# Weighted approximation for limsup sets

Gerardo González Robert  G. González Robert, Department of Mathematical and
Physical Sciences, La Trobe University, Bendigo 3552, Australia.
<EMAIL_ADDRESS>,  Mumtaz Hussain  Mumtaz Hussain, Department of
Mathematical and Physical Sciences, La Trobe University, Bendigo 3552,
Australia<EMAIL_ADDRESS>,  Nikita Shulga  Nikita Shulga,
Department of Mathematical and Physical Sciences, La Trobe University, Bendigo
3552, Australia<EMAIL_ADDRESS>and  Benjamin Ward  Ben Ward,
Department of Mathematical and Physical Sciences, La Trobe University, Bendigo
3552, Australia<EMAIL_ADDRESS>
###### Abstract.

Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine
approximation are fundamental results on the metric properties of $\Psi$-well
approximable sets. These foundational results have since been generalised to
the framework of weighted Diophantine approximation for systems of real linear
forms (matrices). In this article, we prove analogues of these weighted
results in a range of settings including the $p$-adics (Theorems 7 and 8),
complex numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and
formal power series (Theorems 13 and 14). We also consider approximation by
uniformly distributed sequences. Under some assumptions on the approximation
functions, we prove a 0-1 dichotomy law (Theorem 15). We obtain divergence
results for any approximation function under some natural restrictions on the
discrepancy (Theorems 16, 17, and 19).

The key tools in proving the main parts of these results are the weighted
ubiquitous systems and weighted mass transference principle introduced
recently by Kleinbock and Wang [Adv. Math. 428 (2023), Paper No. 109154], and
Wang & Wu [Math. Ann. 381 (2021), no. 1-2, 243–317] respectively.

This research is supported by the ARC Discovery Project 200100994.

###### Contents

  1. 1 Introduction
  2. 2 Toolbox
    1. 2.1 Dirichlet and Minkowski’s theorems
    2. 2.2 Weighted Ubiquitous systems
    3. 2.3 Ambient measure statements
    4. 2.4 Hausdorff measure and dimension statements
  3. 3 A motivating example
    1. 3.1 Proof of Corollary 1
    2. 3.2 Proof of Corollary 2
  4. 4 $p$-adic approximation
    1. 4.1 Ubiquity for $p$-adics
    2. 4.2 Proof of Theorem 7
    3. 4.3 Proof of Theorem 8
  5. 5 Complex approximation
    1. 5.1 A Minkowski-type theorem
    2. 5.2 Proof of Theorem 9
    3. 5.3 Proof of Theorem 10
  6. 6 Quaternion approximation
    1. 6.1 Ubiquity for quaternions
    2. 6.2 Proof of Theorem 11
    3. 6.3 Proof of Theorem 12
  7. 7 Formal power series approximation
    1. 7.1 Proof of Theorem 13
    2. 7.2 Proof of Theorem 14
      1. 7.2.1 Upper bound.
      2. 7.2.2 Lower bound.
  8. 8 Uniformly distributed sequences
    1. 8.1 The ubiquity property
    2. 8.2 Proof of Theorem 15
    3. 8.3 Proof of Theorem 16
    4. 8.4 Proof of Theorem 17
    5. 8.5 Proof of Theorems 19 and 20
      1. 8.5.1 Upper bound
      2. 8.5.2 Lower bound

##  1\. Introduction

Let $n,m\geq 1$ be integers and $\Psi=(\psi_{1},\dots,\psi_{n})$ be an
$n$-tuple of multivariable approximation functions of the form
$\psi_{i}:{\mathbb{Z}}^{m}\to{\mathbb{R}}_{+}$ with

$\psi_{i}(\textbf{q})\to 0\quad\text{ as
}\|\textbf{q}\|:=\max(|q_{1}|,\ldots,|q_{m}|)\to\infty.$

For $\mathbf{p}\in{\mathbb{Z}}^{m}$ and $i\in\\{1,\ldots,m\\}$, let $p_{i}$ be
the $i$-th coordinate of $\mathbf{p}$. Let

$W_{n,m}(\Psi):=\left\\{X\in[0,1]^{m\times
n}:\begin{array}[]{l}|\textbf{q}X_{i}+p_{i}|<\psi_{i}(\textbf{q})\quad 1\leq
i\leq n,\\\ \text{ for i. m.
}(\mathbf{p},\textbf{q})\in{\mathbb{Z}}^{n}\times{\mathbb{Z}}^{m}\backslash\\{\textbf{0}\\}\end{array}\right\\}.$

Here and in what follows “i. m.” stands for “infinitely many”.
$X=(x_{i,j})_{1\leq i\leq m,1\leq j\leq n}$ is an $m\times n$ matrix with
entries $x_{i,j}\in[0,1]$, and $X_{i}$ denotes the $i$th column vector of $X$.
So

$\textbf{q}X+\mathbf{p}=\left(\begin{array}[]{c}\textbf{q}X_{1}+p_{1}\\\
\vdots\\\
\textbf{q}X_{n}+p_{n}\end{array}\right)=\left(\begin{array}[]{c}q_{1}x_{1,1}+\dots+q_{m}x_{m,1}+p_{1}\\\
\vdots\\\ q_{1}x_{1,n}+\dots+q_{m}x_{m,n}+p_{n}\end{array}\right).$

A Khintchine-Groshev type theorem tells us the $nm$-dimensional Lebesgue
measure of $W_{n,m}(\Psi)$ which is either zero or full depending upon the
convergence or divergence of a certain series respectively. Naturally, the
series is dependent upon the nature of the approximation functions.

There are many variations of Khintchine-Groshev type theorems, to highlight a
few we recall the following definitions. When $\psi_{1}=\dots=\psi_{n}$ we say
the approximation function $\Psi$ is non-weighted and simply denote it by
$\psi$, otherwise, it is called weighted. If the approximation functions
$\psi_{i}$ are of the form $\phi_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ with
$\psi_{i}(\textbf{a})=\phi_{i}(\|\textbf{a}\|)$ for each $1\leq i\leq n$ we
say the approximation function is univariable. A generalisation of univariable
approximation function is when the sup norm is replaced by a quasi-norm
$\|\cdot\|_{v}=\max_{1\leq i\leq m}|\cdot|^{1/v_{i}}$ with vector
$v=(v_{1},\dots,v_{m})$ for each $v_{i}>0$ and $\sum v_{i}=m$. If each
approximation function is of the form
$\psi_{i}(\textbf{a})=\phi_{i}(\|\textbf{a}\|_{v})$ we follow [6] and say
$\Psi$ has property P. If each approximation function $\psi_{i}$ is monotonic
decreasing we say $\Psi$ is monotonic decreasing. In the case of multivariable
approximation, this means

$\psi_{i}(\textbf{a})<\psi_{i}(\textbf{b})\quad\forall\,\,\,\|\textbf{b}\|<\|\textbf{a}\|.$

The following result, due to Kleinbock and Wang [57, Theorem 2.7] provides the
most modern version of the Khintchine-Groshev theorem in the weighted
monotonic univariable setting. For the sake of brevity, we have stated this
result for the setting satisfying property $P$, but it should be noted that
Kleinbock & Wang’s result was more general still, see [57] for more details.

######  Theorem 1 ([57]).

Let $\Psi=(\psi_{1},\dots,\psi_{n})$ be an $n$-tuple of monotonic decreasing
univariable approximation functions satisfying property P. Then

$\mu^{{\mathbb{R}}}_{m\times
n}\left(W_{n,m}(\Psi)\right)=\begin{cases}0\quad\text{\rm if
}\quad\sum\limits_{r=1}^{\infty}r^{m-1}\prod_{i=1}^{n}\psi_{i}(r)<\infty,\\\
1\quad\text{\rm if
}\quad\sum\limits_{r=1}^{\infty}r^{m-1}\prod_{i=1}^{n}\psi_{i}(r)=\infty.\end{cases}$

Here and throughout we denote the $m\times n$ dimensional Lebesgue measure to
be $\mu_{m\times n}^{{\mathbb{R}}}$. Below we briefly highlight the results
preceding this. These include but are not limited to

  * •

$n=m=1$, $\psi$ monotonic, Khintchine [49].

  * •

$n=m=1$, $\psi$ non-monotonic, conjectured by Duffin and Schaeffer (1941) and
proven by Maynard and Koukoulopoulos [58].

  * •

$n>1,m=1$, $\psi$ is monotonic, Khintchine [50].

  * •

$nm\geq 1$, $\psi$ is univariable, proven by Groshev [37]. In fact, in the
original theorem, there were some stronger assumptions on the approximating
functions $\psi$, that is, $r^{\max(1,m-1)}\psi(r)^{n}$ to be monotonic. The
reason that this assumption is unnecessary is due to [9].

  * •

$nm>1$, $\psi$ is univariable, non-monotonic, by Beresnevich & Velani [13].

  * •

$n\geq 1$ $m>1$, for univariable non-monotic weighted $\Psi$ by Hussain &
Yusupova [45].

  * •

In slightly different settings, for the small linear forms, the analogue of
Theorem 1 was established by Fischler, Hussain, Kirstensen, & Levesley [35].

Similar results for the more general inhomogeneous approximation are known
but, as far as the authors are aware the case of weighted multivariable
inhomogeneous approximation remains unexplored.

When the $n$-tuple of approximation functions $\Psi$ decreases sufficiently
fast such that $\mu^{{\mathbb{R}}}_{m\times n}\left(W_{n,m}(\Psi)\right)=0$ we
desire a more delicate notion of ”size”. The Hausdorff measure and dimension
provide us with such a tool. The following Hausdorff measure and dimension
results are known for the set $W_{n,m}(\Psi)$.

######  Theorem 2 ([76, Theorem 10.2]).

Let $\Psi=(\psi_{1},\dots,\psi_{n})$ be an $n$-tuple of approximation
functions of the form

$\psi_{i}(q)=q^{-\tau_{i}}\,,\quad(1\leq i\leq n)$

for $\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in{\mathbb{R}}^{n}_{+}$ with

$\sum\limits_{i=1}^{n}\tau_{i}>m.$

Then

$\operatorname{\dim_{H}}W_{n,m}(\Psi)=\min_{1\leq k\leq
n}\left\\{n(m-1)+\frac{n+m+\sum\limits_{j:\tau_{j}<\tau_{k}}(\tau_{k}-\tau_{j})}{1+\tau_{k}}\right\\}.$

It should be noted that the above result was proven for a more general range
of non-increasing approximation functions, with the $\tau_{i}$’s in the
dimension result being replaced by limit points of vectors related to
$\psi_{i}$. For example, in the univariable case $\tau$ is replaced by the
lower order at infinity of $\psi$, that is,
$\tau=\liminf_{q\to\infty}\tfrac{-\log\psi(q)}{\log q}$.

