In this paper we are interested in the following well-known conjecture of Hindman.

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
    Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}.  The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors  have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations.  Our main result is the following, which was previously only known in the case $k=1.$

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
 Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}. The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations. Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}. The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{theorem}\label{t2}
 For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

\begin{theorem}\label{rational.moreira}
 Let $P\subseteq\pol$ be finite. For any finite coloring of $\mathbb{N}$ there exist $x,y\in\mathbb{N}$ and a color class $C$ such that $\{x,xy,x+p(y):p\in P\}\subseteq C$.
\end{theorem}

\begin{prop}\label{central*}
 For any finite set of polynomials $P\subseteq\pol$ and any finite coloring of $\mathbb{N}$ the set $$\{y: \exists x \textnormal{ with } \{x,xy,x+p(y):p\in P\} \textnormal{ monochromatic} \} $$

\begin{proof}[Second proof of Corollary \ref{all_syndetic}]
 Fix a finite set $P\subseteq \pol.$ By definition, for each $i$ there is a finite set $F_i\subset \mathbb{N}$ such that $S_i/F_i\supseteq \mathbb{N}.$ Let $k=\max_{i\leq r}|F_i|$ and consider the coloring $c:\mathbb{N}\rightarrow [k]^r$ where an $n\in\mathbb{N}$ is colored based on the $r$-tuple listing the minimal $f_i\in F_i$ such that $f_in\in S_i.$ Let $P'=\{p/f: p\in P, f\in \bigcup_{i\leq r}F_i\}$. By Theorem \ref{rational.moreira}, there are integers $x',y$ such that $S=\{x', x'y, x'+p'(y): p'\in P'\}$ is monochromatic according to the coloring $c.$ Now if $y\in S_i$, by the definition of $c$ there is an $f_i$ such that $f_iS\subseteq S_i.$ Consequently, setting $x=x'f_i$ is as desired.
\end{proof}

\begin{lemma}\label{all thick}
 Suppose $k\in\mathbb{N}$ and $\mathbb{N}=R\cup B$ with both $R$ and $B$ multiplicatively thick. Then one of the colors contains a set $\{x,y,xy,x+iy:i\leq k\}$.
\end{lemma}
\begin{proof}
Suppose for the sake of contradiction that the lemma is false.
 Without loss of generality assume that $R$ is additively piecewise syndetic. Choose $N\in\mathbb{N}$ large enough depending on $k$.

\begin{lemma}\label{log reduction}
 If any finite coloring of $\mathbb{N}$ contains a monochromatic set $\{x,y, x2^y, x+iy: i\leq k\}$ then any finite coloring contains a monochromatic set of the form $\{x,y,x^y,xy^i: i\leq k\}.$
\end{lemma}

\begin{theorem}\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.
\end{theorem}

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations.  Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}.  The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{enumerate}
    \item First, show that the result holds when the color of the step size $y$ is ignored.  In this case, we are interested in finding monochromatic sets $\{x,xy,x+iy: i\leq k\},$ which can be done either through Moreira's work \cite{moreira2017monochromatic} or the first author's \cite{alweiss2022monochromatic}.

    \item Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.

    \item Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well. \end{enumerate}

In order to illustrate this technique we give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.  These are presented in the two subsections of Section \ref{section +}.

We also show that the same basic approach can be adapted to prove the following exponential version of the result.

\begin{theorem}\label{t2}
    For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

The $k=1$ case of this result was proven by Sahasrabudhe \cite{sahasrabudhe2018monochromatic}, even for arbitrary finite colorings.  However, the result seems to be new for $k>1.$  Along the way, we also prove that any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,x^y,xy^i: i\leq k\},$ which gives a new exponential version of Moreira's theorem.

The authors state that “the basic strategy used in the proof of Theorem \ref{t1} … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

The authors state that “the basic strategy used in the proof of the main theorem … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of the main theorem based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.,

Question: Let $k\in\mathbb{N}$, and let $\mathbb{N}$ be colored with two colors (equivalently, let $c:\mathbb{N}\to\{1,2\}$ be any 2-coloring). Which statement holds?

There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,y,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.

Choices:
B. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.
C. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x+y,\, x+2y,\dots,x+ky\}$ have the same color.
D. For every color class $C$ of the 2-coloring, there exist $x,y\in\mathbb{N}$ such that $\{x,\, y,\, xy,\, x+iy: i\le k\}\subseteq C$.
E. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x^y,\, xy^i: i\le k\}$ have the same color; equivalently, $\{x,y,x^y,xy,xy^2,\dots,xy^k\}$ is monochromatic.

The stem only gives the coloring setup and asks which conclusion is valid; it does not state or strongly hint at the specific monochromatic configuration in the correct option.

This is not a direct restatement in the stem. The item asks the learner to distinguish the actual theorem-strength conclusion from weaker, stronger, and altered alternatives.

Some reasoning is required to compare quantifiers and structural differences among the options, especially against the weaker true statement and the overly strong version. However, the item still mainly tests theorem recognition rather than substantial generative derivation.

The distractors are mathematically meaningful and varied: one is a weaker true statement, one improperly strengthens the quantifier, one omits a needed term, and one introduces an unrelated exponential pattern. These reflect realistic failure modes.

A strong MCQ overall: it avoids answer leakage, uses plausible and diagnostically useful distractors, and is not tautological, though it leans more toward theorem identification than deep generative reasoning.


\begin{thm}\label{thm: main}
Consider \eqref{eq: main} with random initial data \eqref{eq: rmain}. Let $c_{n}$ in \eqref{eq: rmain} be $\langle n\rangle^{-(\frac{1}{2}+\theta)},\theta>0$. Let $T_{\epsilon}\approx \epsilon^{-1}$. One has the following large deviation principle
\begin{equation}\label{eq: estimatemain}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon\ln \mathbb{P}(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}|c_{n}|^{2}}.
\end{equation}
\end{thm}

\subsection{Statement of results}
Consider cubic NLS on 1d torus $\mathbb{T}$,
\begin{equation}\label{eq: main}
\begin{cases}
iu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation}
here $\epsilon>0$ is a parameter.

We will focus on the random initial data of type
\begin{equation}\label{eq: rmain}
u^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}
where $c_{n}=\<n\>^{-(\frac{1}{2}+\theta)}, \theta>0$, and $g_{n}^{\omega}$ be i.i.d standard complex Gaussian.\\

Note that $u^{\omega}_{0}\in L_{x}^{2}$ almost surely, and it is well known that all $L^{2}$ initial data generates a global flow, \cite{bourgain1993fourier}. Our main result is

\begin{equation}\label{eq: main}
\begin{cases}
iu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation}

We will focus on the random initial data of type
\begin{equation}\label{eq: rmain}
u^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}
where $c_{n}=\<n\>^{-(\frac{1}{2}+\theta)}, \theta>0$, and $g_{n}^{\omega}$ be i.i.d standard complex Gaussian.\\

Both \cite{garrido2023large}, \cite{liang2025random}, and this work actually work for time scale for $c\epsilon^{-1}|\ln \epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \cite{liang2025random}, they state the time scale as $O(\epsilon^{-1}|\ln \epsilon|)$, however, it seems hard to cover time scale for $C\epsilon^{-1}|\ln \epsilon|$ for $C$ large, see the line below (3.30) in \cite{liang2025random}.
\end{rem}

\begin{lem}\label{lem: modlinearldp}
Let $z_{0}>0$, $T_{\epsilon}=O(\epsilon^{-1})$ and $u_{app}$ be defined as \eqref{eq: uapp}, one has the large deviation principle,
\begin{equation}\label{eq: modlinearldp}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon \ln \mathbb{P}(\|u_{app}\|_{L_{t,x}^{\infty}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}c_{n}^{2}}
\end{equation}

and
\begin{lem}\label{lem: errorcontrol}
Let $u_{0}$ be as in \eqref{eq: u0}, and $u_{app}$ be as in \eqref{eq: uapp}. Let $u$ be the associated solution to \eqref{eq: main} with initial data $u_{0}$. Let $\theta>0$ be fixed. Let $T_{\epsilon}=O(\epsilon^{-1})$. Let $C_{0}\gg 1$. Then there exists $s>\frac{1}{2}$, and $\delta_{1}>0$, so that
\begin{equation}\label{eq: errorcontrol}
-\ln\mathbb{P}(\|u-u_{app}\|_{L_{t}^{\infty}H^{s}([0,T_{\epsilon}]\times \mathbb{T})}\geq \epsilon^{-\frac{1}{2}+\delta_{1}})\geq C_{0}\epsilon^{-1}
\end{equation}

