\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying \eqref{eqn:Nunidimensional}, and $(S_n, n \geq 1)$ the ERW defined by \eqref{eqn:defERWC}.
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\mathbb{C}(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\mathbb{C}$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\mathrm{e}^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \mathrm{e}^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize}

\begin{theorem}
\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying \eqref{eqn:Nunidimensional}, and $(S_n, n \geq 1)$ the ERW defined by \eqref{eqn:defERWC}.
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\C(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\C$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\erm^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \erm^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize}
\end{theorem}

The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \cite{ScT} to investigate the influence of memory on random walk dynamics. In $\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by
\begin{equation}
  \label{eqn:defERW1}
  S_0 = 0, \quad S_1 = 1, \quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \text{ for $n \geq 2$,}
\end{equation}
where $(R_n, n \geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are
further independent from the sequence $(I_n, n \geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\{1,\ldots,k-1\}$.

Provided that $p>1/2$, this dynamic can be rephrased as follows. At each step, the walker remembers, with probability $2p-1$, one of its past steps chosen uniformly at random, and repeats this step. With complementary probability $2(1-p)$, it performs an independent step equal to $1$ or $-1$. An analogous definition can be taken when $p<1/2$ considering $1-2p$ as the probability of performing the opposite remembered step. The parameter $p$ is usually referred to as the memory parameter of the ERW, but we see from the previous rephrasing the effective memory is more accurately described by $a := 2p-1$. The value $a=0$ (i.e. $p=1/2$) corresponds to the absence of memory, while $|a|$ measures the strength of the dependence on past increments. When $a>0$, the walker tends to repeat the chosen past step (“positive” memory), whereas for $a<0$ it tends to do the opposite (“negative” memory). In both cases, the memory is strong when $|a|$ is close to $1$.

From this perspective, the walk appears to exhibit long-range dependence, as the walk tends for example to reinforce previous directions when $a>0$ -- the ERW is a special case of \textit{step-reinforced random walk}. Consequently, the walker’s trajectory can deviate significantly from a simple random walk, leading to superdiffusive behavior when $a>1/2$ while for $a\in(-1,1/2)$ the process remains diffusive but with different variance compared to the classical case.

A natural generalisation for this process would be to define a version of the elephant random walk in dimension $d$. A $\mathbb{Z}^d$ version of this model was first introduced in \cite{BercuLaulin2019}, while the works \cite{CuL,Qin} further investigate its properties, notably recurrence and transience.
The process considered was the following: at each time $n$, after selecting uniformly at random a previous step, the walker either moves according to this step with probability $p$, or with probability $1-p$ moves towards one of the $2d-1$ other neighbour of $S_n$ chosen uniformly at random. It can be shown in this situation that the ERW undergoes a phase transition when the memory parameter crosses $p_c = \frac{2d+1}{4d}$. If $p \leq p_c$, the ERW is in the diffusive regime, while it drifts at some sub-linear rate if $p > p_c$. This behaviour is therefore very close to the one observed in dimension $1$.

The objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \geq 2)$ be i.i.d. random rotations of $\mathbb{R}^d$ (i.e. random variables in $\mathcal{O}(\mathbb{R}^d)$) independent of the sequence $(I_n, n \geq 2)$ defined above, we then define the ERW in $\mathbb{R}^d$ as
\begin{equation}
  \label{eqn:defERWd}
    S_0 = (0,\ldots, 0), \quad S_1 = (1,0,\ldots, 0) \quad \text{and} \quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).
\end{equation}
This model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\mathbb{P}(R_n = \mathrm{Id}) = p$ and $\mathcal{L}\big(R_n | R_n \neq \mathrm{Id}\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\mathbb{R}^d$.

In the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\mathbb{R}^2$ with the complex plane $\mathbb{C}$ in the usual fashion, and we define the ERW in $\mathbb{C}$ as the following complex-valued stochastic process. Let $(\theta_n, n \geq 1)$ be i.i.d. $[0,2\pi)$-valued random variables, further independent from the sequence $(I_n, n \geq 2)$ defined above, we set
\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \mathrm{e}^{i \theta_n}.
\end{equation}
Some sample trajectories of $S$ are represented in Figure~\ref{fig:ex} when $\theta_1$ is a.s. a constant.

