\begin{aligned}
		&j,&&x=i,\\
		&i,&&x=j,\\
		&x,&&\text{otherwise.}
	\end{aligned}
	\right.
	\]
	And set $S^2(n)=\{\phi\circ\phi':\phi,\phi'\in S(n)\}$.
	Then the cardinality of $S^2(n)$ is not greater than $n^4$, which ensures that our example has positive entropy.
	It is easy to see that for large enough $n$ and any $\phi\in S^2(n)$, there is some $1\le m_\phi \le n$ such that $\phi(m_\phi)=m_\phi$, which admits the factor map corresponding to the maximal equicontinuous factor is proximal in our example.

	\section{Combinatorial lemmas}\label{s:com_lem}
	In this section, we give a combinatorial lemma which is used throughout the construction of our example.
	This lemma is used to ensure the multiple minimality of our example.
	And it is induced by an example of minimal subshifts with the properties that all points are multiply minimal, or moreover, this example is doubly minimal.

	\begin{theorem}\label{l:mul_min_shi}
		\cite[Theorem 4.3]{Opr19}
		For any two distinct words $A,B\in\{0,1\}^s$, where $s\ge 1$, and for any $K>0$, there exists a minimal subshift $X\subset \{0,1\}^{\Z}$ and $N$ such that
		\begin{itemize}
			\item[1.] $(X,\sigma)$ is doubly minimal, weakly mixing and has zero topological entropy,
			\item[2.] $A,B\in\mathcal{L}(X)$,
			\item[3.] $\frac1n|\{j:x_{i+j}=1,j<n\}|\le 1/K$ for every $x\in X$, $i\in\Z$ and $n\ge N$,
			\item[4.] each $x\in X$ is a concatenation of words $A,B$ separated by sequences of zeros.
		\end{itemize}
	\end{theorem}

	Now, by this theorem, we give an example as following, which is more convenient for us to prove some combinatorial lemmas.
	\newcommand{\NN}[1]{\{1,2,3,\dots,#1\}}
	\newcommand{\NNo}[1]{\{0,1,2,\dots,#1\}}

	\begin{lemma}\label{l:mul_min_shi2}
		Let integer $Q\ge 1$.
		There exists a minimal subshift $Z\subset \NN{Q}^{\Z}$ such that
		\begin{itemize}
			\item[(1)] $(Z^d,\sigma_d)$ is minimal for all $d\ge 1$,
			\item[(2)] For any $a,b\in \NN{Q}$, $[ab]\cap Z\neq\emptyset$.
		\end{itemize}
	\end{lemma}

	\begin{proof}
		Take an integer $s'\ge 1$ with $2^{s'}>Q$.
		Then choose $s\ge 1$ large enough and distinct $A,B\in\{0,1\}^s$ which are concatenations of words in $\{0,1\}^{s'}$ and include all concatenations of any two words in $\{0,1\}^{s'}$, that is, for each $w=A,B$,
		\[
		\{w|_{[is',(i+2)s')}:0\le i\le\frac{s}{s'}-2\}=\{uv:u,v\in\{0,1\}^{s'}\}.
		\]

		By Theorem \ref{l:mul_min_shi}, there is a minimal subshift $(X,\sigma)$ which is doubly minimal and $A,B\in\mathcal{L}(X)$.
		Choose a bijection $\xi: \{0,1\}^{s'}\to \NN{2^{s'}}$.
		Define $\xi^\Z: \{0,1\}^{\Z}\to \NN{2^{s'}}^{\Z}$ by
		\[
		\xi^\Z((x_i)_{i\in\Z})=(\xi(x|_{[is',is'+s')}))_{i\in\Z}.
		\]
		So $\xi^\Z$ is a continuous bijection and $\xi^\Z\circ\sigma^{s'}=\sigma\circ\xi^\Z$.
		Choose $x_*\in[A]\cap X$ and let $y_*=\xi^\Z(x_*)\in \NN{2^{s'}}^{\Z}$ and $Y=\overline{\mathcal{O}}(y_*,\sigma)\subset \NN{2^{s'}}^{\Z}$.
		Since $(X,\sigma)$ is doubly minimal, by \cite{AM85}, $(X^d,\sigma_d)$ is minimal for all $d$, which implies that $(Y^d,\sigma_d)$ is also minimal for all $d$.

		Finally, choose a surjection $\eta: \NN{2^{s'}}\to \NN{Q}$, define $\eta^\Z: \NN{2^{s'}}^\Z \to \NN{Q}^\Z$ by $\eta^\Z((x_i)_{i\in\Z})=(\eta(x_i))_{i\in\Z}$ and let $Z:=\eta^\Z(Y)$.
		By multiple minimality of $(Y,\sigma)$, (1) is proved.
		Since $A$ is a concatenations of all words in $\{0,1\}^{s'}$ and includes all concatenations of any two words in $\{0,1\}^{s'}$, (2) is proved.
		So $(Z,\sigma)$ is as required.
	\end{proof}
	Next, we apply Lemma \ref{l:mul_min_shi2} to establish the following weaker combinatorial result, which encapsulates the core idea of the proof strategy.

	\begin{lemma}
		Let $d\ge 1$ and $Q\ge 1$. Then there are $n> Q$ and a partition
		\[
		\NNo{n}=\biguplus_{q=1}^{Q}R_q
		\]
		such that for any $0\le l\le n$ and $1\le q\le Q$, there is $K$ such that for $1\le i\le d$,
		\[
		iK+l\in R_q\cup(R_q+n+1).
		\]
	\end{lemma}

	\begin{proof}
		Fix $d\ge 1$ and $Q\ge 1$.
		Let $(Z,\sigma)$ be as in Lemma \ref{l:mul_min_shi2} for $Q$.
		We will use the multiple minimality of $(Z,\sigma)$ to prove the lemma.
		For any $x\in Z$, there is $n_x>0$ such that
		\[
		x_0=x_{n_x}=\cdots=x_{dn_x}.
		\]
		Then
		\[
		\bigcup_{x\in Z}[x|_{[0,dn_x]}]=Z.
		\]
		So there is a finite subset $I\subset Z$ such that
		\[
		\bigcup_{x\in I}[x|_{[0,dn_x]}]=Z.
		\]
		Since for any $1\le a\le Q$, $[a]\cap Z\neq\emptyset$, we have $\{x_0:x\in I\}=\NN{Q}$.

		Fix $z_*\in Z$.
		For each $x\in I$, there is $l_x>0$ such that $\sigma^{l_x}z_*,\dots,\sigma^{dl_x}z_*\in[x|_{[0,dn_x]}]$.
		Let $L=\max\{dl_x+dn_x:x\in I\}$.
		Since $(Z^d,\sigma_d)$ is minimal, there is $M>0$ such that
		\[
		\bigcup_{n=0}^{M}\sigma_d^{-n}[z_*|_{[0,L]}]^d=Z^d.
		\]
		For integer $H\geq 1$, there exists $m_{t,H}\in [0,M]\cap \N$ such that $(\sigma^{H-t}z_*)^{(d)}\in\sigma_d^{-m_{t,H}}[z_*|_{[0,L]}]^d$ for all $1\le t \le dM+L$. Let $\mathbf{m_H}=(m_{1,H},m_{2,H},\cdots,m_{dM+l,H})$. Then we can find $L_1<L_2$ such that $L_2-L_1>dM+L$ and $\mathbf{m_{L_1}}=\mathbf{m_{L_2}}$. We denote $(m_{1},m_{2},\cdots,m_{dM+l})=\mathbf{m_{L_1}}=\mathbf{m_{L_2}}$. Then
		\[
		(\sigma^{L_1-t}z_*)^{(d)},(\sigma^{L_2-t}z_*)^{(d)}\in\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\text{ for all }1\le t\le dM+L.
		\]