By \cite[Lemma 4.2]{garrido2023large} or \cite[Lemma 4.2]{Tzvetkov2017quasi}, $\eta$ is a standard complex Gaussian if and only $\eta e^{it|\eta|^{2}}$ is a standard complex Gaussian. Our proof for Lemma \ref{lem: modlinearldp} actually yields to the same result\footnote{We will not use this result in the proof, and strictly speaking those two results, though same, are not equivalent, but we feel it is good to record this estimate for linear flow here.} for the linear free flow.
\begin{lem}\label{lem: ldpforlinearflow}
 Let $z_{0}>0$, $T_{\epsilon}=O(\epsilon^{-1})$ and $u_{0}$ be as in \eqref{eq: u0}, one has the large deviation principle
\begin{equation}\label{eq: linearldp}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon \ln \mathbb{P}(\|e^{it\Delta}u_{0}\|_{L_{t,x}^{\infty}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}c_{n}^{2}}
\end{equation}
\end{lem}

In this case, we we reduce \eqref{eq: crucialreduce} to proving that for all $\omega \in F_\epsilon - G_\epsilon$, one has
 \begin{equation}\label{eq: I-I-II}
 \sup_{0\leq \tau\leq t} \sum_{R_{n}(n_1,n_2,n_3)} \frac{|g_{n_1}g_{n_2}h_{n_3}|}{\<n_1\>^{\frac{1}{2}+\theta }\<n_2\>^{\frac{1}{2}+\theta }}
 \lesssim \<n\>^{-\theta + \delta_5} \epsilon ^{-3/2-3\delta_1}.
 \end{equation}
 Using Cauchy-Schwarz inequality and \eqref{eq: gepsic}, for all $\omega \in F_\epsilon - G_\epsilon$, we have
 \begin{equation}\label{eq: uniformI-I-II}
 \begin{aligned}
 &\sup_{0\leq \tau\leq t} \sum_{R_{n}(n_1,n_2,n_3)} \frac{|g_{n_1}g_{n_2}h_{n_3}|}{\<n_1\>^{\frac{1}{2}+\theta }\<n_2\>^{\frac{1}{2}+\theta }}\\
 &\lesssim \|h\|_{C_t[0,T]H^s} \left(\sum_{n_3} \<n_3\>^{-2s} \left|\sum_{R_{n_3,n}(n_1,n_2)} \frac{\epsilon ^{-1-2\delta_1 } }{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}\right|^2\right)^\frac{1}{2}\\
 &\lesssim \epsilon ^{-3/2 -3\delta_1 } \left(\sum_{n_3} \<n_3\>^{-2s} \left|\sum_{R_{n_3,n}(n_1,n_2)} \frac{1}{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}\right|^2\right)^\frac{1}{2},
 \end{aligned}
 \end{equation}
 where for fixed $n,n_3$, we denote $R_{n_3,n}(n_1,n_2) := \{(n_1,n_2)\in\mathbb{Z}^2:n=n_1-n_2+n_3,n_2\neq n_1,n_3\}$.
 Next, we aim to reduce \eqref{eq: I-I-II} to following
 \begin{equation}\label{eq: mainreductionI-I-II}
 \sum_{R_{n_3,n}(n_1,n_2)} \frac{1}{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}
 \lesssim \left\langle n-n_3 \right\rangle ^{-2\theta +2\delta_5} .
 \end{equation}
 If \eqref{eq: mainreductionI-I-II} holds, noting that
 \begin{equation*}
 \sum_{|n_3|\ll |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_3|\gg |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \sim \sum_{|n_3|\gg |n|} \<n_3\>^{-2s-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_3|\sim |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \sim \<n\>^{-2s} \sum_{|n_3|\sim |n|} \<n_3-n\>^{-4\theta +4\delta_5}
 \lesssim \<n\>^{1-2s-4\theta +4\delta_5},
 \end{equation*}
 then we have
 \begin{equation*}
 \sum_{n_3} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5},
 \end{equation*}
 which, along with \eqref{eq: mainreductionI-I-II} and \eqref{eq: uniformI-I-II}, yields \eqref{eq: I-I-II}. So we reduce \eqref{eq: I-I-II} to \eqref{eq: mainreductionI-I-II} if we take \fbox{$\delta_5< \theta $}. Similarly,
 noting that
 \begin{equation*}
 \sum_{|n_1|\ll |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \lesssim \left\langle n-n_3\right\rangle ^{\delta_4-\frac{1}{2}-\theta} \sum_{|n_1|\ll |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta }
 \lesssim \left\langle n-n_3\right\rangle ^{2\delta_4-2\theta}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_1|\gg |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \sim \sum_{|n_1|\gg |n-n_3|} \<n_1\>^{2\delta_4-1-2\theta }
 \lesssim \<n-n_3\>^{-2\theta +2\delta_4}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_1|\sim |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \sim \left\langle n-n_3\right\rangle ^{\delta_4-\frac{1}{2}-\theta} \sum_{|n_1|\sim |n-n_3|} \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \lesssim \<n-n_3\>^{-2\theta +2\delta_4} ,
 \end{equation*}
 then we have \eqref{eq: mainreductionI-I-II}
 if we take \fbox{$\delta_5\leq \delta_4$}.

\begin{rem}
Upon finishing of this work, we notice a recent preprint of Liang-Wang, \cite{liang2025random},  which handles Theorem \ref{thm: main} in the case $\theta>\frac{1}{2}$. It should be noted that when $\theta>\frac{1}{2}$, the Fourier coefficient is in $l^{1}$, which forms an algebra. We handle all $\theta>0$.\\

Both \cite{garrido2023large}, \cite{liang2025random}, and this work actually work for time scale for $c\epsilon^{-1}|\ln \epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \cite{liang2025random}, they state the time scale as $O(\epsilon^{-1}|\ln \epsilon|)$, however, it seems hard to cover time scale for $C\epsilon^{-1}|\ln \epsilon|$ for $C$ large, see the line below (3.30) in \cite{liang2025random}.
\end{rem}

\subsection{Background and motivation}
This work is inspired by the pioneering work \cite{garrido2023large}. Motivated by rogue wave phenomena, Garrido, Grande, Kurianski and Staffilani study \eqref{eq: main}, with random initial data of form \eqref{eq: rmain}, but for $c_{n}=ae^{-b|n|}$. They essentially
\footnote{They state the result for pointwise in $t\leq T_{\epsilon}\sim \epsilon^{-1}$. We take this chance to cover $\|u\|_{L_{t,x}^{\infty}}$ rather than estimate pointwise in $t$.} obtained \eqref{eq: estimatemain} for such types of random initial data. It has been remarked in \cite[Remark 1.3]{garrido2023large}, \textit{finding optimal family of coefficients (in terms of decay)}, rather than considering exponential decay $c_{n}$, remains an interesting open question.

By combining random data type techniques in \cite{bourgain1994periodic,bourgain1996invariant} and some ideas in \cite{compaan2021pointwise} (which also goes back to Bourgain's 1990s seminal work), we are able to extend the result in \cite{garrido2023large} to coefficients with polynomial decay $\langle n\rangle^{-\frac{1}{2}-\theta}$, $\theta>0$. Such constraint is optimal in the sense when $
\theta\leq 0$, even the initial data will leave $L_{x}^{\infty}$. It remains an interesting question to go beyond the time scale $T_{\epsilon}$. We note that this $\epsilon^{-1}$ time scale (neglecting logarithm corrections) is in some sense critical and we refer to \cite[Remark 1.7]{garrido2023large} for more details.
We note that both current work and \cite{garrido2023large},  can cover time scale of form $c\frac{|\ln \epsilon|}{\epsilon}$, where $c$ is some small universal number.\\

The overall proof scheme of current article follows \cite{garrido2023large}, which contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part, (there is some subtle part here since one needs to first slightly perturb the free linear flow, we will discuss it later).\\

 Our main contributions in this work are
\begin{itemize}
\item We find a rather straightforward proof for the large deviation principle for the linear flow, which simplifies the arguments in \cite{garrido2023large}, and more importantly, extends to random initial data of form $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$, $\forall \theta>0$. Our techniques for this part are crucial for our further analysis in the nonlinear smoothing part.
\item We adapt the $X^{s,b}$ analysis in \cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \cite{garrido2023large}, which gives the desired nonlinear smoothing\footnote{On one hand, we will use the $X^{s,b}$ analysis in the local random data theory as a black box;  on the other hand, the long time analysis we use to get nonlinear smoothing does explicitly use computations of $X^{s,b}$ type, but without going to the explicit form of $X^{s,b}$ analysis.}.
\end{itemize}

We end this session by pointing out ever since the seminal work \cite{bourgain1994periodic,bourgain1996invariant}, random initial data theory in nonlinear dispersive PDEs has become a very active research field. Many researchers contribute to this field and it is impossible to survey the field here. We simply mention that  in the perspective of  regularity in NLS model, the recent breakthrough \cite{deng2022random} gives a quite complete answer. We refer to \cite{deng2022random}, \cite{garrido2023large} and reference therein for more related reference.