\begin{figure}[ht]
\centering
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d1}
    \caption{$\theta_1= \frac{\pi}{3} - 0.1$ a.s.}
  \end{subfigure}
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d3}
    \caption{$\theta_1 = \frac{\pi}{3}$ a.s.}
  \end{subfigure}
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d5}
    \caption{$\theta_1 = \frac{\pi}{3} + 0.1$ a.s.}
  \end{subfigure}
\caption{Sample path of $1000$ steps of an elephant random walks in $\mathbb{R}^2$ as defined in \eqref{eqn:defERWC}, in which $\theta_1$ is a.s. a constant.}
\label{fig:ex}
\end{figure}

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equation}
  \Phi_k := \mathbb{E}(\mathrm{e}^{i k \theta_1}), \quad \text{for $k \geq 1$},
\end{equation}
the Fourier coefficient of the law of $\theta_1$ and, more specifically,  of $\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\theta_1$ takes values in $\{0,\pi\}$ with $\mathbb{P}(\theta_1=0)=p$, one recovers $\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\Phi_1| \in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\Phi_1=0)$, and maximal in the situation represented in Figure~\ref{fig:ex} in which $\Phi_1 \in \mathbb{S}^1$.

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
  \label{eqn:Nunidimensional}
  \mathbb{P}(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\mathbb{C}$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \cite{ScT} to investigate the influence of memory on random walk dynamics. In $\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by
\begin{equation}
 \label{eqn:defERW1}
 S_0 = 0, \quad S_1 = 1, \quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \text{ for $n \geq 2$,}
\end{equation}
where $(R_n, n \geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are
further independent from the sequence $(I_n, n \geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\{1,\ldots,k-1\}$.

The objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \geq 2)$ be i.i.d. random rotations of $\mathbb{R}^d$ (i.e. random variables in $\mathcal{O}(\mathbb{R}^d)$) independent of the sequence $(I_n, n \geq 2)$ defined above, we then define the ERW in $\mathbb{R}^d$ as
\begin{equation}
 \label{eqn:defERWd}
 S_0 = (0,\ldots, 0), \quad S_1 = (1,0,\ldots, 0) \quad \text{and} \quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).
\end{equation}
This model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\mathbb{P}(R_n = \mathrm{Id}) = p$ and $\mathcal{L}\big(R_n | R_n \neq \mathrm{Id}\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\mathbb{R}^d$.

In the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\mathbb{R}^2$ with the complex plane $\mathbb{C}$ in the usual fashion, and we define the ERW in $\mathbb{C}$ as the following complex-valued stochastic process. Let $(\theta_n, n \geq 1)$ be i.i.d. $[0,2\pi)$-valued random variables, further independent from the sequence $(I_n, n \geq 2)$ defined above, we set
\begin{equation}
 \label{eqn:defERWC}
 S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \mathrm{e}^{i \theta_n}.
\end{equation}
Some sample trajectories of $S$ are represented in Figure~\ref{fig:ex} when $\theta_1$ is a.s. a constant.

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equation}
 \Phi_k := \mathbb{E}(\mathrm{e}^{i k \theta_1}), \quad \text{for $k \geq 1$},
\end{equation}
the Fourier coefficient of the law of $\theta_1$ and, more specifically, of $\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\theta_1$ takes values in $\{0,\pi\}$ with $\mathbb{P}(\theta_1=0)=p$, one recovers $\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\Phi_1| \in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\Phi_1=0)$, and maximal in the situation represented in Figure~\ref{fig:ex} in which $\Phi_1 \in \mathbb{S}^1$.

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
 \label{eqn:Nunidimensional}
 \mathbb{P}(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\mathbb{C}$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

\begin{equation}
  \label{eqn:Nunidimensional}
  \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}

\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \erm^{i \theta_n}.
\end{equation}

\label{fig:ex}
\end{figure}

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equat

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
 \label{eqn:Nunidimensional}
 \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\C(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\C(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\C$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

In order to provide variety to this article, we propose to prove Theorem~\ref{thm:main} in two different ways. We study the regime $\Re(\Phi_1) \leq 1/2$ in Section~\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that
\begin{equation}
 \label{eqn:defMartingale}
 \left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)}, n\geq 1 \right) \text{ is a martingale,}
\end{equation}
its asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \cite{BaB,BercuLaulin2019}.