		Then let $n=L_2-L_1-1$,
		\[
		\NNo{n}=\biguplus_{a=1}^{Q}\{k\in \NNo{n}: \sigma^{L_1+k}z_*\in [a]\}:=\biguplus_{a=1}^{Q}R_a.
		\]

		Fix $1\le a\le Q$ and $0\le l\le n$.
		Divide $l$ into 2 cases:
		\begin{itemize}
			\item[Case 1:] $0\le l\le n-dM-L$.
			Since
			\[
			\bigcup_{n=0}^{M}\sigma_d^{-n}[z_*|_{[0,L]}]^d=Z^d,
			\]
			there is $0\le m\le M$,
			\[
			(\sigma^{L_1+l}z_*)^{(d)}\in\sigma_d^{-m}[z_*|_{[0,L]}]^d,
			\]
			that is, for $1\le i\le d$,
			\[\sigma^{L_1+l+im}z_*\in [z_*|_{[0,L]}].\]
			Then for $x\in I$ with $x_0=a$, we have
			\[\sigma^{L_1+l+im+i(l_x+n_x)}z_*\in[x_0]=[a].\]
			Since $l\le n-dM-L$, we have
			$l+im+i(l_x+n_x)\in R_a$.

			\item[Case 2:] $n-dM-L<l\le n$.
			Let $t=n+1-l\le dM+L$.
			Then we have
			\[\sigma^{L_1+l+im_t}z_*\in [z_*|_{[0,L]}]\]
			for $1\le i\le d$.
			Then for $x\in I$ with $x_0=a$, we have
			\[\sigma^{L_1+l+im_t+i(l_x+n_x)}z_*\in[x_0]=[a]\]
			for $1\le i\le d$.
			For $1\le i\le d$ with $l+im_t+i(l_x+n_x)\le n$, we have already know $l+im_t+i(l_x+n_x)\in R_a$.
			For $1\le i\le d$ with $l+im_t+i(l_x+n_x)> n$, noticing that
			\[(\sigma^{L_1-t}z_*)^{(d)},(\sigma^{L_2-t}z_*)^{(d)}\in\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\]
			and
			$L_1-t=L_1+l-(L_2-L_1)=L_1+l-n-1$,
			we have
			\[\sigma^{L_1+l-n-1}z_*\in\sigma_d^{-m_t}[z_*|_{[0,L]}],\]
			that is,
			\[\sigma^{L_1+l-n-1+im_t+i(l_x+n_x)}z_*\in[x_0]=[a].\]
			So we have $l-n-1+im_t+i(l_x+n_x)\in R_a$.
		\end{itemize}

		Sum up with above cases, there are $0\le m\le M$ such that for $1\le i\le d$,
		\[l+i(m+l_x+n_x)\in R_a\cup(R_a+n+1)\]
		where $x\in I$ with $x_0=a$.

	\end{proof}

	Indeed, by choosing more $n_x$ that $x$ hits cylinder set $[x_0]$ and longer $L_2-L_1$, we can prove the following stronger combinatorial lemma.
	For $c\ge 1$ and finite subset $R\subset \N$, write
	\[\mathcal{R}_c(R)=\{R'\subset \N: \#R-\#(R\cap R')<c.\}\]

	\begin{lemma}\label{l:combin}
		Let positive integers $d\ge 1$, $c\ge 1$, $N\ge 1$ and $Q\ge 1$.
		Then there are $n\ge N$ and a partition
		\[\NNo{n}=\biguplus_{q=1}^{Q}R_q\]
		satisfying the following property:
		\begin{itemize}
			\item[(i)] $\#R_q\ge \sqrt{n}-1$ for all $1\le q\le Q$, and
			\item[(ii)] for any $0\le l\le n$, $1\le q,q'\le Q$, $R^{(1)}_q,R^{(2)}_q,\dots,R^{(2d)}_q\in \mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\dots,R^{(2d)}_{q'}\in \mathcal{R}_c(R_{q'})$,
			there is an positive integer $K$ such that for $1\le i\le d$,
			\[iK+l\in R^{(2i-1)}_q\cup (R^{(2i)}_q+n+1)\]
			and
			\[iK+l+1\in R^{(2i-1)}_{q'}\cup (R^{(2i)}_{q'}+n+1).\]
		\end{itemize}
	\end{lemma}

	\begin{proof}
		Fix $d\ge 1$, $c\ge 1$, $N\ge 1$ and $Q\ge 1$.
		Let $(Z,\sigma)$ be as in Lemma \ref{l:mul_min_shi2} for $d$ and $Q$.

		For any $x\in Z$, since $(Z^d,\sigma_d)$ is minimal,
		choose $4cd+1$ positive integers $n^{(1)}_x,n^{(2)}_x,\dots,n^{(4cd+1)}_x\in N_{\sigma_d}(x^{(d)},[x_0x_1]^d)$ such that
		\[dn^{(i)}_x<n^{(i+1)}_x,\,1\le i\le 4cd.\]

		Then we have
		\[\bigcup_{x\in Z}[x|_{[0,dn^{(4cd+1)}_x]}]=Z.\]
		So there is a finite $I\subset Z$ such that
		\[\bigcup_{x\in I}[x|_{[0,dn^{(4cd+1)}_x]}]=Z.\]
		Since for any word $w\in\NN{Q}^2$, $[w]\cap Z\neq\emptyset$, we have $\{[x_0x_1]:x\in I\}=\{[w]:w\in\NN{Q}^2\}$.

		Fix $z_*\in Z$.
		For $x\in I$, by minimality of $(Z^d,\sigma_d)$, there is $l_x>dn^{(4cd+1)}_x$ such that
		\[\sigma^{l_x}_d\left((z_*)^{(d)}\right)\in[x|_{[0,dn^{(4cd+1)}_x]}]^d.\]
		Then for any $n^{(i)}_x$, $1\le i\le 4cd+1$, we have $\sigma^{l_x+n^{(i)}_x}_d((z_*)^{(d)})\in[x_0x_1]^d$.

		Let $L=2+\max\{d(l_x+n^{(4cd+1)}_x):x\in I\}$.
		By minimality of $(Z^d,\sigma_d)$, there is $M>0$ such that
		\begin{equation}\label{e:M}
			\bigcup_{m=0}^{M}\sigma_d^{-m}[z_*|_{[0,L]}]=Z.
		\end{equation}

		Then there are $0\le m_1,m_2,\dots,m_{dM+L}\le M$ and $L_1<L_2$ such that $L_2-L_1>\max\{(dM+L)^2,N\}$ and
		\begin{equation}\label{e:m_t}
			(\sigma^{L_1-t}z_*)^{(d)},(\sigma^{L_2-t}z_*)^{(d)}\in\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\text{ for all }1\le t\le dM+L.
		\end{equation}

		Then let $n=L_2-L_1-1$ and for $1\le q\le Q$,
		\[R_q=\{k\in \NNo{n}: \sigma^{L_1+k}z_*\in [q]\}.\]
		So we have $\NNo{n}=\biguplus_{q=1}^{Q}R_q$.

		We will show this partition is as required.

