# The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Lorenz Frühwirth and Manuel Hauke

###### Abstract

Given a monotonically decreasing
$\psi:\operatorname{\mathbb{N}}\to[0,\infty)$, Khintchine’s Theorem provides
an efficient tool to decide whether, for almost every
$\alpha\in\operatorname{\mathbb{R}}$, there are infinitely many
$(p,q)\in\operatorname{\mathbb{Z}}^{2}$ such that
$\left\lvert\alpha-\frac{p}{q}\right\rvert\leq\frac{\psi(q)}{q}$. The recent
result of Koukoulopoulos and Maynard provides an elegant way of removing
monotonicity when only counting reduced fractions. Gallagher showed a
multiplicative higher-dimensional generalization to Khintchine’s Theorem,
again assuming monotonicity. In this article, we prove the following Duffin-
Schaeffer-type result for multiplicative approximations: For any $k\geq 1$,
any function $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ (not necessarily
monotonic) and almost every $\alpha\in\mathbb{R}^{k}$, there exist infinitely
many $q$ such that
$\prod\limits_{i=1}^{k}\left\lvert\alpha_{i}-\frac{p_{i}}{q}\right\rvert\leq\frac{\psi(q)}{q^{k}},p_{1},\ldots,p_{k}$
all coprime to $q$, if and only if

$\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\left(\frac{\varphi(q)}{q}\right)^{k}\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}=\infty.$

This settles a conjecture of Beresnevich, Haynes, and Velani.

> Dedicated to the memory of Aleksandr Khintchine, on the occasion of the
> 100th anniversary of his foundational theorem in metric Diophantine
> approximation.

##  1 Introduction and main results

###  1.1 The metric theory of Diophantine approximation

Given a function $\psi:\operatorname{\mathbb{N}}\to[0,\infty)$ and a
(typically irrational) number $\alpha\in[0,1]$, the main goal of Diophantine
approximation is to try to understand whether there are infinitely many
$(a,q)\in\operatorname{\mathbb{Z}}\times\operatorname{\mathbb{N}}$ such that

$\left\lvert\alpha-\frac{a}{q}\right\rvert\leq\frac{\psi(q)}{q}.$ (1)

In case of $\psi(q)=\frac{1}{q}$, Dirichlet’s Theorem shows that (1) holds for
all $\alpha\in[0,1]$. This is known to be asymptotically sharp for badly
approximable numbers, but Khintchine’s Theorem tells us that we can improve
upon Dirichlet’s Theorem for Lebesgue almost every
$\alpha\in\operatorname{\mathbb{R}}$:

######  Theorem (Khintchine, 1924).

Let $\psi:\operatorname{\mathbb{N}}\to[0,\infty)$ be a monotonically
decreasing function. Writing

$S(\psi):=\left\\{\alpha\in[0,1]:\lVert q\alpha\rVert\leq\psi(q),\text{ for
i.m. }q\in\operatorname{\mathbb{N}}\right\\},$

we have

$\lambda_{1}\left(S(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)=\infty,\\\ 0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)<\infty.\end{cases}$

Here and in what follows “i.m.” stands for “infinitely many”, $\lambda_{k}$
denotes the $k$-dimensional Lebesgue measure and $\lVert.\rVert$ is the
distance to the next integer (see Section 1.6 for proper definitions and
notations). In their seminal article, Duffin and Schaeffer [16] constructed a
(non-monotonic) function $\psi$ such that
$\sum_{q\in\operatorname{\mathbb{N}}}\psi(q)=\infty$, but for almost every
$\alpha$, (1) has only finitely many solutions. Their example makes heavy use
of the fact that for given $x\in\operatorname{\mathbb{Q}}$, there are many
different pairs $(a,q)\in\operatorname{\mathbb{Z}}^{2}$ such that
$\frac{a}{q}=x$. To resolve this issue, Duffin and Schaeffer suggested to only
consider reduced fractions to make this representation unique. To formalize
this, let

$\lVert qx\rVert^{\prime}:=\min_{\gcd(p,q)=1}\lvert qx-p\rvert$

denote the best approximation by coprime fractions and for given
$\psi:\operatorname{\mathbb{N}}\to[0,\infty)$, consider the set

$D(\psi):=\left\\{\alpha\in[0,1]:\lVert
q\alpha\rVert^{\prime}\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\}.$

Consequently, Duffin and Schaeffer conjectured that for any
$\psi:\operatorname{\mathbb{N}}\to[0,\infty)$, we have

$\lambda_{1}\left(D(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\frac{\varphi(q)}{q}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\frac{\varphi(q)}{q}<\infty.\end{cases}$

This became one of the most famous open questions in Diophantine approximation
and after contributions made in, e.g., [1, 3, 6, 18, 19, 23, 27, 33], it was
finally proven by a breakthrough result of Koukoulopoulos and Maynard [25].

###  1.2 The approximation theory in higher dimensions

When looking at Diophantine approximation in dimension $k\geq 1$, there are
several classical and well-studied measures of approximation quality
considered. We will highlight the two topics that are likely the most
extensively studied: One the one hand there is the simultaneous approximation
where one considers the set

$S_{k}(\psi):=\left\\{\alpha\in[0,1]^{k}:\max\limits_{1\leq i\leq k}\lVert
q\alpha_{i}\rVert\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\},$

and its coprime counterpart

$D_{k}(\psi):=\left\\{\alpha\in[0,1]^{k}:\max\limits_{1\leq i\leq k}\lVert
q\alpha_{i}\rVert^{\prime}\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\}.$

Another well-researched area concerns multiplicative Diophantine
approximation, where one examines the set

$S_{k}^{\times}(\psi):=\left\\{\alpha\in[0,1]^{k}:\prod\limits_{i=1}^{k}\lVert
q\alpha_{i}\rVert\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\},$

and its coprime counterpart

$D_{k}^{\times}(\psi):=\left\\{\alpha\in[0,1]^{k}:\prod\limits_{i=1}^{k}\lVert
q\alpha_{i}\rVert^{\prime}\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\}.$

The case of simultaneous approximation is metrically well understood: A
straightforward adaptation of Khintchine’s Theorem shows that for monotonic
$\psi$ and any $k\in\operatorname{\mathbb{N}}$,

$\lambda_{k}(S_{k}(\psi))=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)^{k}=\infty,\\\ 0,&\text{
if }\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)^{k}<\infty.\end{cases}$

Gallagher [20] proved that for $k\geq 2$, the monotonicity assumption can be
removed, which is, considering the Duffin-Schaeffer counterexample, in
contrast to the case $k=1$. In the coprime setup, Pollington and Vaughan [27]
proved that for any (not necessarily monotonic) function $\psi$ and $k\geq 2$,

$\lambda_{k}(D_{k}(\psi))=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)\psi(q)}{q}\right)^{k}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)\psi(q)}{q}\right)^{k}<\infty,\end{cases}$

and by the much later result of Koukoulopoulos and Maynard [25], this also
holds true in the case of $k=1$.

We now direct our focus towards the investigation of multiplicative
Diophantine approximation, a subject that has attracted considerable attention
in recent years, largely due to its connection to a well-known unsolved
problem: Littlewood’s conjecture postulates that for any $\varepsilon>0$ and
$\psi_{\varepsilon}(q):=\frac{\varepsilon}{q}$, we have $\lim_{\varepsilon\to
0}S_{2}^{\times}(\psi_{\varepsilon})=\emptyset$. Although Einsiedler, Katok,
and Lindenstrauss [17] could show that the Hausdorff dimension of
$\lim_{\varepsilon\to 0}S_{2}^{\times}(\psi_{\varepsilon})$ is $0$, the
conjecture itself remains widely open. Clearly, the result in [17] implies
that for almost every $(\alpha,\beta)\in[0,1]^{2}$,
$\liminf_{n\to\infty}n\lVert n\alpha\rVert\lVert n\beta\rVert=0$ (which can
already be deduced from Khintchine’s Theorem), but the rate of approximation
can be improved in the multiplicative setup when allowing an exceptional set
of (two-dimensional) Lebesgue measure $0$. For various adaptations of this
result, we refer the reader to [5] and the references therein. In the metric
question, Gallagher [20] proved the following, assuming monotonicity of the
approximation function111We remark that in the actual statement of Gallagher,
there is no restriction on $\psi$ (clearly for $\psi(q)\geq 1/2$, the question
becomes trivial), and $\log(1/\psi(q))$ is replaced by $\log(q)$. This is
because in the monotonic case, the critical regime is
$\log(q)\asymp\log(1/\psi(q))$, which is no longer true when $\psi(q)$ fails
to be monotonic. The statement in the form given above was used already in
several subsequent works such as [7, 26, 34]. Actually, Beresnevich, Haynes,
and Velani [7] constructed a non-monotonic counterexample where for $k\geq 2$,
$\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)^{k}\log(q)^{k}=\infty$,
but $\lambda_{k}(S_{k}^{\times}(\psi))=0$..

