# Metric bootstraps for limsup sets

Felipe A. Ramírez
Wesleyan University

###### Abstract

In metric Diophantine approximation, one frequently encounters the problem of
showing that a limsup set has positive or full measure. Often it is a set of
points in $m$-dimensional Euclidean space, or a set of $n$-by-$m$ systems of
linear forms, satisfying some approximation condition infinitely often. The
main results of this paper are bootstraps: if one can establish positive
measure for such a limsup set in $m$-dimensional Euclidean space, then one can
establish positive or full measure for an associated limsup set in the setting
of $n$-by-$m$ systems of linear forms. Consequently, a class of
$m$-dimensional results in Diophantine approximation can be bootstrapped to
corresponding $n$-by-$m$-dimensional results. This leads to short proofs of
existing, new, and hypothetical theorems for limsup sets that arise in the
theory of systems of linear forms. We present several of these.

###### Contents

  1. 1 Introduction and results
    1. 1.1 Setting
    2. 1.2 Metric dichotomies
    3. 1.3 Main results
  2. 2 Applications
    1. 2.1 Khintchine 1924, 1926 $\implies$ Khintchine–Groshev 1938
    2. 2.2 Szüsz 1958 and Schmidt 1964 $\implies$ Sprindzuk 1979
    3. 2.3 Gallagher 1965 $\implies$ Beresnevich–Velani 2010
    4. 2.4 Yu 2019 $\implies$ multivariate version of Allen–Ramírez 2023 ($m>2$)
    5. 2.5 Nesharim–Rühr–Shi 2020 ($n=1$) $\implies$ Nesharim–Rühr–Shi 2020
    6. 2.6 Inhomogeneous Duffin–Schaeffer $\implies$ same for systems of linear forms
    7. 2.7 Chow–Technau 2023 $\implies$ a version for dual approximation
  3. 3 Examples
  4. 4 Measure-theoretic and geometric tools
  5. 5 Properties of approximation sets
  6. 6 Reductions
  7. 7 Proof of þ1.1(a)
  8. 8 Proof of þ1.1(b)
  9. 9 Proof of þ1.2(a)
  10. 10 Proof of þ1.2(b)

## Notation and conventions

  * •

$\mathbb{N}$ denotes the positive integers.

  * •

$\mathbb{Z}_{+}$ and $\mathbb{R}_{+}$ denote the nonnegative integers and the
nonnnegative reals.

  * •

$\operatorname{m}$ denotes Lebesgue measure, possibly with a subscript when
the dimension is not clear from the context.

  * •

$\lvert\cdot\rvert$ denotes the maximum norm, with respect to which all
distances are to be understood.

##  1 Introduction and results

###  1.1 Setting

Consider the collection of $n$-by-$m$ systems of linear forms that send
infinitely many integer vectors into some prescribed union of target balls in
$\mathbb{R}^{m}$, that is, the set of matrices $\mathbf{X}\in M_{n\times
m}(\mathbb{R})$ for which there are infinitely many
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ such that

$\mathbf{q}\mathbf{X}\in B(\mathbf{0},\psi(\mathbf{q}))+P(\mathbf{q}),$ (1)

where $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ is a given function,
$B(\mathbf{0},\psi(\mathbf{q}))\subseteq\mathbb{R}^{m}$ is the ball of radius
$\psi(\mathbf{q})\geqslant 0$ centered at the origin
$\mathbf{0}\in\mathbb{R}^{m}$, and $P(\mathbf{q})\subseteq\mathbb{R}^{m}$ is a
discrete set determining which translates of the ball are acceptable targets.
That set, which is denoted here by

$W_{n,m}^{P}(\psi)=\left\\{\mathbf{X}\in M_{n\times m}(\mathbb{R}):\textrm{
(\ref{eq:one}) holds for infinitely many
}\mathbf{q}\in\mathbb{Z}_{+}^{n}\right\\},$

is at the heart of _asymptotic metric Diophantine approximation_. Many
problems in the field concern sets of the form $W_{n,m}^{P}(\psi)$ for various
choices of $P$. For classical _homogeneous approximation_ one has
$P(\mathbf{q})$ a subset of $\mathbb{Z}^{m}$, while for _inhomogeneous
approximation_ $P(\mathbf{q})$ is a subset of $\mathbf{y}+\mathbb{Z}^{m}$
where $\mathbf{y}\in\mathbb{R}^{m}$ is a fixed inhomogeneous parameter. Often,
the definition of $P$ involves coprimality or congruence conditions on the
entries of its members.

The main question addressed here is _what is the Lebesgue measure of
$W_{n,m}^{P}(\psi)$?_ It should be noted that $W_{n,m}^{P}(\psi)$ is the
limsup set corresponding to the sets

$A(\mathbf{q})=\left\\{\mathbf{X}\in M_{n\times m}(\mathbb{R}):\textrm{
(\ref{eq:one}) holds}\right\\}.$

As such, there are cases where the question is answered immediately by the
Borel–Cantelli lemma. Namely, if for every ball $B\subseteq M_{n\times
m}(\mathbb{R})$, the series
$\sum_{\mathbf{q}}\operatorname{m}(A(\mathbf{q})\cap B)$ converges, then by
the Borel–Cantelli lemma, $W_{n,m}^{P}(\psi)$ has zero measure.

The focus, therefore, is on situations where the series diverges, and the goal
is always to show that $W_{n,m}^{P}(\psi)$ has positive or full measure. Our
main result is that if that goal can be achieved in the $1$-by-$m$ setting,
then it can also be achieved in the $n$-by-$m$ setting for all $n\geqslant 1$.

###  1.2 Metric dichotomies

The prototypical result for sets of the form $W_{n,m}^{P}(\psi)$ is a zero-
full metric dichotomy: an assertion that the measure is either zero or full,
together with a criterion to determine which of the two it is. It is usually
determined by the convergence or divergence of a series associated to $\psi$
and $P$—a series which is understood as a proxy for the measure sum mentioned
above.

The seminal example is Khintchine’s theorem (1924, 1926, [19, 20]). It says
that if $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ is nonincreasing, and
$P(q)=\mathbb{Z}^{m}$ for all $q\in\mathbb{Z}_{+}$, then
$\operatorname{m}(W_{1,m}^{P}(\psi))$ is zero or full depending on the
convergence or divergence of the series $\sum_{q}\psi(q)^{m}$. The $n$-by-$m$
version of Khintchine’s theorem is known as the Khintchine–Groshev Theorem
(1938, [16]). It says, for functions
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ only depending on
$\lvert\mathbf{q}\rvert$ and satisfying a monotonicity condition, and with
$P(\mathbf{q})=\mathbb{Z}^{m}$ as before, that
$\operatorname{m}(W_{n,m}^{P}(\psi))$ is zero or full depending on the
convergence or divergence of the series $\sum_{q}q^{n-1}\psi(q)^{m}$.

An inhomogeneous version of the Khintchine–Groshev theorem can be found in the
work of Sprindzuk (1979, [28]). The statement is exactly that of the
(homogeneous) Khintchine–Groshev theorem, except that
$P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$ for every
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$, where $\mathbf{y}\in\mathbb{R}^{m}$ is a
fixed inhomogeneous parameter. The $1$-by-$1$ case had been proved in 1958 by
Szüsz [29] and the $1$-by-$m$ case had been proved in 1964 by Schmidt [27];
these constitute the inhomogeneous Khintchine theorem.

Note that when $P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$, all integer
translates of the ball $B(\mathbf{y},\psi(\mathbf{q}))$ are admissible targets
for $\mathbf{q}\mathbf{X}$. Other classical settings impose restrictions
involving coprimality, such as in the Duffin–Schaeffer conjecture (1941,
[13]), now a theorem due to Koukoulopoulos–Maynard (2020, [21]). It says that
if $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ and
$P(q)=\\{p\in\mathbb{Z}:\gcd(p,q)=1\\}$ for all $q\in\mathbb{Z}_{+}$, then
$\operatorname{m}(W_{1,1}^{P}(\psi))$ is zero or full depending on the
convergence or divergence of the series $\sum_{q}\varphi(q)\psi(q)/q$, where
$\varphi$ is Euler’s totient function. The $m$-dimensional ($m>1$) version of
this was proved by Pollington and Vaughan in 1990 [23] and the corresponding
$n$-by-$m$ version was proved recently by the author [26], after having been
conjectured by Beresnevich, Bernik, Dodson, and Velani in 2009 [4].

There are many other such dichotomies. For example, the monotonicity
assumption appearing in the Khintchine–Groshev theorem has been the subject of
much research, and versions where that assumption is relaxed have appeared in
[7, 28]. In 2010, Beresnevich–Velani proved a multivariate—meaning
$\psi(\mathbf{q})$ does not have to depend only on
$\lvert\mathbf{q}\rvert$—version with no monotonicity assumption whenever
$m>1$ [7]. The $n=1$ cases of this had been established by Gallagher in 1965
[15]. In 2023 Allen and the author proved a version of the (univariate)
inhomogeneous Khintchine–Groshev theorem without monotonicity assumptions
whenever $nm>2$ [2]. The $n=1$ cases of this had been proved by Yu in 2019
[30].

###  1.3 Main results

The purpose of this article is to show that for metric dichotomies, the
$1$-by-$m$ theory underpins the $n$-by-$m$ theory: the $n$-by-$m$ theorems in
the previous section follow from their $1$-by-$m$ cases; several $n$-by-$m$
theorems that do not already appear in the literature follow from $1$-by-$m$
theorems that do; and certain open conjectures in the $1$-by-$m$ setting give
rise to their $n$-by-$m$ counterparts.

All of this follows from applying our main results,
þLABEL:thm:bootstrap,thm:bootstrap11. Briefly, these state that for a given
function $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$, there is an associated
family of functions $\Psi_{Q}:\mathbb{N}\to[0,\infty]$ ($Q\geqslant 1$) such
that if $W_{1,m}^{P}(\Psi_{1})$ has positive measure then so does
$W_{n,m}^{P}(\psi)$, and if $W_{1,m}^{P}(\Psi_{Q})$ is bounded away from $0$
for $Q\geqslant 1$ then $W_{n,m}^{P}(\psi)$ has full measure.

þLABEL:thm:bootstrap,thm:bootstrap11 apply when $P$ conforms to the following
definition.

######  Definition 1.1.

þ For each $d\geqslant 1$, let $P(d)\subseteq(\mathbb{R}/d\mathbb{Z})^{m}$ be
a finite set, and define for each $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ the set
$P(\mathbf{q})$ to be the lift of $P(\gcd(\mathbf{q}))$ to $\mathbb{R}^{m}$.
We say $P$ is _relatively uniformly discrete_ if there exist positive real
numbers $b,c>0$ such that for every $d\geqslant 1$ there is a subset
$P^{\prime}(d)\subseteq P(d)$ with

$\th\\#P^{\prime}(d)\geqslant c\\#P(d)\qquad\textrm{ and }\qquad
b\leqslant\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\rvert\qquad\textrm{ for all
distinct }\mathbf{p}_{1},\mathbf{p}_{2}\in P^{\prime}(\mathbf{q}),$ (2)

where $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ is any vector with
$\gcd(\mathbf{q})=d$ and $P^{\prime}(\mathbf{q})$ denotes the lift of
$P^{\prime}(d)$ to $\mathbb{R}^{m}$. If, moreover, there exists $a>0$ such
that for all $d\geqslant 1$,

$\frac{ad}{(\\#P(d))^{1/m}}\leqslant\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\rvert\qquad\textrm{
for all distinct }\mathbf{p}_{1},\mathbf{p}_{2}\in
P^{\prime}(\mathbf{q})\quad(\gcd(\mathbf{q})=d),$ (3)

then we will say $P$ is _relatively well-spread_. If these conditions can be
accomplished with $c=1$, then we drop ‘relatively’ from the terminology.

######  Remark.

Well-spreadedness means that it is possible to occupy a fixed positive
proportion of $(\mathbb{R}/d\mathbb{Z})^{m}$ by placing disjoint balls around
the points in $P(d)$. The condition of relative well-spreadedness is satisfied
in all the metric dichotomies discussed in §1.2.

Notice that if $P(\mathbf{q})$ is the lift to $\mathbb{R}^{m}$ of a finite
subset of $(\mathbb{R}/\gcd(\mathbf{q})\mathbb{Z})^{m}$, then (1) is a
$\mathbb{Z}^{mn}$-periodic condition on $\mathbf{X}$. If this is true of each
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$, then $W_{n,m}^{P}(\psi)$ is a
$\mathbb{Z}^{mn}$-periodic set. In this case, it is convenient to redefine it
as

$W_{n,m}^{P}(\psi)=\left\\{\mathbf{X}\in\mathbb{T}^{nm}\subseteq M_{n\times
m}(\mathbb{R}):\textrm{ (\ref{eq:one}) holds for infinitely many
}\mathbf{q}\in\mathbb{Z}_{+}^{n}\right\\},$

where $\mathbb{T}^{nm}$ is identified with the elements of $M_{n\times
m}(\mathbb{R})$ whose entries lie in $[0,1]$. The main theorems are stated
with respect to this view of $W_{n,m}^{P}(\psi)$.

The last definition to be introduced before stating the results is that of the
auxiliary functions $\Psi_{Q}$. Given $m,n\in\mathbb{N}$,
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$, and $Q\geqslant 1$, define
$\Psi_{Q}:\mathbb{N}\to[0,\infty]$ by

$\Psi_{Q}(d)=\left\lparen\sum_{\begin{subarray}{c}\mathbf{q}\in\mathbb{Z}_{+}^{n}\\\
\gcd(\mathbf{q})=d\\\
\left\lvert\mathbf{q}/\gcd(\mathbf{q})\right\rvert\geqslant
Q\end{subarray}}\psi(\mathbf{q})^{m}\right\rparen^{1/m}.$ (4)

Let $\Psi:=\Psi_{1}$.

######  Theorem 1.1.

þ Fix $m,n\in\mathbb{N}$.

  1. a)

If $P$ is relatively well-spread, then for functions
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$,

$\operatorname{m}(W_{1,m}^{P}(\Psi))>0\qquad\implies\qquad\operatorname{m}(W_{n,m}^{P}(\psi))>0.$

  2. b)

If $P$ is relatively uniformly discrete, then for bounded functions
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$,

$\operatorname{m}(W_{1,m}^{P}(\Psi))>0\qquad\implies\qquad\operatorname{m}(W_{n,m}^{P}(\psi))>0.$

######  Remark.

It is possible that $\Psi(d)=\infty$ for some $d\geqslant 1$. We will see that
in such cases one automatically has $\operatorname{m}(W_{n,m}^{P}(\psi))=1$,
so there is nothing to prove (þ2.1). In the meantime, let it be understood
that a ball of infinite radius contains all of $\mathbb{R}^{m}$. Notice, also,
that if $n=1$, then $\psi(q)=\Psi(q)$ for all $q\geqslant 1$, and the theorem
is trivial.

One might suspect that if $W_{1,m}^{P}(\Psi)$ has full measure rather than
just positive measure, then it is possible to conclude that
$W_{n,m}^{P}(\psi)$ also has full measure. However, þ 3.3 in §3 shows that
this is not true in general.

The next result achieves full measure for $W_{n,m}^{P}(\psi)$ from positive
measure of $W_{1,m}^{P}(\Psi_{Q})$ for all $Q\geqslant 1$.

######  Theorem 1.2.

þ Fix $m,n\in\mathbb{N}$ and assume $\lim_{d\to\infty}\\#P(d)=\infty$.

  1. a)

If $P$ is relatively well-spread, then for functions
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$,

$\inf_{Q}\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))>0\qquad\implies\qquad\operatorname{m}(W_{n,m}^{P}(\psi))=1.$

  2. b)

If $P$ is relatively uniformly discrete, then for bounded functions
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$,

$\inf_{Q}\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))>0\qquad\implies\qquad\operatorname{m}(W_{n,m}^{P}(\psi))=1.$

The theorem is vacuously true when $n=1$. This is because for all integers
$q\geqslant 1$, one has $\lvert q/\gcd(q)\rvert=1$, and so if $n=1$ then
$\Psi_{Q}$ is defined by an empty sum as soon as $Q>1$. Consequently, if
$Q>1$, then $W_{1,m}^{P}(\Psi_{Q})$ has zero measure. To put it another way,
þ1.2 is only meaningful if one increases dimension from $n=1$ to $n>1$. In
fact, the condition on $\Psi_{Q}$ ensures that the set $W_{n,m}^{P}(\psi)$ is
genuinely higher-dimensional. Without it, one can reproduce a
$1$-by-$m$-dimensional set in $\mathbb{T}^{nm}$ by choosing $\psi$ to be
supported on a line in $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ (see þ3.2).

The applications in §2 all begin with some $1$-by-$m$-dimensional knowledge—a
_simultaneous theorem_. For the applications that require þ1.2, the
simultaneous theorem already takes care of the cases where the condition on
$\Psi_{Q}$ does not hold. In other words, the $1$-by-$m$-dimensional artifice
described in the previous paragraph represents a problem that has already been
solved in the $1$-by-$m$ setting.

##  2 Applications

The general outline for the applications of
þLABEL:thm:bootstrap,thm:bootstrap11 is this: Given a function $\psi$
satisfying a divergence condition in the $n$-by-$m$-setting, define the
associated functions $\Psi_{Q}$. If there exists $d,Q\geqslant 1$ for which
$\Psi_{Q}(d)=\infty$, then the following proposition gives the desired result.

