[Communication-efficient similarity search for signals]\label{thm:jl_signals_receiver}
Fix $d\ge 1$ and $n \ge 8,$ and let $f_1,\dots,f_n\colon [0,1]^d\to[-1,1]$ be continuous.
A \emph{sketching protocol} consists of an encoding map
\[
\Enc\colon C([0,1]^d)\to\mathbb R^k
\]
and a decoding rule
\[
\Dec\colon \mathbb R^k\times \mathbb R^k\to\mathbb R,
\]
where $\Dec(\Enc(f_i),\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance
\[
D(i,j):=\int_{[0,1]^d}\abs{f_i(x)-f_j(x)}^2\,\,\mathrm{d} x.
\]

Then for every $\varepsilon\in(0,1)$ there exist $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\Enc,\Dec$
such that for all $i,j\in[n]$,
\[
\abs{\Dec(\Enc(f_i),\Enc(f_j)) - D(i,j)} \le \varepsilon.
\]
Moreover, one may take $\Dec(u,v)=\left|u-v\right|^2$ (i.e.\ decoding is just squared Euclidean distance $\left|\cdot\right|^2$) and $\Enc$ to be linear.

The theorems of Helly~\cite{helly1923mengen} and Carath\'eodory~\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.
At a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.
Both results have given rise to extensive and largely independent lines of research; we refer to~\cite{barany2022helly, barany2021combinatorial} for recent surveys.

In a breakthrough paper~\cite{adiprasito2020theorems}, Adiprasito, B\'ar\'any, Mustafa, and Terpai obtained \emph{no-dimensional} (or \emph{approximate}) versions of Helly's and Carath\'eodory's theorems in Euclidean spaces.
These results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,
one gets approximate conclusions controlled by a single natural parameter.
Motivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.

The goal of this note is to bring attention to this circle of problems.
We formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.
At the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,
which allows one to pass from $\ell_1$-type or $\ell_\infty$-type geometry to an appropriately chosen $\ell_p$-space with desired geometric properties.

We will treat the needed Banach space facts as a black box.
For convenience, we recall all relevant definitions and inequalities in \Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.
Then we explain in detail what we mean by no-dimensional versions of Helly's and Carath\'eodory's theorems, survey recent results on the topic, and formulate several open problems.
Finally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.

\subsection{Applications of no-dimensional results}

In many applications \cite{Garofalakis2013,Wang2016} one stores or transmits large objects
(signals, images, fields), while the downstream task uses only pairwise similarity queries,
often measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).
The following theorem shows that, for a fixed collection of $n$ signals, one can replace each
signal by a short ``fingerprint'' in $\mathbb R^k$ for sufficiently small $k$, while preserving all
pairwise squared $L_2$-distances up to additive error~$\varepsilon$.

The result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss
flattening lemma \cite{johnson1984extensions}; see also \cite{vershynin2018high} for modern
expositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random
projection argument, our proof is based on a deterministic construction arising from a
no-dimensional Carath\'eodory-type lemma. This provides a simple illustration of the method.

The theorems of Helly~\cite{helly1923mengen} and Carath\'eodory~\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.
At a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.
Both results have given rise to extensive and largely independent lines of research; we refer to~\cite{barany2022helly, barany2021combinatorial} for recent surveys.

In a breakthrough paper~\cite{adiprasito2020theorems}, Adiprasito, B\'ar\'any, Mustafa, and Terpai obtained \emph{no-dimensional} (or \emph{approximate}) versions of Helly's and Carath\'eodory's theorems in Euclidean spaces.
These results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,
one gets approximate conclusions controlled by a single natural parameter.
Motivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.

The goal of this note is to bring attention to this circle of problems.
We formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.
At the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,
which allows one to pass from $\ell_1$-type or $\ell_\infty$-type geometry to an appropriately chosen $\ell_p$-space with desired geometric properties.

We will treat the needed Banach space facts as a black box.
For convenience, we recall all relevant definitions and inequalities in \Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.
Then we explain in detail what we mean by no-dimensional versions of Helly's and Carath\'eodory's theorems, survey recent results on the topic, and formulate several open problems.
Finally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.

In many applications \cite{Garofalakis2013,Wang2016} one stores or transmits large objects
(signals, images, fields), while the downstream task uses only pairwise similarity queries,
often measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).
The following theorem shows that, for a fixed collection of $n$ signals, one can replace each
signal by a short ``fingerprint'' in $\mathbb R^k$ for sufficiently small $k$, while preserving all
pairwise squared $L_2$-distances up to additive error~$\varepsilon$.

The result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss
flattening lemma \cite{johnson1984extensions}; see also \cite{vershynin2018high} for modern
expositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random
projection argument, our proof is based on a deterministic construction arising from a
no-dimensional Carath\'eodory-type lemma. This provides a simple illustration of the method.

The result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss
flattening lemma \cite{johnson1984extensions}; see also \cite{vershynin2018high} for modern
expositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random
projection argument, our proof is based on a deterministic construction arising from a
no-dimensional Carath\'eodory-type lemma. This provides a simple illustration of the method.

We emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.
First, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range
of the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit
this in the proof). Third, the functions are defined on a domain of measure one, which is also
important for our argument since we work on a probability space.

\begin{thm}[Local-to-global Chebyshev regression]\label{thm:chebyshev_ball}
Let $a_1,\dots,a_m\in [-1,1]^d$ and $b_1,\dots,b_m\in \R$.
Fix $k\in[m]$ and parameters $R\ge 1$ and $r\ge 0$.
Suppose that for every subset $J\subset[m]$ with $\card{J}=k$ there exists a point $x_J\in R\ball{}_{\ell_1^d}$ such that
\[
\max_{j\in J}\abs{\iprod{a_j}{x_J}-b_j}\ \le\ r.
\]
Then there exists a point $x\in eR\ball{}_{\ell_1^d}$ such that
\[
\max_{i\in[m]}\abs{\iprod{a_i}{x}-b_i}
\ \le\
r + 21\,R\,\sqrt{\frac{\ln d}{k}}.
\]
\end{thm}

\begin{thm}[Quantum feasibility from local consistency]\label{thm:quantum_psd_local_to_global}
Let $d\ge 3$ and $k\in[m]$. Let $A_1,\dots,A_m\in\mathcal{H}_d$ be Hermitian matrices satisfying
\[
\normsch[\infty]{A_i}\le 1 \qquad\text{for all } i\in[m],
\]
and let $b_1,\dots,b_m\in\R$. Fix $t\ge 0$.
Assume that for every subset $J\subset[m]$ with $\abs{J}=k$, there exists a density matrix
$\rho_J\succeq 0$ with $\mathrm{Tr}(\rho_J)=1$ such that
\[
\abs{\iprod{A_j}{\rho_J} - b_j}
\le t \qquad\text{for all } j\in J.
\]
Then there exists a density matrix $\rho\succeq 0$ with $\mathrm{Tr}(\rho)=1$ such that
\[
\abs{\iprod{A_i}{\rho} - b_i}
\ \le\
t + 21\,\sqrt{\frac{\ln d}{k}},
\]
for all $i\in[m]$.
\end{thm}

\begin{lem}\label{lem:Caratheodory_sequence_l_p}
Let $2\le p<+\infty$.
Then the sequence $\braces{21\sqrt{\frac{p-1}{k}}}_{k\ge 1}$ is a Carath\'eodory sequence for $\ell_p$ and for $S_p$.
Moreover, whenever the convex hull of a subset $Q$ of the unit ball $\ball{}_X$ contains the origin,
there exists a greedy algorithm that selects points $x_1,\dots,x_k\in Q$ successively so that
\[
\norm{\frac{x_1+\cdots+x_k}{k}}
\le R_k(X)
\]
at each step.
\end{lem}

\section{Dimension strikes back. Carath\'eodory-type results and problems}
\subsection{Johnson--Lindenstrauss-type sketching}
\Href{Theorem}{thm:jl_signals_receiver} follows from the following result
since the measure of $[0,1]^d$ is one. We note that similar results were obtained in \cite{Eskenazis2023}.
\begin{thm}[Johnson--Lindenstrauss sketching for bounded functions]\label{thm:JL_for_functions}
Let $(\Omega,\mu)$ be a probability space and let $f_1,\dots,f_n\colon \Omega\to[-1,1]$ be functions in $L_2(\Omega,\mu)$.
Fix $\varepsilon\in(0,1)$. Then there exists an integer
\[
k \ \le\ C\,\frac{\ln n}{\varepsilon^2}
\]
and vectors $x_1,\dots,x_n\in\R^k$ such that for all $i,j\in[n]$,
\[
\enorm{x_i-x_j}^2- \varepsilon \ \le\
\norm{f_i-f_j}_2^2 \ \le\
\enorm{x_i-x_j}^2 + \varepsilon .
\]
Here $C>0$ is an absolute constant.
\end{thm}

Apply \Href{Lemma}{lem:Caratheodory_sequence_l_p} in the space $\ell_\infty^{ {n\choose 2}}$ (via the substitution from \Href{Proposition}{prp:l_p_trick_bounds})
to obtain points $\omega_1,\dots,\omega_k\in\Omega$ such that the empirical average
\[
\frac{1}{k}\sum_{s=1}^k v(\omega_s)
\]
approximates $\int_\Omega v(\omega)\,d\mu(\omega)$ in $\ell_\infty^{ {n\choose 2}}$ up to error $\varepsilon$.
Equivalently, for all $i<j$,
\[
\enorm{
\frac{1}{k}\sum_{s=1}^k \abs{f_i(\omega_s)-f_j(\omega_s)}^2
-
\norm{f_i-f_j}_2^2
}
\le \varepsilon.
\]
Now define
\[
x_i:=\frac{1}{\sqrt{k}}\big(f_i(\omega_1),\dots,f_i(\omega_k)\big)\in\R^k.
\]
Then
\[
\enorm{x_i-x_j}^2=\frac{1}{k}\sum_{s=1}^k \abs{f_i(\omega_s)-f_j(\omega_s)}^2,
\]
and the desired inequality follows.
\end{proof}
\subsection{Positive semidefinite matrix sparsification}

\begin{conj}\label{conj:deterministic_rudelson}
Let $A_1,\dots,A_n\in \mathcal{M}_d$ be positive semidefinite and suppose that
\[
\lambda_1A_1+\cdots+\lambda_nA_n=\id_d,
\]
where the coefficients $\lambda_i$ are nonnegative and summing up to one.
Then for every $\varepsilon\in(0,1)$ there exists a deterministic algorithm that selects indices
$i_1,\dots,i_k\in[n]$ with $k\le C(\varepsilon)\,d\ln d$ and outputs positive coefficients $\beta_1,\dots,\beta_k$
such that
\[
\Bigl\|\beta_1A_{i_1}+\cdots+\beta_kA_{i_k}-\id_d\Bigr\|_{S_\infty}\le \varepsilon.
\]
\end{conj}

We emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.
First, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range
of the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit
this in the proof). Third, the functions are defined on a domain of measure one, which is also
important for our argument since we work on a probability space.

The main point is that the statement admits a deterministic proof based on a no-dimensional
Carath\'eodory lemma.

Also, we obtain several local-to-global estimates for $\ell_1$-minimization problems using no-dimensional Helly-type results.
We write $[m]:=\braces{1,\dots,m}$ for $m\in\mathbb N$, and we equip $\mathbb R^d$ with the standard $\ell_p$ norm $\left\|\cdot\right\|_p$.
We denote by $\ell_p^d:=(\mathbb R^d,\left\|\cdot\right\|_p)$ the resulting normed space and by $\ball{}_{\ell_p^d}$ its unit ball.