\subsection{Notation}
Many terms $g_{n}^{\omega}, u^{\omega}, u_{0}^{\omega}$ are random, and thus depend on $\omega$. For simplicity, we will hide the implicit $\omega$ and simply denote them as $g_{n}, u,u_{0}$.\\

We introduce $\NNN_{1}, \NNN_{2}$ as
\begin{equation}\label{eq: n1}
\NNN_{1}(f_{1},f_{2},f_{3})=\sum_{n_{2}\neq n_{1}, n_{3}}\hat{f}_{1}(n_{1})\overline{\hat{f}_{2}}(n_{2})\hat{f}_{3}(n_{3})e^{i(n_{1}-n_{2}+n_{3})x},
\end{equation}
\begin{equation}\label{eq: n_{2}}
\NNN_{2}(f_{1},f_{2},f_{3})=\sum_{n}\hat{f}_{1}(n)\overline{\hat{f}_{2}}(n)\hat{f}_{3}(n)e^{inx}
\end{equation}
and one has
\begin{equation}\label{eq: algebra}
|f|^{2}f-2\frac{1}{2\pi}(\int |f|^{2})f=\NNN_{1}(f,f,f)-\NNN_{2}(f,f,f)
\end{equation}
We sometimes short $\NNN_{i}(f,f,f)$ as $\NNN_{i}(f)$.

The proof of Theorem~\ref{thm: main} is described as following the scheme of \cite{garrido2023large}: it “contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in \cite{garrido2023large}” and extends to initial data $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$ for all $\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in \cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \cite{garrido2023large}, which gives the desired nonlinear smoothing” for the Duhamel part.

The proof of the main theorem is described as following the scheme of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,”” and extends to initial data $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$ for all $\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in Bourgain, “Periodic nonlinear Schrödinger equation and invariant measures,” and Bourgain, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation,” in the setting of long time analysis of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” which gives the desired nonlinear smoothing” for the Duhamel part.,

\label{thm: main}
Consider \begin{equation}\label{eq: main}
\begin{cases}\niu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation} with random initial data \begin{equation}\label{eq: rmain}\nu^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}. Let $c_{n}$ in the equation above be $\langle n\rangle^{-(\frac{1}{2}+\theta)},\theta>0$. Let $T_{\epsilon}\approx \epsilon^{-1}$. One has the following large deviation principle
\begin{equation}\label{eq: estimatemain}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon\ln \mathbb{P}(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}|c_{n}|^{2}}.
\end{equation},

Question:
Let \(\theta>0\), \(\langle n\rangle=(1+|n|^{2})^{1/2}\), and for each \(\epsilon>0\) consider the cubic nonlinear Schr\"odinger equation on the one-dimensional torus \(\mathbb T\):
\[
\begin{cases}
iu_t+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_0^{\omega}(x),
\end{cases}
\]
with random initial data
\[
u_0^{\omega}(x)=\sum_{n\in\mathbb Z} c_n g_n^{\omega} e^{inx}, \qquad c_n=\langle n\rangle^{-(\frac12+\theta)},
\]
where \(\{g_n^{\omega}\}_{n\in\mathbb Z}\) are i.i.d. standard complex Gaussian random variables. Let \(u^{\omega}\) denote the corresponding solution, and let \(T_{\epsilon}\) be comparable to \(\epsilon^{-1}\). For a fixed \(z_0>0\), which large-deviation estimate holds for
\(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}\) as \(\epsilon\to 0^+\)?

\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]

Choices:
B.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sup_{n\in\mathbb Z}|c_n|^2}.
\]
C.
\[
\limsup_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
\le -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]
D.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{2\sum_{n\in\mathbb Z}|c_n|^2}.
\]
E.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\Big)
= -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]

Gaussian variance from full \ell^2 mass \sum_n |c_n|^2

drop the full limit identity to the corresponding upper large-deviation bound

exact rate constant inherited from the complex Gaussian tail

critical threshold scaling z_0\epsilon^{-1/2} for the supremum norm

The stem does not explicitly reveal the correct option. It gives the PDE, the random initial data, and the scaling regime, but not the exact large-deviation rate or the precise form of the limit.

This is essentially a direct theorem-recall item: under the theorem's hypotheses, the student is asked to identify the exact conclusion. It mainly tests recognition of the stated result rather than deriving a new consequence.

There is some reasoning pressure because the student must distinguish between an exact limit, a weaker upper bound, an incorrect variance surrogate, and an overextended time-scale claim. However, the item still primarily rewards recall of the theorem's exact statement.

The distractors are strong and mathematically meaningful: a wrong Gaussian constant, a weaker-but-true style statement, confusion between total variance and single-mode size, and an invalid quantifier extension in the time scale. These reflect realistic failure modes.

A technically well-constructed theorem-identification MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a specialized result rather than a genuinely generative reasoning task.


\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\mathbbm{P}[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}

\begin{theorem}\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\P[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}
\end{theorem}

\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\D}] \text{ joining } \varepsilon\S^1 \text{ and } \frac{1}{2}\S^1\}.
\end{equation}

\begin{aligned}
F(\varepsilon)&\le \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n\\
&\le  \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-\frac{\log2}{|\log q_0|}(\log\log|\log\varepsilon|-\log C')}+\frac{C'}{\log|\log\varepsilon|}
\le\frac{C_2}{\log|\log\varepsilon|}
\end{aligned}
$$
for some positive constant $C_2$, as desired.
\end{proof}

\appendix
\section{Proof of Theorem~\ref{thm-cr-mod-relation}}
\label{appendix}

In this appendix, we provide the proof of Theorem~\ref{thm-cr-mod-relation}, which is implicit in~\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\hH:=\{z\in\C:\text{Im}(z)>0\}$, and let $\sm$ be the restriction of the $\SLE_{8/3}$ loop measure on $\C$ to the loops that are contained in $\hH$ and surround $i$. According to the conformal restriction~\cite{werner-loops}, if $\Phi:\D\to\hH$ is a conformal map with $\Phi(0)=i$, then $\sm=\Phi_*\SLE_{8/3,\D}^{\lp}$. Therefore, to prove Theorem~\ref{thm-cr-mod-relation}, it suffices to show the following
\begin{theorem}
\label{thm-appendix}
There is a constant $C>0$ such that for any measurable function $f:\R_+\to\R_+$ and $\lambda\ge0$, we have
$$
\int f\left(\Mod(\hH\backslash\overline{D(\eta)})\right)\psi_\eta'(i)^\lambda\sm(d\eta)=
C\frac{\sqrt{12\lambda-1}}{\sinh\left(\frac{\pi}{3}\sqrt{12\lambda-1}\right)}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\eta(2i\tau)f(\tau)d \tau.
$$
Here $\psi_\eta$ is the conformal map from $\hH$ to $D(\eta)$ such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$.
\end{theorem}

The proof of Theorem~\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\sqrt{\frac{8}{3}}$. In the following we fix $\gamma=\sqrt{\frac{8}{3}}$ and let $Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Let $\P_{\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\hH$ with mean zero on the upper semi-circle $\S^1\cap\hH$.

\begin{definition}[Liouville field on $\hH$]
\label{def-LF-H}
Let $(h,c)$ be sampled from $\P_{\hH}\times[e^{-Qc}dc]$, and let $\phi(z)=h(z)-2Q\log|z|_++c$ (here $|z|_+:=\max\{|z|,1\}$). Denote the law of $\phi$ by $\LF_{\hH}$.

For $\alpha\in\R$, let $\LF_{\hH}^{(\alpha,i)}(d\phi):=\lim_{\varepsilon\to0}\varepsilon^{\frac{\alpha^2}{2}}e^{\alpha\phi_\varepsilon(i)}\LF_{\hH}(d\phi)$ (the limit exists in the vague topology, see~\cite[Lemma 2.2]{ARS-FZZ}). Here $\phi_\varepsilon(i)$ is the average of $\phi$ over the circle $\partial B_\varepsilon(i)$.
\end{definition}

For a sample $\phi$ from $\LF_{\hH}^{(\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\nu_{\phi}:=\lim\limits_{\varepsilon\to0}\varepsilon^{\frac{\gamma^2}{4}}e^{\frac{\gamma}{2}\phi_\varepsilon(x)}dx$, where $\phi_\varepsilon(x)$ is the average of $\phi$ over the semi-circle $\partial B_\varepsilon(x)\cap\hH$.
Then denote $\{\LF_{\hH}^{(\alpha,i)}(\ell)\}_{\ell>0}$ to be the disintegration of $\LF_{\hH}^{(\alpha,i)}$ over $\nu_\phi(\R)$; i.e.~$\LF_{\hH}^{(\alpha,i)}=\int_0^\infty\LF_{\hH}^{(\alpha,i)}(\ell)d\ell$. We refer readers to see e.g.~\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\cite[Lemma 2.7]{ARS-FZZ}, for $\alpha>\frac{\gamma}{2}$,
there are constants $C_{\alpha,\gamma}>0$ such that $|\LF_{\hH}^{(\alpha,i)}(\ell)|=C_{\alpha,\gamma}\ell^{\frac{2}{\gamma}(\alpha-Q)-1}$ for any $\ell>0$.