######  Proposition 2.1.

þ Fix $m,n\in\mathbb{N}$. For each $d\geqslant 1$, let
$P(d)\subseteq(\mathbb{R}/d\mathbb{Z})^{m}$ be a finite set, and for each
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q})=d$, let
$P(\mathbf{q})$ be the lift to $\mathbb{R}^{m}$ of $P(d)$. Let
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$. If there exist $d,Q\geqslant 1$
for which $\Psi_{Q}(d)=\infty$, then $\operatorname{m}(W_{n,m}^{P}(\psi))=1$.

######  Remark.

The proof is in §6. The result can also be deduced from [28, Chapter 1,
Theorem 14].

If, on the other hand, $\Psi_{Q}(d)$ is always finite, then we show that
$\Psi_{1}$ (or $\Psi_{Q}$) satisfies the divergence condition from a
_simultaneous theorem_ in the $1$-by-$m$-setting. The simultaneous theorem
then gives $\operatorname{m}(W_{1,m}^{P}(\Psi_{1}))>0$ (or
$\inf_{Q}\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))>0$), and þ1.1 (or þ1.2)
bootstraps this to $\operatorname{m}(W_{n,m}^{P}(\psi))>0$ (or
$\operatorname{m}(W_{n,m}^{P}(\psi))=1$).

Some of these applications are short proofs of existing results, some are new
results, and some are hypothetical in the sense that they rely on hypothetical
$1$-by-$m$-dimensional theorems. The new theorems are numbered.

###  2.1 Khintchine 1924, 1926 $\implies$ Khintchine–Groshev 1938

Khintchine’s theorem was proved in 1924 in dimension $1$ and in 1926 for
higher dimensions [20, 19]. In 1938, Groshev proved what is now known as the
Khintchine–Groshev theorem [16]—the analogue of Khintchine’s theorem for
$n$-by-$m$ systems of linear forms. We now use þ1.1 and a zero-one law due to
Beresnevich and Velani [6, Theorem 1] to show that the Khintchine–Groshev
theorem follows from Khintchine’s theorem.

######  Theorem (Khintchine–Groshev theorem).

Fix $m,n\in\mathbb{N}$. With $P(\mathbf{q})=\mathbb{Z}^{m}$ for every
$d\geqslant 1$, and $\psi(\mathbf{q})$ depending only on
$\lvert\mathbf{q}\rvert$,

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}<\infty\\\ 1&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}=\infty\textrm{ and
}q^{n-1}\psi(q)^{m}\textrm{ is nonincreasing.}\\\ \end{cases}$

###### Proof modulo the $n=1$ case.

The convergence case of this theorem is a simple application of the
Borel–Cantelli lemma, so only the divergence case needs to be addressed.

Let $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ be a function such that
$q^{n-1}\psi(q)^{m}$ is nonincreasing and

$\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}=\infty.$ (5)

Note that in particular $\psi$ is a nonincreasing function. For each
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$, let $P(\mathbf{q})=\mathbb{Z}^{m}$. It is
the lift to $\mathbb{R}^{m}$ of

$P(\gcd(\mathbf{q}))=\left\lparen\mathbb{Z}/\gcd(\mathbf{q})\mathbb{Z}\right\rparen^{m}\subseteq\left\lparen\mathbb{R}/\gcd(\mathbf{q})\mathbb{Z}\right\rparen^{m}.$

Clearly, $P$ is well-spread in the sense of Definition 1.1. Extend $\psi$ to a
function $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ by setting
$\psi(\mathbf{q}):=\psi(\lvert\mathbf{q}\rvert)$. For $d\geqslant 1$ define
$\Psi:=\Psi_{1}$ by (4). If there exists $d\geqslant 1$ for which
$\Psi(d)=\infty$, then þ2.1 gives that
$\operatorname{m}(W_{n,m}^{P}(\psi))=1$. Otherwise, (5) implies that the
series $\sum_{d}\Psi(d)^{m}$ diverges. To see this, compute

$\sum_{d\geqslant
1}\Psi(d)^{m}=\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}=\sum_{q=1}^{\infty}\sum_{\lvert\mathbf{q}\rvert=q}\psi(\mathbf{q})^{m}\asymp\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}.$

And the fact that $\psi$ is nonincreasing implies that $\Psi(d)$ is
nonincreasing. To see this, observe that

$\Psi(d)=\left\lparen\sum_{\gcd(\mathbf{q})=1}\psi(d\mathbf{q})^{m}\right\rparen^{1/m}$

and note that for every $\mathbf{q}\in\mathbb{Z}_{+}^{n}$, one has that
$\psi(d\mathbf{q}):=\psi(d\lvert\mathbf{q}\rvert)$ is a nonincreasing function
of $d\geqslant 1$. Now Khintchine’s theorem, gives
$\operatorname{m}(W_{1,m}^{P}(\Psi))=1$, and þ1.1 gives
$\operatorname{m}(W_{n,m}^{P}(\psi))>0$. Full measure follows from the zero-
one law [6, Theorem 1]. ∎

The next application shows how þ1.2 can be used in cases where there is no
pre-existing zero-one law.

###  2.2 Szüsz 1958 and Schmidt 1964 $\implies$ Sprindzuk 1979

The inhomogeneous version of Khintchine’s theorem was proved in 1958 by Szüsz
in dimension $m=1$, and in 1964 by Schmidt in higher dimensions [29, 27].
Combining these with þ1.2 gives the following inhomogeneous version of
Khintchine–Groshev, which can be found in Sprindzuk [28, 1979].

######  Theorem (Inhomogeneous Khintchine–Groshev theorem).

Fix $m,n\in\mathbb{N}$. With $P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$ for
every $d\geqslant 1$, where $\mathbf{y}\in\mathbb{R}^{m}$ a constant, and
$\psi(\mathbf{q})$ depending only on $\lvert\mathbf{q}\rvert$,

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}<\infty\\\ 1&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}=\infty\textrm{ and
}q^{n-1}\psi(q)^{m}\textrm{ is nonincreasing.}\\\ \end{cases}$

###### Proof modulo the $n=1$ case.

Only the divergence case requires attention, since the convergence case is an
application of the Borel–Cantelli lemma.

Let $\mathbf{y}\in\mathbb{R}^{m}$. Let $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$
be a function such that $q^{n-1}\psi(q)^{m}$ is nonincreasing and such that
(5) holds. Notice that $\psi$ is a nonincreasing function. Let
$P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$, the lift to $\mathbb{R}^{m}$ of

$P(\gcd(\mathbf{q}))=\mathbf{y}+\left\lparen\mathbb{Z}/\gcd(\mathbf{q})\mathbb{Z}\right\rparen^{m}\subseteq\left\lparen\mathbb{R}/\gcd(\mathbf{q})\mathbb{Z}\right\rparen^{m}$

for all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$. As in the proof in the previous
section, $P$ is well-spread in the sense of Definition 1.1. Extend $\psi$ to a
function $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ by setting
$\psi(\mathbf{q}):=\psi(\lvert\mathbf{q}\rvert)$. For $d,Q\geqslant 1$ define
$\Psi_{Q}(d)$ by (4). If there exists $d,Q\geqslant 1$ for which
$\Psi_{Q}(d)=\infty$, then þ2.1 gives that
$\operatorname{m}(W_{n,m}^{P}(\psi))=1$. Otherwise, we have as a consequence
of (5) that the series $\sum_{d}\Psi(d)^{m}$ diverges. Notice now that

$\sum_{d\geqslant
1}\Psi_{Q}(d)^{m}=\sum_{\begin{subarray}{c}\mathbf{q}\in\mathbb{Z}_{+}^{n}\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\psi(\mathbf{q})^{m}\\\
=\sum_{q=1}^{\infty}\sum_{\begin{subarray}{c}\lvert\mathbf{q}\rvert=q\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\psi(\mathbf{q})^{m}.$

Let
$\mathbb{P}_{Q}^{n}=\\{\mathbf{q}\in\mathbb{Z}_{+}^{n}:\gcd(\mathbf{q})=1,\lvert\mathbf{q}\rvert<Q\\}$.
It is a finite set. Let $M$ be large enough that
$\\#\\{\mathbf{q}\in\mathbb{Z}_{+}^{n}:\lvert\mathbf{q}\rvert=q\\}\geqslant
2\\#\mathbb{P}_{Q}^{n}$ for all $q\geqslant M$. Then, continuing the above
calculations,

$\displaystyle\sum_{d\geqslant 1}\Psi_{Q}(d)$
$\displaystyle\geqslant\sum_{q=M}^{\infty}\sum_{\begin{subarray}{c}\lvert\mathbf{q}\rvert=q\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\psi(\mathbf{q})^{m}$
$\displaystyle\geqslant\sum_{q=M}^{\infty}\psi(q)^{m}\left\lparen\\#\\{\mathbf{q}\in\mathbb{Z}_{+}^{n}:\lvert\mathbf{q}\rvert=q\\}-\\#\mathbb{P}_{Q}^{n}\right\rparen$
$\displaystyle\geqslant\frac{1}{2}\sum_{q=M}^{\infty}\psi(q)^{m}\\#\\{\mathbf{q}\in\mathbb{Z}_{+}^{n}:\lvert\mathbf{q}\rvert=q\\}$
$\displaystyle\asymp\sum_{q=M}^{\infty}q^{n-1}\psi(q)^{m}.$

Now, by (5) we have that $\sum_{d}\Psi_{Q}(d)$ diverges. The monotonicity of
$\psi$ implies that

$\Psi_{Q}(d)=\left\lparen\sum_{\mathbf{q}\in\mathbb{P}_{\infty}^{n}\setminus\mathbb{P}_{Q}^{n}}\psi(d\mathbf{q})^{m}\right\rparen^{1/m}$

is nonincreasing, since for every fixed
$\mathbf{q}\in\mathbb{P}_{\infty}^{n}\setminus\mathbb{P}_{Q}^{n}$, the
function $\psi(d\mathbf{q})=\psi(d\lvert\mathbf{q}\rvert)$ is nonincreasing.
The inhomogeneous version of Khintchine’s theorem now gives
$\operatorname{m}(W_{1,m}(\Psi_{Q}))=1$, and by þ1.2,
$\operatorname{m}(W_{n,m}^{P}(\psi))=1$. ∎

###  2.3 Gallagher 1965 $\implies$ Beresnevich–Velani 2010

Famously, Khintchine’s theorem requires a monotonicity assumption in dimension
$1$. But in 1965 Gallagher [15] showed that that assumption is unnecessary in
higher dimensions. Sprindzuk removed the monotonicity assumption from the
Khintchine–Groshev theorem in the cases where $n>2$, and in 2010
Beresnevich–Velani were able to remove the monotonicity assumption from the
Khintchine–Groshev theorem in all cases where $mn>1$ [28, 7]. They also proved
a multivariate analog of the Khintchine–Groshev theorem without monotonicity
assumptions, for $m>1$. þ1.2 can be used to derive this result from its $n=1$
cases, that is, from Gallagher’s 1965 result. In fact, þ1.1 is sufficient for
this purpose, thanks again to the zero-one law [6, Theorem 1].

######  Theorem (Beresnevich–Velani, [7, Theorem 5]).

Fix $m,n\in\mathbb{N}$ with $m>1$. Let $P(\mathbf{q})=\mathbb{Z}^{m}$ for
every $\mathbf{q}$. Then

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}<\infty\\\
1&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}=\infty.\end{cases}$

###### Proof modulo the $n=1$ case.

The convergence case of this theorem is an application of the Borel–Cantelli
lemma, so we only need to discuss the divergence case.

Let $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ and assume that
$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}$ diverges. Define
$\Psi=\Psi_{1}$ by (4). If there exists $d\geqslant 1$ for which
$\Psi(d)=\infty$, then the theorem follows from þ2.1. Otherwise, observe that

$\sum_{d\geqslant
1}\Psi(d)^{m}=\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}$

diverges. Since $m>1$, Gallagher’s extension of Khintchine’s theorem gives
$\operatorname{m}(W_{1,m}^{P}(\Psi))=1$. By þ1.1,
$\operatorname{m}(W_{n,m}^{P}(\psi))>0$. Applying once again Beresnevich and
Velani’s zero-one law for systems of linear forms [6, Theorem 1], we have
$\operatorname{m}(W_{n,m}^{P}(\psi))=1$. ∎

As an immediate corollary, one gets the $m>1$ cases of [7, Theorem 1], the
version where $\psi(\mathbf{q})$ is constant on spheres
$\lvert\mathbf{q}\rvert=q$.

###  2.4 Yu 2019 $\implies$ multivariate version of Allen–Ramírez 2023 ($m>2$)

In 2019, Yu proved that the inhomogeneous Khintchine theorem does not need a
monotonicity assumption in dimension $m>2$ [30, Theorem 1.8]. In 2023, Allen
and the author showed that one can remove the monotonicity from the
inhomogeneous Khintchine–Groshev theorem whenever $nm>2$ [2, Theorem 1]. The
$m>2$ cases of this result can now be proved by applying þ1.2 to Yu’s result.
In fact, one can prove the following multivariate theorem, an inhomogeneous
version of Beresnevich–Velani [7, Theorem 5] for $m>2$.

######  Theorem 2.2.

þ Fix $m,n\in\mathbb{N}$ with $m>2$. Let $\mathbf{y}\in\mathbb{R}^{m}$ and
$P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$ for every
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$. Then

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}<\infty\\\
1&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}=\infty.\end{cases}$

###### Proof.

The convergence part is a standard application of the Borel–Cantelli lemma, so
we will concentrate on the divergence part.

Let $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ such that
$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}$ diverges, and let
$\Psi(d)$ be defined as in þ1.1. If there exists $d\geqslant 1$ for which
$\Psi(d)=\infty$, then we are done by þ2.1. Assume then that $\Psi(d)$ is
always finite.

If there exists $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q})=1$
such that

$\sum_{d\geqslant 1}\psi(d\mathbf{q})^{m}=\infty,$ (6)

then by Yu’s result, $\operatorname{m}(W_{1,m}^{P}(\psi_{\mathbf{q}}))=1$,
where $\psi_{\mathbf{q}}:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ is defined by
$\psi_{\mathbf{q}}(d)=\psi(d\mathbf{q})$. But in this case

$T^{-1}\left\lparen W_{1,m}^{P}(\psi_{\mathbf{q}})\right\rparen\subseteq
W_{n,m}^{P}(\psi)$

where $T:\mathbb{T}^{mn}\to\mathbb{T}^{m}$ is the map
$\mathbf{X}\mapsto\mathbf{q}\mathbf{X}\pmod{1}$. That map preserves measure,
therefore, $\operatorname{m}(W_{n,m}^{P}(\psi))=1$.

On the other hand, suppose there is no $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with
$\gcd(\mathbf{q})=1$ for which (6) holds. Then for every $Q\geqslant 1$,

$\sum_{d\geqslant 1}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert<Q\end{subarray}}\psi(d\mathbf{q})^{m}<\infty$

therefore

$\sum_{d\geqslant 1}\Psi_{Q}(d)^{m}=\sum_{d\geqslant
1}\Psi(d)^{m}-\sum_{d\geqslant
1}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert<Q\end{subarray}}\psi(d\mathbf{q})^{m}=\infty$

where $\Psi_{Q}$ is defined by (4). Now, by [30, Theorem 1.8],
$\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))=1$. Since this holds for every
$Q\geqslant 1$, þ1.2 implies that $\operatorname{m}\left\lparen
W_{n,m}^{P}(\psi)\right\rparen=1$, finishing the proof. ∎

As a corollary, one gets the univariate version of þ 2.2, where $\psi$ depends
only on $\lvert\mathbf{q}\rvert$—that is, the inhomogeneous Khintchine–Groshev
theorem without monotonicity in all cases where $m>2$. This was proved for all
$nm>2$ in [2, Theorem 1]. In fact, one can prove a version where the
inhomogeneous parameter $\mathbf{y}\in\mathbb{R}^{m}$ can vary with
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$.

In [2, Conjecture 1] it is conjectured that it should be possible to remove
the monotonicity condition from the inhomogeneous univariate
Khintchine–Groshev theorem in the remaining $1$-by-$2$ and $2$-by-$1$ cases.
The $1$-by-$2$ case of the conjecture, combined with þ1.2, would imply that þ
2.2 also holds for $m=2$, bringing it in line with the homogeneous theorem of
Beresnevich–Velani in §2.3.

###  2.5 Nesharim–Rühr–Shi 2020 ($n=1$) $\implies$ Nesharim–Rühr–Shi 2020

In 1988 Harman considered approximation of real numbers by rational numbers
$p/q$ whose numerators and denominators lie in prescribed arithmetic
sequences, say $p\equiv r\bmod a$ and $q\equiv s\bmod b$ where $a,b\geqslant
1$, $0\leqslant r<a-1$, and $0\leqslant s<b-1$ are fixed integers. He found
asymptotics for the number of such rational approximates to typical real
numbers [17], which yield in particular a Khintchine-type metric dichotomy
like the ones discussed here. That dichotomy was extended to the linear forms
setting in 2020 by Nesharim, Rühr, and Shi [22].