As with the result of Kleinbock and Wang various results preceded this. We
highlight a few below. For brevity, we stick to considering the approximation
function of the form considered in the above theorem.

  * •

$n=m=1$ was independently proven by Jarník and Besicovitch.

  * •

for all linear forms but for the non-weighted case, it was proven by Bovey and
Dodson in [15]. This result was further generalised to the lower order
settings by Dodson in [23].

  * •

for $m=1$ and $n\geq 1$ it was proven in the weighted case by Rynne [69]. In
fact, the result of Rynne was more general, as he restricted the rational
denominators to an infinite subset of integers. This result was further
generalised to dual settings by Dickinson and Rynne [70].

The aim of this paper is to prove analogies of the above two theorems in a
variety of settings. In order to prove Lebesgue dichotomy statements (analogue
of Theorem 1) in a variety of settings we use the recently developed notion of
weighted ubiquitous systems. The notion of weighted ubiquity was introduced by
Wang and Wu in [76] and was developed further by Kleinbock and Wang in [57].
In the paper [76], the authors established a very powerful weighted mass
transference principle that, under weighted ubiquity assumption, allows for
the Hausdorff measure/dimension results in the weighted settings. We refer the
reader to [4, 5] for a comprehensive survey of the mass transference
principle.

In the following section, we recall this theory. In the subsequent sections,
along with some other things, we apply this framework to a variety of settings
including Diophantine approximation in $p$-adics, formal power series, complex
numbers, quaternions, and uniform distribution theory.

##  2\. Toolbox

This section consists of a range of tools used to determine metric properties
of $\limsup$ sets. The first subsection recalls Dirichlet’s and Minkowski’s
theorems on linear forms. The second subsection provides the generalised setup
of weighted ubiquitous systems into which our applications fall. The third
subsection gives the techniques required to determine ambient measure results
on $\limsup$ sets. The last subsection recalls the definition of Hausdorff
measure and dimension and gives the tools required to prove Hausdorff measure
and dimension results on $\limsup$ sets.

###  2.1. Dirichlet and Minkowski’s theorems

A basic problem of Diophantine approximation is to approximate a given real
number by rational numbers to a certain degree. For example, it is obvious
that for a given $\alpha\in{\mathbb{R}}$ and any $q\in{\mathbb{N}}$, there is
some integer $p$ such that

$|q\alpha-p|<\frac{1}{2}.$

A classical result by Dirichlet improves this observation. Namely, given
$\alpha\in{\mathbb{R}}$, for each real number $Q>1$ there is some
$(q,p)\in{\mathbb{Z}}^{2}$ such that

$1\leq q\leq Q\quad\text{ and }\quad|q\alpha-p|<\frac{1}{Q}.$

As a consequence, there are infinitely many pairs
$(q,p)\in{\mathbb{N}}\times{\mathbb{Z}}$ satisfying

$|q\alpha-p|<\frac{1}{q}.$

Dirichlet’s result can be proven using the pigeonhole principle. Furthermore,
the argument can be easily extended to linear forms (see, for example, [71,
Chapter II, Theorem 1E]).

######  Theorem 3 (Dirichlet, 1842).

Let $m,n\in{\mathbb{N}}$ and $A\in{\mathbb{R}}^{m\times n}$ be given. For any
$Q>1$, there exists a non-zero
$\mathbf{q}=(q_{1},\ldots,q_{m})\in{\mathbb{Z}}^{m}$ and some
$\mathbf{p}=(p_{1},\ldots,p_{n})\in{\mathbb{Z}}^{n}$ such that

$\displaystyle|\mathbf{q}A_{i}-p_{i}|$ $\displaystyle<Q^{-1}\quad(1\leq i\leq
n),$ $\displaystyle|q_{j}|$ $\displaystyle\leq Q^{\frac{n}{m}}\quad(1\leq
j\leq m).$

Take $m,n\in{\mathbb{N}}$. Define the function
$\Psi:{\mathbb{N}}\to{\mathbb{R}}_{+}$ by $\Psi(r)=r^{-m/n}$. For every
$\mathbf{q}=(q_{1},\ldots,q_{m})\in{\mathbb{Z}}^{m}$, write
$\|\mathbf{q}\|=\max\\{|q_{1}|,\ldots,|q_{m}|\\}$. By Theorem 3, for every
matrix $A\in{\mathbb{R}}^{m\times n}$ there are infinitely many vectors
$0\neq\mathbf{q}\in{\mathbb{Z}}^{m}$ and $\mathbf{p}\in{\mathbb{Z}}^{n}$ such
that

(2.1) $|\mathbf{q}A_{i}-p_{i}|<\Psi(\|\mathbf{q}\|)\quad(1\leq i\leq n).$

Dirichlet’s theorem and its consequences, despite their foundational
character, are not strong enough for many applications including the ones we
study in this article. This is because we might need to replace the function
$\Psi$ in (2.1) by positive and non-increasing functions $\Psi_{i}$ in each
coordinate axis $i\in\\{1,\ldots,n\\}$. Minkowski’s theorem for linear forms
(Theorem 4 below) is a key tool towards this goal.

Recall that a lattice $\Lambda$ on ${\mathbb{R}}^{n}$, $n\in{\mathbb{N}}$, is
a set of the form

$\Lambda=\left\\{a_{1}\mathbf{v}_{1}+\ldots+a_{n}\mathbf{v}_{n}:a_{1},\ldots,a_{n}\in{\mathbb{Z}}\right\\},$

where $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\in{\mathbb{R}}^{n}$ are $n$ given
linearly independent row vectors. The determinant of $\Lambda$, denoted
$\det(\Lambda)$, is the determinant of the matrix formed by
$\mathbf{v}_{1},\ldots,\mathbf{v}_{n}$.

######  Theorem 4 (Minkowski, 1896).

Let $N\in{\mathbb{N}}$, $\Lambda\subseteq{\mathbb{R}}^{N}$ a lattice of
determinant $d$, and $A\in{\mathbb{R}}^{N\times N}$. If the positive real
numbers $c_{1},\ldots,c_{N}$ satisfy

$\det(\Lambda)|\det(A)|\leq c_{1}\cdots c_{N},$

then, for $j_{0}\in\\{1,\ldots,N\\}$, there is some non-zero
$\mathbf{u}\in\Lambda$ such that

$\displaystyle\left|\mathbf{u}A_{j}\right|$ $\displaystyle\leq
c_{j}\quad(1\leq j\leq j_{0}),$ $\displaystyle\left|\mathbf{u}A_{j}\right|$
$\displaystyle<c_{j}\quad(j_{0}+1\leq j\leq N).$

A proof of Theorem 4 can be found in [18, Chapter III, Theorem III]. Our
formulation is slightly different from the one in the reference, but the proof
can be easily adapted to our case.

In relation to Diophantine approximation Theorem 4 allows us to deduce that
for vector $v=(v_{1},\dots,v_{m})$ with $\sum_{i=1}^{m}v_{i}=m$, for any
$A\in{\mathbb{R}}^{m\times n}$

$|\textbf{q}A_{i}-p_{i}|<\psi_{i}(\|\textbf{q}\|_{v})\quad(1\leq i\leq n)\,,$

is solved for infinitely many integer vectors
$(\mathbf{p},\textbf{q})\in{\mathbb{Z}}^{n}\times({\mathbb{Z}}^{m}\backslash\\{0\\})$
provided the $n$-tuple of approximation functions satisfy

$\prod_{i=1}^{n}\psi_{i}(r)=r^{-1}\quad\text{ and
}\quad\psi_{i}(r)<\tfrac{1}{2}\quad(1\leq i\leq n)\,\text{ monotonic
decreasing}$

for all $r\in{\mathbb{R}}_{+}$.

###  2.2. Weighted Ubiquitous systems

In this section we give the definition of local ubiquity for rectangles as
given in [57]. This definition is a generalisation of ubiquity for rectangles
as found in [76]. The notion of an “Ubiquitous system” for balls was
introduced by Dodson, Rynne, and Vickers [26] which was then generalised to
the abstract metric space settings in [9].

Fix an integer $n\geq 1$, and for each $1\leq i\leq n$ let
$(\Omega_{i},|\cdot|_{i},\mu_{i})$ be a bounded locally compact metric space,
where $|\cdot|_{i}$ denotes a metric on $\Omega_{i}$, and $\mu_{i}$ denotes a
Borel probability measure over $\Omega_{i}$. Further, we assume that the
measure $\mu_{i}$ is $\delta_{i}$-Ahlfors regular probability measure. That
is, there exists constants $0<c_{1}<c_{2}<\infty$ such that for any ball
$B_{i}(x_{i},r)$ with centre $x_{i}\in\Omega_{i}$ and radius $0<r<r_{0}$ for
some $r_{0}\in{\mathbb{R}}_{+}$ we have that

$c_{1}r^{\delta_{i}}\leq\mu_{i}\left(B_{i}(x_{i},r)\right)\leq
c_{2}r^{\delta_{i}}\,.$

Consider the product space $(\Omega,\|\cdot\|,\mu)$, where

$\Omega=\prod_{i=1}^{n}\Omega_{i},\quad\mu=\prod_{i=1}^{n}\mu_{i},\quad\|\cdot\|=\max_{1\leq
i\leq n}|\cdot|_{i}$

are defined in the usual way. For any $x\in\Omega$ and $r\in{\mathbb{R}}_{+}$
define the open ball

$B(x,r)=\left\\{y\in\Omega:\max_{1\leq i\leq
n}|x_{i}-y_{i}|_{i}<r\right\\}=\prod_{i=1}^{n}B_{i}(x_{i},r),$

where $B_{i}$ are the open $r$-balls associated with the $i^{\text{th}}$
metric space $\Omega_{i}$. Let $J$ be a countably infinite index set, and
$\beta:J\to{\mathbb{R}}_{+}$, $\alpha\mapsto\beta_{\alpha}$ a positive
function satisfying the condition that for any $N\in{\mathbb{N}}$

$\\#\left\\{\alpha\in J:\beta_{\alpha}<N\right\\}<\infty.$

Let $l_{k},u_{k}$ be two sequences in ${\mathbb{R}}_{+}$ such that $u_{k}\geq
l_{k}$ with $l_{k}\to\infty$ as $k\to\infty$. Define

$J_{k}=\\{\alpha\in J:l_{k}\leq\beta_{\alpha}\leq u_{k}\\}.$

Let $\rho=(\rho_{1},\dots,\rho_{n})$ be an $n$-tuple of non-increasing
functions $\rho_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that each
$\rho_{i}(x)\to 0$ as $x\to\infty$. For each $1\leq i\leq n$, let
$(R_{\alpha,i})_{\alpha\in J}$ be a sequence of subsets in $\Omega_{i}$. The
family of sets $(R_{\alpha})_{\alpha\in J}$ where

$R_{\alpha}=\prod_{i=1}^{n}R_{\alpha,i},$

for each $\alpha\in J$, are called resonant sets.

