or $X(t),Y(\lambda(t))<0$. Therefore for any $s\in [0,\tilde{T}]$ \begin{align}\label{firstLambdaBound} \left\|\int_0^s \is_{X(u)\geq 0}du - \int_0^s \is_{Y(\lambda(u))\geq 0}du\right\| \leq \int_0^s \|\is_{X(t)\geq 0}-\is_{Y(\lambda(t))\geq 0}\|du \leq \int_0^{\tilde{T}} \is_{\{u\notin \bigcup I_i\}}du = \tilde{T}-Leb \left(\bigcup I_i\right) < \epsilon.\end{align} Similarly, since $Y$ is $\alpha$-H\"{o}lder continuous, and thus uniformly continuous on $[0,\tilde{T}]$, thus choosing $\tilde{\delta}$ sufficiently small, we can make $\|Y(u)-Y(\lambda(u))\|$ arbitrarily small, and therefore, as in equation (\ref{firstLambdaBound}), we can make $\int_0^s\|\is_{Y(u)\geq 0}-\is_{Y(\lambda(u))\geq 0}\|du$ arbitrarily small. It then follows by the triangle inequality that for any $s\in [0,\tilde{T}]$ \begin{align} \label{secondLambdaBound} \left\| \int_0^s \is_{X(u)\geq 0}du - \int_0^s \is_{Y(u)\geq 0}du \right\| < \epsilon. \end{align} Now we define a function $\tilde{\lambda}:[0,T]\to \B{R}$ as follows: $$s(t):=\inf\left\{u\geq 0:\int_0^u\is_{X(v)\geq 0}dv\geq t\right\}\quad \text{and}\quad \tilde{\lambda}(t):=\int_0^{\lambda(s(t))}\is_{Y(u)\geq 0}du.$$ Since $\int_0^{\tilde{T}}\is_{X(v)\geq 0}du > T$, we have that $s:[0,T]\to [0,\tilde{T}]$, and so the function $\tilde{\lambda}$ is well defined on $[0,T]$. We can also observe that $\tilde{\lambda}$ is by definition non-decreasing. Now by the triangle inequality \begin{align} \|t-\tilde{\lambda}(t)\| &= \left\| t - \int_0^{\lambda(s(t))}\is_{Y\geq 0}\right\| = \left\| \int_0^{s(t)} \is_{X\geq 0} - \int_0^{s(t)}\is_{Y\geq 0} + \int_0^{s(t)}\is_{Y\geq 0} - \int_0^{\lambda(s(t))}\is_{Y\geq 0} \right\| \notag \\ & \label{thirdLineSko}\leq \left\|\int_0^{s(t)}\is_{X\geq 0}-\int_0^{s(t)}\is_{Y\geq 0}\right\| + \left\|\int_0^{s(t)}\is_{Y\geq 0}-\int_0^{\lambda(s(t))}\is_{Y\geq 0}\right\| \end{align} where we use the fact that by definition $\int_0^{s(t)} \is_{X \geq 0} = t$. Then first term of (\ref{thirdLineSko}) can be made arbitrarily small by (\ref{secondLambdaBound}) and the second term is bounded by $\tilde{\delta}$, therefore we can choose $\tilde{\delta}$ sufficiently small so that $\sup_{t\in [0,T]}\|t-\tilde{\lambda}(t)\|<\epsilon$. Next we prove that for all $t\in [0,T]$, we can make $\|g(X)(t)-g(Y)(\tilde{\lambda}(t))\|$ arbitrarily small by choosing $\delta$ sufficiently small. So fix $t\in [0,T]$, and define $$v(t):=\inf\left\{s\geq 0: \int_0^s \is_{Y\geq 0} \geq \int_0^{\lambda(s(t))} \is_{Y\geq 0}\right\}.$$ Observe that by definition of $g,\lambda, \tilde{\lambda}$ we have $g(X)(t)=X(s(t))$ and $g(Y)(\tilde{\lambda}(t))=Y(v(t))$. We can note also that by definition, $X(s(t))\geq 0$, and if $Y(\lambda(s(t)))\geq 0$, then $v(t)=\lambda(s(t))$. If $Y(\lambda(s(t)))< 0$, then by (\ref{firstLambdaBound}), $\|v(t)-\lambda(s(t))\|<\epsilon$. Therefore as $Y$ is, by assumption, $1/4$-H\"{o}lder continuous, and by definition of $\lambda$, $\|X(s(t))-Y(\lambda(s(t))\|<\epsilon$, we have that: \begin{align} \label{gSmall}\|g(X)(t) - g(Y)(\tilde{\lambda}(t))\| \leq \|X(s(t))-Y(\lambda(s(t))\| + \|Y(\lambda(s(t))-Y(v(t))\|\leq \epsilon + C\|\lambda(s(t))-v(t)\|^{1/4} < \epsilon + C\epsilon^{1/4},\end{align} which can be made arbitrarily small. Hence we can see that $\tilde{\lambda}$ is such that $\sup_{t\in [0,T]}\|t-\tilde{\lambda}(t)\|$ and $\sup_{t\in [0,T]}\|X(t)-Y(\tilde{\lambda}(t))\|$ can be made arbitrarily small. However for Skorokhod continuity, we also require $\tilde{\lambda}$ to be increasing and bijective $[0,T]\to [0,T]$, so it remains to show that we can approximate $\tilde{\lambda}$ by an increasing bijection $[0,T]\to [0,T]$. Since $\tilde{\lambda}:[0,T]\to \B{R}$ is increasing, there are at most countably many discontinuities. Call the points of discontinuity $t_1,t_2,\ldots$ and suppose they have jumps of size $J_i:=\tilde{\lambda}(t_i)-\tilde{\lambda}(t_i-)$. Let $N$ be such that $\sum_{i=1}^{N-1}J_i>1-\epsilon$. Then define $\hat{\lambda}_{N,n}$, $n\geq 0$, to be the function $\tilde{\lambda}$ with the discontinuities at $t_{N+1},t_{N+2},\ldots t_{N+n}$ removed: $$ \hat{\lambda}_{N,n}(t):= \tilde{\lambda}(t) - \sum_{i=1}^n J_{N+i}\is_{t\geq t_{N+i}} \leq \hat{\lambda}_{N,n-1}.$$ Then since $(\hat{\lambda}_{N,n})_{n\in \B{N}}$ is a decreasing sequence of non-decreasing functions which is bounded below by $0$, thus $\hat{\lambda}_N:=\lim_{n\to\infty} \hat{\lambda}_{N,n}$ exists and is non-decreasing with $N$ jump discontinuities at $t_1,t_2,\ldots t_N$. Furthermore, by construction we have that $\sup_{t\in [0,T]} \|\tilde{\lambda}(t)-\hat{\lambda}_N(t)\|\leq \epsilon$. Now consider $\hat{\delta}>0$ taken sufficiently small so that the intervals $[t_i-\hat{\delta},t_i + \hat{\delta}]$, $i=1,\ldots, N$ are pairwise disjoint. Then define $\hat{\lambda}(t)$ to be the function which takes value $\hat{\lambda}_N(t)$ on $[0,T]\setminus\bigcup_{i=1}^N [t_i-\hat{\delta},t_i+\hat{\delta}]$ and which linearly interpolates across each interval. Therefore $\hat{\lambda}$ is continuous, and choosing $\hat{\delta}$ sufficiently small, we can make $\sup_{t\in [0,T]}\|\tilde{\lambda}(t)-\hat{\lambda}(t)\|$ arbitrarily small. Finally, define $\lambda_{\text{cand}}:=(\hat{\lambda}(t)+\epsilon t)/(\hat{\lambda}(T)+T\epsilon)$. So $\lambda_{\text{cand}}$ is continuous and strictly increasing bijection $\lambda_{\text{cand}}:[0,T]\to [0,T]$, and we can choose $\delta$ sufficiently small so that $\sup_{0\leq t\leq T}\|\lambda_{\text{cand}}(t)-\tilde{\lambda}(t)\|<\epsilon$. Therefore by the triangle inequality, we can make $\sup_{t\in [0,T]}\|\lambda_{\text{cand}}(t)-t\|$ arbitrarily small. Moreover, since $g(Y)(t)$ is $1/4$-H\"{o}lder continuous as a function of $t$, thus by the triangle inequality and (\ref{gSmall}): $$\sup_{0\leq t\leq T}\|g(X)(t)-g(Y)(\lambda_{\text{cand}}(t))\|\leq \sup_{0\leq t\leq T}\|g(X)(t)-g(Y)(\tilde{\lambda}(t))\| + \sup_{0\leq t\leq T}\|g(Y)(\tilde{\lambda}(t))-g(Y)(\lambda_{\text{cand}}(t))\|,$$ can be made arbitrarily small. Therefore for $\delta$ sufficiently small, we have that $d_\infty(X,Y)<\delta$ implies $d_T(g(X),g(Y))<\epsilon$. Specifically, say that for each $T\in \B{N}$, $d_\infty(X,Y)<\delta_T \implies d_T(g(X),g(Y))<\epsilon$. Thus for $d_\infty(X,Y)<\min_{T\leq N}\delta_T$, we have: $$ d_\infty(g(X),g(Y))=\sum_{T=1}^\infty 2^{-T}(1\wedge d_T(g(X),g(Y)) < \sum_{T=1}^N 2^{-T}\epsilon + \sum_{T=N+1}^\infty 1 < \epsilon + 2^{-N}. $$ Since $\epsilon,N$ are arbitrary, this can be made arbitrarily small. Hence we can finally conclude that, for all $\epsilon > 0$, there exists $\delta>0$ such that $d_\infty(X,Y)<\delta$ implies that $d_\infty(g(X),g(Y))<\epsilon$. Therefore $g$ is continuous at $Y$. Since Brownian motion almost surely has the properties we asked of $Y$, Brownian motion $B$ is almost surely not in the discontinuity set of the transformation $g$; that is, $\B{P}(B\in D_g)=0$, as required. \end{proof} Recall Proposition \ref{asympSmallTC}: \begin{propn-non} Let $(W(t))_{t\geq 0}$ be a stochastic process such that $(m^{-1/2}W(mt))_{t\geq 0}$ converges in the distribution as $m\to\infty$ to $(B(t))_{t\geq 0}$ (resp. $(\|B(t)\|)_{t\geq 0}$) in the Skorohod topology on $\C{D}([0,\infty),\B{R})$, where $B$ is a standard Brownian motion. Suppose that for every $T>0$ we have $$\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|\xrightarrow[m\to\infty]{\B{P}} 0.$$ Then $\left(m^{-1/2}W(mt+\alpha(mt))\right)_{t\geq 0}\xrightarrow[m\to\infty]{d}(B(t))_{t\geq 0}$ (resp. $(\|B(t)\|)_{t\geq 0}$) in the Skorohod topology on $\C{D}([0,\infty),\B{R})$ \end{propn-non} \begin{proof} We begin by showing that the result holds for convergence in $\C{D}([0,T],\B{R})$. That is, fix $T>0$ and assume that $(m^{-1/2}X(mt))_{0\leq t\leq T} \xrightarrow[m\to\infty]{d}(B(t))_{0\leq t\leq T}$ in the Skorohod topology on $\C{D}([0,T],\B{R})$. In order to prove that $\left(m^{-1/2}X(mt+\alpha(mt))\right)_{0\leq t\leq T}\xrightarrow[m\to\infty]{d}(B(t))_{0\leq t\leq T}$, it suffices to prove that $$(U_m(t))_{0\leq t\leq T}:=\left( m^{-1/2}X(mt+\alpha(mt))-m^{-1/2}X(mt)\right)_{0\leq t\leq T} \xrightarrow[m\to\infty]{\B{P}} 0$$ in the Skorohod topology on $\C{D}([0,T],\B{R})$, since by Slutsky's theorem, if $A_n\xrightarrow{d} A$ and $B_n\xrightarrow{\B{P}}0$, then $A_n + B_n\xrightarrow{d}A$. Therefore we want to prove that $\forall \; \varepsilon > 0$, there exists $M$ such that $m>M$ implies \begin{equation}\label{aimConv}\B{P}\left(\sup_{0\leq t\leq T}\|U_m(t)\|>\varepsilon\right)< \varepsilon.\end{equation} Now define the modulus of continuity of a process $Y(t)$ as $w(Y,\delta):=\underset{\|s-t\|\leq \delta}{\sup}\|Y(s)-Y(t)\|$. Note that $w(Y,\delta)=\underset{\|s-t\|\leq \delta}{\sup}\|Y(s)-Z(s)+Z(s)-Z(t)+Z(t)-Y(t)\|$ so using the triangle inequality, we have that $w(Y,\delta)\leq 2\sup_{0\leq t\leq 1}\|Y(t)-Z(t)\| + w(Z,\delta)$. Therefore by symmetry, it follows that $\|w(Y,\delta)-w(Z,\delta)\|\leq 2\sup_{0\leq t\leq 1}\|Y(t)-Z(t)\|$. Hence $w(Y,\delta)$ is continuous in $Y$ with respect to the Skorohod metric. Therefore: $$\lim_{m\to\infty}w\left((m^{-1/2}X(mt))_{0\leq t\leq T},\delta\right)=w\left(\lim_{m\to\infty}(m^{-1/2}X(mt))_{0\leq t\leq T},\delta\right)\overset{d}{=}w((B(t))_{0\leq t\leq T},\delta).$$ The Portmanteau theorem (see Lemma 2.2 of \cite{asympStats}) and Levy's modulus of continuity theorem then imply that $$\underset{m\to\infty}\limsup \,\B{P}\left(w\left((m^{-1/2}X(mt))_{0\leq t\leq T},\delta\right)\geq \varepsilon\right)\leq \B{P}\left(w\left((B(t))_{0\leq t\leq T},\delta\right)\geq \varepsilon\right)\xrightarrow[\delta\to 0]{}0.$$ Now consider the event $E(m,\delta)$ on which $\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|>\delta$. By definition of modulus of continuity, we have that $\sup_{0\leq t\leq 1}\|U_m(t)\|\leq w\left((m^{-1/2}X(mt))_{0\leq t\leq T},\delta\right)$ on $E(m,\delta)^c$, and therefore: $$ \limsup_{m\to\infty} \, \B{P}\left( \sup_{0\leq t\leq T} \|U_m(t)\| \mathbbm{1}_{E(m,\delta)^c}>\varepsilon\right)\xrightarrow[\delta\to 0]{}0 $$ Alternatively, since $\sup_{0\leq t\leq T}\|m^{-1}\alpha(mt)\|\xrightarrow[m\to\infty]{\B{P}}0$, we have that $\B{P}(E(m,\delta))\to 0$ as $m\to \infty$, and hence $\B{P}\left(\sup_{0\leq t\leq T} \|U_m(t)\|\mathbbm{1}_{E(m,\delta)}>\varepsilon\right)\leq \B{P}\left(E(m,\delta) \right) \xrightarrow[m\to\infty]{}0$. Therefore $$ \lim_{m\to\infty}\B{P}\left( \sup_{0\leq t\leq T} \|U_m(t)\|\mathbbm{1}_{E(m\delta)}>\varepsilon\right) = 0 $$ So combining the convergence on $E(m,\delta)$ and $E(m,\delta)^c$, equation (\ref{aimConv}) is proven as required, and hence by Slutsky's theorem, we have $$\left(m^{-1/2}X(mt+\alpha(mt))\right)_{0\leq t\leq T}\xrightarrow[m\to\infty]{d}(B(t))_{0\leq t\leq T}.$$ Finally, since the convergence holds for all $T$, thus by Lemma 3 of section 16 of \cite{bills}, it holds that $$\left(m^{-1/2}X(mt+\alpha(mt))\right)_{t\geq 0}\xrightarrow[m\to\infty]{d}(B(t))_{t\geq 0}$$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R})$. \end{proof} The aim of this section is to prove the following theorem about the velocity of the $N$-BBM system. \begin{theorem} \label{asympVeloc} $$v_N = \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left( \frac{1}{(\log N)^2}\right).$$ \end{theorem} Our proof will use Theorem 7.6 of \cite{barakErdos} of Mallein and Ramassamy, which we state here for completeness. The theorem gives the asymptotic speed of discrete-time branching random walks with selection ($N$-BRW). In the $N$-BRW, we have $N$ particles and at each discrete time, every particle branches ito a random number of offspring, located at random distances from their parents, and we simultaneously delete all but the $N$ rightmost particles. The offspring is described by a distribution $\C{M}$, which is a random point process. \begin{theorem} \textit{(Theorem 7.6 in \cite{barakErdos})} Let $Z^N$ be an $N$-BRW and let $M\sim \C{M}$ satisfy: \begin{itemize} \item $\kappa(\theta):=\log\B{E}\left(\sum_{m\in M}e^{\theta m}\right) < \infty \quad \forall \theta >0 $ \item $\exists \theta^* > 0$ such that $\theta^* \kappa'(\theta^)=\kappa(\theta^)$ \item $\B{E}\left(\|\max_{m\in M}m\|^2\right)<\infty$ \item $\B{E}\left(\sum_{m\in M}e^{\theta^* m}\left(\log \sum_{m\in M}e^{\theta^* m}\right)^2\right)<\infty$ \end{itemize} Then $v_N:=\lim_{t\to \infty} t^{-1}Z^N_1(t)=\lim_{t\to\infty}t^{-1}Z^N_N(t)$ exists and $v_N=\kappa'(\theta^) + \frac{\pi^2 \theta^ \kappa''(\theta^)}{2(\log N)^2} + o\left((\log N)^{-2}\right)$ \end{theorem} We can thus couple $N$-BBM to appropriately chosen $N$-BRWs to determine the speed of $N$-BBM. \begin{lemma} \label{speedUB} $v_N \leq \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left( \frac{1}{(\log N)^2}\right)$ \end{lemma} \begin{proof} Define $Z^{N,1}(t)$ to be a branching Brownian motion process in which at each integer time we delete all but the $N$ rightmost particles. Therefore $Z^{N,1}$ may have more than $N$ particles at some time $t\in \B{R}\setminus \B{Z}$, but has exactly $N$ particles at each time in $\B{Z}$. By Proposition 3 of \cite{hydroNBBM}, there is a natural coupling between $Z^N$ and $Z^{N,1}$ such that $Z^N(0)\preccurlyeq Z^{N,1}(0) \implies Z^N(t)\preccurlyeq Z^{N,1}(t)$ for all subsequent times $t>0$. Therefore, defining $v_{N,1}:=\lim_k k^{-1}Z_1^{N,1}(k)$, then certainly $v_{N}\leq v_{N,1}$. By construction, taking $Z^{N,1}$ at integer times, $(Z^{N,1}(n))_{n\in \B{N}}$ is an $N$-BRW with reproduction law $\C{M}_1$, say. The law $\C{M}_1$ describes the distribution of particles at time $1$ of a branching Brownian motion process started from a single particle at the origin. Let $M_1$ be a random point process of law $\C{M}_1$. It is easy to check that $\C{M}_1$ satisfies the condition of the above Theorem, and we can calculate $\kappa(\theta) = 1+\theta^2/2$. Therefore the conclusion of the Theorem gives $\theta^=\sqrt{2}$ and $$ v_N \leq v_{N,1} = \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left( \frac{1}{(\log N)^2}\right), $$ as required. \end{proof} For the lower bound, we will define another branching random walk $(Y^{N,\delta}(t))_{t\geq 0}$ with selection at discrete times $(k\delta)_{k\in\B{N}}$. Consider a branching Brownian motion process $\hat{Y}^1$ starting with a single particle at the origin in which we permit at most one branching event. Therefore there at most 2 particles in $\hat{Y}^1$ at any time. Let the distribution of the particles of $\hat{Y}^1(\delta)$ be called $\C{\hat{M}}_\delta$. Now define $(Y^{N,\delta}(n\delta))_{n\in\B{N}}$ to be the $N$-BRW starting with $\lfloor \frac{N}{2} \rfloor$ particles with offspring distribution $\C{\hat{M}}_\delta$ and with selection of the right-most $\lfloor \frac{N}{2} \rfloor$ particles. Then since there are always fewer particles in the process $Y^{N,\delta}$ than in $Z^{N}$, Lemma 7.4 of \cite{barakErdos} proves that there exists a coupling such that $Y^{N,\delta}(0)\oop Z^{N}(0) \implies Y^{N,\delta}(t)\oop Z^N(t)$ for all $t\geq 0$. Denoting the speed of the process $Y^{N,\delta}$ by $\hat{v}_{N,\delta}$, it immediately follows that $\hat{v}_{N,\delta}\leq v_N$ Thus it only remains give the velocity $\hat{v}_{N,\delta}$. \begin{lemma} $v_N \geq \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left( \frac{1}{(\log N)^2}\right)$ \end{lemma} \begin{proof} Let $\hat{M}_\delta$ be a random point process with law $\C{\hat{M}}_\delta$. Conditioning on the branching time, we calculate: \begin{align} \hat{\kappa}_\delta(\theta)&:=\log\B{E}\left(\sum_{m\in \hat{M}_\delta}e^{\theta m} \right) = \log\left( \int_0^\infty \B{E}\left[\sum_{m\in \hat{M}_\delta}e^{\theta m}\middle\| \tau=t\right]e^{-t}dt\right) \\ &=\log \left( \int_0^\delta 2e^{-t}\B{E}[e^{\theta B_\delta}]dt + \int_\delta^\infty e^{-t}\B{E}[e^{\theta B_\delta}]dt\right) \\ &=\frac{\theta^2 \delta}{2} + \log (2-e^{-\delta}) \end{align} where $(B_t)_{t\geq 0}$ is a standard Brownian motion, and therefore $\theta^* = \sqrt{2\delta^{-1}\log(2-e^{-\delta})}$. So the above Theorem gives $$ v_N \geq \hat{v}_{N,\delta} = \sqrt{2\delta^{-1}\log(2-e^{-\delta})} - \frac{\pi^2 \sqrt{2\delta^{-1}\log(2-e^{-\delta})}}{\sqrt{2}(\log N)^2} + o\left(\frac{1}{(\log N)^2}\right) $$ and by L'H\^{o}pital's rule, $\sqrt{2\delta^{-1}\log(2-e^{-\delta})} \to \sqrt{2}$ as $\delta \downarrow 0$, so $v_N \geq \sqrt{2}-\frac{\pi^2}{2(\log N)^2} + o((\log N)^{-2})$ as required. \end{proof} \begin{proof} \textit{(of Theorem \ref{asympVeloc})} Putting together the preceding two lemmas gives the desired result immediately. \end{proof} Results about the $N$-BBM are often easier to prove than results about the $N$-Brownian bees system due to the fact that the $N$-BBM is translation invariant. A number of our proofs will use properties of the $N$-BBM and then construct a coupling between $N$-BBM and $N$-Brownian bees in order to yield the desired results. Therefore in the first section, we introduce a number of couplings which we will need in the proof of Theorem \ref{mainThem}. We begin by defining the $N$-BBM and $N$-Brownian bees processes exactly. In different proofs, it will make sense to either describe the process by $N$ particles with intrinsic labels which do not change, or to label the particles by order, so we give both definitions here. For an $\B{R}^N$-process, define $\Theta^N:\B{R}^N \to \B{R}^N$ to be the function which ranks the components of the vector, breaking ties arbitrarily. So the $N$-particle processes $(A(t))_{t\geq 0}$ and $(\Theta^N(A(t))_{t\geq 0}$ have the same empirical measure but different labellings of components. \vspace{1em} \begin{defn} \label{nbbmDef} (\textbf{$N$-BBM}) Let $(W^N(t))_{t\geq 0}=\left((W^N_1(t))_{t\geq 0},\ldots,(W^N_N(t))_{t\geq 0}\right)$ be the trajectories of $N$ particles on the real line. The particles behave as $N$ independent Brownian motions, and at rate $N$ the leftmost particle in the system jumps to the position of a uniformly randomly chosen particle. At any time $t\geq 0$, let $Z^N(t):=\Theta^N(W^N(t))$; so $Z^N$ is the process with ordered components. In the rest of this paper, we will call an $N$-BBM any cadlag stochastic process which has the same empirical distribution as $W^N$ or $Z^N$ \end{defn} \vspace{1em} \begin{defn} \label{beesDef} (\textbf{$N$-Brownian Bees}) Let $(V^N(t))_{t\geq 0} = \left((V^N_1(t))_{t\geq 0},\ldots,(V^N_N(t))_{t\geq 0}\right)$ be the trajectories of $N$ particles on the real line. The particles behave as $N$ independent Brownian motions, and at rate $N$ the particle furthest from the origin jumps to the position of a uniformly randomly chosen particle. At any time $t\geq 0$, let $X^N(t):=\Theta^N(V^N(t))$; so $X^N$ is the process with ordered components. In the rest of this paper, we will call an $N$-Brownian bees any cadlag stochastic process which has the same empirical distribution as $V^N$ or $X^N$. \end{defn} \paragraph{Remark:} We could equivalently think of the $N$-BBM (resp. $N$-Brownian bees) as being a process in which each particle branches at unit rate and at each branching event the leftmost particle (resp. furthest particle from the origin) is deleted, giving its label to the newly born particle. We also define a notion of `ordering' for particle systems. \vspace{1em} \begin{defn} Let $A=(a_1,\ldots,a_m)\in \B{R}^m$ and $B=(b_1,\ldots, b_n)\in \B{R}^n$. Then $A\oop B$ (we say that ``$A$ lies to the left of $B$'', or ``$A$ is less than $B$ in the sense of $\oop$'') if and only if $$\|\{i:a_i \geq c\}\| \leq \|\{i: b_i \geq c\}\| \; \forall c\in \B{R}.$$ When $m=n$, this means that the empirical cumulative distribution functions are pointwise ordered. \end{defn} Now our first result will prove that both the $N$-BBM and the $N$-Brownian bees can be constructed from i.i.d. Brownian motions, a Poisson point process, and i.i.d. uniform random variables on $\{1,2,\ldots,N\}$. This representation of $N$-BBM and $N$-Brownian bees will then make proving couplings with certain properties very straightforward. The first natural coupling this result gives is that the $N$-BBM process is monotonic as a function of its initial configuration, and the second is that an $N$-Brownian bees process can be coupled to always stay to the left of an $N$-BBM process. \vspace{1em} \begin{propn} \label{generalCouple} Let $B=((B^{j}(t))_{t\geq 0})_{1\leq j\leq N}$ be a family of $N$ i.i.d Brownian motions, $\nu$ be an initial configuration of $N$ particles, $Q=(Q(t))_{t\geq 0}$ be a Poisson point process of intensity $N$, and $I=(I_i)_{i\in \B{N}}$ be a sequence of i.i.d random variables which are uniform on $\{1,2,\ldots,N\}$. Then there exist functions $\Upsilon$ and $\Xi$ with values in $\mathbb R^N$ such that: $$(X^N(t))_{t\geq 0}=\Theta^N\left( \Upsilon\left(\nu,B, Q, I,t\right)_{t\geq 0}\right)$$ is an $N$-Brownian bees process and $$(Z^N(t))_{t\geq 0}=\Theta^N\left( \Xi\left(\nu,B, Q, I,t\right)_{t\geq 0}\right)$$ is an $N$-BBM process. Moreover, given configurations $\nu$ and $\nu'$ such that $\nu \oop \nu'$, we have $$\Upsilon(\nu, B, Q, I,t) \oop \Xi(\nu, B, Q, I,t)\oop \Xi(\nu', B, Q, I,t) \; \forall t\geq 0$$ \end{propn} \begin{proof} We will construct the processes inductively. Let $\tau_0=0$, and recursively, define $$\tau_{i+1}:=\inf\{t> \tau_i:Q(t)\neq Q(t-)\}$$ to be the discontinuities of the Poisson point process $Q$. These times will be the branching times of the processes. For any initial configuration $\eta$, define $$\Upsilon(\eta, B,Q,I,0) = \Xi(\eta, B,Q,I,0)=\eta .$$ Now suppose for induction that $\Upsilon$ and $\Xi$ are defined up to a time $\tau_i$ so that $\Upsilon\left(\nu,B, Q, I,t\right)_{0\leq t\leq \tau_i}$ is an $N$-Brownian bees process and $\Xi\left(\nu,B, Q, I,t\right)_{0\leq t\leq \tau_i}$ and $\Xi\left(\nu',B,Q,I,t\right)_{0\leq t \leq \tau_i}$ are $N$-BBM processes with $$\Upsilon\left(\nu,B, Q, I,t\right)\oop \Xi\left(\nu,B, Q, I,t\right)\oop \Xi\left(\nu',B,Q,I,t\right) \text{ for } t\in [0,\tau_i].$$ We will then define $\Upsilon$ and $\Xi$ on $[\tau_i,\tau_{i+1}]$. Let $\Upsilon_j(\nu, B,Q,I,\tau_i)$ (resp. $\Xi_j(\nu, B,Q,I,\tau_i)$) denote the $j$\textsuperscript{th} smallest particle of $\Upsilon(\nu, B,Q,I,\tau_i)$ (resp. $\Xi(\nu,B,Q,I,\tau_i)$). Then for $t\in [\tau_i,\tau_{i+1})$, we will drive the particles $\Upsilon_j(\nu,B,Q,I,t)$, $\Xi_j(\nu,B,Q,I,t)$, and $\Xi_j(\nu',B,Q,I,t)$ by the same Brownian motion $(B^{j}(t))_{\tau_i \leq t\leq \tau_{i+1}}$. Since $\Upsilon(\nu,B,Q,I,\tau_i)\oop \Xi(\nu,B,Q,I,\tau_i)\oop \Xi(\nu',B,Q,I,\tau_i)$ implies that $\Upsilon_j(\nu,B,Q,I,\tau_i)\leq \Xi_j(\nu,B,Q,I,\tau_i)\leq \Xi_j(\nu',B,Q,I,\tau_i)$ for all $j\in \{1,2,\ldots,N\}$, then certainly: $$\Upsilon_j(\nu,B,Q,I,\tau_i)+B^{j}(t)-B^{j}(\tau_i)\leq \Xi_j(\nu,B,Q,I,\tau_i)+B^{j}(t)-B^{j}(\tau_i)\leq \Xi_j(\nu',B,Q,I,\tau_i)+B^{j}(t)-B^{j}(\tau_i),$$ for all $j\in \{1,2,\ldots,N\}$, and so we have $\Upsilon(\nu,B,Q,I,t)\oop \Xi(\nu,B,Q,I,t)\oop \Xi(\nu',B,Q,I,t)$ for $t\in [\tau_i,\tau_{i+1})$. Now let us define two operators $k,l:\B{R}^N \times \{1,2,\ldots,N\} \to \B{R}^N$. For a vector $v=(v_1,v_2,\ldots,v_N)$ with ordered components $v_1\leq v_2 \leq \cdots \leq v_N$ we define: $$l(v,i)=(v_2,v_3,\ldots,v_{i-1},v_i,v_i,v_{i+1},\ldots,v_N),$$ and $$k(v,i)=\begin{cases} (v_2,v_3,\ldots,v_{i-1},v_i,v_i,v_{i+1},\ldots,v_N) & \text{ if }\|v_1\|\geq \|v_N\| \\ (v_1,v_2,\ldots, v_{i-1},v_i,v_i,v_{i+1},\ldots,v_{N-1}) & \text{ if }\|v_N\| >\|v_1\| \end{cases}$$ In words, if $v$ describes the positions of $n$ particles, then $l(v,i)$ duplicates the $i$\textsuperscript{th} smallest particle of $v$ whilst killing the leftmost particle, and $k(v,i)$ duplicates the $i$\textsuperscript{th} smallest particle of $v$ whilst killing the particle of largest magnitude. Note that by definition, we certainly have that $k(v,i)\oop l(v,i)\oop l(v',i)$ for any $v, v' \in \B{R}^N$ such that $v\oop v'$ and $i\in \{1,2,\ldots,N\}$. Finally, we can define $$\Upsilon(\nu,B,Q,I,\tau_{i+1})=k(\Theta^N(\Upsilon(\nu,B,Q,I,\tau_{i+1}-)),I_{i+1}),$$ and $$\Xi(\nu,B,Q,I,\tau_{i+1})=l(\Theta^N(\Xi(\nu,B,Q,I,\tau_{i+1}-)),I_{i+1}),$$ so that it clearly follows that: $$\Upsilon(\nu,B,Q,I,\tau_{i+1})\oop \Xi(\nu,B,Q,I,\tau_{i+1})\oop \Xi(\nu',B,Q,I,\tau_{i+1}).$$ Hence by induction, this ordering holds for all $t\geq 0$. It is obvious from the construction that $\Upsilon(\nu,B,Q,I,t)_{t\geq 0}$ is indeed an $N$-Brownian bees process and $\Xi(\nu,B,Q,I,t)_{t\geq 0}$ is indeed an $N$-BBM process. \end{proof} Now we note that if we want to consider an $N$-BBM or an $N$-Brownian bees process with added drift of $\mu$, we can simply consider it as $\Upsilon(\nu,B^\mu,Q,I,t)_{t\geq 0}$ or $\Xi(\nu,B^\mu,Q,I,t)_{t\geq 0}$ respectively, where $B^\mu$ is a family of i.i.d Brownian motions each with drift $\mu$. It immediately follows that the coupling still holds. From here on, $Z^{N,\mu}$ will be an $N$-BBM process in which each particle has an additional drift of $\mu\in\B{R}$ on each particle, and similarly $X^{N,\mu}$ will be an $N$-Brownian bees process with an additional drift of $\mu \in \B{R}$ on each particle. The next result, Lemma \ref{absCouple}, gives another coupling between an $N$-Brownian bees process and an $N$-BBM. It essentially says that, on time intervals when no particle hits the origin, we can couple the absolute values of $N$-Brownian bees with $N$-BBM. To prove this, we will give an alternative way to represent the $N$-Brownian bees as a function of an initial configuration, a Poisson point process, $N$ i.i.d Brownian motions, and a sequence of uniform random variables. The ordering of this coupling will break down as soon as a particle hits the origin, but this will not be a problem for how we will use the result. \vspace{1em} \begin{lemma} \label{absCouple} Let $\mu \leq 0$, and let $B$, $\nu$, $Q$, $I$, and $\Xi$ be as in Proposition \ref{generalCouple}. Then there exists a function $\tilde{\Upsilon}$ such that we can write: $$(X^{N,\mu}(t))_{t\geq 0}=\Theta^N\left(\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)_{t\geq 0}\right).$$ so that, if $T:=\inf\{t\geq 0:X_i^{N,\mu}(t)=0 \text{ for some i}\}$ is the first hitting time of the origin by a particle of $X^{N,\mu}$, and $\tilde{\nu}\oop -\|\nu\|$ (where $\|\nu\|$ indicates taking absolute value element-wise), then: $$\Xi(\tilde{\nu},B^\mu,Q,I,t) \oop -\|\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)\|\; \text{ for }t\leq T.$$ \end{lemma} \begin{proof} As with $\Upsilon(\nu,B^\mu,Q,I,t)$, we will define $\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)_{t\geq 0}$ recursively. Let $\tilde{\Upsilon}(\nu,B^\mu,Q,I,0)=\nu$, and suppose for induction that we have constructed $\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)$ up to time $\tau_i$ so that $\Xi(\tilde{\nu},B^\mu,Q,I,t)\oop -\|\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)\|$ for $t\in [0,\tau_i]$. We will now construct the process up to time $\tau_{i+1}$. Similarly to the previous proof, $\tilde{\Upsilon}_j(\nu,B^\mu,Q,I,t)$ denotes the $j$\textsuperscript{th} smallest particle of $\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)$ and define \[S_{i,j} := \text{sign} \left( \tilde{\Upsilon}_j(\nu,B^\mu,Q,I,\tau_i) \right) \in\{-1,+1\}.\] Let us write the Brownian motions with drift $\mu$, $B^\mu$ as $B^{j}(t)+\mu t$ for each $1\leq j\leq N$. Then for $t\in [\tau_i,\tau_{i+1})$, we will drive the particle $\tilde{\Upsilon}_j(\nu,B^\mu,Q,I,\tau_i)$ by the Brownian motion $$ (-S_{i,j}B^{j}(t)+\mu t)_{\tau_i \leq t< \tau_{i+1}\wedge T};$$ that is, for $t \in [\tau_i, \tau_{i+1}\wedge T)$ we define \[ \tilde \Upsilon_j(t) = \tilde \Upsilon_j(\tau_i) - S_{i,j}(B^j(t)-B^j(\tau_i))+\mu (t-\tau_i) \] Then we observe that before time $T$, $\tilde{\Upsilon}_j(t)$ and $\tilde{\Upsilon}_j(\tau_i)$ have the same sign, and therefore \[ - \|\tilde \Upsilon_j(t) \| = -S_{i,j}\tilde{\Upsilon}_j(t) = -\|\tilde \Upsilon_j(\tau_i)\| + (B^{j}(t)-B^j(\tau_i)) - S_{i,j} \mu (t-\tau_i), \quad t \in [\tau_i, \tau_{i+1}\wedge T). \] Since we have $\Xi_j(\tilde{\nu},B^\mu,Q,I,\tau_i)\leq -\|\tilde{\Upsilon}_j(\nu,B^\mu,Q,I,\tau_i)\|$, we must also have that for $t\in [\tau_i,\tau_{i+1}\wedge T)$: \begin{align} \Xi_j(\nu,B^\mu,Q,I,t) &=\Xi_j(\nu,B^\mu,Q,I,\tau_i)+B^{j}(t)-B^{j}(\tau_i)+\mu(t-\tau_i) \\ &\leq -\|\tilde{\Upsilon}_j(\nu,B^\mu , Q,I,\tau_i)\| + B^{j}(t)-B^{j}(\tau_i) - S_{i,j}\mu(t-\tau_i) \\& =-\|\tilde{\Upsilon}_j(\nu,B^\mu , Q,I,t)\| \end{align} where we have used that $\mu\leq 0$ and $S_{i,j}\in \{-1,+1\}$. It remains to describe what happens at the branching times. Recall the branching-selection operators $k,l$ from the previous proof, and let $\tilde{I}_{i+1}$ be the index such that $-\|\tilde{\Upsilon}_{\tilde{I}_{i+1}}(\nu,B^\mu,Q,I,\tau_{i+1})\|$ is the $I_{i+1}$\textsuperscript{th} smallest element of $-\|\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1})\|$. So then if we define: $$\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1})=k\left(\Theta^N\left(\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1}-)\right),\tilde{I}_{i+1}\right),$$ it immediately follows that if $\tau_{i+1}<T$, then \begin{align}\Xi(\tilde{\nu},B^\mu,Q,I,\tau_{i+1})=l\left(\Theta^N\left(\Xi(\tilde{\nu},B^\mu,Q,I,\tau_{i+1}-)\right),I_{i+1}\right) \oop l\left(-\left\|\Theta^N\left(\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1}-)\right)\right\|,I_{i+1}\right)\\ =k\left(\Theta^N\left(\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1}-)\right),\tilde{I}_{i+1}\right)=\tilde{\Upsilon}(\nu,B^\mu,Q,I,\tau_{i+1}). \end{align} Finally, all that remains is to observe that $\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)_{t\geq 0}$ is a cadlag stochastic process with particles moving like Brownian motions with drift $\mu$. Moreover, at rate $N$, a uniformly randomly chosen particle branches, and by the definition of $k$, the particle furthest from the origin is removed. Therefore $\Theta^N(\tilde{\Upsilon}(\nu,B^\mu,Q,I,t)_{t\geq 0}$ is an $N$-Brownian bees process with drift $\mu$. \end{proof} Recall that for the sub-critical case of Theorem \ref{mainThem}, we want to prove the following: \begin{theorem} Let $(X^{N,\mu}(t))_{t\geq 0}$ be a one dimensional Brownian bee system with sub-critical drift $\mu<\mu_c^N$. Then $(X^{N,\mu}(nt_0))_{n\in\B{N}}$ is positive Harris recurrent for any $t_0>0$ and has a unique stationary measure $\pi^{N,\mu}$ on $\B{R}^N$ so that for any $\chi\in\B{R}^N$ $$\lim_{t\to\infty}\sup_{C}\|\B{P}(X^{N,\mu}(t)\in C\|X^{N,\mu}(0)=\chi)-\pi^{N,\mu}(C)\|_{TV}=0.$$ \end{theorem} Therefore let us begin this section by defining positive Harris recurrence. We use Definition (2.2) in Athreya and Ney (\hspace{1sp}\cite{athreyaNey}), which is later used in the context of branching-selection particle systems by Berestycki et al. and Durrett \& Remenik (Proposition 6.5 in \cite{beesBBNP} and Proposition 3.1 in \cite{durrRemenik} respectively). As in \cite{beesBBNP}, this allows us to apply Theorems 4.1 and 6.1 of \cite{athreyaNey} which essentially say that the stochastic process has a unique invariant probability distribution towards which it converges. \begin{defn} A stochastic process $(Y_n)_{n\in\B{N}}$ with state space $\C{Y}$ is called a \textbf{recurrent and strongly aperiodic Harris chain} if $\exists A\subseteq \C{Y}$ such that: \begin{itemize} \item $\exists \varepsilon >0$ and probability measure $q$ on $A$ such that $\B{P}_\xi(Y_1\in C)\geq \varepsilon q(C) \; \; \forall \xi \in A, C\subseteq A$ \item the hitting time of $A$ is almost surely finite for any initial $Y_0 \in \C{Y}$; that is $\B{P}_\xi(\tau_A < \infty)=1 \; \; \forall \xi \in \C{Y}$ where $\tau_A := \inf\{ n\geq 1: Y_n\in A\} $. \end{itemize} Further, we call the proces \textbf{positive Harris recurrent} if $\sup_{\xi \in A}\B{E}_\xi[\tau_A]<\infty $, and \textbf{null Harris recurrent} otherwise. \end{defn} We begin by proving the additional condition for \textit{positive} Harris recurrence. In particular, we will first prove the follow proposition: \begin{propn} \label{finExpNBBM} Let $Z^N$ be an N-BBM system with drift $\mu > -\mu_c^N$. Let $\mathfrak{X}^-_{N,s}$ be the set of initial configurations of $Z^N$ in which the rightmost particle is at position $s\in \B{R}$; that is, $Z_N^N(0)=s$. Let the stopping time $T^\mu_0$ be defined by $$T^\mu_0:=\inf\{t\geq 0:Z^N_N(t)\geq 0\}$$ Then there exist finite constants $a,b$ depending on $\mu$ such that $\sup_{x\in\mathfrak{X}^-_{N,-s}}\B{E}_x[T^\mu_0]\leq a+bs<\infty$. \end{propn} \begin{proof} For the purpose of this proof, it will be helpful to consider, for $k\in \B{N}$, the $N$-BBM process $(Z^N(t))_{k\leq t\leq k+1}$ as a function of $Z^N(k)$ and $N$ independent binary branching Brownian motions (BBMs), $(\C{B}_{k}^1(t))_{0\leq t\leq 1},\ldots$,$(\C{B}_k^N(t))_{0\leq t\leq 1}$. The BBM $(\C{B}_k^i(t))_{0\leq t\leq 1}$ starts from a single particle at the origin and has constant drift $\mu$. We let $\C{N}_k^i(t)$ denote the number of particles in the BBM $\C{B}_k^i$ at time $t$, and label its particles, in the order in which they are born, by $B_k^{i,1}(t),\ldots B^{i,\C{N}_k^i(t)}_k(t)$. We also choose that $(\C{B}_k^i(t))_{0\leq t\leq 1}$ will be the BBM which drives the particle which is $i$\textsuperscript{th} smallest at time $k$, $Z_i^N(k)$ Then given $Z^N(k)$, we can construct $(Z^N(t))_{k\leq t\leq k+1}$ as a subset of $$\bigcup_{i=1}^N \bigcup_{j=1}^{\C{N}_k^i(t)}\{Z_i^N(k)+B^{i,j}_j(t)\}=:U(t)$$ as follows. Particles in $U(t)$ are of two different types, `alive' and `ghost': \begin{itemize} \item At time $k$, all particles are `alive'. \item When an `alive' particle branches, it branches into two `alive' particles, and simultaneously the leftmost `alive' particle (which may be one of the particles involved in the branching event) changes to `ghost' \item When a `ghost' particle branches, it branches into two `ghost' particles. \end{itemize} Then at time $t$, the $N$-BBM $Z^N(t)$ is described by the positions of the $N$ `alive' particles in $U(t)$. For $t\in [k,k+1]$, write \begin{align} Z^N(t)=\Phi\left(Z^N(k),\{(\C{B}^i_k(t))_{0\leq t\leq 1}:i=1,\ldots,N\}\right) \end{align} to see explicitly $Z^N(t)$ as a function of $Z^N(k)$ and $N$ independent BBMs. Now for $k\in \B{N}$, we will consider the following event $A_k := A^{(1)}_k \cap A^{(2)}_k \cap A^{(3)}_k \cap A^{(4)}_k$, where \begin{align} A^{(1)}_k &=\{\C{N}^N_{k-1}(1)=N\}\\ A^{(2)}_k &=\{\C{N}^1_{k-1}(1)=\C{N}^2_{k-1}(1)=\ldots =\C{N}^{N-1}_{k-1}(1)=1\}\\ A^{(3)}_k &=\{\text{At each branching time }T_1,\ldots,T_{N-1}\text{ of }(\C{B}^N_{k-1}(t))_{0\leq t\leq 1}\text{ we have }B^{i,1}_{k-1}(T_j)<0\\ &\quad \quad \quad\text{for }i,j=1,\ldots,N-1\}\\ A^{(4)}_t &=\{B^{N,i}_{k-1}(T_j)>0\text{ and }B^{N,i}_{k-1}(t)<1\text{ for }i=1,\ldots,\C{N}^{N}_{k-1}(T_j),\;j=1,\ldots,N-1\text{ and }t\in[0,1]\} \end{align} and define the sequence of stopping times by $\tau_0=0$ and for $i\geq 1$: $$ \tau_i = \inf\{t\in \B{N}: t> \tau_{i-1},\; A_t\text{ occurs}\} $$ In plain English, these are essentially the integer times when the rightmost particle of the system `regenerates' the system by becoming an ancestor of every particle alive in one unit of time. Conditions $A_k^{(3)},A_k^{(4)}$ ensure that at each branching time, the leftmost `alive' particle is one of the particles $Z^N_i(k-1)+B^{i,1}_{k-1}(T_j)$ for $i=1,\ldots, N-1$ and therefore the $N$ `alive' particles at time $k$ are the $N$ particles of the BBM $\C{B}_{k-1}^{N}$. Therefore, the particles alive at time $\tau_i$ are located at \begin{equation}\label{pclPosns}\{Z^N_N(\tau_i-1)+B^{N,j}_{\tau_i-1}(1), j=1,\ldots,\C{N}^N_{\tau_i-1}(1)\}.\end{equation} This construction of the $N$-BBM process then makes it explicitly clear that the event $A_t$ is independent of $Z^N(t-1)$; that is, we can verify whether $A_t$ occurs given only the driving BBMs. Therefore it follows that $\tau_{i+1}-\tau_i$ are i.i.d., and by definition of $\tau_i$, $\B{E}[\tau_{i+1}-\tau_i]\geq 1$. The definition of $A_t$ also makes it clear that the events $A_1,A_2,A_3,\ldots$ are independent and $\B{P}(A_1)=\B{P}(A_2)=\ldots$, therefore $\inf\{k\in \B{N}:A_k\text{ occurs}\}$ is a geometric random variable which bounds $\tau_1$ from above, therefore $\B{E}[\tau_{i+1}-\tau_i]<\infty$. Furthermore, since the processes $(\C{B}_{\tau_i-1}^{N}(t))_{0\leq t\leq 1}$ are i.i.d, we clearly have that $Z^N(\tau_{i+1})-Z^N(\tau_i)$ are i.i.d for $i\geq 1$ (although $Z^N(\tau_1)-Z^N(\tau_0)$ will depend on the initial condition). Then since $\tau_i\xrightarrow[i\to\infty]{a.s.}\infty$, we have $\lim_{i\to\infty}Z(\tau_i)/\tau_i= \lim_{t\to\infty}Z(t)/t = \mu+\mu_c^N$. So by \cite{janson} (equation (1.5)), we have that there exist finite constants $a',b'\geq 0$ such that $$\B{E}\left[\inf\{i\in \B{N}:Z^N_N(\tau_{i+1})-Z^N_N(\tau_1)\geq R\}\right]\leq a'+b'R.$$ Now observe that $Z^N_N(\tau_1)-Z^N_N(0)$ is stochastically dominated by the maximum of a BBM started from $N$ particles. Also, since $\tau_1$ is bounded above by a geometric random variable, $\B{E}e^{\tau_1} < \infty$, therefore by the many-to-one lemma (see for example Lemma 2.1 in \cite{kim}), we have that $$\B{E}[Z_N^N(\tau_1)-Z^N_N(0)]<N\B{E}[e^{\tau_1}]\B{E}\left[\sup_{t<\tau_1}B_t\right]<\infty.$$ Thus it follows that for any $\xi \in \mathfrak{X}^-_{N,-s}$ we have \begin{align}\B{E}_\xi\left[\inf\{i\in \B{N}:Z_N^N(\tau_{i+1})\geq 0\}\right]&=\B{E}_\xi\left[\inf\{i\in \B{N}:Z^N_N(\tau_{i+1})-Z^N_N(\tau_1)\geq s-Z^N_N(\tau_1)+Z^N_N(0)\}\right]\\ &\leq a' + b'\left(s-\B{E}_\xi[Z^N_N(\tau_1)-Z^N_N(0)]\right).\end{align} The inequality follows by conditioning on $Z^N_N(\tau_1)$, and noting that $Z^N_N(\tau_{i+1})-Z^N_N(\tau_i)$ is dependent on the distribution of particles $Z^N_N(\tau_i)$ as seen from the rightmost particle $Z^N_N(\tau_i)$, but independent of $Z^N_N(\tau_i)$ itself; therefore independently of $Z^N_N(\tau_1)$, $Z^N_N(\tau_{i+1})-Z^N_N(\tau_i)$ are i.i.d. for $i\geq 1$. Thus $$\B{E}_\xi[T_0^\mu] \leq \B{E}[\tau_1-\tau_0]\B{E}_\xi\left[\inf\{i\in \B{N}:Z_N^N(\tau_{i+1})\geq 0\}\right] \leq \B{E}[\tau_1-\tau_0]\left(a' + b'(s-\B{E}_\xi[Z^N_N(\tau_1)-Z^N_N(0)])\right)=: a+bs,$$ thus completing the proof. \end{proof} Note that the restriction to the set $x\in \mathfrak{X}^-_{N,-s}$ is necessary and the result is not true for general starting configurations, since we could start from a position in which the rightmost particle is arbitrarily far from the origin and therefore has arbitrarily large hitting time. We can start with an initial condition in which all particles except for the rightmost are arbitrarily far from the origin, but the proposition is essentially saying that, since the rightmost particle will eventually have $N$ descendants, the arbitrarily far left positions will not ultimately matter. Note also that the drift $-\mu_c^N$ is the critical drift at which the expected time becomes infinite. We are now ready to prove the positive Harris recurrence of the Brownian bees: \begin{theorem} \label{BeesPosRec} Let $X^{N,\mu}$ be an $N$ Brownian bee system with drift $\mu \in(-\mu_c^N,\mu_c^N)$. Then $(X^{N,\mu}(t_0 n))_{n\in \B{N}}$ is positive Harris recurrent for any $t_0 > 0$ \end{theorem} \begin{proof} Assume without loss of generality $\mu_c^N < \mu < 0$. In the case of $\mu=0$, Theorem \ref{mainThem} is proven in \cite{beesBBNP}. Let $A=[-1,1]^N$ and $\tau_A:=\inf\{n\geq 1:X^{N,\mu}(t_0n)\in A\} $ be the hitting time to $A$ by $(X^{N,\mu}(t_0n))_{n\in\B{N}}$. Define $\tilde{X}(t)$ to be the position of the particle of $X^{N,\mu}(t)$ closest to $0$. Let $\tilde{\sigma}_0:=0$ and recursively define $\tilde{\sigma}_{i+1}:=\inf\{n \in \B{N}: n > \tilde{\sigma}_{i}: \tilde{X}(s)=0\text{ for some }s\in [(n-1)t_0,nt_0]\}$. Define $\mathfrak{X}_{N,M}$ to be the set of configurations of $N$ particles with the particle closest to zero in $[-M,M]$. Now our first observation is that as $\tilde{X}(s)=0$ for some $s \in ((\tilde{\sigma}_i-1)t_0,\tilde{\sigma}_i t_0)$, then $X^{N,\mu}((\tilde{\sigma}_i+1) t_0) \in A$ with non-zero probability bounded below by $p_0$. This can happen by the particle closest to zero branching $N-1$ times in $(\tilde{\sigma}_i t_0,(\tilde{\sigma}_i+1)t_0)$ and the Brownian motions driving the particles having sufficiently small displacements. Next we want to control the `bad' events where $\tilde{X}(s)$ hits $0$ in $s \in ((\tilde{\sigma}_i-1)t_0,\tilde{\sigma}_i t_0)$ but we fail to have $X^{N,\mu}((\tilde{\sigma}_i+1)t_0)\in A$. Using crude bounds on Brownian motion $B$, we can see that \begin{align}\B{P}\Big(\|\tilde{X}((\tilde{\sigma}_1+1)t_0)\| &\in (M,M+1]\Big)\leq \B{P}\left(\sup_{t\leq 2t_0}\|B_{2t_0}-B_t\| > M\right) \\ =&\B{P}\left(\sup_{t\leq 2t_0}B_t + \mu_t > M\right)+\B{P}\left(\inf_{t\leq 2t_0}B_t + \mu_t < -M\right)=2\B{P}(\tilde{B}_{2t_0}>M-\mu t_0)+2\B{P}(\tilde{B}_{2t_0}>M+\mu t_0)\end{align} where $\tilde{B}$ is a standard Brownian motion and $M \in \B{N}$. This follows from the fact that $\|\tilde{X}(t)\|$ is bounded above by $\|B(t)\|$, and therefore $\B{P}(\|\tilde{X}\|\in (M,M+1])\leq \B{P}(\|B(t)\|>M)$. The final equality is an immediate application of the reflection principle. Then using the Chernoff bound, we can bound this above by $$ \B{P}\left(\|\tilde{X}((\tilde{\sigma}_1+1)t_0)\| \in (M,M+1]\right)\leq \bar{C}\exp\left(-\frac{(M-\mu t_0)^2}{4t_0}\right) + \bar{C}\exp\left(-\frac{(M+\mu t_0)^2}{4t_0}\right), $$ for some constant $\bar{C}$. Then by Lemma \ref{absCouple}, there is a coupling between drifted $N$-BBM, $Z^{N,\mu}$, and $X^{N,\mu}$ such that before $\tilde{X}(t)$ hits zero, we have $Z^{N,\mu}(0)=-\|X^{N,\mu}(0)\| \implies Z^{N,\mu}(t)\oop -\|X^{N,\mu}(t)\|$. Thus $\tilde{X}(s)$ must hit $0$ before $Z^{N,\mu}_N(s)$ hits $0$, and we can use the coupling of Lemma \ref{absCouple} and Proposition \ref{finExpNBBM} to give the upper bound for $M\in \B{N}$ $$\sup_{x\in \mathfrak{X}_{N,M}}\B{E}_x[\tilde{\sigma}_1]<a + bM \leq C_\mu M,$$ for some constant $C_\mu<\infty$ which depends on $\mu$ and $t_0$. Therefore combining our two bounds, we show that the return time of $\tilde{X}$ to $0$ has finite expectation: $$ \B{E}[\tilde{\sigma}_{i+1}-\tilde{\sigma}_i]\leq \sum_{M=1}^\infty \sup_{x\in\mathfrak{X}_{N,-M-1}} \B{E}_x[\tilde{\sigma}_1]\B{P}(\|\tilde{X}(\tilde{\sigma}_i)\|\in (M,M+1]) =: C < \infty $$ Then noting that at each time $(\tilde{\sigma}_i+1)t_0$, $\tilde{X}((\tilde{\sigma}_i+1)t_0)\in A$ with probability at least $p_0$, and therefore $\sup_{x\in A}\B{E}_x[\tau_A]\leq C\B{E}[Geo(p_0)]<\infty$, where $Geo(p_0)$ is a geometric random variable with parameter $p_0$. Therefore to prove positive Harris recurrence it remains only to show that $\exists \varepsilon>0$ and measure $q$ such that $$\B{P}_a(X^{N,\mu}(t_0)\in S)\geq \varepsilon q(S)\quad \text{for all }a\in A, S\subseteq A.$$ Fix $a=(a_1,...,a_N)\in A$, $S \subseteq A$. Then conditioning on the event that no branching occurs in $[0,t_0]$ (which happens with probability $e^{-t_0N}$), we have: \begin{align} \label{recurrenceLeb} \B{P}(X^N(t_0)\in S&\|X^N(0)=a) \geq e^{-t_0N} \int_S \prod_{i=1}^N(2\pi t_0)^{-1/2}e^{-\frac{(x_i-a_i)^2}{2t_0}}dx_N...dx_1 \\ &\geq e^{-t_0N} (2\pi t_0)^{-N/2}\int_S\prod_{i=1}^Ne^{-1/t_0}dx_N...dx_1 = e^{-t_0N}(2\pi t_0)^{-N/2}e^{-N/t_0}Leb(S) \end{align} So normalising on $A$, which has finite Lebesgue measure, the condition is met. Thus $(X^{N,\mu}(t_0 n))_{n\in\B{N}}$ is positive Harris recurrent for $\mu \in (-\mu_c^N,\mu_c^N)$ \end{proof} \begin{proof} \textit{(Of Theorem \ref{mainThem}.1)} We have proven that for any $t_0>0$ the process $(X^{N,\mu}(nt_0))_{n\in \B{N}}$ is a positive recurrent and strongly aperiodic Harris chain. Therefore by Theorems 4.1 and 6.1 of \cite{athreyaNey}, the conclusion of the Theorem \ref{mainThem}.1 holds. \end{proof} In the early 2000's, the physicists Brunet and Derrida pioneered the study of branching particle systems with selection. In these so called \textit{Brunet-Derrida} or \textit{Branching-Selection} systems, we have a number of particles which can branch and have offspring, and also some \textit{selection mechanism/rule} which removes individuals from the population, keeping the population size constant. The particles removed at branching events are typically the particle of least `fitness' according to some fitness function; for example $f(x)=-\\|x\\|$, in which case it is always the particle furthest from the origin which is removed. We can also remove particles from the system by measuring fitness with respect to the whole system; for example, removing the particle furthest from the barycentre (as in \cite{bbb}). The selection mechanism therefore introduces a `survival-of-the-fittest' phenomenon into the model. A well known branching-selection particle system is the $N$-branching Brownian motion ($N$-BBM) which is studied in \cite{selecJBOT}, \cite{hydroNBBM}, \cite{maillard}. In the $N$-BBM, $N$ particles behave as independent Brownian motions on the real line, each branching into 2 particles at rate 1, and at each branching event, the leftmost particle is removed in order to keep a fixed-size population of $N$ particles. This mechanism introduces a \textit{selective pressure} which `pushes' the cloud of particles to the right. Writing $Z(t)=(Z_1(t),...,Z_N(t))\in\B{R}^N$ for the positions of the particles at time $t$, then we say that $Z$ has velocity $v_N$ if the following limits exist and are equal: $$ v_N = \lim_{t\to\infty}\min_{1\leq i\leq N} \frac{Z_i(t)}{t} = \lim_{t\to\infty}\max_{1\leq i\leq N} \frac{Z_i(t)}{t} $$ It is well known in the literature (see for example Section 1.3 of \cite{maillard} or Remark 1.6 of \cite{shape}) that the velocity of the $N$-BBM is given by: $v_N = \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left(\frac{1}{(\log N)^2}\right).$ This is very similar to B\'{e}rard and Gou\'{e}r\'{e} (\hspace{1sp}\cite{berardGouere}), who prove an analogous result for the $N$-branching random walk. A complete proof for the case of the $N$-BBM is given in the appendix. Another branching-selection particle system which has attracted attention lately is called `Brownian bees', and is studied by Berestycki, Brunet, Nolen, and Pennington in \cite{afree}, \cite{beesBBNP}. In the $N$-Brownian bees particle system, $N$ particles behave as Brownian motions, each branching at rate $1$. At each branching event, we delete the particle which is furthest from the origin. The $N$-Brownian bees system is defined in $\B{R}^d$ for any number of dimensions, $d$, but in this paper we specifically study $d=1$. A number of things are known about the $N$-Brownian bees system. Firstly, the system is positive Harris recurrent (Proposition 6.5 in \cite{beesBBNP}). It is also known that in the large population limit, the empirical distribution of particles converges to the solution of a free boundary problem. The aim of this paper is to study the $N$-Brownian bees system in which each particle also has a drift of $\mu$. Intuitively, when the drift is small, the system shouldn't behave much differently to the $N$-Brownian bees system without drift. Essentially, the selection rule which deletes the furthest particle from the origin outweighs the effect of the drift and succeeds in keeping all particles near to the origin. Alternatively, when the drift is sufficiently large, it will win against the selection mechanism and particles will be drifting off to $\pm\infty$. We will therefore prove that there is a critical drift $\mu_c^N$ such that when $\|\mu\| < \mu_c^N$, the $N$-Brownian bee system is positive Harris recurrent, and when $\|\mu\| > \mu_c^N$, the system is transient in the sense that each particle diverges to $\frac{\mu}{\|\mu\|}\infty$ as $t\to\infty$. Now we can observe that when all $N$ particles of a Brownian bees system lie in $(-\infty,0]$, the system has the same behaviour as $N$-BBM with added drift $\mu$, which has asymptotic velocity $\mu + v_N$. Therefore we should expect the critical drift $\mu_c^N$ to be: $$\mu_c^N = v_N = \sqrt{2} - \frac{\pi^2}{\sqrt{2}(\log N)^2} + o\left(\frac{1}{(\log N)^2}\right)$$ In the critical case where $\|\mu\|=\mu_c^N$, the system becomes null-recurrent and, under a suitable rescaling, the $N$ particles act like a single point mass behaving as a reflected Brownian motion. Our main result is the following theorem: \begin{theorem} \label{mainThem} Let $X^{N,\mu}=(X^{N,\mu}(t))_{t\geq 0}$ be a one dimensional $N$-Brownian bee system with drift $\mu\in \B{R}$, and let $\mu_c^N=v_N$. Then: \begin{enumerate} \item \underline{Sub-Critical Case}: when $\|\mu\| < \mu_c^N$, then for any $t_0>0$, the process $(X^{N,\mu}(nt_0))_{n\in \B{N}}$ is positive Harris recurrent and therefore has a unique stationary probability measure $\pi^{N,\mu}$ on $\B{R}^N$ so that for any $\chi\in \B{R}^N$ \begin{equation} \lim_{t\to\infty}\sup_{C}\|\B{P}_{\chi}(X^{N,\mu}(t)\in C)-\pi^{N,\mu}(C)\|_{TV}=0. \end{equation*} where $\|\cdot\|_{TV}$ represents the total variation norm, the supremum is taken over all measurable sets, and $\B{P}_\chi(\cdot):=\B{P}(\cdot\|X^{N,\mu}(0)=\chi)$. \item \underline{Critical Case}: When $\|\mu \|=\mu_c^N$, there is an invariance principle $$ \left(m^{-1/2}X^{N,\mu}(mt)\right)_{t\geq 0} \xrightarrow[m\to\infty]{d} \left(\frac{\mu}{\|\mu\|}\beta^{-1/2}\sigma \|B(t)\|\underline{1}\right)_{t\geq 0}. $$ in the Skorokhod topology on $\C{D}([0,\infty),\B{R}^N)$, the space of cadlag functions $[0,\infty)\to \B{R}$, where $B$ is a standard Brownian motion, $\underline{1}=(1,\ldots,1)\in \B{R}^N$, and $\beta,\sigma$ are constants which we will define later. \item \underline{Super-Critical Case}: When $\|\mu\|>\mu_c^N$, $X^{N,\mu}$ is transient in the sense that each particle $X_i^{N,\mu}$ diverges to $\frac{\mu}{\|\mu\|}\infty$ as $t\to\infty$, and further $$\lim_{t\to\infty}\frac{X^{N,\mu}_1(t)}{t}=\lim_{t\to\infty}\frac{X^{N,\mu}_N(t)}{t} = \left(\|\mu\| - \mu_c^N\right)\frac{\mu}{\|\mu\|} \quad \text{a.s.}$$ where $X_i^{N,\mu}$ denotes the $i$\textsuperscript{th} smallest particle of $X^{N,\mu}$. \end{enumerate} \end{theorem} The subcritical and supercritical cases of Theorem \ref{mainThem} were also independently proven in the master's thesis of Flynn (\hspace{1sp}\cite{flynn}). As we noted above, the behaviour of the $N$-Brownian bees system without drift is known when the number of particles tends to $\infty$. In particular for $d=1$, it is known that as $N\to\infty$, the empirical distribution of particles converges to the solution of a free boundary problem given by \begin{align} \label{beesFBP1} \begin{cases} u_t=\frac{1}{2}u_{xx} + u & x\in(-R_t,R_t), t>0, \\ \int_{-R_t}^{R_t} u(s,t)ds=1 & t>0, \\ u(x,t)=0 & x\notin (R_t,R_t), t>0, \\ u \text{ continuous}. \end{cases} \end{align} with an initial condition based on the initial distribution of particles. This result is proven by Berestycki et al. in \cite{afree}, \cite{beesBBNP}. We conjecture here that for Brownian bees with sub-critical drift $\mu$, the empirical distribution converges to the solution of the free boundary problem: \begin{align} \begin{cases} u_t=\frac{1}{2}u_{xx} - \mu u_x + u & x\in(-R_t,R_t), t>0, \\ \int_{-R_t}^{R_t} u(s,t)ds=1 & t>0, \\ u(x,t)=0 & x\notin (R_t,R_t), t>0, \\ u \text{ continuous}. \end{cases} \end{align} The methods of proof used by Berestycki et al. rely on the symmetry of the system, and view it essentially as a system of reflected Brownian motions on $[0,\infty)$. However these methods no longer work when drift is introduced, since $\|B_t + \mu t\|$ does not behave like a reflected Brownian motion and in fact is not even Markovian. Therefore this conjecture remains unproven. Another property that we will need in the course of our proof is that $N$-BBM is associated. Association is a property of random variables which can essentially be thought of as meaning that `an FKG-type inequality holds' (see definition 1.1 of \cite{assoc}). \begin{defn} Let $X$ be a random variable taking values in a partially ordered space $\C{E}$ with partial ordering $\leq_o$. A function $f:\C{E}\to \B{R}$ is called \textbf{increasing} if for $e_1,e_2\in \C{E}$, we have $e_1 \leq_o e_2 \implies f(e_1)\leq f(e_2)$. An event $A$ is called increasing if the indicator of $A$ is an increasing function. We call the random variable $X$ \textbf{associated} if we have $\B{E}[f(X)g(X)]\geq \B{E}[f(X)]\B{E}[g(X)]$ for all increasing functions $f,g$. \end{defn} An immediate consequence of the definition of association is that if $h:\C{E}\to \C{E}$ is a monotonic increasing function, then $h(X)$ is also associated. This follows because for any increasing functions $f,g$, the monotonicity of $h$ implies that $f\circ h$ and $ g\circ h$ are also increasing functions, so $\B{E}[f(h(X))g(h(X))]\geq \B{E}[f(h(X))]\B{E}[g(h(X))]$. Associated random variables were first studied by Esary, Proschan, and Walkup in \cite{assoc}, and much has been written about them in the literature since then, and we will show here that, by the construction of the $N$-BBM process, this property holds for the $N$-BBM process as well. In particular, we will prove that: \begin{lemma} \label{nbbmAssoc} Fix $T\in (0,\infty)$. Then the $N$-BBM $(Z(t))_{t\in [0,T]}$, viewed as a $\C{D}([0,T],\B{R}^N)$-valued random variable, is associated for all $T$, under the partial order relation $\leq_o$ where $(X_t)_{t\in [0,T]} \leq_o (Y_t)_{t\in [0,T]} \iff X_t \oop Y_t \; \forall t\in [0,T]$. \end{lemma} \begin{proof} We will prove this fact by using the construction: $$(Z^N(t))_{t\geq 0}=\Theta^N\left( \Xi\left(\nu,B, Q, I,t\right)_{t\geq 0}\right)$$ from Proposition \ref{generalCouple}. The space $\C{D}([0,T],\B{R})$ has a natural partial ordering $\leq_T$ where for $x,y\in \C{D}([0,T],\B{R})$, we have $x\leq_T y$ if and only if $x(t)\leq y(t) \; \forall t\in [0,T]$. It is proven by Alexandre Legrand in \cite{levyAssoc} that any $1$-dimensional L\'{e}vy process when viewed as a $\C{D}([0,T],\B{R})$-valued random variable is associated under the partial ordering $\leq_T$ for any $T$. In particular, this includes the Brownian motions and Poisson process used to construct $Z$ above. The association of Brownian motion was already proven by David Barbato in \cite{barbato} under a different partial ordering, but Legrand's work extends this to the more natural partial ordering $\leq_T$. We also know that a uniform random variable on the set $\{1,2,\ldots,N\}$ is associated; Chebyshev's sum inequality states that for increasing functions $f,g:{1,\ldots,N}\to \B{R}$, $\frac1N \sum_{i=1}^N f(i)g(i) \geq \left(\frac1N \sum_{i=1}^N f(i)\right)\left(\frac1N \sum_{j=1}^N g(j)\right)$. Another important property of associated random variables is that association of random variables is preserved under taking products. In particular, if $E_i$ is an $\C{E}_i$-valued associated random variables with partial ordering $\leq_i$ in probability space $(\Omega_i,\C{F}_i,\B{P}_i)$ for $i\in \Gamma$, then $(E_i)_{i\in \Gamma}$ is an $\bigotimes_{i\in \Gamma}\C{E}_i$-valued associated random variable in the product probability space, with the product partial order $\leq_p$ (ie. $(a_i)_{i\in \Gamma} \leq_p (b_i)_{i\in \Gamma} \iff a_i \leq_i b_i$ for all $i\in \Gamma$). This is Lemma 2.5 in \cite{levyAssoc} for finite $\Gamma$, or Theorem 3 in \cite{barbato} for more general $\Gamma$. Therefore, as in the statement of Proposition \ref{generalCouple}, define a family of i.i.d Brownian motions $B=((B^{j}(t)){t\geq 0})_{1\leq j\leq N}$, a Poisson point process with intensity $N$, $Q=(Q(t))_{t\geq 0}$, and a family $I=(I_i)_{i\in \B{N}}$ of uniform random variables. Then $(B,Q,I)_{t\in [0,T]}=((B^{j}(t))_{1\leq j\leq N,t\in [0,T]},(Q(t))_{t\in [0,T]},(I_i)_{i\in \B{N}})$ is an associated random variable under the product partial ordering. Finally it remains to show that the $N$-BBM $(Z^N(t))_{t\in [0,T]}=\Theta^N(\Xi(\nu,B,Q,I,t))_{t\in [0,T]}$ is a monotonic function of the associated random variable $(B,Q,I)_{t\in [0,T]}$ and therefore is also associated. The monotonicity of $Z^N$ as a function of $B$ is obvious; increasing the Brownian motions driving the particles increases $Z^N$. It is also obvious that increasing $Q$ increases $Z^N$, as an increased $Q$ means that positive jumps of particles happen sooner. Finally to see that $\Xi(\nu,B,Q,I,t)_{t\in [0,T]}$ is also monotonic as a function $I$, we note that clearly by definition $l(v,i)\oop l(v,j)$ for $i\leq j$. Hence the $N$-BBM is a monotonic increasing function of the associated random variable $(B,Q,I)_{t\in [0,T]}$ and hence $(Z^N(t)_{t\in [0,T]}=\Theta^N(\Xi(\nu,B,Q,I,t))_{t\in [0,T]}$ is associated. \end{proof}
2412.04316v1	http://arxiv.org/abs/2412.04316v1	Stealthy Optimal Range-Sensor Placement for Target Localization	\documentclass[letterpaper,10pt,conference]{ieeeconf} \IEEEoverridecommandlockouts \overrideIEEEmargins \usepackage[compress]{cite} \usepackage{amsmath,amssymb,amsfonts} \usepackage{algorithmic} \usepackage{graphicx} \usepackage{textcomp} \usepackage[dvipsnames]{xcolor} \usepackage{comment} \usepackage{soul} \usepackage[hidelinks]{hyperref} \usepackage{enumerate} \usepackage{mathtools} \newcommand{\defeq}{\coloneqq} \usepackage[capitalize]{cleveref} \usepackage[font=footnotesize]{caption} \usepackage{subcaption} \usepackage{multirow} \usepackage{gensymb} \usepackage{optidef} \usepackage{booktabs} \usepackage{commath} \usepackage{soul} \usepackage{pgfplots} \usepgfplotslibrary{colormaps} \usepgfplotslibrary{patchplots} \pgfplotsset{compat=1.16} \usepackage{matlab-prettifier} \usepackage{cuted} \usepackage{flushend} \usepackage{accents} \usepackage{lipsum} \newcommand{\unbar}[1]{\underaccent{\bar}{#1}} \DeclareRobustCommand{\hlyellow}[1]{#1} \DeclareRobustCommand{\hlorange}[1]{#1} \DeclareRobustCommand{\hlgreen}[1]{#1} \DeclareRobustCommand{\hlcyan}[1]{#1} \newcommand{\cross}[1]{{#1}^{\times}} \newcommand{\tr}[1]{{#1}^{\top}} \newcommand{\R}{\mathbb{R}} \renewcommand{\norm}[1]{\\|#1\\|} \newcommand{\normm}[1]{\left\\|#1\right\\|} \newcommand{\bb}[1]{\mathbf{#1}} \newcommand{\bbb}[1]{\boldsymbol{#1}} \renewcommand{\arraystretch}{1.7} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\sat}{sat} \DeclareMathOperator{\sig}{sig} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\argmax}{argmax} \DeclareMathOperator{\argmin}{argmin} \DeclarePairedDelimiter\ceil{\lceil}{\rceil} \DeclarePairedDelimiter\floor{\lfloor}{\rfloor} \newtheorem{theorem}{Theorem} \newtheorem{property}{Property} \newtheorem{lemma}{Lemma} \newtheorem{definition}{Definition} \newtheorem{remark}{Remark} \newtheorem{problem}{Problem} \newtheorem{assumption}{Assumption} \newtheorem{proposition}{Proposition} \newcommand{\LL}[1]{\textcolor{blue}{#1}} \newcommand{\LLL}[1]{\textcolor{red}{#1}} \newcommand{\MHY}[1]{\textcolor{orange}{#1}} \title{\LARGE \bf Stealthy Optimal Range-Sensor Placement for Target Localization } \author{Mohammad Hussein Yoosefian Nooshabadi, Rifat Sipahi, and Laurent Lessard\thanks{Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-23-2-0014. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.}\thanks{All authors are with the Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA.\newline {\tt\footnotesize\{yoosefiannooshabad.m, r.sipahi, l.lessard\}\newline @northeastern.edu}} } \begin{document} \maketitle \thispagestyle{empty} \pagestyle{empty} \begin{abstract} We study a stealthy range-sensor placement problem where a set of range sensors are to be placed with respect to targets to effectively localize them while maintaining a degree of stealthiness from the targets. This is an open and challenging problem since two competing objectives must be balanced: (a) optimally placing the sensors to maximize their ability to localize the targets and (b) minimizing the information the targets gather regarding the sensors. We provide analytical solutions in 2D for the case of any number of sensors that localize two targets. \end{abstract} \section{INTRODUCTION}\label{sec: intro} We consider the problem of optimal sensor placement subject to stealthiness constraints. In this problem we have a network of range-only sensors and another network of stationary \emph{targets} (also equipped with range-only sensors). The goal is to obtain spatial configurations of the sensors that maximize their ability to localize the targets while limiting the targets' ability to localize the sensors. This problem concerns two competing objectives, \hlorange{each of which has been studied extensively.} The first competing objective is \emph{target localization} where the goal is to solely optimize the sensors' localization performance \cite{bishop2010optimality, moreno2013optimal, sadeghi2020optimal}. In \cite{moreno2013optimal} the optimal relative sensor-target configuration is derived for multiple sensors and multiple targets in 2D settings. In \cite{sadeghi2020optimal} the scenario with multiple sensors and a single target is considered where the sensors are constrained to lie inside a connected region. Both aforementioned works characterize localization performance using the Fisher Information Matrix (FIM) \cite[\S2]{barfoot2024state}. Target localization is a special case of \textit{optimal sensor placement} where the goal is to find the optimal location of a network of sensors such that some notion of information they obtain is maximized; \hlyellow{this problem has many applications, including structural health monitoring} \cite{ostachowicz2019optimization} \hlyellow{and experiment design} \cite{zayats2010optimal}. \looseness=-1 The second competing objective is \textit{stealthy sensor placement} where the goal is to place the sensors in spatial configurations such that the localization performance of the targets is limited. Various measures of localization performance for the adversary sensors are introduced in the literature, including entropy \cite{molloy2023smoother}, predictability exponent \cite{xu2022predictability} and FIM \cite{farokhi2020privacy}. Stealthiness has also been studied in the context of mobile sensors for \hlyellow{different applications} such as adversarial search-and-rescue \cite{rahman2022adversar}, information acquisition \cite{schlotfeldt2018adversarial}, pursuit-evasion \cite{chung2011search} and covert surveillance \cite{huang2022decentralized}. The work in \cite{karabag2019least} uses the FIM to make the policy of a single mobile sensor difficult to infer for an adversary Bayesian estimator. In \cite{khojasteh2022location} a sensor equipped with a range sensor is allowed to deviate from its prescribed trajectory to enhance its location secrecy encoded via the FIM. The notion of stealthiness is also related to the topics of \textit{privacy} and \textit{security}, which have \hlyellow{applications in} numerical optimization \cite{farokhi2020privacy}, machine learning \cite{zhang2016dynamic}, \hlyellow{smart grids} \cite{li2018information}, and communication \cite{wang2021physical}. In the present paper, \hlorange{we combine the two aforementioned objectives by optimizing localization performance subject to a stealthiness constraint, quantifying each using a min-max formulation of the FIM. To the best of our knowledge, the present study is the first to consider this combination of objectives.} \hlyellow{Possible applications include cooperative distributed sensing and cyber-physical system security; detecting suspicious activities without alerting the attackers.} The paper is organized as follows. In \cref{sec: problem setup}, we describe our min-max FIM formulation. The general case of arbitrarily many sensors and targets leads to a difficult non-convex problem, thus we focus on more tractable special cases. In \cref{sec: Problem formulation 2t2a} we provide a complete solution in the case of two sensors and two targets in 2D. Next, in \cref{sec:Extension}, we treat the case of arbitrarily many sensors and two targets and provide various analytic performance bounds. In \cref{sec: Conclusion and Future Work}, we conclude and discuss future directions. \section{PROBLEM SETUP} \label{sec: problem setup} Consider a 2D arrangement of $m$ sensors $s_1,\dots,s_m$ and $n$ targets $t_1,\dots,t_n$. A possible configuration of these sensors and targets is depicted in \cref{fig: problemDefinition}. We let $\theta_{ij,k}$ denote the angle between $s_i$ and $s_j$ as viewed from $t_k$. Similarly, we let $\beta_{k\ell,j}$ denote the angle between $t_k$ and $t_\ell$ as viewed from $s_j$. \begin{figure}[b!] \centering \includegraphics{fig_problem_defn} \caption{The problem setup in this paper. A set of $m$ sensors (in green) are to be placed such that their ability to localize a set of $n$ targets (in red) is maximized while the targets' ability to localize the sensors is limited.} \label{fig: problemDefinition} \end{figure} We assume sensors and targets use \emph{range-only sensing} and each \hlorange{measurement is subject to additive zero-mean Gaussian noise with variance $\sigma^2$.} Due to the noise, the spatial configuration of the sensors relative to the targets is critical to effectively fuse sensor measurements for localization purposes. The FIM is broadly used to quantify the quality of localization \cite{martinez2006optimal}; it is the inverse of the covariance matrix of the target's position conditioned on the observed measurements. We use the so-called \emph{D-optimality} criterion, which is the determinant of the FIM. Other FIM-based criteria include the A-optimality or E-optimality \cite[\S7]{boyd2004convex}. Since we are using range-only sensing in a 2D setting, the D-optimality, A-optimality, and E-optimality criteria are equivalent \cite{sadeghi2020optimal}. We denote by $\mathcal{I}_k$ the D-optimality criterion for sensors $s_1,\dots,s_m$ localizing a target $t_k$ using their collective range measurements. Similarly we denote by $\mathcal{J}_i$ the D-optimality criterion for targets $t_1,\dots,t_n$ localizing a sensor $s_i$, where \begin{align}\label{eq: det FIM of target k} \mathcal{I}_k & = \frac{1}{\sigma^4}\sum_{1 \leq i < j \leq m}\mkern-18mu\sin^2{\theta_{ij, k}}, & \mathcal{J}_i & = \frac{1}{\sigma^4}\sum_{1\leq k < \ell \leq n}\mkern-18mu\sin^2{\beta_{k\ell, i}}. \end{align} For a detailed derivation of \eqref{eq: det FIM of target k}, see \cite{martinez2006optimal}. \hlgreen{Assuming Gaussian noise is critical in obtaining {\eqref{eq: det FIM of target k}}. Intuitively, information from two range measurements is maximized when the range vectors are perpendicular and minimized when they are parallel.} Motivated by the goal of obtaining the best possible localization of a set of targets while avoiding detection by those same targets, we formulate our problem as follows. \begin{problem}[\textit{min-max optimal stealthy sensor placement}]\label{problem 1} Given target locations $t_1,\dots,t_n$ find angles $\theta_{ij, k}$ and $\beta_{k\ell, i}$ shown in \cref{fig: problemDefinition} corresponding to a feasible arrangement of the sensors $s_1,\dots,s_m$ such that the minimum information that the sensors obtain about the targets is maximized while the maximum information that the targets obtain about the sensors is less than some prescribed level $\gamma^2$. We call $\gamma$ the \textit{information leakage level}. \end{problem} We can formulate \cref{problem 1} as the optimization problem \begin{align}\label{eq: problem def general} \underset{\theta,\, \beta}{\text{maximize}} \quad & \min_{1 \leq k \leq n} \mathcal{I}_k\\ \text{subject to:} \quad & \max_{1\leq i \leq m} \mathcal{J}_i \leq \gamma^2 \notag\\ & (\theta, \beta) \in \mathcal{F}, \notag \end{align} \hlgreen{where $\mathcal{F}$ is the set of all geometrically feasible $(\theta,\beta)$. This constraint ensures that the spatial arrangement of sensors and targets with angles $\theta$ and $\beta$ is realizable. We compute $\mathcal{F}$ for the special case $m=n=2$ in} \cref{prop:cases}. \begin{remark} Instead of constraining the maximum of $\mathcal{J}_i$ (the most information the targets have about any particular sensor), one could constrain the sum or the product of $\mathcal{J}_i$ or some other norm of the vector $(\mathcal{J}_1,\dots,\mathcal{J}_m)$. It is also possible to apply different norms to the expression of $\mathcal{I}_k$. \end{remark} In the next section, we solve \cref{problem 1} for the special case of $m=2$ sensors and $n=2$ targets. \section{TWO SENSORS AND TWO TARGETS}\label{sec: Problem formulation 2t2a} \hlcyan{Substituting {\eqref{eq: det FIM of target k}} into {\eqref{eq: problem def general}} with $m=n=2$ yields} \begin{subequations}\label{eq: problem def for 2s2t} \begin{align} \underset{\theta_1, \theta_2, \beta_1, \beta_2}{\text{maximize}} \quad & \min \bigl(\sin^2\theta_1, \, \sin^2\theta_2\bigr) \label{opt:obj}\\ \text{subject to:} \quad & \max\bigl(\sin^2\beta_1, \, \sin^2\beta_2\bigr) \leq \gamma^2 \label{opt:gamma}\\ & (\theta_1, \theta_2, \beta_1, \beta_2) \in \mathcal{F}, \label{opt:h} \end{align} \end{subequations} where we have used the simplified notation $\theta_k \defeq \theta_{12,k}$ and similarly for $\beta_k$. \hlorange{We also assumed without loss of generality that $\sigma = 1$.} We analyze \eqref{eq: problem def for 2s2t} in three steps: \paragraph{The objective \eqref{opt:obj}} In epigraph form, the level sets $\min\bigl(\sin^2\theta_1, \, \sin^2\theta_2\bigr) \geq \eta^2$ consist of configurations where $\sin\theta_k \geq \eta$ for $k=1,2$. In other words, \begin{equation}\label{eq:thetarange} \theta_k \in [ \arcsin\eta, \pi-\arcsin\eta ] \qquad\text{for }k=1,2. \end{equation} The set of points $t_k$ for which $\theta_k$ is fixed is the arc of a circle passing through $s_1$ and $s_2$. Therefore, given $\eta$, sublevel sets for all $\theta_k$ in \eqref{eq:thetarange} are the exclusive-OR between two congruent discs whose boundaries intersect at $s_1$ and $s_2$ (as shown in \cref{fig: level sets} Left). The objective is maximized at the value 1 when $\theta_1=\theta_2=\tfrac{\pi}{2}$. This corresponds to the configuration when both circles coincide with the dotted circle and $t_k$ lies on this circle. This is logical since localization error is minimized when the range vectors are orthogonal. \begin{figure}[ht] \centering \includegraphics[page=1]{fig_level_sets} \includegraphics[page=2]{fig_level_sets} \caption{\textbf{Left:} Shaded region that must contain $t_1$ and $t_2$ (relative to $s_1$ and $s_2$) so that the objective \eqref{opt:obj} is at least $\eta^2$. The region is formed by two intersecting circles ($\eta=0.7$ shown). The dotted circle shows $\eta=1$. \textbf{Right:} Shaded region that must contain $s_1$ and $s_2$ (relative to $t_1$ and $t_2$) so that information leakage level is at most $\gamma$ ($\gamma=0.7$ shown).} \label{fig: level sets} \end{figure} \paragraph{Stealthiness constraint \eqref{opt:gamma}} Similar to \eqref{eq:thetarange} we have \begin{equation}\label{eq:betarange} \beta_i \in [0,\arcsin\gamma] \cup [\pi-\arcsin\gamma, \pi] \quad\text{for }i=1,2. \end{equation} The set of admissible $\beta_i$ is shown in \cref{fig: level sets} Right. Intuitively, the targets have a large localization error for a sensor when the targets' range vectors are close to being parallel ($\beta_i$ is close to $0$ or to $\pi$). This splits the feasible set into two disjoint regions: \emph{between} $t_1$ and $t_2$ or \emph{outside}. \paragraph{Feasible configurations \eqref{opt:h}} Considering a quadrilateral whose vertices are formed by the sensors and the targets, there are several cases to consider; depending on whether any interior angles are greater than $\pi$ or whether the quadrilateral is self-intersecting. By inspection we have seven distinctive cases, illustrated in \cref{fig: seven cases}. Each case corresponds to a set of constraints given next in \cref{prop:cases}. \begin{figure} \vspace{2mm} \centering \includegraphics[page=1]{fig_seven_cases}\hfill \includegraphics[page=2]{fig_seven_cases}\hfill \includegraphics[page=3]{fig_seven_cases}\hfill \includegraphics[page=4]{fig_seven_cases}\hfill \includegraphics[page=5]{fig_seven_cases}\hfill \includegraphics[page=6]{fig_seven_cases}\hfill \includegraphics[page=7]{fig_seven_cases} \caption{ The seven possible configurations for two sensors and two targets. Each case is characterized by different constraints relating $\theta_1$, $\theta_2$, $\beta_1$ and $\beta_2$ which are given in \cref{prop:cases}. If sensors $s_1$ and $s_2$ are interchangeable then $\mathcal{C}_4 = \mathcal{C}_5$ and $\mathcal{C}_6 = \mathcal{C}_7$ and there are only five distinct cases. } \vspace{-2mm} \label{fig: seven cases} \end{figure} \begin{proposition}\label{prop:cases} The feasible configurations $\mathcal{F}$ in \eqref{opt:h} is the union of the seven cases shown in \cref{fig: seven cases}. In other words, \begin{equation} \mathcal{F} = \biggl\{(\theta_1, \theta_2, \beta_1, \beta_2) \in [0,\pi]^4 \;\bigg\|\; \bigcup_{i=1}^7 \mathcal{C}_i\biggr\}, \end{equation} where $\mathcal{C}_i$ for $i=1,\dots,7$ is the constraint set corresponding to the $i\textsuperscript{th}$ case shown in \cref{fig: seven cases} and expressed below: \begin{subequations} \begin{align} \mathcal{C}_1 &: &&\theta_1 + \theta_2 + \beta_1 + \beta_2 = 2\pi\\ \mathcal{C}_2 &: &-&\theta_1 + \theta_2 + \beta_1 + \beta_2 = 0,\;\; \theta_2 \leq \theta_1\\ \mathcal{C}_3 &: &&\theta_1 - \theta_2 + \beta_1 + \beta_2 = 0,\;\; \theta_1 \leq \theta_2\\ \mathcal{C}_4 &: &&\theta_1 + \theta_2 - \beta_1 + \beta_2 = 0,\;\; \beta_2 \leq \beta_1\\ \mathcal{C}_5 &: &&\theta_1 + \theta_2 + \beta_1 - \beta_2 = 0,\;\; \beta_1 \leq \beta_2\\ \mathcal{C}_6 &: &&\theta_1 - \theta_2 + \beta_1 - \beta_2 = 0,\;\; \theta_1 + \beta_1 \leq \pi \\ \mathcal{C}_7 &: &&\theta_1 - \theta_2 - \beta_1 + \beta_2 = 0,\;\; \theta_1 + \beta_2 \leq \pi \end{align} \end{subequations} If the sensors $s_1$ and $s_2$ are interchangeable (e.g., if there are no additional constraints that distinguish $s_1$ from $s_2$) then swapping $\beta_1$ and $\beta_2$ leads to $\mathcal{C}_4 = \mathcal{C}_5$ and $\mathcal{C}_6 = \mathcal{C}_7$. \end{proposition} \medskip The following result is useful in the sequel and is a straightforward consequence of the constraint equations $\mathcal{C}_i$. \medskip \begin{lemma}\label{lem:C1} Suppose $(\theta_1, \theta_2, \beta_1, \beta_2) \in \mathcal{F}$, where $\mathcal{F}$ is defined in \cref{prop:cases}. Then $\theta_1 + \theta_2 + \beta_1 + \beta_2 \leq 2\pi$, where equality is achievable in a non-degenerate configuration (no sensor is placed arbitrarily close to a target) if and only if $(\theta_1, \theta_2, \beta_1, \beta_2) \in \mathcal{C}_1$. \end{lemma} \medskip We provide the analytical results in two theorems. Our first theorem considers the unconstrained case where the sensors may be placed anywhere in relation to the targets. \begin{theorem}\label{theorem 1} Consider the optimization problem \eqref{eq: problem def for 2s2t}. If the sensors $s_1$ and $s_2$ can be freely placed anywhere then a configuration of sensors is globally optimal if and only if: \begin{enumerate}[(i)] \item $s_1$, $s_2$, $t_1$, $t_2$ are \emph{cyclic} (they lie on a common circle), \item $s_1$ and $s_2$ are diametrically opposed on this circle, and \item The common circle has diameter at least $d/\gamma$; where $d$ is the distance between $t_1$ and $t_2$. \end{enumerate} Moreover, any such configuration satisfies $\theta_1 = \theta_2 = \frac{\pi}{2}$ and $\sin\beta_1 = \sin\beta_2$ and has an optimal objective value of $1$. \end{theorem} \begin{proof} The objective \eqref{opt:obj} is upper-bounded by $1$, which is achieved if and only if $\theta_1=\theta_2=\frac{\pi}{2}$. This is possible if and only if $s_1$, $s_2$, $t_1$ and $t_2$ are on a common circle with diameter $\abs{s_1s_2}$. If $s_1$ and $s_2$ lie on alternating sides of $t_1$ and $t_2$, we recover $\mathcal{C}_1$ from \cref{fig: seven cases}. Otherwise, we recover cases $\mathcal{C}_6$ or $\mathcal{C}_7$. Given such a configuration where the common circle has diameter $D$ apply the Law of Sines to $\triangle t_1 t_2 s_i$ and obtain $d/\sin\beta_i = D$. Now \eqref{opt:gamma} is equivalent to $\sin\beta_i \leq \gamma$ and therefore $D \geq d/\gamma$. \end{proof} Examples of optimal configurations proved in Theorem \ref{theorem 1} for the cases $\mathcal{C}_1$ and $\mathcal{C}_6$ or $\mathcal{C}_7$ are illustrated in \cref{fig: 2t2s Theorem1}. \begin{figure}[ht] \centering \includegraphics[page=1]{fig_thm1} \includegraphics[page=2]{fig_thm1} \caption{Examples of optimal sensor configurations when positions are unconstrained (\cref{theorem 1}). The solid circles delineate the feasible set (see \cref{fig: level sets}). The dotted circle is any larger circle passing through $t_1$ and $t_2$. A configuration is optimal if and only if the sensors $s_1$ and $s_2$ lie on this larger circle and are diametrically opposed. This can happen with alternating sensors and targets (left) or with both sensors on the same side (right).} \label{fig: 2t2s Theorem1} \end{figure} Recall that the feasible set is split into two disjoint regions (see \cref{fig: level sets} Right). \cref{theorem 1} confirms that optimal configurations exist with one sensor between $t_1$ and $t_2$ and one outside or with both sensors outside (see \cref{fig: 2t2s Theorem1}). We next investigate the scenarios where \emph{both} sensors are between $t_1$ and $t_2$; that is $\beta_i \in [\pi-\arcsin\gamma,\pi]$ for $i=1,2$. \begin{theorem}\label{theorem 2} Consider \eqref{eq: problem def for 2s2t} with the extra constraint $\beta_i \in [\pi-\arcsin\gamma,\pi]$ for $i=1,2$. Then, a non-degenerate configuration of sensors is globally optimal if and only if: \begin{enumerate}[(i)] \item $s_1$, $t_1$, $s_2$ and $t_2$ are vertices of a parallelogram, and \item $s_1$ and $s_2$ are on different circles through $t_1$ and $t_2$ with diameter $d/\gamma$; where $d$ is the distance between $t_1$, $t_2$. \end{enumerate} Any such configuration satisfies $\theta_1=\theta_2 = \arcsin\gamma$ and $\beta_1=\beta_2 = \pi-\arcsin\gamma$ and has an optimal objective value of $\gamma^2$. \end{theorem} \begin{proof} We prove sufficiency first. Incorporating the constraints on $\beta_1$ and $\beta_2$ and using the fact that $\sin(\cdot)$ is nonnegative on $[0,\pi]$, we can rewrite \eqref{eq: problem def for 2s2t} as \begin{align}\label{eq: problem def for 2s2t simplified} \underset{\theta_1, \theta_2, \beta_1, \beta_2}{\text{maximize}} \quad & \min (\sin\theta_1, \, \sin\theta_2)\\ \textrm{subject to:} \quad &\pi-\arcsin\gamma \leq \beta_i \leq \pi & i&\in\{1,2\}\notag\\ & 0 \leq \theta_k \leq \pi & k&\in\{1,2\}\notag\\ & (\theta_1, \theta_2, \beta_1, \beta_2) \in \mathcal{F}.\notag \end{align} By concavity of $\sin(\cdot)$ on $[0,\pi]$ and Jensen's inequality, \begin{equation}\label{ineq1} \min(\sin\theta_1, \, \sin\theta_2) \leq \frac{\sin\theta_1 \!+\! \sin\theta_2}{2} \leq \sin\biggl( \frac{\theta_1+\theta_2}{2} \biggr). \end{equation} From \cref{lem:C1} we have $\theta_1 + \theta_2 + \beta_1 + \beta_2 \leq 2\pi$ with equality only in case $\mathcal{C}_1$. Therefore, \begin{equation}\label{ineq2} 0 \leq \frac{\theta_1+\theta_2}{2} \leq \frac{2\pi-\beta_1-\beta_2}{2} \leq \arcsin\gamma \leq \frac{\pi}{2}. \end{equation} Since $\sin(\cdot)$ is monotonically increasing on $[0,\tfrac{\pi}{2}]$ we may combine \eqref{ineq1} and \eqref{ineq2} to upper-bound the objective of \eqref{eq: problem def for 2s2t simplified} \[ \min(\sin\theta_1, \, \sin\theta_2) \leq \sin\biggl( \frac{\theta_1+\theta_2}{2} \biggr) \leq \gamma, \] with equality only possible for case $\mathcal{C}_1$. Indeed setting $\theta_1=\theta_2=\arcsin\gamma$ and $\beta_1=\beta_2=\pi-\arcsin\gamma$ renders the bound tight and satisfies the constraint for case $\mathcal{C}_1$ in \cref{prop:cases} and is therefore optimal. This solution is illustrated in \cref{fig: 2t2s Theorem2}. \begin{figure}[ht] \centering \includegraphics{fig_thm2} \caption{Example of an optimal sensor configuration when sensors are constrained to lie in the region between both targets (\cref{theorem 2}). Solid circles delineate the feasible set (see \cref{fig: level sets}). A configuration is optimal if $s_1$ and $s_2$ lie on opposite arcs and $t_1$, $s_1$, $t_2$ and $s_2$ form a parallelogram.} \label{fig: 2t2s Theorem2} \end{figure} Necessity follows from convexity of \eqref{eq: problem def for 2s2t simplified}. Indeed, from the derivation above, any optimal non-degenerate configuration must be of case $\mathcal{C}_1$ (see \cref{fig: seven cases}), so constraint $\mathcal{C}_1$ holds (all constraints are linear). Moreover, the objective of \eqref{eq: problem def for 2s2t simplified} is the minimum of two concave functions, so it is concave. \end{proof} \begin{remark} \hlcyan{Letting $s_1$ and $s_2$ approach $t_1$ and $t_2$, respectively, yields in the limit $\beta_1 = \beta_2 = \pi - \arcsin\gamma$ and $\theta_1 = \theta_2 = \arcsin\gamma$, which is a globally optimal (but degenerate) configuration of cases $\mathcal{C}_1$, $\mathcal{C}_6$, or $\mathcal{C}_7$.} \end{remark} \section{ARBITRARILY MANY SENSORS}\label{sec:Extension} In this section we extend our results to the case with $n=2$ targets and \hlorange{$m\geq 3$} sensors. When the sensors can be freely placed (as considered in \cref{theorem 1}) a globally optimal solution is to place the sensors infinitely far from the targets in a circular arrangement. Under these conditions the stealthiness constraint is automatically satisfied and the problem reduces to optimal sensor placement with a single target \cite{moreno2013optimal, sadeghi2020optimal}. \begin{figure}[ht] \vspace{1mm} \centering \includegraphics{fig_ms2t} \caption{The problem with $n=2$ targets and $m \geq 3$ sensors, where sensors are constrained to be inside the green region ($\gamma = 0.95$ shown). Sensor $s_i$'s position is characterized by heading angles $\alpha_{i,1}$ and $\alpha_{i,2}$ from targets $t_1$ and $t_2$, respectively. Sensors have positive heading angles when right of the center-line and negative when left.} \vspace{-1mm} \label{fig: ms2t} \end{figure} The more practical, yet challenging, scenario is when the sensors are constrained to lie in the region between $t_1$ and $t_2$ (as considered in \cref{theorem 2}). It is convenient to re-parameterize this problem in terms of the \emph{heading angles} $\alpha_{i,k}$ of sensor $s_i$ viewed from target $t_k$ and measured relative to the center-line joining $t_1$ and $t_2$ (as illustrated in \cref{fig: ms2t}). By convention $\alpha_{i,k}>0$ if $s_i$ is to the right of the center-line and $\alpha_{i,k}<0$ if $s_i$ is to the left. Thus we may set $\theta_{ij,k} = \|\alpha_{i,k}-\alpha_{j,k}\|$ and $\beta_{i} = \pi-\|\alpha_{i,1}+\alpha_{i,2}\|$ and using \eqref{eq:betarange}, write \eqref{eq: problem def general} in epigraph form as \begin{align}\label{eq: problem def for ms2t} \underset{\alpha, \eta^2}{\text{maximize}} \quad & \eta^2\\ \textrm{subject to:} \quad & \mkern-18mu\sum_{1 \leq i < j \leq m}\mkern-18mu\sin^2(\alpha_{i,k}-\alpha_{j,k}) \geq \eta^2 & k&\in\{1,2\}\notag\\ & \abs{\alpha_{i, 1} + \alpha_{i, 2}} \leq \arcsin \gamma & i&\in\{1,\dots,m\}\notag\\ & \alpha_{i,1}\alpha_{i,2} \geq 0 & i&\in\{1,\dots,m\}.\notag \end{align} The final constraint ensures that $\alpha_{i,1}$ and $\alpha_{i,2}$ have the same sign, so that the $i\textsuperscript{th}$ rays emanating from $t_1$ and $t_2$ are guaranteed to intersect at $s_i$. \cref{eq: problem def for ms2t} is non-convex due to the first and last constraints. A closed-form solution remains elusive, so in the sections that follow, we turn our attention to deriving analytic upper and lower bounds to the optimal $\eta^2$. \subsection{Analytic lower bounds}\label{subsec: lower bound analytical} For the maximization problem \eqref{eq: problem def for ms2t}, any feasible configuration of sensors yields a lower bound on the optimal objective value. Based on the results in \cref{sec: Problem formulation 2t2a}, we focus on two important configurations with associated lower bound \begin{equation}\label{eq: lower bound symmetry} \eta^2 \; = \sum_{1 \leq i < j \leq m}\mkern-18mu\sin^2(\alpha_{i,k}-\alpha_{j,k})\quad\text{for }k\in\{1,2\}. \end{equation} \paragraph{Degenerate configuration}\label{section:degenerate_configuration} Refer to \cref{fig: dgnrt and unfrm} Left and consider the two arcs that form the boundaries of the green region. First, place two sensors on the left arc arbitrarily close to $t_1$ and $t_2$, respectively ($s_1$ and $s_2$). Then similarly place two sensors on the right arc ($s_3$ and $s_4$). Continue placing two sensors at a time on alternating arcs until all sensors are placed. If there is an odd number of sensors then place the last sensor in the center of the arc ($s_5$). Since the sensors are either arbitrarily close to the targets or in the middle of one of the arcs, the angles $\alpha_{i, k}$ are equal to $\pm\psi$ or $\pm\phi$ (see \cref{fig: dgnrt and unfrm}, Left). By straightforward calculations we obtain $\psi = 2\phi = \arcsin\gamma$. As a result, using \eqref{eq: lower bound symmetry}, we can compute the lower bound analytically \begin{equation}\label{eq: analytical lb degenerate} \eta^2_\textup{opt} \geq \left\{ \begin{aligned} &\tfrac{1}{4} m^2 \gamma^2 (2-\gamma^2) \\ &\tfrac{1}{4} (m\!-\!1) \bigl(2-2 (1\!-\!\gamma ^2)^{3/2}+(m\!-\!1)\gamma ^2 (2\!-\!\gamma ^2)\bigr) \\ &\tfrac{1}{4} m^2 \gamma ^2 (2-\gamma ^2) - \gamma ^2 + \gamma^4 \\ &\tfrac{1}{4} (m\!-\!1) \bigl(2-2 (1\!-\!\gamma ^2)^{3/2}+(m\!-\!1)\gamma ^2 (2\!-\!\gamma ^2)\bigr) \\ &\hspace{3.2cm} -\gamma^2+\gamma^4+\gamma ^2\sqrt{1-\gamma ^2}\hspace{-0.3cm} \end{aligned} \right. \end{equation} \noindent where the four lower bounds in \eqref{eq: analytical lb degenerate} correspond to the cases $m\equiv 0,1,2,3 \pmod{4}$, respectively. \begin{figure}[t] \centering \includegraphics[page=1]{fig_dgnrt_unfrm} \includegraphics[page=2]{fig_dgnrt_unfrm} \caption{Two configurations used to lower-bound \eqref{eq: problem def for ms2t}. \textbf{Left:} \emph{Degenerate} configuration. Sensors are placed on the boundary of the green region (arbitrarily close to the targets) forming an angle $\psi$ with the center line. When $m$ is odd the last sensor is placed in the middle of the arc with fewest sensors on it; forming an angle $\phi = \frac{1}{2}\psi$ with the center line. \textbf{Right:} \emph{Uniform} configuration. Sensors are placed such that the lengths of the arcs between neighboring sensors (shown in brown) are equal.} \label{fig: dgnrt and unfrm} \end{figure} \paragraph{Uniform configuration} In this configuration, the sensors are placed on the boundaries of the green region such that the arc length between neighboring sensors is equal to $2\delta$ (see \cref{fig: dgnrt and unfrm} Right). Straightforward computation yields $\delta = \frac{2}{m}\arcsin\gamma$. It can be seen from \cref{fig: dgnrt and unfrm} (Right) that the $(\alpha_{i,k}-\alpha_{j,k})$ are multiples of $\delta$. Direct calculation gives \begin{multline}\label{eq: analytical lb uniform} \eta_\textup{opt}^2 \geq \sum_{1 \leq i < j \leq m}\mkern-18mu\sin^2(\alpha_{i,1}-\alpha_{j,1}) = \mkern-18mu\sum_{1 \leq k \leq m-1}\mkern-18mu(m-k)\sin^2(k\delta)\\ = \frac{1}{8\sin^2\delta}\left( m^2 - 1 + \cos(2m\delta) - m^2\cos(2\delta) \right), \end{multline} \cref{eq: analytical lb uniform} together with $\delta = \frac{2}{m}\arcsin\gamma$ gives the second analytical lower bound to \eqref{eq: problem def for ms2t}. \subsection{Analytic upper bound}\label{subsec: upper bound analytical} We provide two upper-bounds for \eqref{eq: problem def for ms2t}. \paragraph{Constraint approach} \hlorange{Let $\mathcal{S}$ and $\mathcal{D}$ denote the pairs of indices corresponding to agents on the \emph{same} side and \emph{different} side of the centerline between both targets.} \hlorange{Since $\|\alpha_{i,k}\|\leq\arcsin\gamma$ we can upper-bound $\|\alpha_{i,k}-\alpha_{j,k}\|$ based on whether $(i,j)$ are on the same side (in $\mathcal{S}$) or on different sides (in $\mathcal{D}$).} As a result, we have for $k\in\{1,2\}$ \begin{equation} \sin^2(\alpha_{i,k}-\alpha_{j,k}) \leq \begin{cases} \gamma^2 & (i,j) \in \mathcal{S}\\ 4\gamma^2(1-\gamma^2) & (i,j) \in \mathcal{D},\;\gamma \leq \frac{1}{\sqrt{2}} \\ 1 & (i,j) \in \mathcal{D},\;\gamma > \frac{1}{\sqrt{2}}. \end{cases} \end{equation} Assuming we have $p$ agents on the right side and $m-p$ on the left, we may apply the bound above and obtain: \begin{align}\label{eq: upper bound p dependent} \eta^2 &= \sum_{1\leq i < j \leq m} \mkern-18mu\sin^2(\alpha_{i,k}-\alpha_{j,k}) \leq \biggl(\binom{p}{2} + \binom{m-p}{2}\biggr)\gamma^2 \notag\\ &\hspace{1cm} + m(m-p) \begin{cases} 4\gamma^2(1-\gamma^2) & \gamma \leq \frac{1}{\sqrt{2}} \\ 1 & \gamma > \frac{1}{\sqrt{2}}. \end{cases} \end{align} Maximizing the right-hand side of \eqref{eq: upper bound p dependent} over $p$ gives $p^\star = \frac{m}{2}$ for even and $p^\star = \frac{m-1}{2}$ for odd values of $m$ and \begin{equation}\label{eq: upper bound first method} \eta^2_{\text{opt}} \leq \begin{cases} \frac{1}{4}m\gamma^2\bigl(m(5 -4\gamma^2) - 2\bigr) & \gamma \leq \frac{1}{\sqrt{2}}\\ \frac{1}{4}m\bigl(m - 2\gamma^2 + m \gamma^2 \bigr) & \gamma > \frac{1}{\sqrt{2}}. \end{cases} \end{equation} \cref{eq: upper bound first method} is an analytical upper bound to \eqref{eq: problem def for ms2t}. \paragraph{Jensen approach} Let $g(\theta)$ be a \hlorange{\emph{concave non-decreasing}} upper bound $\sin^2\theta \leq g(\theta)$. Applying Jensen's inequality to \eqref{eq: lower bound symmetry}, we obtain for $k\in\{1,2\}$ \begin{align}\label{eta2 upper bound jensen} \eta^2 &= \sum_{1 \leq i < j \leq m}\mkern-10mu\sin^2(\alpha_{i,k}-\alpha_{j,k}) \notag \\ &\leq \binom{m}{2}g\Biggl(\binom{m}{2}^{-1}\mkern-18mu\sum_{1 \leq i < j \leq m}\mkern-10mu (\alpha_{i,k}-\alpha_{j,k})\Biggr). \end{align} The upper bound is determined by choosing $\alpha_{i,k}$ in order to maximize the minimum of the right-hand side of \eqref{eta2 upper bound jensen} for $k\in\{1,2\}$. We next sketch the derivation. Start by assuming without loss of generality that $\alpha_{1, 1} \geq \dots \geq \alpha_{m, 1}$, then \begin{equation}\label{eq: sum theta 1} \sum_{1 \leq i < j \leq m} \mkern-18mu (\alpha_{i,1}-\alpha_{j,1}) = \sum_{i=1}^m (m - 2i + 1)\alpha_{i, 1}. \end{equation} The steps involve showing that: (1) half of the $\alpha_{i,1}$ must be nonegative and the other half nonpositive, (2) The second constraint of \eqref{eq: problem def for ms2t} is tight leading to $\alpha_{i,2}+\alpha_{i,1} = \pm\arcsin\gamma$ and $\alpha_{1,2}\leq \cdots \leq \alpha_{m,2}$, and (3) The positive-coefficient and negative-coefficient terms of \eqref{eq: sum theta 1} can be separately upper-bounded using $\min(a,b) \leq \tfrac{1}{2}(a+b)$, which has a trivial solution of letting half the $\alpha_{i,k}$ equal to 0 and the other half equal to $\pm \arcsin\gamma$. This solution leads to \begin{equation}\label{eq: upper bound second method} \eta^2_{\text{opt}} \leq \begin{cases} \binom{m}{2}g\Bigl( \frac{3m}{4(m-1)}\arcsin\gamma \Bigr) & m\text{ even}\\ \binom{m}{2}g\Bigl( \frac{3(m+1)}{4m}\arcsin\gamma \Bigr) & m\text{ odd} \end{cases} \end{equation} \cref{eq: upper bound second method} is the second analytical upper bound to \eqref{eq: problem def for ms2t}. For our simulations, we used a tight concave upper-bound: \[ g(\theta) = \begin{cases} 0.724611\cdot \theta & 0\leq \theta\leq 1.16556 \\ \sin^2(\theta) & 1.16556 < \theta < \frac{\pi}{2} \\ 1 & \tfrac{\pi}{2} \leq \theta \leq \pi. \end{cases} \] \subsection{Combining the bounds}\label{sec: results} In \cref{fig: sim results} we plot the lower bounds of \eqref{eq: analytical lb degenerate} and \eqref{eq: analytical lb uniform} and upper bounds of \eqref{eq: upper bound first method} and \eqref{eq: upper bound second method} normalized by $m^2$ for $m=3,6,9$ and $m\to\infty$. Different bounds are tighter for different values of $\gamma$ and $m$. The gap between our best upper and lower bounds (shaded region) is quite small for all $m$. We also solved \eqref{eq: problem def for ms2t} directly with a local optimizer\footnote{\hlyellow{We used MATLAB's {\footnotesize\tt fmincon} local optimizer with the default {\footnotesize\tt interior-point} algorithm and initialized the decision variables with $\alpha_{i,k} \sim \mathrm{Uniform}[-\arcsin\gamma,\arcsin\gamma]$ and $\eta=0$ for each $\gamma,m,i,k$. Using the algorithm {\footnotesize\tt trust-region-reflective} yielded similar results, but {\footnotesize\tt sqp} did not work at all.}}. As seen in \cref{fig: sim results}, the optimizer is often trapped in local minima, leading to solutions that are worse than our lower bounds. Moreover, the optimizer never beats our lower bound, suggesting that our lower bound may be globally optimal. \begin{figure}[!ht] \centering \includegraphics{Figures/fig_bounds.pdf} \caption{Normalized upper and lower bounds provided in \cref{subsec: lower bound analytical} and \cref{subsec: upper bound analytical} for $m=3,6,9$ and $m\to\infty$. In the first three simulations, the exact problem in \eqref{eq: problem def for ms2t} has been solved using {\footnotesize\tt fmincon} in MATLAB. The shaded region indicates the gap between best upper and lower bounds.} \label{fig: sim results} \end{figure} \section{CONCLUSION AND FUTURE RESEARCH}\label{sec: Conclusion and Future Work} We considered the problem of stealthy optimal sensor placement, where a network of range sensors attempts to maximize the localization information obtained about a set of targets, while limiting the information revealed to the targets (which also have range sensors). We provided analytical solutions for $m=2$ sensors and $n=2$ targets. We also found upper and lower bounds for $m\geq 3$ and $n=2$. One direction for future research is to explore different sensor models such as bearing-only sensors or sensors with range-dependent noise. \hlcyan{Another area of exploration is the case of $n\geq 3$ targets.} In \cref{fig: feasible 4 targets}, we show the region satisfying a stealthiness constraint for a particular arrangement of $n=4$ targets. This is far more complex than the $n=2$ case (see \cref{fig: level sets}). Small changes in $\gamma$ or the target locations can cause dramatic changes to the topology and connectedness of the feasible set.\looseness=-1 \begin{figure}[!ht] \centering \includegraphics{Figures/fig_4target_feasible_set.pdf} \caption{Feasible set corresponding to $\sum_{1\leq k < \ell \leq 4}\sin^2{\beta_{k\ell}} \leq \gamma^2$ for different $\gamma$ and $n=4$ targets. Compared to the $n=2$ case (see \cref{fig: level sets}), the feasible set is complicated; small changes in $\gamma$ cause dramatic changes in shape and connectedness.} \label{fig: feasible 4 targets} \end{figure} \section{ACKNOWLEDGMENTS} The authors wish to thank Dr. Christopher M. Kroninger (DEVCOM ARL) for support and insights on this topic. \bibliographystyle{IEEEtran} \bibliography{references} \end{document}