######  Theorem (Gallagher, 1962).

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be a monotonically decreasing
function. Then for any $k\geq 1$, we have

$\lambda_{k}\left(S_{k}^{\times}(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\log(1/\psi(q))^{k-1}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\log(1/\psi(q))^{k-1}<\infty.\end{cases}$

Roughly speaking, Gallagher’s Theorem shows that for almost all pairs
$(\alpha,\beta)$, we can surpass Littlewood’s conjecture by approximately a
factor of $\log(q)^{2}$, and this bound is optimal up to double-logarithmic
terms. Furthermore, for each added variable, we almost surely gain another
power of $\log(q)$.

In view of the counterexample of Duffin and Schaeffer [16] for a non-monotonic
version of Khintchine’s Theorem, the question arises whether monotonicity is a
necessary assumption in Gallagher’s Theorem. Since in dimension $k=1$, the
multiplicative and simultaneous setups coincide, i.e.
$S_{1}(\psi)=S_{1}^{\times}(\psi),D_{1}(\psi)=D_{1}^{\times}(\psi)$, we know
that in the $1$-dimensional case, monotonicity cannot be removed from
Gallagher’s Theorem for $S_{1}^{\times}(\psi)$, but it is possible to remove
it for $D_{1}^{\times}(\psi)$ when changing the divergence condition by a
factor of $\frac{\varphi(q)}{q}$. Thus, the open question reduces to the case
where $k\geq 2$. Inspired by a different result of Gallagher [20] and
Pollington-Vaughan [27] for the simultaneous approximation, the following two
conjectures (for coprime and non-coprime setup) were posed by Beresnevich,
Haynes and Velani [7]:

######  Conjecture 1.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function. Then
for any $k\geq 2$, we have

$\lambda_{k}\left(S_{k}^{\times}(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\log(1/\psi(q))^{k-1}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\log(1/\psi(q))^{k-1}<\infty.\end{cases}$

######  Conjecture 2.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function. Then
for any $k\geq 1$, we have

$\lambda_{k}\left(D_{k}^{\times}(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log(1/\psi(q))^{k-1}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log(1/\psi(q))^{k-1}<\infty.\end{cases}$

The main purpose of this article is to (dis-)prove Conjecture 2:

######  Theorem 1.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function. Then
for any $k\geq 1$, we have

$\lambda_{k}\left(D_{k}^{\times}(\psi)\right)=\begin{cases}1,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}=\infty,\\\
0,&\text{ if
}\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}<\infty.\end{cases}$

Clearly, Theorem 1 is in the vein of Conjecture 2, since this is a Khintchine-
type $0$-$1$-result that only takes into account the divergence property of a
series solely depending on $\psi$. However, the condition is altered by a
factor of $q/\varphi(q)$ in the log-powers and it is easy to construct
functions $\psi$ such that this factor determines the convergence of the sum,
thus strictly speaking, we disprove Conjecture 2. At first glance, it may
appear surprising that the correct criterion is not immediately evident.
However, unlike the simultaneous case, the divergence property cannot be
simply inferred from a sum of measures argument. This is because the
underlying hyperbolic regions exhibit a complex (and partially overlapping)
structure, in contrast to the hyper-cubes in the simultaneous counterpart. We
will explain and illustrate how to come up with the correct quantities in
Section 2.1.

Prior to this article, asking for $\lVert q\alpha\rVert\in A_{q}$ for i.m. $q$
and some measurable sets
$(A_{q})_{q\in\operatorname{\mathbb{N}}}\subseteq[0,1/2]^{k}$ in the metric
setup required (to the best of the authors’ knowledge) at least one of the
following assumptions (see, e.g., [7, 21] for a more comprehensive
discussion):

  * •

The sequence of measures
$(\lambda_{k}(A_{q}))_{q\in\operatorname{\mathbb{N}}}$ is monotonically
decreasing, or

  * •

The shape of every $A_{q}$ is convex.

We provide a non-exhaustive list of examples illustrating the application of
monotonicity in related contexts: The actual statement of Gallagher [20]
allows general sets $(A_{q})_{q\in\operatorname{\mathbb{N}}}$ with the
property $(y_{1},\ldots,y_{k})\in
A_{q}\Rightarrow(y_{1}^{\prime},\ldots,y_{k}^{\prime})\in A_{q}$ for all
$0\leq|y_{i}^{\prime}|\leq y_{i}\;\forall 1\leq i\leq k$, as long as
$(\lambda_{k}(A_{q}))_{q\in\operatorname{\mathbb{N}}}$ is monotonically
decreasing. This includes both the cases considered in $S_{k}(\psi)$ and
$S_{k}^{\times}(\psi)$. Under the same assumptions on $A_{q}$, Wang and Yu
[34] even established an asymptotic Schmidt-type formula for the number of
solutions to $\\{q\leq Q:\lVert q\alpha\rVert\in A_{q}\\}$ for almost every
$\alpha$ when $Q\to\infty$. The above-mentioned property was further relaxed
in the works of Dodsen and Kristensen [15, 26] where star-shaped domains with
respect to $0$ were considered. Nevertheless, monotonicity remained a crucial
assumption in the proofs of all mentioned results.

In the regime of convex shapes, the most famous example is the already
mentioned work of Pollington and Vaughan [27] where the corresponding
$A_{q}$’s are $k$-dimensional hypercubes. The shape of $A_{q}$ can be altered,
as long as the convexity property is preserved. The usage of convex bodies
helps to estimate the overlap since counting rational points in convex domains
is much easier than in arbitrary shapes. For a detailed discussion, we refer
the reader to the monograph of Harman [21, Chapter 3].

While the assumption of monotonicity is not required in Theorem 1, the sets
under examination are of hyperbolic shape, making them highly non-convex. This
non-convexity significantly complicates the analysis of overlaps. We regard
the capability to address hyperbolic shapes, which arise naturally within the
context of multiplicative Diophantine approximation, as the primary technical
novelty of Theorem 1.

The new idea employed in this article is to establish not a classical overlap
estimate, but to bound the correlation of functions
$\int_{0}^{1}\gamma_{q}(x)\gamma_{r}(x)\,\mathrm{d}x$ in terms of
$\left(\int_{0}^{1}\gamma_{q}(x)\,\mathrm{d}x\right)\left(\int_{0}^{1}\gamma_{r}(x)\,\mathrm{d}x\right)$,
where $\gamma_{q}$ consists of small bumps in the shape of $1/x$ around
rationals $a/q$ with $(a,q)=1$, which is different from the simultaneous
setting where the bumps are just indicators (or in the convex generalization,
can be approximated well enough with indicators). After establishing a nuanced
upper bound for $\gamma_{q}$, we approximate the corresponding bumps using a
sequence of step-functions carefully chosen in the spirit of the Lebesgue
integral construction. The estimation of controlling the correlation among
these step-functions can be roughly traced back to the case of indicator
functions, aligning with the framework of the Duffin-Schaeffer conjecture. By
applying methods established in [2, 25] in a refined manner, we obtain the
final ingredient.

###  1.3 Applications and related work

In [7], both a $0-1$-law and a “Duffin-Schaeffer-Theorem”-type result were
proved. It was shown that assuming the sizes of $\psi(q)$ and
$\frac{q}{\varphi(q)}$ do not have a too large (positive) correlation, i.e. if

$\limsup_{Q\to\infty}\sum_{q=1}^{Q}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\left(\log(1/\psi(q))\right)^{k-1}\left(\sum_{q=1}^{Q}\psi(q)\left(\log(1/\psi(q))\right)^{k-1}\right)^{-1}>0,$
(2)

then Conjectures 1 and 2 are true. We note that in this regime, the
conjectured divergence criteria from both Conjectures coincide with the one
established in Theorem 1, thus there is no contradiction in all statements
being true. The following result is immediately implied by Theorem 1, which
makes progress towards Conjecture 1 by demanding some extra-divergence.

######  Corollary 2.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function such
that

$\sum\limits_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}=\infty.$
(3)

Then

$\lambda_{k}\left(S_{k}^{\times}(\psi)\right)=1.$

In particular, (3) holds whenever

$\sum_{q\in\operatorname{\mathbb{N}}}\frac{\psi(q)}{(\log\log
q)^{k}}\log\left(\frac{\log\log q}{\psi(q)}\right)^{k-1}=\infty.$

For many applications, it is particularly interesting when $\psi$ is of the
form $\psi(q)=\mathbbm{1}_{A}(q)\theta(q)$, where
$A\subseteq\operatorname{\mathbb{N}}$ and $\theta$ is a monotonically
decreasing function. For various results in this context, we refer the
interested reader again to Chapter 3 of the excellent book of Harman [21].
Given an asymptotic density of $A$, i.e. a function
$f:\operatorname{\mathbb{N}}\to[0,1]$ such that

$\frac{\\#\\{n\leq N:n\in A\\}}{N}\sim f(N),$

we ask whether the expected quality of approximation with $q\in A$ only loses
a factor of $f(N)$ in comparison to Khinchine’s Theorem. If we have some
control of the average size of $\frac{q}{\varphi(q)}$ for $q\in A$, i.e.