Define

$\Delta(R_{\alpha},\rho(r))=\prod_{i=1}^{n}\Delta_{i}(R_{\alpha,i},\rho_{i}(r)),$

where for any set $A\subset\Omega_{i}$ and $b\in{\mathbb{R}}_{+}$

$\Delta_{i}(A,b)=\bigcup_{a\in A}B_{i}(a,b)$

is the union of balls in $\Omega_{i}$ of radius $b$ centred at all possible
points in $A$. Generally $\Delta(R_{\alpha},\rho(r))$ is the product of
$\rho_{i}(r)$-neighbourhoods of $R_{a,i}$ for each coordinate
$i\in\\{1,\dots,n\\}$.

The following definition, which was introduced in [2] as a generalisation of
the intersection properties of [9], on the properties of the resonant sets is
required.

######  Definition 1 ($\kappa$-scaling property).

Let $0\leq\kappa_{i}<1$ and $1\leq i\leq n$. The sequence
$(R_{\alpha,i})_{\alpha\in J}$ has a $\kappa_{i}$-scaling property if for any
$\alpha\in J$, any ball $B_{i}(x_{i},r)\subset\Omega_{i}$ with centre
$x_{i}\in R_{\alpha,i}$, and $0<\epsilon<r$ we have

$c_{2}r^{\delta_{i}\kappa_{i}}\epsilon^{\delta_{i}(1-\kappa_{i})}\leq\mu_{i}\left(B_{i}(x_{i},r)\cap\Delta(R_{\alpha,i},\epsilon)\right)\leq
c_{3}r^{\delta_{i}\kappa_{i}}\epsilon^{\delta_{i}(1-\kappa_{i})},$

for some constants $c_{2},c_{3}>0$.

See [2, Section 2] for calculations of $\kappa_{i}$ for various resonant sets.
Intuitively, one can think of $\kappa_{i}$ being the value such that
$\delta_{i}\kappa_{i}$ is the box dimension of $R_{\alpha,i}$. As an example
note that $\kappa=0$ when $R_{\alpha,i}$ is a finite collection of points and
for $R_{\alpha,i}$ $(m-1)$-dimensional affine hyperplanes
$\kappa=\frac{m-1}{m}$. Although the definition considers the
$\kappa_{i}$-scaling property in each coordinate axis, for our purpose we take
$\kappa_{1}=\cdots=\kappa_{n}=\kappa$, and refer to it as $\kappa$-scaling
property. In particular this is the $\kappa$-scaling property considered in
[76].

The following notion of ubiquity for rectangles can be found in [57, Section
2.2].

######  Definition 2 (Local ubiquitous system for rectangles).

Call the pair $\big{(}(R_{\alpha})_{\alpha\in J},\beta\big{)}$ a local
ubiquitous system for rectangles with respect to $\rho$ if there exists a
constant $c>0$ such that for any ball $B\subset\Omega$ and all sufficiently
large $k\in{\mathbb{N}}$

$\mu\left(B\cap\bigcup_{\alpha\in
J_{k}}\Delta(R_{\alpha},\rho(u_{k}))\right)\geq cm(B).$

For an $n$-tuple of approximation functions $\Psi=(\psi_{1},\dots,\psi_{n})$
with each $\psi_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ define

$\displaystyle W^{\Omega}(\Psi)$
$\displaystyle=\left\\{x\in\Omega:x\in\Delta\left(R_{\alpha},\Psi(\beta_{\alpha})\right)\,\text{
for infinitely many }\alpha\in J\right\\}$ $\displaystyle=\limsup_{\alpha\in
J}\Delta\left(R_{\alpha},\Psi(\beta_{\alpha})\right).$

For all the applications listed below, the corresponding sets will be
described by the $\limsup$ set outlined above.

###  2.3. Ambient measure statements

The following is the well-known Borel-Cantelli Lemma usually used to prove the
convergence cases for Lebesgue dichotomy statements, see for example [47,
Theorem 4.18] for a proof.

######  Lemma 1 (Borel-Cantelli lemma, convergence part).

Let $(\Omega,\mathcal{A},\mu)$ be a measure space and let $(A_{k})_{k\geq 1}$
be a sequence of measurable sets. If

$\sum_{k\geq
1}\mu(A_{k})<\infty,\,\quad\text{then}\quad\mu\left(\limsup_{k\to\infty}A_{k}\right)=0\,.$

The following theorem from [57] provides the ambient measure theory for
$W^{\Omega}(\Psi)$ in the divergence case. Prior to stating the result we need
one more definition on functions. For constant $0<c<1$ a function $f$ is said
to be $c$-regular with respect to a sequence $\\{r_{i}\\}_{i\in{\mathbb{N}}}$
if

$f(r_{i+1})\leq cf(r_{i})$

for all sufficiently large $i$.

######  Theorem 5 ([57]).

Let $W^{\Omega}(\Psi)$ be defined as above and assume that
$\left((R_{\alpha})_{\alpha\in J},\beta\right)$ is a local ubiquitous system
for rectangles with respect to $\rho$, and that the resonant sets
$(R_{\alpha,i})$ have $\kappa_{i}$-scaling property and each measure $\mu_{i}$
is $\delta_{i}$-Ahlfors regular. Suppose that

  1. (I)

each $\psi_{i}$ is decreasing,

  2. (II)

for each $1\leq i\leq n$, $\psi_{i}(r)\leq\rho_{i}(r)$ for all
$r\in{\mathbb{R}}_{+}$ and $\rho_{i}(r)\to 0$ as $r\to\infty$,

  3. (III)

either $\rho_{i}$ is $c$-regular on $(u_{k})_{k\geq 1}$ for all $1\leq i\leq
n$ or $\psi_{i}$ is $c$-regular on $(u_{k})_{k\geq 1}$ for all $1\leq i\leq n$
for some $0<c<1$.

Then,

$\mu(W(\Psi))=\mu(\Omega)\quad\text{\rm if
}\quad\quad\sum\limits_{k=1}^{\infty}\prod_{i=1}^{n}\left(\frac{\psi_{i}(u_{k})}{\rho_{i}(u_{k})}\right)^{\delta_{i}(1-\kappa_{i})}\,=\infty.$

It will often be convenient at times to multiply the approximating function
$\Psi$ by some constant. The following lemma, see [7, Lemma 5.7], shows that,
in terms of ambient measure, the measure of the $\limsup$ set remains
unchanged.

######  Lemma 2 ([7]).

Let $(\Omega,\|\cdot\|,\mu)$ be the product space defined above. Let
$(S_{i})_{i\in{\mathbb{N}}}$ be a sequence of subsets in the support of $\mu$
and $(\mathbf{\delta}_{i})_{i\in{\mathbb{N}}}$ be a sequence of positive
$n$-tuples $\mathbf{\delta}_{i}=(\delta^{(1)}_{i},\dots,\delta^{(n)}_{i})$
such that $\delta^{(j)}_{i}\to 0$ as $i\to\infty$ for each $1\leq j\leq n$.
Then, for any $\mathbf{C}=(C_{1},\dots,C_{n})$ and
$\mathbf{c}=(c_{1},\dots,c_{n})$ with $0<c_{j}\leq C_{j}$ for each $1\leq
j\leq n$

(2.2)
$\mu\left(\limsup_{i\to\infty}\Delta(S_{i},\mathbf{C}\mathbf{\delta}_{i})\;\setminus\;\limsup_{i\to\infty}\Delta(S_{i},\mathbf{c}\mathbf{\delta}_{i})\right)=0\,,$

where
$\mathbf{c}\mathbf{\delta}_{i}=(c_{1}\delta^{(1)}_{i},\dots,c_{n}\delta^{(n)}_{i})$
and similarly
$\mathbf{C}\mathbf{\delta}_{i}=(C_{1}\delta^{(1)}_{i},\dots,C_{n}\delta^{(n)}_{i})$.

###  2.4. Hausdorff measure and dimension statements

For completeness, we give below a very brief introduction to Hausdorff
measures and dimensions. For further details see [31].

Let $(\Omega,d)$ be a metric space and $F\subset\Omega$. Then for any
$0<\rho\leq\infty$, any finite or countable collection $\\{B_{i}\\}$ of
subsets of $\Omega$ such that $F\subset\bigcup_{i}B_{i}$ and

$r(B_{i})=\frac{1}{2}\inf\\{r\geq 0:d(x,y)\leq r\quad(x,y\in
B_{i})\\}\leq\rho$

is called a _$\rho$ -cover_ of $F$. Let

$\mathcal{H}_{\rho}^{f}(F)=\inf\sum\limits_{i}f\left(r(B_{i})\right),$

where the infimum is taken over all possible $\rho$-covers $\\{B_{i}\\}$ of
$F$. The $f$-dimensional Hausdorff measure of $F$ is defined to be

$\mathcal{H}^{f}(F)=\lim_{\rho\to 0}\mathcal{H}_{\rho}^{f}(F).$

In the case that $f(r)=r^{s}\;\;(s\geq 0)$, the measure $\mathcal{H}^{f}$ is
denoted $\mathcal{H}^{s}$ and is called _$s$ -dimensional Hausdorff measure_.
For any set $F\subset\Omega$ one can easily verify that there exists a unique
critical value of $s$ at which the function $s\mapsto\mathcal{H}^{s}(F)$
“jumps” from infinity to zero. The value taken by $s$ at this discontinuity is
referred to as the Hausdorff dimension of $F$ and is denoted by
$\operatorname{\dim_{H}}F$; i.e.

$\operatorname{\dim_{H}}F:=\inf\left\\{s\geq
0\;:\;\mathcal{H}^{s}(F)=0\right\\}.$

A countable collection $\\{B_{i}\\}$ is called a _fine cover_ of $F$ if for
every $\rho>0$ it contains a sub-collection that is a $\rho$-cover of $F$.

Below is the Hausdorff measure analogue of the Borel-Cantelli lemma in the
convergence case, which will allow us to estimate the convergence case
Hausdorff measure of certain sets via calculating the Hausdorff $f$-sum of a
fine cover. Recall a collection ${\mathcal{B}}=\\{B_{i}\\}$ of subsets
covering $F$ is called a fine cover if for any $x\in F$ and any $r>0$ there
exists $B_{i}\in{\mathcal{B}}$ such that $r(B_{i})<r$ and $x\in B_{i}$.
Trivially this implies that for any $\rho>0$ there exists a subset of
${\mathcal{B}}$ that is a $\rho$-cover of $F$.