\medskip
One basic example is \emph{Chebyshev regression} (also known as \emph{minimax absolute residual fitting} or $\ell_\infty$-regression), the classical problem of fitting a linear model by minimizing the worst absolute residual:
\[
\min_{x\in\mathbb R^d}\ \max_{i\in[m]} \abs{\iprod{a_i}{x}-b_i}.
\]
It admits an immediate linear programming formulation by introducing an auxiliary variable $t$ and enforcing the two-sided constraints
$-t\le \iprod{a_i}{x}-b_i\le t$ for all $i\in[m]$; see, e.g., \cite{boyd2004convex,ben2001lectures}.
From an algorithmic viewpoint, this connects Chebyshev regression to the general toolbox of convex optimization and linear programming, while its geometry is governed by small active constraint sets, a perspective formalized in the LP-type framework and its randomized algorithms \cite{clarkson1995vegas,matouvsek1992subexponential}.
More recently, Chebyshev/$\ell_\infty$ guarantees have also been studied via modern randomized linear algebra and ``sketch-and-solve'' methods, where one seeks fast approximate solutions together with explicit $\ell_\infty$ control \cite{price2017fast,song2023nearly}.
In the present work, we pursue a different direction: using no-dimensional Helly-type results to derive \emph{local-to-global} approximation guarantees from $k$-wise feasibility information, with explicit dimension dependence.

Our first local-to-global statement shows that approximate global feasibility over the $\ell_1$ ball follows from approximate feasibility on all $k$-subcollections of constraints, with an explicit  dependence on the dimension.

\begin{thm}[Local-to-global Chebyshev regression]\label{thm:chebyshev_ball}
Let $a_1,\dots,a_m\in [-1,1]^d$ and $b_1,\dots,b_m\in \mathbb R$.
Fix $k\in[m]$ and parameters $R\ge 1$ and $r\ge 0$.
Suppose that for every subset $J\subset[m]$ with $\card{J}=k$ there exists a point $x_J\in R\ball{}_{\ell_1^d}$ such that
\[
\max_{j\in J}\abs{\iprod{a_j}{x_J}-b_j}\ \le\ r.
\]
Then there exists a point $x\in eR\ball{}_{\ell_1^d}$ such that
\[
\max_{i\in[m]}\abs{\iprod{a_i}{x}-b_i}
\ \le\
r + 21\,R\,\sqrt{\frac{\ln d}{k}}.
\]
\end{thm}

\medskip
Our second application of no-dimensional Helly-type results is motivated by feasibility and consistency questions in quantum information theory.
A prototypical instance is the \emph{consistency of local density matrices} (also known as the \emph{quantum marginal problem}), which asks whether a collection of local constraints can arise from a single global quantum state; this problem is known to be computationally intractable in general; it was first shown to be in QMA and QMA-complete under Turing reductions by Liu \cite{liu2006consistency}, and subsequently proven to be QMA-complete under Karp reductions by Broadbent and Grilo \cite{broadbent2022qma}.
From a geometric viewpoint, the natural ambient spaces are Schatten classes and their convexity properties, and we refer to \cite{aubrun2017alice} for background on the interface between asymptotic geometric analysis and quantum information.

Fix $d\ge 1$ and let $\mathbb{F}\in\{\mathbb R,\mathbb C\}$.
Write $\mathcal{M}_d:=\mathbb{F}^{d\times d}$ for the space of $d\times d$ matrices over $\mathbb{F}$.
We say that $A\in\mathcal{M}_d$ is \emph{self-adjoint} if $A=A^\ast$, where $A^\ast$ denotes the transpose when $\mathbb{F}=\mathbb R$
and the conjugate transpose when $\mathbb{F}=\mathbb C$.
If $\mathbb{F}=\mathbb C$, let
\[
\mathcal{H}_d:=\{A\in\mathbb C^{d\times d}:\ A=A^\ast\}
\]
denote the  vector space of Hermitian matrices.

For $1\le p\le\infty$ we write $\normsch[p]{A}$ for the Schatten $p$-norm (the $\ell_p$-norm of the singular value vector),
and we equip $\mathcal{M}_d$ with the Frobenius (Hilbert--Schmidt) inner product
\[
\iprod{A}{B}:=\mathrm{Tr}(A^\ast B).
\]

A matrix $\rho\in\mathcal{M}_d$ is called \emph{positive semidefinite}, written $\rho\succeq 0$, if it is self-adjoint and has
nonnegative spectrum. A \emph{density matrix} is a positive semidefinite matrix of unit trace:
\[
\rho\succeq 0,
\qquad
\mathrm{Tr}(\rho)=1.
\]

Given self-adjoint measurement operators $A_1,\dots,A_m\in\mathcal{M}_d$ and target values $b_1,\dots,b_m\in\mathbb R$,
we consider the feasibility problem of finding a density matrix whose linear measurements $\iprod{A_i}{\rho}$ are simultaneously
consistent with $b_i$ up to a tolerance level.
In this work we study a \emph{local-to-global} version: assuming that every subcollection of $k$ constraints is feasible up to error $t$,
we ask whether this implies the existence of a single density matrix that approximately satisfies all $m$ constraints.

\begin{thm}[Quantum feasibility from local consistency]\label{thm:quantum_psd_local_to_global}
Let $d\ge 3$ and $k\in[m]$. Let $A_1,\dots,A_m\in\mathcal{H}_d$ be Hermitian matrices satisfying
\[
\normsch[\infty]{A_i}\le 1 \qquad\text{for all } i\in[m],
\]
and let $b_1,\dots,b_m\in\mathbb R$. Fix $t\ge 0$.
Assume that for every subset $J\subset[m]$ with $\abs{J}=k$, there exists a density matrix
$\rho_J\succeq 0$ with $\mathrm{Tr}(\rho_J)=1$ such that
\[
\abs{\iprod{A_j}{\rho_J} - b_j}
\le t \qquad\text{for all } j\in J.
\]
Then there exists a density matrix $\rho\succeq 0$ with $\mathrm{Tr}(\rho)=1$ such that
\[
\abs{\iprod{A_i}{\rho} - b_i}
\ \le\
t + 21\,\sqrt{\frac{\ln d}{k}},
\]
for all $i\in[m]$.
\end{thm}

The post-theorem introduction gives only a high-level indication of the proof method for Theorem~\ref{thm:jl_signals_receiver}: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\'eodory lemma.” No further steps or structure of the argument are described there.

The post-theorem introduction gives only a high-level indication of the proof method for the main theorem: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\'eodory lemma.” No further steps or structure of the argument are described there.

[Communication-efficient similarity search for signals]\label{thm:jl_signals_receiver}
Fix $d\ge 1$ and $n \ge 8,$ and let $f_1,\dots,f_n\colon [0,1]^d\to[-1,1]$ be continuous.
A \emph{sketching protocol} consists of an encoding map
\[
\Enc\colon C([0,1]^d)\to\mathbb R^k
\]
and a decoding rule
\[
\Dec\colon \mathbb R^k\times \mathbb R^k\to\mathbb R,
\]
where $\Dec(\Enc(f_i),\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance
\[
D(i,j):=\int_{[0,1]^d}\abs{f_i(x)-f_j(x)}^2\,\,\mathrm{d} x.
\]

Then for every $\varepsilon\in(0,1)$ there exist $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\Enc,\Dec$
such that for all $i,j\in[n],$
\[
\abs{\Dec(\Enc(f_i),\Enc(f_j)) - D(i,j)} \le \varepsilon.
\]
Moreover, one may take $\Dec(u,v)=\left|u-v\right|^2$ (i.e.\ decoding is just squared Euclidean distance $\left|\cdot\right|^2$) and $\Enc$ to be linear.

Existential–Universal
Algorithmic or Constructive

Question:
Let $d\ge 1$ and $n\ge 8$, and let $f_1,\dots,f_n\colon [0,1]^d\to[-1,1]$ be continuous functions. Define the squared $L_2$-distance between two signals by
\[
D(i,j):=\int_{[0,1]^d} |f_i(x)-f_j(x)|^2\,dx.
\]
A sketching protocol consists of an encoding map $\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k$ and a decoding rule $\mathrm{Dec}\colon \mathbb R^k\times\mathbb R^k\to\mathbb R$, where $\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j))$ is used to estimate $D(i,j)$. For a given $\varepsilon\in(0,1)$, which statement holds for such a family of signals?

There exist an integer $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\mathrm{Enc}$ and $\mathrm{Dec}$ such that for every $i,j\in[n]$,
\[
\big|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\big|\le \varepsilon.
\]
Moreover, one may choose $\mathrm{Dec}(u,v)=|u-v|^2$ (the squared Euclidean distance in $\mathbb R^k$) and choose $\mathrm{Enc}$ to be linear.

Choices:
B.
There exist an integer $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\mathrm{Enc}$ and $\mathrm{Dec}$ such that for every $i,j\in[n]$,
\[
\big|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\big|\le \varepsilon.
\]
Moreover, one may choose $\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\mathrm{Enc}$ to be linear, with $C>0$ an absolute constant independent of $d$, $n$, $\varepsilon$, and of the range bound on the functions.
C.
There exist an integer $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\mathrm{Enc}$ and $\mathrm{Dec}$ such that for every $i,j\in[n]$,
\[
\big|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\big|\le \varepsilon.
\]
D.
There exist an integer $k\le C\,\frac{\ln n}{\varepsilon^2}$ and maps $\mathrm{Enc}$ and $\mathrm{Dec}$ such that for every $i,j\in[n]$,
\[
\big|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\big|\le \varepsilon.
\]
Moreover, one may choose $\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\mathrm{Enc}$ to be linear even when the functions $f_1,\dots,f_n\colon [0,1]^d\to\mathbb R$ are arbitrary continuous functions, with no assumption that their values lie in $[-1,1]$.
E.
There exist maps $\mathrm{Enc}$ and $\mathrm{Dec}$ such that for every $\varepsilon\in(0,1)$ and every $i,j\in[n]$,
\[
\big|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\big|\le \varepsilon,
\]
with a sketch dimension satisfying
\[
k\le C\,\frac{\ln n}{\varepsilon},
\]
and one may take $\mathrm{Dec}(u,v)=|u-v|^2$ and $\mathrm{Enc}$ linear.

hidden dependence on bounded range normalization

dropped the additional conclusion specifying Euclidean decoding and linear encoding

uniform bounded-range hypothesis $f_i\in[-1,1]$

the $\varepsilon^{-2}$ dependence in the sketch dimension bound


Question:
Let d\ge 1 and n\ge 8, and let f_1,\dots,f_n\colon [0,1]^d\to[-1,1] be continuous functions. For i,j\in\{1,\dots,n\}, define their squared L_2-distance by
\[
D(i,j):=\int_{[0,1]^d} |f_i(x)-f_j(x)|^2\,dx.
\]
A sketching protocol consists of an encoding map \(\mathrm{Enc}\colon C([0,1]^d)\to \mathbb R^k\) and a decoding rule \(\mathrm{Dec}\colon \mathbb R^k\times \mathbb R^k\to \mathbb R\), where \(\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j))\) is used to estimate \(D(i,j)\). Which statement holds for every \(\varepsilon\in(0,1)\)?

