\label{thm:main-intro} Let $q$ be a prime power and $N \geqslant 1$ be an integer. Let $\mathcal{A} = \mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\mathbb{F}_q[t]$ with degree $N$ with size
\begin{equation} \label{main-minimum-size}
|\mathcal{A}| \gg  q^N \exp\big(-\tfrac{1}{3}\sqrt{N \log q}\big).
\end{equation}
Assume there exists a subset $\mathcal{S}=\mathcal{S}_{N,q} \subset \mathcal{A}$ satisfying
\begin{equation} \label{main-asymptotic-subset}
	|\mathcal{S}| = (1+o(1)) |\mathcal{A}| \quad \text{ as } q^N \to \infty.
\end{equation}
and also
\begin{equation} \label{main-mult-energy}
|\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:\,\,\,F_1F_2=G_1G_2\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } q^N \to \infty.
\end{equation}
If $f$ is a Steinhaus multiplicative function over $\mathbb{F}_q[t]$ then
\begin{equation}   \label{eqn:CLT} \tag{$\star$}
\frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F \in \mathcal{A}} f(F)  \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } q^N \to \infty.
\end{equation}

\begin{theorem}\label{thm:CLT-shiftedprimes}
    Let $q \geq 3$ be a prime power and let $N \geq 1$ be an integer. Let $Z \in \mathcal{M}$ satisfy $\deg Z \leq N-1$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
    \[
    \mathcal{A} = \{ P + Z : P \text{ monic irreducible and } \deg P = N \}
    \]
	 as $N \to \infty$.
\end{theorem}

\begin{theorem}\label{thm:main-intro} Let $q$ be a prime power and $N \geq 1$ be an integer. Let $\mathcal{A} = \mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\mathbb{F}_q[t]$ with degree $N$ with size
\begin{equation} \label{main-minimum-size}
|\mathcal{A}| \gg  q^N \exp\big(-\tfrac{1}{3}\sqrt{N \log q}\big).
\end{equation}
Assume there exists a subset $\mathcal{S}=\mathcal{S}_{N,q} \subset \mathcal{A}$ satisfying
\begin{equation} \label{main-asymptotic-subset}
	|\mathcal{S}| = (1+o(1)) |\mathcal{A}| \quad \text{ as } q^N \to \infty.
\end{equation}
and also
\begin{equation} \label{main-mult-energy}
|\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:\,\,\,F_1F_2=G_1G_2\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } q^N \to \infty.
\end{equation}
If $f$ is a Steinhaus multiplicative function over $\mathbb{F}_q[t]$ then
\begin{equation}   \label{eqn:CLT} \tag{$\star$}
\frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F \in \mathcal{A}} f(F)  \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } q^N \to \infty.
\end{equation}
\end{theorem}

\begin{theorem}[Uniform Shiu bound]\label{thm:shiu} Let $q \geq 2$ be a prime power and $N \geq 1$ be an integer. Let $g : \mathcal{M} \to [0,\infty)$ be a non-negative multiplicative function on $\mathcal{M}$ satisfying $\log g(P^{\ell}) \ll \ell $  for every integer $\ell$ and monic irreducible $P$, and $g(F) \ll_{\epsilon} 2^{\epsilon \deg F}$ for every $\epsilon > 0$ and any $F \in \mathcal{M}$.  If  $A \in \mathcal{M}_N$,  $0< \beta < 1/2$, $\beta N < h \leq N-1$, and $N$ is sufficiently large depending only on $\beta$, then
    \begin{align*}
        \sum_{\substack{F\in \mathcal{I}(A,h)}}g(F)\ll_{\beta} \frac{q^{h+1}}{N}\exp\Big(\sum_{\substack{P\in\mathcal{P}_{\leq N}}}\frac{g(P)}{q^{\deg P}}\Big).
    \end{align*}
\end{theorem}

\begin{theorem}\label{thm:CLT-kprimes} Let $q \geq 2$ be a prime power and let $N \geq 1$ be an integer. Assume  $k=o(\log N)$ as $N \to \infty$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds:
\begin{enumerate}[label=(\alph*)]
	\item As $N \to \infty$, for degree $N$ polynomials with $k$ irreducible factors, namely
	\[
	\mathcal{A} = \mathcal{P}_k(N) := \{ F \in \mathcal{M}_N : \Omega(F) = k \}.
	\]
	\item As $N \to \infty$, for   degree $N$ squarefree polynomials   with $k$ irreducible factors,  namely
	\[
	\mathcal{A} = \mathcal{S}_k(N) := \{ F \in \mathcal{M}_N : \Omega(F) = \omega(F) = k\}.
	\]
	\item As $N \to \infty$, for degree $N$  polynomials   with $k$ distinct irreducible factors,  namely
	\[
	\mathcal{A} = \mathcal{D}_k(N) := \{ F \in \mathcal{M}_N : \omega(F) = k\}.
	\]
\end{enumerate}
\end{theorem}

\begin{theorem}\label{thm:CLT-intervals}
    Let $q \geq 2$ be a prime power and let $N\geq 3$ be an integer. Let $K \in \mathcal{M}_N$  and $1 \leq h \leq N-2$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
    \[
    \mathcal{A} = \mathcal{I}(K,h) := \{ F \in \mathcal{M}_N : \deg(F-K) \leq h\}
    \]
    in both of the following cases:
    \begin{enumerate}[label=(\alph*)]
    	\item As $q^{h} \to \infty$ provided $q^{h+1} = o(q^N/N)$.
    	\item As $h \to \infty$ provided $q^{h+1} = o(q^N/N^c)$ for some fixed $c > 2\log 2-1$.
    \end{enumerate}