In this model, remark that $\Phi_1= p - r + i (q-s)$, therefore Theorem~\ref{thm:main} implies the following behaviour for $(S_n, n \geq 1)$, depending on the sign of $p-r-1/2$.
\begin{corollary}
Let $(S_n, n \geq 1) $ be the ERW in $\Z^2$ with parameters $p,q,r,s$ defined above.
\begin{itemize}
 \item If $p < r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}(0, \frac{1}{2 - 4 (p-r)} I_2)$ in law.
 \item If $p=r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{(n\log )^{1/2}}= \mathcal{N}(0, \frac{1}{2} I_2)$ in law.
 \item If $p > r+1/2$, then there exists a random vector $W \in \R^2$ such that
 \begin{multline*}
 \lim_{n \to \infty} \frac{\mathcal{R}_{-(q-s)\log n}S_n}{n^{p-r}} = W \quad \text{a.s.} \\ \text{and}\quad \lim_{n \to \infty} \frac{S_n - n^{p-r} \mathcal{R}_{(q-s)\log n}W}{n^{1/2}} = \mathcal{N}(0,\frac{1}{4(p-r)-2}I_2) \quad \text{ in law},
 \end{multline*}
 writing $\mathcal{R}_\theta$ for a rotation of angle $\theta$.
\end{itemize}
\end{corollary}

We denote by $(S_n, n \geq 1)$ an ERW defined by \eqref{eqn:defERWC}, and we assume in this section that
\begin{equation}
 \label{eqn:cdSub}
 \Re(\Phi_1) = \Re\left(\E\left( \erm^{i\theta_1} \right)\right) < \frac{1}{2}.
\end{equation}
We prove in this section that
\[
 \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C\left(0,\frac{1}{1- 2 \Re(\Phi_1)}\right) \quad \text{ in distribution.}
\]
The proof of \eqref{eqn:mainCrit} follows a similar path with minor modifications, and will be sketched at the end of the section. Let us mention that \eqref{eqn:mainSup} also could have be obtained using the same techniques as developed here.

We denote by $\Gamma$ Euler's Gamma function defined on $\C \backslash \Z_-$, and observe that we can rewrite
\begin{align*}
 a^{(k)}_n &= \prod_{j=1}^{n-1} \left( 1 + \frac{\Phi_k}{j} \right) = \prod_{j=1}^{n-1}\frac{j+\Phi_k}{j} \frac{\Gamma(j+\Phi_k)}{\Gamma(j)} \frac{\Gamma(j)}{\Gamma(j+\Phi_k)}\\
 &= \prod_{j=1}^{n-1} \frac{\Gamma(j + \Phi_k+ 1)}{\Gamma(j+\Phi_k)} \frac{\Gamma(j)}{\Gamma(j + 1)} = \frac{\Gamma(n+\Phi_k)}{\Gamma(n)\Gamma(\Phi_k)},
\end{align*}
from which we deduce there exists $c_k > 0$ such that
\begin{equation}
 \label{eqn:data}
 a_n^{(k)} \sim C_k n^{\Phi_k}, \text{ and } |a_n^{(k)}|^2 \sim |C_k|^2 n^{2 \Re(\Phi_k)} \text{ as $n \to \infty$.}
\end{equation}
Using that $\Re(\Phi_k) \in (-1,1)$ due to \eqref{eqn:Nunidimensional} we conclude that
\[
 \sum_{j=1}^n \frac{1}{|a_j^{(k)}|^2} \begin{cases}
 = O(1) & \text{ if } \Re(\Phi_k)>1/2\\
 \sim |C_k|^2 \log n & \text{ if } \Re(\Phi_k) = 1/2\\
 \sim \frac{|C_k|^2}{1 - 2 \Re(\Phi_k)} n^{1 - 2 \Re(\Phi_k)} &\text{ if } \Re(\Phi_k) < 1/2
 \end{cases}
 \quad \text{ as $n \to \infty$.}
\]
Solving the recursion for $u$ in all three cases, we conclude the proof.
\end{proof}