$\limsup_{N\to\infty}\frac{\\#\\{q\leq N:q\in A,\frac{q}{\varphi(q)}\leq
M\\}}{\\#\\{q\leq N:q\in A\\}}>0,$ (4)

then the result in [7] is enough to obtain the precise information: To name
just some easy examples, if $(p_{n})_{n\in\operatorname{\mathbb{N}}}$ is the
sequence of all primes, then $f(N)=1/\log N$ and $M=2$ satisfy the above
assumptions. We thus obtain the very simple criterion of whether
$\sum_{q\in\operatorname{\mathbb{N}}}\theta(q)=\infty$ (which coincides with
the classical Khintchine Theorem). Hence, we have $\lVert p\alpha\rVert\lVert
p\beta\rVert\leq\frac{1}{p\log p}$ infinitely often, but $\lVert
p\alpha\rVert\lVert p\beta\rVert\leq\frac{1}{p(\log p)^{1+\varepsilon}}$ only
finitely often. If $A$ is a set of positive density, i.e. $f(N)=\delta>0$, we
can approximate with the same quality as in Gallagher’s Theorem, that is,
$\lVert q\alpha\rVert\lVert q\beta\rVert\leq\frac{1}{q(\log q)^{2}}$. If
$A=\\{b^{n},n\in\operatorname{\mathbb{N}}\\}$ for some $b\geq 2$, the above
shows that

$\lVert b^{n}\alpha\rVert\lVert b^{n}\beta\rVert\leq\frac{1}{n^{2}(\log
n)^{2}}$

holds infinitely often, which is sharp up to triple-logarithmic factors.

However, such assertions cannot be inferred for arbitrary sequences
$(a_{n})_{n\in\mathbb{N}}$. Theorem 1 serves as a tool for deriving various
statements in cases where only density estimates are available, without
relying on number-theoretic conditions such as (4). To illustrate this, we
present Corollary 3 below, which provides a good estimate on the required
density of a given sequence $(a_{n})_{n\in\operatorname{\mathbb{N}}}$ such
that Littlewood’s conjecture is true along this sequence for almost every
$(\alpha,\beta)$. More precisely, let $A=\\{a_{1}<a_{2}<\ldots\\}$ be an
infinite set of positive integers and let

$LC(A):=\left\\{(\alpha,\beta)\in[0,1]^{2}:\liminf_{n\to\infty}a_{n}\lVert
a_{n}\alpha\rVert\lVert a_{n}\beta\rVert=0\right\\},$

i.e. the set of irrationals such that Littlewood’s conjecture is true along
the sequence $(a_{n})_{n\in\operatorname{\mathbb{N}}}$ (we refer the
interested reader to [29, Section 1.2] for related results). The following is
an immediate application of Corollary 2.

######  Corollary 3.

For any $A\subseteq\operatorname{\mathbb{N}}$, we have

$\lambda_{2}(LC(A))=\begin{cases}1,\text{ if }\quad\frac{\\#\\{n\leq N:n\in
A\\}}{N}\gg\frac{\log\log N}{(\log N)^{2}},\\\ 0,\text{ if
}\quad\frac{\\#\\{n\leq N:n\in A\\}}{N}\ll\frac{1}{(\log N)^{2}}.\end{cases}$

To the best of the authors’ knowledge, this improves the currently best-known
result that stems from a direct application of the Koukoulopoulos-Maynard
Theorem, where the condition

$\frac{\\#\\{n\leq N:n\in A\\}}{N}\gg\frac{1}{\log N},$

is (up to triple-logarithms), the best bound known to guarantee
$\lambda_{2}(LC(A))=1$. Thus Theorem 3 gives an improvement of a factor of
(roughly) $\log N/(\log\log N)$, with the quality of gain increasing by
another factor of $\log N/(\log\log N)$ when we consider more variables, i.e.
$\lVert n\alpha\rVert\lVert n\beta\rVert\lVert n\gamma\rVert$. One way to get
rid of the double logarithms and therefore obtain a sharp statement would be
to prove (at least some form of) Conjecture 1, a question that remains open.

###  1.4 Hausdorff dimension analogue

Many statements in metric Diophantine approximation have a Hausdorff measure
equivalent: By assuming a fast decay of the approximation function $\psi$, the
sets $S_{k}(\psi),S_{k}^{\times}(\psi),D_{k}(\psi),D_{k}^{\times}(\psi)$ are
all of Lebesgue measure $0$; however, by a phenomenon now widely known as
“mass transference principle”, which was popularized in the seminal work of
Beresnevich-Velani [8], one can immediately deduce a Hausdorff dimension
version of Lebesgue measure theorems in many different setups. The Hausdorff
version for $S_{k}(\psi)$ with $\psi$ monotone is the classical Jarník-
Besicovitch-Theorem, and Beresnevich-Velani [8] established222Strictly
speaking, the result of Beresnevich-Velani proved that the Duffin-Schaeffer
Conjecture would imply a Hausdorff version of itself, thus the actual proof of
the Hausdorff version was implied by the later work of Koukoulopoulos and
Maynard in [25]. the analogue for $D_{k}(\psi)$, again without assuming
monotonicity. In the setting of multiplicative Diophantine approximation, the
picture is not yet that clear, although progress was made in [7, 9, 14, 24]
for $S_{k}^{\times}(\psi)$ and assuming monotonicity, at least in the
divergent case. The best result known so far is due to Hussain and Simmons
[24]: For monotonic $\psi$ with $\psi(q)\to 0$, they showed that
$\dim_{H}(S_{k}^{\times}(\psi))=(k-1)+\min\\{d(\psi),1\\}$ where

$d(\psi)=\inf\left\\{s\in[0,1]:\sum_{q\in\operatorname{\mathbb{N}}}q\left(\frac{\psi(q)}{q}\right)^{s}<\infty\right\\}.$
(5)

As a Corollary of Theorem 1, it turns out that the Hausdorff dimension
statement remains unchanged for $S_{k}^{\times}(\psi)),D_{k}^{\times}(\psi))$
even upon removing the monotonicity assumption:

######  Corollary 4.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function and
let $d(\psi)$ be as in (5). Then we have

$\dim_{H}(S_{k}^{\times}(\psi))=\dim_{H}(D_{k}^{\times}(\psi))=(k-1)+\min\\{d(\psi),1\\}.$

We note that Corollary 4 is not only making a statement on the coprime
approximations as considered in Conjecture 2, but also about the non-coprime
setting from Conjecture 1. Thus, as an immediate Corollary, we also get a
Hausdorff version of Conjecture 1:

######  Corollary 5.

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be an arbitrary function and
assume that

$\sum\limits_{q\in\operatorname{\mathbb{N}}}\psi(q)\log(1/\psi(q))^{k-1}=\infty.$

Then $\dim_{H}(S_{k}^{\times}(\psi))=k$.

###  1.5 Open questions

Besides the obvious question of proving also Conjecture 1, there are natural
generalizations resp. quantifications of Theorem 1. One way to do so is to
allow an inhomogeneous parameter $\gamma\in\mathbb{R}^{k}$ and ask the same
questions about the sets

$S_{k}^{\times}(\psi,\gamma):=\left\\{\alpha\in[0,1]^{k}:\prod\limits_{i=1}^{k}\lVert
q\alpha_{i}-\gamma_{i}\rVert\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\}$

and their coprime counterparts

$D_{k}^{\times}(\psi,\gamma):=\left\\{\alpha\in[0,1]^{k}:\prod\limits_{i=1}^{k}\lVert
q\alpha_{i}-\gamma_{i}\rVert^{\prime}\leq\psi(q)\text{ for i.m.
}q\in\operatorname{\mathbb{N}}\right\\}.$

Assuming monotonicity of $\psi$, the inhomogeneous variant of Gallagher’s
result was proven for $k=1$ by Szüsz [32] and by Chow and Technau [10, 11] for
$k\geq 2$. The authors of the latter obtained actually a stronger result since
they only need to have one metric parameter $\alpha_{k}$ assuming a mild
Diophantine condition for $(\alpha_{1},\ldots,\alpha_{k-1})$. However, all
these proofs made use of the monotonicity of $\psi$, thus the statement
remains open in the non-monotonic setup. Recently a first result for non-
monotonic $\psi$ was given by Yu [37], albeit a strong form of regularity in
the form of $\psi(q)=O((q\log q(\log\log q)^{2})^{-1})$ is assumed.

By generalizing the Duffin-Schaeffer counterexample, Ramírez [30] showed that
for any $\gamma$, the monotonicity assumption is necessary when $k=1$, but as
in the homogeneous setup, the counterexample does not generalize to higher
dimensions. Therefore, the theorem of Chow-Technau [11] or a coprime analogue
in the form of Theorem 1 in the inhomogeneous setup might still be true
without monotonicity assumption; however, the proof given in this article
relies on the proven (homogeneous) Duffin-Schaeffer conjecture and the
inhomogeneous counterpart is still a widely open question, for recent progress
see [4, 22, 35, 36]. In this context, the current methods do not offer a
viable approach to addressing these questions until a deeper understanding of
the one-dimensional inhomogeneous theory is attained. We also remark that
Theorem 1 does not yield any information about the set $D_{k}^{\times}(\psi)$
other than having Lebesque measure $1$ or $0$. In particular, it does not
provide a Diophantine condition that ensures
$\alpha\in\operatorname{\mathbb{R}}^{k}$ to lie in $D_{k}^{\times}(\psi)$.
There is no possibility to provide such a “fibered” result as in [11] with the
metric tools applied in this article, and the approach in [11] does not carry
over to the non-monotonic setup. For fibered results in the well-studied
regime of Littlewood’s conjecture, we refer the reader to, e.g., [12, 13, 28].