		To prove (i), fix $1\le q\le Q$.
		Since $n\ge (dM+L)^2$, we have
		\[D:=\left\lfloor\frac{n}{dM+L}\right\rfloor\ge\left\lfloor\sqrt{n}\right\rfloor\ge \sqrt{n}-1\]
		where $\lfloor x\rfloor$ denote the integer part of real number $x$.
		And then for $0\le D'< D$, $[D'(dM+L),(D'+1)(dM+L))\cap\N\subset \NNo{n}$.
		For each $0\le D'< D$, there is $0\le m\le M$ such that  $(\sigma^{L_1+D'(dM+L)}z_*)^{(d)}\in\sigma_d^{-m}[z_*|_{[0,L]}]^d$.
		So we have $R_q\cap [D'(dM+L),(D'+1)(dM+L))\neq \emptyset$ for all $0\le D'< D$.
		So $\#R_q\ge D\ge \sqrt{n}-1$.
		Therefore, (i) is proved.

		To prove (ii), fix $0\le l\le n$, $1\le q,q'\le Q$, $R^{(1)}_q,R^{(2)}_q,\dots,R^{(2d)}_q\in \mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\dots,R^{(2d)}_{q'}\in \mathcal{R}_c(R_{q'})$.
		Divide $l$ into 2 cases:
		\begin{itemize}
			\item[Case 1:] $0\le l\le n-dM-L$.
			Since \[\bigcup_{m=0}^{M}\sigma_d^{-m}[z_*|_{[0,L]}]^d=Z^d,\]
			there is $0\le m\le M$,
			\[(\sigma^{L_1+l}z_*)^{(d)}\in\sigma_d^{-m}[z_*|_{[0,L]}]^d,\]
			that is, for $1\le i\le d$,
			\[\sigma^{L_1+l+im}z_*\in [z_*|_{[0,L]}].\]
			Then for $x\in I$ with $x_0=q$ and $x_1=q'$, we have
			\[\sigma_d^{m+l_x+n^{(j)}_x}(\sigma^{L_1+l}z_*)^{(d)}\in[x_0x_1]^d=[qq']^d\]
			for $1\le j\le 4cd+1$.
			Since $d(m+l_x+n^{(4cd+1)}_x)<dM+L$,
			we have
			\[
			\begin{aligned}
				&\{m+l_x+n^{(j)}_x:1\le j\le 4cd+1\}\\
				\subset&\{m'\in\NNo{n}:l+im'\in R_{q},\,l+im'+1\in R_{q'}\text{ for all }1\le i\le d\}.
			\end{aligned}

\begin{theorem}\label{t:main}
		There is a minimal subshift $(X^*,\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\ge 2$ and $x\in X^*$, $x^{(d)}$ is $\sigma_d$-minimal.
	\end{theorem}

	Recall that a map $\pi$ is \emph{open} if the image under $\pi$ of each open set is also an open set.
	Our example is an extension of an odometer, and the corresponding factor map is open and not a bijection.
	This fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \cite{Wil84}.

	\subsection{Main ideas}

	The idea of our example is to realize the ``multiple recurrence" in a finite segment.
	Roughly speaking, in the setting of symbolic dynamics, let all the words ``multiply" appear in each longer word.
	This can be done by Lemma \ref{l:combin} in Section \ref{s:com_lem}.
	The proof is mainly based on the example constructed by Oprocha \cite{Opr19}.
	He give an example of doubly minimal subshift $(X,\sigma)$ which admits any given two words of two symbols.
	And by \cite{AM85}, $(X^d,\sigma_d)$ is minimal.
	By this fact, we can get a minimal subshift $(Z,\sigma)$ of more symbols such that $(Z^d,\sigma_d)$ is minimal and all the pair of symbols appear in the subshift.
	By those property, we get Lemma \ref{l:combin}.
	Turning to the setting of symbolic dynamics, this lemma lets all the pair of symbols ``multiply" appear in a word.

	Now, we can construct a subshift of $\{0,1,2\}^{\N}$. Using Lemma \ref{l:combin}, we construct a sequence of sets of words inductively.
	We construct the first set of words.
	Let symbol $2$ be the prefix of those words, and the suffix consists of symbols $0$ and $1$.
	Temporarily add one more symbol $*$, and use Lemma \ref{l:combin} for symbols $0$, $1$ and $*$.
	Then we can get a word whose first symbol is symbol $2$ as prefix and the rest consists of symbols $0$, $1$ and $*$ as suffix.
	Replacing $*$ by $0$ or $1$, we can get a set of words.
	Choose a word of this set called a marker word.
	To ensure the openness of the factor map corresponding to the maximal equicontinuous factor, we should make some adjustment, that is, exchanging two symbols in the suffix of words in this set twice.
	Then we get a larger set of words, which is the first set of words.
	The words generated by adjusting the marker word are called prefix words, which will be the prefix of the words in the next step of constructing longer words.
	The rest words are called suffix words, which form the suffix of words in the next step.
	View the suffix words as symbols, temporarily add one more symbol $*$ and use Lemma \ref{l:combin} again.
	Then we get some words whose prefixes are the prefix words and the rest consist of suffix words and $*$.
	Replace $*$ by suffix words, we can get a new set of words.
	In this set, choose a word whose prefix is the marker word as a new marker word, and make a similar adjustment to words in the new set by exchanging two suffix words twice.
	Then we get a larger set of words, which is the second set of words.
	The words generated by adjusting the new marker word are the new prefix words, and the rest are the new suffix words.
	After finishing this procedure inductively, we get sequences of marker words and sets of words.
	The sequence of marker words give a point in $\{0,1,2\}^{\N}$, and the orbit closure of this point is our constructed subshift.

	The prefix words can induce the factor map corresponding the maximal equicontinuous factor of the constructed subshift.
	For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine.
	This leads to a factor map from the constructed subshift to an adding machine.
	Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.
	Through iterative adjustments of word constructions, we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words.
	This fact leads to the openness of the factor map.
	Finally, by Lemma \ref{l:combin}, each words can ``multiply" appear in a longer word so many times that the adjustment of the words can preserve the ``multiple recurrence".

	\section{Preliminaries}
	\newcommand{\eps}{\epsilon}
	\newcommand{\bb}[1]{\mathbf{#1}}
	\newcommand{\subword}[1]{\left.\left(#1\right)\right|}

	In this section, we give some definitions and notations.

	\subsection{Structure theorem of minimal systems}\label{s:PI}

	For two topological dynamical systems $(X,T),(Y,S)$, a map $\pi:(X,T)\to (Y,S)$ is called a \emph{factor map} if $\pi$ is a continuous surjection and satisfies $\pi\circ T=S\circ \pi$.
	In this case, call $(X,T)$ is an \emph{extension} of $(Y,S)$ and $(Y,S)$ is a \emph{factor} of $(X,T)$.

	Let $\pi:(X,T)\to (Y,S)$ be a factor map of minimal systems.
	The factor map $\pi$ is called \emph{proximal} if every two points $x,y\in X$ with same image under $\pi$ are \emph{proximal}, that is, $\liminf_{n\to\infty}d(T^nx,T^ny)=0$.
	A topological dynamical system is called \emph{equiconituous} if for any $\epsilon>0$, there is $\delta>0$ such that $\sup_{n\in\N}\rho(T^nx,T^ny)<\epsilon$ whenever $\rho(x,y)<\delta$.
	Since each topological dynamical system has a maximal equicontinuous factor, we denote by $(X_{eq},T_{eq})$ the \emph{maximal equicontinuous factor} of a topological dynamical system $(X,T)$.
	If a topological dynamical system is a proximal extension of some equicontinuous system, then this equicontinuous system is a maximal equicontinuous factor of the topological dynamical system.