2412.04389v1	http://arxiv.org/abs/2412.04389v1	Hypergraph burning, matchings, and zero forcing	\documentclass[12pt]{amsart} \usepackage{graphicx} \usepackage{amsmath,amsthm,amssymb,epsfig} \usepackage[table]{xcolor} \usepackage{url} \usepackage{tikz} \usepackage{float} \usepackage{mathtools} \usepackage{subcaption} \usepackage[linesnumbered,lined,boxed,commentsnumbered]{algorithm2e} \usepackage{MnSymbol} \usepackage{xcolor} \usepackage{comment} \usepackage{tkz-graph} \usetikzlibrary{fit} \usetikzlibrary{shapes} \usepackage{enumitem} \usepackage[left=3.5cm,top=3.5cm,right=3.5cm,bottom=3.5cm]{geometry} \usepackage{mathrsfs} \newcommand{\calB}{\mathcal{B}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calS}{\mathcal{S}} \newcommand{\IG}{\mathrm{IG}} \newcommand{\core}{\mathrm{core}} \newcommand{\Ind}[1]{\langle #1 \rangle} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{question}{Question} \title{Hypergraph burning, matchings, and zero forcing} \author[A.\ Bonato]{Anthony Bonato} \author[C.\ Jones]{Caleb Jones} \author[T.G.\ Marbach]{Trent G.\ Marbach} \author[T.\ Mishura]{Teddy Mishura} \author[Z.\ Zhang]{Zhiyuan Zhang} \address[A1,A2,A3,A4,A5]{Toronto Metropolitan University, Toronto, Canada} \email[A1]{(A1) [email protected]} \email[A2]{(A2) [email protected]} \email[A3]{(A3) [email protected]} \email[A4]{(A4) [email protected]} \email[A5]{(A5) [email protected]} \keywords{hypergraphs, burning, matchings, zero forcing} \subjclass{05C65,05C70,05C85} \begin{document} \begin{abstract} Lazy burning is a recently introduced variation of burning where only one set of vertices is chosen to burn in the first round. In hypergraphs, lazy burning spreads when all but one vertex in a hyperedge is burned. The lazy burning number is the minimum number of initially burned vertices that eventually burns all vertices. We give several equivalent characterizations of lazy burning on hypergraphs using matchings and zero forcing, and then apply these to give new bounds and complexity results. We prove that the lazy burning number of a hypergraph $H$ equals its order minus the maximum cardinality of a certain matching on its incidence graph. Using this characterization, we give a formula for the lazy burning number of a dual hypergraph and give new bounds on the lazy burning number based on various hypergraph parameters. We show that the lazy burning number of a hypergraph may be characterized by a maximal subhypergraph that results from iteratively deleting vertices in singleton hyperedges. We prove that lazy burning on a hypergraph is equivalent to zero forcing on its incidence graph and show an equivalence between skew zero forcing on a graph and lazy burning on its neighborhood hypergraph. As a result, we show that finding an upper bound on the lazy burning number of a hypergraph is NP-complete, which resolves a conjecture from \cite{BJR}. By applying lazy burning, we show that computing an upper bound on the skew zero forcing number for bipartite graphs is NP-complete. We finish with open problems. \end{abstract} \maketitle \section{Introduction} Graph burning is a simplified model for the spread of influence in a network. Associated with the process is a parameter introduced in \cite{BJR,BJR1}, the burning number, which quantifies the speed at which the influence spreads to every vertex. Given a graph $G$, the burning process on $G$ is a discrete-time process. At the beginning of the first round, all vertices are unburned. In each round, first, all unburned vertices that have a burned neighbor become burned, and then one new unburned vertex is chosen to burn if such a vertex is available. If at the end of round $k$ every vertex of $G$ is burned, then $G$ is $k$-\emph{burnable}. The \emph{burning number} of $G,$ written $b(G),$ is the least $k$ such that $G$ is $k$-burnable. For further background on graph burning, see the survey \cite{survey} and the book \cite{bbook}. \emph{Hypergraph burning} was introduced in \cite{BJP,JT} as a natural variant of graph burning. The rules for hypergraph burning are identical to those of burning graphs, except for how burning propagates within a hyperedge. Recall that in a hypergraph, a \emph{singleton} edge contains exactly one vertex. In hypergraphs, the burning spreads to a vertex $v$ in round $r$ if and only if there is a non-singleton hyperedge $\{v,u_1,\ldots,u_k\}$ such that $v$ was not burned and each of $u_1,u_2,\ldots,u_k$ was burned at the end of round $r-1$. In particular, a vertex becomes burned if it is the only unburned vertex in a hyperedge. A natural variation of burning that is our principal focus here is \emph{lazy hypergraph burning}, where a set of vertices is chosen to burn in the first round only; no other vertices are chosen to burn in later rounds. The \textit{lazy burning number} of $H$, denoted $b_L(H)$, is the minimum cardinality of a set of vertices burned in the first round that eventually burn all vertices. Note that for a hypergraph $H,$ we have that $b_L(H) \le b(H).$ We refer to the set of vertices chosen to burn in the first round as a \emph{lazy burning set}. A lazy burning set $S$ such that $\|S\|=b_L(H)$ is called \emph{optimal}. If a vertex is burned and not part of a lazy burning set, then we say it is burned by \emph{propagation}; all rounds other than the first where a lazy burning set is chosen are called \emph{propagation rounds}. We note that we consider a slightly modified rule for lazy burning than the one given in \cite{BJP}, which simplifies our discussion. If a hyperedge $h$ contains exactly one unburned vertex $v$, then $v$ burns. The only difference between this rule and the original one is that now singleton hyperedges \emph{spontaneously burn} in the sense that they cause the vertices they contain to burn immediately. Note that this cannot increase the lazy burning number. Further, no optimal lazy burning set will contain a vertex belonging to a singleton edge, and various bounds (such as $\|V(H)\|-\|E(H)\|\leq b_L(H)$ from \cite{BJP}) remain unchanged with only minor variations in their proof. In the \emph{zero forcing} coloring process, a vertex subset of a graph $G$ is initially chosen to be black, while the remaining vertices are white (some authors alternatively refer to the colors as blue and white, respectively). Consider the following \emph{zero forcing color change rule}: a black vertex $u$ can change the color of a white vertex $v$ if $v$ is the only white neighbor of $u$; we say that $u$ \emph{forces} $v$ or that $v$ is \emph{forced}, and write $u \rightarrow v.$ A \emph{zero forcing set} of $G$ is a set $Z \subseteq V(G)$ of vertices such that if the vertices of $Z$ are black and the rest are white, then every vertex eventually becomes black after repeated applications of the zero forcing color change rule. The minimum cardinality of a zero forcing set for a graph is known as the \textit{zero forcing number} $z(G)$ of the graph $G$. The {\em skew zero forcing color change rule} requires that if a vertex $u$ has exactly one white neighbor $v$, then $u$ forces $v$. The only difference between the two processes is that in skew zero forcing, a vertex does not need to be black to force one of its neighbors. A {\em skew zero forcing set} and the {\em skew zero forcing number $z_0(G)$} are defined analogously. As shown in \cite{IMA}, we have that $z_0(G)\leq z(G)$. For more background on these coloring processes, see \cite{AIM, FHsurvey, hlsbook, IMA}. The paper is organized as follows. In Section~2, we define chronological lists of lazy burning on a hypergraph, which leads to the notion of a $C$-matching. In Corollary~\ref{cor:Burning_equals_CMatching}, we prove that $b_L(H)=\|V(H)\|-m(H)$, where $m(H)$ is the maximum cardinality of a $C$-matching on the incidence graph of $H$. This provides a formula for the lazy burning number of the dual hypergraph in Corollary~\ref{cor1} and provides various new lower and upper bounds on the lazy burning number. For example, Corollary~\ref{cor:delta-delta} gives that in a linear hypergraph with minimum degree $\delta$ with at least $\delta$ hyperedges, $b_L(H) \geq \|V(H)\| -\|E(H)\| + \binom{\delta}{2}$, while Theorem~\ref{upp} shows that $b_L(H) \leq \|V(H)\| - \left\lceil \frac{\|E(H)\|}{\Delta}\right\rceil,$ where $\Delta$ is the maximum vertex degree. In Section~3, we characterize the lazy burning number of a hypergraph using its core, which is the hypergraph that results by iteratively deleting vertices in singleton hyperedges; see Theorem~\ref{thm: empty core}. Section~4 focuses on connections with zero forcing, and Theorem~\ref{zero_set_equivalence} shows that lazy burning is equivalent to zero forcing on incidence graphs. We prove in Theorem~\ref{skew_forcing_equals_lazy_burning} that skew zero forcing on a graph is equivalent to lazy burning on its neighborhood hypergraph. Using this result, we prove that computing an upper bound on the lazy burning number of a hypergraph is NP-complete, which answers Problem 11 from \cite{BJP}. In Theorem~\ref{szf on bpt is npc}, we find applications of lazy burning to skew zero forcing and show the skew zero forcing problem on bipartite graphs is NP-complete. The final section lists open problems. For more background on hypergraphs, see \cite{berge1,berge2,bretto}, and for more background on graph theory, see~\cite{west}. \section{Chronological Lists and $C$-Matchings} A concept gaining in popularity in the literature is a \emph{chronological list of zero forcing}. Given an initial zero forcing set $S_0$, a list of forces $(u_1 \rightarrow v_1, u_2 \rightarrow v_2, \ldots, u_k \rightarrow v_k)$ is defined so that the list of sets $S_0, S_1, S_2, \ldots, S_k$ with $S_{i+1} = S_i \cup \{v_{i+1}\}$ satisfies $N[u_i]\setminus\{v_i\} \subseteq S_{i-1}$. Intuitively, if the set $S_{i-1}$ is a subset of the black vertices at any time during the zero forcing process, then the vertex $u_i$ will force the vertex $v_i$ in the next round if $v_i$ is not already black. It follows that $S_i$ will be a subset of the black vertices at some time in the process. We note that we may have two vertices $v_i$ and $v_j$, with $ v_i$ occurring earlier than $v_j$ in the chronological list, but where vertex $v_j$ would be forced by some black vertex in the zero forcing process in an earlier round than $v_i$. A chronological list of skew zero forces may be defined analogously. If the context is clear, then we abbreviate the lists of either type as a chronological list of forces. We illustrate chronological lists in Figure~\ref{fig:ZF_on_C}, where two adjacent black vertices, $a$ and $b$, are initially chosen as our zero forcing set. In the first zero forcing propagation round, $c$ and $d$ are forced by $a$ and $b,$ respectively. In the second round, $e$ is forced by either $c$ or $d$. However, one possible chronological list of forces is $(b \rightarrow d, d \rightarrow e, e \rightarrow c)$, so $c$ is the last to be forced in the chronological list of forces, even though $c$ is forced before $e$ in the zero forcing process. \begin{figure}[htpb!] \centering \includegraphics[width=\linewidth/2]{pentagon.png} \caption{A zero forcing set for the $5$-cycle, with a chronological list of zero forces given by the vertices labeled 1, 2, and 3.} \label{fig:ZF_on_C} \end{figure} We now introduce an analogous concept for the lazy burning of a hypergraph $H$. A \emph{chronological list of lazy burnings}, $(\{h_1,v_1\},\{h_2,v_2\}, \ldots, \{h_k,v_k\})$ with $v_i\in h_i \in E(H)$, is such that the list of sets $$B_0, B_1=B_0\cup\{v_1\}, B_2=B_1\cup \{v_2\}, \ldots, B_k=B_{k-1}\cup\{v_k\}$$ with $B_{i+1} = B_i\cup \{v_{i+1}\}$ satisfying $h_i\setminus\{v_i\} \subseteq B_{i-1}$. We note that, if $B_k = V(H)$, then $B_0$ is a burning set for $H$ and $k = n - \|B_0\|$. In the case that we take $B_0$ to be a lazy burning set, if $B_{i-1}$ is burned at any time during the lazy burning process, then each vertex in $h_i\setminus \{v_i\}$ is burned. Hence, the vertex $v_i$ will be burned in the next round if it is not already burned. It follows that the entire hypergraph will be burned at some point and that vertex $v_i$ will be burned by round $i$ within the lazy burning process on $H$. We summarize these observations in the following lemma. \begin{lemma} \label{lem:ChronList_equals_Burning} If $(\{h_1,v_1\},\{h_2,v_2\}, \ldots, \{h_k,v_k\})$ is a chronological list of lazy burnings of $H$, then $V(H) \setminus \{v_1, v_2, \ldots, v_k\}$ is a lazy burning set for $H$, and the vertex $v_i$ will be burned during or before the $i$-th lazy burning propagation round. \end{lemma} A \emph{matching} in a graph is a set of edges such that no two edges in the set share a common vertex. The \emph{incidence graph} $\IG(H)$, also known as a \emph{Levi graph}, of a hypergraph $H$ is the bipartite graph with vertex set $V(H)\cup E(H)$, and with an edge in $\mathrm{IG}(H)$ between vertex $v\in V(H)$ and vertex $h\in E(H)$ whenever $v\in h$. We note that there is an implicit ordering on the parts $V(H)$ and $E(H)$ of the bipartition. We will write an edge of $\IG(H)$ as an ordered pair, say $vh$, where $v\in V(H)$ and $h \in E(H)$. A chronological list of lazy burning on a hypergraph $H$ can be converted to a matching in $\mathrm{IG}(H)$, although we need a more restrictive structure than this for our purposes. We define a \emph{$C$-matching} in $\mathrm{IG}(H)$ to be a list of edges from $\mathrm{IG}(H)$, say $M=(v_1h_1, v_2h_2, \ldots, v_kh_k)$, such that $$ h_i \cap V_{i}=\emptyset,$$ where $V_{i} = \{v_{i+1}, v_{i+2}, \ldots, v_{k}\}$ and $1\leq i\leq k$. Define $m(H)$ to be the maximum cardinality of a $C$-matching on $\mathrm{IG}(H)$. \begin{figure}[h!] \centering \begin{subfigure}{0.40\textwidth} \centering \includegraphics[width=\linewidth]{hypergraph.png} \caption{A hypergraph with hyperedges $\{a,b,c\},\{b,c, d\},\{c,d,e\}$.\ A lazy burning set $\{a,b\}$ is indicated by the filled vertices, and a chronological list of lazy burnings is indicated by the vertices containing numbers.} \label{fig:image1} \end{subfigure} \hfill \begin{subfigure}{0.50\textwidth} \centering \includegraphics[width=\linewidth]{incidence.png} \caption{The corresponding incidence graph of the hypergraph.\ A $C$-matching is indicated by the circled edges, labeled from 1 to 3 based on their appearance in the $C$-matching.} \label{fig:image2} \end{subfigure} \caption{Comparison of a chronological list for lazy burning (A) to a $C$-matching on the corresponding incidence graph (B).} \label{fig:two_images} \end{figure} As the following lemma demonstrates, chronological lists and $C$-matchings are equivalent. \begin{lemma} \label{lem:ChronList_equals_CMatching} The list $(\{h_1,v_1\},\{h_2,v_2\}, \ldots, \{h_k,v_k\})$ is a chronological list of lazy burning on $H$ if and only if $(v_1h_1, v_2h_2, \ldots, v_kh_k)$ is a $C$-matching on $\mathrm{IG}(H)$. \end{lemma} \begin{proof} For the forward implication, suppose that $$(\{h_1,v_1\},\{h_2,v_2\}, \ldots, \{h_k,v_k\})$$ is a chronological list of lazy burning on $H$. For $ 0\leq i\leq k $, we define $V_{i} = \{v_{i+1}, v_{i+2}, \ldots, v_{k}\}$ and $B_{i} = V(H) \setminus V_{i}$. From the definition of a chronological list, for each $1\leq i\leq k$, $h_i\setminus \{v_i\} \subseteq B_{i-1} = V(H) \setminus V_{i-1}$. This is equivalent to $(h_i\setminus \{v_i\})\cap V_{i-1} = \emptyset,$ which itself is equivalent to $h_i \cap V_{i} = \emptyset$. This is what is required to make $(v_1h_1, v_2h_2, \ldots, v_kh_k)$ a $C$-matching of $H$. By reversing the chain of implications in the previous paragraph, the proof of the reverse direction follows. \end{proof} Lemmas~\ref{lem:ChronList_equals_Burning} and \ref{lem:ChronList_equals_CMatching} can then be combined to yield the following equality characterizing the lazy burning number of a hypergraph in terms of its order and a maximum cardinality of a $C$-matching. For a $C$-matching $M = (v_1h_1, v_2h_2,\dots, v_mh_m)$, define $M_V= \{v_1,v_2,\dots, v_m\}$, which is the set of vertices that occur in an edge of $M$. \begin{corollary} \label{cor:Burning_equals_CMatching} Let $H$ be a hypergraph and $M$ be a $C$-matching in $\IG(H)$. We have that $V(H)\setminus M_V$ is a lazy burning set for $H$, and $$m(H) = \|V(H)\| - b_L(H).$$ \end{corollary} Define the \emph{dual} of a hypergraph $H=(V,E)$ to be the hypergraph $H^$ with vertex set $E$ and with hyperedge set $\{N(v) : v \in V\}$, where we use $N(v)$ to denote the set of hyperedges in $E$ that contain vertex $v$. The bipartite graph formed by dropping the order of the parts of $\mathrm{IG}(H)$ is the same as that from dropping the order of the parts of $\mathrm{IG}(H^)$. Given a $C$-matching $M=(v_1h_1, v_2h_2, \ldots, v_kh_k)$ in $\mathrm{IG}(H)$, the \emph{retrograde of $M$}, written $M^R$, is the list of edges of $M$ in reverse order and with the ordering of the vertices in each edge swapped; that is, $(h_kv_k, h_{k-1}v_{k-1}, \ldots, h_1v_1)$. The retrograde of $M$ is an ordered set of ordered edges of $\IG(H^)$. We will show that this retrograde is a $C$-matching of $\IG(H^)$. \begin{lemma} \label{lem:retrograde_dual} Let $H=(V,E)$ be a hypergraph. For a $C$-matching $M$ in a bipartite graph $\mathrm{IG}(H)$, the retrograde of $M$ is a $C$-matching in the bipartite graph $\mathrm{IG}(H^)$. \end{lemma} \begin{proof} Let $M=(v_1h_1, v_2h_2, \ldots, v_mh_m)$ be a $C$-matching of the bipartite graph $(V\cup E,I)$, and define $V_i = \{v_{i+1}, v_{i+2}, \ldots, v_{m}\}$ and $E_i = \{h_1, h_2, \ldots, h_{i-1}\}$. By the definition of incidence graphs and $C$-matchings, we have $h_i \cap V_{i}=\emptyset$. For contradiction, we suppose that the retrograde of $M$, $(h_mv_m, h_{m-1}v_{m-1}, \ldots, h_1v_1)$, is not a $C$-matching of the bipartite graph $\IG(H^)=(E\cup V,\overline{I})$. Note that this implies that there is some $j$ such that $N(v_j) \cap E_{j} \neq \emptyset$, and so there exists some $i \leq j-1$ such that $h_i \in N(v_j)$. However, this implies that $v_j \in h_i$. Since $j \geq i+1$, we also have that $v_j \in V_{i}$. It then follows that $v_j \in h_i \cap V_{i}$, but this contradicts the fact that $M$ is a $C$-matching. This completes the proof. \end{proof} Lemma~\ref{lem:retrograde_dual} immediately yields the following result. \begin{corollary} \label{cor:CMatching_Duals} For a hypergraph $H$, $m(H) = m(H^)$. \end{corollary} By combining Corollary~\ref{cor:Burning_equals_CMatching} and Corollary~\ref{cor:CMatching_Duals}, we derive the following. \begin{corollary}\label{cor1} For a hypergraph $H$, $\|V(H)\| - b_L(H) = \|V(H^)\| - b_L(H^)$. \end{corollary} Alternatively, we may describe the lazy burning number of the dual hypergraph as $ b_L(H^) = \|E(H)\| - \|V(H)\| + b_L(H)$. \subsection{Lower Bounds} Consider a $C$-matching $(v_1h_1, v_2h_2, \ldots, v_k h_k)$ in $\IG(H)$. The vertices in $\bigcup_{j \leq i} h_j$ must contain only vertices that are either in the lazy burning set for the corresponding lazy burning process or in $\{v_1, v_2, \ldots, v_i\}$. As such, describing the cardinalities of the hyperedges and the size of the intersection of hyperedges yields lower bounds on the lazy burning number. \begin{theorem}\label{thm:general_lazy_lowerbound} Let $H$ be a hypergraph such that each pair of hyperedges intersects in at most $\overline{\lambda}$ vertices. For $t$ a positive integer, define $\overline{D}_t$ as the minimum sum of the cardinalities of $t$ distinct hyperedges in $H$. For every $t$ with $1 \leq t \leq m(H)$, we have that \[ b_L(H) \geq \overline{D}_t - \overline{\lambda} \binom{t}{2} - t. \] \end{theorem} \begin{proof} Suppose $B$ is an optimal burning set for $H$. As seen in Corollary~\ref{cor:Burning_equals_CMatching}, we may define a $C$-matching in $\mathrm{IG}(H)$, say $M=(v_1h_1, v_2h_2, \ldots, v_mh_m)$, where $m = m(H)$ and $B = V(H) \setminus M_V$. Define $V_{i} = \{v_{i+1}, v_{i+2}, \dots, v_m\}$ for $1\leq i\leq m$. By the definition of a $C$-matching, we have that $$h_i \cap V_{i}=\emptyset.$$ Fix $t$ for $1\leq t\leq m$. The set $H_t=\left( \bigcup_{1 \leq j \leq t} h_j\right) \setminus\{v_1, v_2,\ldots, v_t\}$ satisfies $H_t\cap \{v_1, v_2,\ldots, v_m\} = \emptyset$ and, therefore, $H_t\subseteq B$. For $1\leq i\leq t$, define the set $h'_i = h_i \setminus \bigcup_{1 \leq j < i} h_{j}$, so that $H'_t=\bigcup_{1 \leq j \leq t} h'_{j}$ is a union of disjoint sets. Since each pair of hyperedges may intersect for at most $\overline{\lambda}$ vertices, we have that $\|H'_t\|\geq \sum_{j=1}^{t} (\|h_{j}\| - \overline{\lambda} (j-1))$. Also, $H_t = H'_t\setminus \{v_1, v_2, \ldots, v_m\} \subseteq H'_t\setminus \{v_1, v_2, \ldots, v_t\}$ and, thus, we have \begin{eqnarray} \|H_t\| &\geq & \left( \sum_{j=1}^{t} (\|h_j\| - \overline{\lambda} (j-1)) \right) - t \\ &= & \left( \sum_{j=1}^{t} \|h_j\|\right) - \overline{\lambda} \binom{t}{2} - t\\ &\geq & \overline{D}_t - \overline{\lambda} \binom{t}{2} - t. \end{eqnarray} Since $H_t \subseteq B$, the proof is complete. \end{proof} By Corollary~\ref{cor:Burning_equals_CMatching} and Corollary~\ref{cor:CMatching_Duals}, we may consider a dual argument that swaps the roles of the vertices and hyperedges in Theorem~\ref{thm:general_lazy_lowerbound}, which yields the following corollary. \begin{corollary} Let $H$ be a hypergraph such that each pair of vertices occurs together in at most $\lambda$ hyperedges and suppose that the sum of the degrees of the $t$ smallest-degree vertices is $D_t$. For each $t$ with $1 \leq t \leq m(H)$, \[ b_L(H) \geq \|V(H)\| -\|E(H)\| + D_t - \lambda \binom{t}{2} - t. \] \end{corollary} If the graph has bounds placed on these parameters, then we may provide a more descriptive outcome. \begin{theorem} \label{thm:BurningLowerUniformity} Let $H$ be a hypergraph such that each pair of hyperedges intersects in at most $\overline{\lambda}$ vertices, each hyperedge has cardinality at least $r$, and there are at least $\lceil \frac{r-1}{\overline{\lambda}} + \frac{1}{2} \rceil$ vertices. We then have that $$b_L(H) \geq \frac{(r-1)^2}{2\overline{\lambda}} + \frac{r -1}{2} - \frac{3\overline{\lambda}}{8}.$$ \end{theorem} \begin{proof} From Theorem~\ref{thm:general_lazy_lowerbound}, we have that $b_L(H) \geq rt - \overline{\lambda}\binom{t}{2} - t$ for $1\leq t\leq m(H)$. We may treat this lower bound as an integer-valued function $f(t)$ and then extend it to a real-valued function $f(x)$, where $x\in\mathbb{R}$. The function $f$ is maximized when the derivative is zero or at the endpoints of its domain. This occurs when $x=\frac{r-1}{\overline{\lambda}} + \frac{1}{2},$ since we have assumed that there are at least $\lceil \frac{r-1}{\overline{\lambda}} + \frac{1}{2} \rceil$ many vertices. As $t\in \mathbb{Z}$, the maximum must then occur at either $\lceil \frac{r-1}{\overline{\lambda}} + \frac{1}{2} \rceil$ or $\lfloor \frac{r-1}{\overline{\lambda}} + \frac{1}{2} \rfloor$. We may therefore assume that $t$ has the form $t = \frac{r-1}{\overline{\lambda}} + \frac{1}{2}+\varepsilon$ for some $-1 < \varepsilon < 1$. Substituting this value back in to $rt - \overline{\lambda}\binom{t}{2} - t$ yields $\frac{(r-1+\overline{\lambda}/2)^2}{2\overline{\lambda}} - \frac{\varepsilon^2\overline{\lambda}}{2}$. We thus find that $b_L(H) \geq \frac{(r-1+\overline{\lambda}/2)^2}{2\overline{\lambda}}-\frac{\overline{\lambda}}{2}$, from which the result follows. \end{proof} We may again make use of Corollary~\ref{cor:Burning_equals_CMatching} to obtain a dual result. \begin{corollary}\label{app} Let $H$ be a hypergraph such that each pair of vertices occurs together in at most $\lambda$ hyperedges. Suppose that the minimum degree of the vertices is $\delta$, and $H$ has at least $\lceil \frac{\delta-1} {\lambda} +\frac{1}{2} \rceil$ hyperedges. We have that \[ b_L(H) \geq \|V(H)\| -\|E(H)\| + \frac{\delta-1}{2 \lambda} - \frac{\delta-1}{2} - \frac{3\lambda}{8}. \] \end{corollary} In the case that the hypergraph is linear, meaning $\overline{\lambda}=1$, we may repeat the proof of Theorem~\ref{thm:BurningLowerUniformity}, but where we find that the function is maximized with either $t=r-1$ or $t=r$. This gives the following result. \begin{theorem} If $H$ is a linear hypergraph such that each hyperedge has cardinality at least $r$ and there are at least $r$ hyperedges, then $$b_L(H) \geq \binom{r}{2}.$$ \end{theorem} Corollary~\ref{cor:Burning_equals_CMatching} again can be used when deriving the corresponding dual result. \begin{corollary}\label{cor:delta-delta} If $H$ is a linear hypergraph with minimum degree $\delta$, and there are at least $\delta$ hyperedges, then $$ b_L(H) \geq \|V(H)\| -\|E(H)\| + \binom{\delta}{2}.$$ \end{corollary} \subsection{Upper Bounds} By decomposing a $C$-matching into the set of vertices or the set of edges that are contained in the matching, we find that we have a vertex cover or edge cover of $H$, respectively. This also provides us with a number of upper bounds on the lazy burning number of $H$. \begin{lemma} \label{lem:Cmatching_MinEdgeCover} If $H$ is a hypergraph, then we have that any maximum $C$-matching has cardinality at least that of a minimum vertex cover. \end{lemma} \begin{proof} Let $M=(v_1h_1, v_2h_2, \ldots, v_kh_k)$ be a $C$-matching of the maximum possible length $k=\|V(H)\|-b_L(H)$. We will show that the set of vertices given by $M_V=\{v_1, v_2,\ldots, v_k\}$ forms a vertex cover of $H$. Suppose for contradiction that $M_V$ does not form a vertex cover. There must then be some hyperedge $h$ with none of its vertices in $M_V$. For any vertex $v\in h$, the matched pair $vh$ may be appended to the $C$-matching to form a longer $C$-matching, contradicting maximality. \end{proof} The proof to show that the cardinality of a maximum $C$-matching has at least that of the cardinality of a minimum edge cover of $H$ is analogous with dual terms interchanged. An \emph{isolated vertex} is not in any hyperedge. \begin{lemma} \label{lem:Cmatching_MinVertexCover} If $H$ is a hypergraph with no isolated vertices, then any maximum $C$-matching has a cardinality at least the cardinality of a minimum edge cover of $H$. \end{lemma} There are straightforward upper bounds on the minimum size of a vertex cover and an edge cover in terms of the maximum degree $\Delta(H)$ and maximum hyperedge cardinality $\overline{\Delta}(H)$, respectively. \begin{theorem}\label{upp} If $H$ is a hypergraph with maximum vertex degree $\Delta$, then $$b_L(H) \leq \|V(H)\| - \left\lceil \frac{\|E(H)\|}{\Delta}\right\rceil .$$ \end{theorem} \begin{proof} We find a lower bound on the size of a vertex cover of $H$, which combines with Lemma~\ref{lem:Cmatching_MinVertexCover} to complete the proof. If a vertex cover of $H$ contains $t$ vertices, then since each vertex is contained in at most $\Delta$ hyperedges, at most $t\Delta$ hyperedges contain vertices from the vertex cover. Since each hyperedge of $E(H)$ contains a vertex in the cover, we then have $\|E(H)\| \leq t\Delta$. We then have that $b_L(H) = \|V(H)\| - m(H) \leq \|V(H)\| - t$, and the result follows. \end{proof} In an analogous fashion, the hyperedges from a $C$-matching form an edge cover of $H$, from which we derive the following result using an edge cover instead of a vertex cover. \begin{theorem} If $H$ is a hypergraph with maximum hyperedge cardinality $\overline{\Delta}$ that contains no isolated vertices, then $$b_L(H) \leq \|V(H)\| - \left\lceil \frac{\|V(H)\|}{\overline{\Delta}}\right\rceil .$$ \end{theorem} In the above two results, we found that the set of vertices (respectively, edges) contained within a $C$-matching formed a vertex (respectively, edge) cover of $H$. Instead, if we consider the ordered list of vertices $(v_1, v_2, \ldots, v_k)$ coming from the restriction of a maximum $C$-matching to just its vertices, then we arrive at a topic previously studied by the present authors in the context of burning Latin square hypergraphs; see \cite{LS}. An $n$-uniform Latin square hypergraph, written $H_L$, has vertex set as the entries of the Latin square $L$ and hyperedges containing all vertices that share a given row, a given column, or a given symbol. A \emph{cover-sequence} of $H_L$ is a sequence of vertices $v_1, v_2,\ldots, v_k$ such that $\{v_1, v_2,\ldots, v_k\}$ is a vertex cover of $H_L$ and each $v_i$ does not share all three of its incident hyperedges with vertices that proceeded it in the sequence. It is straightforward to see that the restriction of a $C$-matching to its vertices satisfies these conditions. Further, if we match a vertex $v_i$ to a hyperedge $h_i$ such that $h_i$ does not contain a vertex earlier in the sequence, then the constructed sequence of vertex-edge pairs forms a $C$-matching on $\mathrm{IG}(H_L)$. As such, $C$-matchings of $\mathrm{IG}(H_L)$ are equivalent to cover-sequences of Latin squares, and so results in this section generalize results on the lazy burning of Latin squares in \cite{LS}. \section{Hypergraph Cores} We introduce another characterization of lazy burning sets in terms of certain induced subhypergraphs. We are interested in the maximum induced subhypergraph so that every hyperedge is of cardinality greater than one. In the literature, this would correspond to the so-called 2-core of the dual hypergraph. For ease of notation, we refer to this simply as the \emph{core} of $H$, written $\core(H)$ (this is not to be confused with the core of a graph as used in graph homomorphism theory). The core can be obtained by continually deleting every hyperedge with cardinality one, along with the vertex it contains, until all remaining hyperedges have cardinality greater than one. Let $H$ be a hypergraph and $U \subseteq V(H)$. The \emph{subhypergraph of $H$ weakly induced by $U$}, denoted $H[U]$, is the hypergraph with $V(H[U]) = U$ and $$ E(H[U]) = \{ h\cap U: h\in E(H)\text{ and } h\cap U\neq\emptyset\}. $$ This is also not to be confused with the strongly induced subhypergraphs; in the following, we mean an {\em induced subhypergraph} as a weakly induced subhypergraph since it is the only type of subhypergraph we are concerned with in this paper. We define the subhypergraph formed by vertex removal as $H\setminus U = H[V(H)\setminus U]$. We now provide an algorithm for determining the core of a hypergraph and a theorem on its correctness. \setlength{\algomargin}{20pt} \begin{algorithm}[!htbp] \SetKwInOut{Input}{input}\SetKwInOut{Output}{output} \Input{A hypergraph $H$.} \Output{The core of $H$, an ordered list of vertex removals $R$.} \BlankLine Set $H' \leftarrow H$;\\ Set $R \leftarrow []$ (an empty list);\\ Set $S \leftarrow \{e\in E(H'): \|e\| = 1\}$;\\ \While{$S\neq\emptyset$}{ Pick any $\{v\}\in S$; \\ Set $S\leftarrow S\setminus\{\{v\}\}$;\\ Add $v$ to $R$;\\ Set $S \leftarrow S\cup\{\{u\}: \{u,v\}\in E(H')\}$;\\ Set $H' \leftarrow H'\setminus\{v\}$; } \Return{$H'$ and $R$.} \caption{An algorithm returning the core of $H$ and an ordered list of vertex removals.} \label{core_H} \end{algorithm} \begin{theorem} After inputting a hypergraph $H$ to Algorithm~\ref{core_H}, it returns $\core(H)$ and a vertex set $R = V(H)\setminus V(\core(H))$ with an indexing. \end{theorem} \begin{proof} Suppose there are two induced subhypergraphs of $H$, say $H_1$ and $H_2$, such that all their hyperedges are of cardinality greater than one. The subhypergraph $H[V(H_1)\cup V(H_2)]$ also has the cardinality of all its hyperedges greater than one. Therefore, the core of a hypergraph is unique. From Algorithm~\ref{core_H}, we obtain an induced subhypergraph $J$ of $\core(H)$ with no hyperedge of cardinality one. Let $R = V(H)\setminus V(J) = \{v_1,v_2,\dots, v_r\}$ be the removed vertices, indexed by the algorithm. Suppose to the contrary that $J$ is not the core, and hence there is $A \subseteq R$ so that $H' = H[V(J)\cup A] = \core(H)$. Let $m$, with $1\leq m\leq r,$ be the least index so that $v_{m}\in A$. If $R' = \{v_1,v_2,\dots, v_{m-1}\}$, then $R'\subseteq R\setminus A$ and, therefore, $H'$ is an induced subhypergraph of $H\setminus R'$. By Algorithm~\ref{core_H}, we have that $\{v_{m}\}\in E(H\setminus R')$, which implies that $\{v_{m}\}\in E(H')$. This contradicts the assumption that $H'$ contains no singleton hyperedge. \end{proof} We note that Algorithm 1 produces the core and a list of vertex removals in polynomial time. \begin{theorem}\label{thm: core complexity} If $H$ is a hypergraph with order $n$ and $m$ edges, then the complexity of Algorithm~\ref{core_H} is $O(nm)$. \end{theorem} \begin{proof} Line~3 takes $O(m)$-time. The \textbf{while} loop on Line~4 repeats for at most $n$ times, say $O(n)$. For each iteration of the \textbf{while} loop, both Line~8 and 9 can be computed in $O(m)$-time. All other lines take $O(1)$-time. \end{proof} In the following lemma, we collect some properties of cores. The proofs either follow from the definitions or are straightforward, arguing via a chronological list of lazy burning. A hypergraph is {\em degenerate} if its core contains no vertex. \begin{lemma}\label{lemmac} Let $H$ be a hypergraph and $R =\{r_1,r_2,\dots, r_k\} = V(H)\setminus V(\core(H))$ be the vertex set returned by Algorithm~\ref{core_H}. Suppose $R\neq\emptyset$. We have the following. \begin{enumerate} \item For every $1\leq i\leq k$, $\{r_i\}$ is a singleton hyperedge in $H\setminus\{r_1,r_2,\dots, r_{i-1}\}$. In particular, $R\neq\emptyset$ implies $H$ contains a singleton hyperedge. \item If $L\subsetneq L'\subseteq V(H)$ then $\core(H\setminus L')$ is an induced subhypergraph of $\core(H\setminus L)$. If $L' = L\cup\{v\}$ for some $v\in V(H)\setminus L$ such that $\{v\}\in E(H\setminus L)$, then $\core(H\setminus L') = \core(H\setminus L)$, and $H\setminus L'$ is degenerate if and only if $H\setminus L$ is degenerate. \item For all $1\leq i\leq k$, $\core(H) = \core(H\setminus\{r_1,r_2,\dots, r_i\})$. \end{enumerate} \end{lemma} We also need the following lemma. \begin{lemma}\label{lemma: backward closure} Let $H$ be a hypergraph, $h\in E(H)$, and $B\subseteq V(H)$. Assume $h\subseteq B$ and choose any vertex $u\in h$. The vertex set $B$ is a lazy burning set for $H$ if and only if $B\setminus\{u\}$ is a lazy burning set for $H$. \end{lemma} \begin{proof} Fix a chronological list $S$ of $B$ and insert $\{h,u\}$ at the beginning of $S$. One may verify that the new list is a chronological list of a lazy burning set $B\setminus\{u\}$. The reverse implication follows immediately from the fact that $B\setminus \{u\}\subseteq B$. \end{proof} We provide another characterization of lazy burning in the following, which is the main result of the section. \begin{theorem}\label{thm: empty core} A vertex subset $B$ of a hypergraph $H$ is a lazy burning set if and only if the hypergraph $H\setminus B$ is degenerate. \end{theorem} \begin{proof} Let $B\subseteq V(H)$. We proceed by induction on the order of $H\setminus B$, say $k = \|V(H\setminus B)\|$. The statement holds when $k=0$, where $V(H) = B$. For the induction hypothesis, fix $k\geq 1$, and assume the following holds for any hypergraph $H_0$ and vertex set $B_0\subseteq V(H_0)$ satisfying $\|V(H_0\setminus B_0)\|< k$: $B_0$ is a lazy burning set for $H_0$ if and only if $H_0\setminus B_0$ is degenerate. Let $H$ be a hypergraph and $B\subseteq V(H)$ such that $\|V(H\setminus B)\|=k$. We will prove the statement holds for $H$ and $B$. For the forward direction, suppose $B$ is a lazy burning set for $H$. Fix a chronological list of $B$ and let $\{v_1, h_1\}$ be the first element in the list. Let $B_1 = B\cup\{v_1\}$; we have that $B_1$ is a lazy burning set for $H$ since $B\subseteq B_1$. By the induction hypothesis and that $\|V(H\setminus B_1)\|<k$, we have that $H\setminus B_1$ is degenerate. Finally, since $\{v_1\} = h_1\setminus B$ is a hyperedge in $H\setminus B$, we have that $H\setminus B$ is degenerate by Lemma~\ref{lemmac} (2). For the reverse direction, suppose $H\setminus B$ is degenerate. By Lemma~\ref{lemmac} (1), let $r_1\in R$ be such that $\{r_1\}\in E(H\setminus B)$. This implies that there is a hyperedge $h\in E(H)$ so that $\{r_1\} = h\setminus B$. By Lemma~\ref{lemmac} (2), $H\setminus(B\cup\{r_1\})$ is degenerate. Therefore, by the induction hypothesis, $B\cup \{r_1\}$ is a lazy burning set for $H$. Since $h\subseteq B\cup\{r_1\}$, $B$ is a lazy burning set for $H$ by Lemma~\ref{lemma: backward closure}. \end{proof} The next result reduces the burning number of a hypergraph to the burning number of its core. \begin{theorem}\label{thm: also burns the core} For any hypergraph $H$ and $B\subseteq V(\core(H))$, $B$ is a lazy burning set for $\core(H)$ if and only if it is a lazy burning set for $H$. \end{theorem} \begin{proof} We proceed with the proof by induction on the order of $R = V(H)\setminus V(\core(H))$. When $\|R\| = 0$, we have $H = \core(H)$, and the statement holds. Let $H$ be a hypergraph so that $\|R\|>0$ and suppose the statement holds for all hypergraphs $H_0$ such that $\|V(H_0)\setminus V(\core(H_0))\|<\|R\|$. Assume an indexing on $R$ by Algorithm~\ref{core_H}. The first element $r_1\in R$ is a singleton hyperedge according to Lemma~\ref{lemmac} (1), so $\core(H) = \core(H\setminus\{r_1\})$ by Lemma~\ref{lemmac} (3). We then have that $$\|R\setminus\{r_1\}\| = \|V(H\setminus\{r_1\})\setminus V(\core(H))\|<\|R\|.$$ By the induction hypothesis, $B$ is a lazy burning set for $\core(H)$ if and only if $B$ is a lazy burning set for $H\setminus\{r_1\}$. We finish the proof by a series of equivalent statements. The set $B$ being a lazy burning set for $\core(H)$ is equivalent to $B$ being a lazy burning set for $\core(H\setminus\{r_1\})$ because they are the same hypergraph according to Lemma~\ref{lemmac} (3). The latter is equivalent to $H\setminus(B\cup \{r_1\})$ being degenerate by Theorem~\ref{thm: empty core}, which is equivalent to $H\setminus B$ being degenerate by applying Lemma~\ref{lemmac} (2) with $L = B$ and $v = r_1$. The latter condition is equivalent to $B$ being a lazy burning set for $H$ by Theorem~\ref{thm: empty core}. The proof follows. \end{proof} As an immediate corollary, we have the following. \begin{corollary} \label{H_equals_core} For any hypergraph $H$, $b_L(H) = b_L(\core(H))$. \end{corollary} By applying hypergraph cores, we show the monotonicity of lazy burning numbers with respect to vertex removal. The following theorem contrasts with zero forcing, which is not monotone. \begin{theorem} The lazy burning number is monotone under vertex removal; that is, for any vertex $v\in V(H)$, $b_L(H)-1\leq b_L(H\setminus\{v\})\leq b_L(H).$ \end{theorem} \begin{proof} For the upper bound, let $B$ be an optimal lazy burning set for $H$. Suppose first that $v\notin B$. Observe that $B\cup\{v\}$ is a lazy burning set for $H$, so $H\setminus(B\cup\{v\})$ is degenerate by Theorem \ref{thm: empty core}. Also note that $H\setminus(B\cup\{v\})=(H\setminus\{v\})\setminus B$ and $B\subseteq V(H\setminus\{v\})$, so we have that $B$ is a lazy burning set for $H\setminus\{v\}$ by Theorem \ref{thm: empty core}. We can therefore conclude that $b_L(H\setminus\{v\})\leq \|B\|=b_L(H)$ in this case. Now, suppose $v\in B$. Observe that $H\setminus B=(H\setminus\{v\})\setminus (B\setminus\{v\})$, and hence, they have the same core. By Theorem~\ref{thm: empty core}, $H\setminus B$ is degenerate, so $(H\setminus\{v\})\setminus (B\setminus\{v\})$ is also degenerate. This implies $B\setminus\{v\}$ is a lazy burning set for $H\setminus\{v\}$ by Theorem \ref{thm: empty core}. Hence, $b_L(H\setminus\{v\})\leq \|B\setminus\{v\}\|<\|B\|=b_L(H)$ in this case. For the lower bound, let $B$ be an optimal lazy burning set for $H\setminus\{v\}$. By Theorem~\ref{thm: empty core}, we have that $(H\setminus\{v\})\setminus B$ is degenerate. Observe that $(H\setminus\{v\})\setminus B = H\setminus(B\cup\{v\})$. Since this hypergraph is degenerate, $B\cup\{v\}$ is a lazy burning set for $H$ by Theorem~\ref{thm: empty core}. We therefore have $b_L(H)\leq \|B\cup\{v\}\|=\|B\|+1=b_L(H\setminus\{v\})+1$, and the lower bound follows.\end{proof} \section{Complexity via Zero Forcing} The principal goal of this section is to show that computing an upper bound on the lazy burning number is NP-complete, which answers a conjecture in \cite{BJP}. We first provide another characterization of lazy burning sets in hypergraphs via zero forcing sets in their incidence graphs. \begin{theorem}\label{zero_set_equivalence} For a hypergraph $H$, a subset $B\subseteq V(H)$ is a lazy burning set for $H$ if and only if $B\cup E(H)$ is a zero forcing set for $\mathrm{IG}(H)$. \end{theorem} \begin{proof} For the forward direction, suppose that $B$ is a lazy burning set for $H$. Let $k = \|V(H)\| - \|B\|$ and consider a chronological list of lazy burning of $S = (\{v_1,h_1\},\{v_2,h_2\},\dots, \{v_k, h_k\})$ with the corresponding list of burned vertices $B = B_0, B_1,\dots, B_k = V(H)$. For $0\leq i\leq k$, define $B_i' = B_i\cup E(H)\subseteq V(\IG(H))$. For $1\leq i\leq k$, we have that $h_i\setminus\{v_i\}\subseteq B_{i-1}$. It follows that $N_{\IG(H)}(h_i)\setminus\{v_i\}\subseteq B_{i-1} \subseteq B_{i-1}'$ in $\IG(H)$, and hence, the list $(h_1\to v_1, h_1\to v_2, \dots, h_k\to v_k)$ is a chronological list of zero forcing in $\IG(H)$, where $B_0', B_1', \dots, B_k'$ is the corresponding list of black vertices. We may conclude that $B\cup E(H)$ is a zero forcing set for $\IG(H)$. The reverse direction follows analogously by reversing the implications in the forward direction above. \end{proof} We note that Theorem~\ref{zero_set_equivalence} gives the new inequality $$z(\mathrm{IG}(H))\leq b_L(H)+\|E(H)\|.$$ Denote the lazy burning number as defined in \cite{JT}, where singleton hyperedges do not spontaneously burn, by $b_L^o(H)$. Consider the following decision problems. \medskip \noindent PROBLEM: Lazy burning \\ INSTANCE: A hypergraph $H$ and a positive integer $k \leq \|V(H)\|$.\\ QUESTION: Is $b_L(H)\leq k$? \medskip \noindent PROBLEM: Lazy burning without spontaneous burning \\ INSTANCE: A hypergraph $H$ and a positive integer $k \leq \|V(H)\|$.\\ QUESTION: Is $b_L^o(H)\leq k$? \medskip To show the NP-completeness of the two problems, we need the following theorem. \begin{theorem}\label{thm 0} For every hypergraph $H$, if $H$ contains no singleton hyperedge, then $b_L^o(H) = b_L(H)$. Further, $b_L^o(\core(H)) = b_L(H)$. \end{theorem} \begin{proof} For the first statement, if a list is a chronological list of lazy burning in one process, then it is also a chronological list of lazy burning in the other one. For the second statement, we have that $b_L^o(\core(H)) = b_L(\core(H)) = b_L(H)$, where the first equality holds since the core of a hypergraph contains no singleton hyperedge, and the second equality holds by Corollary~\ref{H_equals_core}. \end{proof} The symmetric and skew zero forcing decision problems are as follows. \medskip \noindent PROBLEM: Zero forcing \\ INSTANCE: A graph $G$ and a positive integer $k \leq \|V(G)\|$.\\ QUESTION: Is $z(G)\leq k$? \medskip \noindent PROBLEM: Skew zero forcing \\ INSTANCE: A graph $G$ and a positive integer $k \leq \|V(G)\|$.\\ QUESTION: Is $z_0(G)\leq k$? \medskip Let $G$ be a graph and $N(v) = N_G(v)$ be the \textit{neighborhood} of $v\in V(G)$. The {\em open neighborhood hypergraph} ${\calN}(G)$ is a hypergraph with $V({\calN}(G)) = V(G)$ and $E(\calN(G)) = \{N_G(v):v\in V(G)\}$. The {\em closed neighborhood hypergraph ${\calN}[G]$} is defined analogously with the hyperedge set consisting of the {\em closed neighborhood} $N[v] = N_G[v] = N(v)\cup\{v\}$ of each vertex in $G$. The following theorem characterizes skew zero forcing as lazy burning on the open neighborhood hypergraph. \begin{theorem} \label{skew_forcing_equals_lazy_burning} Let $G$ be a graph and $B\subseteq V(G)$. The set $B$ is a skew zero forcing set for $G$ if and only if $B$ is a lazy burning set for $\calN(G)$. In particular, $z_0(G) = b_L(\calN(G))$. \end{theorem} \begin{proof} Let $\calB = (B = B_0, B_1, \dots, B_k = V(G))$ be a list of vertex subsets, where $k = \|V(H)\| - \|B\|$, and $V(H)\setminus B = \{v_1, v_2, \dots, v_k\}$ is indexed so that $B_i = B_{i-1}\cup \{v_i\}$ for $1\leq i\leq k$. Suppose that $\calB$ is the corresponding list of black vertices of a chronological list of skew zero forces of $B$, say $S = (u_1\to v_1, u_2\to v_2, \dots, u_k\to v_k)$. For $1\leq i\leq k$, we have that $N(u_i)\setminus\{v_i\}\subseteq B_{i-1}$. Hence, $S$ is a chronological list of skew zero forcing if and only if the list $$(\{v_1, N(u_1)\}, \{v_2, N(u_2)\},\dots, \{v_k, N(u_k)\})$$ is a chronological list of lazy burning of $B$ on $\calN(G)$. \end{proof} An analogous and so omitted argument proves the following. \begin{theorem}\label{thm: closed neighborhood and zero forcing} Let $G$ be a graph and $B\subseteq V(G)$. If $B$ is a zero forcing set for $G$, then $B$ is a lazy burning set for $\calN[G]$. In particular, $b_L(\calN[G])\leq z(G)$. \end{theorem} The converse of Theorem~\ref{thm: closed neighborhood and zero forcing} does not hold. Let $K_{1,j}$ be the star graph, where $V(K_{1,j}) = \{v,u_1, u_2,\dots, u_j\}$ and $v$ is the universal vertex. Note that $N_{K_{1,j}}[v] = V(K_{1,j})$ and $N_{K_{1,j}}[u_i] = \{v, u_i\}$ for each leaf $u_i$. We then have that $$E(\calN[K_{1,j}]) = \{V(K_{1,j}), v u_1, v u_2, \dots, v u_j\}.$$ The set $\{v\}$ is a lazy burning set for ${\calN}[K_{1,j}]$ since burning spreads via each edge $vu_i\in E({\calN}[K_{1,j}])$. Meanwhile, the zero forcing number of a star graph is $j -1 $ by initially forcing all leaves except one. For completeness, we restate results on the complexity of skew zero forcing from \cite{CF}. We first need some terminology introduced there. For a graph $G$, let $T(G)$ denote the graph obtained by ``gluing a triangle'' to each vertex of $G$. More precisely, $V(T(G)) = V(G)\times\{1,2,3\}$ and \begin{equation} \begin{split} E(T(G)) = &\ \{e\times\{1\}: e\in E(G)\}\ \cup \\ &\ \{(v,i)(v, j): v\in V(G), i,j\in\{1,2,3\}, i\neq j\}. \end{split} \end{equation} In particular, the induced subgraph $T(G)[V(G)\times\{1\}]$ is isomorphic to $G$ using a canonical map such that $(v,1)\mapsto v$, for $v\in V(G)$. \begin{theorem}[{\cite[Theorem~3.7]{CF}}]\label{red} For a graph $G$ and $B\subseteq V(G)$, $B$ is a zero forcing set if and only if $B\times\{1\}$ is a skew zero forcing set for $T(G)$. Further, $B$ is optimal if and only if $B\times\{1\}$ is optimal; that is $z(G) = z_0(T(G))$. \end{theorem} By Theorem~\ref{red}, the skew zero forcing problem is at least as hard as the zero forcing problem, which is shown to be NP-complete in \cite{A, Y}. We then have the following result. \begin{theorem}[{\cite[Corollary~3.8]{CF}}]\label{szf is NPC} The skew zero forcing problem is NP-complete. \end{theorem} We now state the main result of this section. \begin{theorem} \label{lbsc is npc} The lazy burning problem is NP-complete. \end{theorem} \begin{proof} Given a hypergraph $H$ and a vertex set $B\subseteq V(H)$, determining whether or not $B$ is a lazy burning set for $H$ takes polynomial time by determining whether or not $H\setminus B$ is degenerate, as in Theorem~\ref{thm: empty core}. Theorem~\ref{thm: core complexity} states that Algorithm~\ref{core_H} can verify it in polynomial time. Hence, the lazy burning problem is in NP. By Theorem~\ref{skew_forcing_equals_lazy_burning}, we know that $\calN(\cdot)$ is a transformation from the skew zero forcing problem to the lazy burning problem. That is, if a vertex set $B\subseteq V(G)$ is a skew forcing a set of a graph $G$, then $B\subseteq V(\calN(G)) = V(G)$ is a lazy burning set for the hypergraph $\calN(G)$. Note that $\calN(\cdot)$ is a polynomial-time transformation. Therefore, the lazy burning problem is NP-hard by Theorem~\ref{szf is NPC}, and hence, is NP-complete. \end{proof} We also derive the following. \begin{theorem} The lazy burning problem without spontaneous burning is NP-complete. \end{theorem} \begin{proof} Given a hypergraph $H$ and a vertex subset $B$, we first show that verifying whether or not $B$ is a lazy burning set can take polynomial time. This can be done by considering first taking another hypergraph $H'$ with $V(H') = V(H)$ and $E(H') = E(H)\setminus\{e\in E(H): \|e\|=1\}$. We know that $B$ is a lazy burning set without spontaneous burning if and only if $ H'\setminus B$ is degenerate. The first step takes $O(m)$-time, and the second step takes $O(nm)$-time by Theorem~\ref{thm: core complexity}. In particular, the lazy burning problem without spontaneous burning is in NP. The function $\core(\cdot)$ is a transformation from the lazy burning problem with spontaneous burning to the lazy burning problem without spontaneous burning, which is proved in Theorem~\ref{thm 0}. Hence, the lazy burning problem without spontaneous burning is NP-hard and, hence, is NP-complete. \end{proof} We may use the complexity results from lazy burning to analyze the complexity of the skew zero forcing numbers of bipartite graphs. Given a bipartite graph $G = (X\cup Y, I)$, we may construct two hypergraphs $H_1$ and $H_2$, where $V(H_1) = X$, $E(H_1) = \{N_G(y): y\in Y\}$, and $H_2 = H_1^$; that is, we may view every bipartite graph as a hypergraph $H = (V, E)$ by considering $V = X$ and $E = \{N_G(y): y\in Y\}$. Note that the hyperedge set $E=E(H_1)$ may be a multiset, and lazy burning performs identically on hypergraphs whenever $E(H_1)$ is a set or a multiset. \begin{theorem}\label{thm: lb-lbdual is szf} Let $H$ be a hypergraph. We then have that $B$ is a lazy burning set for $H$ and $B^$ is a lazy burning set for $H^$ if and only if $B\cup B^$ is a skew zero forcing set for $\IG(H)$. In particular, $$ z_0(\IG(H)) = b_L(H) + b_L(H^). $$ \end{theorem} \begin{proof} The proof is straightforward by considering that the neighborhood hypergraph of an incidence graph $G = \IG(H)$ is precisely the disjoint union of $H\cup H^$. The statement follows by Theorem~\ref{skew_forcing_equals_lazy_burning}. \end{proof} We have the following by the alternative form of Corollary~\ref{cor1}. \begin{corollary}\label{cor: z=2bl} For any hypergraph $H$, $ z_0(\IG(H)) = 2b_L(H) + \|E(H)\| - \|V(H)\|$. \end{corollary} \medskip\noindent PROBLEM: Skew zero forcing on bipartite graphs\\ INSTANCE: A bipartite graph $G$ and an integer $k\leq \|V(G)\|$. \\ QUESTION: Is $z_0(G)\leq k$? \medskip We arrive at the following new result on skew zero forcing. The results in \cite{DeAlba} also give a similar characterization of skew zero forcing on bipartite graphs using special matchings, but our technique is essentially different. \begin{theorem}\label{szf on bpt is npc} The skew zero forcing problem on bipartite graphs is NP-complete. \end{theorem} \begin{proof} For a hypergraph $H$, it takes polynomial time to generate the bipartite graph $\IG(H)$. By Corollary~\ref{cor:Burning_equals_CMatching}, from a lazy burning set $B$ for $H$, we obtain a $C$-matching by performing the lazy burning process and then obtain a lazy burning set for $H^*$. Since verifying if a set of vertices is a skew zero forcing set for a graph takes polynomial time, and the lazy burning problem is NP-complete by Theorem~\ref{lbsc is npc}, we have that the skew zero forcing problem on bipartite graphs is NP-complete. \end{proof} \section{Further Directions} We presented several characterizations of lazy burning on hypergraphs via matchings and zero forcing. It would be interesting to see what new insights can be provided by these connections, especially in the case of zero forcing on graphs. The lazy burning of hypergraphs associated with Steiner Triple Systems was shown to equal their dimension; see \cite{STS}. One question is to determine if there are relationships between the lazy burning number of hypergraphs associated with other designs, such as balanced incomplete block designs, and a natural notion of dimension. A hypergraph is {\em critical $k$-burnable} if $b_L(H) = k$, and removing any vertex results in the decrease of its lazy burning number. It would be interesting to characterize critical $k$-burnable hypergraphs. \section{Acknowledgements} The first author was supported by an NSERC grant, and the second author was supported by an NSERC PGS-D scholarship. \begin{thebibliography}{99} \bibitem{A} A.\ Aazami, \emph{Hardness results and approximation algorithms for some problems on graphs}, PhD thesis, University of Waterloo, 2008. \bibitem{AIM} AIM Minimum Rank\ -\ Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, \textit{Linear Algebra and its Applications} \textbf{428} (2008) 1628--1648. \bibitem{berge1} C.\ Berge, \emph{Graphs and Hypergraphs}, Elsevier, New York, 1973. \bibitem{berge2} C.\ Berge, \emph{Hypergraphs: The Theory of Finite Sets}, North-Holland, Amsterdam, 1989. \bibitem{survey} A.\ Bonato, A survey of graph burning, \emph{Contributions to Discrete Mathematics} \textbf{16} (2021) 185--197. \bibitem{bbook} A.\ Bonato, \emph{An Invitation to Pursuit-Evasion Games and Graph Theory}, American Mathematical Society, Providence, Rhode Island, 2022. \bibitem{BJR} A.\ Bonato, J.\ Janssen, E.\ Roshanbin, Burning a graph as a model of social contagion, In: \emph{Proceedings of WAW'14}, 2014. \bibitem{BJR1} A.\ Bonato, J.\ Janssen, E.\ Roshanbin, How to burn a graph, \emph{Internet Mathematics} \textbf{1}--\textbf{2} (2016) 85--100. \bibitem{LS} A.\ Bonato, C.\ Jones, T.G.\ Marbach, T.\ Mishura, How to burn a Latin square, Preprint 2024. \bibitem{bretto} A.\ Bretto, \emph{Hypergraph Theory: an Introduction}, Springer, 2013. \bibitem{STS} A.\ Burgess, P.\ Danziger, C.\ Jones, T.G.\ Marbach, D.\ Pike, Burning Steiner Triple Systems, Preprint 2024. \bibitem{BJP} A.\ Burgess, C.\ Jones, D.\ Pike, Extending graph burning to hypergraphs, Preprint 2024. \bibitem{CF} J.\ Cooper, G.\ Fickes, Zero loci of nullvectors and skew zero forcing in graphs and hypergraphs, Preprint 2024. \bibitem{DeAlba} L.M.\ DeAlba, Some results on minimum skew zero forcing sets, and skew zero forcing number, Preprint 2024. \bibitem{FHsurvey} S.M.\ Fallat, L.\ Hogben, The minimum rank of symmetric matrices described by a graph: A survey, \textit{Linear Algebra and its Applications} \textbf{426} (2007) 558--582. \bibitem{hlsbook} L.\ Hogben, J.C.-H.\ Lin, B.L.\ Shader, \textit{Inverse Problems and Zero Forcing for Graphs}, \textbf{270}, American Mathematical Society, 2022. \bibitem{IMA} IMA-ISU Research Group on Minimum Rank, Minimum rank of skew-symmetric matrices described by a graph, \textit{Linear Algebra and its Applications} \textbf{432} (2010) 2457--2472. \bibitem{JT} C.\ Jones, \emph{Hypergraph Burning}, Masters Thesis, Memorial University of Newfoundland, 2023. \bibitem{west} D.B.\ West, \emph{Introduction to Graph Theory}, 2nd edition, Prentice Hall, 2001. \bibitem{Y} B.\ Yang, Fast-mixed searching and related problems on graphs, \emph{Theoretical Computer Science} \textbf{507} (2013) 100--113. \end{thebibliography} \end{document} Given a hypergraph $H$ and $L\subseteq V(H)$, the {\em weak induced subhypergraph} is the hypergraph $H\setminus L$ with vertex set $V(H)\setminus L$ and the edge set consists of $e\setminus L$ for $e\in E(H)$ and $e\not\subseteq L$. \begin{algorithm}[H] \label{core_H} \SetKwInOut{Input}{input}\SetKwInOut{Output}{output} \Input{a hypergraph $H=(V,E)$} \Output{the core of $H$, $C_H$} \BlankLine initialize $C_H=H$ \Repeat{$C_H$ is unchanged}{ \ForAll{$e\in E(C_H)$}{ \If{$\|e\|=1$}{ Delete both $e$ and the vertex it contains from $C_H$ } } } \Return{$C_H$} \text{(the core of $H$)} \caption{Construct the core of $H$.} \end{algorithm} \begin{lemma} Algorithm \ref{core_H} runs in time at most $\mathcal{O}(nm^2),$ where $\|V(H)\|=n$ and $\|E(H)\|=m$. \end{lemma} \begin{proof} Each iteration of the outer loop (lines 2-8) deletes at least one vertex from $C_H$ except for the last iteration. Hence, the outer loop executes at most $n+1$ times. Each instance of the inner loop (lines 3-7) iterates over at most $m$ objects. The operation in line 4 takes constant time. Finally, consider the operation in line 5. Denote the unique vertex in $e$ by $v$. Deleting $v$ from $V(C_H)$ and $e$ from $E(C_H)$ takes constant time. However, the algorithm must also delete $v$ from each other edge in $H$ that contained it, which accounts for at most $m$ more operations. \end{proof} Perhaps unsurprisingly, we can characterize lazy burning sets using the concept of cores. This is interesting because $C_H$ is a rigorously defined set-theoretic object that is independent of the rules of the lazy hypergraph burning game. \begin{theorem} \label{core_equals_lazy_burning} If $H$ is a hypergraph and $L\subseteq V(H)$, then $L$ is a lazy burning set for $H$ if and only if the core of $H\setminus L$ is empty. \end{theorem} \begin{proof} For the forward implication, assume $L$ is a lazy burning set for $H$ and let $u_1,\ldots,u_k$ be an ordering of $V(H)\setminus L$ such that the earlier a vertex burns in the lazy burning process, the earlier it appears in the list (with ties broken arbitrarily). We will prove that none of the vertices $u_i$ are in the core of $H\setminus L$ by induction on $i$. For the base case, $u_1$ burns in the first propagation step because it belongs to some edge $e$ that contains no other unburned vertices. If $e$ was a singleton edge, then $u_1$ gets deleted and hence does not belong to $C_{H\setminus L}$. Otherwise, each other vertex in $e$ belongs to $L$, and hence the corresponding edge in $H\setminus L$ contains only $u_1$. Hence, $u_1$ gets deleted in this case as well. Now, assume each of $u_1,u_2,\ldots,u_i$ gets deleted while constructing $C_{H\setminus L}$. Consider the edge $e$ that causes $u_{i+1}$ to burn in the lazy burning game. By the definition of our ordering, no vertex in $e$ is of the form $u_j$ with $j>i+1$. Now, consider the corresponding edge in $H\setminus L$. It contains only vertices from the set $\{ u_1,u_2,\ldots,u_{i+1} \}$ (that is, no vertices from $L$ or with a higher index than $i+1$). Since each of $u_1,\ldots,u_i$ gets deleted while constructing $C_{H\setminus L}$, $e$ eventually contains only the vertex $u_{i+1}$, so $u_{i+1}$ gets deleted. For the reverse implication, assume the core of $H\setminus L$ is empty and let $u_1,\ldots,u_k$ be an ordering of $V(H)\setminus L$ such that the earlier a vertex gets deleted (via Algorithm \ref{core_H}), the earlier it appears in the list (with ties broken arbitrarily). We will prove that each of the vertices $u_i$ burn in $H$ when $L$ is burned as a lazy burning set by induction on $i$. For the base case, since $u_1$ was deleted from $H\setminus L$ in the first step of the algorithm, it must belong to some singleton edge $e$ in $H\setminus L$. Consider the corresponding edge in $H$. If it is still a singleton (that is, it contains no vertices from $L$), then $u_1$ burns via spontaneous combustion. Otherwise, the corresponding edge in $H$ only contains $u_1$ as well as vertices from $L$, so $u_1$ burns via propagation. Now, assume each of $u_1,u_2,\ldots,u_i$ is burned. Consider the edge $e$ that contains $u_{i+1}$ and causes it to be deleted from $H\setminus L$ (via Algorithm \ref{core_H}). Denote the corresponding edge in $H$ by $e^\prime$. If $e^\prime$ is also a singleton, then $u_{i+1}$ burns via spontaneous combustion. Otherwise, $e^\prime$ becomes the singleton edge $e$ (containing only $u_{i+1}$) after deleting $L$ and a series of vertex deletions via the algorithm. Suppose $e^\prime$ contains vertices of the form $u_j$ with $j>i+1$. For this edge to become a singleton containing only $u_{i+1}$, $u_j$ must be deleted strictly before $u_{i+1}$, which contradicts our ordering of $V(H)\setminus L$. Hence, $e^\prime$ is an edge in $H$ that contains only vertices from the set $L\cup\{ u_1,\ldots, u_{i+1}\}$. Observe that $u_{i+1}\in e^\prime$. Since each of $u_1,\ldots,u_i$ eventually burns, at some point in the lazy burning process $e^\prime$ contains only burned vertices and $u_{i+1}$, and hence $e^\prime$ causes $u_{i+1}$ to burn via propagation. \end{proof} \textcolor{blue}{Use above corollary (whose proof is very similar to the above thm) in complexity result!} Note that the lazy burning number of $C_H$ is the same regardless of whether or not spontaneous combustion is allowed, as $C_H$ contains no singletons. \begin{corollary} \label{no_SC_NP_comp} Let $H$ be a hypergraph. The decision problem \emph{``Is $b_L(H)\leq k$?''} is NP-complete when spontaneous combustion is not allowed. \end{corollary} \begin{proof} Consider an arbitrary hypergraph $H$. If $H$ has no singleton edges, then this decision problem is equivalent to the one from Theorem~\ref{SC_NP_complete}, so assume $H$ has singleton edges. We will show that a solution to the original model of lazy burning yields a solution to lazy burning with spontaneous combustion on $H$, and hence, the original model is ``at least as hard.'' The proof will follow, as the original model of lazy burning is known to be in NP \cite{burgess2024extendinggraphburninghypergraphs}. Given $H$, we first compute $C_H$, which notably only takes polynomial time. Let $L$ be an optimal lazy burning set for $C_H$ in the original model. We then note that $L$ is also an optimal lazy burning set for $C_H$ when spontaneous combustion is allowed. Finally, by Theorem~\ref{core_bl_equality} and its proof, $L$ is an optimal lazy burning set for $H$ when spontaneous combustion is allowed. \end{proof} \textcolor{blue}{Caleb: is the above proof formal enough? Maybe use the following proof instead: Suppose we wish to answer the question ``is $b_L(H)\leq k$?'' (with spontaneous combustion), and we have the answer to the question ``is $b_L(C_H)\leq k$?'' (without spontaneous combustion). If $b_L(C_H)\leq k$, then $b_L(H)\leq k$ (with spontaneous combustion) since any lazy burning set for $C_H$ is a lazy burning set for $H$ (see the proof of Theorem~\ref{core_bl_equality}). Otherwise, $b_L(C_H)> k$ (without spontaneous combustion). Suppose $b_L(H)\leq k$ (with spontaneous combustion). There exists a lazy burning set $L$ for $H$ of size at most $k$. By the same argument as the proof of Theorem~\ref{core_bl_equality}, $L\cap C_H$ is a lazy burning set for $C_H$ (without spontaneous combustion). Since $\|L\cap C_H\|\leq \|L\|\leq k$, we have that $b_L(C_H)\leq k$ (without spontaneous combustion), which is a contradiction. Thus, $b_L(H)> k$ (with spontaneous combustion), so a solution to ``is $b_L(C_H)\leq k$?'' yields a solution to the NP-complete decision problem from Theorem~\ref{SC_NP_complete}. Therefore, original lazy burning is NP-complete on the family of hypergraphs that are the cores of other hypergraphs. Since it is in NP \cite{burgess2024extendinggraphburninghypergraphs}, it is NP-complete in general. } \begin{theorem}\label{zero_set_equivalence} If $H$ is a hypergraph and $L\subseteq V(H)$, then $L$ is a lazy burning set for $H$ if and only if $L\cup E(H)$ is a zero forcing set for $\mathrm{IG}(H)$. \end{theorem} \begin{proof} For the forward implication, assume $L$ is a lazy burning set for $H$ and let $u_1,u_2,\ldots,u_k$ be an ordering of $V(H)\setminus L$ such that the earlier a vertex burns in the lazy burning process, the earlier it appears in the list (with ties broken arbitrarily). We claim that $L\cup E(H)$ is a zero forcing set in $\mathrm{IG}(H)$. In particular, we will prove that the vertices corresponding to $u_1,u_2,\ldots,u_k$ in $\mathrm{IG}(H)$ will become forced using induction on $i$. This will prove the claim, seeing as these are the only unforced vertices in $\mathrm{IG}(H)$ at the start of the game. For the base case, $u_1$ burns in the first propagation step because it belongs to some edge $e$ that contains no other unburned vertices. Hence, in $\mathrm{IG}(H)$, $e$ is a forced vertex whose only unforced neighbor is $u_1$, so $e$ forces $u_1$. Now assume each of $u_1,u_2,\ldots,u_i$ becomes forced in $\mathrm{IG}(H)$. Consider the edge $e$ that causes $u_{i+1}$ to burn in the lazy burning game. By the definition of our ordering, no vertex in $e$ is of the form $u_j$ with $j>i+1$. Hence, in $\mathrm{IG}(H)$, $e$ is a forced vertex whose open neighborhood is a subset of $\{ u_1,u_2,\ldots,u_{i+1}\}$ (and $eu_{i+1}\in E(\mathrm{IG}(H))$ is guaranteed). Since $u_{i+1}$ is unforced and each of $u_1,u_2,\ldots,u_i$ is forced, $u_{i+1}$ is the unique unforced neighbor of $e$. Hence, $e$ forces $u_{i+1}$. For the reverse implication, assume $L\cup E(H)$ is a zero forcing set for $\mathrm{IG}(H)$. Let $u_1,\ldots,u_k$ be an ordering of $V(H)\setminus L$ such that the earlier a vertex becomes forced, the earlier it appears in the list (with ties broken arbitrarily). We will prove that each of the vertices $u_i$ burns in $H$ when $L$ is burned as a lazy burning set by induction on $i$. For the base case, since $u_1$ becomes forced in the first time step, it must be the unique unforced neighbor of some vertex $e$, where $e$ belongs to the partite set of vertices that is indexed by $E(H)$. Consider the edge in $H$ corresponding to $e$ (and call it $e$ also for simplicity). We have that $u_1\in e$ in $H$. In $\mathrm{IG}(H)$, each neighbor of $e$ belongs to the zero forcing set except for $u_1$. Thus, in $H$, each vertex in $e$ belongs to the lazy burning set except for $u_1$, so $u_1$ burns via propagation. Now assume each of $u_1,u_2,\ldots,u_i$ is burned in $H$. Consider the vertex $e$ in $\mathrm{IG}(H)$ that forces $u_{i+1}$. Since $e$ is adjacent to $u_{i+1}$, $e$ must belong to the partite set in $\mathrm{IG}(H)$ whose vertices are indexed by $E(H)$. Also, each other neighbor of $e$ must be forced in a time step strictly earlier than $u_{i+1}$, so the open neighborhood of $e$ in $\mathrm{IG}(H)$ is a subset of $\{ u_1,u_2,\ldots,u_{i+1}\}$. Hence, in $H$, $e$ is an edge that is a subset of $\{ u_1,u_2,\ldots,u_{i+1}\}$. Since $u_{i+1}$ is the only unburned vertex in this set and $u_{i+1}\in e$, $u_{i+1}$ burns due to propagation within $e$. \end{proof} \begin{theorem} \label{inf_fam_tight_1} There exists an infinite family of hypergraphs for which the bound in Corollary~\ref{another_zero_bound} is tight. \end{theorem} \begin{proof} Consider the family of generalized stars constructed as follows. Given $k,\ell\geq 2$, define $H_{k,\ell}$ as the hypergraph with $\ell$ edges, each of which contains $k$ vertices of degree one, such that all edges intersect in a central vertex. More formally, $E(H_{k,\ell})=\{e_1,e_2,\ldots,e_\ell\}$ and $V(H_{k,\ell})=\{v,u_1^1,\ldots,u_k^1,\ldots,u_1^\ell,\ldots,u_k^\ell\}$ where $e_i=\{v,u_1^i,\ldots,u_k^i\}$ for each $i$. Note that $H_{k,\ell}$ is $(k+1)$-uniform. First, we show that $b_L(H_{k,\ell})=(k-1)\ell+1$. Any lazy burning set for $H_{k,\ell}$ must contain at least $k-1$ vertices of degree one in each edge. If any edge $e$ contains fewer than $k-1$ burned degree-one vertices at the start of the game, then that edge contains two or more unburned vertices of degree one. These vertices will never burn because $e$ will never contain exactly one unburned vertex. Hence, $b_L(H_{k,\ell})\geq(k-1)\ell$. However, if we choose $k-1$ degree-one vertices in each edge as their lazy burning set, then every edge will contain an unburned vertex of degree one, as well as $v$, which is also unburned, so no burning will spread. Hence, $b_L(H_{k,\ell})>(k-1)\ell$. Indeed, if we initially burn $k-1$ degree-one vertices per edge as well as $v$ then $H_{k,\ell}$ will burn in one round, so $b_L(H_{k,\ell})\leq(k-1)\ell+1$. Since any optimal lazy burning set for $H_{k,\ell}$ contains a degree-one vertex in each edge, the maximum number of non-adjacent vertices one can choose from an optimal lazy burning set is $\|E(H_{k,\ell})\|=\ell.$ Hence, for this family of hypergraphs, the upper bound from Corollary~\ref{another_zero_bound} is $$b_L(H_{k,\ell})+\|E(H_{k,\ell})\|-x=b_L(H_{k,\ell})+\ell-\ell=b_L(H_{k,\ell}).$$ We now consider the corresponding incidence graph $I_{H_{k,\ell}}$. Observe that $I_{H_{k,\ell}}$ may be viewed as a rooted tree, with $v$ as the root. The vertices $e_1$ through $e_\ell$ are all incident to $v$ and in the second layer of the tree, and the vertices $u_i^j$ are in the third layer of the tree (and they are all leaves). Now that $e_j$ is adjacent to each vertex $u_1^j,u_2^j,\ldots,u_k^j$. We will prove that $z(I_{H_{k,\ell}})=(k-1)\ell+1$, and the proof will be complete. First, for each vertex $e_j$, at least $k-1$ of the $k$ leaves incident to it must be initially forced. Otherwise, there are two or more unforced leaves incident to $e_j$, and hence $e_j$ will never have a unique unforced neighbor. Thus, the unforced leaves incident to $e_j$ will never become forced, so $z(I_{H_{k,\ell}})\geq (k-1)\ell$. Indeed, if we force exactly $k-1$ leaves per vertex $e_j$ (and force no other vertices), then the vertices $e_j$ will all immediately become forced. However, no more forcing will occur, as each vertex $e_j$ will be incident to two unforced vertices; $v$ and a leaf. Thus, $z(I_{H_{k,\ell}})> (k-1)\ell$. Indeed, if one forces $k-1$ leaves per vertex $e_j$ as well as $v$, then $I_{H_{k,\ell}}$ will become completely forced in two steps, so $z(I_{H_{k,\ell}})\leq (k-1)\ell+1$ and the proof is complete. \end{proof} \begin{proof} Let $L$ be an optimal lazy burning set for $H$, and let $S$ be the corresponding zero forcing set of size $b_L(H)+\|E(H)\|$ for $\mathrm{IG}(H)$. Denote the chosen vertices in $L$ by $v_1,v_2,\ldots,v_x$ and let $e_1,e_2,\ldots,e_x$ be a list of edges in $H$ such that $v_i\in e_i$ for each $i$. Observe that the neighborhoods of the vertices $v_i$ in $\mathrm{IG}(H)$ are disjoint. Hence, in $\mathrm{IG}(H)$, the vertices $v_i$ and $e_i$ are adjacent, and if $i\neq j$, then the vertices $v_i$ and $e_j$ are not adjacent. We claim that $S\setminus\{e_1,e_2,\ldots,e_k\}$ is a zero forcing set in $\mathrm{IG}(H)$. Each $v_i$ is part of the initially forced set, and $e_i$ is the only neighbor of $v_i$ that has been removed from the forcing set. As every neighbor of $v_i$ in $\mathrm{IG}(H)$ is forced except for $e_i$, $v_i$ will immediately force $e_i$. The proof follows, as $S$ is a forcing set for $\mathrm{IG}(H)$. \end{proof}