Fix $m,n\in\mathbb{N}$. For a function
$\bar{\psi}:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ and fixed vectors
$(\mathbf{a},\mathbf{b})\in\mathbb{N}^{m}\times\mathbb{N}^{n}$ and
$(\mathbf{r},\mathbf{s})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$, define

$\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})=\left\\{\mathbf{X}\in
M_{n\times m}(\mathbb{R}):\begin{subarray}{c}\textrm{ there are infinitely
many }(\mathbf{p},\mathbf{q})\in\mathbb{Z}^{m}\times\mathbb{Z}_{+}^{n}\\\
\textrm{ satisfying
}\lvert\mathbf{q}\mathbf{X}-\mathbf{p}\rvert<\bar{\psi}(\mathbf{q})\textrm{
such that}\\\ \mathbf{p}\equiv\mathbf{r}\,(\bmod\,\mathbf{a})\textrm{ and
}\mathbf{q}\equiv\mathbf{s}\,(\bmod\,\mathbf{b})\end{subarray}\right\\}$

where $\mathbf{p}\equiv\mathbf{r}\,(\bmod\,\mathbf{a})$ means entrywise
congruence. Nesharim, Rühr, and Shi prove the following theorem. Its $n>1$
cases can be obtained from its $n=1$ case using þ1.2.

######  Theorem (Nesharim–Rühr–Shi [22, Theorem 1.2]).

Fix $m,n\in\mathbb{N}$. Let $\bar{\psi}:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ be
nonincreasing, and fix
$(\mathbf{a},\mathbf{b})\in\mathbb{N}^{m}\times\mathbb{N}^{n}$ and
$(\mathbf{r},\mathbf{s})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$. Then

$\operatorname{m}\left\lparen\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})\right\rparen=\begin{cases}\textsc{zero}&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\bar{\psi}(q)^{m}<\infty\\\
\textsc{full}&\textrm{if
}\sum_{q=1}^{\infty}q^{n-1}\bar{\psi}(q)^{m}=\infty\end{cases}$

where $\bar{\psi}(\mathbf{q}):=\bar{\psi}(\lvert\mathbf{q}\rvert)$.

###### Proof modulo the $n=1$ case.

The convergence case follows from the convergence case of the
Khintchine–Groshev theorem, so only the divergence case needs proving.

Let $a=\operatorname{lcm}(\mathbf{a})$ and $b=\operatorname{lcm}(\mathbf{b})$.
Note that no generality is lost if every entry of the vector $\mathbf{a}$ is
replaced with $a$, that is, if it is assumed that $\mathbf{a}=(a,\dots,a)$.

For each $\mathbf{q}\in\mathbb{Z}_{+}^{n}$, put

$P(\mathbf{q})=\frac{\mathbf{r}}{a}+\mathbb{Z}^{m}$

and note that this defines a well-spread $P$, as in Definition 1.1. Define
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ by

$\psi(\mathbf{q})=\begin{cases}\frac{1}{a}\bar{\psi}(\lvert\mathbf{q}\rvert)&\textrm{if
}\mathbf{q}\equiv\mathbf{s}\,(\bmod\mathbf{b})\\\
0&\textrm{otherwise}\end{cases}$

and for each $0\leqslant s\leqslant b-1$, define

$\psi_{s}(\mathbf{q})=\begin{cases}\psi(\mathbf{q})&\textrm{if
}\gcd(\mathbf{q})\equiv s\,(\bmod b)\\\ 0&\textrm{otherwise,}\end{cases}$

so that $\psi=\psi_{0}+\dots+\psi_{b-1}$. Since $\bar{\psi}$ is nonincreasing
and $\sum q^{n-1}\bar{\psi}(q)^{m}$ diverges, it is readily verified that

$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi(\mathbf{q})^{m}$

also diverges.

Suppose there is some $s\in\\{0,\dots,b-1\\}$ and
$\mathbf{q}^{\prime}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q}^{\prime})=1$
such that

$\sum_{d\geqslant 1}\psi_{s}(d\mathbf{q}^{\prime})^{m}=\infty.$ (7)

Then

$\sum_{d\geqslant 1}\bar{\psi}(d\lvert\mathbf{q}^{\prime}\rvert)^{m}=\infty,$

and by the $n=1$ case,
$\operatorname{m}(\mathcal{K}_{1,m}(\bar{\psi}_{\mathbf{q}^{\prime}},\mathbf{a},b,\mathbf{r},s))=\textsc{full}$,
where
$\bar{\psi}_{\mathbf{q}^{\prime}}(d):=\bar{\psi}(d\lvert\mathbf{q}^{\prime}\rvert)$.
But note that the scaled set
$\frac{1}{a}\mathcal{K}_{1,m}(\bar{\psi}_{\mathbf{q}^{\prime}},\mathbf{a},b,\mathbf{r},s)$
is the lift to $\mathbb{R}^{m}$ of
$W_{1,m}^{P}(\psi_{s,\mathbf{q}^{\prime}})$, where
$\psi_{s,\mathbf{q}^{\prime}}(d):=\psi_{s}(d\lvert\mathbf{q}^{\prime}\rvert)$,
and so $\operatorname{m}\lparen
W_{1,m}^{P}(\psi_{s,\mathbf{q}^{\prime}})\rparen=1$. But

$T_{\mathbf{q}^{\prime}}^{-1}\lparen
W_{1,m}^{P}(\psi_{s,\mathbf{q}^{\prime}})\rparen\subseteq
W_{n,m}^{P}(\psi_{s}),$

therefore, $\operatorname{m}\lparen W_{n,m}^{P}(\psi_{s})\rparen=1$, since
$T_{\mathbf{q}^{\prime}}$ is measure-preserving. And the lift of
$W_{n,m}^{P}(\psi_{s})$ is contained in the scaled set
$\frac{1}{a}\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})$,
so
$\operatorname{m}\lparen\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})\rparen=\textsc{full}$.

Suppose, on the other hand, that there are no $s\in\\{0,\dots,b-1\\}$ and
$\gcd(\mathbf{q}^{\prime})=1$ such that (7) holds. Choose $s$ so that

$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\psi_{s}(\mathbf{q})^{m}=\infty,$

and let $Q\geqslant 1$. Define $\Psi_{s,Q}$ by (4). Since (7) holds for no
$\mathbf{q}^{\prime}\in\mathbb{Z}_{+}^{n}$, it follows that

$\sum_{d\geqslant 1}\Psi_{s,Q}(d)^{m}=\infty.$

Note that $\Psi_{s,Q}$ is supported on $d\equiv s\,(\bmod\,b)$, but can be
extended to a nonincreasing function supported on all $d\geqslant 1$. The
divergence condition persists, so the $n=1$ case of this theorem implies that
$\operatorname{m}\lparen\mathcal{K}_{1,m}(a\Psi_{s,Q},\mathbf{a},b,\mathbf{r},s)\rparen=\textsc{full}$.
But now the scaled set
$\frac{1}{a}\mathcal{K}_{1,m}(a\Psi_{s,Q},\mathbf{a},b,\mathbf{r},s)$ is the
lift to $\mathbb{R}^{m}$ of $W_{1,m}^{P}(\Psi_{s,Q})$, so
$\operatorname{m}\lparen W_{1,m}^{P}(\Psi_{s,Q})\rparen=1$. Since $Q\geqslant
1$ was arbitrary, þ1.2 implies that $\operatorname{m}\lparen
W_{n,m}^{P}(\psi_{s})\rparen=1$, so $\operatorname{m}\lparen
W_{n,m}^{P}(\psi)\rparen=1$. Finally, the lift of $W_{n,m}^{P}(\psi)$ to
$M_{n\times m}(\mathbb{R})$ is a scaling of
$\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})$ by
$1/a$, so it follows that
$\operatorname{m}\lparen\mathcal{K}_{n,m}(\bar{\psi},\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s})\rparen=\textsc{full}$
and the proof is finished. ∎

Adiceam proves the $m=n=1$ case of this theorem with the additional condition
$\gcd(p,q)=\gcd(a,b,r,s)$ imposed in the definition of
$\mathcal{K}_{1,1}(\psi,a,b,r,s)$ (see [1, Theorem 2]). þ1.2 can be used to
prove the dual version of that statement, that is, the $m=1$ cases of the
above theorem, with the additional condition
$\gcd(p,\mathbf{q})=\gcd(a,\mathbf{b},r,\mathbf{s})$.

###  2.6 Inhomogeneous Duffin–Schaeffer $\implies$ same for systems of linear
forms

In 1941, Duffin and Schaeffer showed by counterexample that the monotonicity
condition cannot be removed from the one-dimensional version of Khintchine’s
theorem. In the same paper, they formulated what became known as the
Duffin–Schaeffer conjecture [13], a problem that stood eight decades. It was
finally proved in a 2020 breakthrough by Koukoulopoulos and Maynard [21]. A
higher-dimensional version had been proved in 1990 by Pollington and Vaughan
[23]. The following theorem is the version of the Duffin–Schaeffer conjecture
for systems of linear forms. It was conjectured by
Beresnevich–Bernik–Dodson–Velani [4] in 2009 and recently proved by the author
[26].

######  Theorem (Duffin–Schaeffer conjecture for systems of linear forms).

Fix $m,n\in\mathbb{N}$. Let

$P(\mathbf{q})=\\{\mathbf{p}\in\mathbb{Z}^{m}:\gcd(p_{i},\mathbf{q})=1,\,i=1,\dots,m\\}$

Then

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}<\infty\\\
1&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}=\infty.\end{cases}$

This theorem also follows easily now, by applying þ1.2 with the results of
Pollington–Vaughan and Koukoulopoulos–Maynard as input. The relative well-
spreadedness of $P$ follows from [26, Lemma 5]. In fact, the proofs of
þLABEL:thm:bootstrap,thm:bootstrap11 are abstractions of the arguments in
[26].

In [24], Duffin–Schaeffer-style counterexamples are presented, showing that
the monotonicity condition cannot be removed from the one-dimensional
inhomogeneous Khintchine theorem, and an inhomogeneous version of the
Duffin–Schaeffer conjecture is discussed. Specifically, it is the $m=n=1$
version of the above stated theorem, but with

$P(q)=\\{p+y:p\in\mathbb{Z},\gcd(p,q)=1,\,i=1,\dots,m\\}$

for every $q\in\mathbb{Z}_{+}$, where $y\in\mathbb{R}$ is an arbitrarily fixed
inhomogeneous parameter. A version of that conjecture for systems of linear
forms would go as follows.

######  Conjecture 2.3 (Inhomogeneous Duffin–Schaeffer conjecture for systems
of linear forms).

þ Fix $m,n\in\mathbb{N}$. Let $\mathbf{y}\in\mathbb{R}^{m}$ and let

$P(\mathbf{q})=\\{\mathbf{p}+\mathbf{y}:\mathbf{p}\in\mathbb{Z}^{m},\gcd(p_{i},\mathbf{q})=1,\,i=1,\dots,m\\}.$

Then

$\operatorname{m}\left\lparen
W_{n,m}^{P}(\psi)\right\rparen=\begin{cases}0&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}<\infty\\\
1&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}=\infty.\end{cases}$

There has not been much progress on þ 2.3 in the $m=n=1$ setting, although
some evidence in favor of it is presented in [24] and special cases of a weak
version of it (stated below as þ2.7) have been established by Chow and
Technau. Part of what makes the problem difficult is that the coprimality
requirement no longer accomplishes what it accomplished in the homogeneous
setting. In the homogeneous setting, the coprimality gives $P(q_{1})/q_{1}\cap
P(q_{2})/q_{2}=\emptyset$ for all $q_{1}\neq q_{2}$, that is, it prevents
certain overlaps of intervals that would have occurred without coprimality.
Those overlaps are not prevented by coprimality once an inhomogeneous
parameter $y\neq 0$ is introduced. One could take this as a sign that þ 2.3
may not be the “correct” inhomogeneous generalization of Duffin–Schaeffer.
(Another is proposed by Chow–Technau [11, Question 1.24].) Nevertheless, this
author harbors some hope for it.

Regarding the higher-dimensional cases of þ 2.3, a univariate version—meaning
that $\psi$ only depends on $\lvert\mathbf{q}\rvert$—has been established for
$n>2$ by Allen and the author [3, Theorem 3], lending further support for the
plausibility of the conjecture.

In any case, þ1.2 allows one to prove the following conditional result.

######  Theorem 2.4.

þ Fix $m\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{R}^{m}$. If þ 2.3 holds with
inhomogeneous parameter $\mathbf{y}$ and with $n=1$, then it holds with
inhomogeneous parameter $\mathbf{y}$ for all $n\geqslant 1$. In particular, if
the $m$-dimensional inhomogeneous Duffin–Schaeffer conjecture is true, then so
is the $n$-by-$m$-dimensional inhomogeneous Duffin–Schaeffer conjecture.

###### Proof.

As usual, the convergence part is not at issue. Let
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ satisfy

$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}=\infty.$

If there exists $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q})=1$
such that

$\sum_{d\geqslant
1}\left\lparen\frac{\varphi(d)\psi(d\mathbf{q})}{d}\right\rparen^{m}=\infty,$
(8)

then by assumption $\operatorname{m}\lparen
W_{1,m}^{P}(\psi_{\mathbf{q}})\rparen=1$, where
$\psi_{\mathbf{q}}:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ by
$\psi_{\mathbf{q}}(d)=\psi(d\mathbf{q})$. But in this case

$T^{-1}\left\lparen W_{1,m}^{P}(\psi_{\mathbf{q}})\right\rparen\subseteq
W_{n,m}^{P}(\psi),$

where $T:\mathbb{T}^{mn}\to\mathbb{T}^{m}$ is the map
$\mathbf{X}\mapsto\mathbf{q}\mathbf{X}\pmod{1}$. That map preserves measure,
therefore, $\operatorname{m}(W_{n,m}^{P}(\psi))=1$.

On the other hand, suppose there is no $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with
$\gcd(\mathbf{q})=1$ for which (8) holds. Then for every $Q\geqslant 1$,

$\sum_{d\geqslant 1}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert<Q\end{subarray}}\left\lparen\frac{\varphi(d)\psi(d\mathbf{q})}{d}\right\rparen^{m}<\infty$
(9)

therefore

$\sum_{d\geqslant
1}\left\lparen\frac{\varphi(d)\Psi_{Q}(d)}{d}\right\rparen^{m}=\sum_{d\geqslant
1}\left\lparen\frac{\varphi(d)\Psi(d)}{d}\right\rparen^{m}-\sum_{d\geqslant
1}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert<Q\end{subarray}}\left\lparen\frac{\varphi(d)\psi(d\mathbf{q})}{d}\right\rparen^{m}=\infty$

where $\Psi_{Q}$ is defined by (4). Now, by assumption,
$\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))=1$. Since $Q\geqslant 1$ was
arbitrary, þ1.2 gives $\operatorname{m}(W_{n,m}^{P}(\psi))=1$, finishing the
proof. ∎

###  2.7 Chow–Technau 2023 $\implies$ a version for dual approximation

For many of the years during which the Duffin–Schaeffer conjecture was open,
there was a seemingly weaker related problem, Catlin’s conjecture (1976,
[10]), that also remained open and was only established as a consequence of
Duffin–Schaeffer once Koukoulopoulos and Maynard proved it [21]. The following
is the version of Catlin’s conjecture for systems of linear forms, proved in
dimensions $m>1$ by Beresnevich–Velani [7] and $m=1,n>1$ by the author [26].

######  Theorem (Catlin’s conjecture for systems of linear forms).

Fix $m,n\in\mathbb{N}$. For $\mathbf{q}\in\mathbb{Z}^{n}$, let

$\Phi_{m}(\mathbf{q})=\\#\\{\mathbf{p}\in\mathbb{Z}^{m}:\lvert\mathbf{p}\rvert\leqslant\lvert\mathbf{q}\rvert,\gcd(\mathbf{p},\mathbf{q})=1\\},$

and $P(\mathbf{q})=\mathbb{Z}$. Then

$\operatorname{m}(W_{n,m}^{P}(\psi))=\begin{cases}0&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}^{n}\setminus\\{0\\}}\Phi_{m}(\mathbf{q})\sup_{t\geqslant
1}\left\lparen\frac{\psi(t\mathbf{q})}{t\lvert\mathbf{q}\rvert}\right\rparen^{m}<\infty\\\
1&\textrm{if
}\sum_{\mathbf{q}\in\mathbb{Z}^{n}\setminus\\{0\\}}\Phi_{m}(\mathbf{q})\sup_{t\geqslant
1}\left\lparen\frac{\psi(t\mathbf{q})}{t\lvert\mathbf{q}\rvert}\right\rparen^{m}=\infty.\end{cases}$

It is not clear what the appropriate version of Catlin’s conjecture should be
in the inhomogeneous setting. In the classical statement, the series has a
clear purpose: it ensures that each rational number $p/q$ contributes exactly
once to the measure sum, with the maximal measure of an approximation interval
of which it is the center. The series does something similar in higher
dimensions $n,m\geqslant 1$. But if an inhomogeneous parameter
$\mathbf{y}\neq\mathbf{0}$ is introduced, then this effect is lost.

On the other hand, in [25, 26] it is shown that in the cases where $m=1$ (the
“dual” cases), the divergence part of Catlin’s conjecture for systems of
linear forms is equivalent to the following weak version of the
Duffin–Schaeffer conjecture.