We also need the Liouville field on the annulus. For $\tau>0$, let $\mathcal{C}_\tau=[0,\tau]\times[0,1]/\mathord\sim$ be the horizontal cylinder with modulus $\tau$, where under $\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\in[0,\tau]$. Let $\P_{\tau}$ be the free boundary GFF on $\mathcal{C}_\tau$
with zero mean on the circle $\{\frac{\tau}{2}\}\times[0,1]/\mathord\sim$.
\begin{definition}[{\cite[Definition 2.2]{ARS-Annulus}}]
\label{def-LF-annulus}
Let $(h,c)$ be sampled from $\P_\tau\times dc$. Then denote the law of $\phi=h+c$ by $\LF_\tau$ (which is an infinite measure on $H^{-1}(\mathcal{C}_\tau)$).
\end{definition}

For a sample $\phi$ from $\LF_\tau$,
we can similarly define its boundary GMC measures $\nu^1_\phi, \nu_\phi^2$ on the two boundaries $\partial_1\mathcal{C}_\tau=\{0\}\times[0,1]\mathord\sim$ and $\partial_2\mathcal{C}_\tau=\{1\}\times[0,1]\mathord\sim$, respectively. We also let $\{\LF_\tau(\ell_1,\ell_2)\}_{\ell_1,\ell_2>0}$ be the disintegration of $\LF_\tau$ over $\nu_\phi^1(\partial\cC_1)$ and $\nu_\phi^2(\partial\cC_2)$, i.e.~$\LF_\tau=\iint_{\R_+^2}\LF_\tau(\ell_1,\ell_2)d\ell_1d\ell_2$. The following result from~\cite{ARS-Annulus} gives the exact solvability of $|\LF_\tau(\ell_1,\ell_2)|$, which is based on~\cite{wu-annulus}.

\begin{proposition}[{\cite[Equation (3.6)]{ARS-Annulus}}]
 \label{prop-LF-integrability}
For $\tau>0$ and $y\in(-1,\frac{4}{\gamma^2})$, we have
$$
\iint_{\R_+^2}\ell_1e^{-\ell_1}\ell_2^{y}|\LF_\tau(\ell_1,\ell_2)|d\ell_1d\ell_2=\frac{\pi\gamma y\Gamma(1+y)}{2\sin(\frac{\gamma^2}{4}\pi y)}e^{\frac{\pi}{4}\gamma^2\tau y^2}.
$$
\end{proposition}

For two domains $D,\wt{D}\subset\C$ and a conformal map $g:D\to\wt{D}$, when $h$ is a distribution on $D$, we define $g\bullet h:=h\circ g^{-1}+Q\log|(g^{-1})'|$ (which is a distribution on $\wt D$). Let $\Omega$ be the space of simple loops in $\hH$ surrounding $i$, and $\Conf(\hH,i)$ be the group of conformal automorphisms of $\hH$ that fix $i$. For $(\phi,\eta,g,\theta)\in H^{-1}(\hH)\times\Omega\times \Conf(\hH,i)\times[0,1]$, define a measurable map $F$ by
$$
F(\phi,\eta,g,\theta):=\left((g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}, f_{\eta}^\theta\bullet \phi|_{\hH\backslash\overline{D(\eta)}},\Mod(\hH\backslash\overline{D(\eta)})\right).
$$
Here $\psi_\eta:\hH\to D(\eta)$ is the conformal map such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$, and
$f_{\eta}^\theta:\hH\backslash\overline{D(\eta)}\to\mathcal{C}_\tau$ is the conformal map such that $f_{\eta}^\theta(0)=\theta i$ with $\tau=\Mod(\hH\backslash\overline{D(\eta)})$.

Let $\Haar_{(\hH,i)}$ be the Haar measure on $\Conf(\hH,i)$ such that $|\Haar_{(\hH,i)}|=1$, and let $\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.
The following proposition, which is essentially from~\cite{ARS-Annulus}, describes the law of $\LF_\hH^{(\gamma,i)}$ when cut by a simple loop sampled from $\sm$.

\begin{proposition}\label{prop-sle-weld}
There is a constant $C>0$ such that for any $\ell_1>0$,
$$
F_*\left(\LF_{\hH}^{(\gamma,i)}(\ell_1)\times\sm\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
Here $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}
\begin{proof}
By~\cite{ARS-Annulus}, the $\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\hH$ gives the Liouville field on $\hH$~\cite[Theorem 3.4]{ARS-FZZ}.
\end{proof}

For $\alpha\in\R$, let $\Delta_\alpha=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$. Define the measure $\sm^\alpha$ via $\frac{d\sm^\alpha}{d\sm}(\eta)=\psi_\eta'(i)^{2\Delta_\alpha-2}$, where $\psi_\eta:\hH\to D(\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}.

\begin{proposition}
 \label{prop-sle-reweight}
There is a constant $C>0$ such that for any $\ell_1>0$ and $\alpha\in\R$,
$$
F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_1)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\alpha,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
As in Proposition~\ref{prop-sle-weld}, we view the right side above as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}

\begin{proof}
For $\phi\in H^{-1}(\hH)$, $\eta\in\Omega$ and $g\in\Conf(\hH,i)$, let $\phi_1=(g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}$. Recall that for $\varepsilon>0$, $(\phi_1)_\varepsilon(i)$ is the average of $\phi_1$ over $\partial B_\varepsilon(i)$.
By~\cite[Lemma 4.8]{ARS-FZZ},
as $\varepsilon\to0$, we have
$$
\begin{aligned}
&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi_1)_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
=~&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi\circ\psi_\eta\circ g^{-1})_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times|\psi_\eta'(i)|^{(\alpha-\gamma)Q}\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
\to~& F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_2)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right).
\end{aligned}

\label{section-intro}
\subsection{Overview and the main result}

There is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\ge 1$, the Brownian intersection exponent $\zeta_n=\frac{4n^2-1}{12}$~\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\cite{lsw-restriction}: both of the \emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\kappa=\frac{8}{3}$.

Note that in the critical percolation, one can also define the \emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.

To be precise, let $(B_t)_{t\ge0}$ be a planar Brownian motion starting from $0$. Denote $\mathbbm{S}^1:=\{z\in\mathbbm{C}:|z|=1\}$ and $\mathbbm{D}:=\{z\in\mathbbm{C}:|z|<1\}$ to be the unit circle and the unit disk, respectively. Let $\tau_\mathbbm{D}$ be the first hitting time of $\mathbbm{S}^1$ for $(B_t)_{t\ge0}$.
For each $\varepsilon\in(0,\frac{1}{2})$, consider the event
\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\mathbbm{D}}] \text{ joining } \varepsilon\mathbbm{S}^1 \text{ and } \frac{1}{2}\mathbbm{S}^1\}.
\end{equation}
Namely, $\Bac_\varepsilon$ happens if there are two continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbbm{C}$ such that $\gamma^1[0,1]\cap\gamma^2[0,1]=\emptyset$, $\gamma^i[0,1]\subset B[0,\tau_{\mathbbm{D}}]$, $\gamma^i(0)\in\varepsilon\mathbbm{S}^1$, and $\gamma^i(1)\in\frac{1}{2}\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\Bac_\varepsilon$ the \emph{backbone event} for the Brownian trajectory $B[0,\tau_\mathbbm{D}]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\mathbbm{P}[\Bac_\varepsilon]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\P[\Bac_\varepsilon]$.