\end{theorem}

\label{sec:intro}

\subsection{Number field setting} \label{subsec:intro-nf} Given a sequence $(f(p))_{p}$ indexed by primes $p$ of independent random variables uniformly distributed on the complex unit circle $\mathbb{S}^1 = \{z \in \mathbb{C}: |z| =1\}$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{N}$) is a random variable $f : \mathbb{N} \to \mathbb{C}$ defined by $f(1) = 1$ and
$f(n) = f(p_1)^{a_1} \cdots f(p_r)^{a_r},$
where $n = p_1^{a_1} \cdots p_r^{a_r}$ is the prime factorization of $n \geqslant 2$.  The study of these random functions originated with Wintner \cite{Wintner-1944} in 1944 and have re-emerged as a very active area of interest in the past decades; see a recent survey by Harper \cite{Harper-2024} for related discussion.

Given any finite subset $\mathcal{A} \subseteq \mathbb{N}$, the complex random variable
$|\mathcal{A}|^{-1/2} \sum_{n \in \mathcal{A}} f(n)$
has mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? The answer depends on the subsets $\mathcal{A}$. Central limit theorems have been established in many cases such as  integers   with few prime factors, short intervals, and polynomial values. More precisely, denoting $\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\mathcal{A}$ satisfies a CLT for
\begin{itemize}
	\item $\mathcal{A} = \{ 1 \leqslant n \leqslant x : \omega(n) = k\}$ with $k = o(\log \log x)$ as $x \to \infty$,
	\item $\mathcal{A} = \{ x-y \leqslant n \leqslant x \}$ with $y \leqslant x/(\log x)^{2\log 2-1+\varepsilon}$ as $x \to \infty$,
	\item $\mathcal{A} = \{ Q(n) : 1 \leqslant n \leqslant x \}$ as $x \to \infty$, where $Q \in \mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \in \mathbb{Z}$.
\end{itemize}
These results are respectively due to Harper \cite{Harper-2013},  Soundararajan--Xu \cite{SoundXu-2023}, and Klurman--Shkredov--Xu \cite{RandomChowla}. On the other hand, Harper   surprisingly showed that the natural choice
$
\mathcal{A} = [1,x] \cap \mathbb{N}
$
does \textit{not} converge to the standard complex normal \cite{Harper-2013} and, in fact, the sum converges to zero in distribution \cite{Harper-2020}; see Gorodetsky--Wong \cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \cite{AtherfoldNajnudel-2025}  for a closely related breakthrough. A natural question, then, is over which subsets $\mathcal{A}$ does such a central limit theorem hold?   Soundararajan and Xu \cite{SoundXu-2023} recently established a flexible criterion.

\hypertarget{customlabel}{\label{thm:SoundXu-integer}}
\begin{theorem*}[Soundararajan--Xu]
    Let $x \geqslant 10$ be large. Let $\mathcal{A}=\mathcal{A}_x$ be a subset of $[1,x] \cap \mathbb{N}$ with size
    \[
    |\mathcal{A}|\geqslant x \exp(-\tfrac13\sqrt{\log x\log\log x}).
    \]
   Assume there exists a subset $\mathcal{S}=\mathcal{S}_x\subset\mathcal{A}_x$ with size $|\mathcal{S}|=(1+o(1))|\mathcal{A}|$ as $x \to \infty$ satisfying
    \begin{align*}
        |\{(s_1,s_2,s_3,s_4)\in\mathcal{S}^4:\,\,\, s_1s_2=s_3s_4\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } x \to \infty.
    \end{align*}
   If $f$ is a Steinhaus random multiplicative function over $\mathbb{Z}$, then
    \begin{align*}
        \frac{1}{\sqrt{|\mathcal{A}|}}\sum_{n\in\mathcal{A}}f(n) \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } x \to \infty.
    \end{align*}
\end{theorem*}
\begin{remark*}
    Here and throughout, the notation $\,\xrightarrow{d}\,$ indicates convergence in distribution, and $\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.
\end{remark*}

The principal goal of this paper is to extend this central limit theorem criterion and some applications from the \textit{number field setting} over $\mathbb{Z}$ to the \textit{function field setting} over the polynomial ring $\mathbb{F}_q[t]$, where $q$ is a prime power and $\mathbb{F}_q$ is the unique finite field of $q$ elements (see, e.g., \cite{Gran-Harp-Sound-2015} for an introduction to multiplicative functions over $\mathbb{F}_q[t]$).