\begin{proof}
Let $n \geq 1$, we observe that
\begin{align*}
 \Delta M_k &= M_{k+1} - M_k = \frac{S_n + \Delta S_n}{a_{n+1}} - \frac{S_n}{a_n} = \frac{1}{a_{n+1}}\left( S_n + \Delta S_n - \left(1+ \frac{\Phi_1}{n}\right) S_n \right)\\
 &= \frac{1}{a_{n+1}} \left( \Delta S_n - \frac{\Phi_1}{n}S_n \right).
\end{align*}
Therefore,
\begin{align*}
 \E\left( |\Delta M_k|^2 \middle| \mathcal{F}_k \right)
 &= \frac{1}{|a_{k+1}|^2}\E\left( |\Delta S_k|^2 + \left| \frac{\Phi_1}{k} S_k \right|^2 - 2 \Re\left( \frac{\Phi_1}{k} S_k \overline{\Delta S_k} \right) \middle| \mathcal{F}_k \right)\\
 &= \frac{1}{|a_{k+1}|^2} \left( 1 + \left| \frac{\Phi_1}{k} S_k \right|^2 - 2 \Re\left( \left|\frac{\Phi_1}{k} S_k\right|^2 \right)\right) \text{ a.s.},
\end{align*}
using that $|\Delta S_k| = 1$ and $\E(\Delta S_{k}| \mathcal{F}_k) = \Phi_1\frac{S_k}{k}$ a.s. This yields
\begin{equation*}
 \E\left( |\Delta M_k|^2 \middle| \mathcal{F}_k \right) = \frac{1}{|a_{k+1}|^2}\left( 1 - \left| \frac{\Phi_1}{k} S_k \right|^2\right) \text{ a.s.}
\end{equation*}
As a consequence, we have
\begin{equation}
 \label{eqn:formula}
 \crochet{M,\bar{M}}_n = \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} - \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2 \quad \text{a.s.}
\end{equation}
We now study the asymptotic behaviour of these two sums.
We deduce from \eqref{eqn:data} that
\begin{equation}
 \label{eqn:asymp}
 v_n := \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \sim |C_1|^2 \frac{n^{1 - 2 \Re(\Phi_{1})}}{1 - 2 \Re(\Phi_1)},
\end{equation}
and from Lemma~\ref{lem:martingale} that
\[
 \frac{1}{|a_{n+1}|^2 n^2}\E\left( |S_n|^2\right) \sim_{n \to \infty} |C_1|^2 n^{-(1+ 2\Re(\Phi_1))},
\]
as $n \to \infty$. Therefore, there exists $C > 0$ such that for all $n$ large enough, we have
\begin{equation}
 \label{eqn:intermediate}
 \E\left( \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2\right) \leq C n^{-2\Re(\Phi_1)} = o(v_n).
\end{equation}
It shows that $\lim_{n \to \infty} \frac{1}{v_n}\sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2 = 0$ in probability, which completes the proof.
\end{proof}

We now turn to the central limit theorem result, writing
\begin{align*}
 &\phantom{=}\frac{S_n - n^{\Re(\Phi) \erm^{i \Im(\Phi) \log n}} W \exp(-\Phi_1 \log E) }{\sqrt{n}}\\
 &= \frac{Z_1(\tau_n) - W \exp(\Phi_1 \log (N_{\tau_n}/E))}{\sqrt{n}}\\
 &= \erm^{-\tau_n/2}\frac{Z_1(\tau_n) - W \erm^{\Phi_1 \tau_n} + W \erm^{\Phi_1 \tau_n} - W \exp(\Phi_1 \log (N_{\tau_n}/E)}{\sqrt{N_{\tau_n} \erm^{-\tau_n}}}.
\end{align*}
By Corollary~\ref{cor:obvious}, we observe that
\[
 \lim_{t \to \infty} \erm^{-t/2}\frac{Z_1(t) - W \erm^{\Phi_1 t}}{\sqrt{N_t \erm^{-t}}} = \mathcal{N}_\C(0,\frac{1}{2 \Re(\Phi_1) - 1}) \quad \text{in law},
\]
from which we deduce straightforwardly that $\erm^{-\tau_n/2}\frac{Z_1(\tau_n) - W \erm^{\Phi_1 \tau_n} }{\sqrt{N_{\tau_n} \erm^{-\tau_n}}}$ admit the same limit in distribution as $n \to \infty$, using that $\tau_n - \log n$ is tight. In addition, we have
\[
 \frac{\erm^{\Phi_1 t} - \erm^{\Phi_1 \log (N_t / E)}}{\sqrt{N_t}} = \frac{1+o(1)}{\sqrt{E}}\erm^{(\Phi_1 - 1/2)t}(1 - \erm^{\Phi_1 \log (N_t \erm^{-t}/E)}) \quad \text{a.s.}
\]
As $ 1/2 < \Re(\Phi_1)<1$ and $\lim_{t \to \infty} \frac{N_t - E \erm^{t}}{\erm^{t/2}}$ converges in distribution towards a Gaussian random variable, we conclude that this quantity converges to $0$ in probability.