Let $f$ be the conformal map from $D(\ell_1)$ to $\D$ with $f(0)=0$, $f'(0)>0$ and let $\rho=f(\wt\ell_1)$. Note that $\CR(\rho,0)=\frac{\CR(\wt\ell_1,0)}{\CR(\ell_1,0)}$.
The main result of this section is the following exact law of $\CR(\rho,0)$, which is crucial to the final proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.
\begin{proposition}
\label{prop-cr-rho}
For $\lambda\ge0$, we have
\begin{equation}\label{eq:cr-rho}
\E[\CR(\rho,0)^\lambda]=\frac{\sinh(\frac{\pi}{2}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}}-\frac{2\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right)}.
\end{equation}
When $\lambda\in[0,\frac{1}{12})$, the right side of~\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\ref{section-intro}.
\end{proposition}

Let $\nu_\D$ be the law of the loop chosen from the counting measure over $\{\ell_i\}_{i\ge1}$; namely, $\nu_\D$ is such that $\int F(\eta)\nu_\D(d\eta)=\P\left[\sum_{i\ge1}F(\ell_i)\right]$ for any bounded measurable function $F$. The following result, based on~\cite{cfsx} and Theorem~\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\nu_\D$.
\begin{lemma}\label{lem-CR-BM-counting-ell}
There is a constant $C>0$ such that for $\lambda\ge0$, we have
\begin{equation}
 \label{eq-CR-BM-counting-ell}
 \int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C\frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\pi\sqrt{12\lambda-1})}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right).
 \end{equation}
When $\lambda\in(0,\frac{1}{12})$, the right side of~\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.
\end{lemma}
\begin{proof}
By combining~\cite[Lemma 5.3]{cfsx} with~\cite[Theorem 1.3]{cfsx}, we have $\frac{d\nu_{\D}}{d\SLE^{\lp}_{8/3,\D}}(\eta)=\frac{C_1}{\Mod(\eta,\S^1)}$ for some constant $C_1>0$.
Then by Theorem~\ref{thm-cr-mod-relation}, we find
\begin{equation}\label{eq:cr-integral}
\int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C
 \frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\frac{\eta(2i\tau)}{\tau}d\tau.
\end{equation}
for some constant $C>0$.
Note that for $a>-\frac{\pi}{6}$, we have $\int_0^\infty e^{-a\tau}\eta(2i\tau)d\tau=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}$ (see e.g.~\cite[Equation (A.4)]{ARS-Annulus}) and
 $$
 \frac{d}{da}\left[-\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{(2-\sqrt{3})^2+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right)\right]=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}.
 $$
Therefore, we obtain
 $$
 \int_0^\infty e^{-a\tau}\frac{\eta(2i\tau)}{\tau}d\tau=\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right).
 $$
for $a>-\frac{\pi}{6}$. Combined with the right side of~\eqref{eq:cr-integral}, we conclude.
\end{proof}

The upper bound of $\P[\mathsf{Bac}_\varepsilon]$ then relies on the following iterative inequality.
\begin{proposition}
 \label{prop-upper-bound-backbone}
 Let $q_0=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
 there is a constant $C_0>0$ such that for any $\varepsilon\in(0,10^{-4})$,
 $$
 \P[\Bac_\varepsilon]\le \frac{C_0}{\log|\log\varepsilon|}+q_0\P[\Bac_{5\sqrt{\varepsilon}}].
 $$
\end{proposition}
\begin{proof}
Note that
 \begin{align}
 \P[\Bac_\varepsilon]&\le\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &\le \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &= \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset, \Bac_\varepsilon]\label{eq:upper-1}
 \end{align}
Here the last line follows from that $(\wt\ell_1\cap(5\sqrt{\varepsilon}\D\cup\A_{1/2})=\emptyset)\cap\Bac_\varepsilon=\emptyset$.
By Lemma~\ref{lem-CR-BM-ell1}, Corollary~\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that
\begin{align}
\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]&\le\P[\CR(\ell_1,0)<20\sqrt{\varepsilon}]+\P[\CR(\wt\ell_1,0)<20\sqrt{\varepsilon}]\nonumber\\
 & \le C\varepsilon^{\frac{1}{8}}+C'\frac{1}{\log|\log\varepsilon|}\le \frac{C_0}{\log|\log\varepsilon|}.\label{eq:upper-2}
\end{align}

Let $q_0:=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
\begin{equation}\label{eq:upper-3}
\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset,\Bac_\varepsilon]\le q_0\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset].
\end{equation}
Let $g:D(\wt\ell_1)\to\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.
By Lemma~\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\in[0,s_1]}$ is independent of $B[s_1,\tau_\D]$, and has the same law (up to a time change) as $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$. Hence, if we denote $\wt\P$ to be the law of $(g(B_t))_{t\in[0,s_1]}$ conditioned on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$, then $\wt\P$ is still the same as (up to a time change) the law of $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$.
By Lemma~\ref{lem-variant-Koebe-1/4},
on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$,
we have $\overline{g(\varepsilon\D)}\subset5\sqrt{\varepsilon}\D$ and
$g(\frac{1}{2}\D\cap D(\wt\ell_1))\supset \frac{1}{2}\D$. Then we find
\begin{equation}\label{eq:upper-4}
\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset]\le \wt\P[\Bac_{5\sqrt{\varepsilon}}]=\P[\Bac_{5\sqrt{\varepsilon}}],
\end{equation}
where the last equality follows from the rotational invariance of $B[0,\tau_\D]$. The result then follows from combining~\eqref{eq:upper-1},~\eqref{eq:upper-2},~\eqref{eq:upper-3} and~\eqref{eq:upper-4}.
\end{proof}

Now we prove the upper bound for $\P[\Bac_\varepsilon]$, finishing the proof of Theorem~\ref{thm:backbone}.
\begin{proof}[Proof of Theorem~\ref{thm:backbone}, the upper bound]
Let $\varepsilon\in(0,10^{-4})$ and $F(\varepsilon):=\P[\Bac_\varepsilon]$. Note that the function $x\mapsto\frac{1}{\log|\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\frac{1}{\log|\log(25x)|}\le\frac{C}{\log|\log x|}$ for any $x\in(0,10^{-4})$. Then for each $n\in\mathbbm{Z}_+$ with $25\varepsilon^{\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\ref{prop-upper-bound-backbone} for $n$ times, we find
\begin{align}
 F(\varepsilon)
 &\le C_0\sum\limits_{k=0}^{n-1}\frac{q_0^k}{\log|\log(5^{2-\frac{1}{2^{k-1}}}\varepsilon^{\frac{1}{2^k}})|}+q_0^nF(5^{2-\frac{1}{2^{n-1}}}\varepsilon^{\frac{1}{2^n}})\nonumber\\
 &\le \frac{C_0}{1-q_0}\frac{1}{\log|\log(25\varepsilon^{\frac{1}{2^n}})|}+q_0^n\le\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon^{\frac{1}{2^n}})|}+q_0^n\nonumber\\
 &=\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n.\label{eq-proof-upper-bound-backbone}
\end{align}

\begin{remark}
The same result also holds when $\frac{1}{2}\mathbbm{S}^1$ in~\eqref{eq:def-backbone} is changed to any fixed $r\mathbbm{S}^1$ with $r\in(0,1)$ (and for sufficiently small $\varepsilon$), except that the corresponding constants in~\eqref{eq:bb-proba} will depend on $r$.
\end{remark}

Note that the backbone event $\Bac_\varepsilon$ is closely related to the spatial distribution of the \emph{cut points} of $B[0,\tau_\mathbbm{D}]$. Indeed, our proof of Theorem~\ref{thm:backbone} is based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}. We mention that~\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\ref{thm:backbone} without relying on LQG.

Theorem~\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let
\[
\mathcal{B}:=\{B_s:s\in[0,\tau_\mathbbm{D}) \text{ and there exists } \varepsilon>0 \text{ such that } (B_{s+u})_{0\le u\le\varepsilon} \text{ does not have a cut point}\}.
\]
Then Theorem~\ref{thm:backbone} indicates that $\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\mathcal{B}$ with the gauge $r\mapsto r^2\log\frac{1}{r}\log\log\frac{1}{r}\log\log\log\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\tau_\mathbbm{D}]$ itself is $r\mapsto r^2\log\frac{1}{r}\log\log\log\frac{1}{r}$, see e.g.~\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\cite{BL}, we know that $\mathcal{B}$ contains the set of double points of $(B_t)_{0\le t\le\tau_\mathbbm{D}}$, which has the Hausdorff gauge $r\mapsto r^2(\log\frac{1}{r}\log\log\log\frac{1}{r})^2$~\cite{LG87}.

\subsection{Outlook and discussions}
\label{section-discussion}

Here we give several remarks and related questions before going into the proof.

\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}

\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\tau_\mathbbm{D}]$ joining $\varepsilon\mathbbm{S}^1$ and $\frac{1}{2}\mathbbm{S}^1$ with $n\ge2$. Note that for $n\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\tau_\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\tau_\mathbbm{D}]$, $0$ and $B_{\tau_\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\ge3$, see~\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.

\item
There is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\ge0}$ be the Brownian excursion on the upper half plane $\mathbbm{H}$ from $0$ to $\infty$, and define $R\mathbbm{S}^+:=R\mathbbm{S}^1\cap\mathbbm{H}$ for $R>0$. Then for $n\ge1$, consider the asymptotic probability that
there exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\mathbbm{S}^+$ and $R\mathbbm{S}^+$ as $R\to\infty$.
Note that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\cite[Theorem 4]{brownian-beads}.