There exists an absolute constant \(C>0\) such that, for every \(\varepsilon\in(0,1)\), there exist an integer \(k\le C\,\frac{\ln n}{\varepsilon^2}\), a linear map \(\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k\), and a decoding rule that may be chosen as \(\mathrm{Dec}(u,v)=|u-v|^2\), with the property that for all \(i,j\in\{1,\dots,n\}\),
\[
\bigl|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\bigr|\le \varepsilon.
\]
Equivalently, all pairwise squared \(L_2\)-distances among \(f_1,\dots,f_n\) can be approximated within additive error \(\varepsilon\) by squared Euclidean distances between their \(k\)-dimensional sketches.

Choices:
B.
There exists an absolute constant \(C>0\) such that, for every \(\varepsilon\in(0,1)\), there exist an integer \(k\le C\,\frac{\ln n}{\varepsilon^2}\), a linear map \(\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k\), and a decoding rule that may be chosen as \(\mathrm{Dec}(u,v)=|u-v|^2\), with the property that for all \(i,j\in\{1,\dots,n\}\),
\[
(1-\varepsilon)D(i,j)\le \mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j))\le (1+\varepsilon)D(i,j).
\]
Equivalently, all pairwise squared \(L_2\)-distances among \(f_1,\dots,f_n\) can be approximated within relative error \(\varepsilon\) by squared Euclidean distances between their \(k\)-dimensional sketches.
C.
There exists an absolute constant \(C>0\) such that, for every \(\varepsilon\in(0,1)\), there exist an integer \(k\le C\,\frac{\ln n}{\varepsilon^2}\), an encoding map \(\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k\), and a decoding rule \(\mathrm{Dec}\colon\mathbb R^k\times\mathbb R^k\to\mathbb R\) such that for all \(i,j\in\{1,\dots,n\}\),
\[
\bigl|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\bigr|\le \varepsilon.
\]
In particular, all pairwise squared \(L_2\)-distances among \(f_1,\dots,f_n\) admit additive \(\varepsilon\)-approximation from \(k\)-dimensional sketches.
D.
There exists an absolute constant \(C>0\) such that, for every \(\varepsilon\in(0,1)\), one can choose an integer \(k\le C\,\frac{\ln n}{\varepsilon^2}\) and a single linear map \(\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k\), depending only on \(d,n,\varepsilon\) and not on the particular collection \(f_1,\dots,f_n\), together with \(\mathrm{Dec}(u,v)=|u-v|^2\), so that for every family of continuous functions \(f_1,\dots,f_n\colon[0,1]^d\to[-1,1]\) and all \(i,j\in\{1,\dots,n\}\),
\[
\bigl|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\bigr|\le \varepsilon.
\]
E.
There exists an absolute constant \(C>0\) such that, for every \(\varepsilon\in(0,1)\), there exist an integer \(k\le C\,\frac{\ln n}{\varepsilon^2}\), a linear map \(\mathrm{Enc}\colon C([0,1]^d)\to\mathbb R^k\), and a decoding rule that may be chosen as \(\mathrm{Dec}(u,v)=|u-v|^2\), with the property that for all \(i,j\in\{1,\dots,n\}\),
\[
\bigl|\mathrm{Dec}(\mathrm{Enc}(f_i),\mathrm{Enc}(f_j)) - D(i,j)\bigr|\le \varepsilon,
\]
and moreover one may always take \(k\le C\,\frac{\ln n}{\varepsilon}\).


\label{theorem:k-dim-ds}
   Let $k\geq 1$, and let $\map{\psi}{\mathbb{N}}{\left[0,1/2\right]}$ be such that $\sum_{q=1}^\infty \left(\varphi(q)\psi(q)/q\right)^k=\infty$. For $\alpha\in \left[0,1\right]^k$, denote by $S_k(\alpha,Q)$ the number of solutions $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ to the inequality:
    \begin{align*}
        \pn{q\alpha-a}{\infty} < \psi(q), \quad \text{given } q \leq Q,
    \end{align*}
    satisfying $\gcd(a_i, q)=1$, for $i\in\{1,\dots, k\}$, $a=(a_1,\dots, a_k)$. Moreover, given $Q\in \mathbb{N}$, define:
    \begin{align*}
        \Psi_k(Q)\coloneq \sum_{q\leq Q}\left(\frac{2\psi(q)\varphi(q)}{q}\right)^k.
    \end{align*}
    Then for all $\varepsilon>0$, as $Q\to \infty$, we have for almost all $\alpha$ that:
    \begin{align*}
        S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}\left( \Psi_k(Q)^{1/2+\varepsilon} \right).
    \end{align*}

\begin{theorem}\label{theorem:k-dim-ds}
   Let $k\geq 1$, and let $\map{\psi}{\N}{\left[0,1/2\right]}$ be such that $\sum_{q=1}^\infty \left(\varphi(q)\psi(q)/q\right)^k=\infty$. For $\alpha\in \left[0,1\right]^k$, denote by $S_k(\alpha,Q)$ the number of solutions $(a,q)\in\Z^k\times \N$ to the inequality:
    \begin{align*}
        \pn{q\alpha-a}{\infty} < \psi(q), \quad \text{given } q \leq Q,
    \end{align*}
    satisfying $\gcd(a_i, q)=1$, for $i\in\{1,\dots, k\}$, $a=(a_1,\dots, a_k)$. Moreover, given $Q\in \N$, define:
    \begin{align*}
        \Psi_k(Q)\coloneq \sum_{q\leq Q}\left(\frac{2\psi(q)\varphi(q)}{q}\right)^k.
    \end{align*}
    Then for all $\ep>0$, as $Q\to \infty$, we have for almost all $\alpha$ that:
    \begin{align*}
        S_k(\alpha,Q)=\Psi_k(Q)+O_{\ep,k}\left( \Psi_k(Q)^{1/2+\ep} \right).
    \end{align*}
\end{theorem}

Simultaneous Diophantine approximation concerns the existence of rational approximations of multidimensional irrational points: a classical result of Dirichlet states that, given $k\geq 1$, for every $\alpha\in \mathbb{R}^k$, there are infinitely many solutions $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ to the inequality \cite[Chapter~2]{schmidt1980diophantine}:
\begin{align*}
    \pn{q\alpha - a}{\infty}<q^{-1/k}.
\end{align*}
Moreover, this bound is optimal up to a constant for numbers with bounded partial quotients when represented as continued fractions; this class of numbers are known as badly approximable, and contains e.g. for $k=1$ the golden ratio $\varphi$, and $\sqrt{2}$. However, for typical $\alpha$, in the sense of Lebesgue measure, it turns out that this bound can be significantly improved: metric Diophantine approximation asks the question of by exactly how much. For an overview of metric Diophantine approximation (and metric number theory, in general), one should read the monograph \cite{harman1998metricnumbertheory}.

A notable foundational result of metric Diophantine approximation is Khintchine's theorem: given a function $\map{\psi}{\mathbb{N}}{[0,\infty)}$ monotonically decreasing, Khintchine's theorem states that for almost every $\alpha\in \mathbb{R}^k$, there exists infinitely many pairs $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ satisfying the inequality:
\begin{align*}
    \pn{q\alpha-a}{\infty}< \psi(q),
\end{align*}
if and only if $\sum_{q\in\mathbb{N}}(\psi(q))^k$ diverges \cite[Section~3]{beresnevich2009classical}. Intuitively, one can choose $\psi(q)=1/q\log q$ to see that on average the bound from Dirichlet's theorem can be improved by a logarithmic factor. In the case of only finitely many solutions, Khintchine's theorem is immediate from the convergence Borel-Cantelli lemma. The divergence lemma, however, can not be applied directly as there is no stochastic independence, but when $\psi$ is monotonic, the set system under consideration behaves approximately independent in a sense that is sufficient for the statement to hold.

In 1941, Duffin and Schaeffer \cite{duffin1941khintchinesproblem} showed that Khintchine's theorem in 1-dimension fails without the monotonicity condition, by constructing a counterexample where the series $\sum_{q\in \mathbb{N}} \psi(q)$ diverges, yet, for almost every $\alpha$, only finitely many solutions exist. Their work relies on the multiple representations of fractions: if $p/q$ is a close approximation for $\alpha$, then so is $np/nq$ for all $n\in\mathbb{N}$. As a result, if the function $\psi$ is constructed to have values disproportionately high on integer multiples of $q$, the sum may still diverge, despite the $\alpha$ admitting good approximations being relatively rare.
This led them to conjecture that a variant of Khintchine's theorem would hold when non-reduced fractions are excluded, as follows: denoting by $\varphi$ the Euler totient function, if $\map{\psi}{\mathbb{N}}{[0,\infty)}$ is a function such that $\sum_{q\in\mathbb{N}}\psi(q)\varphi(q)/q$ diverges, then for almost all $\alpha\in\mathbb{R}$, there exists infinitely many solutions to the inequality:
\begin{align}
    \left\lvert \alpha - \frac{p}{q}\right\rvert < \frac{\psi(q)}{q}.\label{eq:approx-ineq}
\end{align}
The Duffin-Schaeffer conjecture was open for 78 years before it was proven by Koukoulopoulos and Maynard \cite{koukoulopoulos2020duffin}. A full survey on the conjecture and an exposition of its proof can be found in \cite{hauke2025centurymetricdiophantineapproximation}.

The work of \cite{koukoulopoulos2020duffin}, however, left open the question of quantitative results on the number of approximations, assuming some bound on $q$. A quantitative version of Khintchine's theorem was proven by Schmidt in 1960 \cite{schmidt1960ametricaltheorem}; one can also view \cite[Chapter~4]{harman1998metricnumbertheory} for an overview of Schmidt's method. Schmidt-type results on the Duffin-Schaeffer conjecture have been the subject of various recent works \cite{aistleitner2023metrictheoryofapproximations, hauke2024provingduffinschaefferconjecturegcd}, with the sharpest known being of Koukoulopoulos, Maynard and Yang \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}.
Towards this quantitative question, given $\map{\psi}{\mathbb{N}}{[0,1/2]}$ and $Q\geq 1$, we define:
\begin{align*}
    \Psi(Q) \coloneq \sum_{q\leq Q}\frac{2\psi(q)\varphi(q)}{q}.
\end{align*}
Moreover, denote by $S(\alpha,Q)$ the number of pairs $(p,q)\in \mathbb{Z} \times \mathbb{N}$ with $q\leq Q,\ \gcd(p,q)=1$ that satisfy the inequality \eqref{eq:approx-ineq}. The work of Koukoulopoulos, Maynard and Yang \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} then states that, assuming $\lim_{Q\to\infty}\Psi(Q)=\infty$, for every $\varepsilon>0$ and almost all $\alpha\in\mathbb{R}$, we have as $Q\to \infty$:
\begin{align*}
    S(\alpha,Q)=\Psi(Q)+O_\varepsilon\left(\Psi(Q)^{\frac{1}{2}+\varepsilon}\right).
\end{align*}

In this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\geq 1$, given $\map{\psi}{\mathbb{N}}{[0,\infty)}$ and $\alpha\in \mathbb{R}^k$, we ask whether there exists infinitely many solutions $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ satisfying the inequality:
\begin{align}
    \pn{q\alpha-a}{\infty}< \psi(q),\label{eq:kdim-approx-ineq}
\end{align}
where $a=(a_1,\dots,a_k)$ satisfies $\gcd(a_i, q)=1$ for $i\in\{1,\dots, k\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\geq 2$, the question was first raised by Sprind\v{z}uk \cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \eqref{eq:kdim-approx-ineq} if and only if the sum $\sum_{q\in\mathbb{N}}(\psi(q)\varphi(q)/q)^k$ diverges.