######  Theorem (Weak dual Duffin–Schaeffer conjecture).

Fix $n\in\mathbb{N}$ and for every $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ let
$P(\mathbf{q})=\mathbb{Z}$. Then

$\operatorname{m}(W_{n,1}^{P}(\psi))=1\qquad\textrm{if}\qquad\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}=\infty.$

This theorem is a direct consequence of the $n$-by-$1$ Duffin–Schaeffer
conjecture. Of course, its statement can be made inhomogeneous exactly as in þ
2.3.

######  Conjecture 2.5 (Weak inhomogeneous Duffin–Schaeffer conjecture for
systems of linear forms).

þ Fix $m,n\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{R}^{m}$. For every
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ let
$P(\mathbf{q})=\mathbf{y}+\mathbb{Z}^{m}$. Then

$\operatorname{m}(W_{n,m}^{P}(\psi))=1\qquad\textrm{if}\qquad\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\left\lparen\frac{\varphi(\gcd(\mathbf{q}))\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}=\infty.$

The set $P(\mathbf{q})$ appearing in þ 2.5 is a subset of the one appearing in
þ 2.3, which is why þ 2.5 is referred to as weaker. But, of course, that may
not mean it is easier to prove.

Again, þ1.2 allows one to prove a conditional result.

######  Theorem 2.6.

þ Fix $m\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{R}^{m}$. If þ 2.5 holds with
inhomogeneous parameter $\mathbf{y}$ and with $n=1$, then it holds with
inhomogeneous parameter $\mathbf{y}$ for all $n\geqslant 1$. In particular, if
the $m$-dimensional weak inhomogeneous Duffin–Schaeffer conjecture is true,
then so is the $n$-by-$m$-dimensional weak inhomogeneous Duffin–Schaeffer
conjecture.

###### Proof.

The proof is identical to that of þ 2.4. ∎

In [11], Chow and Technau prove þ 2.5 in dimension $m=n=1$ for a special class
of functions.

######  Theorem 2.7 (Chow–Technau, [11, Theorem 1.23]).

þ Let $k\geqslant 1$. Fix
${\boldsymbol{\alpha}}=({\alpha}_{1},\dots,{\alpha}_{k})\in\mathbb{R}^{k}$ and
$\gamma_{1},\dots,\gamma_{k}\in\mathbb{R}$. For $k=1$, suppose that
$\alpha_{1}$ is an irrational non-Liouville number, and for $k\geqslant 2$
assume

$\sup\left\\{w>0:\lVert q{\alpha}_{1}\rVert\cdot\lVert
q{\alpha}_{2}\rVert\cdots\lVert q{\alpha}_{k}\rVert<q^{-w}\textrm{ for
infinitely many }q\in\mathbb{N}\right\\}<\frac{k}{k-1}.$

Let $\bar{\Psi}:\mathbb{N}\to\mathbb{R}_{+}$ be a nonincreasing function such
that $\sum\bar{\Psi}(q)(\log q)^{k}$ diverges, and let

$\Psi(q)=\frac{\bar{\Psi}(q)}{\lVert q\alpha_{1}-\gamma_{1}\rVert\cdots\lVert
q\alpha_{k}-\gamma_{k}\rVert}\qquad(q\in\mathbb{N}).$

Then þ 2.5 holds in dimension $m=n=1$ for $\Psi$.

Using þ 1.2, þ 2.7 can be bootstrapped to prove a special case of the dual
version of þ 2.5.

######  Theorem 2.8 (Special case of dual weak inhomogeneous Duffin–Schaeffer
conjecture).

þ Fix $n\geqslant 1$. Let ${\boldsymbol{\alpha}}$ and
$\gamma_{1},\dots,\gamma_{k}$ be as in þ 2.7. Let
$\bar{\psi}:\mathbb{N}\to\mathbb{R}_{+}$ be nonincreasing and such that

$\sum_{\mathbf{q}\in\mathbb{Z}_{+}^{n}}\bar{\psi}(\lvert\mathbf{q}\rvert)(\log\gcd(\mathbf{q}))^{k}=\infty.$
(10)

Let $\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ be defined by

$\psi(\mathbf{q})=\frac{\bar{\psi}(\lvert\mathbf{q}\rvert)}{\lVert
d\alpha_{1}-\gamma_{1}\rVert\cdots\lVert
d\alpha_{k}-\gamma_{k}\rVert}\qquad(d=\gcd(\mathbf{q})).$

Then the $n$-by-$1$ þ 2.5 holds for $\psi$.

###### Proof.

Let $y\in\mathbb{R}$ and $P\equiv y+\mathbb{Z}$. The aim is to show that
$\operatorname{m}\lparen W_{n,1}^{P}(\psi)\rparen=1$. Suppose there exists
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q})=1$ such that

$\sum_{d\geqslant 1}\bar{\psi}(\lvert
d\mathbf{q}\rvert)(\log\gcd(d\mathbf{q}))^{k}=\sum_{d\geqslant
1}\bar{\psi}(\lvert d\mathbf{q}\rvert)(\log d)^{k}=\infty.$ (11)

Then with

$\bar{\Psi}_{\mathbf{q}}(d):=\bar{\psi}(\lvert
d\mathbf{q}\rvert)\qquad\textrm{and}\qquad\tilde{\Psi}(d):=\psi(d\mathbf{q})$

playing the roles of $\bar{\Psi}$ and $\Psi$ from þ 2.7, that theorem gives
$W_{1,1}^{P}(\tilde{\Psi})=1$. But

$T^{-1}\left\lparen W_{1,1}^{P}(\tilde{\Psi})\right\rparen\subseteq
W_{n,1}^{P}(\psi)$

where $T:\mathbb{T}^{mn}\to\mathbb{T}$ is the measure-preserving
$\mathbf{X}\mapsto\mathbf{q}\mathbf{X}$, so $\operatorname{m}\lparen
W_{n,1}^{P}(\psi)\rparen=1$ and the proof is finished.

Assume, therefore, that there is no $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with
$\gcd(\mathbf{q})=1$ for which (11) holds. Consequently, for every $Q\geqslant
1$, we have

$\sum_{\begin{subarray}{c}\mathbf{q}\in\mathbb{Z}_{+}^{n}\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\bar{\psi}(\lvert\mathbf{q}\rvert)(\log\gcd(\mathbf{q}))^{k}=\infty.$
(12)

Define $\Psi$ and $\Psi_{Q}$ as in þ LABEL:thm:bootstrap,thm:bootstrap11.
Notice that

$\displaystyle\Psi_{Q}(d)$
$\displaystyle=\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=d\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\frac{\bar{\psi}(\lvert\mathbf{q}\rvert)}{\lVert
d\alpha_{1}-\gamma_{1}\rVert\cdots\lVert d\alpha_{k}-\gamma_{k}\rVert}$
$\displaystyle=\frac{\bar{\Psi}_{Q}(d)}{\lVert
d\alpha_{1}-\gamma_{1}\rVert\cdots\lVert d\alpha_{k}-\gamma_{k}\rVert}$

where

$\bar{\Psi}_{Q}(d)=\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert\geqslant Q\end{subarray}}\bar{\psi}(\lvert
d\mathbf{q}\rvert).$

Since $\bar{\psi}$ is nonincreasing, so is $\bar{\Psi}_{Q}(d)$. Observe that

$\displaystyle\sum_{d\geqslant 1}\bar{\Psi}_{Q}(d)(\log d)^{k}$
$\displaystyle=\sum_{d\geqslant
1}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=1\\\
\lvert\mathbf{q}\rvert\geqslant Q\end{subarray}}(\log d)^{k}\bar{\psi}(\lvert
d\mathbf{q}\rvert)$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{q}\in\mathbb{Z}_{+}^{n}\\\
\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant
Q\end{subarray}}\bar{\psi}(\lvert\mathbf{q}\rvert)(\log\gcd(\mathbf{q}))^{k},$

which diverges, per (12). Therefore, $\Psi_{Q}$ satisfies the conditions from
þ 2.7, hence $\operatorname{m}(W_{1,1}^{P}(\Psi_{Q}))=1$. Finally, since
$Q\geqslant 1$ was arbitrary, þ 1.2 implies that
$\operatorname{m}(W_{n,1}^{P}(\psi))=1$, as desired. ∎

##  3 Examples

In each of the applications in §2, a full-measure set in $\mathbb{T}^{m}$ is
bootstrapped to a full-measure set in $\mathbb{T}^{mn}$. Yet
þLABEL:thm:bootstrap,thm:bootstrap11 do not require full-measure in
$\mathbb{T}^{m}$ in order to apply. The next example presents an application
of þ 1.2 where a set of positive and copositive measure in $\mathbb{T}^{m}$ is
bootstrapped to a full-measure set in $\mathbb{T}^{nm}$.

######  Example 3.1 (Positive, copositive $\longrightarrow$ full).

þ Let $m>1$. For $d\geqslant 1$, let

$P(d)=\left\\{\mathbf{p}:0\leqslant
p_{i}\leqslant\frac{d}{2},\,i=1,\dots,m\right\\}.$ (13)

Let $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ be such that $\sum
q^{n-1}\psi(q)^{m}$ diverges, where $n>1$, and extend $\psi$ to
$\mathbb{Z}_{+}^{n}$ by setting
$\psi(\mathbf{q})=\psi(\lvert\mathbf{q}\rvert)$. Let $Q\geqslant 1$. Then, as
was shown in §2.2, the series $\sum\Psi_{Q}(d)^{m}$ also diverges, where
$\Psi_{Q}$ is as in þ 1.2. By Gallagher’s extension of Khintchine’s theorem,
$\operatorname{m}\lparen W_{1,m}^{\mathbb{Z}}(\Psi_{Q})\rparen=1$, therefore,
$\operatorname{m}\lparen W_{1,m}^{P}(\Psi_{Q})\rparen=2^{-m}>0$. Since $P$ is
well-spread in the sense of þ 1.1, þ 1.2 applies to give
$\operatorname{m}\lparen W_{n,m}^{P}(\psi)\rparen=1$.

The next is an example where þ1.2 does not apply but þ1.1 does. A positive,
copositive measure set in $\mathbb{T}^{m}$ is bootstrapped to a positive,
copositive measure set in $\mathbb{T}^{mn}$, illustrating the importance of
the assumption on $\Psi_{Q}$ appearing in þ1.2.

######  Example 3.2 (Positive, copositive $\longrightarrow$ positive,
copositive).

þ Fix $m,n\in\mathbb{N}$ and for $d\geqslant 1$, define $P(d)$ as in (13). Let
$\psi:\mathbb{Z}_{+}^{n}\to\mathbb{R}_{+}$ be a function supported on vectors
of the form $\mathbf{q}=(d,0,0,\dots,0)$ and such that
$\sum\psi(\mathbf{q})^{m}$ diverges and
$\psi(\mathbf{q})=o(\lvert\mathbf{q}\rvert)$. Then the associated function
$\Psi$ obviously satisfies $\sum_{d}\Psi(d)^{m}=\infty$, and Khintchine’s
theorem can be used to show that $W_{1,m}^{P}(\Psi)=2^{-m}$. (It has full
measure in $[0,1/2]^{m}$.) þ1.1 gives $\operatorname{m}\lparen
W_{n,m}^{P}(\psi)\rparen>0$. But

$W_{n,m}^{P}(\psi)=W_{1,m}^{P}(\Psi)\times[0,1]^{(m-n)n},$

and so $\operatorname{m}\lparen W_{n,m}^{P}(\psi)\rparen=2^{-m}$.

This example is easily generalized to situations where $\psi$ is supported on
points $d\mathbf{q}^{\prime}\,(d\geqslant 1)$ for some fixed
$\mathbf{q}^{\prime}$ with $\gcd(\mathbf{q}^{\prime})=1$. This is the
artificially $1$-by-$m$-dimensional construction mentioned after the statement
of þ1.2, and it is what the assumption on $\Psi_{Q}$ avoids.

One may wonder whether þ 1.1 might achieve a full-measure set in
$\mathbb{T}^{mn}$ if it is further assumed that $\operatorname{m}\lparen
W_{1,m}^{P}(\Psi)\rparen=1$. The next example shows that þ 1.1 does not
necessarily bootstrap full measure to full measure.

######  Example 3.3 (Full $\longrightarrow$ positive, copositive).

þ For $d\geqslant 1$ odd, let

$P(d)=\left\\{\mathbf{p}:0\leqslant
p_{i}\leqslant\frac{d}{2},\,i=1,\dots,m\right\\}.$

and for $d\geqslant 1$ even, let

$P(d)=\left\\{\mathbf{p}:\frac{d}{2}\leqslant p_{i}\leqslant
d,\,i=1,\dots,m\right\\}.$

Let $\psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ be nonincreasing and such that
$\sum\psi(q)^{m}$ diverges. Extend $\psi$ to $\mathbb{Z}_{+}^{2}$ by setting

$\psi(\mathbf{q})=\begin{cases}\psi(d)&\textrm{if
}\mathbf{q}=\begin{pmatrix}d&0\end{pmatrix},d\textrm{ odd}\\\
\psi(d)&\textrm{if }\mathbf{q}=\begin{pmatrix}0&d\end{pmatrix},d\textrm{
even}\\\ 0&\textrm{otherwise.}\end{cases}$

Notice that $\Psi(d)=\psi(d)$, where $\Psi$ is defined as in þ 1.1. By
Khintchine’s theorem, $\operatorname{m}\lparen
W_{1,m}^{\mathbb{Z}}(\Psi)\rparen=1$, and since $\Psi$ is nonincreasing it can
be shown that $\operatorname{m}\lparen W_{1,m}^{P}(\Psi)\rparen=1$. But $P$ is
well-spread, so þ 1.1 implies that $\operatorname{m}\lparen
W_{n,m}^{P}(\psi)\rparen>0$. Evidently, this cannot be improved to full
measure. After all, if $\psi(q)=o(q)$, then

$W_{n,m}^{P}(\psi)\subseteq\left\lparen[0,1/2]\times[0,1]\right\rparen\cup\left\lparen[0,1]\times[1/2,1]\right\rparen\subseteq\mathbb{T}^{2}.$

It has measure at most $3/4$.

##  4 Measure-theoretic and geometric tools

This section is a collection of important measure-theoretic and geometric
facts.

The first few illustrate a theme that is prevalent in metric number theory: in
order to show that the limsup set for a sequence $(A_{q})_{q=1}^{\infty}$ of
subsets in a probability space has positive measure, one should try to show
that the sets $A_{q}$ are in some sense independent. For example, the second
Borel–Cantelli lemma says that if they are pairwise independent and have
diverging measure-sum, then the limsup set has full measure. In practice, one
must make do with weaker forms of independence. The ones that are most useful
in metric number theory are averaged forms of independence, like the ones
exhibited in the following propositions.

######  Proposition 4.1 (Chung–Erdős Lemma, [12]).

þ If $(X,\mu)$ is a probability space and $(A_{q})_{q\in\mathbb{N}}\subseteq
X$ is a sequence of measurable subsets such that
$0<\sum_{q}\mu(A_{q})<\infty$, then

$\mu\left\lparen\bigcup_{q=1}^{\infty}A_{q}\right\rparen\geqslant\frac{\left\lparen\sum_{q=1}^{\infty}\mu(A_{q})\right\rparen^{2}}{\sum_{q,r=1}^{\infty}\mu(A_{q}\cap
A_{r})}.$

###### Proof.

This result is proved in [12] for finitely many sets $A_{1},\dots,A_{Q}$, with
the conclusion

$\mu\left\lparen\bigcup_{q=1}^{Q}A_{q}\right\rparen\geqslant\frac{\left\lparen\sum_{q=1}^{Q}\mu(A_{q})\right\rparen^{2}}{\sum_{q,r=1}^{Q}\mu(A_{q}\cap
A_{r})}.$

This immediately implies

$\mu\left\lparen\bigcup_{q=1}^{\infty}A_{q}\right\rparen\geqslant\frac{\left\lparen\sum_{q=1}^{Q}\mu(A_{q})\right\rparen^{2}}{\sum_{q,r=1}^{Q}\mu(A_{q}\cap
A_{r})}.$

for every $Q\geqslant 1$. The result stated here follows upon taking
$Q\to\infty$. ∎

The next result is an extension of the second Borel–Cantelli lemma showing
that a sequence of sets that are _quasi-independent on average_ and that have
diverging measure sum must have a positive-measure limsup set.

######  Proposition 4.2 (Erdős–Renyi, [14]).

þ Let $(X,\mu)$ be a probability space and $(A_{k})_{k\geqslant 1}$ a sequence
of measurable sets such that for every $\sum_{k\geqslant 1}\mu(A_{k})$
diverges. If there exists a constant $C\geqslant 0$ such that for infinitely
many $D$,

$\sum_{k,\ell\leqslant D}\mu(A_{k}\cap A_{\ell})\leqslant
C\left\lparen\sum_{k\leqslant D}\mu(A_{k})\right\rparen^{2}$ (14)

then

$\mu\left\lparen\limsup_{k\to\infty}A_{k}\right\rparen\geqslant\frac{1}{C}.$

######  Remark.

Erdős and Renyi proved this in [14] with $C=1$. The proof of the more general
statement above is found in many places, for example [18, Lemma 2.3], [28,
Chapter 1, Lemma 5].

