\label{se:2}

We denote by $c$ and $C$ positive and finite constants whose values are universal but may change from line to line.
If $\{a_n\}$ and $\{b_n\}$ are non-negative sequences, then we write
$

\begin{equation}\label{eq:1.10}
        \p\left(\max\limits_{\tau_i\leq m\leq \sigma_i-1}d_i(m)\geq C\log^2 i\right)=\exp\left({-(1+o(1))2\sqrt{C}\log i}\right).
    \end{equation}
The threshold for this property is thus precisely determined to be $C=\frac14$. Then plugging \eqref{eq:1.10} and the Borel-Cantelli lemma into \eqref{eq:1.9} gives Theorem \ref{main result}.
\end{proof}

\section{Proof of Proposition \ref{prop 1.3}}\label{tech-section}
The rest of the paper is devoted to proving Proposition \ref{prop 1.3}. We begin in Section \ref{Tech-esti} by establishing the required technical estimates. The main steps of the proof of the lower and upper bounds are then presented in Sections \ref{se:3} and \ref{se:4}, respectively.

\subsection{Technical Estimates}\label{Tech-esti}
Let $(G_n)$ be a SRW start at $G_1=1$, conditioned on never hit $0$.
That is, $(G_n)$ is an $h-$process.  In other words, starting from $G_1=1$ and with respect to its natural filtration $\mathcal{F}^{G}_{n}$, its transition probabilities are explicitly given as follows:
\begin{align*}
    &\p\left(G_{n+1}=G_n+1\mid \mathcal{F}^{G}_{n}\right)=\frac{G_n+1}{2G_n},\\
    &\p\left(G_{n+1}=G_n-1\mid \mathcal{F}^{G}_{n}\right)=\frac{G_n-1}{2G_n}.
\end{align*}
Define $\tau^G_s:=\inf\{n:G_n=s\}$ is the stopping time when  $(G_n)$ hits $x$ for the first time. Define
\begin{equation*}
    \mathcal{G}_s:=\left\{ y : \exists \ k\in[1, \tau^G_s] \text{ such that } G_k = y, \text{ and } G_l \neq y \text{ for } 1 \leq l \leq \tau^G_s,\ l \neq k \right\}
\end{equation*}
which is the once visited set at time $\tau^G_s$. Let $g(s)=\#\mathcal{G}_s$.
\begin{proposition}\label{prop1.5}
For any $\epsilon>0$, $0<c_1<c_2$, there exists $N=N(\epsilon,c_1,c_2)>0$ such that for any $s\geq N$, $C\in [c_1,c_2]$, we have
    \begin{equation*}
       \exp\left({-(2+\epsilon)C\log s}\right)\leq \p\left(g(s)\geq C\log^2 s\right)\leq \exp\left({-(2-\epsilon)C\log s}\right).
    \end{equation*}
\end{proposition}
By adapting ideas from \cite[Theorem 2]{Ma88}, Proposition \ref{prop1.5} can be proved by verifying the following two lemmas. We remark that the first lemma is similar to \cite[Lemma 4]{Ma88}.
\begin{lemma}\label{lem1.6}
For any $0<\epsilon<1$, $0<c_1<c_2$, there exists $N=N(\epsilon,c_1,c_2)>0$ such that for any $n\geq N$, $k\in [c_1\log s,c_2\log s]$, we have
\begin{equation*}
    \left(\frac{(1-\epsilon)\log s}{2}\right)^k\leq \e \left[\binom{g(s)}{k}\right]\leq \left(\frac{(1+\epsilon)\log s}{2}\right)^k.
\end{equation*}
\end{lemma}

\begin{lemma}\label{lem1.7}
    If a sequence of random variables $X_s\in \N$ satisfy that for any $0<\epsilon<1$, $0<c_1<c_2$, there exists $N=N(\epsilon,c_1,c_2)>0$ such that for any $n\geq N$, $k\in [c_1\log s,c_2\log s]$,
\begin{equation*}
    \left(\frac{(1-\epsilon)\log s}{2}\right)^k\leq \e \left[\binom{X_s}{k}\right]\leq \left(\frac{(1+\epsilon)\log s}{2}\right)^k,
\end{equation*}
then for any constants $\epsilon>0$, $0<c_1<c_2$, there exists $N=N(\epsilon,c_1,c_2)>0$ such that for any $n\geq N$, $C\in [c_1,c_2]$, it holds that
    \begin{equation*}
       \exp\left(-(2+\epsilon)C\log s\right)\leq \p\left(X_s\geq C\log^2 s\right)\leq \exp\left(-(2-\epsilon)C\log s\right).
    \end{equation*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem1.6}]
    For any set $A=\{x_1,x_2,\dots ,x_k\}\subset \{1,2,\dots ,s-1\}$, suppose $x_1<x_2<\dots<x_k$ and $k\le s-1$.
    Then $A\subset \mathcal{G}_s$ if and only if $G_{n_1}>x_i$ for any $1\leq i\leq k-1$ and $\tau^G_{x_i}+1\leq {n_1}\leq \tau^G_{x_{i+1}}$, and $G_{n_2}>x_k$ for any $\tau^G_{x_k}+1\leq n_2\leq \tau^G_{s}$.
    Similar to our previous techniques \cite[(4.7)]{FH25}, we see that $(\frac{1}{G_n})_{n\ge1}$ is martingale. By applying the Optional Stopping theorem at $\tau^G_{a}\wedge\tau^G_{b}$, with the initial state $\frac{1}{G_1}=\frac{1}{a+1}$. One has,
    \begin{equation}\label{mart-anal}
        \p\left(G_n>a,\ \mbox{ for any } \tau^G_{a}+1\leq n\leq \tau^G_{b}\,\Big|\, \mathcal{F}^{G}_{\tau^G_{a}}\right)=\frac{b}{2a(b-a)}.
    \end{equation}

\begin{align}\label{Stopping-times}
    &\sigma_i = \inf \left\{ k > \tau_i : S_k \in [N_{\tau_i - 1}, M_{\tau_i - 1}] \right\}, \nonumber\\
    &\tau_{i+1} = \inf \left\{ k > \sigma_i : S_k \notin [N_{\sigma_i}, M_{\sigma_i}] \right\}.
\end{align}
An important observation is that the time $\sigma_i$ can also be characterized as the first return to the pre-$\tau_i$ level:
\begin{equation*}
    \sigma_i = \inf \{ k > \tau_i : S_k = S_{\tau_i - 1} \}.
\end{equation*}

Finally, let $(T_n)$ be a SRW on $\mathbb{Z}$ starting at $T_1 = 1$ and stopped at $\sigma := \inf \{ n : T_n = 0 \}$. We then define the process
\[
T^i_n = | S_{\tau_i - 1 + n} - S_{\tau_i - 1} |, \quad \text{for } 1 \leq n \leq \sigma_i - \tau_i + 1.
\]
The process $\left( T^i_n \right)_{1 \leq n \leq \sigma_i - \tau_i + 1}$ is equal in distribution to $\left( T_n \right)_{1 \leq n \leq \sigma}$ and is independent of $\mathcal{F}_{\tau_i}$.

Given that the indices $M,j,k$ and $x$ (possibly random) are considered in the limit as they tend to infinity, it is to be understood throughout that all assertions hold for sufficiently large index values, at times without explicit mention.

\section{Proof of Theorem \ref{main result}}\label{Proof of main result}

We divide the proof of Theorem \ref{main result} into two lemmas and a proposition.

\begin{lemma}\label{lem1.2}
    \begin{equation*}
        \lim_{j\rightarrow \infty}\frac{\log(\sigma_j)}{\log j}=2\quad a.s.
    \end{equation*}
\end{lemma}
\begin{proof}
First, we connect the growth of $\sigma_j$ with the range of the random walk. Note that the increase of the range of $(S_n)$ during $[\tau_j,\sigma_j]$ is identical to the range of $(T_n)_{1\le n\le \sigma}$. Therefore, for $k\in\mathbb{Z}^+$,
\begin{equation}\label{T-distribution}
        \p\left( \r_{\sigma_{j+1}}\geq \r_{\sigma_j}+k \right)=\p\left( \max\limits_{1\leq s\leq \sigma}T_s\geq k\right)=\frac{1}{k}.
\end{equation}
Suppose $X_i$, $i\in \N$ are i.i.d. random variables with distribution
\begin{equation}\label{eq:logvariable}
        \p(X_i=k)=\frac{1}{k(k+1)}\text{~for~} k\in \z^+.
\end{equation}
Hence, the distribution of $\{\r_{\sigma_j}\}$ is identical to the distribution of $\left\{\sum\limits_{i=1}^{j}X_i\right\}$ with $j\in\z^+$. Furthermore, for any $C,\epsilon>0$,
    \begin{align}
        \p\left(\sum\limits_{i=1}^{j}X_i\geq Cj^{1+\epsilon}\right)
        &\le\p\left(\mathop{\cup}_{i=1}^{j}\left\{X_i>j^{1+\epsilon}\right\}\right)+\p\left(\sum_{i=1}^{j}X_i\cdot1_{\{X_i\le j^{1+\epsilon}\}}\ge Cj^{1+\epsilon}\right)\nonumber\\
        &\leq j\ \p(X_i\geq j^{1+\epsilon})+\frac{j\ \e(X_i\cdot1_{X_i\leq j^{1+\epsilon}})}{Cj^{1+\epsilon}}\nonumber\\
        &\leq j^{-\epsilon}(1+C^{-1}(1+\epsilon)\log j).\label{eq:1.3}
    \end{align}

\begin{equation}\label{eq:3.5}
    \tilde{\mathcal{D}}_{K_M}\cup  \widehat{\mathcal{D}}_{n_1}\subset\mathcal{D}_{n_1}.
\end{equation}
Combining \eqref{eq:3.2}, \eqref{eq:3.3}, \eqref{eq:3.4} and \eqref{eq:3.5} gives
\begin{align*}
    &\p\left(\max_{1\leq m\leq \sigma-1}\curvevisitonenumber{m}\geq M\right)\\
    \geq& \;\,\p(\beta_0<\infty)\cdot\p\left(\left.\#\tilde{\mathcal{D}}_{K_M}\geq \left(1-c_3\right)M\right|\beta_0<\infty \right)
    \cdot\p\left(\left.\exists\ i\geq 1,\ \gamma_i<\beta_{\rm fail},\ Z_i\geq c_3 M\right|\mathcal{F}_{\beta_0}\right)\\
    \geq& \;\,\exp\left({-\left(-2(\Lambda+\epsilon_0)+(2+\epsilon)\left(\frac{\epsilon_0+\epsilon_1}{\Lambda(\Lambda+\epsilon_0)}\right)\right)\sqrt{M}+O(1)}\right).
\end{align*}
Taking $\epsilon_1$ and $\epsilon$ close to $0$,
we obtain
\begin{align*}
    \Lambda\geq \Lambda+\epsilon_0-\frac{\epsilon_0}{\Lambda(\Lambda+\epsilon_0)}
\end{align*}
for any $\epsilon_0>0$, which gives $\Lambda\geq -1$ and completes the proof of \eqref{lower-bound}.
\end{proof}

\subsection{The Upper Bound}\label{se:4}
In this subsection, we prove the upper bound of Proposition \ref{prop 1.3}, namely:
\begin{proposition}\label{prop4.1}
\begin{equation*}
    \limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq m\leq \sigma-1}\curvevisitonenumber{m}\geq M\right)\right)}{2\sqrt{M}}= -1.
\end{equation*}
\end{proposition}
The upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \ref{prop4.1}. Define, for any $c>0$,
\begin{equation}
    h(c):=\limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq s\leq \text{e}^{c\sqrt{M}}}g(s)\geq M\right)\right)}{2\sqrt{M}}.\label{eq:4.0}
\end{equation}

\begin{thm}\label{main result}
With probability $1$,
\begin{equation*}
\limsup_{n\to\infty} \dfrac{g_1(n)}{\log^2 n}=\frac{1}{16}.
\end{equation*}
\end{thm}

Let $(S_n)_{n\geq 0}$ be a discrete-time simple random walk (SRW) on $\mathbb{Z}$ starting at $S_0 = 0$.
For any $n\in{\mathbb N}$ and $s\in{\mathbb Z}$, write
\begin{equation*}
\xi(s,n)=\#\{0\le j\le n : S_j=s\},\qquad \mathcal{R}(n)=\{s\in\mathbb{Z}:\exists\ i\in[0,n],\ S_i=s\},
\end{equation*}
for the local time (number of visits) at site $s$ up to time $n$, and the range (set of visited sites) of the random walk by time $n$ respectively. Let $g_k(n)$ be the number of sites visited exactly $k$ times up to time $n$, that is, $$g_k(n):=\#\{s:\xi(s,n)=k\}.$$
Among these, $g_1$ is particularly interesting and has been the focus of significant research interest.
Intuitively, the set of once-visited sites can be linked to the ``points of increase'' of the SRW (see Dvoretzky, Erd\H{o}s and Kakutani \cite{DEK61} and Peres \cite{Pe96}).
An interesting result was established by Newman \cite{Ne84}, who proved that ${\mathbb E}\, g_1(n)=2$ for all $n\in{\mathbb N}$. Furthermore, Han \cite{CH25} proved the convergence $\mathbb{E}[g_k(n)] \to 2$ for any fixed $k \ge 2$.
\footnote{During C.H.'s visit to Sichuan University in September 2025, Yinshan Chang kindly informed C.H. that this convergence result was obtained in Yuhang Han's undergraduate thesis \cite{CH25} of which Y.C. is the supervisor.}

However, is it possible that as $n$ tends to infinity, $g_1(n)$ attains atypically large values at some $n$'s? Motivated by this question, Erd\H{o}s and R\'{e}v\'{e}sz (see \cite{Ma88} for details) posed the problem of identifying the scaling function $\kappa(n)\to\infty$ for which $\frac{g_1(n)}{\kappa(n)}$ admits a non-degenerate limit, i.e.,
$$\limsup\limits_{n\to\infty}\dfrac{g_1(n)}{\kappa(n)}=C\in(0,\infty)\quad\quad a.s.$$
This question was resolved by Major as follows.
\begin{thmA*}[\cite{Ma88}]\label{OVS-limsup-behavior-unknown-constant}
There exists a constant $0<C<\infty$ such that
\begin{eqnarray*}
\limsup_{n\to\infty} \dfrac{g_1(n)}{\log^2 n}=C~~~~~~a.s.
\end{eqnarray*}
\end{thmA*}
However, the precise value of the constant $C$ in the above theorem remained unknown. In this paper, we determine this constant exactly. Our main result is the following.