######  Lemma 3 (Hausdorff-Cantelli lemma).

Let $\\{B_{i}\\}\subset\Omega$ be a fine cover of a set $F$ and let $f$ be a
dimension function such that

(2.3) $\sum_{i}f\left(r(B_{i})\right)\,<\,\infty.$

Then $\mathcal{H}^{f}(F)=0$. In particular, if $f(r)=r^{s}$ and we have (2.3)
then

$\operatorname{\dim_{H}}F\leq s.$

For the lower bound of the Hausdorff dimension and the divergent counterpart
of the Hausdorff measure theory, we have the following theorem due to Wang and
Wu [76, Theorem 3.1-3.2], which appeals to the notion of weighted ubiquitous
systems.

######  Theorem 6 ([76]).

Let $W(\Psi)$ be defined as above and assume that
$\left((R_{\alpha})_{\alpha\in J},\beta\right)$ is a local ubiquitous system
for rectangles with respect to $\rho=(\rho^{a_{1}},\dots,\rho^{a_{n}})$ for
some function $\rho:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ and
$(a_{1},\dots,a_{n})\in{\mathbb{R}}^{n}_{+}$, where
$\rho^{a_{j}}(x):=\rho(x)^{a_{j}}$. Assume, for each $1\leq i\leq n$, the
resonant sets $(R_{\alpha,i})$ have $\kappa$-scaling property. Assume each
measure $\mu_{i}$ is $\delta_{i}$-Ahlfors regular. Then, if
$\Psi=(\rho^{a_{1}+t_{1}},\dots,\rho^{a_{n}+t_{n}})$ for some
$\textbf{t}=(t_{1},\dots,t_{n})\in{\mathbb{R}}^{n}_{+}$,

$\operatorname{\dim_{H}}W(\Psi)\geq\min_{A_{i}\in
A}\left\\{\sum\limits_{j\in\mathcal{K}_{1}}\delta_{j}+\sum\limits_{j\in\mathcal{K}_{2}}\delta_{j}+\kappa\sum\limits_{j\in\mathcal{K}_{3}}\delta_{j}+(1-\kappa)\frac{\sum\limits_{j\in\mathcal{K}_{3}}a_{j}\delta_{j}-\sum\limits_{j\in\mathcal{K}_{2}}t_{j}\delta_{j}}{A_{i}}\right\\}=s,$

where $A=\\{a_{i},a_{i}+t_{i},1\leq i\leq n\\}$ and
$\mathcal{K}_{1},\mathcal{K}_{2},\mathcal{K}_{3}$ are a partition of
$\\{1,\dots,n\\}$ defined as

$\mathcal{K}_{1}=\\{j:a_{j}\geq
A_{i}\\},\quad\mathcal{K}_{2}=\\{j:a_{j}+t_{j}\leq
A_{i}\\}\backslash\mathcal{K}_{1},\quad\mathcal{K}_{3}=\\{1,\dots
n\\}\backslash(\mathcal{K}_{1}\cup\mathcal{K}_{2}).$

Furthermore, for any ball $B\subset\Omega$ we have

(2.4) $\mathcal{H}^{s}(B\cap W(\Psi))=\mathcal{H}^{s}(B).$

The main motivation for this result came from the landmark paper of
Beresnevich and Velani [12] in which they developed the mass transference
principle from balls to balls. This is surprising as the Hausdorff measure
theory underpins the Lebesgue measure theory. This powerful tool has since
been generalised to various settings, see for instance [2, 3, 4, 43, 76, 77]
and references therein.

We clarify some notations that will be used throughout. For real quantities
$A,B$ and a parameter $t$, we write $A\ll_{t}B$ if $A\leq c(t)B$ for a
constant $c(t)>0$ that depends on $t$ only (while $A$ and $B$ may depend on
other parameters). We write $A\asymp_{t}B$ if $A\ll_{t}B\ll_{t}A$. If the
constant $c>0$ is absolute, we simply write $A\ll B$ and $A\asymp B$.

##  3\. A motivating example

As a motivating example, and a warm-up to later applications, let us consider
the set

$W^{{\mathbb{R}}}_{1,2}(\psi_{1},\psi_{2})=\left\\{(x_{1},x_{2})\in
I^{2}:|qx_{i}-p_{i}|<\psi_{i}(q)\quad(i=1,2)\quad\text{ for i.m.}\
(p_{1},p_{2},q)\in{\mathbb{Z}}^{2}\times{\mathbb{N}}\right\\}\,,$

where $I^{2}=[0,1]^{2}$. Theorem 1 allows us to deduce the following result.

######  Corollary 1.

For $\psi_{1},\psi_{2}:{\mathbb{N}}\to{\mathbb{R}}_{+}$ monotonic decreasing
functions, we have that

$\mu^{{\mathbb{R}}}_{2}\left(W^{{\mathbb{R}}}_{1,2}(\psi_{1},\psi_{2})\right)=\begin{cases}0\quad\text{\rm
if
}\quad\sum\limits_{q=1}^{\infty}\psi_{1}(q)\psi_{2}(q)<\infty\,,\\\\[8.61108pt]
1\quad\text{\rm if
}\quad\sum\limits_{q=1}^{\infty}\psi_{1}(q)\psi_{2}(q)=\infty\,.\end{cases}$

Furthermore, from Theorem 2 we can deduce the following

######  Corollary 2.

For

$\psi_{1}(q)=q^{-\tau_{1}}\quad\text{\rm and }\quad\psi_{2}(q)=q^{-\tau_{2}}$

with $\tau_{1}+\tau_{2}>1$ and $\tau_{1},\tau_{2}>0$, we have that

$\operatorname{\dim_{H}}W^{{\mathbb{R}}}_{1,2}(\psi_{1},\psi_{2})=\min_{i=1,2}\left\\{\frac{3+(\tau_{i}-\min\\{\tau_{1},\tau_{2}\\})}{1+\tau_{i}}\right\\}\,.$

We now show how both of these results can be derived by constructing a
suitable ubiquitous system of rectangles and then applying the theorems of the
previous section. The weighted ubiquitous system is setup in much the same was
as the classical ubiquitous system, with the key difference being that Theorem
4 is used in place of Theorem 3. See for example [11, Theorem 1.1.4] for the
statement in the classical one dimensional setting. However, the application
of the weighted ubiquitous system is slightly more complex. In particular it
requires a careful choice of the $\rho$ function.

To begin, let

$\displaystyle\Omega$
$\displaystyle=[0,1]^{2},\quad\mu=\mu^{{\mathbb{R}}}_{2},$ $\displaystyle J$
$\displaystyle={\mathbb{N}},\quad\beta_{\alpha}=\alpha,$ $\displaystyle l_{k}$
$\displaystyle=M^{k-1},\quad u_{k}=M^{k},$ $\displaystyle J_{k}$
$\displaystyle=\left\\{q\in{\mathbb{N}}:M^{k-1}\leq q\leq M^{k}\right\\},$
$\displaystyle R_{q}$
$\displaystyle=\left\\{\left(\tfrac{p_{1}}{q},\tfrac{p_{2}}{q}\right):0\leq
p_{1},p_{2}\leq q\right\\}\,.$

Note that $R_{q}$ are collections of points and so $(R_{q})_{q\geq 1}$ has
$\kappa$-scaling property $\kappa=0$. Since
$\mu_{2}^{{\mathbb{R}}}=\mu_{1}^{{\mathbb{R}}}\times\mu_{1}^{{\mathbb{R}}}$
trivially $\delta_{1}=\delta_{2}=1$. Here $M$ in the definitions of $u_{k}$
and $l_{k}$ is some large constant (we can take any integer $M\geq 64$).
Observe that

$W^{{\mathbb{R}}}_{1,2}(\psi_{1},\psi_{2})=\limsup_{q\to\infty}\Delta\left(R_{q},\left(\tfrac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}\right)\right)\,.$

We prove the following.

######  Proposition 1.

Let $\rho_{1},\rho_{2}:{\mathbb{N}}\to{\mathbb{R}}_{+}$ be functions of the
form

(3.1) $\rho_{i}(q)=M\frac{\phi_{i}(q)}{q}\,\quad(i=1,2)\quad\text{ with
}\quad\phi_{1}(q)\phi_{2}(q)=q^{-1}\,,$

and each $\phi_{i}(q)\to 0$ as $q\to\infty$. Then for any ball $B\subset
I^{2}$ there exists $k_{0}\in{\mathbb{N}}$ such that for all $k>k_{0}$ we have

$\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{q\in
J_{k}}\Delta\left(R_{q}\left(\rho_{1}(M^{k}),\rho_{2}(M^{k})\right)\right)\right)\geq\frac{1}{2}\mu^{{\mathbb{R}}}_{2}(B)\,.$

###### Proof.

The following is a standard method of proving such a result. Initially let us
give some bounds on $k_{0}$ that will be used later. Choose $k_{0}$ such that
for all $k>k_{0}$ we have

$\phi_{i}(M^{k})<\frac{1}{2}\qquad(i=1,2)\quad\text{ and }\quad
M^{k}>\frac{32}{3}\pi^{2}\mu^{{\mathbb{R}}}_{2}(B)^{-1}\,.$

It will become clear later why such bounds are chosen. Now pick any $k>k_{0}$.
Firstly, by Minkowski’s Theorem for systems of linear forms (Theorem 4) for
any $(x_{1},x_{2})\in[0,1]^{2}$ the system

$\begin{cases}|qx_{1}-p_{1}|<\phi_{1}(M^{k}),\\\
|qx_{2}-p_{2}|<\phi_{2}(M^{k}),\\\ |q|\leq M^{k}\end{cases}$

has a non-zero integer solution $(p_{1},p_{2},q)\in{\mathbb{Z}}^{3}$. Note by
our choice of $k_{0}$ that $q\neq 0$, since otherwise we would have $p_{1}$
and $p_{2}$ non-zero solving $|p_{i}|<\tfrac{1}{2}$ which is clearly false.
So, multiplying the solution $(p_{1},p_{2},q)$ through by $-1$ if necessary,
we have that $(p_{1},p_{2},q)\in{\mathbb{Z}}^{2}\times{\mathbb{N}}$. Dividing
through by $q$ in the first two inequalities allows us to see that

(3.2) $\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{1\leq q\leq
M^{k}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)=\mu^{{\mathbb{R}}}_{2}(B)\,.$