In order to provide variety to this article, we propose to prove Theorem~\ref{thm:main} in two different ways. We study the regime $\Re(\Phi_1) \leq 1/2$ in Section~\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that
\begin{equation}
  \label{eqn:defMartingale}
  \left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)}, n\geq 1 \right) \text{ is a martingale,}
\end{equation}
its asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \cite{BaB,BercuLaulin2019}.

We then turn to the regime $\Re(\Phi_1)> 1/2$ in Section~\ref{sec:spiraling}. We use there the known connection between ERW and P\'olya urns, which was notably exploited by Baur and Bertoin in \cite{BaB} to obtain a functional central limit theorem, as well as the connection between P\'olya urns and multitype Galton-Watson processes, that was exploited by Janson~\cite{Jan} to obtain asymptotics on the behaviour of these urns. These results cannot be applied immediately as we do not necessarily have a finite number of colors\footnote{Understand here directions that steps of the ERW might align with.}. P\'olya urns with infinite number have been already studied in the literature \cite{JMV} however, considering that we are only interested with a single martingale of that process, the analysis can be simplified. Specifically, borrowing arguments from the study of the additive martingales of branching random walks at complex parameters \cite{IKM} will allow us to conclude.

Before turning to the proofs though, we begin by stating explicitly the behaviour of the ERW in $\mathbb{Z}^2$. This process is parametrized by four positive numbers $p,q,r,s$ such that $p+q+r+s=1$.  These parameters correspond respectively to the probability for a walker oriented according to one of its uniformly sampled previous steps to move forward (repeating the step), to the left (making a rotation of angle $\pi/2$), backward (making a rotation of angle $\pi$) or to the right (making a rotation of angle $3\pi/2$). The case $q=r=s=(1-p)/3$ reduces to the previously studied standard ERW in the plane.

In this model, remark that $\Phi_1= p - r + i (q-s)$, therefore Theorem~\ref{thm:main} implies the following behaviour for $(S_n, n \geq 1)$, depending on the sign of $p-r-1/2$.
\begin{corollary}
Let $(S_n, n \geq 1) $ be the ERW in $\mathbb{Z}^2$ with parameters $p,q,r,s$ defined above.
\begin{itemize}
  \item If $p < r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}(0, \frac{1}{2 - 4 (p-r)} I_2)$ in law.
  \item If $p=r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{(n\log )^{1/2}}= \mathcal{N}(0, \frac{1}{2} I_2)$ in law.
  \item If $p > r+1/2$, then there exists a random vector $W \in \mathbb{R}^2$ such that
  \begin{multline*}
    \lim_{n \to \infty}  \frac{\mathcal{R}_{-(q-s)\log n}S_n}{n^{p-r}} = W \quad \text{a.s.} \\ \text{and}\quad \lim_{n \to \infty} \frac{S_n - n^{p-r} \mathcal{R}_{(q-s)\log n}W}{n^{1/2}} = \mathcal{N}(0,\frac{1}{4(p-r)-2}I_2) \quad \text{ in law},
  \end{multline*}
  writing $\mathcal{R}_\theta$ for a rotation of angle $\theta$.
\end{itemize}
\end{corollary}

Finally, remark that the ERW in $\mathbb{Z}^2$ can be seen as a reinforced version of the Markov chains studied by Lopusanschi and Simon in \cite{LoS}, studying the asymptotic behaviour of the Lévy area of this path would therefore be of interest, especially studying the convergence of the ERW under a rough path topology.

\medskip
We now turn to the proof of Theorem~\ref{thm:main} in the two following sections. The proof of \eqref{eqn:mainSub} is given in the following section using Lindenberg's central limit theorem, with a sketch of the proof of \eqref{eqn:mainCrit}. The proof of \eqref{eqn:mainSup} is completed in Section~\ref{sec:spiraling}, using Poissonization methods and the analysis of a branching process.

To prove Theorem~\ref{thm:main}, the article proposes \emph{two different ways} depending on the regime of $\Re(\Phi_1)$.