Following the recent progress in the 1-dimensional case, we now ask the question of quantitative results for the higher-dimensional setup of Pollington and Vaughan. By periodicity, we consider only $\alpha \in [0,1]^k$ and thus $0\leq a \leq q$, and we assume $\psi(q)\leq 1/2$, following \cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. The main theorem of this paper is the following. We state the result for all $k\geq 1$ for completeness; our contribution, of course, is the case $k\geq 2$.

A notable foundational result of metric Diophantine approximation is Khintchine's theorem: given a function $\map{\psi}{\mathbb{N}}{[0,\infty)}$ monotonically decreasing, Khintchine's theorem states that for almost every $\alpha\in \mathbb{R}^k$, there exists infinitely many pairs $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ satisfying the inequality:
\begin{align*}
 \pn{q\alpha-a}{\infty}< \psi(q),
\end{align*}
if and only if $\sum_{q\in\mathbb{N}}(\psi(q))^k$ diverges \cite[Section~3]{beresnevich2009classical}. Intuitively, one can choose $\psi(q)=1/q\log q$ to see that on average the bound from Dirichlet's theorem can be improved by a logarithmic factor. In the case of only finitely many solutions, Khintchine's theorem is immediate from the convergence Borel-Cantelli lemma. The divergence lemma, however, can not be applied directly as there is no stochastic independence, but when $\psi$ is monotonic, the set system under consideration behaves approximately independent in a sense that is sufficient for the statement to hold.

In 1941, Duffin and Schaeffer \cite{duffin1941khintchinesproblem} showed that Khintchine's theorem in 1-dimension fails without the monotonicity condition, by constructing a counterexample where the series $\sum_{q\in \mathbb{N}} \psi(q)$ diverges, yet, for almost every $\alpha$, only finitely many solutions exist. Their work relies on the multiple representations of fractions: if $p/q$ is a close approximation for $\alpha$, then so is $np/nq$ for all $n\in\mathbb{N}$. As a result, if the function $\psi$ is constructed to have values disproportionately high on integer multiples of $q$, the sum may still diverge, despite the $\alpha$ admitting good approximations being relatively rare.
This led them to conjecture that a variant of Khintchine's theorem would hold when non-reduced fractions are excluded, as follows: denoting by $\varphi$ the Euler totient function, if $\map{\psi}{\mathbb{N}}{[0,\infty)}$ is a function such that $\sum_{q\in\mathbb{N}}\psi(q)\varphi(q)/q$ diverges, then for almost all $\alpha\in\mathbb{R}$, there exists infinitely many solutions to the inequality:
\begin{align}
 \left\lvert \alpha - \frac{p}{q}\right\rvert < \frac{\psi(q)}{q}.\label{eq:approx-ineq}
\end{align}
The Duffin-Schaeffer conjecture was open for 78 years before it was proven by Koukoulopoulos and Maynard \cite{koukoulopoulos2020duffin}. A full survey on the conjecture and an exposition of its proof can be found in \cite{hauke2025centurymetricdiophantineapproximation}.

The work of \cite{koukoulopoulos2020duffin}, however, left open the question of quantitative results on the number of approximations, assuming some bound on $q$. A quantitative version of Khintchine's theorem was proven by Schmidt in 1960 \cite{schmidt1960ametricaltheorem}; one can also view \cite[Chapter~4]{harman1998metricnumbertheory} for an overview of Schmidt's method. Schmidt-type results on the Duffin-Schaeffer conjecture have been the subject of various recent works \cite{aistleitner2023metrictheoryofapproximations, hauke2024provingduffinschaefferconjecturegcd}, with the sharpest known being of Koukoulopoulos, Maynard and Yang \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}.
Towards this quantitative question, given $\map{\psi}{\mathbb{N}}{[0,1/2]}$ and $Q\geq 1$, we define:
\begin{align*}
 \Psi(Q) \coloneq \sum_{q\leq Q}\frac{2\psi(q)\varphi(q)}{q}.
\end{align*}
Moreover, denote by $S(\alpha,Q)$ the number of pairs $(p,q)\in \mathbb{Z} \times \mathbb{N}$ with $q\leq Q,\ \gcd(p,q)=1$ that satisfy the inequality \eqref{eq:approx-ineq}. The work of Koukoulopoulos, Maynard and Yang \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} then states that, assuming $\lim_{Q\to\infty}\Psi(Q)=\infty$, for every $\varepsilon>0$ and almost all $\alpha\in\mathbb{R}$, we have as $Q\to \infty$:
\begin{align*}
 S(\alpha,Q)=\Psi(Q)+O_\varepsilon\left(\Psi(Q)^{\frac{1}{2}+\varepsilon}\right).
\end{align*}

In this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\geq 1$, given $\map{\psi}{\mathbb{N}}{[0,\infty)}$ and $\alpha\in \mathbb{R}^k$, we ask whether there exists infinitely many solutions $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ satisfying the inequality:
\begin{align}
 \pn{q\alpha-a}{\infty}< \psi(q),\label{eq:kdim-approx-ineq}
\end{align}
where $a=(a_1,\dots,a_k)$ satisfies $\gcd(a_i, q)=1$ for $i\in\{1,\dots, k\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\geq 2$, the question was first raised by Sprind\v{z}uk \cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \eqref{eq:kdim-approx-ineq} if and only if the sum $\sum_{q\in\mathbb{N}}(\psi(q)\varphi(q)/q)^k$ diverges.

Following the recent progress in the 1-dimensional case, we now ask the question of quantitative results for the higher-dimensional setup of Pollington and Vaughan. By periodicity, we consider only $\alpha \in [0,1]^k$ and thus $0\leq a \leq q$, and we assume $\psi(q)\leq 1/2$, following \cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. The main theorem of this paper is the following. We state the result for all $k\geq 1$ for completeness; our contribution, of course, is the case $k\geq 2$.

In this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\geq 1$, given $\map{\psi}{\N}{[0,\infty)}$ and $\alpha\in \R^k$, we ask whether there exists infinitely many solutions $(a,q)\in\Z^k\times \N$ satisfying the inequality:
\begin{align}
 \pn{q\alpha-a}{\infty}< \psi(q),\label{eq:kdim-approx-ineq}
\end{align}
where $a=(a_1,\dots,a_k)$ satisfies $\gcd(a_i, q)=1$ for $i\in\{1,\dots, k\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\geq 2$, the question was first raised by Sprind\v{z}uk \cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \eqref{eq:kdim-approx-ineq} if and only if the sum $\sum_{q\in\N}(\psi(q)\varphi(q)/q)^k$ diverges.