In the theory of Hausdorff measures, the question about Hausdorff dimensions
is coarse enough to absorb all $\log$-factors and
$\frac{q}{\varphi(q)}$-factors that appear in the Lebesgue measure statements.
However, when we consider arbitrary dimension functions $f$ that are not
necessarily of the form $f(r)=r^{s}$, the picture is different. Here, a
difference between $H^{f}(D_{k}^{\times}(\psi))$ and
$H^{f}(S_{k}^{\times}(\psi))$ for specific $f$ seems plausible and the correct
thresholds remain open.

Finally, another natural open question is to ask about an asymptotic formula
that counts the number of coprime solutions up to a given threshold. This was
successfully established by Aistleitner, Borda, and the second-named author in
[2] for the case $k=1$, but is yet unclear for $k\geq 2$. Achieving this goal
necessitates a deeper understanding of the variance estimate introduced in
this article. Furthermore, and probably even more challenging, one cannot
afford to truncate the sets $A_{q}$ in a way that loses a constant factor of
the corresponding measures, a procedure that is applied at various occasions
in the proof of Theorem 1.

###  1.6 Notation

Throughout the text, we use the standard O- and o- notations, as well as the
Vinogradov notations $\ll,\gg$. If $f\ll g$ and $g\ll f$, we write $f\asymp g$
or $f=\Theta(g)$. For $x\in\mathbb{R}$, $\lVert
x\rVert:=\min\limits_{n\in\mathbb{Z}}\lvert x-n\rvert$ denotes the distance to
the nearest integer. For $a,q\in\operatorname{\mathbb{Z}}$, we write
$(a,q):=\gcd(a,q)$. For $q\in\operatorname{\mathbb{N}}$, we denote
$\operatorname{\mathbb{Z}}_{q}:=\\{0,\ldots,q-1\\}$ and
$\operatorname{\mathbb{Z}}_{q}^{*}:=\\{a\in\operatorname{\mathbb{Z}}_{q}:(a,q)=1\\}$.
For $k$ being an integer, we denote the $k$-dimensional Lebesgue-measure as
$\lambda_{k}$. Given two positive real numbers $x,y$, we write $x\land
y:=\min\left\\{x,y\right\\}$. Furthermore, for any
$k\in\operatorname{\mathbb{N}}$, we define $x\log^{k}(1/x)\mid_{x=0}:=0.$

##  2 Proof of the main result

###  2.1 The convergent case and a discussion on the correct divergence
condition

Before we prove the convergent case (which is easier than the divergent case,
but still not as trivial as in most other scenarios), we discuss the
particular divergence condition of Theorem 1 and its deviation from the
conjectured threshold in [7]. We will focus on the case $k=2$ here since in
the case $k=1$ the conditions coincide and for $k\geq 3$, an analogous
phenomenon to $k=2$ takes place. Starting with the non-coprime case, given
some function $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$, let

$A_{q}=A_{q}(\psi):=\left\\{(\alpha,\beta)\in[0,1]^{2}:\lVert
q\alpha\rVert\lVert q\beta\rVert\leq\psi(q)\right\\}.$

Note that we can write $S_{2}^{\times}(\psi)=\limsup_{q\to\infty}A_{q}$. We
observe that $A_{q}$ consists of a regular pattern of $q^{2}$ "star-shaped"
forms with their origins at the rational points $(a/q,b/q)$ (see Figure 1). It
is easy to check that we have a partition into

$A_{q}=\bigcup\limits_{(a,b)\in\mathbb{Z}_{q}}^{\bm{\cdot}}A_{q,a,b}$ (6)

where

$A_{q,a,b}=\left\\{(\alpha,\beta)\in[0,1]^{2}:\left\lvert\alpha-\frac{a}{q}\right\rvert\left\lvert\beta-\frac{b}{q}\right\rvert\leq\frac{\psi(q)}{q^{2}},\left\lvert\alpha-\frac{a}{q}\right\rvert\leq\frac{1}{2q},\left\lvert\alpha-\frac{b}{q}\right\rvert\leq\frac{1}{2q}\right\\}.$

By elementary calculus, we obtain for any $(a,b)\in\mathbb{Z}_{q}^{2}$ that

$\lambda_{2}(A_{q,a,b})\asymp\frac{\psi(q)}{q^{2}}\log\left(\frac{1}{\psi(q)}\right)$

with an absolute implied constant. Thus by (6) we obtain
$\lambda_{2}(A_{q})\asymp\psi(q)\log\left(\frac{1}{\psi(q)}\right)$, which
coincides with the divergence condition in Conjecture 1 and in Gallagher’s
Theorem.

Figure 1: An illustration of the set $A_{4}$ (the non-coprime setting). The
red area corresponds to $A_{4,1,1}$. Note that for illustration purposes, the
lower-left corner has the coordinate $(1/8,1/8)$.

Changing the attention now to the coprime setting, the question about the
Lebesgue measure of
$A_{q}^{\prime}=A_{q}^{\prime}(\psi):=\left\\{(\alpha,\beta)\in[0,1]^{2}:\lVert
q\alpha\rVert^{\prime}\lVert q\beta\rVert^{\prime}\leq\psi(q)\right\\}$
becomes non-trivial: One is tempted to apply the same partition argument as in
the non-coprime setup in the form of

$\bigcup\limits_{(a,b)\in\mathbb{Z}_{q}^{*}}^{.}A_{q,a,b}=:A_{q}^{\prime\prime},$
(7)

and one would assume $A_{q}^{\prime}=A_{q}^{\prime\prime}$. This would give us
$\lambda_{2}(A_{q}^{\prime})\asymp\frac{\varphi(q)^{2}}{q^{2}}\log\left(\frac{1}{\psi(q)}\right)$
which would coincide with the measures that lead to the condition in
Conjecture 2.

However, (7) is not true: Assuming the contrary, this would imply that for
$(a,b)\in\mathbb{Z}_{q}$ such that $(a,q)>1$ or $(b,q)>1$, the sets
$A_{q}^{\prime}$ and
$B_{a,b,q}:=\left\\{(\alpha,\beta)\in[0,1]^{2}:\left\lvert\alpha-\frac{a}{q}\right\rvert\leq\frac{1}{2q},\left\lvert\alpha-\frac{b}{q}\right\rvert\leq\frac{1}{2q}\right\\}$
are disjoint, but this is wrong: Assuming, e.g., $(a-1,q)=1,(b,q)=1$, then a
very good approximation of $\beta$ by $b/q$ implies that even for $\alpha$
close to $\frac{a-1}{q}$ (which implies
$\left\lvert\alpha-\frac{a}{q}\right\rvert\geq\frac{1}{2q}$), we have
$\left\lvert\alpha-\frac{a}{q}\right\rvert\left\lvert\beta-\frac{b}{q}\right\rvert\leq\frac{\psi(q)}{q^{2}}$.
To put this in simple words, the “arm” of the “star” with its center at
$((a-1)/q,b/q)$ “reaches” into $B_{a,b,q}$ and if
$(a,b)\notin{(\mathbb{Z}_{q}^{*})}^{2}$, then this part of the arm is in
$A_{q}^{\prime}$, but not in $A_{q}^{\prime\prime}$ (see Figure 2). Actually,
the “arm” reaches as far as half of the distance to another coprime element in
the same direction. It can be verified by a maximization problem (which is
reflected in the equality case of the arithmetic-geometric mean inequality,
which appears in the proof of Proposition 2.1) that the extremal configuration
happens when the coprime elements are equally spaced among $\mathbb{Z}_{q}$,
which leads to each such “arm” to be of length $\asymp\frac{q}{\varphi(q)}$.
Computing the mass of one such “star” with “arm lengths”
$\asymp\frac{q}{\varphi(q)}$ yields the measure
$\frac{\psi(q)}{q^{2}}\log\left(\frac{\frac{q}{\varphi(q)}}{\psi(q)}\right)$
and since there are $\varphi(q)^{2}$ such coprime “stars”, we obtain the
divergence condition from Theorem 1.

Having the heuristic explanation established, we turn to the actual proof of
the convergent case of Theorem 1.

Figure 2: To the left, we see $A_{4}^{\prime}$, which is the proper set in the
coprime setting. To the right, we have $A_{4}^{\prime\prime}$, which would
give the conjectured divergence criterion in [7]. We note that in both
figures, the lower left corner has the coordinate $(1/8,1/8)$.

######  Proposition 2.1.

Let $k\geq 1$ and $\psi:\operatorname{\mathbb{N}}\rightarrow[0,1/2]$ be a
function such that

$\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\psi(q)\varphi(q)}\right)^{k-1}<\infty.$

Then $\lambda_{k}(D_{k}^{\times}(\psi))=0$.