The proofs of þ LABEL:thm:bootstrap,thm:bootstrap11 make direct use of the
following variant of þ 4.2.

######  Proposition 4.3.

þ Let $(X,\mu)$ be a probability space and $(A_{d,q})_{d,q\geqslant 1}$ a
collection of measurable sets such that for every $d\geqslant 1$,

$\sum_{q=1}^{\infty}\mu(A_{d,q})<\infty\qquad\textrm{ while
}\qquad\sum_{d=1}^{\infty}\sum_{q=1}^{\infty}\mu(A_{d,q})=\infty.$ (15)

If there exists a constant $C\geqslant 0$ such that for infinitely many $D$,

$\sum_{\begin{subarray}{c}(k,q),(\ell,r)\\\ k,\ell\leqslant
D\end{subarray}}\mu(A_{k,q}\cap A_{\ell,r})\leqslant
C\left\lparen\sum_{\begin{subarray}{c}(d,q)\\\ d\leqslant
D\end{subarray}}\mu(A_{d,q})\right\rparen^{2}$ (16)

then

$\mu\left\lparen\limsup_{d,q\to\infty}A_{d,q}\right\rparen\geqslant\frac{1}{C}.$

###### Proof.

Let ${\varepsilon}>0$. For each $d\geqslant 1$, let $q_{d}\geqslant 1$ be such
that

$\sum_{q=1}^{q_{d}}\mu(A_{d,q})\geqslant(1-{\varepsilon})\sum_{q=1}^{\infty}\mu(A_{d},q).$
(17)

Let $(A_{k})$ be the sequence

$\left\\{\\{A_{d,q}\\}_{q=1}^{q_{d}}\right\\}_{d=1}^{\infty}=\left\\{A_{1,1},\dots,A_{1,q_{1}},A_{2,1},A_{2,2},\dots,A_{2,q_{2}},A_{3,1},\dots\right\\}$

For each $D\geqslant 1$ for which (16) holds let $K=q_{1}+q_{2}+\dots+q_{D}$,
so that

$\sum_{k,\ell\leqslant K}\mu(A_{k}\cap A_{\ell})\leqslant
C\left\lparen\sum_{\begin{subarray}{c}(d,q)\\\ d\leqslant
D\end{subarray}}\mu(A_{d,q})\right\rparen^{2}\overset{(\ref{eq:31})}{\leqslant}\frac{C}{(1-{\varepsilon})^{2}}\left\lparen\sum_{k\leqslant
K}\mu(A_{k})\right\rparen^{2}.$

By þ4.2,

$\mu\left\lparen\limsup_{k\to\infty}A_{k}\right\rparen\geqslant\frac{(1-{\varepsilon})^{2}}{C},$

therefore

$\mu\left\lparen\limsup_{d,q\to\infty}A_{d,q}\right\rparen\geqslant\frac{(1-{\varepsilon})^{2}}{C}.$

Since ${\varepsilon}>0$ was arbitrary, the same holds with ${\varepsilon}=0$.
∎

Beresnevich and Velani have recently proved a collection of results
establishing partial converses to þ 4.2. The following theorem states that if
a sequence of balls has a positive-measure limsup set, then it must contain a
quasi-independent subsequence.

######  Theorem 4.4 (Beresnevich–Velani, [8]).

þ Let $(\Omega,\mathcal{A},\mu,d)$ be a metric measure space equipped with a
doubling Borel probability measure $\mu$. Let $\\{B_{i}\\}_{i\in\mathbb{N}}$
be a sequence of balls in $\Omega$ with $r(B_{i})\to 0$ as $i\to\infty$ and
such that

$\exists a,b>1\quad\textrm{ such that}\quad\mu(aB_{i})\leqslant
b\mu(B_{i})\quad\textrm{for all $i$ sufficiently large.}$

There exists a constant $K>0$ depending only on $a,b$, and the doubling
constant of the measure $\mu$ such that the following are equivalent:

  * •

$\mu(\limsup_{i\to\infty}B_{i})>0$

  * •

There exists a subsequence $\\{L_{i}\\}_{i\in\mathbb{N}}$ of
$\\{B_{i}\\}_{i\in\mathbb{N}}$ with

$\sum_{i=1}^{\infty}\mu(L_{i})=\infty$

and for infinitely many $Q\in\mathbb{N}$

$\sum_{s,t=1}^{Q}\mu(L_{s}\cap L_{t})\leqslant
C\left\lparen\sum_{s=1}^{Q}\mu(L_{s})\right\rparen^{2},$

where $C=K\mu(\limsup_{i\to\infty}B_{i})^{-2}$.

######  Remark.

This statement is a combination of [8, Theorem 3], where the quasi-
independence is asserted, and [8, Proposition 2], where the constant $C>0$ is
specified. It is especially important to the proof of þ1.2 that the constant
$C>0$ is understood.

The next lemma was proved by Cassels [9] in the course of proving his zero-one
law for sets of the form $W_{1,m}^{\mathbb{Z}}(\Psi)$. The lemma says that the
measure of the limsup set of a sequence of balls is not altered if the balls
are uniformly scaled. Cassels’ lemma has enjoyed frequent use in metric
Diophantine approximation. Its role in §7 and §8 is in allowing us to scale
the function $\psi$ (and, correspondingly, $\Psi$) until the balls making up
certain approximation sets, to be discussed in §5, are disjoint.

######  Lemma 4.5 (Cassels’ lemma, [9]).

þ Let for each $k$, let $B_{k}$ be a ball in the torus $\mathbb{T}^{d}$ having
radius $c\psi_{k}$ where $\psi_{k}>0$ and $c>0$ is a constant. Then
$\operatorname{m}(\limsup_{k\to\infty}B_{k})$ is independent of the value of
the constant $c>0$.

A well-known consequence of the Lebesgue density theorem asserts that if a set
occupies a fixed positive proportion of every ball, then it must have full
measure. In the next lemma of Beresnevich, Dickinson, and Velani, the
requirement of a fixed positive proportion is relaxed. The lemma plays a
crucial role in the proof of þ1.2; it allows us to prove full measure of
$W_{n,m}^{P}(\psi)$ after having established positive measure on small scales.

######  Lemma 4.6 (Beresnevich–Dickinson–Velani, [5, Lemma 6]).

þ Let $(X,d)$ be a metric space with a finite measure $\mu$ such that every
open set is $\mu$-measurable. Let $A$ be a Borel subset of $X$ and let
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be an increasing function with $f(x)\to 0$
as $x\to 0$. If for every open set $U\subseteq X$ we have

$\mu(A\cap U)\geqslant f(\mu(U)),$

then $\mu(A)=\mu(X)$.

The next is Vitali’s covering lemma. There are several versions, all of which
have a similar flavor: a collection of balls can be refined to a subcollection
of disjoint balls without losing too much measure from the original
collection. We state it for finitely many balls in the torus. The lemma is
used in the proofs of parts (b) of þ LABEL:thm:bootstrap,thm:bootstrap11 in §8
and §10.

######  Lemma 4.7 (Vitali’s covering lemma).

þ For every finite collection $B_{1},\dots,B_{k}$ of balls in the torus
$\mathbb{T}^{d}$, there exists a subcollection
$\tilde{B}_{1},\dots,\tilde{B}_{\ell}$ such that

$\bigcup_{i=1}^{k}B_{i}\subseteq\bigcup_{j=1}^{\ell}3\bullet\tilde{B}_{j},$

(where $3\bullet B$ denotes the threefold concentric dilation of a ball $B$)
and such that $\tilde{B}_{i}\cap\tilde{B}_{j}=\emptyset$ for all $i\neq j$.

The next lemma is useful for understanding the behavior under scaling of
intersections of the approximation sets to be discussed in §5. In §7 it
facilitates a comparison between certain pairwise intersections arising in the
$n$-by-$m$ setting and the pairwise intersections in the $1$-by-$m$ setting.

######  Lemma 4.8 ([26, Lemma 4]).

þ Suppose $I_{1},I_{2},\dots,I_{r}\subseteq\mathbb{T}^{m}$ are disjoint balls
and $J_{1},J_{2},\dots,J_{s}\subseteq\mathbb{T}^{m}$ are disjoint balls. Then
for any $0<\Sigma\leqslant 1$,

$\operatorname{m}\left\lparen\bigcup_{i=1}^{r}\Sigma\bullet
I_{i}\cap\bigcup_{j=1}^{s}\Sigma\bullet
J_{j}\right\rparen\leqslant\Sigma^{m}\operatorname{m}\left\lparen\bigcup_{i=1}^{r}I_{i}\cap\bigcup_{j=1}^{s}J_{j}\right\rparen,$

where $\Sigma\bullet I_{i}$ denotes the concentric contraction of the ball
$I_{i}$ by $\Sigma$, and similar for $\Sigma\bullet J_{j}$.

###### Proof.

The disjointness assumptions implies

$\operatorname{m}\left\lparen\bigcup_{i=1}^{r}I_{i}\cap\bigcup_{j=1}^{s}J_{j}\right\rparen=\sum_{i,j}\operatorname{m}\left\lparen
I_{i}\cap J_{j}\right\rparen$ (18)

and

$\operatorname{m}\left\lparen\bigcup_{i=1}^{r}\Sigma\bullet
I_{i}\cap\bigcup_{j=1}^{s}\Sigma\bullet
J_{j}\right\rparen=\sum_{i,j}\operatorname{m}\left\lparen\Sigma\bullet
I_{i}\cap\Sigma\bullet J_{j}\right\rparen.$ (19)

Let $1\leqslant i\leqslant r$ and $1\leqslant j\leqslant s$ and denote by
$\bar{I}_{i}$ and $\bar{J}_{j}$ the images of $I_{i}$ and $J_{j}$ under a
scaling of the metric in $\mathbb{T}^{m}$ by $\Sigma$, and let
$\left\lparen\bar{\mathbb{T}}^{m},\bar{\operatorname{m}}\right\rparen$ denote
the accordingly scaled measure space, namely,
$\bar{\mathbb{T}}^{m}=\mathbb{T}^{m}$ as a set and
$\bar{\operatorname{m}}=\Sigma^{m}\operatorname{m}$. Then we have

$\bar{\operatorname{m}}(\bar{I}_{i})=\operatorname{m}(\Sigma\bullet
I_{i})=\Sigma^{m}\operatorname{m}(I_{i})\quad\textrm{and}\quad\bar{\operatorname{m}}(\bar{J}_{j})=\operatorname{m}(\Sigma\bullet
J_{j})=\Sigma^{m}\operatorname{m}(J_{j}).$

Note that the centers of $\Sigma\bullet I_{i}$ and $\Sigma\bullet J_{j}$ in
$\mathbb{T}^{m}$ are farther apart that the centers of $\bar{I}_{i}$ and
$\bar{J}_{j}$ in $\bar{\mathbb{T}}^{m}$. Therefore,

$\operatorname{m}\left\lparen\Sigma\bullet I_{i}\cap\Sigma\bullet
J_{j}\right\rparen\leqslant\bar{\operatorname{m}}\left\lparen\bar{I}_{i}\cap\bar{J}_{j}\right\rparen=\Sigma^{m}\operatorname{m}(I_{i}\cap
J_{j}).$

Putting this into (18) and (19) proves the lemma. ∎

##  5 Properties of approximation sets

þLABEL:thm:bootstrap,thm:bootstrap11 concern limsups sets of the form

$W_{n,m}^{P}(\psi)=\limsup_{\lvert\mathbf{q}\rvert\to\infty}A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q})),$

where, for $\mathbf{q}\in\mathbb{Z}_{+}^{n}$, and $r\geqslant 0$,

$A_{n,m}^{P}(\mathbf{q},r)=\left\\{\mathbf{X}\in\mathbb{T}^{nm}:\mathbf{q}\mathbf{X}\in
B(r)+P(\mathbf{q})\right\\}.$

For the proofs, it is essential that we understand the measures of the sets
$A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ and their pairwise overlaps. This
section is a collection of lemmas about
$A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ and $A_{1,m}^{P}(d,\Psi(d))$.

The following lemma concerns the measures of
$A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$. Essentially, if $\psi(\mathbf{q})$
is not too large, then $A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ can be
viewed as an array of disjoint parallelepipeds whose bases form a union of
disjoint balls in $\mathbb{T}^{m}$.

######  Lemma 5.1.

þ For each $d\geqslant 1$, let $P(d)\subseteq(\mathbb{R}/d\mathbb{Z})^{m}$ be
a finite set, and for each $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ let
$P(\mathbf{q})$ denote the lift to $\mathbb{R}^{m}$ of $P(\gcd(\mathbf{q}))$.
If

$r\leqslant\frac{1}{2}\min\left\\{\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\rvert:\mathbf{p}_{1},\mathbf{p}_{2}\in
P(\mathbf{q}),\mathbf{p}_{1}\neq\mathbf{p}_{2}\right\\},$ (20)

then

$\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},r)\right\rparen=\\#P(\gcd(\mathbf{q}))\left\lparen\frac{2r}{\gcd(\mathbf{q})}\right\rparen^{m}.$
(21)

In particular, if $P$ is relatively well-spread as in Definition 1.1 and

$r\leqslant\frac{ad}{2\\#(P(d))^{1/m}},$ (22)

where $a$ is as in (3), then

$\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},r)\right\rparen=\\#P^{\prime}(\gcd(\mathbf{q}))\left\lparen\frac{2r}{\gcd(\mathbf{q})}\right\rparen^{m},$
(23)

hence

$\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},r)\right\rparen\asymp\\#P(\gcd(\mathbf{q}))\left\lparen\frac{2r}{\gcd(\mathbf{q})}\right\rparen^{m}.$
(24)

###### Proof.

Denote $d=\gcd(\mathbf{q})$ and let $\mathbf{q}^{\prime}\in\mathbb{Z}_{+}^{n}$
be the unique primitive vector such that $\mathbf{q}=d\mathbf{q}^{\prime}$.
The mapping $T:\mathbb{T}^{nm}\to\mathbb{T}^{m}$ defined by
$\mathbf{X}\mapsto\mathbf{q}^{\prime}\mathbf{X}\pmod{1}$ is Lebesgue measure-
preserving, in the sense that for every measurable set
$A\subseteq\mathbb{T}^{m}$,

$\operatorname{m}\left\lparen
T^{-1}(A)\right\rparen=\operatorname{m}\left\lparen A\right\rparen,$ (25)

where $\operatorname{m}$ is to be understood as $nm$-dimensional Lebesgue
measure on the left-hand side, and $m$-dimensional Lebesgue measure on the
right-hand side.

Now, notice that

$A_{n,m}^{P}(\mathbf{q},r)=T^{-1}\left\lparen A_{1,m}^{P}(d,r)\right\rparen$
(26)

hence it suffices to compute the measure of $A_{1,m}^{P}(d,r)$. This set is a
union of $\\#P(d)$ balls in $\mathbb{T}^{m}$ having radius $r/d$ and centers
in the set

$\left\\{\frac{\mathbf{p}}{d}:\mathbf{p}\in
P(\gcd(\mathbf{q}))\right\\}+\mathbb{Z}^{m}\subseteq\mathbb{T}^{m}.$

If (20) holds, then those balls are disjoint, therefore

$\operatorname{m}\left\lparen
A_{1,m}^{P}(d,r)\right\rparen=\\#P(d)\left\lparen\frac{2r}{d}\right\rparen^{m},$

and (21) is proved after taking (25) and (26) into account.

Suppose $P$ is relatively well-spread and (22) holds. Let
$P^{\prime}(d)\subseteq P(d)$ be as in Definition 1.1. Now
$A_{1,m}^{P^{\prime}}(d,r)$ is a union of disjoint balls, and so

$\operatorname{m}\left\lparen
A_{1,m}^{P^{\prime}}(d,r)\right\rparen=\\#P^{\prime}(d)\left\lparen\frac{2r}{d}\right\rparen^{m}\geqslant
c\\#P(d)\left\lparen\frac{2r}{d}\right\rparen^{m},$

proving (23). Since $A_{1,m}^{P^{\prime}}(d,r)\subseteq A_{1,m}^{P}(d,r)$, it
follows that

$c\\#P(d)\left\lparen\frac{2r}{d}\right\rparen^{m}\leqslant\operatorname{m}\left\lparen
A_{1,m}^{P^{\prime}}(d,r)\right\rparen\leqslant\\#P(d)\left\lparen\frac{2r}{d}\right\rparen^{m},$

and (24) follows after applying (25). ∎

The next lemma implies that if $\psi(\mathbf{q})$ is not too large, then the
sets $A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ are pairwise independent for
pairs $\mathbf{q}_{1},\mathbf{q}_{2}$ that are linearly independent.

######  Lemma 5.2.

þ Let
$P(\gcd(\mathbf{q}))\subseteq(\mathbb{R}/\gcd(\mathbf{q})\mathbb{Z})^{m}$ be a
finite set for all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$. If
$\mathbf{q}_{1},\mathbf{q}_{2}\in\mathbb{Z}_{+}^{n}$ are linearly independent
and each set $A_{1,m}^{P}(\gcd(\mathbf{q}_{i}),r_{i}),\,(i=1,2)$ is a union of
disjoint balls, then

$\operatorname{m}\left\lparen A_{n,m}^{P}(\mathbf{q}_{1},r_{1})\cap
A_{n,m}^{P}(\mathbf{q}_{2},r_{2})\right\rparen=\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q}_{1},r_{1})\right\rparen\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q}_{2},r_{2})\right\rparen,$

that is, the sets $A_{n,m}^{P}(\mathbf{q}_{1},r_{1})$ and
$A_{n,m}^{P}(\mathbf{q}_{2},r_{2})$ are independent.