\subsection{Function field setting} \label{subsec:intro-ff} Let $\mathcal{M}$ be the set of monic polynomials of $\mathbb{F}_q[t]$. Given a sequence  $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables  uniformly distributed on $\mathbb{S}^1$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{F}_q[t]$) is a random variable $f :  \mathcal{M} \to \mathbb{C}$ defined by $f(1) = 1$ and
\begin{align*}
 	f(F)= f(P_1)^{a_1} \cdots f(P_r)^{a_r},
\end{align*}
where $F = P_1^{a_1} \cdots P_r^{a_r}$ is the unique factorization of $F \in \mathcal{M}$ into powers of distinct monic irreducibles $P_1,\dots,P_r$.  Given any finite subset $\mathcal{A} \subseteq \mathcal{M}$, the complex random variable
\begin{align}\label{eq:partialsums}
    \frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F\in\mathcal{A}}f(F)
\end{align}
has mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$?  This question has strong parallels with the number field setting, but there are fewer examples over function fields.

Corresponding to Harper's result \cite{Harper-2013,Harper-2020} over $\mathbb{Z}$ from \S\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \cite{SoundZaman-2022} implies that the random sum \eqref{eq:partialsums} with the natural choice
	\[
	\mathcal{A} = \mathcal{M}_N := \{ F \in \mathcal{M} : \deg F = N \}
	\]
	does \textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \to \infty$ and then $N \to \infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} for Rademacher random multiplicative functions  over polynomials with few irreducible factors, namely when
\[
\mathcal{A} = \{ F \in \mathcal{M}_N : \omega(F) = k\} \text{ with  } k = o(\log N)  \text{ as }N \to \infty.
\]
Here $\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper  \cite{Harper-2013} in \S\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\mathbb{F}_q[t]$.

\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.

Given any finite subset $\mathcal{A} \subseteq \mathbb{N}$, the complex random variable
$|\mathcal{A}|^{-1/2} \sum_{n \in \mathcal{A}} f(n)$
has mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? The answer depends on the subsets $\mathcal{A}$. Central limit theorems have been established in many cases such as integers with few prime factors, short intervals, and polynomial values. More precisely, denoting $\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\mathcal{A}$ satisfies a CLT for
\begin{itemize}
 \item $\mathcal{A} = \{ 1 \leqslant n \leqslant x : \omega(n) = k\}$ with $k = o(\log \log x)$ as $x \to \infty$,
 \item $\mathcal{A} = \{ x-y \leqslant n \leqslant x \}$ with $y \leqslant x/(\log x)^{2\log 2-1+\varepsilon}$ as $x \to \infty$,
 \item $\mathcal{A} = \{ Q(n) : 1 \leqslant n \leqslant x \}$ as $x \to \infty$, where $Q \in \mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \in \mathbb{Z}$.
\end{itemize}
These results are respectively due to Harper \cite{Harper-2013}, Soundararajan--Xu \cite{SoundXu-2023}, and Klurman--Shkredov--Xu \cite{RandomChowla}. On the other hand, Harper surprisingly showed that the natural choice
$
\mathcal{A} = [1,x] \cap \mathbb{N}
$
does \textit{not} converge to the standard complex normal \cite{Harper-2013} and, in fact, the sum converges to zero in distribution \cite{Harper-2020}; see Gorodetsky--Wong \cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \cite{AtherfoldNajnudel-2025} for a closely related breakthrough. A natural question, then, is over which subsets $\mathcal{A}$ does such a central limit theorem hold? Soundararajan and Xu \cite{SoundXu-2023} recently established a flexible criterion.

\hypertarget{customlabel}{\label{thm:SoundXu-integer}}
\begin{theorem*}[Soundararajan--Xu]
 Let $x \geqslant 10$ be large. Let $\mathcal{A}=\mathcal{A}_x$ be a subset of $[1,x] \cap \mathbb{N}$ with size
 \[
 |\mathcal{A}|\geqslant x \exp(-\tfrac13\sqrt{\log x\log\log x}).
 \]
 Assume there exists a subset $\mathcal{S}=\mathcal{S}_x\subset\mathcal{A}_x$ with size $|\mathcal{S}|=(1+o(1))|\mathcal{A}|$ as $x \to \infty$ satisfying
 \begin{align*}
 |\{(s_1,s_2,s_3,s_4)\in\mathcal{S}^4:\,\,\, s_1s_2=s_3s_4\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } x \to \infty.
 \end{align*}
 If $f$ is a Steinhaus random multiplicative function over $\mathbb{Z}$, then
 \begin{align*}
 \frac{1}{\sqrt{|\mathcal{A}|}}\sum_{n\in\mathcal{A}}f(n) \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } x \to \infty.
 \end{align*}
\end{theorem*}
\begin{remark*}
 Here and throughout, the notation $\,\xrightarrow{d}\,$ indicates convergence in distribution, and $\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.
\end{remark*}