A related quantity, the minimal local time defined as $f(n)=\min\{\xi(s,n):s\in\mathcal{R}(n)\}$, was introduced by Erd\H{o}s and R\'{e}v\'{e}sz \cite{ER87,ER91} to study its $\limsup$ behavior. After a series of developments by T\'{o}th \cite{To96} and R\'{e}v\'{e}sz \cite{Re13}, our previous work \cite{FH25} ultimately established that
\begin{equation*}
\limsup_{n\to\infty} \frac{f(n)}{\log\log n} = \frac{1}{\log 2}~~~~~~a.s.
\end{equation*}
This result implies that
$$\liminf\limits_{n\to\infty}g_1(n)=0~~~~~~a.s.$$

Finally, we now briefly discuss a similar result for $g_k(n)$ with $k\ge2$ that springs from Lemma~\ref{lem1.6}.
\begin{rem}\label{gk-result}
There exist a constant $0<C<\infty$ such that
$$\limsup_{n\to\infty} \dfrac{g_k(n)}{\log^2 n}=C~~~~a.s.$$
The key idea is to adapt Lemma~\ref{lem1.6} by redefining
\begin{equation*}
 g_{k}(s):=\#\mathcal{G}_{s,k}=\#\left\{ x : \xi(x,\tau^G_s)=k\right\}
\end{equation*}
which counts the number of sites visited exactly $k$ times by $(G_n)$ at time $\tau^G_s$. Under the assumptions of Lemma~\ref{prop1.5}, a similar bound holds,
\begin{equation*}
 \left(\frac{(1-\epsilon)\log s}{2}\right)^j\leq \e \left[\binom{g_{k}(s)}{j}\right]\leq \left(\frac{(1+\epsilon)\log s}{2}\right)^j.
\end{equation*}
The remainder of the proof follows similarly to \cite{Ma88}.
The definition of $g_k$ (sites visited exactly $k$ times) creates correlations between different parts of the path, unlike the "point of increase." This inherent correlation prevents a decomposition into i.i.d.~excursions.
\end{rem}

\subsection{The Upper Bound}\label{se:4}
In this subsection, we prove the upper bound of Proposition \ref{prop 1.3}, namely:
\begin{proposition}\label{prop4.1}
\begin{equation*}
 \limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq m\leq \sigma-1}\curvevisitonenumber{m}\geq M\right)\right)}{2\sqrt{M}}= -1.
\end{equation*}
\end{proposition}
The upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \ref{prop4.1}. Define, for any $c>0$,
\begin{equation}
 h(c):=\limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq s\leq \text{e}^{c\sqrt{M}}}g(s)\geq M\right)\right)}{2\sqrt{M}}.\label{eq:4.0}
\end{equation}
\begin{lemma}\label{lem4.2}
 For any $0<N_1<N_2$ such that $\p\left(\max\limits_{1\leq s\leq N_2}g(s)\geq M\right)\leq 0.1$, we have
 \begin{equation}
 \p\left(\max\limits_{1\leq s\leq N_2}g(s)\geq M\right)\geq \frac{N_2-N_1}{8N_1(\log N_1+1)}\p\left(\max\limits_{1\leq s\leq N_1}g(s)\geq M\right).\label{eq:lem4.2}
 \end{equation}
\end{lemma}
Suppose
\begin{equation}\label{eq:3.1}
 \Theta:=\limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq m\leq \sigma-1}\curvevisitonenumber{m}\geq M\right)\right)}{2\sqrt{M}}.
\end{equation}
\begin{lemma}\label{lem4.3}
 For any $0<c<-2\Theta+\frac{(1+\Theta)^{2}}{2}$,
 \footnote{The motivation for writing the expression as $-2\Theta+\frac{(1+\Theta)^2}{2}$ is to clearly show that it exceeds $-2\Theta$ by precisely $\frac{(1+\Theta)^2}{2}$. The same logic applies to the analogous expressions that follow.}
 \begin{equation}
 h(c)\leq \frac{c}{2}+\Theta-\frac{(1+\Theta)^2}{4}.\label{eq:lem4.3}
 \end{equation}
\end{lemma}
\begin{lemma}\label{lem4.4}
 If \eqref{eq:lem4.3} holds for $c\leq c_1$, then it holds for any
 $$c=c_2<\min\left(1+\frac{1-\Theta}{4}c_1,-2\Theta+\frac{(1+\Theta)^{2}}{2}\right).$$
\end{lemma}
We now proceed to prove Proposition \ref{prop4.1} on the basis of the three lemmas established above.

Since the proof of Lemma \ref{lem4.3} relies on Lemma \ref{lem4.4}, we first prove the latter.
\begin{proof}[Proof of Lemma \ref{lem4.4}]
 Define event
 \begin{equation*}
 \mathcal{G}_{s_1,s_2}:=\left\{ s_1<s\leq s_2 : \exists \ k,\ 1 \leq k \leq \tau^G_{s_2}, \text{ such that } G_k = s, \text{ and } G_l \neq s \text{ for } 1 \leq l \leq \tau^G_{s_1},\ l \neq k \right\}
 \end{equation*}
Then, $\mathcal{G}_{s_2} \subset \mathcal{G}_{s_1} \cup \mathcal{G}_{s_1,s_2}$ for all $s_1<s_2$.
Assume $c_2 > 1$ and let $N_1 = \lfloor e^{(2c_2 - 2)\sqrt{M}} \rfloor$. With this, define$\tilde{\beta}_{1,k}:=\tau_{kN_1+1}^{G}$. Then, for $i \geq 1$, we define the following stopping times:
 \begin{align*}
 &\tilde\gamma_{i,k}:=\inf\{n>\tilde\beta_{i,k}:\ G_n\in \{G_{\tilde\beta_{i,k}}-1,(k+1)N_1\}\};\\ &\tilde\beta_{i+1,k}:=\inf\{n>\gamma_{i,k}':\ G_n=M^G_{\gamma_{i,k}'}+1\}.
 \end{align*}
 Under the condition that $M^G_{\tilde\gamma_{i,k}}-G_{\tilde\beta_{i,k}}=l-1$, we have
 $$(G_{n+{\tilde\beta_{i,k}}-1}-G_{{\tilde\beta_{i,k}}})_{1\leq n\leq \tau^G_{{G_{\tilde\beta_{i,k}}+l-1}}} \stackrel{d}{=}(G_n)_{1\leq n\leq \tau^G_{l}}.$$
 Then for any $M'>0$ and $k\le a_k$ such that $G_{\tilde\gamma_{a_k,k}}=(k+1)N_1$, we have
 \begin{equation}
 \p\left(\left.\max_{G_{\tilde\beta_{i,k}}\leq s<G_{\tilde\beta_{i+1,k}}}|\mathcal{G}_{kN_1,s}|\geq M'\right| M^G_{\tilde\gamma_{i,k}}-G_{\tilde\beta_{i,k}}=l-1,\mathcal{F}^G_{\tilde\beta_{i,k}}\right)=\p\left(\max\limits_{1\le s\le l}g(s)\geq M'\right).\label{eq:4.4}
 \end{equation}
 By Lemma \ref{lem4.2}, for $\p\left(\max\limits_{1\leq s\leq N_1}g(s)\geq M'\right)\leq 0.1$,
 \begin{equation}
 \p(g(l)\geq M')\leq 16(\log N_1+1)lN_1^{-1}\p\left(\max_{1\leq s\leq N_1}g(s)\geq M'\right).\label{eq:4.5}
 \end{equation}
Sum up \eqref{eq:4.4} for $1\leq i\leq a_k$, combining with \eqref{eq:4.5} gives
 \begin{align}\label{mathcal-G-calculate}
 &\p\left(\max_{kN_1+1\leq n\leq (k+1)N_1}\mathcal{G}_{kN_1,n}\geq M'\,\Big|\,\mathcal{F}^G_{\tau^G_{kN_1}}\right)\notag\\
 \leq& \sum_{i=1}^{a_k}\p\left(g(M^G_{\tilde\gamma_{i,k}}-G_{\tilde\beta_{i,k}}+1)\geq M'\Big|\mathcal{F}^G_{\tilde\beta_{i,k}}\right)\notag\\
 \leq& 16(\log N_1+1)N_1^{-1}\sum_{i=1}^{a_k}(M^G_{\tilde\gamma_{i,k}}-G_{\tilde\beta_{i,k}}+1)\cdot\p\left(\max_{1\leq s\leq N_1}g(s)\geq M'\right)\notag\\
 =&16(\log N_1+1)\cdot\p\left(\max_{1\leq s\leq N_1}g(s)\geq M'\right).
 \end{align}
For any $\epsilon > 0$, assume that $\max\limits_{kN_1+1 \leq s \leq (k+1)N_1} g(s) \geq M$. Then there exists an integer $0 \leq i \leq \epsilon^{-1}$ with the property that
\begin{itemize}
 \item $g(kN_1) \geq \epsilon i M$,
 \item $\max\limits_{kN_1+1 \leq s \leq (k+1)N_1} \mathcal{G}_{kN_1,s} \geq (1 - \epsilon(i+1)) M$.
\end{itemize}
 It follows that
 \begin{eqnarray}\label{liminf-cal01}
 &&\p\left(\max_{kN_1+1\leq s\leq (k+1)N_1}g(s)\geq M\right)\notag\\
 &\leq& \sum_{i=0}^{\lfloor \epsilon^{-1}\rfloor} \p(g(kN_1)\geq \epsilon iM)\cdot\p\left(\max_{kN_1+1\leq s\leq (k+1)N_1}\mathcal{G}_{kN_1,s}\geq (1-\epsilon(i+1))M\,\Big|\,\mathcal{F}^G_{\tau^G_{kN_1}}\right)\notag\\
 &\overset{\eqref{mathcal-G-calculate}}{\le} & M\cdot\sum_{i=0}^{\lfloor \epsilon^{-1}\rfloor} \p(g(kN_1)\geq \epsilon iM)\cdot\p\left(\max_{1\leq s\leq N_1}g(s)\geq (1-\epsilon(i+1))M\right).
 \end{eqnarray}
 By Proposition \ref{prop1.5}, for any $0\leq k\leq N_1^{-1}e^{c_2\sqrt{M}}$ and any $\epsilon'>0$,
 \begin{equation}\label{liminf-cal02}
 \p(g(kN_1)\geq \epsilon iM)\leq \exp\left({-(2-\epsilon')\epsilon i c_2^{-1}\sqrt{M}}\right).
 \end{equation}
 Putting together \eqref{liminf-cal01} and \eqref{liminf-cal02}, we obtain
 \begin{align}
&\limsup_{M\to\infty}\frac{\log\left(\p\left(\max\limits_{kN_1+1\leq s\leq (k+1)N_1}g(s)\geq M\right)\right)}{2\sqrt{M}}\notag\\
\leq& \max\limits_{c\in [0,1]}\left\{-cc_2^{-1}+\limsup_{M\to\infty}\frac{\log\left(\p\left(\max\limits_{1\leq s\leq N_1}g(s)\geq (1-c)M\right)\right)}{2\sqrt{M}}\right\}.\label{eq:4.7}
 \end{align}

It can be shown that $\limsup_{n\to\infty} g_k(n)$ is on the order of $\log^2 n$; nevertheless, our method does not yield the exact leading-order constant. See Remark \ref{gk-result} for details.
In comparison, the corresponding quantity $g_1^{(d)}(n)$ for the $d$-dimensional simple random walk $S_n^{(d)}$, $d \geq 2$, by Erd\H{o}s and Taylor \cite{ET60} and Flatto \cite{Fl76}, it is known that almost surely,
\begin{equation*}
\lim_{n\to\infty} \dfrac{g_1^{(2)}(n)\cdot\log^2 n}{\pi^2n}=\frac{1}{16}\quad\quad\mbox{and}\quad\quad\lim_{n\to\infty}\frac{g_1^{(d)}(n)}{n}=\gamma_d^2(1-\gamma_d)^{k-1},
\end{equation*}
where $d\ge3$ and $\gamma_d$ is the escape probability of a simple random walk on $\mathbb{Z}^d$.