Now observe that

$\displaystyle\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{1\leq q\leq
M^{k}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)$
$\displaystyle\leq\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{1\leq
q<M^{k-1}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)+\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{M^{k-1}\leq
q\leq
M^{k}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{M^{k-1}},\tfrac{\phi_{2}(M^{k})}{M^{k-1}}\right)\right)\right).$

The second summation on the right-hand side of the above inequality is
precisely what we are trying to calculate (since
$\tfrac{\phi_{i}(M^{k})}{M^{k-1}}=\rho_{i}(M^{k})$ for $i=1,2.$). Note (3.2)
gives us that the left hand side of the above inequality is
$\mu^{{\mathbb{R}}}_{2}(B)$, so if we can show that

$\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{1\leq
q<M^{k-1}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)<\tfrac{1}{2}\mu^{{\mathbb{R}}}_{2}(B)$

we are done. To see this we compute that

$\displaystyle\mu^{{\mathbb{R}}}_{2}\left(B\cap\bigcup_{1\leq
q<M^{k-1}}\Delta\left(R_{q},\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)$
$\displaystyle\leq\sum_{1\leq
q<M^{k-1}}\sum_{\left(\tfrac{p_{1}}{q},\tfrac{p_{2}}{q}\right)\in R_{q}\cap
B}\mu^{{\mathbb{R}}}_{2}\left(\Delta\left((\tfrac{p_{1}}{q},\tfrac{p_{2}}{q}),\left(\tfrac{\phi_{1}(M^{k})}{q},\tfrac{\phi_{2}(M^{k})}{q}\right)\right)\right)$
$\displaystyle\leq\sum_{1\leq
q<M^{k-1}}\left(2qr(B)+1\right)^{2}4q^{-2}\phi_{1}(M^{k})\phi_{2}(M^{k})$
$\displaystyle\overset{(*)}{\leq}\sum_{1\leq
q<M^{k-1}}\left(4q^{2}\mu^{{\mathbb{R}}}_{2}(B)+4\right)4q^{-2}M^{-k}$
$\displaystyle\leq 16M^{-1}\mu^{{\mathbb{R}}}_{2}(B)+16M^{-k}\sum_{1\leq
q<M^{k-1}}q^{-2}$ $\displaystyle\overset{(M\geq
64)}{\leq}\tfrac{1}{4}\mu^{{\mathbb{R}}}_{2}(B)+M^{-k}16\tfrac{\pi^{2}}{6}$
$\displaystyle\leq\tfrac{1}{2}\mu^{{\mathbb{R}}}_{2}(B),$

where $(*)$ follows on using that $(a+b)^{2}\leq(2a)^{2}+(2b)^{2}$ for all
$a,b\geq R_{+}$. Hence the proof is complete. ∎

###  3.1. Proof of Corollary 1

Begin with the convergence case of Corollary 1. By the Borel-Cantelli
convergence lemma, and the above formulation of
$W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})$ in terms of a $\limsup$ set, we
have that

(3.3)
$\mu^{{\mathbb{R}}}_{2}\left(W^{{\mathbb{R}}}_{1,2}(\psi_{1},\psi_{2})\right)=0\quad\text{
if}\quad\sum_{q=1}^{\infty}\mu^{{\mathbb{R}}}_{2}\left(\Delta\left(R_{q},\left(\tfrac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}\right)\right)\right)<\infty\,.$

A quick calculation of the Lebesgue measure of the collection of rectangles
$\Delta\left(R_{q},\left(\tfrac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}\right)\right)$
yields

$\mu^{{\mathbb{R}}}_{2}\left(\Delta\left(R_{q},\left(\tfrac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}\right)\right)\right)\leq
4\psi_{1}(q)\psi_{2}(q)\,,$

and inputting this into (3.3) completes the convergence case.

In order to prove the divergence case of Corollary 1 we now need to show
conditions (I)-(III) of Theorem 5 are verified for some suitably chosen
functions $\rho_{1},\rho_{2}$ satisfying (3.1). Notice in this setting, our
functions are of the form

$\Psi(q)=\left(\tfrac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}\right),$

So the conditions (I)-(III) correspond to

  * I(${\mathbb{R}}$)

$\frac{\psi_{1}(q)}{q},\tfrac{\psi_{2}(q)}{q}$ monotonic decreasing as
$q\to\infty$,

  * II(${\mathbb{R}}$)

$\rho_{1}(q)\geq\tfrac{\psi_{1}(q)}{q}$ and
$\rho_{2}(q)\geq\tfrac{\psi_{2}(q)}{q}$ for all $q\in{\mathbb{N}}$, and
$\rho_{1}(q),\rho_{2}(q)\to 0$ as $q\to\infty$.

  * III(${\mathbb{R}}$)

$\frac{\psi_{1}(q)}{q}$ and $\frac{\psi_{2}(q)}{q}$ are $c$-regular on the
sequence $(M^{k})_{k\geq 1}$.

Since $\psi_{1},\psi_{2}$ are decreasing I(${\mathbb{R}}$) is immediately
satisfied. For III(${\mathbb{R}}$) note that for each $i=1,2$

$\frac{\psi_{i}(M^{k+1})}{M^{k+1}}=M^{-1}\frac{\psi_{i}(M^{k+1})}{M^{k}}\leq
M^{-1}\frac{\psi_{i}(M^{k})}{M^{k}}$

where the last inequality follows due to the monotonic decreasing property of
$\psi_{i}$. Thus it remains to choose functions $\rho_{1},\rho_{2}$ so that
II(${\mathbb{R}}$) is satisfied.

For each $q\in{\mathbb{N}}$, let $l_{1}(q),l_{2}(q)\in\\{1,2\\}$ be the
ordering such that

$\psi_{l_{1}(q)}(q)\geq\psi_{l_{2}(q)}(q)\,.$

We can assume without loss of generality that

(3.4) $\psi_{1}(q)\psi_{2}(q)<q^{-1}$

for all sufficiently large $q\in{\mathbb{N}}$, say $q>q_{0}$. Otherwise by
Theorem 4 and the monotonicity of each $\psi_{i}$ we have that
$W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})=[0,1]^{2}$. For $q>q_{0}$ consider
the function

$\phi^{*}(q)=q^{-1}\psi_{l_{1}(q)}(q)^{-1}\,.$

Observe that $\phi^{*}(q)>\psi_{l_{2}(q)}(q)$, since otherwise we would not
have (3.4). Now, for each $q>q_{0}$ choose functions $\rho_{1},\rho_{2}$ to be

$\displaystyle\phi_{l_{1}(q)}(q)$
$\displaystyle=\begin{cases}\psi_{l_{1}(q)}(q)\quad\text{ if
}\quad\psi_{l_{1}(q)}(q)>q^{-\tfrac{1}{2}}\,,\\\\[8.61108pt]
q^{-\tfrac{1}{2}}\quad\quad\quad\text{ otherwise.}\end{cases}$
$\displaystyle\phi_{l_{2}(q)}(q)$
$\displaystyle=\begin{cases}\phi^{*}(q)\,\,\,\quad\text{ if
}\quad\psi_{l_{1}(q)}(q)>q^{-\tfrac{1}{2}}\,,\\\\[8.61108pt]
q^{-\tfrac{1}{2}}\quad\quad\text{ otherwise.}\end{cases}$

Note $\phi_{1},\phi_{2}$ satisfy condition (3.1), and furthermore

$\rho_{i}(q)=M\frac{\phi_{i}(q)}{q}>\frac{\psi_{i}(q)}{q}\quad(i=1,2)\,,$

where the inequality follows by our choice of each $\phi_{i}$. So
II(${\mathbb{R}}$) is satisfied, thus Theorem 4.2 is applicable. It remains to
see, via Cauchy condensation, that

$\sum_{q=1}^{\infty}\psi_{1}(q)\psi_{2}(q)\asymp\sum_{k=1}^{\infty}M^{k}\psi_{1}(M^{k})\psi_{2}(M^{k}),$

and so this completes the divergence case.

###  3.2. Proof of Corollary 2

For the upper bound consider the standard cover

$\bigcup_{q>N}\Delta\left(R_{q},\left(q^{-1-\tau_{1}},q^{-1-\tau_{2}}\right)\right)$

for any $N\in{\mathbb{N}}$. Letting $N$ tend to infinity naturally gives rise
to a better cover of $W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})$. Each layer
$\Delta\left(R_{q},(q^{-1-\tau_{1}},q^{-1-\tau_{2}})\right)$ is composed of
$q^{2}$ rectangles, each of which can be covered by either one large ball with
the diameter equal to the longest sidelength, or by several smaller balls with
diameters equal to the shortest sidelenght of the rectangle. For now take the
balls with larger diameter. Say $\tau_{1}\geq\tau_{2}$ and so
$2q^{-1-\tau_{2}}$ is the larger sidelenght. Then we have that

$\mathcal{H}^{s}\left(W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\right)\leq\sum_{q>N}q^{2}2^{s}q^{-(1+\tau_{2})s}\leq
2^{s}\sum_{q>N}q^{2-(1+\tau_{2})s}\to 0$

as $N\to\infty$ for any $s>\frac{3}{1+\tau_{2}}$. Hence

$\operatorname{\dim_{H}}W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\leq\frac{3}{1+\tau_{2}}$

when $\tau_{1}\geq\tau_{2}$. Now consider the other case. Then each rectangle
in $\Delta\left(R_{q},(q^{-1-\tau_{1}},q^{-1-\tau_{2}})\right)$ can be covered
by

$\frac{2q^{-(1+\tau_{2})}}{2q^{-(1+\tau_{1}})}=q^{(\tau_{1}-\tau_{2})}$

balls of diameter $2q^{-(1+\tau_{1})}$. Hence

$\mathcal{H}^{s}\left(W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\right)\leq\sum_{q>N}q^{2}q^{(\tau_{1}-\tau_{2}}2^{s}q^{-(1+\tau_{1})s}\leq
2^{s}\sum_{q>N}q^{2+(\tau_{1}-\tau_{2})-(1+\tau_{1})s}\to 0$

as $N\to\infty$ for any $s>\frac{3+(\tau_{1}-\tau_{2})}{1+\tau_{1}}$. Hence

$\operatorname{\dim_{H}}W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\leq\frac{3+(\tau_{1}-\tau_{2})}{1+\tau_{1}}\,.$

Taking the minimum over the two cases completes the upper bound of Corollary
2.