\subsection{Function field setting} \label{subsec:intro-ff} Let $\mathcal{M}$ be the set of monic polynomials of $\mathbb{F}_q[t]$. Given a sequence $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables uniformly distributed on $\mathbb{S}^1$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{F}_q[t]$) is a random variable $f : \mathcal{M} \to \mathbb{C}$ defined by $f(1) = 1$ and
\begin{align*}
 f(F)= f(P_1)^{a_1} \cdots f(P_r)^{a_r},
\end{align*}
where $F = P_1^{a_1} \cdots P_r^{a_r}$ is the unique factorization of $F \in \mathcal{M}$ into powers of distinct monic irreducibles $P_1,\dots,P_r$. Given any finite subset $\mathcal{A} \subseteq \mathcal{M}$, the complex random variable
\begin{align}\label{eq:partialsums}
 \frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F\in\mathcal{A}}f(F)
\end{align}
has mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? This question has strong parallels with the number field setting, but there are fewer examples over function fields.

Corresponding to Harper's result \cite{Harper-2013,Harper-2020} over $\mathbb{Z}$ from \S\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \cite{SoundZaman-2022} implies that the random sum \eqref{eq:partialsums} with the natural choice
 \[
 \mathcal{A} = \mathcal{M}_N := \{ F \in \mathcal{M} : \deg F = N \}
 \]
 does \textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \to \infty$ and then $N \to \infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} for Rademacher random multiplicative functions over polynomials with few irreducible factors, namely when
\[
\mathcal{A} = \{ F \in \mathcal{M}_N : \omega(F) = k\} \text{ with } k = o(\log N) \text{ as }N \to \infty.
\]
Here $\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper \cite{Harper-2013} in \S\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\mathbb{F}_q[t]$.

\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.

\begin{remark} \label[remark]{rem:limit-convention}
Here and throughout, any limit of $q$ or $N$ permits the parameters to vary and depend on each other arbitrarily unless explicitly stated otherwise. For example, the limit $q^N \to \infty$ includes the cases $q$ fixed with $N \to \infty$, $N$ fixed with $q \to \infty$, and more.
\end{remark}

We refer to the subset of polynomials whose prime factors all have degree exceeding an integer $z \geq 1$ as \textit{$z$-rough polynomials}. In other words, polynomials $F$ such that
\begin{equation} \label{def:min-degree}
P^-(F) := \min\{ \deg P : P \mid F, \, P \text{ monic irreducible} \}
\end{equation}
exceeds $z$.
 As suggested by Xu \cite{Xu-2022}, one can expect a CLT when summing over sufficiently rough polynomials. As a brief final application of \cref{thm:main-intro}, we confirm this observation in \S\ref{sec:roughpolyCLT}.
\begin{theorem}\label{thm:CLT-rough}
 Let $q \geq 2$ be a prime power and $N\geq 1$ an integer. Assume $N^{1/2} \leq z \leq N-1$ is an integer satisfying $N^{1/2} = o(z)$ as $N \to \infty$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
 \[
 \mathcal{A} = \{ F \in \mathcal{M}_N : P^-(F) > z \}
 \]
 as $N \to \infty$.
\end{theorem}

\begin{enumerate}[label=(\roman*)]
 \item $|\mathcal{A} \backslash \mathcal{S}| = o(|\mathcal{A}|)$. \label{thmmain:i}
 \item $|\{ (F_1,F_2,G_1,G_2) \in \mathcal{S}^4 : F_1 F_2=G_1 G_2 , F_1 \neq G_1, F_1 \neq G_2 \}| = o(|\mathcal{A}|^2)$. \label{thmmain:ii}
 \item $|\{F\in \mathcal{S}:\ P^+_\prec(F)=Q\}| = o(|\mathcal{A}|)$ for every monic irreducible $Q$. \label{thmmain:iii}
\end{enumerate}
If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$ then, as $q^N\to\infty$,
\[
Z=\frac{1}{\sqrt{|\mathcal{A}|}}\sum_{F\in \mathcal{A}} f(F)\ \xrightarrow{d}\ \mathcal{CN}(0,1).
\]
\end{theorem}
\begin{remark}\label[remark]{rem:martingale} The advantage of our refined filtration with $P_{\prec}^+(F)$ can be seen from condition (iii). Observe that
\begin{align*}
\max_{Q \in \mathcal{P}} | \{ F \in \mathcal{M}_N : P^+_{\prec}(F) = Q \} | & \ll q^N \exp\Big( - \tfrac{1}{2} \sqrt{N \log q} \Big), \\
\max_{d \geq 1} | \{ F \in \mathcal{M}_N : P^+(F) = d \} | & \gg q^N /N,
\end{align*}
upon applying \cref{lem:filtration} with $h=N-1$ for the first bound, and noting $d=N$ corresponds to the total number of monic irreducibles for the second bound. The latter filtration is based purely on the degree $P^+(F)$ and would require us to assume $|\mathcal{A}| \gg q^N/N$ in \cref{thm:main-intro}. Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} successfully used this filtration in their case, because their desired application for $\mathcal{A}$ was sufficiently large. However, the constraint $|\mathcal{A}| \gg q^N/N$ is too strict for our intended applications. Our refined filtration with $P^+_{\prec}(F)$ allows us to only require $|\mathcal{A}| \gg q^N \exp(- \tfrac{1}{3}\sqrt{N \log q})$ in \cref{thm:main-intro}, which is notably more flexible. It also more closely mimics Harper \cite{Harper-2013} and his filtration based on $\{ m \in [1,n] \cap \mathbb{Z} : P^+(m) = p\}$ for any prime $p$, where $P^+(m)$ denotes the largest prime factor of $m \in \mathbb{Z}$.
\end{remark}