\item In our forthcoming work~\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\le t\le\tau_\mathbbm{D}}$ be as before, and let $\mathcal{L}$ be an independent Brownian loop soup on $\mathbbm{D}$ with intensity $\frac{c}{2}$ for $c\in(0,1]$.
For a subset $A\subset\ol\mathbbm{D}$, denote $\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\mathcal{L}$ intersecting with $A$,
and let $H:=\mathcal{C}(B[0,\tau_\mathbbm{D}])$.
However, unlike the Brownian motion case,
there is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$
connected to $\frac{1}{2}\mathbbm{S}^1$. Moreover, in~\cite{cx} we will show the following
\begin{theorem}
We say $t\in(0,\tau_\mathbbm{D}]$ is a cut time of $H$ if $\mathcal{C}(B[0,t])\cap \mathcal{C}(B[t,\tau_\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$.
Then for $c\in(0,1)$,
\begin{equation}\label{eq:p_e}
\mathbbm{P}[E]=2^{\frac{c+1}{2}}\sqrt{\frac{6}{1-c}}\left(\int_0^\infty\tau^{-1-\frac{c}{2}}e^{\frac{1-c}{6}\pi\tau}\eta(2i\tau)^{1-c}d\tau\right)^{-1}
\end{equation}
where $\eta(2i\tau):=e^{-\frac{\pi}{6}\tau}\prod_{n=1}^{\infty}\left(1-e^{-4\pi n\tau}\right)$ is the Dedekind eta function.
\end{theorem}
Note that the right side of~\eqref{eq:p_e} tends to $1$ as $c\uparrow1$, and tends to $0$ as $c\downarrow0$.

\end{itemize}

\medskip
\noindent\textbf{Organization of the paper.} In Section~\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops. Such relation was implicitly established in~\cite{ARS-Annulus}, and we will provide its proof in Appendix~\ref{appendix} for completeness. Then in Section~\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and solve it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.

\medskip
\noindent\textbf{Basic notations.}
For two compact sets $A,B\subset\mathbbm{C}$, let $\mathrm{dist}(A,B):=\inf\{|a-b|:a\in A,b\in B\}$.
For a simple loop $\ell\subset\mathbbm{C}$, let $D(\ell)$ be the bounded connected component of $\mathbbm{C}\backslash\ell$. For a simple loop $\ell\subset\mathbbm{C}$ with $0\in D(\ell)$, we denote the conformal radius of $D(\ell)$ seen from 0 by $\mathrm{CR}(\ell,0)$, i.e. if $f:\mathbbm{D}\to D(\ell)$ is a conformal map that fixes the origin, then $\mathrm{CR}(\ell,0)=|f'(0)|$.

For $0<r<1$, let $\A_r$ be the standard annulus $\{z\in\mathbbm{C}: r<|z|<1\}$. For each annular domain $A\subset\mathbbm{C}$,
there exists a unique $\tau>0$ such that $A$ and $\A_{e^{-2\pi\tau}}$ are conformally equivalent. We call $\tau$ the \textit{modulus} of $A$ and write $\mathrm{Mod}(A):=\tau$. For two simple loops $\eta_1,\eta_2$ such that $\overline{D(\eta_1)}\subset D(\eta_2)$, we also write the modulus of the annular domain $D(\eta_2)\setminus\overline{D(\eta_1)}$ as $\mathrm{Mod}(\eta_1,\eta_2)$ for simplicity.

We will frequently deal with elementary functions such as $\sqrt{x},\sinh(\sqrt{x}),\cosh(\sqrt{x})$, and $\tanh(\sqrt{x})$. In the following, we view them as functions defined on $\mathbbm{R}$ by taking $i\sqrt{|x|},i\sin(\sqrt{|x|}),\cos(\sqrt{|x|})$, and $i\tan(\sqrt{|x|})$ for $x<0$, respectively.

\medskip
\noindent\textbf{Acknowledgment.}
We are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.
G.C. and Z.X.\ were partially supported by National Key R\&D Program of China (No.\ 2023YFA1010700). G.C. was partially supported by National Key R\&D Program of China (No. 2021YFA1002700).

Our proof of Theorem~\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; Section~\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.”

Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”

Existential–Universal
Inequality or Bound

Question:
Let $(B_t)_{t\ge 0}$ be planar Brownian motion started from $0$, let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let $\tau_{\mathbb D}$ be the first time that $B_t$ hits the unit circle $\mathbb S^1=\{z\in\mathbb C:|z|=1\}$. For $\varepsilon\in(0,\tfrac12)$, define the backbone event
\[
\mathrm{Bac}_\varepsilon=
\Big\{\text{there exist two disjoint subpaths of }B[0,\tau_{\mathbb D}]\text{ joining }\varepsilon\mathbb S^1\text{ and }\tfrac12\mathbb S^1\Big\}.
\]
Equivalently, there exist continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\tau_{\mathbb D}]$, such that $\gamma^i(0)\in \varepsilon\mathbb S^1$ and $\gamma^i(1)\in \tfrac12\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\mathbb P[\mathrm{Bac}_\varepsilon]$ for small $\varepsilon$?

There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]

Choices:
B.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
C_1\,\frac{1}{|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le C_2\,\frac{1}{|\log \varepsilon|}.
\]
C.
There exists a constant $C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
D.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac12)$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
E.
For every $\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{(\log|\log \varepsilon|)^{1+\delta}}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{(\log|\log \varepsilon|)^{1-\delta}}.
\]

The stem defines the event carefully but does not reveal the asymptotic form of the probability. The correct iterated-log scale is not stated or strongly hinted at in the setup.

This is essentially a theorem-recall item: it asks which explicit probabilistic estimate holds, and the correct choice is the exact theorem statement up to constants. That makes it close to a direct restatement rather than a conceptually transformed application.

There is some reasoning pressure because the student must distinguish exact truth from nearby alternatives (single log vs iterated log, two-sided vs one-sided bounds, validity range). However, the item mainly rewards recognition of the precise result rather than generating a conclusion from underlying principles.

The distractors are strong: they are mathematically close to the true statement and reflect realistic confusions such as wrong decay scale, a weaker true statement, overextended parameter range, and an unjustified stronger upper bound.

A solid recognition-based theorem-checking MCQ with high-quality distractors and no answer leakage, but it is largely a direct restatement of a known result and only moderately tests generative reasoning.


Question:
Let $(B_t)_{t\ge 0}$ be planar Brownian motion started from $0$, let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let $\tau_{\mathbb D}$ be the first time that $B_t$ hits the unit circle $\mathbb S^1=\{z\in\mathbb C:|z|=1\}$. For $\varepsilon\in(0,\tfrac12)$, define the backbone event
\[
\mathrm{Bac}_\varepsilon=
\Big\{\text{there exist two disjoint subpaths of }B[0,\tau_{\mathbb D}]\text{ joining }\varepsilon\mathbb S^1\text{ and }\tfrac12\mathbb S^1\Big\}.
\]
Equivalently, there exist continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\tau_{\mathbb D}]$, such that $\gamma^i(0)\in \varepsilon\mathbb S^1$ and $\gamma^i(1)\in \tfrac12\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\mathbb P[\mathrm{Bac}_\varepsilon]$ for small $\varepsilon$?

Choices:
B.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
C_1\,\frac{1}{|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le C_2\,\frac{1}{|\log \varepsilon|}.
\]
C.
There exists a constant $C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
D.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac12)$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
E.
For every $\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{(\log|\log \varepsilon|)^{1+\delta}}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{(\log|\log \varepsilon|)^{1-\delta}}.
\]

The stem defines the event carefully but does not reveal the asymptotic form or explicitly hint at the iterated-log behavior. The correct answer is not leaked by wording in the question.

This is close to theorem recall: the stem asks directly which quantitative estimate holds, and the correct option is essentially the theorem statement. However, the alternatives introduce competing nearby formulations, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish the exact two-sided estimate from weaker, overextended, or slightly perturbed variants. Still, the item mainly tests recognition/recall of the precise result rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: confusing log with iterated log, accepting a weaker true statement, overextending the epsilon range, or allowing flexible exponents. They are distinct and well-calibrated.

A strong technical MCQ with no answer leakage and high-quality distractors, but it is somewhat theorem-recall-heavy and only moderately tests generative reasoning.


[Rigidity of lattices]
    If $\Gamma$ is a finitely generated group quasi-isometric to an irreducible lattice in a semisimple Lie group $G$, then there is a short exact sequence
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    where $\Lambda$ is a lattice in $G$, and $F$ is a finite group.

\subsection{Background}

One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One theme of geometric group theory is the rich relationship between the algebraic structure of groups and the large-scale geometry of spaces on which they act. By the ``large-scale geometry" of a space we mean the metric structure that is preserved by quasi-isometries, maps which preserve distances up to a controlled error. Gromov proposed in a 1983 ICM address \cite{gromov-icm} a broad research program of studying finitely generated groups as geometric objects and classifying them up to quasi-isometry. One aspect involves identifying instances of quasi-isometric rigidity, the phenomenon which occurs when algebraic properties of a finitely generated group are determined by its large-scale geometry. For example, a celebrated theorem by Gromov \cite{Gromov-polynomial} states that finitely generated groups of polynomial growth are virtually nilpotent (the converse is also true \cite{wolf1968growth}). Since growth rate is invariant under quasi-isometry, it follows that any finitely generated group which is quasi-isometric to a space of polynomial growth necessarily has a nilpotent subgroup of finite index. Thus we have an instance of when the quasi-isometry type of a group determines some of its algebraic structure.