###### Proof.

The set $A_{n,m}^{P}(\mathbf{q}_{i},r_{i})$ is a union of finitely many
disjoint sets

$A_{n,m}^{P}(\mathbf{q}_{i},r_{i})=\bigcup_{E\in\mathcal{E}_{i}}E$ (27)

where each $E\in\mathcal{E}_{i}$ is of the form

$E_{n,m}(\mathbf{q}_{i},r_{i},\mathbf{v}_{i})=\left\\{\mathbf{X}\in M_{n\times
m}(\mathbb{R}):\mathbf{q}_{i}\mathbf{X}\in
B(\mathbf{v}_{i},r_{i})\right\\}+\mathbb{Z}^{nm}\subseteq\mathbb{T}^{nm},$

where $\mathbf{v}_{i}=(v_{1}^{(i)},\dots,v_{m}^{(i)})\subseteq\mathbb{Z}^{m}$
need not be specified in this argument. In turn, each such set is naturally a
product of its projections,

$E_{n,m}(\mathbf{q}_{i},r_{i},\mathbf{v}_{i})=\prod_{j=1}^{m}E_{n,1}(\mathbf{q}_{i},r_{i},v_{j}^{(i)}).$

It is not hard to show (see for example [26, Lemma 1]) that the sets
$E_{n,1}(\mathbf{q}_{1})$ and $E_{n,1}(\mathbf{q}_{2})$ are independent if
$\mathbf{q}_{1}$ and $\mathbf{q}_{2}$ are linearly independent, hence the same
holds for the sets $E_{n,m}(\mathbf{q}_{i},r_{i},\mathbf{v}_{i})\,(i=1,2)$.
And since the unions (27) are disjoint, we may conclude that the sets
$A_{n,m}^{P}(\mathbf{q}_{i},r_{i})\,(i=1,2)$ are independent. ∎

The next lemma concerns sets $E_{1,m}(q,r,\mathbf{v})$ having the form
discussed in the proof of þ 5.2. With $n=1$, the set $E_{1,m}(q,r,\mathbf{v})$
is a grid of balls in $\mathbb{T}^{m}$. The lemma observes that as
$q\to\infty$, that grid of balls becomes uniformly distributed in
$\mathbb{T}^{m}$.

######  Lemma 5.3.

þ Let $m\geqslant 1$. For every ball $W\subseteq\mathbb{T}^{m}$, there exists
$Q\geqslant 1$ such that for all $q\geqslant Q$, $r\geqslant 0$, and
$\mathbf{v}\in\mathbb{R}^{m}$,

$\operatorname{m}\left\lparen E_{1,m}(q,r,\mathbf{v})\cap
W\right\rparen\geqslant\frac{1}{2}\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{v})\right\rparen\operatorname{m}\left\lparen
W\right\rparen.$ (28)

######  Remark.

The precise value of the constant $1/2$ is not important. It can easily be
increased to anything less than $1$, although such an improvement will not be
needed here.

###### Proof.

If $r\geqslant 1/2$, then $E_{1,m}(q,r,\mathbf{v})=\mathbb{T}^{m}$, so (28)
obviously holds. Therefore, the lemma only needs to be proved for that
$0\leqslant r<1/2$. In this case, $E_{1,m}(q,r,\mathbf{v})$ is a union of
$q^{m}$ disjoint balls of radius $r/q$, so

$\operatorname{m}(E_{1,m}(q,r,\mathbf{v}))=(2r)^{m}.$

Notice also that

$E_{1,m}(q,r,\mathbf{v})\cap
W=E_{1,m}(q,r,\mathbf{0})\cap(W+\mathbf{v}^{\prime})$

for some $\mathbf{v}^{\prime}\in\mathbb{R}^{m}$.

Let

$\mu_{q}=\frac{1}{q^{m}}\sum_{\mathbf{p}\in(\mathbb{Z}/q\mathbb{Z})^{m}}\delta_{\mathbf{p}/q}$

be the measure uniformly supported on the rational points in $\mathbb{T}^{m}$
having denominator $q$. It is well-known that these measures equidistribute,
that is,

$\mu_{q}\overset{w^{*}}{\to}\operatorname{m}\qquad\textrm{as}\qquad
q\to\infty.$

Furthermore, the rate of equidistribution is uniform, in the sense that for
every ball $V\subseteq\mathbb{T}^{m}$ and ${\varepsilon}>0$ one has

$\lvert\mu_{q}(V+\mathbf{v})-\operatorname{m}(V+\mathbf{v})\rvert<{\varepsilon}\qquad(\mathbf{v}\in\mathbb{R}^{m})$

for all $q$ sufficiently large.

Let $W^{\prime}$ be the concentric scaling of $W$ such that
$\operatorname{m}(W^{\prime})=\frac{1}{\sqrt{2}}\operatorname{m}(W)$. Then
there exists $Q\geqslant 1$ such that the following hold for all $q\geqslant
Q$:

  1. 1.

$\mu_{q}(W^{\prime}+\mathbf{w})\geqslant\frac{1}{\sqrt{2}}\operatorname{m}(W^{\prime}+\mathbf{w})=\frac{1}{2}\operatorname{m}(W+\mathbf{w})$
for all $\mathbf{w}\in\mathbb{R}^{m}$.

  2. 2.

Every ball of diameter $1/q$ centered in $W^{\prime}$ is fully contained in
$W$.

The second point implies that for every $q\geqslant Q$, and every $0\leqslant
r<1/2$,

$\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{0})\cap(W+\mathbf{v}^{\prime}\right\rparen\geqslant(2r)^{m}\mu_{q}(W^{\prime}+\mathbf{v}^{\prime})=\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{0})\right\rparen\mu_{q}(W^{\prime}+\mathbf{v}^{\prime}),$

and the first point implies that

$\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{0})\cap(W+\mathbf{v}^{\prime})\right\rparen\geqslant\frac{1}{2}\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{0})\right\rparen\operatorname{m}(W+\mathbf{v}^{\prime}).$

Equivalently,

$\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{v})\cap(W)\right\rparen\geqslant\frac{1}{2}\operatorname{m}\left\lparen
E_{1,m}(q,r,\mathbf{v})\right\rparen\operatorname{m}(W).$

The lemma is proved. ∎

In the next lemma, it is observed that if
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert$ is large, then a natural projection
of $A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ to $\mathbb{T}^{m}$ consists of
sets to which þ 5.3 applies, and therefore the sets
$A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$ inherit the uniformity of that
lemma. This is crucial for the proof of þ1.2.

######  Lemma 5.4.

þ For every open set $U\subseteq\mathbb{T}^{nm}$ there exists $Q_{0}\geqslant
1$ such that the following holds. For each $\mathbf{q}$, let
$P(\gcd(\mathbf{q}))\subseteq(\mathbb{R}/\gcd(\mathbf{q})\mathbb{Z})^{m}$ be a
finite set. If $\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$ and
(20) holds, then

$\operatorname{m}\left\lparen A_{n,m}^{P}(\mathbf{q},r)\cap
U\right\rparen\geqslant\frac{1}{3}\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},r)\right\rparen\operatorname{m}\left\lparen
U\right\rparen.$ (29)

######  Remark.

As in þ5.3, the precise value $1/3$ is not important.

###### Proof.

Since $r\geqslant 0$ satisfies (20), $A_{n,m}^{P}(\mathbf{q},r)$ is a union of
finitely many disjoint sets,

$A_{n,m}^{P}(\mathbf{q},r)=\bigcup_{E\in\mathcal{E}}E$ (30)

where each $E\in\mathcal{E}$ is of the form

$E_{n,m}(\mathbf{q},r,\mathbf{v})=\left\\{\mathbf{X}\in M_{n\times
m}(\mathbb{R}):\mathbf{q}\mathbf{X}\in
B(\mathbf{v},r)\right\\}+\mathbb{Z}^{nm}\subseteq\mathbb{T}^{nm},$

as in the proof of þ5.2. It is therefore enough to prove this lemma with
$E_{n,m}(\mathbf{q},r,\mathbf{v})$ in place of $A_{n,m}^{P}(\mathbf{q},r)$.

Since $U$ is open, there are finitely many disjoint balls
$V_{1},V_{2},\dots,V_{L}\subseteq U$ such that

$\sum_{i=1}^{L}\operatorname{m}(V_{i})\geqslant\frac{2}{3}\operatorname{m}(U).$

The lemma will be proved after establishing that there exists $Q_{0}\geqslant
1$ such that

$\operatorname{m}\left\lparen E_{n,m}(\mathbf{q},r,\mathbf{v})\cap
V\right\rparen\geqslant\frac{1}{2}\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\right\rparen\operatorname{m}\left\lparen
V\right\rparen\qquad(V=V_{1},\dots,V_{L})$ (31)

for all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$.

Write

$\mathbb{T}^{mn}=\pi(\mathbb{T}^{mn})\times\pi(\mathbb{T}^{mn})^{\perp}\cong\mathbb{T}^{m}\times\mathbb{T}^{mn-m}$
(32)

where $\pi$ is the orthogonal projection of $\mathbb{T}^{nm}$ to the row
corresponding to the coordinate in which $\lvert\mathbf{q}\rvert$ is achieved,
and $\pi(\mathbb{T}^{mn})^{\perp}\cong\mathbb{T}^{mn-n}$ is the orthogonal
complement, so that every point in $\mathbb{T}^{mn}$ has a unique expression
as $\mathbf{X}=(\mathbf{Y},\mathbf{Z})$ in the product. Denote
$\pi^{\perp}(\mathbf{Y},\mathbf{Z})=\mathbf{Z}$. The Lebesgue measure of
$\mathbb{T}^{mn}$ splits accordingly as

$\operatorname{m}_{nm}=\operatorname{m}_{m}\times\operatorname{m}_{mn-m},$

which for the rest of this proof we will write as
$\operatorname{m}=\operatorname{m}_{1}\times\operatorname{m}_{2}$ to avoid
cumbersome subscripts. For a set $S\subseteq\mathbb{T}^{mn}$, let

$S_{\mathbf{Z}}=\pi\left\lparen
S\cap(\mathbb{T}^{m}\times\\{\mathbf{Z}\\})\right\rparen$

be the cross-section of $S$ through
$\mathbf{Z}\in\pi(\mathbb{T}^{mn})^{\perp}$, so that by Fubini’s theorem, the
measure of $S$ is

$\operatorname{m}(S)=\int\operatorname{m}_{1}(S_{\mathbf{Z}})\,\operatorname{d}\operatorname{m}_{2}(\mathbf{Z}).$

Then

$\operatorname{m}\left\lparen E_{n,m}(\mathbf{q},r,\mathbf{v})\cap
V\right\rparen=\int\operatorname{m}_{1}\left\lparen(E_{n,m}(\mathbf{q},r,\mathbf{v})\cap
V)_{\mathbf{Z}}\right\rparen\,\mathop{}\\!\mathrm{d}\operatorname{m}_{2}(\mathbf{Z}).$
(33)

But for every $\mathbf{Z}\in\mathbb{T}^{mn-m}$ there is some
$\mathbf{v}_{\mathbf{Z}}\in\mathbb{R}^{m}$ such that

$\displaystyle(E_{n,m}(\mathbf{q},r,\mathbf{v})\cap V)_{\mathbf{Z}}$
$\displaystyle=E_{1,m}(\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert,r/\gcd(\mathbf{q}),\mathbf{v}_{\mathbf{Z}})\cap
V_{\mathbf{Z}}$
$\displaystyle=\begin{cases}E_{1,m}(\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert,r/\gcd(\mathbf{q}),\mathbf{v}_{\mathbf{Z}})\cap\pi(V)&\textrm{if
}\mathbf{Z}\in\pi^{\perp}(V)\\\ \emptyset&\textrm{if
}\mathbf{Z}\notin\pi^{\perp}(V).\end{cases}$

The cross-sections are sets of the kind treated in þ5.3.

Let $W_{i}=\pi(V_{i})$ for $i=1,\dots,L$, and let $Q_{i}\geqslant 1$ be as in
þ5.3. Put $Q_{0}=\max_{i}Q_{i}$. Then, as long as
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$, we have

$\displaystyle\operatorname{m}_{1}\left\lparen
E_{1,m}(\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert,r/\gcd(\mathbf{q}),\mathbf{v}_{\mathbf{Z}})\cap\pi(V)\right\rparen$
$\displaystyle\geqslant\frac{1}{2}\operatorname{m}_{1}\left\lparen
E_{1,m}(\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert,r/\gcd(\mathbf{q}),\mathbf{v}_{\mathbf{Z}})\right\rparen\operatorname{m}_{1}(\pi(V))$
$\displaystyle=\frac{1}{2}\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\right\rparen\operatorname{m}_{1}(\pi(V)).$

Returning to (33),

$\displaystyle\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\cap V\right\rparen$
$\displaystyle=\int\operatorname{m}_{1}\left\lparen(E_{n,m}(\mathbf{q},r,\mathbf{v})\cap
V)_{\mathbf{Z}}\right\rparen\,\mathop{}\\!\mathrm{d}\operatorname{m}_{2}(\mathbf{Z})$
$\displaystyle\geqslant\frac{1}{2}\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\right\rparen\int\operatorname{m}_{1}(\pi(V))\mathbf{1}_{\pi^{\perp}(V)}(\mathbf{Z})\,\mathop{}\\!\mathrm{d}\operatorname{m}_{2}(\mathbf{Z})$
$\displaystyle=\frac{1}{2}\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\right\rparen\int\operatorname{m}_{1}(V_{\mathbf{Z}})\,\mathop{}\\!\mathrm{d}\operatorname{m}_{2}(\mathbf{Z})$
$\displaystyle=\frac{1}{2}\operatorname{m}\left\lparen
E_{n,m}(\mathbf{q},r,\mathbf{v})\right\rparen\operatorname{m}(V),$

giving (31), which was the goal. ∎

The next lemma is essentially a restatement of þ4.4 (Beresnevich–Velani [8])
that applies directly in the proofs of þLABEL:thm:bootstrap,thm:bootstrap11.
Under the assumption that $\operatorname{m}\lparen
W_{1,m}^{P}(\Psi)\rparen>0$, þ4.4 provides a (sub)sequence of balls contained
in the sets $A_{1,m}^{P}(d,\Psi(d))$ which is quasi-independent on average.
The point of the following lemma is to show, using þ4.4, that there is a
refinement $R\subseteq P$ such that the sets $A_{1,m}^{R}(d,\Psi(d))$
themselves are quasi-independent on average.

######  Lemma 5.5.

þ Let $m\in\mathbb{N}$. Suppose for every $d\geqslant 1$,
$P(d)\subseteq(\mathbb{R}/d\mathbb{Z})^{m}$ is a finite set and for each
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ let $P(\mathbf{q})$ be the lift to
$\mathbb{R}^{m}$ of $P(\gcd(\mathbf{q}))$. Suppose
$\Psi:\mathbb{Z}_{+}\to\mathbb{R}_{+}$ with

$\Psi(d)=o(d)\quad\textrm{and}\quad(\\#P(d))^{1/m}\frac{\Psi(d)}{d}=O(1),$
(34)

and for each $d\geqslant 1$,

$\Psi(d)<\frac{1}{2}\min\left\\{\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\rvert:\mathbf{p}_{1},\mathbf{p}_{2}\in\bar{P}(d),\mathbf{p}_{1}\neq\mathbf{p}_{2}\right\\},$
(35)

where $\bar{P}(d)$ is the lift of $P(d)$ to $\mathbb{R}^{m}$. If
$\operatorname{m}(W_{1,m}^{P}(\Psi))>0$, then there exists a refinement
$R\subseteq P$ such that

$\sum_{d=1}^{\infty}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen=\infty$ (36)

and such that for infinitely many $D\geqslant 1$,

$\sum_{k,\ell=1}^{D}\operatorname{m}\left\lparen A_{1,m}^{R}(k,\Psi(k))\cap
A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen\leqslant
C\left\lparen\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen\right\rparen^{2}$ (37)

where $C=K\operatorname{m}\lparen W_{1,m}^{P}(\Psi)\rparen^{-2}$ and $K>0$ is
an absolute constant.