	Before introducing the structure theorem for minimal systems \cite{EGS75}, let us give some definitions.
	For two topological dynamical systems $(X,T)$ and $(Y,T)$, let $\pi:X\to Y$ be the factor map from $(X,T)$ to $(Y,T)$.
	The factor map $\pi$ is called \emph{equicontinuous} if for any $\epsilon>0$, there is $\delta>0$ such that $\sup_{n\in\N}\rho(T^nx,T^ny)<\epsilon$ whenever $\rho(x,y)<\delta$ and $\pi(x)=\pi(y)$.
	A topological dynamical system $(X,T)$ is called \emph{transitive} if for any two non-empty open subset $U,V$, there is $n\in\N$ such that $U\cap T^{-n}V\neq \emptyset$.
	The factor map $\pi$ is called \emph{weakly mixing} if the subsystem $R_\pi:=\{(x,y)\in X\times X:\pi(x)=\pi(y)\}$ of $(X\times X, T\times T)$ is transitive.
	The factor map $\pi$ is called \emph{relatively incontractible} (RIC) if it is open and for every $n\ge 1$, let $R^n_\pi:=\{(x_1,\dots,x_n)\in X^n:\pi(x_1)=\cdots=\pi(x_n)\}$, and the minimal points of the system $(R^n_\pi,T\times T\times \cdots \times T)$ are dense in $R^n_\pi$.

	A minimal system $(X,T)$ is called a \emph{strictly PI-system} if there is an ordinal $\alpha$ and a family of systems $\{(X_{\beta},T_{\beta})\}_{\beta\le \alpha}$ such that the followings hold:
	\begin{itemize}
		\item $W_0$ is a singleton;
		\item for any $\beta<\alpha$, there is a factor map from $(X_{\beta+1},T_{\beta+1})$ to $(X_{\beta},T_{\beta})$ which is either proximal or equicontinuous;
		\item for any limit ordinal $\beta\le \alpha$, $(X_{\beta},T_{\beta})$ is the inverse limit of system $\{(X_{\gamma},T_{\gamma})\}_{\gamma<\beta}$;
		\item $W_\alpha=X$.
	\end{itemize}
	The system $(X,T)$ is called a \emph{PI-system} if there is a proximal factor map from some strictly PI-system to $(X,T)$.

	Now, we introduce the structure theorem for minimal systems \cite{EGS75}.

	\begin{theorem}[{Structure theorem for minimal systems}]
		Let $(X,T)$ be a minimal system.
		Then there are a proximal extension $X_{\infty}$ of $X$ and a RIC weakly mixing extension $\pi_{\infty}$ from $X_{\infty}$ to a strictly PI-system $Y_{\infty}$.
		The extension $\pi_{\infty}$ is a bijection if and only if $X$ is a PI-system.
	\end{theorem}

Denote by $\mathbb{N}$ the set of all natural numbers including $0$, and $\Z_+$ the set of all positive integers.
 Let $X$ be a compact metric space with a metric $\rho$, and $T:X\rightarrow X$ be a continuous surjection.
 The pair $(X,T)$ is called a \emph{topological dynamical system}.
 For $d\ge 2$, write $x^{(d)}:=(x,x\dots,x)\in X^d$ and $\tau_d:=T\times T^2\times \cdots \times T^d$.

Recurrence and minimality are important in the study of topological dynamical systems.
 The Birkhoff recurrence theorem shows that each topological dynamical system has at least one recurrent point.
 For a topological dynamical system $(X,T)$, a point $x\in X$ is called \emph{recurrent} for the map $T$ if for some strictly increasing sequence $\{n_i\}$ of $\Z_+$, $T^{n_i}x\to x$ as $i\to \infty$.
 Recall that the \emph{orbit closure} of a point $x\in X$ under the map $T$ is the closure of its orbit $\mathcal{O}(x,T):=\{T^nx:n\in\mathbb{N}\}$.
 A point $x\in X$ is called \emph{minimal} or \emph{uniform recurrent} for a map $T$ if its orbit closure under the map $T$ is \emph{minimal}, that is, does not contain any closed $T$-invariant subset.
 It is known that each topological dynamical system has at least one minimal point.
 Furstenberg \cite{Fur77} proves the multiple recurrence theorem, which gives a dynamical proof of Szemer\'edi theorem.
 The multiple recurrence theorem can shows the multiple Birkhoff recurrence theorem, that is, the existence of multiply recurrent points.
 A point $x\in X$ is called \emph{multiply recurrent} if the point $x^{(d)}\in X^d$ is recurrent for $\tau_d$, that is, there is a strictly increasing sequence $\{n_i\}$ of $\Z_+$ such that $T^{jn_i}x\to x$ as $i\to \infty$ for $1\le j\le d$.
 Furstenberg and Weiss \cite{FW78} show the existence of multiply recurrent points by topologically dynamical tools, where Furstenberg \cite{Fur77} uses ergodic theory.
 Furstenberg \cite{Fur81} asks whether there is always a \emph{multiple minimal} point, that is, a point $x\in X$ such that $x^{(d)}$ is a minimal point for $\tau_d$.
 However, Huang, Shao and Ye \cite{HSY22} give a counterexample to this question.
 They show that there is a minimal weakly mixing system $(X,T)$ such that for all $x\in X$, $(x,x)$ is not minimal for $T\times T^2$.

On the other hand, there are some minimal systems whose points are all multiply minimal.
 For a \emph{doubly minimal} system $(X,T)$, that is, the orbit closure of any pair $(x,y)$ under $T\times T$ is either $X\times X$ or the graph of a power of $T$,
 Auslander and Markley \cite{AM85} prove that $(X^d,\tau_d)$ is minimal for all $d\in \Z_+$.
 This show that all the points of a doubly minimal system are multiply minimal.
 Besides, a \emph{distal} system is a topological system $(X,T)$ such that $\inf_{n\in \mathbb{N}}\rho(T^nx,T^ny)>0$ for any $x\neq y\in X$.
 All points in a distal system are multiply minimal.
 Weiss \cite{Wei98} show that any ergodic system with zero entropy has a uniquely ergodic model that is doubly minimal.
 Huang and Ye \cite{HY15} show that all the doubly minimal systems are subshifts.

In \cite{HSY22}, the authors ask the following question:
 \begin{question}
 Is there a minimal PI-system $(X,T)$ which is a non-trivial open proximal extension of an equicontinuous system $(Y,T)$ such that for each $x\in X$, $(x,x)$ is minimal for $T\times T^2$?
 \end{question}

This question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \ref{s:PI}).
 We give a positive answer to this question by constructing a minimal subshift $(X^*,\sigma)$. Write $\sigma_d:=\sigma\times \sigma^2\times \cdots \times \sigma^d$.

\begin{abstract}
 We construct a minimal subshift \((X^{*},\sigma)\) that serves as an open proximal extension of its maximal equicontinuous factor. We establish that every point in this subshift is multiply recurrent minimal. This work solves an open problem raised by Huang, Shao and Ye regarding the existence of minimal PI-systems such that each point is multiply minimal.
 \end{abstract}
 \noindent
 {\bf Keywords.} minimal systems, symbolic systems, multiply minimal.\\
 {\bf MSC2020:} 37B05, 37B10, 37B20

This question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \ref{s:PI}).
 We give a positive answer to this question by constructing a minimal subshift $(X^*,\sigma)$. Write $\sigma_d:=\sigma\times \sigma^2\times \cdots \times \sigma^d$.

Recall that a map $\pi$ is \emph{open} if the image under $\pi$ of each open set is also an open set.
 Our example is an extension of an odometer, and the corresponding factor map is open and not a bijection.
 This fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \cite{Wil84}.