2412.04579v1	http://arxiv.org/abs/2412.04579v1	Solvable families of random block tridiagonal matrices	\documentclass[12pt]{article} \title{Solvable families of random block tridiagonal matrices} \date{} \author{Brian Rider and Benedek Valk\'o} \oddsidemargin 0in \topmargin 0in \headheight 0in \headsep 0in \textheight 9in \textwidth 6.7in \renewcommand{\baselinestretch}{1.3} \usepackage{amsfonts,color} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb, url} \usepackage{hyperref} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}[theorem]{Claim} \newtheorem{fact}[theorem]{Fact} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{examples}[theorem]{Examples} \newcommand{\eps}{\varepsilon} \newcommand{\Z}{{\mathbb Z}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\FF}{{\mathbb{F}}} \newcommand{\UU}{{\mathbb U}} \newcommand{\R}{{\mathbb R}} \newcommand{\CC}{{\mathbb C}} \newcommand{\ud}{{\mathbb U}} \newcommand{\Rnn}{{\R_{\geq 0}}} \newcommand{\N}{{\mathbb N}} \newcommand{\cP}{{\mathcal P}} \newcommand{\cC}{{\mathcal C}} \newcommand{\ev}{{\rm E}} \newcommand{\pr}{\mbox{\rm P}} \newcommand{\lstar}{{\raise-0.15ex\hbox{$\scriptstyle \ast$}}} \newcommand{\ldot}{.} \newcommand{\vfi}{\varphi} \newcommand{\cN}{\mathcal{N}} \newcommand{\var}{\text{Var }} \newcommand{\mat}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4 \\ \end{array} \right)} \theoremstyle{remark} \newcommand{\Balpha}{\underline{\alpha}} \newcommand{\Btheta}{\underline{\theta}} \newcommand{\Blambda}{\underline{\lambda}} \newcommand{\Bq}{\underline{q}} \newcommand{\Bx}{\underline{x}} \newcommand{\By}{\underline{y}} \newcommand{\Ba}{\underline{a}} \newcommand{\Bb}{\underline{b}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\cc}{\mathbb{C}} \newcommand{\rr}{\mathbb{R}} \newcommand{\ind}{{\bf{1}}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cFF}{\widetilde {\mathcal{F}}} \newcommand{\cL}{\mathcal{L}} \newcommand{\qq}{\mathbb{Q}} \newcommand{\hh}{\mathbb{H}} \newcommand{\oo}{\mathbb{O}} \newcommand{\cX}{\mathcal{X}} \newcommand{\re}{\text{Re}} \newcommand{\sech}{\text{ sech }} \newcommand{\Tr}{\textup{Tr}} \def\eqd{\stackrel{d}{=}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\Pf}{\operatorname{Pf}} \newcommand{\Hf}{\operatorname{Hf}} \newcommand{\ww}{\boldsymbol\omega} \newcommand{\nn}{\boldsymbol\eta} \newcommand{\cA}{\mathcal{A}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cD}{\mathcal{D}} \newcommand{\dd}{\Theta} \newcommand{\T}{\dag} \newcommand{\lst}[1]{[\![#1 ]\!]} \newcommand{\nint}[2]{\lfloor #1 \rfloor_{#2}} \newcommand{\nfr}[2]{\left\{ #1 \right\}_{#2}} \newcommand{\mbf}[1]{\mathbf{#1}} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\HH}{\mathtt{H}_{\beta, n}} \newcommand{\WW}{\mathtt{W}_{\beta, n,m}} \newcommand{\SQW}{\mathtt{SqW}_\beta} \newcommand{\benedek}[1]{\textcolor{red}{#1}} \newcommand{\brian}[1]{\textcolor{blue}{#1}} \bibliographystyle{plain} \begin{document} \maketitle \abstract{We introduce two families of random tridiagonal block matrices for which the joint eigenvalue distributions can be computed explicitly. These distributions are novel within random matrix theory, and exhibit interactions among eigenvalue coordinates beyond the typical mean-field log-gas type. Leveraging the matrix models, we go on to describe the point process limits at the edges of the spectrum in two ways: through certain random differential operators, and also in terms of coupled systems of diffusions. Along the way we establish several algebraic identities involving sums of Vandermonde determinant products. } \section{Introduction} Trotter observed that if one applies the Householder tridiagonalization process to a GOE or GUE random matrix then the resulting real symmetric tridiagonal matrix will have independent entries (up to symmetry) with normal and chi distributions \cite{Trotter}. In \cite{DE} Dumitriu and Edelman presented a far reaching generalization of this result. They show that, for any $\beta > 0$, the $ n \times n$ random Jacobi matrix with independent $N(0,\frac{2}{\beta})$ random variables along the diagonal, and independent $ \frac{1}{\sqrt{\beta}} \chi_{\beta(n-1)}, \frac{1}{\sqrt{\beta}} \chi_{\beta(n-2)}, \dots, \frac{1}{\beta} \chi_\beta$ random variables along the off-diagonals, has joint eigenvalue density proportional to: \begin{equation} \label{eig_DE} \left\|\Delta(\lambda)\right\|^\beta e^{-\frac{\beta}{4} \sum_{j=1}^n \lambda_j^2}. \end{equation} Here $\Delta(\lambda)$ denotes the usual Vandermonde determinant of the eigenvalues. This includes Trotter's result for GOE or GUE upon setting $\beta=1$ or $2$. The Dumitriu-Edelman model for the Gaussian, or ``Hermite", beta ensemble, along with their Laguerre counterparts, initiated an immense amount of activity in the study of the scaling limits of beta ensembles. See for instance, \cite{ES}, \cite{KillipNenciu}, \cite{RRV}, \cite{RR}, \cite{KS}, \cite{BVBV}, \cite{KRV}, and \cite{BVBV_sbo}. Motivated both by the original construction of \cite{DE} along with its ensuing impact, here we establish two families of similarly solvable block-tridiagonal matrix models. Let $\HH (r,s)$ denote the distribution of the $rn \times rn$ symmetric or Hermitian block tridiagonal matrix with $r \times r$ diagonal blocks distributed as independent copies of G(O/U)E, and descending upper diagonal blocks distributed as independent copies of the (lower triangular) positive square root of a real/complex Wishart with parameters $(r, (r+s)(n-i))$. Here $i$ is the index of the offdiagonal block entry, and $\beta=1$ and 2 corresponds to the real and complex case, respectively. As in the $r=1$ case, the diagonal and and offdiagonal variables are also independent of each other. A more detailed description of these ensembles is provided in Section \ref{subs:matrix_distr}. Note of course that the Wishart distribution is the natural multivariate analog of the $\chi^2$ distribution, and that $\HH(1,s)$ is just the original Dumitriu-Edelman model, after a reparameterization. Further, when $s=0$, our model may in fact be arrived by a suitable block tridiagonalization procedure of the corresponding $rn \times rn$ G(O/U)E, {\`a} la Trotter. This has already been noticed in \cite{Spike2} in the context of eigenvalue spiking. Finding a suitable general beta version of the spiked Tracy-Widom laws introduced in that paper was another motivation for our work. Our main result is: \begin{theorem} \label{thm:main} For $\beta =1$ and $2$, the symmetrized joint eigenvalue density of $\HH(r,s)$ can be computed explicitly in the following cases: \begin{align} \label{density1} \frac{1}{Z_{n, \beta, r, 2}} \|\Delta({\lambda})\|^{\beta} \left( \sum_{(\mathcal{A}_1,\dots,\mathcal{A}_r)\in \cP_{r,n}} \prod_{j=1}^r \Delta(\cA_j)^2 \right) e^{- \frac{\beta}{4}\sum_{i=1}^{rn} \lambda_i^2}, \quad \mbox{ for } r \ge 2, \ \beta s=2, \end{align} and \begin{align} \label{density2} \frac{2^n}{Z_{n, \beta, 2, \beta s}} \Delta({\lambda})^{\beta+\frac{\beta s}{2}} \left\|\Pf \left(\frac{{\bf{1}}_{i \neq j}}{\lambda_i -\lambda_j} \right)\right\|^{\frac{\beta s}{2}} e^{- \frac{\beta}{4}\sum_{i=1}^{2n}\lambda_i^2} \quad \mbox{ for } r = 2, \ \beta s = 2,4. \end{align} It further holds that \begin{align} &Z_{n, \beta, r, \beta s} =(n r)! (2\pi)^{\frac{nr}{2}} \left(\tfrac{\beta}{2}\right)^{a_{n,\beta,r,s}} \Gamma\left(\tfrac{\beta}{2}\right)^{-nr} \prod_{k=1}^{nr} \Gamma\left(\tfrac{\beta}{2}\left(k+s \lceil\tfrac{k}{r}\rceil\right)\right) \times \begin{cases} 1, \quad &\beta s=2,\\ (\beta/12)^n, \quad &\beta s=4, \end{cases} \end{align} with $a_{n,\beta,r,s}=%-\frac{\beta}{4} n r (n (r+s)+s-1)-\frac{nr}{2}= -\frac{\beta}{4} n r (n (r+s)+s)+\left(\tfrac{\beta}{4}-\tfrac{1}{2}\right){nr}$ for all $n$, $\beta = 1$ and $2$, and combinations of $r$ and $s$ in \eqref{density1} and \eqref{density2}. \end{theorem} Here for $r\ge 2$ and $n\ge 1$, $\cP_{r,n}$ denotes the set of size $r$ equipartitions of $\lst{rn} := \{ 1,2, \dots rn\}$. That is, $\{\cA_1, \dots\cA_r\}\in \cP_{r,n}$ if $\|\cA_i\|=n$ for all $i$ and the $\cA_i$ form a partition of $\lst{rn}$. With that, for any $\cA \subset \lst{rn}$, we write $\Delta(\cA)$ as shorthand for the Vandermonde determinant in the $\|\cA\|$ ordered eigenvalue variables with indices drawn from $\cA$ (suppressing the explicit dependence on $\lambda_i, i \in \cA$). Finally, $\Pf(M)$ denotes the Pfaffian of $M$. In both \eqref{density1} and \eqref{density2} we see novel types of interactions among the points beyond the usual $\|\Delta({\lambda})\|$ to some power. The formulas for the overlapping $r=2$, $\beta s = 2$ cases are shown to agree by a Pfaffian/Vandermonde identity, see Lemma \ref{lem:det4_identities} below. This is one of several identities involving sums of powers of Vandermonde determinants that we prove in Section \ref{sec:det_identities}. We also note that \eqref{density1} is consistent with \eqref{eig_DE} upon taking $r=1$, as then the sum over equipartitions reduces to $\Delta(\lambda)^2 = \Delta(\lambda)^{\beta s}$. One might anticipate that the form of the $r=2$ family should generalize to all even integer $\beta s$. However, computer assisted calculations for small $n$ values indicate that the Pffafian structure in \eqref{density2} breaks down for $\beta s=6$. Understanding what happens for larger block size $r$ beyond $\beta s=2$ also remains open. Our difficulty in extending exact formulas to either parameter regime is tied to our approach to proving Theorem \ref{thm:main}. This rests on computing the absolute $\beta s$-moment of a certain structured determinant over the Haar distributed Orthogonal or Unitary group (in dimension $rn$). We do this by expansion and re-summation, the underlying complexity of which grows in both $r$ and $\beta s$. In another direction, our block model could certainly be constructed using quaternion ingredients, leading to $\HH(r,s)$ with $\beta=4$. The non-commutativity of the quaternion variables poses additional technical challenges in extending Theorem \ref{thm:main} to that setting, though we expect these are not insurmountable. Next, a natural question is whether densities of the form \eqref{density1} or \eqref{density2} appear ``in the wild". In fact, the $r=2$ family bears close resemblance to what is known as the Moore-Read, or Pfaffian, state for the fractional quantum Hall effect, see \cite{MR_1991}. In that theory the points lie in the complex plane, so \eqref{density2} might be viewed as a one-dimensional caricature of these states in the same way that the Gaussian (and other) beta ensembles are one-dimensional caricatures of a true coulomb gas. The eigenvalues of random block matrices have of course been studied in a number of capacities, most notably perhaps as structured band matrices connected to the Anderson or Wegner orbital models, see for example \cite{SchSch} and the references therein. Motivated by the theory of matrix orthogonal polynomials, \cite{Dette1} and \cite{Dette2} introduce families of ``block beta" Hermite, Laguerre and Jacobi ensembles built out of Gaussian and/or $\chi$ variables, and study their limiting density of states. The large deviations of related ensembles have been considered in \cite{Rouault1} and \cite{Rouault2}. Our work though is the first to provide a systematic approach to finding solvable block models. We close the introduction with descriptions of: (i) the soft edge asymptotics for $\HH(r,s)$, and (ii), how the results stated through that point, including the associated asymptotics, extend to a family of block Wishart (or Laguerre) ensembles. After this, Section 2 lays out some basic facts on the spectral theory of block tridiagonal matrices along with the detailed definitions of our various matrix models. Section 3 provides an overview of the eigenvalue density derivations, identifying a certain moment calculation as fundamental (see Theorem \ref{thm:moment}). That calculation is spread over Sections 4 and 5, for moments $\beta s =2$ and $\beta s = 4$ respectively. Section 6 establishes a number of identities (and presents a conjecture in a related spirit) involving sums of Vandermonde determinant powers required in the preceding. Finally, Section 7 is devoted to asymptotics. \subsection{Soft edge asymptotics of $\HH(r,s)$} While it does not appear possible to compute correlations directly from the formulas \eqref{density1} or \eqref{density2}, the random operator approach is available. In the block setting this was developed by Bloemendal and Vir\'ag for the soft edge in \cite{Spike2}, and their approach applies to our case for any values of $r$ and $s$. In fact, it even applies in the $\beta=4$ case where we do not have statements about the joint eigenvalue densities. Introduce the $\beta =1,2,$ or $4$ matrix Brownian motion $B_x$ in dimension $r$: the independent, stationary increment process for which $B_y- B_x \sim B_{y-x}$ is distributed as $\sqrt{y-x}$ times a copy of $r \times r$ G(O/U/S)E. Next, for $\gamma > 0$, bring in the differential operator acting on $r$-dimensional vector valued functions on $\R_{+}$, \begin{equation}\label{eq:H_op} \mathcal{H}_{\beta, \gamma} = - \frac{d^2}{dx^2} + rx + \sqrt{\frac{2}{\gamma}} B'_x. \end{equation} When $\gamma=1$ this is the multivariate Stochastic Airy Operator of \cite{Spike2}. In particular, with a Dirichlet boundary condition at the origin, the spectrum of $-\mathcal{H}_{\beta} = -\mathcal{H}_{\beta, 1}$ is given by the $\operatorname{Airy}_\beta$ process, the edge scaling limit of the Gaussian beta ensemble. The largest value of this process (which is minus the ground state eigenvalue of $\mathcal{H}_{\beta}$), has classical Tracy-Widom distribution $TW_\beta$ with $\beta =1,2, 4$. \begin{theorem} \label{thm:limit_op} For any $r, s$ and $\beta=1,2,4$, let $\mathbf{T}_n \sim \HH(r,s)$. Denote by $\lambda_0^{(n)} < \lambda_1^{(n)} < \cdots $ the eigenvalues of the renormalized \begin{equation} \mathbf{H}_n = \gamma^{-1/2} (rn)^{1/6} \Bigl(2 \sqrt{(r+s)n} {I}_{rn} - \mathbf{T}_n \Bigr), \end{equation} and by $\Lambda_0 < \Lambda_1 < \cdots$ the Dirichlet eigenvalues of $ \mathcal{H}_{\beta, \gamma}$ with the choice $\gamma = \frac{r+s}{r}$ . Then the point process $\{ \lambda_0^{(n)} ,\lambda_1^{(n)} , \dots\}$ converges in law to $\{\Lambda_0, \Lambda_1, \dots \} $ as $n\to \infty$. \end{theorem} The proof of Theorem \ref{thm:limit_op} follows that of the main result of \cite{Spike2}, though we sketch an overview of the ideas in Section \ref {sec:asymptotics}. Similarly, Theorem 1.5 of \cite{Spike2} provides a second description of the limiting point process $\{ \Lambda_i \}_{i \ge 0}$ via matrix oscillation theory. Applying the same here yields: \begin{corollary} \label{cor:osc} Define the measure $\mathbb{P}$ on paths $\mbf{p}=(p_1, \dots p_r):[0,\infty) \mapsto ( -\infty, \infty]$ induced by the stochastic differential equation system \begin{equation} \label{mult_sde} dp_i = \frac{2}{\sqrt{\beta \gamma}} db_i + \left(\lambda + rx - p_i^2 + \sum_{j \neq i} \frac{2}{p_i - p_j} \right)dx,\qquad 1\le i \le r, \end{equation} starting from $(p_1(0), \cdots , p_r(0)) = \{\infty\}^r$ and entering $\{ p_1 < \cdots < p_r\}$ at $x>0$. Here $(b_1, \cdots b_k)$ is a standard real $r$-dimensional Brownian motion; $p_1$ can hit $-\infty$ in finite time, whereupon it is placed at $+\infty$ and the re-indexed process starts afresh. Then with $\Lambda_0< \Lambda_1< \cdots $ defined as in Theorem \ref{thm:limit_op}, it holds that \begin{align} P( \Lambda_k \le \lambda ) = \mathbb{P} ( x\mapsto \mbf{p}(x) \mbox{ explodes at most } {k} \mbox{ times } ) \end{align} for all $k \ge 0$. \end{corollary} The above corollary immediately implies that, whenever $\beta \gamma$ equals a classical value, {\em{i.e.}} $1,2,$ or $4$, we can deduce that the limiting edge point process corresponds to that of the G(O/U/S)E. In particular, in this case $\Lambda_0$ will have $TW_{\beta \gamma}$ distribution. This again is one of the primary take-aways of \cite{Spike2}. Due to the equivalence of the pre-limit models across different values of $r$, it is known that, again when the diffusion parameter is classical, the explosion times of \eqref{mult_sde} are equal in law for all $r\ge 1$. No direct proof of this striking fact is known. Specifying to the cases for which we have novel explicit joint eigenvalue densities, this implies: \begin{corollary} \label{cor:betalimit} Consider the random point process defined by the $r=2$, $\beta s = 2$ joint density \eqref{density1} in Theorem \ref{thm:main}. When $\beta=1$, the appropriately rescaled point process converges in law to the $\operatorname{Airy}_2$ point process. In the case of $r=2$ and $\beta s= 4$ the appropriately scaled process determined by \eqref{density2} in Theorem \ref{thm:main}converges in law to the $\operatorname{Airy}_4$ point process when $\beta=2$. In particular, in these cases the largest eigenvalues (after rescaling) converge to the classical $TW_2$ and $TW_4$ distributions, respectively. \end{corollary} Conjecturing that the $r$-fold diffusion characterization of Corollary \ref{cor:osc} provides the description of the $\operatorname{Airy}_{\beta \gamma}$ process for any $\beta \gamma>0$ we arrive to the following. \begin{conjecture} \label{con:betalimit} More generally, the point process scaling limit of \eqref{density1} is distributed as $\operatorname{Airy}_{\beta+2/r}$ for all $r \ge 2$ and $\beta =1$ or $2$. In the case of \eqref{density2} with $\beta s = 4$ and $\beta=1$, the point process scaling limit is $\operatorname{Airy}_{3}$. \end{conjecture} \subsection{Block Laguerre ensembles} In \cite{DE} the authors also produce $\beta$ generalizations of the classical Laguerre (Wishart) ensemble, showing that there is an $n\times n$ tridiagonal matrix model built out of independent $\chi$ variables for which the eigenvalue density is proportional to \begin{equation} \label{eig_DE1} \left\|\Delta(\lambda)\right\|^\beta \prod_{i=1}^n \lambda_i^{\frac{\beta}{2}(m-n+1) -1} e^{-\frac{\beta}{2} \sum_{i=1}^n \lambda_i} \mathbf{1}_{\R_+^n}. \end{equation} When $\beta =1$ or $2$ this coincides with that of the law of a sample covariance matrix for $m\ge n$ independent real or complex normal samples in dimension $n$. Along with $\beta$ now taking any positive value, the model behind \eqref{eig_DE1} allows $m$ to be generalized to any real number greater than $n-1$. We define the distribution $\mathtt{W}_{n,m, \beta}(r, s)$ on nonnegative definite block tridiagonals as follows. Let $\mathbf{L}_n$ be an $rn \times rn$ block bidiagonal matrix with independent $r\times r$ diagonal and upper offdiagonal blocks denoted by $\{\mbf{D}_i\}_{i=1,n}$ and $\{\mbf{O}_i\}_{i=1, n-1}$, that are lower and upper triangular matrices, respectively. Distribute these according to square-root Wishart matrices with parameters $(r, (r+s)(m+1 -i))$ and $(r, (r+s)(n-i))$, respectively. Then $\mathtt{W}_{n, ,m, \beta}(r, s)$ has the law $\mbf{L}_n \mbf{L}_n^\dagger$. Full details are provided in Definition \ref{def:BlockW}. Again, when $s=0$ this model has been considered previously in \cite{Spike2} and \cite{RR} in connection to eigenvalue spiking. In that case the underlying random matrix $\mbf{L}_n$ arises from an explicit block bi-diagonalization of an $rn \times rm$ matrix of independent Gaussians. Effectively the same considerations behind Theorem \ref{thm:main} imply the following. \begin{theorem}\label{thm:main_W} The joint eigenvalue density of $\mathtt{W}_{n, m, \beta}(r, s)$ for $\beta=1$ or $2$ has the form \eqref{density1} for general $r\ge 2$ and $\beta s=2$ and \eqref{density2} for $r =2$ and $\beta s =2$ or $4$ with an explicitly computable normalizing constant, the only change being that the Gaussian weight $ e^{-\frac{\beta}{4} \sum_{i=1}^{rn} \lambda_i^2}$ is replaced by $ \prod_{i=1}^{rn} \lambda_i^{\frac{\beta}{2}( (r+s)(m-n)+1)-1} e^{-\frac{\beta}{2} \lambda_i}$, restricted to $\R_{+}^{rn}$. \end{theorem} In terms of asymptotics, we focus on the choice $m = n +a $ for fixed $a > -1/(r+s)$ as $n \rightarrow \infty$ and look at the scaling limit of the smallest eigenvalues, which end up being in the vicinity of the origin. This is the random matrix hard edge, and introduces novel limiting phenomena beyond what we have seen for $\mathtt{H}_{n, \beta}(r, s)$. Note that it may proved along the same lines to Theorem \ref{thm:limit_op} that the suitably centered and scaled largest eigenvalues under $\mathtt{W}_{n, m, \beta}(r, s)$ will converge to those of $\mathcal{H}_{\beta, \gamma}$, for an appropriate $\gamma$, and the same is in fact true for the smallest eigenvalues when $\liminf_{n\to \infty} m/n>1$. For the hard edge, the characterizing limit operator is now of Sturm-Liouville type: again acting on $r$-dimensional vector valued functions, \begin{equation} \label{matrixgenerator} \mathcal{G}_{\beta, \gamma} = - e^{rx} \, {\bf{Z}_x} \frac{d}{dx} {\mbf{Z}_x^{-1} } \frac{d}{dx}. \end{equation} Here $x \mapsto {{\mbf{Z}}_x} $ is a symmetrized version of drifted Brownian on the general real or complex linear group dimension $r$, the parameters $\gamma$ and $a$ coefficients of the defining stochastic differential equation (see \eqref{WandA} below). Similar to $\mathcal{H}_{\beta, \gamma}$, the operator $\mathcal{G}_{\beta, \gamma}$ for $\gamma =1$ has previously been shown to characterize multi-spiked hard edge laws \cite{RR2} for $\beta =1,2,4$. For $\gamma=1$ and $r=1$ this is the Stochastic Bessel Operator introduced by Ram\'{\i}rez and Rider in \cite{RR}. In analogy with Theorem \ref{thm:limit_op} and Corollary \ref{cor:osc}, we have: \begin{theorem} \label{thm:limit_op1} For $\mbf{W}_n \sim \mathtt{W}_{ n, n+a, n}(r, s)$ denote by $0 < {\lambda}_0^{(n)} < {\lambda}_1^{(n)} < \cdots $ the point process of eigenvalues of $ \frac{rn}{\gamma} \, \mbf{W}_n$. As $n \rightarrow \infty$ this converges in law to the point process $0 < \hat{\Lambda}_0< \hat{\Lambda}_1 <\cdots $ of Dirichlet eigenvalues of $ \mathcal{G}_{\beta, \gamma}$ with $\gamma = \frac{r+s}{r}$. \end{theorem} The dependence on the many underlying parameters is made more explicit in the Riccati picture. \begin{corollary} \label{cor:osc1} Let $\mathbb{P}$ be the measure on (non-intersecting) paths ${\mathbf{q}}: [\mu, \infty) \mapsto [-\infty, \infty]^r$ defined by \begin{equation} \label{rrq} d q_{i} = \frac{2}{\sqrt{\beta \gamma}} q_{i} db_i + \left( \left(\frac{a}{\gamma} + \frac{2}{\beta \gamma}\right) q_{i} - q_{i}^2 - e^{-r x} + q_{i} \sum_{j \neq i} \frac{ q_{i} + q_{j}}{ q_{i}- q_{j} } \right) dx, \end{equation} started from $\{ \infty\}^r$ with the same ordering and re-indexing conventions upon possible passages to $-\infty$ described in Corollary \ref{cor:osc}. With $0 < \hat{\Lambda}_0< \hat{\Lambda}_1 <\cdots $ defined in Theorem \ref{thm:limit_op1} it holds \begin{equation} \label{HardEdge_zeros} P (\hat{\Lambda}_k > \lambda) = \mathbb{P} ( x \mapsto \mbf{q}(x) \mbox{ vanishes at most } k \mbox{ times } ) \end{equation} for any given $k = 0,1,\dots$. \end{corollary} And again, whenever $\beta \gamma = 1, 2$ or $4$ we conclude that the point process scaling limit of the smallest eigenvalues of ${\mathtt{W}}_{n, \beta} (r, s)$ is the classical hard edge, or Bessel, point process. More generally, we conjecture that these limits are given by the general $\beta \gamma$ hard edge process defined in \cite{RR}. In particular, versions of Corollary \ref{cor:betalimit} and Conjecture \ref{con:betalimit} are readily formulated. We record these at the end of Section 7. Having dealt with the soft and hard edge scaling limit of our models, it is natural to ask if the same can be done in the bulk case. The analogous results to \cite{Spike2} and \cite{RR2} for the bulk have not though yet been developed. Another natural future direction is to extend our results to circular ensembles using the results of \cite{KillipNenciu} as a starting point. \medskip \noindent\textbf{Acknowledgements.} The authors thank Philippe Di Francesco for pointing out reference \cite{DSZ}. B.V.~was partially supported by the University of Wisconsin – Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation and by the National Science Foundation award DMS-2246435. \section{Preliminaries} We start by outlining some basic facts on the spectral theory of block Jacobi matrices, then introduce the various distributions which we will work with. Throughout the paper we will use $\FF$ to denote $\R$ ($\beta=1$) or $\CC$ ($\beta=2$). In particular, we use $\FF$-hermitian and $\FF$-unitary for real symmetric/hermitian and orthogonal/unitary matrices. We use $\mbf{X}^\T$ to denote the transpose/conjugate transpose of an $\FF$-matrix $\mbf{X}$. \subsection{Block Jacobi matrices} We work with the following block generalization of tridiagonal Jacobi matrices. \begin{definition} Let $r, n\ge 1$. An $(rn)\times(rn)$ matrix $\mbf{T}$ is called an $\FF$-valued $r$-block Jacobi matrix if it is a $\FF$-hermitian block tridiagonal matrix built from $r\times r$ blocks satisfying the following conditions. The diagonal blocks $\mbf{A}_1, \dots, \mbf{A}_n$ are $r\times r$ $\FF$-hermitian matrices. The off-diagonal blocks $\mbf{B}_1, \dots, \mbf{B}_{n-1}$ above the diagonal are lower triangular with positive diagonal entries, see \eqref{eq:T}. We denote the set of such matrices by $\mathfrak{M}_{n,\beta, r}$. \begin{align}\label{eq:T} \mbf{T}= \left[\begin{array}{ccccc} \mbf{A}_1& \mbf{B}_1 & 0 &\dots & \\ \mbf{B}_1^{\dag} & \mbf{A}_2 &\mbf{B}_2 &\dots \\ 0&\ddots & \ddots & \ddots &0 \\ & 0 & \mbf{B}_{n-2}^\dag &\mbf{A}_{n-1} &\mbf{B}_{n-1} \\ & & 0 & \mbf{B}_{n-1}^\dag & \mbf{A}_n\\ \end{array} \right] \end{align} \end{definition} Note that an $r$-block Jacobi matrix can be viewed $(2r+1)$-diagonal band matrix with positive entries at the boundaries of the band. Let $\mbf{e}_{\lst{r}}=[\mbf{I}_r,\mbf{0}_{r\times (n-1)r}]^{\T}$ denote $(rn)\times r$ matrix built from the first $r$ coordinate vectors. (We do not explicitly denote the $n$-dependence.) The proof of the following theorem can be found for example in \cite{Spike2}, it relies on the Householder tridiagonalization algorithm in a block setting. \begin{theorem}[\cite{Spike2}]\label{thm:block_basic_1} Suppose that $\mbf{M}$ is an $\FF$-hermitian $rn\times rn$ matrix for which the matrix \begin{align}\label{eq:S1234} \mbf{S}=[\mbf{e}_{\lst{r}}, \mbf{M}\mbf{e}_{\lst{r}},\dots, \mbf{M}^{n-1}\mbf{e}_{\lst{r}}] \end{align} is invertible. Then there is an $\FF$-unitary matrix $\mbf{O}$ of the form $\mbf{I}_r\oplus \widetilde{\mbf{O}}$ and a unique $\mbf{T}\in \mathfrak{M}_{n,\beta, r}$, so that $\mbf{T}=\mbf{O}^{\T} \mbf{M} \mbf{O}$. The matrix $\mbf{O}$ can be chosen as the $\mbf{Q}$ in the unique QR decomposition $\mbf{S}=\mbf{Q}\mbf{R}$ for which $\mbf{R}$ has positive diagonal entries. \end{theorem} For $r=1$ the spectral measure of an $n\times n$ tridiagonal hermitian matrix $\mbf{T}$ with respect to the first coordinate vector $\mbf{e}_1$ is defined as the probability measure \begin{align}\label{eq:spec_m} \mu=\sum_{j=1}^n \|\mbf{v}_{j,1}\|^2 \delta_{\lambda_j}. \end{align} Here $\mbf{v}_{j,1}$ is the first coordinate of the normalized eigenvector corresponding to $\lambda_j$. Our next definition provides a natural extension of the spectral measure for $r$-block Jacobi matrices. \begin{definition} Suppose that $\mbf{M}$ is an $\FF$-hermitian $rn\times rn$ matrix. We define the spectral measure of $\mbf{M}$ with respect to $\mbf{e}_{\lst{r}}$ as the $r\times r$ matrix-valued measure \begin{align} \mu_{\lst{r}}=\sum_{j=1}^{rn} \mbf{v}_{j,\lst{r}} \cdot \mbf{v}_{j,\lst{r}}^{\T} \,\delta_{\lambda_j}. \end{align} Here $\mbf{v}_{j}$ is the normalized eigenvector corresponding to $\lambda_j$, and $\mbf{v}_{j,\lst{r}}\in \FF^r$ is the projection of $\mbf{v}_j$ to the first $r$ coordinates. \end{definition} Note that $\mu_{\lst{r}}$ only depends on the eigenspaces, so it is well-defined even though the choice of $\mbf{v}$ is not unique. If $\mbf{T}$ is the $r$-block Jacobi matrix obtained from an $\FF$-hermitian $\mbf{M}$ via Theorem \ref{thm:block_basic_1} then we have \begin{align} \int x^j d\mu_{\lst{r}}=\mbf{e}_{\lst{r}}^{\T} \mbf{M}^j \mbf{e}_{\lst{r}}= \mbf{e}_{\lst{r}}^{\T} \mbf{T}^j \mbf{e}_{\lst{r}}. \end{align} It can be shown that there is a one-to-one correspondence between the $r$-block Jacobi matrices and possible $r\times r$ matrix valued `probability' measures, see Section 2 of \cite{MOPUC}. \subsection{Random block matrices}\label{subs:matrix_distr} We start with an overview of the various distributions that serve as building blocks for our models, and then provide a precise definition of the $\HH(r,s)$ and $\WW(r,s)$ distributions. \begin{definition} The $\FF$-valued standard normal is denoted by $\FF N(0,1)$. The components are independent mean zero normals with variance $\frac{1}{\beta}$. The probability density function is proportional to $e^{-\frac{\beta}{2} \|x\|^2}$. \end{definition} We record the fact that if $\mbf{x}$ is a $d$-dimensional random vector with i.i.d.~$\FF N(0,1)$ entries then the distribution of $\|\mbf{x}\|$ is $\frac{1}{\sqrt{\beta}}\chi_{\beta d}$. The probability density function of $\|\mbf{x}\|$ is \[ 2\, \frac{ (\beta/2)^{\frac{\beta d}{2}}}{\Gamma(\beta d/2)} x^{\beta d-1} e^{-\frac{\beta}{2} x^2}. \] \begin{definition} Let $\mbf{Y}$ be an $n\times n$ matrix with i.i.d.~$\FF N(0,1)$ entries, and set $\mbf{X}=\frac1{\sqrt{2}} (\mbf{Y}+\mbf{Y}^{\T})$. The distribution of $\mbf{X}$ is called the $\FF$-valued Gaussian ensemble, or G$\FF$E$(n)$. For $\beta=1$ this is the Gaussian Orthogonal Ensemble (GOE), and for $\beta=2$ this is the Gaussian Unitary Ensemble (GOE). \end{definition} The diagonal entries of G$\FF$E are $N(0,\tfrac{2}{\beta})$ distributed, while the off-diagonal entries are i.i.d.~$\FF N(0,1)$. The entries are independent up to the real/hermitian symmetry. In the matrix variables the probability density function of G$\FF$E is proportional to $ e^{-\frac{\beta}{4} \Tr \mbf{X}\mbf{X}^{\T}}$. \begin{definition} Let $\mbf{Y}$ be an $n\times m$ (with $n\le m$) matrix with i.i.d.~$\FF N(0,1)$ entries. The distribution of the matrix $\mbf{X}=\mbf{Y}\mbf{Y}^T$ is called the $\FF$-valued Wishart distribution with parameters $(n,m)$. \end{definition} The following is a classical result in random matrix theory. \begin{theorem} The joint eigenvalue density of the $\FF$-valued $n\times n$ Gaussian ensemble is given by \eqref{eig_DE}. The distribution is called the Gaussian beta ensemble, and it is denoted by $G{\beta}E(n)$. The joint eigenvalue density of the $\FF$-valued Wishart distribution with parameters $(n,m)$ is given by \eqref{eig_DE1}. The distribution is called the Laguerre beta ensemble, and it is denoted by $L{\beta}E(n,m)$. In both cases the normalized eigenvectors can be chosen in a way so that the eigenvector matrix is Haar-distributed on the $n\times n$ $\FF$-unitary matrices while being independent of the eigenvalues. \end{theorem} \begin{definition} The $\FF$-valued square root Wishart matrix with parameters $n\le m$ is the distribution of the $n\times n$ lower triangular matrix $\mbf{X}$ with the following independent entries: \begin{align} x_{i,j}\sim \begin{cases} \FF N(0,1),& \qquad \text{if $i>j$},\\ \frac{1}{\sqrt{\beta}} \chi_{\beta (m+1-i)},& \qquad \text{if $i=j$},\\ 0,& \qquad \text{if $i<j$}. \end{cases} \end{align} We denote this distribution by $\SQW(n,m)$. \end{definition} We note that the joint probability density function of the non-zero entries of $\SQW(n,m)$ is proportional to \begin{align}\label{eq:SqW_pdf} \prod_{i>j} e^{-\frac{\beta}{2} \|x_{i,j}\|^2} \prod_{i=1}^n x_{i,i}^{\beta (m+1-i)-1} e^{-\frac{\beta}{2} x_{i,i}^2}=e^{-\frac{\beta}{2} \Tr \mbf{X}\mbf{X}^\T} \det(\mbf{X})^{\beta (m+1)-1} \prod_{i=1}^n x_{i,i}^{-\beta i}. \end{align} As the following classical result due to Bartlett \cite{Bartlett1933} shows, $\SQW(n,m)$ gives the distribution of the Cholesky factor of the Wishart distribution. \begin{theorem}[\cite{Bartlett1933}]\label{thm:bartlett} Suppose that the matrix $\mbf{X}$ has $\FF$-valued Wishart distribution with parameters $(n,m)$. Let $\mbf{R}$ be the lower triangular square root of $\mbf{X}$ with almost surely positive diagonal entries: $\mbf{X}=\mbf{R} \mbf{R}^{\T}$. Then $\mbf{R}$ has $\SQW(n,m)$ distribution. \end{theorem} We can now introduce the family of random block matrices that we study. \begin{definition} \label{def:BlockH} Let $r,n\ge 1$ and $s\ge 0$. We denote by $\HH(r,s)$ the distribution of the $\FF$-valued random $r$-block Jacobi matrix of size $(rn)\times(rn)$ with independent blocks $\mbf{A}_k, \mbf{B}_k$ where $\mbf{A}_k\sim$ G$\FF$E$(r)$ and $\mbf{B}_k\sim \SQW(r,(r+s)(n-k))$. \end{definition} Note that $\HH(1,0)$ is just the distribution of the tridiagonal matrix of Dumitriu and Edelman (and Trotter) given for the Gaussian beta ensemble. As the following theorem shows, for $r\ge 1$ the $\HH(r,0)$ distribution is the result of the $r$-block Householder process applied to G$\FF$E$(rn)$. \begin{theorem}[\cite{Spike2}]\label{thm:GFE_block} Let $\mbf{M}$ have G$\FF$E$(rn)$ distribution, and consider the matrix $\mbf{S}$ defined via \eqref{eq:S1234}. Then $\mbf{S}$ is a.s.~invertible, and the $r$-block Jacobi matrix $\mbf{T}$ produced by Theorem \ref{thm:block_basic_1} has $\HH(r,0)$ distribution. The eigenvalues of $\mbf{T}$ are distributed as $G\beta E(rn)$, and the normalized eigenvector matrix $\mbf{V}=[\mbf{v}_{i,j}]_{i,j\in \lst{rn}}$ can be chosen in a way so that the first $r$ rows of $\mbf{V}$ are independent of the eigenvalues and have the same distribution as the first $r$ rows of an $rn\times rn$ Haar $\FF$-unitary matrix. \end{theorem} Theorem \ref{thm:GFE_block} fully describes the distribution of the matrix valued spectral measure $\mu_{\lst{r}}$ of $\mbf{T}$. In particular, it shows that the weights and the support are independent of each other, and the weights can be obtained from a Haar $\FF$-unitary matrix. \begin{definition}\label{def:BlockW} Let $r,n\ge 1$, $m>-1/r$, and $s\ge 0$. Let $\mathbf{L}$ be an $rn \times rn$ block bidiagonal matrix with independent $r\times r$ diagonal and upper offdiagonal blocks denoted by $\{\mbf{D}_i\}_{i=1,n}$ and $\{\mbf{O}_i\}_{i=1, n-1}$ with $\mbf{D}_i^{\T}\sim \SQW(r,(r+s)(m+1-i))$ and $\mbf{O}_i\sim \SQW(r,(r+s)(n-i))$. We denote the distribution of $\mbf{W}=\mbf{L}\mbf{L}^{\T}$ by $\WW(r,s)$. \end{definition} Again, $\WW(1,0)$ is just the tridiagonal model given by Dumitriu and Edelman for the Laguerre beta ensemble. The analogue of Theorem \ref{thm:GFE_block} holds. \begin{theorem}[\cite{Spike2}]\label{thm:W_block} Let $\mbf{M}$ have $\FF$-valued Wishart distribution with parameters $(rn,rm)$, and consider the matrix $\mbf{S}$ defined via \eqref{eq:S1234}. Then $\mbf{S}$ is a.s.~invertible, and the $r$-block Jacobi matrix $\mbf{T}$ produced by Theorem \ref{thm:block_basic_1} has $\WW(r,0)$ distribution. The eigenvalues of $\mbf{T}$ are distributed as $L\beta E(rn,rm)$, and the normalized eigenvectors can be chosen in a way that the first $r$ rows are independent of the eigenvalues and have the same distribution as the first $r$ rows of an $rn\times rn$ Haar $\FF$-unitary matrix. \end{theorem} \section{New distributions via biasing} We start this section with a brief review of the Dumitriu-Edelman result \cite{DE}. We introduce the key tools for our block generalization and provide the proofs of our main theorems modulo a certain moment computation that is delayed to the subsequent sections. \subsection{Revisiting the Hermite beta ensemble} For completeness, we state the Dumitriu-Edelman result in full and provide a proof which foreshadows the techniques used to prove Theorem \ref{thm:main}. \begin{theorem}[\cite{DE}]\label{thm:DE} Fix $\beta>0$ and an integer $n\ge 1$. Let $a_1,\dots, a_n, b_1, \dots, b_{n-1}$ be independent random variables with $a_j\sim N(0,\tfrac{2}{\beta})$, $b_j\sim \frac{1}{\sqrt{\beta}}\chi_{\beta (n-j)}$. Then the symmetric tridiagonal matrix $\mbf{T}$ with diagonal $a_1,a_2,\dots$ and off-diagonal $b_1,b_2, \dots$ has a joint symmetrized eigenvalue density on $\R^n$ given by \ \begin{align}\label{eq:GbE} \frac{1}{Z_{n,\beta}} \left\|\Delta(\lambda)\right\|^\beta e^{-\frac{\beta}{4} \sum_{j=1}^n \lambda_j^2}, \end{align} with \begin{align}\label{eq:GbE_constant} Z_{n,\beta}={n!} (2\pi)^{n/2} (\beta/2)^{-\frac{\beta}{4}n(n-1)-\frac{n}{2}} \,\Gamma(\beta/2)^{-n} \prod_{j=1}^n \Gamma(\beta j/2). \end{align} Moreover, the spectral weights of $\mbf{T}$ corresponding to the first coordinate vector have Dirichlet$(\beta/2,\dots, \beta/2)$ joint distribution, and this weight vector is independent of the eigenvalues. \end{theorem} \begin{proof} Consider an $n\times n$ Jacobi matrix $\mbf{T}$ with diagonal entries $a_1,\dots, a_n$ and off-diagonal positive entries $b_1, \dots, b_{n-1}$. Denote by $p_j$ the spectral weight of $\lambda_j$ in the spectral measure \eqref{eq:spec_m}. It is well known that \begin{align}\label{eq:magic_Delta_p} \|\Delta({\lambda})\|= \prod_{k=1}^n p_k^{-1/2} \prod_{k=1}^{n-1} b_k^{(n-k)}, \end{align} see for instance eq.~1.148 of \cite{ForBook}. We also take as given that the theorem holds for $\beta=1$ due to \cite{Trotter}, and the fact that the Householder tridiagonalization process does not change the spectral measure with respect to the first coordinate. Next, for $\mbf{T}$ be a random tridiagonal matrix defined in the statement with $\beta=1$, introduce a biased version of the distribution of $\mbf{T}$ with the biasing function \[ g_\beta(\mbf{b})=\prod_{k=1}^{n-1} b_k^{(\beta-1)(n-k)}. \] The biasing produces a random tridiagonal matrix $\mbf{\wt{T}}$ where the diagonal and off-diagonal entries are still independent, the distribution of the diagonal entries is still $N(0,2)$, but the distribution of the $k$th off-diagonal entry has changed from $\chi_{n-k}$ to $\chi_{\beta(n-k)}$. By \eqref{eq:magic_Delta_p} we have \begin{align}\label{eq:bias_DE} g_\beta(\mbf{b})=\|\Delta({\lambda})\|^{\beta-1} \prod_{k=1}^n p_k^{-\frac{\beta-1}{2}}, \end{align} hence biasing the entries of $\mbf{T}$ with $g_\beta(\mbf{b})$ is the same as biasing the spectral variables $\lambda, \mbf{p}$ with the appropriate product on the right hand side of \eqref{eq:bias_DE}. This immediately implies that the eigenvalues and spectral weights of $\mbf{\wt{T}}$ are still independent of each other, that the joint eigenvalue density of $\mbf{\wt{T}}$ is proportional to $\|\Delta(\lambda)\|^\beta e^{-\frac{1}{4}\sum_{k=1}^n \lambda_k^2}$, and that its spectral weights have Dirichlet$(\beta/2,\dots,\beta/2)$ distribution. The complete statement of the theorem now follows after scaling $\mbf{\wt{T}}$ by $ \frac{1}{\sqrt{\beta}}$. The value of the normalizing constant $Z_{n,\beta}$ follows from the known $\beta=1$ factor (see eq.~1.160 of \cite{ForBook}) along with an evaluation of $E[g_\beta(\mbf{b})]$. \end{proof} \begin{remark} The original proof of Theorem \ref{thm:DE} given in \cite{DE} is slightly different. It uses the $\beta=1$ case as a starting point to derive an expression for the Jacobian of the one-to-one transformation $(\mbf{a},\mbf{b}) \mapsto ({\lambda}, \mbf{p})$. It then uses the identity \eqref{eq:magic_Delta_p} to compute the joint density of the spectral variables $({\lambda}, \mbf{p})$ for the tridiagonal matrix $\mbf{T}$. As it was already remarked in \cite{DE}, the expression for the Jacobian of the transformation $(\mbf{a},\mbf{b}) \mapsto ({\lambda}, \mbf{p})$ can also be computed directly, without relying on Theorem \ref{thm:DE} being true in the $\beta=1$ case. \end{remark} \subsection{Key spectral identity} We establish a block Jacobi matrix equivalent of the identity \eqref{eq:magic_Delta_p} which relates products of powers of the determinants of the off-diagonal blocks to eigenvalues and eigenvector entries. First we need the following definition. \begin{definition} For $\lambda = \{ \lambda_j\}_{j \in \lst{rn} }$ and an $r\times rn $ matrix $\mbf{X}$ define the $rn \times rn$ matrix $\mbf{M}=\mbf{M}({\lambda}, \mbf{X})$ entry-wise as \begin{align}\label{eq:MagicM_ij} M_{i,j}=\lambda_i^{\nint{j}{r}} x_{ \nfr{j}{r},i}, \end{align} in which \begin{align} \nint{j}{r} :=\left\lfloor \frac{j-1}{r} \right\rfloor, \qquad \nfr{j}{r}:=j-r \nint{j}{r}. \end{align} Note that the shift in $ \nint{j}{r}$ means that $\lfloor r \rfloor_r = 0$ and hence $\nfr{j}{r}\in \lst{r}$. For an $rn \times rn$ matrix $\mbf{X}$ we define $\mbf{M}({\lambda}, \mbf{X})$ using the $r \times rn$ submatrix of the first $r$ rows of $\mbf{X}$. \end{definition} The operation $({\lambda}, \mbf{X})\mapsto \mbf{M}$ can be realized as a block-matrix built from the row-by-row Kronecker products of the $rn\times r$ matrix $\mbf{X}^T$ and the $rn\times n$ Vandermonde type matrix $\mbf{\Lambda}=(\lambda_{i}^{j-1})_{i\in \lst{rn},j\in \lst{n}}$. Similar constructions show up in the literature as the ``face-splitting product'' or Khatri–Rao product \cite{KR_1968}. As an example, \begin{align}\label{eq:M6} \mbf{M}({\lambda}, \mbf{X})= \left( \begin{array}{cccccc} x_{1,1} & x_{2,1} & \lambda_1 x_{1,1} & \lambda_1 x_{2,1} & \lambda_1^2 x_{1,1} & \lambda_1^2 x_{2,1} \\ x_{1,2} & x_{2,2} & \lambda_2 x_{1,2} & \lambda_2 x_{2,2} & \lambda_2^2 x_{1,2} & \lambda_2^2 x_{2,2} \\ x_{1,3} & x_{2,3} & \lambda_3 x_{1,3} & \lambda_3 x_{2,3} & \lambda_3^2 x_{1,3} & \lambda_3^2 x_{2,3} \\ x_{1,4} & x_{2,4} & \lambda_4 x_{1,4} & \lambda_4 x_{2,4} & \lambda_4^2 x_{1,4} & \lambda_4^2 x_{2,4} \\ x_{1,5} & x_{2,5} & \lambda_5 x_{1,5} & \lambda_5 x_{2,5} & \lambda_5^2 x_{1,5} & \lambda_5^2 x_{2,5} \\ x_{1,6} & x_{2,6} & \lambda_6 x_{1,6} & \lambda_6 x_{2,6} & \lambda_6^2 x_{1,6} & \lambda_6^2 x_{2,6} \\ \end{array} \right), \end{align} illustrates the $n=3$, $r=2$ case. The advertised formula, recorded next, involves the determinant of $\mbf{M}$. \begin{proposition}\label{prop:magic} Suppose that $\mbf{T}\in \mathfrak{M}_{n,\beta, r}$ with blocks $\mbf{A}_j, \mbf{B}_j$. Then \begin{align}\label{eq:magic11} \prod_{j=1}^{n-1} \det(\mbf{B}_j)^{n-j}=\|\det \mbf{M}({\lambda}, \mbf{Q})\|, \end{align} where $\mbf{\lambda}$ are the eigenvalues of $\mbf{T}$, and $\mbf{Q}$ is the matrix of the normalized eigenvectors of $\mbf{T}$ ordered according to $\lambda$. \end{proposition} When $r=1$ one can easily see from \eqref{eq:M6} that $\det {\mbf{M}(\lambda, \mbf{Q})} = \Delta(\lambda) \prod_{i=1}^n q_{1,i}$ upon factoring out the eigenvector coordinates from each row, recovering the identity \eqref{eq:magic_Delta_p}. We remark that for a particular $\mbf{T}$ the choice of $\mbf{Q}$ is not unique: eigenvectors could be multiplied by a unit phase, and there is even more freedom if $\mbf{T}$ has eigenvalues with multiplicity higher than one. The proof below shows that the expression on the right hand side of \eqref{eq:magic11} does not depend on the particular choice of $\mbf{Q}$. \begin{proof}[Proof of Proposition \ref{prop:magic}] Consider the $rn \times rn $ matrix $\mbf{S}$ from \eqref{eq:S1234} built from $\mbf{T}$. This is an upper triangular block matrix where the $j$th diagonal block is $\mbf{B}_{j-1}^{\T}\cdots \mbf{B}_1^{\T}$, the first diagonal block being $\mbf{I}_r$. The determinant of $\mbf{S}$ is the product of the determinants of its diagonal blocks: \[ \det \mbf{S}=\prod_{j=2}^n \det(\mbf{B}_{j-1}^{\T}\cdots \mbf{B}_1^{\T})=\prod_{j=1}^{n-1} \det(\mbf{B}_j)^{n-j}. \] Denote by $\boldsymbol{\Lambda}$ the diagonal matrix built from $\lambda$. Then we have $\mbf{T}=\mbf{Q}\boldsymbol{\Lambda} \mbf{Q}^{\T}$ and \begin{align} \mbf{T}^{j} \mbf{e}_{\lst{r}}= \mbf{Q} \boldsymbol{\Lambda} \mbf{Q}^{\T}\mbf{e}_{\lst{r}}. \end{align} Hence $\mbf{S}= {\mbf{Q}} \widetilde{\mbf{S}}$, and $\|\det \mbf{S}\|=\|\det \wt{\mbf{S}}\|$ with \begin{equation} \widetilde{\mbf{S}}=[\mbf{Q}^{\T}\mbf{e}_{\lst{r}}, \boldsymbol{\Lambda}\mbf{Q}^{\T}\mbf{e}_{\lst{r}}, \dots, \boldsymbol{\Lambda}^{n-1}\mbf{Q}^{\T}\mbf{e}_{\lst{r}}]=\mbf{M}(\mbf{\lambda}, \overline{\mbf{Q}}). \end{equation} Since $\|\det \mbf{M}(\mbf{\lambda}, \overline{\mbf{Q}})\|=\|\det \mbf{M}(\mbf{\lambda}, {\overline{}\mbf{Q}})\|$, the proposition follows. \end{proof} We remark that for $r=2$ and $\lambda_j>0$ the following remarkable identity holds for $\det \mbf{M}({\lambda}, \mbf{Q})$ (see equation (52) in \cite{LT2006}): \begin{align}\label{eq:Pfaffian_detM} \det \mbf{M}({\lambda}, \mbf{Q})=\prod_{1\le i<j\le 2n}(\sqrt{\lambda_i}+\sqrt{\lambda_j}) \, \operatorname{Pf}\left(\frac{q_{1,i}q_{2,j}-q_{1,j}q_{2,i}}{\sqrt{\lambda_i}+\sqrt{\lambda_j}}\right). \end{align} We are not aware of extensions of this result for $r>2$. \subsection{Proofs of Theorem \ref{thm:main} and \ref{thm:main_W}} We are now ready to present the proofs of our main theorems, modulo one key ingredient. \begin{proof}[Proof of Theorem \ref{thm:main} -- first steps] The joint density of the entries of a matrix $\mbf{T}$ distributed as $\HH(r,s)$ is given by \begin{align} f(\mbf{A},\mbf{B})=C_{\beta,n,r,s} \exp(-\frac{\beta}{4} \Tr \mbf{T} \mbf{T}^{\T}) \prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta(r+s)(n-m)+\beta-1} \prod_{m=1}^{n-1} \prod_{j=1}^r (b_{j,j}^{(m)})^{-\beta j}, \end{align} where $C_{\beta, n, r, s}$ is an explicitly computable normalizing constant. Note that the distribution $\HH(r,s)$ can be realized as the biased version of the distribution $\HH(r,0)$ with the biasing function $ \prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta s(n-m)}$. Using $E_{s}$ for the expectation with respect to $\HH(r,s)$ (suppressing the dependence on $\beta, n, r$) we have, for any (bounded) test function $h$, \begin{align} E_{s}[h(\mbf{T})] & =\frac{E_{0}\left[h(\mbf{T})\prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta s(n-m)} \right]} {E_{0}\left[\prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta s(n-m)} \right]} \\ & := \frac{1}{Z_n} E_{0}\left[h(\mbf{T})\prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta s(n-m)} \right]. \nonumber \end{align} In particular, this is true if $h(\mbf{T})=g({\lambda})$ with a (bounded) test function $g$. By Proposition \ref{prop:magic} we have \begin{align}\label{eq:magic_identity} \prod_{m=1}^{n-1} (\det \mbf{B}_m)^{\beta s(n-m)}=\|\det \mbf{M}({\lambda}, \mbf{Q})\|^{\beta s} \end{align} where $\mbf{Q}$ is the $r\times (rn)$ matrix of the first $r$ coordinates of the normalized eigenvectors of $\mbf{T}$. By Theorem \ref{thm:GFE_block}, $\mbf{Q}$ and ${\lambda}$ are independent under the distribution $\HH(r,0)$, and $\mbf{Q}$ is distributed as the first $r$ rows of a Haar $\FF$-unitary matrix. This implies that \begin{align} E_{s}[g({\lambda})] & = \frac{1}{Z_n} E_{0}\left[g({\lambda}) F_{\beta,r,n, \beta s}({\lambda})\right] \end{align} where \begin{align} \label{Reweight} F_{\beta, r,n, \beta s}({\lambda}) & = E_{\boldsymbol{Q}} \left[\|\det \mbf{M}({\lambda}, \mbf{Q})\|^{\beta s}\right]. \end{align} In other words, the distribution of ${\lambda}$ is just G$\beta$E biased by the functional $ F_{\beta, r,n,\beta s}({\lambda})$. As for the normalizer, the left hand side of \eqref{eq:magic_identity} readily yields \begin{align}\nonumber Z_n &= E_{0} \left[ \prod_{m=1}^{n-1}(\det \mbf{B}_m)^{(s)(n-m)} \right] = \prod_{m=1}^{n-1} \prod_{i=1}^r E (b^{(m)}_{i,i})^{\beta s(n-m)} \\ \nonumber & = \prod_{m=1}^{n-1} \prod_{i=1}^r (\beta/2) ^{-\frac{\beta s}{2}(n-m)}\frac{ \Gamma \left(\frac{\beta}{2}((r+s)(n-m)-i+1 ))\right)}{\Gamma \left(\frac{\beta}{2}{((r(n-m)-i+1) }\right)}\\&=(\beta/2)^{-\frac{\beta }{4} r s n(n-1)} \prod_{m=1}^{n-1} \prod_{i=1}^r \frac{ \Gamma \left(\frac{\beta}{2}((r+s)(n-m)-i+1 )\right)}{\Gamma \left(\frac{\beta}{2}(r(n-m)-i+1) \right)}.\label{eq:bias_const} \end{align} since the $b^{(m)}_{i,i}$ are independent $ \frac{1}{\sqrt{\beta}} \times \chi_{\beta (r(n-m)+1-i)}$ variables. \end{proof} Identifying the joint eigenvalue density for any $\HH(r,s)$ model then comes down to computing the determinant moments indicated in \eqref{Reweight}. We have been able to do this only for general $r\ge 2$ and for $\beta s=2$, and $r=2, \beta s=4$. In particular, the proof of Theorem \ref{thm:main} is finished by establishing the following. \begin{theorem} \label{thm:moment} With $\boldsymbol{Q}$ Haar distributed $\FF$-unitary matrix we have \begin{align} E_{\mbf{Q}} \| \det \mbf{M}({\lambda}, \mbf{Q}) \|^{\beta s} = c_{n, \beta, r,\beta s} \times \begin{cases} \sum\limits_{(\cA_1,\dots, \cA_r)\in \cP_{r,n}} \prod_{j=1}^r \Delta(\cA_j)^2, & \mbox{ for } r \ge2, \beta s=2,\\[10pt] \sum\limits_{(\cA,\cA')\in \cP_{2,n}} \Delta(\cA)^4 \Delta(\cA')^4, & \mbox{ for } r=2, \beta s = 4, \end{cases} \end{align} where \begin{align} c_{n,\beta, r, \beta s} =(\beta/2) ^{\frac{\beta}{2}rsn} \prod_{i=1}^{r} \frac{\Gamma \left(\frac{\beta }{2}(rn+1-i)\right)}{ \Gamma \left(\frac{\beta}{2}((r+s)n+1-i) \right)} \times \begin{cases} 1, \qquad &r\ge2, \beta s=2,\\ (12/\beta)^n, \qquad &r=2, \beta s=4. \end{cases} \end{align} \end{theorem} One immediately recognizes the interaction term reported in \eqref{density1}. To arrive at the form of $r=2$, $s=2,4$ densities appearing in \eqref{density2} we will also prove: \begin{equation} \label{id_quad_to_square} \sum_{ (\cA,\cA')\in \cP_{2,n} } \Delta (\cA)^4 \Delta (\cA')^4 = 2^{-n} \left(\sum_{ (\cA,\cA')\in \cP_{2,n} } \Delta (\cA)^2 \Delta (\cA')^2\right)^2, \end{equation} and \begin{align} \label{id_pfaff} \sum_{ (\cA,\cA')\in \cP_{2,n} } \Delta(\cA)^2 \Delta (\cA')^2&=(-2)^n \Delta (\lambda) \Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right). \end{align} The proof of Theorem \ref{thm:moment} is divided between Sections \ref{sec:2moment} and \ref{sec:4moment}, for $s=2$ and $s=4$ respectively. The identities \eqref{id_quad_to_square} and \eqref{id_pfaff} are proven in Section \ref{sec:det_identities} along with a number of related determinantal formulas used throughout. \begin{proof}[Proof of Theorem \ref{thm:main} -- final steps] We have shown that the joint eigenvalue distribution of $\HH(r,s)$ is just $G\beta{E}(rn)$ biased by the functional $ F_{\beta, r,n,\beta s}({\lambda})$ given in \eqref{Reweight}. Theorem \ref{thm:moment} provides $ F_{\beta, r,n,\beta s}({\lambda})$ for $r\ge 2$ and $\beta s=2$ and for $r=2$, $\beta s=4$. When $r\ge 2$, $\beta s=2$ this gives the joint density function \eqref{density1}. The normalizing constant $Z_{n,\beta, r, \beta s}$ satisfies \begin{align}\label{eq:norm_const} Z_{n,\beta,r,\beta s}=Z_{nr, \beta} Z_n c_{n,\beta, r, \beta s}^{-1}, \end{align} where $Z_{nr,\beta}$ is the normalizing constant in \eqref{eq:GbE_constant}, $Z_n$ is given in \eqref{eq:bias_const} and $c_{n,\beta, r, \beta s}$ is given in Theorem \ref{thm:moment}. When $r=2$ and $\beta s=2$ or 4 then we also use \eqref{id_quad_to_square} and \eqref{id_quad_to_square} to rewrite expression in Theorem \ref{thm:moment} with a Pfaffian to obtain the joint density function \eqref{density2}. The normalizing constant is again given by \eqref{eq:norm_const}, note that we needed the additional $2^n$ factor in \eqref{density2} to match the two forms of the $r=2$, $\beta s=2$ case. The reported constant in Theorem \ref{thm:main} now follows after some algebra, noting that \begin{align} \prod_{m=1}^{n-1}\prod_{i=1}^r \frac{ \Gamma \left(\frac{\beta}{2}((r+s)(n-m)-i+1 )\right)}{\Gamma \left(\frac{\beta}{2}(r(n-m)-i+1) \right)} \prod_{i=1}^{r} \frac{ \Gamma \left(\frac{\beta}{2}((r+s)n+1-i) \right)}{\Gamma \left(\frac{\beta }{2}(rn+1-i)\right)} = \prod_{k=1}^{rn} \frac{\Gamma\left(\frac{\beta}{2} \left(k+s \lceil\frac{k}{r} \rceil\right)\right)}{\Gamma\left(\frac{\beta}{2} k\right)}. \end{align} \end{proof} The proof of Theorem \ref{thm:main_W} follows the same biasing idea as the proof of Theorem \ref{thm:main}. \begin{proof}[Proof of Theorem \ref{thm:main_W}] Let $\mbf{T}=\mbf{L}\mbf{L}^{\T}\sim \WW(r,s)$, with $\mbf{L}$ as in Definition \ref{def:BlockW}. (Note that $\mbf{L}$ is a function of $\mbf{T}$.) The joint density of the entries $\mbf{O}_m, \mbf{D}_m$ of $\mbf{L}$ are given by \begin{align} f(\mbf{D}, \mbf{O})=&C_{\beta, n,m,r,s} \exp(-\frac{\beta}{2}\Tr \mbf{T} \mbf{T}^{\T}))\prod_{j=1}^n (\det \mbf{D}_j)^{\beta (r+s)(m-j)+\beta-1} \\ & \times \prod_{j=1}^{n-1} (\det \mbf{O}_j)^{\beta (r+s)(n-j)+\beta-1}\prod_{j=1}^n \prod_{k=1}^r (d^{(j)}_{k, k})^{-\beta k} \prod_{j=1}^{n-1} \prod_{k=1}^r (o^{(j)}_{k, k})^{-\beta k}, \end{align} where $C_{\beta,n,m,r,s}$ is an explicitly computable normalizing constant. As before, we can realize $\WW(r,s)$ using a biased version of $\WW(r,0)$. Using $E_s$ for the expectation with respect to $\WW(r,s)$ we have, for any bounded test function $h$, \begin{align} E_{s}[h(\mbf{T})] & =\frac{E_{0}\left[h(\mbf{T})\prod_{j=1}^n (\det \mbf{D}_j)^{\beta s(m-j)} \prod_{j=1}^{n-1} (\det \mbf{O}_j)^{\beta s(n-j)} \right]} {E_{0}\left[\prod_{j=1}^n (\det \mbf{D}_j)^{\beta s(m-j)} \prod_{j=1}^{n-1} (\det \mbf{O}_j)^{\beta s(n-j)} \right]} \\ & := \frac{1}{Z_{n,m}} E_{0}\left[h(\mbf{T})\prod_{j=1}^n (\det \mbf{D}_j)^{\beta s(m-j)} \prod_{j=1}^{n-1} (\det \mbf{O}_j)^{\beta s(n-j)} \right]. \nonumber \end{align} The matrix $\mbf{T}=\mbf{L} \mbf{L}^{\T}$ has off-diagonal blocks $\mbf{B}_j=\mbf{D}_j \mbf{O}_j$, hence by Proposition \ref{prop:magic} we have \begin{align} \prod_{j=1}^{n-1} \det(\mbf{D}_j \mbf{O}_j)^{n-j}=\prod_{j=1}^{n} \det(\mbf{D}_j)^{n-j} \prod_{j=1}^{n-1} (\det \mbf{O}_j)^{n-j}=\|\det \mbf{M}({\lambda}, \mbf{Q})\|, \end{align} where $\mbf{Q}$ is the matrix of the normalized eigenvalue matrix of $\mbf{T}$. We also have \begin{align} \prod_{j=1}^{rn} \lambda_j=\det \mbf{T}=(\det \mbf{L})^2=\prod_{j=1}^n (\det \mbf{D}_j)^2. \end{align} Using $h(\mbf{T})=g({\lambda})$ with a (bounded) test function $g$ we now get \begin{align} E_{s}[g({\lambda})] & = \frac{1}{Z_n} E_{0}\left[g({\lambda}) F_{\beta,r,n,m, s}({\lambda})\right] \end{align} where \begin{align} \label{Reweight1} F_{\beta, r,n, m,s}({\lambda}) & =\prod_{j=1}^{rn} \lambda_j^{\frac{\beta}{2}s(m-n)} \times E_{\boldsymbol{Q}} \left[\|\det \mbf{M}({\lambda}, \mbf{Q})\|^{\beta s} \right] \end{align} Hence the distribution of $\lambda$ under $\WW(r,s)$ is just L$\beta$E$(rn,rm)$ biased by $F_{\beta, r,n, m,s}({\lambda})$. The statement of the theorem now follows from Theorem \ref{thm:moment} and the identities \eqref{id_quad_to_square} and \eqref{id_pfaff}. The normalizing constant can be explicitly evaluated using a similar argument as in Theorem \ref{thm:main}. \end{proof} \subsection{A reduction and a representation} We end this section with two lemmas that play significant roles in our computations of the moments of $\det \mbf{M}({\lambda}, \mbf{Q})$. The first observation is that in computing moments of $\det \mbf{M}({\lambda}, \mbf{Q})$ we may replace the Haar distributed $\mbf{Q}$ with a matrix of independent Gaussians. \begin{lemma}[Gaussian reduction lemma]\label{lem:Gauss_red} Let $\mbf{X}$ be an $rn \times rn$ matrix of $i.i.d.$ $\mathbb{F} N(0,1)$ entries, and $\mbf{Q}$ a Haar distributed $\mathbb{F}$-unitary matrix. Let $R_{i,i}\sim \frac{1}{\sqrt{\beta}} \chi_{\beta(rn+1-i)}, 1\le i\le r$ be independent of each other and of $\mbf{Q}$. Then for any fixed collection $\lambda_i\in \R, i \in \lst{rn}$ we have the following distributional identity: \begin{align}\det \mbf{M}({\lambda}, \mbf{X})\eqd \prod_{i=1}^{r} R_{i,i}^n \times \det \mbf{M}({\lambda}, \mbf{Q}). \end{align} It follows that for any $s>0$ we have \begin{align} E \|\det \mbf{M}({\lambda}, \mbf{Q})\|^{\beta s}=\kappa_{n,\beta, n,r, \beta s} \cdot E \|\det \mbf{M}({\lambda}, \mbf{X})\|^{\beta s}, \end{align} with $\kappa_{n,\beta, n,r, \beta s} =(\beta/2) ^{\frac{1}{2}rsn}\times \prod_{i=1}^{r} \frac{\Gamma \left(\frac{\beta }{2}(rn+1-i)\right)}{ \Gamma \left(\frac{\beta}{2}((r+s)n+1-i) \right)}$. \end{lemma} \begin{proof} Denote the columns of $\mbf{X}$ by $\mbf{X}_i, i\in \lst{rn}$. Consider the QR decomposition $\mathbf{X}=\mathbf{Q}\mathbf{R}$ of the matrix $\mathbf{X}=[\mathbf{X}_1, \dots, \mathbf{X}_{nr}]$ where the diagonal elements of $\mbf{R}$ are chosen to be positive real numbers. From the $\FF$-unitary invariance of $\mathbf{X}$ it follows that $\mathbf{Q}$ and $\mathbf{R}$ are independent and $\mathbf{Q}$ is a Haar distributed $\mathbb{F}$-unitary matrix. (See e.g.~\cite{Meckes2019}.) By Theorem \ref{thm:bartlett} we also know that the diagonal entries $R_{i,i}, i\in \lst{rn}$ of $\mathbf{R}$ are independent with ${R}_{i,i}\sim \frac{1}{\sqrt{\beta}}\chi_{\beta(rn+1-i)}$. From $\mbf{X}=\mbf{Q}\mbf{R}$ we have \[ \mbf{X}_i=R_{i,i} \mbf{Q}_i+\sum_{j=1}^{i-1} R_{j,i} \mbf{Q}_j, \] and using elementary row operations we get \begin{align}\label{eq:magic_dist} \det\mbf{M}({\lambda}, \mbf{X})=\prod_{a=1}^r R_{a,a}^n \cdot \det\mbf{M}({\lambda}, \mbf{Q}). \end{align} Using the independence of $\mbf{R}$ and $\mbf{Q}$ the statement of the lemma now follows. \end{proof} Our second observation is that, by regrouping terms in its Laplace expansion, the determinant of $\mbf{M}(\lambda, \mbf{Q})$ can be written as a weighted sum of Vandermonde determinant products. We will index the permutations in this expansion in a particular way, and this appears frequently enough that we isolate its definition. \begin{definition} For $\underline{\cA}=(\cA_1,\dots,\cA_r)\in \cP_{r,n}$ let $\sigma_{\underline{\cA}}\in S_{rn}$ denote the following permutation of the set $\lst{rn}$: \begin{align} \sigma_{\underline{\cA}}(a+(m-1)n)= \text{the } a^{th} \text{ largest element of } \cA_m, \qquad \text{for}\, a\in \lst{n}, m\in \lst{r}. \end{align} Now when $r=2$ the elements of $(\cA_1, \cA_2)$ of $\cP_{2,n}$ can be identified just with the subsets $\cA\subset \lst{2n}$ of cardinality $n$. Using $\cA'$ for the complement of $\cA$ in $\lst{2n}$ throughout, we further denote: \begin{equation} \label{shorthand} \sigma_{{\cA}} := \sigma_{\cA, \cA'} \in S_{2n}. \end{equation} \end{definition} With this set, we have the following representation. \begin{lemma}\label{prop:M_expansion} Let $\lambda_i, i \in \lst{rn}$, and $Q_{a,b}, a\in \lst{r}, b\in \lst{rn}$ be fixed numbers. For $m\in \lst{r}$, $\cA\subset \lst{rn}$ let $Q_m(\cA)=\prod_{a\in \cA} Q_{m,a}$. Then \begin{align}\label{eq:detM} \det \mbf{M}({\lambda}, \mbf{Q})&= \sum_{(\cA_1,\dots,\cA_r)\in \cP_{r,n}} \!\!\sgn(\sigma_{\underline{\cA}}) \prod_{m=1}^r Q_{m}(\cA_m) \Delta(\cA_m). \end{align} \end{lemma} \begin{proof} First, any permutation $\sigma\in S_{rn}$ of $\lst{rn}$ can be mapped in a one-to-one way to a pair consisting of an element of $\cP_{r,n}$ and a vector $(\sigma_1, \dots, \sigma_r)$ of $r$ permutations of $S_n$. For $\sigma\in S_{rn}, m\in \lst{r}$ let \begin{align} \cA_m=\{\sigma(i); i\in \lst{rn}, \nfr{i}{r}=m\} =\{\sigma(m), \sigma(m+r), \dots, \sigma(m+r(n-1))\}, \end{align} and let $\sigma_m$ be the permutation taking the ordered version of $\cA_m$ to the sequence \[(\sigma(m), \sigma(m+r), \dots, \sigma(m+r(n-1))). \] (Strictly speaking, $\sigma_m$ is the permutation that is generated by this mapping on the relative ranks of the elements of $\cA_m$ within the set.) Note that $\sigma$ can be reconstructed from the pair $(\cA_1,\dots, \cA_r), (\sigma_1, \dots, \sigma_r)$, and we have that \begin{align} \sgn(\sigma)=\sgn(\sigma_{\underline{\cA}}) \prod_{m=1}^r \sgn(\sigma_m). \end{align} Next, recall the expression of the entries $M_{i,j}$ of $\mbf{M}=\mbf{M}({\lambda}, \mbf{Q})$ from \eqref{eq:MagicM_ij}, and write out the determinant expansion: \begin{align} \det \mbf{M}&= \det \mbf{M}^{\dagger}=\sum_{\sigma\in S_{rn}} \sgn(\sigma) \prod_{i=1}^{rn} M_{\sigma(j),i}\\ &=\sum_{\sigma\in S_{rn}} \sgn(\sigma) \prod_{m=1}^{r} \prod_{a=0}^{n-1} \lambda_{\sigma(m+ar)}^{a} Q_{m,\sigma(m+ar)} \\ &=\sum_{(\cA_1,\dots,\cA_r)\in \cP_{r,n}}\!\!\sgn(\sigma_{\underline{\cA}}) \prod_{m=1}^r Q_{m}(\cA_m) \sum_{\sigma_1, \dots, \sigma_r} \prod_{m=1}^r \sgn(\sigma_m) \prod_{a=0}^{n-1} \lambda_{\sigma_m(m+a r)}^{a}. \end{align} Note that the second summation in the last line is over the possible vector of permutations $(\sigma_1,\dots,\sigma_r)\in S_n^r$ corresponding to a particular partition $(\cA_1,\dots,\cA_r)\in \cP_{r,n}$. For a fixed $m\in \lst{r}$ and $\cA_m$ the sum $\sum_{\sigma_m} \sgn(\sigma_m) \prod_{a=0}^{n-1} \lambda_{\sigma_m(m+a r)}^{a}$ is just the Vandermonde determinant $\Delta(\cA_m)$ which concludes the proof. \end{proof} \section{Evaluating $E_{\boldsymbol{Q}} \left[\|\det \mbf{M}({\lambda}, \mbf{Q})\|^2\right]$} \label{sec:2moment} The next proposition gives the proof of the first statement of Theorem \ref{thm:moment}. \begin{proposition}\label{prop:det_l=2} Fix $r\ge 2, n\ge 2$. Let $\mbf{Q}$ be an $rn\times rn$ Haar distributed $\mathbb{F}$-unitary matrix. Then, for all ${\lambda}\in \R^{rn}$ we have \begin{align}\label{eq:det2_general} E_{\boldsymbol{Q}} \left[\|\det \mbf{M}( {\lambda}, \mbf{Q})\|^2\right] = \kappa_{n,\beta, r,2} \sum_{(\cA_1,\dots, \cA_r)\in \cP_{r,n}} \prod_{j=1}^r \Delta (\cA_j)^2. \end{align} Here $\kappa_{n,\beta,r,2}$ is as defined in Lemma \ref{lem:Gauss_red}. \end{proposition} We provide two proofs. The first relies on the Gaussian reduction of Lemma \ref{lem:Gauss_red}. The second is provided for historical interest: it only uses the exchangeability properties of the Haar distribution, together with a determinantal identity from 1851 due to Sylvester \cite{Sylvester1851}. As the notation involved in the latter approach gets fairly heavy, we only present this second proof for $r=2$, $\beta =1$. We also note that in the $r=2$ case one can use the identity \eqref{eq:Pfaffian_detM} to obtain yet another independent proof. \begin{proof}[Proof of Proposition \ref{prop:det_l=2} using Gaussian reduction] Given Lemma \ref{lem:Gauss_red} the statement is equivalent to the following. Let $\mbf{X}$ be an $r\times (rn)$ matrix with i.i.d.~standard real or complex Gaussian entries $X_{a,b}, a\in\lst{r}, n\in \lst{rn}$, then \begin{align}\label{eq:detM_2_Gauss} E \left[\|\det \mbf{M}({\lambda}, \mbf{X})\|^2\right] = \sum_{(\cA_1,\dots, \cA_r)\in \cP_{r,n}} \prod_{j=1}^r \Delta(\cA_j)^2. \end{align} For a subset $\cA\subset \lst{rn}$ and $m\in \lst{r}$ we denote by $X_{m}(\cA)$ the product $\prod_{j\in \cA} X_{m,j}$. From Proposition \ref{prop:M_expansion} we get \begin{align} E[\|\det \mbf{M}\|^2]=\sum_{\underline{\cA}\in \cP_{r,n}}\sum_{\underline{\cB}\in \cP_{r,n}} \sgn(\sigma_{\underline{\cA}})\sgn(\sigma_{\underline{\cB}}) E[\prod_{m=1}^r X_{m}(\cA_m) \overline{X}_{m}(\cB_m)] \prod_{m=1}^r \Delta(\cA_m) \Delta(\cB_m) \end{align} Since $X_{a,b}$ are i.i.d.~$\FF N(0,1)$ random variables, we have \begin{align} E[\prod_{m=1}^r X_{m}(\cA_m) \overline{X}_{m}(\cB_m)] =\begin{cases} 1, \qquad &\text{if $\cA_m=\cB_m$ for all $m\in \lst{r}$,}\\ 0, \qquad &\text{otherwise.} \end{cases} \end{align} From this the identity \eqref{eq:detM_2_Gauss} and the statement of the proposition follows. \end{proof} \begin{proof}[Proof of Proposition \ref{prop:det_l=2} for $r=2$ and $\beta=1$ using exchangeability] Using again Proposition \ref{prop:M_expansion}, we have that \begin{align} E[\det \mbf{M}^2]=&\sum_{\|\cA\|=n, \cA\subset\lst{2n}}\sum_{\|\cB\|=n, \cB\subset\lst{2n}} \sgn(\sigma_{\cA}) \sgn(\sigma_{\cB}) E[ Q_{1}(\cA) Q_{2}(\cA')Q_{1}(\cB) Q_{2}(\cB')] \\ &\hskip 130pt \times \Delta(\cA) \Delta(\cA')\Delta(\cB) \Delta(\cB'). \end{align} Recall the shorthand \eqref{shorthand}. The expression $Q_{1}(\cA) Q_{2}(\cA')Q_{1}(\cB) Q_{2}(\cB')$ is a monomial of the form $\prod_{j=1}^{2n} q_{1,j}^{a_j} q_{2,j}^{2-a_j}$ where for each $j$ we have $a_j\in \{0,1,2\}$. By the exchangeability of the columns of a Haar orthogonal matrix we have \begin{align}\label{eq:def_g} E[\prod_{j=1}^{2n} q_{1,j}^{a_j} q_{2,j}^{2-a_j}]=g(c_0,c_1,c_2) \end{align} for some function $g$ where $c_i=\|\{a_j=i: j\in \lst{2n}\}\|$. Moreover, \begin{align} E[Q_{1}(\cA) Q_{2}(\cA')Q_{1}(\cB) Q_{2}(\cB')]=g(a,a,2n-2a), \qquad a=\|\cA\cap \cB\|=\|\cA' \cap \cB'\|. \end{align} Thus \begin{align} E[\det \mbf{M}^2]=&\sum_{\|\cA\|=n, \cA\subset\lst{2n}}\sgn(\sigma_{\cA}) \Delta(\cA) \Delta(\cA')\\ &\hskip 80pt \times \sum_{a=0}^n g(a,a,2n-a) \sum_{\|\cB\|=n, \|\cA\cap \cB\|=a} \sgn(\sigma_{\cB}) \Delta(\cB) \Delta(\cB'). \end{align} We will show that for any fixed $0\le a\le n$, $\cA\subset \lst{2n}$ with $\|\cA\|=n$, and $\cS\subset \cA$ with $\|\cS\|=a$ we have \begin{align}\label{eq:det_sylv} (-1)^{n-a} \Delta(\cA) \Delta(\cA') = \sgn(\sigma_{\cA}) \sum_{\|\cB\|=n, \cA\cap \cB=\cS} \sgn(\sigma_{\cB}) \Delta(\cB) \Delta(\cB'). \end{align} This will imply \begin{align} E[\det \mbf{M}^2]=\left(\sum_{a=0}^{n}(-1)^{n-a} \binom{n}{a}g(a,a,2n-2a)\right) \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \Delta(\cA)^2 \Delta(\cA')^2, \end{align} which proves the proposition for $r=2$ with the constant left in the form \begin{align} c_{n,2,2}&=\sum_{a=0}^{n}(-1)^{n-a} \binom{n}{a}g(a,a,2n-2a)=E\left[\prod_{j=1}^n \left(q_{1,2j-1}q_{2,2j-1}q_{1,2j}q_{2,2j}-q_{1,2j-1}^2 q_{2,2j}^2\right)\right], \end{align} by \eqref{eq:def_g}. Returning to \eqref{eq:det_sylv}, when $a=n$ the sum on the right side of that identity only contains the term corresponding to $\cB=\cA$, so the two sides are equal. When $a=0$ then again the right side only contains one term, the one corresponding to $\cB=\cA'$. Since $\sigma_{\cA'}$ can be obtained from $\sigma_{\cA}$ using $n$ transpositions, \eqref{eq:det_sylv} holds in this case as well. To prove \eqref{eq:det_sylv} in the $1\le a\le n-1$ case we use the following identity due to Sylvester \cite{Sylvester1851}. Suppose that $\mbf{R}_1$, $\mbf{R}_2$ are $n\times n$ matrices and $1\le m\le n$. Let $I$ be a fixed subset of column indices of $\mbf{R}_1$ with $\|I\|=m$. Then \begin{align}\label{eq:Sylvester} \det(\mbf{R}_1) \det(\mbf{R}_2)=\sum_{(\widehat{\mbf{R}}_1, \widehat{\mbf{R}}_2)} \det(\widehat{\mbf{R}}_1) \det( \widehat{\mbf{R}}_2) \end{align} where the sum is over all pairs of $n\times n$ matrices $(\widehat{\mbf{R}}_1, \widehat{\mbf{R}}_2)$ which can be obtained from $(\mbf{R}_1, \mbf{R}_2)$ by interchanging the $m$ columns of $\mbf{R}_1$ corresponding to $I$ with $m$ columns of $\mbf{R}_2$, while preserving the ordering of the columns. Fix $1\le a\le n-1$, $\cA\subset \lst{2n}$ with $\|\cA\|=n$, and $\cS\subset \cA$ with $\|\cS\|=a$. Let $\mbf{R}_1$ be the $n\times n$ Vandermonde matrix corresponding to $\cA$, $\mbf{R}_2$ the Vandermonde matrix corresponding to $\cA'$, and consider Sylvester's identity applied for $m=n-a$ where the fixed set of coordinates of $\mbf{R}_1$ correspond to $\widetilde{\cS}:=\cA\setminus \cS$. For each $(\widehat{\mbf{R}}_1,\widehat{\mbf{R}}_2)$ in the sum, let $\cB$ be the set of indices corresponding to the columns of $\widehat{\mbf{R}}_1$, then $\|\cB\|=n, \cA\cap \cB=\cS$. We set $\cT=\cA' \cap \cB'$ and $\widetilde{\cT}=\cA'\setminus \cT$. The left side of \eqref{eq:Sylvester} is $\Delta(\cA)\Delta(\cA')$, while the sum on the right side is \begin{align} \sum_{\|\cB\|=n, \cA\cap \cB=\cS}\sgn(\sigma_{\cA,\widetilde \cS\to \widetilde \cT}) \sgn(\sigma_{\cA',\widetilde \cT\to \widetilde \cS}) \Delta(\cB)\Delta(\cB'). \end{align} Here $\sigma_{\cA,\widetilde \cS\to \widetilde \cT}$ is the permutation of the columns of $\cA$ that we get after replacing the columns corresponding to $\wt \cS$ with the columns corresponding to $\wt \cT$ from $\cA' $ (keeping their order), and then reordering them based on their indices. The permutation $\sigma_{\cA',\widetilde \cT\to \widetilde \cS}$ is defined similarly. The identity \eqref{eq:det_sylv} now follows if we show that \begin{align}\label{eq:perm_sgn} \sgn(\sigma_{\cA,\widetilde \cS\to \widetilde \cT}) \sgn(\sigma_{\cA',\widetilde \cT\to \widetilde \cS})=(-1)^{n-a} \sgn(\sigma_{\cA}) \sgn(\sigma_{\cB}). \end{align} We will do this by transforming the permutation $\sigma_{\cA}$ into $\sigma_{\cB}$ while keeping track of the signature of the required steps. Consider the permutation $\sigma_{\cA}$. Interchange the indices in $\wt \cS$ with $\wt \cT$ (keeping their respective order), since $\|\wt \cS\|=\|\wt \cT\|=n-a$, we can achieve this with $n-a$ transpositions. Then reorder the elements in positions $1,3,\dots, 2n-1$, and then in the positions $2,4, \dots, 2n$. We can do this using the permutations $\sigma_{\cA,\widetilde \cS\to \widetilde \cT}$ and $\sigma_{\cA',\widetilde \cT\to \widetilde \cS}$ applied to these collections of indices, so the signature of this step is exactly $\sgn(\sigma_{\cA,\widetilde \cS\to \widetilde \cT}) \sgn(\sigma_{\cA',\widetilde \cT\to \widetilde \cS})$. The resulting permutation is exactly $\sigma_{\cB}$, and the signature of the steps we have taken is $(-1)^{n-a}\sgn(\sigma_{\cA,\widetilde \cS\to \widetilde \cT}) \sgn(\sigma_{\cA',\widetilde \cT\to \widetilde \cS})$, which proves \eqref{eq:perm_sgn}. \end{proof} \section{Evaluating $E_{\boldsymbol{Q}} \left[\|\det \mbf{M}({\lambda}, \mbf{Q})\|^4\right]$ for $r=2$} \label{sec:4moment} We show: \begin{proposition}\label{prop:det_l=4} Fix $n\ge 2$. Let $\mbf{X}$ be a $2\times (2n)$ matrix of independent $\FF N(0,1)$ random variables. For all ${\lambda}\in \R^{2n}$ it holds that \begin{align}\label{eq:det4} E \|\det \mbf{M}({\lambda}, \mbf{X})\|^4 = (12/\beta)^n \sum_{\cA\subset \lst{2n}, \|\cA\|=n} \Delta (\cA)^4 \Delta(\cA')^4. \end{align} \end{proposition} This establishes the second statement of Theorem \ref{thm:moment} via the Gaussian reduction Lemma \ref{lem:Gauss_red}. \begin{proof} We carry out the proof in the $\beta=1$ case, when the entries of $\mbf{X}$ are real standard Gaussians, commenting on the necessary modifications when $\beta =2$ at the end. As both sides of the claimed identity are continuous in $\lambda_{i}, i\in \lst{2n}$ , we can assume that those variables are distinct. Next, by Proposition \ref{prop:M_expansion} we have \[ \det \mbf{M}({\lambda},\mbf{X})=\sum_{\cA\subset \lst{2n}, \|\cA\|=n} \sgn(\sigma_{\cA}) X_1(\cA)X_2(\cA') \Delta(\cA)\Delta(\cA') \] and so also \begin{align}\label{eq:det^4} E[ \det \mbf{M}^4]&=\sum_{\substack{\cA_i\subset \lst{2n}, \|\cA_i\|=n\\ 1\le i\le 4}} \prod_{i=1}^4 \sgn(\sigma_{\cA_i}) E \left [\prod_{i=1}^4 X_1(\cA_i) X_2(\cA_i') \right] \prod_{i=1}^4 \Delta(\cA_i)\Delta(\cA_i'). \end{align} For a given choice of $\cA_1,\cA_2, \cA_3, \cA_4$ and $(j_1,j_2,j_3,j_4)\in \{0,1\}^4$ let \begin{equation} \cB_{(j_1,j_2,j_3,j_4)}=\cA_1^* \cA_2^* \cA_3^* \cA_4^, \quad \text{with} \quad \cA_i^=\begin{cases} \cA_i,& \qquad \text{if } j_i=1,\\ \cA_i',& \qquad \text{if } j_i=0. \label{overlaps} \end{cases} \end{equation} Also set $b_{(j_1,j_2,j_3,j_4)}=\|\cB_{(j_1,j_2,j_3,j_4)}\|$. Because the entries of $\mbf{X}$ are i.i.d.~standard normals, the expected value of $\prod_{i=1}^4 X_1(\cA_i) X_2(\cA_i')$ is zero if any entry of $\mbf{X}$ appears an odd number of times. Since the fourth moment of a standard normal is 3, this yields \begin{align}\label{eq:det4_111} E\left[\prod_{i=1}^4 X_1(\cA_i) X_2(\cA_i')\right]=3^{b_{(1,1,1,1)}+b_{(0,0,0,0)}} \ind(b_{(j_1,j_2,j_3,j_4)}=0 \text{ if $\sum_{i=1}^4 j_i$ is odd}). \end{align} In other words, after we take expected value in \eqref{eq:det^4}, only those quadruples $(\cA_1,\dots, \cA_4)$ contribute for which intersections of the form $\cA_{i_1}' \cA_{i_2} \cA_{i_3} \cA_{i_4}$ and $\cA_{i_1} \cA_{i_2}' \cA_{i_3}' \cA_{i_4}'$ are empty (for distinct $i_1,\dots, i_4$). Let $\cG_n$ denote the set of quadruples $(\cA_1,\dots, \cA_4)$ that satisfy this compatibility condition together with $\|\cA_i\|=n$, $\cA_i\subset \lst{2n}$. To simplify notation, we encode the sequences $(j_1,j_2,j_3,j_4)$ where $\sum_{i=1}^4 j_i$ is even with the integers 1, \dots, 8 as follows. \begin{align} \label{setlist} \begin{array}{c\|c\|c\|c\|c\|c\|c\|c} 1111 & 1100 & 1010 & 1001 & 0110 & 0101 & 0011 & 0000\\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \end{array} \end{align} Hence $\cB_{(1,0,1,0)}$ and $\cB_3$ both denote the set $\cA_1 \cA_2' \cA_3 \cA_4'$, and $b_3=b_{(1,0,1,0)}$. At the same time, if $(\cA_1,\dots, \cA_4)\in \cG_n$, then each $\cA_i$ and $\cA_i'$ can be written as the disjoint union of four of the sets $\cB_j, 1\le j\le 8$. For example, $\cA_1=\cB_1\cup \cB_2\cup \cB_3\cup \cB_4$. From $\|\cA_i\|=\|\cA_i'\|$, $1\le i\le 4$, we get four equations with the sum of four terms on both sides. For instance, $\|\cA_1\|=\|\cA_1'\|$ yields \[ b_1+b_2+b_3+b_4=b_5+b_6+b_7+b_8, \] and so on. These four equations can be readily reduced to the simpler identities \begin{align} b_j=b_{9-j}, \qquad 1\le j\le 4. \end{align} Equations \eqref{eq:det4} and \eqref{eq:det4_111} together with $b_1=b_8$ lead to \begin{align}\label{eq:det_4_2} E\left[ \det \mbf{M}^4\right]=\sum_{(\cA_1,\dots,\cA_4)\in \cG_n} 9^{ b_1} \prod_{i=1}^4 \sgn(\sigma_{\cA_i}) \prod_{i=1}^4 \Delta(\cA_i)\Delta(\cA_i'). \end{align} Next, for a particular $(\cA_1,\dots, \cA_4)\in \cG_n$ consider the following ``special'' ordering of the elements of $\lst{2n}$: $x\prec y$ if $x\in \cB_i, y\in \cB_j$ with $i<j$ or if $x<y$ with $x,y\in \cB_i$. Denote by $\wt \Delta(\cA)$ the Vandermonde determinant corresponding to the elements $\lambda_i, i\in \cA$ using our special ordering, and let $\sigma_i$ be the permutation that maps $1,3,\dots, 2n-1$ and $2,4,\dots, 2n$ into the elements of $\cA_i$ and $\cA_i'$ according to the special order. These permutations can be visualized using the following table: \begin{align} \begin{array}{r\|c\|c} &1,3,\dots,2n-1&2,4,\dots,2n \\ \hline \sigma_1&\cB_1, \cB_2,\cB_3,\cB_4&\cB_5, \cB_6,\cB_7,\cB_8\\ \hline \sigma_2&\cB_1, \cB_2,\cB_5,\cB_7&\cB_3, \cB_4,\cB_7,\cB_8\\ \hline \sigma_3&\cB_1, \cB_3,\cB_5,\cB_7&\cB_2, \cB_4,\cB_6,\cB_8\\ \hline \sigma_4&\cB_1, \cB_4,\cB_6,\cB_7&\cB_2, \cB_3,\cB_5,\cB_8 \end{array} \end{align} Note that for each $1\le i\le 4$ the permutation $\sigma_{\cA_i}$ can be obtained from $\sigma_{i}$ by reordering the elements $\sigma(2i-1),1\le i\le n$ and $\sigma(2i),1\le i\le n$, respectively. These two permutations correspond to the column permutations that take the Vandermonde matrices corresponding to $\wt \Delta(\cA_i)$ and $\wt \Delta(\cA_i')$ into the Vandermonde matrices of $\Delta(\cA_i)$ and $\Delta(\cA_i')$. From this observation we get \begin{align} \sgn(\sigma_{\cA_i}) \Delta(\cA_i)\Delta(\cA_i')= \sgn(\sigma_i) \wt \Delta(\cA_i)\wt \Delta(\cA_i')\qquad 1\le i\le 4, \end{align} and hence \begin{align}\label{eq:det_4_3} E\left[ \det \mbf{M}^4\right]=\sum_{(\cA_1,\dots,\cA_4)\in \cG_n} 9^{ b_1} \prod_{i=1}^4 \sgn(\sigma_i) \prod_{i=1}^4 \wt\Delta(\cA_i)\wt\Delta(\cA_i'), \end{align} as an equivalent form of \eqref{eq:det_4_2}. One reason for introducing the above ordering is that: for every $(\cA_1,\dots,\cA_4)\in \cG_n$, \[ \prod_{i=1}^4 \sgn(\sigma_i)=1. \] To see this, first note that since $\|\cB_3\|=\|\cB_6\|=b_3$ and $\|\cB_4\|=\|\cB_5\|=b_4$, we can obtain $\sigma_2$ from $\sigma_1$ using the following steps: switch the respective elements of $\cB_3$ with $\cB_6$, and then the elements of $\cB_4$ with those of $\cB_5$ (this takes $b_3+b_4$ transpositions). Then switch the order of the `blocks' $\cB_6$ and $\cB_5$, and the blocks $\cB_4$ and $\cB_3$ (these two steps can be done with the same number of transpositions). We can visualize these steps as follows: \[ \left(\cB_1, \cB_2,\cB_3,\cB_4\|\cB_5, \cB_6,\cB_7,\cB_8\right)\to \left(\cB_1, \cB_2,\cB_6,\cB_5\|\cB_4, \cB_3,\cB_7,\cB_8\right) \to \left(\cB_1, \cB_2,\cB_5,\cB_6\|\cB_3, \cB_4,\cB_7,\cB_8\right). \] This shows that \begin{align}\label{eq:sign_1} \sgn(\sigma_1)=\sgn(\sigma_2) (-1)^{b_3+b_4}. \end{align} Similarly, we can obtain $\sigma_4$ from $\sigma_3$ by switching the respecting elements of $\cB_i$ with $\cB_{9-i}$ for $i=3,4$, and then switch the order of $\cB_6, \cB_4$ and $\cB_5, \cB_3$: \[ \left(\cB_1, \cB_3,\cB_5,\cB_7\|\cB_2, \cB_4,\cB_6,\cB_8\right)\to \left(\cB_1, \cB_6,\cB_4,\cB_7\|\cB_2, \cB_5,\cB_3,\cB_8\right) \to \left(\cB_1, \cB_4,\cB_6,\cB_7\|\cB_2, \cB_3,\cB_5,\cB_8\right). \] This yields \begin{align}\label{eq:sign_2} \sgn(\sigma_3)=\sgn(\sigma_4) (-1)^{b_3+b_4}. \end{align} Combining \eqref{eq:sign_1} and \eqref{eq:sign_2} produces the claimed $\prod_{i=1}^4 \sgn(\sigma_i)=1$. At this point then we have the identity, \begin{align} \label{eq:det_4_4} E\left[ \det \mbf{M}^4\right]=\sum_{(\cA_1,\dots,\cA_4)\in \cG_n} 9^{b_1} \prod_{i=1}^4 \wt\Delta(\cA_i)\wt\Delta(\cA_i'), \end{align} and our next step is to rewrite the right hand side of \eqref{eq:det_4_4} as a product of Cauchy determinants. Towards this, for two disjoint sets $\cS_1, \cS_2\subset \lst{2n}$ we introduce the notation \begin{align}\label{eq:defdelta} \dd(\cS_1,\cS_2):= \prod_{s_1\in \cS_1} \prod_{s_2\in \cS_2}(\lambda_{s_1}-\lambda_{s_2}), \end{align} with the empty product defined as 1. For a particular $(\cA_1,\dots, \cA_4)\in \cG_n$ for each $1\le i\le 4$ we can group the variables into the sets $\cB_j$ and write \begin{align}\label{eq:prod_det_0} \wt\Delta(\cA_i)\wt\Delta(\cA_i')=\prod_{j=1}^8 \Delta(\cB_j)\prod_{a_1>a_2} \dd(\cB_{a_1}, \cB_{a_2})\prod_{b_1>b_2} \dd(\cB_{b_1}, \cB_{b_2}). \end{align} Here $a_1,a_2$ are indices from the ``$\cB$-partition'' of $\cA_i$ and $b_1,b_2$ are indices from the the ``$\cB$-partition'' of $\cA_i'$. Multiplying these equations for $1\le i\le 4$ and some bookkeeping yields \begin{align}\label{eq:prod_det_44} \prod_{i=1}^4\wt\Delta(\cA_i)\wt\Delta(\cA_i')=\prod_{j=1}^8 \Delta(\cB_j)^4 \prod_{\substack{9- j_1\neq j_2\\1\le j_2< j_1\le 8}} \dd(\cB_{j_1}, \cB_{j_2})^2. \end{align} A similar argument gives \begin{align}\label{eq:Delta_prod} \Delta(\lambda)^2=\prod_{j=1}^8 \Delta(\cB_j)^2 \prod_{1\le a<b\le 8} \dd(\cB_b, \cB_a)^2. \end{align} Recall next the Cauchy determinant formula. For two disjoint sets $\cS_1, \cS_2\subset \lst{2n}$ with $\|\cS_1\|=\|\cS_2\|$ we denote \begin{align}\label{eq:Cauchy_det} C_{\cS_1,\cS_2}=\det \left(\frac{1}{\lambda_{s_1}-\lambda_{s_2}}\right)_{s_1\in \cS_1, s_2, \cS_2} = (-1)^{\binom{\|\cS_1\|}{2}} \cdot \frac{\Delta(\cS_1)\Delta(\cS_2)}{\dd(\cS_1,\cS_2)}. \end{align} (If $\cS_1, \cS_2$ are empty we set $C_{\cS_1,\cS_2}=1$). From \eqref{eq:prod_det_44} and \eqref{eq:Delta_prod} we may deduce that \begin{align}\label{eq:prod_det_4} \Delta(\lambda)^{-2} \prod_{i=1}^4\wt\Delta(\cA_i)\wt\Delta(\cA_i')=\prod_{j=1}^4 C_{\cB_j,\cB_{9-j}}^2, \end{align} for again any $(\cA_1,\dots, \cA_4)\in \cG_n$ The point is that by Lemma \ref{lem:Cauchy_cycle} below we have the following. Suppose that $b_i, 1\le i\le 8$ are fixed nonnegative integers with $b_i=b_{9-i}$ and $\sum_{i=1}^4 b_i=n$. Let $\cP_{\underline{b}}$ be the collection of partitions $\underline{\cB}=(\cB_1,\dots, \cB_8)$ of $\lst{2n}$ with $\|\cB_i\|=b_i, 1\le i\le 8$. Then \begin{align}\label{eq:Cauchy_sum} \sum_{\underline{\cB}\in \cP_{\underline{b}}} \prod_{i=1}^4 C_{\cB_i,\cB_{9-i}}^2 =\binom{n}{b_1,b_2,b_3,b_4} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} C_{\cA,\cA'}^2. \end{align} Taking \eqref{eq:Cauchy_sum} for granted, \eqref{eq:prod_det_4} allows us to rewrite \eqref{eq:det_4_4} as in \begin{align}\notag E\left[ \det \mbf{M}^4\right]&=\Delta(\lambda)^2\sum_{b_1+\cdots+b_4=n, b_i\ge 0} 9^{b_1} \binom{n}{b_1,b_2,b_3,b_4} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} C_{\cA,\cA'}^2\\ &=12^n \times \Delta(\lambda)^2 \sum_{\cA\subset \lst{2n}, \|\cA\|=n} C_{\cA,\cA'}^2\label{eq:det4_9}, \end{align} having used the multinomial theorem in the second line. We can now finish the proof. For any $\cA\subset \lst{2n}$ with $\|\cA\|=n$, \begin{align}\label{eq:det2n_C} \Delta(\lambda)^2= \Delta(\cA)^2 \Delta(\cA')^2 \dd(\cA,\cA')^2, \end{align} which by the Cauchy determinant formula implies \[ \Delta(\lambda)^2 C_{\cA,\cA'}^2= \Delta(\cA)^4 \Delta(\cA')^4. \] Substituting the above into \eqref{eq:det4_9} produces \[ E\left[ \det \mbf{M}^4\right]=12^n \sum_{\cA\subset \lst{2n}, \|\cA\|=n} \Delta(\cA)^4 \Delta(\cA')^4, \] which is the claimed statement for $\beta=1$. For $\beta =2$, the key change comes in equation \eqref{eq:det4_111}, where one now gets \begin{align}\label{eq:new_det4} &E \left [\prod_{i=1}^2 X_1(\cA_i) X_2(\cA_i') \prod_{i=3}^4 \overline{X}_1(\cA_i) \overline{X}_2(\cA_i')\right]\\ &\hskip100pt \notag=2^{b_{(1,1,1,1)}+b_{(0,0,0,0)}} \ind(b_{(j_1,j_2,j_3,j_4)}=0 \text{ if $j_1+j_2\neq j_3+j_4$}). \end{align} The coefficient $2$ and the updated condition in the indicator function result from the fourth absolute moment and the rotational invariance of the standard complex Gaussian distribution, respectively. Equation \eqref{eq:new_det4} now implies that if the sets $\cB_{(1,0,1,0)}$ and $\cB_{(0,1,0,1)}$ are not empty then the corresponding quadruple $(\cA_1,\dots,\cA_4)$ does not contribute in the sum analogous to \eqref{eq:det^4}. This translates to having $b_3=b_6=0$ in all subsequent calculations. In particular, the multinomial factor in \eqref{eq:det4_9} is replaced by $ 4^{b_1} \binom{n}{b_1,b_2,b_4}$, which explains the reported constant of $6^n$ for $\beta=2$. \end{proof} \section{Determinantal identities} \label{sec:det_identities} The main goal of the section is to prove the following identity which is crucial in the proof of Proposition \ref{prop:det_l=4}. \begin{lemma}\label{lem:Cauchy_cycle} Let $n\ge 2$, and suppose that $b_i, 1\le i\le 8$ are fixed nonnegative integers with $b_i=b_{9-i}$ and $\sum_{i=1}^4 b_i=n$. Let $\cP_{\underline{b}}$ be the set of partitions $\underline{\cB}=(\cB_1,\dots, \cB_8)$ of $\lst{2n}$ with $\|\cB_i\|=b_i, 1\le i\le 8$. Then \begin{align}\label{eq:Cauchy_sum_1} \sum_{\underline{\cB}\in \cP_{\underline{b}}} \prod_{j=1}^4 C_{\cB_j,\cB_{9-j}}^2 =\binom{n}{b_1,b_2,b_3,b_4} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} C_{\cA,\cA'}^2. \end{align} \end{lemma} We also establish the following which unifies the various descriptions of our $r=2$ eigenvalue densities. \begin{lemma}\label{lem:det4_identities} Fix $n\ge 2$, and let ${\lambda}\in \R^{2n}$. Then the following identities hold: \begin{align}\label{eq:det4_id_1} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} \Delta (\cA)^4 \Delta (\cA')^4 & =2^{-n} \left(\sum_{\cA\subset \lst{2n}, \|\cA\|=n} \Delta (\cA)^2 \Delta (\cA')^2\right)^2 \\ & =2^n \Delta(\lambda)^2 \left\|\Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right)\right\|^2. \label{eq:det4_id_2} \end{align} \end{lemma} First though we present a conjectured generalization of Lemma \ref{lem:Cauchy_cycle} motivated by the proof of Proposition \ref{prop:det_l=4}. \subsection{A conjecture on certain Vandermonde sums} The Gaussian reduction was key to computing $E_{\boldsymbol{Q}} \left[\|\det \mbf{M}({\lambda}, \mbf{Q})\|^4\right]$ as it vastly cuts down the number of terms on the right hand side of the basic formula \eqref{eq:det^4}. Assuming however that the end result is only a function of the exchangeability of the rows and columns of $\mbf{Q}$, leads to an interesting set of conjectures involving sums of squared Vandermonde determinants. To explain, recall the enumeration of ``overlaps" $\cB_{(j_1,j_2, j_3, j_4)}$ defined in \eqref{overlaps}. Here $j_i \in \{0,1\}$, and previously all overlaps for which $\sum_{i=1}^4 j_i$ is odd immediately dropped from consideration. Restoring these, we have sixteen possible overlaps $(j_1, j_2, j_3, j_4)$ which we again list in reverse lexicographic order, and now denote their sizes by $x_i$, $i \in \{1, \dots, 16\}$. (Compare with \eqref{setlist}.) In particular, we now have present for example $\cB_{(1,1,0,1)}$ as the third element on the list for which we set $x_3 = \|B_{(1,1,0,1)}\| $. With these conventions set we can state: \begin{conjecture} Fix nonnegative integers $x_1, \dots, x_{16}$ with $x_1+\dots+x_8=n$. Then \begin{align} \label{conjecture} &\sum_{(\cA_1,\dots,\cA_4)\subset \mathcal{S}_{\mbf{x}}} \prod_{i=1}^4 \sgn(\sigma_{\cA_i}) \Delta_{\cA_i}^2 \Delta_{\cA_i'}^2 \\ & \qquad\qquad = \ind(x_i=x_{17-i}) \, (-1)^{x_2+x_3+x_5+x_7} { n \choose x_1, \dots , x_8 } \sum_{\cA \subset \lst{2n}, \|\cA\|=n} \Delta_{\cA}^4 \Delta_{\cA'}^4. \nonumber \end{align} Here $\mathcal{S}_{\mbf{x}}$ is the collection of all quadruples $(\cA_1,\dots,\cA_4)$ where the overlap sizes are given by $x_1,\dots, x_{16}$. \end{conjecture} The identity \eqref{eq:Cauchy_sum_1} of Lemma \ref{lem:Cauchy_cycle}, which can be recast in terms of Vandermonde instead of Cauchy determinants, is in fact the special case of \eqref{conjecture} where one restricts the sum on the left hand side to ``even" symmetric overlaps (for which $x_2 = x_3 = x_5=x_8 = 0$). \subsection{Proofs of Lemmas \ref{lem:Cauchy_cycle} and \ref{lem:det4_identities}} We start by recalling the definitions of the Pfaffian and Hafnian of a matrix. \begin{definition}\label{def:PfaffHaf} For a $2n \times 2n$ skew-symmetric matrix $M$, the Pfaffian of $M$ is defined by \begin{align}\label{eq:Pfaffiandef} \Pf(M)=\frac{1}{2^n n!} \sum_{\sigma\in S_{2n}} \sgn(\sigma) \prod_{i=1}^n M_{{\sigma(2i-1)},{\sigma(2i)}} = \sum_{\alpha\in \Pi_{2n}} \sgn(\pi_\alpha) \prod_{i=1}^n M_{\alpha_{i,1},\alpha_{i,2}}, \end{align} while for a $2n \times 2n$ symmetric matrix $M$, the Hafnian of $M$ is defined by \begin{align}\label{eq:Hafniandef} \Hf(M)=\frac{1}{2^n n!} \sum_{\sigma\in S_{2n}} \prod_{i=1}^n M_{{\sigma(2i-1)},{\sigma(2i)}} =\sum_{\alpha\in \Pi_{2n}} \prod_{i=1}^n M_{\alpha_{i,1},\alpha_{i,2}}. \end{align} Here $\Pi_{2n}$ refers to the set of matchings on $\lst{2n}$, {\em{i.e}}, partitions $(\alpha_{1,1},\alpha_{1,2}), \dots (\alpha_{n,1},\alpha_{n,2})$ of $\lst{2n}$ with $\alpha_{i,1}<\alpha_{i,2}$ and $\alpha_{1,1}<\cdots<\alpha_{n,1}$. In the Pfaffian formula, $\pi_\alpha$ denotes the corresponding permutation $(\alpha_{1,1},\alpha_{1,2}, \alpha_{2,1},\alpha_{2,2}\dots \alpha_{n,1},\alpha_{n,2})$. \end{definition} Breaking the proof of Lemma \ref{lem:det4_identities} into two parts,, we start with: \begin{proposition}\label{prop:det2+Pf} Let ${\lambda}\in \R^{2n}$. Then we have \begin{align}\label{eq:det2+Pf} \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \Delta(\cA)^2 \Delta (\cA')^2&=(-1)^n 2^n \Delta (\lambda) \Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right). \end{align} The right side is defined as the appropriate limit if the $\lambda_i$'s are not all distinct. \end{proposition} \begin{proof} Assuming that all the $\lambda_i$'s are distinct, the definition \eqref{eq:Pfaffiandef} gives \begin{align} \Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right)& =\sum_{\alpha\in \Pi_{2n}} \sgn(\pi_\alpha) \prod_{i=1}^n \frac{1}{\lambda_{\alpha_{i,1}}-\lambda_{\alpha_{i,2}}}. \end{align} A matching $\alpha \in \Pi_{2n}$ can be encoded with a pair $(\cA, \tau)$ where $\|\cA\|=n, \cA\subset \lst{2n}$ and $\tau\in S_n$. Indeed, $\cA=\{\alpha_{1,1}, \dots, \alpha_{n,1}\}$ and the permutation $\tau\in S_n$ that takes the ordered version of $\cA'$ into $(\alpha_{1,2}, \dots, \alpha_{n,2})$ identifies the matching $\alpha$. Note that $\sgn(\pi_\alpha)$ is equal to $\sgn(\sigma_{\cA} )\sgn(\eta_n)$ where $\sigma_{\cA}$ is defined in \eqref{shorthand}, and $\eta_n$ is the permutation $(1,n+1,2, n+2, \dots, n, 2n)$. This leads to \begin{align} \label{eq:Pf4_2} \Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right)=\sgn(\eta_n) \sum_{\|\cA\|=n, \cA\subset \lst{2n}} \sgn(\sigma_{\cA}) \sum_{\tau\in S_n} \sgn(\tau) \prod_{i=1}^n \frac{1}{\lambda_{a'_{\tau(i)}}-\lambda_{a_i}}, \end{align} where we used $a_i, a_i', i\in \lst{n}$ to denote the (ordered) elements of $\cA$ and $ \cA'$, respectively. For a given $\|\cA\|=n, \cA\subset \lst{2n}$ we have \[ \Delta(\lambda)=\Delta(\cA)\Delta(\cA') \prod_{\substack{1\le i<j\le 2n\\ \left\|\{i,j\}\cap \cA\right\|=1}} (\lambda_j-\lambda_i). \] By \eqref{eq:defdelta} we have \[ \prod_{\substack{1\le i<j\le 2n\\ \left\|\{i,j\}\cap \cA\right\|=1}} (\lambda_j-\lambda_i)=(-1)^{k_\cA} \dd(\cA, \cA') \] with $k_{\cA}$ the number of pairs $(i,j)$ with $1\le i<j\le 2n$ and $i\in \cA, j\in \cA'$. Observe then that $n^2-k_{\cA}$ is exactly the number of inversions of the permutation $\sigma_{\cA}$. Hence, \begin{align} \label{eq:det_exp_A_Ac} \Delta (\lambda)=(-1)^n \sgn(\sigma_{\cA}) \Delta (\cA)\Delta (\cA') \dd(\cA, \cA'). \end{align} Using this identity we get \begin{align}\notag \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \frac{\Delta (\cA)^2 \Delta (\cA')^2}{ \Delta(\lambda)}&=(-1)^n \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \sgn(\sigma_{\cA}) \frac{\Delta(\cA) \Delta(\cA')}{\dd(\cA, \cA')} \\ &=(-1)^{n+\binom{n}{2}}\sum_{\|\cA\|=n, \cA\subset\lst{2n}} \sgn(\sigma_{\cA}) C_{\cA, \cA'} \, \label{eq:det4_123} \\ &=(-1)^{n+\binom{n}{2}}\sum_{\|\cA\|=n, \cA\subset\lst{2n}} \sgn(\sigma_{\cA}) \sum_{\tau\in S_n} \sgn(\tau) \frac{1}{\lambda_{a_i}-\lambda_{a'_{\tau(i)}}}, \notag \end{align} where in the last line we expanded the Cauchy determinant, and used that $a_i, a_i', i\in \lst{n}$ for the ordered elements of $\cA, \cA'$, respectively. Note that $\sgn(\eta_n)=(-1)^{\binom{n}{2}}$, hence by comparing \eqref{eq:det4_123} with \eqref{eq:Pf4_2} the statement follows. \end{proof} \begin{proposition}\label{prop:det4+det2} Let ${\lambda}\in \R^{2n}$. Then we have \begin{align}\label{eq:det4+det2} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} \Delta (\cA)^4 \Delta (\cA')^4&=2^{n} \Delta(\lambda)^2 \Hf\left(\frac{\ind_{i\neq j}}{(\lambda_i-\lambda_j)^2}\right). \end{align} Again, the right side is defined as the appropriate limit if the $\lambda_i$'s are not all distinct. \end{proposition} To prove Proposition \ref{prop:det4+det2} we will use the following lemma, which is well-known in the statistical physics community. (See e.g.~(D.29) in \cite{FGM_1995}.) We include the proof for completeness. \begin{lemma}[Cycle cancellation]\label{lem:cycle} Suppose that $k\ge 3$, and $z_1, \dots, z_k$ are distinct numbers. Then \begin{align}\label{eq:cycle} \sum^_{\sigma} \prod_{i=1}^k \frac{1}{z_i-z_{\sigma(i)}}=0, \end{align} where the sum is over all cycles $\sigma$ supported on $\{1,2,\dots,k\}$. We also have \begin{align}\label{eq:cycle_full} \sum_{\sigma\in S_k} \prod_{i=1}^k \frac{1}{z_{\sigma(i)}-z_{\sigma(i+1)}}=0, \end{align} where $\sigma(k+1)=\sigma(1)$. \end{lemma} \begin{proof} We have \begin{align}\label{eq:cycle_3} \frac{1}{(z_1-z_a)(z_b-z_1)}=\frac{1}{z_b-z_a}\cdot \left(\frac{1}{z_b-z_1}-\frac{1}{z_a-z_1}\right). \end{align} Fix a length $k-1$ cycle $(b_1, \dots, b_{k-1})$ on $\{1,\dots, k-1\}$, and consider all length $k$ cycles that we get by inserting $k$ at some point. Then by \eqref{eq:cycle_3} the contribution of all of these cycles in the sum in \eqref{eq:cycle} (using $b_k=b_1$) is \[ \left(\sum_{j=1}^{k-1} \frac{1}{z_{b_j}-z_k}-\frac{1}{z_{b_{j+1}}-z_k}\right) \prod_{j=1}^{k-1} \frac{1}{z_{b_j}-z_{b_{j+1}}}=0. \] This proves \eqref{eq:cycle}. The identity \eqref{eq:cycle_full} now also follows by observing that the sum in \eqref{eq:cycle_full} is exactly $k$ times the sum in \eqref{eq:cycle}. \end{proof} \begin{proof}[Proof of Proposition \ref{prop:det4+det2}] It is sufficient to prove the statement in the case where all the $\lambda_i$'s are distinct. We first write \begin{align}\label{eq:det4det2_0} \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \frac{\Delta(\cA)^4 \Delta(\cA')^4}{ \Delta(\lambda)^2}& = \sum_{\|\cA\|=n, \cA\subset\lst{2n}} C_{\cA, \cA'}^2 \\ &=\frac{1}{(n!)^2} \sum_{\substack{a_1,\dots, a_n, a_1', \dots, a_n'\\\{a_1,\dots, a_n, a_1', \dots, a_n'\}=\lst{2n}}} C_{(a_1,\dots,a_n), (a_1', \dots, a_n')}^2.\label{eq:det4det2_1} \end{align} Here we used \eqref{eq:det_exp_A_Ac}, and in the second step we reordered the rows and columns of the Cauchy determinants in all possible ways. Next we expand the Cauchy determinants to get \begin{align}\nonumber & \sum_{\|\cA\|=n, \cA\subset\lst{2n}} \frac{ \Delta (\cA)^4 \Delta (\cA')^4}{ \Delta (\lambda)^2}=\\ &\hskip100pt\frac{1}{(n!)^2}\sum_{a_i, a_i'} \sum_{\sigma, \tau\in S_n} \sgn(\sigma\circ \tau^{-1}) \prod_{i=1}^n \frac{1}{(\lambda_{a_i} - \lambda_{a'_{\sigma(i)}})(\lambda_{a_i} - \lambda_{a'_{\tau(i)}})}. \label{eq:det4_sum} \end{align} (The restrictions on $a_i, a_i'$ are the same as in \eqref{eq:det4det2_1}.) We first evaluate the diagonal part of the double sum in \eqref{eq:det4_sum} by showing that \begin{align}\label{eq:det4_diag} \frac{1}{(n!)^2}\sum_{a_i, a_i'} \sum_{\sigma\in S_{n}} \prod_{i=1}^n \frac{1}{(\lambda_{a_i} - \lambda_{a'_{\sigma(i)}})^2}=2^n \Hf \left( \frac{\ind_{i \neq j }}{(\lambda_i -\lambda_j)^{2}} \right). \end{align} Consider the mapping that takes a particular choice of $a_1,\dots, a_n, a_1', \dots, a_n'$ and $\sigma\in S_n$ into the matching of $\lst{2n}$ that matches $a_i$ with $a_{\sigma(i)}'$ for $1\le i\le n$. Each particular matching $\alpha\in \Pi_{2n}$ shows up exactly $2^n (n!)^2$ times as the result of this mapping, which proves \eqref{eq:det4_diag}. To complete the proof of our proposition we just need to show that the `non-diagonal' terms in the sum \eqref{eq:det4_sum} cancel out. For this it is sufficient to show that if $\sigma\neq \tau$ are fixed elements of $S_n$ then \begin{equation} \label{eq:off_diag_vanish} \sum_{\substack{a_1,\dots, a_n, a_1', \dots, a_n'\\\{a_1,\dots, a_n, a_1', \dots, a_n'\}=\lst{2n}}} \prod_{i=1}^n \frac{1}{(\lambda_{a_i} - \lambda_{a'_{\sigma(i)}})(\lambda_{a_i} - \lambda_{a'_{\tau(i)}})}=0. \end{equation} For a particular pair $\sigma, \tau$ consider the permutation on $\lst{2n}=\{a_1,\dots, a_n, a_1', \dots, a_n'\}$ that takes $a_i$ to $a_{\sigma_i}'$ and $a_i'$ to $a_{\tau^{-1}(i)}$. This permutation has even cycles, and because $\sigma\neq \tau$ it must have at least one cycle of length at least 4. Let one of these cycles be \begin{align}\label{eq:cycle_123} a_{i_1}\to a'_{j_1} \to a_{i_2} \to a'_{j_2} \to \dots a'_{j_k}\to a_{i_1}. \end{align} Here $1\le i_1,\dots,i_k\le n$ are distinct, and the same holds for $j_1, \dots, j_k$. Let $\cB\subset \lst{2n}$ with $\|\cB\|=2k$. We claim that \begin{align}\label{eq:off_diag_vanish_1} \sum_{\substack{a_1,\dots, a_n, a_1', \dots, a_n'\\\{a_1,\dots, a_n, a_1', \dots, a_n'\}=\lst{2n}\\ \{a_{i_1},a'_{j_1}, a_{i_2}, a'_{j_2} \dots, a_{i_k}, a'_{j_k}\}=\cB }} \prod_{i=1}^n \frac{1}{(\lambda_{a_i} - \lambda_{a'_{\sigma(i)}})(\lambda_{a_i} - \lambda_{a'_{\tau(i)}})}=0. \end{align} The sum on the left can be rewritten as a double sum where we first sum over all possible assignments of the values of $a_{i_1},a'_{j_1}, a_{i_2}, a'_{j_2} \dots, a_{i_k}, a'_{j_k}$, and then in the second sum we sum over the remaining $2n-2k$ variables. Factoring out the terms corresponding to the indices not in $\cB$, the inner sum becomes \[ \sum_{\{a_{i_1},a'_{j_1}, a_{i_2}, a'_{j_2} \dots, a_{i_k}, a'_{j_k}\}=\cB} \prod_{\ell=1}^k \frac{1}{\lambda_{a_{i_\ell}}-\lambda_{a'_{j_\ell}}}\cdot \frac{1}{\lambda_{a'_{j_\ell}}-\lambda_{a_{i_{\ell+1}}}} \] with $i_{k+1}=i_1$. But this is equal to 0 by \eqref{eq:cycle_full} of Lemma \ref{lem:cycle}, as applied to the sum over the permutations of elements of $\cB$ and the values $\lambda_{a_{i_1}}, \lambda_{a'_{j_1}}, \dots, \lambda_{a_{i_k}}, \lambda_{a'_{j_k}}$. This proves \eqref{eq:off_diag_vanish_1}, which gives \eqref{eq:off_diag_vanish} and the statement of the proposition. \end{proof} We now have all the components to prove Lemma \ref{lem:det4_identities}. \begin{proof}[Proof of Lemma \ref{lem:det4_identities}] The proof follows from the statements of Proposition \ref{prop:det2+Pf} and \ref{prop:det4+det2} once we establish that \begin{align}\label{eq:Pf2=Hf} \Pf\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right)^2= \det\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right) = \Hf\left(\frac{\ind_{i\neq j}}{(\lambda_i-\lambda_j)^2}\right). \end{align} The first equality here is due the standard fact that square of the Pfaffian of a skew-symmetric matrix is equal to its determinant. The second has in fact been noticed before in \cite{DSZ}, and can be seen by expansion: \[ \det\left(\frac{\ind_{i\neq j}}{\lambda_i-\lambda_j}\right)=\sum_{\sigma\in S_{2n}, \sigma(j)\neq j} \sgn(\sigma) \prod_{j=1}^{2n} \frac{1}{\lambda_j-\lambda_{\sigma(j)}}, \] where the sum is over all the permutations of $\lst{2n}$ that have no fixed elements. Because of Lemma \ref{lem:cycle}, the contribution of the permutations that have a cycle of length at least 3 cancels out. The remaining terms correspond to the permutations that only have 2-cycles, that is, the permutations that corresponding to perfect matchings of $\lst{2n}$. For such a permutation $\sigma$ we have \[ \sgn(\sigma) \prod_{j=1}^{2n} \frac{1}{\lambda_j-\lambda_{\sigma(j)}}=\prod_{i\in \cA} \frac{1}{(\lambda_{i}-\lambda_{\sigma(i)})^2} \] where $\cA$ is a set containing an element from each 2-cycle. By \eqref{eq:Hafniandef} the sum of these terms is exactly the Hafnian of the $n\times n$ matrix with $(i,j)$ entry $\frac{\ind_{i\neq j}}{(\lambda_i-\lambda_j)^2}$. \end{proof} We are now ready to prove Lemma \ref{lem:Cauchy_cycle}. \begin{proof}[Proof of Lemma \ref{lem:Cauchy_cycle}] We first note that by \eqref{eq:det4det2_0} and Proposition \ref{prop:det4+det2} we have \begin{align}\label{eq:C_Hf} \sum_{\cA\subset \lst{2n}, \|\cA\|=n} C_{\cA,\cA'}^2=2^n \Hf\left(\frac{\ind_{i\neq j}}{(\lambda_i-\lambda_j)^2}\right). \end{align} Fix $b_1, \dots, b_4$ with $\sum_{i=1}^4 b_i=n$ and set $b_i=b_{9-i}$ for $5\le j\le 8$. For a particular $\sigma\in S_{2n}$ denote by $\cB_{\sigma,1}, \dots, \cB_{\sigma,4},\cB'_{\sigma,1}, \dots, \cB'_{\sigma,4}$ the ordered lists we obtain once we partition $(\sigma(1),\dots, \sigma(2n))$ into parts of lengths $b_1,b_2, b_3, b_4, b_8, b_7, b_6, b_5$. Then we have \begin{align}\label{eq:Cauchy_B4_1} \sum_{\underline{\cB}\in \cP_{\underline{b}}} \prod_{i=1}^4 C_{\cB_i,\cB_{9-i}^2} =\frac{1}{(b_1! b_2! b_3! b_4!)