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One of the landmark results in research on quasi-isometric rigidity is the complete quasi-isometry classification of lattices in semisimple Lie groups. A large body of work in the 1980s and 1990s by several people over many papers culminated in a general theorem on the rigidity of the class of lattices among all finitely generated groups (see \cite{farb} for a detailed survey). Informally, this theorem states that any group quasi-isometric to a lattice in a semisimple Lie group is almost a lattice in that Lie group. More precisely,

One of the major breakthroughs leading to this general classification was the work of Schwartz \cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.

\begin{theorem}[Schwartz]\label{Schwartz-theorem}
 Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\bH^2$. If $\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\Lambda$ in $\Isom{X}$, then there exists a short exact sequence
 \[
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \]
 where $\Lambda' \le \Isom{X}$ is a non-uniform lattice commensurable to $\Lambda$, and $F$ is a finite group.
\end{theorem}

\begin{theorem}[Frigerio--Lafont--Sisto]\label{FLS}
 If $\Gamma$ is a finitely generated group quasi-isometric to $\pi_1(M) \times \bZ^d$, then there exist short exact sequences
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & \bZ^d \ar[r,"j"] & \Gamma' \ar[r] & \Delta \ar[r] & 1
 \end{tikzcd}
 \end{equation*}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \pi_1(M') \ar[r] & 1
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\bZ^d)$ is contained in the center of $\Gamma'$. In other words, $\Gamma$ is virtually a central extension by $\bZ^d$ of a finite extension of $\pi_1(M')$.
\end{theorem}

\begin{thmx}\label{Theorem A}
 Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
 \begin{equation}\label{virtually nilcentral extension}
 \begin{tikzcd}
 1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
 \end{tikzcd}
 \end{equation}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

\begin{theorem}
Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
 \begin{equation}
 \begin{tikzcd}
 1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
 \tag{\ref{virtually nilcentral extension}}
 \end{tikzcd}
 \end{equation}
 \begin{equation*}
 \begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
 \end{tikzcd}
 \end{equation*}
 where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{theorem}

\begin{proof}
By Proposition \ref{kernel qi to N}, $\ker\theta$ is quasi-isometric to $N$, where $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. In particular $\ker\theta$ has polynomial growth. Then by Gromov's polynomial growth theorem, $\ker\theta$ is virtually nilpotent. So $\ker\theta$ has a finite-index, hence finitely generated, nilpotent subgroup $K$. Such groups are virtually torsion-free, so take a finite-index torsion-free subgroup $K' \le K$. Since $\ker\theta$ is finitely generated, it has only finitely many subgroups with the same index as $K'$. Let $L'$ denote their intersection. Then $L'$ is a finite-index characteristic subgroup of $\ker\theta$. Moreover, $L'$ is finitely generated, nilpotent, and torsion-free. A theorem of Malcev \cite{malcev} then says that $L'$ embeds as a lattice in a simply connected nilpotent Lie group $N'$, known as the (real) Malcev completion of $L'$ (see also \cite[Theorem 2.18]{raghunathan1972discrete}). Since $L'$ is characteristic in $\ker\theta$, it is normal in $\Gamma$. By construction, $\Gamma/L'$ is an extension of $\Lambda_\Gamma = \Gamma/\ker\theta$ by the finite group $F = \ker\theta/L'$. By Corollary \ref{commensurable}, $\Lambda_\Gamma$ is virtually a finite-index subgroup $\Lambda' \le \Lambda$ which is also a non-uniform lattice in $\text{Isom}(X)$. Let $\Gamma' = \theta^{-1}(\Lambda')$ and $\Delta = \Gamma'/L'$. Then we obtain the desired short exact sequences.

Now suppose we are in the situation of Theorem \ref{Theorem A} and we have the following two short exact sequences.
\begin{equation}\label{vne}
\begin{tikzcd}
 1 \ar[r] & N \ar[r] & \Gamma \ar[r] & \Delta \ar[r] & 1
\end{tikzcd}
\end{equation}
\begin{equation}\label{finite extension}
\begin{tikzcd}
 1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda \ar[r] & 1
\end{tikzcd}
\end{equation}
where $N$ is a nilpotent lattice (we are now using $N$ to denote the lattice, not the ambient Lie group), $F$ is finite, and $\Lambda$ is a non-uniform lattice in $\text{Isom}(X)$, where $X \ne \bH^2$ is a negatively curved symmetric space. Since $N$ is finitely generated, nilpotent, and torsion-free, $Z(N/Z_i) = Z_{i+1}/Z_i$ is a finitely generated free abelian group. That is, $Z(N/Z_i) = \bZ^{d_i}$ for some $d_i$, and so $\Sigma(N) = \max_i d_i$. We now focus on the situation when $X$ is quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\text{Isom}(X)$ has sufficiently large dimension.

\begin{thmx}\label{Theorem A}
    Let $X \ne \bH^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
    \begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

\begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}

One of the major breakthroughs leading to this general classification was the work of Schwartz \cite{Schwartz} on non-uniform lattices in rank one semisimple Lie groups. These Lie groups agree, up to index 2, with the isometry groups of the negatively curved symmetric spaces: real, complex, quaternionic hyperbolic space, and the Cayley hyperbolic plane.

\begin{theorem}[Schwartz]\label{Schwartz-theorem}
    Let $X$ be a negatively curved symmetric space other than the real hyperbolic plane $\mathbb{H}^2$. If $\Gamma$ is a finitely generated group quasi-isometric to a non-uniform lattice $\Lambda$ in $\textup{Isom}({X})$, then there exists a short exact sequence
    \[
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \]
    where $\Lambda' \le \textup{Isom}({X})$ is a non-uniform lattice commensurable to $\Lambda$, and $F$ is a finite group.
\end{theorem}

In the case of $X = \mathbb{H}^2$, non-uniform lattices in $\Isom{\mathbb{H}^2} = \text{PSL}(2,\mathbb{R})$ are virtually the fundamental groups of complete hyperbolic surfaces with finitely many punctures, and hence are virtually free. Any group quasi-isometric to a virtually free group is also virtually free, and every free group can be realized as a non-uniform lattice in $\text{PSL}(2,\mathbb{R})$, so quasi-isometric rigidity does hold in this case. However, the additional conclusion of commensurability fails. One reason for this failure is that commensurability preserves arithmeticity, and $\text{PSL}(2,\mathbb{R})$ contains both arithmetic and non-arithmetic non-uniform lattices. But since these lattices are virtually free, they are quasi-isometric to each other.

Following Schwartz, lattices such as $\Lambda$ and $\Lambda'$ shall from now on be called \textit{non-uniform rank one lattices}. Then an informal summary might be: any group quasi-isometric to a non-uniform rank one lattice is almost a commensurable non-uniform rank one lattice.

In an extensive study on high-dimensional graph manifolds \cite{FLS}, Frigerio--Lafont--Sisto prove, among many other things, a quasi-isometric rigidity result for products of the form $\pi_1(M) \times \mathbb{Z}^d$, where $M$ is a complete non-compact finite-volume hyperbolic $m$-manifold, $m\ge3$.

\begin{theorem}[Frigerio--Lafont--Sisto]\label{FLS}
    If $\Gamma$ is a finitely generated group quasi-isometric to $\pi_1(M) \times \mathbb{Z}^d$, then there exist short exact sequences
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & \mathbb{Z}^d \ar[r,"j"] & \Gamma' \ar[r] & \Delta \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \pi_1(M') \ar[r] & 1
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ has finite index, $M'$ is a finite-sheeted covering of $M$, $\Delta$ is a group, and $F$ is a finite group. Moreover, $j(\mathbb{Z}^d)$ is contained in the center of $\Gamma'$. In other words, $\Gamma$ is virtually a central extension by $\mathbb{Z}^d$ of a finite extension of $\pi_1(M')$.
\end{theorem}

Observe that the case $d = 0$ is covered by Schwartz' theorem, and indeed, the proof of this theorem applies many of the ideas and results from \cite{Schwartz}.