Finally, we prove the variance estimate, and thus conclude the proof of our main result. The proof follows \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} with minor modifications.
\begin{proof}[Proof of Theorem~\ref{theorem:k-dim-variance}]
 Note that we have:
 \begin{align*}
 \lint{\left[0,1\right]^k}{}{\left(S_k(\alpha, Q)-\Psi_k(Q)\right)^2}{\alpha}= \sum_{q,r\leq Q} \lambda_k(\A_q^k \cap \A_r^k) - \Psi_k(Q)^2,
 \end{align*}
 and so we will show instead that:
 \begin{align*}
 \sum_{q,r\leq Q} \lambda_k(\A_q^k \cap \A_r^k) \leq \Psi_k(Q)^2 + \Psi_k(Q) + O_{\ep,k}(\Psi_k(Q)^{1+\ep}).
 \end{align*}
 When $q=r$, we have simply $\sum_{q\leq Q}\lambda_k(\A_q^k) = \Psi_k(Q)$. Thus, consider the sum of terms $q\neq r$, and assume that $D=D(q,r)\geq 1/2$. We consider the contributions over the following partition:
 \begin{align*}
 \E^{(1)} &\coloneq\set{(q,r)\in \left[1,Q\right]^k}{L_{D^2}(q,r)\leq \frac{1}{D},\ \omega_{D^2}(q,r)\leq \frac{\ep}{4k}\log(2D)},\\
 \E^{(2)} &\coloneq\set{(q,r)\in \left[1,Q\right]^k}{L_{D^2}(q,r)> \frac{1}{D}},\\
 \E^{(3)} &\coloneq\set{(q,r)\in \left[1,Q\right]^k}{L_{D^2}(q,r)\leq \frac{1}{D},\ \omega_{D^2}(q,r)> \frac{\ep}{4k}\log(2D)}.
 \end{align*}
 First, consider the sum over $\E^{(1)}$. Then applying Lemma~\ref{lem:k-dim-overlap} with $t=D^2$ gives:
 \begin{align*}
 \lambda_k(\A_q^k\cap\A_r^k) &\leq \lambda_k(\A_q^k)\lambda_k(\A_r^k)\me^{2k/D}\left(1+O\left(\frac{2^{\ep\log(2D)/4}\log(4D)}{D}\right)\right)\\
 &\leq \lambda_k(\A_q^k)\lambda_k(\A_r^k)\left(1+O_k(D^{-1+\ep/2})\right)
 \end{align*}
 Therefore:
 \begin{align*}
 \sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k\cap\A_r^k) &\leq \sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k) + O_k\left(\sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k)D^{-1+\ep/2}\right)\\
 &\leq \Psi_k(Q)^2+O_k\left(\sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k)D^{-1+\ep/2}\right).
 \end{align*}
 Focusing on the error term, define $R\coloneq\sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k)D^{-1+\ep/2}$, and let $I$ be the smallest non-negative integer such that $2^I\geq \Psi_k(Q)$. We then split $R$ via dyadic partition on $D$, writing $R_i$ as the part of the sum with $D\in [2^{i-1},2^i)$, and $R'$ the part with $D\geq 2^I$. Then:
 \begin{align*}
 R'&\leq \Psi_k(Q)^{-1+\ep/2}\sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k) \leq \Psi_k(Q)^{1+\ep/2},
 \end{align*}
 and for $i\in \{0,1,\dots,I\}$ we have by Proposition~\ref{proposition:overlap-sum-1} with $y=2^i$ that:
 \begin{align*}
 R_i &\leq (2^{i-1})^{-1+\ep/2}\sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k)\\
 &\ll_{\ep,k} (2^i)^{-1+\ep/2}(2^i)^{1-\ep}\Psi_k(Q)^{1+\ep}\\
 &= 2^{-i\epsilon/2}\Psi_k(Q)^{1+\ep}.
 \end{align*}
 We thus have $R \ll_{\ep,k}\Psi_k(Q)^{1+\ep}$ and so:
 \begin{align*}
 \sum_{(q,r)\in\E^{(1)}}\lambda_k(\A_q^k\cap\A_r^k) \leq \Psi_k(Q)^k + O_{\ep,k}(\Psi_k(Q)^{1+\ep}).
 \end{align*}
 Considering $\E^{(2)}$, we use Lemma~\ref{lem:pv-overlap} to say:
 \begin{align*}
 \sum_{(q,r)\in\E^{(2)}}\lambda_k(\A_q^k\cap\A_r^k) &\ll_k \sum_{(q,r)\in\E^{(2)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k)\me^{kL_D(q,r)}.
 \end{align*}
 If $D\in [2^{i-1},2^i)$, we have $L_{4^{i-1}}(q,r)\geq L_{D^2}(q,r)>1/D\geq 2^{-i}$. We then write:
 \begin{align*}
 \E^{(2)}=\bigcup_{i=0}^\infty \bigcup_{j=i}^\infty \E^{(2)}_{i,j},
 \end{align*}
 where:
 \begin{align*}
 \E^{(2)}_{i,i}&\coloneq\set{(q,r)\in \E^{(2)}}{D(q,r)\in[2^{i-1},2^i),\ L_{4^{i-1}}(q,r)\leq 1},\text{ and}\\
 \E^{(2)}_{i,j}&\coloneq\set{(q,r)\in \E^{(2)}}{D(q,r)\in[2^{i-1},2^i),\ L_{4^{j-1}}(q,r)\leq 1<L_{4^{j-2}}(q,r)},\text{ for }j>i.
 \end{align*}
 Then, if $(q,r)\in\E^{(2)}_{i,j}$, we have:
 \begin{align*}
 L_D(q,r)\leq L_{2^{i-1}}(q,r)\leq \sum_{2^{i-1}< p\leq 4^{j-1}}\frac{1}{p}+L_{4^{j-1}}(q,r) \leq \log\frac{j+1}{i+1} + O(1),
 \end{align*}
 and so:
 \begin{align*}
 \sum_{(q,r)\in\E^{(2)}}\lambda_k(\A_q^k\cap\A_r^k) \ll_k \sum_{j\geq i\geq 0}\left(\frac{j+1}{i+1}\right)^k\sum_{(q,r)\in\E^{(2)}_{i,j}}\lambda_k(\A_q^k)\lambda_k(\A_r^k).
 \end{align*}
 Then for $i\geq 0$, Proposition~\ref{proposition:overlap-sum-2} with $y=2^i$, $t=\max\{1, 4^{i-1}\}$, $s=2^i$ and $C=8$ gives:
 \begin{align*}
 \sum_{(q,r)\in\E^{(2)}_{i,i}}\lambda_k(\A_q^k)\lambda_k(\A_r^k) &\ll_{\ep,k} \exp(-2^i)\Psi_k(Q)^{1+\ep}.
 \end{align*}
 Similarly, for $j>i$, choosing $y=2^i$, $t=\max\{1, 4^{j-2}\}$, $s=1$ and $C=17$ gives:
 \begin{align*}
 \sum_{(q,r)\in\E^{(2)}_{i,j}}\lambda_k(\A_q^k)\lambda_k(\A_r^k) \ll_{\ep,k} \exp(-4^j)\Psi_k(Q)^{1+\ep}.
 \end{align*}
 Together, this gives:
 \begin{align*}
 \sum_{(q,r)\in\E^{(2)}}\lambda_k(\A_q^k\cap\A_r^k) &\ll_{\ep,k} \Psi_k(Q)^{1+\ep}\left(\sum_{i\geq 0}\exp(-2^i) +\sum_{j>i}\left(\frac{j+1}{i+1}\right)^k\exp(-4^j)\right)\\
 &\ll_{\ep,k} \Psi_k(Q)^{1+\ep}.
 \end{align*}
 Finally, considering $\E^{(3)}$, we have that:
 \begin{align*}
 L_D(q,r)\leq \sum_{D<p\leq D^2}\frac{1}{p} + L_{D^2}(q,r) \ll \log\log D^2-\log\log D + \frac{1}{D} \ll 1.
 \end{align*}
 We thus have by Lemma~\ref{lem:pv-overlap} that:
 \begin{align*}
 \sum_{(q,r)\in\E^{(3)}}\lambda_k(\A_q^k\cap\A_r^k) &\ll_k \sum_{(q,r)\in\E^{(3)}}\lambda_k(\A_q^k)\lambda_k(\A_r^k).
 \end{align*}
 If $D\in [2^{i-1},2^i)$, we have:
 \begin{align*}
 \omega_{4^i}(q,r)\geq \omega_{D^2}(q,r)&> \frac{\ep}{4k}\log(2D) \geq \frac{\ep}{8k}\log(4^i).
 \end{align*}
 We can thus write:
 \begin{align*}
 \sum_{(q,r)\in\E^{(3)}}\lambda_k(\A_q^k\cap\A_r^k) &\ll_k \sum_{i\geq 0}\sum_{(q,r)\in\E^{(3)}_i}\lambda_k(\A_q^k)\lambda_k(\A_r^k),
 \end{align*}
 where, for each $i\geq 0$:
 \begin{align*}
 \E^{(3)}_{i}&\coloneq\set{(q,r)\in \E^{(3)}}{D(q,r)\in[2^{i-1},2^i),\ \omega_{4^i}(q,r)> \frac{\ep}{8k}\log(4^i)}.
 \end{align*}
 Applying Proposition~\ref{proposition:overlap-sum-3} with $y=2^i$, $t=4^i$, $\kappa=\ep/8k$ and $C=2$ gives:
 \begin{align*}
 \sum_{(q,r)\in\E^{(3)}_i}\lambda_k(\A_q^k)\lambda_k(\A_r^k) &\ll_{\ep,k} (4^i)^{-2}(2^i)^{1-\ep}\Psi_k(Q)^{1-\ep} \ll_\ep 2^{-i}\Psi_k(Q)^{1+\ep},
 \end{align*}
 and so:
 \begin{align*}
 \sum_{(q,r)\in\E^{(3)}}\lambda_k(\A_q^k\cap\A_r^k) &\ll_{\ep,k} \sum_{i\geq 0}2^{-i}\Psi_k(Q)^{1+\ep} \ll_\ep\Psi_k(Q)^{1+\ep}.
 \end{align*}
 Putting everything together:
 \begin{align*}
 \sum_{q,r\leq Q}\lambda_k(\A_q^k\cap\A_r^k) &= \sum_{q\leq Q}\lambda_k(\A_q^k) + \sum_{i=1}^3\sum_{(q,r)\in\E^{(i)}}\lambda_k(\A_q^k\cap\A_r^k)\\
 &\leq \Psi_k(Q) + \Psi_k(Q)^2 + O_{\ep,k}(\Psi_k(Q)^{1+\ep}),
 \end{align*}
 as required.
\end{proof}

This result establishes a quantitative estimate for the number of solutions for the $k$-dimensional Duffin-Schaeffer conjecture for all $k\geq 2$, matching the best-known error term in the 1-dimensional case from \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. We note that, in the original, non-reduced fraction case of Khintchine, without assuming monotonicity, asymptotic formulae exist only in the case $k\geq 3$. For $k=1,2$, we have such formulae with monotonicity; for $k=1$, we know that this monotonicity is essential, but for $k=2$, it remains an open question: see \cite[Chapter~3]{harman1998metricnumbertheory} or \cite[Section~5]{beresnevich2010khintchinegroshev}.

The method of proving Theorem~\ref{theorem:k-dim-ds} and similar results in metric Diophantine approximation is a standard procedure in metric number theory, with ideas dating back to at least \cite{cassels1950somemetricaltheorems, schmidt1964metricaltheorems}. We omit the proof, referring the reader to \cite[Lemma~1.5]{harman1998metricnumbertheory} for the general procedure and to \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} for the details in our case. The key is a variance estimate of random variables describing the number of solutions for a given $\alpha$. From a probabilistic perspective, the 1-dimensional Duffin-Schaeffer conjecture asks whether a uniformly randomly chosen irrational $\alpha$ is contained within the set system:
\begin{align*}
    \A_q \coloneq \bigcup_{\substack{0\leq a \leq q,\\\gcd(a,q)=1}}\left(\frac{a}{q}-\frac{\psi(q)}{q}, \frac{a}{q} + \frac{\psi(q)}{q}\right)\cap \left[0,1 \right]
\end{align*}
for infinitely or finitely many $q$; for $k\geq2$, we ask about $\A_q^k\coloneq \A_q^{k-1}\times \A_q$. Denoting then by $\lambda_k$ the $k$-dimensional Lebesgue measure restricted to $[0,1]^k$, we note that $\Psi_k(Q) = \sum_{q=1}^Q\lambda_k(\A_q^k)$ as:
\begin{align*}
        \lambda_k(\A_q^k) = \left(\frac{2\psi(q)\varphi(q)}{q}\right)^k.
\end{align*}
Following this, one can also see that $S_k(\alpha,Q)=\sum_{q=1}^Q\ind{\A_q^k}(\alpha)$ for every $\alpha$. The result of Theorem~\ref{theorem:k-dim-ds} can then be understood as describing the concentration of $S_k(\alpha,Q)$ around its mean. As before, the result would follow immediately from the divergence Borel-Cantelli lemma with pairwise independence of the random variables $\ind{\A_q}$, which we do not have here. In fact, the overlap $\lambda_k(\A_q^k\cap \mathcal{A}^k_r)$ can be significantly larger than $\lambda_k(\A_q^k)\lambda_k(\A_r^k)$ for pairs $(q,r)$ where the prime factorisation of $Q$ contains a large number of small primes not dividing $r$, or vice versa. The crux of the argument involves showing that such pairs are relatively rare, and thus on average we have a form of quasi-independence. In fact, one can show that the set system moves towards pairwise independence on average as the total mass tends towards infinity. Our key result below establishes exactly this for the case $k\geq 2$; again, stated as $k\geq 1$ for completeness.

\begin{theorem}\label{theorem:k-dim-variance}
    Let $k\geq 1$ and $Q\in \mathbb{N}$, and define $S_k(\alpha,Q)$ and $\Psi_k(Q)$ as in Theorem~\ref{theorem:k-dim-ds}. Then, for every fixed $\varepsilon>0$, we have:
    \begin{align*}
         \lint{\left[0,1\right]^k}{}{\left(S_k(\alpha, Q)-\Psi_k(Q)\right)^2}{\alpha} \leq \Psi_k(Q) + O_{\varepsilon,k}(\Psi_k(Q)^{1+\varepsilon}).
     \end{align*}
\end{theorem}

The rest of this paper is dedicated to constructing the proof of Theorem~\ref{theorem:k-dim-variance}. The standard approach involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition. In the $k$-dimensional problem, one can simply adapt the $1$-dimensional overlap estimates proven in \cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer, pollington1990kdimensionalduffinschaeffer}, as shown in Section~\ref{sec:overlap}. From here, we construct three bilinear estimates, allowing us to bound the variance contributions of the problematic pairs of integers $(q,r)$ with many prime factors. In the $1$-dimensional case, the approach of \cite{koukoulopoulos2020duffin, koukoulopoulos2024sharpquantitativeversionduffinschaeffer} is a technical iterative construction via GCD graphs. This approach was generalised in \cite{hauke2024provingduffinschaefferconjecturegcd, vazquez2024almostsharpquantitativeduffinschaeffergcd}, providing a framework for similar bounds on general multiplicative functions. In Section~\ref{sec:bilinear}, we show how this framework can be employed to construct equivalent bilinear bounds in all dimensions $k\geq 2$. From here, the variance estimate follows with an approach adapted from \cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}, as shown in Section~\ref{sec:variance}.

\subsection*{Acknowledgements}
The author would like to thank Manuel Hauke for introducing them to the topic and for guidance and discussion throughout the work. The work was conducted during a research stay in TU Graz, as part of the author's master's thesis in NTNU; the author would thus also like to thank the Institute of Analysis and Number Theory in TU Graz for their hospitality.

The proof method for Theorem~\ref{theorem:k-dim-ds} is described as a “standard procedure in metric number theory” and is omitted (with references). The sketch given is:

- Recast the counting function as a sum of indicator random variables: define the set system
\[
\A_q \coloneq \bigcup_{\substack{0\leq a \leq q,\\\gcd(a,q)=1}}\left(\frac{a}{q}-\frac{\psi(q)}{q}, \frac{a}{q} + \frac{\psi(q)}{q}\right)\cap \left[0,1 \right],
\]
(and for $k\ge2$ use $\A_q^k$), so that
\[\Psi_k(Q)=\sum_{q=1}^Q \lambda_k(\A_q^k), \qquad S_k(\alpha,Q)=\sum_{q=1}^Q \ind{\A_q^k}(\alpha).\]
Thus Theorem~\ref{theorem:k-dim-ds} “describ[es] the concentration of $S_k(\alpha,Q)$ around its mean.”