^2}\sum_{\sigma\in S_{2n}} \prod_{j=1}^4 C^2_{\cB_{\sigma,j},\cB'_{\sigma,j}}. \end{align} We introduce the temporary notation $z_i=\lambda_{\sigma(i)}$ and $z'_i=\lambda_{\sigma(n+i)}$ for $1\le i\le n$, and also $\cD_1=\{1,\dots, b_1\}$ $\cD_2=\{b_1+1,\dots, b_1+b_2\}$, $\cD_3=\{b_1+b_2+1, \dots, b_1+b_2+b_3\}$, $\cD_4=\{b_1+b_2+b_3+1,\dots,n\}$. For a given $1\le j\le 4$ we can expand the square of the appropriate Cauchy determinant as \begin{align} C^2_{\cB_{\sigma,j},\cB'_{\sigma,j}}=\sum_{\eta_j, \tilde \eta_j\in S(\cD_j)} \sgn(\eta_j)\sgn(\tilde \eta_j) \prod_{i\in \cD_j} \frac{1}{z_i-z'_{\eta_j(i)}} \cdot \frac{1}{z_i-z'_{\tilde \eta_j(i)}}, \end{align} where $S(\cD_j)$ is the set of permutations of $\cD_j$. We can represent $\eta_1,\dots, \eta_4$ as a single permutation of $\lst{n}$ that preserves $\cD_1,\cD_2,\cD_3,\cD_4$. Denoting the set of these permutations $S_{b_1,b_2,b_3,b_4}$ we get \begin{align}\label{eq:Cauchy_B4_1_2} \sum_{\underline{\cB}\in \cP_{\underline{b}}} \prod_{j=1}^4 C_{\cB_j,\cB_{9-j}}^2 =\frac{1}{(b_1! b_2! b_3! b_4!)^2}\sum_{\sigma\in S_{2n}} \sum_{\eta,\tilde \eta \in S_{b_1,b_2,b_3,b_4}} \sgn(\eta) \sgn(\tilde \eta) \prod_{i=1}^n \frac{1}{z_i-z'_{\eta(i)}}\cdot \frac{1}{z_i-z'_{\tilde \eta(i)}}. \end{align} Just as in previous computations, we consider the diagonal and off-diagonal terms of the resulting sum separately, and show that the off-diagonal terms cancel. The diagonal terms correspond to the cases $\eta=\tilde \eta$, this gives \begin{align}\label{eq:Cauchy_B4_2} \frac{1}{(b_1! b_2! b_3! b_4!)^2}\sum_{\sigma\in S_{2n}} \sum_{\eta \in S_{b_1,b_2,b_3,b_4}} \prod_{i=1}^n \frac{1}{(z_i-z'_{\eta(i)})^2}. \end{align} Note that \[ \prod_{i=1}^n \frac{1}{(z_i-z'_{\eta(i)})^2}=\prod_{\ell=1}^n \frac{1}{(\lambda_{\alpha_{i,1}}-\lambda_{\alpha_{i,2}})^2} \] where $(\alpha_{1,1},\alpha_{1,2}), \dots (\alpha_{n,1},\alpha_{n,2})$ is a matching of $\lst{2n}$, i.e.~an element of $\Pi_{2n}$. The term corresponding to a particular matching $\alpha\in \Pi_{2n}$ shows up in the sum in \eqref{eq:Cauchy_B4_2} exactly $2^n n! b_1! b_2! b_3! b_4!$ times. Indeed:\\ -- for each pair $(\alpha_{i,1},\alpha_{i,2})$ in the matching we can choose which element will be a $z_i$ ($2^n$ choices), this identifies the index sets $\{\sigma(1),\dots,\sigma(n)\}$ and $\{\sigma(n+1),\dots,\sigma(2n)\}$,\\ -- we can choose the values $\sigma(1), \dots, \sigma(n)$ by choosing one of the $n!$ permutations of the corresponding set, this will identify the lists $\cB_{\sigma,j}, 1\le j\le 4$, and the \emph{sets} corresponding to the lists $\cB'_{\sigma,j}, 1\le j\le 4$,\\ -- we can choose the ordering of elements within the sets corresponding to $\cB'_{\sigma,j}, 1\le j\le 4$, this can be done $b_1!b_2!b_3!b_4!$ ways, this completely identifies $\sigma$ and $\eta$. Our calculations show that the sum in \eqref{eq:Cauchy_B4_2} is equal to \[ \frac{n!}{b_1!b_2!b_3!b_4!} 2^n \sum_{\alpha\in \Pi_{2n}} \prod_{i=1}^n \frac{1}{(\lambda_{\alpha_{i,1}}-\lambda_{\alpha_{i,2}})^2}=\binom{n}{b_1,b_2,b_3,b_4} 2^n \Hf\left(\frac{\ind_{i\neq j}}{(\lambda_i-\lambda_j)^2}\right). \] This, together with \eqref{eq:C_Hf}, shows that the contribution of the diagonal terms is indeed equal to the expression on the right side of \eqref{eq:Cauchy_sum_1}. To finish the proof of our lemma we need to show that the off-diagonal terms in \eqref{eq:Cauchy_B4_1_2} cancel out. For this it is enough to show that if $\eta\neq \tilde \eta$ are fixed permutations in $S_{b_1,b_2,b_3,b_4}$ then the following sum is equal to zero: \begin{align}\label{eq:Cauchy_B4_3} \sum_{\sigma\in S_{2n}} \prod_{i=1}^n \frac{1}{z_i-z'_{\eta(i)}}\cdot \frac{1}{z_i-z'_{\tilde \eta(i)}}. \end{align} To prove this statement, we first note that the pair $\eta, \tilde \eta$ generates a permutation of $\lst{2n}$ with $i\to n+\eta(i)$, $n+i\to \tilde \eta^{-1}(i)$, $1\le i\le n$. This permutation has only even length cycles, and since $\eta\neq \tilde \eta$, it has at least one cycle with length $2\ell\ge 4$. Let one of these cycles be \[ i_1\to n+j_1\to i_2\to n+j_2\to \dots n+j_\ell\to i_1. \] The contribution of this cycle to the term corresponding to a particular $\sigma\in S_{2n}$ in \eqref{eq:Cauchy_B4_3} is \begin{align}\label{eq:Cauchy_B4_4} (-1)^k \prod_{k=1}^\ell \frac{1}{z_{i_k}-z'_{j_k}}\cdot \frac{1}{z'_{j_k}-z_{z_{i_{k+1}}}}. \end{align} (With $i_{n+1}=i_1$.) The proof now follows along the line of the proof of \eqref{eq:off_diag_vanish_1}. Let \[ \cF=\{i_1, \dots, i_\ell, n+j_1, \dots, n+j_\ell\}. \] We can evaluate the sum in \eqref{eq:Cauchy_B4_3} in two steps by first fixing the values of $\sigma$ outside the index set $\cF$, and then assigning the values of \[ \sigma(i_1), \dots, \sigma(i_\ell), \sigma(n+j_1), \dots, \sigma(n+j_\ell) \] out of the remaining $2\ell$ indices in all possible ways. By Lemma \ref{lem:cycle} the sum of the terms \eqref{eq:Cauchy_B4_4} in this second step will be 0, which proves that the sum in \eqref{eq:Cauchy_B4_3} is also 0. This shows that the contribution of the off-diagonal terms in \eqref{eq:Cauchy_B4_1_2} vanishes, proving our lemma. \end{proof} \section{Asymptotics} \label{sec:asymptotics} We provide sketches of the proofs of Theorem \ref{thm:limit_op} and Corollary \ref{cor:osc}, along with the complementary Theorem \ref{thm:limit_op1} and Corollary \ref{cor:osc1}. \subsection{The ${\tt{H}}_{\beta, n}(r,s)$ soft edge} The edges of eigenvalue support for ${\tt{H}}_{\beta,n} (r,s)$ can be tracked through its limiting density of states. The method of Trotter \cite{Trotter} gives the following (which we state without proof): \begin{proposition} Denote by $\mu_\gamma$ the semi-circle distribution with density $ \frac{1}{\pi \gamma} \sqrt{(4 \gamma - \lambda^2)_+}$. For any integer $r$ any $s > 1-r$, the empirical spectral measure $\frac{1}{rn} \sum_{i=1}^{rn} \delta_{\lambda_i}$ of eigenvalues of $\HH(r,s)$ converges weakly in law to $\mu_\gamma$ with $\gamma = \frac{r+s}{r}$. \end{proposition} To study the scaling limit of the maximal eigenvalues let $\mbf{T}_n$ be distributed as ${\tt{H}}_{\beta, n}(r,s)$ and consider the centered and scaled matrix model \begin{align} \label{eq:Hn} \mathbf{H}_n & = \gamma^{-1/2} (rn)^{1/6} (2 \sqrt{(r+s)n} {I}_{rn} - \mathbf{T}_n ) \\ & = 2 m_n^{2} \mbf{I}_{rn} - \sqrt{\frac{m_n}{{\gamma}}} \mbf{T}_n, \quad \quad \quad \quad m_n = (rn)^{1/3}. \nonumber \end{align} We identify $m_n$ as a continuum scale and view $v \in \mathbb{F}^{nr}$, on which $\mathbf{H}_n$ acts, as a vector-valued step function: \begin{equation} \label{embed} (v_0, v_1, \dots) \in \ell_{nr}^2 \mapsto v(x) = v_{\lfloor m_n x \rfloor} \in L^2(\R_+, \mathbb{F}^r). \end{equation} Said another way, for integers $i\in [0, r-1] $ and $k \in [0,n]$, $v_{i + k r}$ is identified with $v_i( k/m_{n})$ With this embedding in mind, write $ \mathbf{H}_n = m_n^2 \Delta_n + m_n {\mathbf{V}}_n$. Here $\Delta_n$ denotes discrete Laplacian in the underlying basis: it has diagonal blocks $2{I}_r$ and off-diagonal blocks $- {I}_r$. The $\mbf{V}_n$ is then considered to define a matrix-valued potential. The main technical result of \cite{Spike2} provides criteria for the convergence of the spectrum of $\mathbf{H}_n$ (in law) in terms of the convergence of the integrated components of $\mathbf{V}_n$. A version of this, tailored to the particular setting considered here, is the following. \begin{proposition} \label{Model_SoftEdgeConv} Suppose that $\mbf{T}_n$ is an $rn\times rn$ Jacobi $r$-block matrix with blocks $\mbf{A}_{n,i}, \mbf{B}_{n,i}$. Let $\mbf{H}_n$ and $m_n$ be defined as in \eqref{eq:Hn}. Expressing $\mathbf{H}_n$ as $ m_n^2 \Delta_n + m_n {\mathbf{V}}_n$, denote the running sums of the diagonal and upper diagonal block components of the potential $\mbf{V}_n$ by: for $0 \le x \le (rn)^{2/3}$, \begin{align} \label{eq:potential_int} {\bf{Y}}_{n,1}(x) = \sum_{i=1}^{[m_n x]} \left( - \frac{1}{\sqrt{\gamma m_n}} \mathbf{A}_{n,i} \right) \quad {\bf{Y}}_{n,2}(x) = \sum_{i=1}^{[m_n x]} \left( m_n^2 \mbf{I}_r - \frac{1}{\sqrt{\gamma m_n}} \mathbf{B}_{n,i} \right). \end{align} Now assume that: \medskip \noindent {(i)} Both $\{ {\bf{Y}}_{n,1}(x)\}_{x \ge 0}$ and $\{ {\bf{Y}}_{n,2}(x)\}_{x \ge 0}$ are tight and \begin{equation} \label{potential_convergence} {\bf{Y}}_{n,1}(x) + ({\bf{Y}}_{n,2}(x) + {\bf{Y}}_{n,2}^{\dagger}(x) ) \Rightarrow \frac{1}{2} r x^2 \mbf{I}_r+ \sqrt{\frac{2}{\gamma}} B(x) \end{equation} in law in the uniform-on-compacts topology; \medskip \noindent {(ii)} For $j=1,2$ there is a decomposition of the increments, \begin{equation} \label{potential_decomp} (\Delta {\bf{Y}})_{n, j} = \nn_{n,j} + (\Delta {\ww})_{n,j}, \end{equation} where: (1) the $\nn_{n,j, i} $ are diagonal and are bounded entry-wise as in \begin{align} \kappa^{-1} x - \kappa \le \nn_{n,1}(x) + \nn_{n,2}(x) \le \kappa x + \kappa, & \ \ \ \nn_{n,2}(x) \le 2 m_n, \label{compact1} \end{align} for some $\kappa \ge 1$, and (2), with $\\|\cdot\\|$ the spectral norm, there exists an $\epsilon > 0$ so that \begin{align} \\| \ww_{n,1}(x) - \ww_{n,1}(y)\\|^2 + \\| \ww_{n,2}(x) - \ww_{n,2}(y)\\|^2 & \le \kappa_n x^{1-\epsilon} + \kappa_n, \label{compact3} \end{align} for all $\|x-y\| \le 1$ and $\kappa_n \ge 1$ a sequence of tight random constants. \medskip Then, with $\mathcal{H}_{\beta, \gamma}$ defined in \eqref{eq:H_op}, we have that any finite collection of ordered eigenvalues of $\mathbf{H}_n$, along with their associated eigenfunctions as elements in $L^2 (\mathbb{F}^r)$, converge jointly in law to the corresponding eigenvalues/eigenfunctions of $ \mathcal{H}_{\beta, \gamma}$. \end{proposition} Condition (i) simply identifies the correct limit potential. Condition (ii) provides an almost sure lower bound on $\langle v, \mathbf{H}_n v \rangle$ independent of $n$ which is essential for extracting eigenvalue limits. The bound \eqref{compact3} controls the random oscillations of the potential by the growth of $\nn_{n,j}$, which by \eqref{compact1} is well-controlled in terms of its natural limit. Both conditions are readily checked in our case. The independence of the entries of ${\bf{Y}}_{n,1}$ and ${\bf{Y}}_{n,2}$ allows one to establish the functional limit theorem in (i) component-wise. For (ii) one can take $\nn_{n,0} =0$ and $\nn_{n,2}$ the vector of expectations of centered $\chi$ variables appearing on the diagonal of ${\bf{Y}}_{n,2}$. In both (i) and (ii) that the sequence $\chi_p -\sqrt{p}$ converges in law to a $N(0,\frac{1}{2})$ random variable as $p \uparrow \infty$ and satisfies a uniform in $p$ subgaussian bound plays a fundamental role. With the matrix models considered in \cite{Spike2} so similar to $\mathbf{H}_n$, the necessary details are effectively identical to what has already been done there. The proof of Corollary \ref{cor:osc} is also similar to that of the corresponding statement in \cite{Spike2}, though here is a sketch of the main idea. Fix $\lambda\in \R$ and consider the system: \begin{align} \label{Eig:system} dF(x) & = F'(x) dx, \\ dF'(x) & = (\lambda - r x) F(x) + \sqrt{\frac{2}{\gamma}} F (x) dB(x), \nonumber \end{align} with initial condition $F$ satisfying $(F(0), F'(0)) = (0, \mbf{I}_r)$. Here $F$ is an $r\times r$ matrix valued function. It can be shown that the number of eigenvalues of the Dirichlet problem for $\mathcal{H}_{\beta, \gamma}$ less than $\lambda$ coincides with the number of zeros of $\det F$ on $\R_+$ Next define $P(x) = F'(x) F(x)^{-1} $, the matrix Riccati substitution. This satisfies \begin{equation} \label{matrix_ricatti} d P(x) = ((\lambda - r x)\mbf{I}_r - P^2(x))dx + \sqrt{\frac{2}{\gamma}} dB(x). \end{equation} One may then verify that points $x'$ where $\det F$ vanishes correspond to $P(x)$ possessing an eigenvalue $p(x)$ that explodes to $-\infty$ as $x \rightarrow x'$. This is exactly the content of Theorem \ref{cor:osc}: the stochastic differential equation \eqref{mult_sde}, as can be derived from \eqref{matrix_ricatti} by an application of It\^{o}'s Lemma, describes the evolution of the eigenvalues $(p_1(x), \dots, p_r(x))$ of $P(x)$, continued through explosion times to $-\infty$. (Again, see \cite{Spike2} for additional details.) \subsection{Hard edge for ${\tt{W}}_{\beta, n, n+a}(r,s)$} To give a precise definition of $\mathcal{G}_{\beta, \gamma}$ from \eqref{matrixgenerator} we first have to define $\mbf{Z}_x$. We introduce a new type of $r \times r$ matrix Brownian motion $x \mapsto B_x$ in which all entries are independent, the off-diagonal entries are standard ${\mathbb{F}}$-Brownian motions and the diagonal entries are real Brownian motions with common diffusion coefficient $\frac{1}{\beta}$. Then the coefficient matrix $\mbf{Z}_x$ is given by \begin{equation} \label{WandA} \mbf{Z}_x = \mbf{Y}_x \mbf{Y}_x^{\dagger}, \quad \mbf{Y}_x^{-1} d \mbf{Y}_x = \frac{1}{\sqrt{\gamma}} dB_x + \left( \frac{a}{2 \gamma} - \frac{1}{2\beta \gamma} \right) \mbf{I}_r dx, \end{equation} in which $\mbf{Y}_0 = 0.$ Notice that in the $r=1$ and $\gamma =1$ setting in which the Stochastic Bessel Operator was first introduced, $Z_x = e^{ \frac{2}{\sqrt{\beta}} b(x) + a x}$ with a standard one-dimensional Brownian motion $b(x)$. While \eqref{matrixgenerator} is a nice format in which to package the limiting operator $\mathcal{G}_{\beta, \gamma}$, we will actually identify this operator via its inverse. An exercise shows that, specifying a Dirichlet condition at the origin, $\mathcal{G}_{\beta, \gamma} = \mathcal{L}_{\beta, \gamma} \mathcal{L}_{\beta, \gamma}^{\dagger} $ where \begin{equation} \label{kernel_ops} \mathcal{L}_{\beta, \gamma}^{-1} f (x) = \int_x^\infty {\ell}(x,y) f(y) dy, \quad {\ell}(x,y) = e^{-rx/2} \mbf{Y}_x^{-1} \mbf{Y}_y. \end{equation} This structure is natural as $\mathtt{W}_{n,m, \beta}(r,s)$ is realized as a matrix model via $\mbf{L}_n \mbf{L}_n^{\dagger}$, where $\mbf{L}_n$ is block bi-diagonal, and hence explicitly invertible. The results of \cite{RR2} imply that $\mathcal{L}_{\beta, \gamma}^{-1}$ is Hilbert-Schmidt for any $\gamma >0$ and $a>-1$, and the main convergence result there can be summarized, in the spirit of Proposition \ref{Model_SoftEdgeConv}, as follows. \begin{proposition}\label{prop:hard_lim} Given a block bidiagonal matrix $\mbf{L}_n$ with independent diagonal and upper diagonal $r \times r$ blocks $\mbf{D}_k=\mbf{D}_{n,k}$ and $\mbf{O}_k=\mbf{O}_{n,k}$, embed $ \mbf{ L}_n^{-1}$ into $L^2([0,1], \mathbb{F}^r)$ in the manner of \eqref{embed} with now $m_n = n$. In particular, define the piecewise step kernel \begin{equation} \label{explicit_inv} \ell_{n}(x,y) = \sqrt{ \frac{ {n} \gamma}{r}} {\mbf{D}}_{\lfloor nx \rfloor}^{-1} {\mbf{X}}_n(x)^{-1} \mbf{X}_n(y) \mathbf{1}_{0 \le x < y \le 1}, \quad {\mbf{X}}_n(x) = \prod_{k= 1}^{ \lfloor nx \rfloor} {\mbf{O}}_k {\mbf{D}}_{k+1}^{-1}. \end{equation} Assume that: \medskip (i) As $n \rightarrow \infty$, in the uniform-on-compacts topology $$ \Bigl( \sqrt{\frac{{n} \gamma}{r}} \, \mbf{D}_{\lfloor nx \rfloor}^{-1}, \ \mbf{X}_n(x) \Bigr) \Rightarrow \Bigl( \frac{1}{r \sqrt{ 1-x}} \mbf{I}_r, \, \mbf{X}_x \Bigr), \qquad \text{for } x\in [0,1), $$ where $\mbf{X}_x$ satisfies the matrix stochastic differential equation $$ \, \mbf{X}_x^{-1} d \mbf{X}_x = \frac{1}{\sqrt{r \gamma (1-x)}} d B_x - \frac{a -\frac{1}{ \beta}}{2 r\gamma (1- x)} {I}_r dx, \qquad \mbf{X}_0=0. $$ (ii) There is the bound $$ \int_0^1 \int_0^y \| \ell_n(x,y) \|^2 dx dy \le \kappa_n $$ with $\kappa_n$ a sequence of tight random constants. \medskip Then $spec( \frac{rn}{ \gamma} \mbf{L}_n \mbf{L}_n^\dagger) \rightarrow spec(\mathcal{G}_{\beta, \gamma})$ in the manner described in Proposition \ref{Model_SoftEdgeConv}. \end{proposition} The point is that the integral kernel $\ell_n(x,y)$ is an exact representation of $(\sqrt{rn / \gamma} \, \mbf{L}_n)^{-1}$. The convergence in (i) identifies the pointwise limit of $\ell_n$ while (ii) implies (subsequential) convergence of the corresponding integral operator in Hilbert-Schmidt norm. Combined, we have that $ (\frac{rn}{ \gamma} \mbf{L}_n \mbf{L}_n^{\dagger})^{-1}$ converges in norm in the same subsequential sense; convergence of the finite parts of the spectrum follows. Checking the conditions of Proposition \ref{prop:hard_lim} for ${\tt{W}}_{\beta, n, n+a}(r,s)$ follows the arguments of \cite{RR2}. For (i), by an explicit expansion of the inverse of the diagonal blocks, the increments of $\mbf{X}_n$ have the form $$ {\mbf{O}}_k {\mbf{D}}_{k+1}^{-1} = \mbf{I}_r + \frac{1}{\sqrt{ (r+s)(n-k)}} \mbf{G}_k + \frac{-a+ \frac{1}{\beta} }{2(r+s) (n-k)} \mbf{I}_r + \boldsymbol{\varepsilon}_k. $$ Here the $\mbf{G}_k$ are independent and have independent entries which are $\FF$-normals off diagonal and centered $\chi$ variables on the diagonal. As such, the $\mbf{G}_k$ form the increments of the limiting matrix Brownian motion $B_x$. The $\boldsymbol{\varepsilon}_k$ matrices are error terms for which one can derive the estimate $E \\|\boldsymbol{\eps}_k \\|^p = O( (n-k)^{-\frac{3}{2} p})$. The proof of (ii) is far more technical, as it requires sharp control of the paths of $x \mapsto \mbf{X}_n(x)$ in a vicinity of $x=1$. Note that while the embedding of the matrix $\mbf{L}_n^{-1}$ naturally takes place as an operator on $[0,1]$, the advertised limit $\mathcal{L}^{-1}_{\beta, \gamma}$ lives on $[0, \infty)$. This is just more convenient for eventual comparison to the soft edge operator $\mathcal{H}_{\beta, \gamma}$, and the kernel $\ell(x,y)$ is related to the limit of $\ell_n$, which is constructed from the limit objects defined in (i) above, by the simple change of variables $x \mapsto 1 - e^{-rx} =\varphi(x)$. As one can readily check, $ \mbf{X}_{\varphi(x)} = \mbf{Y}_x$ and $ \frac{1}{r\sqrt{1-\varphi(x)}} d\varphi(x) = e^{-rx/2} dx$. This establishes Theorem \ref{thm:limit_op1} From here the proof of Corollary \ref{cor:osc1} amounts to writing out the eigenvalue problem for $\mathcal{G}_{\beta, \gamma}$ as a system, then invoking the matrix Riccati correspondence, in analogy with equations \ref{Eig:system} and \ref{matrix_ricatti}. Section 4 of \cite{RR2} provides the details. We conclude by recording the hard edge versions of Corollary \ref{cor:betalimit} and Conjecture \ref{con:betalimit}. Denote by $\operatorname{Bessel}_{\beta, a}$ the random point process defined by the $r=1$ and $\gamma =1$ case of Theorem \ref{thm:limit_op1} (and Corollary \ref{cor:osc1}). To be totally concrete, the points of $\operatorname{Bessel}_{\beta, a}$ are the Dirichlet eigenvalues of the one-dimensional operator \begin{align} -e^{(a+1)x +\frac{2}{\sqrt{\beta}} b(x)} \frac{d}{dx} e^{-ax -\frac{2}{\sqrt{\beta}} b(x)} \frac{d}{dx} \cdot \end{align*} acting on the positive half-line. This is well defined for all $\beta > 0$ and $a> -1$. \begin{corollary} Consider the point processes of eigenvalues for our solvable instances of ${\tt{W}}_{\beta, n, n+a}(r,s)$. These have explicit joint densities proportional to \begin{align} \label{density3} \|\Delta({\lambda})\|^{\beta} \left( \sum_{(\mathcal{A}_1,\dots,\mathcal{A}_r)\in \cP_{r,n}} \prod_{j=1}^r \Delta(\cA_j)^2 \right) \prod_{i=1}^{rn} \lambda_i^{\frac{\beta}{2}( (r+s)a+1)-1} e^{-\frac{\beta}{2} \lambda_i} \ind_{\R_n^+} \end{align} for $r \ge 2$ and $\beta s=2$, and to \begin{align} \label{density4} \Delta({\lambda})^{\beta+\frac{\beta s}{2}} \left\|\Pf \left(\frac{{\bf{1}}_{i \neq j}}{\lambda_i -\lambda_j} \right)\right\|^{\frac{\beta s}{2}} \prod_{i=1}^{rn} \lambda_i^{\frac{\beta}{2}( (r+s) a+1)-1} e^{-\frac{\beta}{2} \lambda_i} \ind_{\R_n^+} \end{align} for $r=2$ and $\beta s =2$ or $4$. When $r=2$, $\beta s =2 $ and $\beta=1$, the scaling limit of the minimal points is given by ${\operatorname{Bessel}}_{2, a/2}$. When $r=2$, $\beta s = 4$, and $\beta = 2$, the scaling limit is $\operatorname{Bessel}_{4, a/2}$. \end{corollary} \begin{conjecture} More generally, the minimal points under \eqref{density3} have scaling limit given by ${\operatorname{Bessel}}_{\beta +\frac{2}{r}, a/(1+\frac{2}{\beta r})}$ for $r \ge 2$ and $\beta =1$ or $2$. For the minimal points under \eqref{density4} with $\beta s =4$ and $\beta =1$ the scaling limit is instead ${\operatorname{Bessel}}_{3, a/3}$. \end{conjecture} \def\cprime{$'$} \begin{thebibliography}{10} \bibitem{Bartlett1933} M.~S. Bartlett. \newblock On the theory of statistical regression. \newblock {\em Proc. R. Soc. 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2412.12130v1	http://arxiv.org/abs/2412.12130v1	On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers	\documentclass[11pt,a4paper]{article} \usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amsfonts,amsthm,latexsym,amsmath,amssymb,amscd,epsfig,psfrag,enumerate} \usepackage{graphics,graphicx, bezier, float, color, hyperref} \usepackage{amssymb,url} \usepackage{multienum} \usepackage[table]{xcolor} \usepackage{multicol,multirow} \usepackage{graphicx} \usepackage{fancyvrb} \usepackage{parskip} \usepackage[toc,page]{appendix} \sloppy \setlength{\parindent}{0pt} \setlength\parskip{0.1in} \usepackage[top=2.7cm, bottom=2.7cm, left=1.5cm, right=1.5cm]{geometry} \usepackage{xcolor} \usepackage{blkarray} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{cor}{Corollary}[section] \newtheorem{remark}{Remark}[section] \usepackage[none]{hyphenat}[section] \newtheorem{definition}{Definition}[section] \newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} \numberwithin{equation}{section} \numberwithin{table}{section} \numberwithin{figure}{section} \title{On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers} \author{Herbert Batte$^{1,} $ and Florian Luca$^{1,2}$} \date{} \begin{document} \maketitle \abstract{ Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x=L_m^{(k)}$, in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$.} {\bf Keywords and phrases}: $k$--generalized Lucas numbers; linear forms in logarithms; Baker--Davenport reduction method, LLL--algorithm. {\bf 2020 Mathematics Subject Classification}: 11B39, 11D61, 11D45. \thanks{$ ^{} $ Corresponding author} \section{Statements and Declarations} \textbf{Competing Interests:} The authors declare that they have no conflict of interest. \\ \\ \textbf{Ethical Approval:} This article does not contain any studies involving human participants or animals performed by any of the authors. \\ \\ \textbf{Data Availability Statement:} Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. \\ \\ \textbf{Author Contributions:} The authors contributed equally to the conceptualization, analysis, and writing of the manuscript. \section{Introduction}\label{intro} \subsection{Background}\label{sec:1.1} For any integer \( k \ge 2 \), the sequence of $k$--generalized Lucas numbers is defined by the recurrence relation: \[ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)}, \quad \text{for all} \ n \ge 2, \] with the initial terms set as \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all \( k \ge 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \ge 3 \). For \( k = 2 \), this reduces to the classical sequence of Lucas numbers, and the superscript ${}^{(k)}$ is typically omitted. The study of sequences such as the Fibonacci and Lucas numbers, and their generalizations, has led to the discovery of numerous interesting identities and Diophantine equations. Similar to how various authors have explored identities involving the $k$--Fibonacci numbers, the $k$--Lucas numbers also exhibit a rich structure that can be investigated through the lens of number theory. In \cite{GGL}, Gmez et al. studied the Diophantine equation \begin{align}\label{eqGGL} \left(F_{n+1}^{(k)}\right)^x+\left(F_{n}^{(k)}\right)^x-\left(F_{n-1}^{(k)}\right)^x=F_m^{(k)}, \end{align} in nonnegative integers $n$, $m$, $k$, $x$, with $n\ge 1$ and $k\ge 2$, where $F_r^{(k)}$ is the $r^{\text{th}}$ term of the $k$--Fibonacci sequence. They showed that the Diophantine eq. \eqref{eqGGL} does not have non--trivial solutions $(k, n, m, x)$ with $k \ge 2$, $n \ge 1$, $m \ge 2$ and $x \ge 1$. Motivated by the work in \cite{GGL}, we study the same Diophantine equation but with $k$--Lucas numbers. Specifically, we solve the nonlinear Diophantine equation \begin{align}\label{eq:main} \left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x=L_m^{(k)}, \end{align} in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$. We prove the following result. \subsection{Main Result}\label{sec:1.2} \begin{theorem}\label{thm1.1} Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers. Then, the only integer solutions $(n,m,k,x)$ to Eq. \eqref{eq:main} with $n\ge 0,~m\ge 0,~k\ge 2$ and $x\ge 0$ (with the convention that $0^0:=1$ in case $(n,x)=(0,0)$ and $k\ge 3$) are \begin{align} (n,m,k,x)\in \{(n,1,k,0),(1,0,k,1):k\ge 2\}\cup\{(0,2,k,1),(1,3,k,2):k\ge 3\}\cup \{(0,3,2,1),(0,3,2,2)\}. \end{align} \end{theorem} \section{Methods} \subsection{Preliminaries} It is known that \begin{align}\label{eq2.2} L_n^{(k)} = 3 \cdot 2^{n-2},\qquad \text{for all}\qquad 2 \le n \le k. \end{align} Additionally, $L_{k+1}^{(k)}=3\cdot 2^{k-1}-2$ and by induction one proves that \begin{equation} \label{eq:32} L_n^{(k)}<3\cdot 2^{n-2}\qquad {\text{\rm holds for all}}\qquad n\ge k+1. \end{equation} Next, we revisit some properties of the $k$--generalized Lucas numbers. They form a linearly recurrent sequence of characteristic polynomial \[ \Psi_k(x) = x^k - x^{k-1} - \cdots - x - 1, \] which is irreducible over $\mathbb{Q}[x]$. The polynomial $\Psi_k(x)$ possesses a unique real root $\alpha(k)>1$ and all the other roots are inside the unit circle, see \cite{MIL}. The root $\alpha(k):=\alpha$ is in the interval \begin{align}\label{eq2.3} 2(1 - 2^{-k} ) < \alpha < 2 \end{align} as noted in \cite{WOL}. As in the classical case when $k=2$, it was shown in \cite{BRL} that \begin{align}\label{eq2.4} \alpha^{n-1} \le L_n^{(k)}\le2\alpha^n, \quad \text{holds for all} \quad n\ge0, ~k\ge 2. \end{align} Also, $L_{-1}^{(k)}=-1$ if $k=2$ and $0$ otherwise. In particular, combining \eqref{eq2.4} and \eqref{eq:main}, we have, if $n\ge 1$, \begin{align} \alpha^{m-1} \le L_m^{(k)}=\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x<2\left(L_{n+1}^{(k)}\right)^x\le \alpha^{(n+3)x+2}, \end{align} where we have used the fact that $2<\alpha^2$, for all $k\ge 2$. The above inequality also holds for $n=0$ since in this case $$ (L_{n+1}^{(k)})^x+(L_n^{(k)})^x-(L_{n-1}^{(k)})^x\le 1+2^x+1\le 2^{x+1}<\alpha^{2x+2}<\alpha^{(n+3)x+2}. $$ This gives $m<(n+3)x+3$. On the other hand, if $n\ge 1$, and $x\ge 2$, we have \begin{align} \alpha^{m+2}>2\alpha^{m} \ge L_m^{(k)}=\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x>\alpha^{nx}+\alpha^{(n-1)x}-2^x\alpha^{(n-1)x}\ge \alpha^{nx-1}, \end{align} from which we have $m>nx-3$. The above inequality also holds for $x=1$ since in this case $$ \alpha^{m+2}>2\alpha^m\ge L_m^{(k)}=L_{n+1}^{(k)}+L_n^{(k)}-L_{n-1}^{(k)}\ge 2L_{n}^{(k)}> \alpha^{n},\quad {\text{\rm so}}\qquad m>n-3. $$ Finally, the above inequality holds for $n=0$ as well since we assume that $m\ge 0$. Therefore, we have \begin{align}\label{m_b} nx-3 <m<(n+3)x+3. \end{align} Let $k\ge 2$ and define \begin{equation} \label{eq:fk} f_k(x):=\dfrac{x-1}{2+(k+1)(x-2)}. \end{equation} We have $$ \frac{df_k(x)}{dx}=-\frac{(k-1)}{(2+(k+1)(x-2))^2}<0\qquad {\text{\rm for~all}}\qquad x>0. $$ In particular, inequality \eqref{eq2.3} implies that \begin{align}\label{eq2.5} \dfrac{1}{2}=f_k(2)<f_k(\alpha)<f_k(2(1 - 2^{-k} ))\le \dfrac{3}{4}, \end{align} for all $k\ge 3$. It is easy to check that the above inequality holds for $k=2$ as well. Further, it is easy to verify that $\|f_k(\alpha_i)\|<1$, for all $2\le i\le k$, where $\alpha_i$ are the remaining roots of $\Psi_k(x)$ for $i=2,\ldots,k$. Moreover, it was shown in \cite{BRL} that \begin{align}\label{eq2.6} L_n^{(k)}=\displaystyle\sum_{i=1}^{k}(2\alpha_i-1)f_k(\alpha_i)\alpha_i^{n-1}~~\text{and}~~\left\|L_n^{(k)}-f_k(\alpha)(2\alpha-1)\alpha^{n-1}\right\|<\dfrac{3}{2}, \end{align} for all $k\ge 2$ and $n\ge 2-k$. This means that \begin{equation} \label{eq:Lnwitherror} L_n^{(k)}=f_k(\alpha)(2\alpha-1)\alpha^{n-1}+e_k(n), \qquad {\text{\rm where}}\qquad \|e_k(n)\|<1.5. \end{equation} The left expression in \eqref{eq2.6} is known as the Binet formula for $L_n^{(k)}$. Furthermore, the right inequality in \eqref{eq2.6} shows that the contribution of the zeros that are inside the unit circle to $L_n^{(k)}$ is small. Next, we state the following result which we shall use later in the proof of the main result. It is stated as Lemma 3 in \cite{Fay}. \begin{lemma}[Lemma 3 in \cite{Fay}]\label{fay} Let $k \ge 2$, $c \in (0,1)$ and $2\le n < 2^{ck}$. Then \[ L_n^{(k)} = 3 \cdot 2^{n-2}(1 + \zeta'_n), \quad \text{with} \quad \|\zeta'_n\| < \begin{cases} \dfrac{4}{2^{(1-c)k}}, & \text{if} \ c \le 0.693, \\ \dfrac{8.1}{2^{(1-c)k}}, & \text{otherwise}. \end{cases} \] \end{lemma} Next, we prove the following result. \begin{lemma}\label{lem:Lnx} Let \( k \ge 2 \), \( x \), \( n \) be positive integers. Then \[ \left(L_n^{(k)}\right)^x = f_k(\alpha)^x (2\alpha-1)^x\alpha^{(n-1)x}(1 + \eta_n), \] with \[ \|\eta_n\| < \dfrac{1.5xe^{1.5x/\alpha^{n-1}}}{\alpha^{n-1}}. \] \end{lemma} \begin{proof} By the Binet formula in \eqref{eq:Lnwitherror} and inequalities \eqref{eq2.5}, it follows that \[ \left(L_n^{(k)}\right)^x= f_k(\alpha)^x (2\alpha-1)^x\alpha^{(n-1)x} \left( 1 + \dfrac{e_k(n)}{f_k(\alpha)(2\alpha-1) \alpha^{n-1}} \right)^x. \] So, if we define \[ \eta_n := \left( 1 + \dfrac{e_k(n)}{f_k(\alpha)(2\alpha-1) \alpha^{n-1}} \right)^x - 1 = \sum_{j=1}^x \binom{x}{j} \left( \frac{e_k(n)}{f_k(\alpha)(2\alpha-1) \alpha^{n-1}} \right)^j; \] we get \[ \|\eta_n\| < \sum_{j=1}^x \dfrac{(1.5x/\alpha^{n-1})^j}{j!} < \dfrac{1.5x}{\alpha^{n-1}} \sum_{j=1}^x \dfrac{(1.5x/\alpha^{n-1})^{j-1}}{(j-1)!} \le \dfrac{1.5xe^{1.5x/\alpha^{n-1}}}{\alpha^{n-1}}, \] where we have used the fact $ \|e_k(n)\| < 1.5$ and $ (2\alpha-1)f_k(\alpha)>1$. \end{proof} Lastly here, we prove the following. \begin{lemma}\label{Ln:x} Let $x \geq 1$, $k \geq 2$, $i \in \{-1,0, 1\}$, $n + i \geq k + 2$ and $\max\{n + i, 16x\} < 2^{ck}$ for some $c \in (0, 0.25)$. Then, the estimate \[ \left(L_{n+i}^{(k)}\right)^x = 3^x\cdot 2^{(n+i-2)x} \left(1 + \xi_{n+i} \right) \] holds with \[ \|\xi_{n+i}\| <\dfrac{2}{2^{(1-2c)k}}. \] \end{lemma} \begin{proof} By Lemma \ref{fay} with $c \in (0, 0.25)$, we have \[ L_{n+i}^{(k)} = 3 \cdot 2^{n+i-2} \left( 1 + \zeta'_{n+i} \right), \quad \text{with} \quad \|\zeta'_{n+i}\| < \dfrac{4}{2^{(1-c)k}}. \] Hence, \begin{eqnarray} \left(L_{n+i}^{(k)}\right)^x & = & 3^x\cdot 2^{(n+i-2)x} \left(1 +\zeta_{n+i}' \right)^x\\ & = & 3^x 2^{(n+i-2)x} \exp(x\log(1+\zeta_{n+i}'))\\ & = & 3^x 2^{(n+i-2)x} \exp(\eta_{n+i}')\qquad \|\eta_{n+i}'\|<2x\|\zeta_{n+i}'\|\\ & = & 3^x2^{(n+i-2)x}(1+\xi_{n+i}')\qquad \|\xi_{n+i}'\|<2\|\eta_{n+i}'\|<4x\|\zeta_{n+i}'\|<\frac{16x}{2^{(1-c)k}}<\frac{1}{2^{(1-2c)k}}\left(<\frac{1}{2}\right) \end{eqnarray} The above calculations are justified since both inequalities $\|\log(1+y)\|<2\|y\|$ and $\|\exp y-1\|<2\|y\|$ hold for $y\in (-1/2,1/2)$, and the fact that all the intermediate quantities $\zeta_{n+i}',~\eta_{n+i}',\xi_{n+i'}$ are in the range $(-1/2,1/2)$ follows from the very last inequality above. \end{proof} \subsection{Linear forms in logarithms} We use Baker--type lower bounds for nonzero linear forms in logarithms of algebraic numbers. There are many such bounds mentioned in the literature but we use one of Matveev from \cite{MAT}. Before we can formulate such inequalities, we need the notion of height of an algebraic number recalled below. \begin{definition}\label{def2.1} Let $ \gamma $ be an algebraic number of degree $ d $ with minimal primitive polynomial over the integers $$ a_{0}x^{d}+a_{1}x^{d-1}+\cdots+a_{d}=a_{0}\prod_{i=1}^{d}(x-\gamma^{(i)}), $$ where the leading coefficient $ a_{0} $ is positive. Then, the logarithmic height of $ \gamma$ is given by $$ h(\gamma):= \dfrac{1}{d}\Big(\log a_{0}+\sum_{i=1}^{d}\log \max\{\|\gamma^{(i)}\|,1\} \Big). $$ \end{definition} In particular, if $ \gamma$ is a rational number represented as $\gamma=p/q$ with coprime integers $p$ and $ q\ge 1$, then $ h(\gamma ) = \log \max\{\|p\|, q\} $. The following properties of the logarithmic height function $ h(\cdot) $ will be used in the rest of the paper without further reference: \begin{equation}\nonumber \begin{aligned} h(\gamma_{1}\pm\gamma_{2}) &\leq h(\gamma_{1})+h(\gamma_{2})+\log 2;\\ h(\gamma_{1}\gamma_{2}^{\pm 1} ) &\leq h(\gamma_{1})+h(\gamma_{2});\\ h(\gamma^{s}) &= \|s\|h(\gamma) \quad {\text{\rm valid for}}\quad s\in \mathbb{Z}. \end{aligned} \end{equation} In Section 3, equation (12) of \cite{Brl} these properties were used to show the following inequality: \begin{align}\label{eq2.9} h\left(f_k(\alpha)\right)<3\log k, ~~\text{for all}~~k\ge 2. \end{align} A linear form in logarithms is an expression \begin{equation} \label{eq:Lambda} \Lambda:=b_1\log \gamma_1+\cdots+b_t\log \gamma_t, \end{equation} where $\gamma_1,\ldots,\gamma_t$ are positive real algebraic numbers and $b_1,\ldots,b_t$ are integers. We assume, $\Lambda\ne 0$. We need lower bounds for $\|\Lambda\|$. We write ${\mathbb K}:={\mathbb Q}(\gamma_1,\ldots,\gamma_t)$ and $D$ for the degree of ${\mathbb K}$ over ${\mathbb Q}$. We give a direct consequence of Matveev's inequality from \cite{MAT}. That is, we quote it in a form which we shall use. \begin{theorem}[Matveev, \cite{MAT}] \label{thm:Mat} Put $\Gamma:=\gamma_1^{b_1}\cdots \gamma_t^{b_t}-1=e^{\Lambda}-1$. Then $$ \log \|\Gamma\|>-1.4\cdot 30^{t+3}\cdot t^{4.5} \cdot D^2 (1+\log D)(1+\log B)A_1\cdots A_t, $$ where $B\ge \max\{\|b_1\|,\ldots,\|b_t\|\}$ and $A_i\ge \max\{Dh(\gamma_i),\|\log \gamma_i\|,0.16\}$ for $i=1,\ldots,t$. \end{theorem} In our application of Matveev's result (Theorem \ref{thm:Mat}), we need to ensure that the linear forms in logarithms are indeed nonzero. To ensure this, we shall need the following result given as Lemma 2.8 in \cite{GGL1}. \begin{lemma}[Lemma 2.8 in \cite{GGL1}]\label{lemGLm} Let \( N := N_{\mathbb{K}/\mathbb{Q}}, \) where \( \mathbb{K} = \mathbb{Q}(\alpha) \). Then \begin{enumerate}[(i)] \item For \( n, m \geq 1 \) and \( k \geq 2 \), \( \|N(\alpha)\| = 1 \). \item \( N(2\alpha - 1) = 2^{k+1} - 3 \) and \( N(f_k(\alpha)) = (k - 1)^2 / (2^{k+1}k^k - (k + 1)^{k+1}) \). \item For \( k \geq 2 \), \( N((2\alpha - 1)f_k(\alpha)) \le 1 \). This inequality is strict for $k\ge 3$ (and is an equality for $k=2$). \end{enumerate} \end{lemma} At some point, we treat cases with $t=2$, that is, linear forms in two logarithms. Let $A_1,~A_2>1$ be real numbers such that \begin{equation} \label{eq:Aih} \log A_i\ge \max\left\{h(\gamma_i),\frac{\|\log \gamma_i\|}{D},\frac{1}{D}\right\}\quad {\text{\rm for}}\quad i=1,2. \end{equation} Put $$ b':=\frac{\|b_1\|}{D\log A_2}+\frac{\|b_2\|}{D\log A_1}. $$ The following result is Corollary 2 in \cite{LMN}. \begin{theorem}[Laurent et al., \cite{LMN}] \label{thm:LMNh} In case $t=2$, we put $$\Lambda:=b_1\log \gamma_1-b_2\log \gamma_2, $$ where $\|\gamma_{1}\|, \|\gamma_{2}\|\ge 1$ are two multiplicatively independent real algebraic numbers and $b_1, b_2$ are positive integers. Then, we have $$ \log \|\Lambda\|\ge -24.34 D^4\left(\max\left\{\log b'+0.14,\frac{21}{D},\frac{1}{2}\right\}\right)^2\log A_1\log A_2. $$ \end{theorem} During the calculations, upper bounds on the variables are obtained which are too large, thus there is need to reduce them. To do so, we use some results from approximation lattices and the so--called LLL--reduction method from \cite{LLL}. We explain this in the following subsection. \subsection{Reduced Bases for Lattices and LLL--reduction methods}\label{sec2.3} Let $k$ be a positive integer. A subset $\mathcal{L}$ of the $k$--dimensional real vector space ${ \mathbb{R}^k}$ is called a lattice if there exists a basis $\{b_1, b_2, \ldots, b_k \}$ of $\mathbb{R}^k$ such that \begin{align} \mathcal{L} = \sum_{i=1}^{k} \mathbb{Z} b_i = \left\{ \sum_{i=1}^{k} r_i b_i \mid r_i \in \mathbb{Z} \right\}. \end{align} We say that $b_1, b_2, \ldots, b_k$ form a basis for $\mathcal{L}$, or that they span $\mathcal{L}$. We call $k$ the rank of $ \mathcal{L}$. The determinant $\text{det}(\mathcal{L})$, of $\mathcal{L}$ is defined by \begin{align} \text{det}(\mathcal{L}) = \| \det(b_1, b_2, \ldots, b_k) \|, \end{align} with the $b_i$'s being written as column vectors. This is a positive real number that does not depend on the choice of the basis (see \cite{Cas}, Section 1.2). Given linearly independent vectors $b_1, b_2, \ldots, b_k$ in $ \mathbb{R}^k$, we refer back to the Gram--Schmidt orthogonalization technique. This method allows us to inductively define vectors $b^_i$ (with $1 \leq i \leq k$) and real coefficients $\mu_{i,j}$ (for $1 \leq j \leq i \leq k$). Specifically, \begin{align} b^_i &= b_i - \sum_{j=1}^{i-1} \mu_{i,j} b^_j,~~~ \mu_{i,j} = \dfrac{\langle b_i, b^_j\rangle }{\langle b^_j, b^_j\rangle}, \end{align} where \( \langle \cdot , \cdot \rangle \) denotes the ordinary inner product on \( \mathbb{R}^k \). Notice that \( b^_i \) is the orthogonal projection of \( b_i \) on the orthogonal complement of the span of \( b_1, \ldots, b_{i-1} \), and that \( \mathbb{R}b_i \) is orthogonal to the span of \( b^_1, \ldots, b^_{i-1} \) for \( 1 \leq i \leq k \). It follows that \( b^_1, b^_2, \ldots, b^_k \) is an orthogonal basis of \( \mathbb{R}^k \). \begin{definition} The basis $b_1, b_2, \ldots, b_n$ for the lattice $\mathcal{L}$ is called reduced if \begin{align} \\| \mu_{i,j} \\| &\leq \frac{1}{2}, \quad \text{for} \quad 1 \leq j < i \leq n,~~ \text{and}\\ \\|b^_{i}+\mu_{i,i-1} b^_{i-1}\\|^2 &\geq \frac{3}{4}\\|b^_{i-1}\\|^2, \quad \text{for} \quad 1 < i \leq n, \end{align} where $ \\| \cdot \\| $ denotes the ordinary Euclidean length. The constant $ {3}/{4}$ above is arbitrarily chosen, and may be replaced by any fixed real number $ y $ in the interval ${1}/{4} < y < 1$ (see \cite{LLL}, Section 1). \end{definition} Let $\mathcal{L}\subseteq\mathbb{R}^k$ be a $k-$dimensional lattice with reduced basis $b_1,\ldots,b_k$ and denote by $B$ the matrix with columns $b_1,\ldots,b_k$. We define \[ l\left( \mathcal{L},y\right):= \left\{ \begin{array}{c} \min_{x\in \mathcal{L}}\|\|x-y\|\| \quad ;~~ y\not\in \mathcal{L}\\ \min_{0\ne x\in \mathcal{L}}\|\|x\|\| \quad ;~~ y\in \mathcal{L} \end{array} \right., \] where $\|\|\cdot\|\|$ denotes the Euclidean norm on $\mathbb{R}^k$. It is well known that, by applying the LLL--algorithm, it is possible to give in polynomial time a lower bound for $l\left( \mathcal{L},y\right)$, namely a positive constant $c_1$ such that $l\left(\mathcal{L},y\right)\ge c_1$ holds (see \cite{SMA}, Section V.4). \begin{lemma}\label{lem2.5} Let $y\in\mathbb{R}^k$ and $z=B^{-1}y$ with $z=(z_1,\ldots,z_k)^T$. Furthermore, \begin{enumerate}[(i)] \item if $y\not \in \mathcal{L}$, let $i_0$ be the largest index such that $z_{i_0}\ne 0$ and put $\sigma:=\{z_{i_0}\}$, where $\{\cdot\}$ denotes the distance to the nearest integer. \item if $y\in \mathcal{L}$, put $\sigma:=1$. \end{enumerate} \noindent Finally, let \[ c_2:=\max\limits_{1\le j\le k}\left\{\dfrac{\|\|b_1\|\|^2}{\|\|b_j^\|\|^2}\right\}. \] Then, \[ l\left( \mathcal{L},y\right)^2\ge c_2^{-1}\sigma^2\|\|b_1\|\|^2:=c_1^2. \] \end{lemma} In our application, we are given real numbers $\eta_0,\eta_1,\ldots,\eta_k$ which are linearly independent over $\mathbb{Q}$ and two positive constants $c_3$ and $c_4$ such that \begin{align}\label{2.9} \|\eta_0+a_1\eta_1+\cdots +a_k \eta_k\|\le c_3 \exp(-c_4 H), \end{align} where the integers $a_i$ are bounded as $\|a_i\|\le A_i$ with $A_i$ given upper bounds for $1\le i\le k$. We write $A_0:=\max\limits_{1\le i\le k}\{A_i\}$. The basic idea in such a situation, from \cite{Weg}, is to approximate the linear form \eqref{2.9} by an approximation lattice. So, we consider the lattice $\mathcal{L}$ generated by the columns of the matrix $$ \mathcal{A}=\begin{pmatrix} 1 & 0 &\ldots& 0 & 0 \\ 0 & 1 &\ldots& 0 & 0 \\ \vdots & \vdots &\vdots& \vdots & \vdots \\ 0 & 0 &\ldots& 1 & 0 \\ \lfloor C\eta_1\rfloor & \lfloor C\eta_2\rfloor&\ldots & \lfloor C\eta_{k-1}\rfloor& \lfloor C\eta_{k} \rfloor \end{pmatrix} ,$$ where $C$ is a large constant usually of the size of about $A_0^k$ . Let us assume that we have an LLL--reduced basis $b_1,\ldots, b_k$ of $\mathcal{L}$ and that we have a lower bound $l\left(\mathcal{L},y\right)\ge c_1$ with $y:=(0,0,\ldots,-\lfloor C\eta_0\rfloor)$. Note that $ c_1$ can be computed by using the results of Lemma \ref{lem2.5}. Then, with these notations the following result is Lemma VI.1 in \cite{SMA}. \begin{lemma}[Lemma VI.1 in \cite{SMA}]\label{lem2.6} Let $S:=\displaystyle\sum_{i=1}^{k-1}A_i^2$ and $T:=\dfrac{1+\sum_{i=1}^{k}A_i}{2}$. If $c_1^2\ge T^2+S$, then inequality \eqref{2.9} implies that we either have $a_1=a_2=\cdots=a_{k-1}=0$ and $a_k=-\dfrac{\lfloor C\eta_0 \rfloor}{\lfloor C\eta_k \rfloor}$, or \[ H\le \dfrac{1}{c_4}\left(\log(Cc_3)-\log\left(\sqrt{c_1^2-S}-T\right)\right). \] \end{lemma} Finally, we present an analytic argument which is Lemma 7 in \cite{GL}. \begin{lemma}[Lemma 7 in \cite{GL}]\label{Guz} If $ r \geq 1 $, $T > (4r^2)^r$ and $T > \dfrac{p}{(\log p)^r}$, then $$p < 2^r T (\log T)^r.$$ \end{lemma} SageMath 9.5 is used to perform all computations in this work. \section{Proof of Theorem \ref{thm1.1}.}\label{Sec3} In this section, we prove Theorem \ref{thm1.1}. To do this, we first find some trivial solutions. \subsection{Trivial solutions} Here, we study equation \eqref{eq:main} in the following trivial scenarios. \begin{enumerate}[(a)] \item If $x=0$, then (assuming $0^0:=1$ when $n=0$ and $k\ge 3$) \eqref{eq:main} becomes $L_m^{(k)}=1$, for which $m=1$. We therefore get $(n,m,k,x)=(n,1,k,0)$ with $n\ge 1$ for all $k\ge 2$. \item If $x=1$, then the Diophantine equation \eqref{eq:main} becomes \begin{align}\label{x1} L_{n+1}^{(k)}+L_{n}^{(k)}-L_{n-1}^{(k)}=L_m^{(k)}. \end{align} Moreover, the inequalities in \eqref{m_b} tells us that for $x=1$, we have $n-3 <m<n+6$, for all $k\ge 2$. This implies that $ m\in\{n-2, n-1,n,n+1,\ldots,n+5\}$. A straightforward verification shows that the only solutions to \eqref{x1} with these options are $(n,m,k,x)=(0,2,k,1)$ with $k\ge 3$, $(n,m,k,x)=(1,0,k,1)$ for all $k\ge 2$ and also the sporadic solution $(n,m,k,x)=(0,3,2,1)$. \end{enumerate} Assume from now on that $x\ge 2$. Suppose next that $m\le 2$. Since $m\ge nx-2$, we get $nx\le 4$ and since $x\ge 2$, we get $n\in \{0,1,2\}$. Thus, $L_m^{(k)}\in \{2,1,3\}$, while $$ \left(L_{n+1}^{(k)}\right)^x+\left(L_n^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x\ge 1+2^x-1=2^x\ge 4>L_m^{(k)}, $$ so there are no solutions in this range. For the remaining part of the proof, we assume $k\ge 2$, $x\ge 2$, $m\ge 3$. \subsection{The case $k=2$ and $n=0,2$} Assume $k=2$. In the case $n=0$ we get the equation $$ 1+2^x-(-1)^x=L_m. $$ When $x$ is even, we get $L_m=2^x$ and it is known (for example, by Carmichael's primitive divisor theorem \cite{Car}) that the only powers of $2$ in the Lucas sequence are $m=0,~3$ for which $L_0=2,~L_3=2^2$ and the only one with even exponent $x$ is $L_3=2^2$. We get the solution $(n,m,k,x)=(0,3,2,2)$. When $x$ is odd, we get $L_m=2^x+2$. When $x=1$, we get the solution $m=3$ which leads to $(n,m,k,x)=(0,3,2,1)$. When $x=2$, we get $L_m=6$, which is false since $6$ is not a member of the sequence of Lucas numbers. When $x\ge 3$, we get that $2\\| L_m$ so $m$ is even (and multiple of $3$). In particular, $L_m=L_{m/2}^2\pm 2$. We thus get $$ L_{m/2}^2\pm 2=2^x+2,\qquad {\text{\rm so}}\qquad L_{m/2}^2\in \{2^x,2^x+4\}. $$ Since $x$ is odd, the equation $L_{m/2}^2=2^x$ is not possible. Thus, $L_{m/2}^2=2^x+4$ leading to $2\\| L_{m/2}$ and $(L_{m/2}^2/2)^2=2^{x-2}+1$. This is a particular instance of the Catalan equation and its only solution is $x-2=3$ and $L_{m/2}/2=3$, leading to $L_{m/2}=6$, but this is wrong since $6$ is not a member of the sequence of Lucas numbers. Assume $n=2$. In this case we get the equation $$ 4^x+3^x-1=L_m. $$ The left--hand side is a multiple of $3$, so $m$ is even. Considering the values of $x$ modulo $4$, we get that the left--hand side is congruent to $1$ modulo $5$ (if $x\equiv 0,1\pmod 5$), to $4\pmod 5$ (if $x\equiv 2\pmod 4$) and to $0\pmod 5$ (if $x\equiv 3\pmod 4$). Since the Lucas sequence is periodic modulo $5$ with period $4$ and achieves the values $2,1,3,4,2,1,3,4,\ldots$, we get that only the cases $L_m\equiv 1,4\pmod 5$ are possible and this implies $m\equiv 1,3\pmod 4$. This contradicts the fact that $m$ is even. \subsection{The case $n\le k$}\label{subsec3.2} \subsubsection{Bounding $x$ and $m$ in terms of $k$} We start by revisiting \eqref{eq2.2}. Assume $n\ge 3$. Then we rewrite \eqref{eq:main} as \begin{align} L_m^{(k)}&= \left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x\\ &=\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x. \end{align} Assume $m\le k$. Then, since $m\ge 3$, we have $L_m^{(k)}=3\cdot 2^{m-2}$. If $n\ge 3$, then the equation above is \begin{align} 3\cdot2^{m-2}=\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x. \end{align} The right--hand side above is divisible by $3^x$. So, $3^x$ must divide $3\cdot2^{m-2}$, which is false for $x\ge 2$. This was in case $n\ge 3$. We must consider the cases of small $n$ as well. If $n=0,1,2$, then since $k\ne 2$ when $n=0,2$, we have $$ 3\cdot 2^{m-2}=L_m^{(k)}=(L_{n+1}^{(k)})^x+(L_n^{(k)})^x-(L_{n-1}^{(k)})^x\in \{1+2^x,3^x+1-2^x,6^x+3^x-1\} $$ (the first element above corresponds to $n=0$ and $k\ge 3$, the next element corresponds to $n=1$ and the third element corresponds to $n=2$ and $k\ge 3$, respectively). We need to find out when the elements of the above set are of the form $3\cdot 2^{m-2}$. The first is odd and the third is not a multiple of $3$, so they cannot be of the form $3\cdot 2^{m-2}$. Finally, for the second element, we get the equation $3\cdot 2^{m-2}=3^x+1-2^x$. This has the solution $(x,m)=(2,3)$ and no other solution since for $x\ge 3$, the right--hand side is at least $20$ and not a multiple of $8$, so it cannot be of the form $3\cdot 2^{m-2}$. Thus, we get the additional solution $(n,m,k,x)=(1,3,k,2)$. Note that $k\ge 3$ since $3=m\le k$. For the rest of this section, we assume $k\ge 2$, $x\ge 2$, $m\ge \max\{k+1,nx-2\}$. To proceed, we prove the following result. \begin{lemma}\label{lem3.1} Let $n$, $m$, $k$, $x$ be integer solutions to Eq. \eqref{eq:main} with $k\ge 2$, $0\le n\le k$, $k\ge 2$ and $m\ge \max\{k+1,nx-2\}$, then \begin{align} x< 3.1\cdot 10^{14} k^5(\log k)^2 \log m. \end{align} \end{lemma} \begin{proof} Assume first that $n\ge 3$. We go back to \eqref{eq:main} and rewrite it as \begin{align} L_m^{(k)}&=\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x. \end{align} In case $n\in \{0,1,2\}$, then again using the fact that $k\ge 3$ when $n\in \{0,2\}$, the right--hand side above is \begin{equation} \label{eq:r} 2^x+1,\quad 3^x+1-2^x, \quad 6^x+3^x-1. \end{equation} Now, using the Binet formula in \eqref{eq:Lnwitherror}, we have $$ L_m^{(k)}-f_k(\alpha)(2\alpha-1)\alpha^{m-1}=e_k(m). $$ Thus, assuming $n\ge 3$, we get \begin{align} L_m^{(k)}-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=e_k(m)\\ \left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=e_k(m)\\ \left(3\cdot 2^{n-1}\right)^x-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=-\left(3\cdot 2^{n-2}\right)^x+\left(3\cdot 2^{n-3}\right)^x+e_k(m)\\ 1-3^{-x}2^{-(n-1)x}f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=-\dfrac{1}{2^x}+\dfrac{1}{2^{2x}}+\dfrac{e_k(m)}{\left(3\cdot 2^{n-1}\right)^x}. \end{align} Taking absolute values and simplifying, we get \begin{align} \label{eq:gen} \|\Gamma_1\|:= \left\|3^{-x}2^{-(n-1)x}f_k(\alpha)(2\alpha-1)\alpha^{m-1}-1\right\|<\dfrac{3}{2^x}. \end{align} This was for the case $n\ge 3$. For the cases when $n=0,1,2$, the amount $3\cdot 2^{n-1}$ above gets replaced by $2,3,6$, respectively and the right--hand sides get replaced by $$ \frac{1}{2^x}+\frac{\|e_k(m)\|}{2^x},\quad \left(\frac{2}{3}\right)^x+\frac{1}{3^x}+\frac{\|e_k(m)\|}{3^x},\quad \left(\frac{3}{6}\right)^x+\left(\frac{1}{6}\right)^x+\frac{\|e_k(m)\|}{6^x}, $$ respectively. All these expressions are bounded by $3(2/3)^x$. Thus, we get \begin{equation} \label{eq:gen1} \|\Gamma_1\|<3\left(\frac{2}{3}\right)^x, \end{equation} where now \begin{equation} \label{g1} \Gamma_1:=\delta^{-x} f_k(\alpha)(2\alpha-1)\alpha^{m-1}-1,\qquad \delta\in \{2,3,6\}. \end{equation} Note that $\Gamma_1\ne 0$ in all cases, otherwise we would have $$ f_k(\alpha)(2\alpha-1)\alpha^{m-1}\in \{(3\cdot 2^{n-1})^x,2^x,3^x,6^x\}. $$ Applying norms in ${\mathbb K}={\mathbb Q}(\alpha)$ and using $\|N(\alpha)\|=1$, and item $(ii)$ and $(iii)$ of Lemma \ref{lemGLm}, the above equation becomes $$ N\left(f_k(\alpha)\right)\cdot N(2\alpha-1)\in \{(3\cdot 2^{n-1})^{kx},2^{kx}, 3^{kx},6^{kx}\}. $$ which implies that \begin{align} 1\ge \dfrac{(k - 1)^2 }{ 2^{k+1}k^k - (k + 1)^{k+1}}\cdot \left(2^{k+1}-3\right)=\left(2\right)^{kx}\ge \left(2\right)^{2\cdot 2}=16, \end{align} a contradiction, for $x\ge 2$ and $k\ge 2$. Thus, $\Gamma_1\ne 0$. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\alpha)$. When $n\ge 3$, we have $D = k$, $t :=4$, \begin{equation}\nonumber \begin{aligned} \gamma_{1}&:=(2\alpha-1)f_k(\alpha),\qquad \gamma_{2}:=\alpha, \qquad\gamma_{3}:=3,\qquad \gamma_{4}:=2,\\ b_{1}&:=1,\qquad b_{2}:=m-1,\qquad b_{3}:=-x,\qquad b_{4}:=-(n-1)x. \end{aligned} \end{equation} We can take $A_3:=k \log 3$ and $A_4:=k \log 2$. Additionally, $h(\gamma_{2})=(\log \alpha)/k <0.7/k$, so we take $A_{2}:=0.7$. For $A_1$, we first compute \begin{align} h(\gamma_{1}):=h\left((2\alpha-1)f_k(\alpha)\right)\le h\left((2\alpha-1)\right)+h\left(f_k(\alpha)\right)<\log 3+3\log k<6\log k, \end{align} for all $k\ge 2$. So, we can take $A_1:=6k\log k$. Next, $B \geq \max\{\|b_i\|:i=1,2,3, 4\}$, and by relation \eqref{m_b}, we can take $B:=m$. Now, by Theorem \ref{thm:Mat}, \begin{align}\label{eq3.4} \log \|\Gamma_1\| &> -1.4\cdot 30^{7} \cdot 4^{4.5}\cdot k^2 (1+\log k)(1+\log m)\cdot 6k\log k \cdot (0.7) \cdot (k\log 3)\cdot (k\log 3)\nonumber\\ &> -2.1\cdot 10^{14} k^5(\log k)^2 \log m. \end{align} For the cases $n=0,1,2$, we take $t:=3$ instead, keep the same $\gamma_1,~\gamma_2$ but $\gamma_3:= \delta$. Thus, we can take $A_3:=k\log 6$. We then get \begin{align}\label{eq3.41} \log \|\Gamma_1\| &> -1.4\cdot 30^{6} \cdot 4^{3.5}\cdot k^2 (1+\log k)(1+\log m)\cdot 6k\log k \cdot (0.7) \cdot (k\log 6))\nonumber\\ &> -2.7\cdot 10^{12} k^4(\log k)^2 \log m. \end{align} When $n\ge 3$, comparing \eqref{eq:gen} and \eqref{eq3.4}, we get \begin{align} x\log 2-\log 3&<2.1\cdot 10^{14} k^5(\log k)^2 \log m, \end{align} which leads to $x<3.1\cdot 10^{14} k^5(\log k)^2 \log m$. When $n=0,1,2$, applying instead \eqref{eq:gen1} we get $$ x\log(3/2)-\log 3<2.7\cdot 10^{12} k^4(\log k)^2 \log m, $$ and this gives a smaller bound on $x$. \end{proof} Next, we prove the following. \begin{lemma}\label{lem3.2} Let $n$, $m$, $k$, $x$ be integer solutions to Eq. \eqref{eq:main} with $k\ge 2$, $0\le n\le k$, $k\ge 2$ and $m\ge \max\{k+1,5\}$, then \begin{align} m< 6.3\cdot 10^{32}k^{10}(\log k)^5 . \end{align} \end{lemma} \begin{proof} If $n=0,1,2$, then $$ m< (n+3)x+3\le 5x+3<5\cdot 3.1\cdot 10^{14} k^5 (\log k)^2\log m+3<2\cdot 10^{15} k^5 (\log k)^2 \log m. $$ Thus, $$ \frac{m}{\log m}<2\times 10^{15} k^5 (\log k)^2. $$ Applying Lemma \ref{Guz} with $p:=m$, $r:=2$ and $T:=2\times 10^{15} k^5 (\log k)^2$ we get a better bound than the one from the statement of the lemma. From now on, we assume $n\ge 3$. We go back to \eqref{eq:main} and rewrite it as \begin{align} L_m^{(k)}&=\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x=\left(3\cdot 2^{n-2}\right)^x\left(2^x+1-2^{-x}\right). \end{align} As before, we use the Binet formula in \eqref{eq:Lnwitherror} and have \begin{align} L_m^{(k)}-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=e_k(m)\\ \left(3\cdot 2^{n-2}\right)^x\left(2^x+1-2^{-x}\right)-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=e_k(m)\\ \left(3\cdot 2^{n-2}\right)^{x}\left(2^x+1-2^{-x}\right)\left(f_k(\alpha)\right)^{-1}(2\alpha-1)^{-1}\alpha^{-(m-1)}-1&=\dfrac{e_k(m)}{f_k(\alpha)(2\alpha-1)\alpha^{m-1}}. \end{align} Taking absolute values and simplifying, we get \begin{align}\label{g2} \|\Gamma_2\|&:=\left\|\left(3\cdot 2^{n-2}\right)^{x}\left(2^x+1-2^{-x}\right)\left(f_k(\alpha)\right)^{-1}(2\alpha-1)^{-1}\alpha^{-(m-1)}-1\right\| <\dfrac{1.5}{0.5\cdot1.5\alpha^{m-1}} =\dfrac{2}{\alpha^{m-1}}, \end{align} where we have used the fact that $f_k(\alpha)>0.5$ and $2\alpha-1>1.5$, for all $k\ge 2$. Clearly, $\Gamma_2\ne 0$, otherwise we would have \begin{align} f_k(\alpha)= \dfrac{\left(3\cdot 2^{n-2}\right)^x\left(2^x+1-2^{-x}\right)}{(2\alpha-1)\alpha^{m-1}}. \end{align} Taking norms in ${\mathbb K}={\mathbb Q}(\alpha)$ as before, the above equation becomes \begin{align} N\left(f_k(\alpha)\right)\cdot N(2\alpha-1)=\left(3\cdot 2^{n-2}\right)^{kx}\left(2^x+1-2^{-x}\right)^k. \end{align} We have already shown before that the left--hand side above is $\le 1$. Therefore, \begin{align} 1\ge N\left(f_k(\alpha)\right)\cdot N(2\alpha-1)=\left(3\cdot 2^{n-2}\right)^{kx}\left(2^x+1-2^{-x}\right)^k>500, \end{align} a contradiction, for all $n\ge 3$, $x\ge 2$ and $k\ge 2$. Note that in the deduction above, we have used the fact that the function $\left(2^x+1-2^{-x}\right)$ is increasing and it is at least $ 4$ for all $x\ge 2$. Thus, $\Gamma_2\ne 0$. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\alpha)$, so $D = k$, $t :=4$, \begin{equation}\nonumber \begin{aligned} \gamma_{1}&:=3,\qquad \gamma_{2}:=2, \qquad\gamma_{3}:=\left(2^x+1-2^{-x}\right)/\left((2\alpha-1)f_k(\alpha)\right),\qquad \gamma_{4}:=\alpha,\\ b_{1}&:=x\qquad b_{2}:=(n-2)x,\qquad b_{3}:=1,\qquad b_{4}:=-(m-1). \end{aligned} \end{equation} As before, $A_1:=k \log 3$, $A_2:=k \log 2$ and $A_{4}:=0.7$. To determine what $A_3$ could be, we first compute \begin{align} h(\gamma_{3})&=h\left(\left(2^x+1-2^{-x}\right)/\left((2\alpha-1)f_k(\alpha)\right)\right)\le h\left(2^x+1-2^{-x}\right)+h(2\alpha-1)+h\left(f_k(\alpha)\right)\\ &<2x\log 2+2\log 2+\log 3+3\log k\\ &<2\left(3.1\cdot 10^{14} k^5(\log k)^2 \log m\right)\log 2+2\log 2+6\log k\\ &<4.5\cdot 10^{14} k^5(\log k)^2 \log m, \end{align} where we have used Lemma \ref{lem3.1}. Thus, we can take $A_3:=4.5\cdot 10^{14} k^6(\log k)^2 \log m$. Next, $B \geq \max\{\|b_i\|:i=1,2,3,4\}$, and by relation \eqref{m_b}, we can take $B:=m$. Now, by Theorem \ref{thm:Mat}, \begin{align}\label{eq3.8} \log \|\Gamma_2\| &> -1.4\cdot 30^{7} \cdot 4^{4.5}\cdot k^2 (1+\log k)(1+\log m)\cdot k \log 3\cdot k \log 2\cdot 0.7\cdot 4.5\cdot 10^{14} k^6(\log k)^2 \log m\nonumber\\ &> -6.8\cdot 10^{27} k^{10}(\log k)^3 (\log m)^2. \end{align} Comparing \eqref{g2} and \eqref{eq3.8}, we get \begin{align} (m-1)\log\alpha-\log 2&<6.8\cdot 10^{27} k^{10}(\log k)^3 (\log m)^2, \end{align} which leads to $m<1.5\cdot 10^{28} k^{10}(\log k)^3 (\log m)^2$. We now apply Lemma \ref{Guz} with $p:=m$, $r:=2$, $T:=1.5\cdot 10^{28} k^{10}(\log k)^3 >(4r^2)^r=256$ for all $k\ge 2$. We get \begin{align} m&<2^2\cdot 1.5\cdot 10^{28} k^{10}(\log k)^3\left(\log 1.5\cdot 10^{28} k^{10}(\log k)^3\right)^2\\ &=6\cdot 10^{28} k^{10}(\log k)^3 \left(\log (1.5\cdot 10^{28})+10\log k+3\log\log k\right)^2\\ &<6\cdot 10^{28} k^{10}(\log k)^5 \left(\dfrac{65}{\log k}+10+\dfrac{3\log\log k}{\log k}\right)^2, \end{align} so $m<6.3\cdot 10^{32}k^{10}(\log k)^5 $. \end{proof} \subsubsection{The case $k>800$}\label{sub322} Here, we proceed by assuming for a moment that $k>800$. We get $m<6.3\cdot 10^{32}k^{10}(\log k)^5 <2^{0.28k}$, for all $k>800$. Assume $n\ge 3$. We can rewrite \eqref{eq:main} as \begin{align} L_m^{(k)} &=\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x. \end{align} By part $(i)$ of Lemma \ref{fay} with $c:=0.28$, we have \begin{align} \left\|\left(3\cdot 2^{n-1}\right)^x+\left(3\cdot 2^{n-2}\right)^x-\left(3\cdot 2^{n-3}\right)^x-3\cdot 2^{m-2}\right\|<3\cdot 2^{m-2}\cdot \dfrac{4}{2^{0.72k}}, \end{align} which can be rewritten as \begin{align}\label{eqc} \left\|3^{x-1}(2^{2x}+2^x-1)-2^{m-2-(n-3)x}\right\|<2^{m-2-(n-3)x} \cdot\frac{ 4}{2^{0.72 k}}. \end{align} Now, putting $y:=m-2-(n-3)x$ in \eqref{eqc} and dividing through by $2^y$, we get \begin{align}\label{eq:0} \left\|\frac{3^{x-1} (2^{2x}+2^x-1)}{2^y}-1\right\|<\frac{4}{2^{0.72 k}}. \end{align} This was for $n\ge 3$. For $n=0,1,2$ (since $k\ge 3$ for $n=0,2$), the same argument leads to the following inequalities \begin{eqnarray} \left\|\frac{2^{x-(m-2)} (1+2^{-x})}{3}-1\right\| & < & \frac{4}{2^{0.72k}},\qquad n=0;\\ \left\|\frac{3^{x-1}(1-(2^x-1)/3^x)}{2^{m-2}}-1\right\| & < & \frac{4}{2^{0.72k}},\qquad n=1;\\ \left\|\frac{3^{x-1}(1+(3^x-1)/6^x)}{2^{m-2-x}}-1\right\| & < & \frac{4}{2^{0.72k}}, \qquad n=2. \end{eqnarray} The first one is false which shows that the case $n=0$ cannot hold for $k>800$. Indeed, either $x-(m-2)\le 1$ or $x-(m-2)\ge 2$. If $x-(m-2)\le 1$, then $$ \frac{2^{x-(m-2)} (1+2^{-x})}{3}\le \frac{2(1+1/2^2)}{3}=\frac{5}{6},\qquad {\text{\rm so}}\qquad \left\|\frac{2^{x-(m-2)}(1+2^{-x})}{3}-1\right\|\ge \frac{1}{6}, $$ while if $x-(m-2)\ge 2$, then $$ \frac{2^{x-(m-2)} (1+2^{-x})}{3}\ge \frac{2^2}{3}=\frac{4}{3},\qquad {\text{\rm so}}\qquad \left\|\frac{2^{x-(m-2)}(1+2^{-x})}{3}-1\right\|\ge \frac{1}{3}. $$ Thus, in all cases, when $n=0$, we get $1/6<4/2^{0.72k}$, so $2^{0.72k}<24$, which is false for $k>800$. Going back to the case $n\ge 3$, the left--hand side in \eqref{eq:0} is \begin{align} \left\|e^{(x-1)\log 3+\log(2^{2x}+2^x+1)-y\log 2}-1\right\|, \end{align} and the right--hand side is $<1/2$. So, we get by a simple argument in calculus that \begin{align} \left\|(x-1)\log 3+\log(2^{2x}+2^x-1)-y\log 2\right\|<\frac{8}{2^{0.72 k}}. \end{align} This can be rewritten as \begin{align} \left\|(x-1)\log 3+2x\log 2+\log\left(1+\frac{2^x-1}{2^{2x}}\right)-y\log 2\right\|<\frac{8}{2^{0.72k}}, \end{align} or \begin{equation}\label{eq:1} \|(x-1)\log 3-z\log 2\|<\frac{8}{2^{0.72k}}+\log\left(1+\frac{2^x-1}{2^{2x}}\right)<\frac{8}{2^{0.72k}}+\frac{2^x-1}{2^{2x}}, \end{equation} with $z:=y-2x$. Since $x\ge 2$, the right--hand side in \eqref{eq:1} is $<6/16$, so $z>0$. The cases $n=1,~2$ lead, by similar arguments, to \begin{equation}\label{eq:1prime} \|(x-1)\log 3-z\log 2\|<\frac{8}{2^{0.72k}}+\delta(x),\qquad \delta(x):=\begin{cases} -\log\left(1-\dfrac{2^x-1}{3^x}\right), & \text{if} \ n=1, \\ \log\left(1+\dfrac{3^x-1}{6^x}\right), & \text{if} \ n=2,\end{cases} \end{equation} and $z:=m-2$ when $n=1$ and $z:=m-2-x$, when $n=2$. In both cases above the right--hand sides are $<3/4$ so $z>0$. We get that $$ \delta(x)\le \frac{2(2^x-1)}{3^x}\quad (n=1)\qquad {\text{\rm and}}\qquad \delta(x)<\frac{3^x-1}{6^x}\qquad (n=2). $$ Further, $$ z\log 2\le (x-1)\log 3+\|(x-1)\log 3-z\log 2\|<(x-1)\log 3+\frac{3}{4}, $$ so \begin{equation} \label{eq:2} \frac{z}{\log 3}<\frac{x-1}{\log 2} +\frac{3}{4(\log 2\log 3)}<\frac{x-1}{\log 2}+1<2x. \end{equation} The left--hand side in \eqref{eq:1} (or \eqref{eq:1prime}) is of the form \begin{align} \|b_1\log \gamma_1-b_2\log \gamma_2\|, \end{align} where $\gamma_1:=3,~\gamma_2:=2,~b_1:=x-1$ and $b_2:=z$. The numbers $\gamma_1$, $\gamma_2$ are rational so $D=1$. We take \begin{align} \log A_1\ge \max\left\{h(\gamma_{1}), \|\log \gamma_1\|, 1\right\},\quad \log A_2\ge \max\left\{h(\gamma_{2}), \|\log \gamma_2\|, 1\right\}, \end{align} so that $\log A_1=\log 3$ and $\log A_2=1$. Further, \begin{align} b'=\frac{b_1}{D\log A_2}+\frac{b_2}{D\log A_1}=x-1+\frac{z}{\log 3}<x-1+2x<3x, \end{align} by \eqref{eq:2}. Since $\gamma_1,~\gamma_2$ are real, positive and multiplicatively independent, Theorem \ref{thm:LMNh} shows that \begin{align}\label{eq:b} \|(x-1)\log 3-z\log 2\|>\exp\left(-24.34 \left(\max\left\{21, \log b'+0.14\right\}\right)^2\log 3\right). \end{align} Since \begin{align} \frac{8}{2^{0.72k}}+\delta(x)\le \frac{8}{2^{0.72k}}+\frac{2}{2^{x/2}}\le \frac{10}{2^{\min\{0.72k,rx\}}}, \end{align} where \begin{equation} \label{eq:delta} \delta(x)\in\left\{\frac{2^x-1}{2^{2x}},\frac{2(2^x-1)}{3^x},\frac{3^x-1}{6^x}\right\}, \end{equation} and we take $r=1$ except if $n=1$ when $r=1/2$, we compare \eqref{eq:1} and \eqref{eq:b} to conclude that \begin{equation} \label{eq:3} \min \{0.72 k,rx\} \log 2-\log 10<24.34 (\max\{21,\log(3x)+0.14\})^2\log 3. \end{equation} We proceed in two cases, when $\min \{0.72 k,x\}$ is $0.72k$ or $rx$. \medskip \begin{enumerate}[(i)] \item Assume first that $\min \{0.72 k,rx\}=0.72k$, then $$ (0.72 \log 2)k<24.34( \max\{21, \log(3x)+0.14\})^2\log 3+\log 10. $$ If $\max\{21, \log(3x)+0.14\}=21$, we get $ (0.72\log 2)k<24.34\cdot (21)^2\log 3+\log 10$, so $k<24000$. If $\max\{21, \log(3x)+0.14\}= \log(3x)+0.14$, we get $$ (0.72\log 2)k <24.34\log 3 (\log(3x)+0.14)^2+\log 10, $$ so that $$ k<54 (\log(3x)+0.14)^2+5<59(\log(3x)+0.14))^2, $$ for all $x\ge 2$. By Lemma \ref{lem3.2}, we have $$ \log m<\log 6.3+32\log 10+10(\log k)+5\log\log k, $$ and by Lemma \ref{lem3.1}, $$ x<3.1\cdot 10^{14} k^5 (\log k)^2 \log m. $$ Thus, \begin{align} x &< 3.1\cdot 10^{14} (59(\log(3x)+0.14)^2)^{5} (\log 59+2\log(\log(3x)+0.14))^2\\ &\times (\log 6.3+32\log 10+10(\log(59(\log(3x)+0.14)^2))+5\log(\log 59+2\log(\log(3x)+0.14))). \end{align} We get $x<10^{47}$. \medskip \item Next, if $\min \{0.72 k,rx\}=rx$, we then get that $$ rx\log 2-\log 10<24.34(\max\{21,\log(3x)+0.10\})^2\log 3. $$ If the maximum in the right above is in $21$, we then get $rx<18000$, while if the maximum is in $\log(3x)+0.14$, we then get $$ rx\log 2<24.34\log 3 (\log(3x)+0.14)^2+\log 10\le 24.34\log 3(\log(3rx)+0.14+\log 2)^2+\log 10, $$ so that $rx<4000$. \end{enumerate} \medskip So, in all cases $x<10^{47}$. We now return to \eqref{eq:1} which we write as \begin{align} \left\|\frac{\log 3}{\log 2}-\frac{z}{x-1}\right\|<\frac{8/2^{0.72k}+\delta(x)}{(x-1)\log 2}<\frac{12/2^{0.72k}+2\delta(x)}{x-1}, \end{align} where $\delta(x)$ is one of the three functions appearing in \eqref{eq:delta} (and $\delta(x)=(2^x-1)/2^{2x}$ if $n\ge 3$). We generated the $97$th convergent $p_{97}/q_{97}=[a_0,a_1,\ldots,a_{97}]=[1,1,\ldots,3]$ of $\log 3/\log 2$ and we got that $q_{97}>10^{47}$ and that $\max\{a_k: 0\le k\le 97\}=55$. By well--known properties of continued fractions, we get \begin{align} \frac{1}{57(x-1)^2}<\left\|\frac{\log 3}{\log 2}-\frac{y-2x}{x-1}\right\|<\frac{12/2^{0.72k}+2\delta(x)}{x-1}. \end{align} Therefore, we have \begin{align} \frac{1}{57(x-1)}<\frac{12}{2^{0.72k}}+2\delta(x). \end{align} The left--hand side is at least $1/(57\cdot 10^{47})$, while $12/2^{0.72k}<10^{-170}$ since $k>800$. So, we get $$ \frac{1}{58 (x-1)}<2\delta(x), $$ which gives $x\le 10$ except if $n=1$ for which $x\le 20$. The cases $n=1,2$ lead to $m<(n+3)x+3<100<k$, a contradiction. Thus, $x\le 10$, $n\ge 3$, $z=y-2x$ and \begin{equation} \label{eq:4} \|(x-1)\log 3-(y-2x)\log 2\|<\frac{8}{2^{0.72k}}+\frac{2^x-1}{2^{2x}}. \end{equation} We used a simple code to compute the minimum values of the expression \[ \|(x-1)\log 3 - (y-2x)\log 2\|, \] for integer values of \( y - 2x \). By assigning values to \( x \) ranging from 2 to 10, the code found the \( y \) that minimized the expression for each \( x \). The results showed that the left--hand side of \eqref{eq:4} was at least \[ 0.28,~0.11,~0.16,~0.23,~0.05,~0.33,~0.06,~0.22,~0.18, \] for \( x = 2, 3, \ldots, 10 \), respectively. It turns out that for no $x\in \{2,3,\ldots,10\}$ inequality \eqref{eq:4} is satisfied. This finishes the argument that the case $k>800$ is not possible. \subsubsection{The case $k\le800$} To proceed, we go back to \eqref{eq:gen} with the assumption that $3\le n \le k$ and write \begin{align} \|\Lambda_1\|:=\|\log(\Gamma_1+1)\|=\left\|\log f_k(\alpha)+\log(2\alpha-1)+(m-1)\log\alpha-x\log 3-(n-1)x\log 2\right\|<\dfrac{4.5}{2^x}, \end{align} where $a_1 := 1$, $a_2 := 1$, $a_3 := m-1$, $a_4 := -x$, $a_5 := -(n-1)x$ are integers with \[ \max\{\|a_i\| : 1 \leq i \leq 5\} < m<6.3 \cdot 10^{32} k^{10} (\log k)^5< 10^{66}, \] where we used Lemma \ref{lem3.2}, for all $k\le 800$. For each $k \in [2, 800]$, we used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form $\|\Lambda_1\|$, with integer coefficients $a_i$ not exceeding $6.3 \cdot 10^{32} k^{10} (\log k)^5$ in absolute value. Specifically, we consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor& \lfloor C\log 3 \rfloor & \lfloor C\log2 \rfloor \end{pmatrix} ,$$ with $C:= 10^{331}$ and choose $y:=\left(0,0,0,0,0\right)$. Now, by Lemma \ref{lem2.5}, we get $$c_2=10^{-69}\qquad\text{and}\qquad l\left(\mathcal{L},y\right)\ge c_1:=3.13\cdot10^{68}.$$ So, Lemma \ref{lem2.6} gives $S=5\cdot 10^{132}$ and $T=2.5\cdot 10^{66}$. Since $c_1^2\ge T^2+S$, then choosing $c_3:=4.5$ and $c_4:=\log2$, we get $x\le 872$. Next, if $n\in\{0,1,2\}$, we instead go back to \eqref{g1} together with \eqref{eq:gen1} and write \begin{align} \left\|\log f_k(\alpha)+\log(2\alpha-1)+(m-1)\log\alpha-x\log \delta\right\|<\dfrac{4.5}{1.5^x}, \end{align} where $a_1 := 1$, $a_2 := 1$, $a_3 := m-1$, $a_4 := -x$ are integers with $\max\{\|a_i\| : 1 \leq i \leq 5\} < 10^{66}$, as before. Again, for each $k \in [2, 800]$ and each $\delta\in\{2,3,6\}$, we used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form above and consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor& \lfloor C\log \delta \rfloor \end{pmatrix} ,$$ with $C:= 10^{265}$ and choose $y:=\left(0,0,0,0\right)$. So, Lemma \ref{lem2.5} gives $c_2=10^{-70}$ and $l\left(\mathcal{L},y\right)\ge c_1:=2.94\cdot10^{68}$. Moreover, Lemma \ref{lem2.6} gives $S=4\cdot 10^{132}$ and $T=2\cdot 10^{66}$. Since $c_1^2\ge T^2+S$, then choosing $c_3:=4.5$ and $c_4:=\log1.5$, we get $x\le 1119$. Thus, $x\le 1119$ for all cases when $0\le n\le k\le 800$. Finally in this subsection, we go back to \eqref{g2} and write $\|\Lambda_2\|:=\|\log(\Gamma_2+1)\|$ as \begin{align} \left\|\log f_k(\alpha)+\log(2\alpha-1)+(m-1)\log\alpha-x\log 3-(n-2)x\log 2-\log \left(2^x+1-2^{-x}\right)\right\|<\dfrac{3}{\alpha^{m-1}} \end{align} where $a_1 := 1$, $a_2 := 1$, $a_3 := m-1$, $a_4 := -x$, $a_5 := -(n-2)x$ and $a_6:=-1$ (in case $n\ge 3$) are integers with \[ \max\{\|a_i\| : 1 \leq i \leq 5\} < m<6.3 \cdot 10^{32} k^{10} (\log k)^5< 10^{66}, \] where we used Lemma \ref{lem3.2}, for all $k\le 800$. So, for each $k \in [2, 800]$ and each $x\in [1,1119]$ we used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form $\|\Lambda_2\|$, with integer coefficients $a_i$ not exceeding $6.3 \cdot 10^{32} k^{10} (\log k)^5$ in absolute value. That is, we instead consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor& \lfloor C\log 3 \rfloor & \lfloor C\log2 \rfloor & \lfloor C\log\left(2^x+1-2^{-x}\right) \rfloor \end{pmatrix} ,$$ with $C:= 10^{396}$ and choose $y:=\left(0,0,0,0,0,0\right)$. Now, by Lemma \ref{lem2.5}, we get $$c_2:=10^{-69}\qquad\text{and}\qquad l\left(\mathcal{L},y\right)\ge c_1:=3.87\cdot10^{68}.$$ So, Lemma \ref{lem2.6} gives $S=6\cdot 10^{132}$ and $T=3\cdot 10^{66}$. Since $c_1^2\ge T^2+S$, then choosing $c_3:=3$ and $c_4:=\log\alpha$, we get $m\le 1474$. This was for $n\ge 3$. If $n\in \{0,1,2\}$, then $m<(n+3)x+3\le 5x+3\le 5598$. Therefore, for the case when $0\le n\le k$, we have that all solutions $n$, $m$, $k$, $x$ to \eqref{eq:main} satisfy $k\in [2,800]$, $m\in [3,5597]$, $n\in [0,\min\{k,\lfloor(m+3)/x\rfloor\}]$ and $x\in[1,1119]$. A computational search in SageMath within these specified ranges for solutions to the Diophantine equation \eqref{eq:main} yielded only the solutions presented in the Theorem \ref{thm1.1}. To efficiently handle large \( L_n^{(k)} \) values, our SageMath 9.5 code utilized batch processing, iterating through all \( (n, m,k, x) \) combinations within these ranges. \subsection{The case $n> k$} \subsubsection{Bounding $x$, $n$ and $m$ in terms of $k$} Again, we proceed as in Subsection \ref{subsec3.2}. We start by proving a series of results. \begin{lemma}\label{lem3.3} Let $n$, $m$, $k$, $x$ be integer solutions to Eq. \eqref{eq:main} with $n>k\ge 2$, $x\ge 2$ and $m\ge 3$, then \begin{align} x<2.6\cdot 10^{15} n k^4(\log k)^3\log n . \end{align} \end{lemma} \begin{proof} We go back to \eqref{eq:main} and rewrite it using the Binet formula in \eqref{eq:Lnwitherror} as \begin{align} L_m^{(k)}-f_k(\alpha)(2\alpha-1)\alpha^{m-1}&=e_k(m)\\ f_k(\alpha)(2\alpha-1)\alpha^{m-1}-\left(L_{n+1}^{(k)}\right)^{x}&=\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x-e_k(m)\\ 0<f_k(\alpha)(2\alpha-1)\alpha^{m-1}\left(L_{n+1}^{(k)}\right)^{-x}-1&=\left(\dfrac{L_{n}^{(k)}}{L_{n+1}^{(k)}}\right)^x-\left(\dfrac{L_{n-1}^{(k)}}{L_{n+1}^{(k)}}\right)^x-\dfrac{e_k(m)}{\left(L_{n+1}^{(k)}\right)^x}\\ &<\left(\dfrac{L_{n}^{(k)}}{L_{n+1}^{(k)}}\right)^x<0.7^x<\dfrac{1}{1.4^x}, \end{align} The above calculation is justified for $n>k\ge 2$ and $x\ge 2$, because then $n\ge 3$ so $L_n^{(k)}>L_{n-1}^{(k)}> 1$, so since $x\ge 2$, $$ \left(L_{n}^{(k)}\right)^x\ge \left(L_{n-1}^{(k)}+1\right)^x>\left(L_{n-1}^{(k)}\right)^x+x\ge \left(L_{n-1}^{(k)}\right)^x+2>\left(L_{n-1}^{(k)}\right)^x+e_k(m) $$ and $$ \left(L_{n-1}^{(k)}\right)^x\ge 2^x\ge 4>1.5>-e_{k}(m),\qquad {\text{\rm so}}\qquad (L_{n-1}^{(k)})^x+e_k(m)>0. $$ We thus get \begin{align}\label{g3} \|\Gamma_3\|:=\left\|f_k(\alpha)(2\alpha-1)\alpha^{m-1}\left(L_{n+1}^{(k)}\right)^{-x}-1\right\|<\dfrac{1}{1.4^x}. \end{align} Note that $\Gamma_3\ne 0$, otherwise we would have \begin{align} f_k(\alpha)= \dfrac{\left(L_{n+1}^{(k)}\right)^x}{(2\alpha-1)\alpha^{m-1}}. \end{align} Taking norms in ${\mathbb K}={\mathbb Q}(\alpha)$ as before, the above equation becomes \begin{align} N\left(f_k(\alpha)\right)\cdot N(2\alpha-1)=\left(L_{n+1}^{(k)}\right)^{kx}. \end{align} We have already shown before that the left--hand side above is $\le 1$. Therefore, \begin{align} 1\ge N\left(f_k(\alpha)\right)\cdot N(2\alpha-1)=\left(L_{n+1}^{(k)}\right)^{kx}\ge \left(L_{4}^{(2)}\right)^4=2401, \end{align} a contradiction, for all $n>k\ge 2$ and $x\ge 2$. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\alpha)$. We have $D = k$, $t :=3$, \begin{equation}\nonumber \begin{aligned} \gamma_{1}&:=L_{n+1}^{(k)}, \qquad\gamma_{2}:=(2\alpha-1)f_k(\alpha),\qquad \gamma_{3}:=\alpha,\\ b_{1}&:=-x,\qquad b_{2}:=1,\qquad b_{3}:=m-1. \end{aligned} \end{equation} Since $h\left(L_{n+1}^{(k)}\right)\le h(2\alpha^n)<n\log2\alpha<n\log 4<1.4n$, we can take $A_1:=1.4kn$. As before, we take $A_{2}:=6k\log k$, $A_{3}:=0.7$ and $B:=m$. Now, by Theorem \ref{thm:Mat}, \begin{align}\label{eq3.12} \log \|\Gamma_3\| &> -1.4\cdot 30^{6} \cdot 3^{4.5}\cdot k^2 (1+\log k)(1+\log m)\cdot 1.4kn\cdot 0.7\cdot 6k\log k\nonumber\\ &> -4.0\cdot 10^{12} n k^4(\log k)^2 \log m. \end{align} Comparing \eqref{g3} and \eqref{eq3.12}, we get \begin{align} x\log 1.4&<4.0\cdot 10^{12} n k^4(\log k)^2 \log m,\\ x&<1.2\cdot 10^{13} n k^4(\log k)^2 \log m. \end{align} Now, by inequality \eqref{m_b}, $m<(n+3)x+3$, hence \begin{align} x&<1.2\cdot 10^{13} n k^4(\log k)^2 \log ((n+3)x+3)\\ &<2.7\cdot 10^{13} n k^4(\log k)^2 \log (nx). \end{align} Now, multiplying $n$ on both sides of the above inequality, we get $nx<2.7\cdot 10^{13} n^2 k^4(\log k)^2 \log (nx)$. We now apply Lemma \ref{Guz} with $p:=nx$, $r:=1$, $T:=2.7\cdot 10^{13} n^2 k^4(\log k)^2$ and have \begin{align} nx&<2\cdot 2.7\cdot 10^{13} n^2 k^4(\log k)^2\log 2.7\cdot 10^{13} n^2 k^4(\log k)^2\\ &=5.4\cdot 10^{13} n^2 k^4(\log k)^2 \left(\log (2.7\cdot 10^{13})+2\log n+4\log k+2\log\log k\right)\\ &<5.4\cdot 10^{13} n^2 k^4(\log k)^3\log n \left(\dfrac{31}{\log n\log k}+\dfrac{2}{\log k}+\dfrac{4}{\log n}+\dfrac{2\log\log k}{\log n\log k}\right), \end{align} so that $nx<2.6\cdot 10^{15} n^2 k^4(\log k)^3\log n $. Dividing both sides by $n$, we get $x<2.6\cdot 10^{15} n k^4(\log k)^3\log n $. \end{proof} To proceed, we now work under the assumption that $n > 800$, so that we can explicitly determine an upper bound for $n$, $m$ and $x$ in terms of $k$ only. We prove the following result. \begin{lemma}\label{lem3.4} Let $n$, $m$, $k$, $x$ be integer solutions to Eq. \eqref{eq:main} with $n>\max\{800,k\}$, $x\ge 2$ and $m\ge 3$, then \begin{align} n<5.5\cdot 10^{27} k^6(\log k)^6, \qquad x< 8.7\cdot 10^{30}k^7(\log k)^8\qquad\text{and}\qquad m<5.6\cdot 10^{44}k^{10}(\log k)^{12}. \end{align} \end{lemma} \begin{proof} Since $n>k$, then Lemma \ref{lem3.3} tells us that \begin{align}\label{x_bound1} x<2.6\cdot 10^{15} n^5(\log n)^4. \end{align} So, for each $i\in\{-1,0,1\}$, the inequality \begin{align} \dfrac{x}{\alpha^{n+i-1}}<\dfrac{2.6\cdot 10^{15} n^5(\log n)^4}{\alpha^{n-2}}<\dfrac{1}{\alpha^{0.7n}}, \end{align} holds for all $n>800$. Now, Lemma \ref{lem:Lnx} tells us that \begin{align}\label{eq3.14} \left(L_{n+i}^{(k)}\right)^x = f_k(\alpha)^x (2\alpha-1)^x\alpha^{(n+i-1)x}(1 + \eta_{n}),\qquad\text{with}\qquad \|\eta_{n}\| < \dfrac{1.5xe^{1.5x/\alpha^{n+i-1}}}{\alpha^{n+i-1}}<\dfrac{10}{\alpha^{0.7n}}. \end{align} So, we rewrite \eqref{eq:main} using \eqref{eq3.14} as \begin{align} f_k(\alpha)(2\alpha-1)\alpha^{m-1}&-f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}\left(1+\alpha^{-x}-\alpha^{-2x}\right)\\ &=f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}\left(1+\alpha^{-x}-\alpha^{-2x}\right)\eta_n+e_k(m). \end{align} Dividing both sides of the above equality by $f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}$ and taking absolute values, we get \begin{align} &\left\|f_k(\alpha)(2\alpha-1)\alpha^{m-1}f_k(\alpha)^{-x}(2\alpha-1)^{-x}\alpha^{-(n-1)x}-\left(1+\alpha^{-x}-\alpha^{-2x}\right)\right\|\\ &<\|\eta_n\|\left(1+\alpha^{-x}-\alpha^{-2x}\right)+\dfrac{1.5}{f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}}\\ &<\dfrac{15}{\alpha^{0.7n}}+\dfrac{1.5}{\alpha^{(n-1)x}}<\dfrac{18}{\alpha^{0.7n}}, \end{align} where we have used the fact that $0<1+\alpha^{-x}-\alpha^{-2x}<1.5$, $f_k(\alpha)(2\alpha-1)\alpha^{n-1}>\alpha^{n-1}$ and $(n-1)x>0.7n$, for all $n>800$, $x\ge 2$ and $k\ge 2$. This means that \begin{align}\label{g4} \|\Gamma_4\|:= \left\|f_k(\alpha)^{1-x}(2\alpha-1)^{1-x}\alpha^{m-1-(n-1)x}-1\right\|<\dfrac{18}{\alpha^{0.7n}}+\dfrac{1}{\alpha^x}+\dfrac{1}{\alpha^{2x}}<\dfrac{27}{\alpha^{\min\{0.7n,x\}}}. \end{align} Note that $\Gamma_4\ne 0$, otherwise we would have \begin{align}\label{fk} \left((2\alpha-1)f_k(\alpha)\right)^{x-1}=\alpha^{m-1-(n-1)x}. \end{align} Applying norms in ${\mathbb K}={\mathbb Q}(\alpha)$ and using $\|N(\alpha)\|=1$, equation \eqref{fk} becomes \begin{align} \|N\left((2\alpha-1)f_k(\alpha)\right)\|^{x-1}=1, \end{align} which implies that $x=1$, for $k\ge 3$, contradicting the working assumption that $x\ge 2$. For the special case when $k=2$, we have that $(2\alpha-1)f_k(\alpha)=\alpha$, so that \eqref{fk} becomes $\alpha^{x-1}=\alpha^{m-1-(n-1)x}$, giving $m=nx$. At this point, we go back and rewrite \eqref{eq:main} with $k=2$ and $m=nx$ as \begin{align} \left(L_{n+1}\right)^x+\left(L_{n}\right)^x-\left(L_{n-1}\right)^x=L_{nx}. \end{align} Now, since $\left(L_{n}\right)^x-\left(L_{n-1}\right)^x>0$ for all $x\ge 2$ and $n\ge 4$, we have by the Binet formula $L_n=\alpha^n+\beta^n$, that \begin{align} \left(\alpha^{n+1}-1\right)^x<\left(\alpha^{n+1}+\beta^{n+1}\right)^x= \left(L_{n+1}\right)^x\le \left(L_{n+1}\right)^x+\left(L_{n}\right)^x-\left(L_{n-1}\right)^x=L_{nx}< \alpha^{nx}+1. \end{align} This leads to \begin{align} \alpha^x\left(1-\frac{1}{\alpha^{n+1}}\right)^x< 1+\frac{1}{\alpha^{nx}}, \end{align} so that \begin{align} 2< \alpha^2\left(1-\frac{1}{\alpha^{4+1}}\right)^2\le \alpha^x\left(1-\frac{1}{\alpha^{n+1}}\right)^x< 1+\frac{1}{\alpha^{nx}}\le 1+\frac{1}{\alpha^{4\cdot 2}}<1.1, \end{align} which is also a contradiction. Thus, $\Gamma_4\ne 0$. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\alpha)$. We have $D = k$, $t :=2$, \begin{equation}\nonumber \begin{aligned} \gamma_{1}&:=(2\alpha-1)f_k(\alpha),\qquad \gamma_{2}:=\alpha,\\ b_{1}&:=1-x,\qquad b_{2}:=m-1-(n-1)x. \end{aligned} \end{equation} As before, we take $A_{1}:=6k\log k$ and $A_{2}:=0.7$. By the second inequality in \eqref{m_b}, we can take $B:=x$. So, Theorem \ref{thm:Mat} gives \begin{align}\label{eq3.16} \log \|\Gamma_4\| &> -1.4\cdot 30^{5} \cdot 2^{4.5}\cdot k^2 (1+\log k)(1+\log x)\cdot 0.7\cdot 6k\log k\nonumber\\ &> -2.0\cdot 10^{10} k^3(\log k)^2 \log x. \end{align} Comparing \eqref{g4} and \eqref{eq3.16}, we get \begin{align} \min\{0.7n,x\}\log\alpha-\log 27&<2.0\cdot 10^{10} k^3(\log k)^2 \log x,\\ \min\{0.7n,x\}&<5.0\cdot 10^{10} k^3(\log k)^2 \log x. \end{align} We distinguish between two cases. \textbf{Case I}: If the $\min\{0.7n,x\}=0.7n$, then $n<7.2\cdot 10^{10} k^3(\log k)^2 \log x$, and using Lemma \ref{lem3.3}, we get \begin{align} n&<7.2\cdot 10^{10} k^3(\log k)^2 \log \left( 2.6\cdot 10^{15} n k^4(\log k)^3\log n \right) \\ &=7.2\cdot 10^{10} k^3(\log k)^2 \left(\log 2.6\cdot 10^{15} +\log n +4\log k +3\log\log k+\log\log n \right)\\ &<7.2\cdot 10^{10} k^3(\log k)^3\log n \left(\dfrac{36}{\log n\log k} +\dfrac{1}{\log k} +\dfrac{4}{\log n} +3\dfrac{\log\log k}{\log n\log k}+\dfrac{\log\log n}{\log n\log k} \right)\\ &<7.3\cdot 10^{11} k^3(\log k)^3\log n. \end{align} This tells us that $n/\log n<7.3\cdot 10^{11} k^3(\log k)^3$. Applying Lemma \ref{Guz} with $$ p:=n, \quad r:=1, \quad T:=7.3\cdot 10^{11} k^3(\log k)^3, $$ we get \begin{align} n&<2\cdot7.3\cdot 10^{11} k^3(\log k)^3\log \left(7.3\cdot 10^{11} k^3(\log k)^3\right)\\ &<1.5\cdot 10^{12} k^3(\log k)^3 \left(\log (7.3\cdot 10^{11})+3\log k+3\log\log k\right)\\ &<1.5\cdot 10^{12} k^3(\log k)^4 \left(\dfrac{28}{\log k}+3+\dfrac{3\log\log k}{\log k}\right), \end{align} so that $n<6.3\cdot 10^{13} k^3(\log k)^4$. Inserting this upper bound of $n$ in Lemma \ref{lem3.3}, we have \begin{align} x&<2.6\cdot 10^{15} \left(6.3\cdot 10^{13} k^3(\log k)^4\right) k^4(\log k)^3\log \left(6.3\cdot 10^{13} k^3(\log k)^4\right)\\ &< 1.8\cdot 10^{29}k^7(\log k)^8\left(\dfrac{32}{\log k}+3+\dfrac{4\log\log k}{\log k}\right)\\ &< 8.7\cdot 10^{30}k^7(\log k)^8. \end{align} By inequality \eqref{m_b}, $m<(n+3)x+3<5.6\cdot 10^{44}k^{10}(\log k)^{12}$. \medskip \textbf{Case II}: If the $\min\{0.7n,x\}=x$, then $x<5.0\cdot 10^{10} k^3(\log k)^2 \log x$. Applying Lemma \ref{Guz} with $p:=x$, $r:=1$, $T:=5.0\cdot 10^{10} k^3(\log k)^2$, we get \begin{align} x&<2\cdot 5.0\cdot 10^{10} k^3(\log k)^2\log \left(5.0\cdot 10^{10} k^3(\log k)^2\right)\\ &=1.0\cdot 10^{11} k^3(\log k)^2 \left(\log (5.0\cdot 10^{10})+3\log k+2\log\log k\right)\\ &< 10^{11} k^3(\log k)^3 \left(\dfrac{25}{\log k}+3+\dfrac{2\log\log k}{\log k}\right), \end{align} so that \begin{align}\label{xc} x<4.0\cdot 10^{12} k^3(\log k)^3. \end{align} Now, since $x\le 0.7n$, then for each $i\in\{-1,0,1\}$, the inequality \begin{align} \dfrac{x}{\alpha^{n+i-1}}<\dfrac{0.7n}{\alpha^{n-2}}<\dfrac{1}{\alpha^{0.97n}}, \end{align} holds for all $n>800$. Now, Lemma \ref{lem:Lnx} tells us that \begin{align}\label{eq3.17} \left(L_{n+i}^{(k)}\right)^x = f_k(\alpha)^x (2\alpha-1)^x\alpha^{(n+i-1)x}(1 + \eta_{n}),\qquad\text{with}\qquad \|\eta_{n}\| < \dfrac{1.5xe^{1.5x/\alpha^{n+i-1}}}{\alpha^{n+i-1}}<\dfrac{2}{\alpha^{0.97n}}. \end{align} So, we rewrite \eqref{eq:main} using \eqref{eq3.17} as \begin{align} f_k(\alpha)(2\alpha-1)\alpha^{m-1}&-f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}\left(1+\alpha^{-x}-\alpha^{-2x}\right)\\ &=f_k(\alpha)^x(2\alpha-1)^x\alpha^{(n-1)x}\left(1+\alpha^{-x}-\alpha^{-2x}\right)\eta_n+e_k(m). \end{align} Dividing both sides of the above equality by $ f_k(\alpha)(2\alpha-1)\alpha^{m-1}$ and taking absolute values, we get \begin{align} &\left\|f_k(\alpha)^{x-1}(2\alpha-1)^{x-1}\alpha^{(n-1)x-(m-1)}\left(1+\alpha^{-x}-\alpha^{-2x}\right)-1\right\|\\ &<\|\eta_n\|f_k(\alpha)^{x-1}(2\alpha-1)^{x-1}\alpha^{(n-1)x-(m-1)}\left(1+\alpha^{-x}-\alpha^{-2x}\right)+\dfrac{1.5}{f_k(\alpha)(2\alpha-1)\alpha^{m-1}}\\ &<\dfrac{2}{\alpha^{0.97n}}\cdot \dfrac{1}{\left(f_k(\alpha)(2\alpha-1)\alpha\right)^{1-x}}\cdot \dfrac{1}{\alpha^{(m-2)-(n-2)x}} +\dfrac{1.5}{\alpha^{m-1}}\\ &<\dfrac{2}{\alpha^{0.97n}}\cdot \dfrac{1/\left(0.5\cdot 2\cdot1\right)^{1-x}}{\alpha^{-5}} +\dfrac{1.5}{\alpha^{n-4}}<\dfrac{2}{\alpha^{0.97n-5}} +\dfrac{1.5}{\alpha^{n-4}}\\ & <\dfrac{3}{\alpha^{0.97n-5}}, \end{align} for $n>800$, $x\ge 2$ and $k\ge 2$. This means that \begin{align}\label{g5} \|\Gamma_5\|:= \left\|f_k(\alpha)^{x-1}(2\alpha-1)^{x-1}\alpha^{(n-1)x-(m-1)}\left(1+\alpha^{-x}-\alpha^{-2x}\right)-1\right\|<\dfrac{3}{\alpha^{0.97n-5}}. \end{align} Note that $\Gamma_5\ne 0$, otherwise we would have \begin{align} \left((2\alpha-1)f_k(\alpha)\right)^{x-1}\alpha^{(n-1)x-(m-1)}\left(1+\alpha^{-x}-\alpha^{-2x}\right)=1. \end{align} Taking norms in ${\mathbb K}={\mathbb Q}(\alpha)$ as before and using parts $(i)$ and $(ii)$ of Lemma \ref{lemGLm}, the above equation becomes \begin{align}\label{prod} \left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot N\left(1+\alpha^{-x}-\alpha^{-2x}\right)&=1,\nonumber\\ \left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot \left(1+\alpha^{-x}-\alpha^{-2x}\right)\prod_{j= 2}^k\left\|1+(\alpha^{(j)})^{-x}-(\alpha^{(j)})^{-2x}\right\|&=1. \end{align} Since $1+\alpha^{-x}-\alpha^{-2x}<2$, for all $x\ge 2$, we have from \eqref{prod} that \begin{align} 1&= \left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot \left(1+\alpha^{-x}-\alpha^{-2x}\right)\prod_{j= 2}^k\left\|1+(\alpha^{(j)})^{-x}-(\alpha^{(j)})^{-2x}\right\|\\ &<\left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot 2\cdot \prod_{j= 2}^k\left\|(\alpha^{(j)})^{-2x}-(\alpha^{(j)})^{-x}-1\right\|\\ &<\left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot 2\prod_{j= 2}^k3\left\|(\alpha^{(j)})^{-2x}\right\| <\left(\dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1}\cdot 2\cdot 3^{k-1}\alpha^{2x}\\ &<\left(8^{x/(x-1)}\cdot3^{(k-1)/(x-1)} \dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1} \\ &<\left(8^{2}\cdot3^{k-1} \dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}\right)^{x-1} <1, \end{align} for all $k\ge 8$ and $x\ge 2$, a contradiction. When $k=3,4,5,6,7$, we have that the expression \begin{align} \dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}, \end{align} in \eqref{prod} is equal to $$ 13/44,\quad 29/563, \qquad 61/9584, \quad 125/205937\quad 253/5390272, $$ respectively. Since $13\nmid 44$, $29\nmid 563$, $61\nmid 9584$, $125\nmid 205937$ and $253\nmid 5390272$, equation \eqref{prod} can not hold when $k=3,4,5,6,7$. When $k=2$, we have that \begin{align} \dfrac{\left( 2^{k+1} - 3\right) (k - 1)^2}{ 2^{k+1}k^k - (k + 1)^{k+1}}=1. \end{align} In this case, equation \eqref{prod} becomes \begin{align} \left(1+\alpha^{-x}-\alpha^{-2x}\right)\left(1+\beta^{-x}-\beta^{-2x}\right)=\pm 1, \end{align} where $\alpha=\left(1+\sqrt5\right)/2$ and $\beta=\left(1-\sqrt5\right)/2 $. We multiply both sides of the above equation by $(\alpha\beta)^{2x}=1$ and get \begin{align}\label{lx} \left(\alpha^{2x}+\alpha^{x}-1\right)\left(\beta^{2x}+\beta^{x}-1\right)&=\pm 1,\nonumber\\ 1+ (-1)^x(\alpha^x+\beta^x)-\left(\alpha^{2x}+\beta^{2x}\right)+(-1)^x+1-(\alpha^x+\beta^x) &=\pm 1,\nonumber\\ 2+ ((-1)^x-1)L_x-L_{2x}+(-1)^x &=\pm 1. \end{align} Now, if $x$ is even in \eqref{lx}, then we have $L_{2x} =2,4$ leading to $x=0$ which contradicts the working assumption that $x\ge 2$. On the other hand, if $x$ is odd in \eqref{lx}, then $L_{2x}+L_x =0,2$, therefore we get that $L_{2x} =-L_x,~2-L_x$, which are both negative numbers for $x\ge 2$, another contradiction. Thus, in all cases, $\Gamma_5\ne 0$. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\alpha)$. We have $D = k$, $t :=3$, \begin{equation}\nonumber \begin{aligned} \gamma_{1}&:=(2\alpha-1)f_k(\alpha),\qquad \gamma_{2}:=\alpha,\qquad\gamma_{3}:=\left(1+\alpha^{-x}-\alpha^{-2x}\right),\\ b_{1}&:=x-1,\qquad b_{2}:=(n-1)x-(m-1),\qquad b_3:=1. \end{aligned} \end{equation} As before, we take $A_{1}:=6k\log k$ and $A_{2}:=0.7$. For $A_3$, we first compute \begin{align} Dh(\gamma_3)=kh\left(1+\alpha^{-x}-\alpha^{-2x}\right)<3kx(\log\alpha)/k, \end{align} so that we can take $A_3:=3x$. By the second inequality in \eqref{m_b}, we can take $B:=x$. So, Theorem \ref{thm:Mat} gives \begin{align}\label{eq3.19} \log \|\Gamma_5\| &> -1.4\cdot 30^{6} \cdot 3^{4.5}\cdot k^2 (1+\log k)(1+\log x)\cdot 0.7\cdot 6k\log k\cdot 3x\nonumber\\ &> -1.1\cdot 10^{13} x k^3(\log k)^2 \log x. \end{align} Comparing \eqref{g5} and \eqref{eq3.19}, we get \begin{align} (0.97n-5)\log\alpha-\log 3&<1.1\cdot 10^{13} x k^3(\log k)^2 \log x,\\ n&<2.8\cdot 10^{13} k^3(\log k)^2 x\log x. \end{align} Moreover, by \eqref{xc}, $x<4.0\cdot 10^{12} k^3(\log k)^3$, so we have \begin{align} n&<2.8\cdot 10^{13} k^3(\log k)^2 \cdot 4.0\cdot 10^{12} k^3(\log k)^3\cdot \log \left(4.0\cdot 10^{12} k^3(\log k)^3\right)\\ &<1.2\cdot 10^{26} k^6(\log k)^6\left(\dfrac{30}{\log k}+3+3\dfrac{\log\log k}{\log k}\right)\\ &<5.5\cdot 10^{27} k^6(\log k)^6. \end{align} Lastly, by inequality \eqref{m_b}, $m<(n+3)x+3<2.3\cdot 10^{40}k^9(\log k)^{9}$. Comparing all inequalities of $n$, $m$ and $x$ in both cases, we have the inequalities in Lemma \ref{lem3.4}. \end{proof} \subsubsection{The case $k>800$} The inequalities in Lemma \ref{lem3.4} were obtained under the assumptions that $n > 800$. However, when $n \le 800$, the inequalities \eqref{m_b} and \eqref{x_bound1} yield even smaller upper bounds for $x$ and $m$ in terms of $k$. From now on, let us assume that $k > 800$. By Lemma \ref{lem3.4}, we get \begin{align} n+i<5.51\cdot 10^{27} k^6(\log k)^6<2^{0.24k} \qquad\text{and}\qquad m<5.6\cdot 10^{44}k^{10}(\log k)^{12} <2^{0.39k}, \end{align} for all $k>800$ and $i\in\{-1,0,1\}$. By Lemma \ref{Ln:x} with $c=0.39$ and $x=1$, we have \begin{align} L_m^{(k)} = 3 \cdot 2^{m-2} \left( 1 + \xi_m \right), \quad \text{with} \quad \|\xi_m\| < \dfrac{2}{2^{0.22k}}. \end{align} Again, by Lemma \ref{Ln:x} with $c=0.24$ and $x\ge 2$, we have \begin{align} \left(L_{n+i}^{(k)}\right)^x = 3^x\cdot 2^{(n+i-2)x} \left(1 +\xi_{n+i} \right), \quad \text{with} \quad \|\xi_{n+i}\| <\dfrac{2}{2^{0.52k}}. \end{align} Now, for all $n>k\ge 2$, we rewrite \eqref{eq:main} as \begin{align} \left\|3^x\cdot 2^{(n-1)x}\left(1+2^{-x}-2^{-2x}\right) - 3 \cdot 2^{m-2} \right\|&\le\left( 3^x\cdot 2^{(n-1)x} \left(1+2^{-x}+2^{-2x}\right)+3 \cdot 2^{m-2} \right)\cdot \dfrac{2}{2^{0.22k}}, \end{align} where we chose $2/2^{0.22k}$ over $2/2^{0.52k}$ since $2/2^{0.52k}<2/2^{0.22k}$. In the above, we divide through by the $\max\{3^x\cdot 2^{(n-1)x}\left(1+2^{-x}-2^{-2x}\right),~ 3 \cdot 2^{m-2}\}$. In the left--hand side, we have $$\left\|3^{x-1}\cdot2^{-(m-2)}\cdot 2^{(n-1)x}\left(1+2^{-x}-2^{-2x}\right) -1\right\|\quad\text{or}\quad \left\|3^{1-x}\cdot2^{(m-2)}\cdot 2^{-(n-1)x}\left(1+2^{-x}-2^{-2x}\right)^{-1} -1\right\|,$$ while on the right--hand side, we have $$ \left(\dfrac{1+2^{-x}+2^{-2x}}{1+2^{-x}-2^{-2x}}+\dfrac{3 \cdot 2^{m-2}}{3^x\cdot 2^{(n-1)x}\left(1+2^{-x}-2^{-2x}\right)} \right)\cdot \dfrac{2}{2^{0.22k}}<(1.25+1)\cdot \dfrac{2}{2^{0.22k}}= \dfrac{4.5}{2^{0.22k}}, $$ or $$ \left(\dfrac{3^x\cdot 2^{(n-1)x}(1+2^{-x}+2^{-2x})}{3 \cdot 2^{m-2}}+1 \right)\cdot \dfrac{2}{2^{0.22k}}\le(1+1)\cdot \dfrac{2}{2^{0.22k}}= \dfrac{4}{2^{0.22k}}, $$ respectively. Thus, in both cases, we have \begin{align}\label{eqc1} \left\|3^{x-1}\cdot2^{-(m-2)}\cdot 2^{(n-1)x}\left(1+2^{-x}-2^{-2x}\right) -1\right\|< \dfrac{4.5}{2^{0.22k}}. \end{align} Again, putting $y:=m-2-(n-3)x$ in \eqref{eqc1} and dividing through by $2^y$, we get \begin{align}\label{eq:01} \left\|\frac{3^{x-1} (2^{2x}+2^x-1)}{2^y}-1\right\|<\dfrac{4.5}{2^{0.22k}}. \end{align} Again, the left--hand side in \eqref{eq:01} is \begin{align} \left\|e^{(x-1)\log 3+\log(2^{2x}+2^x+1)-y\log 2}-1\right\|, \end{align} and the right--hand side is $<1/2$. So, we get by a simple argument in calculus that \begin{align} \left\|(x-1)\log 3+\log(2^{2x}+2^x-1)-y\log 2\right\|<\dfrac{9}{2^{0.22k}}. \end{align} This can be rewritten as \begin{align} \left\|(x-1)\log 3+2x\log 2+\log\left(1+\frac{2^x-1}{2^{2x}}\right)-y\log 2\right\|<\dfrac{9}{2^{0.22k}}, \end{align} or \begin{equation}\label{eq:11} \|(x-1)\log 3-(y-2x)\log 2\|<\dfrac{9}{2^{0.22k}}+\log\left(1+\frac{2^x+1}{2^{2x}}\right)<\dfrac{9}{2^{0.22k}}+\frac{2^x-1}{2^{2x}}. \end{equation} Since $x\ge 2$, the right--hand side in \eqref{eq:11} is still $<6/16$, so $z:=y-2x>0$ as before. This means that the conclusion in \eqref{eq:2} still holds even in this case. The left--hand side in \eqref{eq:11} is of the form \begin{align} \|b_1\log \gamma_1-b_2\log \gamma_2\|, \end{align} where $\gamma_1:=3,~\gamma_2:=2,~b_1:=x-1$ and $b_2:=z$. So, we take as before that $D=1$, $\log A_1=\log 3$, $\log A_2=1$ and $b'<3x$, by \eqref{eq:2}. Now, Theorem \ref{thm:LMNh} shows that \begin{align}\label{eq:b1} \|(x-1)\log 3-(y-2x)\log 2\|>\exp\left(-24.34 \left(\max\left\{21, \log b'+0.14\right\}\right)^2\log 3\right). \end{align} Since \begin{align} \frac{9}{2^{0.22k}}+\frac{2^x-1}{2^{2x}}\le \frac{9}{2^{0.22k}}+\frac{1}{2^x}\le \frac{10}{2^{\min\{0.22k,x\}}}, \end{align} we compare \eqref{eq:11} and \eqref{eq:b1} to conclude that \begin{equation} \label{eq:31} \min \{0.22 k,x\} \log 2-\log 10<24.34 (\max\{21,\log(3x)+0.14\})^2\log 3. \end{equation} We proceed in two cases, when $\min \{0.22 k,x\}$ is $0.22k$ or $x$. \begin{enumerate}[(i)] \item Assume first that $\min \{0.22 k,x\}=0.22k$, then $$ (0.22 \log 2)k<24.34( \max\{21, \log(3x)+0.14\})^2\log 3+\log 10. $$ If $\max\{21, \log(3x)+0.14\}=21$, we get $ (0.22\log 2)k<24.34\cdot (21)^2\log 3+\log 260$, so $k<77350$. If $\max\{21, \log(3x)+0.14\}= \log(3x)+0.14$, we get $$ (0.22\log 2)k <24.34\log 3 (\log(3x)+0.14)^2+\log 10, $$ so that $$ k<176 (\log(3x)+0.14)^2+16<187(\log(3x)+0.14))^2, $$ for all $x\ge 1$. By Lemma \ref{lem3.4}, we have $$ x<8.7\cdot 10^{30} k^7 (\log k)^8. $$ Thus, \begin{align} x &< 8.7\cdot 10^{30} (187(\log(3x)+0.14)^2)^{7} (\log 187+2\log(\log(3x)+0.14))^8. \end{align} We get $x<10^{89}$. \item Next, if $\min \{0.22 k,x\}=x$, we then get $$ x\log 2-\log 10<24.34(\max\{21,\log(3x)+0.10\})^2\log 3. $$ If the maximum in the right above is in $21$, we then get $x<18000$, while if the maximum is in $\log(3x)+0.14$, we then get $$ x\log 2<24.34\log 3 (\log(3x)+0.14)^2+\log 10, $$ so that $x<4000$. \end{enumerate} So, in all cases $x<10^{89}$. We now return to \eqref{eq:11} which we write as \begin{align} \left\|\frac{\log 3}{\log 2}-\frac{y-2x}{x-1}\right\|<\frac{9/2^{0.22k}+(2^x-1)/(2^{2x})}{(x-1)\log 2}<\frac{13/2^{0.22k}+2(2^x-1)/(2^{2x})}{x-1}. \end{align} Now, we generated the $187$th convergent $p_{187}/q_{187}=[a_0,a_1,\ldots,a_{187}]=[1,1,\ldots,2]$ of $\log 3/\log 2$ and we got that $q_{187}>10^{89}$ and that $\max\{a_k: 0\le k\le 187\}=55$. By well--known properties of continued fractions, we get \begin{align} \frac{1}{57(x-1)^2}<\left\|\frac{\log 3}{\log 2}-\frac{y-2x}{x-1}\right\|<\frac{13/2^{0.22k}+2(2^x-1)/(2^{2x})}{x-1}. \end{align} Therefore, we have \begin{align} \frac{1}{57(x-1)}<\frac{13}{2^{0.22k}}+\frac{2^x-1}{2^{2x-1}}. \end{align} The left--hand side is at least $1/(57\cdot 10^{89})$, while $13/2^{0.22k}<10^{-50}$ since $k>800$. So, we get $$ \frac{1}{58 (x-1)}<\frac{2^x-1}{2^{2x-1}}, $$ which gives $x\le 10$, as before in Subsection \ref{sub322}. Thus, \begin{equation} \label{eq:41} \|(x-1)\log 3-(y-2x)\log 2\|<\frac{9}{2^{0.22k}}+\frac{2^x-1}{2^{2x}}. \end{equation} Again, we used a simple code to compute the minimum values of the expression \[ \|(x-1)\log 3 - (y-2x)\log 2\|, \] for integer values of \( y - 2x \). As before, by assigning values to \( x \) ranging from 2 to 10, the code found the \( y \) that minimized the expression for each \( x \). The results still showed that the left--hand side of \eqref{eq:41} was at least \[ 0.28,~0.11,~0.16,~0.23,~0.05,~0.33,~0.06,~0.22,~0.18, \] for \( x = 2, 3, \ldots, 10 \), respectively. Again, it turned out that for no $x\in \{2,3,\ldots,10\}$, inequality \eqref{eq:41} is satisfied. This finishes the argument that the case $k>800$ is not possible here. \subsubsection{The case $k\le800$} To proceed, let us assume that $n > 800$, so, we can use Lemma \ref{lem3.4} to obtain upper bounds on $n$, $x$, and $m$. We assume that $x \leq 10$. Using inequality \eqref{g5}, we write $\|\Lambda_5\|:=\|\log (\Gamma_5+1)\|$ as \begin{align} \|\Lambda_5\|&:= \left\|(x-1)\log f_k(\alpha)+(x-1)\log(2\alpha-1)+\left((n-1)x-(m-1)\right)\log\alpha+\log\left(1+\alpha^{-x}-\alpha^{-2x}\right)\right\|\\ &<\dfrac{4.5}{\alpha^{0.97n-5}}, \end{align} where $a_1 := x-1$, $a_2 := x-1$, $a_3 := (n-1)x-(m-1)$, $a_4 := 1$ are integers with \[ \max\{\|a_i\| : 1 \leq i \leq 4\} < m<5.6\cdot 10^{44}k^{10}(\log k)^{12}<4.8\cdot 10^{83}, \] where we used Lemma \ref{lem3.4}, for all $k\le 800$. For each $k \in [2, 800]$ and $x\in [2,10]$, we used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form $\|\Lambda_5\|$, with integer coefficients $a_i$ not exceeding $5.6\cdot 10^{44}k^{10}(\log k)^{12}$ in absolute value. Specifically, we consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor& \lfloor C\log \left(1+\alpha^{-x}-\alpha^{-2x}\right) \rfloor \end{pmatrix} ,$$ with $C:= 10^{335}$ and choose $y:=\left(0,0,0,0\right)$. Now, by Lemma \ref{lem2.5}, we get $$c_2:=10^{-85}\qquad\text{and}\qquad l\left(\mathcal{L},y\right)\ge c_1:=1.56\cdot10^{84}.$$ So, Lemma \ref{lem2.6} gives $S=9.3\cdot 10^{167}$ and $T=9.6\cdot 10^{83}$. Since $c_1^2\ge T^2+S$, then choosing $c_3:=4.5$ and $c_4:=\log\alpha$, we get $0.97n-5\le 1235$, or $n\le 1278$. Now, we work with $x>10$. We go to inequality \eqref{g4} and write $\|\Lambda_4\|:=\|\log (\Gamma_4+1)\|$ as \begin{align} \|\Lambda_4\|:= \left\|\left(m-1-(n-1)x\right)\log\alpha-(x-1)\log \left((2\alpha-1)f_k(\alpha)\right)\right\|<\dfrac{40.5}{\alpha^{\min\{0.7n,x\}}}. \end{align} where we have used the fact that $\min\{0.7n,x\} \geq 10$. Dividing both sides of the above inequality by $(x - 1) \log \alpha$, we obtain \begin{align}\label{eqr2} \left\| \frac{\log \left((2\alpha-1)f_k(\alpha)\right)}{\log \alpha} - \frac{m-1-(n-1)x}{x - 1} \right\| < \frac{40.5}{\alpha^{\min\{0.7n,x\}}(x - 1) \log \alpha} < \frac{10}{\alpha^{\min\{0.7n,x\}}}, \end{align} for all $x>10$. We proceed by examining two cases from \eqref{eqr2}. \textbf{Case I}: If $m-1 = (n-1)x$, then inequality \eqref{eqr2} becomes \[ \left\| \frac{\log \left((2\alpha-1)f_k(\alpha)\right)}{\log \alpha} \right\| < \frac{10}{\alpha^{\min\{0.7n,x\}}}. \] We get \begin{align} \frac{10}{\alpha^{\min\{0.7n,x\}}}> \frac{\log \left((2\alpha-1)f_k(\alpha)\right)}{\log \alpha}>\frac{\log (2.2\cdot 0.5)}{\log 2}>0.1 \end{align} for all $k \in [2, 800]$. Thus, $\alpha^{\min\{0.7n,x\}}<100$, or equivalently, $\min\{0.7n,x\}<10$. This contradicts our assumptions that $n > 800$ and $x > 10$. So this case is not possible. \textbf{Case II}: If $m-1 \ne (n-1)x$, we have \begin{align} \|\Lambda_4\|:= \left\|(x-1)\log f_k(\alpha)+(x-1)\log(2\alpha-1)+\left((n-1)x-(m-1)\right)\log\alpha\right\|<\dfrac{40.5}{\alpha^{\min\{0.7n,x\}}}, \end{align} where $a_1 := x-1$, $a_2 := x-1$, $a_3 := (n-1)x-(m-1)$ are integers with \[ \max\{\|a_i\| : 1 \leq i \leq 3\} < m<5.6\cdot 10^{44}k^{10}(\log k)^{12}<4.8\cdot 10^{83}, \] For each $k \in [2, 800]$, we again used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form $\|\Lambda_4\|$, with integer coefficients $a_i$ not exceeding $5.6\cdot 10^{44}k^{10}(\log k)^{12}$ in absolute value. We consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor \end{pmatrix} ,$$ with $C:= 10^{252}$ and choose $y:=\left(0,0,0\right)$. So, by Lemma \ref{lem2.5}, we get $$c_2=10^{-87}\qquad\text{and}\qquad l\left(\mathcal{L},y\right)\ge c_1:=1.2\cdot10^{85}.$$ So, Lemma \ref{lem2.6} gives $S=7.0\cdot 10^{167}$ and $T=7.2\cdot 10^{83}$. Since $c_1^2\ge T^2+S$, then choosing $c_3:=40.5$ and $c_4:=\log\alpha$, we get $\min\{0.7n,x\}\le 825$. This implies that $n\le 1178$ or $x\le 825$. Let us first assume that $n\le 1178$ holds, that is $\min\{0.7n,x\}=0.7n$. We go to inequality \eqref{g3} and write $\|\Lambda_3\|:=\|\log (\Gamma_3+1)\|$ as \begin{align} \|\Lambda_3\|:=\left\|\log f_k(\alpha)+\log (2\alpha-1)+(m-1)\log\alpha-x\log\left(L_{n+1}^{(k)}\right)\right\|<\dfrac{1.5}{1.4^x}, \end{align} where $a_1 := 1$, $a_2 := 1$, $a_3 := m-1$, $a_4 := -x$ are integers with \[ \max\{\|a_i\| : 1 \leq i \leq 4\} < m<5.6\cdot 10^{44}k^{10}(\log k)^{12}<4.8\cdot 10^{83}, \] where we used Lemma \ref{lem3.4}, for all $k\le 800$. For each $k \in [2, 800]$ and $n\in [801,1178]$, we used the LLL--algorithm to compute a lower bound for the smallest nonzero number of the form $\|\Lambda_3\|$, with integer coefficients $a_i$ not exceeding $5.6\cdot 10^{44}k^{10}(\log k)^{12}$ in absolute value. Specifically, we consider the approximation lattice $$ \mathcal{A}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \lfloor C\log f_k(\alpha)\rfloor & \lfloor C\log (2\alpha-1) \rfloor& \lfloor C\log \alpha \rfloor& \left\lfloor C\log \left(L_{n+1}^{(k)}\right) \right\rfloor \end{pmatrix} ,$$ with $C:= 10^{335}$, $y:=\left(0,0,0,0\right)$, $c_2:=10^{-85}$ and $l\left(\mathcal{L},y\right)\ge c_1:=1.56\cdot10^{84}$. As before with $\Lambda_5$, Lemma \ref{lem2.6} gives $S=9.3\cdot 10^{167}$ and $T=9.6\cdot 10^{83}$. Choosing $c_3:=1.5$ and $c_4:=\log 1.4$, we get $x\le 1722$. Next, if $x\le 825$ holds, that is $\min\{0.7n,x\}=x$, we go back to $\|\Lambda_5\|$ and apply LLL--algorithm. We obtain a similar conclusion as before that $n\le 1278$. To sum up, we have that the integer solutions $n$, $m$, $k$, $x$ to \eqref{eq:main} with $n > k \ge 2$, $k \le 800$, $n > 800$ and $x \ge 2$, satisfy $n \le 1278$ and $x \le 1722$. Finally, we consider the case when $n \le 800$ and since we are working with $k \le 800$ and $n > k$, we get $k \in [2,799]$. Now, we use $\Lambda_3$ as defined before. We apply the LLL-- algorithm with \[ \max\{\|a_i\| : 1 \leq i \leq 4\} < m<(n+3)x+3<803\cdot 2.6\cdot 10^{15} n k^4(\log k)^3\log n+3<1.4\cdot 10^{36}, \] where we have used Lemma \ref{lem3.3}. We consider the same approximation lattice as for $\Lambda_3$, but with $C:= 10^{145}$. Choosing $y:=\left(0,0,0,0\right)$, we get $c_2:=10^{-39}$ and $l\left(\mathcal{L},y\right)\ge c_1:=4.0\cdot10^{37}$. Lemma \ref{lem2.6} gives $S=7.84\cdot 10^{72}$ and $T=2.8\cdot 10^{36}$. Choosing $c_3:=1.5$ and $c_4:=\log 1.4$, we get $x\le 736$. To conclude the case $n>k$, we look for solutions to \eqref{eq:main} in the ranges $k\in [2, 800]$, $n \in [k + 1, 800]$ and $x \in [2, 736]$ or $n \in [801, 1278]$ and $x \in [2, 1722]$, all with $nx-3 <m<(n+3)x+3$. A computational search in SageMath 9.5 allows us to conclude that there are no other integral solutions to \eqref{eq:main} apart from those given in Theorem \ref{thm1.1}. To efficiently handle large \( L_n^{(k)} \) values, our SageMath 9.5 code utilized batch processing, iterating through all \( (n, m,k, x) \) combinations within these ranges. \qed \section{Acknowledgments} We thank the school of mathematics at Stellenbosch university for providing the necessary resources and support during this research. Additionally, we are grateful for the use of virtual SageMath 9.5 libraries that made computations in this work possible. \begin{thebibliography}{99} \bibitem{Brl} Bravo, J. J., \& Luca, F. (2013). 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2412.04740v1	http://arxiv.org/abs/2412.04740v1	Estimates for the first eigenvalue of the one-dimensional $p$-Laplacian	\documentclass[a4paper,12pt]{article} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{color} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{problem}[theorem]{Problem} \renewcommand{\theequation}{{\rm \thesection.\arabic{equation}}} \renewcommand{\labelenumi}{\rm (\roman{enumi})} \begin{document} \title{Estimates for the first eigenvalue of the one-dimensional $p$-Laplacian} \author{ Ryuji \ Kajikiya${}^{1}$ and \ Shingo \ Takeuchi${}^{*2}$ \\[1ex] {\small\itshape ${}^$ Center for Physics and Mathematics,}\\ {\small\itshape Osaka Electro-Communication University,}\\ {\small\itshape Neyagawa, Osaka 572-8530, Japan} \\ {\small\upshape E-mail: [email protected]} \\[1ex] {\small\itshape ${}^{*}$ Department of Mathematical Sciences,}\\ {\small\itshape Shibaura Institute of Technology,}\\ {\small\itshape 307 Fukasaku, Minuma-ku, Saitama-shi,}\\ {\small\itshape Saitama 337-8570, Japan} \\ {\small\upshape E-mail: [email protected]} } \footnotetext[1]{The first author was supported by JSPS KAKENHI Grant Number 20K03686.} \footnotetext[2]{The second author was supported by JSPS KAKENHI Grant Number 22K03392.} \date{} \maketitle \begin{abstract} In the present paper, we study the first eigenvalue $\lambda(p)$ of the one-dimensional $p$-Laplacian in the interval $(-1,1)$. We give an upper and lower estimate of $\lambda(p)$ and study its asymptotic behavior as $p\to 1+0$ or $p\to\infty$. \end{abstract} {\itshape Key words and phrases.} $p$-Laplacian, first eigenvalue, estimate. \newline 2020 {\itshape Mathematical Subject Classification.} 34B09, 34L30, 26D05, 33B10. \section{Introduction and main result}\label{section-1} \setcounter{equation}{0} We study the first eigenvalue $\lambda(p)$ of the one-dimensional $p$-Laplacian, \begin{equation}\label{eq:1.1} (\|u'\|^{p-2}u')' + \lambda(p)\|u\|^{p-2}u =0 \quad \mbox{in } (-1,1), \quad u(-1)=u(1)=0, \end{equation} where $1<p<\infty$. Then $\lambda(p)$ is represented as \begin{equation}\label{eq:1.2} \lambda(p)=(p-1)\left(\dfrac{\pi}{p\sin(\pi/p)} \right)^p. \end{equation} For the proof of \eqref{eq:1.2}, we refer the readers to \cite{DM, KTT} or \cite[pp.4--5]{DR}. In the problem \eqref{eq:1.1}, if the interval $(-1,1)$ is replaced by $(-L,L)$ with $L>0$, then the first eigenvalue is written as $\lambda(p,L)=\lambda(p)/L^p$. Kajikiya, Tanaka and Tanaka~\cite{KTT} proved the next theorem. \begin{theorem}[\cite{KTT}]\label{th:1.1} \begin{enumerate} \item If $0<L\leq 1$, $\lambda_p(p,L)>0$ for $1<p<\infty$, where $\lambda_p(p,L)$ denotes the partial derivative with respect to $p$. Therefore $\lambda(p,L)$ is strictly increasing with respect to $p$. Moreover, $\lambda(p,L)$ diverges to infinity as $p\to\infty$. \item If $L>1$, then there exists a unique $p_(L)>0$ such that $\lambda_p(p,L)>0$ for $p\in (1,p_(L))$ and $\lambda_p(p,L)<0$ for $p\in(p_(L),\infty)$ and $\lambda(p,L)$ converges to zero as $p \to\infty$. \end{enumerate} \end{theorem} The theorem above gives an information on the monotonicity or non-monotonicity of the eigenvalue. In the present paper, we concentrate on $\lambda(p)$ because the properties of $\lambda(p,L)$ follow from those of $\lambda(p)$ by the relation $\lambda(p,L)=\lambda(p)/L^p$. The eigenvalue $\lambda(p)$ in \eqref{eq:1.2} seems complicated and difficult to understand. Therefore we shall give a simple and easy estimate for it. This is our purpose of the present paper. Our another interest is to investigate how $\lambda(p)$ and its derivative behave as $p\to 1+0$ or $p \to \infty$. In the present paper, we give the estimate and the asymptotic behavior of the first eigenvalue $\lambda(p)$. Our main result is as follows. \begin{theorem}\label{th:1.2} The first eigenvalue $\lambda(p)$ is estimated as \begin{equation}\label{eq:1.3} p<\lambda(p)<p+\frac{\pi^2}{6}-1 \quad \mbox{for } 2\leq p<\infty, \end{equation} \begin{equation}\label{eq:1.4} \left(\frac{p}{p-1}\right)^{p-1}<\lambda(p)<(p-1)^{1-p}\left(1+\frac{\pi^2}{6}(p-1)\right)^{p-1} \quad \mbox{for } 1<p<2. \end{equation} \end{theorem} In the theorem above, we give the lower and upper estimates of $\lambda(p)$. These terms satisfy the following inequalities. \begin{lemma}\label{le:1.3} For $1<p<2$, it holds that \begin{equation}\label{eq:1.5} p<\left(\frac{p}{p-1}\right)^{p-1}, \end{equation} \begin{equation}\label{eq:1.6} (p-1)^{1-p}\left(1+\frac{\pi^2}{6}(p-1)\right)^{p-1} <p+\frac{\pi^2}{6}-1. \end{equation} \end{lemma} Observing Theorem \ref{th:1.2} and Lemma \ref{le:1.3}, we have the next result, which is an easy and simple estimate for $\lambda(p)$. \begin{corollary}\label{co:1.4} The first eigenvalue $\lambda(p)$ satisfies \eqref{eq:1.3} for all $1<p<\infty$. \end{corollary} We shall show that $\lambda(p)$ is analytic for $p \in (1,\infty)$. We put $$ p:=\pi/x, \quad y:=\left(\dfrac{\pi}{p\sin(\pi/p)} \right)^p= \left(\dfrac{x}{\sin x} \right)^{\pi/x}. $$ Then $\lambda(p)=(p-1)y$. We compute $\log y$ as $$ \log y=-\frac{\pi}{x}\log\left(\frac{\sin x}{x}\right). $$ Since $\sin x/x$ is positive and analytic in $(0,\pi)$, the function $\log y$ is analytic with respect to $x \in (0,\pi)$, and so is $y=e^{\log y}$. Accordingly, $y$ (hence $\lambda(p)$) is analytic with respect to $p$ because $p=\pi/x$. We observe that $$ \log \lambda(p)=\log\left(\frac{\pi-x}{x}\right) -\frac{\pi}{x}\log\left(\frac{\sin x}{x}\right). $$ The function above is not well defined at $x=0$. However, we shall show that $\lambda(p)-p=\lambda(\pi/x)-\pi/x$ is analytic for $x\in (-\pi,\pi)$. Moreover we shall give its Maclaurin expansion, from which we derive the behavior of $\lambda(p)$ near $p=\infty$. \begin{theorem}\label{th:1.5} The function $\lambda(\pi/x)-\pi/x$ is analytic in $(-\pi,\pi)$ and its Maclaurin series is written as \begin{equation}\label{eq:1.7} \lambda\left(\frac{\pi}{x}\right)-\frac{\pi}{x} = \frac{\pi^2}{6}-1 +\left(\frac{\pi^3}{72}-\frac{\pi}{6}\right)x +\left(\frac{\pi^4}{1296}-\frac{\pi^2}{120}\right)x^2+\cdots. \end{equation} The expansion above is rewritten as, for $1<p<\infty$, \begin{equation}\label{eq:1.8} \lambda(p) = p+ \frac{\pi^2}{6}-1 +\left(\frac{\pi^3}{72}-\frac{\pi}{6}\right)\frac{\pi}{p} +\left(\frac{\pi^4}{1296}-\frac{\pi^2}{120}\right)\left(\frac{\pi}{p}\right)^2 +\cdots. \end{equation} \end{theorem} Denote the derivative of the first eigenvalue $\lambda(p)$ by $\lambda'(p)$. We shall compute their limits as $p\to 1+0$ or $p\to \infty$ by using the theorem above. \begin{theorem}\label{th:1.6} The first eigenvalue $\lambda(p)$ and its derivative $\lambda'(p)$ satisfy \begin{equation}\label{eq:1.9} \lim_{p\to 1+0}\lambda(p)=1, \quad \lim_{p\to 1+0}\lambda'(p)=\infty, \end{equation} \begin{equation}\label{eq:1.10} \lim_{p\to\infty}(\lambda(p)-p)=\frac{\pi^2}{6}-1, \quad \lim_{p\to\infty}\lambda'(p)=1. \end{equation} \end{theorem} As a byproduct of \eqref{eq:1.3}, we immediately obtain the following inequality for the sinc function $\sin{(\pi x)}/(\pi x)$. This inequality may already be known, but at least we could not find any literature with its proof. \begin{corollary}\label{co:1.7} It holds that $$ \left(\frac{1-x}{1+((\pi^2/6)-1)x}\right)^x <\frac{\sin{(\pi x)}}{\pi x} <(1-x)^x \quad \mbox{for } 0<x<1. $$ \end{corollary} \section{Proof of the theorem} \setcounter{equation}{0} We shall prove Theorem \ref{th:1.2}. We first prove the next proposition, the lower estimate of $\lambda(p)$ for $2\leq p<\infty$. \begin{proposition}\label{pr:2.1} It holds that \begin{equation}\label{eq:2.1} p<(p-1)\left(\dfrac{\pi}{p\sin(\pi/p)} \right)^p \quad \mbox{for } 2 \leq p<\infty. \end{equation} \end{proposition} \begin{proof} We shall use the inequality below, which is proved by Zhu~\cite{L}, $$ \dfrac{\sin x}{x}\leq \dfrac{2}{\pi}+\dfrac{\pi-2}{\pi^3}(\pi^2 - 4x^2) \quad \mbox{for } 0<x\leq \pi/2. $$ Let $p\geq 2$. Substituting $x=\pi/p$ in the inequality above, we have $$ \dfrac{\sin (\pi/p)}{\pi/p}\leq \dfrac{2}{\pi}+\dfrac{\pi-2}{\pi^3}(\pi^2 - 4\pi^2/p^2) =1-\dfrac{4(\pi-2)}{\pi p^2}. $$ Put $A:=4(\pi-2)/(\pi p^2)$. We note that $0<A<1$ because $p\geq 2$. We rewrite the inequality above as $$ \dfrac{\pi}{p\sin(\pi/p)} \geq (1-A)^{-1}. $$ It is enough to prove the inequality below, \begin{equation}\label{eq:2.2} p<(p-1)(1-A)^{-p}. \end{equation} Indeed, combining the two inequalities above, we obtain \eqref{eq:2.1}. Taking the logarithm of \eqref{eq:2.2}, we have \begin{equation}\label{eq:2.3} \log p < \log(p-1) - p\log (1-A). \end{equation} We shall show the inequality above. The Maclaurin series shows that $$ \log(1-x)=-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}-\cdots < -x \quad \mbox{for } 0<x<1. $$ Substituting $x=A$ in the inequality above, we obtain $$ -\log(1-A) > A. $$ Therefore, to show \eqref{eq:2.3}, we have only to prove that $$ \log p < \log(p-1) + pA, $$ i.e., \begin{equation}\label{eq:2.4} \log p < \log(p-1) + \dfrac{4(\pi-2)}{\pi p}. \end{equation} We define $$ f(p):=\log(p-1)- \log p + \dfrac{4(\pi-2)}{\pi p}. $$ We shall show that $f(p)>0$ for $p\geq2$. Differentiating it, we obtain $$ f'(p)=\dfrac{1}{p-1} - \dfrac{1}{p} - \dfrac{4(\pi-2)}{\pi p^2}, $$ i.e., $$ \pi p^2(p-1)f'(p)=- (3\pi - 8)p + 4(\pi-2). $$ Thus $f(p)$ achieves its maximum at $p^:=4(\pi-2)/(3\pi-8)$. Observe that $p^>2$. Hence $f(p)$ is increasing for $p\in(2,p^)$ and decreasing for $p\in(p^,\infty)$. Moreover, we have $$ f(2)=\dfrac{2(\pi-2)}{\pi} -\log 2=0.033613\cdots >0, \quad \lim_{p\to\infty}f(p)=0, $$ which shows that $f(p)>0$ for $p\in[2,\infty)$. Therefore \eqref{eq:2.4} holds for $p\in[2,\infty)$. The proof is complete. \end{proof} We next prove the upper estimate of the eigenvalue $\lambda(p)$ for $p\geq 2$, \begin{equation}\label{eq:2.5} \lambda(p)=(p-1)\left(\dfrac{\pi}{p\sin(\pi/p)}\right)^p < p+\dfrac{\pi^2}{6}-1. \end{equation} Taking the logarithm of \eqref{eq:2.5}, we have $$ \log(p-1)+p\log(\pi/(p\sin (\pi/p)))<\log(p+(\pi^2/6)-1). $$ Putting \begin{equation}\label{eq:2.6} p:=\pi/x, \quad a:=(\pi^2/6)-1, \end{equation} we obtain $$ \log((\pi/x)-1)+(\pi/x)\log(x/\sin x) <\log((\pi/x)+a), $$ which is rewritten as \begin{equation}\label{eq:2.7} x\log\left(\dfrac{\pi+ax}{\pi-x}\right) + \pi\log\left(\dfrac{\sin x}{x}\right)>0. \end{equation} Note that \eqref{eq:2.5} is equivalent to \eqref{eq:2.7}. To prove \eqref{eq:2.7}, we estimate its first term from below in the next lemma. \begin{lemma}\label{le:2.2} Let $a$ be defined by \eqref{eq:2.6}. For $0<x<\pi$, it holds that $$ \log\left(\dfrac{\pi+ax}{\pi-x}\right) >\dfrac{\pi}{6}x + \dfrac{1}{72}(12-\pi^2)x^2 +\dfrac{108-18\pi^2+\pi^4}{648\pi}x^3. $$ \end{lemma} \begin{proof} The right hand side is the first three terms in the Maclaurin series of the left hand side. We put $$ g(x):=\log\left(\dfrac{\pi+ax}{\pi-x}\right) -\dfrac{\pi}{6}x - \dfrac{1}{72}(12-\pi^2)x^2 -\dfrac{108-18\pi^2+\pi^4}{648\pi}x^3. $$ We shall show that $g(x)>0$ in $(0,\pi)$. The derivative of $g(x)$ is computed as $$ g'(x) =\dfrac{(a+1)\pi}{(\pi+ax)(\pi-x)} -\dfrac{\pi}{6}-\dfrac{1}{36}(12-\pi^2)x -\dfrac{108-18\pi^2+\pi^4}{216\pi}x^2. $$ Putting $G(x):=6(\pi+ax)(\pi-x)g'(x)$ and using $a=(\pi^2/6)-1$, we obtain \begin{align} G(x) & =\dfrac{1}{216}(864-216\pi^2+24\pi^4-\pi^6)x^3 \\ & \quad + \dfrac{1}{216\pi}(-648+216\pi^2-24\pi^4+\pi^6)x^4. \end{align} We compute that \begin{align} & 864-216\pi^2+24\pi^4-\pi^6=108.594\cdots>0, \\ & -648+216\pi^2-24\pi^4+\pi^6 =107.405\cdots>0. \end{align} Hence $G(x)>0$ for $x>0$. Thus $g'(x)>0$ for $0<x<\pi$ and therefore $g(x)$ is increasing in $(0,\pi)$. Since $g(0)=0$, $g(x)$ is positive for $0<x<\pi$. The proof is complete. \end{proof} The first term in \eqref{eq:2.7} is estimated from below by Lemma~\ref{le:2.2}. We shall evaluate the second term in the next lemma. \begin{lemma}\label{le:2.3} For $0<x\leq \pi/2$, it holds that $$ \log\left(\dfrac{\sin x}{x}\right) > -\dfrac{1}{6}x^2 -\dfrac{1}{180}x^4 -\dfrac{17}{15120}x^6 -\frac{41}{604800}x^8. $$ \end{lemma} \begin{proof} We first show that \begin{equation}\label{eq:2.8} \log(1-t)+t+\dfrac{1}{2}t^2+ \dfrac{1}{2}t^3>0 \quad \mbox{for } 0<t \leq 2/5. \end{equation} We denote the left hand side by $h(t)$, that is, $$ h(t):=\log(1-t)+t+\dfrac{1}{2}t^2+ \dfrac{1}{2}t^3. $$ Differentiating it, we obtain $$ h'(t)=-\frac{1}{1-t}+1+t+ \frac{3}{2}t^2 =\frac{t^2(1-3t)}{2(1-t)}. $$ Therefore $h(t)$ is increasing in $(0,1/3)$ and decreasing in $(1/3,1)$. Since $h(0)=0$, it is positive in $(0,1/3]$. For $1/3<t<2/5$, we get $$ h(t)> h(2/5)=\log(3/5)+\frac{64}{125}=0.0011743\cdots>0. $$ Thus $h(t)$ is positive for $0<t\leq 2/5$, i.e., \eqref{eq:2.8} holds. We rewrite \eqref{eq:2.8} as \begin{equation}\label{eq:2.9} \log(1-t)>-t-\dfrac{1}{2}t^2- \dfrac{1}{2}t^3 \quad \mbox{for } 0<t \leq 2/5. \end{equation} Put $k(x):=(x-\sin x)/x$. Then it is increasing for $x\in(0,\pi)$. We have $$ k(x)\leq k(\pi/2)=\frac{\pi-2}{\pi}=0.36338\cdots<2/5 \quad \mbox{for } 0<x\leq \pi/2. $$ Accordingly, we obtain \begin{equation}\label{eq:2.10} 0<\frac{x-\sin x}{x}<\frac{2}{5} \quad \mbox{if } 0<x\leq \frac{\pi}{2}. \end{equation} Computing the derivatives, we easily prove that $$ \sin x >x -\dfrac{1}{6} x^3 \quad \mbox{for } x>0, $$ $$ \sin x >x -\dfrac{1}{3!} x^3 + \frac{1}{5!}x^5 - \frac{1}{7!}x^7 \quad \mbox{for } x>0. $$ From the inequalities above, it follows that \begin{equation}\label{eq:2.11} 0<\dfrac{x-\sin x}{x}< \frac{1}{6}x^2 \quad \mbox{for } x>0, \end{equation} \begin{equation}\label{eq:2.12} 0<\dfrac{x-\sin x}{x}< \frac{1}{6}x^2 - \frac{1}{5!}x^4 + \frac{1}{7!}x^6 \quad \mbox{for } x>0. \end{equation} By \eqref{eq:2.12}, we get \begin{align}\label{eq:2.13} -\frac{1}{2}\left(\dfrac{x-\sin x}{x}\right)^2 & >-\frac{1}{2}\left(\frac{1}{6}x^2 - \frac{1}{5!}x^4 + \frac{1}{7!}x^6\right)^2 \nonumber \\ & =-\frac{1}{72}x^4-\frac{1}{2}\left(\frac{1}{5!}\right)^2x^8-\frac{1}{2}\left(\frac{1}{7!}\right)^2x^{12} \nonumber \\ &\quad +\frac{1}{6\cdot 5!}x^6+\frac{1}{5!\cdot 7!}x^{10}-\frac{1}{6\cdot 7!}x^8. \end{align} Observe that \begin{equation}\label{eq:2.14} -\frac{1}{2}\left(\frac{1}{7!}\right)^2x^{12}+ \frac{1}{5!\cdot 7!}x^{10}>0 \quad \mbox{for } 0<x<\pi/2. \end{equation} By \eqref{eq:2.13} and \eqref{eq:2.14}, we obtain \begin{equation}\label{eq:2.15} -\frac{1}{2}\left(\dfrac{x-\sin x}{x}\right)^2 >-\frac{1}{72}x^4+\frac{1}{720}x^6 -\frac{41}{604800}x^8. \end{equation} It follows from \eqref{eq:2.11} that \begin{equation}\label{eq:2.16} -\frac{1}{2}\left(\dfrac{x-\sin x}{x}\right)^3>-\frac{1}{2}\left(\frac{1}{6}x^2\right)^3. \end{equation} By \eqref{eq:2.12}, \eqref{eq:2.15} and \eqref{eq:2.16}, we obtain \begin{align}\label{eq:2.17} & -\dfrac{x-\sin x}{x} -\frac{1}{2}\left(\dfrac{x-\sin x}{x}\right)^2 -\frac{1}{2}\left(\dfrac{x-\sin x}{x}\right)^3 \nonumber \\ & >-\frac{1}{6}x^2 + \frac{1}{5!}x^4 - \frac{1}{7!}x^6 -\frac{1}{72}x^4 + \frac{1}{720}x^6 -\frac{41}{604800}x^8 -\dfrac{1}{432}x^6 \nonumber \\ & = -\frac{1}{6}x^2 -\frac{1}{180}x^4 - \frac{17}{15120}x^6 -\frac{41}{604800}x^8. \end{align} Observe \eqref{eq:2.9} and \eqref{eq:2.10}. Put $t:=(x-\sin x)/x$. Since $\log(\sin x/x)=\log(1-t)$, we substitute $t=(x-\sin x)/x$ in \eqref{eq:2.9} and use \eqref{eq:2.17}. Then we obtain $$ \log(\sin x/x)=\log(1-t)> -\frac{1}{6}x^2 -\frac{1}{180}x^4 - \frac{17}{15120}x^6 -\frac{41}{604800}x^8. $$ The proof is complete. \end{proof} Combining Lemmas \ref{le:2.2} and \ref{le:2.3}, we shall prove \eqref{eq:2.7}, which is equivalent to \eqref{eq:2.5}. \begin{proposition}\label{pr:2.4} The inequality \eqref{eq:2.7} holds for $0<x\leq \pi/2$, that is, \eqref{eq:2.5} remains valid for $2\leq p <\infty$. \end{proposition} \begin{proof} By Lemmas \ref{le:2.2} and \ref{le:2.3}, we have for $0<x\leq \pi/2$, \begin{align} & x\log\left(\dfrac{\pi+ax}{\pi-x}\right) + \pi\log\left(\dfrac{\sin x}{x}\right) \\ & >\dfrac{\pi}{6}x^2 + \dfrac{1}{72}(12-\pi^2)x^3 +\dfrac{108-18\pi^2+\pi^4}{648\pi}x^4 -\dfrac{\pi}{6}x^2 -\dfrac{\pi}{180}x^4 \\ & \quad -\dfrac{17\pi}{15120}x^6 - \frac{41\pi}{604800}x^8\\ & = \left[\dfrac{1}{72}(12-\pi^2) -\frac{108\pi^2-5\pi^4-540}{3240\pi}x -\dfrac{17\pi}{15120}x^3 - \frac{41\pi}{604800}x^5 \right]x^3. \end{align} Here we note that $$ 108\pi^2-5\pi^4-540=38.8718\cdots>0. $$ Therefore $$ \phi(x):=\dfrac{1}{72}(12-\pi^2) -\frac{108\pi^2-5\pi^4-540}{3240\pi}x -\dfrac{17\pi}{15120}x^3 - \frac{41\pi}{604800}x^5, $$ is a decreasing function of $x$. For $0<x\leq \pi/2$, it holds that \begin{align} \phi(x) & \geq \phi(\pi/2)=\dfrac{1}{72}(12-\pi^2) -\frac{108\pi^2-5\pi^4-540}{3240\pi}\frac{\pi}{2} -\dfrac{17\pi}{15120}(\pi/2)^3 \\ & \quad - \frac{41\pi}{604800}(\pi/2)^5 \\ & = \frac{1}{4} -\frac{11\pi^2}{360} + \frac{229\pi^4}{362880} - \frac{41\pi^6}{19353600} =0.0078633\cdots>0. \end{align} Therefore, for $0<x \leq \pi/2$, we have $$ x\log\left(\dfrac{\pi+ax}{\pi-x}\right) + \pi\log\left(\dfrac{\sin x}{x}\right) >0. $$ \end{proof} Using Propositions \ref{pr:2.1} and \ref{pr:2.4}, we shall show Theorem \ref{th:1.2}. \begin{proof}[Proof of Theorem \ref{th:1.2}] In Propositions \ref{pr:2.1} and \ref{pr:2.4}, we have already proved \eqref{eq:1.3}. By using the duality $1/p+1/q=1$, we shall derive \eqref{eq:1.4} from \eqref{eq:1.3}. Let $q\in (1,2)$ be any number. Put $p:=q/(q-1)$. Then $2<p<\infty$. Substituting $p=q/(q-1)$ in the first inequality in \eqref{eq:1.3}, we have \begin{equation}\label{eq:2.18} \dfrac{q}{q-1}<\dfrac{1}{q-1}\left(\dfrac{(q-1)\pi}{q\sin((q-1)\pi/q)}\right)^{q/(q-1)}. \end{equation} Using the relation \begin{equation}\label{eq:2.19} \sin((q-1)\pi/q)=\sin(\pi - \pi/q)=\sin(\pi/q), \end{equation} we reduce \eqref{eq:2.18} to $$ q<\left(\dfrac{(q-1)\pi}{q\sin(\pi/q)}\right)^{q/(q-1)}. $$ or equivalently, $$ \left(\frac{q}{q-1}\right)^{q-1} < (q-1)\left(\dfrac{\pi}{q\sin(\pi/q)}\right)^q=\lambda(q). $$ This is the lower estimate of $\lambda(p)$ in \eqref{eq:1.4} with $p$ replaced by $q$. We shall show the upper estimate in \eqref{eq:1.4}. Let $q\in (1,2)$ be any number. Put $p:=q/(q-1)$. Then $p\in(2,\infty)$. Substituting $p=q/(q-1)$ in the second inequality in \eqref{eq:1.3}, we have $$ \dfrac{1}{q-1}\left(\frac{(q-1)\pi}{q\sin((q-1)\pi/q)}\right)^{q/(q-1)} < \frac{1}{q-1}+\frac{\pi^2}{6}. $$ By \eqref{eq:2.19}, the inequality above is rewritten as $$ (q-1) \left(\frac{\pi}{q\sin(\pi/q)}\right)^q < (q-1)^{1-q}\left(1+ \frac{\pi^2}{6}(q-1)\right)^{q-1}. $$ Consequently, we have the upper estimate of $\lambda(p)$ in \eqref{eq:1.4}. The proof is complete. \end{proof} \begin{proof}[Proof of Lemma \ref{le:1.3}] We rewrite \eqref{eq:1.5} as $$ (p-1)^{p-1}<p^{p-2} \quad \mbox{for } 1<p<2. $$ Taking the logarithm of the inequality above, we have $$ (p-1)\log(p-1)<(p-2)\log p. $$ We shall show the above inequality. Put $$ f(p):=(p-2)\log p -(p-1) \log(p-1). $$ It is enough to show that $f(p)$ is positive. Differentiating $f(p)$, we have $$ f'(p)=\log p -\log (p-1) - \frac{2}{p}, $$ $$ f''(p)=\frac{1}{p} - \frac{1}{p-1} +\frac{2}{p^2}=-\dfrac{2-p}{p^2(p-1)}<0 \quad \mbox{for } p\in(1,2). $$ Hence $f(p)$ is concave. Furthermore, we have $$ \lim_{p\to 1+0}f(p)=0, \quad f(2)=0, $$ which means that $f(p)>0$ for $1<p<2$. Accordingly, \eqref{eq:1.5} holds. We shall prove \eqref{eq:1.6}. Let $1<p<2$. Put $x:=p-1$. Since $1<p<2$, we have $0<x<1$. Then \eqref{eq:1.6} is rewritten as $$ x^{-x}(1+(\pi^2/6)x)^x < x+ \pi^2/6 \quad \mbox{for } 0<x<1. $$ Taking the logarithm of the inequality above, we have $$ -x\log x +x\log(1+(\pi^2/6)x)<\log(x+\pi^2/6), $$ or equivalently, \begin{equation}\label{eq:2.20} \log(x+\pi^2/6) + x\log x -x\log(1+(\pi^2/6)x)>0. \end{equation} Observe that \eqref{eq:2.20} is equivalent to \eqref{eq:1.6}. We define $$ g(x):=\log(x+\pi^2/6) + x\log x -x\log(1+(\pi^2/6)x). $$ We have only to prove that $g(x)$ is positive in $(0,1)$. The first and second derivatives of $g(x)$ are computed as $$ g'(x)=\dfrac{1}{x+(\pi^2/6)} + \log x +1 -\log(1+(\pi^2/6)x) -\frac{\pi^2 x}{\pi^2 x + 6}. $$ $$ g''(x)=-\frac{36}{(6x+\pi^2)^2} + \frac{1}{x} - \frac{\pi^2}{\pi^2 x + 6} -\frac{6\pi^2}{(\pi^2 x +6)^2}. $$ Then we have \begin{align} & x(6x+\pi^2)^2(\pi^2 x +6)^2g''(x) \\ & = 36(-\pi^4x^3+(-12\pi^2+36)x^2+(12\pi^2-36)x+\pi^4) \\ & =36[\pi^4(1-x^3)+(12\pi^2-36)x(1-x)]>0. \end{align} Consequently, $g''(x)>0$ for $0<x<1$, and therefore $g(x)$ is convex in $(0,1)$. Moreover, \begin{align} g'(1) & =\frac{6}{\pi^2+6} +1 -\log(1+(\pi^2/6))-\frac{\pi^2}{\pi^2+6} \\ & =\frac{12}{\pi^2+6} -\log(1+(\pi^2/6))=-0.21648\cdots <0. \end{align} Thus $g'(x)<0$ for $0<x<1$ and hence $g(x)$ is decreasing. Since $g(1)=0$, $g(x)$ is positive for $0<x<1$. The proof is complete. \end{proof} We shall prove Theorem \ref{th:1.5}, the analyticity of $\lambda(\pi/x)-\pi/x$. \begin{proof}[Proof of Theorem \ref{th:1.5}] Putting $p:=\pi/x$, we write $\lambda(p)-p$ as \begin{align}\label{eq:2.21} \lambda(p)-p & =p\left[\left(\frac{\pi}{p\sin(\pi/p)}\right)^p-1\right]- \left(\frac{\pi}{p\sin(\pi/p)}\right)^p \nonumber \\ & =\frac{\pi(y-1)}{x}- y, \end{align} where $y$ is defined by $$ y:=\left(\frac{\pi}{p\sin(\pi/p)}\right)^p=\left(\frac{x}{\sin x}\right)^{\pi/x}. $$ Then we have \begin{equation}\label{eq:2.22} \log y=-\frac{\pi}{x}\log(\sin x/x)=-\frac{\pi}{x}\log(1-(x-\sin x)/x). \end{equation} The function $\sin x/x$ is analytic in $\mathbb{R}$. More precisely, the point $x=0$ is a removable singularity. After defining $\sin x/x =1$ at $x=0$, it is analytic in $\mathbb{R}$. Since $0<\sin x/x\leq 1$ for $x \in (-\pi,\pi)$, \eqref{eq:2.22} shows that $\log y$ is analytic in $(-\pi,\pi)$, and so is $y=e^{\log y}$. We shall use the Maclaurin series: \begin{equation}\label{eq:2.23} \frac{x-\sin x}{x}=\frac{1}{3!}x^2-\frac{1}{5!}x^4+\frac{1}{7!}x^6 - \cdots, \end{equation} \begin{equation}\label{eq:2.24} \log(1-t)=-t-\frac{1}{2}t^2-\frac{1}{3}t^3+\cdots. \end{equation} Putting $t=(x-\sin x)/x$ and using \eqref{eq:2.23} with \eqref{eq:2.24}, we obtain \begin{align}\label{eq:2.25} & \log(1-(x-\sin x)/x) \nonumber \\ & =-\frac{x-\sin x}{x} -\frac{1}{2}\left(\frac{x-\sin x}{x}\right)^2 - \frac{1}{3}\left(\frac{x-\sin x}{x}\right)^3 - \cdots \nonumber \\ & =-\frac{1}{6}x^2 - \frac{1}{180}x^4 + \cdots. \end{align} Substituting \eqref{eq:2.25} in \eqref{eq:2.22}, we have \begin{align} \log y & =-\frac{\pi}{x}\left(-\frac{1}{6}x^2 - \frac{1}{180}x^4 + \cdots \right) \\ & =\frac{\pi}{6}x + \frac{\pi}{180}x^3 + \cdots. \end{align} Using the Maclaurin expansion of $e^t$ with the formula above, we have \begin{align}\label{eq:2.26} y & =\exp\left(\frac{\pi}{6}x + \frac{\pi}{180}x^3 + \cdots\right) \nonumber \\ & =1+ \left(\frac{\pi}{6}x + \frac{\pi}{180}x^3 + \cdots\right) + \frac{1}{2!}\left(\frac{\pi}{6}x + \frac{\pi}{180}x^3 + \cdots\right)^2 \nonumber \\ & \quad + \frac{1}{3!}\left(\frac{\pi}{6}x + \frac{\pi}{180}x^3 + \cdots\right)^3 +\cdots \nonumber \\ & =1+ \frac{\pi}{6}x + \frac{\pi^2}{72}x^2 + \left(\frac{\pi}{180} + \frac{\pi^3}{1296}\right)x^3+ \cdots. \end{align} Accordingly, $(\pi/x)(y-1)$ is analytic for $x\in (-\pi,\pi)$. Using \eqref{eq:2.26} in \eqref{eq:2.21}, we obtain \begin{align} \lambda(p)-p & =\frac{\pi}{x}(y-1)-y \\ & = \frac{\pi^2}{6} + \frac{\pi^3}{72}x + \left(\frac{\pi^2}{180} + \frac{\pi^4}{1296}\right)x^2+\cdots \\ & \quad -\left[1+ \frac{\pi}{6}x + \frac{\pi^2}{72}x^2 + \left(\frac{\pi}{180} + \frac{\pi^3}{1296}\right)x^3 + \cdots\right] \\ & =\frac{\pi^2}{6}-1 + \left(\frac{\pi^3}{72}-\frac{\pi}{6}\right)x + \left(\frac{\pi^4}{1296}-\frac{\pi^2}{120}\right)x^2 +\cdots. \end{align} Consequently, $\lambda(\pi/x)-\pi/x$ is analytic and \eqref{eq:1.7} holds true. The proof is complete. \end{proof} We investigate the limits of $\lambda(p)$ and $\lambda'(p)$ as $p\to 1$. \begin{proposition}\label{pr:2.5} The assertion \eqref{eq:1.9} holds. \end{proposition} \begin{proof} We shall show the first claim in \eqref{eq:1.9}. Put $p:=\pi/x$. Then $p\to 1+0$ if and only if $x\to \pi-0$. The eigenvalue $\lambda(p)$ is rewritten as \begin{align} \lambda(p) & =(p-1)\left(\frac{\pi}{p\sin(\pi/p)}\right)^p =(p-1)^{1-p}\left(\frac{(p-1)\pi}{p\sin(\pi/p)}\right)^p \\ & =\left(\frac{\pi-x}{x}\right)^{(x-\pi)/x}\left(\frac{\pi-x}{\sin x}\right)^{\pi/x}. \end{align} Then the first limit in \eqref{eq:1.9} follows from the facts that $$ \lim_{x\to \pi-0} \left(\frac{\pi-x}{x}\right)^{(x-\pi)/x} =\lim_{x\to\pi-0}\left(\frac{\pi-x}{\sin x}\right)^{\pi/x}=1. $$ To compute the limit of $\lambda'(p)$ as $p \to 1$, we shall prove the formula, \begin{equation}\label{eq:2.27} \lambda'(p)=\lambda(p) \left[\frac{x}{\pi-x} - \log(\sin x/x) + \frac{x\cos x-\sin x}{\sin x}\right], \end{equation} with $p:=\pi/x$. Taking the logarithm of $\lambda(p)$ in \eqref{eq:1.2}, we obtain $$ \log \lambda(p)=\log(p-1)+ p\log(\pi/(p\sin (\pi/p))). $$ Putting $p:=\pi/x$, we have \begin{equation}\label{eq:2.28} \log \lambda(p)=\log((\pi-x)/x) - \frac{\pi}{x}\log(\sin x/x). \end{equation} Differentiating $\log \lambda(p)$ with respect to $x$, we get $$ \frac{d \log \lambda(p)}{dx} = \frac{\lambda'(p)}{\lambda(p)}\frac{dp}{dx}=-\frac{\pi}{x^2}\frac{\lambda'(p)}{\lambda(p)}. $$ Differentiating both sides of \eqref{eq:2.28}, we have $$ -\frac{\pi}{x^2}\frac{\lambda'(p)}{\lambda(p)} =-\frac{\pi}{x(\pi-x)} + \frac{\pi}{x^2}\log(\sin x/x) -\frac{\pi(x\cos x-\sin x)}{x^2\sin x}. $$ Multiplying both sides by $-(x^2/\pi)\lambda(p)$, we obtain \eqref{eq:2.27}. Rewrite \eqref{eq:2.27} as \begin{equation}\label{eq:2.29} \lambda'(p)=\lambda(p) \left[ -\log(\sin x/x) + x\left(\frac{1}{\pi-x} + \frac{\cos x}{\sin x}\right)-1\right]. \end{equation} Note that $p\to 1+0$ is equivalent to $x\to \pi-0$. We shall prove that \begin{equation}\label{eq:2.30} \lim_{x\to\pi-0}\left(\frac{1}{\pi-x} + \frac{\cos x}{\sin x}\right)=0. \end{equation} We easily find that $$ \frac{1}{\pi-x} + \frac{\cos x}{\sin x}= \frac{\sin x + (\pi-x)\cos x}{(\pi-x)\sin x}. $$ Using L'Hospital's rule twice, we get \begin{align} \lim_{x\to\pi-0}\frac{\sin x + (\pi-x)\cos x}{(\pi-x)\sin x} & =\lim_{x\to\pi-0}\frac{-(\pi-x)\sin x}{-\sin x + (\pi-x)\cos x} \\ & =\lim_{x\to\pi-0}\frac{\sin x - (\pi-x)\cos x}{-2\cos x - (\pi-x)\sin x}=0. \end{align} Therefore we obtain \eqref{eq:2.30}. Observe that $\log(\sin x/x)$ diverges to $-\infty$ as $x\to \pi-0$. Using \eqref{eq:2.30} in \eqref{eq:2.29} and noting that $\lambda(p)$ converges to $1$ as $p\to 1+0$, we find that $\lambda'(p)\to \infty$ as $p\to 1+0$. Consequently, we obtain the second claim in \eqref{eq:1.9}. \end{proof} Using Proposition \ref{pr:2.5} and Theorem \ref{th:1.5}, we shall prove Theorem \ref{th:1.6}. \begin{proof}[Proof of Theorem \ref{th:1.6}] The assertion \eqref{eq:1.9} has already been proved in Proposition \ref{pr:2.5}. We shall show \eqref{eq:1.10}. Letting $p\to\infty$ in \eqref{eq:1.8}, we have the first limit in \eqref{eq:1.10}. Denote the right hand side in \eqref{eq:1.7} by $f(x)$. Then we have $$ \lambda(\pi/x) -\pi/x=f(x). $$ Differentiating it, we have $$ -\frac{\pi}{x^2}(\lambda'(\pi/x)-1)=f'(x), $$ which is rewritten as $$ \lambda'(\pi/x) -1 =-\frac{x^2}{\pi}f'(x). $$ Letting $x\to +0$ (or $p\to \infty$), we have $\lim_{p\to\infty}(\lambda'(p)-1)=0$. The proof is complete. \end{proof} We conclude the present paper by proving Corollary~\ref{co:1.7}. \begin{proof}[Proof of Corollary~\ref{co:1.7}] By \eqref{eq:1.2} and \eqref{eq:1.3}, we obtain $$ p<(p-1)\left(\frac{\pi}{p\sin(\pi/p)}\right)^p <p+a, \quad \mbox{with } a:=\frac{\pi^2}{6}-1. $$ We transform the inequality above to $$ \left(\frac{p}{p-1}\right)^{1/p} < \frac{\pi}{p\sin(\pi/p)}< \left(\frac{p+a}{p-1}\right)^{1/p}. $$ Putting $p:=1/x$, we have $$ \left(\frac{1}{1-x}\right)^x < \frac{\pi x}{\sin(\pi x)}< \left(\frac{1+ax}{1-x}\right)^x. $$ Taking the reciprocal of both sides, we obtain the conclusion. \end{proof} \begin{thebibliography}{99} \bibitem{DR} O. Do\v{s}l\'{y} and P. \v{R}eh\'{a}k, Half-linear differential equations, North-Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. \bibitem{DM} P. Dr\'{a}bek and R. Man\'{a}sevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian, Differential Integral Equations 12, 773--788, (1999). \bibitem{KTT} R. Kajikiya, M. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one-dimensional $(p,q)$-Laplace equation. Electronic Journal of Differential Equations, 2017, Paper No. 107, (2017). \bibitem{L} L. Zhu, Sharpening Jordan's inequality and the Yang Le inequality, Appl. Math. Lett. (3) 19, 240--243, (2006). \end{thebibliography} \end{document}
2412.04848v2	http://arxiv.org/abs/2412.04848v2	Quadratic Modelings of Syndrome Decoding	\documentclass[runningheads]{llncs} \usepackage[utf8]{inputenc} \usepackage{amssymb} \usepackage{listings} \usepackage{amsfonts} \usepackage{float} \usepackage{amsmath,latexsym} \usepackage{graphicx} \usepackage{fancyvrb} \usepackage{authblk} \usepackage{paralist} \usepackage{makecell} \usepackage{comment} \usepackage{cite} \DeclareMathOperator{\lcm}{lcm} \usepackage[table,xcdraw]{xcolor} \newif\ifanonymous \anonymousfalse \usepackage{xcolor} \usepackage{tikz-cd} \usepackage{xcolor} \definecolor{linkcolor}{rgb}{0.65,0,0} \definecolor{citecolor}{rgb}{0,0.4,0} \definecolor{urlcolor}{rgb}{0,0,0.65} \usepackage[colorlinks=true, linkcolor=linkcolor, urlcolor=urlcolor, citecolor=citecolor]{hyperref} \definecolor{darkblue}{RGB}{0,0,160} \definecolor{darkdarkred}{RGB}{180,0,0} \definecolor{darkgreen}{RGB}{0,140,0} \newcommand{\FF}{\mathbb{F}} \newcommand{\FFt}{\mathbb{F}_2} \newcommand{\FFq}{\mathbb{F}_q} \newcommand{\FFqm}{\mathbb{F}_{q^m}} \newcommand{\K}{\mathbb{K}} \newcommand{\vh}{\mathbf{h}} \newcommand{\vs}{\mathbf{s}} \newcommand{\vb}{\mathbf{b}} \newcommand{\vc}{\mathbf{c}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vu}{\mathbf{u}} \newcommand{\vv}{\mathbf{v}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vy}{\mathbf{y}} \newcommand{\vt}{\mathbf{t}} \newcommand{\vz}{\mathbf{z}} \newcommand{\vH}{\mathbf{H}} \newcommand{\parts}[2]{\left\{{#1 \atop #2}\right\}} \newcommand{\htop}{{\mathrm{top}}} \newtheorem{algorithm}{Algorithm} \newtheorem{modeling}{Modeling} \newtheorem{notation}{Notation} \newcommand{\Cf}{\mathbf{C}_f} \newcommand{\HH}{\mathbf{H}} \newcommand{\X}{\mathcal{X}} \newcommand{\CC}{\mathcal{C}} \newcommand{\OO}{\mathcal{O}} \newcommand{\GG}{\mathcal{G}} \newcommand{\LL}{\mathcal{L}} \newcommand{\Fqm}{\mathbb{F}_{q^m}} \newcommand{\Fq}{\mathbb{F}_2} \newcommand{\supp}{\mathsf{supp}} \newcommand{\Span}{\mathsf{span}} \newcommand{\rk}{\mathsf{rk}} \newcommand{\hash}{\mathsf{hash}} \newcommand{\wt}{\mathsf{wt}} \newcommand{\lm}{\mathsf{lm}} \newcommand{\Mat}{\mathsf{Mat}} \newcommand{\pk}{\mathsf{pk}} \newcommand{\sk}{\mathsf{sk}} \newcommand{\fail}{\mathsf{fail}} \newcommand{\init}{\mathsf{init}} \newcommand{\GL}{{\sf GL}} \newcommand{\ireg}[1]{i_{\mathrm{reg}}(#1)} \newcommand{\dreg}[1]{d_{\mathrm{reg}}(#1)} \newcommand{\pr}{{\mathbb{P}}} \newcommand{\ord}{\mathsf{ord}} \newcommand{\alec}[1]{{\color{red} $\clubsuit\clubsuit\clubsuit$ Alessio C.: [#1]}} \newcommand{\alem}[1]{{\color{blue} $\clubsuit\clubsuit\clubsuit$ Alessio M.: [#1]}} \newcommand{\alex}[1]{{\color{orange} $\clubsuit\clubsuit\clubsuit$ Alex: [#1]}} \newcommand{\rocco}[1]{{\color{purple} $\clubsuit\clubsuit\clubsuit$ Rocco: [#1]}} \newcommand{\ryann}[1]{{\color{darkgreen} $\clubsuit\clubsuit\clubsuit$ Ryann: [#1]}} \newcommand{\todo}[1]{{\color{magenta} $\star$ \underline{To do:} [#1]}} \begin{document} \title{Quadratic Modelings of Syndrome Decoding} \author{Alessio Caminata \inst{1} \and Ryann Cartor \inst{2}\and Alessio Meneghetti \inst{3}\and Rocco Mora \inst{4} \and Alex Pellegrini \inst{5}} \authorrunning{A. Caminata et al.} \institute{Universit\`a di Genova \and Clemson University \and Universit\`a di Trento \and CISPA Helmholtz Center for Information Security \and Eindhoven University of Technology } \maketitle \begin{abstract} This paper presents enhanced reductions of the bounded-weight and exact-weight Syndrome Decoding Problem (SDP) to a system of quadratic equations. Over $\FFt$, we improve on a previous work and study the degree of regularity of the modeling of the exact weight SDP. Additionally, we introduce a novel technique that transforms SDP instances over $\FF_q$ into systems of polynomial equations and thoroughly investigate the dimension of their varieties. Experimental results are provided to evaluate the complexity of solving SDP instances using our models through Gr\"obner bases techniques. \keywords{Syndrome Decoding \and Gr\"obner Basis \and Cryptanalysis \and Code-Based Cryptography \and Multivariate Cryptography} \end{abstract} \section{Introduction}\label{sec:intro} As widespread quantum computing becomes closer to reality, accurate cryptanalysis of post-quantum cryptosystems is of the utmost importance. Code-based cryptography is one of the main areas of focus in the search for quantum-secure cryptosystems. This is well represented by the NIST Post-Quantum Standardization Process, where as many as three finalists, namely Classic McEliece \cite{bernstein2017classic} (an IND-CCA2 secure variation of McEliece's very first code-based scheme \cite{mceliece1978public}), HQC \cite{melchor2018hamming} and BIKE \cite{aragon2022bike}, belong to this family. Similarly, NIST's additional call for digital signatures has numerous proposals that make use of linear codes. Many of the proposed schemes are based on the hardness of (sometimes structured variants of) the syndrome decoding problem. The parameters of many code-based schemes are carefully chosen to align with the latest advancements with respect to this computational problem. Despite decades of intensive research in this direction, all the algorithms developed so far exhibit exponential complexity. This is not surprising, since the problem has been shown to be NP-hard \cite{berlekamp1978inherent}. In particular, after more than 60 years of investigation since the groundbreaking paper of Prange \cite{DBLP:journals/tit/Prange62}, the reduction in the exponent for most parameters of interest has been minimal \cite{stern1989method, D89, finiasz2009security, bernstein2011smaller, may2011decoding, becker2012decoding, may2015computing, both2018decoding}. All the works mentioned fall into the family of Information Set Decoding (ISD) algorithms, whose basic observation is that it is easier to guess error-free positions, and guessing enough of them is sufficient to decode. This resistance to ISD algorithms makes the syndrome decoding problem a reliable foundation for code-based cryptosystems. To comprehensively assess security, it is imperative to consider attacks stemming from various other realms of post-quantum cryptography. For instance, attacks typically associated with multivariate or lattice-based schemes should also be taken into account for code-based schemes, when applicable. A remarkable example is offered by dual attacks, originally introduced in lattice-based cryptography, where, however, they have been strongly questioned. In contrast, their code-based counterpart \cite{carrier2022statistical, carrier2024reduction} has recently outperformed ISD techniques for a non-negligible regime of parameters, by reducing the decoding problem to the closely related Learning Parity with Noise problem. Concerning polynomial system solving strategies, another notable illustration of this is the algebraic MinRank attack, which broke the rank-metric code-based schemes RQC and Rollo \cite{bardet2020algebraic, DBLP:conf/asiacrypt/BardetBCGPSTV20} and now represents the state-of-the-art for MinRank cryptanalysis, beating combinatorial approaches. In the Hamming metric, a reduction that transforms an instance of the syndrome decoding problem into a system of quadratic equations over $\mathbb{F}_2$ was introduced in \cite{2021/meneghetti}. The most expensive step of the transformation, in terms of numbers of new variables and new equations introduced, is the so-called \textit{Hamming-weight computation encoding}. Indeed, for a binary linear code of length $n$, the procedure dominates the overall complexity of the reduction with a complexity of $\mathcal{O}(n\log_2(n)^2)$. Despite the considerable theoretical interest in this transformation, the latter is too inefficient to be of practical interest in solving the syndrome decoding problem. Thus, the problem of improving the reduction in order to obtain a more effectively solvable system remains open. Moreover, \cite{2021/meneghetti} covers only the binary case, leaving unanswered the challenge of modeling through algebraic equations the decoding problem for codes defined over finite fields with more than two elements. \paragraph{Our contribution.} In this work, we improve on the reduction presented in \cite{2021/meneghetti} by a factor of \(\log_2(n)\), thereby reducing the number of introduced variables and equations and achieving an overall reduction cost of \(\mathcal{O}(n\log_2(n))\). This improvement is achieved by leveraging the recursive structure of the equations generated by the Hamming-weight computation encoding and by transforming the equations similarly to the reduction procedure in Buchberger's algorithm \cite{1965/buchberger} for Gröbner basis computation. When considering a version of the syndrome decoding problem that requires an error vector with a specified Hamming weight, we derive a further improved modeling, for which we study the degree of regularity. As a second contribution, we present a novel approach that transforms an instance of the syndrome decoding problem over \(\mathbb{F}_{q}\) for \(q \geq 2\) into a system of polynomial equations. This significantly broadens the applicability of our methods to a wider range of code-based cryptosystems. A common feature of our algebraic modelings is that if the decoding problem admits multiple solutions, the Gröbner basis naturally determines all of them. We also provide theoretical and experimental data to analyze the complexity of solving syndrome decoding instances using our modelings, demonstrating that, at least for small parameters, our new strategy is practical and successful. Software (MAGMA scripts) supporting this work can be found \href{https://github.com/rexos/phd-cryptography-code/tree/main/modelings}{here}. \paragraph{Structure of the paper.} The next section recalls the background and notions necessary for this work. In Section~\ref{sec:mps}, we review the reduction described in \cite{2021/meneghetti} from the syndrome decoding problem to that of finding the zeroes of a set of polynomials. In Section~\ref{sec:EWM}, we describe two modelings that improve upon \cite{2021/meneghetti}. We study the degree of regularity of the modeling for the exact weight syndrome decoding problem, along with experimental results, in Section~\ref{sec:complexity-analysis}. Finally, in Section~\ref{sec:Fq}, we present a novel modeling of the syndrome decoding problem over $\mathbb{F}_{q}$ with $q \geq 2$, for which we provide a theoretical study of the variety and experimental analysis of the solving complexity with Gr\"obner bases techniques. \section{Preliminaries} \label{sec:prelim} This paper investigates the reduction of the Syndrome Decoding Problem (SDP) into a Polynomial System Solving Problem (PoSSo). In this section, we briefly recall the definitions of both problems, as well as the notions of solving degree and degree of regularity, which are commonly used to estimate the computational complexity of the PoSSo problem. \subsection{The Syndrome Decoding Problem} An $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\FF_q^n$. We call $n$ the length of the code, and $k$ its dimension. An element $\mathbf{x}\in\FF_q^n$ is called a codeword if $\mathbf{x}\in\mathcal{C}$. The number of nonzero entries in $\mathbf{x}$ is called the Hamming weight of $\mathbf{x}$ and we denote it as $\wt(\mathbf{x})$. Given a code $\mathcal{C}$ we define a parity check matrix of $\mathcal{C}$ as $\mathbf{H}\in\FF_q^{(n-k)\times n}$ such that the right kernel of $\mathbf{H}$ is the code $\mathcal{C}$. The subspace spanned by the rows of $\HH$ is called the dual code of $\mathcal{C}$. Many code-based cryptosystems rely on the hardness of solving the Syndrome Decoding Problem (SDP), see Problems~\ref{BSDP} and~\ref{EWSDP} described below. \begin{problem}[SDP: Syndrome Decoding Problem]\label{BSDP} Given integers $n,k,t$ such that $k\leq n$ and $t\leq n$, an instance of the problem SD$(\HH,\mathbf{s},t)$ consists of a parity check matrix $\mathbf{H}\in\FF_q^{(n-k)\times n}$ and a vector $\mathbf{s}\in\FF_q^{n-k}$ (called the syndrome). A solution to the problem is a vector $\mathbf{e}\in \mathbb{F}_q^n$ such that $\mathbf{He}^\top=\mathbf{s}^\top$ and $\wt(\mathbf{e})\leq t$. \end{problem} \noindent In later sections, we will also refer to Problem~\ref{BSDP} as the ``Bounded Syndrome Decoding" Problem. We will also consider the following variant of SDP. \begin{problem}[ESDP: Exact Weight Syndrome Decoding Problem]\label{EWSDP} Given integers $n,k,t$ such that $k\leq n$ and $t\leq n$, an instance of the problem ESD$(\HH,\mathbf{s},t)$ consists of a parity check matrix $\mathbf{H}\in\FF_q^{(n-k)\times n}$ and a vector $\mathbf{s}\in\FF_q^{n-k}$ (called the syndrome). A solution to the problem is a vector $\mathbf{e}\in \mathbb{F}_q^n$ such that $\mathbf{He}^\top=\mathbf{s}^\top$ and $\wt(\mathbf{e})= t$. \end{problem} Additionally, a close variant of the Syndrome Decoding Problem is the \textit{Codeword Finding Problem}, where the syndrome $\vs$ is the zero vector ${\mathbf{0}}$. Since the null vector is always a solution of the parity-check equations $\mathbf{He}^\top=\mathbf{0}^\top$, a nonzero $\ve$ of weight at most (or exactly) $t$ is sought. The name of the problem refers to the fact that any element in the right kernel of $\mathbf{H}$ belongs to the code $\mathcal{C}$ having $\HH$ as parity-check matrix. We will later need to distinguish this variant in the analysis of one of our modelings. In addition to length and dimension, a fundamental notion in coding theory and consequently in code-based cryptography is the minimum distance $d$ of an $\FF_q$-linear code, i.e. the Hamming weight of the smallest nonzero codeword in the code. Such a quantity is strictly related to the number of solutions to the syndrome decoding problem. Knowing the expected number of solutions from given parameters is extremely important in cryptography, in order to assess the security correctly. It is guaranteed that the problem does not admit more than one solution as long as the number of errors is upper bounded by $\frac{d-1}{2}$. However, in practice, much better can be done for randomly generated codes. Indeed, it turns out that random codes achieve the so-called Gilbert-Varshamov (GV) distance $d_{GV}$, defined as the largest integer such that \[ \sum_{i=0}^{d_{GV}-1} \binom{n}{i}(q-1)^i \le q^{n-k}. \] It can be shown that, as long as the number of errors is below the Gilbert-Varshamov distance, the Syndrome Decoding problem \textit{typically} has a unique solution. Moreover, the instances where the number of errors attains the GV distance are those supposed to be the most difficult. \subsection{The Polynomial System Solving Problem} The Polynomial System Solving Problem (PoSSo) is the following. We define it over a finite field $\FF_q$, athough it can be more generally considered over any field. \begin{problem}[PoSSo: Polynomial System Solving]\label{PoSSo} Given integers $N,r\geq2$, an instance of the PoSSo problem consists of a system of polynomials $\mathcal{F}=\{f_1,\dots,f_r\}$ in $R=\FF_q[x_1,\dots,x_N]$ with $N$ variables and coefficients in $\FF_q$. A solution to the problem is a vector $\mathbf{a}\in\FF_q^N$ such that $f_1(\mathbf{a})=\cdots=f_r(\mathbf{a})=0$. \end{problem} \begin{remark}A special case of PoSSo when $\deg(f_i)=2$ for $1\leq i\leq r$ is called MQ (Multivariate Quadratic) and is the basis for multivaritate cryptography. \end{remark} The following outlines a standard strategy for finding the solutions of a polynomial system $\mathcal{F}$ by means of Gr\"obner bases. \begin{compactenum} \item Find a degree reverse lexicographic ($\mathsf{degrevlex}$) Gr\"obner basis of the ideal $\langle\mathcal{F}\rangle$; \item Convert the obtained $\mathsf{degrevlex}$ Gr\"obner basis into a lexicographic ($\mathsf{lex}$) Gr\"obner basis, where the solutions of the system can be easily read from the ideal in this form. \end{compactenum} The second step can be done by FGLM \cite{FGLM93}, or a similar algorithm, whose complexity depends on the degree of the ideal. This is usually faster than the first step, especially when the system $\mathcal{F}$ has few solutions. Therefore, we focus on the first step. The fastest known algorithms to compute a $\mathsf{degrevlex}$ Gr\"obner basis are the linear algebra based algorithms such as F4 \cite{faugereF4}, F5 \cite{F5paper}, or XL \cite{XL00}. These transform the problem of computing a Gr\"obner basis into one or more instances of Gaussian elimination of the Macaulay matrices. The complexity of these algorithms is dominated by the Gaussian elimination on the largest Macaulay matrix encountered during the process. The size of a Macaulay matrix depends on the degrees of the input polynomials $f_1,\dots,f_r$, on the number of variables $N$, and on a degree $d$. In a nutshell, the \emph{Macaulay matrix} $M_{\leq d}$ of degree $d$ of $\mathcal{F}$ has columns indexed by the monic monomials of degree $\leq d$, sorted in decreasing order from left to right (with respect to the chosen $\mathsf{degrevlex}$ term order). The rows of $M_{\leq d}$ are indexed by the polynomials $m_{i,j}f_j$, where $m_{i,j}$ is a monic monomial such that $\deg(m_{i,j}f_j)\leq d$. The entry $(i,j)$ of $M_{\leq d}$ is the coefficient of the monomial of column $j$ in the polynomial corresponding to the $i$-th row. The \emph{solving degree} of $\mathcal{F}$ is defined as the least degree $d$ such that Gaussian elimination on the Macaulay matrix $M_{\leq d}$ produces a $\mathsf{degrevlex}$ Gr\"obner basis of $\mathcal{F}$. We denote the solving degree of $\mathcal{F}$ by $d_{\mathrm{sol}}(\mathcal{F})$. We have to compute Macaulay matrices up to degree $d_{\mathrm{sol}}=d_{\mathrm{sol}}(\mathcal{F})$, and the largest one we encounter has $a=\sum_{i=1}^r{{N+d_{\mathrm{sol}}-d_i}\choose{d_{\mathrm{sol}}-d_i}}$ many rows and $b={{N+d_{\mathrm{sol}}}\choose{d_{\mathrm{sol}}}}$ many columns, where $d_i=\deg f_i$. Therefore, taking into account the complexity of Gaussian elimination of this matrix, an upper bound on the complexity of solving the system $\mathcal{F}$ with this method is \begin{equation}\label{eq:GBcomplexity} \OO\left({{N+d_{\mathrm{sol}}}\choose{d_{\mathrm{sol}}}}^\omega\right), \end{equation} with $2\leq\omega\leq3$. \begin{remark} If $\mathcal{F}$ is not homogeneous, Gaussian elimination on $M_{\leq d}$ may produce a row corresponding to a polynomial $f$ with $\deg f<d$, where the leading term of $f$ was not the leading term of any row in $M_{\leq d}$. Some algorithms, for example $F4$, address this by adding rows for polynomials $mf$ ($\deg(mf)\leq d$) for some monomial $m$ and recomputing the reduced row echelon form. If no Gr\"obner basis is found in degree $\leq d$, they proceed to higher degrees, potentially enlarging the span of $M_{\leq d}$ and reducing the solving degree. Throughout this paper, we consider only the case where no extra rows are added. Note that the solving degree as defined above is an upper bound on the degree at which algorithms using this variation terminate. \end{remark} Since the solving degree of a polynomial system may be difficult to estimate, several invariants related to the solving degree (that are hopefully easier to compute) have been introduced. One of the most important is the \emph{degree of regularity} introduced by Bardet, Faug\`ere, and Salvy \cite{bardet2004complexity}. We briefly recall its definition and connection with the solving degree. Let $\langle\mathcal{F}^{\mathrm{top}}\rangle=\langle f_1^{\mathrm{top}},\dots,f_r^{\mathrm{top}}\rangle$ be the ideal of the polynomial ring $R$ generated by the homogeneous part of highest degree of the polynomial system $\mathcal{F}$. Assume that $\langle\mathcal{F}^{\mathrm{top}}\rangle_d=R_d$ for $d\gg0$. The \emph{degree of regularity} of $\mathcal{F}$ is \begin{equation} \dreg{\mathcal{F}}=\min\{d\in\mathbb{N}\mid \langle\mathcal{F}^{\mathrm{top}}\rangle_e=R_e \ \forall e\geq d\}. \end{equation} The degree of regularity can be read off from the Hilbert series of $\langle\mathcal{F}^{\mathrm{top}}\rangle$. Let $I$ be a homogeneous ideal of $R$, and let $A=R/I$. For an integer $d\geq 0$, we denote by $A_d$ the homogeneous component of degree $d$ of $A$. The function $\mathrm{HF}_A(-):\mathbb{N}\rightarrow\mathbb{N}$, $\mathrm{HF}_A(d)=\dim_{\FF_q}A_d$ is called \emph{Hilbert function} of $A$. The generating series of $\mathrm{HF}_A$ is called \emph{Hilbert series} of $A$. We denote it by $\mathrm{HS}_A(z)=\sum_{d\in\mathbb{N}}\mathrm{HF}_A(d)z^d$. \begin{remark}\label{rem:polyHS} Under the assumption that $\langle\mathcal{F}^{\mathrm{top}}\rangle_d=R_d$ for $d\gg0$, the Hilbert series of $A=R/\langle\mathcal{F}^{\mathrm{top}}\rangle$ is a polynomial. Then, the degree of regularity of $\mathcal{F}$ is given by $\dreg{\mathcal{F}}=\deg \mathrm{HS}_A(z)+1$ (see \cite[Theorem~12]{2021/caminatagorla}). \end{remark} \noindent Under suitable assumptions, the degree of regularity provides an upper bound for the solving degree \cite{CaminataG23, 2023/salizzoni, Semaev2021651}. Moreover, it is often assumed that the two values are close. Although this occurs in many relevant situations, there are examples where these two invariants can be arbitrarily far apart (see \cite{2021/caminatagorla, 2013/dingschmidt, Bigdeli202175}). We will see in Section~\ref{sec:dreg-EWM} that the degree of regularity of the system presented in Section~\ref{subsec:f2ESD} seems to yield a much higher value than the solving degree achieved during the Gr\"obner basis algorithm. \section{The MPS Modeling}\label{sec:mps} This section is devoted to an overview of the algebraic modeling of the syndrome decoding problem proposed in~\cite{2021/meneghetti} (referred to as the MPS modeling). We fix the following notation for this section. \begin{notation}\label{MPSnotation} Let $n\ge 2$ and let $\CC \subseteq \FF_2^n$ be a $[n,k,d]$-linear code having a parity check matrix $\HH \in \FF_2^{(n-k) \times n}$. We define $\ell = \lfloor \log_2(n) \rfloor + 1$. Let $\vs \in \FF_2^{n-k}$ play the role of the syndrome and let $0\le t \le \lfloor (d-1)/2 \rfloor$ be the target error weight. Let $X = \left(x_1,\ldots,x_n\right)$ and $Y=(Y_1,\dots,Y_n)$ with $Y_j=(y_{j,1}, \dots, y_{j,\ell})$ be two sets of variables and we consider the polynomial ring $\FF_2[X,Y]$. \end{notation} We define the following maps $\pi_i$ for $i=1,\ldots,n$, \begin{align} \pi_i : \FFt^{n} &\rightarrow \FFt^i \\ (v_1,\ldots,v_n) &\mapsto (v_1,\ldots,v_i). \end{align} The construction of the proposed algebraic modeling consists of four steps and uses the variables contained in $X$ and $Y$ to express relations and dependencies. Each of these steps produces a set of polynomials in $\FF_2[X,Y]$. An extra step of the construction reduces the aforementioned polynomials to quadratic polynomials. The idea is to construct an algebraic system having a variety containing elements $(\vx \mid \vy_1 \mid \cdots \mid \vy_n)\in \FFt^{n(\ell + 1)}$ whose first $n$ entries represent an element $\vx$ of $\FFt^n$ such that $\HH\vx^\top = \vs^\top$. The remaining $n\ell$ entries are considered to be the concatenation of $n$ elements $\vy_i \in \FFt^{\ell}$ where the elements of $\vy_i$ represent the binary expansion of $\wt(\pi_i(\vx))$ for every $i=1,\ldots,n$, with $\pi_i(\vx)=(x_1,\dots,x_i)$. By this definition, the list $\vy_n$ represents the binary expansion of $\wt(\vx)$. The system finally enforces that $\vy_n$ represents the binary expansion of an integer $t^\prime$ such that $t^\prime \le t$. The elements of the variety of solutions of this algebraic modeling are finally projected onto their first $n$ coordinates, revealing the solutions to the original syndrome decoding problem. Here is a description of the four steps of reduction of the MPS modeling. We describe the set obtained in each step as a set of polynomials in $\FFt[X,Y]$. \begin{itemize} \item \textit{Parity check encoding.} This step ensures that the solution of the algebraic system satisfies the parity check equations imposed by the parity check matrix $\HH$ and the syndrome vector $\vs$. Here, we compute the set of $n-k$ linear polynomials \begin{equation}\label{eq:pce} \left\{\sum_{i=1}^n h_{i,j}x_i + s_j \mid j\in\{1,\ldots,n-k\}\right\}. \end{equation} \item \textit{Hamming weight computation encoding.} This part of the modeling provides a set of polynomials that describes the binary encoding of $\wt(\pi_i(\vx))$ for every $i=1,\ldots,n$ described above. The set of polynomials achieving this goal, is given by the union of the three following sets consisting of the $\ell+n-1$ polynomials in the sets \begin{equation} \begin{split}\label{eq:lineareqs} &\left\{ f_{1,1}=x_1 + y_{1,1}, f_{1,2}=y_{1,2}, \ldots, f_{1,\ell}=y_{1,\ell} \right\},\\ &\left\{f_{i,1}=x_i + y_{i, 1} + y_{i-1,1} \mid i=2,\ldots,n \right\} \end{split} \end{equation} and the $(n-1)(\ell -1)$ polynomials \begin{equation}\label{eq:othereqs} \left\{ f_{i,j}=\left(\prod_{h=1}^{j-1}y_{i-1, h}\right)x_i + y_{i,j} + y_{i-1,j} \mid i=2,\ldots,n,\ j=2,\ldots,\ell \right\}. \end{equation} We labeled the polynomials of the sets in~\eqref{eq:lineareqs} and in~\eqref{eq:othereqs} because the improvements in the next sections will mainly involve them. \item \textit{Weight constraint encoding.} This part produces a set consisting of a single polynomial that enforces the constraint $\wt(\vx) \le t$ by dealing with the variables in $Y_n$. Let $\vv \in \FFt^\ell$ represent the binary expansion of $t$. Consider the $\ell$ polynomials in $\FFt[X,Y]$ defined as $$f_j = (y_{n, j} +v_j)\prod_{h=j+1}^\ell (y_{n, h} + v_h + 1) $$ for $j=1,\ldots,\ell$. The set is the singleton \begin{equation}\label{eq:MPSwce} \left\{ \sum_{j=1}^\ell (v_j + 1)f_j \right\}. \end{equation} \item \textit{Finite field equations.} The set of $n + n\ell$ finite field polynomials of $\FFt[X,Y]$ is \begin{equation} \label{eq:ffe} \left\{x_i^2- x_i \mid i=1,\ldots,n\right\} \cup \left\{y_{i,j}^2- y_{i,j} \mid i=1,\ldots,n,\ j=1,\ldots,\ell\right\}, \end{equation} and ensures that the elements of the variety are restricted to elements of $\FFt^{n(\ell + 1)}$. \end{itemize} The algebraic system corresponding to an instance of the syndrome decoding problem is then the union of the four sets described above. Clearly, this is not a quadratic system; thus the authors apply a linearization strategy that introduces a number of auxiliary variables used to label monomials of degree $2$. This eventually results in a large quadratic system in many more than just $n(\ell + 1)$ variables. In fact, the final quadratic system ends up having equations and variables bounded by $\OO(n\log_2(n)^2)$. \section{Improving the MPS Modeling}\label{sec:EWM} In this section, we provide improvements of the MPS modeling that reduce the number of equations and variables in the final algebraic system. We keep the same notation as in Notation~\ref{MPSnotation}. First, we consider the case of the syndrome decoding problem, i.e. with a bounded weight error. We then consider the case of the exact weight syndrome decoding problem. We observe that one can avoid the linearization step as the resulting system is already quadratic. \subsection{Improved Modeling for the Case of SDP}\label{subsec:f2SD} We consider the $\mathsf{degrevlex}$ monomial ordering on $\FFt[X,Y]$ with the $X$ variables greater than the $Y$ variables, and denote by $\lm(p)$ the leading monomial of a polynomial $p$. Notice that since we are in the binary case, the notions of leading monomial and that of leading term coincide. Denote by $F = \{f_{i,j} \mid i=1,\ldots,n,\ j=1,\ldots,\ell\} \subset \FFt[X,Y]$ the set of polynomials of cardinality $n\ell$ given by \eqref{eq:lineareqs} and \eqref{eq:othereqs} for a code of length $n$. We aim at building a set $G=\{g_{i,j} \mid i=1,\ldots,n,\ j=1,\ldots,\ell\}\subset \FFt[X,Y]$ consisting of polynomials of degree at most $2$ such that $\langle G \rangle = \langle F \rangle$. Denote with $F[i,j]$ the polynomial $f_{i,j}$, similarly for $G$. We first give a description of the set $G$ and then formally describe the new modeling. Construct $G$ as follows: \begin{itemize} \item Put $G[1,1] = x_1 + y_{1,1}$ and $G[1,h] = y_{1,h}$ for $h = 2,\ldots, \ell$; \item Set $G[i,1] = F[i,1] = x_i + y_{i, 1} + y_{i-1,1}$ for every $i = 2,\ldots,n$; \item Compute \begin{align} G[i,j] &= F[i,j] + y_{i-1, j-1}F[i,j-1]\\ &= F[i,j] + \lm(F[i,j]) + y_{i-1, j-1}(y_{i,j-1} + y_{i-1,j-1})\\ &= y_{i,j} + y_{i-1,j} + y_{i-1,j-1}^2 + y_{i,j-1}y_{i-1,j-1}. \end{align} for every $i=2,\ldots,n$ and $j = 2,\ldots,\ell$, where equality holds because $\lm(F[i,j]) = y_{i-1,j-1}\lm(F[i,j-1])$. \end{itemize} \begin{remark} The algebraic system we are going to construct contains the field polynomials $x_i^2- x_i$ for each $i=1,\ldots,n$ and $y_{i,j}^2- y_{i,j}$ for every $i=1,\ldots,n$ and $j=1,\ldots,\ell$. Therefore, in terms of generating elements of the ideal, any squared term in $G[i,j]$ can be reduced to a linear term. \end{remark} The set $G \subset \FFt[X,Y] $ contains $n\ell$ polynomials of degree at most two. The following proposition proves that the set $G \subset \FFt[X,Y]$ computed as above and $F$ generate the same ideal of $\FFt[X,Y]$. \begin{proposition} We have $\langle G \rangle = \langle F \rangle$. \end{proposition} \begin{proof} The inclusion $\langle G \rangle \subseteq\langle F \rangle$ is trivial. To prove the other inclusion, we show that we can write any element of the basis $F$ as an $\FFt[X,Y]$-linear combination of elements of the basis $G$. By construction, $G[1,j] = F[1,j]$ for every $j=1,\ldots,\ell$. For every $i = 2,\ldots,n$ we prove $F[i,j]\in \langle G \rangle$ by induction on $j$.\\ For $j=1$ we have $F[i,1] = G[i,1]$.\\ Assume that $F[i,j] = \sum_{h=1}^j p_{i,j,h} G[i,h]$ with $p_{i,j,h}\in \FFt[X,Y]$. Then by construction we have \begin{align} F[i,j+1] &= G[i,j+1] - y_{i-1, j}F[i,j]\\ &= G[i,j+1] - y_{i-1, j} \sum_{h=1}^j p_{i,j,h} G[i,h] \end{align} proving the claim. \qed \end{proof} We thus redefine the Hamming weight computation encoding as follows: \begin{itemize} \item \textit{Hamming weight computation encoding.} Compute the following union of subsets of $\FFt[X,Y]$: \begin{align} &\left\{ x_1 + y_{1,1}, y_{1,2}, \ldots, y_{1,\ell} \right\} \cup \left\{x_i + y_{i, 1} + y_{i-1,1} \mid i=2,\ldots,n \right\}\\ &\cup \big\{ y_{i,j-1}y_{i-1,j-1} + y_{i,j} + y_{i-1,j-1} + y_{i-1,j} \\ & \ \ \ \mid i=2,\ldots,n,\ j=2,\ldots,\ell \big\}, \end{align} \end{itemize} \subsubsection{Further improvement.} Set now $\ell_t = \lfloor \log_2 (t) \rfloor + 1$. A further improvement to the MPS modeling (described in Equation~\eqref{eq:SDhwce}) follows by observing that in the non-trivial case where $t < n$, we can impose that the last $\ell-\ell_t$ entries of $\vy_i$ must be $0$ for every $i=1,\ldots,n$. This means that we can add the linear equations $y_{i, j} = 0$ for every $i=1,\ldots,n$ and $j=\ell_t+1,\ldots,\ell$. By inspection, setting the aforementioned variables to $0$ will make part of the equations of the Hamming weight computation encoding vanish. We can equivalently simply consider the equations that remain, and get rid of the variables which have been set to $0$. Consider the following updated notation. \begin{notation}\label{ImprovedMPSnotation} Let $n\ge 2$ and let $\CC \subseteq \FF_2^n$ be a $[n,k,d]$-linear code having a parity check matrix $\HH \in \FF_2^{(n-k) \times n}$. Let $\vs \in \FF_2^{n-k}$ play the role of the syndrome and let $0\le t \le \lfloor (d-1)/2 \rfloor$ be the target error weight. We define $\ell_t = \lfloor \log_2(t) \rfloor + 1$. Let $X = \left(x_1,\ldots,x_n\right)$ and $Y=(Y_1,\dots,Y_n)$ with $Y_j=(y_{j,1}, \dots, y_{j,\ell_t})$ be two sets of variables and consider the polynomial ring $\FF_2[X,Y]$. \end{notation} Under Notation~\ref{ImprovedMPSnotation}, the effect of our improvement on the set of polynomials produced by the Hamming weight computation encoding is the following. \begin{itemize} \item \textit{Hamming weight computation encoding.} Compute the following union of subsets of $\FFt[X,Y]$: \begin{equation}\label{eq:SDhwce} \begin{split} &\left\{ x_1 + y_{1,1}, y_{1,2}, \ldots, y_{1,\ell_t} \right\} \cup \left\{x_i + y_{i, 1} + y_{i-1,1} \mid i=2,\ldots,n \right\}\\ &\cup \big\{ y_{i,j-1}y_{i-1,j-1} + y_{i,j} + y_{i-1,j-1} + y_{i-1,j} \\ & \ \ \ \mid i=2,\ldots,n,\ j=2,\ldots,\ell_t \big\} \cup \left\{ y_{i,\ell_t}y_{i-1,\ell_t} + y_{i-1,\ell_t} \mid i=2,\ldots,n\right\}. \end{split} \end{equation} \end{itemize} The effect on the weight constraint encoding is simply the decrease in the degree from $\ell$ to $\ell_t$ of the produced polynomial. This is the only non-quadratic polynomial left in the modeling. We can turn this polynomial into a set of $\OO(t\ell_t)$ polynomials of degree up to $2$ in $\OO(t\ell_t)$ variables with the same linearization techniques described in~\cite[Fact 1 and Lemma 11]{2021/meneghetti}. To summarize, our modeling is defined in the following way. \begin{modeling}[Improved Modeling for the SDP over $\FF_2$] \label{modeling: improvedSD_F2} Given an instance $(\HH,\mathbf{s},t)$ of Problem~\ref{BSDP} over $\FF_2$, Modeling~\ref{modeling: improvedSD_F2} is the union of the sets of polynomials \eqref{eq:pce},\eqref{eq:MPSwce}, \eqref{eq:ffe} and \eqref{eq:SDhwce}. \end{modeling} The improved modeling is an algebraic system of $\OO(n(\ell_t+2) -k + t\ell_t)$ polynomials of degree at most $2$ in $\OO(n(\ell_t+1) + t\ell_t)$ variables. Note that most applications of the SDP to code-based cryptography, for instance in the McEliece scheme, choose $t \ll n$, hence the asymptotic bounds on the number of polynomials and variables in the improved modeling are both $\OO(n\ell_t)$. As shown in Table \ref{table: improvement}, our modeling improves over MPS by a factor of $\log_2(n) \log_t(n)$. \begin{table}[H] \centering \begin{tabular}{\|c\|c\|c\|} \hline & \# Polynomials & \# Variables\\ \hline \cite{2021/meneghetti} & $\mathcal{O}( n \log_2(n)^2)$ & $\mathcal{O}( n \log_2(n)^2)$ \\ \hline Modeling~\ref{modeling: improvedSD_F2} & $\OO(n\log_2(t))$ & $\OO(n\log_2(t))$\\ \hline \end{tabular} \vspace{2mm} \caption{Comparison with the asymptotic size of the polynomial system in \cite[Theorem 13]{2021/meneghetti}, where $n$ is the length of the code and $t$ the bound on the weight of the target vector, that is $\wt(\ve)\leq t$.} \label{table: improvement} \end{table} \subsection{Improved Modeling for the Case of ESDP}\label{subsec:f2ESD} It is possible to obtain an algebraic modeling for the ESDP by tweaking the modeling described in the previous section. In fact, it is enough to redefine the weight constraint encoding to enforce that $\vy_n$ represents the binary expansion of an integer $t^\prime$ such that $t^\prime=t$ exactly. To this end, let $\vv \in \FFt^{\ell_t}$ represent the binary expansion of an integer $t$. Under the same notation as in Notation~\ref{ImprovedMPSnotation}, the following version of the weight constraint encoding describes the ESDP modeling with $\wt(\ve) = t$. \begin{itemize} \item \textit{Weight constraint encoding.} Compute the following set of linear polynomials: \begin{equation}\label{eq:ESDwce} \left\{ y_{n, j} + v_j \mid j=1,\ldots,\ell_t \right\}. \end{equation} \end{itemize} Using these polynomials leads to Modeling \begin{modeling}[Improved Modeling for the ESDP over $\FF_2$] \label{modeling: improvedESD_F2} Given an instance $(\HH,\mathbf{s},t)$ of Problem~\ref{EWSDP} over $\FF_2$, Modeling~\ref{modeling: improvedESD_F2} is the union of the sets of polynomials \eqref{eq:pce}, \eqref{eq:ffe}, \eqref{eq:SDhwce} and \eqref{eq:ESDwce}. \end{modeling} Observe that, replacing the original Hamming weight computation encoding with that in~\eqref{eq:SDhwce} and the weight constraint encoding with that in~\eqref{eq:ESDwce}, we obtain an algebraic system of polynomials of degree at most $2$ for ESDP. Hence, linearization is not needed, moreover, we can give the exact number of equations and variables of this system. We report these values in Table~\ref{table:esd-model-sizes}. \begin{table}[H] \centering \begin{tabular}{\|c\|c\|c\|} \hline & \# Polynomials & \# Variables\\ \hline Modeling~\ref{modeling: improvedESD_F2} & $2n\ell_t + 3n + \ell_t - k - 1$ & $n(\ell_t + 1)$\\ \hline \end{tabular} \vspace{2mm} \caption{Number of equations and variables of the algebraic modeling of ESDP with $\wt(\ve)=t$. The value of $\ell_t$ is $\lfloor \log_2(t) \rfloor + 1$.} \label{table:esd-model-sizes} \end{table} \section{Complexity Analysis of Modeling~\ref{modeling: improvedESD_F2}}\label{sec:complexity-analysis} \label{sec:dreg-EWM} In this section, we investigate the complexity of solving the algebraic system for the ESDP given in Modeling~\ref{modeling: improvedESD_F2} using standard Gröbner basis methods. An upper bound on the complexity is given by the formula \eqref{eq:GBcomplexity} which depends on both the number of variables and the solving degree. Typically, the solving degree of the system is estimated by assessing its degree of regularity. However, in our analysis, we experimentally show that the degree of regularity often significantly exceeds the solving degree for systems given in Section~\ref{subsec:f2ESD} (see the results in Table~\ref{Tab:q2-SolveDeg}). This distinction is crucial in cryptography, where these concepts are frequently used interchangeably. Our findings underscore the importance of thoroughly verifying such claims to ensure accurate security assessments and parameter selection. \begin{remark} We point out that the study in \cite{2023/briaud} investigates a particular case of the problem that this paper deals with, that is the \emph{regular} syndrome decoding problem. The regular syndrome decoding problem considers error vectors having a regular distribution of non-zero entries. The algebraic modeling proposed in~\cite{2023/briaud} is conjectured to exhibit semi-regular behavior when the linear parity-check constraints and the fixed, structured quadratic polynomials are considered separately. This suggests that, to some extent, their model behaves like a random polynomial system. Despite the fact that the problem tackled in~\cite{2023/briaud} is a particular case of the problem we consider, our modeling has not been devised as a generalization of their modeling. Furthermore, we show that for the more general case, our modeling yields different results. \end{remark} For the rest of this section, we retain the notation defined in Notation~\ref{ImprovedMPSnotation}. We consider the polynomial ring $\FFt[X,Y]$ with the $\mathsf{degrevlex}$ term order with the $X$ variables greater than the $Y$ variables. Let $S \subset \FFt[X,Y]$ be the set of polynomials of Modeling~\ref{modeling: improvedESD_F2} as described in Section~\ref{subsec:f2ESD}. Let $L$ and $Q$ denote the sets of linear and quadratic polynomials, respectively. Clearly $S = L \cup Q$. Write also $L = L_\vH \cup P$, where $L_\vH$ denotes the set of linear polynomials in~\eqref{eq:pce} introduced with the parity check matrix $\vH$, and $P$ denotes the remaining linear polynomials in $S$. In other words, $P$ is the following set \[\begin{split} P = &\left\{ x_1 + y_{1,1}, y_{1,2}, \ldots, y_{1,\ell_t} \right\} \cup \left\{x_i + y_{i, 1} + y_{i-1,1} \mid i=2,\ldots,n \right\} \\ \cup &\left\{ y_{n, j} + v_j \mid j=1,\ldots,\ell_t \right\}. \end{split} \] We want to estimate the degree of regularity of $S$. Since we do not know $L_\vH$ a priori, we consider the set $S\setminus L_\vH = Q \cup P$ and compute its degree of regularity. Indeed, we found that analyzing the degree of regularity or solving degree of the system with the linear equations \eqref{eq:pce} of $L_\vH$ included was too challenging and unpredictable, as it heavily depends on the specific instance of the parity check matrix $\vH$. For this reason, we chose to establish mathematical results for the system without $L_{\vH}$, with the aim of providing a clearer foundation. Notice that the degree of regularity of $S\setminus L_\vH = Q \cup P$ gives an upper bound to the degree of regularity of the whole system $S$ (see Remark~\ref{rem:range fordregS}). We break down the problem by first computing the degree of regularity of $Q$ and then that of $Q \cup P$. We take advantage of the fact that the Hilbert series of $Q$ and of $Q \cup P$ are polynomials and compute their degree, i.e. for instance, $\dreg{Q}=\deg \mathrm{HS}_{\FFt[X,Y]/\langle Q^\htop\rangle}(z)+1$ as per Remark~\ref{rem:polyHS}, similarly for $Q\cup P$. To this end, we are going to compute the maximum degree of a monomial in $\FFt[X,Y]/\langle Q^\htop\rangle$, similarly we do for $Q \cup P$. \subsubsection{The quadratic polynomials.}\label{subsec:quad-polys} We begin by studying the degree of regularity of the quadratic part $Q$ of the system $S$ of Modeling~\ref{modeling: improvedESD_F2}. The highest degree part of $Q$ has a very nice structure, as explained in the following remark. \begin{remark}\label{rem:qtopdef} The set $Q^\htop$ is the union of the following three sets $$\left\{x_i^2 \mid i=1,\ldots,n\right\}, \left\{y_{i,j}^2 \mid i=1,\ldots,n,\ j=1,\ldots,\ell_t\right\}$$ and $$\left\{ y_{i-1,j}y_{i,j} \mid i=2,\ldots,n,\ j=1,\ldots,\ell_t \right\}.$$ The ideal $\langle Q^\htop \rangle \subseteq \FFt[X,Y]$ is thus a monomial ideal. \end{remark} The following lemma gives the structure of the quotient ring $\FFt[X,Y]/\langle Q^\htop \rangle$. \begin{lemma}\label{lem:groebnerQh} The set $Q^\htop$ is a Gr\"obner basis of the ideal $\langle Q^\htop\rangle$. \end{lemma} \begin{proof} As observed in Remark~\ref{rem:qtopdef}, $Q^\htop$ is a monomial ideal. Given any two elements of $m_1,m_2 \in Q^\htop$ it is clear that for $a = \lcm (m_1,m_2)/m_1 \in \FFt[X,Y]$ and $b = \lcm (m_1,m_2)/m_2 \in \FFt[X,Y]$ we have that $am_1 - bm_2 = 0$. \qed \end{proof} \ifodd0 We can exploit the knowledge of the Gr\"obner basis of $\langle Q^\htop \rangle$ given in Lemma \ref{lem:groebnerQh} to compute the coefficients of the Hilbert series $\mathcal{H}_R$. The $(k+1)$-th coefficient of $\mathcal{H}_R$ is given by $\dim_{\FFq}(\FFt[X,Y]_k/I_k)$, in other words, the number of monomials of degree $k$ in $R$. This coincides with the number of monomials of $\FFt[X,Y]$ of degree $k$ that are not a multiple of any monomial in $\GG$. We can model this problem in terms of subsets of $[n(l+1)]$, or equivalently, elements of $2^{[n(l+1)]}$. Let $B_1,\ldots B_{n\ell -n-\ell +1}$ be the sets of two elements indexing the variables of each mixed monomial in $\GG$ (monomials in the third set). Counting monomials of degree $k$ in $R$ boils down to counting the number of subsets of $[n(l+1)]$ of cardinality $k$ not containing any $B_i$. \begin{example}\label{ex:n4} Let $n=4$ be the length of a code, then $\ell_t = 2$. A Gr\"obner basis of $\langle Q^\htop \rangle$ is the union of \begin{equation} \left\{ y_{1,1}y_{2,1}, y_{1,2}y_{2,2}, y_{2,1}y_{3,1}, y_{2,2}y_{3,2}, y_{3,1}y_{4,1}, y_{3,2}y_{4,2}\right\} \end{equation} and \begin{equation} \left\{ x_{1}^2, x_{2}^2, x_{3}^2, x_{4}^2, y_{1,1}^2, y_{1,2}^2, y_{2,1}^2, y_{2,2}^2, y_{3,1}^2, y_{3,2}^2, y_{4,1}^2, y_{4,2}^2 \right\}. \end{equation} \ifodd0 Following our argument we obtain the $(n-1)\cdot(l-1) = n\ell -n-\ell+1 = 6$ sets $B_i$, indexing mixed monomials, are \begin{align} B_1 = \{1,4\},&B_2 = \{4,7\},B_3 = \{7,11\},\\ B_4 = \{2,5\},&B_5 = \{5,8\},B_6 = \{8,11\}. \end{align} \end{example} \noindent The following simple lemma is crucial for computing the degree of regularity of $Q$. For the sake of simplicity, we state it in terms of sets, and it ultimately provides a method to construct maximal monomials in the quotient ring $\FFt[X,Y]/\langle Q^\htop \rangle$. \begin{lemma}\label{lem:maximalset} Let $ \mathcal{N} = \{1, 2, 3, \dots, n\} $ and $ \mathcal{P} = \{\{1,2\}, \{2,3\}, \dots, \{n-1, n\}\} $, where $ \mathcal{P} $ consists of consecutive pairs of elements from $ \mathcal{N} $. Then: \begin{itemize} \item If $ n $ is even, there are exactly two sets of maximal cardinality $ \mathcal{S}_1, \mathcal{S}_2 \subseteq \mathcal{N} $ such that no set in $ \mathcal{P} $ is a subset of $ \mathcal{S} $. \item If $ n $ is odd, there is exactly one set of maximal cardinality $ \mathcal{S} \subseteq \mathcal{N} $ such that no set in $ \mathcal{P} $ is a subset of $ \mathcal{S} $. \end{itemize} \end{lemma} \begin{proof} We aim to find the number of sets of maximal cardinality $ \mathcal{S} \subseteq \mathcal{N} $ such that no pair from $ \mathcal{P} $ (i.e., no two consecutive elements) appears in $ \mathcal{S} $. In order to avoid pairs of consecutive elements, we can only select non-consecutive elements from $ \mathcal{N} $. To maximize the size of $ \mathcal{S} $, we select every other element from $ \mathcal{N} $. The size of such a set of maximal cardinality $ \mathcal{S} $ is: $\left\lceil \frac{n}{2} \right\rceil$. Thus: \begin{itemize} \item If $ n $ is even, a set of maximal cardinality contains $ \frac{n}{2} $ elements. \item If $ n $ is odd, a set of maximal cardinality contains $ \frac{n+1}{2} $ elements. \end{itemize} \textbf{Case 1: $ n $ is even.