1. Regime $\Re(\Phi_1)\le 1/2$ (Section~\ref{sec:diffusive}): use Lindeberg's central limit theorem. The \emph{key observation} is that
\[
\left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)},\; n\geq 1 \right)\text{ is a martingale,}
\]
and its asymptotic behaviour is deduced by studying its \emph{quadratic variation}. Aside from the need to handle a \emph{complex martingale}, the argument "follows closely standard methods" from the 1D or $d$-dimensional case. The proof of \eqref{eqn:mainSub} is given there via Lindeberg's CLT, with a \emph{sketch} for \eqref{eqn:mainCrit}.

2. Regime $\Re(\Phi_1)>1/2$ (Section~\ref{sec:spiraling}): use the "known connection between ERW and P\'olya urns" and also the connection between P\'olya urns and \emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \emph{finite number of colors}, the cited results "cannot be applied immediately"; however, because the focus is on "a single martingale" of the process, "the analysis can be simplified". Concretely, "borrowing arguments from the study of the additive martingales of branching random walks at complex parameters" allows one to conclude \eqref{eqn:mainSup}; this part uses "Poissonization methods and the analysis of a branching process."

To prove the main theorem, the article proposes \emph{two different ways} depending on the regime of $\Re(\Phi_1)$.

1. Regime $\Re(\Phi_1)\le 1/2$ (Next): use Lindeberg's central limit theorem. The \emph{key observation} is that
\[
\left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)},\; n\geq 1 \right)\text{ is a martingale,}
\]
and its asymptotic behaviour is deduced by studying its \emph{quadratic variation}. Aside from the need to handle a \emph{complex martingale}, the argument "follows closely standard methods" from the 1D or $d$-dimensional case. The proof of
\begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
is given there via Lindeberg's CLT, with a \emph{sketch} for
\begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\C(0, 1)\quad \text{ in law}.
  \end{equation}

2. Regime $\Re(\Phi_1)>1/2$ (Next): use the "known connection between ERW and P\'olya urns" and also the connection between P\'olya urns and \emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \emph{finite number of colors}, the cited results "cannot be applied immediately"; however, because the focus is on "a single martingale" of the process, "the analysis can be simplified". Concretely, "borrowing arguments from the study of the additive martingales of branching random walks at complex parameters" allows one to conclude
\begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\erm^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \erm^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
this part uses "Poissonization methods and the analysis of a branching process."

\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying
\begin{equation}
  \label{eqn:Nunidimensional}
  \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
and $(S_n, n \geq 1)$ the ERW defined by
\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \erm^{i \theta_n}.
\end{equation}
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\mathbb{C}(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\mathbb{C}$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\mathrm{e}^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \mathrm{e}^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize},

Question:
Let $(\theta_n)_{n\ge 1}$ be i.i.d. random variables taking values in $[0,2\pi)$, and for $k\ge 1$ set $\Phi_k:=\mathbb E(e^{ik\theta_1})$. Assume
\[
\mathbb P(\theta_1\in\{0,\pi\})<1 \qquad\text{(equivalently, }\Re(\Phi_2)<1\text{).}
\]
Let $(I_n)_{n\ge 2}$ be independent random variables, independent of $(\theta_n)$, with $I_n$ uniformly distributed on $\{1,\dots,n-1\}$. Define the planar elephant random walk $(S_n)_{n\ge 0}$ in $\mathbb C$ by
\[
S_0=0,\qquad S_1=1,\qquad S_n=S_{n-1}+(S_{I_n}-S_{I_n-1})e^{i\theta_n}\quad(n\ge 2).
\]
Here $\mathcal N_{\mathbb C}(0,\sigma^2)$ denotes the centered complex normal law for which the real and imaginary parts are independent real Gaussian random variables with variance $\sigma^2/2$. Which statement holds for this process?

The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)<\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}\in(\tfrac13,\infty)$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}\in(1,\infty)$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]

Choices:
B.
The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)\le\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]
C.
If $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W.
\]
D.
The asymptotic behavior is determined by $|\Phi_1|$ as follows: (i) if $|\Phi_1|<\tfrac12$, then with $\sigma^2=\frac{1}{1-2|\Phi_1|}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $|\Phi_1|=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $|\Phi_1|>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2|\Phi_1|-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]
E.
The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)<\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $\Re(\Phi_1)>\tfrac12$, then there exists a deterministic constant $w\in\mathbb C$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} w
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}w}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]

drops the fluctuation CLT and variance formula in the superdiffusive regime

dependence on Re(Phi_1) replaced by dependence on |Phi_1|

random limit W replaced by deterministic constant


\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


\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