###### Proof.

Defining $\Psi(q):=\psi(q)\left(\frac{\varphi(q)}{q}\right)^{k-1}$, we observe
that

$\displaystyle\\#\left\\{q\in\operatorname{\mathbb{N}}:\prod_{i=1}^{k}||q\alpha_{i}||^{\prime}\leq\psi(q)\right\\}$
$\displaystyle\leq\\#\left\\{q\in\operatorname{\mathbb{N}}:||q\alpha_{k}||^{\prime}\leq\frac{\psi(q)}{\prod_{i=1}^{k-1}||q\alpha_{i}||^{\prime}},\forall
i\in\\{1,\ldots,k-1\\}:||q\alpha_{i}||^{\prime}\geq\Psi(q)\right\\}$
$\displaystyle\qquad+\\#\left\\{q\in\operatorname{\mathbb{N}}:\exists
i\in\\{1,\ldots,k-1\\}:||q\alpha_{i}||^{\prime}\leq\Psi(q)\right\\}$
$\displaystyle\leq\\#\left\\{q\in\operatorname{\mathbb{N}}:||q\alpha_{k}||^{\prime}\leq\psi(q)\prod_{i=1}^{k-1}\left[\frac{1}{||q\alpha_{i}||^{\prime}}\land\frac{1}{\Psi(q)}\right]\right\\}$
$\displaystyle\qquad+\sum_{i=1}^{k-1}\\#\left\\{q\in\operatorname{\mathbb{N}}:||q\alpha_{i}||^{\prime}\leq\Psi(q)\right\\}$
$\displaystyle=:M_{1}(q)+M_{2}(q).$

We observe that $M_{2}(q)$ is almost surely finite by the convergence part of
the Koukoulopoulos-Maynard-Theorem, since by assumption we have

$\sum_{q\in\operatorname{\mathbb{N}}}\frac{\varphi(q)}{q}\Psi(q)=\sum_{q\in\operatorname{\mathbb{N}}}\psi(q)\left(\frac{\varphi(q)}{q}\right)^{k}<\infty.$

Thus, we are left with $M_{1}(q)$, which is almost surely finite by the
convergence part of the Koukoulopoulos-Maynard theorem, provided that

$\sum_{q\in\operatorname{\mathbb{N}}}\frac{\varphi(q)}{q}\psi(q)\psi_{\alpha}(q)<\infty$
(8)

holds for almost all $\alpha\in[0,1]^{k-1}$, where

$\psi_{\alpha}(q):=\prod_{i=1}^{k-1}\left[\frac{1}{||q\alpha_{i}||^{\prime}}\land\frac{1}{\Psi(q)}\right].$

The condition in (8) is certainly true if

$\int_{[0,1]^{k-1}}\sum_{q\in\operatorname{\mathbb{N}}}\frac{\varphi(q)}{q}\psi(q)\psi_{\alpha}(q)\,\mathrm{d}\alpha=\sum_{q\in\operatorname{\mathbb{N}}}\frac{\varphi(q)}{q}\psi(q)\int_{[0,1]^{k-1}}\psi_{\alpha}(q)\,\mathrm{d}\alpha<\infty,$

where we are allowed to interchange integral and summation by Fubini’s theorem
for non-negative integrands. We further get

$\displaystyle\int_{[0,1]^{k-1}}\psi_{\alpha}(q)\,\mathrm{d}\alpha=\left(\int_{0}^{1}\tilde{\psi}_{\beta}(q)\,\mathrm{d}\beta\right)^{k-1},$

with
$\tilde{\psi}_{\beta}(q):=\frac{1}{||q\beta||^{\prime}}\land\frac{1}{\Psi(q)}$
for $\beta\in[0,1]$. In the following, we compute an upper bound for the
expected value of $\tilde{\psi}_{\beta}(q)$, where we obtain

$\displaystyle\int_{0}^{1}\tilde{\psi}_{\beta}(q)\,\mathrm{d}\beta=\sum_{b=0}^{q-1}\int_{b/q}^{(b+1)/q}\left(\max_{a\in\operatorname{\mathbb{Z}}_{q}^{*}}\frac{1}{|q\beta-a|}\land\frac{1}{\Psi(q)}\right)\,\mathrm{d}\beta=:I_{q}.$

For any $b\in\operatorname{\mathbb{Z}}_{q}$, we introduce the following sets

$J_{b}:=\left(\frac{b}{q}-\frac{1}{2q},\frac{b}{q}+\frac{1}{2q}\right)\setminus\left\\{\frac{b}{q}\right\\}=\left(\frac{b}{q}-\frac{1}{2q},\frac{b}{q}\right)\cup\left(\frac{b}{q},\frac{b}{q}+\frac{1}{2q}\right)=:J_{b}^{\prime}\cup
J_{b}^{\prime\prime}.$

Then, $\arg\min_{c\in\operatorname{\mathbb{Z}}_{q}^{*}}|q\beta-c|$ is constant
on $J_{b}^{\prime}$ and $J_{b}^{\prime\prime}$ respectively. This allows us to
define the following covering of $\operatorname{\mathbb{Z}}_{q}$ by

$A_{a}:=\left\\{b\in\operatorname{\mathbb{Z}}_{q}:a=\arg\min_{c\in\operatorname{\mathbb{Z}}_{q}^{*}}|q\beta-c|,\forall\beta\in
J_{b}^{\prime}\right\\}\cup\left\\{b\in\operatorname{\mathbb{Z}}_{q}:a=\arg\min_{c\in\operatorname{\mathbb{Z}}_{q}^{*}}|q\beta-c|,\forall\beta\in
J_{b}^{\prime\prime}\right\\},\quad a\in\operatorname{\mathbb{Z}}_{q}^{*}.$

Note that any $b\in\operatorname{\mathbb{Z}}_{q}$ can be contained in at most
two different sets $A_{a}$, which leads to

$\displaystyle I_{q}$
$\displaystyle=\sum_{b=0}^{q-1}\int_{b/q-1/2q}^{b/q+1/2q}\left(\max_{a\in\operatorname{\mathbb{Z}}_{q}^{*}}\frac{1}{|q\beta-a|}\land\frac{1}{\Psi(q)}\right)\,\mathrm{d}\beta$
$\displaystyle\leq 2\sum_{\begin{subarray}{c}a=0\\\
(a,q)=1\end{subarray}}^{q-1}\sum_{b\in
A_{a}}\int_{J_{b}}\frac{1}{|q\beta-a|}\land\frac{1}{\Psi(q)}\,\mathrm{d}\beta.$

Now we observe that $A_{a}\subseteq[a-f_{1}(a),a+f_{2}(a)]$, where the
$f_{i}(a)\in\operatorname{\mathbb{N}}$ are such that

$\sum_{\begin{subarray}{c}a=0\\\
(a,q)=1\end{subarray}}^{q-1}f_{1}(a)+f_{2}(a)\leq 2q.$

Therefore, we obtain

$\displaystyle I_{q}$
$\displaystyle\leq\frac{1}{q}\sum_{(a,q)=1}\int_{-f_{1}(a)}^{f_{2}(a)}\frac{1}{|\beta|}\land\frac{1}{\Psi(q)}\,\mathrm{d}\beta$
$\displaystyle\ll\frac{1}{q}\sum_{(a,q)=1}\left[\log\left(\frac{f_{1}(a)}{\Psi(q)}\right)+\log\left(\frac{f_{2}(a)}{\Psi(q)}\right)\right]$
$\displaystyle\ll\frac{\varphi(q)}{q}\log\left(\left(\frac{q}{\varphi(q)}\right)^{k}\frac{1}{\psi(q)}\right)$
$\displaystyle\ll\frac{\varphi(q)}{q}\log\left(\frac{q}{\varphi(q)\psi(q)}\right),$

where the estimate in the second last line uses the arithmetic-geometric mean
inequality. This implies

$\sum_{q\in\operatorname{\mathbb{N}}}\frac{\varphi(q)}{q}\psi(q)\int_{[0,1]^{k-1}}\psi_{\alpha}(q)\,\mathrm{d}\alpha\ll\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}<\infty,$

as desired. ∎

###  2.2 The divergent case

####  2.2.1 Proof of Theorem 1 assuming a variance estimate

Let $k\geq 2$ be fixed and recall that, under the assumption of

$\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1}=\infty,$
(9)

we aim to show that for almost all
$\alpha=(\alpha_{1},\ldots,\alpha_{k})\in\operatorname{\mathbb{R}}^{k}$, there
are infinitely many $q\in\operatorname{\mathbb{N}}$ such that

$||q\alpha_{1}||^{\prime}||q\alpha_{2}||^{\prime}\cdots||q\alpha_{k}||^{\prime}\leq\psi(q).$

For the quantity in (9), for any fixed small $\rho>0$, we have at least one of
the two following:

$\displaystyle\sum_{\begin{subarray}{c}q\in\operatorname{\mathbb{N}}\\\
\frac{\varphi(q)}{q}\leq\rho\end{subarray}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\psi(q)\varphi(q)}\right)^{k-1}=\infty\quad\text{or}\quad\sum_{\begin{subarray}{c}q\in\operatorname{\mathbb{N}}\\\
\frac{\varphi(q)}{q}>\rho\end{subarray}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\psi(q)\varphi(q)}\right)^{k-1}=\infty.$