A related quantity, the minimal local time defined as $f(n)=\min\{\xi(s,n):s\in\mathcal{R}(n)\}$, was introduced by Erd\H{o}s and R\'{e}v\'{e}sz \cite{ER87,ER91} to study its $\limsup$ behavior. After a series of developments by T\'{o}th \cite{To96} and R\'{e}v\'{e}sz \cite{Re13}, our previous work \cite{FH25} ultimately established that
\begin{equation*}
\limsup_{n\to\infty} \frac{f(n)}{\log\log n} = \frac{1}{\log 2}~~~~~~a.s.
\end{equation*}
This result implies that
$$\liminf\limits_{n\to\infty}g_1(n)=0~~~~~~a.s.$$

Turning our attention from the number of least visited to the most frequently visited sites (also called {\it favorite sites}), we find a rich body of results.
For instance, T\'{o}th \cite{To01} and Ding and Shen \cite{DS18} established that a SRW on $\mathbb{Z}$ almost surely has exactly three favorite sites infinitely often, but never more. Extensions to higher dimensions have been studied by Erd\H{o}s and R\'{e}v\'{e}sz \cite{ER91}, as well as by the second author of this paper together with Li, Okada and Zheng \cite{HLOZ24}. There are many results on the favorite sites of SRW. See Erd\H{o}s and R\'{e}v\'{e}sz \cite{ER84}, Bass and Griffin \cite{BG85}, Lifshits and Shi \cite{LS04}, Bass \cite{Ba23} and Dembo, Peres, Rosen and Zeitouni \cite{DPRZ01} and for comprehensive surveys, we refer to Shi and T\'{o}th \cite{ST00} and Okada \cite{Oka16}.

Our overall strategy is to decompose the entire path of the random walk into alternating ``inward'' and ``outward'' excursions by stopping times (Inspired by our previous work \cite{FH25} and referring to \eqref{Stopping-times} for the precise definitions, we capture the transitions between exploration and return to a previous range). This allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions.
More precisely, we study the probability of the {\it rare event} that each excursion contains unusually large number of once-visited sites.
An intuitive observation is that the set of sites visited only once is closely related to the ``points of increase'' at each excursion.
We finalize the proof by employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion. The power of this method lies in its ability to handle both the lower and upper bounds of the probability of {\it rare events}, which are separately derived in Sections \ref{se:3} and \ref{se:4}.

We now briefly outline the organization of this paper. Section \ref{se:2} contains the necessary preliminaries. The proof of our main result, Theorem \ref{main result}, is presented in Section \ref{Proof of main result}. Finally, a key proposition for Theorem \ref{main result} is proved in Section \ref{tech-section}.

To prove Theorem~\ref{main result}, the strategy is to "decompose the entire path of the random walk into alternating `inward' and `outward' excursions by stopping times" (see \eqref{Stopping-times} for definitions), which "allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions." More precisely, the proof studies "the probability of the {\it rare event} that each excursion contains unusually large number of once-visited sites," using the observation that "the set of sites visited only once is closely related to the `points of increase' at each excursion." The proof is finalized by "employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion," and this method "handle[s] both the lower and upper bounds of the probability of {\it rare events}, which are separately derived in Sections \ref{se:3} and \ref{se:4}."

To prove the main theorem, the strategy is to "decompose the entire path of the random walk into alternating `inward' and `outward' excursions by stopping times" (see
\begin{align}\label{Stopping-times}
    &\sigma_i = \inf \left\{ k > \tau_i : S_k \in [N_{\tau_i - 1}, M_{\tau_i - 1}] \right\}, \nonumber\\
    &\tau_{i+1} = \inf \left\{ k > \sigma_i : S_k \notin [N_{\sigma_i}, M_{\sigma_i}] \right\}.
\end{align}
for definitions), which "allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions." More precisely, the proof studies "the probability of the {\it rare event} that each excursion contains unusually large number of once-visited sites," using the observation that "the set of sites visited only once is closely related to the `points of increase' at each excursion." The proof is finalized by "employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion," and this method "handle[s] both the lower and upper bounds of the probability of {\it rare events}," which are separately derived in the part beginning
\label{se:3}
We now prove the lower bound in Proposition \ref{prop 1.3}. Specifically,
\begin{proposition}\label{prop3.1}
\begin{equation}\label{lower-bound}
    \liminf_{M\to\infty} \frac{\log\left(\

and in the part beginning
\subsection{The Upper Bound}\label{se:4}
In this subsection, we prove the upper bound of Proposition \ref{prop 1.3}, namely:
\begin{proposition}\label{prop4.1}
\begin{equation*}
    \limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq m\leq \sigma-1}\curvevisitonenumber{m}\geq M\right)\right)}{2\sqrt{M}}= -1.
\end{equation*}
\end{proposition}
The upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \ref{prop4.1}. Define, for any $c>0$,
\begin{equation}
    h(c):=\limsup_{M\to\infty} \frac{\log\left(\p\left(\max\limits_{1\leq s\leq \text{e}^{c\sqrt{M}}}g(s)\geq M\right)\right)}{2\sqrt{M}}.\label{eq:4.0}
\end{equation}
.

Question:
Let $(S_n)_{n\ge 0}$ be a discrete-time simple random walk on $\mathbb{Z}$ with $S_0=0$. For $s\in\mathbb{Z}$ and $n\in\mathbb{N}$, define the local time at $s$ up to time $n$ by
\[
\xi(s,n)=\#\{0\le j\le n:S_j=s\},
\]
and let
\[
g_1(n)=\#\{s\in\mathbb{Z}:\xi(s,n)=1\}
\]
be the number of sites visited exactly once by time $n$. Which statement holds with probability $1$ for this random walk?

\[\limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=\frac{1}{16}.\]

Choices:
B. \[\limsup_{n\to\infty}\frac{g_1(n)}{\log n}=\frac{1}{16}.\]
C. \[\limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}<\infty.\]
D. \[\lim_{n\to\infty}\frac{g_1(n)}{\log^2 n}=\frac{1}{16}.\]
E. \[\limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=\frac{1}{8}.\]

correct logarithmic scaling from rare-event excursion analysis

exact value and equality in the limsup, keeping only boundedness on the \log^2 n scale

limsup-only conclusion produced by atypical large excursions replaced by full limit

constant determined by the upper/lower rare-event exponent and excursion decomposition

The stem only defines the random-walk quantities and asks which asymptotic statement is true; it does not reveal the correct scaling, constant, or limsup form.

This is essentially a direct theorem-recall item: the correct option is the precise theorem statement itself rather than a derived consequence from intermediate facts given in the stem.

There is some reasoning pressure in distinguishing limsup vs limit, exact vs weaker conclusions, and the logarithmic scale, but selecting the exact constant largely depends on prior recall rather than generating a conclusion from supplied information.

The distractors are plausible and mathematically meaningful: wrong scaling (B), weaker true statement (C), unjustified strengthening to a full limit (D), and wrong constant (E). These reflect common theorem-misremembering and asymptotic-reasoning errors.

A solid recall-based theorem MCQ with strong distractors and no answer leakage, but it is mostly a direct restatement of a known result rather than a question that genuinely elicits generative mathematical reasoning.


\begin{theorem}[Lorea, 1979~\cite{lorea}]
\label{theorem:matroid}
Let $G = (V, E)$ be a graph, and define $\mathcal{I} = \{F \subseteq E : (V, F) \text{ is } (k, \ell)\text{-sparse} \}$.
Then $M = (E, \mathcal{I})$ is a matroid.
\FBOX
\end{theorem}

\label{sec:intro}
Throughout this paper, let $k$ and $\ell$ be non-negative integers with $\ell < 2k$, let $G = (V, E)$ be a loopless multigraph with $n = |V|$ vertices and $m = |E|$ edges, and assume that the edges $e_1, \dots, e_m$ are sorted in non-increasing order with respect to a given weight function.
The graph $G = (V, E)$ is called \emph{$(k, \ell)$-sparse} if, for each subset $X \subseteq V$, the number $i(X)$ of edges induced by $X$ is at most $\max\{k|X| - \ell, 0\}$.
Furthermore, if $G$ is $(k, \ell)$-sparse and has exactly $\max\{k |V| - \ell, 0\}$ edges, then we say that $G$ is \emph{$(k, \ell)$-tight}.
A graph is \emph{$(k, \ell)$-spanning} if it contains a $(k, \ell)$-tight subgraph that spans the entire vertex set.
A \emph{$(k, \ell)$-block} of a $(k, \ell)$-sparse graph is a subset $X \subseteq V$ that induces a $(k, \ell)$-tight subgraph.
A \emph{$(k, \ell)$-component} is an inclusion-wise maximal $(k, \ell)$-block.
These definitions form the foundation of a rich theory, closely connected to matroids and rigidity, and they naturally lead to fundamental optimization problems that are the focus of this paper.

\smallskip
The concept of $(k, \ell)$-sparse graphs was first introduced by Lorea~\cite{lorea} as part of his work on matroidal families.
Since then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science.
For example, $(k, k)$-tight graphs appeared in the work of Nash-Williams~\cite{nash1961edgeDisjointST} and Tutte~\cite{tutte} as a characterization of graphs that can be decomposed into $k$ edge-disjoint spanning trees.
A classical result in rigidity theory, due to Laman~\cite{laman}, characterizes generic minimally rigid bar-joint frameworks in the plane as $(2, 3)$-tight graphs, while the rigid graphs correspond to the $(2, 3)$-spanning graphs.
For a comprehensive overview of rigidity theory, see~\cite{jordan2016combinatorial,schulze2017rigidity}.

A wide range of optimization and decision problems can be reduced to the \emph{maximum-size $(k, \ell)$-sparse subgraph problem}, where $k$ and $\ell$ are fixed non-negative integers determined by the specific task.
The goal is to find a $(k, \ell)$-sparse subgraph of a given graph $G = (V, E)$ that contains the maximum number of edges.
A natural generalization is the \emph{maximum-weight $(k, \ell)$-sparse subgraph problem}, where the goal is to find a $(k, \ell)$-sparse subgraph with maximum total weight under a given weight function on the edges.
Many classical problems in graph theory can be viewed as special cases of this framework --- for example, finding a maximum-weight spanning tree, a maximum-weight subgraph that can be decomposed into $k$ forests, or a maximum-weight rigid spanning subgraph in the plane.

Both the maximum-size and the maximum-weight $(k, \ell)$-sparse subgraph problems can be solved using the pebble game algorithms introduced in~\cite{bergPhD, berg2003algorithms, hendrickson, pebble, pebbleDS}.
The formulation in~\cite{berg2003algorithms} provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper.
The naive implementation of these algorithms runs in $O(nm)$ time, where $n$ and $m$ denote the number of vertices and edges, respectively.
A widely accepted approach in the literature claimed to improve this to $O(n^2 + m)$ by employing a more sophisticated data structure~\cite{pebble,pebbleDS}, building on ideas in~\cite{berg2003algorithms}.
In recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed~\cite{madarasi2023klSparse, mihalyko2022augmentation}, and the question of whether such a time bound is truly achievable has remained open.

\smallskip
In our paper, we provide a positive answer to this question: we present an algorithm that solves the maximum-weight $(k, \ell)$-sparse subgraph problem in $O(n^2 + m)$ time.
Our approach combines a carefully designed data structure with a refined analysis, leading to both theoretical and practical advances.
As a direct corollary, the maximum-weight rigid subgraph problem --- corresponding to the well-known case $k = 2$, $\ell = 3$ --- can now be solved in quadratic time.
Beyond its standalone significance, the maximum-weight $(k, \ell)$-sparse subgraph problem also frequently arises as a subroutine in more complex combinatorial optimization problems.
For example, our algorithm substantially improves the running time of approximation algorithms for the minimum-weight redundantly rigid and globally rigid subgraph problems~\cite{jordan2020minimum} and their generalizations~\cite{mihalyko2022augmentation} in the metric case.
Further applications include enumerating non-crossing minimally rigid frameworks~\cite{avis2008enumerating}, as well as recognizing kinematic joints~\cite{john2012kinematic}.
An efficient implementation of the algorithms discussed in this paper is publicly available online~\cite{githubSparse}, and a detailed empirical evaluation can be found in~\cite{madarasi2025efficientKL}.