- Identify the main obstacle: the divergence Borel–Cantelli lemma “would follow immediately” under pairwise independence of $\ind{\A_q}$, but this fails because overlaps can be much larger than products:
\(\lambda_k(\A_q^k\cap \mathcal{A}^k_r)\) can be “significantly larger” than \(\lambda_k(\A_q^k)\lambda_k(\A_r^k)\) for certain $(q,r)$ related to their prime factorizations.

- State the crux: show that such highly dependent pairs are “relatively rare,” giving “on average… quasi-independence,” and in fact “the set system moves towards pairwise independence on average as the total mass tends towards infinity.”

- Reduce Theorem~\ref{theorem:k-dim-ds} to a variance bound: “The key is a variance estimate” for $S_k(\alpha,Q)$, formalized as Theorem~\ref{theorem:k-dim-variance}, which bounds
\(\int_{[0,1]^k} (S_k(\alpha,Q)-\Psi_k(Q))^2\, d\alpha\).

- How the variance bound is proved (high level): it “involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition,” using adapted $1$-dimensional overlap estimates, then “three bilinear estimates” to control contributions from “problematic pairs… with many prime factors,” after which “the variance estimate follows with an approach adapted from” the cited 1D work.

The proof method for the main theorem is described as a “standard procedure in metric number theory” and is omitted (with references). The sketch given is:

- Recast the counting function as a sum of indicator random variables: define the set system
\[
\A_q \coloneq \bigcup_{\substack{0\leq a \leq q,\\\gcd(a,q)=1}}\left(\frac{a}{q}-\frac{\psi(q)}{q}, \frac{a}{q} + \frac{\psi(q)}{q}\right)\cap \left[0,1 \right],
\]
(and for $k\ge2$ use $\A_q^k$), so that
\[\Psi_k(Q)=\sum_{q=1}^Q \lambda_k(\A_q^k), \qquad S_k(\alpha,Q)=\sum_{q=1}^Q \ind{\A_q^k}(\alpha).\]
Thus the main theorem “describ[es] the concentration of $S_k(\alpha,Q)$ around its mean.”

- Identify the main obstacle: the divergence Borel–Cantelli lemma “would follow immediately” under pairwise independence of $\ind{\A_q}$, but this fails because overlaps can be much larger than products:
\(\lambda_k(\A_q^k\cap \mathcal{A}^k_r)\) can be “significantly larger” than \(\lambda_k(\A_q^k)\lambda_k(\A_r^k)\) for certain $(q,r)$ related to their prime factorizations.

- State the crux: show that such highly dependent pairs are “relatively rare,” giving “on average… quasi-independence,” and in fact “the set system moves towards pairwise independence on average as the total mass tends towards infinity.”

- Reduce the main theorem to a variance bound: “The key is a variance estimate” for $S_k(\alpha,Q)$, formalized in the following result.

\begin{theorem}\label{theorem:k-dim-variance}
    Let $k\geq 1$ and $Q\in \N$, and define $S_k(\alpha,Q)$ and $\Psi_k(Q)$ as in Theorem~\ref{theorem:k-dim-ds}. Then, for every fixed $\ep>0$, we have:
    \begin{align*}
         \lint{\left[0,1\right]^k}{}{\left(S_k(\alpha, Q)-\Psi_k(Q)\right)^2}{\alpha} \leq \Psi_k(Q) + O_{\ep,k}(\Psi_k(Q)^{1+\ep}).
     \end{align*}
\end{theorem}

This bounds
\(\int_{[0,1]^k} (S_k(\alpha,Q)-\Psi_k(Q))^2\, d\alpha\).

- How the variance bound is proved (high level): it “involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition,” using adapted $1$-dimensional overlap estimates, then “three bilinear estimates” to control contributions from “problematic pairs… with many prime factors,” after which “the variance estimate follows with an approach adapted from” the cited 1D work.

Question:
Let $k\ge 1$, let $\psi:\mathbb N\to[0,1/2]$ satisfy
\[
\sum_{q=1}^\infty \left(\frac{\varphi(q)\psi(q)}{q}\right)^k=\infty,
\]
where $\varphi$ is Euler's totient function. For $\alpha=(\alpha_1,\dots,\alpha_k)\in[0,1]^k$, let $S_k(\alpha,Q)$ be the number of pairs $(a,q)\in\mathbb Z^k\times\mathbb N$ with $q\le Q$, $a=(a_1,\dots,a_k)$, and $\gcd(a_i,q)=1$ for each $i\in\{1,\dots,k\}$, such that
\[
\|q\alpha-a\|_\infty<\psi(q),
\]
where $\|x\|_\infty=\max_{1\le i\le k}|x_i|$. Also define
\[
\Psi_k(Q):=\sum_{q\le Q}\left(\frac{2\psi(q)\varphi(q)}{q}\right)^k.
\]
As $Q\to\infty$, which asymptotic statement holds for almost all $\alpha\in[0,1]^k$ (with respect to Lebesgue measure)?

For every $\varepsilon>0$, one has
\[
S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}\big(\Psi_k(Q)^{1/2+\varepsilon}\big)
\]
as $Q\to\infty$, for almost all $\alpha\in[0,1]^k$.

Choices:
B.
For every $\varepsilon>0$, one has
\[
S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}\big(\Psi_k(Q)^{1+\varepsilon}\big)
\]
as $Q\to\infty$, for almost all $\alpha\in[0,1]^k$.
C.
For almost all $\alpha\in[0,1]^k$, one has
\[
S_k(\alpha,Q)=\Psi_k(Q)+o\big(\Psi_k(Q)\big)
\]
as $Q\to\infty$.
D.
There exists $\varepsilon>0$ such that one has
\[
S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}\big(\Psi_k(Q)^{1/2+\varepsilon}\big)
\]
as $Q\to\infty$, for almost all $\alpha\in[0,1]^k$.
E.
For every $\varepsilon>0$, one has
\[
S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}(1)
\]
as $Q\to\infty$, for almost all $\alpha\in[0,1]^k$.

dropped quantitative power-saving error term

uniform bounded error replacing fluctuation scale from variance bound


\begin{theoremL}\label{thm:asboundsPoly}
    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\in\mathbb Z[x]$ is a product of at least two distinct linear factors over $\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\in\mathbb Z, c\in\mathbb Q$. Then almost surely we have $$\left\lvert\sum_{n\leq N} f(P(n))\right\rvert \ll \sqrt{N\log\log N}.$$ Moreover, almost surely there exists a sequence $N_k\to\infty$ such that
    \begin{equation*}
        \left\lvert\sum_{n\leq N_k} f(P(n))\right\rvert\gg \sqrt{N_k\log\log N_k}.
    \end{equation*}
\end{theoremL}

\begin{lemma}\label{lemma:unSmoothSquarefree}
    Let $N$ be large, $\alpha>\frac{1}{2}$, and let $N\geq H=H(N)\geq N^{\frac{4\alpha + 1}{5}}$. The number of square-free integers in the short interval $(N-H, N]$ such that $P^+(n)>N^{\alpha}$ is
    \begin{equation*}
        \gg H+O\left(\frac{H}{\log N} \right).
    \end{equation*}
\end{lemma}

Since their introduction by Wintner \cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory.
\begin{definition}
    A \emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.

    A \emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\ldots$  supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\{-1, 1\}$.
\end{definition}
Although random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\'asz \cite{Halasz1977} and Hal\'asz and R\'enyi \cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\sum_{n\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\frac{1}{\sqrt N}\sum_{n\leq N} f(n)$ has an approximately Gaussian distribution as $N\to\infty$.
Harper \cite{Harper2013} showed that this is not the case and in fact, he later showed \cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\sqrt N/(\log\log N)^{\frac14}$. One might therefore hope to obtain
an interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.

From an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include:
\begin{itemize}\item[(i)] integers with an atypically small number of prime factors (Hough \cite{Hough2011}, Harper \cite{Harper2013}),
\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \cite{soundararajan2023clt}, Pandey, Wang, and Xu \cite{PandeyWangXu2024}),
\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\neq 0$ (Soundararajan and Xu \cite{soundararajan2023clt}),
\item[(iv)] polynomial images (Najnudel \cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \cite{KlurmanShkredovXu2023}, Chinis and the author \cite{ChinisShala2025}).
\end{itemize}
In each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.

There has been considerable work in understanding almost sure large fluctuations for the full sum $\sum_{n\leq N} f(n)$ and related variants. Harper \cite{harper2020large} showed that for any (slowly growing) function $V(N)\to\infty$, almost surely there is a sequence of positive integers $N_k\to\infty$ such that
\begin{equation*}
    \left\lvert\sum_{n\leq N_k}f(n)\right\rvert\gg\sqrt{N_k}\frac{(\log\log N_k)^{\frac14}}{V(N_k)}.
\end{equation*}
This matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\sqrt N$ times a power of $\log\log N$) have been proved by Lau, Tenenbaum and Wu \cite{LauTenenbaumWu2012}, as well as most recently by Caich \cite{Caich2023} with the current best known almost sure bound
\begin{equation*}
    \left\lvert\sum_{n\leq N} f(n)\right\rvert\ll_{\epsilon} \sqrt N(\log\log N)^{\frac{3}{4}+\epsilon}.
\end{equation*}
In the case when the summands are demanded to have a prime factor bigger than $\sqrt N$, Mastrostefano \cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\frac{1}{4}+\epsilon$ in place of $\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \cite{harper2020large}; see also the work of Hardy \cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\sum_{n\leq N} \frac{f(n)}{\sqrt n}$
has been considered by Aymone, Heap and Zhao \cite{aymone2021partial}, and Hardy \cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.

Recently, Hoban, Shah, Ismail, Verreault, and Zaman \cite{HobanShahIsmailVerreaultZaman2025} have extended the framework of Soundararajan and Xu \cite{soundararajan2023clt} in a different direction, namely in the function field setting. It is likely that the methods of this paper, combined with the work in \cite{HobanShahIsmailVerreaultZaman2025}, can be used to prove analogous results over function fields.

We are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \ref{section:generalSetup} for the general but rather technical results.

\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.

Since their introduction by Wintner \cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory.
\begin{definition}
 A \emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.

A \emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\ldots$ supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\{-1, 1\}$.
\end{definition}
Although random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\'asz \cite{Halasz1977} and Hal\'asz and R\'enyi \cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\sum_{n\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\frac{1}{\sqrt N}\sum_{n\leq N} f(n)$ has an approximately Gaussian distribution as $N\to\infty$.
Harper \cite{Harper2013} showed that this is not the case and in fact, he later showed \cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\sqrt N/(\log\log N)^{\frac14}$. One might therefore hope to obtain
an interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.

From an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include:
\begin{itemize}\item[(i)] integers with an atypically small number of prime factors (Hough \cite{Hough2011}, Harper \cite{Harper2013}),
\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \cite{soundararajan2023clt}, Pandey, Wang, and Xu \cite{PandeyWangXu2024}),
\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\neq 0$ (Soundararajan and Xu \cite{soundararajan2023clt}),
\item[(iv)] polynomial images (Najnudel \cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \cite{KlurmanShkredovXu2023}, Chinis and the author \cite{ChinisShala2025}).
\end{itemize}
In each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.