Let $\pi:(X,T)\to (Y,S)$ be a factor map of minimal systems.
 The factor map $\pi$ is called \emph{proximal} if every two points $x,y\in X$ with same image under $\pi$ are \emph{proximal}, that is, $\liminf_{n\to\infty}d(T^nx,T^ny)=0$.
 A topological dynamical system is called \emph{equiconituous} if for any $\epsilon>0$, there is $\delta>0$ such that $\sup_{n\in\N}\rho(T^nx,T^ny)<\epsilon$ whenever $\rho(x,y)<\delta$.
 Since each topological dynamical system has a maximal equicontinuous factor, we denote by $(X_{eq},T_{eq})$ the \emph{maximal equicontinuous factor} of a topological dynamical system $(X,T)$.
 If a topological dynamical system is a proximal extension of some equicontinuous system, then this equicontinuous system is a maximal equicontinuous factor of the topological dynamical system.

\begin{theorem}\label{l:mul_min_shi}
 \cite[Theorem 4.3]{Opr19}
 For any two distinct words $A,B\in\{0,1\}^s$, where $s\ge 1$, and for any $K>0$, there exists a minimal subshift $X\subset \{0,1\}^{\Z}$ and $N$ such that
 \begin{itemize}
 \item[1.] $(X,\sigma)$ is doubly minimal, weakly mixing and has zero topological entropy,
 \item[2.] $A,B\in\mathcal{L}(X)$,
 \item[3.] $\frac1n|\{j:x_{i+j}=1,j<n\}|\le 1/K$ for every $x\in X$, $i\in\Z$ and $n\ge N$,
 \item[4.] each $x\in X$ is a concatenation of words $A,B$ separated by sequences of zeros.
 \end{itemize}
 \end{theorem}

\begin{lemma}\label{l:mul_min_shi2}
 Let integer $Q\ge 1$.
 There exists a minimal subshift $Z\subset \NN{Q}^{\Z}$ such that
 \begin{itemize}
 \item[(1)] $(Z^d,\sigma_d)$ is minimal for all $d\ge 1$,
 \item[(2)] For any $a,b\in \NN{Q}$, $[ab]\cap Z\neq\emptyset$.
 \end{itemize}
 \end{lemma}

\section{Proof of Theorem 1.2}In this section, we will construct a minimal subshift which is an open proximal extension of its maximal equicontinuous factor.
 We will also show that all points in this subshift are multiply minimal.
 \subsection{Inductive construction of words}In this subsection, we construct a minimal subshift inductively.
 Given a finite alphabet $\Sigma=\{0,1,2\}$, our example is a subshift of $\{0,1,2\}^\N$.

\begin{proof}[Proof of Theorem 1.2]
 Sum up with Proposition \ref{p:open} and Proposition \ref{p:proximal}, we show that $(X^*,\sigma)$ is an open proximal extension of its maximal equicontinuous factor.
 And by Proposition \ref{p:mul_min}, for any $d\ge 2$ and $x\in X^*$, $x^{(d)}$ is $\sigma_d$-minimal.
 \end{proof}

Recall that a map $\pi$ is \emph{open} if the image under $\pi$ of each open set is also an open set.
	Our example is an extension of an odometer, and the corresponding factor map is open and not a bijection.
	This fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \cite{Wil84}.

	\subsection{Main ideas}

	The idea of our example is to realize the ``multiple recurrence" in a finite segment.
	Roughly speaking, in the setting of symbolic dynamics, let all the words ``multiply" appear in each longer word.
	This can be done by Lemma \ref{l:combin} in Section \ref{s:com_lem}.
	The proof is mainly based on the example constructed by Oprocha \cite{Opr19}.
	He give an example of doubly minimal subshift $(X,\sigma)$ which admits any given two words of two symbols.
	And by \cite{AM85}, $(X^d,\sigma_d)$ is minimal.
	By this fact, we can get a minimal subshift $(Z,\sigma)$ of more symbols such that $(Z^d,\sigma_d)$ is minimal and all the pair of symbols appear in the subshift.
	By those property, we get Lemma \ref{l:combin}.
	Turning to the setting of symbolic dynamics, this lemma lets all the pair of symbols ``multiply" appear in a word.

	Now, we can construct a subshift of $\{0,1,2\}^{\mathbb{N}}$. Using Lemma \ref{l:combin}, we construct a sequence of sets of words inductively.
	We construct the first set of words.
	Let symbol $2$ be the prefix of those words, and the suffix consists of symbols $0$ and $1$.
	Temporarily add one more symbol $*$, and use Lemma \ref{l:combin} for symbols $0$, $1$ and $*$.
	Then we can get a word whose first symbol is symbol $2$ as prefix and the rest consists of symbols $0$, $1$ and $*$ as suffix.
	Replacing $*$ by $0$ or $1$, we can get a set of words.
	Choose a word of this set called a marker word.
	To ensure the openness of the factor map corresponding to the maximal equicontinuous factor, we should make some adjustment, that is, exchanging two symbols in the suffix of words in this set twice.
	Then we get a larger set of words, which is the first set of words.
	The words generated by adjusting the marker word are called prefix words, which will be the prefix of the words in the next step of constructing longer words.
	The rest words are called suffix words, which form the suffix of words in the next step.
	View the suffix words as symbols, temporarily add one more symbol $*$ and use Lemma \ref{l:combin} again.
	Then we get some words whose prefixes are the prefix words and the rest consist of suffix words and $*$.
	Replace $*$ by suffix words, we can get a new set of words.
	In this set, choose a word whose prefix is the marker word as a new marker word, and make a similar adjustment to words in the new set by exchanging two suffix words twice.
	Then we get a larger set of words, which is the second set of words.
	The words generated by adjusting the new marker word are the new prefix words, and the rest are the new suffix words.
	After finishing this procedure inductively, we get sequences of marker words and sets of words.
	The sequence of marker words give a point in $\{0,1,2\}^{\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.

	The prefix words can induce the factor map corresponding the maximal equicontinuous factor of the constructed subshift.
	For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine.
	This leads to a factor map from the constructed subshift to an adding machine.
	Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.
	Through iterative adjustments of word constructions, we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words.
	This fact leads to the openness of the factor map.
	Finally, by Lemma \ref{l:combin}, each words can ``multiply" appear in a longer word so many times that the adjustment of the words can preserve the ``multiple recurrence".

To prove Theorem~\ref{t:main}, the construction aims to realize “multiple recurrence” in a finite segment: “let all the words ‘multiply’ appear in each longer word.” This is enabled by Lemma~\ref{l:combin}, whose input comes from Oprocha’s example of a doubly minimal subshift together with the fact (from \cite{AM85}) that “$(X^d,\sigma_d)$ is minimal”; this yields a minimal subshift $(Z,\sigma)$ with “$(Z^d,\sigma_d)$ is minimal and all the pair of symbols appear,” and “by those property, we get Lemma~\ref{l:combin}.”

Using Lemma~\ref{l:combin}, one “construct[s] a sequence of sets of words inductively” in $\{0,1,2\}^{\mathbb{N}}$: start with words having prefix $2$ and suffix over $\{0,1\}$, “temporarily add one more symbol $*$, and use Lemma~\ref{l:combin}$,” then “replac[e] $*$ by $0$ or $1$” to get a set of words and choose a “marker word.” To ensure openness later, one makes an “adjustment, that is, exchanging two symbols in the suffix of words in this set twice,” producing “prefix words” (generated from adjusting the marker) and “suffix words.” Iterating: view suffix words as symbols, add $*$, apply Lemma~\ref{l:combin} again to get longer words with specified prefixes, replace $*$ by suffix words, choose a new marker, and again “make a similar adjustment … by exchanging two suffix words twice.” The inductive procedure produces “sequences of marker words and sets of words”; “the sequence of marker words give a point in $\{0,1,2\}^{\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.”