\smallskip
The rest of the paper is organized as follows.
Section~\ref{sec:naive_algo} provides an overview of the classical augmenting-path algorithm --- often referred to as the ``naive'' pebble game --- for computing maximum-size and maximum-weight $(k, \ell)$-sparse subgraphs.
In Section~\ref{sec:comp_algo}, we present our component-based variant of this algorithm, which forms the core of our contributions.
This high-level approach relies on two auxiliary procedures for managing $(k, \ell)$-components, whose implementation can be tailored to the specific setting of the problem.
Our main contribution is presented in Section~\ref{sec:alg_general}, where we give a complete implementation of these subroutines for the general range $0 \leq \ell < 2k$.
Combining structural properties of $(k, \ell)$-components with an efficient data structure, we prove that the resulting algorithm achieves a running time of $O(n^2 + m)$, improving on the $O(nm)$ bound of the naive approach.
Finally, Section~\ref{sec:spec_cases} addresses two notable special cases.
First, we examine the case $0 \leq \ell \leq k$, where the disjointness of $(k, \ell)$-components allows for a simplified implementation reducing the space complexity from $O(n^2)$ to $O(n)$.
Second, we address the unweighted variant for the general range, in which a vertex-wise edge-processing strategy retains the $O(n^2 + m)$ running time while requiring only $O(n)$ space.
These results together establish a unified algorithmic framework for efficiently solving the maximum-size and maximum-weight $(k, \ell)$-sparse subgraph problem across the full range of parameters.

\subsection{The naive augmenting-path algorithm}\label{sec:naive_algo}
We overview the classical, or ``naive'', version of the pebble game algorithm for finding a maximum-size or maximum-weight $(k, \ell)$-sparse subgraph.
We begin by stating two fundamental results that are essential for proving the correctness of the algorithm, and then present the algorithm itself.

\smallskip
The concept of $(k, \ell)$-sparse graphs was first introduced by Lorea~\cite{lorea} as part of his work on matroidal families.
Since then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science.
For example, $(k, k)$-tight graphs appeared in the work of Nash-Williams~\cite{nash1961edgeDisjointST} and Tutte~\cite{tutte} as a characterization of graphs that can be decomposed into $k$ edge-disjoint spanning trees.
A classical result in rigidity theory, due to Laman~\cite{laman}, characterizes generic minimally rigid bar-joint frameworks in the plane as $(2, 3)$-tight graphs, while the rigid graphs correspond to the $(2, 3)$-spanning graphs.
For a comprehensive overview of rigidity theory, see~\cite{jordan2016combinatorial,schulze2017rigidity}.

Both the maximum-size and the maximum-weight $(k, \ell)$-sparse subgraph problems can be solved using the pebble game algorithms introduced in~\cite{bergPhD, berg2003algorithms, hendrickson, pebble, pebbleDS}.
The formulation in~\cite{berg2003algorithms} provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper.
The naive implementation of these algorithms runs in $O(nm)$ time, where $n$ and $m$ denote the number of vertices and edges, respectively.
A widely accepted approach in the literature claimed to improve this to $O(n^2 + m)$ by employing a more sophisticated data structure~\cite{pebble,pebbleDS}, building on ideas in~\cite{berg2003algorithms}.
In recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed~\cite{madarasi2023klSparse, mihalyko2022augmentation}, and the question of whether such a time bound is truly achievable has remained open.

\smallskip
In our paper, we provide a positive answer to this question: we present an algorithm that solves the maximum-weight $(k, \ell)$-sparse subgraph problem in $O(n^2 + m)$ time.
Our approach combines a carefully designed data structure with a refined analysis, leading to both theoretical and practical advances.
As a direct corollary, the maximum-weight rigid subgraph problem --- corresponding to the well-known case $k = 2$, $\ell = 3$ --- can now be solved in quadratic time.
Beyond its standalone significance, the maximum-weight $(k, \ell)$-sparse subgraph problem also frequently arises as a subroutine in more complex combinatorial optimization problems.
For example, our algorithm substantially improves the running time of approximation algorithms for the minimum-weight redundantly rigid and globally rigid subgraph problems~\cite{jordan2020minimum} and their generalizations~\cite{mihalyko2022augmentation} in the metric case.
Further applications include enumerating non-crossing minimally rigid frameworks~\cite{avis2008enumerating}, as well as recognizing kinematic joints~\cite{john2012kinematic}.
An efficient implementation of the algorithms discussed in this paper is publicly available online~\cite{githubSparse}, and a detailed empirical evaluation can be found in~\cite{madarasi2025efficientKL}.

\smallskip
The rest of the paper is organized as follows.
Section~\ref{sec:naive_algo} provides an overview of the classical augmenting-path algorithm --- often referred to as the ``naive'' pebble game --- for computing maximum-size and maximum-weight $(k, \ell)$-sparse subgraphs.
In Section~\ref{sec:comp_algo}, we present our component-based variant of this algorithm, which forms the core of our contributions.
This high-level approach relies on two auxiliary procedures for managing $(k, \ell)$-components, whose implementation can be tailored to the specific setting of the problem.
Our main contribution is presented in Section~\ref{sec:alg_general}, where we give a complete implementation of these subroutines for the general range $0 \leq \ell < 2k$.
Combining structural properties of $(k, \ell)$-components with an efficient data structure, we prove that the resulting algorithm achieves a running time of $O(n^2 + m)$, improving on the $O(nm)$ bound of the naive approach.
Finally, Section~\ref{sec:spec_cases} addresses two notable special cases.
First, we examine the case $0 \leq \ell \leq k$, where the disjointness of $(k, \ell)$-components allows for a simplified implementation reducing the space complexity from $O(n^2)$ to $O(n)$.
Second, we address the unweighted variant for the general range, in which a vertex-wise edge-processing strategy retains the $O(n^2 + m)$ running time while requiring only $O(n)$ space.
These results together establish a unified algorithmic framework for efficiently solving the maximum-size and maximum-weight $(k, \ell)$-sparse subgraph problem across the full range of parameters.

\subsection{The naive augmenting-path algorithm}\label{sec:naive_algo}
We overview the classical, or ``naive'', version of the pebble game algorithm for finding a maximum-size or maximum-weight $(k, \ell)$-sparse subgraph.
We begin by stating two fundamental results that are essential for proving the correctness of the algorithm, and then present the algorithm itself.

We also need the following technical result, which can be derived from Hakimi's Orientation Lemma~\cite{SLHOrientationLemma}.
Here we give a direct proof.
Note that a similar lemma appeared as Lemma~1 in~\cite{berg2003algorithms} for $(k, \ell)=(2, 3)$.
\begin{lemma}\label{lemma:sparse_orientation}
Let $H = (V, F)$ be a $(k, \ell)$-sparse graph, and let $u,v \in V$ be two distinct vertices.
Suppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.
Then:
\begin{enumerate}
 \item If $\varrho_D(u) + \varrho_D(v) < 2k - \ell$, then $H' = (V, F \cup \{uv\})$ is also $(k, \ell)$-sparse.
 \item If $\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$, and there is no directed path in $D$ from any vertex $w \in V \setminus \{u, v\}$ with $\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \cup \{uv\})$ is not $(k, \ell)$-sparse.
\end{enumerate}
\end{lemma}
\begin{proof}
 We prove the two parts separately.

The correctness of this method immediately follows by Theorem~\ref{theorem:matroid} and Lemma~\ref{lemma:sparse_orientation}.
Algorithm~\ref{alg:basic} provides an implementation of this iterative approach.
\begin{algorithm}[H]
 \caption{Naive Pebble Game for $0 \leq \ell < 2k$}\label{alg:basic}
 \begin{algorithmic}[1]
 \Input An undirected graph $G = (V, E)$.
 \Require The edges $e_1, \dots, e_m$ of $G$ are sorted in non-increasing order of weight.
 \Output The edge set of a maximum-weight $(k, \ell)$-sparse subgraph.
 \vspace{0.2em}
 \hrule
 \vspace{0.2em}
 \Procedure{PebbleGame$_{k,\ell}$}{$G = (V, E)$}
 \State $F \gets \emptyset$ \Comment{Initialize the edge set of the optimal subgraph}
 \State $D \gets (V, \emptyset)$ \Comment{Initialize a directed graph on $V$ without arcs}
 \For{$e \gets e_1, \dots, e_m$} \Comment{Process the edges in non-increasing order of weight}\label{alg:basic:for}
 \State $u, v \gets \operatorname{endpoints}(e)$
 \While{$\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$}\label{alg:basic:while}
 \State find a path $P$ in $D$ from $\{w \in V \setminus \{u, v\} : \varrho_D(w) < k\}$ to $\{u, v\}$\label{alg:basic:findPath}
 \If{no such path exists}
 \State \Break \Comment{Exit the \texttt{while} loop}\label{alg:basic:reject}
 \EndIf
 \State reverse the arcs of $P$ in $D$ \Comment{Decrease the indegree of $u$ or $v$}\label{alg:basic:reverse}
 \EndWhile
 \If{$\varrho_D(u) + \varrho_D(v) < 2k - \ell$}\label{alg:basic:acceptIf}
 \State $F \gets F \cup \{e\}$ \Comment{Accept edge $e$}
 \If{$\varrho_D(v) < k$}
 \State $D \gets D \cup \{uv\}$ \Comment{Insert an arc from $u$ to $v$ in $D$}
 \Else
 \State $D \gets D \cup \{vu\}$ \Comment{Insert an arc from $v$ to $u$ in $D$}
 \EndIf
 \EndIf\label{alg:basic:acceptEndIf}
 \EndFor
 \State \Return $F$ \Comment{Return the edge set of the optimal subgraph}
 \EndProcedure
 \end{algorithmic}
\end{algorithm}
\begin{proposition}
Algorithm~\ref{alg:basic} runs in $O(nm)$ time.
\end{proposition}
\begin{proof}
The \texttt{while} loop in line~\ref{alg:basic:while} takes at most $\ell + 1$ iterations per edge, and each augmenting path $P$ can be found via a single traversal of $D$ in $O(n)$ time.
Thus, each edge is processed in $O(n)$ time overall, yielding a total running time of $O(nm)$.
\end{proof}

\begin{lemma}
\label{lemma:intersection}
Let $X$ and $Y$ be $(k, \ell)$-blocks of a $(k, \ell)$-sparse graph $H = (V, F)$ such that $|X \cap Y| \geq 2$.
Then both $X \cap Y$ and $X \cup Y$ are $(k, \ell)$-blocks.
\end{lemma}
\begin{proof}
Let $X, Y \subseteq V$ be distinct blocks of $H$ such that $|X \cap Y| \geq 2$.
Recall that $i_H(X)$ denotes the number of edges induced by $X$.
By the supermodularity of this function, we have
\[
 k(|X| + |Y|) - 2 \ell = i_H(X) + i_H(Y) \leq i_H(X \cap Y) + i_H(X \cup Y) \leq k|X \cap Y| + k|X \cup Y| - 2 \ell = k(|X| + |Y|) - 2 \ell.
\]
Since the left- and right-hand sides are equal, all inequalities must hold with equality, and hence $X \cap Y$ and $X \cup Y$ are blocks.
\end{proof}

The subroutines \textsc{InCommonComponent} and \textsc{UpdateComponents} will be described in full detail in the next section.
In Section~\ref{sec:spec_cases}, we present more efficient implementations tailored to two special cases.
Before presenting the implementation of \textsc{FindComponent} --- which is shared across all special cases --- we first analyze how the $(k, \ell)$-blocks of a $(k, \ell)$-sparse graph change when a new edge is inserted.
This analysis relies on the following two lemmas.
\begin{lemma}
\label{lemma:opt_cond}
Let $D$ be a $k$-indegree-bounded orientation of a $(k, \ell)$-sparse graph $H = (V, F)$ and let $u, v \in V$ be distinct vertices such that $\varrho_D(u) + \varrho_D(v) \leq 2k - \ell$.
A set $X \subseteq V$ containing both $u$ and $v$ is a $(k, \ell)$-block if and only if $\varrho_D(u) + \varrho_D(v) = 2k - \ell$, $\varrho_D(X) = 0$, and each vertex in $X \setminus \{u, v\}$ has indegree $k$.
\end{lemma}
\begin{proof}
 A set $X$ containing $u$ and $v$ is a block if and only if $i_H(X) = k|X| - \ell$.
 By the definition of $i_H(X)$, we have
 $$
 i_H(X) = \sum_{w \in X} \varrho_D(w) - \varrho_D(X) \leq \sum_{w \in X} \varrho_D(w) \leq k(|X| - 2) + 2k - \ell = k|X| - \ell.
 $$
Equality holds if and only if $\varrho_D(X) = 0$, $\varrho_D(u) + \varrho_D(v) = 2k - \ell$, and each vertex in $X \setminus \{u, v\}$ has indegree~$k$.
\end{proof}