\subsection{Main results}

Our main contribution is a generalization of Theorem \ref{FLS} to products $\Lambda \times L$, where $\Lambda$ is a non-uniform rank one lattice and $L$ is a lattice in a simply connected nilpotent Lie group. From now on, a lattice such as $L$ is called a \textit{nilpotent lattice}. Our first theorem says that up to finite noise, any group quasi-isometric to $\Lambda \times L$ is an extension of a non-uniform rank one lattice commensurable to $\Lambda$ by a nilpotent lattice quasi-isometric to $L$.

\begin{thmx}\label{Theorem A}
    Let $X \ne \mathbb{H}^2$ be a negatively curved symmetric space. Let $\Lambda$ be a non-uniform lattice in $\textup{Isom}(X)$ and $L$ be a nilpotent lattice. If $\Gamma$ is a finitely generated group quasi-isometric to $\Lambda \times L$, then there exist short exact sequences
    \begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
    \begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
    where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.
\end{thmx}

The general outline of the proof is similar to that of the proof of Theorem \ref{FLS}. First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} guarantees that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$, up to bounded error. Thus the quasi-action of $\Gamma$ on $B\times N$ induces a quasi-action of $\Gamma$ on $B$. A key result in \cite{Schwartz} is that quasi-isometries of the neutered space $B$ have finite distance (with respect to the sup norm) from isometries of $X$. Thus we obtain a homomorphism $\theta : \Gamma \to \text{Isom}(X)$, and we show that the image $\text{im} \hspace{.8mm} \theta$ is a non-uniform lattice commensurable to $\Lambda$. The action $\theta : \Gamma \to \text{Isom}(X)$ came from a quasi-action on $B$, which itself was coarsely projected from a quasi-action on $B \times N$. Hence we are able to show that the kernel of $\theta$ is quasi-isometric to $N$. Then $\Gamma = \text{im} \hspace{.8mm} \theta / \ker \theta$, so we pass to finite-index subgroups as necessary to obtain the desired short exact sequences.

Theorem \ref{FLS} asserts that $\mathbb{Z}^d$  can be made central in the group extension. In our more general setting, however, the extension is by a nilpotent group. So we define the notion of a nilcentral extension, analogous to that of a central extension, and find conditions which are sufficient to guarantee that (\ref{virtually nilcentral extension}) may be taken to be a nilcentral extension. Given a nilpotent group $G$ with upper central series $1 = Z_0 \lhd Z_1 \lhd \dots \lhd Z_n = G$, define
\[
    \Sigma(G) \vcentcolon= \max_i \text{rank}(Z_{i+1}/Z_i).
\]

\begin{thmx}\label{Theorem B}
    Assume the hypotheses of Theorem \ref{Theorem A} and let $L'$ be the nilpotent lattice obtained from the conclusion of the theorem. If $X$ is either quaternionic hyperbolic space or the Cayley hyperbolic plane, and $\dim\textup{Isom}(X) > \Sigma(L')$, then the group extension (\ref{virtually nilcentral extension}) in the conclusion of Theorem \ref{Theorem A} is virtually nilcentral.
\end{thmx}

Our proof of this theorem relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T). The precise statement we apply is in \cite{fisher2012strengthening}.

First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} implies that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$ (up to bounded error), so the quasi-action on $B\times N$ induces a quasi-action on $B$.

A key result in \cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\theta: \Gamma \to \Isom(X)$, and one shows $\mathrm{im}\,\theta$ is a non-uniform lattice commensurable to $\Lambda$.

Because $\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\times N$, one can show $\ker\theta$ is quasi-isometric to $N$. Since “$\Gamma = \mathrm{im}\,\theta / \ker\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences (as in Theorem~\ref{Theorem A}).

For Theorem~\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\Sigma(L')$ and $\dim\Isom(X)$) ensuring that \eqref{virtually nilcentral extension} is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”

First, the quasi-isometry between $\Gamma$ and $\Lambda \times L$ induces a quasi-action of $\Gamma$ on $B \times N$, where $B \subset X$ is the neutered space associated to $\Lambda$, and $N$ is the simply connected nilpotent Lie group in which $L$ is a lattice. A theorem of Kapovich--Kleiner--Leeb \cite{KKL} implies that quasi-isometries $B \times N \to B \times N$ project to quasi-isometries $B \to B$ (up to bounded error), so the quasi-action on $B\times N$ induces a quasi-action on $B$.

A key result in \cite{Schwartz} says quasi-isometries of the neutered space $B$ have finite distance (in the sup norm) from isometries of $X$. Therefore one obtains a homomorphism $\theta: \Gamma \to \Isom(X)$, and one shows $\mathrm{im}\,\theta$ is a non-uniform lattice commensurable to $\Lambda$.

Because $\theta$ comes from the quasi-action on $B$ that was coarsely projected from the quasi-action on $B\times N$, one can show $\ker\theta$ is quasi-isometric to $N$. Since “$\Gamma = \mathrm{im}\,\theta / \ker\theta$”, one then passes to finite-index subgroups as necessary to obtain the desired short exact sequences, namely the conclusion that there exist short exact sequences
\begin{equation}\label{virtually nilcentral extension}
    \begin{tikzcd}
        1 \ar[r] & L' \ar[r] & \Gamma' \ar[r] & \Delta \ar[r] & 1,
    \end{tikzcd}
    \end{equation}
\begin{equation*}
    \begin{tikzcd}
        1 \ar[r] & F \ar[r] & \Delta \ar[r] & \Lambda' \ar[r] & 1.
    \end{tikzcd}
    \end{equation*}
where $\Gamma' \le \Gamma$ and $\Lambda' \le \Lambda$ have finite index, $L'$ is a nilpotent lattice quasi-isometric to $L$, $\Delta$ is a group, and $F$ is a finite group.

For Theorem~\ref{Theorem B}, after noting that in this setting the extension is “by a nilpotent group” (not necessarily central), the authors “define the notion of a nilcentral extension” and “find conditions” (involving $\Sigma(L')$ and $\dim\Isom(X)$) ensuring that the equation above is “virtually nilcentral”; the proof “relies on a form of Margulis--Corlette--Gromov--Schoen superrigidity for Lie groups with Kazhdan's property (T).”

Question: Let $G$ be a semisimple Lie group, and let an irreducible lattice in $G$ mean a discrete subgroup of finite covolume that is irreducible with respect to the semisimple decomposition of $G$. Suppose $\Gamma$ is a finitely generated group that is quasi-isometric to such an irreducible lattice in $G$. Which of the following conclusions about $\Gamma$ holds?

There exists a short exact sequence
\[
\begin{tikzcd}
1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda \ar[r] & 1,
\end{tikzcd}
\]
where $\Lambda$ is a lattice in $G$ and $F$ is a finite group.

Choices:
B.
There exists a short exact sequence
\[
\begin{tikzcd}
1 \ar[r] & F \ar[r] & \Gamma' \ar[r] & \Lambda \ar[r] & 1,
\end{tikzcd}
\]
where $\Gamma' \leq \Gamma$ has finite index, $\Lambda$ is an irreducible lattice in $G$, and $F$ is a finite group.
C.
$\Gamma$ is virtually a finite extension of a lattice in $G$; equivalently, there exist a finite-index subgroup $\Gamma' \leq \Gamma$, a lattice $\Lambda \leq G$, and a finite group $F$ fitting into a short exact sequence
\[
\begin{tikzcd}
1 \ar[r] & F \ar[r] & \Gamma' \ar[r] & \Lambda \ar[r] & 1.
\end{tikzcd}
\]
D.
There exists a short exact sequence
\[
\begin{tikzcd}
1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda \ar[r] & 1,
\end{tikzcd}
\]
where $\Lambda$ is an irreducible lattice in $G$ commensurable with the given irreducible lattice, and $F$ is a finitely generated nilpotent group.
E.
There exists a short exact sequence
\[
\begin{tikzcd}
1 \ar[r] & F \ar[r] & \Gamma \ar[r] & \Lambda \ar[r] & 1,
\end{tikzcd}
\]
where $\Lambda$ is a lattice in $G$ and $F$ is finite; moreover, one may always choose $\Lambda$ to be commensurable to the original irreducible lattice in $G$.

need for exact conclusion on all of Gamma rather than only after passing to finite index in the product-lattice argument

dropped the requirement that the short exact sequence hold for \Gamma itself, allowing passage to a finite-index subgroup

kernel is finite, not merely virtually nilpotent or finitely generated nilpotent

asserted commensurability to the original lattice, which is not stated in the main rigidity theorem

The stem states only the hypotheses and asks for the correct conclusion; it does not explicitly reveal the exact finite-extension conclusion or uniquely telegraph choice A.

This is essentially a direct recall/application of a quasi-isometric rigidity theorem: the hypotheses in the stem closely match the theorem, and the correct option states its conclusion almost verbatim.

There is some reasoning pressure because the options differ by subtle quantifiers and strength (exact sequence for all of Γ vs. after finite index, finite vs. nilpotent kernel, commensurability claims). Still, a student who knows the theorem can answer mainly by recognition rather than substantial derivation.

The distractors are mathematically close, distinct, and plausible: one is a weaker true-looking variant, others alter the kernel type or add an unjustified commensurability claim. These reflect realistic failure modes in reading rigidity statements.

A solid precision-testing theorem-recall MCQ with strong distractors, but it is quite close to a direct restatement of the underlying rigidity theorem rather than a deeply generative reasoning task.