Fix $\epsilon > 0$ sufficiently small. As in the proof of \cref{prop:short-interval-mult-energy}, we again count off-diagonal solutions to $F_1F_2 = G_1G_2$ for $F_1,F_2,G_1,G_2 \in \mathcal{S}$ with the same GCD parametrization. Write $F_1 = GA, G_1 = GB, F_2 = HB,$ and $G_2 = HA$. Denoting $d = \deg G = \deg H$, it follows by \cref{prop:short-interval-mult-energy} and its proof that $N-h \leq d \leq h$, and we may assume $h \geq N/2$ without loss. Set $m=(1+ \epsilon)\log N$ so that $\Omega(GABH)\leq 2m$. This yields the following estimate for the off-diagonal solutions:
\begin{align}
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}|
 &\ll \sum_{d=N-h}^{h} \, \ssum_{\substack{G, H \in \mathcal{M}_d}} \, \ssum_{\substack{A, B \in \mathcal{M}_{N-d}\\ GA, HB, GB, HA\in \mathcal{I}(K,h)}}2^{2m-\Omega(GABH)} \nonumber\\
 &=2^{2m}\sum_{d=N-h}^{h} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}\ssum_{\substack{A,B\in \mathcal{I}(K,h)/G \\ A,B \in \mathcal{M}_{N-d}}}2^{-\Omega(A)-\Omega(B)}\sum_{\substack{ H\in \mathcal{I}(K,h)/A \\ H \in \mathcal{M}_d}}2^{-\Omega(H)}. \label{eqn:short-intervals-off-diag}
\end{align}
As described at the end of \S\ref{sec:intervals}, for each $A \in \mathcal{M}_{N-d}$, the interval quotient $\mathcal{I}(K,h)/A$ is \textit{equal} to the interval $\mathcal{I}(\widetilde{K},h+d-N)$ for some $\widetilde{K}$ (depending on $A$) of degree $d = N - (N-d)$ because $h+d-N \geq 0$ in our sum. Thus, by \cref{thm:shiu} with $g(H) = 2^{-\Omega(H)}$, it follows that
 \[
 \sum_{\substack{ H\in \mathcal{I}(K,h)/A \\ H \in \mathcal{M}_d}}2^{-\Omega(H)} = \sum_{\substack{ H\in \mathcal{I}(\widetilde{K},h+d-N) }}2^{-\Omega(H)} \ll \frac{q^{h+d-N+1}}{d} \exp\Big( \sum_{P \in \mathcal{P}_{\leq d} } \frac{1}{2q^{\deg(P)}} \Big) \ll \frac{q^{h+d-N+1}}{d^{1/2}}
 \]
 for each $A \in \mathcal{M}_{N-d}$, since by the prime number theorem $\sum_{P \in \mathcal{P}_{\leq d} } q^{-\deg(P)} = \log d +O(1)$. By \eqref{eqn:short-intervals-off-diag}, this implies that
 \[
\mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m}\sum_{d=N-h}^{h} \frac{q^{h+d-N+1}}{d^{1/2}} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}\Big( \sum_{\substack{A\in \mathcal{I}(K,h)/G \\ A \in \mathcal{M}_{N-d}}}2^{-\Omega(A)} \Big)^2
 \]
 since the sums over $A$ and $B$ are identical. For the sum over $A$, the same argument with \cref{thm:shiu} applies. Indeed, for each $G \in \mathcal{M}_d$, the interval quotient $\mathcal{I}(K,h)/G$ is \textit{equal} to the interval $\mathcal{I}(\widetilde{K}',h-d)$ for some $\widetilde{K}'$ (depending on $G$) of degree $N-d$ because $h-d \geq 0$ in our sum. Thus, by \cref{thm:shiu} with $g(A) = 2^{-\Omega(A)}$, we similarly have that
 \[
 \sum_{\substack{A\in \mathcal{I}(K,h)/G \\ A \in \mathcal{M}_{N-d}}}2^{-\Omega(A)}
 =
 \sum_{\substack{A\in \mathcal{I}(\widetilde{K}',h-d)}}2^{-\Omega(A)} \ll \frac{q^{h-d+1}}{N-d} \exp\Big( \sum_{P \in \mathcal{P}_{\leq N-d} } \frac{1}{2 q^{\deg(P)}} \Big) \ll \frac{q^{h-d+1}}{(N-d)^{1/2}}.
 \]
Inserting this into the previous equation, we deduce that
 \[
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m}\sum_{d=N-h}^{h} \frac{q^{3h-N-d+3}}{d^{1/2} (N-d)} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}.
 \]
Upon noting $\mathcal{I}(G,d-1) = \mathcal{M}_d$, we again apply \cref{thm:shiu} on the sum over $G$ to see that
\[
\sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)} \ll \frac{q^{d+1}}{d^{1/2}}
\]
for each $N-h \leq d \leq h$. Therefore, we may conclude that
\[
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m} q^{3h+3-N}\sum_{d=N-h}^h \frac{1}{d(N-d)} \ll 2^{2m} q^{3h+3-N} \frac{\log N}{N}
\]
since $N-d \geq h \geq N/2$ and $\sum_{d=N-h}^h \frac{1}{d} \leq \sum_{d=1}^N \frac{1}{d} \ll \log N$. Using our choice $m = (1+\epsilon) N$ and noting $|\mathcal{S}| \asymp |\mathcal{I}(K,h)| = q^{h+1}$ by \cref{lem:short-interval-hardy-ramanujan}, we have overall that
\[
2|\mathcal{S}|^2 - |\mathcal{S}| \leq \mathsf{E}_\times(\mathcal{S}) \leq 2|\mathcal{S}|^2 - |\mathcal{S}| + O\Big( |\mathcal{S}|^2 q^{h+1-N} N^{2\log 2 (1+\epsilon) - 1} \log N \Big).
\]
The error term is $o(|\mathcal{S}|^2)$ provided $q^{h+1} = o(q^N / N^{2\log 2-1+2\epsilon} )$. Our assumption states $q^{h+1} = o(q^N/N^c)$ as $h \to \infty$ for some fixed $c > 2\log 2-1$. By fixing $\epsilon = \epsilon(c) > 0$ sufficiently small, we deduce that $\mathsf{E}_\times(\mathcal{S}) = 2 |\mathcal{S}|^2(1+o(1))$ as $h \to \infty$. This verifies (ii) and establishes case (b).
\end{proof}