There has been considerable work in understanding almost sure large fluctuations for the full sum $\sum_{n\leq N} f(n)$ and related variants. Harper \cite{harper2020large} showed that for any (slowly growing) function $V(N)\to\infty$, almost surely there is a sequence of positive integers $N_k\to\infty$ such that
\begin{equation*}
 \left\lvert\sum_{n\leq N_k}f(n)\right\rvert\gg\sqrt{N_k}\frac{(\log\log N_k)^{\frac14}}{V(N_k)}.
\end{equation*}
This matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\sqrt N$ times a power of $\log\log N$) have been proved by Lau, Tenenbaum and Wu \cite{LauTenenbaumWu2012}, as well as most recently by Caich \cite{Caich2023} with the current best known almost sure bound
\begin{equation*}
 \left\lvert\sum_{n\leq N} f(n)\right\rvert\ll_{\epsilon} \sqrt N(\log\log N)^{\frac{3}{4}+\epsilon}.
\end{equation*}
In the case when the summands are demanded to have a prime factor bigger than $\sqrt N$, Mastrostefano \cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\frac{1}{4}+\epsilon$ in place of $\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \cite{harper2020large}; see also the work of Hardy \cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\sum_{n\leq N} \frac{f(n)}{\sqrt n}$
has been considered by Aymone, Heap and Zhao \cite{aymone2021partial}, and Hardy \cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.

We are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \ref{section:generalSetup} for the general but rather technical results.

\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.

The almost sure upper bound in Theorem \ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\mathbb Q$. Indeed, the method used in \cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\log n$.

\begin{theoremL}\label{thm:asBoundsShort}
 Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\frac{11}{15}}\leq H(N)\leq N/(\log N)^c$
 for some constant $c>2\log 2-1$.
 Then almost surely there exists a sequence $N_k\to\infty$ with corresponding $H_k = H(N_k)$ such that
 \begin{equation*}
 \left\lvert\sum_{N_k-H_k < n \leq N_k}f(n)\right\rvert\gg \sqrt{H_k\log\frac{N_k}{H_k}}.
 \end{equation*}
\end{theoremL}

\section{Overview of the proofs}
\subsection{Martingale structure} Our starting point is the observation of Harper \cite{Harper2013} that sums of a random multiplicative function have the structure of a martingale with respect to the filtration induced by the largest prime factor, namely
\begin{equation*}
 \sum_{n\in\mathcal A}f(n) = \sum_{p^{\alpha}\leq N}f(p^{\alpha})\sum_{\substack{n\in\mathcal A \\ P^+(n)=p \\ p^{\alpha}\parallel n}}f\left(\frac{n}{p^{\alpha}}\right).
\end{equation*}
This observation was utilized in almost all subsequent works on random multiplicative functions. For us, its usefulness lies in the fact that it morally reduces the understanding of $\sum_{n\in\mathcal A}f(n)$ to evaluating the second and fourth moments of the sum, provided they do not blow up. More precisely, we will apply the following quantitative result from the book of Hall and Heyde \cite{HallHeyde1980}, yielding a bound on the Kolmogorov distance between our martingale and a Gaussian.
\begin{prop}[\cite{HallHeyde1980}, Theorem 3.9]\label{prop:quantCLT}
 Let $X_1, X_2, \ldots, X_n$ be a real square-integrable martingale difference sequence.
 Let $S_n = \sum_{i=1}^n X_i$. Then
 \begin{equation*}
\left\lvert\mathbb P(S_n\leq x) - \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\emph{d}t\right\rvert\ll \frac{\left(\sum_{i=1}^n \mathbb E\lvert X_i\rvert^4 + \mathbb E\left\lvert\sum_{i=1}^n X_i^2 - 1\right\rvert^2\right)^{1/5}}{1+\lvert x\rvert^{\frac{16}{5}}}.
 \end{equation*}
\end{prop}
\subsection{Almost sure upper bound} Equipped with the above result, we may readily prove an almost sure upper bound for $\sum_{n\in\mathcal A} f(n)$, provided our approximation is suitably strong. In order to access the tails at $x\approx C\sqrt{\log k}$ which occur with probability $\approx k^{-C^2/2}$ for the Gaussian, we will need an approximation that is at least this good. We are able to achieve this in the case of polynomial images, but not short intervals -- we will discuss this in more detail in the following subsections. Using our quantitative approximation at several spread out scales together with a slow variation property between the scales (see Proposition \ref{prop:slowVar} and Lemma \ref{lemma:slowVarPoly}), the Borel-Cantelli lemma establishes the almost sure upper bound.

\begin{lemma}\label{lemma:slowVarPoly} Let $f$ be a Rademacher (or Steinhaus) random multiplicative function, and let $P\in\mathbb Z[x]$ be a polynomial as in Theorem \ref{thm:asboundsPoly}.
For any constant $A>0$, there exists a constant $c>0$ such that for $N_l = \lfloor e^{l^c}\rfloor$, we have that
 \begin{equation*}
 \max_{N_{l}<N\leq N_{l+1}} \left\lvert\sum_{n\leq N} f(P(n)) - \sum_{n\leq N_{l}} f(P(n)) \right\rvert \ll \frac{\sqrt{N_{l+1}}}{(\log N_{l+1})^A}
 \end{equation*} holds almost surely for every $l\in\mathbb N$.
\end{lemma}

We finally prove the almost sure upper bound. By Lemma \ref{lemma:slowVarPoly} with $A=1$, it suffices to show that almost surely there exists a constant $C>0$ such that for all $l\in\mathbb N$ we have
 \begin{equation}\label{eq:ubPoly}
 \left\lvert\sum_{n\leq N_l} f(P(n))\right\rvert\leq C \sqrt{N_l\log\log N_l},
 \end{equation}
 where $N_l$ is as in the statement of the lemma (in the Steinhaus case, we do this separately for the real and imaginary parts). By Corollary \ref{cor:quantCLTpoly}, the probability that the above fails for a fixed $l$ and large enough implied constant $C$ is $\ll (\log N_l)^{-\frac{C^2}{2}}$. Making $C$ larger (in terms of $c$) if necessary, we have that the series of probabilities $$\sum_{l=1}^\infty (\log N_l)^{-\frac{C^2}{2}} = \sum_{l=1}^\infty \frac{1}{l^{\frac{cC^2}{2}}}<\infty,$$ thus \eqref{eq:ubPoly} holds almost surely by Borel-Cantelli (with a possibly even larger constant $C$ to account for the finitely many failures).

\begin{theorem}\label{thm:localizedBoundPoly}
 Let $f$ be a Rademacher (or Steinhaus) random multiplicative function and let $P\in\mathbb Z[x]$ be a polynomial as in Theorem \ref{thm:asboundsPoly}. There exists a constant $c>0$ such that for any large enough $X$, we have that
 \begin{equation*}
 \max_{X<N\leq X^2} \frac{1}{\sqrt N} \left\lvert\sum_{n\leq N}f(P(n)) \right\rvert \gg \sqrt{\log\log X}
 \end{equation*} holds with a suitably small implied constant with probability $1-O(\exp(-(\log X)^{c}))$.
\end{theorem}

\section{Proof of Theorem \ref{thm:asBoundsShort}: Short intervals} Here we will require the more sophisticated part of Corollary \ref{cor:normalcomparison}. Let $H$ be as in the statement of Theorem \ref{thm:asBoundsShort} and consider $\mathcal A_N = \{n\in\mathbb N : N-H<n\leq N\}$. Note that $\lvert\mathcal A_N\rvert\sim \kappa H$ for some\footnote{In the Steinhaus case we have $\kappa = 1$, whereas in the Rademacher case $\kappa = \frac{6}{\pi^2}$.} $\kappa>0$. Recall
\begin{equation*}
 S_N = \frac{1}{\sqrt{\lvert\mathcal A_N\rvert}}\sum_{n\in\mathcal A_N} \mathfrak Tf(n)\sim \frac{1}{\sqrt{\kappa H}}\sum_{N-H<n\leq n }\mathfrak Tf(n).
\end{equation*}
Moreover, let $X$ be a large parameter. For some small $\delta>0$ depending on the function $H$ and then even smaller $\varepsilon_0>0$, find $k= (X/H(X))^{\varepsilon_0}$ primes $l$ in the range $h(X)/2<l\leq h(X)$, where $h(X) := (X/H(X))^{\delta}$. We will not fix $\delta$ or $\varepsilon_0$ for now, as there will be various (but finitely many) places where we may have make them smaller. For these values of $l$, let $N_l = lX$ with corresponding $H_l = H(N_l)$.
Observe that for large enough $X$ and small enough $\delta>0$, the sets $\mathcal A_{N_l}$ are disjoint. It is not these sets that we will apply Theorem \ref{thm: complicatedapproxJointG} to. If we did this, the very smooth integers in the sets $\mathcal A_{N_l}$ would prevent us from obtaining a suitably small value for $\varepsilon_2$. We bypass this by splitting $S_{N_l}$ as
\begin{equation}\label{eq:splitting1}
 \sum_{N_l - H_l<n\leq N_l}f(n) = \sum_{(Xh(X))^{\frac{2}{3}}< p}f(p)\sum_{\substack{N_l-H_l < n\leq N_l \\ p\mid n}}f\left(\frac np\right) + \sum_{\substack{N_l-H_l<n\leq N_l \\ P^+(n)\leq (Xh(X))^{\frac{2}{3}}}}f(n).
\end{equation}
If $H_1> N_1/(\log N_1)^2$ (hence since $H$ is increasing we have $H_l \gg Xh(X)/(\log X)^2$ for all values of $l$), we choose some $\varepsilon>0$ to be specified later and additionally split the first part as
\begin{equation}\label{eq:splitting2}
 \sum_{N_l - H_l<n\leq N_l}f(n) = \sum_{(Xh(X))^{\frac{2}{3}}< p }f(p)\sum_{\substack{N_l-H_l < n\leq N_l \\ p\mid n \\ \Omega(n)\leq (1+\varepsilon)\log\log X}}f\left(\frac np\right) \\ + \sum_{\substack{N_l-H_l<n\leq N_l \\ P^+(n)\leq (Xh(X))^{\frac{2}{3}} \\ \text{or } \Omega(n)> (1+\varepsilon)\log\log X }}f(n).
\end{equation}
We first show that the sum over integers with $\Omega(n)> (1+\varepsilon)\log\log X$ above may be typically ignored for all values of $l$.

The almost sure upper bound in Theorem \ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\mathbb Q$. Indeed, the method used in \cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\log n$.

To demonstrate how this was used, let us consider $P(n) = n^2+1$. By the aforementioned fact, one can ``pull out'' a unique large prime factor for such $n$ and still capture a good ``bulk'' of the sum for infinitely many (sparse) $N$, that is
\begin{equation*}
    \sum_{n\leq N}f(n^2+1)\approx \sum_{N\log N < p\leq N^2}f(p)\sum_{\substack{n\leq N \\ p\mid n^2+1}}f\left(\frac{n^2+1}{p}\right).
\end{equation*}

Upon conditioning on the small primes, this is a weighted sum of independent random variables, which is then handled using intricate combinatorial arguments and Gaussian approximation.