For the lower bound we use Theorem 6 combined with Proposition 1. Within the
notation of Theorem 6 set

$a_{i}=a_{i}^{*}+1,\quad t_{i}=\tau_{i}-a_{i}^{*}\quad(i=1,2),$

and consider the function $\rho^{*}(q)=q^{-1}$. Then provided
$a_{1}^{*}+a_{2}^{*}=1$ and each $a_{i}^{*}>0$ the functions
$\rho_{1}(q)=\rho^{*}(q)^{a_{1}}$, $\rho_{2}(q)=\rho^{*}(q)^{a_{2}}$ are
applicable to Proposition 1. The constant $M$ appearing in the conditions of
$\rho$ in Proposition 1 can clearly be omitted.

Suppose that $\tau_{1}\geq\tau_{2}$. If

  1. (i)

$\tau_{2}>\tfrac{1}{2}$: Then set $a_{1}^{*}=a_{2}^{*}=\tfrac{1}{2}$. So
$a_{1}^{*}+a_{2}^{*}=1$. Now the sets
$\mathcal{K}_{1},\mathcal{K}_{2},\mathcal{K}_{3}$ as defined in Theorem 6
become

$\begin{cases}\mathcal{K}_{1}=\\{1,2\\},\quad\mathcal{K}_{2}=\emptyset,\quad\mathcal{K}_{3}=\emptyset&\text{if}\quad
A=a_{1}^{*}=a_{2}^{*},\\\\[8.61108pt]
\mathcal{K}_{1}=\emptyset,\quad\mathcal{K}_{2}=\\{2\\},\quad\mathcal{K}_{3}=\\{1\\}\,&\text{if}\quad
A=a_{2}+t_{2}=\tau_{2}+1,\\\\[8.61108pt]
\mathcal{K}_{1}=\emptyset,\quad\mathcal{K}_{2}=\\{1,2\\},\quad\mathcal{K}_{3}=\emptyset&\text{if}\quad
A=a_{1}+t_{1}=\tau_{1}+1.\end{cases}$

Inputting each of these into the formula for $s$ as in Theorem 6 we get the
values

$\operatorname{\dim_{H}}W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\geq\min\left\\{2,\frac{3}{1+\tau_{2}},\frac{3+(\tau_{1}-\tau_{2})}{1+\tau_{1}}\right\\}\,.$

  2. (ii)

$\tau_{2}<\tfrac{1}{2}$: Then set $a_{1}^{*}=1-\tau_{2}$ and
$a_{2}^{*}=\tau_{2}$. Again, note $a_{1}^{*}+a_{2}^{*}=1$. Now the sets
$\mathcal{K}_{1},\mathcal{K}_{2},\mathcal{K}_{3}$ as defined in Theorem 6
become

$\begin{cases}\mathcal{K}_{1}=\\{1,2\\}\,,\quad\mathcal{K}_{2}=\emptyset\,,\quad\
\mathcal{K}_{3}=\emptyset\,,\quad\quad\text{\rm if }\quad
A=a_{2}^{*}+1=\tau_{2}+1,\\\ \mathcal{K}_{1}=\\{1\\},\ \ \
\quad\mathcal{K}_{2}=\\{2\\},\quad\ \mathcal{K}_{3}=\emptyset\quad\quad\ {\rm
if}\ \quad A=a_{1}^{*}+1=2-\tau_{2},\\\
\mathcal{K}_{1}=\\{1,2\\},\quad\mathcal{K}_{2}=\emptyset,\quad\ \
\mathcal{K}_{3}=\emptyset\quad\quad\ \ {\rm if}\quad
A=a_{2}+t_{2}=\tau_{2}+1,\\\ \mathcal{K}_{1}=\emptyset,\quad\quad\
\mathcal{K}_{2}=\\{1,2\\},\ \mathcal{K}_{3}=\emptyset\quad\quad\ \ {\rm
if}\quad A=a_{1}+t_{1}=\tau_{1}+1\,.\end{cases}$

Inputting each of these into the formula for $s$ as in Theorem 6 we get the
values

$\operatorname{\dim_{H}}W_{1,2}^{{\mathbb{R}}}(\psi_{1},\psi_{2})\geq\min\left\\{2,2,2,\frac{3+(\tau_{1}-\tau_{2})}{1+\tau_{1}}\right\\}\,=\min\left\\{2,\frac{3+(\tau_{1}-\tau_{2})}{1+\tau_{1}}\right\\}.$

Combining the two cases gives us the lower bound of Corollary 2 completing the
proof.

##  4\. $p$-adic approximation

Fix a prime $p$ and $n,m\in{\mathbb{N}}$. Let $|\cdot|_{p}$ be the $p$-adic
norm, ${\mathbb{Q}}_{p}^{m\times n}$ the set of $m\times n$ dimensional
matrices with entries from the $p$-adic numbers ${\mathbb{Q}}_{p}$, and
$I_{p}^{m\times n}:={\mathbb{Z}}_{p}^{m\times n}$ the $m\times n$ matrices
with entries from the $p$-adic integers
${\mathbb{Z}}_{p}:=\\{x\in{\mathbb{Q}}_{p}:|x|_{p}\leq 1\\}$. For matrix $X\in
I_{p}^{m\times n}$ we denote by $X_{i}$ the $i$th column vector of $X$. Let
$\mu_{m\times n}^{{\mathbb{Q}}_{p}}$ denote the $m\times n$-dimensional Haar
measure on ${\mathbb{Q}}_{p}^{m\times n}$ normalised by
$\mu^{{\mathbb{Q}}_{p}}_{m\times n}(I_{p}^{m\times n})=1$.

Let $\Psi:{\mathbb{Z}}^{n+m}\to{\mathbb{R}}_{+}^{n}$ be an $n$-tuple of
approximation functions of the form

$\Psi(\textbf{a})=\left(\frac{\psi_{1}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}},\dots,\frac{\psi_{n}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}\right)$

with functions $\psi_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$. Throughout this
section let $\|\cdot\|_{v}$ be the quasi-norm $\|\textbf{a}\|_{v}=\max_{1\leq
i\leq n+m}|a_{i}|^{1/v_{i}}$ for vector $v=(v_{1},\dots,v_{n+m})$ with

(4.1) $v_{i}>0\quad(1\leq i\leq m),\quad\sum\limits_{i=1}^{m}v_{i}=m\,,\quad
v_{i+m}=1\quad(1\leq i\leq n).$

Define

$W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi):=\left\\{X\in I_{p}^{m\times
n}:\begin{array}[]{l}|\textbf{a}_{0}X_{i}-a_{i}|_{p}<\frac{\psi_{i}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}\quad(1\leq
i\leq n)\\\\[8.61108pt] \text{\rm for infinitely many
}\textbf{a}=(\textbf{a}_{0},a_{1},\dots,a_{n})\in{\mathbb{Z}}^{m+n}\end{array}\right\\}.$

Note that the approximation function depends on $n+m$ values, rather than $m$
values in the real setting. This is because in the $p$-adic setting, for any
$x\in{\mathbb{Z}}_{p}$, one can make the value $|x-z|_{p}$ arbitrarily small
for a sufficiently large choice of $z\in{\mathbb{Z}}$, that is, ${\mathbb{Z}}$
is dense in ${\mathbb{Z}}_{p}$. This is also why we additionally require each
approximation function to decrease faster than $\frac{1}{\|\textbf{a}\|_{v}}$,
and the condition that the last $n$ digits of the vector $v$ are each less
than or equal to $1$ is needed. For instance consider the simplified case
$n=2$, $m=1$, $\|\cdot\|_{v}$ the sup norm (i.e. $v=(1,1,1)$), and suppose
that $\frac{\psi_{1}(r)}{r}>r^{-1}$ for all sufficiently large
$r\in{\mathbb{N}}$. Then
$\mathcal{W}_{2,1}((\psi_{1},\psi_{2}))={\mathbb{Z}}_{p}^{2}$ for any choice
of function $\psi_{2}$, since for any $X\in{\mathbb{Z}}_{p}^{2}$ there are
infinitely many integers vectors (of the form
$(a_{0},a_{1},a_{2})=(0,p^{k},0)$, for $k\in{\mathbb{N}}$ sufficiently large)
that $(\psi_{1},\psi_{2})$-approximate $X$.

We prove the following $p$-adic analogue to the weighted Khintchine-Groshev
Theorem 1.

######  Theorem 7.

Let $\Psi$ be an $n$-tuple of approximation functions defined as above and
suppose each $\psi_{i}$ is monotonically decreasing. Then

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\right)=\begin{cases}0\quad\text{\rm
if
}\quad\sum\limits_{r=1}^{\infty}r^{m-1}\prod\limits_{i=1}^{n}\psi_{i}(r)<\infty,\\\\[8.61108pt]
1\quad\text{\rm if
}\quad\sum\limits_{r=1}^{\infty}r^{m-1}\prod\limits_{i=1}^{n}\psi_{i}(r)=\infty.\end{cases}$

We again highlight previously known results, these include

  * •

$n=m=1$, $\psi$ monotonic Jarník [46].

  * •

$n=m=1$, $\psi$ non-monotonic proven by Haynes [39] via Maynard &
Koukoulopolous [58]. In fact, in this paper, Haynes proved that if the
variance method from probability theory can be used to solve the Duffin-
Schaeffer conjecture, then almost the entire (classical) Duffin-Schaeffer
conjecture will follow. Conversely, if the variance method can be used to
solve completely the classical Duffin-Schaeffer conjecture, then the
corresponding conjecture is true in every field $\mathbb{Q}_{p}$.

  * •

$nm\geq 1$, $\psi$ monotonic univariable proven by Lutz [66]. See also
Beresnevich, Dickinson, Velani [9] for proof via ubiquitous systems.

  * •

$n>1$ $m=1$, $\Psi$ weighted monotonic univariable, proven by Beresnevich,
Levesley & Ward [10].

  * •

$n=1$ $m\geq 1$, $\psi$ monotonic univariable and for the inhomogeneous
settings by Datta & Ghosh [20]. This result with property $P$ in place of the
univariable condition is claimed true in [20, Remark 1.4 (5)].

We also provide the complimentary Hausdorff dimension result for the same
setup.

######  Theorem 8.

Let $\Psi$ be of the form

$\Psi(\textbf{a})=\left(\|\textbf{a}\|_{v}^{-\tau_{1}},\dots,\|\textbf{a}\|_{v}^{-\tau_{n}}\right)$

for vectors $\boldsymbol{v}=(1,\dots,1)$ and
$\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in{\mathbb{R}}^{n}_{+}$ with
$\sum\limits_{i}\tau_{i}>m+n$ and each $\tau_{i}>1$. Then

$\operatorname{\dim_{H}}W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)=\min_{1\leq j\leq
n}\left\\{n(m-1)+\frac{n+m-\sum\limits_{i:\tau_{i}<\tau_{j}}(\tau_{i}-\tau_{j})}{\tau_{j}}\right\\}=s$

and

$\mathcal{H}^{s}\left(W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\right)=\infty.$

The condition that $\sum\limits_{i}\tau_{i}>m+n$ is standard. For
$\sum\limits_{i}\tau_{i}\leq m+n$ by the standard version of Dirichlet’s
approximation theorem in the $p$-adic setting, see Lemma 4 below, we have that
$W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)=I_{p}^{m\times n}$.