\begin{remark} \label[remark]{rem:limit-convention}
Here and throughout, any limit of $q$ or $N$ permits the parameters  to vary and depend on each other arbitrarily unless explicitly stated otherwise.  For example, the limit $q^N \to \infty$ includes the cases $q$ fixed with $N \to \infty$, $N$ fixed with $q \to \infty$, and more.
\end{remark}

As remarked in \cite{SoundXu-2023}, the size constraints on $|\mathcal{A}|$ and $|\mathcal{S}|$ are mild, but \eqref{main-mult-energy} is  pivotal. This condition  concerns the \textit{multiplicative energy} $\mathsf{E}_{\times}(\mathcal{S})$ of a finite set $\mathcal{S} \subseteq \mathcal{M}$, defined by
\begin{equation} \label{eqn:mult-energy}
    \mathsf{E}_{\times}(\mathcal{S}) := |\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:F_1F_2=G_1G_2\}| = \mathbb{E} \Big| \sum_{F \in \mathcal{S} } f(F) \Big|^4.
\end{equation}
We say a family of sets $\mathcal{S}$ have \textit{asymptotically trivial multiplicative energy} when
\begin{equation}\label{trivialenergy}
\mathsf{E}_{\times}(\mathcal{S}) =(2+o(1))|\mathcal{S}|^2 \quad \text{ as } |\mathcal{S}| \to \infty.
\end{equation}
The trivial (or diagonal) solutions $(F_1, F_2, F_1, F_2)$ and $(F_1, F_2, F_2, F_1)$ for \eqref{eqn:mult-energy} always exist, so it is immediate that $\mathsf{E}_{\times}(\mathcal{S}) \geqslant 2|\mathcal{S}|^2 - |\mathcal{S}|$. Statement \eqref{trivialenergy} therefore requires that the multiplicative energy is dominated by the trivial solutions in the limit.

The proof of \cref{thm:main-intro} largely follows Soundararajan--Xu by invoking a central limit theorem for martingale difference sequences due to McLeish (\cref{thm:mcleishclt}). This approach was utilized in \cite{FUSRP2020} for function fields but they applied a coarse filtration depending only on
\begin{equation} \label{def:max-degree}
P^+(F) := \max\{ \deg P : P \mid F, P \text{ monic irreducible } \},
\end{equation}
the maximum degree of any monic irreducible dividing $F \in \mathcal{M}$. We require a more refined filtration for each individual prime $P$, necessitating a new estimate for smooth polynomials in short intervals (\cref{lem:filtration}) in addition to a related estimate of Gorodetsky \cite{OfirSmoothPolys23}; see \cref{rem:martingale} for details.

From \cref{thm:main-intro}, we deduce  four central limit theorems:
\begin{itemize}
	\item (\cref{thm:CLT-intervals})  CLT for short intervals
	\item (\cref{thm:CLT-kprimes}) CLT for restricted number of prime factors
	\item (\cref{thm:CLT-shiftedprimes}) CLT for shifted primes
	\item (\cref{thm:CLT-rough}) CLT for rough polynomials
\end{itemize}
Each application requires a variety of function field number theoretic lemmas, many of which we could not find in the literature with sufficient uniformity. We established several new results which we expect to be of independent interest: a uniform Hildebrand bound for smooth polynomials in short intervals (\cref{prop:intervalshift}), a uniform Shiu's theorem for multiplicative functions (\cref{thm:shiu}), and a uniform Chebyshev bound for rough polynomials in short intervals (\cref{lem:chebyshev}). See  \S\ref{sec:CLTs} for details on these four central limit theorems and number theoretic lemmas.