Indeed, the method described above cannot be directly applied to the case when $P$ is a product of linear factors, for all its prime factors are of size at most linear in $N$.
In this paper, we will use a different method that does not involve any combinatorial splitting of the sum or conditioning. We will rather fully utilize the martingale structure of the entire sum -- we will elaborate on this in the next section. In particular, provided enough number-theoretic input (see the work of Martin \cite{Martin2002}), our method would also work to prove almost sure large fluctuations of $N$-smooth\footnote{An integer is called $y$-smooth if all of the prime factors dividing it are at most $y$.} values of irreducible polynomials.

\subsection{Integers in a short interval} A central limit theorem for sums of a random multiplicative function over integers in a short interval $(N-H, N]$ was first proved by Chatterjeee and Soundararajan \cite{ChatterjeeSoundararajan2012} when $H=o\left(\frac{N}{\log N}\right)$, using essentially only second and fourth moment information. This was later extended to $H\leq \frac{N}{(\log N)^{c}}$ with $c>2\log2-1$ by Soundararajan and Xu \cite{soundararajan2023clt}, entering a regime of $H$ where naively the fourth moment of the sum blows up. However, this is bypassed by throwing out $0\%$ of the integers in the short interval with atypically many prime factors, resulting in a controlled fourth moment.

We prove an almost sure lower bound in the same regime.

\begin{theoremL}\label{thm:asBoundsShort}
    Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\frac{11}{15}}\leq H(N)\leq N/(\log N)^c$
    for some constant $c>2\log 2-1$.
    Then almost surely there exists a sequence $N_k\to\infty$ with corresponding $H_k = H(N_k)$ such that
    \begin{equation*}
        \left\lvert\sum_{N_k-H_k < n \leq N_k}f(n)\right\rvert\gg \sqrt{H_k\log\frac{N_k}{H_k}}.
    \end{equation*}
\end{theoremL}

\begin{remark}
    The fluctuations of order $\sqrt{\log\frac{N}{H}}$ may seem surprisingly large at first, especially in a naive comparison with the law of iterated logarithm. However, this is explained by the fact that short sums of length $H$ decorrelate much faster, and indeed roughly $N/H$ essentially independent sums fit in a ``window'' of size $N$. In agreement with the law of iterated logarithm, the maximum of these sums has size roughly $\sqrt{\log\frac{N}{H}}$.
\end{remark}

\begin{remark}
    The conditions on $H$ are imposed simply for convenience -- any standard choices such as $H=N^{\alpha}$ for $\alpha < 1$, $H = N/\exp\left(\sqrt{\log N}\right)$, or $H = N/(\log N)^A$ for some constant $A>2\log 2-1$, satisfy these conditions (for sufficiently large $N$).

    We have not attempted to optimize the exponent $\frac{11}{15}$ -- it is likely possible to bring it down to $\frac{3}{5}$. However, the lower bound on $H$ should not be essential: for very small $H\ll \log N$ the large fluctuations are captured by almost sure long runs of ones, of length $\approx \log N$ (see the work of Erdős and Rényi \cite{ErdosRenyi1970}), whereas if $\log N\ll H\ll N^{\frac{11}{15}}$, our method should work (with significant simplifications when $H\ll N^{\frac{1}{2}}$), but the restriction stems from a result on square-free smooth integers in short intervals; see Lemma \ref{lemma:unSmoothSquarefree}. Since we are only proving a lower bound and therefore can pick convenient scales to work with, it is likely possible to circumvent this using the Matom\"aki--Radziwiłł machinery; see the work of Jain \cite{jain2025smooth} and the references within. However, we have opted for simplicity, especially since our result captures the most interesting regime where there is a transition of the large fluctuations from $\sqrt{\log N}$ down to $\sqrt{\log \log N}$.\footnote{We thank Sarvagya Jain for discussions pertaining to this remark.}
\end{remark}

It would be interesting to prove the corresponding almost sure upper bound in any regime of $H$. For rather large $H$, it is likely that the sophisticated ideas in \cite{LauTenenbaumWu2012, Mastrostefano2022, Hardy2023AlmostSure, Atherfold2025} relating the short sum $\sum_{N-H<n\leq N} f(n)$ to a random Euler product could be useful. It is unclear, however, whether this would result in a sharp (or almost sharp) upper bound in any regime of $H$ -- this is an ongoing investigation.

Upcoming work of Harper, Soundararajan, and Xu
establishes a central limit theorem in the range $H = o(N)$, although with a normalization of the sum that differs from just the standard deviation. It would be interesting to determine whether our method would succeed to prove almost sure lower bounds of the right order of magnitude in this extended range. See also the work of Caich \cite{Caich2024randomshort} for more on the intricate behavior of short sums in this regime.

The introduction explains that prior lower-bound methods (e.g. \cite{KlurmanShkredovXu2023, ChinisShala2025}) for handling \(\sum_{n\le N} f(P(n))\) when \(P\) has an irreducible factor of degree \(\ge 2\) use the fact that for many \(n\), \(P(n)\) has a prime factor \(> n\log n\). For example, with \(P(n)=n^2+1\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \(N\):
\[
\sum_{n\le N} f(n^2+1)\approx \sum_{N\log N < p\le N^2} f(p)\sum_{\substack{n\le N\\ p\mid n^2+1}} f\!\left(\frac{n^2+1}{p}\right).
\]
After “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”

It then notes this approach “cannot be directly applied” when \(P\) is a product of linear factors, since then “all its prime factors are of size at most linear in \(N\).” For Theorem~\ref{thm:asboundsPoly}, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin \cite{Martin2002}).”

The introduction explains that prior lower-bound methods (e.g. J. Klurman, I. D. Shkredov, and J. Xu, *Title not found in ref_dict* (2023), and K. Chinis and A. Shala, *Title not found in ref_dict* (2025)) for handling \(\sum_{n\le N} f(P(n))\) when \(P\) has an irreducible factor of degree \(\ge 2\) use the fact that for many \(n\), \(P(n)\) has a prime factor \(> n\log n\). For example, with \(P(n)=n^2+1\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \(N\):
\[
\sum_{n\le N} f(n^2+1)\approx \sum_{N\log N < p\le N^2} f(p)\sum_{\substack{n\le N\\ p\mid n^2+1}} f\!\left(\frac{n^2+1}{p}\right).
\]
After “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”

It then notes this approach “cannot be directly applied” when \(P\) is a product of linear factors, since then “all its prime factors are of size at most linear in \(N\).” In establishing the main theorem, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin C. A. Martin, *Title not found in ref_dict* (2002)).”,

\label{thm:asboundsPoly}
    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\in\mathbb Z[x]$ is a product of at least two distinct linear factors over $\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\in\mathbb Z, c\in\mathbb Q$. Then almost surely we have $$\left\lvert\sum_{n\leq N} f(P(n))\right\rvert \ll \sqrt{N\log\log N}.$$ Moreover, almost surely there exists a sequence $N_k\to\infty$ such that
    \begin{equation*}
        \left\lvert\sum_{n\leq N_k} f(P(n))\right\rvert\gg \sqrt{N_k\log\log N_k}.
    \end{equation*},

Question: Let $f$ be either a Steinhaus or a Rademacher random multiplicative function. Here, a Steinhaus random multiplicative function satisfies $f(mn)=f(m)f(n)$ for all positive integers $m,n$, and for each prime $p$, the random variable $f(p)$ is uniformly distributed on the unit circle. A Rademacher random multiplicative function is supported on square-free integers, satisfies $f(mn)=f(m)f(n)$ whenever $(m,n)=1$, and for each prime $p$, $f(p)$ is uniformly distributed on $\{-1,1\}$. Let $P\in \mathbb Z[x]$. In the Rademacher case, assume that $P$ is either a product of at least two distinct linear factors over $\mathbb Q$, or an irreducible polynomial of degree $2$, and that $P$ takes infinitely many square-free values. In the Steinhaus case, assume that $P(x)$ is not of the form $w(x+c)^d$ with $w\in \mathbb Z$ and $c\in \mathbb Q$. Under these assumptions, which quantitative estimate holds for the polynomially sampled sums $\sum_{n\le N} f(P(n))$?

Almost surely, one has
$$\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log\log N}.$$
Moreover, almost surely there exists a sequence $N_k\to\infty$ such that
$$\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k\log\log N_k}.$$

Choices:
B.
Almost surely, one has
$$\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt N\,(\log\log N)^{\frac14}.$$
Moreover, almost surely there exists a sequence $N_k\to\infty$ such that
$$\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k}\,(\log\log N_k)^{\frac14}.$$
C.
Almost surely, one has
$$\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log\log N}.$$
D.
For every sequence $N_k\to\infty$, almost surely one has
$$\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k\log\log N_k}$$
for all sufficiently large $k$, and in particular
$$\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log\log N}.$$
E.
Almost surely, one has
$$\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log N}.$$
Moreover, almost surely there exists a sequence $N_k\to\infty$ such that
$$\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k\log N_k}.$$


Question: Let $f$ be either a Steinhaus or a Rademacher random multiplicative function. Here a Steinhaus random multiplicative function satisfies $f(mn)=f(m)f(n)$ for all positive integers $m,n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle; a Rademacher random multiplicative function is supported on square-free integers, satisfies $f(mn)=f(m)f(n)$ whenever $(m,n)=1$, and for each prime $p$, $f(p)$ is uniformly distributed on $\{-1,1\}$. Let $P\in\mathbb Z[x]$. In the Rademacher case, assume that $P$ is either a product of at least two distinct linear factors over $\mathbb Q$ or an irreducible polynomial of degree $2$, and that $P$ takes infinitely many square-free values. In the Steinhaus case, assume that $P(x)$ is not of the form $w(x+c)^d$ with $w\in\mathbb Z$ and $c\in\mathbb Q$. Under these assumptions, which asymptotic statement about the partial sums $\sum_{n\le N} f(P(n))$ holds almost surely as $N\to\infty$?

Almost surely, one has the uniform upper bound
\[
\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log\log N}
\]
for all sufficiently large $N$, and moreover almost surely there exists a sequence $N_k\to\infty$ such that
\[
\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k\log\log N_k}.
\]

Choices:
B.
Almost surely, one has the uniform upper bound
\[
\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt N\, (\log\log N)^{\frac14}
\]
for all sufficiently large $N$, and moreover almost surely there exists a sequence $N_k\to\infty$ such that
\[
\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k}\, (\log\log N_k)^{\frac14}.
\]
C.
Almost surely, there exists a sequence $N_k\to\infty$ such that
\[
\left|\sum_{n\le N_k} f(P(n))\right| \ll \sqrt{N_k\log\log N_k}.
\]
D.
Almost surely, for every $\varepsilon>0$ one has the uniform upper bound
\[
\left|\sum_{n\le N} f(P(n))\right| \ll_{\varepsilon} \sqrt N\, (\log N)^{\varepsilon}
\]
for all sufficiently large $N$, and moreover almost surely there exists a sequence $N_k\to\infty$ such that
\[
\left|\sum_{n\le N_k} f(P(n))\right| \gg \sqrt{N_k\log\log N_k}.
\]
E.
Almost surely, one has the uniform upper bound
\[
\left|\sum_{n\le N} f(P(n))\right| \ll \sqrt{N\log\log N}
\]
for all sufficiently large $N$, and moreover almost surely there exists a sequence $N_k\to\infty$ such that
\[
\sum_{n\le N_k} f(P(n)) \gg \sqrt{N_k\log\log N_k}.
\]

dropped substantive lower-bound content and uniformity in N

stronger uniform upper bound replacing \sqrt{N\log\log N} by near-\sqrt N

absolute value in the lower bound along a subsequence