Now we derive a necessary and sufficient condition for the existence of a component containing two chosen vertices $u$ and $v$.
A similar lemma was proven as Lemma~2 in~\cite{berg2003algorithms}.
\begin{lemma}
 \label{lemma:new_comp_arises}
 Let $D$ be a $k$-indegree-bounded orientation of a $(k, \ell)$-sparse graph $H = (V, F)$ and let $u, v \in V$ be distinct vertices such that $\varrho_D(u) + \varrho_D(v) = 2k - \ell$.
 Define $S = \{w \in V \setminus \{u, v\} : \varrho_D(w) < k\}$, and let $T$ be the set of vertices not reachable from $S$ in $D$.
 There exists a $(k, \ell)$-component containing $u$ and $v$ if and only if both $u$ and $v$ lie in $T$.
 If such a component exists, then it is exactly $T$.
\end{lemma}
\begin{proof}
Suppose that $u, v \in T$.
Since $S$ and $T$ are disjoint, we have $\varrho_D(w) = k$ for each $w \in T \setminus \{u,v\}$.
Furthermore, $T$ has no incoming edges, so $\varrho_D(T)=0$.
By Lemma~\ref{lemma:opt_cond}, $T$ is a $(k, \ell)$-block, and hence $u$ and $v$ lie in a common component.

\begin{algorithm}[H]
\caption{Update Components for $0 \leq \ell < 2k$}
\label{alg:update1}
 \begin{algorithmic}[1]
 \Input A newly formed $(k, \ell)$-component $C$.
 \Effect Updates $M$ and $\mathcal{C}$ to restore the invariants above.
 \vspace{0.2em}
 \hrule
 \vspace{0.2em}
 \Procedure{UpdateComponents$_{k, \ell}$}{$C$}
 \State $U \gets \emptyset$ \Comment{Initialize the union of the components contained in $C$}
 \State $\mathcal{C}' \gets \{C\}$ \Comment{Initialize the new set of components}
 \For{$X \in \mathcal{C}$}
 \If{$X \subseteq C$} \Comment{Component contained in $C$?}
 \label{alg:update1:if_subset}
 \For{$(u, v) \in (U \setminus X) \times (X \setminus U)$} \Comment{Mark vertex pairs between $U$ and $X$}
 \label{alg:update1:for_descartes}
 \State $M_{u, v} \gets M_{v, u} \gets 1$
 \EndFor
 \State $U \gets U \cup X$ \Comment{Merge $X$ into $U$}
 \label{alg:update1:merge}
 \Else
 \State $\mathcal{C}' \gets \mathcal{C}' \cup \{X\}$ \Comment{Append $X$ to $\mathcal{C}'$}
 \label{alg:update1:append_comp}
 \EndIf
 \EndFor
 \For{$(u, v) \in U \times (C \setminus U)$} \Comment{Mark pairs with exactly one vertex in $U$}
 \label{alg:update1:for_descartes_2}
 \State $M_{u, v} \gets M_{v, u} \gets 1$
 \EndFor
 \For{$(u, v) \in (C \setminus U) \times (C \setminus U)$} \Comment{Mark pairs completely outside $U$}
 \label{alg:update1:for_descartes_3}
 \State $M_{u, v} \gets 1$
 \EndFor
 \State $\mathcal{C} \gets \mathcal{C}'$ \Comment{Update $\mathcal{C}$ to contain the components of the new graph}
 \EndProcedure
 \end{algorithmic}
\end{algorithm}

We also need the following technical result, which can be derived from Hakimi's Orientation Lemma~\cite{SLHOrientationLemma}.
Here we give a direct proof.
Note that a similar lemma appeared as Lemma~1 in~\cite{berg2003algorithms} for $(k, \ell)=(2, 3)$.
\begin{lemma}\label{lemma:sparse_orientation}
Let $H = (V, F)$ be a $(k, \ell)$-sparse graph, and let $u,v \in V$ be two distinct vertices.
Suppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.
Then:
\begin{enumerate}
    \item If $\varrho_D(u) + \varrho_D(v) < 2k - \ell$, then $H' = (V, F \cup \{uv\})$ is also $(k, \ell)$-sparse.
    \item If $\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$, and there is no directed path in $D$ from any vertex $w \in V \setminus \{u, v\}$ with $\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \cup \{uv\})$ is not $(k, \ell)$-sparse.
\end{enumerate}
\end{lemma}
\begin{proof}
  We prove the two parts separately.

  \smallskip
  1.\ Since $H$ is $(k, \ell)$-sparse, it suffices to verify the sparsity condition for subsets $X \subseteq V$ containing both $u$ and $v$, as these are the only sets affected by the insertion of the edge $uv$.
  For any such $X$,
  \[
    i_{H'}(X) = i_H(X) + 1 \leq \sum_{w \in X} \varrho_D(w) + 1 \leq k(|X| - 2) + (2k - \ell - 1) + 1 = k|X| - \ell,
  \]
  which confirms that $H'$ remains $(k, \ell)$-sparse.

  \smallskip
  2.\ Let $T \subseteq V$ denote the set of vertices from which $u$ or $v$ is reachable in $D$.
  Clearly, $\varrho_D(T) = 0$, and by assumption, $\varrho_D(w) = k$ for each $w \in T \setminus \{u,v\}$.
  Hence,
  \[
    i_{H'}(T) = i_H(T) + 1 = \sum_{w \in T} \varrho_D(w) - \varrho_D(T) + 1 \geq (2k - \ell) + k(|T| - 2) + 1 = k|T| - \ell + 1,
  \]
  which violates the sparsity condition, proving that $H'$ is not $(k, \ell)$-sparse.
\end{proof}

\paragraph{Checking edge acceptability via orientations.}
Given a $k$-indegree-bounded orientation $D$ of $H$, Lemma~\ref{lemma:sparse_orientation} yields an algorithm for checking whether the insertion of an edge $uv$ preserves $(k, \ell)$-sparsity.
While $\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$, we find a directed path $P$ in $D$ from a vertex in $\{w \in V \setminus \{u, v\} : \varrho_D(w) < k\}$ to either $u$ or $v$.
If no such path exists, then the insertion of $uv$ violates the sparsity condition.
Otherwise, we reverse the arcs of $P$, which decreases $\varrho_D(u) + \varrho_D(v)$ without violating the $k$-indegree bound.
Repeating this process at most $\ell + 1$ times determines whether the edge $uv$ can be inserted.
$\bullet$
\medskip

We are now ready to describe the augmenting-path algorithm for finding a maximum-weight $(k, \ell)$-sparse subgraph.
The algorithm is built on two key ideas:
\begin{itemize}
\item We iteratively construct the optimal subgraph $H = (V, F)$, considering the edges one by one in the order $e_1, \dots, e_m$.
  Each edge $e$ is inserted into $H$ if and only if the resulting subgraph $H' = (V, F \cup \{e\})$ remains $(k, \ell)$-sparse.
\item We maintain a $k$-indegree-bounded orientation $D$ of the current subgraph $F$.
  For each edge $e$, we apply Lemma~\ref{lemma:sparse_orientation} together with the orientation-based procedure described above to determine whether the edge can be accepted.
  If $H' = (V, F \cup \{e\})$ is $(k, \ell)$-sparse, then we insert $e$ into $H$ and update $D$ by inserting $e$ with an orientation that preserves the $k$-indegree bound.
\end{itemize}

The correctness of this method immediately follows by Theorem~\ref{theorem:matroid} and Lemma~\ref{lemma:sparse_orientation}.
Algorithm~\ref{alg:basic} provides an implementation of this iterative approach.
\begin{algorithm}[H]
  \caption{Naive Pebble Game for $0 \leq \ell < 2k$}\label{alg:basic}
  \begin{algorithmic}[1]
    \item[\textbf{Input:}] An undirected graph $G = (V, E)$.
    \Require The edges $e_1, \dots, e_m$ of $G$ are sorted in non-increasing order of weight.
    \item[\textbf{Output:}] The edge set of a maximum-weight $(k, \ell)$-sparse subgraph.
    \vspace{0.2em}
    \hrule
    \vspace{0.2em}
    \Procedure{PebbleGame$_{k,\ell}$}{$G = (V, E)$}
      \State $F \gets \emptyset$ \Comment{Initialize the edge set of the optimal subgraph}
      \State $D \gets (V, \emptyset)$ \Comment{Initialize a directed graph on $V$ without arcs}
      \For{$e \gets e_1, \dots, e_m$} \Comment{Process the edges in non-increasing order of weight}\label{alg:basic:for}
        \State $u, v \gets \operatorname{endpoints}(e)$
      	\While{$\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$}\label{alg:basic:while}
      	  \State find a path $P$ in $D$ from $\{w \in V \setminus \{u, v\} : \varrho_D(w) < k\}$ to $\{u, v\}$\label{alg:basic:findPath}
      	  \If{no such path exists}
      	    \State \Break \Comment{Exit the \texttt{while} loop}\label{alg:basic:reject}
      	  \EndIf
      	  \State reverse the arcs of $P$ in $D$ \Comment{Decrease the indegree of $u$ or $v$}\label{alg:basic:reverse}
      	\EndWhile
      	\If{$\varrho_D(u) + \varrho_D(v) < 2k - \ell$}\label{alg:basic:acceptIf}
          \State $F \gets F \cup \{e\}$ \Comment{Accept edge $e$}
      	  \If{$\varrho_D(v) < k$}
      	    \State $D \gets D \cup \{uv\}$ \Comment{Insert an arc from $u$ to $v$ in $D$}
      	  \Else
      	    \State $D \gets D \cup \{vu\}$ \Comment{Insert an arc from $v$ to $u$ in $D$}
      	  \EndIf
      	\EndIf\label{alg:basic:acceptEndIf}
      \EndFor
      \State \Return $F$ \Comment{Return the edge set of the optimal subgraph}
    \EndProcedure
  \end{algorithmic}
\end{algorithm}
\begin{proposition}
Algorithm~\ref{alg:basic} runs in $O(nm)$ time.
\end{proposition}
\begin{proof}
The \texttt{while} loop in line~\ref{alg:basic:while} takes at most $\ell + 1$ iterations per edge, and each augmenting path $P$ can be found via a single traversal of $D$ in $O(n)$ time.
Thus, each edge is processed in $O(n)$ time overall, yielding a total running time of $O(nm)$.
\end{proof}

The post-theorem text does not give a proof sketch of Theorem~\ref{theorem:matroid}. It only states that “The correctness of this method immediately follows by Theorem~\ref{theorem:matroid} and Lemma~\ref{lemma:sparse_orientation},” and then describes an algorithmic procedure based on maintaining a $k$-indegree-bounded orientation and using Lemma~\ref{lemma:sparse_orientation} to decide whether adding an edge preserves $(k,\ell)$-sparsity.

The post-theorem text does not give a proof sketch of Theorem~\ref{theorem:matroid}. It only states that “The correctness of this method immediately follows in establishing the main theorem, and we first use the following lemma.

\begin{lemma}\label{lemma:sparse_orientation}
Let $H = (V, F)$ be a $(k, \ell)$-sparse graph, and let $u,v \in V$ be two distinct vertices.
Suppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.
Then:
\begin{enumerate}
    \item If $\varrho_D(u) + \varrho_D(v) < 2k - \ell$, then $H' = (V, F \cup \{uv\})$ is also $(k, \ell)$-sparse.
    \item If $\varrho_D(u) + \varrho_D(v) \geq 2k - \ell$, and there is no directed path in $D$ from any vertex $w \in V \setminus \{u, v\}$ with $\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \cup \{uv\})$ is not $(k, \ell)$-sparse.
\end{enumerate}
\end{lemma},”

and then describes an algorithmic procedure based on maintaining a $k$-indegree-bounded orientation and using the lemma above to decide whether adding an edge preserves $(k,\ell)$-sparsity.,

[Lorea, 1979~\cite{lorea}]
\label{theorem:matroid}
Let $G = (V, E)$ be a graph, and define $\mathcal{I} = \{F \subseteq E : (V, F) \text{ is } (k, \ell)\text{-sparse} \}$.
Then $M = (E, \mathcal{I})$ is a matroid.
\FBOX,

Question:
Fix integers $k$ and $\ell$ with $0 \le \ell < 2k$. For a graph $G=(V,E)$, define
\[
\mathcal I=\{F\subseteq E : (V,F)\text{ is }(k,\ell)\text{-sparse}\},
\]
where a graph $(V,F)$ is $(k,\ell)$-sparse if every vertex subset $X\subseteq V$ with $|X|\ge 2$ spans at most $k|X|-\ell$ edges of $F$. Which statement holds for every graph $G$ under this construction?