Suppose first the latter to be true. Then
$\tilde{\psi}(q):=\psi(q)\mathds{1}_{[\frac{\varphi(q)}{q}>\rho]}$ satisfies
(2). Thus by [7, Theorem 2], it follows immediately that
$\lambda_{k}(D_{k}^{\times}(\psi))=1$. Thus, we may restrict ourselves to the
case where for every $q$ in the support of $\psi$, it holds that

$\frac{\varphi(q)}{q}\leq\rho$ (10)

for some small $\rho>0$ that is specified below. We will show that almost
surely,

$\\#\left\\{q\in\operatorname{\mathbb{N}}:||q\alpha_{1}||^{\prime}\leq\psi_{\alpha}(q)\right\\}=\infty,$
(11)

where
$\psi_{\alpha}(q):=\frac{\psi(q)}{\prod_{i=2}^{k}||q\alpha_{i}||^{\prime}}$.
By the Koukoulopoulos-Maynard-Theorem, (11) is true whenever

$\sum_{q\leq Q}\frac{\varphi(q)}{q}\psi_{\mathbf{\alpha}}(q)=\infty.$

Defining

$\rho=\rho(k):=\frac{1}{4^{k-1}}$ (12)

and assuming (10), we obtain
$g_{k}(q):=\left(\frac{\psi(q)\varphi(q)}{q}\right)^{1/(k-1)}\leq 1/2$ for all
$q\in\operatorname{\mathbb{N}}$. Using this, a straightforward computation
shows that

$\frac{\varphi(q)}{q}\psi_{\mathbf{\alpha}}(q)\geq\gamma_{q}^{(k)}(\alpha),$
(13)

where

$\gamma_{q}^{(k)}(x_{1},\ldots,x_{k-1}):=\prod_{i=1}^{k-1}\sum_{(a,q)=1}f_{q}^{(k)}(qx_{i}-a),$

with

$f_{q}^{(k)}(y)=\begin{cases}1,&\text{if}\quad|y|\leq g_{k}(q),\\\
\frac{g_{k}(q)}{|y|},&\text{if}\quad|y|\in\left[g_{k}(q),g_{k}(q)^{1/2}\right],\\\
0,&\text{else}.\end{cases}$

By elementary calculus and Fubini’s Theorem, we obtain

$\int_{[0,1]^{k-1}}\sum_{q\in\operatorname{\mathbb{N}}}\gamma_{q}^{(k)}(\alpha)\,\mathrm{d}\alpha\gg\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\log\left(\frac{q}{\psi(q)\varphi(q)}\right)^{k-1}=\infty,$
(14)

where the last equality is assumption (9). Next, we make use of the following
variance estimate for $\gamma_{q}^{(k)}$, which is the remaining key
ingredient in the proof of Theorem 1:

######  Proposition 2.2.

Let $\gamma_{q}^{(k)}$ be defined as above. Then for all $\delta>0$, there
exists a $Q(\delta)\in\operatorname{\mathbb{N}}$ such that for all $Q\geq
Q(\delta)$ we have

$\sum_{q,r\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x\leq(1+\delta)\left(\sum_{q\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x\right)^{2}.$

Assuming Proposition 2.2 (which is proven down below) to hold, we finish the
proof of Theorem 1 in the following way by a standard argument: Using (13), it
suffices to show that for almost all
$\alpha=(\alpha_{1},\ldots,\alpha_{k-1})\in\operatorname{\mathbb{R}}^{k-1}$,
we have

$S_{\alpha}:=\sum_{q\in\operatorname{\mathbb{N}}}\gamma_{q}^{(k)}(\alpha)=\infty.$
(15)

For $c>0$, let
$A_{c}:=\\{\alpha\in[0,1]^{k-1}:S_{\alpha}>c\\},B_{c}=[0,1]\setminus A_{c}$
and $A_{\infty}=\lim_{c\to\infty}A_{c}$. Assuming to the contrary of (15) that
$\lambda_{k-1}(A_{\infty})<1$, by continuity of measures, there are
$C>0,\varepsilon>0$ such that $\lambda_{k-1}(B_{C})>\varepsilon$. We write

$S_{\alpha}^{Q}:=\sum_{q\leq Q}\gamma^{(k)}_{q}(\alpha)$

and we note that

$\lim_{Q\rightarrow\infty}\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha=\infty$

by (14). By Proposition 2.2, for any $\delta>0$, we find a $Q=Q(\delta)$ such
that

$\int_{[0,1]^{k-1}}\left[S_{\alpha}^{Q}\right]^{2}\,\mathrm{d}\alpha-\left(\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha\right)^{2}\leq\delta\left(\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha\right)^{2},$

for all $Q\geq Q(\delta)$. We choose $\delta=\frac{\varepsilon}{4}$ and apply
Chebyshev’s inequality which yields

$\lambda_{k-1}\left(\left\\{\alpha\in[0,1]^{k-1}:\left\lvert
S_{\alpha}^{Q}-\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha\right\rvert>\frac{1}{2}\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha\right\\}\right)\leq
4\delta,$

provided that $Q\geq Q(\delta)$. For such $Q$, we have
$S_{\alpha}^{Q}>\frac{1}{2}\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha$
on a set of Lebesgue measure of at least $1-4\delta$. Choosing $Q$ large
enough such that $\int_{[0,1]^{k-1}}S_{\alpha}^{Q}\,\mathrm{d}\alpha>2C$
implies $\lambda_{k-1}(B_{C})\leq 4\delta<\varepsilon$, a contradiction.

####  2.2.2 The variance estimate

###### Proof of Proposition 2.2.

Let $\varepsilon>0$ and define

$f_{q,\varepsilon}^{(k)}(y):=\begin{cases}1,&\text{ if }0\leq|y|\leq
g_{k}(q)\left(1+\varepsilon\log\left(g_{k}(q)^{-1}\right)\right),\\\
f_{q}^{(k)}(y),&\text{ otherwise}.\end{cases}$

Moreover, for $x\in\operatorname{\mathbb{R}}^{k-1}$ we set
$\gamma_{q,\varepsilon}^{(k)}(x):=\prod_{i=1}^{k-1}\sum_{(a,q)=1}f_{q,\varepsilon}^{(k)}(qx_{i}-a)$
and recall that $g_{k}(q)=\left(\frac{\psi(q)\varphi(q)}{q}\right)^{1/(k-1)}$.
Clearly, for all $y\in\operatorname{\mathbb{R}}$ we have $f_{q}^{(k)}(y)\leq
f_{q,\varepsilon}^{(k)}(y)$ and hence we get the estimate

$\displaystyle\sum_{q,r\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x$
$\displaystyle\leq\sum_{q,r\leq
Q}\left(\sum_{(a,q)=1}\sum_{(b,r)=1}\int_{0}^{1}f_{q,\varepsilon}^{(k)}(qy-a)f_{r,\varepsilon}^{(k)}(ry-b)\,\mathrm{d}y\right)^{k-1}.$

For every $K\in\operatorname{\mathbb{N}}$ and every
$q\in\operatorname{\mathbb{N}}$, we can choose a finite sequence of points

$g_{k}(q)\left(1+\varepsilon\log\left(g_{k}(q)^{-1}\right)\right)\leq
x_{q,0}\leq\ldots\leq x_{q,K-1}\leq 2g_{k}(q)^{1/2}$

such that

$f_{q,\varepsilon}^{(k)}(y)\leq\frac{1}{K}\sum_{i=0}^{K-1}\mathds{1}_{[-x_{q,i},x_{q,i}]}(x),\qquad
f_{q,\varepsilon}^{(k)}(y)=\lim_{K\rightarrow\infty}\frac{1}{K}\sum_{i=0}^{K-1}\mathds{1}_{[-x_{q,i},x_{q,i}]}(x).$

$g_{k}(q)$$g_{k}(q)^{1/2}$$-x_{q,K-1}$ Figure 3: The picture illustrates the
relation of $f_{q}^{(k)}(y)$, $f_{q,\varepsilon}^{(k)}(y)$ and
$\frac{1}{K}\sum_{i=0}^{K-1}\mathds{1}_{[-x_{q,i},x_{q,i}]}(y)$.