The maximal equicontinuous factor is obtained from prefix structure: “For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine,” giving a factor map to an odometer. “Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.” Openness comes from the iterative adjustments: “we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words. This fact leads to the openness of the factor map.” Finally, “by Lemma~\ref{l:combin}, each words can ‘multiply’ appear in a longer word so many times that the adjustment of the words can preserve the ‘multiple recurrence’,” yielding the claimed $\sigma_d$-minimality for $x^{(d)}$.

To prove the main theorem, the construction aims to realize “multiple recurrence” in a finite segment: “let all the words ‘multiply’ appear in each longer word.” This is enabled by the following lemma.

\begin{lemma}\label{l:combin}
		Let positive integers $d\ge 1$, $c\ge 1$, $N\ge 1$ and $Q\ge 1$.
		Then there are $n\ge N$ and a partition
		\[\NNo{n}=\biguplus_{q=1}^{Q}R_q\]
		satisfying the following property:
		\begin{itemize}
			\item[(i)] $\#R_q\ge \sqrt{n}-1$ for all $1\le q\le Q$, and
			\item[(ii)] for any $0\le l\le n$, $1\le q,q'\le Q$, $R^{(1)}_q,R^{(2)}_q,\dots,R^{(2d)}_q\in \mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\dots,R^{(2d)}_{q'}\in \mathcal{R}_c(R_{q'})$,
			there is an positive integer $K$ such that for $1\le i\le d$,
			\[iK+l\in R^{(2i-1)}_q\cup (R^{(2i)}_q+n+1)\]
			and
			\[iK+l+1\in R^{(2i-1)}_{q'}\cup (R^{(2i)}_{q'}+n+1).\]
		\end{itemize}
	\end{lemma}

Its input comes from Oprocha’s example of a doubly minimal subshift together with the fact (from \cite{AM85}) that “$(X^d,\sigma_d)$ is minimal”; this yields a minimal subshift $(Z,\sigma)$ with “$(Z^d,\sigma_d)$ is minimal and all the pair of symbols appear,” and “by those property, we get” the lemma above.

Using the lemma above, one “construct[s] a sequence of sets of words inductively” in $\{0,1,2\}^{\mathbb{N}}$: start with words having prefix $2$ and suffix over $\{0,1\}$, “temporarily add one more symbol $*$, and use” the lemma above, then “replac[e] $*$ by $0$ or $1$” to get a set of words and choose a “marker word.” To ensure openness later, one makes an “adjustment, that is, exchanging two symbols in the suffix of words in this set twice,” producing “prefix words” (generated from adjusting the marker) and “suffix words.” Iterating: view suffix words as symbols, add $*$, apply the lemma above again to get longer words with specified prefixes, replace $*$ by suffix words, choose a new marker, and again “make a similar adjustment … by exchanging two suffix words twice.” The inductive procedure produces “sequences of marker words and sets of words”; “the sequence of marker words give a point in $\{0,1,2\}^{\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.”

The maximal equicontinuous factor is obtained from prefix structure: “For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine,” giving a factor map to an odometer. “Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.” Openness comes from the iterative adjustments: “we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words. This fact leads to the openness of the factor map.” Finally, “by” the lemma above, each words can ‘multiply’ appear in a longer word so many times that the adjustment of the words can preserve the ‘multiple recurrence’, yielding the claimed $\sigma_d$-minimality for $x^{(d)}$.

\label{t:main}
		There is a minimal subshift $(X^*,\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\ge 2$ and $x\in X^*$, $x^{(d)}$ is $\sigma_d$-minimal.,

Question: Let (X,\sigma) be a subshift, meaning a symbolic topological dynamical system that is closed and shift-invariant, and write \(x^{(d)}=(x,\dots,x)\) for the diagonal point in \(X^d\). For \(d\ge 2\), define \(\sigma_d:=\sigma\times \sigma^2\times \cdots \times \sigma^d\). A point is \(\sigma_d\)-minimal if its orbit closure under \(\sigma_d\) is a minimal subsystem. A factor map is called open if it sends open sets to open sets, and proximal if any two points in the same fiber are proximal. The maximal equicontinuous factor of a system is its largest equicontinuous factor. Which existence statement holds?

There exists a minimal subshift \((X^*,\sigma)\) such that the factor map from \((X^*,\sigma)\) onto its maximal equicontinuous factor is open and proximal, and for every integer \(d\ge 2\) and every \(x\in X^*\), the diagonal point \(x^{(d)}\) is \(\sigma_d\)-minimal.

Choices:
B. There exists a minimal subshift \((X^*,\sigma)\) such that the factor map from \((X^*,\sigma)\) onto its maximal equicontinuous factor is open and proximal, and there exists an integer \(d\ge 2\) for which, for every \(x\in X^*\), the diagonal point \(x^{(d)}\) is \(\sigma_d\)-minimal.
C. There exists a minimal subshift \((X^*,\sigma)\) such that the factor map from \((X^*,\sigma)\) onto its maximal equicontinuous factor is open and proximal, and for \(d=2\) and every \(x\in X^*\), the diagonal point \((x,x)\) is \((\sigma\times\sigma^2)\)-minimal.
D. There exists a minimal subshift \((X^*,\sigma)\) such that the factor map from \((X^*,\sigma)\) onto its maximal equicontinuous factor is open, and for every integer \(d\ge 2\) and every \(x\in X^*\), the orbit closure of \(x^{(d)}\) under \(\sigma_d\) is the whole space \((X^*)^d\).
E. There exists a minimal subshift \((X^*,\sigma)\) such that the factor map from \((X^*,\sigma)\) onto its maximal equicontinuous factor is proximal, and for every integer \(d\ge 2\) and every \(x\in X^*\), the diagonal point \(x^{(d)}\) is \(\sigma_d\)-minimal; moreover, this factor map can be chosen to be one-to-one.

minimality_of_diagonal_orbit replaced by full-product minimality

The stem gives definitions and asks for an existence statement, but it does not explicitly or implicitly reveal the correct option. No direct wording in the stem singles out choice A.

The item is close to a theorem-recall/restatement question: after introducing the relevant notions, it asks which existence theorem holds. The variants differ by quantifiers and extra properties, so it is not purely verbatim, but it is still only a mild reformulation.

There is some reasoning pressure in comparing 'for every d' versus 'there exists d', minimality versus full-product orbit closure, and open/proximal versus one-to-one. However, the task mainly tests recognition of the target theorem, and the presence of a weaker true option reduces genuine generative reasoning.

Several distractors are plausible and reflect common mistakes with quantifier weakening/strengthening and confusion of dynamical properties. But choice C is a weaker statement that is also true if A is true, so it is not a clean distractor and makes the single-correct-answer format problematic.

Good on avoiding answer leakage, but only moderate as an MCQ: it is largely theorem recognition, and the weaker true option C creates ambiguity and weakens the distractor set.


\begin{theorem}\label{thm:largeclass}
    There exist uncountably many finitely generated pairwise non-isomorphic torsion-free groups which are all quasi-isometric.
\end{theorem}

A central theme within Geometric Group Theory is the study of coarse geometry.
\emph{Quasi-isometries} provide a natural framework for working with large-scale geometric features of finitely generated groups.
Much attention has been given to studying the possible nature of quasi-isometry classes. With regards to large sets, the first \emph{uncountable} family of different quasi-isometry classes of finitely generated groups was constructed by Grigorchuk \cite{grigorchuk}. This used periodic groups and hence contained torsion; the first such family consisting of torsion-free groups was built by Bowditch a decade later \cite{bowditch}.

More recently, uncountable families of quasi-isometrically distinguished finitely generated groups have been exhibited with an abundance of other properties, including: solvable groups \cite{cornuliertessera}, groups containing expanders (\cite{hume}, \cite{grubersisto}), groups with exotic finiteness properties \cite{krophollerlearysoroko}, groups admitting interesting acylindrical actions (\cite{abbotthume}, \cite{abbotthume2}), residually finite groups \cite{chongwise}, and groups with various combinations of several different kinds of dimension \cite{martinezpedrozasanchezsaldana}.