The set system $M=(E,\mathcal I)$ is a matroid; equivalently, the edge sets of $(k,\ell)$-sparse spanning subgraphs of $G$ form the independent sets of a matroid on ground set $E$.

Choices:
B. The set system $M=(E,\mathcal I)$ is a matroid whenever $0\le \ell\le 2k$; equivalently, the edge sets of $(k,\ell)$-sparse spanning subgraphs of $G$ form the independent sets of a matroid on ground set $E$ for the entire boundary range $\ell=2k$ as well.
C. The set system $M=(E,\mathcal I)$ is hereditary: if $F\in\mathcal I$ and $F'\subseteq F$, then $F'\in\mathcal I$.
D. The set system $M=(E,\mathcal I)$ is a greedoid on ground set $E$; equivalently, every nonempty $(k,\ell)$-sparse edge set admits an ordering in which each initial segment is again $(k,\ell)$-sparse.
E. For every graph $G=(V,E)$, the inclusion-maximal members of $\mathcal I$ all have the same cardinality, and hence every maximal $(k,\ell)$-sparse spanning subgraph of $G$ has the same number of edges.

strict parameter range $0\le \ell<2k$ replaced by boundary case $\ell=2k$

drops augmentation/exchange structure and keeps only downward closure of sparsity

confuses matroid independence with greedoid accessibility/feasible-order property

replaces full matroid conclusion by equicardinality of maximal sparse sets, omitting augmentation and overstating consequence as standalone theorem

The stem defines the sparsity system and asks which universal statement is true, but it does not explicitly reveal that the matroid conclusion is correct. There is no direct textual leakage of choice A.

Choice A is essentially the standard theorem for $(k,\ell)$-sparse graphs under the exact hypothesis $0\le \ell<2k$. The item mainly asks the test-taker to recognize the theorem rather than infer a new consequence.

Some reasoning is needed to separate the full matroid statement from weaker true properties like heredity and from tempting near-misses such as the boundary case $\ell=2k$. However, the item is still driven mostly by theorem recall rather than substantial generative reasoning.

The distractors are strong and mathematically meaningful: B tests the sharp parameter range, C is a weaker true property, D confuses matroid and greedoid structure, and E isolates a matroid-like consequence without full justification. These reflect realistic failure modes.

A mathematically well-constructed MCQ with strong distractors, but it is largely a direct theorem-recognition question and therefore weak on tautology avoidance and only moderate on generative reasoning.


Let $\mathbb{D}$ be the unit disk, let $f \colon \mathbb{D} \to \mathbb{D}$ be
a holomorphic self map of $\mathbb{D}$, and consider the dynamical system given by its iterates $\{ f^n \}_{n\in \mathbb{N}}$.
The Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \in \overline{\mathbb{D}}$ such that the orbit of every point in $\mathbb{D}$ converges to $p$, that is, $f^n(z) \to p$ as $n \to \infty$ for every $z \in \mathbb{D}$. We say that $p$ is the {\em Denjoy-Wolff point} of $f$.

In the case where $f$ is an {\em inner function}, that is, for $\lambda$-almost every $\zeta\in\partial \mathbb{D}$ the radial limit \[ f^*(\zeta)=\lim_{r\to1} f(r\zeta)\] belongs to $\partial\mathbb{D}$ (where $\lambda$ denotes the normalized Lebesgue measure on $\partial\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension
$$
f^{*} \colon \partial\mathbb{D} \longrightarrow \partial\mathbb{D}.
$$
Assuming that the Denjoy-Wolff point $p$ lies in $\partial\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.
In the seminal work of Aaronson, Doreing and Mañé \cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\lambda$-almost every point on $\partial\mathbb{D}$ converges to the Denjoy-Wolff point $p\in\partial \mathbb{D}$ if and only if \[\sum_{n\geq 0} 1-|f^n(0|<\infty.\]

It is well-known that {\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\in (0,1)$, see Section \ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which
rate do these orbits approach the Denjoy–Wolff point.

If the Denjoy-Wolff point $p\in\partial \mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\partial \mathbb{D}$ in a neighborhood of $p$.

The case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \coloneqq \{ z \in \mathbb{C}\colon a <|z-p| < b\}$, we prove the following. \\

In the case where $f$ is an {\em inner function}, that is, for $\lambda$-almost every $\zeta\in\partial \mathbb{D}$ the radial limit \[ f^*(\zeta)=\lim_{r\to1} f(r\zeta)\] belongs to $\partial\mathbb{D}$ (where $\lambda$ denotes the normalized Lebesgue measure on $\partial\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension
$$
f^{*} \colon \partial\mathbb{D} \longrightarrow \partial\mathbb{D}.
$$
Assuming that the Denjoy-Wolff point $p$ lies in $\partial\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.
In the seminal work of Aaronson, Doreing and Mañé \cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\lambda$-almost every point on $\partial\mathbb{D}$ converges to the Denjoy-Wolff point $p\in\partial \mathbb{D}$ if and only if \[\sum_{n\geq 0} 1-|f^n(0|<\infty.\]

If the Denjoy-Wolff point $p\in\partial \mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\partial \mathbb{D}$ in a neighborhood of $p$.

Without being precise, we say that a shrinking target is a collection of arcs of $\partial\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n})$ in Theorem \ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.
\

In this section, we prove Theorem \ref{A}, which states that if $f \colon \mathbb{D} \to \mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$, $\alpha = f'(p) \in (0,1)$, then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.

\begin{lemma}\label{cotas+}
 Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic self map of $\mathbb{D}$ and let $p \in \partial\mathbb{D}$ be its Denjoy-Wolff point. Let $\alpha = f'(p)$. Then, for every $\delta > 0$, there exists a real constant $C \geq 1$ such that $$\frac{1}{C} (\alpha - \delta)^n \leq |f^n(0) - p| \leq C \alpha^n,$$ for $n$ large enough.

We now prove that given $0 < \varepsilon < 1$, $$(f^*)^n(\zeta) \in D(p, \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$, where $\alpha = f'(p) \in (0,1)$.

Consider the arc $J_n(\varepsilon) = D(p,\alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$, $n \in \mathbb{N}$, and define
 \begin{align*}
 E(\varepsilon) &\coloneqq \{ \zeta \in \partial\mathbb{D}\colon f^n(\zeta) \in J_n(\varepsilon) \text{\ for all $n$ large enough}\}\\
 &= \{ \zeta \in \partial\mathbb{D} \colon (f^n(\zeta)) \text{\ fails to hit\ } (J_n(\varepsilon)^c)\},
 \end{align*}
 where $J_n(\varepsilon)^c = \partial\mathbb{D}\setminus J_n(\varepsilon)$. We will prove that $E(\varepsilon)$ has full measure.

We finally prove that given $0 < \varepsilon < 1$, $$(f^*)^n(\zeta) \not \in D(p, \alpha^{(1+\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.

\begin{teo}\label{A}
        Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$. Let $\alpha = f'(p) \in (0,1)$. Then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.
\end{teo}

\
The proof of the theorem is  based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties (see Section \ref{sec:shrinking}), developed in \cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0.

Without being precise, we say that a shrinking target is a collection of  arcs of $\partial\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs  $A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n})$ in Theorem \ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.
\

{\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.

The proof of Theorem~\ref{A} is "based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties" (Section~\ref{sec:shrinking}), as developed in \cite{benini_shrinking_2024}. The arcs $A(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n})$ appearing in Theorem~\ref{A} "form a shrinking target", and the goal is to "conclude that it is hit almost surely" by boundary orbits. "Via Möbius transformations", the authors "convert" the autonomous system into "a non-autonomous one fixing $0$" in order to "apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely". The argument also "make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions."

The proof of Theorem~\ref{A} is "based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties" (Section~\ref{sec:shrinking}), as developed in \cite{benini_shrinking_2024}. The arcs $A(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n})$ appearing in the main theorem "form a shrinking target", and the goal is to "conclude that it is hit almost surely" by boundary orbits. "Via Möbius transformations", the authors "convert" the autonomous system into "a non-autonomous one fixing $0$" in order to "apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely". The argument also "make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions."

\label{A}
        Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$. Let $\alpha = f'(p) \in (0,1)$. Then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.,

Question: Let \(\mathbb D\) be the unit disk, let \(\partial\mathbb D\) be its boundary, and let \(\lambda\) denote the normalized Lebesgue measure on \(\partial\mathbb D\). Suppose \(f:\mathbb D\to\mathbb D\) is a holomorphic self-map that is an inner function, meaning that for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) the radial limit \(f^*(\zeta)=\lim_{r\to 1^-}f(r\zeta)\) exists and lies in \(\partial\mathbb D\). Assume that \(f\) is hyperbolic with Denjoy-Wolff point \(p\in\partial\mathbb D\), so that \(f^n(z)\to p\) for every \(z\in\mathbb D\), and let \(\alpha=f'(p)\in(0,1)\). For \(a,b>0\), define \(A(p;a,b)=\{z\in\mathbb C: a<|z-p|<b\}\). Which statement holds for every such map \(f\)?

For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n}\bigr)\cap\partial\mathbb D.
\]
Equivalently, for all sufficiently large \(n\),
\[
\alpha^{(1+\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}.
\]

Choices:
B.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1-\varepsilon)n},\alpha^{(1+\varepsilon)n}\bigr)\cap\partial\mathbb D.
\]
Equivalently, for all sufficiently large \(n\),
\[
\alpha^{(1-\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1+\varepsilon)n}.
\]
C.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N\) such that for all \(n\ge N\),
\[
\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}.
\]
Equivalently, \((f^*)^n(\zeta)\) eventually lies in the shrinking arc \(D\bigl(p,\alpha^{(1-\varepsilon)n}\bigr)\cap\partial\mathbb D\).
D.
There exists \(\varepsilon_0\in(0,1)\) such that for every hyperbolic inner function \(f\) as above and for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\), there exists \(N\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1+\varepsilon_0)n},\alpha^{(1-\varepsilon_0)n}\bigr)\cap\partial\mathbb D.
\]
Equivalently, one can choose the same exponent gap \(\varepsilon_0\) uniformly for all such maps.
E.
For every \(\varepsilon\in(0,1)\), for every \(\zeta\in\partial\mathbb D\) there exists \(N\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n}\bigr)\cap\partial\mathbb D.
\]
Equivalently, every boundary orbit eventually satisfies
\[
\alpha^{(1+\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}.
\]

correct annulus orientation from shrinking target scales

dropped the lower bound excluding points too close to the Denjoy-Wolff point

quantification over epsilon depending on the map versus a universal epsilon

almost-sure hitting replaced by pointwise convergence for every boundary point

The stem provides background and hypotheses but does not explicitly reveal the asymptotic annulus estimate. The correct answer is not stated or trivially implied by the wording.

The item is not tautological relative to the stem, but the correct option is essentially a precise theorem statement under the given hypotheses. It mainly asks the student to recognize the exact conclusion rather than derive a novel one.

There is moderate reasoning pressure: the student must distinguish subtle asymptotic, measure-theoretic, and quantifier differences ('almost every' vs 'every', two-sided annulus bounds vs one-sided bound, map-dependent epsilon vs universal epsilon, correct exponent ordering). However, it still leans more toward theorem recognition than genuine construction or derivation.

The distractors are strong and mathematically meaningful. They reflect realistic failure modes: reversed annulus orientation, a weaker true statement, an invalid uniformity claim, and an overstrong pointwise claim.

A solid high-level MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than fully generative mathematical reasoning.


\begin{theorem}
\label{thm-main}
	 For $k\geq 1$, the Bergman kernel $\rho_k$ of $H^0(X,L^k)$ at each $p\in X$ satisfies
	 \[\rho_k(p)=(\frac{k}{2\pi})^n\left(1+\sum_{v\in \Lambda\setminus \{0\}}e^{-\frac{k}{4}(|v|_{H})^2}\cos\big(2\pi k\alpha_v(p)\big)\right). \]

\end{theorem}

\begin{proposition}\label{prop-surj}
	 $\Phi_\vcal$ is surjective.
\end{proposition}

Let $(M,\omega)$ be a K\"{a}hler  manifold of complex dimension $n$, and let $L\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\omega$. The Bergman space $\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:
\[
\int_M |s|_{h}^2\,\frac{\omega^n}{n!}<\infty.
\]
With the $L^2$--inner product induced by $h$ and $\omega$, $\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\to\hcal_k$, characterized by
\[
s(y)=\int_M \langle s(x),K_k(x,y)\rangle_{h^k}\,\frac{\omega^n(x)}{n!}\qquad(\forall s\in\hcal_k).
\]
The diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,
\[
\rho_k(x)=|K_k(x,x)|_{h^k}.
\]

For trivial bundles over domains in $\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.