} Let $ n = 2k $. The largest possible set $ \mathcal{S} $ will contain $ k = \frac{n}{2} $ elements. There are exactly two ways to construct such a set: \begin{enumerate} \item Start with 1 and select every other element: $\mathcal{S}_1 = \{1, 3, 5, \dots, n-1\}.$ This set contains all the odd-numbered elements of $ \mathcal{N} $, and its size is $ k $. \item Start with 2 and select every other element: $\mathcal{S}_2 = \{2, 4, 6, \dots, n\}.$ This set contains all the even-numbered elements of $ \mathcal{N} $, and its size is also $ k $. \end{enumerate} Since there are no other ways to select $ k $ elements without picking consecutive elements, these are the only two sets of maximal cardinality for $ n $ even.\\ \textbf{Case 2: $ n $ is odd.} Let $ n = 2k + 1 $. The largest possible set $ \mathcal{S} $ contains $ k + 1 = \frac{n+1}{2} $ elements. In this case, there is only one way to construct a set of size $ k + 1 $ that avoids consecutive elements, i.e. start with 1 and select every other element: $\mathcal{S}_1 = \{1, 3, 5, \dots, n\}.$ This set contains $ k + 1 $ elements and avoids consecutive pairs. If we were to start with 2 and select every other element, we would only get $ k $ elements: $\mathcal{S}_2 = \{2, 4, 6, \dots, n-1\}.$ This is not maximal, as it contains fewer than $ k + 1 $ elements. Thus, for $ n $ odd, there is exactly one maximal set. \qed \end{proof} Lemma~\ref{lem:maximalset} can be used to prove the following corollary, which we will use to construct a maximal degree monomial in $\FFt[X,Y]/\langle Q^\htop \rangle$. The idea behind the construction lies in the observation that a Gr\"obner basis of $Q^\htop$ can be written as the union of disjoint subsets $Q^\htop_{j,n}$ for $j=1,\ldots,\ell_t$, see Theorem~\ref{Thm:Dreg-of-Qtop}, which we describe in the next corollary. Also, the next corollary computes a maximal degree monomial with respect to $Q^\htop_{j,n}$ for every $j=1,\ldots,\ell_t$. Given these monomials, computing a maximal degree monomial in $\FFt[X,Y]/\langle Q^\htop \cup P^\htop\rangle$, or equivalently, the degree of its Hilbert series, becomes feasible with a slight modification of the subsets due to the presence of linear polynomials in $P^\htop$. \begin{corollary}\label{cor:maximalmonomial} Let $n\in \mathbb{N}$ with $n\ge 2$, and define $$Q^\htop_{j,n} := \left\{ y_{1,j}y_{2,j}, y_{2,j}y_{3,j}, \ldots, y_{n-1,j}y_{n,j}\right\} \cup \left\{y_{i,j}^2 \mid i=1,\ldots,n\right\} \subset \FFt[y_{1,j},\ldots,y_{n,j}],$$ for some $j\in \mathbb{N}$. If $n$ is even then there exists two monomials of maximal degree $\left\lceil\frac{n}{2} \right\rceil$ in $\FFt[y_{1,j},\ldots,y_{n,j}]/\langle Q^\htop_{j,n} \rangle$, namely \[ m_1 = \prod_{\substack{i=1,\ldots,n-1,\\ i\ \text{odd}}}y_{i,j} \quad \textnormal{and}\quad m_2 =\prod_{\substack{i=2,\ldots,n,\\ i\ \text{even}}}y_{i,j}. \] If $n$ is odd, then there exists a unique monomial of maximal degree $\left\lceil\frac{n}{2} \right\rceil$ in $\FFt[y_{1,j},\ldots,y_{n,j}]/\langle Q^\htop_{j,n} \rangle$, namely \[ m = \prod_{\substack{i=1,\ldots,n,\\ i\ \text{odd}}}y_{i,j}. \] \end{corollary} \noindent We are ready to prove the following theorem, which provides the degree of regularity of $Q$. \begin{theorem}\label{Thm:Dreg-of-Qtop} $$\dreg{Q}= \begin{cases} n + \ell_t n/2 + 1 \quad &\text{ if } n \equiv 0 \bmod 2\\ n + \ell_t(n+1)/2 + 1 \quad &\text{ if } n \equiv 1 \bmod 2 \end{cases}.$$ Equivalently, $$\dreg{Q} = n + \ell_t\lceil n/2 \rceil + 1.$$ \end{theorem} \begin{proof} Let $Q^\htop_{j,n} \subset \FFt[y_{1,j},\ldots,y_{n,j}]$ as in Corollary~\ref{cor:maximalmonomial}, for every $j=1,\ldots,\ell_t$. Observe that \begin{equation}\label{eq:qtopasunion} Q^\htop = \bigcup_{j=1}^{\ell_t} Q^\htop_{j,n} \cup \left\{x_i^2 \mid i=1,\ldots,n\right\}. \end{equation} Corollary~\ref{cor:maximalmonomial} computes a monomial $m_j \in \FFt[y_{1,j},\ldots,y_{n,j}]$ of maximal degree $\lceil n/2 \rceil$ such that $m_j \not \in \langle Q^\htop_h\rangle$ for every $j=1,\ldots,\ell_t$ and every $h=1,\ldots,\ell_t$. This implies that $m_j \not \in \langle Q^\htop \rangle$ for every $j$. It is now clear that the monomial \[ m:= \prod_{i=1}^n x_i \prod_{j=1}^{\ell_t}m_j \in \FFt[X,Y] \] is such that $m \not \in \langle Q^\htop \rangle$. Note that the the set $\left\{x_i^2 \mid i=1,\ldots,n\right\}$ in \eqref{eq:qtopasunion} enforces that $m$ must be squarefree in the variables $x_1,\ldots,x_n$. By the maximality of each $m_j$ and that of $\prod_{i=1}^n x_i$, any multiple of $m$ by a non-constant term would trivially be in $\langle Q^\htop \rangle$. Since $$d:=\deg m = n + \ell_t\lceil n/2 \rceil,$$ we have that the $(d+1)$-th coefficient of the Hilbert series of $\FFt[X,Y]/\langle Q^\htop \rangle$ is $0$. The result on the degree of regularity $\dreg{Q}$ follows. \qed \end{proof} \ifodd0 \begin{theorem}\label{Thm:Dreg-of-Qtop} $$\dreg{Q}= \begin{cases} 2n + (\ell-1)n/2 + 1 \quad &\text{ if } n \equiv 0 \bmod 2\\ 2n + (\ell-1)(n+1)/2 + 1 \quad &\text{ if } n \equiv 1 \bmod 2 \end{cases}.$$ Equivalently, $$\dreg{Q} = 2n + (\ell-1)\lceil n/2 \rceil + 1.$$ \end{theorem} \begin{proof} We set $R=\FFt[X,Y]/\langle Q^\htop \rangle$. We show that the maximum degree of a monomial of $R$ is $d := 2n + (l-1)\lceil n/2 \rceil$ via a constructive argument and show that any monomial of degree $d+1$ lives in $\langle Q^\htop \rangle$. From Lemma~\ref{lem:groebnerQh} and Remark~\ref{rem:qtopdef} a Gr\"obner basis of $\langle Q^\htop \rangle$ is $Q^\htop$ itself, i.e. the union of three sets of quadratic monomials, the first two being the squares of all the variables $\vx$ and $\vy$ and the third being the set of mixed monomials $$\left\{ y_{(i-1)\ell+j-1}y_{(i-2)\ell+j-1} \mid i=2,\ldots,n,\ j=2,\ldots,\ell\right\}.$$ According to the first two sets, to construct a monomial of $R$ we need to build a square free monomial, i.e. which is at most linear in each variable. The third set imposes restrictions on the variables that can be chosen to build the monomial. Assume that $n \equiv 0 \bmod 2$, the case with $n$ odd can be proven similarly, by a different choice of variables. Consider the set of variables $$V_1 := \left\{y_{(h(\ell-1)-2)\ell +j - 1} \mid h=1,\ldots,n/2,\ j=2,\ldots,\ell\right\}$$ and let $M_1:= \prod_{v\in V_1}v$. Furthermore, none of the variables in the set $$V_2 := \left\{x_i,y_{i\ell} \mid i=1,\ldots,n\right\}$$ appears in a mixed monomial. Let $M_2 := \prod_{v\in V_2}v$. Then $M := M_1M_2$ is a square-free monomial in $R$ of degree $d$. It is easy to verify that $M$ is not a multiple of any of the mixed monomials in the Gr\"obner basis of $\langle Q^\htop \rangle$ and that $d$ is maximal. In addition, any monomial of maximal degree can be constructed similarly. Indeed, any multiple of $M$ is also the multiple of a monomial in the Gr\"obner basis of $\langle Q^\htop \rangle$. To see this, note that, on the one hand, if we take a multiple of $yM$ with $v\| M$, then $M$ is no longer square-free and thus $v^2 \| M$ with $v^2$ lying on the Gr\"obner basis of $\langle Q^\htop \rangle$. On the other hand, the variables that do not appear in $M$ are exactly those in the set $$V_3 := \left\{y_{(h(\ell-1)-1)\ell +j - 1} \mid h=1,\ldots,n/2,\ j=2,\ldots,\ell\right\}.$$ Let $v \in V_3$, and consider the monomial $vM$. We can write $v = y_{(h(\ell-1)-1)\ell +j - 1}$ for some $h\in [1,n/2]$ and $j=2,\ldots,\ell$. By construction of $M$, we have that $vw \| vM$ where $w = y_{(h(\ell-1)-2)\ell +j - 1}$. Setting $i = h(\ell -1)$ we find that the product $vw = y_{(i-1)\ell+j-1}y_{(i-2)\ell+j-1}$ is a mixed monomial on the Gr"obner basis of $\langle Q^\htop \rangle$ meaning that $vM \equiv 0$ in $R$. Finally, notice that any divisor of degree $d' < d$ of $M$ is a nonzero monomial of $R$. This proves that the $d$-th coefficient of the Hilbert series of $R$ is positive and that, by maximality of $d$, the $(d+1)$-th coefficient is the first $0$ coefficient of the series, proving our claim. \qed \end{proof} \begin{example} Let $n=8$ and $\ell_t=3$. According to Theorem~\ref{Thm:Dreg-of-Qtop} the degree of regularity of $ Q$ is $21$. \noindent Using MAGMA, we compute and report the Hilbert series of the quotient ring $\FFt[X,Y]/\langle Q^\htop\rangle$, i.e. \begin{align} \mathrm{HS}_{\FFt[X,Y]/\langle Q^\htop\rangle} (z) =&\ 125z^{20} + 2500z^{19} + 23075z^{18} + 130800z^{17} + \\ &\ 511140z^{16} + 1465020z^{15} + 3198081z^{14} +\\ &\ 5448312z^{13} + 7360635z^{12} + 7966528z^{11} + \\ &\ 6946904z^{10} + 4889800z^9 + 2773415z^8 + \\ &\ 1260580z^7 + 454625z^6 + 128080z^5 + 27524z^4 + \\ &\ 4348z^3 + 475z^2 + 32z + 1, \end{align} thus $\dreg{Q}=\deg \mathrm{HS}_{\FFt[X,Y]/\langle Q^\htop\rangle}+1=21$, matching our results. \end{example} \subsubsection{The linear polynomials.} In this section, we study how the degree of regularity computed in Theorem~\ref{Thm:Dreg-of-Qtop} changes when we add to the quadratic equations $Q$ also the fixed linear equations of $P$, which do not depend on the specific instance of the problem. Specifically, we compute the degree of regularity of $Q \cup P$. For this, we need to consider the ideal $\langle Q^\htop \cup P^\htop\rangle$. Note that this ideal contains $\langle Q^\htop \rangle$, which means that the variety of the former is a subset of the variety of the latter. In particular, the ideal $\langle Q^\htop \cup P^\htop\rangle$ is also zero-dimensional, so its degree of regularity is well-defined. We will use similar arguments to those applied to $\langle Q^\htop \rangle$ to study it. \begin{remark}\label{rem:qtopptopdef} The set $Q^\htop \cup P^\htop$ is the union of the following sets $$\left\{x_i^2 \mid i=1,\ldots,n\right\},\left\{x_i \mid i=1,\ldots,n\right\}, \left\{y_{i,j}^2 \mid i=1,\ldots,n,\ j=1,\ldots,\ell_t\right\},$$ $$\left\{y_{1,j} \mid j=2,\ldots,\ell_t\right\},\left\{y_{n,j} \mid j=1,\ldots,\ell_t\right\}$$ and $$\left\{ y_{i-1,j}y_{i,j} \mid i=2,\ldots,n,\ j=1,\ldots,\ell_t \right\}.$$ and the ideal $\langle Q^\htop \cup P^\htop \rangle \subseteq \FFt[X,Y]$ is thus a monomial ideal. \end{remark} Next lemma provides a Gr\"obner basis of the ideal $\langle Q^\htop \cup P^\htop \rangle \subseteq \FFt[X,Y]$. \begin{lemma}\label{lem:gbqtopptop} A Gr\"obner basis $G$ for $\langle Q^\htop \cup P^\htop \rangle \subseteq \FFt[X,Y]$ is $$G = \left\{x_i \mid i=1,\ldots,n\right\} \cup \left\{ y_{i-1,j}y_{i,j} \mid i=3,\ldots,n-1,\ j=1,\ldots,\ell_t \right\}\cup $$ $$\left\{ y_{1,1}y_{2,1}\right\}\cup\left\{y_{1,j} \mid j=2,\ldots,\ell_t\right\}\cup\left\{y_{n,j} \mid j=1,\ldots,\ell_t\right\}\cup$$ $$\left\{y_{i,j}^2 \mid i=2,\ldots,n-1,\ j=1,\ldots,\ell_t\right\} \cup \left\{y_{1,1}^2\right\}. $$ \end{lemma} \begin{proof} The proof of this statements follows directly from inspecting Remark~\ref{rem:qtopptopdef} and the same observations as in proof of Lemma~\ref{lem:groebnerQh}. \qed \end{proof} The next theorem gives the exact value of the degree of regularity of the system $Q \cup P$. The proof uses similar arguments to those used for the proof of Theorem~\ref{Thm:Dreg-of-Qtop}. \begin{theorem}\label{thm:dregQtopPtop} The degree of regularity of $Q \cup P$ is $$\dreg{Q \cup P} = \left\lceil\frac{n-1}{2} \right\rceil + (\ell_t - 1)\left\lceil\frac{n-2}{2} \right\rceil + 1.$$ \end{theorem} \begin{proof} Define the set \[ \tilde{Q}^\htop_{j,n} := Q^\htop_{j,n-1} \setminus \{y_{1,j}y_{2,j}\} \subset \FFt[y_{2,j},\ldots,y_{n-1,j}], \] for every $j=1,\ldots,\ell_t$. Let $G$ be a Gr\"obner basis of $\langle Q^\htop \cup P^\htop\rangle$ as in Lemma~\ref{lem:gbqtopptop}. Due to the presence of the linear monomials contributed by $P^\htop$, we observe \begin{equation}\label{eq:qtopptopasunion} G = Q^\htop_{1,n-1} \cup \bigcup_{j=2}^{\ell_t} \tilde{Q}^\htop_{j,n-1} \cup \left\{x_i^2 \mid i=1,\ldots,n\right\}. \end{equation} Applying Corollary~\ref{cor:maximalmonomial}, we can get a monomial $m_1 \in \FFt[y_{1,1},\ldots,y_{n-1,1}]$ of maximal degree $\deg m_1 = \lceil (n-1)/2 \rceil$ such that $m_1 \not \in \FFt[y_{1,1},\ldots,y_{n-1,1}]/\langle Q^\htop_{1,n-1} \rangle$. We can obtain other $\ell_t - 1$ monomials $m_j$ of maximal degree $d = \lceil (n-2)/2 \rceil$, such that $m_j \not \in \langle \tilde{Q}^\htop_{h,n-1} \rangle$ for every $h=1,\ldots,\ell_t$ and every $j = 2,\ldots,\ell_t$. Let now $$m := \prod_{j=1}^{\ell_t}m_j \in \FFt[X,Y]/\langle G\rangle$$ then $$d:=\deg m = \left\lceil\frac{n-1}{2} \right\rceil + (\ell_t - 1)\left\lceil\frac{n-2}{2} \right\rceil,$$ meaning that the $(d+1)$-th coefficient of the Hilbert series of $\FFt[X,Y]/\langle G \rangle$ is $0$. The result on the degree of regularity $\dreg{Q\cup P}$ follows. \qed \end{proof} \begin{example} Let $n=8$ and $\ell_t=3$. According to Theorem~\ref{thm:dregQtopPtop} the degree of regularity of $ Q \cup P$ is $11$. \noindent Using MAGMA, we compute and report the Hilbert series of the quotient ring $\FFt[X,Y]/\langle Q^\htop \cup P^\htop\rangle$, i.e. \begin{align} \mathrm{HS}_{\FFt[X,Y]/\langle Q^\htop \cup P^\htop\rangle} (z) =&\ 16z^{10} + 240z^9 + 1188z^8 + 2920z^7 + 4132z^6 + 3608z^5 +\\ &\ 2005z^4 + 710z^3 + 155z^2 + 19z + 1, \end{align} thus $\dreg{Q\cup P}=\deg \mathrm{HS}_{\FFt[X,Y]/\langle Q^\htop \cup P^\htop\rangle}+1=11$, matching our results. \end{example} \begin{remark}\label{rem:range fordregS} Since Theorem~\ref{thm:dregQtopPtop} considers the set $Q^\htop \cup P^\htop = S^\htop \setminus L_\vH^\htop$, it only gives an upper bound to the degree of regularity of $S$, that is $$\dreg{S}\leq \left\lceil\frac{n-1}{2} \right\rceil + (\ell_t - 1)\left\lceil\frac{n-2}{2} \right\rceil + 1.$$ \end{remark} In the next section, we provide some experimental data showing the gap between the value computed in Theorem~\ref{thm:dregQtopPtop} and that of the actual solving degree. \subsection{Experimental results} We performed several experiments for Modeling~\ref{modeling: improvedESD_F2} taking as input both random and Goppa codes, and we obtained a solving degree which is much smaller than the upper bound for the degree of regularity computed in Theorem~\ref{thm:dregQtopPtop}. This results in a much lower complexity estimate. We provide a selection of our experiments in Table~\ref{Tab:q2-SolveDeg}. The MAGMA code used for our experiments can be found \href{https://github.com/rexos/phd-cryptography-code/blob/main/modelings/Modeling2.txt}{here}. \vspace{-2mm} \input{q2-SD-examples} \subsubsection{Comparison with combinatorial attacks.} We also compare the complexity of solving ESDP using Gr\"obner basis techniques with the more traditional combinatorial attack proposed by Prange. The latter is widely regarded as a reference algorithm for decoding linear codes in the Hamming metric, and forms the basis of many state-of-the-art attacks on code-based cryptosystems. However, Gröbner basis computations are known to be computationally expensive, particularly as the number of variables and the degree of the polynomials grow. In contrast, combinatorial methods like Prange exploit structured randomization and search to solve the problem more efficiently, albeit with large memory requirements. In Table~\ref{Tab:q2-SolveDeg}, the complexity of the Prange algorithm has been computed using the Esser-Bellini cryptographic estimator~\cite{PKC:EssBel22}, see also this~\href{https://estimators.crypto.tii.ae/configuration?id=SDEstimator}{link}. On the other hand, the complexity of solving the polynomial system of Modeling~\ref{modeling: improvedESD_F2} with Gr\"obner bases methods has been computed using formula~\eqref{eq:GBcomplexity} with $N = n(\lfloor\log_2(t)\rfloor + 2)$, the highest step degree $d_M$ achieved in the MAGMA experiments as a proxy for the solving degree $d_{\mathrm{sol}}$ and the conservative matrix multiplication constant $\omega = 2.807$. These results confirm that direct combinatorial attacks like Prange outperform algebraic methods when solving the syndrome decoding problem, particularly for parameters of practical interest in code-based cryptography. Despite this, algebraic approaches remain valuable from a theoretical perspective and may offer insights into alternative solution strategies that could be leveraged in other contexts. We include this comparison to emphasize that the purpose of our Gröbner basis approach is not to compete with combinatorial attacks in efficiency, but rather to provide an alternative algebraic perspective on the syndrome decoding problem. Such perspectives can contribute to understanding the hardness of related problems in post-quantum cryptography. \input{ExtensionFields.tex} \subsection{Experimental results.} We solved the quadratic system associated with Modeling~\ref{modeling: quadBSD_Fq} for several random codes. We show in Table~\ref{Tab:qNEQ2-SolveDeg} that, similarly to the case over $\FFt$, the solving degree is surprisingly small. MAGMA code used for our experiments with Modeling~\ref{modeling: quadBSD_Fq} can be found at \href{https://github.com/rexos/phd-cryptography-code/blob/main/modelings/Modeling5.magma}{this link}. \input{qNOT2-SD-examples} \section{Conclusion and Future Directions} We have presented a new algebraic cryptanalysis for both the bounded and the exact versions of the Syndrome Decoding problem. In the binary case, our modelings significantly improved the previous attempt of \cite{2021/meneghetti}, by capturing the weight condition on the solution vector with quadratic polynomials. We have also experimentally shown that the behavior of the associated Gr\"obner basis is very different from that of a random system with the same number of variables and equations, \textit{leading to a much better complexity}. We have thus taken an important step towards making algebraic algorithms potentially competitive for the decoding problem. We introduced algebraic modelings for the first time in the case of the general syndrome decoding problem over larger finite fields. Notably, one of them is quadratic with a number of variables and equations that is linear or quasi-linear in the code length, \textit{independently from the field size}. We have analyzed that, despite the constant degree of the equations involved, the system correctly solves the decoding problem and with high probability does not have spurious solutions for all parameters that are relevant to the problem. \vskip 0.5cm An open question to this work is to understand more clearly the behavior of the Gr\"obner basis computation both in the binary and in the general finite field cases and to get a theoretical estimate of the complexity that better matches with the one obtained from the experiments. This is a difficult task, as it is often the case for very structured algebraic systems, and probably requires to develop dedicated tools to analyze such behavior. Another interesting and natural follow-up to this work can be to analyze the impact of hybrid strategies on solving the proposed systems. Since the weight of the solution sought is relatively low, a convenient choice is to set most of the variables of $X$ to 0. It is not difficult to see that this approach is reminiscent of the guess part in Prange or later ISD algorithms. In the case of binary systems, we have verified experimentally that the best hybrid trade-off actually boils down to the Prange algorithm, the best complexity being indeed obtained when enough zeros to linearize the system are guessed. However, the system hides a lot of structure and offers many different ways to specialize variables. For example, the auxiliary variables from the vector $\vy$ can also be fixed and they too have different probabilities of taking a value equal to 0 or 1. It is therefore not at all unrealistic to speculate that an ad-hoc and smart hybridization technique may lead to a better trade-off than a fully combinatorial approach. \subsection{Acknowledgments.} We would like to thank the reviewers for their detailed and valuable feedback. Additionally, special thanks to Magali Bardet, Tanja Lange and Alberto Ravagnani for the fruitful discussions and insights. This publication was created with the co-financing of the European Union FSE-REACT-EU, PON Research and Innovation 2014-2020 DM1062/2021. A. Caminata is supported by the PRIN 2020 grant 2020355B8Y ``Squarefree Gr\"obner degenerations, special varieties and related topics'', by the PRIN PNRR 2022 grant P2022J4HRR ``Mathematical Primitives for Post Quantum Digital Signatures'', by the MUR Excellence Department Project awarded to Dipartimento di Matematica, Università di Genova, CUP D33C23001110001, and by the European Union within the program NextGenerationEU. A. Meneghetti acknowledges support from Ripple's University Blockchain Research Initiative. A. Caminata and A. Meneghetti are members of the INdAM Research Group GNSAGA. \bibliography{biblio.bib,crypto,abbrev0} \bibliographystyle{splncs04} \appendix \section{Section~\ref{Sec:Dim_of_Var} Bounds on the zero-dimensionality}\label{app:Dim_Exp} \begin{table}[H] \centering \resizebox{0.9\textwidth}{!}{\begin{tabular}{\|c \|\|c c c c\|\|} \hline $[n,k]_q$& $t=\lfloor (d_{GV}-1)/2\rfloor$ & $t=\lfloor (n-k)/2\rfloor$ & $t=d_{GV}$ & $t=d_{GV}+1$ \\ \hline $[100,50]_{2}$ & \makecell{$t=5,$ \\ $\pr\le 2.32\cdot 10^{-20} $} & \makecell{$t= 25,$ \\ $\pr\le 1 $} & \makecell{$t=12 ,$ \\ $\pr\le 6.73\cdot 10^{-9} $} & \makecell{$t=13 ,$ \\ $\pr\le 1.84\cdot 10^{-7} $} \\ \hline $[100,50]_{7}$ & \makecell{$t= 12 , $ \\ $ \pr\le 8.44\cdot 10^{-51} $} & \makecell{$t= 25 , $ \\ $ \pr\le 1.89 \cdot 10^{-20} $} & \makecell{$t= 25 , $ \\ $ \pr\le 1.89 \cdot 10^{-20} $} & \makecell{$t= 26 , $ \\ $ \pr\le 2.68 \cdot 10^{-18}$} \\ \hline $[100,50]_{127}$ & \makecell{$t= 18 , $ \\ $ \pr\le 5.48 \cdot 10^{-118} $} & \makecell{$t= 25 , $ \\ $ \pr\le 1.23 \cdot 10^{-84} $} & \makecell{$t= 37 , $ \\ $ \pr\le 5.38 \cdot 10^{-30} $} & \makecell{$t= 38 , $ \\ $ \pr\le 1.44 \cdot 10^{-25} $} \\ \hline $[100,80]_{2}$ & \makecell{$t= 1 , $ \\ $ \pr\le 9.09 \cdot 10^{-11} $} & \makecell{$t= 10 , $ \\ $ \pr\le 1 $} & \makecell{$t= 4 , $ \\ $ \pr\le 3.14\cdot 10^{-4} $} & \makecell{$t= 5 , $ \\ $ \pr\le 0.0268$} \\ \hline $[100,80]_{7}$ & \makecell{$t= 3 , $ \\ $ \pr\le 4.06 \cdot 10^{-25} $} & \makecell{$t= 10 , $ \\ $ \pr\le 2.92 \cdot 10^{-5} $} & \makecell{$t= 8 , $ \\ $ \pr\le 1.30 \cdot 10^{-10} $} & \makecell{$t= 9 , $ \\ $ \pr\le 6.55 \cdot 10^{-8} $} \\ \hline $[100,80]_{127}$ & \makecell{$t= 6 , $ \\ $ \pr\le 1.16 \cdot 10^{-52} $} & \makecell{$t= 10 , $ \\ $ \pr\le 1.14 \cdot 10^{-31} $} & \makecell{$t= 13 , $ \\ $ \pr\le 1.97 \cdot 10^{-16} $} & \makecell{$t= 14 , $ \\ $ \pr\le 1.97 \cdot 10^{-11} $} \\ \hline \end{tabular} } \vspace{2mm} \caption{Bound on the probability $\pr$ that the ideal associated with the system has a strictly positive dimension for the decoding problem with a randomly sampled syndrome.} \label{table: bound_SD} \end{table} \vspace{-.5in} \begin{table}[H] \centering \resizebox{0.8\textwidth}{!}{\begin{tabular}{\|c \|\|c c c c c\|\|} \hline $[n,k]_q$& $t=\lfloor (d_{GV}-1)/2\rfloor$ & $t=\lfloor (n-k)/2\rfloor$ & $t=d_{GV}-2$ & $t=d_{GV}-1$ & $t=d_{GV}$\\ \hline $[ 100 , 50 ]_{ 2 }$ & \makecell{$t= 5 ,$ \\ $ \pr\le 2.07\cdot 10^{-6} $} & \makecell{$t= 25 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 10 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 11 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 12 ,$ \\ $ \pr\le 1 $} \\ \hline $[ 100 , 50 ]_{ 7 }$ & \makecell{$t= 12 ,$ \\ $ \pr\le 8.08\cdot 10^{-18} $} & \makecell{$t= 25 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 23 ,$ \\ $ \pr\le 0.378 $} & \makecell{$t= 24 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 25 ,$ \\ $ \pr\le 1 $} \\ \hline $[ 100 , 50 ]_{ 127 }$ & \makecell{$t= 18 ,$ \\ $ \pr\le 1.46\cdot 10^{-48} $} & \makecell{$t= 25 ,$ \\ $ \pr\le 6.16\cdot 10^{-30} $} & \makecell{$t= 35 ,$ \\ $ \pr\le 3.04\cdot 10^{-5} $} & \makecell{$t= 36 ,$ \\ $ \pr\le 0.00696 $} & \makecell{$t= 37 ,$ \\ $ \pr\le 1 $} \\ \hline $[ 100 , 80 ]_{ 2 }$ & \makecell{$t= 1 ,$ \\ $ \pr\le 9.54\cdot 10^{-5} $} & \makecell{$t= 10 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 2 ,$ \\ $ \pr\le 0.0142 $} & \makecell{$t= 3 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 4 ,$ \\ $ \pr\le 1 $} \\ \hline $[ 100 , 80 ]_{ 7 }$ & \makecell{$t= 3 ,$ \\ $ \pr\le 6.93\cdot 10^{-10} $} & \makecell{$t= 10 ,$ \\ $ \pr\le 1 $} & \makecell{$t= 6 ,$ \\ $ \pr\le 0.00176 $} & \makecell{$t= 7 ,$ \\ $ \pr\le 0.165 $} & \makecell{$t= 8 ,$ \\ $ \pr\le 1 $} \\ \hline $[ 100 , 80 ]_{ 127 }$ & \makecell{$t= 6 ,$ \\ $ \pr\le 4.20\cdot 10^{-21} $} & \makecell{$t= 10 ,$ \\ $ \pr\le 1.59\cdot 10^{-8} $} & \makecell{$t= 11 ,$ \\ $ \pr\le 1.65\cdot 10^{-5} $} & \makecell{$t= 12 ,$ \\ $ \pr\le 0.0155$} & \makecell{$t= 13 ,$ \\ $ \pr\le 1 $} \\ \hline \end{tabular} } \vspace{2mm} \caption{Bound on the probability $\pr$ that the ideal associated with the system has strictly positive dimension for the codeword finding problem (i.e. with null syndrome).} \label{table: bound_CF} \end{table} \end{document} \begin{table}[H] \centering \begin{tabular}{\|c\|c\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline \cellcolor{gray!20!}$n$& \cellcolor{gray!20!}$k$&\cellcolor{gray!20!}$t$& \cellcolor{gray!20!}Code Type&\cellcolor{gray!20!} \# Eqs &\cellcolor{gray!20!} \# Vars& \cellcolor{gray!20!}$d_{\mathrm{reg}}$ &\cellcolor{gray!20!} SR $d_{\mathrm{reg}}$&\cellcolor{gray!20!} $d_{\mathrm{M}}$&\cellcolor{gray!20!}Prange&\cellcolor{gray!20!} Modeling~\ref{modeling: improvedESD_F2} \\ \hline 8 & 2 & 2 & Goppa & (17,38) & 24 & $\leq8$ & 3 & 2 & 9 & 23 \\ \hline 10 & 5 & 4 & Random & (20,67) & 40 & $\leq14$ & 5 & 3 & 10 & 38 \\ \hline 16 & 8 & 2 & Goppa & (27,78) & 48 & $\leq16$ & 5 & 3 & 12 & 40 \\ \hline 20 & 10 & 5 & Random* & (35,137) & 80 & $\leq29$ & 7 & 3 & 12 & 46 \\ \hline 30 & 15 & 7 & Random* & (50,207) & 120 & $\leq44$ & 10 & 4 & 15 & 65 \\ \hline 32 & 12 & 4 & Goppa & (57,221) & 128 & $\leq 47$ & 10 & 3 & 16 & 52 \\ \hline 32 & 17 & 3 & Goppa & (50,158) & 96 & $\leq32$ & 5 & 4 & 16 & 61 \\ \hline 32 & 22 & 2 & Goppa & (45,158) & 96 & $\leq 32$ & 7 & 3 & 15 & 48 \\ \hline 40 & 20 & 8 & Random* & (67,356) & 160 & $\leq78$ & 16 & 5 & 17 & 88 \\ \hline 50 & 30 & 5 & Random & (75,347) & 200 & $\leq74$ & 15& 4 & 20 & 73\\ \hline 50 & 40 & 4 & Random & (65,347) & 200 & $\leq74$ & 17& 4 & 14 & 73\\ \hline 64 & 52 & 2 & Goppa & (79,318) & 192 & $\leq64$ & 14 & 4 & 18 & 72\\ \hline 64 & 40 & 4 & Goppa & (93,445) & 256 & $\leq95$ & 19 & 4 & 21 & 77\\ \hline 64 & 16 & 8 & Goppa & (119,572) & 320 & $\leq126$ & 21 & 4 & 19 & 81\\ \hline \end{tabular} \vspace{2mm} \caption{This table shows experimental results for $\mathbb{F}_2$-linear codes using Modeling~\ref{modeling: improvedESD_F2} for the ESDP. The number of equations is given in the format (\#linear equations, \#quadratic equations). The values in the $d_{\mathrm{M}}$ column represent the smallest degree D such that MAGMA function GroebnerBasis(F,D) gives the Gr\"obner basis, i.e. highest step degree achieved when directly computing the Gröbner basis of the system in MAGMA, but ignoring the unnecessary steps in high degree that MAGMA F4 algorithm may do to insure termination. The SR $d_{reg}$ column gives the degree of regularity of a semi-regular system of equations with the associated number of linear equations, quadratic equations, and variables, using \cite[Corollary 3.3.8]{B04}. The values in $d_{\mathrm{reg}}$ column are upper bounds for the degree of regularity of the system as provided by Theorem~\ref{thm:dregQtopPtop} and Remark~\ref{rem:range fordregS}. Random codes with ``'' are decoding challenges from \url{https://decodingchallenge.org/syndrome}, with a number of errors slightly above Gilbert-Varshamov distance. The other random code instances are below GV distance instead. Instances with ``Goppa'' Code Type are random full-length binary Goppa codes with a number of errors equal to the Goppa polynomial degree. The Prange and Modeling 2 columns state the $\log_2$ of the complexities of the Prange algorithm (Esser-Bellini estimator~\cite{PKC:EssBel22}) and Gr\"obner basis computations \eqref{eq:GBcomplexity} with $d_{\mathrm{M}}$, respectively. } \label{Tab:q2-SolveDeg} \end{table}\vspace{-.2in} \section{Modelings over $\FF_q$} \label{sec:Fq} Each of the modelings we have discussed thus far (MPS, Modeling \ref{modeling: improvedSD_F2}, and Modeling \ref{modeling: improvedESD_F2}) are limited to the binary case. To the best of our knowledge, there is no modeling of the general syndrome decoding problem over $\FF_q$ for $q>2$ in the literature. In this section, we adapt the previous modelings to a generic finite field $\FF_q$, for some prime power $q\ge 2$, and explain how to efficiently (i.e. in polynomial time) obtain a polynomial system encoding an instance of the Syndrome Decoding Problem. We will adopt the following notation throughout this section. \begin{notation}\label{FQnotation} Let $n\geq2$ and let $\CC \subseteq \FF_q^n$ be a $[n,k,d]$-linear code having a parity check matrix $\HH \in \FF_q^{(n-k) \times n}$. The vector $\vs \in \FF_q^{n-k}$ denotes the syndrome and $0\le t \le \lfloor (d-1)/2 \rfloor$ is the target error weight. Let $r_1,r_2>0$ be two integers. We will work over the polynomial ring $\FF_{q^{r_1}}[X,Y,Z]$, where $X=(x_1,\dots,x_n)$, $Y=(Y_1,\dots,Y_n)$, $Y_j=(y_{j,1}, \dots, y_{j,r_2})$, and $Z=(z_1,\dots,z_n)$ are variables. \end{notation} As in the previous sections, $\vx=(x_1,\dots,x_n)$ is the vector of variables corresponding to the solution of the syndrome decoding problem. On the other hand, the role of the integers $r_1,r_2$ and of the variables $Y$ and $Z$ will be illustrated later. We separately describe and explain the sets of polynomials that, together, model Problems~\ref{BSDP} and \ref{EWSDP}. Then we provide an analysis of the correctness of our modelings. \subsection{Construction of the Equations} \subsubsection{Identifying the support of a vector of $\mathbb{F}_q^n$.} Unlike the Boolean case, the value of an element in the support of $\vx$ is not uniquely determined when $q\ge 3$. In order to count the number of nonzero coordinates with algebraic equations, we first need to map all nonzero elements to a unique element of $\FFq^$, say 1. Thus, in addition to the $Y$'s variables encoding the partial Hamming weights, we introduce here another length-$n$ vector of variables $Z=(z_1,\dots,z_n)$, each of which can only assume two values, 0 or 1, depending on whether the corresponding $X$ coordinate is nonzero. First, we tackle the problem of describing the relation between $X$ and $Z$ through algebraic equations. We distinguish two cases, depending on the target version of the problem, and then prove the sets of polynomials correctly describe our target. \begin{itemize} \item \textit{Support constraint encoding for Problem~\ref{BSDP}.} Compute the following set of $2n$ quadratic polynomials \begin{equation} \label{eq:sceBSD} \{ x_j(z_j-1) \mid j=1,\dots,n\} \cup \{ z_j^2-z_j \mid j=1,\dots,n\}. \end{equation} \item \textit{Support constraint encoding for Problem~\ref{EWSDP}.} Compute the following set of $n$ polynomials of degree $q-1$ \begin{equation} \label{eq:sceEWSD} \{z_j-x_j^{q-1}\mid j=1,\dots,n\}. \end{equation} \end{itemize} In the first case, the condition $z_j=1$ if $x_j\ne 0$ is given from the first set of polynomials. Otherwise, the second set implies $z_j\in\{0,1\}$. Therefore, the support of $(z_1,\ldots,z_n)$ contains the support of $(x_1,\ldots,x_n)$ and thus $\wt((z_1,\ldots,z_n))\ge \wt((x_1,\ldots,x_n))$. In the second case, in order for the corresponding equations to be satisfied, $z_j=1$ if and only if $x_j\ne 0$, and $z_j=0$ otherwise. Hence $\wt((z_1,\ldots,z_n))=\wt((x_1,\ldots,x_n))$. From a computational point of view, the support constraint encoding for Problem~\ref{EWSDP} has a strong limitation, that is the high degree of the polynomials. A Gr\"obner basis computation would need to reach at least degree $q-1$ before taking into account such polynomials, leading to infeasible calculations unless $q$ is very small. This is reminiscent of the problem of including field equations in modelings over large fields. Yet, this issue does not appear in the support constraint encoding for Problem~\ref{BSDP}: the polynomials have constant degree 2 regardless of the field size $q$, making a modeling for Problem~\ref{BSDP} more realistic and valuable for effective computations. \subsubsection{Hamming weight computation encoding.} A difficulty arising from a direct generalization to large fields of the previous approach is the update of the weight registers, i.e. of the vectors $\vy_i$'s. In order to overcome this limitation, we introduce a different strategy for encoding the partial weights. More precisely, we substitute their binary expansion with vectors from a linear recurring sequence over an extension of $\FF_q$. As we will see, this approach naturally allows for the choice of different trade-offs between the number of variables and finite field size. We first recall some known results about (univariate) polynomials over finite fields, companion matrices, and linear recurring sequences. We mainly refer to \cite{LN94} for this part. \begin{definition}[Companion matrix, Chapter 2, \S5 \cite{LN94}] Let $f(x)=x^d+f_{d-1}x^{d-1}+\dots+f_0\in \FF_q[x]$ be a monic polynomial. Its companion matrix is \begin{equation} \label{eq: companion} \Cf = \begin{bmatrix} 0 & 0 & \cdots & 0 & -f_0\\ 1 & 0 & \cdots & 0 & -f_1\\ 0 & 1 & \cdots & 0 & -f_2\\ \vdots & \vdots& \ddots &\vdots &\vdots\\ 0 & 0 & \cdots & 1 & -f_{d-1} \end{bmatrix}. \end{equation} \end{definition} It is well known that the equation $f(\Cf)=0$ is satisfied, hence, if $f$ is a monic irreducible polynomial over $\FF_q$, then its companion matrix $\Cf$ plays the role of a root of $f$. It follows that the elements of the extension field $\FF_{q^d}$ can be written, according to this representation, as polynomials in $\Cf$ of degree strictly less than $d$. We also recall that the order of a non-constant polynomial $f$ with $f_0\neq 0$ is the least positive integer $e$ such that $f(x) \mid x^e-1$. A polynomial in $\FF_q[x]$ of degree $d$ is said primitive if it is monic, $f(0)\ne 0$, and $\ord(f)=q^d-1$. Such polynomials can be found from the factorization of $x^{q^d-1}-1$. The theory of linear recurring sequences says that the sequence of vectors $\vy_0,\Cf \vy_0,\Cf^2 \vy_0,\dots$, for some nonzero $\vy_0$ and companion matrix with $f_0\ne 0$, is periodic with least period equal to the order of $f$, when the latter is irreducible (cf. \cite[Theorem 6.28]{LN94}). Therefore, by choosing $f$ primitive, we obtain a sequence of vectors $\vy_0,\Cf \vy_0,\Cf^2 \vy_0,\dots$ of maximal order $q^d-1$. On the other hand, the choice of $\vy_0$ does not seem to affect any property of our modeling. Without loss of generality, from now on we fix the initial state vector \begin{equation} \vy_0=\begin{pmatrix} 1&0& \cdots &0 \end{pmatrix}^\top. \end{equation} By tuning the values $r_1$ and $r_2$ we can choose the number of variables used for the Hamming weight computation encoding, at the cost of working over more or less large field extensions. More precisely, take \[ m:= \min \{i \in \mathbb{N} \mid q^i > \max(t, n-t)+1 \}, \] and let $r_1,r_2$ be two positive integers such that $m\le r_1 r_2$. Then, let $f\in \FF_{q^{r_1}}[x]$ be a primitive polynomial of degree $r_2$ and $\Cf \in \FF_{q^{r_1}}^{r_2\times r_2}$ its companion matrix. For convenience sake, we will use the column vector notation for the $Y_j$'s blocks of variables, i.e. $$Y_j= \begin{pmatrix} y_{j,1}& \cdots& y_{j,r_2} \end{pmatrix}^\top .$$ The polynomial encoding the partial Hamming weight is the following. \begin{itemize} \item \textit{Hamming weight computation encoding.} Compute the $n r_2$ affine bilinear (in $Y$ and $Z$) polynomials from the expansion of \begin{equation} \label{eq: hwceFqinit} Y_1-(1-z_1)\cdot \vy_0-z_1\cdot\Cf\cdot \vy_0 \end{equation} and \begin{equation} \label{eq: hwceFq} \{Y_j-(1-z_j)\cdot Y_{j-1}-z_j\cdot\Cf\cdot Y_{j-1} \;,\qquad \mathrm{for}\;j\in\{2,\ldots,n\}\}. \end{equation} \end{itemize} \begin{remark} Using this approach, $r_1$ determines the finite field over which we define the resulting polynomial system (and thus the Multivariate Quadratic Problem instance), while $r_2$ determines the number of variables required for the weight computation encoding (which is strictly linked to the computational complexity). Observe that, together, equations \eqref{eq: hwceFqinit} and \eqref{eq: hwceFq} correspond to $nr_2$ polynomial equations over $\FF_{q^{r_1}}$ in $nr_2+n$ variables. In several cases, working with a small amount of variables (namely, $r_1$ large and $r_2$ small) is preferable, but there are some instances in which working over small finite fields with a large number of variables (i.e. $r_1$ small and $r_2$ large) is advantageous. \end{remark} The next result shows that the Hamming weight of $\vz$ is correctly computed. \begin{proposition}\label{prop: correctness} Consider the system given by \eqref{eq: hwceFqinit} and \eqref{eq: hwceFq} over $\FF_{q^{r_1}}[Y, Z]$. Any solution $(\vy, \vz)=(\vy_1,\dots,\vy_n,\vz)\in \FF_{q^{r_1}}^{r_2 n}\times \{0,1\}^n$ of the system satisfies \[\vy_j=\Cf^{\wt(\pi_j(\vz))}\vy_0.\] In particular, $\vy_n=\Cf^{\wt(\vz)}\vy_0.$ \end{proposition} \begin{proof} It follows directly from the hypotheses by an inductive argument. \\ The first step is considering $\vy_1:=(y_{1,1},\ldots,y_{1,r_2})^{\top}$, which by definition is $$ \vy_1=(1-z_1)\vy_0 + z_1 \Cf \vy_0= \left\{ \begin{array}{rl} \vy_0&\mathrm{if}\; z_1=0\\ \Cf \vy_0&\mathrm{if}\; z_1\neq 0 \end{array}\;, \right. $$ namely, $\vy_1=\Cf^{\wt(z_1)}\vy_0=\Cf^{\wt(\pi_1(\vz))}\vy_0$. \\ For the inductive step, we consider $\vy_{j-1}:=(y_{j-1,1},\ldots,y_{j-1,r_2})^{\top}$ to be equal to $\Cf^{\wt(\pi_{j-1}(\vz))}\vy_0$, and we look at the definition of $\vy_j$. We have $$ \vy_j=(1-z_j)\vy_{j-1}+z_j \Cf \vy_{j-1}=\left\{ \begin{array}{cl} \vy_{j-1}&\mathrm{if}\; z_j=0\\ \Cf \vy_{j-1}&\mathrm{if}\; z_j\neq 0 \end{array}\;, \right. $$ and either way, we obtain $$ \vy_j=\Cf^{z_j}\cdot \vy_{j-1}=\Cf^{z_j}\Cf^{\wt(\pi_{j-1}(\vz))}\vy_0=\Cf^{\wt(z_j)+\wt(\pi_{j-1}(\vz))}\vy_0=\Cf^{\wt(\pi_{j}(\vz))}\vy_0.$$ \qed \end{proof} \subsubsection{Weight constraint encoding.} As for the previous modelings, the weight constraint encoding simply ensures that the representation of the last partial Hamming weight coincides with the representation of the total Hamming weight. \begin{itemize} \item \textit{Weight constraint encoding.} Compute the $r_2$ affine linear polynomials in $Y$ from the expansion of \begin{equation} \label{eq: wceFq} \vy_n-\Cf^{t} \vy_0 . \end{equation} \end{itemize} \begin{corollary} \label{cor: Z} Consider the system given by \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq} and \eqref{eq: wceFq} over $\FF_{q^{r_1}}[Y, Z]$ and let $z_1,\ldots,z_n$ be either $0$ or $1$. Then \begin{enumerate} \item The number of solutions $(\vy, \vz)\in \FF_{q^{r_1}}^{r_2 n}\times \{0,1\}^n$ of the system is equal to the number of binary vectors of Hamming weight equal to $t$, i.e. $\binom{n}{t}$; \item If $(\bar{\vy},\bar{\vz})$ and $(\tilde{\vy},\tilde{\vz})$ are two distinct solutions, then $\bar{\vz}\ne \tilde{\vz}$. \end{enumerate} \end{corollary} \begin{proof} From Proposition~\ref{prop: correctness} and \eqref{eq: wceFq}, it follows that any solution $(\vy, \vz)$ satisfies \[ \Cf^t\vy_0= \begin{pmatrix} y_{n,1}\\ \vdots\\ y_{n,r_2} \end{pmatrix} =\Cf^{\wt(\vz)}\vy_0, \] which implies $\ord(f) \mid \lvert t-\wt{(\vz)}\rvert$. Since $f$ is primitive, we obtain \[ \ord(f)=(q^{r_1})^{r_2} -1 \ge q^m - 1> \max(t,n-t). \] On the other hand, $0\le \wt(\vz)\le n$, hence $\lvert t-\wt{(\vz)}\rvert \le \min(t,n-t)$. Therefore, the only way $\ord(f)$ can divide $\lvert t-\wt{(\vz)}\rvert$, is that $\lvert t-\wt{(\vz)}\rvert=0$, i.e. $\wt(\vz)=t$. Substituting $\vz$ in \eqref{eq: hwceFqinit} uniquely determines all the values $y_{1,1},\dots,y_{1,r_2}$ by linear equations of the form $y_{1,i}=c_i$. Moreover, substituting $y_{j-1,1},\dots,y_{j-1,r_2}$ in \eqref{eq: hwceFq} recursively determines all the values $y_{j,1},\dots,y_{j,r_2}$ in a similar manner. Therefore, if $\bar{\vz}= \tilde{\vz}$ are the projections over the last $n$ coordinates of two solutions, then also $(\bar{\vy},\bar{\vz})=(\tilde{\vy},\tilde{\vz})$, which concludes the proof. \qed \end{proof} \subsubsection{Field equations.} The field equations concern only the $X$ part of the variables. \begin{itemize} \item \textit{Field equations.} The equations are obtained from the $n$ polynomials \begin{equation} \label{eq: ffe_Fq} \{x_j^q-x_j \mid j =1,\dots,n\}. \end{equation} \end{itemize} Indeed, $\vz$ already lies over $\{0,1\}^n$ because of the support constraint equations (and \eqref{eq: ffe_Fq}), while \eqref{eq: hwceFqinit} and \eqref{eq: hwceFq} force $Y$ to lie over $\FF_{q^{r_1}}$. \subsection{The Modelings} We are finally ready to describe the algebraic systems over $\FF_{q^{r_1}}[X,Y,Z]$ for Problems~\ref{EWSDP} and \ref{BSDP} and prove their correctness. \begin{modeling}[Modeling for the SDP over $\FF_q$] \label{modeling: BSD_Fq} Given an instance $(\HH,\mathbf{s},t)$ of Problem~\ref{BSDP} over $\FF_q$, Modeling~\ref{modeling: BSD_Fq} is the union of the sets of polynomials \eqref{eq:pce}, \eqref{eq:sceBSD}, \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq}, \eqref{eq: wceFq} and \eqref{eq: ffe_Fq}. \end{modeling} \begin{modeling}[Modeling for the ESDP over $\FF_q$] \label{modeling: EWSD_Fq} Given an instance $(\HH,\mathbf{s},t)$ of Problem~\ref{EWSDP} over $\FF_q$, Modeling~\ref{modeling: EWSD_Fq} is the union of the sets of polynomials \eqref{eq:pce}, \eqref{eq:sceEWSD}, \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq}, \eqref{eq: wceFq} and \eqref{eq: ffe_Fq}. \end{modeling} As already said, finite field equations cannot be efficiently taken into account when dealing with large fields. In the exact weight syndrome decoding modeling, the support constraint equations are high-degree as well, so the problem would persist. On the other hand, it becomes convenient to remove the field equations in the bounded syndrome decoding problem. This leads to a new quadratic modeling. \begin{modeling}[Quadratic Modeling for the SDP over $\FF_q$] \label{modeling: quadBSD_Fq} Given an instance $(\HH,\mathbf{s},t)$ of Problem~\ref{BSDP} over $\FF_q$, Modeling~\ref{modeling: quadBSD_Fq} is the union of the sets of polynomials \eqref{eq:pce}, \eqref{eq:sceBSD}, \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq}, \eqref{eq: wceFq}. \end{modeling} In the next section we thoroughly investigate the effect of removing the field equations from Modeling~\ref{modeling: EWSD_Fq}. We find, that at least for the parameter choices interesting for cryptography, the solutions of our modeling without field equations still lie over $\FFq$ with high probability. Table~\ref{table:Fq-model-sizes} provides the number of variables and equations for the three modelings over $\FF_q$. \begin{table}[H] \centering \begin{tabular}{\|c\|c\|c\|c\|} \hline & \# Polynomials & \# Variables & Degree\\ \hline Modeling~\ref{modeling: BSD_Fq} & $4n-k+n r_2+r_2$ & $n(r_2 + 2)$ & $q$\\ \hline Modeling~\ref{modeling: EWSD_Fq} & $3n-k+n r_2+r_2$ & $n(r_2 + 2)$ & $q$\\ \hline Modeling~\ref{modeling: quadBSD_Fq} & $3n-k+n r_2+r_2$ & $n(r_2 + 2)$ & $2$\\ \hline \end{tabular} \vspace{2mm} \caption{Number of equations, number of variables and maximum degree of the algebraic modelings over $\FF_{q^{r_1}}$.} \label{table:Fq-model-sizes} \end{table} \begin{remark} Since $r_2$ can be chosen as at most $m=\OO(\log_q (n))$, both the number of polynomials and variables are quasi-linear in the code-length $n$ in all the three modelings, namely they are $\OO(\log_2 (n))$. At the cost of defining the system over $\FF_{q^{r_1}}=\FF_{q^{m}}$, these quantities become linear in $n$, as the choice $r_2=1$ is possible. \end{remark} The modelings above capture exactly the corresponding syndrome decoding problem variants. \begin{theorem} \label{prop: sol iff} Given an instance $(\HH,\vs,t)$, \begin{enumerate} \item The vector $(\vx, \vy, \vz)$ is a solution of Modeling~\ref{modeling: BSD_Fq} if and only if $\vx$ is a solution of Problem~\ref{BSDP} and $\vx\in\FF_q$; \item The vector $(\vx, \vy, \vz)$ is a solution of Modeling~\ref{modeling: EWSD_Fq} if and only if $\vx$ is a solution of Problem~\ref{EWSDP} and $\vx\in\FF_q$; \item The vector $(\vx, \vy, \vz)$ is a solution of Modeling~\ref{modeling: quadBSD_Fq} if and only if $\vx$ is a solution of Problem~\ref{BSDP} and $\vx\in\overline{\FF_q}$, where $\overline{\FF_q}$ denotes the algebraic closure of $\FF_q$. \end{enumerate} \end{theorem} \begin{proof} Since the parity-check equations \eqref{eq:pce} belong to all three modelings, it remains to prove the conditions on the weight of $\vx$ and to determine the base field over the vector can lie.\\ \textit{Proof of 1.} Modeling~\ref{modeling: BSD_Fq} contains the field equations, therefore $\vx\in \FF_q^n$. It has already been proved that \eqref{eq:sceBSD} implies $\wt(\vx)\le\wt(\vz)$. By Corollary~\ref{cor: Z}, \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq} and \eqref{eq: wceFq} imply that $\wt(\vz)=t$, hence $\wt(\vx)\le t$.\\ \textit{Proof of 2.} Modeling~\ref{modeling: EWSD_Fq} contains the field equations, therefore $\vx\in \FF_q^n$. It has already been proved that \eqref{eq:sceEWSD} implies $\wt(\vx)=\wt(\vz)$. By Corollary~\ref{cor: Z}, \eqref{eq: hwceFqinit}, \eqref{eq: hwceFq} and \eqref{eq: wceFq} imply that $\wt(\vz)=t$, hence $\wt(\vx)= t$.\\ \textit{Proof of 3.} The proof is analogous to the proof of \textit{1.}, with the only exception that Modeling~\ref{modeling: quadBSD_Fq} does not contain the field equations. Hence, the solutions of the system are all the vectors defined over the algebraic closure $\overline{\FF_q}$ that satisfy the parity-check equations and have weight at most $t$. \qed \end{proof} \subsection{The Dimension of the Variety Associated with Modeling~\ref{modeling: quadBSD_Fq}}\label{Sec:Dim_of_Var} Unlike the modelings that include the field equations, Modeling~\ref{modeling: quadBSD_Fq} is not a priori associated with a zero-dimensional ideal. This represents the main drawback of Modeling~\ref{modeling: quadBSD_Fq} compared to Modeling~\ref{modeling: BSD_Fq}. The zero-dimensional property is desirable because it is necessary for defining the degree of regularity and for applying the FGLM algorithm to convert the $\mathsf{degrevlex}$ Gröbner basis into a $\mathsf{lex}$ basis. While it is possible to convert non-zero dimensional ideals using methods such as Gr\"obner walk or others, the process may not be as straightforward \cite{GBwalk1, GBwalk2}. In this subsection, we analyze the dimension of the variety associated with the ideal corresponding to Modeling~\ref{modeling: quadBSD_Fq}. We will explore the conditions under which the variety is zero-dimensional, as well as the probability of this occurring. We begin with the following reduction. \begin{proposition}\label{prop:reductiontoProblem} Let $\vx\in\FF_q^n$ be a vector which is a solution of Problem~\ref{BSDP} for a given instance $(\HH,\vs,t)$. In other words, $\vx$ satisfies $\mathbf{H}\vx^\top=\mathbf{s}^\top$ and $\wt(\vx)\leq t$. Then, there exist finitely many $(\vy,\vz)$ such that $(\vx, \vy, \vz)$ is a solution of Modeling~\ref{modeling: quadBSD_Fq}. \end{proposition} \begin{proof} Let us first consider a vector $\vx$ satisfying the parity-check equations and such that $\wt(\vx)=t$. Then, in Modeling~\ref{modeling: quadBSD_Fq}, the $z_i$ must detect exactly the support of $\vx$, and $\vx$ uniquely determines a solution $(\vx,\vy,\vz)$ of the system. If instead $\wt(\vx)=\bar{t}<t$, then there exist $t-\bar{t}$ indexes where the $Z$ variables can have value 1 while $\vx_i=0$. Thus, for any solution $\bar{\vx}\in\FF_q^n$ of the parity check matrix of weight $\bar{t}$, there exist \[ \binom{n-\bar{t}}{t-\bar{t}} \] different solutions $(\bar{\vx}, \vy, \vz)$. A special case is given by the codeword finding problem, i.e. where the syndrome $\vs$ is the zero vector. Here the null vector is a solution of Problem~\ref{BSDP} and leads to $\binom{n}{t}$ solutions, thus likely increasing a lot the solving degree and the cost of a Gr\"obner basis computation. We can get rid of all these solutions by fixing one variable $z_i=1$, thus forcing any solution to have weight at least 1 and removing the null vector. If the target solution has weight $t$, then the guess has success with probability $t/n$. We will discuss this strategy in more detail at the end of this subsection. \qed \end{proof} Proposition~\ref{prop:reductiontoProblem} implies that each solution to the decoding problem (Problem~\ref{BSDP}) corresponds to a finite number of solutions for Modeling~\ref{modeling: quadBSD_Fq}. This allows us to conduct an analysis that is independent of the specific modeling, as long as it accurately encodes the decoding problem in the sense of Theorem~\ref{prop: sol iff}. Therefore, we will focus on the Krull dimension of the solution set of Problem~\ref{BSDP} and provide a probability estimate for this dimension being zero. First, in the following remark we briefly collect the definition of Krull dimension and some important properties we will use in the sequel. Expanded details and proofs can be found e.g. in \cite[\S4, Chapter~9]{cox1997ideals} or other standard references in commutative algebra and algebraic geometry. \begin{remark}[Krull dimension]\label{prop: Krull=dim}\label{prop: max_dim} Let $\K$ be an algebraically closed field (we will apply the following definitions and results to $\K=\overline{\FF_q}$). An affine variety $V$ is the zero locus in $\K^m$ of a proper ideal $I$ of the polynomial ring $\K[x_1,\dots,x_m]$. We say that $V$ is irreducible if it is not possible to write $V=V_1\cup V_2$ where $V_1,V_2\subsetneq V$ are two proper subvarieties. Irreducibility of $V$ is equivalent to the corresponding ideal $I$ being prime. The \emph{Krull dimension} or simply the dimension of a variety $V$ is defined as the maximal length $d$ of the chains $V_0\subsetneq V_1\subsetneq \cdots \subsetneq V_d$, of distinct nonempty irreducible subvarieties of $V$. This is also equivalent to the supremum of the lengths of all chains of prime ideals containing the defining ideal $I$ of $V$. For example, the Krull dimension of an affine linear space $\mathcal{L}$ generated by $a$ linearly independent affine linear polynomials $L_1,\dots,L_a$ is precisely $m-a$, that is its dimension as an affine space. This can be seen by completing the polynomials to a maximal linearly independent system of $m$ equations (in $m$ variables) $L_1,\dots,L_a,L_{a+1},\dots,L_m$ and then considering the following maximal chain of prime ideals \[ \mathcal{L}=\langle L_1,\dots L_a\rangle\subsetneq\langle L_1,\dots L_a,L_{a+1}\rangle\subsetneq\cdots\subsetneq \langle L_1,\dots L_m\rangle. \] Notice that each ideal in this chain is prime, being generated by linearly independent polynomials of degree $1$. Finally, we mention that, thanks to the Noetherian property of the polynomial ring, a variety $V$ can be written uniquely as the union of irreducible varieties, which are called the irreducible components of $V$. Thus, the dimension of $V$ coincides with the largest of the dimensions of its irreducible components (see \cite[\S 4,Chapter 9, Corollary 9]{cox1997ideals}). \end{remark} Let $S\subseteq [n]$. Given a matrix $\mathbf{H}$ with $n$ columns and a vector $\vx \in \FF_q^n$, we denote by $\mathbf{H}_S$ the submatrix of $\mathbf{H}$ of columns indexed by $S$ and by $\vx_S \in \FF_q^{\|S\|}$ the vector obtained by deleting the coordinates corresponding to $[n]\setminus S$ from $\vx\in\FF_q^n$. On the contrary, let $\mathsf{pad}_S(\vx)\in\FF_q^n$ be the vector obtained from $\vx\in \FF_q^{\|S\|}$ by padding with 0's the positions corresponding to $[n]\setminus S$. \begin{proposition} Let $\mathcal{C}$ be an $[n,k]$ code with parity-check matrix $\mathbf{H}\in \FF_q^{(n-k)\times n}$. Then the set of solutions of Problem~\ref{BSDP} with target weight $t$ and syndrome $\vs$ for the code $\mathcal{C}$ is the finite union of irreducible components, namely \[ \bigcup_{S \subset [n], \|S\|=t} \{ \mathsf{pad}_S(\vx) \in \FF_q^n \mid \mathbf{H}_S \vx= \vs\}. \] \end{proposition} \begin{proof} For any $S$ of cardinality $t$, $\wt(\mathsf{pad}_S(\vx))=\wt(\vx)\le t$ and $\mathbf{H}\cdot \mathsf{pad}_S({\vx}) =\mathbf{H}_S \vx=\vs$. Thus, the set of solutions of the decoding problem contains $$ \bigcup_{S \subset [n], \|S\|=t} \{ \mathsf{pad}_S(\vx) \in \FF_q^n \mid \mathbf{H}_S \vx= \vs\}.$$ On the other hand, for any solution $\vx \in \FF_q^n$ to the decoding problem, let $S$ be a set of cardinality $t$ containing the support of $\vx$. Then, $\vx\in \{ \mathsf{pad}_S(\vx) \in \FF_q^n \mid \mathbf{H}_S \vx= \vs\}$. Finally, all the sets $ \{ \mathsf{pad}_S(\vx) \in \FF_q^n \mid \mathbf{H}_S \vx= \vs\}$ are irreducible being affine linear spaces. \qed \end{proof} The next proposition characterizes the dimension of the irreducible components of the set of solutions of the decoding problem and the finite field extension over which solutions are defined. \begin{proposition} \label{prop: variety} The Krull dimension of the solution set of Problem~\ref{BSDP} with target weight $t$ and syndrome $\vs$ for the linear code with parity-check matrix $\mathbf{H}$ is \begin{equation} \label{eq: dimension_Ideal} t-\min\{\rk(\mathbf{H}_S) \mid S\subseteq [n], \|S\|=t, \rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S)\}. \end{equation} Moreover, if the dimension is 0, then all solutions lie over $\FF_q$. \end{proposition} \begin{proof} Let us fix the support $S$ of $t$ possible error positions. By Remark~\ref{prop: Krull=dim}, the Krull dimension of the irreducible components coincides with their dimensions as affine linear spaces. The case study of the set of solutions of \[ \mathbf{H}_S \vx = \vs, \] seen as a variety, thus becomes the following: \begin{itemize} \item if $\rk((\mathbf{H}_S\mid \vs))>\rk(\mathbf{H}_S) \Rightarrow$ the variety is empty; \item if $\rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S) \Rightarrow$ the variety has dimension $t-\rk(\mathbf{H}_S)$. In particular, if the variety has dimension 0, i.e. $\rk(\mathbf{H}_S)=t$, then it has a unique element, which belongs to $\FF_q$. \end{itemize} By Remark~\ref{prop: max_dim}, the dimension of the solutions set is obtained as the maximum dimension over all the irreducible components corresponding to some $S$ for which $\rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S)$: \begin{align} &\max \{ t - \rk(\mathbf{H}_S) \mid S\subseteq [n], \|S\|=t, \rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S)\}\\=& t-\min\{\rk(\mathbf{H}_S) \mid S\subseteq [n], \|S\|=t, \rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S)\}. \end{align} Let us now consider the case of a zero-dimensional variety. Then for any choice of $S$, there is at most a solution and it must belong to $\FF_q^n$. Hence, all solutions belong to $\FF_q^n$. \qed \end{proof} \begin{corollary} \label{cor: dim_variety} The dimension of the variety associated with Modeling~\ref{modeling: quadBSD_Fq} is \begin{equation} t-\min\{\rk(\mathbf{H}_S) \mid S\subseteq [n], \|S\|=t, \rk((\mathbf{H}_S\mid \vs))=\rk(\mathbf{H}_S)\}. \end{equation} \end{corollary} \begin{proof} It readily follows from Propositions \ref{prop: max_dim} and \ref{prop: variety} and the fact that each solution of Problem~\ref{BSDP} corresponds to a finite number of solutions of Modeling~\ref{modeling: quadBSD_Fq}, thus it does not increase the dimension of the variety. \qed \end{proof} For relevant and not too-small parameters, we usually have $t\ll n-k$. Assuming the weight distribution of a linear code follows closely the Bernoulli one, we can estimate the probability that the ideal is zero-dimensional, and thus, by exploiting the proof of Proposition \ref{prop: variety}, that all the solutions $\vx$ lie over $\FF_q$. \begin{proposition} \label{prop: bound} Let $\mathcal{C}$ be an $\mathbb{F}_q$-linear code and let $W_i(\mathcal{C})$ the number of codewords of weight exactly $i$ in $\mathcal{C}$. Then the probability that Modeling~\ref{modeling: quadBSD_Fq} provides a variety of strictly positive dimension when $t<n-k$ is upper bounded by \[ \sum_{i=1}^t W_i(\mathcal{C})\left(\frac{1}{q^{n-k-i+1}}+\binom{n-i}{t-i}\left(\frac{1}{q^{n-k-t+1}}-\frac{1}{q^{n-k-i+1}}\right)\right) \] for a randomly sampled syndrome. For the codeword finding problem, the same probability is upper bounded by \[ \sum_{i=1}^t W_i(\mathcal{C})\binom{n-i}{t-i}. \] \end{proposition} \begin{proof} It follows from Corollary~\ref{cor: dim_variety} that the variety has positive dimension if and only if there exists a set $S\subseteq [n]$, $\|S\|=t$, such that $\mathbf{H}_S$ is not full-rank and the syndrome $\vs$ belongs to the column space of $\mathbf{H}_S$. The parity-check matrix $\mathbf{H}$ has $i$ linearly dependent columns indexed by the set $S$ if and only if the corresponding code $\mathcal{C}$ has a codeword of weight $\le i$ with support contained in $S$. Hence any codeword of positive weight $i\le t$ with support $S'$ identifies a set of $\binom{n-i}{t-i}$ supersets $S\supseteq S'$. On the other hand, each set $S$ of $t$ dependent columns is associated with \textit{at least} one codeword of weight $\le t$, hence iterating over such codewords is enough to guarantee an upper bound. Let $W_i(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ of weight exactly $i$. By splitting the event $\{\vs \in \mathsf{ColSpace}(\mathbf{H}_{S})\}$ into the union of the two disjoint events $\{\vs \in \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})\}$ and $\{\vs \in \mathsf{ColSpace}(\mathbf{H}_{S})\setminus \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})\}$ and using that $\rk(\mathbf{H}_{\supp(\vc)})\le i-1$ and $\rk(\mathbf{H}_{S})\le \rk(\mathbf{H}_{\supp(\vc)})+(t-i)$, we thus obtain an upper bound on the sought probability: \begin{align} &\mathbb{P}\left(\bigcup_{\substack{S\subseteq [n] \\ \|S\|=t \land \rk(\mathbf{H}_S)<t}} \{\vs\in \mathsf{ColSpace}(\mathbf{H}_S)\}\right)\\ \le &\sum_{\substack{S\subseteq [n] \\ \|S\|=t \land \rk(\mathbf{H}_S)<t}}\mathbb{P}(\{\vs\in \mathsf{ColSpace}(\mathbf{H}_S)\})\\ \le & \sum_{\substack{\vc \in \mathcal{C} \\ \wt(\vc)\le t} } \left( \mathbb{P}(\{\vs\in \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})\}) + \sum_{\substack{\supp(\vc)\subseteq S\subseteq [n] \\ \|S\|=t }} \mathbb{P}(\{\vs\in \mathsf{ColSpace}(\mathbf{H}_S)\setminus \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})\})\right)\\ = & \sum_{i=1}^t W_i(\mathcal{C}) \left( \frac{q^{\dim_{\FF_q} \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}}+ \binom{n-i}{t-i}\frac{q^{\dim_{\FF_q} \mathsf{ColSpace}(\mathbf{H}_{S})}-q^{\dim_{\FF_q} \mathsf{ColSpace}(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}} \right)\\ = & \sum_{i=1}^t W_i(\mathcal{C}) \left( \frac{q^{\rk(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}}+ \binom{n-i}{t-i}\frac{q^{\rk(\mathbf{H}_{S})}-q^{\rk(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}} \right)\\ \le & \sum_{i=1}^t W_i(\mathcal{C}) \left( \frac{q^{\rk(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}}+ \binom{n-i}{t-i}\frac{q^{\rk(\mathbf{H}_{\supp(\vc)})}}{q^{n-k}}(q^{t-i}-1) \right)\\ \le & \sum_{i=1}^t W_i(\mathcal{C}) \left( \frac{1}{q^{n-k-i+1}}+ \binom{n-i}{t-i} \left(\frac{1}{q^{n-k-t+1}}-\frac{1}{q^{n-k-i+1}}\right) \right). \end{align} Finally, in the case of the codeword finding problem, i.e. if the syndrome is the zero vector, the condition on the positive dimension of the variety boils down to the existence of the set $S$, $\|S\|=t$, such that the rank $\mathbf{H}_S$ is defective, as the zero vector belongs to any linear subspace. Therefore, in this case, the calculations are simplified into: \[ \mathbb{P}\left(\bigcup_{\substack{S\subseteq [n] \\ \|S\|=t \land \rk(\mathbf{H}_S)<t}} \{0\in \mathsf{ColSpace}(\mathbf{H}_S)\}\right) \le\sum_{\substack{S\subseteq [n] \\ \|S\|=t \land \rk(\mathbf{H}_S)<t}} 1 \le \sum_{i=1}^t W_i(\mathcal{C}) \binom{n-i}{t-i} . \] \qed\end{proof} \begin{remark} For random codes, the weight distribution follows closely the Bernoulli one, i.e. $W_i(\mathcal{C})\approx \frac{\binom{n}{i}(q-1)^i}{q^{n-k}}$. Under this assumption, the probability of having a zero-dimensional ideal for the decoding modeling with a random syndrome can be estimated as \[ \frac{1}{q^{n-k}}\sum_{i=1}^t \binom{n}{i}(q-1)^i\left(\frac{1}{q^{n-k-i+1}}+\binom{n-i}{t-i}\left(\frac{1}{q^{n-k-t+1}}-\frac{1}{q^{n-k-i+1}}\right)\right), \] while for the codeword finding problem as \[ \frac{1}{q^{n-k}}\sum_{i=1}^t \binom{n}{i}\binom{n-i}{t-i}(q-1)^i. \] We remark in particular that the bound is independent from the choice of $(r_1,r_2)$. Different admissible pairs provide of course different solutions, but the projections of the varieties with respect to the $x_i$'s variables are the same over the field closure, which is what determines the dimension of the associated ideals. \end{remark} In Appendix~\ref{app:Dim_Exp}, Tables \ref{table: bound_SD} and \ref{table: bound_CF}, we provide examples of such bounds for concrete parameters. In the case of syndrome decoding, these probabilities are very small even at Gilbert-Varshamov distance and the issue of having ideals of positive dimension is thus absolutely negligible for the purpose of cryptanalysis. On the opposite, the bound on the probability of the same event for the codeword finding problem becomes completely useless when approaching the Gilbert-Varshamov distance. Indeed, the trivial upper bound ``$\pr \le 1$'' entries from the two tables mean that the bound given by Proposition \ref{prop: bound} gives a number larger than 1. A possible workaround for the described issue with the codeword finding version is to make use of hybrid methods. Indeed, it is enough to guess a number of nonzero positions equal or greater than the ideal dimension to decrease the latter to 0 with high probability. Recalling that the solution space is projective and one the value of one nonzero entry can be chosen arbitrarily, specializing $l$ coordinates has a success probability of $\frac{\binom{n-l}{t-l}}{\binom{n}{t} (q-1)^{l-1}}$. In cryptanalysis, however, it is usually assumed to know the minimal weight of a (nonzero) solution. This is because, if there exists a solution of weight smaller than the target, then the challenge is actually easier. A simple strategy to obtain a zero-dimensional ideal in this setting is thus the following. If we suppose to know a lower bound $d'$ the minimum distance $d(C)$ of the code, then it means that any $d'-1$ columns of the parity-check matrix $\mathbf{H}$ are linearly independent. Therefore, Equation~\eqref{eq: dimension_Ideal} implies that the dimension of the ideal for the bounded weight modeling with target weight $d'$ is exactly 1 and thus it is enough to specialize one variable $x_i$ to any element in $\FF_q$ to obtain a zero-dimensional ideal. \begin{table}[h] \resizebox{\textwidth}{!}{\begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline \cellcolor{gray!20!} & \cellcolor{gray!20!} &\cellcolor{gray!20!} & \cellcolor{gray!20!} & \multicolumn{5}{\|c\|}{\cellcolor{gray!20!}$q=7$} & \multicolumn{5}{\|c\|}{\cellcolor{gray!20!}$q=16,17$}& \multicolumn{5}{\|c\|}{\cellcolor{gray!20!}$q=127$}\\ \hline \cellcolor{gray!20!}$n$ & \cellcolor{gray!20!}$k$ &\cellcolor{gray!20!} $t$ & \cellcolor{gray!20!}\#lin & \cellcolor{gray!20!}$r_1$ & \cellcolor{gray!20!}$r_2$ & \cellcolor{gray!20!}\#quad & \cellcolor{gray!20!}\#vars & \cellcolor{gray!20!}$d_{\mathrm{M}}$ & \cellcolor{gray!20!}$r_1$ & \cellcolor{gray!20!}$r_2$ & \cellcolor{gray!20!}\#quad & \cellcolor{gray!20!}\#vars & \cellcolor{gray!20!}$d_{\mathrm{M}}$ & \cellcolor{gray!20!}$r_1$ & \cellcolor{gray!20!}$r_2$ & \cellcolor{gray!20!}\#quad & \cellcolor{gray!20!}\#vars & \cellcolor{gray!20!}$d_{\mathrm{M}}$ \\ \hline 10 & 5 & 2 & 5 & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=1$} & \multicolumn{5}{\|c\|}{$m=1$} \\ \cline{5-19} &&&& 2 & 1 & 30& 30& 4 & 1 & 1 & 30 & 30 & 4 & 1 & 1 & 30 & 30 & 4 \\ &&&& 1 & 2 & 40& 40& 3 &&&&& &&&&& \\ \hline 15 & 9 & 3 & 6 & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=1$} & \multicolumn{5}{\|c\|}{$m=1$} \\ \cline{5-19} &&&& 2 & 1 & 45& 45& 4 & 1 & 1 & 45 & 45 & 4 & 1 & 1 & 45 & 45 & 4 \\ &&&& 1 & 2 & 60& 60& 4 &&&&& &&&&& \\ \hline 19 & 10 & 5 & 9 & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=1$} & \multicolumn{5}{\|c\|}{$m=1$} \\ \cline{5-19} &&&& 2 & 1 & 57& 57& 5 & 1 & 1 & 57 & 57 & 5 & 1 & 1 & 57 & 57 & 5 \\ &&&& 1 & 2 & 76& 76& 4 &&&&& &&&&& \\ \hline 22 & 14 & 4 & 8 & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=1$} \\ \cline{5-19} &&&& 2 & 1 & 66& 66& 5 & 2 & 1 & 66& 66& 5 & 1 & 1 & 66 & 66 & 4 \\ &&&& 1 & 2 & 88& 88 & 4 & 1 & 2 & 88& 88 & 4 &&&&& \\ \hline 30 & 20 & 4 & 10 & \multicolumn{5}{\|c\|}{$m=3$} & \multicolumn{5}{\|c\|}{$m=2$} & \multicolumn{5}{\|c\|}{$m=1$} \\ \cline{5-19} &&&& 3 & 1 & 90& 90& $\ge 6$ & 2 & 1 & 90& 90& $\ge 6$ & 1 & 1 & 90 & 90 & $\ge 6$ \\ &&&& 2 & 2 & 120& 120 & 4 & 1 & 2 & 120& 120 & 5 &&&&& \\ &&&& 1 & 3 & 150& 150& 4 &&&&&& &&&&\\ \hline \end{tabular} } \vspace{2mm} \caption{This table gives information from experiments using random $\FF_q$-linear codes using Modeling~\ref{modeling: quadBSD_Fq}. The values in the $d_{\mathrm{M}}$ column represent the highest step degree achieved when directly computing the Gröbner basis of the system in MAGMA. The column ``\#lin'' denotes the number of linear equations, i.e. of parity-check equations, which is independent from the field size. The columns ``\#quad'' and ``\#vars'' stand for the number of quadratic equations and the number of variables, which depend on the value $r_2$ instead. The integer $r_1$ is the extension field degree over which the equations are defined. We recall that the value $m$ leads to different possible choices of $(r_1,r_2)$ and we give all minima with respect to the standard partial order on pairs.} \label{Tab:qNEQ2-SolveDeg} \end{table} \begin{table}[] \centering \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \multicolumn{9}{c}{$q=7$} \\ \hline $n$ & $k$ & $t$ & \# Lin Eqs & \# Quad Eqs & \# Vars & SR $d_{reg}$ & SD & \# Solutions \\ \hline 8 & 2 & 2 & 9 & 32 & 33 & 8 & 3 & 1 \\ \hline 8 & 4 & 1 & 7 & 32 & 33 & 10 & 3 & 6 \\ \hline 10 & 5 & 1 & 8 & 40 & 41 & 12 & 3 & 2 \\ \hline 16 & 2 & 6 & 17 & 64 & 65 & 13 & 3 & 1 \\ \hline 16 & 3 & 4 & 16 & 64 & 65 & 14 & 3 & 1 \\ \hline 16 & 4 & 4 & 15 & 64 & 65 & 14 & 3 & 1 \\ \hline 16 & 8 & 2 & 11 & 64 & 65 & 18 & 4 & 1 \\ \hline 16 & 8 & 1 & 11 & 64 & 65 & 18 & 4 & 1 \\ \hline 20 & 10 & 2 & 13 & 80 & 81 & 21 & 4 & 1 \\ \hline \end{tabular} \caption{This table gives information from experiments using Random $\mathbb{F}_7$-linear codes using the {\color{red} do we have a name?} modeling. The value $\mathrm{sd}(S)$ is the solving degree of $S$, computed as the highest step degree achieved when directly computing the Gr\"obner basis of the system in MAGMA. The values in the SD column give the highest step degree achieved when directly computing the Gr\"obner basis of the system in MAGMA. The SR $d_{reg}$ column gives the degree of regularity of a Semi-Regular system of equations with the associated number of linear equations, quadratic equations, and variables. }\label{Tab:q2-SolveDeg} \end{table}
2412.04932v1	http://arxiv.org/abs/2412.04932v1	Trickle groups	\pdfsuppresswarningpagegroup=1 \documentclass{louloupart1} \usepackage{pinlabel} \usepackage{graphicx} \usepackage[all]{xy} \usepackage{tikz} \usetikzlibrary{decorations.markings,arrows,arrows.meta,positioning} \tikzset{->-/.style={decoration={ markings, mark=at position #1 with {\arrow{Computer Modern Rightarrow[length=5pt,width=5pt]}}},postaction={decorate}}} \tikzset{->-rev/.style={decoration={ markings, mark=at position #1 with {\arrow{Computer Modern Rightarrow[length=5pt,width=5pt,reversed]}}},postaction={decorate}}} \title[Trickle groups]{Trickle groups} \author[P Bellingeri]{Paolo Bellingeri} \givenname{Paolo} \surname{Bellingeri} \address{Paolo Bellingeri, Laboratoire Nicolas Oresme, UMR 6139, CNRS, Université de Caen-Normandie, 14032 Caen Cedex, France \vskip 3 pt Eddy Godelle, Laboratoire Nicolas Oresme, UMR 6139, CNRS, Université de Caen-Normandie, 14032 Caen Cedex, France \vskip 3 pt Luis Paris, Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université de Bourgogne, 21000 Dijon, France} \email{[email protected]} \author[E Godelle]{Eddy Godelle} \givenname{Eddy} \surname{Godelle} \email{[email protected]} \author[L Paris]{Luis Paris} \givenname{Luis} \surname{Paris} \email{[email protected]} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{corl}[thm]{Corollary} \theoremstyle{definition} \newtheorem{defin}{Definition} \newtheorem{rem}{Remark} \newtheorem{expl}{Example} \newtheorem{acknow}{Acknowledgments} \newtheorem{expl1}{Example 1} \newtheorem{expl2}{Example 2} \newtheorem{expl3}{Example 3} \numberwithin{equation}{section} \makeatletter \@arabic\c@figure} \@addtoreset{figure}{section} \makeatother \begin{document} \def\N{\mathbb N} \def\R{\mathbb R} \def\SSS{\mathfrak S} \def\SS{\mathcal S} \def\id{{\rm id}} \def\VJ{{\rm VJ}} \def\KJ{{\rm KJ}} \def\link{{\rm link}} \def\starE{{\rm star}} \def\Tr{{\rm Tr}} \def\Z{\mathbb Z} \def\supp{{\rm supp}} \def\st{{\rm st}} \def\pil{{\rm pil}} \def\RR{\mathcal R} \def\LL{\mathcal L} \def\nf{{\rm nf}} \def\syl{{\rm syl}} \def\FF{\mathcal F} \def\AC{{\rm AC}} \def\Div{{\rm Div}} \def\Ker{{\rm Ker}} \def\KVJ{{\rm KVJ}} \def\PVJ{{\rm PVJ}} n{{\rm fi}} \def\linkE{{\rm link}} \def\Aut{{\rm Aut}} \def\AA{\mathcal A} \begin{abstract} A new family of groups, called trickle groups, is presented. These groups generalize right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a presentation with relations of the form $xy = zx$ and $x^\mu = 1$, that are governed by a simplicial graph, called a trickle graph, endowed with a partial ordering on the vertices, a vertex labeling, and an automorphism of the star of each vertex. We show several examples of trickle groups, including extended cactus groups, certain finite-index subgroups of virtual cactus groups, Thompson group F, and ordered quandle groups. A terminating and confluent rewriting system is established for trickle groups, enabling the definition of normal forms and a solution to the word problem. An alternative solution to the word problem is also presented, offering a simpler formulation akin to Tits' approach for Coxeter groups and Green's for graph products of cyclic groups. A natural notion of a parabolic subgraph of a trickle graph is introduced. The subgroup generated by the vertices of such a subgraph is called a standard parabolic subgroup and it is shown to be the trickle group associated with the subgraph itself. The intersection of two standard parabolic subgroups is also proven to be a standard parabolic subgroup. If only relations of the form $xy = zx$ are retained in the definition of a trickle group, then the resulting group is called a preGarside trickle group. Such a group is proved to be a preGarside group, a torsion-free group, and a Garside group if and only if its associated trickle graph is finite and complete. \smallskip\noindent {\bf AMS Subject Classification\ \ } Primary: 20F10, Secondary: 20F05, 20F36, 20F55, 20F65. \smallskip\noindent {\bf Keywords\ \ } Trickle groups, right-angled Coxeter groups, right-angled Artin groups, cactus groups, virtual cactus groups, Thompson group F, word problem, rewriting systems, preGarside groups, Garside groups. \end{abstract} \maketitle \section{Introduction}\label{sec1} There are numerous groups in the literature defined by relations of the form $xy = zx$, often with additional constraints on the orders of generators. Prominent examples include right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products of cyclic groups. The aim of the present paper is to study a specific family of such groups that we call \emph{trickle groups}. Cactus groups are emblematic examples of trickle groups. These groups first appeared as quasi-braid groups in the study of the mosaic operad \cite{Devad1,EHKR1,KhWi1} and they were subsequently generalized to all Coxeter groups \cite{DaJaSc1}. Their significance was further highlighted in their connection to coboundary categories \cite{HenKam1}, mirroring the role of braid groups in braided categories. Note that the term ``cactus groups'' was coined in \cite{HenKam1}. Additionally, cactus groups and their generalizations to Coxeter groups have found applications in representation theory under various guises \cite{KnTaWo1,Bonna1,Losev1,ChGlPy1,RouWhi1}. For $n \in \N_{\ge 2}$, the \emph{cactus group} $J_n$ is defined by the presentation with generators $x_{p,q}$, $1 \le p < q \le n$, and relations: \begin{itemize} \item[(j1)] $x_{p,q}^2 = 1$, for $1 \le p <q \le n$, \item[(j2)] $x_{p,q} x_{m,r} = x_{m,r} x_{p,q}$, for $[p,q] \cap [m,r] = \emptyset$, \item[(j3)] $x_{p,q} x_{m,r} = x_{p+q-r,p+q-m} x_{p,q}$, for $[m,r]\subset[p,q]$. \end{itemize} The elements of $J_n$ are often depicted using planar diagrams. More precisely, an element $g \in J_n$ is represented by an $n$-tuple of smooth paths in the plane, $b=(b_1,\dots,b_n)$, $b_i : [0,1] \to \R^2$, satisfying the following conditions. \begin{itemize} \item There exists a permutation $\sigma \in \SSS_n$ such that $b_i(0)=(0,i)$ and $b_i(1)=(1,\sigma(i))$, for all $i\in \{1,\dots, n\}$. \item For all $t\in [0,1]$ and all $i \in \{1, \dots, n\}$ we have $\pi_1(b_i(t))=t$, where $\pi_1 : \R^2 \to \R$ denotes the projection onto the first coordinate. \item Crossings between the $b_i$'s may be multiple but are always transversal. \end{itemize} The generator $x_{p,q}$ is represented in Figure \ref{fig1_1} and relation (j3) is illustrated in Figure \ref{fig1_2}. \begin{figure}[ht!] \begin{center} \includegraphics[width=2.8cm]{BeGoPaFig1_1.pdf} \caption{Generator of $J_n$}\label{fig1_1} \end{center} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[width=8.4cm]{BeGoPaFig1_2.pdf} \caption{Relation (j3) in the presentation of $J_n$}\label{fig1_2} \end{center} \end{figure} The extension of this definition to Coxeter groups is straightforward. Let $(W,S)$ be a Coxeter system associated with a Coxeter graph $\Upsilon$. For $X \subseteq S$, the subgroup of $W$ generated by $X$ is called a \emph{standard parabolic subgroup} and is denoted by $W_X$, and the full Coxeter subgraph of $\Upsilon$ spanned by $X$ is denoted by $\Upsilon_X$. We say that $X \subseteq S$ is \emph{irreducible} if $\Upsilon_X$ is connected, and we say that $X$ is of \emph{spherical type} if $W_X$ is finite. In the latter case $W_X$ contains a unique element of maximal length (with respect to $S$), denoted by $w_X$, and this element satisfies $w_X X w_X^{-1}=X$ and $w_X^2=\id$ (see \cite{Bourb1}). n}$. n}$, and relations: \begin{itemize} \item[(j1)] n}$, \item[(j2)] $x_X x_Y = x_Y x_X$, if $\Upsilon_{X\cup Y}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Y$, meaning that $X \cap Y = \emptyset$ and $st = ts$ for all $s \in X$ and $t \in Y$, \item[(j3)] $x_X x_Y = x_{w_X(Y)} x_X$, for $Y \subset X$. \end{itemize} Our aim is to investigate the combinatorial properties of these groups within a broader framework that encompasses various more or less natural generalizations of cactus groups. Let $\Gamma$ be a simplicial graph. We denote the vertex set of $\Gamma$ by $V (\Gamma)$ and the edge set by $E (\Gamma)$. The \emph{link} of a vertex $x \in V(\Gamma)$, denoted by $\linkE_x (\Gamma)$, is the full subgraph of $\Gamma$ spanned by $\{y \in V (\Gamma) \mid \{x, y\} \in E(\Gamma)\}$. The \emph{star} of $x$, denoted by $\starE_x (\Gamma)$, is the full subgraph of $\Gamma$ spanned by $\{y\in V (\Gamma) \mid \{x, y\} \in E(\Gamma)\}\cup \{x\}$. A \emph{trickle graph} is a quadruple $(\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$, where: \begin{itemize} \item $\Gamma$ is a simplicial graph, \item $\le$ is a (partial) order on $V (\Gamma)$, \item $\mu: V(\Gamma) \to \N_{\ge 2} \cup \{\infty\}$ is a vertex labeling, \item $\varphi_x: \starE_x (\Gamma) \to \starE_x (\Gamma)$ is an automorphism of $\starE_x (\Gamma)$ for all $x \in V(\Gamma)$, \end{itemize} that must satisfy certain conditions defined in Section \ref{sec2}. The \emph{trickle group} $\Tr (\Gamma)$ associated with a trickle graph $\Gamma = (\Gamma, \le, \mu,(\varphi_x)_{x\in V(\Gamma)})$ is the group defined by the following presentation. \begin{gather} \Tr (\Gamma) = \langle V(\Gamma) \mid x^{\mu(x)} = 1 \text{ for all } x \in V (\Gamma) \text{ such that } \mu (x) \neq \infty\,,\ \varphi_x(y)\,x = \varphi_y(x)\,y \\ \text{ for all } \{x, y\} \in E(\Gamma) \rangle\,. \end{gather} One of the conditions in the definition of a trickle graph in Section \ref{sec2} entails that, for any edge $\{x,y\} \in E (\Gamma)$, either $\varphi_x (y) = y$ or $\varphi_y (x) = x$. Consequently, the above presentation is indeed a presentation with relations of the form $xy=zx$ and $x^\mu=1$. Another immediate consequence of these conditions is that the trivial order is admissible, and therefore right-angled Coxeter groups, right-angled Artin groups and, more generally, graph products of cyclic groups are trickle groups. As mentioned above, cactus groups are trickle groups, where here $\mu(x) = 2$ for all $x \in V (\Gamma)$. Trickle groups include other groups naturally related to cactus groups, such as the ``Artin'' versions of cactus groups (associated with Coxeter groups), where we keep relations (j2) and (j3) and ignore relations (j1). In this case, such a group is also a preGarside group in the sense of \cite{GodPar2} (see Section \ref{sec7} and Subsection \ref{subsec2_5}). More generally, the example of cactus groups associated with Coxeter systems can be naturally extended to families of subgroups of a given group $G$ satisfying certain properties that are presented in Subsection \ref{subsec3_1}. Our study is also an opportunity to investigate \emph{virtual cactus groups} since, as we will see in Subsection \ref{subsec3_2}, they contain finite index subgroups that are trickle groups. Let $S_1, \dots, S_\ell$ be a collection of $\ell$ circles immersed in the plane having only double transverse crossings. We assign to each crossing a ``positive'', ``negative'' or ``virtual'' value that we indicate on the graphical representation of $S_1 \cup \cdots \cup S_\ell$ as in Figure \ref{fig1_3}. Such a figure is called a \emph{virtual link diagram}. We consider the equivalence relation on the set of virtual link diagrams generated by the isotopy and the virtual Reidemeister moves as defined by Kauffman \cite{Kauff1,Kauff2}. An equivalence class of virtual link diagrams is a \emph{virtual link}. \begin{figure}[ht!] \begin{center} \includegraphics[width=4.4cm]{BeGoPaFig1_3.pdf} \caption{Crossings in a virtual link diagram}\label{fig1_3} \end{center} \end{figure} Since the publication of Kauffman's seminal paper \cite{Kauff1}, the theory of virtual knots and links has grown significantly, and this notion has been extended to other combinatorial and/or topological objects represented by planar diagrams. The leitmotif underlying these theories is that two arcs connecting the same points and passing only through virtual crossings are equivalent. Since the elements of the cactus group $J_n$ are represented by planar diagrams, it is natural to extend $J_n$ by adding virtual crossings to the cactus crossings while keeping the principle that two arcs connecting the same points and passing only through virtual crossings are equivalent. Then we obtain the \emph{virtual cactus group}, $\VJ_n$, which will be studied in detail in Subsection \ref{subsec3_2}. Note that these groups are not new, having been introduced in \cite{IKLPR1}, where it is shown that $\VJ_n$ is the $\SSS_n$-equivariant fundamental group of the real form of the ``cactus flower moduli space'' $\bar F_n$. The connection between virtual cactus groups and trickle groups mirrors that between virtual braid groups and Artin groups \cite{GodPar1, BeCiPa1, BelPar1, BePaTh1}. We prove that $\VJ_n$ can be decomposed as a semi-direct product $\VJ_n = \KJ_n \rtimes \SSS_n$, where $\SSS_n$ is the symmetric group on $\{1,\dots, n\}$, and $\KJ_n$ is a trickle group (see Proposition \ref{prop3_7}). This decomposition enables us to address the word problem in $\VJ_n$, to define normal forms for the elements of $\VJ_n$, and to show that $J_n$ embeds into $\VJ_n$. It is probable that numerous other trickle groups exist within the literature. Among these, we have pinpointed two specific examples: one originating from dynamical systems and the other from knot theory. Thompson group $F$, introduced by Richard Thompson in 1965 in an unpublished manuscript, is a group of homeomorphisms of the real line with many unusual properties. For instance, its derived group $F'$ is a simple group, the quotient $F/F'$ is a free abelian group of rank $2$, and it contains no subgroup isomorphic to the rank $2$ free group. We refer to \cite{CaFlPa1} for a general overview on this group. A well-known presentation for $F$ is as follows: \[ \langle x_n\,,\ n\in \N \mid x_k x_n = x_{n+1} x_k \text{ for } k < n \rangle\,, \] which, while featuring relations of the form $xy=zx$, does not fulfill all the requirements of a trickle group presentation. However, in Subsection \ref{subsec3_3} we show another presentation for $F$ that is indeed a trickle presentation (see Theorem \ref{thm3_15}). So, Thompson group $F$ is a trickle group. This structure is particularly noteworthy as any ``natural standard parabolic subgroup'' of $F$ is a copy of $F$ itself (see Proposition \ref{prop3_16}). Moreover, Theorem \ref{thm2_14} will imply that $F$ is a preGarside group. Quandles are algebraic structures whose axioms reproduce the Reidemeister moves in knot theory. Independently introduced by Joyce \cite{Joyce1} and Matveev \cite{Matve1}, they have been frequently used to construct knot or link invariants. As noted by Joyce \cite{Joyce1} and Matveev \cite{Matve1}, a classification of quandles would de facto entail a classification of knots. This explains the difficulty of studying the set of all quandles, leading researchers to focus on specific families of quandles. Ordered quandles were introduced recently in this perspective \cite{BaPaSi1,DDHPV1}, and it turns out that they are a disguised form of trickle graphs. The details of this construction are given in Subsection \ref{subsec3_4}. As previously mentioned, the objective of this paper is a combinatorial study of trickle groups. In Section \ref{sec4} we determine a confluent and terminating rewriting system for these groups (see Theorem \ref{thm2_4}). According to Newman \cite{Newma1}, this enables the definition of normal forms for their elements and, consequently, a solution to the word problem (see Corollary \ref{corl2_5}). Moreover, it yields other immediate results, such as a characterization of finite trickle groups (see Corollary \ref{corl2_7}). Notice that our rewriting system and its associated normal forms are not novel for graph products of cyclic groups \cite{Wyk1,HerMei1,CrGoWi1}, and a similar rewriting system for classical cactus groups was considered in \cite{Genev1}. Note also that the term ``trickle'', used to designate trickle groups, originates from this algorithm because it metaphorically involves pushing down as many syllables as possible using only relations of the form $xy = zx$. The remaining sections of the paper explore how trickle groups behave in a manner analogous to groups studied in the theory of Coxeter, Artin, and Garside groups. Building upon the algorithms and methods introduced in Section \ref{sec4}, we present in Section \ref{sec5} an alternative algorithm for solving the word problem in a trickle group. This new algorithm offers a simpler formulation compared to the one presented in Section \ref{sec4}. Furthermore, it aligns more closely with the algorithms described in \cite{Tits1} for Coxeter groups, in \cite{Green1} for graph products of cyclic groups, and more generally, in \cite{ParSoe1} for Dyer groups. A natural notion of a \emph{parabolic subgraph} emerges in the context of trickle graphs. This leads to the natural question of studying the trickle groups defined by such subgraphs. Drawing a parallel with the theory of Coxeter and Artin groups, we refer to these as \emph{standard parabolic subgroups}. As the name suggests, we establish in Section \ref{sec6} that a standard parabolic subgroup is indeed a subgroup of the trickle group associated with the original graph (see Theorem \ref{thm2_10}). Moreover, we prove that the intersection of two standard parabolic subgroups is a standard parabolic subgroup (see Corollary \ref{corl2_12}). Section \ref{sec7} establishes a connection between trickle groups and Garside theory. A \emph{preGarside trickle graph} is defined as a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ for which $\mu (x) = \infty$ for all $x \in V (\Gamma)$. A \emph{preGarside trickle group} is a trickle group associated with a preGarside trickle graph. Additionally, one can define an associated \emph{preGarside trickle monoid} using the same presentation, but interpreted as a monoid presentation. The term ``preGarside'' originates from \cite{GodPar2}, where the authors investigate monoids and groups that they call preGarside monoids and preGarside groups. Notable examples of such monoids and groups