###### Proof.

Each set $A_{1,m}^{P}(d,\Psi(d))$ is a union of $\\#P(d)$ balls in
$\mathbb{T}^{m}$, which we may enumerate as

$A_{1,m}^{P}(d,\Psi(d))=\bigcup_{\mathbf{p}\in
P(d)}B\left\lparen\frac{\mathbf{p}}{d},\frac{\Psi(d)}{d}\right\rparen:=\bigcup_{j=1}^{\\#P(d)}B_{j}^{(d)},$

concatenate as

$\left\\{\left\\{B_{j}^{(d)}\right\\}_{j=1}^{\\#P(d)}\right\\}_{d\geqslant
1},$ (38)

and relabel as $\\{B_{i}\\}_{i=1}^{\infty}$. The assumption $\Psi(d)=o(d)$
implies that the radii of the balls $B_{i}$ approach $0$ as $d\to\infty$.
Since

$\operatorname{m}(\limsup_{i\to\infty}B_{i})>0,$

þ 4.4 applies. It provides a subsequence $\\{L_{i}\\}$ of $\\{B_{i}\\}$ and a
constant $C>0$ such that

$\sum_{i=1}^{\infty}\operatorname{m}(L_{i})=\infty$ (39)

and for infinitely many $Q\geqslant 1$,

$\sum_{s,t=1}^{Q}\operatorname{m}(L_{s}\cap
L_{t})\leqslant\frac{C}{2}\left\lparen\sum_{s=1}^{Q}\operatorname{m}(L_{s})\right\rparen^{2}.$
(40)

The subsequence $\\{L_{i}\\}$ corresponds naturally to a refinement
$R(d)\subseteq P(d)$ $(d\geqslant 1)$ and a subsequence of (38), which after
reindexing can be written

$\left\\{\left\\{B_{j}^{(d)}\right\\}_{j=1}^{\\#R(d)}\right\\}_{d\geqslant
1}.$ (41)

The divergence (39) is equivalent to (36), so it is only left to establish
(37).

Let $Q\geqslant 1$ be such that (40) holds. There is a corresponding
$D\geqslant 1$ such that

$\left\\{\left\\{B_{j}^{(d)}\right\\}_{j=1}^{\\#R(d)}\right\\}_{d=1}^{D}\subseteq\left\\{L_{i}\right\\}_{i=1}^{Q}\subseteq\left\\{\left\\{B_{j}^{(d)}\right\\}_{j=1}^{\\#R(d)}\right\\}_{d=1}^{D+1}.$

Then

$\displaystyle\sum_{k,\ell=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(k,\Psi(k))\cap A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen$
$\displaystyle\leqslant\sum_{s,t=1}^{Q}\operatorname{m}(L_{s}\cap L_{t})$
$\displaystyle\leqslant\frac{C}{2}\left\lparen\sum_{s=1}^{Q}\operatorname{m}(L_{s})\right\rparen^{2}$
$\displaystyle\leqslant\frac{C}{2}\left\lparen\sum_{d=1}^{D+1}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen\right\rparen^{2}.$

The last inequality follows from the fact that for each $d\geqslant 1$, the
balls $B_{j}^{(d)}$ from (41) are disjoint, which in turn is a consequence of
(35). Finally, it is easily seen that

$\displaystyle\sum_{d=1}^{D+1}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen$
$\displaystyle=\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen+\left\lparen\frac{\Psi(D+1)}{D+1}\right\rparen^{m}\\#R(D+1)$
$\displaystyle\sim\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen$

as $D\to\infty$, by (34) and the now established (36). Therefore, if
$Q\geqslant 1$ is sufficiently large, then (37) holds. Note that the constant
$C>0$ appearing in this lemma is twice the constant coming from þ4.4, which
takes the form as described. ∎

##  6 Reductions

This section contains results showing that the conclusions of
þLABEL:thm:bootstrap,thm:bootstrap11 sometimes follow without having to assume
anything about $\operatorname{m}\lparen W_{1,m}^{P}(\Psi)\rparen$ or
$\operatorname{m}(W_{1,m}^{P}(\Psi_{Q}))$. The task of proving
þLABEL:thm:bootstrap,thm:bootstrap11 then reduces to treating the cases where
the results of this section do not apply.

We first prove þ2.1, a result that appears as [28, Chapter 1, Theorem 14] in a
form that can be translated to the notation used here. We include a proof for
completeness.

###### Proof of þ2.1.

Notice that if $\Psi_{Q}(d)=\infty$, then $\Psi_{1}(d)=\infty$. Let

$b=\min\left\\{\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\rvert:\mathbf{p}_{1},\mathbf{p}_{2}\in
P(d),\mathbf{p}_{1}\neq\mathbf{p}_{2}\right\\}.$

Suppose there are infinitely many $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with
$\gcd(\mathbf{q})=d$ for which $\psi(\mathbf{q})>b/2$, and denote by $I$ the
set of such $\mathbf{q}$. For each $\mathbf{q}\in I$ we have

$A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))\supseteq
A_{n,m}^{P}(\mathbf{q},b/2).$ (42)

By þ5.1,

$\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},b/2)\right\rparen=\frac{\\#P(d)b^{m}}{d^{m}}>0$

for all $\mathbf{q}\in I$. Therefore,

$\sum_{\mathbf{q}\in I}\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},b/2)\right\rparen=\infty.$

By þ5.2 the sets $A_{n,m}^{P}(\mathbf{q},b/2)$ $(\mathbf{q}\in I)$ are
pairwise independent, therefore by the second Borel–Cantelli lemma,

$\operatorname{m}\left\lparen\limsup_{\begin{subarray}{c}\lvert\mathbf{q}\rvert\to\infty\\\
\mathbf{q}\in I\end{subarray}}A_{n,m}^{P}(\mathbf{q},b/2)\right\rparen=1,$

hence, by (42), $\operatorname{m}\lparen W_{n,m}^{P}(\psi)\rparen=1$.

If, on the other hand, $I$ is a finite set, then by $\Psi_{1}(d)=\infty$ we
must have

$\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=d\\\ \mathbf{q}\notin
I\end{subarray}}\psi(\mathbf{q})^{m}=\infty.$

But for each $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ with $\gcd(\mathbf{q})=d$ and
$\mathbf{q}\notin I$ we have

$\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))\right\rparen=\frac{\\#P(d)}{d^{m}}(2\psi(\mathbf{q}))^{m},$

therefore

$\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=d\\\ \mathbf{q}\notin
I\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))\right\rparen=\infty.$

Again, by þ5.2 the sets $A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))$
$(\gcd(\mathbf{q})=d,\,\mathbf{q}\notin I)$ are pairwise independent,
therefore,

$\operatorname{m}\left\lparen\limsup_{\begin{subarray}{c}\lvert\mathbf{q}\rvert\to\infty\\\
\gcd(\mathbf{q})=d\\\ \mathbf{q}\notin
I\end{subarray}}A_{n,m}^{P}(\mathbf{q},\psi(\mathbf{q}))\right\rparen=1,$

hence $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen=1$. ∎

######  Claim 6.1.

þ Suppose $P$ is relatively well-spread and let $a>0$ be as in (3). If there
are infinitely many $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ for which

$\psi(\mathbf{q})\geqslant\frac{a}{2}\left\lparen\frac{\gcd(\mathbf{q})}{(\\#P(\gcd(\mathbf{q})))^{1/m}}\right\rparen$
(43)

holds, then $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen>0$.
If for infinitely many $Q\geqslant 1$ there exists some
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ satisfying (43) as well as
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q$, then then
$\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen=1$.

###### Proof.

Let $P^{\prime}$ be as in Definition 1.1. For each
$\mathbf{q}\in\mathbb{Z}_{+}^{n}$ satisfying (43), þ5.1 implies

$\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}\left\lparen\mathbf{q},\psi(\mathbf{q})\right\rparen\right\rparen\geqslant\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}\left\lparen\mathbf{q},\frac{a}{2}\left\lparen\frac{\gcd(\mathbf{q})}{(\\#P(\gcd(\mathbf{q})))^{1/m}}\right\rparen\right\rparen\right\rparen\geqslant
a^{m}c>0,$ (44)

where $c>0$ is as in (2). This immediately implies that
$\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen\geqslant
a^{m}c>0$, which proves the first part of the claim.

For the second part of the claim, notice that we may form a sequence
$\\{\mathbf{q}_{k}\\}_{k\geqslant 1}\subseteq\mathbb{Z}_{+}^{n}$ of vectors
satisfying (43), such that $\lvert\mathbf{q}_{k}/\gcd(\mathbf{q}_{k})\rvert$
increases strictly. But each $\mathbf{q}_{k}/\gcd(\mathbf{q}_{k})$ is
primitive, so the vectors $\mathbf{q}_{k}$ are pairwise linearly independent.
Therefore, by þ5.2, the sets

$A_{n,m}^{P^{\prime}}\left\lparen\mathbf{q}_{k},\frac{a}{2}\left\lparen\frac{\gcd(\mathbf{q}_{k})}{(\\#P(\gcd(\mathbf{q}_{k})))^{1/m}}\right\rparen\right\rparen$

are pairwise independent. Since they all have measure $\geqslant a^{m}c>0$,
their measure sum diverges and the second Borel–Cantelli lemma implies that

$\limsup_{k\to\infty}A_{n,m}^{P^{\prime}}\left\lparen\mathbf{q}_{k},\frac{a}{2}\left\lparen\frac{\gcd(\mathbf{q}_{k})}{(\\#P(\gcd(\mathbf{q}_{k})))^{1/m}}\right\rparen\right\rparen$

has full measure, hence $\operatorname{m}\left\lparen
W_{n,m}^{P}(\psi)\right\rparen=1$. ∎

######  Claim 6.2.

þ Assume $P$ is relatively well-spread and that

$\psi(\mathbf{q})\leqslant\frac{a}{2}\frac{\gcd(\mathbf{q})}{(\\#P(\gcd(\mathbf{q})))^{1/m}}\qquad(\mathbf{q}\in\mathbb{Z}_{+}^{n}),$
(45)

where $a>0$ is as in (3), and $\Psi(d)<\infty$ for all $d\geqslant 1$. If
there exists ${\delta}>0$ such that

$\limsup_{d\to\infty}(\\#P(d))^{1/m}\frac{\Psi(d)}{d}\geqslant{\delta},$

then $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen>0$.
Furthermore, if for infinitely many $Q\geqslant 1$,

$\limsup_{d\to\infty}(\\#P(d))^{1/m}\frac{\Psi_{Q}(d)}{d}\geqslant{\delta},$
(46)

then $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen=1$.

###### Proof.

Let $P^{\prime}$ be as in Definition 1.1. The assumption (45) guarantees that

$\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen=\\#P^{\prime}(\gcd(\mathbf{q}))\left\lparen\frac{2\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}\geqslant
c\\#P(\gcd(\mathbf{q}))\left\lparen\frac{2\psi(\mathbf{q})}{\gcd(\mathbf{q})}\right\rparen^{m}$

for every $\mathbf{q}$ and therefore

$\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\geqslant\sum_{\gcd(\mathbf{q})=d}c\\#P(d)\left\lparen\frac{2\psi(\mathbf{q})}{d}\right\rparen^{m}=c\\#P(d)\left\lparen\frac{2}{d}\right\rparen^{m}\Psi(d)^{m}.$

The assumption on $\Psi$ implies that there exists a number $M:=M_{\delta}>0$
and an integer sequence $1\leqslant d_{1}<d_{2}<d_{3}<\dots$ such that

$M\leqslant\sum_{\gcd(\mathbf{q})=d_{k}}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen<\infty\qquad(k\geqslant
1).$

By þ4.1, for each $d\in\\{d_{k}\\}_{k\geqslant 1}$ one has

$\operatorname{m}\left\lparen\bigcup_{\gcd(\mathbf{q})=d}A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\geqslant\frac{\left\lparen\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}}{\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen}$ (47)

and by þ5.2,

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen\\\
=\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen+\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})=d\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen\\\
\leqslant\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen+\left\lparen\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})=d\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}.$

Putting this into (47) leads to

$\displaystyle\operatorname{m}\left\lparen\bigcup_{\gcd(\mathbf{q})=d}A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen$
$\displaystyle\geqslant\frac{\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen}{1+\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen}$
$\displaystyle\geqslant\frac{M}{M+1}>0,$

for all $d\in\\{d_{k}\\}_{k\geqslant 1}$. Consequently, the limsup set
associated to the sets $A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))$
with $\gcd(\mathbf{q})\in\\{d_{k}\\}_{k\geqslant 1}$ has measure at least
$M/(M+1)$, and so does $W_{n,m}^{P}(\psi)$. This proves the first part of the
claim.

For the second part, let $U\subseteq\mathbb{T}^{nm}$ be an open set and let
$Q_{0}\geqslant 1$ be such that (29) holds for all
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$, as per þ 5.4.
Suppose $Q\geqslant Q_{0}$ is such that (46) holds and note that

$\Psi_{Q}(d)=\left\lparen\sum_{\gcd(\mathbf{q})=d}\psi_{Q}(\mathbf{q})^{m}\right\rparen^{1/m}$

where

$\psi_{Q}(\mathbf{q})=\begin{cases}\psi(\mathbf{q})&\textrm{ if
}\left\lvert\frac{\mathbf{q}}{\gcd(\mathbf{q})}\right\rvert\geqslant Q\\\
0&\textrm{ otherwise }.\end{cases}$

As in the first part, there is an integer sequence $1\leqslant
d_{1}<d_{2}<\dots$ such that for all $d\in\\{d_{k}\\}_{k\geqslant 1}$,

$\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\right\rparen\geqslant
M,$

and taking þ 5.4 into account,

$\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\geqslant\frac{1}{3}\operatorname{m}(U)M.$

Now

$\operatorname{m}\left\lparen\bigcup_{\gcd(\mathbf{q})=d}A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\\\
\geqslant\frac{\left\lparen\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\right\rparen^{2}}{\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi_{Q}(\mathbf{r})))\cap U\right\rparen}\\\
\geqslant\frac{1}{9}\operatorname{m}(U)^{2}\frac{\left\lparen\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\right\rparen\right\rparen^{2}}{\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi_{Q}(\mathbf{r})))\right\rparen}\\\
\geqslant\frac{M}{9(M+1)}\operatorname{m}(U)^{2},$

after following the same reasoning as in the first part. This implies that

$\operatorname{m}\lparen W_{n,m}^{P^{\prime}}(\psi)\cap
U\rparen\geqslant\frac{1}{9}\left\lparen\frac{M}{M+1}\right\rparen\operatorname{m}(U)^{2},$

and therefore $\operatorname{m}\lparen W_{n,m}^{P}(\psi)\rparen=1$ by an
application of þ 4.6. ∎

######  Claim 6.3.

þ Assume $P$ is relatively uniformly discrete and that

$\psi(\mathbf{q})\leqslant\frac{b}{2}\qquad(\mathbf{q}\in\mathbb{Z}_{+}^{n})$
(48)

where $b>0$ is as in (2), and $\Psi(d)<\infty$ for all $d\geqslant 1$. If
there exists ${\delta}>0$ such that

$\limsup_{d\to\infty}\frac{\Psi(d)}{d}>{\delta},$

then $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen>0$. If, in
addition, $\\#P(d)\to\infty$ as $d\to\infty$, then
$\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen=1$.

###### Proof.

The assumption implies that there are infinitely many $d\geqslant 1$ such that

$\Psi(d)^{m}=\sum_{\gcd(\mathbf{q})=d}\psi(\mathbf{q})^{m}\geqslant(\delta
d)^{m}.$ (49)

Let $P^{\prime}$ be a refinement of $P$ as in þ 1.1. Since
$\psi(\mathbf{q})\leqslant b/2$, the set
$A_{1,m}^{P^{\prime}}(\gcd(\mathbf{q}),\psi(\mathbf{q}))$ is a union of
disjoint balls, and þ5.1 implies

$\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen=\sum_{\gcd(\mathbf{q})=d}(\\#P^{\prime}(d))\left\lparen\frac{\psi(\mathbf{q})}{d}\right\rparen^{m}\geqslant\delta^{m}(\\#P^{\prime}(d))$
(50)

for each $d\geqslant 1$ satisfying (49). By þ4.1, for each $d\geqslant 1$
satisfying (50), one has

$\operatorname{m}\left\lparen\bigcup_{\gcd(\mathbf{q})=d}A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\geqslant\frac{\left\lparen\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}}{\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen}.$ (51)

By þ5.2,

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen\\\
=\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen+\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})=d\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{r},\psi(\mathbf{r})))\right\rparen\\\
\leqslant\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen+\left\lparen\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})=d\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}.$

Putting this into (51) gives

$\displaystyle\operatorname{m}\left\lparen\bigcup_{\gcd(\mathbf{q})=d}A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen$
$\displaystyle\geqslant\frac{\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen}{1+\sum_{\gcd(\mathbf{q})=d}\operatorname{m}\left\lparen
A_{n,m}^{P^{\prime}}(\mathbf{q},\psi(\mathbf{q}))\right\rparen}$
$\displaystyle\geqslant\frac{\delta^{m}(\\#P^{\prime}(d))}{\delta^{m}(\\#P^{\prime}(d))+1}>0,$
(52)

for all $d\geqslant 1$ for which (50) holds. Since there are infinitely many
such $d\geqslant 1$, and since for each such $d\geqslant 1$ one has
$\\#P^{\prime}(d)\geqslant 1$, it follows that in this case
$\operatorname{m}\left\lparen W_{n,m}(\psi)\right\rparen>0$ and the theorem is
proved. Under the additional assumption that
$\\#P(d)\to\infty,\,(d\to\infty)$, we have that $\\#P^{\prime}(d)\to\infty$,
and the expression in (52) approaches $1$. From here, it follows that
$\operatorname{m}\lparen W_{n,m}^{P}(\psi)\rparen=1$. ∎

##  7 Proof of þ1.1(a)

By þ6.1, it may be assumed that (45) holds for all but finitely many
$\mathbf{q}$. No generality is lost in further assuming that (45) holds for
all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$. þ2.1 and þ6.2 show that it is safe to
assume that $\Psi(d)$ is always finite and that

$(\\#P(d))^{1/m}\frac{\Psi(d)}{d}\to 0\qquad(d\to\infty).$ (53)

By scaling $\psi$ if necessary, it may be assumed that

$\Psi(d)\leqslant\frac{a}{2}\left\lparen\frac{d}{(\\#P(d))^{1/m}}\right\rparen\qquad(d\geqslant
1)$ (54)

where $a>0$ is as in (3). Note that such a scaling of $\Psi$ does not alter
the measure of $W_{1,m}^{P}(\Psi)$, by Cassels’ lemma (þ4.5).