This leads to

$\displaystyle\sum_{q,r\leq
Q}\left(\sum_{(a,q)=1}\sum_{(b,r)=1}\int_{0}^{1}f_{q,\varepsilon}^{(k)}(qy-a)f_{r,\varepsilon}^{(k)}(ry-b)\,\mathrm{d}y\right)^{k-1}$
$\displaystyle\leq$ $\displaystyle\sum_{q,r\leq
Q}\left(\sum_{(a,q)=1}\sum_{(b,r)=1}\frac{1}{K^{2}}\sum_{i,j=0}^{K-1}\int_{0}^{1}\mathds{1}_{[-x_{q,i},x_{q,i}]}(qy-a)\mathds{1}_{[-x_{r,j},x_{r,j}]}(ry-b)\,\mathrm{d}y\right)^{k-1}$
$\displaystyle=$ $\displaystyle\sum_{q,r\leq
Q}\left(\frac{1}{K^{2}}\sum_{i,j=0}^{K-1}\int_{0}^{1}\sum_{(a,q)=1}\mathds{1}_{[-x_{q,i},x_{q,i}]}(qy-a)\sum_{(b,r)=1}\mathds{1}_{[-x_{r,j},x_{r,j}]}(ry-b)\,\mathrm{d}y\right)^{k-1}$
$\displaystyle=$ $\displaystyle\sum_{q,r\leq
Q}\left(\frac{1}{K^{2}}\sum_{i,j=0}^{K-1}\lambda_{1}(A_{q,i}\cap
A_{r,j})\right)^{k-1},$

where
$A_{q,i}:=\bigcup_{(a,q)=1}\left[\frac{a}{q}-\frac{x_{q,i}}{q},\frac{a}{q}+\frac{x_{q,i}}{q}\right]$.
In the last step, we used that the intervals
$\left[\frac{a}{q}-\frac{x_{q,i}}{q},\frac{a}{q}+\frac{x_{q,i}}{q}\right]$ are
disjoint for distinct $a$, since all $x_{q,i}\leq 2g_{k}(q)^{1/2}\leq 1/2$,
where the latter is ensured by (12). Applying [2, Lemma 5], we obtain the
overlap estimate

$\lambda_{1}(A_{q,i}\cap
A_{r,j})\leq\left(1+O\left(\log(D+2)^{-1}\right)\right)\lambda_{1}(A_{q,i})\lambda_{1}(A_{r,j})\prod_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\
p>A\end{subarray}}\left(1+\frac{1}{p-1}\right),$

where

$\displaystyle D$
$\displaystyle=D(q,r,i,j)=\frac{\max\\{rx_{q,i},qx_{r,j}\\}}{\gcd(q,r)},$
$\displaystyle A$
$\displaystyle=A(D):=\exp\left(\frac{\log(D+100)\log\log\log(D+100)}{8\log\log(D+100)}+1\right)$
(16)

are is as in [2, Equation $(10)$]. We note that $A$ is monotonically
increasing as a function in $D$, and $D(q,r,i,j)\geq\bar{D}(q,r)$ with

$\displaystyle\bar{D}(q,r)$
$\displaystyle:=\frac{\max\\{r\psi_{\varepsilon}^{(k)}(q),q\psi_{\varepsilon}^{(k)}(r)\\}}{\gcd(q,r)},\quad\psi_{\varepsilon}^{(k)}(q):=\left[g_{k}(q)\left(1+\varepsilon\log\left(g_{k}(q)^{-1}\right)\right)\right]^{k-1}.$

Moreover, we introduce the quantity

$P_{\psi}(q,r):=\prod_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\
p>\bar{A}(q,r)\end{subarray}}\left(1+\frac{1}{p-1}\right),$

where $\bar{A}(q,r):=A(\bar{D})$. This gives us the estimate

$\lambda_{1}(A_{q,i}\cap
A_{r,j})\leq\lambda_{1}(A_{q,i})\lambda_{1}(A_{r,j})\left(1+O\left(\log(\bar{D}(q,r)+2)^{-1}\right)\right)P_{\psi}(q,r).$

Furthermore, note that

$\displaystyle\lim_{K\rightarrow\infty}\left(\frac{1}{K^{2}}\sum_{i,j=0}^{K-1}\lambda_{1}(A_{q,i})\lambda_{1}(A_{r,j})\right)^{k-1}$
$\displaystyle=\int_{[0,1]^{k-1}}\gamma_{q,\varepsilon}^{(k)}(x)\,\mathrm{d}x\int_{[0,1]^{k-1}}\gamma_{r,\varepsilon}^{(k)}(x)\,\mathrm{d}x$
$\displaystyle\leq(1+\varepsilon)^{2(k-1)}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x\int_{[0,1]^{k-1}}\gamma_{r}^{(k)}(x)\,\mathrm{d}x.$

Hence we get

$\displaystyle\sum_{q,r\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x$
$\displaystyle\leq$ $\displaystyle\sum_{q,r\leq
Q}\left(\lim_{K\rightarrow\infty}\frac{1}{K^{2}}\sum_{i,j=0}^{K-1}\lambda_{1}(A_{q,i})\lambda_{1}(A_{r,j})\left(1+O\left(\log(\bar{D}(q,r)+2)^{-1}\right)\right)\prod_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\
p>\bar{A}(q,r)\end{subarray}}\left(1+\frac{1}{p-1}\right)\right)^{k-1}$
$\displaystyle\leq$ $\displaystyle(1+\varepsilon)^{2(k-1)}\sum_{q,r\leq
Q}\left(\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x\right)\left(\int_{[0,1]^{k-1}}\gamma_{r}^{(k)}(x)\,\mathrm{d}x\right)\left[\left(1+O\left(\log(\bar{D}(q,r)+2)^{-1}\right)\right)P_{\psi}(q,r)\right]^{k-1}.$

Now let

$\chi(q):=\frac{q}{\varphi(q)}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x=\left[g_{k}(q)\left(1+\frac{1}{2}\log\left(g_{k}(q)^{-1}\right)\right)\right]^{k-1}\leq\frac{1}{2},$

where the last inequality is true again by (12). We observe that
$\chi(q)\asymp_{\varepsilon}\psi_{\varepsilon}^{(k)}(q)$ and thus Proposition
2.3 (stated below) gives us

$\begin{split}&\sum_{q,r\leq
Q}\int_{0}^{1}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x\\\
&\leq(1+\varepsilon)^{2(k-1)}\sum_{q,r\leq
Q}\chi(q)\frac{\varphi(q)}{q}\chi(r)\frac{\varphi(r)}{r}\left[\left(1+O\left(\log(D_{\chi}(q,r)+2)^{-1}\right)\right)P_{\chi}(q,r)\right]^{k-1},\end{split}$

where

$P_{\chi}(q,r)=\prod\limits_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\
p>A_{\chi}(q,r)\end{subarray}}\left(1+\frac{1}{p-1}\right)$

and $A_{\chi}(q,r)=A(D_{\chi}(q,r))$ with
$D_{\chi}(q,r)=\max\left\\{\frac{r\chi(q)}{(q,r)},\frac{q\chi(r)}{(q,r)}\right\\}$.
Writing

$X(Q):=\sum_{q\leq Q}\chi(q)\frac{\varphi(q)}{q},$

we can now follow, up to insignificant changes, the proof of [2, Theorem 2]
with the same decomposition into sets $\mathcal{E}_{1}-\mathcal{E}_{5}$ for
some $C>11$ to establish

$\sum_{q,r\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x\leq(1+\varepsilon)^{2(k-1)}X(Q)^{2}\left(1+O_{\varepsilon}(\log(X(Q))^{-1})\right).$

Since

$\displaystyle X(Q)$ $\displaystyle=\sum\limits_{q\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x$
$\displaystyle=\sum\limits_{q\leq
Q}\left(\frac{\varphi(q)}{q}\right)^{k-1}\left(\int_{0}^{1}f_{q}^{(k)}(y)\,\mathrm{d}y\right)^{k-1}$
$\displaystyle\gg\sum\limits_{q\leq
Q}\left(\frac{\varphi(q)}{q}\right)^{k-1}g_{k}(q)^{k-1}\left(\log
g_{k}(q)\right)^{k-1}$ $\displaystyle\gg\sum\limits_{q\leq
Q}\left(\frac{\varphi(q)}{q}\right)^{k}\psi(q)\left(\log\frac{q}{\varphi(q)\psi(q)}\right)^{k-1},$

we see that by assumption, $X(Q)$ tends to infinity as $Q\to\infty$.
Therefore, we find $Q_{\varepsilon}\in\operatorname{\mathbb{N}}$ large enough
such that for all $Q\geq Q_{\varepsilon}$, we have

$\sum_{q,r\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\gamma_{r}^{(k)}(x)\,\mathrm{d}x\leq(1+2\varepsilon)^{2(k-1)}\left(\sum\limits_{q\leq
Q}\int_{[0,1]^{k-1}}\gamma_{q}^{(k)}(x)\,\mathrm{d}x\right)^{2}.$

Thus, the statement follows by choosing $\varepsilon$ small enough such that

$(1+2\varepsilon)^{2(k-1)}\leq(1+\delta).$

∎

######  Proposition 2.3.

Let $\psi,\chi:\operatorname{\mathbb{N}}\to[0,1/2]$ be two functions such that
there exist constants $C_{1},C_{2}>0$ with

$C_{1}\chi(q)\leq\psi(q)\leq C_{2}\chi(q),\quad
q\in\operatorname{\mathbb{N}}.$

Further, denote by
$D_{\psi}(q,r)=\max\left\\{\frac{r\psi(q)}{(q,r)},\frac{q\psi(r)}{(q,r)}\right\\},D_{\chi}(q,r)=\max\left\\{\frac{r\chi(q)}{(q,r)},\frac{q\chi(r)}{(q,r)}\right\\}$
and $A_{\psi}=A(D_{\psi}),A_{\chi}=A(D_{\chi})$ with $A$ as in (16). Then we
have

  * $(i)$ $\log(D_{\psi}+2)=\Theta_{C_{1},C_{2}}\left(\log(D_{\chi}+2)\right)$,

  * $(ii)$ $\prod\limits_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\ p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p}\right)=\prod\limits_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\ p>A_{\chi}\end{subarray}}\left(1+\frac{1}{p}\right)\left(1+O_{C_{1},C_{2}}\left(\frac{1}{\log(D_{\psi}+2)}\right)\right),$