Note that in the simultaneous setting, the dimension and Hausdorff measure
have already been proven [1, 9], and for the weighted setting with $m=1$ and
$n\geq 1$ was proven in [10]. Hence the novelty of the above two theorems is
the weighted approximation case of the dual linear forms.

###  4.1. Ubiquity for $p$-adics

Our key statement to prove the above results is the following ubiquity
statement in the $p$-adic setting (Proposition 2). Given the notation for a
ubiquitous system for rectangles as in §2.2 we now define our setup. Let

  1. i.

$J={\mathbb{Z}}^{m+n}$,

  2. ii.

$\beta:J\to{\mathbb{R}}_{+},\,\alpha=\textbf{a}=(\textbf{a}_{0},\textbf{a}_{1})=(a_{0,1},\dots,a_{0,m},a_{1},\dots,a_{n})\mapsto\beta_{\textbf{a}}=\|\textbf{a}\|_{v}$,

  3. iii.

$l_{k+1}=u_{k}=M^{k+1}$ for some $M\in{\mathbb{N}}$,

  4. iv.

$J_{k}=\left\\{\alpha\in J:M^{k}\leq\|\textbf{a}\|_{v}\leq M^{k+1}\right\\}$,

  5. v.

$R_{\textbf{a},i}=\left\\{X_{i}\in{\mathbb{Z}}_{p}^{m}:\textbf{a}_{0}X_{i}-a_{i}=0\right\\}$,

  6. vi.

$R_{\textbf{a}}=\prod_{i=1}^{n}R_{\textbf{a},i}$,

  7. vii.

$\kappa=\frac{m-1}{m}$ and $\delta_{i}=m$ for each $i=1,\dots,n$.

We prove the following key statement and then prove Theorems 7 and 8 in
subsections 4.2 and 4.3 respectively.

######  Proposition 2.

Consider any ball $B=B(X,r)\subset I_{p}^{m\times n}$ with centre $X\in
I_{p}^{m\times n}$ and radius $r>0$. Let $\rho=(\rho_{1},\dots,\rho_{n})$ be
an $n$-tuple of $\rho_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ functions
satisfying

(4.2) $\rho_{i}(h)=\frac{\phi_{i}(h)}{h},\,\,\,\,(1\leq i\leq n)\,\,\,\text{
and
}\quad\prod_{i=1}^{n}\rho_{i}(h)=p^{-n}h^{-(m+n)}\quad(h\in{\mathbb{R}}_{+}),$

for $\phi_{i}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ with each $\phi_{i}(h)\to
0$ as $h\to\infty$. Suppose that

(4.3)
$p^{\lambda_{0}}>2^{m+n+2}p^{n}\frac{p^{m-\frac{1}{2}}}{p^{m-\frac{1}{2}}-1}$

and

(4.4) $M\geq\left(p^{n(\lambda_{0}-1)}3^{n}4\right)^{\frac{1}{m+n}}\,.$

Then for all sufficiently large $k\in{\mathbb{N}}$, we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J_{k}}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)\geq\frac{1}{2}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(B).$

The following lemma, which can be seen as the $p$-adic version of Theorem 4 is
crucial in our ubiquitous system construction.

######  Lemma 4 ([7, Lemma 6.2]).

Let $n,m\in{\mathbb{N}}$ and $\Psi=(\psi)_{1\leq i\leq n}$ be a n-tuple of
approximation functions. Let $H_{1},\dots,H_{m+n}\geq 1$ be positive integers
and let $H^{n+m}=\prod_{i=1}^{n+m}H_{i}$. Suppose that

$\prod_{i=1}^{n}\psi_{i}(H)\geq H^{-(n+m)}p^{-n}$

with $\psi_{i}(H)<p^{-1}$ for all $1\leq i\leq n$. Then for any
$X=(x_{i,j})\in I_{p}^{m\times n}$ there exists $H_{0}$ dependent only on
$\Psi$, such that for all $H\geq H_{0}$,

$\left|a_{0,1}x_{i,1}+\dots+a_{0,m}x_{i,m}-a_{i}\right|_{p}<\psi_{i}(H)\quad(1\leq
i\leq n),$

has a non-zero solution in integers $a_{0,1},\dots,a_{0,m},a_{1},\dots,a_{n}$
satisfying

$|a_{0,i}|\leq H_{i}\,\,\,\,(1\leq i\leq m)\quad\text{and}\quad|a_{j}|\leq
H_{j}\,\,\,\,(1\leq j\leq n).$

######  Remark 1.

Many variants of this result have appeared, see for example [10, 56]. This is
a well-known result that can be obtained in the standard way using the
pigeonhole principle.

We also need the following lemma which generally tells us how often, for each
$\textbf{a}_{0}\in{\mathbb{Z}}^{m}$, the thickened resonant sets
$R_{(\textbf{a}_{0},\textbf{a}_{1})}$ intersect with a ball.

######  Lemma 5.

Let $\lambda,\lambda_{1}\in{\mathbb{N}}_{0}$ and fix some
$\textbf{a}_{0}\in{\mathbb{Z}}^{m}$ with $\|\textbf{a}_{0}\|_{p}\leq
p^{-\lambda}$. Let $B\subset I_{p}^{m\times n}$ with $r(B)=r<p^{-1}$, and
$U\in{\mathbb{N}}$. Then for all $V\in{\mathbb{N}}$ such that

$\rho_{i}(V)p^{\lambda_{1}}<r\quad\text{ and }\quad V\geq U\,,$

the cardinality of the set

$\\#\left\\{\textbf{a}_{1}\in{\mathbb{Z}}^{n}:\begin{cases}\|\textbf{a}_{1}\|\leq
U\\\ \|\textbf{a}_{1}\|_{p}\leq p^{-\lambda}\end{cases}\quad\text{ and
}\quad\Delta\left(R_{(\textbf{a}_{0},\textbf{a}_{1})},\rho_{i}(V)p^{\lambda_{1}}\right)\cap
B\neq\emptyset\right\\}$

is at most

$\left(1+\frac{U}{p^{\lambda-1}}r\right)^{n}\,.$

###### Proof.

Let $B=B(X_{i},r)=\prod_{i=1}^{n}B_{i}(X_{i},r)=\prod_{i=1}^{n}B_{i}$ for
$B_{i}(X_{i},r)$ representing a $m$-dimensional $p$-adic ball with centre
$X_{i}=(x_{i,1},\dots,x_{i,m})\in{\mathbb{Z}}_{p}^{m}$ and radius $r>0$ in the
$i$th coordinate of the $mn$-dimensional ball $B$ with centre $X\in
I_{p}^{m\times n}$ and radius $r(B)=r>0$. Consider one coordinate $1\leq i\leq
n$ at a time. If there exists

$Y_{i}\in\Delta_{i}\left(R_{\textbf{a},i},\rho_{i}(V)p^{\lambda_{1}}\right)\cap
B_{i}\subset{\mathbb{Z}}_{p}^{m}\,,$

then by definition there exists some $Z_{i}=(z_{i,1},\dots,z_{i,m})\in
R_{\textbf{a},i}$ such that

$Y_{i}\in B_{i}\left(Z_{i},\rho_{i}\left(V\right)p^{\lambda_{1}}\right)\cap
B_{i}\,.$

By the ultrametric property of ${\mathbb{Q}}_{p}$ we have that

$B_{i}(X_{i},r)\cup
B_{i}(Z_{i},\rho_{i}(V)p^{\lambda_{1}})=B_{i}\left(Z_{i},\max\left\\{\rho_{i}(V)p^{\lambda_{1}},r\right\\}\right).$

Hence

$\|Z_{i}-X_{i}\|_{p}=\max_{1\leq j\leq
m}|z_{i,j}-x_{i,j}|_{p}\leq\max\left\\{\rho_{i}(V)p^{\lambda_{1}},r\right\\}.$

Then by the strong triangle inequality, we have that

$\displaystyle\left|\sum\limits_{j=1}^{m}a_{0,j}(z_{i,j}-x_{i,j})\right|_{p}$
$\displaystyle\leq\|\textbf{a}_{0}\|_{p}\max\left\\{\rho_{i}(V)p^{\lambda_{1}},r\right\\}$
$\displaystyle\leq\max\left\\{\rho_{i}(V)p^{\lambda_{1}-\lambda},rp^{-\lambda}\right\\}:=f_{i}(V,r,\lambda,\lambda_{1})\,,$

where the second inequality follows from the condition
$\|\textbf{a}_{0}\|_{p}\leq p^{-\lambda}$. Since $Z_{i}\in R_{\textbf{a},i}$
we have that $\left(\sum\limits_{j=1}^{m}a_{0,j}z_{i,j}\right)+a_{i}=0$.
Combining with (4.1) we have

(4.6)
$\left|\left(\sum\limits_{j=1}^{m}a_{0,j}x_{i,j}\right)+a_{i}\right|_{p}\leq
f_{i}(V,r,\lambda,\lambda_{1}),$

Since $\|\textbf{a}_{1}\|_{p}\leq p^{-\lambda}$ we must have that
$p^{\lambda}|a_{i}$ for each $1\leq i\leq n$. Hence

(4.7) $a_{i}\equiv 0\mod p^{\lambda}\,.$

Combining (4.6)-(4.7) and noting that $\|\textbf{a}_{1}\|\leq U$ we have that
each $a_{i}$, written in base $p$-adic expansion, is of the form

$a_{i}=\sum_{k=\lambda+1}^{\max\\{\lambda+1,-\log_{p}f_{i}(V,r,\lambda,\lambda_{1})\\}}d_{k}p^{k}+\sum_{j=\max\\{\lambda+1,-\log_{p}f_{i}(k,r,\lambda,\lambda_{1})\\}+1}^{1+\log_{p}U}d_{j}p^{j}\quad\quad
d_{k}\in\\{0,\dots p-1\\}\,,$

with $d_{k}$ fixed for
$k\in\left\\{\lambda,\dots,\max\\{\lambda+1,-\log_{p}f_{i}(V,r,\lambda,\lambda_{1})\\}\right\\}$
depending on $X_{i}$. Thus there are at most

$1+p^{\left(1+\log_{p}U\,-\max\\{\lambda+1,-\log_{p}f_{i}(V,r,\lambda,\lambda_{1})\\}\right)}=1+pU\min\\{p^{-\lambda-1},f_{i}(V,r,\lambda,\lambda_{1})\\}$

possible values of $a_{i}$. Inputting the value for
$f_{i}(V,r,\lambda,\lambda_{1})$ we have that the cardinality of possible
values of each $a_{i}$ is bounded above by

$1+\min\left\\{\frac{U}{p^{\lambda}},\max\left\\{\frac{U}{p^{\lambda}}\left(\rho_{i}(V)p^{\lambda_{1}+1}\right),\frac{U}{p^{\lambda}}rp\right\\}\right\\}.$

Since $\rho_{i}(V)p^{\lambda_{1}+1}<r$ we have that the cardinality is bounded
from above by

$1+\frac{U}{p^{\lambda}}rp\,.$

Taking the product over each coordinate axis gives us our claimed upper bound.