include all Artin monoids and all Artin groups. Garside monoids and Garside groups are also preGarside monoids and preGarside groups. In Section \ref{sec7}, we prove that a preGarside trickle monoid is indeed a preGarside monoid and that a preGarside trickle group is a preGarside group (see Theorem \ref{thm2_14}). Furthermore, the monoid and the group are respectively a Garside monoid and a Garside group if and only if $\Gamma$ is finite and complete (see Theorem \ref{thm2_15}). This introduces new examples of Garside groups to which we can associate Coxeter-style quotients. Another objective of Section \ref{sec7} is to address certain questions that arise for preGarside monoids and groups within the context of preGarside trickle monoids and groups. First, we already know from Section \ref{sec4} that a preGarisde trickle group has a solution to the word problem. Then, we prove in Section \ref{sec7} that, if the vertex set of the graph is finite, then the preGarside trickle group is torsion-free (see Theorem \ref{thm2_16}). The remaining questions concern the relationship between monoids and groups. These inquiries, posed in \cite{GodPar2}, are specifically focused on preGarside monoids and groups. We prove that a preGarside trickle monoid embeds into its enveloping group (see Theorem \ref{thm2_17}) and we prove several results concerning its parabolic submonoids and subgroups (see Theorem \ref{thm2_18}). Trickle groups appear to be a reasonable generalization of graph products of cyclic groups, and consequently, of right-angled Artin groups and right-angled Coxeter groups. Therefore, it is natural to explore whether results known for some or all graph products of cyclic groups can be extended to some or all trickle groups. Relevant questions in this direction include: \begin{itemize} \item[(1)] Do the normal forms described in Section \ref{sec4} form a regular language? Are trickle groups automatic or bi-automatic? \item[(2)] Which trickle groups admit geometric actions on CAT(0) cube complexes? \item[(3)] Are trickle groups residually finite? Are preGarside trickle groups residually nilpotent without torsion? Can we determine the Lie algebra associated with their lower central series, as done for right-angled Artin groups (see \cite{DucKro1})? \item[(4)] Which preGarside trickle groups are orderable, bi-orderable, or admit isolated orders? \end{itemize} Additionally, it would be valuable to discover new examples of trickle groups, particularly those arising from areas of mathematics beyond group theory. The paper is organized as follows. Section \ref{sec2} presents the fundamental definitions and precise statements of our main results. It is divided into five subsections. In the first subsection we introduce the concepts of trickle graphs and trickle groups, along with illustrative examples like graph products of cyclic groups. In Subsection \ref{subsec2_2} we state the main results of Section \ref{sec4}, which focuses on the trickle algorithm. In Subsection \ref{subsec2_3} we state the main results of Section \ref{sec5}, which focuses on the Tits-style algorithm. In Subsection \ref{subsec2_4} we state the main results of Section \ref{sec6}, which focuses on parabolic subgroups. In Subsection \ref{subsec2_5} we state the main results of Section \ref{sec7}, which focuses on preGarside trickle groups. Section \ref{sec3} is devoted to examples and it is divided into four subsections: Subsection \ref{subsec3_1} for cactus groups in their generalized version, Subsection \ref{subsec3_2} for virtual cactus groups, Subsection \ref{subsec3_3} for Thompson group F, and Subsection \ref{subsec3_4} for ordered quandle groups. As mentioned earlier, Sections \ref{sec4}, \ref{sec5}, \ref{sec6} and \ref{sec7} contain the proofs: those concerning the trickle algorithm in Section \ref{sec4}, those concerning the Tits-style algorithm in Section \ref{sec5}, those concerning parabolic subgroups in Section \ref{sec6}, and those concerning preGarside trickle groups in Section \ref{sec7}. \begin{acknow} This work originated during a research residency program titled ``Cactus and Posets'' at CIRM (Luminy, Marseille, France) from November 7th to 11th, 2022. The three authors extend their sincere gratitude to CIRM for the generous support and resources provided (funding, dedicated workspaces, library access, etc.), without which this project would not have been possible. \end{acknow} \section{Definitions and statements}\label{sec2} \subsection{Definitions and first examples}\label{subsec2_1} The set of vertices of a simplicial graph $\Gamma$ is denoted by $V(\Gamma)$ and the set of its edges is denoted by $E(\Gamma)$. The \emph{link} of a vertex $x \in V(\Gamma)$, denoted by $\link_x(\Gamma)$, is the full subgraph of $\Gamma$ spanned by $\{y \in V (\Gamma) \mid \{x, y\} \in E (\Gamma) \}$, and the \emph{star} of $x$, denoted by $\starE_x (\Gamma)$, is the full subgraph of $\Gamma$ spanned by $\{y \in V(\Gamma) \mid \{x, y\} \in E (\Gamma)\} \cup \{x\}$. \begin{defin} A \emph{trickle graph} is a quadruple $(\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$, where \begin{itemize} \item $\Gamma$ is a simplicial graph, \item $\le$ is a (partial) order on $V (\Gamma)$, \item $\mu$ is a labeling $\mu: V(\Gamma) \to \N_{\ge 2} \cup \{\infty\}$ of the vertices, \item $\varphi_x: \starE_x (\Gamma) \to \starE_x (\Gamma)$ is an automorphism of $\starE_x (\Gamma)$ for all $x \in V (\Gamma)$. \end{itemize} For $x, y \in V (\Gamma)$ the notation $x\|\|y$ means that $x$ and $y$ are not comparable in the sense that $x \not \le y$ and $y \not \le x$. We set $E_{\|\|} (\Gamma) = \{\{x, y\} \in E(\Gamma) \mid x\|\|y \}$. The quadruple $(\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ must satisfy the following conditions. \begin{itemize} \item[(a)] For all $x, y \in V (\Gamma)$, if $x < y$, then $\{x, y\} \in E(\Gamma)$. \item[(b)] For all $x, y, z \in V (\Gamma)$, if $\{x, y\} \in E_{\|\|} (\Gamma)$ and $z \le y$, then $\{ x, z\} \in E_{\|\|} (\Gamma)$. \item[(c)] For all $x \in V(\Gamma)$ and all $y, z \in \starE_x (\Gamma)$, we have $z\le y$ if and only if $\varphi_x(z) \le \varphi_x (y)$. \item[(d)] For all $x \in V(\Gamma)$ and all $ y \in \starE_x (\Gamma)$, if $\varphi_x (y) \neq y$, then $y < x$. \item[(e)] For all $x \in V (\Gamma)$, if $\mu (x)$ is finite, then $\varphi_x$ has finite order and its order divides $\mu (x)$. \item[(f)] For all $x \in V(\Gamma)$ and all $y \in \starE_x (\Gamma)$, $\mu (\varphi_x (y)) = \mu (y)$. \item[(g)] For all $x, y, z \in V (\Gamma)$, if $z < y < x$, then $(\varphi_x \circ \varphi_y) (z) = (\varphi_{y'} \circ \varphi_x) (z)$, where $y' = \varphi_x (y)$. \end{itemize} We will often say that $\Gamma$ is a trickle graph meaning that, implicitly, $\le$, $\mu$ and $(\varphi_x)_{x \in V (\Gamma)}$ are also given. \end{defin} \begin{rem} Let $x, y, z \in V(\Gamma)$ be such that $z \le y \le x$. Then it is easily seen that $(\varphi_x \circ \varphi_y) (z) = (\varphi_{y'} \circ \varphi_x) (z)$, where $ y' = \varphi_x (y)$, if either $z=y$ or $y=x$. So, Condition (g) in the definition of a trickle graph also holds if at least two of the three vertices are equal. \end{rem} \begin{defin} The \emph{trickle group} $\Tr (\Gamma)$ associated with a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma) })$ is the group defined by the following presentation. \[ \Tr (\Gamma) = \langle V(\Gamma) \mid x^{\mu (x)} = 1 \text{ for } x \in V (\Gamma) \text{ such that } \mu ( x) \neq \infty \,, \ \varphi_x (y) \, x = \varphi_y (x) \,y \text{ for } \{x, y\} \in E (\Gamma) \rangle\,. \] \end{defin} \begin{rem} By Condition (d) in the definition of a trickle graph, if $\{x, y\} \in E_{\|\|} (\Gamma)$, then the relation $\varphi_x (y)\, x = \varphi_y(x) \,y$ becomes $y x = x y$. If $x < y$, then this relation becomes $y x = \varphi_y (x) \, y$. So, $\Tr (\Gamma)$ has a presentation with relations of the form $x y = z x$ and $x^\mu=1$. \end{rem} \begin{expl1} Let $\Gamma$ be a simplicial graph and let $\mu : V(\Gamma) \to \N_{\ge 2} \cup \{ \infty\}$ be a labeling of the vertices. To the pair $(\Gamma, \mu)$ we associate the \emph{graph product of cyclic groups} $G(\Gamma, \mu)$ defined by the following presentation. \[ G (\Gamma, \mu) = \langle V (\Gamma) \mid x^{\mu (x)} = 1 \text{ for } x \in V (\Gamma) \text{ such that } \mu (x) \neq \infty\,,\ x y = y x \text{ for } \{x, y\} \in E(\Gamma) \rangle\,. \] If $\mu (x) = \infty$ for all $x \in V(\Gamma)$, then $G (\Gamma, \mu)$ is a \emph{right-angled Artin group}, and if $\mu (x) = 2$ for all $x \in V(\Gamma)$, then $G (\Gamma, \mu)$ is a \emph{right-angled Coxeter group}. It is easily seen that $G (\Gamma, \mu)$ is a trickle group, where $\le$ is the trivial order defined by $x \le y$ if and only if $x = y$, and $ \varphi_x$ is the identity of $\starE_x (\Gamma)$ for all $x \in V(\Gamma)$. \end{expl1} \begin{expl2} Le $\Upsilon$ be a Coxeter graph and let $(W,S)$ be its associated Coxeter system. As in the introduction, for $X \subseteq S$, we denote by $W_X$ the standard parabolic subgroup generated by $X$ and by $\Upsilon_X$ the full Coxeter subgraph of $\Upsilon$ spanned by $X$. Recall that $X \subseteq S$ is called \emph{irreducible} and of \emph{spherical type} if $\Upsilon_X$ is connected and $W_X$ is finite. Recall also that, in this case, $W_X$ contains a unique element of maximal length (with respect to $S$), denoted by $w_X$, and this element satisfies $w_X^2 = 1$ and $w_X X w_X = X$ (see \cite{Bourb1}). n}$ the set of non-empty subsets $X \subseteq S$ that are irreducible and of spherical type. We define $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ as follows. n$. Two vertices $x_X$ and $x_Y$ are connected by an edge if either $X \subset Y$, or $Y \subset X$, or $\Upsilon_{X \cup Y}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Y$ in the sense that $X \cap Y = \emptyset$ and $\{s, t\} \not \in E (\Upsilon)$ for all $s \in X$ and $t \in Y$. We set $x_X \le x_Y$ if $X \subseteq Y$. n}$. Let $x_X \in V (\Gamma)$ and $x_Y \in V (\starE_{x_X} (\Gamma))$. We set $\varphi_{x_X} (x_Y) = x_{w_X(Y)}$ if $Y \subset X$ and $\varphi_{x_X} (x_Y) = x_Y$ otherwise. It is easily verified that $\le$ is a (partial) order on $V (\Gamma)$ and that, for all $x_X \in V (\Gamma)$, $\varphi_{x_X}$ is an automorphism of $\starE_{x_X} (\Gamma)$. Now, we prove that $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ satisfies Conditions (a) to (g) of the definition of a trickle graph. \begin{lem}\label{lem2_1} The quadruple $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ above defined is a trickle graph. \end{lem} \begin{proof} Conditions (a), (d) and (f) are satisfied by definition of $\Gamma$. We show that $\Gamma$ satisfies Condition (b). n}$ be such that $\{x_X, x_Y\} \in E_{\|\|} (\Gamma)$ and $x_Z \le x_Y$. Then $\Upsilon_{X \cup Y}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Y$ and $Z \subset Y$. Thus, $\Upsilon_{X \cup Z}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Z$, hence $\{x_X, x_Z\} \in E_{\|\|} (\Gamma)$. We show that $\Gamma$ satisfies Condition (e). n}$. Since $w_X$ has order $2$, we have $\varphi_{x_X}^2 (x_Y) = x_{w_X^2(Y)} = x_Y$ if $Y \subset X$. We have $\varphi_{x_X}^2 (x_Y) = \varphi_{x_X} (x_Y) = x_Y$ for every other vertex of $\starE_{x_X} (\Gamma)$, hence the order of $\varphi_{x_X}$ divides $\mu(x_X) =2$. We show that $\Gamma$ satisfies Condition (c). n}$ be such that $x_Z, x_Y \in \starE_{x_X} (\Gamma)$. Since, by Condition (e) already proved, $\varphi_{x_X}^2 = \id$, to show the equivalence $x_Z \le x_Y\ \Leftrightarrow\ \varphi_{x_X} (x_Z) \le \varphi_{x_X} (x_Y)$, it suffices to show the implication $x_Z \le x_Y\ \Rightarrow\ \varphi_{x_X} (x_Z) \le \varphi_{x_X} (x_Y)$. Suppose $x_Z \le x_Y$, that is, $Z \subseteq Y$. If $\Upsilon_{X \cup Y}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Y$, then $\Upsilon_{X \cup Z}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Z $, hence $\varphi_{x_X} (x_Z) = x_Z \le x_Y = \varphi_{x_X} (x_Y)$. If $x_X \le x_Z \le x_Y$, then $X \subseteq Z \subseteq Y$, hence $\varphi_{x_X} (x_Z) = x_Z \le x_Y = \varphi_{x_X} (x_Y)$. If $x_Z \le x_X \le x_Y$, then $Z \subseteq X \subseteq Y$ hence $w_X (Z) \subseteq X \subseteq Y$, and therefore $\varphi_{x_X} (x_Z) = x_{w_X(Z)} \le x_X \le x_Y = \varphi_{x_X} (x_Y)$. If $x_X \le x_Y$ and $x_Z \|\| x_X$, then $X \subseteq Y$, $Z \subseteq Y$ and $\Upsilon_{X \cup Z}$ is the disjoint union of $\Upsilon_X$ and $\Upsilon_Z$, hence $\varphi_{x_X} (x_Z) = x_Z \le \varphi_{x_X} (x_Y) = x_Y$. If $x_Z \le x_Y \le x_X$, then $Z \subseteq Y \subseteq X$, hence $w_X(Z) \subseteq w_X(Y) \subseteq X$, and therefore $\varphi_{x_X} (x_Z) = x_{w_X(Z)} \le x_{w_X(Y)} = \varphi_{x_X} (x_Y)$. Finally we show that $\Gamma$ satisfies Condition (g). n}$ be such that $Z \subseteq Y \subseteq X$. Let $Y' = w_X (Y)$. We have \[ (w_X w_Y) (Z) = (w_X w_Y w_X^{-1} w_X) (Z) = (w_{w_X(Y)} w_X) (Z) = (w_{Y'} w_X) (Z)\,, \] hence $(\varphi_{x_X} \circ \varphi_{x_Y}) (x_Z) = (\varphi_{x_{Y'}} \circ \varphi_{x_X}) (x_Z)$. \end{proof} It is obvious that the cactus group $C(W,S)$ is equal to the trickle group $\Tr (\Gamma)$. \end{expl2} \begin{expl3} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph. Notice that $\widetilde{\Gamma} = (\Gamma, \le, \mu, (\varphi_x^{-1})_{x \in V(\Gamma)})$ is also a trickle graph which we call the \emph{dual trickle graph} of $\Gamma$. On the other hand, we can define the \emph{dual trickle group} $\widetilde{\Tr} (\Gamma)$ by the following presentation. \[ \widetilde{\Tr} (\Gamma) = \langle V (\Gamma) \mid x^{\mu(x)} = 1 \text{ for } x \in V (\Gamma) \text{ such that } \mu (x) \neq \infty\,,\ x\, \varphi_x (y) = y\, \varphi_y (x) \text{ for } \{ x, y \} \in E (\Gamma) \rangle\,. \] Trickle groups and dual trickle groups are related by the following. \begin{prop}\label{prop2_2} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph. Then $\widetilde{\Tr} (\widetilde{\Gamma})=\Tr (\Gamma)$. \end{prop} \begin{proof} The group $\widetilde{\Tr} (\widetilde{\Gamma})$ has the following presentation. \begin{gather} \widetilde{\Tr} (\widetilde{\Gamma}) = \langle V (\Gamma) \mid x^{\mu (x)} = 1 \text{ for } x \in V (\Gamma) \text{ such that } \mu (x) \neq \infty\,,\ x\, \varphi_x^{-1} (y) = y \, \varphi_y^{-1} (x)\\ \text{ for } \{x, y\} \in E(\Gamma) \rangle\,. \end{gather} Let $f : V(\Gamma) \to \widetilde{\Tr} (\widetilde{\Gamma})$ be the map defined by $f(x) = x$ for all $x \in V(\Gamma)$. Let $x \in V(\Gamma)$ be such that $\mu (x) \neq \infty$. Then $f(x)^{\mu (x)} = x^{\mu (x)} = 1$. Let $e = \{x, y\} \in E (\Gamma)$. If $x \|\| y$, then with $e$ we associate the relation $xy = yx$ in the presentation of $\Tr (\Gamma)$ as well as in that of $\widetilde{\Tr} (\widetilde{\Gamma})$. So, $f(x) \, f(y) = f(y) \, f(x)$. Suppose $x < y$. The case $y < x$ is treated in the same way. With $e$ we associate the relation $y x = \varphi_y(x)\,y$ in the presentation of $\Tr (\Gamma)$. Let $x' = \varphi_y (x)$. By definition of a trickle graph we have $e' =\{ x' ,y \} \in E(\Gamma)$ and $x' < y$. With $e'$ we associate the relation $y \, \varphi_y^{-1} (x') = x' y$ in the presentation of $\widetilde{\Tr} (\widetilde{\Gamma})$. But, since $x' = \varphi_y (x)$, this relation is also read $y x = \varphi_y(x)\, y$. Thus, $f(y) \, f(x) = f(\varphi_y(x)) \, f(y)$. This shows that $f$ induces a homomorphism $f : \Tr (\Gamma) \to \widetilde{\Tr} (\widetilde{\Gamma})$. We show in the same way that we have a homomorphism $f': \widetilde{\Tr} (\widetilde{\Gamma}) \to \Tr (\Gamma)$ which sends $x$ to $x$ for all $x \in V(\Gamma)$. It is clear that $f'$ is the inverse of $f$, hence $f$ is an isomorphism. \end{proof} \begin{rem} The proof of Proposition \ref{prop2_2} proves more than what is stated: it actually proves that the presentation of $\widetilde{\Tr} (\widetilde{\Gamma})$ is equal to that of $\Tr (\Gamma)$. Nevertheless, even if the two presentations coincide, it will be useful subsequently to call the presentation \[ \langle V (\Gamma) \mid x^{\mu (x)} = 1 \text{ for } x \in V (\Gamma) \text{ such that } \mu (x) \neq \infty\,,\ x\,\varphi_x^{-1} (y) = y \, \varphi_y^{-1} (x) \text{ for } \{x, y\} \in E(\Gamma) \rangle \] the \emph{dual presentation} of $\Tr (\Gamma)$. \end{rem} \end{expl3} \subsection{The trickle algorithm -- Results of Section \ref{sec4}}\label{subsec2_2} Let $A$ be a set, which we call an \emph{alphabet}, and let $A^$ be the free monoid on $A$. The elements of $A^$ are called \emph{words} and they are written as finite sequences. The empty word is denoted by $\epsilon$ and the concatenation of two words $w_1, w_2 \in A^$ is denoted by $w_1 \cdot w_2$. A \emph{rewriting system} on $A^$ is defined to be a subset $R \subseteq A^ \times A^$. Let $w, w' \in A^$. We set $w \stackrel{R}{\to} w'$ or simply $w \to w'$ if there exist $w_1, w_2 \in A^$ and $(u,v) \in R$ such that $w = w_1 \cdot u \cdot w_2$ and $w' = w_1 \cdot v \cdot w_2$. More generally, we set $w \stackrel{R\ }{\to} w'$ or simply $w \to^* w'$ if either $w = w'$ or there exists a finite sequence $w = w_0, w_1, \dots, w_p = w'$ in $A^$ such that $w_{i-1} \to w_i$ for all $i \in \{1, \dots, p\}$. A word $w \in A^$ is said to be \emph{$R$-reducible} if there exists $w' \in A^$ such that $w \to w'$. Otherwise we say that $w$ is \emph{$R$-irreducible}. The pair $(A, R)$ is a \emph{rewriting system for a monoid} $M$ if $\langle A \mid u = v \text{ for } (u,v) \in R \rangle^+$ is a monoid presentation for $M$. A \emph{rewriting system for a group} $G$ is a rewriting system for $G$ viewed as a monoid. In particular, in this case $A$ generates $G$ as a monoid. If $(A, R)$ is a rewriting system for a monoid $M$ and $w=(\alpha_1, \alpha_2, \dots,\alpha_\ell) \in A^$, then we denote by $\overline{w} = \alpha_1 \alpha_2 \dots \alpha_\ell$ the element of $M$ represented by $w$. Let $R$ be a rewriting system on $A^$. We say that $R$ is \emph{terminating} if there is no infinite sequence $\{w_k\}_{k=0}^\infty$ in $A^$ such that $w_{k-1} \to w_k$ for all $k \in \N_{\ge 1}$. We say that $R$ is \emph{confluent} if, for all $u, v_1, v_2 \in A^$ such that $u \to^ v_1$ and $u \to^* v_2$, there exists $w \in A^$ such that $v_1 \to^ w$ and $v_2 \to^* w$. The importance of terminating and confluent rewriting systems comes from the following. \begin{thm}[Newman \cite{Newma1}]\label{thm2_3} Let $(A,R)$ be a terminating and confluent rewriting system for a monoid $M$. \begin{itemize} \item[(1)] For all $w'\in A^$ there exists a unique $R$-irreducible word $w \in A^$ such that $w' \to^* w$. \item[(2)] For all $g \in M$ there exists a unique $R$-irreducible word $w \in A^$ such that $g = \overline{w}$. \end{itemize} \end{thm} Now, we fix a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$, and we turn to describe a rewriting system for $\Tr (\Gamma)$. The alphabet of our rewriting system is not $V(\Gamma) \sqcup V (\Gamma)^{-1}$, as one may expect, but a more complicated set $\Omega = \Omega (\Gamma)$, which is generally infinite, and which is described as follows. Throughout the paper we use the following notations. For $\mu \in \N_{\ge 2} \cup \{\infty\}$ we set $\Z_\mu = \Z / \mu \Z$ if $\mu \neq \infty$ and $\Z_\mu = \Z$ if $\mu = \infty$. The set of \emph{syllables} of $\Gamma$ is the abstract set \[ S (\Gamma) = \{x^a \mid x \in V (\Gamma) \text{ and } a \in \Z_{\mu (x)} \setminus \{0\} \}\,. \] A \emph{stratum} of $\Gamma$ is a finite subset $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \} \subseteq S (\Gamma)$ such that $x_i \neq x_j$ and $\{x_i, x_j\} \in E (\Gamma)$ for all $i, j \in \{1, \dots, p\}$ such that $i \neq j$. The \emph{support} of $U$ is $\supp (U) = \{x_1, x_2, \dots, x_p\} \subseteq V (\Gamma)$ and its \emph{length} is the integer $p$, which is denoted by $\lg_\st (U)$. The empty set $\emptyset$ is assumed to be a stratum whose support is $\emptyset$. The set of strata is denoted by $\Omega = \Omega (\Gamma)$. The set $\Omega$ is the alphabet of our rewriting system. The elements of $\Omega^$ are called \emph{pilings}. If $u = (U_1, \dots, U_p)$ is a piling, then $p$ is the \emph{length} of $u$, which is denoted by $\lg_\pil (u)$. Now, we define three operations on the strata which will be used to define our rewriting system. The first operation consists in removing an element from a stratum. If $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\}$ is a non-empty stratum and $x_i^{a_i} \in U$, then we set \[ L (U, x_i^{a_i}) = U \setminus \{ x_i^{a_i} \} = \{ x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}} , x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}\}\,. \] Note that $L (U, x_i^{a_i})$ is a stratum. The second operation consists in ``extracting'' a syllable from a stratum. Let $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\}$ be a non-empty stratum and let $x_i^{a_i} \in U$. We number the elements of $U$ such that, if $x_j > x_k$, then $j < k$. Then we set \[ \gamma (U, x_i^{a_i}) = \big( (\varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)\big)^{a_i}\,. \] It is easily seen that $\gamma(U, x_i^{a_i})$ is well-defined and belongs to $S(\Gamma)$. Moreover, it will be proved in Section \ref{sec4} (see Lemma \ref{lem4_2}) that the definition of $\gamma (U, x_i^{a_i})$ does not depend on the choice of the numbering of the elements of $U$. The third operation consists in ``adding'' a syllable to a stratum. Let $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \} \in \Omega$ and let $y^b \in S(\Gamma)$. We say that $y^b$ can be \emph{added} to $U$ if either $y \in \supp (U)$ or $\{y, x_i\} \in E(\Gamma)$ for all $i \in \{1, \dots, p\}$. Suppose $y^b$ can be added to $U$. If $y \not \in \supp (U)$, then we set \[ R (U, y^b) = \{ \varphi_y^{-b} (x_1)^{a_1}, \dots, \varphi_y^{-b} (x_p)^{a_p}, y^b\}\,. \] If $y = x_i \in \supp (U)$ and $b + a_i = 0$ (in $\Z_{\mu (y)}$), then we set \[ R(U, y^b) = \{\varphi_y^{-b} (x_1)^{a_1}, \dots, \varphi_y^{-b} (x_{i-1})^{a_{i-1}}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}\}\,. \] If $y = x_i \in \supp (U)$ and $b + a_i \neq 0$ (in $\Z_{\mu (y)}$), then we set \[ R (U, y^b) = \{ \varphi_y^{-b} (x_1)^{a_1}, \dots, \varphi_y^{-b} (x_{i-1})^{a_{i-1}}, y^{a_i+b}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p) ^{a_p}\}\,. \] Note that, in the third case, since $y = x_i = \varphi_y^{-b}(x_i)$, $y^{a_i+b}$ can be replaced by $\varphi_y^{-b} (x_i)^{a_i+b}$. Note also that $R (U, y^b)$ is always a stratum. \begin{defin} Let $(U, V)$ be a pair of strata with $V \neq \emptyset$ and let $x^a \in V$. We set $V' = L(V, x^a)$ and $y^a = \gamma (V, x^a)$. We assume that $y^a$ can be added to $U$ and we set $U' = R(U, y^a)$. Then we say that $r = ((U, V), (U', V')) \in \Omega^* \times \Omega^$ is a \emph{T-transformation}. In this case we write $T (U, V, x^a) = (U', V')$. We denote by $\RR_1$ the set of T-transformations. On the other hand, we set $\RR_0 = \{ ((\emptyset), \epsilon)\} \subset \Omega^ \times \Omega^$, where $(\emptyset)$ is the piling of length $1$ whose only entry is $\emptyset$ and $\epsilon$ is the empty piling of length $0$. Finally, we set $\RR = \RR (\Gamma)= \RR_0 \cup \RR_1$. \end{defin} The following will be proved in Section \ref{sec4}. \begin{thm}\label{thm2_4} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph, let $\Omega = \Omega (\Gamma)$ be the set of strata of $\Gamma$, and let $\RR = \RR (\Gamma) \subseteq \Omega^ \times \Omega^$ be as defined above. Then $\RR$ is a rewriting system for $\Tr (\Gamma)$, and it is terminating and confluent. \end{thm} As a consequence of Theorem \ref{thm2_4} we get that $\langle \Omega \mid u = v \text{ for } (u,v) \in \RR \rangle^+$ is a monoid presentation for $\Tr (\Gamma)$. We explain how to go from this presentation to the standard one and vice versa. Set $M (\Gamma) = \langle \Omega \mid u = v \text{ for } (u,v) \in \RR \rangle^+$. Let $U \in \Omega$. We write $U = \{ x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \}$ so that, if $x_i > x_j$, then $i < j$, and we set $\omega (U) = x_1^{a_1} x_2^{a_2} \dots x_p^{a_p} \in \Tr (\Gamma)$. We will show in Lemma \ref{lem4_13} that the definition of $\omega (U)$ does not depend on the choice of the numbering of the elements of $U$. Then the isomorphism $\Phi: M (\Gamma) \to \Tr (\Gamma)$ sends $U$ to $\omega (U)$ for all $U \in \Omega$. The reverse isomorphism $\Psi: \Tr (\Gamma) \to M (\Gamma)$ sends $x$ (resp. $x^{-1}$) to the piling $(\{x\})$ (resp. $(\{ x^{-1} \})$) of length $1$ for all $x \in V (\Gamma)$. Details and proofs about these isomorphisms will be given in Section \ref{sec4}. Recall that, if $G$ is a group and $V$ is a generating set for $G$, then a \emph{set of normal forms} for $G$ based on $V \sqcup V^{-1}$ is a language $\LL \subseteq (V \sqcup V^{-1})^$ such that, for all $g \in G$, there exists a unique $w \in \LL$ such that $\overline{w} = g$. The above constructions provide normal forms for $\Tr (\Gamma)$ based on $V (\Gamma) \sqcup V(\Gamma)^{-1}$ as well as an algorithm to calculate them when $V (\Gamma)$ is finite, as follows. Let $\mu \in \N_{\ge 2} \cup \{ \infty \}$ and $a \in \Z_\mu \setminus \{ 0\}$. If $\mu = \infty$, then we set $\rho (a) = a$. If $\mu \neq \infty$, then we denote by $\rho (a)$ the unique representative of $a$ sitting inside $\{1, \dots, \mu-1\}$. We fix a total order $\preceq$ on $V (\Gamma)$ which extends the partial order $\le$ in the sense that, if $x \le y$, then $x \preceq y$. Such a total order always exists but it is not unique in general. Let $U$ be a non-empty stratum that we write $U = \{ x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$. Then we set $\hat \omega (U) = x_1^{\rho (a_1)} \cdot x_2^{\rho (a_2)} \cdots x_p^{\rho (a_p)} \in (V (\Gamma) \sqcup V (\Gamma)^{-1})^$. Note that $\hat \omega (U)$ is a representative of $\omega (U)$. Let $g \in \Tr (\Gamma)$. By Theorems \ref{thm2_3} and \ref{thm2_4} there exists a unique piling $w = (U_1, U_2, \dots, U_p) \in \Omega^$ such that $w$ is $\RR$-irreducible and $\overline{w} = \Psi (g)$. Then we set \[ \nf (g) = \hat \omega (U_1) \cdot \hat \omega (U_2) \cdots \hat \omega (U_p) \in (V (\Gamma) \sqcup V (\Gamma)^{-1})^\,. \] The following is a direct consequence of the previous constructions. \begin{corl}\label{corl2_5} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph. \begin{itemize} \item[(1)] The set $\LL = \LL (\Gamma) = \{ \nf (g) \mid g \in \Tr (\Gamma) \}$ is a set of normal forms for $\Tr (\Gamma)$ based on $V (\Gamma) \sqcup V (\Gamma)^{-1}$. \item[(2)] Suppose that $V( \Gamma)$ is finite. Then there exists an algorithm which, given a word $w \in (V(\Gamma) \sqcup V(\Gamma)^{-1})^$, calculates $\nf (\overline{w})$. In particular, this algorithm is a solution to the word problem in $\Tr (\Gamma)$. \end{itemize} \end{corl} \begin{rem} Corollary \ref{corl2_5}\,(2) can be extended to a trickle graph with infinite but countable vertex set $V (\Gamma)$ provided that there exist an algorithm to manipulate the elements of $V (\Gamma)$, an algorithm which, given $x, y \in V (\Gamma)$, decides whether $\{x, y \} \in E (\Gamma)$ or not, an algorithm which, given $x,y \in V (\Gamma)$, decides whether $x \le y$ or not, and an algorithm which, given $x \in V (\Gamma)$ and $y \in \starE_x (\Gamma)$, determines $\varphi_x (y)$. \end{rem} Another straightforward consequence of Theorems \ref{thm2_3} and \ref{thm2_4} is the following. \begin{corl}\label{corl2_6} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph. Then the map $S (\Gamma) \to \Tr (\Gamma)$, $x^a \mapsto x^a$, is injective. In particular, $V(\Gamma)$ is a subset of $\Tr (\Gamma)$. \end{corl} \begin{proof} The result follows from the fact that, for $x^a \in S(\Gamma)$, $x^{\rho (a)} \in (V(\Gamma) \sqcup V (\Gamma)^{-1})^$ is the normal form of $x^a \in \Tr (\Gamma)$. \end{proof} Another more unexpected consequence is the following. \begin{corl}\label{corl2_7} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph. Then $\Tr (\Gamma)$ is finite if and only if $V(\Gamma)$ is finite, $\Gamma$ is complete, and $\mu (x) \neq \infty$ for all $x \in V(\Gamma)$. \end{corl} \begin{proof} Suppose $V (\Gamma)$ is infinite. Then $\Tr (\Gamma)$ is infinite because $V(\Gamma) \subseteq \Tr (\Gamma)$. Suppose there exists $x \in V(\Gamma)$ such that $\mu (x) = \infty$. Then, by Corollary \ref{corl2_6}, $\{ x^a \mid a \in \Z \setminus \{0\} \} \subseteq S (\Gamma) \subseteq \Tr (\Gamma)$, hence $\Tr (\Gamma)$ is infinite. Suppose $\Gamma$ is not complete. Let $x,y \in V(\Gamma)$ be such that $\{x, y\} \not \in E (\Gamma)$. Let $u=(\{x\}, \{y\}) \in \Omega^$. Then, for all $n \in \N$, $u^n$ is an $\RR$-irreducible piling of length $2n$, hence, by Theorems \ref{thm2_3} and \ref{thm2_4}, the set $\{\overline{u ^n} \mid n \in \N\}$ is an infinite subset of $\Tr (\Gamma)$, and therefore $\Tr (\Gamma)$ is infinite. Suppose $V(\Gamma)$ is finite, $\Gamma$ is complete, and $\mu (x) \neq \infty$ for all $x \in V(\Gamma)$. We fix a total order $\preceq$ on $V (\Gamma)$ which extends the partial order $\le$, and we write $V (\Gamma) = \{ x_1, x_2, \dots, x_p \} $ with $x_1 \succ x_2 \succ \cdots \succ x_p$. Then the set of normal forms for $\Tr (\Gamma)$ is \[ \{ x_{i_1}^{a_1} \cdot x_{i_2}^{a_2} \cdots x_{i_q}^{a_q} \mid 1 \le i_1 < i_2 < \cdots < i_q \le p\,,\ 1 \le a_j \le \mu (x_{i_j})-1 \text{ for } 1 \le j \le q\}\,. \] This set is finite and it is in one-to-one correspondence with $\Tr (\Gamma)$, hence $\Tr (\Gamma)$ is finite. \end{proof} \subsection{The Tits-style algorithm -- Results of Section \ref{sec5}}\label{subsec2_3} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph. Recall that the set of \emph{syllables} of $\Tr (\Gamma)$ is $S (\Gamma) = \{x^a \mid x \in V(\Gamma) \text{ and } a \in \Z_{\mu (x)} \setminus\{0\} \}$. Recall also that, by Corollary \ref{corl2_6}, $S (\Gamma)$ is a subset of $\Tr (\Gamma)$. The elements of $S(\Gamma)^$ are called \emph{syllabic words}. For $w=(x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}) \in S(\Gamma)^$ we denote by $\overline{w} = x_1^{a_1} x_2^{a_2} \dots x_p^{a_p}$ the element of $\Tr (\Gamma)$ represented by $w$. The integer $p$ is called the \emph{length} of $w$ and it is denoted by $\lg (w)$. The smallest length of a syllabic word representing an element $g \in \Tr (\Gamma)$ is called the \emph{syllabic length} of $g$ and it is denoted by $\lg_{\syl} (g)$. A syllabic word $w = (x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p})$ is said to be \emph{syllabically reduced} if $\lg (w) = \lg_\syl (\overline{w})$. Our Tits-style algorithm uses a rewriting system on $S (\Gamma)^$, but it does not use Newman's results \cite{Newma1} stated in Theorem \ref{thm2_3}. This rewriting system is defined as follows. We set \begin{gather} \RR_{I} = \{ ((x^a, x^{-a}), \epsilon) \mid x \in V (\Gamma)\,,\ a \in \Z_{\mu (x)} \setminus \{0\} \}\ \cup\\ \{ ((x^a, x^b),(x^{a+b})) \mid x \in V (\Gamma)\,, \ a, b \in \Z_{\mu (x)} \setminus \{0\} \text{ and } a + b \neq 0\}\,,\\ \RR_{II} = \{ ((x^a, y^b), (\varphi_x^a (y)^b, \varphi_y^{-b}(x)^a)) \mid x^a, y^b \in S (\Gamma) \text{ and } \{x,y\} \in E (\Gamma) \}\,,\\ \RR_M = \RR_I \cup \RR_{II}\,. \end{gather} Let $x^a, y^b \in S (\Gamma)$ be such that $\{x,y\} \in E (\Gamma)$. Note that, if $x \|\| y$, then $(\varphi_x^a (y)^b, \varphi_y^{-b}(x)^a) = (y^b, x^a)$, if $x > y$, then $ (\varphi_x^a (y)^b, \varphi_y^{-b}(x)^a) = (\varphi_x^a (y)^b, x^a)$, and, if $x < y$, then $(\varphi_x^a (y)^b, \varphi_y^{-b}(x)^a) = (y^b, \varphi_y^{-b}(x)^a)$. Note also that in all three cases we have the equality $x^a y^b = \varphi_x^a (y)^b\, \varphi_y^{-b}(x)^a$ in $\Tr (\Gamma)$. For $X \in \{I, II, M\}$ and $w, w' \in S (\Gamma)^$, we write $w \stackrel{X}{\to} w'$ in place of $w \stackrel{\RR_X}{\to} w'$ and $w \stackrel{X\ }{\to} w'$ in place of $w \stackrel{\RR_X\, }{\longrightarrow} w'$. Let $w, w' \in S (\Gamma)^$. Notice that, if $w \stackrel{I}{\to} w'$, then $\lg (w) > \lg (w')$. In particular, the operation $\stackrel{I}{\to}$ is not reversible. On the other hand the operation $\stackrel{II}{\to}$ is reversible in the sense that, if $w \stackrel{II}{\to} w'$, then $\lg (w) = \lg (w')$ and $w' \stackrel{II}{\to} w$. We say that a syllabic word $w \in S (\Gamma)^$ is \emph{syllabically $M$-reduced} if there is no word $w' \in S (\Gamma)^$ such that $w \stackrel{M\ }{\to} w'$ and $\lg (w') < \lg (w)$. The following theorem is similar to the classical solution to the word problem for Coxeter groups \cite{Tits1}, for graph products of cyclic groups \cite{Green1}, and, more generally, for Dyer groups \cite{ ParSoe1}. It will be proved in Section \ref{sec5}. \begin{thm}\label{thm2_8} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph. \begin{itemize} \item[(1)] For every $w \in S (\Gamma)^$, $w$ is syllabically reduced if and only if $w$ is syllabically $M$-reduced. \item[(2)] For every $w, v \in S (\Gamma)^$, if $w$ and $v$ are syllabically reduced and $\overline{w} = \overline{v}$, then $w \stackrel{II\, }{\to} v$. \end{itemize} \end{thm} \subsection{Parabolic subgroups -- Results of Section \ref{sec6}}\label{subsec2_4} \begin{defin} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph. A full subgraph $\Gamma_1$ of $\Gamma$ is called \emph{parabolic} if, for every $x \in V(\Gamma_1)$, the map $\varphi_x$ stabilizes $\starE_x (\Gamma_1)$. \end{defin} \begin{expl} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph. \begin{itemize} \item[(1)] If $X$ is a subset of $V(\Gamma)$ whose elements are pairwise incomparable, then the full subgraph of $\Gamma$ spanned by $X$ is parabolic. \item[(2)] If $X$ is a subset of $V(\Gamma)$ and $X_\downarrow = \{y \in V(\Gamma) \mid \exists x \in X, y \leq x\}$, then the full subgraph of $\Gamma$ spanned by $X_\downarrow$ is parabolic. \end{itemize} \end{expl} The following can be easily checked. \begin{lem}\label{lem2_9} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph and let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. We denote by $\le_1$ and by $\mu_1$ the restrictions of $\le$ and $\mu$ to $V(\Gamma_1)$, respectively, and, for $x \in V (\Gamma_1)$, we denote by $\varphi_{1,x}$ the restriction of $\varphi_x$ to $\starE_x (\Gamma_1)$. \begin{itemize} \item[(1)] The quadruple $\Gamma_1 = (\Gamma_1, \le_1, \mu_1, (\varphi_{1,x})_{x\in V(\Gamma_1)})$ is a trickle graph. \item[(2)] The embedding of $V(\Gamma_1)$ into $V(\Gamma)$ induces a group homomorphism $\iota_1: \Tr (\Gamma_1) \to \Tr (\Gamma)$. \end{itemize} \end{lem} In our study in Section \ref{sec6} we will use the normal forms for $\Tr (\Gamma)$ as they are defined in Subsection \ref{subsec2_2}. We recall them. We fix a total order $\preceq$ on $V(\Gamma)$ which extends the partial order $\le$ in the sense that, if $x \le y$, then $x \preceq y$. If $U$ is a non-empty stratum that we write $U = \{ x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$, then we set $\hat \omega (U) = x_1^{\rho (a_1)} \cdot x_2^{\rho (a_2)} \cdots x_p^{\rho (a_p)} \in (V (\Gamma) \sqcup V (\Gamma)^{-1})^$. Let $g \in \Tr (\Gamma)$. By Theorems \ref{thm2_3} and \ref{thm2_4} there exists a unique piling $w = (U_1, U_2, \dots, U_p) \in \Omega^$ such that $w$ is $\RR$-irreducible and $\overline{w} = g$. Then the \emph{normal form} of $g$ is \[ \nf_\Gamma (g) = \nf (g) = \hat \omega (U_1) \cdot \hat \omega (U_2) \cdots \hat \omega (U_p) \in (V (\Gamma) \sqcup V (\Gamma)^{-1})^\,. \] Let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. As before, we assume that $V (\Gamma_1)$ is endowed with the restriction $\le_1$ of $\le$ to $V (\Gamma_1)$ and with the restriction $\mu_1: V (\Gamma_1) \to \N_{\ge 2} \cup \{ \infty\}$ of $\mu$ to $V (\Gamma_1)$. Also, for each $x \in V (\Gamma_1)$, we denote by $\varphi_{1,x}$ the restriction of $\varphi_x$ to $\starE_x (\Gamma_1)$. Thus, as pointed out in Lemma \ref{lem2_9}, $\Gamma_1 = (\Gamma_1, \le_1, \mu_1, (\varphi_{1,x})_{x \in V (\Gamma_1)})$ is a trickle graph. Now, we denote by $\preceq_1$ the restriction of $\preceq$ to $V (\Gamma_1)$ and we observe that $\preceq_1$ extends the partial order $\le_1$, hence it can be used to define normal forms for $\Tr (\Gamma_1)$. Unless otherwise stated, we will always use this total order to define normal forms for $\Tr (\Gamma_1)$. The main result of Section \ref{sec6} is the following. \begin{thm}\label{thm2_10} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph and let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. \begin{itemize} \item[(1)] The homomorphism $\iota_1: \Tr (\Gamma_1) \to \Tr (\Gamma)$ induced by the embedding of $V (\Gamma_1)$ into $V (\Gamma)$ is injective. In particular, we can identify $\Tr (\Gamma_1)$ with its image in $\Tr (\Gamma)$ under $\iota_1$. \item[(2)] Let $g \in \Tr (\Gamma)$. We have $g \in \Tr (\Gamma_1)$ if and only if $\nf (g) \in (V (\Gamma_1) \sqcup V(\Gamma_1)^{-1})^$. Moreover, in this case, $g$ has the same normal form in $\Tr (\Gamma)$ as in $\Tr (\Gamma_1)$, that is, $\nf_\Gamma (g) = \nf_{\Gamma_1} (g)$. \end{itemize} \end{thm} \begin{defin} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ be a trickle graph and let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. Then $\Tr (\Gamma_1)$, viewed as a subgroup of $\Tr (\Gamma)$, is called a \emph{standard parabolic subgroup} of $\Tr (\Gamma)$. \end{defin} \begin{expl} Let $(W,S)$ be a Coxeter system. Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ be the trickle graph such that $\Tr (\Gamma) = C(W, S)$, as defined in Subsection \ref{subsec2_1}. n} \}$. Let $Y \subseteq S$. n} \mid X \subseteq Y \}$ and we denote by $\Gamma_Y$ the full subgraph of $\Gamma$ spanned by $\{ x_X \mid X \in \SS_{Y,\fin}\}$ (Caution: do not confuse $\Gamma_Y$ with the full Coxeter subgraph spanned by $Y$). The proof of the following is left to the reader. \begin{lem}\label{lem2_11} Under the above assumptions, $\Gamma_Y$ is a parabolic subgraph of $\Gamma$ and $\Tr (\Gamma_Y)$ is naturally isomorphic to the cactus group $C (W_Y, Y)$. In particular, $C(W_Y, Y)$ is a standard parabolic subgroup of $C(W, S)$. \end{lem} \end{expl} If $\Gamma_1$ and $\Gamma_2$ are two full subgraphs of a trickle graph $\Gamma$, then we denote by $\Gamma_1 \cap \Gamma_2$ the full subgraph of $\Gamma$ spanned by $V (\Gamma_1) \cap V (\Gamma_2)$. It is easily seen that, if $\Gamma_1$ and $\Gamma_2$ are both parabolic subgraphs, then $\Gamma_1 \cap \Gamma_2$ is also a parabolic subgraph. Now, from Theorem \ref{thm2_10} we deduce the following. \begin{corl}\label{corl2_12} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ be a trickle graph and let $\Gamma_1, \Gamma_2$ be two parabolic subgraphs of $\Gamma$. Then $\Tr (\Gamma_1) \cap \Tr (\Gamma_2) = \Tr (\Gamma_1 \cap \Gamma_2)$. In particular, the intersection of two standard parabolic subgroups of $\Tr (\Gamma)$ is a standard parabolic subgroup. \end{corl} \begin{proof} The inclusion $\Tr (\Gamma_1 \cap \Gamma_2) \subseteq \Tr (\Gamma_1) \cap \Tr (\Gamma_2)$ is obvious. We prove the reverse inclusion. Let $g \in \Tr (\Gamma_1) \cap \Tr (\Gamma_2)$. By Theorem \ref{thm2_10}, $\nf (g) \in (V (\Gamma_1) \sqcup V (\Gamma_1)^{-1})^$ and $\nf (g) \in (V (\Gamma_2 ) \sqcup V (\Gamma_2)^{-1})^$. But \[ (V (\Gamma_1) \sqcup V (\Gamma_1)^{-1})^ \cap (V (\Gamma_2) \sqcup V (\Gamma_2)^{-1})^* = (V (\Gamma_1 \cap \Gamma_2) \sqcup V (\Gamma_1 \cap \Gamma_2)^{-1})^\,, \] hence $\nf (g) \in (V (\Gamma_1 \cap \Gamma_2) \sqcup V (\Gamma_1 \cap \Gamma_2)^{-1})^$, and therefore $g \in \Tr (\Gamma_1 \cap \Gamma_2)$. \end{proof} Another straightforward consequence of Theorem \ref{thm2_10} is the following. \begin{corl}\label{corl2_13} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ be a trickle graph with $V (\Gamma)$ finite, and let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. There exists an algorithm which, given $g \in \Tr (\Gamma)$, decides whether $g$ belongs to $\Tr (\Gamma_1)$ or not. \end{corl} \begin{proof} Let $g \in \Tr (\Gamma)$. Then, by Theorem \ref{thm2_10}, $g \in \Tr (\Gamma_1)$ if and only if $\nf (g) \in (V (\Gamma_1) \sqcup V (\Gamma_1)^{-1})^$. \end{proof} \begin{rem} As with Corollary \ref{corl2_5}\,(2), Corollary \ref{corl2_13} can be extended to the case where $V (\Gamma)$ is infinite but countable, provided that one has an algorithm for manipulating the elements of $V (\Gamma)$, an algorithm which, given $x,y \in V (\Gamma)$, decides whether $\{ x,y\}$ belongs to $E (\Gamma)$ or not, an algorithm which, given $x,y \in V (\Gamma)$, decides whether $x < y$ or not, an algorithm which, given $x \in V (\Gamma)$ and $y \in \starE_x (\Gamma)$, calculates $\varphi_x (y)$, and an algorithm which, given $x \in V (\Gamma)$, decides whether $x \in V (\Gamma_1)$ or not. \end{rem} \subsection{PreGarside trickle groups -- Results of Section \ref{sec7}}\label{subsec2_5} We start by recalling some definitions and notations on monoids. We say that a monoid $M$ is \emph{cancellative} if, for all $a, b, c, d \in M$, if $c a d = c b d$, then $a = b$. We say that $M$ is \emph{atomic} if there exists a map $\nu : M \to \N$, called a \emph{norm}, such that $\nu (a) = 0$ if and only if $a=1$, and $\nu (a b) \ge \nu (a) + \nu (b)$ for all $a, b \in M$. If $M$ is atomic, then we can define two partial orders $\le_L$ and $\le_R$ on $M$ as follows. Let $a, b \in M$. We set $a \le_L b$ if there exists $c \in M$ such that $a c = b$, and we set $a \le_R b$ if there exists $c' \in M$ such that $c' a = b$. For $b \in M$ we set $\Div_L (b) = \{a \in M \mid a \le_L b \}$ and $\Div_R (b) = \{ a \in M \mid a \le_R b\}$. We say that $b$ is \emph{balanced} if $\Div_L (b) = \Div_R (b)$, and, in this case, we set $\Div (b) = \Div_L (b) = \Div_R (b)$. The enveloping group of a monoid $M$ is denoted by $G(M)$ and the natural homomorphism from $M$ to $G (M)$ is denoted by $\iota_M : M \to G(M)$. \begin{defin} A monoid $M$ is called a \emph{preGarside monoid} if it satisfies the following three properties. \begin{itemize} \item[(a)] $M$ is cancellative and atomic. \item[(b)] For all $a,b \in M$, if the set $\{ c \in M \mid a \le_L c \text{ and } b \le_L c \}$ is non-empty, then it contains a least element that we denote by $a \vee_L b$. \item[(c)] For all $a,b \in M$, if the set $\{ c \in M \mid a \le_R c \text{ and } b \le_R c \}$ is non-empty, then it contains a least element that we denote by $a \vee_R b$. \end{itemize} A \emph{preGarside group} is the enveloping group of a preGarside monoid. \end{defin} \begin{defin} Let $M$ be a preGarside monoid. An element $\Delta \in M$ is called a \emph{Garside element} if $\Delta$ is balanced and $\Div (\Delta)$ generates $M$. If $M$ contains a Garside element, then $M$ is called a \emph{Garside monoid} and $G (M)$ is called a \emph{Garside group}. \end{defin} The notions of Garside monoids and Garside groups were introduced by Dehornoy and the third author in \cite{DehPar1,Dehor1}, where they show that these monoids and groups share many properties with spherical type Artin monoids and groups such as solutions to the word problem and to the conjugacy problem. The theory of Garside groups is experiencing a significant growth and many examples and applications have appeared since the publication of \cite{DehPar1,Dehor1}. On the other hand, our definitions of preGarside monoid and group are taken from \cite{GodPar2}. The motivation for their study is that, by \cite{BriSai1}, every Artin monoid is a preGarside monoid and every Artin group is a preGarside group. So, preGarside groups are to Garside groups what Artin groups are to spherical type Artin groups. \begin{defin} We say that a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ is a \emph{preGarside trickle graph} if $\mu (x) = \infty$ for all $x \in V (\Gamma)$. In this case, in addition to the trickle group $\Tr (\Gamma)$, we associate with $\Gamma$ a \emph{preGarside trickle monoid}, $\Tr^+ (\Gamma)$, which is defined by the following monoid presentation. \[ \Tr^+ (\Gamma) = \langle V (\Gamma) \mid \varphi_x (y)\, x = \varphi_y (x)\, y \text{ for all } \{ x, y \} \in E (\Gamma) \rangle^+ \,. \] It is clear that $\Tr (\Gamma)$ is the enveloping group of $\Tr^+ (\Gamma)$. \end{defin} The above definition is motivated by the following theorem which will be proved in Section \ref{sec7}. \begin{thm}\label{thm2_14} Let $\Gamma$ be a preGarside trickle graph. Then $\Tr^+ (\Gamma)$ is a preGarside monoid. \end{thm} The next question is: when is $\Tr^+ (\Gamma)$ a Garside monoid? The answer is given by the following theorem which will be also proved in Section \ref{sec7}. \begin{thm}\label{thm2_15} Let $\Gamma$ be a preGarside trickle graph. Then $\Tr^+ (\Gamma)$ is a Garside monoid if and only if $V(\Gamma)$ is finite and $\Gamma$ is a complete graph. \end{thm} \begin{rem} Any spherical type Artin group $A$ has a natural finite quotient $W$, which is its associated Coxeter group, and the projection $A \to W$ is an important tool in the study of the group. Such a quotient does not exist in general for a given Garside group, and determining which Garside groups have such quotients is a standard question in the theory (see for instance \cite{Gobet1}). Now, notice that Theorem \ref{thm2_15} provides new examples of such Garside groups. Indeed, if $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ is a preGarisde trickle graph with finite $V (\Gamma)$ and $\Gamma$ complete, then, by Theorem \ref{thm2_15}, $\Tr (\Gamma)$ is a Garside group and, by Corollary \ref{corl2_7}, the quotient of $\Tr (\Gamma)$ by the relations $x^{\bar \mu(x)} = 1$, $x \in V (\Gamma)$, where $\bar \mu (x)$ is the order of $\varphi_x$ if $\varphi_x$ is different from the identity and $\bar \mu (x) = 2$ if $\varphi_x$ is the identity, is a finite trickle group. \end{rem} In \cite{GodPar3} four elementary questions are asked on Artin groups, questions which are unsolved to this date, and which also arise naturally for preGarside groups. Two of them are of particular interest in this paper: \begin{itemize} \item[(1)] Is any preGarside group torsion-free? \item[(2)] Does any preGarside group has a solution to the word problem? \end{itemize} Concerning preGarside trickle groups, a positive answer to Question (2) when $V (\Gamma)$ is finite is given by Corollary \ref{corl2_5}. A positive answer to Question (1) in the case where $V (\Gamma)$ is finite is given by the following theorem which will be proved in Section \ref{sec7}. \begin{thm}\label{thm2_16} Let $\Gamma$ be a preGarside trickle graph with $V (\Gamma)$ finite. Then $\Tr (\Gamma)$ is torsion-free. \end{thm} We do not know whether the conclusion of Theorem \ref{thm2_16} holds if $V (\Gamma)$ is infinite. Another question more specific to preGarside groups is the following. \begin{itemize} \item[(3)] Let $M$ be a preGarside monoid. Is the natural homomorphism $\iota_M : M \to G(M)$ injective? \end{itemize} We know from \cite{Paris1} that the answer is yes if the monoid is an Artin monoid. The answer is also yes if the monoid is a preGarside trickle monoid as stated in the following theorem which will be proved in Section \ref{sec7}. \begin{thm}\label{thm2_17} Let $\Gamma$ be a preGarside trickle graph. Then the natural homomorphism $\iota_{\Tr^+ (\Gamma)}: \Tr^+ (\Gamma) \to \Tr (\Gamma)$ is injective. \end{thm} \begin{defin} Let $M$ be a preGarside monoid and let $N$ be a submonoid of $M$. We say that $N$ is \emph{special} if, for all $a,b \in M$, if $ab \in N$, then $a,b \in N$. We say that $N$ is a \emph{parabolic submonoid} of $M$ if it satisfies the following three properties. \begin{itemize} \item[(a)] $N$ is special. \item[(b)] For all $a,b \in N$, if $a \vee_L b$ exists, then $a\vee_L b \in N$. \item[(c)] For all $a,b \in N$, if $a \vee_R b$ exists, then $a\vee_R b \in N$. \end{itemize} \end{defin} In the case of PreGarside trickle monoids, this definition, taken from \cite{GodPar2}, coincides with that given in Subsection \ref{subsec2_4} (see Proposition \ref{prop7_8}). On the other hand, in the case of Garside monoids, it coincides with the definition of \cite{Godel1,Godel2}. In particular, a parabolic submonoid of a Garside monoid is itself a Garside monoid. Now, a consequence of Proposition \ref{prop7_8} combined with the results of Subsection \ref{subsec2_4} is a positive answer in the particular case of preGarside trickle monoids to three questions asked in \cite{GodPar2} for parabolic submonoids of preGarside monoids. \begin{thm}\label{thm2_18} Let $M$ be a preGarside trickle monoid. \begin{itemize} \item[(1)] Let $N$ be a parabolic submonoid of $M$. Then the embedding $N \hookrightarrow M$ induces an embedding $G (N) \hookrightarrow G (M)$. So, we can consider $G(N)$ as a subgroup of $G(M)$. \item[(2)] Let $N$ be a parabolic submonoid of $M$. Then $M \cap G(N) = N$. \item[(3)] Let $N_1$ and $N_2$ be two parabolic submonoids of $M$. Then $N_1 \cap N_2$ is a parabolic submonoid of $M$ and $G(N_1) \cap G (N_2) = G (N_1 \cap N_2)$. \end{itemize} \end{thm} Again, a detailed proof of this theorem will be given in Section \ref{sec7}. \section{Examples}\label{sec3} \subsection{Generalized cactus groups}\label{subsec3_1} Let $G$ be a group. A \emph{cactus basis} based on $G$ is a non-empty family $\FF$ of pairs $X = (G_X, \Delta_X)$, where $G_X$ is a subgroup of $G$ and $\Delta_X$ is an element of $G_X$, such that, \begin{itemize} \item[(a)] $G_X \neq G_Y$ for $X,Y \in \FF$ distinct; \item[(b)] for $X, Y \in \FF$, if $G_Y \subset G_X$, then $(\Delta_X G_Y \Delta_X^{-1}, \Delta_X \Delta_Y \Delta_X^{-1}), (\Delta_X^{-1} G_Y \Delta_X, \Delta_X^{-1} \Delta_Y \Delta_X) \in \FF$. \end{itemize} Recall that, if $H_1, H_2$ are two subgroups of a group $G$, then $[H_1, H_2]$ denotes the subgroup of $G$ generated by $\{h_1 h_2 h_1^{-1} h_2^{-1} \mid h_1 \in H_1\,,\ h_2 \in H_2\}$. In particular, the equality $[H_1, H_2]=\{ 1 \}$ means that $h_1 h_2 = h_2 h_1$ for all $h_1 \in H_1$ and $h_2 \in H_2$. Let $\FF$ be a cactus basis based on $G$. Let $\mu : \FF \to \N \cup \{\infty\}$ be a map which satisfies the following conditions: \begin{itemize} \item for all $X \in \FF$, if $\mu (X)$ is finite, then the order of $\Delta_X$ is finite and divides $\mu (X)$; \item for all $X = (G_X, \Delta_X)$ and $Y = (G_Y, \Delta_Y)$ in $\FF$, if $G_Y \subset G_X$, then $\mu (Y) = \mu (Y')$, where $Y' = (\Delta_X G_Y \Delta_X^{-1}, \Delta_X \Delta_Y \Delta_X^{-1})$. \end{itemize} Then we define a quadruple $\Gamma_{\FF,\mu} = \Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ as follows. The set of vertices of $\Gamma$ is a set $V(\Gamma) = \{ x_X \mid X \in \FF\}$ in one-to-one correspondence with $\FF$. Two vertices $x_X$ and $x_Y$ are connected by an edge if either $G_X \subset G_Y$, or $G_Y \subset G_X$, or $G_X \cap G_Y = \{1\}$ and $[G_X, G_Y]= \{1\}$. We set $x_X \le x_Y$ if $G_X \subseteq G_Y$. The map $\mu : V (\Gamma) \to \N_{\ge 2} \cup \{ \infty\}$ is defined by $\mu (x_X) = \mu (X)$ for all $x_X \in V (\Gamma)$. Let $x_X \in V (\Gamma)$ and $x_Y \in V (\starE_{x_X} (\Gamma))$. If $x_Y \le x_X$, then we set $\varphi_{x_X} (x_Y) = x_{Y'}$, where $X=(G_X,\Delta_X)$, $Y=(G_Y, \Delta_Y)$ and $Y'= (\Delta_X G_Y \Delta_X^{-1}, \Delta_X \Delta_Y \Delta_X^{-1})$. If $x_Y \not \le x_X$, then we set $\varphi_{x_X} (x_Y) = x_Y$. The proof of the following is identical to that of Lemma \ref{lem2_1}, hence it is left to the reader. \begin{lem}\label{lem3_1} Let $\FF$ be a cactus basis based on a group $G$ and let $\mu : \FF \to \N_{\ge 2} \cup \{ \infty \}$ be a map as above. Then $\Gamma_{\FF,\mu} = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ is a trickle graph. \end{lem} \begin{rem} Let $\FF$ be a cactus basis based on a group $G$ and let $\mu : \FF \to \N_{\ge 2} \cup \{ \infty \}$ be a map as above. Then there is a homomorphism $\pi: \Tr (\Gamma_{\FF,\mu}) \to G$ which sends $x_X$ to $\Delta_X$ for all $X \in \FF$. This homomorphism plays an important role in the study of cactus groups associated to Coxeter systems. In that case it is known to be surjective and its kernel is known to embed into a right-angled Coxeter group (see \cite{Mosto1, Yu1}). \end{rem} It is not difficult to produce examples of cactus basis. Below we give three examples which seem particularly interesting to us because they highlight the link between trickle groups and the theory of Coxeter, Artin and Garside groups. \begin{expl1} Let $(W,S)$ be a Coxeter system. n}$, $w_X$ denotes the longest element in $W_X$. n} \}$ is a cactus basis based on $W$. n}$, then $\Tr(\Gamma_{\FF,\mu})$ is the cactus group $C(W, S)$. n}$, then $\Tr (\Gamma_{\FF,\mu})$ is the ``Artin'' version of $C(W,S)$. We call this group the \emph{Artin-cactus group} associated with $(W,S)$ and we denote it by $\AC (W,S)$. Like Artin groups, these groups are preGarside groups (see Theorem \ref{thm2_14}). We also know that, when $S$ is finite, they are torsion-free and they have a solution to the word problem (see Theorem \ref{thm2_16} and Corollary \ref{corl2_5}). \end{expl1} \begin{expl2} Dual structures for Artin groups were introduced by Birman--Ko--Lee \cite{BiKoLe1} for the braid groups and by Bessis \cite{Bessi1} for all Artin groups. They are quite mysterious but they can be extremely useful for understanding some Artin groups. For example, they are the main tool in the solutions to the word problem and to the $K (\pi, 1)$ conjecture for Artin groups of affine type \cite{McCSul1,PaoSal1}. Let $(W,S)$ be a Coxeter system of spherical type (i.e. such that $W$ is finite). Then another cactus basis based on $W$ can be defined as follows. We denote by $T = \{ w s w^{-1} \mid w \in W\,,\ s \in S\}$ the set of reflections of $W$ and by $\lg_T$ the word length on $W$ with respect to $T$. Fix a Coxeter element $c$. Then the triple $(W,T,c)$ is called a \emph{Coxeter dual system} (see \cite[Definition 1.3.2]{Bessi1}). Let $\delta$ be a \emph{standard parabolic element} (relative to $c$), that is, an element of $W$ such that $\lg_T (\delta) + \lg_T (\delta^{-1} c) = \lg_T (c)$. Let $t_1 t_2\dots t_k = \delta$ be a decomposition of $\delta$ on $T$ with $k = \lg_T (\delta)$. Then the \emph{standard (dual) parabolic subgroup} $W_\delta$ associated with $\delta$ is the subgroup generated by $\{t_1, t_2, \dots, t_k\}$. We know from \cite[Corollary 1.6.2, Definition 1.6.3]{Bessi1} that this subgroup depends only on $\delta$ and not on the chosen decomposition and that $(W_\delta, T \cap W_\delta, \delta)$ is a dual Coxeter system. Moreover, by \cite[Lemma 2.5]{BraWat1}, if $\delta, \delta'$ are two standard parabolic elements, then $\delta = \delta'$ if and only if $W_\delta = W_{\delta'}$. We say that a standard parabolic subgroup $W_\delta$ (or $\delta$) is \emph{irreducible} if we cannot decompose $T \cap W_{\delta}$ into a non-trivial partition $ T_1\sqcup T_2$ such that $xy = yx$ for all $x \in T_1$ and $y \in T_2$. Let $(W,T,c)$ be a dual Coxeter system with $W$ finite. Then the family $\FF$ consisting of the pairs $X_\delta = (W_\delta, \delta)$, where $\delta$ is a standard (non-trivial) irreducible parabolic element, is a cactus basis based on $W$. If we denote by $\mu (X_\delta)$ the order of $\delta$ for all $X_\delta \in \FF$, then we call $\Tr(\Gamma_{\FF, \mu})$ the \emph{dual cactus group} associated with $(W,T,c)$ and we denote it by $C^ (W,T,c)$. If we set $\mu (X_\delta) = \infty$ for all $X_\delta \in \FF$, then we call $\Tr (\Gamma_{\FF,\mu})$ the \emph{dual Artin-cactus group} associated with $(W,T,c)$ and we denote it by $\AC^(W,T,c)$. We do not know when $C^ (W, T, \delta)$ is isomorphic to $C (W, S)$ and when $\AC^* (W, T, \delta)$ is isomorphic to $\AC (W, S)$. \begin{expl} Let $W$ be the symmetric group on three letters endowed with its dual presentation \[ W = \langle x ,y ,z \mid x^2 = y^2 = z^2 = 1\,,\ xy = yz = zx \rangle\,. \] For the Coxeter element $c = xy$, there are four standard non-trivial parabolic elements, $x, y, z, u=xy$, and the dual cactus group and dual Artin-cactus group have the following presentations: \begin{gather} C^ = \langle x, y, z, u \mid x^2 = y^2 = z^2 = u^3 = 1\,,\ x u = u z\,,\ y u = u x\,, z u = u y \rangle\,,\\ \AC^* = \langle x, y, z, u \mid x u = u z\,,\ y u = u x\,, z u = u y \rangle\,. \end{gather} \end{expl} \end{expl2} \begin{expl3} Examples 1 and 2 extend naturally to preGarside monoids (or groups) as follows. Let $M$ be a preGarside monoid. An \emph{atom} of $M$ is an element $a$ in $M \setminus \{1\}$ such that, for all $a_1, a_2 \in M$, if $a = a_1 a_2$, then $a_1 = 1$ or $a_2 = 1$. We denote by $\AA$ the set of atoms of $M$. It is easily shown that $\AA$ generates $M$ and, if $N$ is a parabolic submonoid, then $\AA \cap N$ is the set of atoms of $N$ and it generates $N$. Let $N$ be a Garside parabolic submonoid of $M$. By \cite{Dehor1} the submonoid $N$ admits a least Garside element, which we denote by $\Delta_N$, in the sense that, if $\Delta'_N$ is another Garside element of $N$, then $\Delta_N \in \Div (\Delta'_N)$. We know that $N$ embeds into $G(N)$ and that the conjugation by $\Delta_N$ leaves $N$ invariant, that is, the map $N \to N$, $ g \mapsto \Delta_N g \Delta_N^{-1}$, is well-defined and is an automorphism. Moreover, if $N_1$ is a parabolic submonoid of $N$, then $N_1$ and $\Delta_N N_1 \Delta_N^{-1}$ are both Garside parabolic submonoids of $N$ and $\Delta_N \Delta_{N_1} \Delta_N = \Delta_{\Delta_N N_1 \Delta_N^{-1}}$. Let $M$ be a preGarside monoid and let $\AA$ be the set of its atoms. We say that a parabolic submonoid $N$ of $M$ is \emph{irreducible} if we cannot decompose $\AA \cap N$ into a non-trivial partition $\AA_1 \sqcup \AA_2$ such that $ab = ba$ for all $a \in \AA_1$ and $b \in \AA_2$. Then the family $\FF$ consisting of the pairs $X_N = (G(N), \Delta_N)$, with $N$ an irreducible Garside parabolic submonoid of $M$, is a cactus basis. Note that we do not specify that $\FF$ is a cactus basis based on a given group, and actually we do not know if $\FF$ is a cactus basis based on a group ($G(M)$, for example), but the reader will easily extend the definition in this context. To complete the data we also need to define the labeling $\mu: \FF \to \N_{\ge 2} \cup \{\infty\}$. There are two natural choices. The first consists in taking $\mu(X_N)$ the order of the conjugation by $\Delta_N$ in $\Aut (G(N))$ if different from $1$ and $2$ if the conjugation by $\Delta_N$ is the identity. Note that $\Delta_N$ has always infinite order. The second consists in setting $\mu (X_N) = \infty$ for all $X_N \in \FF$. \end{expl3} \subsection{Virtual cactus groups}\label{subsec3_2} \begin{defin} Let $n \ge 2$. The \emph{virtual cactus group} on $n$ strands, denoted by $\VJ_n$, is the group defined by the presentation with generators \[ x_{p,q}\,,\ 1\le p < q \le n \quad \text{and} \quad \rho_i\,,\ 1 \le i \le n-1\,, \] and relations: \begin{itemize} \item[(j1)] $x_{p,q}^2 = 1$ for $1 \le p < q \le n$, \item[(j2)] $x_{p,q} x_{m,r} = x_{m,r} x_{p,q}$ for $[p,q] \cap [m,r] = \emptyset$, \item[(j3)] $x_{p,q} x_{m,r} = x_{p+q-r,p+q-m} x_{p,q}$ for $[m,r] \subset [p,q]$, \item[(s1)] $\rho_i^2 = 1$ for $1 \le i \le n-1$, \item[(s2)] $\rho_i \rho_j \rho_i = \rho_j \rho_i \rho_j$ for $ \|i-j\| = 1$, \item[(s3)] $\rho_i \rho_j = \rho_j \rho_i$ for $\|i-j\| > 1$, \item[(m1)] $\rho_i x_{p,q} = x_{p,q} \rho_i$ for $i < p-1$ and for $i \ge q+1$, \item[(m2)] $x_{p,q} \rho_{q} \rho_{q-1} \dots \rho_p = \rho_{q} \rho_{q-1} \dots \rho_p x_{p+1,q+1} $ for $1 \le p < q <n$. \end{itemize} \end{defin} As pointed out in the introduction, the elements of the virtual cactus group are represented by planar diagrams. To the usual diagrams which represent the elements of $J_n$ we add virtual crossings keeping the principle that two arcs connecting the same points and passing only through virtual crossings are equivalent. For instance, the generators $x_{p,q}$ and $\rho_i$ are depicted in Figure \ref{fig3_1} and the relation (m2) is illustrated in Figure \ref{fig3_2}. \begin{figure}[ht!] \begin{center} \includegraphics[width=6cm]{BeGoPaFig3_1.pdf} \caption{Generators of $\VJ_n$}\label{fig3_1} \end{center} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[width=10.2cm]{BeGoPaFig3_2.pdf} \caption{Relation (m2) in the presentation of $\VJ_n$}\label{fig3_2} \end{center} \end{figure} Relations (j1--j3) are the relations which define the cactus group $J_n$, and they induce a homomorphism $\iota_J : J_n \to \VJ_n$ which sends $x_{p,q}$ to $x_{p,q}$ for all $1 \le p < q \le n$. Relations (s1--s3) are the relations in the standard presentation of the symmetric group $\SSS_n$, and they induce a homomorphism $\iota_S: \SSS_n \to \VJ_n$ which sends $s_i=(i,i+1)$ to $\rho_i$ for all $1 \le i \le n-1$. Relations (m1--m2), called the \emph{mixed relations}, make that two arcs connecting the same points and passing through only virtual crossings are equivalent. There is a homomorphism $\pi_K: \VJ_n \to \SSS_n$ defined by \[ \pi_K (x_{p,q}) = 1 \text{ for } 1 \le p < q \le n \text{ and } \pi_K (\rho_i) = s_i \text{ for } 1 \le i \le n-1\,. \] Note that $\pi_K \circ \iota_S = \id_{\SSS_n}$, hence $\pi_K$ is surjective, $\iota_S$ is injective, and there is a semi-direct product decomposition $\VJ_n = \KVJ_n \rtimes \SSS_n$, where $\KVJ_n = \Ker (\pi_K)$. The group $\VJ_n$ is not a trickle group, but $\KVJ_n$ is (see Proposition \ref{prop3_7}). This enables to solve the word problem in $\VJ_n$ but also to prove other results on $\VJ_n$ such as the fact that $\iota_J : J_n \to \VJ_n$ is injective (see Corollary \ref{corl3_9}). A different proof of this fact can be found in \cite[Corollary 11.4]{IKLPR1}. \begin{rem} For $1 \le p < q \le n$ we denote by $t(p,q)$ the permutation which sends $p$ to $q$ and $q$ to $p$, $p+1$ to $q-1$ and $q-1$ to $p+1$, and so on. More precisely, for all $0 \le i \le q-p$, $t(p,q)$ sends $p+i$ to $q-i$, and $t(p,q)(k) =k$ if $k \not \in [p,q]$. There exists another natural epimorphism from $\VJ_n$ to $\SSS_n$, denoted by $\pi_P: \VJ_n \to \SSS_n$, defined by \[ \pi_P (x_{p,q}) = t(p,q) \text{ for } 1 \le p < q \le n \text{ and } \pi_P (\rho_i) = s_i \text{ for } 1 \le i \le n-1\,. \] The kernel of $\pi_P$ is the \emph{pure virtual cactus group} $\PVJ_n$ studied in \cite{IKLPR1}, and it is different from the group $\KVJ_n$ that we study. A presentation for $\PVJ_n$ is given in \cite[Lemma 10.12]{IKLPR1}. \end{rem} For $2 \le \ell \le n$ we set $V_\ell = \{ (t_1, \dots, t_\ell) \in \{1, \dots, n\}^\ell \mid t_i \neq t_j \text{ for } i \neq j\}$. Let $y = (t_1, \dots, t_\ell) \in V_\ell$. We choose $w \in \SSS_n$ such that $w(i) = t_i$ for all $1 \le i \le \ell$ and we set $\delta_y = \iota_S(w) \,x_{1,\ell}\, \iota_S(w)^{-1}$. Note that $\delta_y \in \KVJ_n$. \begin{lem}\label{lem3_2} Let $2 \le \ell \le n$ and let $y \in V_\ell$. Then the above definition of $\delta_y$ does not depend on the choice of $w \in \SSS_n$. \end{lem} \begin{proof} We set $y = (t_1, \dots, t_\ell)$ and we take $w, w' \in \SSS_n$ such that $w(i) = w'(i) = t_i$ for all $1 \le i \le \ell$. We have $w^{-1} w'(i) = i$ for all $i \in \{1, \dots, \ell\}$, hence $w^{-1} w'$ lies in the subgroup of $\SSS_n$ generated by $s_{\ell+1}, \dots, s_{n-1}$, and therefore $\iota_S (w^{-1} w')$ lies in the subgroup of $\VJ_n$ generated by $\rho_{\ell+1}, \dots, \rho_{n-1}$. Relations (m1) imply that $\rho_i x_{1,\ell} = x_{1,\ell} \rho_i$ for all $i \in \{\ell+1, \dots, n-1\}$, hence $\iota_S (w^{-1} w') \, x_{1,\ell}\, \iota_S (w'^{-1} w) = x_{1,\ell}$, and therefore $\iota_S(w) \,x_{1,\ell}\, \iota_S (w)^{-1} = \iota_S (w') \,x_{1,\ell} \, \iota_S (w')^{-1}$. \end{proof} We set \[ V = \bigcup_{\ell=2}^n V_\ell\,. \] We consider the action of $\SSS_n$ on $V$ defined by, for $w \in \SSS_n$ and $y = (t_1, \dots, t_\ell) \in V$, \[ w \cdot y = (w (t_1) , \dots, w (t_\ell))\,. \] \begin{lem}\label{lem3_3} \begin{itemize} \item[(1)] Let $w \in \SSS_n$ and $y \in V$. Then $\iota_S (w) \, \delta_y \, \iota_S (w)^{-1} = \delta_{w \cdot y}$. \item[(2)] Let $1 \le p < q \le n$. Then $\delta_{(p,p+1,\dots,q)} = x_{p,q}$. \end{itemize} \end{lem} \begin{proof} We set $y = (t_1, \dots, t_\ell)$ and we choose $w' \in \SSS_n$ such that $w' \cdot (1, \dots, \ell) = (t_1, \dots, t_\ell)$. Notice that $w \cdot y = (ww'(1), \dots, ww'(\ell))= ww' \cdot (1, \dots, \ell)$. Then \[ \iota_S (w) \, \delta_y \, \iota_S (w)^{-1} = \iota_S (w) \, \iota_S (w') \, x_{1,\ell} \, \iota_S (w')^{-1} \, \iota_S (w)^{-1} = \iota_S (w w') \, x_{1,\ell} \, \iota_S (w w')^{-1} = \delta_{w \cdot y}\,. \] Let $u = s_1 s_2 \dots s_{n-1} = (1, 2, \dots, n)$. We easily see using Relations (m1) and (m2) that, for all $1 \le m < r < n$, \[ \iota_S (u) \, x_{m,r}\, \iota_S (u)^{-1} = x_{m+1,r+1}\,. \] On the other hand, $u \cdot (m, m+1, \dots, r) = (u(m), u(m+1), \dots, u(r)) = (m+1, m+2, \dots, r+1)$. So, \[ \delta_{(p, p+1, \dots, q)} = \iota_S (u)^{p-1} \, x_{1,q-p+1} \, \iota_S(u)^{-p+1} = x_{p,q}\,. \proved \] \end{proof} \begin{lem}\label{lem3_4} The set $\{ \delta_y \mid y \in V\}$ generates $\KVJ_n$. \end{lem} \begin{proof} Let $g \in \KVJ_n$. Since $g$ belongs to $\VJ_n$, there exist $ k \ge 0$, $p_1, \dots, p_k, q_1, \dots, q_k \in \N$, $w_0, w_1, \dots, w_k \in \SSS_n$ such that $1 \le p_i <q_i \le n$ for all $1 \le i \le k$, and \[ g = \iota_S (w_0) \, x_{p_1,q_1} \, \iota_S (w_1) \dots x_{p_k,q_k} \, \iota_S(w_k)\,. \] For $1 \le i \le k+1$ we set $v_i = w_0 w_1 \dots w_{i-1}$. Then \[ g = \iota_S (v_1) \, x_{p_1,q_1} \, \iota_S (v_1)^{-1} \, \iota_S (v_2) \, x_{p_2,q_2} \, \iota_S (v_2)^{-1} \dots \iota_S(v_k) \, x_{p_k,q_k} \, \iota_S (v_k)^{-1} \, \iota_S (v_{k+1})\,. \] Since $g \in \KVJ_n$, we have $1 = \pi_K (g)= v_{k+1}$, hence \[ g = \iota_S (v_1) \, x_{p_1,q_1} \, \iota_S (v_1)^{-1} \, \iota_S (v_2) \, x_{p_2,q_2} \, \iota_S (v_2)^{-1} \dots \iota_S(v_k) \, x_{p_k,q_k} \, \iota_S (v_k)^{-1} = \delta_{y_1} \delta_{y_2} \dots \delta_{y_k}\,, \] where $y_i = v_i \cdot (p_i, p_i + 1, \dots, q_i)$ for all $1 \le i \le k$. \end{proof} Now we define a quadruple $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ as follows. The set of vertices of $\Gamma$ is $V = \bigcup_{\ell=2}^n V_\ell$. We set $\mu (y) = 2$ for all $y \in V$. For $y = (t_1, \dots, t_\ell)$ and $y' = (t_1', \dots, t_k')$ in $V$, we set $y' \le y$ if there exist $1 \le p < q \le \ell$ such that $y'= (t_p, t_{p+1}, \dots, t_q)$. Let $y = (t_1, \dots, t_\ell)$ and $y' = (t_1', \dots, t_k')$ in $V$ such that $y \neq y'$. Then $\{y, y'\} \in E (\Gamma)$ if and only if either $y \le y'$ or $y' \le y$ or $\{t_1, \dots, t_\ell\} \cap \{t_1', \dots, t_k' \} = \emptyset$. Let $y = (t_1, \dots, t_\ell) \in V$ and $y' \in \starE_y (\Gamma)$. If $y' \not \le y$, then we set $\varphi_y (y') = y'$. Suppose $y' \le y$. We know that there exist $1 \le p < q \le \ell$ such that $y' = (t_p, \dots, t_q)$. Then we set $\varphi_y (y') = (t_{1 + \ell - q}, \dots, t_{1 + \ell - p})$. \begin{lem}\label{lem3_5} The above defined quadruple $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ is a trickle graph. \end{lem} \begin{proof} Conditions (a) and (d) are satisfied by definition, and Conditions (b) and (f) are obviously satisfied. It is easily seen that, for all $x \in V$, $\varphi_x^2=1$, hence Condition (e) is also satisfied. We show that $\Gamma$ satisfies Condition (c). Let $x \in V$ and $y, z \in \starE_x (\Gamma)$ be such that $z \le y$. If $y \|\| x$ then, by Condition (b), $z \|\| x$, hence $\varphi_x (z) = z \le y = \varphi_x (y)$. Suppose $z \le y < x$. Set $x = (t_1, \dots, t_\ell)$. There exist $1 \le p \le k < m \le q \le \ell$ such that $y = (t_p, \dots, t_q)$ and $z = (t_k, \dots, t_m)$. We have $\varphi_x (y) = (t_{1+\ell-q}, \dots, t_{1+\ell-p})$, $\varphi_x (z) = (t_{1+\ell-m}, \dots, t_{1+\ell -k})$ and $1 \le 1+\ell-p \le 1+\ell-m < 1+\ell-k \le 1 + \ell -p \le \ell$, hence $\varphi_x (z) \le \varphi_x (y)$. This shows that, if $z \le y$, then $\varphi_x (z) \le \varphi_x (y)$. But $\varphi_x^2 = 1$, hence we actually have $z \le y$ if and only if $\varphi_x (z) \le \varphi_x (y)$. Now, we show that $\Gamma$ satisfies Condition (g). Let $x, y, z \in V$ be such that $z \le y \le x$. Set $x = (t_1, \dots, t_\ell)$. There exist $1 \le p \le k < m \le q \le \ell$ such that $y = (t_p, \dots, t_q)$ and $z = (t_k, \dots, t_m)$. Let $y' = \varphi_x (y) = (t_{1+\ell-q}, \dots, t_{1+\ell-p})$. Then a direct calculation shows that \[ (\varphi_x \circ \varphi_y) (z) = (t_{1 + \ell - p - q + k}, \dots, t_{1 + \ell - p - q + m}) = (\varphi_{y'} \circ \varphi_x) (z)\,. \proved \] \end{proof} \begin{lem}\label{lem3_6} There is a surjective homomorphism $\Phi: \Tr (\Gamma) \to \KVJ_n$ which sends $x$ to $\delta_x$ for all $x \in V$. \end{lem} \begin{proof} Let $x = (t_1, \dots, t_\ell) \in V$. Let $w \in \SSS_n$ be such that $w(i) = t_i$ for all $1 \le i \le n$. Then \[ \delta_x^2 = \iota_S (w)\, x_{1,\ell}^2\, \iota_S (w)^{-1} = \iota_S (w) \, \iota_S (w)^{-1} = 1\,. \] Let $x, y \in V$ be such that $\{x, y\} \in E_{\|\|} (\Gamma)$. We set $x=(t_1, \dots, t_\ell)$ and $y = (t_1', \dots, t_k')$. By definition we have $\{t_1, \dots, t_\ell\} \cap \{t_1', \dots, t_k'\} = \emptyset$, hence we can choose $w \in \SSS_n$ such that $ w(i)=t_i$ for all $1 \le i \le \ell$ and $w (\ell + j) = t_j'$ for all $1 \le j\le k$. Then, by Lemma \ref{lem3_3}, \[ \delta_x \delta_y = \iota_S(w) \, x_{1,\ell} x_{\ell+1,\ell+k}\, \iota_S(w)^{-1} = \iota_S(w) \, x_{\ell+1,\ell+k} x_{1,\ell} \, \iota_S(w)^{-1} = \delta_y \delta_x\,. \] Let $x, y \in V$ be such that $y \le x$. We set $x = (t_1, \dots, t_\ell)$ and $y = (t_p, \dots, t_q)$, where $1 \le p < q \le \ell$. We choose $w \in \SSS_n$ such that $w (i) = t_i$ for all $1 \le i \le \ell$. Then, by Lemma \ref{lem3_3}, \[ \delta_x \delta_y = \iota_S(w) \, x_{1,\ell} x_{p,q} \, \iota_S(w)^{-1} = \iota_S(w) \, x_{1+\ell-q,1+\ell-p} x_{1,\ell} \, \iota_S(w)^{-1} = \delta_{\varphi_x(y)} \delta_x\,. \] This shows that there is a homomorphism $\Phi: \Tr (\Gamma) \to \KVJ_n$ which sends $x$ to $\delta_x$ for all $x \in V$. This homomorphism is surjective because, by Lemma \ref{lem3_4}, $\{\delta_x \mid x \in V\}$ generates $\KVJ_n$. \end{proof} \begin{prop}\label{prop3_7} The homomorphism $\Phi: \Tr (\Gamma) \to \KVJ_n$ of Lemma \ref{lem3_6} is an isomorphism. \end{prop} \begin{proof} The action of $\SSS_n$ on $V$ extends to an action of $\SSS_n$ on $\Tr (\Gamma)$, hence we can consider the semi-direct product $G = \Tr (\Gamma ) \rtimes \SSS_n$. Furthermore, by Lemma \ref{lem3_3}, the map $\hat \Phi: G \to \VJ_n$ defined by $\hat \Phi (g,w) = \Phi(g) \, \iota_S (w)$ is a group homomorphism. We show that there is a homomorphism $\Psi: \VJ_n \to G$ which sends $x_{p,q}$ to $x_{p,q}' = (p, p+1, \dots, q ) \in V \subseteq \Tr (\Gamma)$ for $1 \le p < q \le n$, and which sends $\rho_i$ to $s_i = (i, i + 1) \in \SSS_n$ for $1 \le i \le n-1$. The relations (j1) $x_{p,q}'^2 = 1$ for $1 \le p < q \le n$, (j2) $x_{p,q}' x_{m,r}' = x_ {m,r}' x_{p,q}'$ for $[p,q] \cap [m,r] = \emptyset$, and (j3) $x_{p,q}' x_{m,r}' = x_{p+q-r,p+q-m}' x_{p,q}'$ for $[m,r] \subset [p,q]$, follow from the definitions of the trickle graph $\Gamma$ and of the associated presentation of $\Tr (\Gamma)$. The relations (s1) $s_i^2 = 1$ for $1 \le i \le n-1$, (s2) $s_i s_j s_i = s_j s_i s_j$ for $\|i - j\| = 1$, and (s3) $s_i s_j = s_j s_i$ for $\|i - j\| > 1$, are the relations of the standard presentation of the symmetric group $\SSS_n$. Finally, the relations (m1) $s_i x_{p,q}' = x_{p,q}' s_i$ for $i < p-1$ and for $i \ge q+1$, and (m2) $ x_{p,q}'s_{q} s_{q-1} \dots s_p=s_{q} s_{q-1} \dots s_p x_{p+1,q+1}'$ for $1 \le p < q <n$, follow from the action of $\SSS_n$ on $V$. For every $w \in \SSS_n$ we have $(\Psi \circ \hat \Phi) (w) = \Psi (\iota_S (w)) = w$. On the other hand, let $x = (t_1, \dots, t_\ell) \in V$. We choose $w \in \SSS_n$ such that $w \cdot (1, \dots, \ell) = x = (t_1, \dots, t_\ell)$. Then \[ (\Psi \circ \hat \Phi) (x) = \Psi (\delta_x) = \Psi (\iota_S (w)\, x_{1,\ell}\, \iota_S (w)^{-1}) = w \, x_{1,\ell}'\, w^{-1} = w \cdot x_{1,\ell}' = x\,. \] This shows that $\Psi \circ \hat \Phi = \id_G$, hence $\hat \Phi$ is injective, and therefore $\Phi$ is injective. Since we already know that $\Phi$ is surjective, this completes the proof of the proposition. \end{proof} Since $\KVJ_n$ is a finite index subgroup of $\VJ_n$, the following follows from Proposition \ref{prop3_7} and Corollary \ref{corl2_5}. \begin{corl}\label{corl3_8} Let $n \ge 2$. Then $\VJ_n$ has a solution to the word problem. \end{corl} Another consequence of Proposition \ref{prop3_7} is the following. \begin{corl}[Ilin--Kamnitzer--Li--Przytycki--Rybnikov \cite{IKLPR1}]\label{corl3_9} Let $n \ge 2$. Then the natural homomorphism $\iota_J: J_n \to \VJ_n$ is injective. \end{corl} \begin{proof} Recall that, for $1 \le p < q \le n$, we set $x_{p,q}' = (p,p+1,\dots,q) \in V$. Set $U = \{x_{p,q}' \mid 1 \le p < q \le n\}$ and denote by $\Gamma_1$ the full subgraph of $\Gamma$ spanned by $U$. It is easily seen that $\Gamma_1$ is a parabolic subgraph of $\Gamma$. We denote by $\le_1$ and $\mu_1$ the restrictions of $\le$ and $\mu$ to $V(\Gamma_1)$, respectively, and, for every $x \in V (\Gamma_1)$, we denote by $\varphi_{1,x}$ the restriction of $\varphi_x$ to $\starE_x (\Gamma_1)$. Then $\Gamma_1 = (\Gamma_1, \le_1 ,\mu_1, (\varphi_{1,x})_{x\in U})$ is a trickle graph and, by Theorem \ref{thm2_10}, the inclusion $U \hookrightarrow V$ induces an injective homomorphism $\iota: \Tr (\Gamma_1) \to \Tr (\Gamma)$. We easily see that the map $\{x_{p,q} \mid 1 \le p < q \le n\} \to U$, $x_{p,q} \mapsto x_{p,q}'$, induces an isomorphism $\phi: J_n \to \Tr (\Gamma_1)$ and that the composition $\iota \circ \phi: J_n \to \Tr (\Gamma) = \KVJ_n \subseteq \VJ_n$ coincides with $\iota_J: J_n \to \VJ_n$. So, $\iota_J: J_n \to \VJ_n$ is injective. \end{proof} \subsection{Thompson group F}\label{subsec3_3} There are several equivalent definitions of Thompson group $F$. In this paper we consider the one as a subgroup of the group of orientation preserving piecewise linear homeomorphisms of the real line (see \cite[Theorem 1.4.1]{Buril1}). \begin{defin} Let $f: \R \to \R$ be an orientation preserving piecewise linear homeomorphism of the real line. We say that $f$ belongs to \emph{Thompson group $F$} if there exist two finite sequences with the same cardinality $u_0, u_1, \dots, u_n$ and $v_0, v_1, \dots, v_n$ in $\Z [\frac{1}{2}]$ such that: \begin{itemize} \item $u_0 < u_1 < \cdots < u_n$ and $v_0 <v_1 < \cdots <v_n$, \item $f$ sends linearly the interval $[u_{i-1}, u_i]$ onto $[v_{i-1}, v_i]$ for all $i \in \{1, \dots, n\}$, \item $f$ sends the interval $(-\infty, u_0]$ onto $(-\infty, v_0]$ via the translation $t \mapsto t + v_0 - u_0$, and $f$ sends the interval $[u_n, +\infty)$ onto $[v_n, +\infty)$ via the translation $t \mapsto t + v_n - u_n$. \end{itemize} \end{defin} Our goal in this subsection is to provide a trickle graph $\Gamma = (\Gamma, \le, \mu,(\varphi_x)_{x \in V(\Gamma)})$ such that $F \simeq \Tr (\Gamma)$. We start by defining the vertex set $V = V (\Gamma)$. For $p \in \N$ we define a subset $V_p \subseteq \Z [\frac{1}{2}]$ and a map $s_p: V_p \to V_p$ which satisfies $x <s_p (x)$ and $[x, s_p(x)] \cap V_p = \{x, s_p(x)\}$ for all $x \in V_p$ by induction on $p$. We set $V_0 = \Z$ and we set $s_0 (x) = x + 1$ for all $x \in V_0$. Suppose $V_p$ and $s_p: V_p \to V_p$ are defined. Let $x \in V_p$. Let $x' = s_p(x)$ and, for $k \in \N$, let $u_k = u_{p+1,k} (x) = x' - \frac{x'-x}{2^k}$ (see Figure \ref{fig3_3}). We set $V_{p+1} (x) = \{u_{p+1,k} (x) \mid k \in \N\}$. Then \[ V_{p+1} = \bigsqcup_{x \in V_p} V_{p+1} (x)\,. \] If $x \in V_p$ and $k \in \N$, then we set $s_{p+1} (u_{p+1, k} (x)) = u_{p+1, k+1} (x)$. So, in Figure \ref{fig3_3}, $s_{p+1}$ sends $u_k$ to $u_{k+1}$. \begin{figure}[ht!] \begin{center} \includegraphics[width=8cm]{BeGoPaFig3_3.pdf} \caption{Definition of $V_{p+1}$}\label{fig3_3} \end{center} \end{figure} Observe that $V_p\subset V_{p+1}$ for all $p \in \N$ and that $\bigcup_{p \in \N} V_p = \Z [\frac{1}{2}]$. So, $\{ V_p \mid p \in \N \}$ is a filtration of $\Z [\frac{1}{2}]$. Now, we set $V = \Z [\frac{1}{2}] \sqcup \{\infty\}$. We endow $\Z [\frac{1}{2}]$ with the order induced by the one of $\R$, and we set $x < \infty$ for all $x \in \Z [\frac{1}{2}]$. So, the order $\le$ on $V$ is a total order and $\Gamma$ is defined to be the complete graph on $V$. We set $\mu (x) = \infty$ for all $x \in V$. It remains to define $\varphi_x$ for all $x \in V$, but for this we first need to define the generating system $\{h_x \mid x\in V\}$ for $F$ corresponding to the trickle structure. In order to define the generating system $\{h_x \mid x\in V\}$, in addition to the sets $V_p$ and to the maps $s_p: V_p \to V_p$, we need other maps $t_p: V_p \to V_p$, $p \in \N$, defined by induction on $p$ as follows. We set $t_0 (x) = x - 1$ for all $x \in V_0 = \Z$. Let $p \ge 1$ and let $x \in V_p$. If $x \in V_{p-1}$, then we set $t_p (x) = t_{p-1} (x)$. If $x \not \in V_{p-1}$, then there exists $y \in V_{p-1}$ and $k \ge 1$ such that $x = u_{p, k} (y)$, and we set $t_p (x) = u_{p, k-1} (y)$. Note that, for all $x \in V_p$, there exists $N \ge 0$ such that $t_p^n (x) \in V_0 = \Z$ for all $n\ge N$. The map $h_\infty$ is defined to be the translation $h_\infty: \R \to \R$, $t \mapsto t-1$. We define $h_x$ for $x \in \Z [\frac{1}{2}]$ as follows. Let $p$ be the least integer $\ge 0$ such that $x \in V_p$. We set $x' = t_p (x)$ (see Figure \ref{fig3_4}). For all $k \ge 0$ we set $v_k = s_{p+1}^k (x')$ and $v_{-k} = t_p^k (x')$. We have \begin{gather} \cdots < v_{-k} < \cdots < v_{-1} < v_0 = x' < v_1 < \cdots < v_k < \cdots\,,\\ \lim\limits_{k \to -\infty} v_k = -\infty \text{ and } \lim\limits_{k \to +\infty} v_k =x\,. \end{gather} Then $h_x: \R \to \R$ sends linearly the interval $[v_k, v_{k+1}]$ onto $[v_{k-1}, v_k]$ (preserving the orientation) for all $k \in \Z$ and it is the identity on $[x, +\infty)$. \begin{figure}[ht!] \begin{center} \includegraphics[width=8.8cm]{BeGoPaFig3_4.pdf} \caption{Definition of $h_x$}\label{fig3_4} \end{center} \end{figure} The proof of the following is left to the reader. \begin{lem}\label{lem3_10} Let $x \in \Z [\frac{1}{2}]$. Let $(v_k)_{k\in \Z}$ be as defined above. \begin{itemize} \item[(1)] $h_x$ sends linearly the interval $[v_1,x]$ onto $[v_0,x]$. \item[(2)] $h_x$ sends the interval $[x,+\infty)$ onto $[x,+\infty)$ via the identity. \item[(3)] Let $N$ be the least integer $\ge 0$ such that $v_{-N} \in V_0 = \Z$. Then $h_x$ sends $(-\infty,v_{-N}]$ onto $(-\infty,v_{-N-1}]=(-\infty,v_{-N}-1]$ via the translation $t \mapsto t-1$. \end{itemize} In particular, $h_x\in F$. \end{lem} Note that we also have $h_\infty \in F$. Furthermore, we know that $F$ is generated by $\{h_0,h_\infty\}$ (see \cite[Corollary 2.6]{CaFlPa1}), hence: \begin{lem}\label{lem3_11} The set $\{h_x \mid x \in V\}$ generates $F$. \end{lem} The next result is an observation. \begin{lem}\label{lem3_12} Let $x \in V$. Then $h_x (\Z [\frac{1}{2}]) = \Z [\frac{1}{2}]$, $h_x$ preserves the order of $\Z [\frac{1}{2}]$, and $h_x(y)=y$ for all $y \in \Z [\frac{1}{2}]$ such that $y \ge x$. \end{lem} For $x \in V$, we set $\varphi_x (\infty) = \infty$ and $\varphi_x (y) = h_x(y)$ for all $y \in \Z [\frac{1}{2}]$. Now we need to show that $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V})$ is a trickle graph and that the map $V \to F$, $ x \mapsto h_x$, induces an isomorphism $\Tr (\Gamma) \to F$. We begin by proving the following lemma which is a key point in the proofs that will follow. \begin{lem}\label{lem3_13} Let $x \in V$ and let $y \in \Z [\frac{1}{2}]$ be such that $y < x$. Let $y' = h_x (y)$. Then $h_x \circ h_y \circ h_x^{-1} = h_{y'}$. \end{lem} \begin{proof} First assume $x=\infty$. Then $y'=y-1$ and by construction $h_{y'} = h_{y-1} = h_\infty \circ h_y \circ h_\infty^{-1}$. Now assume $x \in \Z [\frac{1}{2}]$. Let $p$ be the least integer $\ge 0$ such that $x \in V_p$. Set $x' = t_p (x)$. For all $k \ge 0$ we set $v_k = s_{p+1}^k (x')$ and $v_{-k} = t_p^k (x')$. Then $h_x: \R \to \R$ sends linearly the interval $[v_k, v_{k+1}]$ onto $[v_{k-1},v_k]$ for all $k \in \Z$ and it is the identity on $[x, +\infty)$. We denote by $\alpha: (-\infty, x) \to \R$ the orientation preserving piecewise linear homeomorphism which sends linearly $[v_k, v_{k+1}]$ onto $[k, k+1]$ for all $k \in \Z$ (see Figure \ref{fig3_5}). Since $h_x$, $h_y$ and $h_{y'}$ induce homeomorphisms of $(-\infty, x)$, we can consider the homeomorphisms of the real line $g_x = \alpha \circ h_x\|_{ (-\infty, x)} \circ \alpha^{-1}$, $g_y = \alpha \circ h_y\|_{(-\infty, x)} \circ \alpha^{-1}$ and $ g_{y'} = \alpha \circ h_{y'}\|_{(-\infty, x)}\circ \alpha^{-1}$. The following properties are easily observed: \begin{itemize} \item $\alpha(\Z [\frac{1}{2}] \cap (-\infty, x)) = \Z[\frac{1}{2}]$, \item $g_x = h_\infty$, $g_y = h_{\alpha(y)}$ and $g_{y'} = h_{\alpha(y')}$. \end{itemize} We have $\alpha (y') = (\alpha \circ h_x) (y) = g_x (\alpha (y)) = h_\infty (\alpha (y))$, hence, by the case $x =\infty$ already treated, \[ g_{y'} = h_{\alpha(y')} = h_\infty \circ h_{\alpha(y)} \circ h_\infty^{-1} = g_x \circ g_y \circ g_x^{-1}\,. \] Since $h_x\|_{[x, +\infty)} = h_y\|_{[x, +\infty)} = h_{y'}\|_{[x, +\infty)} = \id_{[x ,+\infty)}$, we conclude that $h_{y'} =h_x \circ h_y \circ h_x^{-1}$. \end{proof} \begin{figure}[ht!] \begin{center} \includegraphics[width=8.8cm]{BeGoPaFig3_5.pdf} \caption{Definition of $\alpha$}\label{fig3_5} \end{center} \end{figure} Now we prove that $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ is a trickle graph. \begin{lem}\label{lem3_14} The above defined quadruple $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V})$ is a trickle graph. \end{lem} \begin{proof} We should prove that $(\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ satisfies Conditions (a) to (g) of the definition of a trickle graph. Condition (a) is obvious because $\Gamma$ is a complete graph. Condition (b) is empty because $E_{\|\|}(\Gamma)=\emptyset$. Condition (c) holds because $\varphi_x: V \to V$ preserves the order of $V$ for all $x \in V$. Condition (d) holds by Lemma \ref{lem3_12}. Conditions (e) and (f) are obvious because $\mu (x) = \infty$ for all $x \in V$. Condition (g) is a direct consequence of Lemma \ref{lem3_13}. \end{proof} Now, we prove that $\Tr (\Gamma)$ is isomorphic to $F$. \begin{thm}\label{thm3_15} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V})$ be the trickle graph of Lemma \ref{lem3_14}. Then $\Tr (\Gamma)$ is isomorphic to $F$. \end{thm} \begin{proof} By Lemma \ref{lem3_13} there exists a homomorphism $\Phi: \Tr (\Gamma) \to F$ which sends $x$ to $h_x$ for all $x \in V$. This homomorphism is surjective because $\{h_x \mid x \in V\}$ generates $F$ (see Lemma \ref{lem3_11}). So, it remains to prove that $\Phi$ is injective. Let $g \in \Tr (\Gamma) \setminus \{1\}$. Consider the rewriting system $\RR = \RR (\Gamma)$ of Theorem \ref{thm2_4}. Since $\Gamma$ is a complete graph, the $\RR$-irreducible pilings are all of length $1$ or $0$. Moreover, by Theorems \ref{thm2_3} and \ref{thm2_4}, there exists a unique $\RR$-irreducible piling which represents $g$. This piling is different from the empty one, because $g \neq 1$, hence it is of the form $(U)$, where $U=\{x_1^{a_1}, \dots, x_p^{a_p}\}$ and $x_1 > x_2 > \cdots >x_p$, where $p \ge 1$. Then $g = \omega (U) = x_1^{a_1} x_2^{a_2} \dots x_p^{a_p}$, hence \[ \Phi(g) = h_{x_1}^{a_1} \circ h_{x_2}^{a_2} \circ \cdots \circ h_{x_p}^{a_p}\,. \] The homeomorphism $h_{x_1}$ is of infinite order, hence, if $p=1$, then $\Phi (g) = h_{x_1}^{a_1} \neq \id$. Suppose $p \ge 2$. Choose $t \in (x_2,x_1)$. Since $h_{x_i} (t) = t$ for all $2 \le i \le p$ and $h_{x_1}$ is strictly decreasing on $(-\infty, x_1)$, \[ \Phi(g)(t) = h_{x_1}^{a_1}(t) \neq t\,, \] hence $\Phi(g) \neq \id$. This shows that $\Ker (\Phi) = \{1\}$, hence $\Phi$ is injective. \end{proof} To finish this subsection we show that some natural parabolic subgroups of $\Tr (\Gamma)$ are isomorphic to $F$ itself. More precisely, let $x\in \Z [\frac{1}{2}]$, let $V_x = \{x\}_\downarrow = \{ y \in V \mid y \le x\}$, and let $\Gamma_x$ be the full subgraph of $\Gamma$ spanned by $V_x$. As observed in Subsection \ref{subsec2_4}, $\Gamma_x$ is a parabolic subgraph of $\Gamma$. \begin{prop}\label{prop3_16} Let $x \in \Z [\frac{1}{2}]$. There exists a bijection $\beta: V_x \to V$ which induces an isomorphism $\beta: \Tr(\Gamma_x) \to \Tr (\Gamma)$. In particular, $\Tr (\Gamma_x)$ is isomorphic to $F$. \end{prop} \begin{proof} We consider the map $\alpha$ defined in the proof of Lemma \ref{lem3_13}. Let $p$ be the least integer $\ge 0$ such that $x \in V_p$. Set $x' = t_p (x)$. For all $k \ge 0$ we set $v_k = s_{p+1}^k (x')$ and $v_{-k} = t_p^k (x')$. Then $\alpha: (-\infty, x) \to \R$ denotes the orientation preserving piecewise linear homeomorphism which sends linearly $[v_k, v_{k+1}]$ onto $[k, k+1]$ for all $k \in \Z$ (see Figure \ref{fig3_5}). We have $\alpha( (-\infty, x) \cap \Z [\frac{1}{2}]) = \alpha (V_x \setminus \{x\}) = \Z [\frac{1} {2}]$. So, there is a bijection $\beta: V_x \to V$ which sends $y$ to $\alpha (y)$ for all $y \in V_x \setminus \{x\}$ and sends $x$ to $\infty$. It is easily checked that, for all $y,y' \in V_x$, we have $y \le y'$ if and only if $\beta(y) \le \beta (y')$, and that, for all $y \in V_x$, $h_{\beta (y)} = \alpha \circ h_y\|_{(-\infty,x)} \circ \alpha^{-1}$. Thus, $\beta$ induces an isomorphism between $\Gamma_x = (\Gamma_x, \le_x, \mu_x, (\varphi_{x,y})_{y \in V_x})$ and $\Gamma = (\Gamma, \le, \mu, (\varphi_y)_{y \in V})$, where $\le_x$ and $\mu_x$ are the restrictions of $\le$ and $\mu$ to $V_x$, respectively, and, for all $y \in V_x$, $\varphi_{x,y}$ is the restriction of $\varphi_y$ to $V_x = \starE_y (\Gamma_x)$. This $\beta$ obviously defines an isomorphism $\beta : \Tr (\Gamma_x) \to \Tr (\Gamma)$. \end{proof} \begin{rem} By identifying $\Tr (\Gamma)$ with $F$ it is easily seen that $\Tr (\Gamma_x)$ is the subgroup of $F$ consisting of the elements whose support is contained in $(-\infty, x)$. It is well-known and easy to prove that this subgroup is a copy $F$. \end{rem} \subsection{Ordered quandle groups}\label{subsec3_4} A \emph{quandle} is a non-empty set $Q$ endowed with an operation $$ which satisfies the following properties. \begin{itemize} \item[(1)] For all $x\in Q$, $x * x = x$. \item[(2)] For all $x, y, z \in Q$, $(z * x) * (y * x) = (z * y) * x$. \item[(3)] For all $x \in Q$, the map $Q \to Q$, $y \mapsto y * x$, is a bijection. \end{itemize} A quandle $Q$ is said to be \emph{ordered} (on the right) if there exists a total order $\le$ on $Q$ such that, for all $x, y, z \in Q$, we have $y \le z$ if and only if $y * x \le z * x$. \begin{expl1} A group $G$ is called \emph{bi-orderable} if there exists a total order $\le$ on $G$ which is invariant under left and right multiplication. Examples of bi-orderable groups are free abelian groups, free groups and, more generally, right-angled Artin groups. Other important examples are the pure braid groups. Let $G$ be a bi-orderable group, let $\le$ be a total order on $G$ invariant under left and right multiplication, and let $$ be the operation on $G$ defined by $y x = xyx^{-1}$. Then $(G, , \le)$ is an ordered quandle. \end{expl1} \begin{expl2} Consider the field $\R$ of real numbers endowed with its natural total order. Then, for a fixed $\alpha>0$, the binary operation $xy = \alpha x + (1-\alpha) y$ endows $\R$ with an ordered quandle structure (see \cite{BaElh1}). \end{expl2} Let $(Q,,\le)$ be an ordered quandle. Let $\Gamma$ be the full graph whose set of vertices is $Q$. For $x \in Q$ we define $\varphi_x: Q \to Q$ by $\varphi_x (y) = y$ if $x \le y$ and $\varphi_x (y) = y x$ if $y \le x$. We set $\mu (x) = \infty$ for all $x \in Q$. \begin{prop}\label{prop3_17} The above defined quadruple $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in Q})$ is a trickle graph. \end{prop} \begin{proof} Since $\Gamma$ is a full graph and $y\mapsto y * x$ is an order-preserving bijection, the map $\varphi_x$ is an order-preserving automorphism of $\Gamma$ for all $x \in Q$. Since $\Gamma$ is a full graph and $\le$ is a total order, Conditions (a) and (b) are trivially satisfied. Conditions (e) and (f) hold because $\mu (x) = \infty$ for all $x \in Q$. Condition (c) follows from the fact that $Q$ is ordered, and Condition (d) follows from the definition of $\varphi_x$. Finally, Condition (g) is satisfied because $Q$ is a quandle. \end{proof} \section{The trickle algorithm}\label{sec4} In this section we fix a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x\in V(\Gamma)})$ and we keep the notations of Subsection \ref{subsec2_2}. Our aim is to prove Theorem \ref{thm2_4}. Let $A$ be an alphabet and let $R$ be a rewriting system on $A^$. A \emph{critical pair} for $R$ is a quintuple $(u_1, u_2, u_3, v_1, v_2)$ of elements of $A^$ satisfying one of the following two conditions. \begin{itemize} \item[(a)] $(u_1 \cdot u_2, v_1) \in R$, $(u_2 \cdot u_3, v_2) \in R$ and $u_2 \neq \epsilon$; \item[(b)] $(u_1 \cdot u_2 \cdot u_3, v_1) \in R$ and $(u_2, v_2)\in R$. \end{itemize} We say that the critical pair $(u_1, u_2, u_3, v_1, v_2)$ is \emph{resolved} if there exists $w \in A^$ such that \begin{itemize} \item $v_1 \cdot u_3 \to^ w$ and $u_1 \cdot v_2 \to^* w$ in case (a), \item $v_1 \to^* w$ and $u_1 \cdot v_2 \cdot u_3 \to^* w$ in case (b). \end{itemize} We will say that the critical pair is of \emph{type (a)} or of \emph{type (b)} depending on which of the above conditions (a) or (b) it satisfies. The following will be used to show that our rewriting system is confluent. \begin{thm}[Newman \cite{Newma1}]\label{thm4_1} Let $A$ be an alphabet and let $R$ be a terminating rewriting system on $A^$. If all critical pairs of $R$ are resolved, then $R$ is confluent. \end{thm} Now, we move on to the proof of Theorem \ref{thm2_4}. We start by showing that the extraction operation is well-defined. \begin{lem}\label{lem4_2} Let $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\}$ be a non-empty stratum and let $x_i^{a_i} \in U$. Set $U (x_i^{a_i}) = \{ x_j^{a_j} \mid x_j \ge x_i \}$. Then $\supp (U (x_i^{a_i}))$ is totally ordered and $\gamma (U,x_i^{a_i}) = \gamma (U (x_i^{a_i}), x_i^{a_i})$. In particular, the definition of $\gamma (U, x_i^{a_i})$ does not depend on the choice of the numbering of the elements of $U$. \end{lem} \begin{proof} As always we assume that, if $x_j > x_k$, then $j <k$. Recall that \[ \gamma (U, x_i^{a_i}) = \big( (\varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1 }}^{a_{i-1}}) (x_i) \big)^{a_i}\,. \] Let $x_j^{a_j}, x_k^{a_k} \in U (x_i^{a_i})$. Since $x_i \le x_j$, $x_i \le x_k$, and $\{x_i, x_k\} \in E (\Gamma)$, Condition (b) in the definition of a trickle graph implies that either $x_j \le x_k$, or $x_k \le x_j$. This shows that $\supp (U (x_i^{a_i}))$ is totally ordered. Then, since $\supp (U (x_i^{a_i}))$ is totally ordered, the definition of the extraction operation in $U (x_i^{a_i})$ is unique, that is, $\gamma (U (x_i^{a_i}), x_i^{a_i})$ is well-defined. It remains to show that $\gamma (U, x_i^{a_i}) = \gamma (U (x_i^{a_i}), x_i^{a_i})$. Notice that this implies that the definition of $\gamma (U, x_i^{a_i})$ does not depend on the choice of the numbering of the elements of $U$. It suffices to show that, if $x_j^{a_j} \not \in U(x_i^{a_i})$, then $\gamma(U, x_i^{a_i}) =\gamma (L(U, x_j^{a_j}), x_i^{a_i})$. If $j > i$, then there is nothing to prove. Suppose $j < i$. Since $x_i \not \le x_j$, according to Condition (c) in the definition of a trickle graph, \begin{gather} \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_i) \not \le \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_j) = x_j\ \ \Rightarrow\\ \varphi_{x_{j}}^{a_j} \circ \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_i) = \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_i)\ \ \Rightarrow\\ \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_i) = \varphi_{x_{1}}^{a_{1}} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}} \circ \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} (x_i)\ \ \Rightarrow\\ \gamma (U, x_i^{a_i}) = \gamma (L (U, x_j^{a_j}), x_i^{a_i})\,. \end{gather} \end{proof} \begin{corl}\label{corl4_3} Let $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\}$ be a non-empty stratum and let $x_i^{a_i} \in U$. Suppose that, if $x_j > x_k$, then $j <k$. \begin{itemize} \item[(1)] Let $j \in \{1, \dots, p\}$ be such that $x_i \not \le x_j$. Then $\gamma (L(U, x_j^{a_j}), x_i^{a_i}) = \gamma (U, x_i^{a_i})$. \item[(2)] Let $j \in \{i-1, i, i+1, \dots, p\}$. Then $\gamma (U, x_i^{a_i}) = \left( \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_j}^{a_j}(x_i) \right)^{a_i}$. \end{itemize} \end{corl} \begin{proof} Part (1) is a straightforward consequence of Lemma \ref{lem4_2}. We prove Part (2). For $j \ge i$, we have $x_i \not \le x_j$, hence $\varphi_{x_j} (x_i) = x_i$. So, if $j \ge i$, then \[ \left( \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_j}^{a_j}(x_i) \right)^{a_i} = \left( \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}(x_i) \right)^{a_i} = \gamma (U, x_i^{a_i})\,. \proved \] \end{proof} \begin{lem}\label{lem4_4} The rewriting system $\RR$ is terminating. \end{lem} \begin{proof} Define the weight of a piling $u = (U_1, \dots, U_r) \in \Omega^$ as $r + \sum_{i = 1}^r i \cdot \lg_\st (U_i)$. It is easily seen that each elementary rewriting step strictly reduces the weight. So, there is no infinite rewriting sequence, and therefore $\RR$ is terminating. \end{proof} \begin{prop}\label{prop4_5} The rewriting system $\RR$ is confluent. \end{prop} We know from Lemma \ref{lem4_4} that $\RR$ is terminating, hence, by Theorem \ref{thm4_1}, in order to prove Proposition \ref{prop4_5} it suffices to show that every critical pair is resolved. Let $(u_1, u_2, u_3, v_1, v_2)$ be a critical pair. Then we have one of the following three situations. \begin{itemize} \item[(C1)] $u_1 = \epsilon$, $u_2 = (\emptyset)$, $u_3 = (W)$, $v_1 = T (\emptyset, W, z^c)$ and $v_2 = \epsilon$, where $W \in \Omega$ and $z^c \in W$. This is a critical pair of type (b). \item[(C2)] $u_1 = (U)$, $u_2 = (V)$, $u_3 = (W)$, $v_1 = T (U, V, y^b)$ and $v_2 = T (V, W, z^c)$, where $U, V, W \in \Omega$, $y^b \in V$ and $z^c \in W$. This is a critical pair of type (a). \item[(C3)] $u_1 = u_3 = \epsilon$, $u_2 = (U, V)$, $v_1 = T (U, V, y^b)$ and $v_2 = T (U, V, y'^{b'})$, where $U, V \in \Omega$, $y^b, y'^{b'} \in V$ and $y^b \neq y'^{b'}$. This is a critical pair of type (b). \end{itemize} We show in each case that the critical pair is resolved. This requires a case-by-case proof through several lemmas. \begin{lem}\label{lem4_6} Let $x, y, z \in V (\Gamma)$ be such that $z \le y \le x$. Then: \begin{itemize} \item[(1)] $(\varphi_x \circ \varphi_y) (z) = (\varphi_{\varphi_x (y)} \circ \varphi_x) (z)$, \item[(2)] $(\varphi_x^{-1} \circ \varphi_y) (z) = (\varphi_{\varphi_x^{-1} (y)}\circ \varphi_x^{-1}) (z)$, \item[(3)] $(\varphi_x \circ \varphi_y^{-1}) (z) = (\varphi_{\varphi_x(y)}^{-1} \circ \varphi_x) (z)$, \item[(4)] $(\varphi_x^{-1} \circ \varphi_y^{-1}) (z) = (\varphi_{\varphi_x^{-1}(y)}^{-1} \circ \varphi_x^{-1}) (z)$. \end{itemize} \end{lem} \begin{proof} Part (1) is Condition (g) in the definition of a trickle graph. For Part (2) we apply Part (1) to $z_1 = \varphi_x^{-1} (z) \le y_1 = \varphi_x^{-1} (y) \le x$. For Part (3) we apply Part (1) to $z_1 = \varphi_y^{-1} (z) \le y \le x$. Finally, for Part (4) we apply Part (2) to $z_1 = \varphi_y^{-1} (z) \le y \le x$. \end{proof} \begin{lem}\label{lem4_7} Let $x, y, z \in V (\Gamma)$ be such that $z \le y \le x$ and let $a, b \in \Z$. Then \[ (\varphi_x^a \circ \varphi_y^b) (z) = (\varphi_{\varphi_x^a (y)}^b \circ \varphi_x^a) (z)\,. \] \end{lem} \begin{proof} If either $a=0$ or $b=0$ then the result is obvious. So, we can assume that $a \neq 0$ and $b \neq 0$. From here the proof is divided into four cases, depending on whether $a > 0$ and $b > 0$, or $a < 0$ and $b > 0$, or $a > 0$ and $b < 0$, or $a < 0$ and $b < 0$. We treat the case $a > 0$ and $b >0$. The other three cases can be treated in the same way. So, we take two integers $a > 0$ and $b >0$. First, we prove by induction on $a$ that \[ (\varphi_x^a \circ \varphi_y) (z) = (\varphi_{\varphi_x^a (y)} \circ \varphi_x^a) (z)\,. \] The case $a = 1$ is Lemma \ref{lem4_6}\,(1). Assume $a \ge 2$ and that the induction hypothesis holds. Let $y_1 = \varphi_x (y)$ and $z_1 = \varphi_x (z)$. By applying the induction hypothesis to $z_1 \le y_1 \le x$ and Lemma \ref{lem4_6}\,(1) we get: \begin{gather} (\varphi_{\varphi_x^a (y)} \circ \varphi_x^a) (z) = (\varphi_{\varphi_x^{a-1} (y_1)} \circ \varphi_x^{a-1}) (z_1) = (\varphi_x^{a-1} \circ \varphi_{y_1}) (z_1) = (\varphi_x^{a-1} \circ \varphi_{\varphi_x (y)} \circ \varphi_x) (z) =\\ (\varphi_x^{a-1} \circ \varphi_x \circ \varphi_y) (z) = (\varphi_x^a \circ \varphi_y) (z)\,. \end{gather} Now we fix $a \ge 1$ and we prove by induction on $b$ that \[ (\varphi_x^a \circ \varphi_y^b) (z) = (\varphi_{\varphi_x^a (y)}^b \circ \varphi_x^a) (z)\,. \] The case $b = 1$ is proved in the previous paragraph. Assume $b \ge 2$ and that the induction hypothesis holds. Let $z_1 = \varphi_y^{b-1} (z)$. By applying the induction hypothesis and the case $b = 1$ to $z_1 \le y \le x$ we get: \begin{gather} (\varphi_{\varphi_x^a (y)}^b \circ \varphi_x^a) (z) = (\varphi_{\varphi_x^a (y)} \circ \varphi_{\varphi_x^a (y)}^{b-1} \circ \varphi_x^a) (z) = (\varphi_{\varphi_x^a (y)} \circ \varphi_x^a \circ \varphi_y^{b-1}) (z) =\\ (\varphi_{\varphi_x^a (y)} \circ \varphi_x^a) (z_1) = (\varphi_x^a \circ \varphi_y) (z_1) = (\varphi_x^a \circ \varphi_y^b) (z)\,. \end{gather} \end{proof} \begin{lem}\label{lem4_8} In Case (C1) every critical pair $(u_1, u_2, u_3, v_1, v_2)$ is resolved. \end{lem} \begin{proof} We have $u_1 = \epsilon$, $u_2 = (\emptyset)$, $u_3 = (W)$, $v_1 = T (\emptyset, W, z^c)$ and $v_2 = \epsilon$, where $W \in \Omega$ and $z^c \in W$. We must show that there exists $w \in \Omega^$ such that $v_1 \to^ w$ and $u_1 \cdot v_2 \cdot u_3 \to^* w$. In fact, we set $w = u_1 \cdot v_2 \cdot u_3 = u_3 = (W)$ and we show that $v_1 \to^* w$. We set $W = \{z_1^{c_1}, \dots, z_p^{c_p}\}$ so that, if $z_j > z_k$, then $j < k$. Let $i \in \{1, \dots, p\}$ be such that $z^c = z_i^{c_i}$. Let $y = (\varphi_{z_1}^{c_1} \circ \cdots \circ \varphi_{z_{i-1}}^{c_{i-1}}) (z_i)$. We have $\gamma (W, z^c) = y^c$, and $v_1 = T (\emptyset, W, z^c) = (\{y^c\}, L(W, z^c))$. We define $(V_j, W_j) \in \Omega^$ for $j \in \{0,1, \dots, p-1\}$ as follows. We set $V_0 = \{y^c\}$ and $W_0 = L(W, z^c)$. For $1 \le j< i$ we set \[ V_j = \{z_1^{c_1}, \dots, z_j^{c_j}, u_j^c \} \text{ and } W_j = \{z_{j+1}^{c_{j+1}}, \dots, z_{i-1}^{c_{i-1}}, z_{i+1}^{c_{i+1}}, \dots, z_p^{c_p}\}\,, \] where \[ u_j = (\varphi_{z_{j+1}}^{c_{j+1}} \circ \cdots \circ \varphi_{z_{i-1}}^{c_{i-1}}) (z_i)\,. \] For $i \le j \le p-1$ we set \[ V_j = \{z_1^{c_1}, \dots, z_{j+1}^{c_{j+1}}\} \text{ and } W_j = \{z_{j+2}^{c_{j+2}}, \dots, z_p^{c_p}\}\,. \] We show that $(V_j, W_j) \to (V_{j+1}, W_{j+1})$ for all $j \in \{0, \dots, p-2\}$. Since $(V_0, W_0) = v_1$ and $(V_{p-1}, W_{p-1}) = (W, \emptyset) \to (W) =w$, this implies that $v_1 \to^w$. For $1 \le j < i$ we have $\gamma (W_{j-1}, z_j^{c_j}) = z_j^{c_j}$, and for $i \le j \le p-2$ we have $\gamma (W_{j-1}, z_{j+1}^{c_{j+1}}) = z_{j+1}^{c_{j+1}}$. For $1 \le j < i$ we have $L (W_{j-1} ,z_j^{c_j}) = W_j$, and for $i \le j \le p-2$ we have $L (W_{j-1}, z_{j+1}^{c_{j+1}}) = W_j$. For $1 \le k < j \le p$ we have $\{z_k, z_j\} \in E(\Gamma)$, because $W$ is a stratum, and for $j < i$ we have $\{u_{j-1}, z_j\} \in E(\Gamma)$, because, by the choice of the numbering of the indices, \[ u_{j-1} = (\varphi_{z_j}^{c_j} \circ \cdots \circ \varphi_{z_{i-1}}^{c_{i-1}}) (z_i) \text{ and } z_j = (\varphi_{z_j}^{c_j} \circ \cdots \circ \varphi_{z_{i-1}}^{c_{i-1}}) (z_j)\,. \] So, for $1 \le j \le i-1$ the syllable $z_j^{c_j}$ can be added to $V_{j-1}$ and $R (V_{j-1}, z_j^{c_j}) = V_j$. Similarly, for $i \le j \le p-2$ the syllable $z_{j+1}^{c_{j+1}}$ can be added to $V_{j-1}$ and $R (V_{j-1}, z_{j+1}^{c_{j+1}}) = V_j$. We deduce that, for every $j \in \{1, \dots, i-1\}$, we have $(V_j, W_j) = T(V_{j-1}, W_{j-1}, z_j^{c_j})$, and, for every $j \in \{i, \dots, p-2\}$, we have $(V_j, W_j) = T (V_{j-1}, W_{j-1}, z_{j+1}^{c_{j+1}})$. So, $(V_j, W_j) \to (V_{j+1}, W_{j+1})$ for all $j \in \{0, \dots, p-2\}$. \end{proof} \begin{lem}\label{lem4_9} Let $U$ be a non-empty stratum and let $x^a \in U$. Let $y^b \in S (\Gamma)$ be a syllable such that $x \neq y$ and $y^b$ can be added to $U$. Then \[ \gamma (R (U, y^b), \varphi_y^{-b} (x)^a) = \gamma (U, x^a)\,. \] \end{lem} \begin{proof} We set $U = \{ x_1^{a_1}, \dots, x_p^{a_p}\}$ and we assume as always that, if $x_j > x_k$, then $j<k$. Let $i \in \{1, \dots, p\}$ be such that $x^a =x_i^{a_i}$. By Lemma \ref{lem4_2} we can assume that $U (x^a) = \{x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_i^{a_i} \}$ and $x_1 > \cdots > x_{i-1} > x_i$. From here the proof of the lemma is divided into two cases depending on whether $x \not < y$ or $x < y$. {\it Case 1: $x \not < y$.} Let $j \in \{1, \dots, p\}$. Note that \[ \varphi_y^{-b} (x_j) \ge \varphi_y^{-b} (x_i)\ \Leftrightarrow\ x_j \ge x_i \Leftrightarrow\ 1\le j \le i\,. \] Suppose $1\le j \le i$. If $x > y$, then $x_j > y$, hence $\varphi_y^{-b} (x_j) = x_j$. Suppose $y \|\| x$. If we had $y > x_j$, then by Condition (b) in the definition of a trickle graph we would have $x_i \|\| x_j$, which would contradict the hypothesis $x_j \ge x_i$. Thus, $y \|\| x_j$ or $y< x_j$, hence $\varphi_y^{-b} (x_j) = x_j$. So, $\varphi_y^{-b} (x_j) = x_j$ for all $j \in \{1, \dots, i-1,i\}$, hence $R (U, y^b) (x^a) = U (x^a)$. It follows that \[ \gamma (R (U, y^b), \varphi_y^{-b} (x)^a) = \gamma (U (x^a), x^a) = \gamma (U, x^a) \,. \] {\it Case 2: $x < y$.} By Lemma \ref{lem4_2} there exists $k \in \{1, \dots, i-1 \}$ such that $x_{k-1} > y > x_k$ if $y \not \in \supp (U)$ (which simply means $y > x_1$ if $k=1$) and $y = x_k$ if $y \in \supp (U)$. For each $j \in \{k, \dots, i-1, i \}$ we set $z_j = \varphi_y^{-b} (x_j)$. In particular, $\varphi_y^{-b} (x) = z_i$. We have \begin{gather} R(U, y^b) (z_i^a) = \{ x_1^{a_1}, \dots, x_{k-1}^{a_{k-1}}, y^b, z_k^{a_k}, z_{k+1}^{a_{k+1}}, \dots, z_{i-1}^{a_{i-1}}, z_i^{a_i} \} \text{ if } y \not \in \supp (U)\,,\\ R(U, y^b) (z_i^a) = \{ x_1^{a_1}, \dots, x_{k-1}^{a_{k-1}}, x_k^{b+a_k}, z_{k+1}^{a_{k+1}}, \dots, z_{i-1}^{a_{i-1}}, z_i^{a_i} \} \text{ if } y = x_k \in \supp (U) \text{ and } b + a_k \neq 0\,,\\ R(U, y^b) (z_i^a) = \{ x_1^{a_1}, \dots, x_{k-1}^{a_{k-1}}, z_{k+1}^{a_{k+1}}, \dots, z_{i-1}^{a_{i-1}}, z_i^{a_i} \} \text{ if } y= x_k \in \supp (U) \text{ and } b + a_k = 0\,. \end{gather} In the three cases we have \[ \gamma (R (U, y^b), \varphi_y^{-b} (x)^a) = \gamma (R (U, y^b) (z_i^{a_i}), z_i^{a_i}) = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{k-1}}^{a_{k-1}} \circ \varphi_y^b \circ \varphi_{z_k}^{a_k} \circ \cdots \circ \varphi_{z_{i-1}}^{a_{i-1}}) (z_i)^a\,. \] By applying Lemma \ref{lem4_7} several times we get \begin{gather} (\varphi_{z_k}^{a_k} \circ \cdots \circ \varphi_{z_{i-1}}^{a_{i-1}}) (z_i) =\\ (\varphi_{z_k}^{a_k} \circ \cdots \circ \varphi_{z_{i-2}}^{a_{i-2}} \circ \varphi_{\varphi_y^{-b} (x_{i-1})}^{a_{i-1}} \circ \varphi_y^{-b}) (x_i) =\\ (\varphi_{z_k}^{a_k} \circ \cdots \circ \varphi_{\varphi_y^{-b} (x_{i-2})}^{a_{i-2}} \circ \varphi_y^{-b} \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i) = \cdots =\\ (\varphi_y^{-b} \circ \varphi_{x_k}^{a_k} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)\,, \end{gather} hence \[ \gamma (R (U, y^b), \varphi_y^{-b} (x)^a) = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{k-1}}^{a_{k-1}} \circ \varphi_{x_k}^{a_k} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)^a = \gamma (U,x^a)\,. \proved \] \end{proof} \begin{lem}\label{lem4_10} In Case (C2) every critical pair $(u_1,u_2,u_3,v_1,v_2)$ is resolved. \end{lem} \begin{proof} There exist $U, V, W \in \Omega$, $y^b \in V$ and $z^c \in W$ such that $u_1 = (U)$, $u_2 = (V)$, $ u_3 = (W)$, $v_1 = T(U, V, y^b)$ and $v_2 = T(V, W, z^c)$. Let $y'^b = \gamma (V, y^b)$. Then $y'^b$ can be added to $U$ and $T(U,V,y^b) = (U_1, V_1)$, where $U_1 = R (U, y'^b)$ and $V_1 = L (V, y^b)$. Let $z'^c = \gamma (W, z^c)$. Then $z'^c$ can be added to $V$ and $T(V, W, z^c) = (V_2, W_2)$, where $V_2 = R(V, z'^c)$ and $W_2 = L(W, z^c)$. We must show that there exists $w \in \Omega^$ such that $(U_1, V_1, W) \to^ w$ and $(U, V_2, W_2) \to^* w$. We set $V=\{ y_1^{b_1}, \dots, y_p^{b_p} \}$ and we assume as always that, if $y_j > y_k$, then $j < k$. Let $i \in \{1, \dots, p\}$ be such that $y^b = y_i^{b_i}$. From here the proof is divided into three cases depending on whether $z' \neq y$, or $z'=y$ and $b+c \neq 0$, or $z'=y$ and $b +c=0$. {\it Case 1: $z' \neq y$.} We set \begin{gather} V_3 = \{ \varphi_{z'}^{-c} (y_k)^{b_k} \mid 1 \le k \le p \text{ and } k \neq i \} \cup \{ z'^c \} \text{ if } z' \not \in \supp (V)\,,\\ V_3 = \{ \varphi_{z'}^{-c} (y_k)^{b_k} \mid 1 \le k \le p \text{ and } k \not\in \{i, j\} \} \cup \{ z'^{b_j +c} \} \text{ if } z' =y_j \in \supp (V) \text{ and } b_j + c \neq 0\,,\\ V_3 = \{ \varphi_{z'}^{-c} (y_k)^{b_k} \mid 1 \le k \le p \text{ and } k \not\in \{i, j\} \} \text{ if } z' =y_j \in \supp (V) \text{ and } b_j + c = 0\,, \end{gather} and we set $w= (U_1, V_3, W_2)$. Observe that $z'^c$ can be added to $V_1$ and $R(V_1,z'^c) = V_3$, hence $T(V_1,W,z^c) = (V_3, W_2)$, and therefore $(U_1, V_1, W) \to (U_1, V_3, W_2) = w$. By Lemma \ref{lem4_9} we have $\gamma (V_2, \varphi_{z'}^{-c} (y)^b) = \gamma (V, y^b) = y'^b$. We also know that $y'^b$ can be added to $U$ and $R(U,y'^b) = U_1$. Moreover, $L (V_2, \varphi_{z'}^{-c} (y)^b) = V_3$, hence $T(U, V_2, \varphi_{z'}^{-c} (y)^b) = (U_1, V_3)$, and therefore $(U, V_2, W_2) \to (U_1, V_3, W_2) = w$. {\it Case 2: $z' = y$ and $b+c \neq 0$.} We set \[ V_4 = \{ y_1^{b_1}, \dots, y_{i-1}^{b_{i-1}}, \varphi_y^{-c} (y_{i+1})^{b_{i+1}}, \dots, \varphi_y^{-c} (y_p)^{b_p} \}\,, \] $U_4= R (U, y'^{b+c})$, and $w = (U_4, V_4, W_2)$. The syllable $z'^c = y^c$ can be added to $V_1$ and $R(V_1, z'^c) = V_3$, where \[ V_3 = \{ y_1^{b_1}, \dots, y_{i-1}^{b_{i-1}}, y^c, \varphi_y^{-c} (y_{i+1})^{ b_{i+1}}, \dots, \varphi_y^{-c} (y_p)^{b_p} \}\,, \] hence $T(V_1,W, z^c) = (V_3,W_2)$, and therefore $(U_1, V_1, W) \to (U_1, V_3, W_2)$. We have $\gamma (V_3, y^c) = y'^c$, which can be added to $U_1$, $R(U_1, y'^c) = R (U, y'^{b+c}) = U_4$, and $L(V_3, y^c) = V_4$, hence $T (U_1, V_3, y^c) = (U_4, V_4)$, and therefore $(U_1, V_3, W_2) \to (U_4, V_4, W_2) = w$. So, $(U_1, V_1, W) \to^* w$. Note that \[ V_2 = \{ y_1^{b_1}, \dots, y_{i-1}^{b_{i-1}}, y^{b+c}, \varphi_y^{-c} (y_{i+1})^{b_{i+1}}, \dots, \varphi_y^{-c} (y_p)^{b_p} \}\,. \] We have $\gamma (V_2, y^{b+c}) = y'^{b+c}$, which can be added to $U$. Moreover, $R (U, y'^{b+c}) = U_4$ and $L (V_2, y^{b+c}) = V_4$, hence $T (U, V_2, y^{b+c}) = (U_4, V_4)$, and therefore $(U,V_2, W_2) \to (U_4, V_4, W_2)=w$. {\it Case 3: $z'=y$ and $b+c=0$.} We set $w = (U, V_2, W_2)$ and we show that $(U_1, V_1, W) \to^* w$. The syllable $z'^c = y^c$ can be added to $V_1$ and $R(V_1,z'^c) = V_3$, where \[ V_3 = \{ y_1^{b_1}, \dots, y_{i-1}^{b_{i-1}}, y^c, \varphi_y^{-c} (y_{i+1})^{b_{i+1}}, \dots, \varphi_y^{-c} (y_p)^{b_p} \}\,, \] hence $T (V_1, W, z^c) = (V_3, W_2)$, and therefore $(U_1, V_1, W) \to (U_1, V_3, W_2)$. We have $\gamma (V_3, y^c) = y'^c$, which can be added to $U_1$. We have $R(U_1, y'^c) = U$, because $c=-b$, and we have $L(V_1,y^c) = V_2$, since \[ V_2 = \{ y_1^{b_1}, \dots, y_{i-1}^{b_{i-1}}, \varphi_y^{-c} (y_{i+1})^{b_{i+1}}, \dots, \varphi_y^{-c} (y_p)^{b_p} \}\,, \] hence $T (U_1, V_3, y^c) = (U, V_2)$, and therefore $(U_1, V_3, W_2) \to (U, V_2, W_2) = w$. So, $(U_1, V_1, W) \to^* w$. \end{proof} \begin{lem}\label{lem4_11} Let $U$ be a non-empty stratum and let $x^a, y^b \in U$ be such that $x \neq y$. Let $x'^a = \gamma (U, x^a)$, $y'^b = \gamma (U, y^b)$, $x''^a = \gamma (L (U, y ^b), x^a)$ and $y''^b = \gamma (L(U, x^a), y^b)$. \begin{itemize} \item[(1)] We have $\{x',y'\}, \{x'',y'\} \in E(\Gamma)$. \item[(2)] We have $x \|\| y$ if and only if $x' \|\| y'$, and we have $x \le y$ if and only if $x' \le y'$. \item[(3)] We have $\varphi_{x'}^{-a} (y') = \varphi_{x''}^{-a} (y') = y''$. \item[(4)] Let $V$ be another stratum. If both $x'^a$ and $y'^b$ can be added to $V$, then $y''^b$ can be added to $R(V,x'^a)$. \end{itemize} \end{lem} \begin{proof} We set $U = \{ x_1^{a_1}, \dots, x_p^{a_p} \}$ and we suppose that, if $x_k > x_\ell$, then $k < \ell$. Let $i,j \in \{1, \dots, p\}$ be such that $x=x_i$ and $y = x_j$. By Lemma \ref{lem4_2} we can assume that $U (y) = \{x_1^{a_1}, \dots, x_{j-1}^{a_{j-1}}, x_j^{a_j} \}$ and $x_1 > \cdots > x_{j-1} > x_j= y$. From here the proof is divided into two cases depending on whether $y \not < x$, or $y < x$. {\it Case 1: $y \not < x$.} Since $U (y) = \{x_1^{a_1}, \dots, x_{j-1}^{a_{j-1}}, x_j^{a_j} \}$, we have $i > j$. By definition, $x' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$ and, by Corollary \ref{corl4_3}\,(2), $y' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_j)$, hence $\{x',y'\} \in E (\Gamma)$, we have $x\|\| y$ if and only if $x' \|\| y'$, and we have $x < y$ if and only if $x' < y'$. Moreover, since $y \not < x$, we have $y' \not < x'$, hence $\varphi_{x'}^{-a} (y') = y'$. We have $L (U, y^b) = \{ x_1^{a_1}, \dots, x_{j-1}^{a_{j-1}}, x_{j+1}^{a_{j+1}}, \dots, x_p^{a_p} \}$, hence $x'' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}} \circ \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$. On the other hand, if $k > j$, then $x_k \not > x_j$, hence $\varphi_{x_k}^{a_k} (x_j) = x_j$. So, \[ y' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j) = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}} \circ \varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_j)\,. \] It follows that $\{x'', y' \} \in E (\Gamma)$ and $y' \not < x''$. In particular, $\varphi_{x''}^{-a} (y') = y'$. We have $L (U, x^a) = \{ x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p} \}$, hence $y'' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j) = y'$. So, $\varphi_{x'}^{-a} (y') = \varphi_{x''}^{-a} (y') = y'' = y'$. Let $V$ be another stratum such that both $x'^a$ and $y'^b$ can be added to $V$. Set $V = \{z_1^{c_1}, \dots, z_q^{c_q} \}$. The support of $R (V, x'^a)$ is contained in $\{\varphi_{x'}^{-a} (z_1), \dots, \varphi_{x'}^{-a} (z_q)\} \cup \{x'\}$. Since $y'^b$ can be added to $V$, $y'' = \varphi_{x'}^{-a}(y')$, and $\{x', y'\} \in E(\Gamma)$, it follows that $y''^b$ can be added to $R (V, x'^a)$. {\it Case 2: $y < x$.} Since $U (y) = \{x_1^{a_1}, \dots, x_{j-1}^{a_{j-1}}, x_j^{a_j} \}$, we have $i < j$. We have $y' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j)$ and, by Corollary \ref{corl4_3}\,(2), $x' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_i)$, hence $\{x', y' \} \in E (\Gamma)$ and $y' < x'$. Furthermore, $L(U, y^b) = \{x_1^{a_1}, \dots, x_{j-1}^{a_{j-1}}, x_{j+1}^{a_{j+1}}, \dots, x_p^{a_p}\}$, hence, $x'' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i) = x'$. In particular, $\{x'', y'\} \in E (\Gamma)$ and $y' < x''$. Since $L (U, x^a) = \{x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+ 1}}, \dots, x_p^{a_p} \}$, we have $y'' = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} \circ \varphi_{x_{i+1}}^{a_{i+1}} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j)$. On the other hand, by applying Lemma \ref{lem4_7} several times we get \begin{gather} \varphi_{x''}^{-a} (y') = \varphi_{x'}^{-a} (y') = \big( \varphi_{(\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)}^{-a} \circ \varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}} \big) (x_j) =\\ \big( \varphi_{x_1}^{a_1} \circ \varphi_{(\varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)}^{-a} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}} \big) (x_j) = \cdots =\\ (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} \circ \varphi_{x_i}^{-a} \circ \varphi_{x_i}^a \circ \varphi_{x_{i+1}}^{a_{i+1}} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j) = \\ (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}} \circ \varphi_{x_{i+1}}^{a_{i+1}} \circ \cdots \circ \varphi_{x_{j-1}}^{a_{j-1}}) (x_j) = y''\,. \end{gather} Let $V$ be another stratum such that both $x'^a$ and $y'^b$ can be added to $V$. Then we show that $y''^b$ can be added to $R (V, x'^a)$ in the same way as in Case 1. \end{proof} \begin{lem}\label{lem4_12} In Case (C3) every critical pair $(u_1, u_2, u_3, v_1, v_2)$ is resolved. \end{lem} \begin{proof} There exist $U, V \in \Omega$ and $y^b, z^c \in V$ such that $y \neq z$, $u_1 = u_3 = \epsilon$, $u_2 = (U,V) $, $v_1 = T (U, V, y^b)$ and $v_2 = T (U, V, z^c)$. Let $y'^b = \gamma (V, y^b)$. Then $y'^b$ can be added to $U$ and $T (U, V, y^b) = (U_1, V_1)$, where $U_1 = R(U, y'^b)$ and $V_1 = L (V, y^b)$. Let $z'^c = \gamma (V, z^c)$. Then $z'^c$ can be added to $U$ and $T (U, V, z^c) = (U_2, V_2)$, where $U_2 = R (U, z'^c)$ and $V_2 = L(V, z^c)$. We must show that there exists $w \in \Omega^$ such that $(U_1, V_1) \to^ w$ and $(U_2, V_2) \to^* w$. Let $z''^c = \gamma (V_1, z^c)$ and $y''^b = \gamma (V_2, y^b)$. We know from Lemma \ref{lem4_11}\,(4) that $z''^c$ can be added to $U_1$ and $y''^b$ can be added to $U_2$. Let $(U_3, V_3) = T (U_1, V_1, z^c)$ and $(U_4, V_4) = T (U_2, V_2, y^b)$, where $U_3 = R (U_1, z''^c)$, $V_3 = L (V_1, z^c)$, $U_4 = R (U_2, y''^b)$ and $V_4 = L(V_2, y^b)$. We will prove that $(U_3, V_3) = (U_4, V_4)$. Then, by setting $w = (U_3, V_3) = (U_4, V_4)$, we will have $(U_1, V_1) \to w$ and $(U_2, V_2) \to w$. Clearly, $V_3 = V_4 = V \setminus \{ y^b, z^c\}$, hence we only need to prove that $U_3 = U_4$. We set $U= \{x_1^{a_1}, \dots, x_p^{a_p} \}$ and we assume as always that, if $x_i > x_j$, then $i < j$. The rest of the proof is divided into two cases depending on whether $y \|\| z$, or $y < z$. The case $z < y$ is treated in the same way as the case $y < z$ by interchanging the roles of $y$ and $z$. {\it Case 1: $y \|\| z$.} By Lemma \ref{lem4_11} we have $y' \|\| z'$, $y'' = \varphi_{z'}^{-c} (y') = y'$, and $z'' = \varphi_{y'}^{-b} (z') = z'$. Let \begin{gather} W_1 = \{ x_i^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y'\}\,, \ W_2 = \{ x_i^{a_i} \mid 1 \le i \le p \text{ and } x_i \le z'\}\,, \\ W_3 = U \setminus (W_1 \cup W_2)\,. \end{gather} Condition (b) in the definition of a trickle graph implies that $W_1 \cap W_2 = \emptyset$, hence we have the disjoint union $U = W_1 \sqcup W_2 \sqcup W_3$. We have $U_1 = R(U, y'^b) = W_1' \sqcup W_2 \sqcup W_3$, where \begin{gather} W_1' = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y' \} \cup \{y'^b \} \text{ if } y' \not \in \supp (W_1)\,,\\ W_1' = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \cup \{y'^{a_k+b} \} \text{ if } y' = x_k \in \supp (W_1)\\ \hskip 5 truecm \text{ and } a_k + b \neq 0\,,\\ W_1' = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \text{ if } y' = x_k \in \supp (W_1) \text{ and } a_k + b = 0\,. \end{gather} Then $U_3 = R (U_1, z''^c) = R(U_1, z'^c) = W_1' \sqcup W_2' \sqcup W_3$, where \begin{gather} W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le z' \} \cup \{z'^c \} \text{ if } z' \not \in \supp (W_2)\,,\\ W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le z' \text{ and } i \neq \ell \} \cup \{z'^{a_\ell+c} \} \text{ if } z' = x_\ell \in \supp (W_2)\\ \hskip 5 truecm \text{ and } a_\ell + c \neq 0\,,\\ W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le z' \text{ and } i \neq \ell \} \text{ if } z' = x_\ell \in \supp (W_2) \text{ and } a_\ell + c = 0\,. \end{gather} By using the same argument we prove that $U_2= R(U, z'^c) = W_1 \sqcup W_2' \sqcup W_3$, and then that $U_4 = R(U_2, y''^b) = R (U_2, y'^b) = W_1' \sqcup W_2' \sqcup W_3$, hence $U_3 = U_4$. {\it Case 2: $y < z$.} By Lemma \ref{lem4_11} we have $y' < z'$, $z'' = \varphi_{y'}^{-b} (z') = z'$, $y''=\varphi_{z'} ^{-c} (y')$ and $y'' < z'$. Let \begin{gather} W_1 = \{ x_i^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y' \}\,,\ W_2 = \{ x_i^{a_i} \mid 1 \le i \le p\,,\ x_i \le z' \text{ and } x_i \not \le y' \}\,,\\ W_3 = U \setminus (W_1 \cup W_2)\,. \end{gather} By construction we have the disjoint union $U = W_1 \sqcup W_2 \sqcup W_3$. We have $U_1 = R (U, y'^b) = W_1^{(1)} \sqcup W_2 \sqcup W_3$, where \begin{gather} W_1^{(1)} = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y' \} \cup \{y'^b \} \text{ if } y' \not \in \supp (W_1)\,,\\ W_1^{(1)} = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \cup \{y'^{a_k+b} \} \text{ if } y' = x_k \in \supp (W_1)\\ \hskip 4 truecm \text{ and } a_k + b \neq 0\,,\\ W_1^{(1)} = \{ \varphi_{y'}^{-b} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \text{ if } y' = x_k \in \supp (W_1) \text{ and } a_k + b = 0\,. \end{gather} Then $U_3 = R (U_1, z''^c) = R(U_1, z'^c) = W_1^{(2)} \sqcup W_2' \sqcup W_3$, where \begin{gather} W_1^{(2)} = \{(\varphi_{z'}^{-c} \circ \varphi_{y'}^{-b}) (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y' \} \cup \{y''^b \} \text{ if } y' \not \in \supp (W_1)\,,\\ W_1^{(2)} = \{ (\varphi_{z'}^{-c} \circ \varphi_{y'}^{-b}) (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \cup \{y''^{a_k+b} \}\\ \hskip 4 truecm \text{ if } y' = x_k \in \supp (W_1) \text{ and } a_k + b \neq 0\,,\\ W_1^{(2)} = \{ (\varphi_{z'}^{-c} \circ \varphi_{y'}^{-b}) (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \text{ if } y' = x_k \in \supp (W_1)\\ \hskip 4 truecm \text{ and } a_k + b = 0\,, \end{gather} and \begin{gather} W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le z' \text{ and } x_i \not \le y' \} \cup \{z'^c \} \text{ if } z' \not \in \supp (W_2)\,,\\ W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le z'\,,\ x_i \not \le y' \text{ and } i \neq \ell \} \cup \{z'^{a_\ell+c} \}\\ \hskip 4 truecm \text{ if } z' = x_\ell \in \supp (W_2) \text{ and } a_\ell + c \neq 0\,,\\ W_2' = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le z'\,,\ x_i \not \le y' \text{ and } i \neq \ell \} \text{ if } z' = x_\ell \in \supp (W_2)\\ \hskip 4 truecm \text{ and } a_\ell + c = 0\,. \end{gather} On the other hand, $U_2 = R (U, z'^c) = W_1^{(3)} \sqcup W_2' \sqcup W_3$, where \[ W_1^{(3)} = \{ \varphi_{z'}^{-c} (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y'\}\,. \] Then $U_4 = R (U_2, y''^b) = W_1^{(4)} \sqcup W_2' \sqcup W_3$, where \begin{gather} W_1^{(4)} = \{(\varphi_{y''}^{-b} \circ \varphi_{z'}^{-c}) (x_i)^{a_i} \mid 1 \le i \le p \text{ and } x_i \le y' \} \cup \{y''^b \} \text{ if } y' \not \in \supp (W_1)\,,\\ W_1^{(4)} = \{ (\varphi_{y''}^{-b} \circ \varphi_{z'}^{-c}) (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \} \cup \{y''^{a_k+b} \}\\ \hskip 4 truecm \text{ if } y' = x_k \in \supp (W_1) \text{ and } a_k + b \neq 0\,,\\ W_1^{(4)} = \{ (\varphi_{y''}^{-b} \circ \varphi_{z'}^{-c}) (x_i)^{a_i} \mid 1 \le i \le p\,,\ x_i \le y' \text{ and } i \neq k \}\text{ if } y' = x_k \in \supp (W_1)\\ \hskip 4 truecm \text{ and } a_k + b = 0\,. \end{gather} Since $y'' = \varphi_{z'}^{-c} (y')$, by Lemma \ref{lem4_7} we have $(\varphi_{y''}^{-b} \circ \varphi_{z'}^{-c}) (x_i) = (\varphi_{z'}^{-c} \circ \varphi_{y'}^{-b}) (x_i)$ for all $i \in \{1, \dots, p\}$ such that $x_i \le y'$, hence $W_1^{(4)} = W_1^{(2)}$, and therefore $U_3 = U_4$. \end{proof} \begin{proof}[Proof of Proposition \ref{prop4_5}] We know from Lemma \ref{lem4_4} that $\RR$ is terminating, hence, by Theorem \ref{thm4_1}, it suffices to prove that every critical pair is resolved. We have seen that there are three types of critical pairs. In Case (1) we know that every critical pair is resolved by Lemma \ref{lem4_8}, in Case (2) we know that every critical pair is resolved by Lemma \ref{lem4_10}, and in Case (3) we know that every critical pair is resolved by Lemma \ref{lem4_12}. \end{proof} Now we know that $\RR$ is terminating and confluent (see Lemma \ref{lem4_4} and Proposition \ref{prop4_5}), hence, in order to prove Theorem \ref{thm2_4} it remains to show that $\RR$ is a rewriting system for $\Tr (\Gamma)$, that is, to prove that $\Tr (\Gamma) \simeq \langle \Omega \mid u=v \text{ for } (u,v) \in \RR \rangle^+$. We set $M (\Gamma) = \langle \Omega \mid u=v \text{ for } (u,v) \in \RR \rangle^+$. We will construct homomorphisms $\Phi: M (\Gamma) \to \Tr (\Gamma)$ and $\Psi: \Tr (\Gamma) \to M (\Gamma)$, and then we will prove that $\Psi \circ \Phi = \id_{M (\Gamma)}$ and $\Phi \circ \Psi = \id_{\Tr (\Gamma)}$. Let $U$ be a non-empty stratum. We write $U = \{ x_1^{a_1}, \dots, x_p^{a_p} \}$ so that, if $x_i > x_j$, then $i < j$, and we set \[ \omega (U) = x_1^{a_1} x_2^{a_2} \dots x_p^{a_p} \in \Tr (\Gamma)\,. \] If $U = \emptyset$, then we set $\omega (U) = 1$. \begin{lem}\label{lem4_13} Let $U \in \Omega$. Then the definition of $\omega (U)$ does not depend on the choice of the numbering of the elements of $U$. \end{lem} \begin{proof} Suppose we have $U = \{ x_1^{a_1}, \dots, x_p^{a_p} \} = \{ {x_1'}^{a_1'}, \dots, {x_p'}^{a_p'} \} $ so that, if $x_i > x_j$, then $i < j$, and if $x_i' > x_j'$, then $i < j$. We prove that $x_1^{a_1} \dots x_p^{a_p} = {x_1'}^{a_1'} \dots {x_p'}^{a_p'}$ by induction on $p = \lg_\st (U )$. If $p=0$, then $U = \emptyset$, and there is nothing to prove. We assume that $p \ge 1$ and that the induction hypothesis holds. Let $i \in \{1, \dots, p \}$ be such that $x_1^{a_1} = {x_i'}^{a_i'}$. Let $j \in \{1, \dots, p\}$. If $j < i$, then either $x_j' > x_i'$ or $x_j' \|\| x_i'$. But, since ${x_i'}^{a_i'} = x_1^{a_1}$, there is no ${x_j'}^{a_j'} \in U$ such that $x_j' > x_i'$, hence, if $j < i$, then $x_j' \|\| x_i'$. It follows that \begin{gather} {x_1'}^{a_1'} \dots {x_{i-1}'}^{a_{i-1}'} {x_i'}^{a_i'} {x_{i+1}'}^{a_{i+1}'} \dots {x_p'}^{a_p'} = {x_i'}^{a_i'} {x_1'}^{a_1'} \dots {x_{i-1}'}^{a_{i-1}'} {x_{i+1}'}^{a_{i+1}'} \dots {x_p'}^{a_p'} =\\ x_1^{a_1} {x_1'}^{a_1'} \dots {x_{i-1}'}^{a_{i-1}'} {x_{i+1}'}^{a_{i+1}'} \dots {x_p'}^{a_p'}\,. \end{gather} By the induction hypothesis applied to $U \setminus \{ x_1^{a_1} \}$, \[ x_2^{a_2} \dots x_p^{a_p} = {x_1'}^{a_1'} \dots {x_{i-1}'}^{a_{i-1}'} {x_{i+1}'}^{a_{i+1}'} \dots {x_p'}^{a_p'}\,, \] hence $x_1^{a_1} \dots x_p^{a_p} = {x_1'}^{a_1'} \dots {x_p'}^{a_p'}$. \end{proof} Let $\Phi^* : \Omega^* \to \Tr (\Gamma)$ be the homomorphism which sends $U$ to $\omega (U)$ for all $U \in \Omega$. It is easily verified that, for all $(u,v) \in \RR$, $\Phi^* (u) = \Phi^* (v)$, hence $\Phi^$ induces a homomorphism $\Phi : M (\Gamma) \to \Tr (\Gamma)$ which sends $U$ to $\omega (U)$ for all $U \in \Omega$. For the construction of $\Psi : \Tr (\Gamma) \to M (\Gamma)$ we use the following monoid presentation for $\Tr (\Gamma)$: \[ \Tr (\Gamma) = \left\langle S (\Gamma) \left\| \begin{array}{ll} x^a \cdot x^b = x^{a+b} &\text{for } x \in V(\Gamma),\ a, b \in \Z_{\mu (x)} \setminus \{0\} \text{ and } a+b \neq 0\,,\\ x^a \cdot x^b = 1 &\text{for } x \in V(\Gamma),\ a, b \in \Z_{\mu (x)} \setminus \{0\} \text{ and } a+b = 0\,,\\ \varphi_x^a(y)^b \cdot x^a = \varphi_y^b (x)^a \cdot y^b &\text{for } \{x, y\} \in E (\Gamma),\ a \in \Z_{\mu (x)} \setminus \{ 0 \},\ b \in \Z_{\mu (y)} \setminus \{ 0 \} \end{array} \right. \right\rangle^+\,. \] \begin{lem}\label{lem4_14} The homomorphism $\Psi^: S (\Gamma)^* \to M (\Gamma)$ which sends $x^a$ to $(\{x^a\})$ induces a homomorphism $ \Psi: \Tr (\Gamma) \to M (\Gamma)$. \end{lem} \begin{proof} Let $x \in V (\Gamma)$ and $a,b \in \Z_{\mu (x)} \setminus \{ 0 \}$ be such that $a + b \neq 0$. We have the following sequence of rewriting transformations \[ (\{x^a\}, \{x^b\}) \stackrel{\RR}{\to} (\{x^{a+b} \}, \emptyset) \stackrel{\RR}{ \to} (\{ x^{a+b} \})\,, \] hence $\Psi^* (x^a \cdot x^b) = \Psi^* (x^{a+b})$. Let $x \in V (\Gamma)$ and $a,b \in \Z_{\mu (x)} \setminus \{ 0 \}$ be such that $a + b = 0$. We have the following sequence of rewriting transformations \[ (\{x^a\}, \{x^b\}) \stackrel{\RR}{\to} (\emptyset, \emptyset) \stackrel{\RR}{\to} (\emptyset) \stackrel {\RR}{\to} \epsilon\,, \] hence $\Psi^* (x^a \cdot x^b) = \Psi^* (1)$. Let $x^a, y^b \in S (\Gamma)$ be such that $\{x, y\} \in E (\Gamma)$ and $x \|\| y$. We have the following sequences of rewriting transformations \begin{gather} ( \{\varphi_x^a(y)^b\}, \{x^a\} ) = ( \{y^b\}, \{x^a\} ) \stackrel{\RR}{\to} ( \{x^a, y^b\}, \emptyset) \stackrel{\RR}{\to} ( \{x^a, y^b\})\,, \\ ( \{ \varphi_y^b (x)^a\}, \{y^b\} ) = ( \{ x^a\}, \{y^b\} ) \stackrel{\RR}{\to} (\{ x^a, y^b\}, \emptyset) \stackrel{\RR}{\to} ( \{x^a, y^b\})\,, \end{gather} hence $\Psi^* (\varphi_x^a(y)^b \cdot x^a) = \Psi^(\varphi_y^b (x)^a \cdot y^b)$. Let $x^a, y^b \in S (\Gamma)$ be such that $\{x, y\} \in E (\Gamma)$ and $x < y$. We have the following sequences of rewriting transformations \begin{gather} ( \{\varphi_x^a(y)^b\}, \{x^a\} ) = ( \{y^b\}, \{x^a\} ) \stackrel{\RR}{\to} ( \{ y^b, x^a\}, \emptyset ) \stackrel{\RR}{\to} ( \{ y^b, x^a\})\,, \\ ( \{ \varphi_y^b (x)^a\}, \{y^b\} ) \stackrel{\RR}{\to} ( \{ y^b, (\varphi_y^{-b} \circ \varphi_y^b)(x)^a \}, \emptyset ) = ( \{ y^b, x^a\}, \emptyset ) \stackrel{\RR}{\to} ( \{ y^b, x^a\})\,, \end{gather} hence $\Psi^ (\varphi_x^a(y)^b \cdot x^a) = \Psi^(\varphi_y^b (x)^a \cdot y^b)$. \end{proof} \begin{lem}\label{lem4_15} We have $\Psi \circ \Phi = \id_{M (\Gamma)}$ and $\Phi \circ \Psi = \id_{\Tr (\Gamma)}$. \end{lem} \begin{proof} Let $U \in \Omega$. We write $U = \{ x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \}$ so that, if $x_i > x_j$, then $i < j$. We have the following sequence of rewriting transformations in $\Omega^$: \begin{gather} ( \{ x_1^{a_1}\}, \{x_2^{a_2}\}, \dots, \{x_p^{a_p} \}) \to ( \{ x_1^{a_1}, x_2^{a_2}\}, \emptyset, \{x_3^{a_3}\}, \dots, \{x_p^{a_p} \}) \to ( \{ x_1^{a_1}, x_2^{a_2}\}, \{x_3^{a_3}\}, \dots, \{x_p^{a_p} \})\\ \to \cdots \to (\{x_1^{a_1}, \dots, x_{p-1}^{a_{p-1}} \}, \{x_p^{a_p}\} ) \to (\{x_1^{a_1}, \dots, x_{p-1}^{a_{p-1}}, x_p^{a_p} \}, \emptyset ) \to (\{x_1^{a_1}, \dots, x_{p-1}^{a_{p-1}}, x_p^{a_p} \}) = (U)\,. \end{gather} So, \[ (\Psi \circ \Phi) ((U)) = \Psi ( x_1^{a_1} x_2 ^{a_2} \dots x_p^{a_p} ) = ( \{ x_1^{a_1}\}, \{x_2^{a_2}\}, \dots, \{x_p^{a_p} \} ) = (U)\,. \] Since $M (\Gamma)$ is generated by $\{ (U) \mid U \in \Omega \}$, this shows that $\Psi \circ \Phi = \id_{M (\Gamma)}$. For $x^a \in S (\Gamma)$ we have $(\Phi \circ \Psi) (x^a) = \Phi (\{x^a\}) = x^a$. Since $\Tr (\Gamma)$ is generated by $S (\Gamma)$, we conclude that $\Phi \circ \Psi = \id_{\Tr (\Gamma)}$. \end{proof} The following is a straightforward consequence of Lemma \ref{lem4_15}, and it ends the proof of Theorem \ref{thm2_4}. \begin{prop}\label{prop4_16} The pair $(\Omega,\RR)$ is a rewriting system for $\Tr (\Gamma)$. \end{prop} \section{The Tits-style algorithm}\label{sec5} In this section we fix a trickle graph $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V(\Gamma)})$ and we keep the notations of Subsection \ref{subsec2_3}. Our goal is to prove Theorem \ref{thm2_8}. We fix a total order $\preceq$ on $V(\Gamma)$ which extends the partial order $\le$ in the sense that, if $x \le y$, then $x \preceq y$. If $U$ is a non-empty stratum written $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$, then we set $\hat \omega^S (U) = (x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}) \in S (\Gamma)^$. On the other hand we set $\hat \omega^S (U) = \epsilon$ if $U = \emptyset$. Notice that $\hat \omega^S (U)$ is a representative of $\omega (U)$. The following three lemmas are preliminaries to the proof of Theorem \ref{thm2_8}. \begin{lem}\label{lem5_1} Let $U = \{x_1^{a_1}, \dots, x_p^{a_p} \}$ be a stratum. We suppose that the numbering of the elements of $U$ is chosen so that, if $x_i > x_j$, then $i < j$. Let $v = (x_1^{a_1}, \dots, x_p^{a_p}) \in S (\Gamma)^$. Then $v \stackrel{II\ }{\to} \hat \omega^S (U)$. \end{lem} \begin{proof} We argue by induction on $p = \lg_\st (U)$. If $p = 0$, then $U = \emptyset$ and $v = \hat \omega^S (U) = \epsilon$. So, we can assume that $p \ge 1$ and that the induction hypothesis holds. We write $U=\{ {x_1'}^{a_1'}, \dots, {x_p'}^{a_p'} \}$ with $x_1' \succ \cdots \succ x_p'$. Let $i \in \{1, \dots, p \}$ be such that ${x_1'}^{a_1'} = x_i^{a_i}$. Let $j \in \{1, \dots, p\}$. If $j < i$, then either $x_j > x_i$ or $x_j \|\| x_i$. But, since $x_i^{a_i} = {x_1'}^{a_1'}$, there is no $x_j^{a_j} \in U$ such that $x_j > x_i=x_1'$, hence, if $j < i$, then $ x_j \|\| x_i$. It follows that \begin{gather} v= (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_i^{a_i}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II}{\to} (x_1^{a_1}, \dots, x_{i-2}^{a_{i-2}}, x_i^{a_i}, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II}{\to} \cdots \stackrel{II}{\to}\\ (x_i^{a_i}, x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) = ({x_1'}^{a_1'}, x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p})\,. \end{gather} By the induction hypothesis applied to $U \setminus \{ x_i^{a_i} \}$ we have \[ (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II\ }{\to} ({x_2'}^{a_2'}, \dots, {x_p'}^{a_p'})\,. \] So, \[ v = (x_1^{a_1}, \dots, x_p^{a_p}) \stackrel{II\ }{\to} ({x_1'}^{a_1'}, \dots, {x_p'}^{a_p'}) = \hat \omega^S (U)\,. \proved \] \end{proof} \begin{lem}\label{lem5_2} Let $U$ be a non-empty stratum and let $x^a \in U$. Let $y^a = \gamma (x^a, U)$. Then $\hat \omega^S (U) \stackrel{II\ }{\to} (y^a) \cdot \hat \omega^S (L (U, x^a))$. \end{lem} \begin{proof} We write $U = \{ x_1^{a_1}, \dots, x_p^{a_p} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$. Let $i \in \{1, \dots, p \}$ be such that $x^a = x_i^{a_i}$. Note that, if $j < i$, then $x_j \|\| x_i$ or $x_j > x_i$. This implies that, if $j < i$, then $x_j \|\| (\varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$ or $x_j > (\varphi_{x_{j+1}}^{a_{j+1}} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$. Then we have the following sequence of rewriting transformations \begin{gather} \hat \omega^S (U) = (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_i^{a_i}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II}{\to}\\ (x_1^{a_1}, \dots, x_{i-2}^{a_{i-2}}, \varphi_{x_{i-1}}^{a_{i-1}} (x_i)^{a_i}, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II}{\to} \cdots \stackrel{II}{\to}\\ (x_1^{a_1}, (\varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)^{a_i}, x_2^{a_2}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) \stackrel{II}{\to}\\ ( (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)^{a_i}, x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_{i+1}^{a_{i+1}}, \dots, x_p^{a_p}) = (y^a) \cdot \hat \omega^S (L (U, x^a))\,. \end{gather} \end{proof} \begin{lem}\label{lem5_3} Let $U \in \Omega$ be a stratum and let $y^b \in S (\Gamma)$ be a syllable which can be added to $U$. Then \[ \hat \omega^S (U) \cdot (y^b) \stackrel{M\ }{\to} \hat \omega^S (R (U, y^b))\,. \] \end{lem} \begin{proof} We write $U = \{x_1^{a_1}, \dots, x_p^{a_p}\}$ with $x_1 \succ \cdots \succ x_p$. Let $i \in \{0, 1, \dots, p\}$ be such that $x_i \succeq y \succ x_{i+1}$. If $i=0$ this means that $y \not \in \supp (U)$ and $y \succ x_1$, and if $i=p$ this means that $x_p \succeq y$. If $y \in \supp (U)$, then $y=x_i$, otherwise $x_i \succ y \succ x_{i+1}$. Let $j \in \{i+1, \dots, p\}$. Then either $y > x_j$ or $y \|\| x_j$. In both cases we have \[ ((x_j^{a_j}, y^b) , (y^b,\varphi_y^{-b} (x_j)^{a_j})) \in \RR_{II}\,. \] It follows that \[ \hat \omega^S (U) \cdot (y^a) = (x_1^{a_1}, \dots, x_p^{a_p}, y^a) \stackrel{II \ }{\to} (x_1^{a_1}, \dots, x_i^{a_i}, y^b, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p})\,. \] If $y \not \in \supp (U)$, then \[ R(U, y^b) = \{x_1^{a_1}, \dots, x_i^{a_i}, y^b, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}\}\,, \] hence, by Lemma \ref{lem5_1}, \[ (x_1^{a_1}, \dots, x_i^{a_i}, y^b, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}) \stackrel{II \ }{\to} \hat \omega^S (R (U, y^b))\,. \] If $y = x_i \in \supp (U)$ and $b+a_i \neq 0$, then \[ R(U, y^b) = \{x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, y^{a_i+b}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}\}\,, \] hence, by Lemma \ref{lem5_1}, \begin{gather} (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_i^{a_i}, y^b, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}) \stackrel{I }{\to}\\ (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, y^{a_i+b}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}) \stackrel{II \ }{\to} \hat \omega^S (R (U, y^b))\,. \end{gather} If $y = x_i \in \supp (U)$ and $b+a_i = 0$, then \[ R(U, y^b) = \{x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}\}\,, \] hence, by Lemma \ref{lem5_1}, \begin{gather} (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, x_i^{a_i}, y^b, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}) \stackrel{I}{\to}\\ (x_1^{a_1}, \dots, x_{i-1}^{a_{i-1}}, \varphi_y^{-b} (x_{i+1})^{a_{i+1}}, \dots, \varphi_y^{-b} (x_p)^{a_p}) \stackrel{II \ }{\to} \hat \omega^S (R (U, y^b))\,. \end{gather} \end{proof} The following definition is a slight modification of the definition of normal form given in Subsection \ref{subsec2_2}. Let $g \in \Tr (\Gamma)$. By Theorems \ref{thm2_3} and \ref{thm2_4} there exists a unique piling $w = (U_1,U_2, \dots, U_p) \in \Omega^$ such that $w$ is $\RR$-irreducible and $w = \Psi (g)$. Then we define the \emph{$S$-normal form} of $g$ to be \[ \nf^S (g) = \hat \omega^S (U_1) \cdot \hat \omega^S (U_2) \cdots \hat \omega^S (U_p) \in S(\Gamma)^\,. \] Theorem \ref{thm2_8} will be a consequence of the following. \begin{prop}\label{prop5_4} Let $v \in S (\Gamma)^$ and let $g = \bar v \in \Tr (\Gamma)$. Then $v \stackrel{M\ }{\to} \nf^S (g)$. \end{prop} \begin{proof} For a piling $w = (U_1, \dots, U_p) \in \Omega^$ we set \[ \hat \omega^S (w) = \hat \omega^S (U_1) \cdot \hat \omega^S (U_2) \cdots \hat \omega^S (U_p) \in S(\Gamma)^\,. \] Let $w, w' \in \Omega^$. It follows from Lemmas \ref{lem5_2} and \ref{lem5_3} that, if $w \stackrel{\RR\ }{\to} w'$, then $\hat \omega^S (w) \stackrel{M\ }{\to} \hat \omega^S (w')$. Let $v = (x_1^{a_1}, \dots, x_p^{a_p}) \in S (\Gamma)^$ and let $g = \bar v \in \Tr (\Gamma)$. Set $w = ( \{ x_1^{a_1} \}, \{ x_2^{a_2} \}, \dots, \{x_p^{a_p} \}) \in \Omega^$ and denote by $w'$ the unique piling in $\Omega^$ such that $w'$ is $\RR$-irreducible and $\overline{w'} = g$. Then, by Theorems \ref{thm2_3} and \ref{thm2_4}, we have $w \stackrel{\RR\ }{\to} w'$, hence, by the above, \[ v = \hat \omega^S (w) \stackrel{M \ }{\to} \hat \omega^S (w') = \nf^S (g)\,. \proved \] \end{proof} \begin{proof}[Proof of Theorem \ref{thm2_8}] To prove Theorem \ref{thm2_8} it suffices to show that, if $w, w' \in S (\Gamma)^$ are syllabically $M$-reduced and $\overline{w} = \overline{w'}$, then $w \stackrel{II\ }{\to} w'$. Let $w, w' \in S (\Gamma)^$ be two syllabically $M$-reduced words and let $g \in \Tr (\Gamma)$ be such that $\overline{w} = \overline{w'} = g$. By Proposition \ref{prop5_4} we have $w \stackrel{M\ }{\to} \nf^S (g)$ and $w' \stackrel{M\ }{\to} \nf^S (g)$. Since $w$ and $w'$ are both syllabically $M$-reduced, we actually must have $w \stackrel{II\ }{\to} \nf^S (g)$ and $w' \stackrel {II\ }{\to} \nf^S (g)$. Finally, since the operation $\stackrel{II}{\to}$ is reversible, we conclude $w \stackrel{II\ }{\to} \nf^S (g) \stackrel{II\ }{\to} w'$. \end{proof} \section{Parabolic subgroups}\label{sec6} \begin{proof}[Proof of Theorem \ref{thm2_10}] Note that $\Omega (\Gamma_1) \subseteq \Omega (\Gamma)$, hence $\Omega (\Gamma_1)^* \subseteq \Omega (\Gamma)^$. It is easily seen that, if $U \in \Omega (\Gamma_1)$ and $x^a \in U$, then $\gamma (U, x^a) \in S (\Gamma_1)$. It is also easily seen that, if $U \in \Omega (\Gamma_1)$ and $x^a \in S (\Gamma_1)$ can be added to $U$, then $R (U, x^a) \in \Omega (\Gamma_1)$. Let $w \in \Omega (\Gamma_1)^$ and $w' \in \Omega (\Gamma)^$. Then from the above observations it follows that, if $w \stackrel{\RR (\Gamma)\ }{\longrightarrow} w'$, then $w' \in \Omega (\Gamma_1)^$ and $w \stackrel{\RR (\Gamma_1)\ }{\longrightarrow} w'$. Let $g \in \Tr (\Gamma_1)$. Write $g = x_1^{a_1} \dots x_p^{a_p}$ with $x_1^{a_1}, \dots, x_p^{a_p} \in S (\Gamma_1)$, and set \[ v = (\{x_1^{a_1}\}, \{x_2^{a_2}\}, \dots, \{x_p^{a_p}\}) \in \Omega (\Gamma_1)^\,. \] Let $w \in \Omega (\Gamma)^$ be the unique $\RR (\Gamma)$-irreducible piling such that $v \stackrel{\RR (\Gamma)\ }{\longrightarrow} w$. From the above we get $w \in \Omega (\Gamma_1)^$ and $v \stackrel{\RR (\Gamma_1)\ }{\longrightarrow} w$. Moreover, $w$ is $\RR (\Gamma_1)$-irreducible because it is $\RR (\Gamma)$-irreducible. Write $w = (U_1, \dots, U_q)$ with $U_1, \dots, U_q \in \Omega (\Gamma_1)$. Then \[ \nf_\Gamma (\iota_1 (g)) =\nf_{\Gamma_1} (g) = \hat \omega (U_1) \cdot \hat \omega (U_2) \cdots \hat \omega (U_q)\,. \] Let $g \in \Ker (\iota_1)$. From the above we get $\nf_{\Gamma_1} (g) = \nf_\Gamma (\iota_1 (g)) = \epsilon$, hence $g=1$. This shows that $\iota_1$ is injective. Let $g \in \Tr (\Gamma)$. We already know that, if $g \in \Tr (\Gamma_1)$, then $\nf_\Gamma (g) = \nf_{\Gamma_1} (g)$, hence $\nf_\Gamma (g) \in (V (\Gamma_1) \sqcup V (\Gamma_1)^{-1})^$. Conversely, it is obvious that, if $\nf_\Gamma (g) \in (V (\Gamma_1) \sqcup V (\Gamma_1)^{-1})^$, then $g = \overline{\nf_\Gamma (g)} \in \Tr (\Gamma_1)$. \end{proof} \section{PreGarside trickle groups}\label{sec7} In this section $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ denotes a fixed preGarside trickle graph. We also fix a total order $\preceq$ on $V (\Gamma)$ which extends the partial order $\le$ in the sense that, for all $x, y \in V (\Gamma)$, if $x \le y$, then $x \preceq y$. The proof of Theorem \ref{thm2_14} is based on the existence of a complemented presentation for $\Tr^+ (\Gamma)$ that satisfies the sharp $\theta$-cube condition. So, we start by recalling these notions and explaining their link with Theorem \ref{thm2_14}. \begin{defin} Let $\Lambda$ be a simplicial graph. As always we denote by $V (\Lambda)$ its vertex set and by $E (\Lambda)$ its edge set. Let $\hat E (\Lambda) = \{ (x,y) \in V (\Lambda) \times V (\Lambda) \mid \{ x, y \} \in E (\Lambda) \}$. A \emph{partial complement} on $V (\Lambda)$ based on $\Lambda$ is a map $f : \hat E (\Lambda) \to V (\Lambda)^$. A presentation $M = \langle V \mid R \rangle^+$ for a monoid $M$ is called \emph{complemented} if there exist two partial complements $f$ on a graph $\Lambda$ and $g$ on a graph $\Lambda'$ such that $V (\Lambda) = V (\Lambda') = V$ and \[ R = \{ x\, f(x,y) = y \, f(y,x) \mid \{x,y\} \in E(\Lambda) \} = \{ g(y,x)\,x = g(x,y)\,y \mid \{x,y\} \in E(\Lambda') \}\,. \] Such a presentation is called \emph{short} if $f(x,y) \in V$ for all $(x,y) \in \hat E (\Lambda)$, that is, if all relations in $R$ are of the form $xy' = yx'$ with $x,y,x',y' \in V$. \end{defin} \begin{rem} The definition of short presentation given here is more restrictive than the one given in \cite{DDGKM1}, but it allows to consider only homogeneous monoid presentations. In particular, this implies that the considered monoids are all atomic (see the remark below). \end{rem} \begin{expl} Let $\Gamma = (\Gamma, \le, \mu, (\varphi_x)_{x \in V (\Gamma)})$ be our preGarside trickle graph. For all $(x, y) \in \hat E (\Gamma)$ we set $f (x, y) = \varphi_x^{-1} (y)$ and $g (x, y) = \varphi_y (x)$. Then, by definition and (the proof of) Proposition \ref{prop2_2}, $\Tr^+ (\Gamma)$ has the presentation $\Tr^+ (\Gamma) = \langle V \mid R \rangle$, where $V = V(\Gamma)$ and \[ R = \{ x\, f (x, y) = y \, f (y, x) \mid \{ x, y \} \in E (\Gamma) \} = \{ g(y, x) \, x = g (x, y) \, y \mid \{ x, y \} \in E (\Gamma) \} \,. \] In particular, this presentation is a complemented short presentation. \end{expl} \begin{rem} Let $M$ be a monoid with a complemented short presentation $M = \langle V \mid R \rangle$. Define a map $\nu : M \to \N$ as follows. If $a \in M$ is written $a=x_1 \dots x_p$ with $x_1, \dots, x_p \in V$, then set $\nu (a) =p$. The map $\nu$ is well-defined because the relations in $R$ are homogeneous, hence the length $p$ does not depend on the choice of the expression of $a$. It is clear that $\nu (a) = 0$ if and only if $a = 1$ and $\nu (ab) = \nu (a) + \nu (b)$ for all $a, b \in M$, hence $\nu$ is a norm and $M$ is atomic. In particular, $\Tr^+ (\Gamma)$ is atomic. \end{rem} The sharp $\theta$-cube condition introduced in \cite[Definition II.4.14]{DDGKM1} is technical, but in our case where the presentations are short, its definition is simpler as, in particular, there is no need of extending the partial map $\star$ to $V^$ as in the general case. \begin{defin} Let $\langle V \mid R \rangle^+$ be a complemented short presentation of a monoid $M$. Recall that there exist a graph $\Lambda$ with $V = V (\Lambda)$ and a partial complement $g : \hat E (\Lambda) \to V$ such that \[ R = \{ g(y,x)\, x = g(x, y)\, y \mid \{ x, y \} \in E (\Lambda) \}\,. \] We define a partial map $(x ,y) \mapsto x \star y$ from $(V \cup\{ \epsilon\}) \times (V \cup\{ \epsilon\})$ to $V \cup \{ \epsilon \}$ as follows. Let $x, y \in V$ be such that $x \neq y$. If $\{x, y\} \in E (\Lambda)$, then we set $x \star y = g (x, y)$. If $\{x, y\} \not \in E (\Lambda)$, then neither $x \star y$ nor $y \star x$ is defined. On the other hand, for $x \in V \cup \{ \epsilon \}$ we set $x \star x = \epsilon \star x = \epsilon$ and $x \star \epsilon = x$. We say that the presentation $\langle V \mid R \rangle^+$ satisfies the \emph{left sharp $\theta$-cube condition} if, for all $x, y, z \in V$ pairwise distinct, we have the following alternative: \begin{itemize} \item either $(z \star x) \star (y \star x)$ and $(z \star y) \star (x \star y)$ are both defined and are equal, \item or neither expression is defined. \end{itemize} We define similarly the \emph{right sharp $\theta$-cube condition}. We say that the presentation satisfies the \emph{sharp $\theta$-cube condition} if it satisfies both, the left and the right sharp $\theta$-cube conditions. \end{defin} The main tool in the proof of Theorem \ref{thm2_14} comes from \cite[Proposition II.4.16]{DDGKM1}. \begin{prop}[Dehornoy--Digne--Godelle--Krammer--Michel \cite{DDGKM1}]\label{prop7_1} If a monoid $M$ admits a short complemented presentation that satisfies the sharp $\theta$-cube condition, then $M$ is a preGarside monoid. \end{prop} \begin{proof}[Proof of Theorem \ref{thm2_14}] We already know that $\Tr^+ (\Gamma)$ has a short complemented presentation. So, in order to prove Theorem \ref{thm2_14}, it suffices to show that the standard presentation of $\Tr^+ (\Gamma)$ satisfies the sharp $\theta$-cube condition. By symmetry and thanks to Proposition \ref{prop2_2}, it suffices to show that this presentation satisfies the left sharp $\theta$-cube condition. Let $x, y \in V (\Gamma)$ be such that $x \neq y$. If $\{x, y\} \in E (\Gamma)$, then we set $x \star y = \varphi_{y} (x)$, and, if $\{x, y\} \not \in E(\Gamma)$, then $x \star y$ is not defined. On the other hand, for $x \in V (\Gamma) \cup \{\epsilon\}$ we set $\epsilon \star x = x \star x = \epsilon$ and $x \star \epsilon = x$. The fact that the standard presentation of $\Tr^+ (\Gamma)$ satisfies the left sharp $\theta$-cube condition is a straightforward consequence of the following two claims. {\it Claim 1.} For all $x, y, z \in V$ pairwise distinct we have the following alternative: \begin{itemize} \item either $( z \star x) \star ( y \star x)$ and $(z \star y) \star (x \star y)$ are both defined, \item or neither expression is defined. \end{itemize} Furthermore, $(z \star x) \star (y \star x)$ and $(z \star y) \star (x \star y)$ are both defined if and only if $\{x, y\}$, $\{x, z\}$, and $\{y, z\}$ belong to $E (\Gamma)$. {\it Proof of Claim 1.} By symmetry between $x$ and $y$, it suffices to show that $(z \star x) \star (y \star x)$ is defined if and only if $\{x, y\}$, $\{x ,z\}$ and $\{y, z\}$ belong to $E (\Gamma)$. Note that, if $y \star x$ is defined, then $y \star x = \varphi_x (y)$ lies in $V (\Gamma)$ (and it is different from $\epsilon$), and, if $y \star x = \varphi_x (y)$ and $z \star x = \varphi_x (z)$ are both defined, then they are different. So, saying that $(z \star x) \star (y \star x)$ is defined is equivalent of saying that $\{x, y\}$, $\{x, z\}$ and $\{(y \star x), (z \star x) \} = \{\varphi_x (y), \varphi_x (z) \}$ belong to $E (\Gamma)$. But $\{\varphi_x (y), \varphi_x (z)\}$ belongs to $E (\Gamma)$ if and only if $\{y, z\}$ belongs to $E (\Gamma)$, hence $(z \star x) \star (y \star x)$ is defined if and only if $\{x, y\}$, $\{x, z\}$ and $\{ y, z \}$ belong to $E (\Gamma)$. This completes the proof of Claim 1. {\it Claim 2.} Let $x, y, z \in V(\Gamma)$ pairwise distinct be such that $\{x, y\}, \{x, z\}, \{y, z\} \in E(\Gamma)$. Then \[ (z \star x) \star (y \star x) = (z \star y) \star (x \star y)\,. \] {\it Proof of Claim 2.} The proof of Claim 2 is divided into two cases depending on whether $x \|\| y$ or $y < x$. By symmetry, the case $x < y$ is treated in the same way as the case $y < x$. {\it Case 1:} $x \|\| y$. By Condition (b) in the definition of a trickle graph, we cannot have together $z < x$ and $z < y$. Thus, we have two subcases: $z<x$ and $z \not < y$ for the first, and $z \not < x$ and $z \not < y$ for the second. The subcase $z \not < x$ and $z < y$ is treated in the same way as the subcase $z<x$ and $z \not < y$. Suppose $z < x$ and $z \not < y$. Then $\varphi_x (z) \not < \varphi_x (y) = y$, hence \[ (z \star x) \star (y \star x) = \varphi_x (z) \star y = \varphi_x (z) = z \star x = (z \star y) \star (x \star y)\,. \] Suppose $z \not < x$ and $z \not < y$. Then \[ (z \star x) \star (y \star x) = z \star y = z = z \star x = (z \star y) \star (x \star y)\,. \] {\it Case 2:} $y < x$. Here we have three subcases: $z \not < x$ (which implies $z \not < y$) for the first, $z < x$ and $z \not < y$ for the second, and $z < y$ (which implies $z < y < x$) for the third. Suppose $z \not < x$. Then $z = \varphi_x (z) \not < \varphi_x (y)$, hence \[ (z \star x) \star (y \star x) = z \star \varphi_x (y) = z = z \star x = (z \star y) \star (x \star y)\,. \] Suppose $z < x$ and $z \not < y$. Then $\varphi_x (z) \not < \varphi_x (y)$, hence \[ (z \star x) \star (y \star x) = \varphi_x (z) \star \varphi_x (y) = \varphi_x (z) = z \star x = (z \star y) \star (x \star y)\,. \] Suppose $z < y$. Then, by Condition (g) in the definition of a trickle graph, \[ (z \star x) \star (y \star x) = \varphi_x (z) \star \varphi_x (y) = (\varphi_{\varphi_x (y)} \circ \varphi_x) (z) = (\varphi_x \circ \varphi_y) (z) = \varphi_y (z) \star x = (z \star y) \star (x \star y)\,. \] This completes the proof of Claim 2 and therefore of Theorem \ref{thm2_14}. \end{proof} For the remaining proofs in this section we need to understand how the rewriting system $\RR$ of Section \ref{sec4} can be restricted to positive words. Then Theorem \ref{thm2_17} will be a straightforward consequence of this analysis, and therefore it will be proved before Theorem \ref{thm2_15} and Theorem \ref{thm2_16}. Set $S^+ (\Gamma) = \{ x^a \mid x \in V (\Gamma) \text{ and } a \in \N_{\ge 1} \}$. A \emph{positive stratum} of $\Gamma$ is a finite subset $U = \{x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p}\} \subseteq S^+ (\Gamma)$ such that $x_i \neq x_j$ and $\{x_i, x_j\} \in E (\Gamma)$ for all $i, j \in \{1, \dots, p\}$, $i \neq j$. The set of positive strata is denoted by $\Omega^+ = \Omega^+ (\Gamma)$. Note that $\Omega^+ \subset \Omega$, hence $(\Omega^+)^ \subset \Omega^$. It is easily seen that, if $U \in \Omega^+$ and $x^a \in U$, then $\gamma (U, x^a) \in S^+ (\Gamma)$. It is also easily seen that, if $U \in \Omega^+$ and $x^a \in S^+ (\Gamma)$ can be added to $U$, then $R (U, x^a) \in \Omega^+$. Let $\RR^+ = \RR \cap ((\Omega^+)^ \times (\Omega^+)^)$. Then the following is a direct consequence of the above observations combined with Theorem \ref{thm2_4}. \begin{lem}\label{lem7_2} \begin{itemize} \item[(1)] $\RR^+$ is a terminating and confluent rewriting system. \item[(2)] Let $w \in (\Omega^+)^$ and $w' \in \Omega^$. If $w \stackrel{\RR\ }{\longrightarrow} w'$, then $w' \in (\Omega^+)^$ and $w \stackrel{\RR^+\ }{\longrightarrow} w'$. \end{itemize} \end{lem} Set $M^+ (\Gamma) = \langle \Omega^+ \mid u = v \text{ for } (u, v) \in \RR^+ \rangle^+$. Let $U\in \Omega^+$, $U \neq \emptyset$. Write $U = \{ x_1^{a_1}, \dots, x_p^{a_p} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$, and set $\omega^+ (U) = x_1^{a_1} \dots x_p^{a_p} \in \Tr^+ (\Gamma)$. If $U = \emptyset$, then set $\omega^+ (U) = 1$. The same proofs as the ones of Lemmas \ref{lem4_14} and \ref{lem4_15} applied to $M^+ (\Gamma)$ and $\Tr^+ (\Gamma)$ prove the following. \begin{lem}\label{lem7_3} We have an isomorphism $\Phi^+ : M^+ (\Gamma) \to \Tr^+ (\Gamma)$ which sends $U$ to $\omega^+ (U)$ for all $U \in \Omega^+$. \end{lem} If $U$ is a non-empty positive stratum that we write $U = \{ x_1^{a_1}, x_2^{a_2}, \dots, x_p^{a_p} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_p$, then we set $\hat \omega^+ (U) = x_1^{a_1} \cdot x_2^{a_2} \cdots x_p^{a_p} \in V (\Gamma)^$. Note that $\hat \omega^+ (U)$ is a representative of $\omega^+(U)$. Let $g \in \Tr^+ (\Gamma)$. By the above there exists a unique piling $w = (U_1, U_2, \dots, U_p) \in (\Omega^+)^$ such that $w$ is $\RR^+$-irreducible and $w$ represents $g$. Then we set \[ \nf^+ (g) = \hat \omega^+ (U_1) \cdot \hat \omega^+ (U_2) \cdots \hat \omega^+ (U_p) \in V (\Gamma)^\,. \] The following is a direct consequence of the above observations. \begin{lem}\label{lem7_4} \begin{itemize} \item[(1)] Let $g \in \Tr^+ (\Gamma)$. Then $\nf^+ (g) = \nf (\iota_{\Tr^+ (\Gamma)} (g))$. \item[(2)] Let $g \in \Tr (\Gamma)$. We have $g \in \iota_{\Tr^+ (\Gamma)} (\Tr^+ (\Gamma))$ if and only if $\nf (g) \in V(\Gamma)^$. \end{itemize} \end{lem} \begin{proof}[Proof of Theorem \ref{thm2_17}] Let $g, h \in \Tr^+ (\Gamma)$. If $\iota_{\Tr^+ (\Gamma)} (g) = \iota_{\Tr^+ (\Gamma)} (h)$, then we have $\nf ( \iota_{\Tr^+ (\Gamma)} (g)) = \nf ( \iota_{\Tr^+ (\Gamma)} (h))$, hence, by Lemma \ref{lem7_4}, $\nf^+ (g) = \nf^+ (h)$, and therefore $g=h$. \end{proof} Now, we turn to the proof of Theorem \ref{thm2_15}. The following two lemmas are preliminaries of it. \begin{lem}\label{lem7_5} Let $g \in \Tr^+ (\Gamma)$. Let $w = (U_1, \dots, U_p)$ be the unique piling in $(\Omega^+)^$ such that $\Phi^+ (\bar w) = g$ and $w$ is $\RR^+$-irreducible. Write $U_1=\{ x_1^{a_1}, \dots, x_q^{a_q} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_q$. Let $\psi = \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_q}^{a_q}$. Then $\Div_L (g) \cap V (\Gamma) = \psi (\{x_1, x_2, \dots, x_q\})$. \end{lem} \begin{proof} Let $y \in \psi (\{x_1, x_2, \dots, x_q\})$. There exists $i \in \{1, \dots, q\}$ such that $y = \psi (x_i)$. Since $x_j \not > x_i$ for $j \ge i$, we have $y = \psi(x_i) =(\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$. Then \begin{gather} g = x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_i^{a_i} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \, \omega (U_2) \dots \omega (U_p) =\\ x_1^{a_1} \dots x_{i-2}^{a_{i-2}} \varphi_{x_{i-1}}^{a_{i-1}} (x_i)^{a_i} x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \, \omega (U_2) \dots \omega (U_p) = \cdots =\\ (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)^{a_i}\, x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \, \omega (U_2) \dots \omega (U_p) =\\ y\,y^{a_i-1} x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \, \omega (U_2) \dots \omega (U_p)\,, \end{gather} hence $y \in \Div_L (g) \cap V (\Gamma)$. Let $y \in \Div_L (g) \cap V (\Gamma)$. Let $g' \in \Tr^+ (\Gamma)$ be such that $g = yg'$. Let $w'$ be the unique piling in $(\Omega^+)^$ such that $\Phi^+ (\overline{w'}) = g'$ and $w'$ is $\RR^+$-irreducible. Let $u = \{y\} \cdot w' \in (\Omega^+)^$. We have $\Phi^+ (\bar u) = g$, hence $u \stackrel{\RR^+\ }{\longrightarrow} w$. In other words, there exists a finite sequence $u_0=u, u_1, \dots, u_\ell=w$ in $(\Omega^+)^$ such that $u_{j-1} \stackrel{\RR^+}{\to} u_j$ for all $j \in \{1, \dots, \ell \}$. For each $j \in \{0,1, \dots, \ell \}$ we denote by $V_j$ the first stratum in the piling $u_j$ and we set $V_j = \{z_{1,j}^{c_{1,j}}, \dots, z_{r_j,j}^{c_{r_j,j}} \}$ with $z_{1,j} \succ \cdots \succ z_{r_j,j}$. {\it Claim.} Let $j \in \{ 0, 1, \dots, \ell\}$. Then there exists $i \in \{1, \dots, r_j \}$ such that $y = (\varphi_{z_{1,j}}^{c_{1,j}} \circ \cdots \circ \varphi_{z_{i-1,j}}^{c_{i-1,j}}) (z_{i,j})$. {\it Proof of the claim.} We argue by induction on $j$. Suppose $j = 0$. Then $V_0 = \{y\}$ and the claim is trivial. Suppose $j > 0$ and that the induction hypothesis holds. By the induction hypothesis there exists $i \in \{1, \dots, r_{j-1} \}$ such that $y = (\varphi_{z_{1,j-1}}^{c_{1,j-1}} \circ \cdots \circ \varphi_{z_{i-1,j-1}}^{c_{i-1,j-1}}) (z_{i,j-1})$. If $V_j = V_{j-1}$, then there is nothing to prove. So, we can assume that $V_j \neq V_{j-1}$. Then there exists a syllable $z^c$ that can be added to $V_{j-1}$ such that $V_j = R(V_{j-1},z^c)$. Let $k \in \{0, \dots, r_{j-1} \}$ be such that $z_{k,j-1} \succeq z \succ z_{k+1,j-1}$. If $k=0$, then this means that $z \succ z_{1,j-1}$. If $k=r_{j-1}$, then this means that $z_{r_{j-1},j-1} \succeq z$. Also, note that, if $z = z_{k,j-1}$, then $c + c_{k,j-1}>0$, hence $c + c_{k,j-1} \neq 0$. If $k \ge i$, then \[ (\varphi_{z_{1,j}}^{c_{1,j}} \circ \cdots \circ \varphi_{z_{i-1,j}}^{c_{i-1,j}}) (z_{i,j}) = (\varphi_{z_{1,j-1}}^{c_{1,j-1}} \circ \cdots \circ \varphi_{z_{i-1,j-1}}^{c_{i-1,j-1}}) (z_{i,j-1}) = y\,. \] If $k < i$ and $z \neq z_{k,j-1}$, then, by applying Lemma \ref{lem4_7} several times, \begin{gather} (\varphi_{z_{1,j}}^{c_{1,j}} \circ \cdots \circ \varphi_{z_{i,j}}^{c_{i,j}}) (z_{i+1,j}) =\\ (\varphi_{z_{1,j-1}}^{c_{1,j-1}} \circ \cdots \circ \varphi_{z_{k,j-1}}^{c_{k,j-1}} \circ \varphi_z^c \circ \varphi_{\varphi_z^{-c} (z_{k+1,j-1})}^{c_{k+1,j-1}} \circ \cdots \circ \varphi_{\varphi_z^{-c} (z_{i-1,j-1})}^{c_{i-1,j-1}}) (\varphi_z^{-c} (z_{i,j-1})) =\\ (\varphi_{z_{1,j-1}}^{c_{1,j-1}} \circ \cdots \circ \varphi_{z_{k,j-1}}^{c_{k,j-1}} \circ \varphi_z^c \circ \varphi_z^{-c} \circ \varphi_{z_{k+1,j-1}}^{c_{k+1,j-1}} \circ \cdots \circ \varphi_{z_{i-1,j-1}}^{c_{i-1,j-1}}) (z_{i,j-1}) =\\ (\varphi_{z_{1,j-1}}^{c_{1,j-1}} \circ \cdots \circ \varphi_{z_{i-1,j-1}}^{c_{i-1,j-1}}) (z_{i,j-1}) = y\,. \end{gather} If $k < i$ and $z= z_{k,j-1}$, then we prove in the same way that \[ (\varphi_{z_{1,j}}^{c_{1,j}} \circ \cdots \circ \varphi_{z_{i-1,j}}^{c_{i-1,j}}) (z_{i,j}) = y\,. \] This completes the proof of the claim. Applying the claim to $j=\ell$ we see that there exists $i \in \{1, \dots, q \}$ such that $y = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i) = \psi (x_i)$. \end{proof} An element $g \in \Tr^+ (\Gamma)$ is called \emph{square-free} if it can be written in the form $g = x_1 x_2 \dots x_p$ with $x_1 \succ x_2 \succ \cdots \succ x_p$. We denote by $\SF (\Gamma)$ the set of square-free elements. \begin{lem}\label{lem7_6} Suppose $V (\Gamma)$ is finite and $\Gamma$ is complete. Write $V (\Gamma) = \{x_1, \dots, x_n\}$ with $x_1 \succ x_2 \succ \cdots \succ x_n$, and set $\Delta = x_1 x_2 \dots x_n$. Then $\Delta = \vee_L V (\Gamma) = \vee_R V (\Gamma)$ and $\Div_L (\Delta) = \Div_R (\Delta) = \SF (\Gamma)$. In particular, $\Delta$ is balanced and $\Div (\Delta)$ generates $\Tr^+ (\Gamma)$, that is, $\Delta$ is a Garside element. \end{lem} \begin{proof} Let $\nu : \Tr^+ (\Gamma) \to \N$ be the length homomorphism which sends $x_i$ to $1$ for all $i \in \{1, \dots, n\}$. For $p \in \{0,1, \dots, n\}$ we denote by $\SF_p (\Gamma)$ the set of square-free elements of length $p$. Let $g \in \SF_p (\Gamma)$. We write $g = x_{i_1} x_{i_2} \dots x_{i_p}$ with $i_1 < i_2 < \cdots < i_p$, we set $\psi_g = \varphi_{x_{i_1}} \circ \cdots \circ \varphi_{x_{i_p}}$ and $X_g = \psi_g (\{x_{i_1}, \dots, x_{i_p} \})$. We know by Lemma \ref{lem7_5} that $\Div_L (g) \cap V (\Gamma) = X_g$. Furthermore, again by Lemma \ref{lem7_5}, if $h \in \Tr^+ (\Gamma)$ satisfies $\Div_L (h) \cap V (\Gamma) \supseteq X_g$, then $\nu (h) \ge p$. This implies that $\vee_L X_g$ exists and $\vee_L X_g = g$. Denote by $\PP_p (V (\Gamma))$ the set of subsets of $V (\Gamma)$ with cardinality $p$. We have an injective map $\SF_p (\Gamma) \to \PP_p (V (\Gamma))$, $g \mapsto X_g$, and $\|\SF_p (\Gamma)\| = \|\PP_p (V (\Gamma))\| = \binom{n}{p}$, hence this map is a bijection. In particular, since $\SF_n (\Gamma) = \{\Delta\}$ and $\PP_n (V (\Gamma)) = \{ V(\Gamma)\}$, we have $\vee_L V(\Gamma) = \Delta$. Let $g \in \Tr^+ (\Gamma)$. If $g \le_L \Delta$, then $g$ is square-free, otherwise by applying the trickle algorithm we see that $\Delta$ would not be square-free. Suppose that $g$ is square-free. Then, since $X_g \subseteq V (\Gamma)$, we have $g = \vee_L X_g \le_L \vee_L V (\Gamma) = \Delta$. This shows that $\Div_L (\Delta) = \SF (\Gamma)$. For $p \in \{0, 1, \dots, n\}$ we denote by $\widetilde{\SF}_p (\Gamma)$ the set of elements of $\Tr^+ (\Gamma)$ that can be written in the form $x_{i_p} \dots x_{i_2} x_{i_1}$ with $i_1 < i_2 < \cdots < i_p$. We show by induction on $p$ that $\widetilde{\SF}_p (\Gamma) = \SF_p (\Gamma)$. The cases $p=0$ and $p=1$ are obvious, hence we can assume that $p \ge 2$ and that the induction hypothesis holds. Let $g \in \SF_p (\Gamma)$. Write $g$ in the form $g = x_{i_1} x_{i_2} \dots x_{i_p}$ with $i_1 < i_2 < \cdots < i_p$. Since, for all $j \in \{2, \dots, p\}$, we have $x_{i_1} > x_{i_j}$ or $x_{i_1} \|\| x_{i_j}$, \[ g= \varphi_{x_{i_1}}(x_{i_2}) \dots \varphi_{x_{i_1}}(x_{i_p}) \, x_{i_1}\,. \] By Lemma \ref{lem4_13}, $\varphi_{x_{i_1}}(x_{i_2}) \dots \varphi_{x_{i_1}}(x_{i_p}) \in \SF_{p-1} (\Gamma)$. Let $V_1 = \{ x \in V (\Gamma) \mid x \not \ge x_{i_1} \}$ and let $\Gamma_1$ be the full subgraph of $\Gamma$ spanned by $V_1$. Observe that $\Gamma_1$ is a parabolic subgraph of $\Gamma$ and $\varphi_{x_{i_1}}(x_{i_2}) \dots \varphi_{x_{i_1}}(x_{i_p}) \in \SF_{p-1} (\Gamma_1)$. By the induction hypothesis, $\SF_{p-1} (\Gamma_1) = \widetilde{\SF}_{p-1} (\Gamma_1)$, hence $\varphi_{x_{i_1}}(x_{i_2}) \dots \varphi_{x_{i_1}}(x_{i_p})$ can be written in the form $x_{k_p} \dots x_{k_3} x_{k_2}$ with $k_2 < k_3 < \cdots < k_p$ and $x_{k_j} \in V_1$ for all $j \in \{2, \dots, p\}$. Note that, for $j \in \{2, \dots, p\}$, as $x_{k_j} \in V_1$, either $x_{i_1} > x_{k_j}$ or $x_{i_1} \|\| x_{k_j}$. Applying Lemma \ref{lem4_13} to the dual presentation of $\Tr^+ (\Gamma)$ we deduce that $g = x_{k_p} \dots x_{k_2} x_{i_1} \in \widetilde{\SF}_p (\Gamma)$. This shows that $\SF_p (\Gamma) \subseteq \widetilde{\SF}_p (\Gamma)$. We show in the same way that $\widetilde{\SF}_p (\Gamma) \subseteq \SF_p (\Gamma)$, hence $\widetilde{\SF}_p (\Gamma) = \SF_p (\Gamma)$. Let $\widetilde{\SF} (\Gamma) = \bigcup_{p=0}^n \widetilde{\SF}_p (\Gamma)$. From the equality $\SF_n (\Gamma) = \widetilde{\SF}_n (\Gamma)$ it follows that $\Delta = x_n \dots x_2 x_1$. Moreover, by applying the previous reasoning to the dual presentation of $\Tr^+ (\Gamma)$ we get $\Div_R (\Delta) = \widetilde{\SF} (\Gamma) = \SF (\Gamma)$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2_15}] Suppose $\Tr^+ (\Gamma)$ is a Garside monoid. Let $\Delta$ be a Garside element of $\Tr^+ (\Gamma)$. By definition, $\Div (\Delta) = \Div_L (\Delta)$ generates $\Tr^+ (\Gamma)$, hence $V (\Gamma) \subseteq \Div_L (\Delta)$, that is, $\Div_L (\Delta) \cap V (\Gamma) = V (\Gamma)$. Now, Lemma \ref{lem7_5} implies that $\Div_L (\Delta) \cap V(\Gamma) = V(\Gamma)$ is a finite stratum, hence $V (\Gamma)$ is finite and $\Gamma$ is complete. Suppose $V (\Gamma)$ is finite and $\Gamma$ is complete. We write $V (\Gamma) = \{ x_1, \dots, x_n \}$ with $x_1 \succ x_2 \succ \cdots \succ x_n$, and we set $\Delta = x_1 x_2 \dots x_n$. Then, by Lemma \ref{lem7_6}, $\Delta$ is a Garside element, hence $\Tr^+ (\Gamma)$ is a Garside monoid. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2_16}] We argue by induction on $\|V (\Gamma)\|$. If $\|V (\Gamma)\| = 1$, then $\Tr (\Gamma) \simeq \Z$, hence $\Tr (\Gamma)$ is torsion-free. So, we can assume that $\|V (\Gamma)\| \ge 2$ and that the induction hypothesis holds. Let $Y$ be the set of maximal elements of $V (\Gamma)$ for the (partial) order $\le$. First, suppose that there exist $y_1, y_2 \in Y$ such that $y_1 \neq y_2$ and $\{ y_1, y_2 \} \not \in E (\Gamma)$. Let $X_1 = V (\Gamma) \setminus \{ y_1 \}$, $X_2 = V (\Gamma) \setminus \{ y_2 \}$ and $X_{12} = V (\Gamma) \setminus \{ y_1, y_2 \}$. We denote by $\Gamma_1$ the full subgraph of $\Gamma$ spanned by $X_1$, by $\Gamma_2$ the full subgraph of $\Gamma$ spanned by $X_2$, and by $\Gamma_{12}$ the full subgraph of $\Gamma$ spanned by $X_{12}$. It is easily seen that $\Gamma_1$, $\Gamma_2$, and $\Gamma_{12}$ are parabolic subgraphs of $\Gamma$. Moreover, by Theorem \ref{thm2_10}, the embedding of $\Gamma_{12}$ into $\Gamma_k$ induces an injective homomorphism $\Tr (\Gamma_{12}) \hookrightarrow \Tr (\Gamma_k)$ and, by the induction hypothesis, $\Tr (\Gamma_k)$ is torsion-free, for $k \in \{1, 2\}$. From the standard presentation of $\Tr (\Gamma)$ we see that $\Tr (\Gamma) = \Tr (\Gamma_1) _{\Tr (\Gamma_{12})} \Tr (\Gamma_2)$, hence, by \cite[Chapter I, Corollary 1]{Serre1}, $\Tr (\Gamma)$ is torsion-free. Suppose $\{y_1, y_2 \} \in E (\Gamma)$ for all $y_1, y_2 \in Y$, $y_1 \neq y_2$. Note that, since all the elements of $Y$ are maximal for the order $\le$, we have $y_1 \|\| y_2$ for all $y_1, y_2 \in Y$, $y_1 \neq y_2$. Let $X = V(\Gamma) \setminus Y$. Denote by $\Gamma_Y$ the full subgraph of $\Gamma$ spanned by $Y$ and by $\Gamma_X$ the full subgraph of $\Gamma$ spanned by $X$. It is easily observed that $\Gamma_X$ and $\Gamma_Y$ are both parabolic subgraphs of $\Gamma$ and that $\Tr (\Gamma_Y) \simeq \Z^{\|Y\|}$. Let $y \in Y$. Let $x \in X$ be such that $x \not < y$. By definition, there exists $y' \in Y$ such that $x < y'$. Since $y \|\| y'$, Condition (b) in the definition of a trickle graph implies that $x\|\| y$. Thus, $X \subseteq \starE_y (\Gamma)$ and $\varphi_y (X) = X$. Let $y,y' \in Y$, $y \neq y'$. Let $x \in X$. If $x \not < y$ and $x \not < y'$, then \[ (\varphi_y \circ \varphi_{y'}) (x) = \varphi_y (x) = x = \varphi_{y'} (x) = (\varphi_{y'} \circ \varphi_y) (x)\,. \] If $x < y$, then, by Condition (b) in the definition of a trickle graph, $x \|\| y'$. Since $\varphi_y$ is an automorphism of $\starE_y (\Gamma)$, we also have $\varphi_y(x) \|\| y'$. So, \[ (\varphi_y \circ \varphi_{y'}) (x) = \varphi_y (x) = \varphi_{y'}(\varphi_y (x)) = (\varphi_{y'} \circ \varphi_y) (x). \] If $x < y'$, then we prove in the same way that \[ (\varphi_y \circ \varphi_{y'}) (x) = (\varphi_{y'} \circ \varphi_y) (x). \] This shows that we have an action of $\Tr (\Gamma_Y) \simeq \Z^{\|Y\|}$ on $\Tr (\Gamma_X)$ defined by $y \cdot x = \varphi_y (x)$ for all $y \in Y$ and $x \in X$, and that $\Tr (\Gamma) = \Tr (\Gamma_X) \rtimes \Tr (\Gamma_Y)$. Since $\Tr (\Gamma_Y) \simeq \Z^{\|Y\|}$ is torsion-free and, by the induction hypothesis, $\Tr (\Gamma_X)$ is torsion-free, it follows that $\Tr (\Gamma)$ is also torsion-free. \end{proof} Now, we turn to the study of parabolic submonoids and subgroups of preGarside trickle monoids and groups. We first observe that, by repeating mutatis mutandis the proof of Theorem \ref{thm2_10}, we get the following. \begin{lem}\label{lem7_7} Let $\Gamma_1$ be a parabolic subgraph of $\Gamma$. \begin{itemize} \item[(1)] The canonical homomorphism $\iota_{\Tr^+ (\Gamma_1)}: \Tr^+ (\Gamma_1) \to \Tr^+ (\Gamma)$ is injective. So, we can identify $\Tr^+ (\Gamma_1)$ with $\iota_{\Tr^+ (\Gamma_1)} (\Tr^+ (\Gamma_1))$. \item[(2)] Let $g \in \Tr^+ (\Gamma)$. We have $g \in \Tr^+ (\Gamma_1)$ if and only if $\nf^+ (g) \in V (\Gamma_1)^$. Moreover, in this case, $g$ has the same normal form in $\Tr^+ (\Gamma)$ as in $\Tr^+ (\Gamma_1)$, that is, $\nf_\Gamma^+ (g) = \nf_{\Gamma_1}^+ (g)$. \end{itemize} \end{lem} As mentioned in Section \ref{sec2}, we have two different definitions of parabolicity, the one coming from the trickle groups and the one coming from the preGarside monoids. Now, we show that these two definitions coincide. \begin{prop}\label{prop7_8} Let $N$ be a submonoid of $\Tr^+ (\Gamma)$. Then $N$ is a parabolic submonoid of $\Tr^+ (\Gamma)$ if and only if there exists a parabolic subgraph $\Gamma_1$ of $\Gamma$ such that $N = \Tr^+ (\Gamma_1)$. \end{prop} \begin{proof} Suppose $N$ is a parabolic submonoid of $\Tr^+ (\Gamma)$. Set $X = V (\Gamma) \cap N$ and denote by $\Gamma_X$ the full subgraph of $\Gamma$ spanned by $X$. Let $g \in N$. Since $V (\Gamma)$ generates $\Tr^+ (\Gamma)$, the element $g$ can be written in the form $g = x_1 x_2 \dots x_p$ with $x_1, \dots, x_p \in V (\Gamma)$. Since $N$ is a parabolic submonoid, $N$ is a special submonoid, hence $x_1, \dots, x_p \in N$, that is, $x_1, \dots, x_p \in X$. This shows that $N$ is the submonoid of $\Tr^+ (\Gamma)$ generated by $X$. Let $x, y \in X$ be such that $y < x$. Since $\varphi_x(y) \, x = x y \in N$ and $N$ is special, we have $\varphi_x(y) \in N \cap V (\Gamma) = X$. This implies that, for all $x \in X$, $\varphi_x$ restricts to an automorphism of $\starE_x (\Gamma_X)$. So, $\Gamma_X$ is a parabolic subgraph of $\Gamma$ and $N = \Tr^+ (\Gamma_X)$. Now, we take a parabolic subgraph $\Gamma_1$ of $\Gamma$ and we show that $\Tr^+ (\Gamma_1)$ is a parabolic submonoid of $\Tr^+ (\Gamma)$. {\it Claim 1.} $\Tr^+ (\Gamma_1)$ is special. {\it Proof of Claim 1.} We prove that, if $x \in V (\Gamma)$ and $h \in \Tr^+ (\Gamma)$ are such that $g= xh \in \Tr^+ (\Gamma_1)$, then $x \in V (\Gamma_1)$ and $h \in \Tr^+ (\Gamma_1)$. Let $w = (U_1, \dots, U_p)$ be the unique piling in $(\Omega^+)^$ such that $\Phi^+ (\bar w) = g$ and $w$ is $\RR^+$-irreducible. Write $U_1=\{ x_1^{a_1}, \dots, x_q^{a_q} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_q$. Let $\psi = \varphi_{x_1}^{a_1} \circ \varphi_{x_2}^{a_2} \circ \cdots \circ \varphi_{x_q}^{a_q}$. We know from Lemma \ref{lem7_5} that $x \in \psi (\{x_1, x_2, \dots, x_q\})$, hence there exists $i \in \{1, \dots, q \}$ such that $x = (\varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_{i-1}}^{a_{i-1}}) (x_i)$. We have \[ g = x x^{a_i-1} x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \omega (U_2) \dots \omega (U_p)\,. \] By Lemma \ref{lem7_7}, $\nf^+ (g) \in V (\Gamma_1)^$, hence $x_1, \dots, x_q \in V (\Gamma_1)$, and therefore $x \in V (\Gamma_1)$. Moreover, the inclusion $\nf^+ (g) \in V (\Gamma_1)^$ also implies that $\omega (U_2), \dots, \omega (U_p) \in \Tr^+ (\Gamma_1)$, hence \[ h = x^{a_i-1} x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_q^{a_q} \omega (U_2) \dots \omega (U_p) \in \Tr^+ (\Gamma_1)\,. \] This completes the proof of Claim 1. {\it Claim 2.} Let $g_1, g_2 \in \Tr^+ (\Gamma_1)$. Suppose there exists $g \in \Tr^+ (\Gamma)$ such that $g_1, g_2 \le_L g$. Then there exists $g' \in \Tr^+ (\Gamma_1)$ such that $g_1, g_2 \le_L g' \le_L g$. {\it Proof of Claim 2.} We argue by induction on the word length $\nu (g)$ of $g$. If $\nu (g_1) = 0$, then $g_1 = 1$ and $g' = g_2$ satisfies $g' \in \Tr^+ (\Gamma_1)$ and $g_1, g_2 \le_L g' \le_L g$. Similarly, if $\nu (g_2) = 0$, then $g_2 = 1$ and $g' = g_1$ satisfies $g' \in \Tr^+ (\Gamma_1)$ and $g_1, g_2 \le_L g' \le_L g$. The case $\nu (g_1)= 0$ contains the case $\nu (g) = 0$, hence we can assume that $\nu (g_1) \ge 1$, $\nu (g_2) \ge 1$, and that the induction hypothesis holds. Let $y_1, y_2 \in V (\Gamma_1)$ be such that $y_1 \le_L g_1$ and $y_2 \le_L g_2$, and let $h_1, h_2 \in \Tr^+ (\Gamma_1)$ be such that $g_1 = y_1 h_1$ and $g_2 = y_2 h_2$. First, assume that $y_1 = y_2$. Let $h \in \Tr^+ (\Gamma)$ be such that $y_1 h = g$. Since $\Tr^+ (\Gamma)$ is cancellative, $h_1, h_2 \le_L h$, hence, by the induction hypothesis, there exists $h' \in \Tr^+ (\Gamma_1)$ such that $h_1, h_2 \le_L h' \le_L h$. Let $g' = y_1 h'$. Then $g' \in \Tr^+ (\Gamma_1)$ and $g_1, g_2 \le_L g' \le_L g$. Now, assume $y_1 \neq y_2$. Let $w = (U_1, \dots, U_p)$ be the unique piling in $(\Omega^+)^$ such that $\Phi^+ (\bar w) = g$ and $w$ is $\RR^+$-irreducible. Write $U_1=\{ x_1^{a_1}, \dots, x_q^{a_q} \}$ with $x_1 \succ x_2 \succ \cdots \succ x_q$. Let $\psi = \varphi_{x_1}^{a_1} \circ \cdots \circ \varphi_{x_q}^{a_q}$. Since $y_1, y_2 \le_L g$, by Lemma \ref{lem7_5} we have $y_1, y_2 \in \psi (\{x_1, x_2, \dots, x_q\})$. Let $i, j \in \{1, \dots, q\}$ be such that $y_1 = \psi(x_i)$ and $y_2 = \psi(x_j)$. We can assume without loss of generality that $i < j$. Then, since $y_1 = \psi (x_i)$ and $y_2 = \psi (x_j)$, we have $\{y_1, y_2\} \in E (\Gamma)$ and $y_1 \not < y_2$. Moreover, \begin{gather} g = x_1^{a_1} \dots x_q^{a_q} \omega(U_2) \dots \omega (U_p) = y_2^{a_j} x_1^{a_1} \dots x_{j-1}^{a_{j-1}} x_{j+1}^{a_{j+1}} \dots x_q^{a_q} \omega(U_2) \dots \omega (U_p) =\\ y_2^{a_j} y_1^{a_i} x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_{j-1}^{a_{j-1}} x_{j+1}^{a_{j+1}} \dots x_q^{a_q} \omega(U_2) \dots \omega (U_p) =\\ y_2 y_1 (\varphi_{y_1}^{-1}(y_2))^{a_j-1} y_1^{a_i-1} x_1^{a_1} \dots x_{i-1}^{a_{i-1}} x_{i+1}^{a_{i+1}} \dots x_{j-1}^{a_{j-1}} x_{j+1}^{a_{j+1}} \dots x_q^{a_q} \omega(U_2) \dots \omega (U_p)\,, \end{gather} hence $y_2 y_1 = y_1\, \varphi_{y_1}^{-1} (y_2) \le_L g$. Let $h \in \Tr^+ (\Gamma)$ be such that $y_2 y_1 h = y_1\, \varphi_{y_1}^{-1} (y_2)\, h = g$. We have $h_2, y_1 \le_L y_1 h$, hence, by the induction hypothesis, there exists $k_2 \in \Tr^+ (\Gamma_1)$ such that $h_2, y_1 \le_L k_2 \le_L y_1 h$. Similarly, there exists $k_1 \in \Tr^+ (\Gamma_1)$ such that $h_1, \varphi_{y_1}^{-1} (y_2) \le_L k_1 \le_L \varphi_{y_1}^{-1}(y_2)\, h$. Let $\ell_1, \ell_2 \in \Tr^+ (\Gamma_1)$ be such that $k_2 = y_1 \ell_2$ and $k_1 = \varphi_{y_1}^{-1} (y_2)\, \ell_1$. We have $\ell_1, \ell_2 \le_L h$, hence, by the induction hypothesis, there exists $h' \in \Tr^+ (\Gamma_1)$ such that $\ell_1, \ell_2 \le_L h' \le_L h$. Let $g' = y_2 y_1 h' = y_1\, \varphi_{y_1}^{-1} (y_2)\, h'$. Then $g' \in \Tr^+ (\Gamma_1)$ and $g_1, g_2 \le_L g' \le_L g$. This completes the proof of Claim 2. Recall that $\Gamma_1$ is a parabolic subgraph of $\Gamma$ and that we want to prove that $\Tr^+ (\Gamma_1)$ is a parabolic submonoid of $\Tr^+ (\Gamma)$. We know from Claim 1 that $\Tr^+ (\Gamma_1)$ is special. Let $g_1, g_2 \in \Tr^+ (\Gamma_1)$ be such that $g_1 \vee_L g_2$ exists in $\Tr^+ (\Gamma)$. By Claim 2 there exists $g' \in \Tr^+ (\Gamma_1)$ such that $g_1, g_2 \le_L g' \le_L g_1 \vee_L g_2$. Then, by definition, $g' = g_1 \vee_L g_2 \in \Tr^+ (\Gamma_1)$. Let $g_1, g_2 \in \Tr^+ (\Gamma_1)$ be such that $g_1 \vee_R g_2$ exists in $\Tr^+ (\Gamma)$. By applying the above argument to the dual presentation of $\Tr^+ (\Gamma)$ we get that $g_1 \vee_R g_2 \in \Tr^+ (\Gamma_1)$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2_18}] Let $N_1$ be a parabolic submonoid of $\Tr^+ (\Gamma)$. By Proposition \ref{prop7_8} there exists a parabolic subgraph $\Gamma_1$ of $\Gamma$ such that $N_1 = \Tr^+ (\Gamma_1)$. The homomorphism $G(N_1) \to \Tr (\Gamma)$ induced by the embedding $N_1 \hookrightarrow \Tr^+ (\Gamma)$ is the homomorphism $\iota_1 : \Tr (\Gamma_1) \to \Tr (\Gamma)$ induced by the embedding of $\Gamma_1$ into $\Gamma$, and we know from Theorem \ref{thm2_10} that this homomorphism is injective. Let $g \in G(N_1) \cap \Tr^+ (\Gamma) = \Tr (\Gamma_1) \cap \Tr^+ (\Gamma)$. By Theorem \ref{thm2_10} we have $\nf (g) \in (V (\Gamma_1) \cup V (\Gamma_1)^{-1})^$ and by Lemma \ref{lem7_4} we have $\nf (g) \in V(\Gamma)^$, hence $\nf (g) \in V (\Gamma_1)^$, and therefore $g \in \Tr^+ (\Gamma_1)$. This shows that $N_1 = \Tr^+ (\Gamma_1) = G (N_1) \cap \Tr^+ (\Gamma)$. Let $N_2$ be another parabolic submonoid of $\Tr^+ (\Gamma)$. As for $N_1$, there exists a parabolic subgraph $\Gamma_2$ of $\Gamma$ such that $N_2 = \Tr^+ (\Gamma_2)$. Let $g \in N_1 \cap N_2 = \Tr^+ (\Gamma_1) \cap \Tr^+ (\Gamma_2)$. By Lemma \ref{lem7_7} we have $\nf^+ (g) \in V(\Gamma_1)^* \cap V (\Gamma_2)^* = V (\Gamma_1 \cap \Gamma_2) ^*$, hence $g \in \Tr^+ (\Gamma_1 \cap \Gamma_2)$. This shows that $N_1 \cap N_2 \subseteq \Tr^+ (\Gamma_1 \cap \Gamma_2)$. Since the inclusion $\Tr^+ (\Gamma_1 \cap \Gamma_2) \subseteq N_1 \cap N_2$ is obvious, it follows that $N_1 \cap N_2 = \Tr^+ (\Gamma_1 \cap \Gamma_2)$. In particular, by Proposition \ref{prop7_8}, $N_1 \cap N_2$ is a parabolic submonoid of $\Tr^+ (\Gamma)$. Finally, applying Corollary \ref{corl2_12} we get \[ G (N_1) \cap G (N_2) = \Tr (\Gamma_1) \cap \Tr (\Gamma_2) = \Tr (\Gamma_1 \cap \Gamma_2) = G (N_1 \cap N_2)\,. \proved \] \end{proof} \frenchspacing \begin{thebibliography}{DDH$\phantom{}^+$07} \bibitem[BE22]{BaElh1} {\bf I Ba, M Elhamdadi,} {\it Circular orderability and quandles,} Preprint, arXiv:2204.09458, 2022. \bibitem[BPS22]{BaPaSi1} {\bf V\,G Bardakov, I\,B\,S Passi, M Singh,} {\it Zero-divisors and idempotents in quandle rings,} Osaka J. Math. 59 (2022), no. 3, 611--637. \bibitem[BCP16]{BeCiPa1} {\bf P Bellingeri, B\,A Cisneros de la Cruz, L Paris,} {\it A simple solution to the word problem for virtual braid groups,} Pacific J. Math. 283 (2016), no. 2, 271--287. \bibitem[BP20]{BelPar1} {\bf P Bellingeri, L Paris,} {\it Virtual braids and permutations,} Ann. Inst. 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