The Bergman kernel has been intensively studied for domains in $\mathbb C^n$ and for K\"{a}hler  manifolds (see, e.g., \cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,
\[
\rho_k(x)\sim\Big(\frac{k}{2\pi}\Big)^n\Big(1+\frac{b_1(x)}{k}+\frac{b_2(x)}{k^2}+\cdots\Big),
\]
where each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \cite{MM,dailiuma}).

In \cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.
 In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.

\

Let $X=\mathbb C^n/\Lambda$ be an abelian variety over $\mathbb C$. Here, $\Lambda$ is a lattice in $\mathbb C^n$. Let $L\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\mathbb C^n$ such that $\Im H(\Lambda,\Lambda)\subset \mathbb Z$. For each vector $v\in \mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.
Let $\omega=\pi \sqrt{-1} \sum H_{ij} dz^i\wedge d\bar{z}^j \in 2\pi c_1(L)$ be a flat K\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \omega$.

For each $p\in X$, $\exists \tilde{p}\in\mathbb C^n$ such that $p=\pi(\tilde{p})$. Such a $\tilde{p}$ is called a lift of $p$.
For each $v\in \Lambda$, let $\gamma_{p,v}\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\frac{\partial}{\partial a}$, where $a$ is the parameter of the line segment $\tilde{p}+av$, $a\in[0,1]$. It is easy to see that $\gamma_{p,v}$ does not depend on the choice of $\tilde{p}$, and that the length of $\gamma_{p,v}$ is $|v|_H$. Let $\nabla$ be the Chern connection of $(L,h)$.
 Let $e^{2\pi i\alpha_v(p)}=\text{Hol}_{L}(\gamma_{p,v})$ be the holonomy of $\nabla$ along $\gamma_{p,v}$.
Our first result is the following formula:

Let $(M,\omega)$ be a K\"{a}hler manifold of complex dimension $n$, and let $L\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\omega$. The Bergman space $\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:
\[
\int_M |s|_{h}^2\,\frac{\omega^n}{n!}<\infty.
\]
With the $L^2$--inner product induced by $h$ and $\omega$, $\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\to\hcal_k$, characterized by
\[
s(y)=\int_M \langle s(x),K_k(x,y)\rangle_{h^k}\,\frac{\omega^n(x)}{n!}\qquad(\forall s\in\hcal_k).
\]
The diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,
\[
\rho_k(x)=|K_k(x,x)|_{h^k}.
\]

For trivial bundles over domains in $\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.

The Bergman kernel has been intensively studied for domains in $\mathbb C^n$ and for K\"{a}hler manifolds (see, e.g., \cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,
\[
\rho_k(x)\sim\Big(\frac{k}{2\pi}\Big)^n\Big(1+\frac{b_1(x)}{k}+\frac{b_2(x)}{k^2}+\cdots\Big),
\]
where each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \cite{MM,dailiuma}).

In \cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.
 In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.

Let $X=\mathbb C^n/\Lambda$ be an abelian variety over $\mathbb C$. Here, $\Lambda$ is a lattice in $\mathbb C^n$. Let $L\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\mathbb C^n$ such that $\Im H(\Lambda,\Lambda)\subset \mathbb Z$. For each vector $v\in \mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.
Let $\omega=\pi \sqrt{-1} \sum H_{ij} dz^i\wedge d\bar{z}^j \in 2\pi c_1(L)$ be a flat K\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \omega$.

For each $p\in X$, $\exists \tilde{p}\in\mathbb C^n$ such that $p=\pi(\tilde{p})$. Such a $\tilde{p}$ is called a lift of $p$.
For each $v\in \Lambda$, let $\gamma_{p,v}\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\frac{\partial}{\partial a}$, where $a$ is the parameter of the line segment $\tilde{p}+av$, $a\in[0,1]$. It is easy to see that $\gamma_{p,v}$ does not depend on the choice of $\tilde{p}$, and that the length of $\gamma_{p,v}$ is $|v|_H$. Let $\nabla$ be the Chern connection of $(L,h)$.
 Let $e^{2\pi i\alpha_v(p)}=\text{Hol}_{L}(\gamma_{p,v})$ be the holonomy of $\nabla$ along $\gamma_{p,v}$.
Our first result is the following formula:

Lastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\in X$ let $\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).
\begin{theorem}\label{thm-off-diag}
 Let $(X,\omega,L,h)$ be a polarized abelian variety as in Theorem \ref{thm-main}. For every $k\ge 1$ and every $x\neq y$,
 \[
 |K_k(x,y)|_{h^k}\;\le\;\Big(\frac{k}{2\pi}\Big)^{\!n}\sum_{\gamma\in\mathfrak{G}_{x,y}} e^{-\frac{k}{4}\,\ell(\gamma)^2},
 \]
 where $\ell(\gamma)$ is the length of $\gamma$.
\end{theorem}

Let $K_k(z,w)$ be the Bergman kernel of $\hcal_{\C^n,k}$. Then we can write \[K_k(z,w)=\sum_{i=1}^{\infty}f_i(z)\otimes \bar{f}_i(w),\] where $\{f_i\}_{i=1}^\infty$ is an orthonormal basis of $\hcal_{\C^n,k}$.
For each $v\in \Lambda$, we define \[K^v_k(z,w)=\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w).\]
\begin{proposition}
 $K^v_k(z,w)=\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w)$ is independent of the choice of orthonormal basis $\{f_i\}_{i=1}^\infty$ of $\hcal_{\C^n,k}$.
\end{proposition}
\begin{proof}
 For any $f\in \hcal_{\C^n,k}$, we have $f=\sum_{i=1}^{\infty}c_i A_v^*f_i$, since $\{A_v^*f_i\}$ is also an orthonormal basis of $\hcal_{\C^n,k}$. Then for any fixed $w$, \[\int_{\C^n} (f,\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w))_{\pi^*h}\frac{w_0^n}{n!}=\sum_{i=1}^{\infty}c_i f_i(w)=(A_v^{-1})^*f(w), \]
 namely, $K^v_k(z,w)$ is the representation element for the functional $\val_w\circ A_{-v}^*$, hence independent of the choice of the orthonormal basis.
\end{proof}
\begin{defandthm}\label{def-and-thm-k-gamma}
 We define
\[K_{k}^\Lambda(z,w)\eqd \sum_{v\in \Lambda}K^v_k(z,w). \]
Then for $k>0$, and for each fixed $w$, it converges locally uniformly and absolutely. Moreover, we have that for $k>0$, $\sum_{v\in \Lambda}K^v_k(z,w)\in W_k$ for each fixed $w$.
\end{defandthm}
\begin{proof}
 Since $K^v_k(z,w)$ is independent of the choice of orthonormal basis, for each fixed $w$, we can choose the orthonormal basis $\{f_i\}$ such that $f_1=s_w$, a peak section at $w$. So $f_i(w)=0$ for $i>1$. So we have \[K^v_k(z,w)=A_v^*s_w(z)\otimes \bar{s}_w(w).\]
 So by Lemma \ref{lem-sum-g-f-0}, we have $\sum_{v\in \Lambda}K^v_k(z,w)$ converges locally uniformly and absolutely. And by Lemma \ref{lem-sum-g-f-0-in-v-k}, $\sum_{v\in \Lambda}K^v_k(z,w)\in W_k$ for each fixed $w$.
\end{proof}
\begin{lemma}\label{lem-int-s-sum-g-s-p}
 For any $s\in W_k$, we have
 \[\int_{F_\Lambda}(s,\sum_{v\in \Lambda}A_v^*s_p)_{\pi^*h}\frac{\omega_0^n}{n!}=(\frac{2\pi}{k})^{n/2}\frac{s(p)}{s_p(p)}.\]
\end{lemma}
\begin{proof}
 It suffices to show this for $p=0$. By Lemmas \ref{lem-sum-g-f-0} and \ref{lem-sum-g-f-0-in-v-k},
 \[ \int_{F_\Lambda}(s,\sum_{v\in \Lambda}A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}=\sum_{v\in \Lambda}\int_{F_\Lambda}(s,A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}.\]
 On the other hand, since $s_0=(\frac{k}{2\pi})^{n/2}\bm{e}_k$ is $L_1$ integrable on $\C^n$, and $s$ is bounded on $\C^n$, we have $(s,s_0)_{\pi^*h}\in L^1(\C^n,\frac{\omega_0^n}{n!})$. So \[\sum_{v\in \Lambda}\int_{F_\Lambda}(s,A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}=\int_{\C^n} (s,s_0)_{\pi^*h}\frac{\omega_0^n}{n!}.\]
 Then by taking the Taylor expansion of $\frac{s}{\bm{e}_k}$, we get the right hand side equals $(\frac{2\pi}{k})^{n/2}\frac{s(0)}{\bm{e}_k(0)}=\frac{s(0)}{s_0(0)}$.
\end{proof}
\begin{theorem}\label{thm-k-gamma-bergman}
 For $k>0$, $K_{k}^\Lambda(z,w)$ is the Bergman kernel of $W_k$.
\end{theorem}
\begin{proof}
 For fixed $w$, we have \[K_{k}^\Lambda(z,w)=(\sum_{v\in \Lambda}A_v^*s_w(z))\otimes \bar{s}_w(w).\]
 So for each $S\in \hcal_{X,k}$, we have \[\int_{F_\Lambda} (S,K_{k}^\Lambda(z,w))_{\pi^*h}\frac{\omega_0^n}{n!}=\frac{S(w)}{s_w(w)}s_w(w)=S(w).\]
 We have proved the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-main}]
 For a point \(p\in X\) choose a lift \(\tilde p\in\C^n\); without loss of generality we may take \(\tilde p=0\). By Theorem \ref{thm-k-gamma-bergman} we have the identity
 \[
 \rho_{X,k}(0)=K_{k}^\Lambda(0,0)
 = (\frac{k}{2\pi})^n \;+\; \sum_{v\in\Lambda\setminus\{0\}} K^v_k(0,0).
 \]
 Recall that an element $v\in \Lambda$ is primitive if it can not be written as $v=mu$ for some $u\in\Lambda$ and $m>1$.
 Let \(\mathcal P\) denote the set of primitive elements of \(\Lambda\). Writing \(\langle v\rangle=\{mv:\,m\in\Z \}\) for the cyclic subgroup generated by \(v\in\mathcal P\), we may reorganise the sum as
 \[
 \sum_{v\in\Lambda\setminus\{0\}} K^v_k(0,0)
 = \frac12\sum_{v\in\mathcal P}\sum_{u\in\langle v\rangle\setminus\{0\}} K^{u}_k(0,0).
 \]

Applying Theorem \ref{thm-cylinder-main} to the quotient \(\C^n/\langle v\rangle\) yields, for each \(v\in\mathcal P\),
 \[
 \sum_{u\in\langle v\rangle\setminus\{1\}} K^{u}_k(0,0)
 = (\frac{k}{2\pi})^n\sum_{u\in\langle u\rangle\setminus\{0\}}
 e^{-k(\frac{|u|}{2})^2}\cos\!\big(2\pi k\alpha_{u}(p)\big).
 \]
 Summing over all primitive elements \(g\in\mathcal P\) gives the formula in Theorem \ref{thm-main}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-off-diag}]
 By Theorem \ref{thm-k-gamma-bergman}, we have
 \[\pi^*K_{X,k}(z,w)=\sum_{v\in \Lambda}(A_v^*s_w(z))\otimes \bar{s}_w(w).\]
 We can assume that $w=0$. Then since $|A_v^*s_0(z)|=(\frac{k}{2\pi})^{n/2}e^{-\frac{k}{4}d^2(-v,z)}$. We have \[|\pi^*K_{X,k}(z,0)|_h\leq (\frac{k}{2\pi})^{n}\sum_{v\in \Lambda}e^{-\frac{k}{4}d^2(-v,z)}.\]
 Since $d(-v,z)$ can be identified with the length of the corresponding geodesic segment in $X$ joining $\pi(z)$ and $\pi(0)$, the theorem follows.
\end{proof}
\subsection{Proof of Theorem \ref{thm-second}}
For a fixed line bundle $L$ on $X$ with $\omega\in 2\pi c_1(L)$, let $h$ be a Hermitian metric on $L$ such that $\Theta_h=-i\omega$. Let $F$ denote the underlying $C^\infty$ complex line bundle of $L$. Let $\ucal$ be the space of unitary connections on $(F,h)$ whose curvatures satisfy the condition that the $(0,2)$-part is $0$. Let $\nabla_0\in \ucal$ be the Chern connection of $(L,h)$.