In addition, Minasyan, Osin and Witzel provided set-theoretic tools to enable the construction of uncountable sets of quasi-isometrically distinguished groups in general \cite{minasyanosinwitzel}.

The above results address the question of distinguishing large sets of \emph{different} quasi-isometry classes.
On the other hand, the (dual) question of studying \emph{one} large class has been relatively unexplored in the literature.
Erschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \cite{erschler}.
However, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).
This means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \(C_2\) in this case), and they contain lots of torsion due to being periodic.
A similar construction was also considered in \cite{weigelzaleskii}, where the finite kernels were taken to be \(C_p\) instead.

It is more difficult to construct quasi-isometries without relying on torsion or commensurability and finite-index subgroups.
Indeed, various classes of groups admit what is known as \textit{quasi-isometric rigidity}.
Any finitely generated groups quasi-isometric to \(\mathbb{Z}^n\) must contain \(\mathbb{Z}^n\) as a finite-index subgroup (\cite{pansu}, \cite{cornuliertesseravalette}).
Many right-angled Artin groups \(G\) have the property that any group \(H\) quasi-isometric to \(G\) must actually be commensurable to \(G\) \cite{huang}.
A stronger version of this is known for certain sub-families of right-angled Coxeter groups \cite{boundsxie} and solvable groups \cite{bourdonremy}\cite{dymarzfisherxie}, where two members of the family are quasi-isometric if and only if they are isomorphic.
Further similar instances, including different kinds of rigidity, are surveyed in \cite{hughesmartinezpedrozasanchezsaldana}.

Any quasi-isometry class of groups contains groups with torsion: for any group \(G\) within a quasi-isometry class, the group \(G\times H\) is also in the same quasi-isometry class for any finite group \(H\).
Nonetheless, one can ask how many finitely generated \emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).
The aim of the present work is to exhibit the existence of a class for which there are many.

\begin{abstract}
We construct uncountably many finitely generated, pairwise non-isomorphic torsion-free groups, all of which fall into the same quasi-isometry class.
This is done by considering Schur covering groups and group cohomology, with the necessary geometric ingredient coming from the theory of bounded-valued cohomology.
\end{abstract}

The above results address the question of distinguishing large sets of \emph{different} quasi-isometry classes.
On the other hand, the (dual) question of studying \emph{one} large class has been relatively unexplored in the literature.
Erschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \cite{erschler}.
However, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).
This means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \(C_2\) in this case), and they contain lots of torsion due to being periodic.
A similar construction was also considered in \cite{weigelzaleskii}, where the finite kernels were taken to be \(C_p\) instead.

Any quasi-isometry class of groups contains groups with torsion: for any group \(G\) within a quasi-isometry class, the group \(G\times H\) is also in the same quasi-isometry class for any finite group \(H\).
Nonetheless, one can ask how many finitely generated \emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).
The aim of the present work is to exhibit the existence of a class for which there are many.

In fact, we provide a method to construct lots more such new examples.
The idea is to construct central extensions with properties controlled by group cohomology.
The cohomological conditions of \emph{distinguished}, \emph{rich} and \emph{weakly bounded} will be defined in the later sections.
Their purposes are to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.

\begin{theorem}\label{thm:manyclasses}
 Suppose that a finitely generated group \(G\) has free abelian abelianisation.
 If \(H^2(G,\mathbb{Z})\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \(G\) by \(\mathbb{Z}\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.
\end{theorem}

In the spirit of the results in the literature featuring uncountably many different quasi-isometry classes of groups with interesting properties, we would like to initiate the study of large quasi-isometry classes of groups with some fixed property.
For which properties \(\mathcal{P}\) do there exist uncountably many quasi-isometric groups all having \(\mathcal{P}\)?
One property of particular interest is that of being \emph{almost finitely presented}, also known as being of type \(FP_2\), characterised by having a finitely generated relation module.
Unlike finitely presented groups, of which there are only countably many, there are uncountably many groups of type \(FP_2\).

\begin{lemma}[\cite{delaharpe}, Lemma 42]\label{lem:quotients}
 Suppose that \(\mathcal{S}\) is a set of normal subgroups of a finitely generated group \(G\), and \(\mathcal{Q}\) is the set of corresponding quotient groups.
 Then \(\mathcal{S}\) has uncountable cardinality if and only if there are uncountably many isomorphism classes of groups represented in \(\mathcal{Q}\).
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm:manyclasses}]
 Let \(\mathcal{S}\) be an uncountable set of distinguished classes from \(H^2(G,\mathbb{Z})\) which are all rich and weakly bounded.
 Because \(G^{\text{ab}}\) is free abelian, \(G\) will have a unique Schur covering group \(E\).
 Given a particular class \([a]\) from \(\mathcal{S}\), we apply Proposition \ref{prop:pushout} and consider the corresponding commutative diagram (\ref{eq:pushout}).
 Since \([a]\) is rich, the corresponding map \(f\) will be surjective.
 Then by the 4-lemma, the map \(\phi_f\) is surjective and \(E_f\) is the quotient of \(E\) by the kernel of \(\phi_f\).
 The kernel of \(\phi_f\) is precisely the image under the map \(H_2(G,\mathbb{Z})\to E\) of the kernel of \(f\).
 Since the classes in \(\mathcal{S}\) are distinguished, these kernels will all be pairwise distinct.
 By Lemma \ref{lem:stem} and Corollary \ref{cor:fingen}, \(E\) is finitely generated because \(G\) is finitely generated.
 Therefore by Lemma \ref{lem:quotients}, among the corresponding central extension groups \(E_f\) of \(G\) by \(\mathbb{Z}\), there will be uncountably many pairwise non-isomorphic groups.
 Finally, as all of the classes are weakly bounded, by Proposition \ref{prop:qi} these will all be quasi-isometric to each other and to \(G\times\mathbb{Z}\).
\end{proof}

\begin{theorem}\label{thm:manyclasses}
    Suppose that a finitely generated group \(G\) has free abelian abelianisation.
    If \(H^2(G,\mathbb{Z})\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \(G\) by \(\mathbb{Z}\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.
\end{theorem}

In fact, we provide a method to construct lots more such new examples.
The idea is to construct central extensions with properties controlled by group cohomology.
The cohomological conditions of \emph{distinguished}, \emph{rich} and \emph{weakly bounded} will be defined in the later sections.
Their purposes are to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.

\begin{theorem}\label{thm:manyclasses}
    Suppose that a finitely generated group \(G\) has free abelian abelianisation.
    If \(H^2(G,\mathbb{Z})\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \(G\) by \(\mathbb{Z}\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.
\end{theorem}

Furthermore, these groups will all be quasi-isometric to \(G\times\mathbb{Z}\) in particular.
A source of such groups \(G\) comes from small-cancellation groups which are not finitely presented, and the particular group used here for Theorem \ref{thm:largeclass} will come from this category.

In the spirit of the results in the literature featuring uncountably many different quasi-isometry classes of groups with interesting properties, we would like to initiate the study of large quasi-isometry classes of groups with some fixed property.
For which properties \(\mathcal{P}\) do there exist uncountably many quasi-isometric groups all having \(\mathcal{P}\)?
One property of particular interest is that of being \emph{almost finitely presented}, also known as being of type \(FP_2\), characterised by having a finitely generated relation module.
Unlike finitely presented groups, of which there are only countably many, there are uncountably many groups of type \(FP_2\).