\subsection*{Organization} \cref{sec:CLTs} describes the CLT applications deduced from \cref{thm:main-intro}  and other number theoretic results over $\mathbb{F}_q[t]$. \cref{sec:prelim} collects our notation and basic terminology regarding function field intervals, and then presents a  simplified version of the Selberg sieve for function fields. \cref{sec:smoothroughpolys} develops novel bounds on the numbers of smooth and rough polynomials in an interval (\cref{prop:intervalshift,lem:chebyshev}). \cref{sec:clts} establishes \cref{thm:main-intro} by applying these lemmas.

The remaining subsections are dedicated to deducing the four central limit theorems described in \cref{sec:CLTs}.   \cref{sec:shiu} prepares a function field analogue of Shiu's theorem (\cref{thm:shiu}), so that \cref{sec:interval-CLT} can establish the short interval CLT (\cref{thm:CLT-intervals}). \cref{sec:almostprimesCLT,sec:shiftedprimesCLT} respectively establish the few prime factors CLT (\cref{thm:CLT-kprimes}) and shifted prime CLT (\cref{thm:CLT-shiftedprimes}). Finally, \cref{sec:roughpolyCLT} presents a short proof of the rough polynomial CLT (\cref{thm:CLT-rough}).

\subsection*{Acknowledgments} The authors thank Lior Bary-Soroker, Ofir Gorodetsky, Dimitris Koukoulopoulos, and Max Xu for helpful discussions and references. This research was conducted as part of the 2024 Fields Undergraduate Summer Research Program, and the authors are deeply grateful for the Fields Institute's support. AZ was partially supported by NSERC grant RGPIN-2022-04982.

The proof of \cref{thm:main-intro} is described as largely following Soundararajan--Xu by "invoking a central limit theorem for martingale difference sequences due to McLeish (\cref{thm:mcleishclt})." The authors note that prior work in function fields \cite{FUSRP2020} used "a coarse filtration depending only on" \(P^+(F)\), but here they "require a more refined filtration for each individual prime \(P\)," which in turn "necessitat[es] a new estimate for smooth polynomials in short intervals (\cref{lem:filtration})" together with "a related estimate of Gorodetsky." Finally, they state that \cref{thm:main-intro} is established in \cref{sec:clts} "by applying these lemmas."

The proof of the main theorem is described as largely following Soundararajan--Xu by invoking the following central limit theorem for martingale difference sequences due to McLeish.

\begin{theorem}[Complex-Valued McLeish CLT]\label{thm:mcleishclt}
Let $Z_1,\dots,Z_N$ be a complex-valued martingale difference sequence with $\mathbb E[|Z_n|^4]<\infty$
for every $n$, and put $S_N=\sum_{n=1}^N Z_n$. Assume
\({
\sum_{n=1}^N \mathbb E[|Z_n|^2] = 1.
}\)
Then for real $t_1,t_2$ with $t^2=(t_1^2+t_2^2)/2$,
\begin{align*}
\mathbb{E}\Big[e^{i t_1\operatorname{Re}(S_N)+i t_2\operatorname{Im}(S_N)}\Big]
= e^{-t^2/2} & + O\Big(e^{t^2}\Big(\Big(\sum_{n=1}^N\mathbb E[|Z_n|^4]\Big)^{1/4} + \Big(\mathbb E\Big[\Big(\sum_{n=1}^N |Z_n|^2 - 1\Big)^2\Big]\Big)^{1/2}\Big) \Big) \\
& \qquad + O\Big( \max_{\phi\in[0,2\pi]}\Big(\mathbb E\Big[\Big(\sum_{n=1}^N(e^{-i\phi}Z_n^2+e^{i\phi}\overline{Z_n}^2)\Big)^2\Big]\Big)^{1/2} \Big).
\end{align*}
\end{theorem}

The authors note that prior work in function fields \cite{FUSRP2020} used a coarse filtration depending only on $P^+(F)$, but here they require a more refined filtration for each individual prime $P$, which in turn necessitates a new estimate for smooth polynomials in short intervals (\cref{lem:filtration}) together with a related estimate of Gorodetsky. Finally, they state that the main theorem is established later by applying these lemmas.