  * $(iii)$ $\prod\limits_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\ p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p-1}\right)=\prod\limits_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\ p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p}\right)\left(1+O\left(\frac{1}{\log(D_{\psi}+2)^{100}}\right)\right).$

###### Proof.

The claim in $(i)$ is obvious, so we start with the proof of $(ii)$. Since
$\psi$ and $\chi$ are (up to constants) equivalent, it suffices to show that

$\prod_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\ A_{C_{1}\chi}\leq p\leq
A_{C_{2}\chi}\end{subarray}}\left(1+\frac{1}{p}\right)=1+O_{C_{1},C_{2}}\left(\frac{1}{\log(D_{\chi}+2)}\right).$

Employing $(i)$, we get

$\prod_{\begin{subarray}{c}p|\frac{qr}{(q,r)^{2}}\\\ A_{C_{1}\chi}\leq p\leq
A_{C_{2}\chi}\end{subarray}}\left(1+\frac{1}{p}\right)\leq\exp\left(\sum_{A_{C_{1}\chi}\leq
p\leq A_{C_{2}\chi}}\frac{1}{p}\right)\leq\exp\left(\log\left(\frac{\log
A_{C_{2}\chi}}{\log
A_{C_{1}\chi}}+O\left(\frac{1}{\log(A_{\chi})^{2}}\right)\right)\right)$

where we have used Mertens’ Theorem in the form of (for the error term see,
e.g., [31])

$\sum_{p\leq x}\frac{1}{p}=\log\log
x+M+O\left(\exp\big{(}-a\sqrt{\log(x)}\big{)}\right),$

with $M$ being an absolute constant and $a>0$. For any fixed constant $c>0$,
it follows from the definition of $A_{\chi}$ that

$\log A_{c\chi}=\log
A_{\chi}\left(1+O\left(\frac{1}{\log(D_{\chi}+2)}\right)\right),\qquad\frac{1}{\log
A_{\chi}^{2}}=O\left(\frac{1}{\log(D_{\chi}+2)}\right)$

and hence,

$\exp\left(\log\left(\frac{\log A_{C_{2}\chi}}{\log
A_{C_{1}\chi}}+O\left(\frac{1}{\log(A_{\chi})^{2}}\right)\right)\right)=\exp\left(\log\left(1+O\left(\frac{1}{\log(D_{\chi}+2)}\right)\right)\right)=1+O\left(\frac{1}{\log(D_{\chi}+2)}\right).$

This shows the claim in $(ii)$. For $(iii)$, we note that

$\prod_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\
p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p-1}\right)=\prod_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\
p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p}\right)\prod_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\
p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p^{2}-1}\right).$

Since

$\displaystyle\prod_{\begin{subarray}{c}p\mid\frac{qr}{(q,r)^{2}}\\\
p>A_{\psi}\end{subarray}}\left(1+\frac{1}{p^{2}-1}\right)$
$\displaystyle=\exp\left(O\left(\sum_{p>A_{\psi}}\frac{1}{p^{2}}\right)\right)$
$\displaystyle=\exp\left(O\left(\frac{1}{A_{\psi}}\right)\right)$
$\displaystyle=1+O\left(\frac{1}{\log(D_{\psi}+2)^{100}}\right),$

with an absolute implied constant, the statement follows. ∎

###  2.3 Proof of Corollaries 3,4 and 5

###### Proof of Corollary 3.

Let $\varepsilon>0$ fixed and let
$\psi(q)=\psi_{A,\varepsilon}(q):=\mathds{1}_{[q\in A]}\frac{\varepsilon}{q}$.
Assume first that $\frac{\\#\\{n\leq N:n\in A\\}}{N}\gg\frac{\log\log N}{(\log
N)^{2}}$. By Corollary 2, it suffices to show that

$\sum_{q\in\operatorname{\mathbb{N}}}\mathds{1}_{[q\in A]}\frac{1}{q(\log\log
q)^{2}}\log(q(\log\log
q)^{2})\gg\sum_{q\in\operatorname{\mathbb{N}}}\mathds{1}_{[q\in
A]}\frac{\log(q)}{q(\log\log q)^{2}}=\infty.$

By summation by parts and the mean value theorem this holds true since

$\sum_{q\in N}\frac{1}{q\log q\log\log q}=\infty.$

If $\frac{\\#\\{n\leq N:n\in A\\}}{N}\gg\frac{\log\log N}{(\log N)^{2}}$, we
use the fact that the convergence part of Conjecture 1 holds true (for a proof
see [7]), and thus we only need to show that

$\sum_{q\in\operatorname{\mathbb{N}}}\mathds{1}_{[q\in
A]}\frac{1}{q}\log(q)<\infty,$

which follows again after applying summation by parts. ∎

###### Proof of Corollaries 4 and 5.

It was already shown by Hussain and Simmons in [24] that even for non-
monotonic $\psi$

$\dim_{H}(S_{k}^{\times})\leq(k-1)+d(\psi),$ (17)

and since clearly $D_{k}^{\times}(\psi)\subseteq S_{k}^{\times}(\psi)$, the
proof of Corollary 4 is reduced to showing
$\dim_{H}(D_{k}^{\times})\geq(k-1)+d(\psi)$. To prove this, we make use of the
Hausdorff version of the Duffin-Schaeffer conjecture due to Beresnevich-Velani
[8], who proved that

$\dim_{H}(D_{1}^{\times}(\psi))=\min\\{d(\psi),1\\}.$ (18)

In order to compute the Hausdorff-dimension of $D_{k}^{\times}(\psi)$ for
arbitrary $k\geq 1$, we need the following claim: For any $\varepsilon>0$,
there exists a constant $c=c_{\varepsilon}>0$ such that for all
$q\in\operatorname{\mathbb{N}}$

$||q\alpha_{i}||^{\prime}\leq c_{\varepsilon}q^{\varepsilon}.$ (19)

This follows from

$\begin{split}\sum_{\begin{subarray}{c}n\leq x\\\
(n,q)=1\end{subarray}}1&=\sum_{n\leq x}\sum_{d\mid(n,q)}\mu(d)=\sum_{d\mid
q}\mu(d)\sum_{\begin{subarray}{c}n\leq x\\\ d\mid
n\end{subarray}}1=\sum_{d\mid q}\mu(d)\left(\frac{x}{d}+O(1)\right)\\\
&=x\sum_{d\mid
q}\frac{\mu(d)}{d}+O(\tau(q))=x\frac{\varphi(q)}{q}+O(\tau(q)),\end{split}$

which implies that for $(y-x)\gg_{\varepsilon}q^{\varepsilon}$, we have

$\sum_{\begin{subarray}{c}x\leq n\leq y\\\ (n,q)=1\end{subarray}}1>0,$

thus (19) follows. Hence we have

$\displaystyle D_{k}^{\times}(\psi)$
$\displaystyle=\left\\{\alpha\in[0,1]^{k}:\prod_{i=1}^{k}||q\alpha_{i}||^{\prime}\leq\psi(q)\right\\}$
$\displaystyle\supseteq\left\\{\alpha\in[0,1]^{k}:||q\alpha_{1}||^{\prime}\leq\frac{\psi(q)}{c^{k-1}q^{(k-1)\varepsilon}}\right\\}$
$\displaystyle=[0,1]^{k-1}\times D_{1}^{\times}(\psi/f_{\varepsilon}),$

where $f_{\varepsilon}(q):=c^{k-1}q^{(k-1)\varepsilon}$. By elementary
properties of the Hausdorff dimension, the definition of $d(\psi)$ and (18),
we get that for all $\varepsilon>0$,

$\dim_{H}(D_{k}^{\times}(\psi))\geq(k-1)+\dim_{H}\left(D_{1}^{\times}\left(\frac{\psi}{f_{\varepsilon}}\right)\right)\geq(k-1)+\dim_{H}(D_{1}^{\times}\left(\psi\right))-\varepsilon,$

where we used (18) and the fact that $\psi$ is absolutely bounded. Therefore,
it follows that $\dim_{H}(D_{k}^{\times}(\psi))\geq(k-1)+d(\psi)$, which
proves Corollary 4. Corollary 5 can now be deduced from
$q/\varphi(q)\ll_{\varepsilon}q^{\varepsilon}$ and an analogous argumentation.
∎

### Acknowledgements

The authors would like to thank Christoph Aistleitner, Bence Borda, Victor
Beresnevich, Niclas Technau, and Sanju Velani for various valuable discussions
and comments on an earlier version of this manuscript. LF was supported by the
Austrian Science Fund (FWF) Project P 35322. MH was supported by the EPSRC
grant EP/X030784/1. A part of this work was supported by the Swedish Research
Council under grant no. 2016-06596 while MH was in residence at Institut
Mittag-Leffler in Djursholm, Sweden in 2024.

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Lorenz Frühwirth

Graz University of Technology

Steyrergasse 30, 8010 Graz, Austria

Email<EMAIL_ADDRESS>
Manuel Hauke

University of York

Department of Mathematics

YO10 5DD York, United Kingdom

Email<EMAIL_ADDRESS>