For $L'$, we let $\text{Hol}_{L'}(\gamma_{p,v})=e^{2\pi i \alpha'_v(p)}$ and $\phi'_p=\alpha'_{v_1}(p)$. Then we also have \[(P_*(\frac{(2\pi)^n}{k^n}\rho_{L',k}-1))(x)=2\nu \sum_{m\geq 1} e^{-\frac{k}{4}|m|^2|v_1|_H^2}\cos(2\pi m (\lambda t+\phi'_p)). \]
Therefore, if $\rho_{L',k}=\rho_{L,k}$, we get \[\sum_{m\geq 1} e^{-\frac{k}{2}|m|^2|v_1|_H^2}\cos(2\pi m (\lambda t+\phi'_p))= \sum_{m\geq 1} e^{-\frac{k}{2}|m|^2\|v_1\|^2}\cos(2\pi m (\lambda t+\phi_p)), \] for $t\in [0,1]$. So we get $\phi_p'=\phi_p, \mod 1$, namely $\text{Hol}_{L'}(\gamma_{p,v_1})=\text{Hol}_{L'}(\gamma_{p,v_1})$. So we have proved the theorem.

\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-max-min}]
 Part (a) follows directly from Theorem \ref{thm-main}. For part (b) and (c), we have that $\exists C_1>0$, $k_0>0$ such that for $k\geq k_0$, \[\sum_{v\in \Lambda, |v|_H>l_1}e^{-\frac{k}{4}|v|_H^2}\leq C_1 e^{-\frac{k}{4}l_2^2}.\]
 Let $p$ be a point satisfying the condition that $\text{Hol}_{L}(\gamma_{p,v})=1$ for all $v\in S_1$. Suppose that $k\geq k_0$ and $\rho_k(q)>\rho_k(p)$, then we have \[\sum_{v\in S_1}e^{-\frac{k}{4}l_1^2}(1-\cos(2\pi k\alpha_v(q)))\leq 2C_1 e^{-\frac{k}{4}l_2^2}, \]namely \[\sum_{v\in S_1}(1-\cos(2\pi k\alpha_v(q)))\leq 2C_1 e^{-\frac{k}{4}(l_2^2-l_1^2)}.\]
 If we require that $\alpha_v(q)\in (-\pi,\pi]$ for all $v$, then this implies that $|\alpha_v(q)|$ is small for $v\in S_1$.
Let $\{v_1, v_2,\cdots, v_m\}$ be a maximally linearly independent subset of $S_1$.
 From the proof of Proposition \ref{prop-surj}, it is easy to see that $\exists C_2>0$, independent of $k$, such that when $\epsilon$ is small enough, for any $q'\in X$, if $1-\cos(2\pi k\alpha_{v_j}(q'))<\epsilon$, $1\leq j\leq m$, then $\exists p'\in X$ such that $d(p',q')<C_2\epsilon$ and $e^{2\pi k\alpha_{v_j}(p')\sqrt{-1}}=1$, $1\leq j\leq m$. Let $q'=q$, we get $p'$ such that $d(p',q)<2C_2C_1 e^{-\frac{k}{4}(l_2^2-l_1^2)}$ and $e^{2\pi k\alpha_{v_j}(p')\sqrt{-1}}=1$, $1\leq j\leq m$.

\begin{remark}
\begin{itemize}
\item This formula is closely parallel to Theorem~1 of \cite{sun-rem-hyp}, and the proof follows the same strategy. Equivalently, the sum may be written over all geodesic loops based at \(p\), as in \cite{sun-rem-hyp}. It is instructive to compare the two settings: in the hyperbolic case the contribution of a loop of length \(l(\gamma)\) is essentially \(\cosh^{-2k}\!\big(\tfrac{l(\gamma)}{2}\big)\), while in the flat (abelian) case it is the Gaussian factor \(e^{-\frac{k}{2}(|v|_{H})^2}\). This contrast reflects the difference between negative curvature and zero curvature geometries.
\item The formula is also valid when the rank of $\Lambda$ is not maximal, i.e., for polarized complex tori that are not necessarily compact. For simplicity, we only state the compact case here, as we think people care most about the compact case.
\end{itemize}
\end{remark}

For a general polarized manifold \((X,L)\) one may ask:
\begin{question}
Given another Hermitian polarization \((L',h')\) with \(\Theta_{h'}=\Theta_h\) (so \(c_1(L')=c_1(L)\)), under what conditions can one detect that the Bergman kernels \(\rho_{L',k}\) and \(\rho_{L,k}\) are different?
\end{question}
Since the Tian--Catlin--Zelditch expansion depends only on local curvature data, it cannot distinguish such global differences. As an application of Theorem \ref{thm-main} we obtain the following rigidity statement.

\begin{theorem}\label{thm-second}
	Let $(X,L)$ be a polarized abelian variety. Let $L'$ be another polarization of $X$ with respect to a Hermitian metric $h'$ on $L'$ such that $\Theta_{h'}=\Theta_h$. Suppose that $$\rho_{L',k}=\rho_{L,k}$$ for some $k\geq 1$. Then $kL'$ is isomorphic to $kL$.
\end{theorem}
Similar to \cite{sun-rem-hyp}, another application of Theorem \ref{thm-main} concerns the locations of the global maxima and minima of $\rho_k$. Set
\[
l_1=\min_{v\in \Lambda\setminus\{0\}} |v|_{H}, \qquad
l_2=\min_{\substack{v\in \Lambda\\ |v|_{H}>l_1}} |v|_{H},
\]
and write $S_1=\{v\in\Lambda:\;|v|_H=l_1\}$.

\begin{theorem}\label{thm-max-min}
\begin{enumerate}
\item[(a)] Assume $\Im H(u,v)\in 2\mathbb{Z}$ for all $u,v\in\Lambda$. Then for every $k\ge1$ the Bergman density $\rho_k$ attains its maximum precisely at those points $p\in X$ for which
\[
\mathrm{Hol}_L(\gamma_{p,v})=1\quad\text{for all }v\in\Lambda.
\]

\item[(b)] Assume $\Im H(u,v)\in 2\mathbb{Z}$ for all $u,v\in S_1$. There exist constants $C>0$ and $k_0\in\mathbb{N}$ such that for every $k\ge k_0$ every global maximum of $\rho_k$ lies within a neighborhood of radius
\[
C\exp\!\Big(\tfrac{k}{4}(l_1^2-l_2^2)\Big)
\]
of some point $p$ satisfying
\[
\mathrm{Hol}_{kL}(\gamma_{p,v})=1\quad\text{for all }v\in S_1.
\]

\item[(c)] Let $\{v_1,\dots,v_m\}$ be a maximally linearly independent subset of $S_1$ and suppose $\#S_1=2m$ (so $S_1=\{\pm v_1,\dots,\pm v_m\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\ge k_0'$ every global minimum of $\rho_k$ lies within a neighborhood of radius
\[
C'\exp\!\Big(\tfrac{k}{4}(l_1^2-l_2^2)\Big)
\]
of some point $p$ with
\[
\mathrm{Hol}_{kL}(\gamma_{p,v})=-1\quad\text{for all }v\in S_1.
\]
\end{enumerate}
\end{theorem}

\begin{remark}
\begin{itemize}
\item If $(v_1,\dots,v_{2n})$ is any basis of $\Lambda$, Proposition \ref{prop-surj} implies there exist points $p$ with $\mathrm{Hol}_L(\gamma_{p,v_j})=1$ for all $j$. Hence the hypothesis $\Im H(u,v)\in2\mathbb{Z}$ on $\Lambda\times\Lambda$ ensures existence of points where $\mathrm{Hol}_L(\gamma_{p,v})=1$ for every $v\in\Lambda$.
\item In part (b) it is enough to verify the holonomy condition $\mathrm{Hol}_{kL}(\gamma_{p,v_j})=1$ for a maximally independent subset $\{v_j\}_{j=1}^m\subset S_1$.
\end{itemize}
\end{remark}
Similar to Conjecture 1.6 in \cite{sun-rem-hyp}, we propose the following rigidity conjecture for polarized abelian varieties:

\begin{conjecture}
	Let $(X,L)$ and $(X',L')$ be polarized abelian varieties of dimension $n$. Suppose there is a diffeomorphism $\Phi:X\to X'$ and an integer $k\geq 1$ such that
	\[
	\rho_{L,k}=\Phi^*\rho_{L',k}.
	\]
	Then $\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover
	\[
	\Phi^*((L')^{\!k})\cong L^{\!k}\qquad\text{or}\qquad \Phi^*((L')^{\!k})\cong\overline{L^{\!k}},
	\]
	respectively. Here $\overline{L}$ denotes the complex-conjugate line bundle of $L$ (transition functions given by complex conjugates).
\end{conjecture}

Theorem \ref{thm-second} gives partial evidence for this conjecture.

Lastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\in X$ let $\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).
\begin{theorem}\label{thm-off-diag}
	Let $(X,\omega,L,h)$ be a polarized abelian variety as in Theorem \ref{thm-main}. For every $k\ge 1$ and every $x\neq y$,
	\[
	|K_k(x,y)|_{h^k}\;\le\;\Big(\frac{k}{2\pi}\Big)^{\!n}\sum_{\gamma\in\mathfrak{G}_{x,y}} e^{-\frac{k}{4}\,\ell(\gamma)^2},
	\]
	where $\ell(\gamma)$ is the length of $\gamma$.
\end{theorem}

The corresponding result in \cite{sun-rem-hyp} was the first to relate the global off-diagonal decay to the geodesic distance (compared to many local results, see the references therein); the theorem above provides a second instance, now in the flat setting.

\medskip
We briefly outline the proofs. The argument for Theorem \ref{thm-main} follows the same two-step strategy as in \cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\Lambda$. This strategy parallels the construction behind the Selberg trace formula.

Theorem \ref{thm-second} is obtained by integrating the kernel along complementary subtori to isolate the contributions of a chosen lattice direction; comparison of these averaged formulas recovers the holonomy data. Theorems \ref{thm-max-min} and \ref{thm-off-diag} are direct consequences of Theorem \ref{thm-main}, while the off-diagonal decay is obtained by estimating the contribution of peak sections and summing over geodesic segments joining two points.

\medskip

\noindent\textbf{Organization of the paper.}
In Section~2 we collect background material and derive the one-dimensional cylinder formula. Section~\ref{section-cylinder} treats the higher-dimensional cylinder and proves the cylinder case of Theorem~\ref{thm-main}. Section~\ref{sec-proof-thm-main} carries out the lattice summation on the universal cover and completes the proof of Theorem~\ref{thm-main}; the remaining theorems are then proved as applications.

\medskip

\noindent\textbf{Acknowledgements.} The author would like to thank Professor Song Sun for many very helpful discussions.

We briefly outline the proofs. The argument for the main theorem follows the same two-step strategy as in Sun, “Removable singularities in hyperbolic geometry”: first establish the formula for the cylinder (quotients of $\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\Lambda$. This strategy parallels the construction behind the Selberg trace formula.,

Question: Let \(X=\mathbb C^n/\Lambda\) be an abelian variety, where \(\Lambda\subset \mathbb C^n\) is a lattice. Let \(L\to X\) be a positive holomorphic line bundle whose first Chern class is determined by a positive definite Hermitian form \(H\) on \(\mathbb C^n\) with \(\operatorname{Im} H(\Lambda,\Lambda)\subset \mathbb Z\). Write \(|v|_H\) for the norm induced by \(H\), and let \(\omega=\pi i\sum H_{ij}\,dz^i\wedge d\bar z^j\in 2\pi c_1(L)\). Choose a Hermitian metric \(h\) on \(L\) with Chern curvature \(-i\omega\). For each \(p\in X\) and \(v\in\Lambda\), let \(\gamma_{p,v}\) be the geodesic loop based at \(p\) determined by \(v\), and define \(\alpha_v(p)\in \mathbb R/\mathbb Z\) by \(e^{2\pi i\alpha_v(p)}=\operatorname{Hol}_L(\gamma_{p,v})\). For each integer \(k\ge 1\), let \(\rho_k(p)\) denote the diagonal Bergman kernel of \(H^0(X,L^k)\) at \(p\). Which statement holds for every such integer \(k\ge 1\) and every point \(p\in X\)?

\[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]

Choices:
B. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda} e^{-\frac{k}{4}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]
C. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}c_v(p)\right),\qquad |c_v(p)|\le 1\ \text{for all }v\in\Lambda\setminus\{0\}.\]
D. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{2}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]
E. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}\,e^{2\pi i k\alpha_v(p)}\right).\]

The stem gives the geometric setup and definitions but does not state or strongly hint at the exact Bergman-kernel formula. The correct answer is not leaked explicitly or trivially.

This is essentially a theorem-identification question: under the full hypotheses, the student must recognize the precise conclusion. It is not a verbatim restatement, since the options differ in subtle but meaningful ways, but it is still close to recall of a known formula.

Some reasoning is required to distinguish the exact identity from plausible near-misses: excluding the zero vector, pairing terms into a cosine, and getting the Gaussian factor right. However, the task is still primarily recognition of the exact theorem statement rather than substantial derivation.

The distractors are strong: one includes the zero lattice vector, one weakens the conclusion, one uses the wrong Gaussian constant, and one forgets the real pairing into cosine. These are plausible, distinct, and mathematically aligned with common failure modes.

A solid high-level MCQ with no answer leakage and strong distractors, but it is still fairly close to theorem recall rather than a deeply generative reasoning task.