It is assumed that $\operatorname{m}\lparen W_{1,m}^{P}(\Psi)\rparen>0$. For
each $d\geqslant 1$, let $P^{\prime}(d)$ be a refinement of $P(d)$ as in þ1.1,
and then augment $P^{\prime}(d)$ by including points of $P(d)$ until
$P^{\prime}(d)$ is maximal with respect to the requirement that the balls

$B\left\lparen\frac{\mathbf{p}}{d},\frac{\Psi(d)}{d}\right\rparen\qquad(\mathbf{p}\in
P^{\prime}(d))$

be disjoint in $\mathbb{T}^{m}$. Now $A_{1,m}^{P^{\prime}}(d,\Psi(d))$ is a
union of $\\#P^{\prime}(d)\geqslant c\\#P(d)$ disjoint balls in
$\mathbb{T}^{m}$ such that every ball from $A_{1,m}^{P}(d,\Psi(d))$ is
intersected. In particular,

$A_{1,m}^{P}(d,\Psi(d))\subseteq A_{1,m}^{P^{\prime}}(d,3\Psi(d)),$

which implies that $\operatorname{m}\lparen
W_{1,m}^{P^{\prime}}(3\Psi)\rparen\geqslant\operatorname{m}\lparen
W_{1,m}^{P}(\Psi)\rparen>0$. Then, by Cassels’ lemma (þ4.5), we have
$\operatorname{m}\lparen
W_{1,m}^{P^{\prime}}(\Psi)\rparen\geqslant\operatorname{m}\lparen
W_{1,m}^{P}(\Psi)\rparen>0.$

By (53) and (54) we have (34) and (35), so þ5.5 applies. It gives a refinement
$R(d)\subseteq P^{\prime}(d)$ such that the sets $A_{1,m}^{R}(d,\Psi(d))$ have
a diverging measure sum and are quasi-independent on average, that is, there
exists a constant $C>0$ and infinitely many $D\geqslant 1$ such that

$\sum_{k,\ell=1}^{D}\operatorname{m}\left\lparen A_{1,m}^{R}(k,\Psi(k))\cap
A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen\leqslant
C\left\lparen\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen\right\rparen^{2}.$ (55)

It is worth recalling that for each $d\geqslant 1$, the balls constituting set
$A_{1,m}^{R}(d,\Psi(d))$ are disjoint.

(We also point out that we may take

$C=K\kappa^{-2}\qquad\textrm{where}\qquad
0<\kappa\leqslant\operatorname{m}(W_{1,m}^{P}(\Psi)),$ (56)

and $K>0$ is a constant. This observation is not needed here, but will be
useful in the proof of þ1.2 in §9.)

The aim now is to show that the sets
$A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))$ are quasi-independent on average. A
bound is needed on

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\
=\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\not\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\
+\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen.$ (57)

By þ5.2,

$\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\not\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\
=\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\not\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\
\leqslant\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}$ (58)

Evidently, only the last term (57) needs to be controlled. We express it as

$\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\parallel\mathbf{r}\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\ =\sum_{1\leqslant
k<\ell\leqslant D}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=k\\\
\gcd(\mathbf{r})=\ell\\\
\mathbf{q}\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen$

Letting
$\mathbf{q}^{\prime}=\mathbf{q}/\gcd(\mathbf{q})=\mathbf{r}/\gcd(\mathbf{r})$,
notice that

$A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))=T^{-1}\left\lparen
A_{1,m}^{R}(k,\psi(\mathbf{q}))\cap
A_{1,m}^{R}(\ell,\psi(\mathbf{r}))\right\rparen,$

where $T:\mathbb{T}^{mn}\to\mathbb{T}^{m}$ is the measure-preserving
projection $\mathbf{X}\mapsto\mathbf{q}^{\prime}\mathbf{X}$. Hence,

$\sum_{\begin{subarray}{c}\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant D\\\
\mathbf{q}\parallel\mathbf{r}\\\
\mathbf{q}\neq\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\ =\sum_{1\leqslant
k<\ell\leqslant D}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=k\\\
\gcd(\mathbf{r})=\ell\\\
\mathbf{q}\parallel\mathbf{r}\end{subarray}}\operatorname{m}\left\lparen
A_{1,m}^{R}(k,\psi(\mathbf{q}))\cap
A_{1,m}^{R}(\ell,\psi(\mathbf{r}))\right\rparen\\\
\overset{\textrm{\th\ref{lem:dilation}}}{\leqslant}\sum_{1\leqslant
k<\ell\leqslant D}\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=k\\\
\gcd(\mathbf{r})=\ell\\\
\mathbf{q}\parallel\mathbf{r}\end{subarray}}\max\left\\{\frac{\psi(\mathbf{q})}{\Psi(k)},\frac{\psi(\mathbf{r})}{\Psi(\ell)}\right\\}^{m}\operatorname{m}\left\lparen
A_{1,m}^{R}(k,\Psi(k))\cap A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen\\\
=\sum_{1\leqslant k<\ell\leqslant D}\operatorname{m}\left\lparen
A_{1,m}^{R}(k,\Psi(k))\cap
A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen\underbrace{\sum_{\begin{subarray}{c}\gcd(\mathbf{q})=k\\\
\gcd(\mathbf{r})=\ell\\\
\mathbf{q}\parallel\mathbf{r}\end{subarray}}\max\left\\{\frac{\psi(\mathbf{q})}{\Psi(k)},\frac{\psi(\mathbf{r})}{\Psi(\ell)}\right\\}^{m}}_{\leqslant
2}$ (59)

Combining (55), (57), (58), and (59), there are infinitely many $D\geqslant 1$
for which

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\\\
\leqslant\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}+2C\left\lparen\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen\right\rparen^{2}.$ (60)

Finally, note that

$\displaystyle\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen$
$\displaystyle=\sum_{d=1}^{D}\\#R(d)\left\lparen\frac{\Psi(d)}{d}\right\rparen^{m}$
$\displaystyle=\sum_{d=1}^{D}\sum_{\gcd(\mathbf{q})=d}\\#R(d)\left\lparen\frac{\psi(\mathbf{q})}{d}\right\rparen^{m}$
$\displaystyle=\sum_{\gcd(\mathbf{q})\leqslant D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\right\rparen.$

Putting this into (60) shows that the sets
$A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))$ are quasi-independent on average,
that is, there exists a constant $C^{\prime}>0$ and infinitely many
$D\geqslant 1$ such that

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi(\mathbf{r}))\right\rparen\leqslant
C^{\prime}\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))\right\rparen\right\rparen^{2}.$ (61)

Finally, by þ4.3, their associated limsup set has positive measure. This
implies that $\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\right\rparen>0$.
∎

######  Remark.

Evidently, we may take

$C^{\prime}=1+2K{\kappa}^{-2}$ (62)

for any $0<{\kappa}\leqslant\operatorname{m}(W_{1,m}^{P}(\Psi))$, where $K>0$
is a constant only depending on $\mathbb{T}^{m}$ as in (56).

##  8 Proof of þ1.1(b)

Since it is assumed that $\psi$ is bounded, no generality is lost in assuming
that $\psi(\mathbf{q})\leqslant b/2$ for all $\mathbf{q}$, where $b>0$ is as
in (2), that is,

$\left\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\right\rvert\geqslant
b\qquad(d\geqslant 1,\mathbf{p}_{1},\mathbf{p}_{2}\in
P^{\prime}(d),\mathbf{p}_{1}\neq\mathbf{p}_{2})$

with $P^{\prime}(d)\subseteq P(d)$ the refinement in Definition 1.1. After
all, one can always scale $\psi$ by a constant, resulting in a correspondingly
scaled $\Psi$. By Cassels’ lemma (þ4.5), $\operatorname{m}(W_{1,m}^{P}(\Psi))$
is unchanged. By þ2.1 it may be assumed that $\Psi(d)$ is finite for every
$d\geqslant 1$, and by þ6.3 it may be assumed that $\Psi(d)=o(d)$. It is
assumed that $\operatorname{m}\lparen W_{1,m}^{P}(\Psi)\rparen>0$.

Since $A_{1,m}^{P}(d,\Psi(d))$ is a union of finitely many balls in the torus
$\mathbb{T}^{m}$, Vitali’s covering lemma (þ4.7) guarantees that that
collection of balls contains a disjoint subcollection whose three-fold dilates
cover the original. In other words, there exists a subset $Q(d)\subseteq P(d)$
such that $A_{1,m}^{Q}(d,\Psi(d))$ is a union of finitely many _disjoint_
balls and such that

$\operatorname{m}\left\lparen A_{1,m}^{Q}(d,\Psi(d))\right\rparen\geqslant
3^{-m}\operatorname{m}\left\lparen A_{1,m}^{P}(d,\Psi(d))\right\rparen.$

In particular,

$\sum_{d\geqslant 1}\operatorname{m}\left\lparen
A_{1,m}^{Q}(d,\Psi(d))\right\rparen=\infty.$

Furthermore, since

$A_{1,m}^{Q}(d,3\Psi(d))\supseteq A_{1,m}^{P}(d,\Psi(d)),$

we have

$\operatorname{m}\left\lparen
W_{1,m}^{Q}(3\Psi)\right\rparen\geqslant\operatorname{m}\left\lparen
W_{1,m}^{P}(\Psi)\right\rparen>0,$

hence, by Cassels’ lemma (þ4.5), $\operatorname{m}\lparen
W_{1,m}^{Q}(\Psi)\rparen\geqslant\operatorname{m}\lparen
W_{1,m}^{P}(\Psi)\rparen>0$.

Now, by þ5.5, we may further refine $Q$ to $R(d)\subseteq Q(d)$ in such a way
that the sets $A_{1,m}^{R}(d,\Psi(d))$ have a diverging measure sum and are
quasi-independent on average, that is, there exists a constant $C>0$ and
infinitely many $D\geqslant 1$ such that

$\sum_{k,\ell=1}^{D}\operatorname{m}\left\lparen A_{1,m}^{R}(k,\Psi(k))\cap
A_{1,m}^{R}(\ell,\Psi(\ell))\right\rparen\leqslant
C\left\lparen\sum_{d=1}^{D}\operatorname{m}\left\lparen
A_{1,m}^{R}(d,\Psi(d))\right\rparen\right\rparen^{2}.$ (63)

The aim now is to show that the sets
$A_{n,m}^{R}(\mathbf{q},\psi(\mathbf{q}))$ are quasi-independent on average.
It is established exactly as in the Proof of þ1.1(a) from (57) on, verbatim. ∎

##  9 Proof of þ1.2(a)

By þ2.1 we may assume that $\Psi(d)$ and $\Psi_{Q}(d)$ are always finite. By þ
6.1, it may be assumed that for $Q_{0}\geqslant 1$ sufficiently large (45)
holds for all but finitely many $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ having
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$. Select such a
$Q_{0}\geqslant 1$, and note that we lose no generality in assuming that (45)
is satisfied by _all_ $\mathbf{q}\in\mathbb{Z}_{+}^{n}$ having
$\lvert\mathbf{q}/\gcd(\mathbf{q})\rvert\geqslant Q_{0}$. (This entails
altering $\psi(\mathbf{q})$ for at most finitely many $\mathbf{q}$.)

Let $U\subseteq\mathbb{T}^{mn}$ be an open set and, if necessary, enlarge
$Q_{0}\geqslant 1$ so that it is large enough that þ5.4 applies. By þ6.2, we
may further enlarge $Q_{0}$ until it is large enough that

$\Psi_{Q}(d)\leqslant\frac{a}{2}\left\lparen\frac{d}{(\\#P(d))^{1/m}}\right\rparen\qquad(d\gg
1),$ (64)

for all $Q\geqslant Q_{0}$, where $a>0$ is as in (3).

Define $\psi_{Q}$ by

$\psi_{Q}(\mathbf{q})=\begin{cases}\psi(\mathbf{q})&\textrm{ if
}\left\lvert\frac{\mathbf{q}}{\gcd(\mathbf{q})}\right\rvert\geqslant Q\\\
0&\textrm{ otherwise }.\end{cases}$ (65)

Then

$\Psi_{Q}(d)=\left\lparen\sum_{\gcd(\mathbf{q})=d}\psi_{Q}(\mathbf{q})^{m}\right\rparen^{1/m}.$
(66)

It has been assumed that $\operatorname{m}\lparen
W_{1,m}^{P}(\Psi_{Q})\rparen\geqslant{\kappa}>0$, where ${\kappa}>0$ is a
constant. By (64) and the assumption that $\\#P(d)\to\infty$, we have
$\Psi_{Q}(d)/d\to 0$. We are therefore in a position to follow the proof of
þ1.1(a) in §7. It leads to a refinement $R\subseteq P$ such that (61) holds
for the sets $A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))$ and such that
their measure sum diverges. Specifically, there are infinitely many
$D\geqslant 1$ for which

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi_{Q}(\mathbf{r}))\right\rparen\leqslant
C^{\prime}\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\right\rparen\right\rparen^{2},$
(67)

where $C^{\prime}=1+K{\kappa}^{-2}$, with $K>0$ a constant, as remarked in
(62). By þ5.4,

$\operatorname{m}\left\lparen A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\geqslant\frac{1}{3}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\right\rparen\operatorname{m}\left\lparen
U\right\rparen$

for all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$. Therefore, there are infinitely
many $D\geqslant 1$ for which

$\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi_{Q}(\mathbf{r}))\cap U\right\rparen\\\
\leqslant\sum_{\gcd(\mathbf{q}),\gcd(\mathbf{r})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
A_{n,m}^{R}(\mathbf{r},\psi_{Q}(\mathbf{r}))\right\rparen\\\
\overset{(\ref{eq:24})}{\leqslant}C^{\prime}\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\right\rparen\right\rparen^{2}\\\
\leqslant\frac{9C^{\prime}}{\operatorname{m}(U)^{2}}\left\lparen\sum_{\gcd(\mathbf{q})\leqslant
D}\operatorname{m}\left\lparen
A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\right\rparen^{2}.$

By þ4.3,

$\operatorname{m}\left\lparen W_{n,m}^{R}(\psi_{Q})\cap
U\right\rparen=\operatorname{m}\left\lparen\limsup_{\lvert\mathbf{q}\rvert\to\infty}A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))\cap
U\right\rparen\geqslant\frac{\operatorname{m}(U)^{2}}{9C^{\prime}},$

and since $W_{n,m}^{R}(\psi_{Q})\subseteq W_{n,m}^{P}(\psi)$, it follows that

$\operatorname{m}\left\lparen W_{n,m}^{P}(\psi)\cap
U\right\rparen\geqslant\frac{\operatorname{m}(U)^{2}}{9C^{\prime}}.$

Finally, þ4.6 implies that $\operatorname{m}\left\lparen
W_{n,m}^{P}(\psi)\right\rparen=1$, completing the proof. ∎

##  10 Proof of þ1.2(b)

As in the proof of þ1.1(b) in §8, it may be assumed that
$\psi(\mathbf{q})\leqslant b/2$ for all $\mathbf{q}$, where

$\left\lvert\mathbf{p}_{1}-\mathbf{p}_{2}\right\rvert\geqslant
b\qquad(d\geqslant 1,\mathbf{p}_{1},\mathbf{p}_{2}\in
P(d),\mathbf{p}_{1}\neq\mathbf{p}_{2}),$

and, by þ2.1, that $\Psi(d)$ is finite for every $d\geqslant 1$. In fact, þ6.3
implies that $\Psi(d)=o(d)$ may be assumed. It is further assumed that
$\operatorname{m}\left\lparen W_{1,m}^{P}(\Psi)\right\rparen>0$.

Let $U\subseteq\mathbb{T}^{nm}$ be an open set, and $Q\geqslant 1$ be as in
þ5.4. Define $\psi_{Q}$ as in (65) so that (66) holds. Note that
$\psi_{Q}(\mathbf{q})\leqslant b/2$ for all $\mathbf{q}\in\mathbb{Z}_{+}^{n}$,
and that $\Psi_{Q}(d)<\infty$ for all $d\geqslant 1$, and that
$\Psi_{Q}(d)=o(d)$. It is further assumed that
$\inf_{Q}\operatorname{m}\lparen W_{1,m}^{P}(\Psi_{Q})\rparen>0$. The
arguments in §8 show that there is a refinement $R\subseteq P$ such that the
sets $A_{n,m}^{R}(\mathbf{q},\psi_{Q}(\mathbf{q}))$ are quasi-independent on
average and have diverging measure sum. The rest of this proof proceeds
exactly as in the proof of þ1.2(a), starting from (67). ∎

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Wesleyan University

Department of Mathematics and Computer Science

Middletown, CT 06459

United States

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