Question:
Let \(\mathcal M\) be the set of monic polynomials in \(\mathbb F_q[t]\), where \(q\) is a prime power. A Steinhaus random multiplicative function over \(\mathbb F_q[t]\) is a random function \(f:\mathcal M\to\mathbb C\) such that the values \(f(P)\) on monic irreducible polynomials \(P\) are independent and uniformly distributed on the unit circle \(\mathbb S^1\), with \(f(1)=1\) and \(f(F)=\prod_i f(P_i)^{a_i}\) whenever \(F=\prod_i P_i^{a_i}\) is the factorization of \(F\) into powers of distinct monic irreducibles. Fix an integer \(N\ge 1\), and let \(\mathcal A=\mathcal A_{N,q}\) be a subset of the monic polynomials of degree \(N\) such that
\[
|\mathcal A|\gg q^N\exp\!\big(-\tfrac13\sqrt{N\log q}\big).
\]
Assume there is a subset \(\mathcal S=\mathcal S_{N,q}\subset \mathcal A\) with
\[
|\mathcal S|=(1+o(1))|\mathcal A| \quad \text{as } q^N\to\infty,
\]
and whose multiplicative energy satisfies
\[
\big|\{(F_1,F_2,G_1,G_2)\in \mathcal S^4: F_1F_2=G_1G_2\}\big|=(2+o(1))|\mathcal S|^2 \quad \text{as } q^N\to\infty.
\]
Here \(\mathcal{CN}(0,1)\) denotes the standard complex normal distribution. As \(q^N\to\infty\), which limiting statement holds?

For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty.
\]

Choices:
B.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } N\to\infty \text{ with } q \text{ fixed}.
\]
C.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\), the normalized sums
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F)
\]
form a tight family of complex-valued random variables with mean \(0\) and variance \(1\) as \(q^N\to\infty\).
D.
There exists a Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\) such that
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty.
\]
E.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty,
\]
provided merely that
\[
\big|\{(F_1,F_2,G_1,G_2)\in \mathcal S^4: F_1F_2=G_1G_2\}\big|\le (2+o(1))|\mathcal S|^2
\quad \text{as } q^N\to\infty.
\]

The stem does not state the conclusion explicitly and does not directly reveal choice A. It presents hypotheses and asks for the resulting limiting statement, so there is no clear answer leakage.

The item is essentially a direct theorem-recall question: the correct option is just the theorem's conclusion under the stated assumptions. This makes it close to a restatement rather than a genuinely new inference task.

Some reasoning is needed to distinguish the exact asymptotic regime, quantifier strength, and sharpness of the multiplicative energy hypothesis from nearby alternatives. However, the main task is still recognizing the theorem statement rather than generating or synthesizing a conclusion.

The distractors are mathematically plausible and target realistic failure modes: changing the limit regime, weakening convergence to tightness, weakening a universal statement to an existential one, and replacing an asymptotic equality by an upper bound. They are distinct and well-designed.

Low-leakage but largely theorem-recall. The distractors are strong and mathematically meaningful, yet the question is fairly tautological and only moderately tests 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, and let \(f:\mathbb D\to\mathbb D\) be a hyperbolic inner function whose Denjoy--Wolff point is \(p\in \partial\mathbb D\). Let \(\alpha=f'(p)\in(0,1)\), and let \(f^*\) denote the radial boundary map, defined for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) by \(f^*(\zeta)=\lim_{r\to1^-}f(r\zeta)\), where \(\lambda\) is normalized Lebesgue measure on \(\partial\mathbb D\). For \(a,b>0\), define the annulus centered at \(p\) by \(A(p;a,b)=\{z\in\mathbb C: a<|z-p|<b\}\). Under these assumptions, which asymptotic statement holds for the boundary iterates \((f^*)^n\) as \(n\to\infty\)?

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

Choices:
B.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N=N(\zeta,\varepsilon)\) 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,
\[
\alpha^{(1-\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1+\varepsilon)n}
\]
for all sufficiently large \(n\).
C.
For \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\), one has
\[
(f^*)^n(\zeta)\to p
\quad\text{as }n\to\infty.
\]
Equivalently, for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) and every neighborhood \(U\) of \(p\) in \(\partial\mathbb D\), there exists \(N=N(\zeta,U)\) such that \((f^*)^n(\zeta)\in U\) for all \(n\ge N\).
D.
There exists \(N=N(\varepsilon)\) such that for every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) and 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,
\[
\alpha^{(1+\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}
\]
for all \(n\ge N\).
E.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N=N(\zeta,\varepsilon)\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in D\bigl(p,\alpha^{(1+\varepsilon)n}\bigr)\cap\partial\mathbb D,
\]
equivalently,
\[
\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1+\varepsilon)n}
\]
for all sufficiently large \(n\).

The stem gives the setup and definitions but does not explicitly state the annular rate estimate or otherwise reveal option A. The correct answer is not directly leaked.

The item is close to a theorem-recall question: it asks which asymptotic statement holds under standard hypotheses, and the correct option is essentially the precise theorem statement. Still, the alternatives differ in rate, quantifiers, and strength, so it is not a pure verbatim restatement.

Some reasoning is required to distinguish the exact asymptotic estimate from weaker, stronger, or quantifier-shifted variants. However, the item mainly tests recognition of the correct theorem statement rather than substantial derivation from first principles.

The distractors are plausible and mathematically meaningful: B reverses the annulus bounds, C gives a weaker true statement, D introduces an incorrect uniform threshold, and E asserts an overly strong rate. These reflect realistic failure modes.

A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward precise recall/comparison than deep generative reasoning.