\medskip

\textbf{Question}: Does there exist an uncountable set of pairwise non-isomorphic groups of type \(FP_2\) which are all quasi-isometric?

\medskip

The strategy behind Theorem \ref{thm:manyclasses} relies on uncountably many cohomology classes in \(H^2(G,\mathbb{Z})\).
Since groups of type \(FP_2\) must necessarily have finitely generated second cohomology (\cite{brown}, Chapter VIII), this does not apply to the question.

Note that any property \(\mathcal{P'}\) for which there can be at most countably many groups satisfying it, will necessarily not be quasi-isometry invariant among groups with \(\mathcal{P}\), if an appropriate uncountable quasi-isometry class of groups with \(\mathcal{P}\) were to exist.
This does not necessarily have to be true if uncountably many groups with \(\mathcal{P}\) exist.
For example, it is not known whether the word problem is a quasi-isometry invariant among groups of type \(FP_2\).

\medskip

\textbf{Outline.}

Section 2 contains the necessary background on quasi-isometries, central extensions and the relationship between them through \(\ell^{\infty}\)-cohomology. Section 3 studies Schur covering groups and contains the proof of Theorem \ref{thm:manyclasses}. Section 4 exhibits a particular finitely generated group with the correct properties to deduce the statement of Theorem \ref{thm:largeclass}.

\medskip

\textbf{Acknowledgements.}

The study of large quasi-isometry classes was first brought to the author's attention after a conversation between Peter Kropholler and Ian Leary, following the release of \cite{krophollerlearysoroko}.
This work was supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research.

To prove Theorem~\ref{thm:largeclass}, the introduction indicates the following strategy: construct “central extensions with properties controlled by group cohomology”. The relevant cohomological conditions (to be defined later) are \emph{distinguished}, \emph{rich} and \emph{weakly bounded}, whose purposes are “to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.” This is formalized in Theorem~\ref{thm:manyclasses}: if a finitely generated group $G$ (with free abelian abelianisation) has $H^2(G,\mathbb{Z})$ containing “uncountably many distinguished classes which are both rich and weakly bounded”, then “among the corresponding central extensions of $G$ by $\mathbb{Z}$, there are uncountably many groups which are quasi-isometric but pairwise not isomorphic,” and “these groups will all be quasi-isometric to $G\times\mathbb{Z}$.” The group $G$ used to deduce Theorem~\ref{thm:largeclass} is then taken from “small-cancellation groups which are not finitely presented,” and the paper’s outline says Section 3 “contains the proof of Theorem~\ref{thm:manyclasses}” and Section 4 “exhibits a particular finitely generated group with the correct properties to deduce the statement of Theorem~\ref{thm:largeclass}$.”

To prove the main theorem, the introduction indicates the following strategy: construct “central extensions with properties controlled by group cohomology”. The relevant cohomological conditions (to be defined later) are \emph{distinguished}, \emph{rich} and \emph{weakly bounded}, whose purposes are “to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.” This is formalized in the following theorem.

\begin{theorem}\label{thm:manyclasses}
    Suppose that a finitely generated group \(G\) has free abelian abelianisation.
    If \(H^2(G,\mathbb{Z})\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \(G\) by \(\mathbb{Z}\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.
\end{theorem}

In particular, if a finitely generated group $G$ (with free abelian abelianisation) has $H^2(G,\mathbb{Z})$ containing “uncountably many distinguished classes which are both rich and weakly bounded”, then “among the corresponding central extensions of $G$ by $\mathbb{Z}$, there are uncountably many groups which are quasi-isometric but pairwise not isomorphic,” and “these groups will all be quasi-isometric to $G\times\mathbb{Z}$.” The group $G$ used to establish the main theorem is then taken from “small-cancellation groups which are not finitely presented,” and the paper’s outline says that next one proves the theorem above, and later one exhibits a particular finitely generated group with the correct properties to deduce the statement of the main theorem.

\label{thm:largeclass}
    There exist uncountably many finitely generated pairwise non-isomorphic torsion-free groups which are all quasi-isometric.,

Question: Here, two finitely generated groups are called quasi-isometric if, when equipped with word metrics coming from finite generating sets, the resulting metric spaces are quasi-isometric. Which existence statement holds?

There exists an uncountable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic.

Choices:
B. There exists an uncountable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and every two groups in the family are commensurable, but no two groups in the family are isomorphic.
C. There exists an uncountable family of finitely generated groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic.
D. There exists a countable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic.
E. There exists an uncountable family of finitely generated pairwise non-isomorphic torsion-free groups such that every two groups in the family are bi-Lipschitz equivalent.

quasi-isometry obtained without relying on commensurability

quasi-isometric conclusion strengthened to bi-Lipschitz equivalence

The stem only gives the definition of quasi-isometric finitely generated groups and does not hint at the specific existence theorem or its exact strength.

The item is close to asking for the statement of a known theorem, but the presence of nearby stronger/weaker variants means it is not a pure verbatim restatement.

There is some reasoning in comparing cardinality, torsion-free, commensurability, and bi-Lipschitz strength, but the pressure is weakened because choice C is also true if A is true, so the item does not cleanly force a unique generative judgment.

B, D, and E are plausible theorem-like distractors, but C is a defective distractor since it is a weaker true consequence of A, making the options non-exclusive.

Reasonably well-structured and free of answer leakage, but flawed as an MCQ because it appears to have more than one true option. It tests theorem recognition more than robust generative reasoning.


\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let $\mu\not\equiv 0$ be a nonnegative locally finite Borel measure on $\mathbb{R}^n$ that is $\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\subset \mathbb{R}^n$ and every $z\in \mathbb{Z}^n$,\n\[\n\mu(E+z)=\mu(E).\n\]\nLet\n\[\nP(x)=\tfrac12 x^\top A x+b\cdot x+c,\n\]\nwhere $A\in \mathcal S_+^{n\times n}$ is positive definite symmetric, $b\in \mathbb{R}^n$, $c\in \mathbb{R}$, and\n\[\n\det A=\mu(\mathbb{T}^n),\qquad \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n.\n\]\nA periodic solution of\n\[\n\det D^2u=\mu\quad\text{in }\mathbb{R}^n\n\]\nmeans a convex function $u\in C(\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?

There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.

Choices:
B. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\mathbb{R}^n$.
C. There exists at most one periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.
D. For every quadratic function $P(x)=\tfrac12 x^\top A x+b\cdot x+c$ with $A\in \mathcal S_+^{n\times n}$ satisfying $\det A\ge \mu(\mathbb{T}^n)$, there exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants.
E. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\mathbb{R}^n$.

constant-shift ambiguity in comparing two solutions

dropped existence while retaining uniqueness up to constants

exact compatibility condition $\det A=\mu(\mathbb{T}^n)$

normalization removing linear parts before applying Harnack to the difference

The stem gives the hypotheses and asks for the valid conclusion, but it does not explicitly reveal the correct option. There is no direct wording that singles out existence plus uniqueness up to constants.

The item is quite close to a theorem-recall question: the assumptions are laid out in theorem style and the correct answer is essentially the theorem’s conclusion. However, the alternatives introduce meaningful nearby variants, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish exact uniqueness up to constants from strict uniqueness, mere uniqueness without existence, and an incorrect compatibility condition. Still, the task is driven more by precise theorem recognition than by substantial generative mathematical reasoning.

The distractors are strong and mathematically targeted: they test common mistakes about constant-shift ambiguity, omission of existence, weakening/altering the determinant compatibility condition, and overgeneralizing the difference of two solutions.

A solid MCQ with little answer leakage and strong distractors, but it leans heavily on theorem recall rather than deeper generative reasoning.