∎

We now proceed with the proof of Proposition 2.

Let $B=B(X,r)$ with $p^{-1}>r>0$ and $X=(X_{1},\dots,X_{n})=(x_{i,j})\in
I_{p}^{m\times n}$. Choose $k_{r}$ sufficiently large such that for all
$k>k_{r}$

(4.8) $\max_{1\leq i\leq n}\phi_{i}(M^{k+1})<r\quad\text{ and
}\quad\rho_{i}(M^{k+1})p^{\lambda_{0}}<r\,.$

Without loss of generality we can assume that
$r\in\\{p^{-t}:t\in{\mathbb{N}}_{0}\\}$. Denote each linear form

$\textbf{a}_{0}X_{i}+a_{i}=\left(\sum\limits_{j=1}^{m}a_{0,j}x_{i,j}\right)+a_{i}\quad(1\leq
i\leq n)$

for
$\textbf{a}=(\textbf{a}_{0},a_{1},\dots,a_{n})=(a_{0,1},\dots,a_{0,m},a_{1},\dots,a_{n})\in{\mathbb{Z}}^{m+n}$.
Then, by Lemma 4 and conditions (4.2) on $\rho$, we have that the system

$\begin{cases}|\textbf{a}_{0}X_{i}+a_{i}|_{p}\leq\rho_{i}(H)\quad(1\leq i\leq
n),\\\\[8.61108pt] |a_{0,i}|\leq H^{v_{i}}\hskip 56.9055pt(1\leq i\leq
m),\\\\[8.61108pt] |a_{i}|\leq H\hskip 54.06006pt(1\leq i\leq n),\end{cases}$

has a non-zero rational integer solution $\textbf{a}\in{\mathbb{Z}}^{m+n}$.
Furthermore, by (4.1) for each $1\leq i\leq n$ we have that $|a_{i}|\leq H$
and since each $\rho_{i}(H)<H^{-1}$ we are forced to conclude that
$\textbf{a}_{0}\neq 0$. Note that if $\|\textbf{a}_{0}\|_{p}=p^{-\lambda}$ for
some $\lambda\in{\mathbb{N}}_{0}$ then $\|\textbf{a}_{1}\|_{p}\leq
p^{-\lambda}$. This follows by applying the strong triangle inequality to
first row of the above system of inequalities along with the previous
observation that each $\rho_{i}(H)<H^{-1}$.

The system of inequalities readily implies that

(4.9) $\mu^{{\mathbb{Q}}_{p}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J:\|\alpha\|_{v}\leq
M^{k+1}}\Delta\left(R_{\alpha},\rho(M^{k+1})\|\textbf{a}_{0}\|_{p}^{-1}\right)\right)=\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(B).$

Since we want the sidelenghts of the rectangles in the above set to be
independent of $\textbf{a}_{0}$ we want to bound the size of
$\|\textbf{a}_{0}\|_{p}$. We need $\lambda_{0}\in{\mathbb{N}}$ to be large
enough to satisfy some technical condition later, hence we have assumed
condition (4.3).

For each $\lambda\in{\mathbb{N}}_{0}$ consider the sets

$\displaystyle\widehat{J}(k,\lambda)$ $\displaystyle:=\left\\{\alpha\in
J:\|\alpha\|_{v}\leq M^{k+1}\quad\text{ and
}\quad\|\textbf{a}_{0}\|_{p}=p^{-\lambda}\right\\}\,,$
$\displaystyle\widetilde{J}(k,\lambda_{0})$ $\displaystyle:=\left\\{\alpha\in
J:\|\alpha\|_{v}<M^{k}\quad\text{ and }\quad\|\textbf{a}_{0}\|_{p}\geq
p^{-\lambda_{0}}\right\\}\,,$ $\displaystyle J_{k}(\lambda_{0})$
$\displaystyle:=\left\\{\alpha\in J:M^{k}\leq\|\alpha\|_{v}\leq
M^{k+1}\quad\text{ and }\quad\|\textbf{a}_{0}\|_{p}\geq
p^{-\lambda_{0}}\right\\}\,,$

and note that

$J_{k}\supseteq
J_{k}(\lambda_{0})\cup\widetilde{J}(k,\lambda_{0})\cup\bigcup_{\lambda>\lambda_{0}}\widehat{J}(k,\lambda)\,.$

Hence, we have that

$\displaystyle\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(B)=\mu^{{\mathbb{Q}}_{p}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J:\|\alpha\|_{v}\leq
M^{k+1}}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)\|\textbf{a}_{0}\|_{p}^{-1}\right)\right)$
$\displaystyle\leq$ $\displaystyle\underset{:=\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{1})}{\underbrace{\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\bigcup_{\lambda>\lambda_{0}}\bigcup_{\alpha\in\widehat{J}(k,\lambda)}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda}\right)\right)}}$
$\displaystyle+\,\,\,\,\underset{:=\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{2})}{\underbrace{\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\bigcup_{\alpha\in\widetilde{J}(k,\lambda_{0})}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)}}$
$\displaystyle+\,\,\,\,\underset{:=\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{3})}{\underbrace{\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\bigcup_{\alpha\in
J_{k}(\lambda_{0})}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)}}\,.$

Considering our set of interest note that $J_{k}\supseteq J_{k}(\lambda_{0})$
and so we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J_{k}}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)\,\geq\,\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{3})\,\geq\,\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(B)-\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{1})-\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{2})\,.$

Thus showing that

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{1})<\tfrac{1}{4}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)\quad\text{ and
}\quad\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{2})<\tfrac{1}{4}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)$

completes the proof.

For each $\alpha\in J$, $\lambda\in{\mathbb{N}}_{0}$ the thickened resonant
sets with non-empty intersection with the ball $B$ has measure at most

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda}\right)\right)\leq
p^{n\lambda}r^{n(m-1)}\prod_{i=1}^{n}\rho_{i}\left(M^{k+1}\right)=p^{n\lambda-n}r^{n(m-1)}M^{-(k+1)(m+n)}\,.$

Now

$\displaystyle\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{1})$
$\displaystyle\leq\sum_{\lambda>\lambda_{0}}\sum_{\alpha\in\widehat{J}(k,\lambda)}p^{n\lambda-n}r^{n(m-1)}M^{-(k+1)(m+n)}$
$\displaystyle\leq\sum_{\lambda>\lambda_{0}}\sum_{\|\textbf{a}_{0}\|_{v}\leq
M^{k+1}:\|\textbf{a}_{0}\|_{p}=p^{-\lambda}}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{n\lambda-n}r^{n(m-1)}M^{-(k+1)(m+n)}$

by Lemma 5 setting $U=V=M^{k+1}$ and $\lambda_{1}=\lambda$. Observe Lemma 5 is
applicable because $\|\textbf{a}_{0}\|_{p}=p^{-\lambda}$ implies that
$\|\textbf{a}_{1}\|_{p}\leq p^{-\lambda}$, and, since $p^{\lambda}\leq
M^{k+1}$, we have $\rho_{i}(M^{k+1})p^{\lambda}<r$ by (4.8). Then

$\displaystyle\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{1})$
$\displaystyle\leq\sum_{\lambda>\lambda_{0}}\left(2\frac{M^{k+1}}{p^{\lambda}}\right)^{m}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{n\lambda-n}r^{n(m-1)}M^{-(k+1)(m+n)}$
(4.11) $\displaystyle\leq
2^{m}p^{-n}\sum_{\lambda>\lambda_{0}}M^{-(k+1)n}r^{n(m-1)}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{(n-m)\lambda}$
(4.12)
$\displaystyle=2^{m}p^{-n}\sum_{\lambda>\lambda_{0}}\left(M^{-(k+1)}p^{\lambda(1-\frac{1}{2n})}\right)^{n}r^{n(m-1)}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{(-m+\frac{1}{2})\lambda}$

Now, suppose $\frac{M^{k+1}}{p^{\lambda}}r<1$. Then since
$p^{\lambda}<M^{k+1}$, and by the choice of $k_{0}$ so that
$M^{-\frac{1}{2n}(k+1)}<r$ for all $k>k_{0}$, we have that

$\left(M^{-(k+1)}p^{\lambda(1-\frac{1}{2n})}\right)^{n}<r^{n}\,.$

If $\frac{M^{k+1}}{p^{\lambda}}r\geq 1$ then trivially

$\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}\leq
2^{n}p^{n}\left(\frac{M^{k+1}}{p^{\lambda}}r\right)^{n}\,.$

So

$\left(M^{-(k+1)}p^{\lambda(1-\frac{1}{2n})}\right)^{n}r^{n(m-1)}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{(-m+\frac{1}{2})\lambda}\leq
2^{n}r^{nm}p^{(-m+\frac{1}{2})\lambda}\quad\text{ if
}\quad\frac{M^{k+1}}{p^{\lambda}}r<1\,,$

and

$M^{-(k+1)n}r^{n(m-1)}\left(1+\frac{M^{k+1}}{p^{\lambda-1}}r\right)^{n}p^{(n-m)\lambda}\leq
2^{n}p^{n}r^{nm}p^{-m\lambda}\quad\text{ if
}\quad\frac{M^{k+1}}{p^{\lambda}}r\geq 1\,.$

Hence

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{1})\leq
2^{m+n}p^{n}r^{nm}\sum_{\lambda>\lambda_{0}}p^{(-m+\frac{1}{2})\lambda}\leq\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(B)2^{m+n}p^{n}\frac{p^{m-\frac{1}{2}}}{p^{m-\frac{1}{2}}-1}p^{-\lambda_{0}}$

and so by (4.3) we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{1})\leq\frac{1}{4}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)$

as required.

To show that $\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{2})<\tfrac{1}{4}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)$ observe that

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{2})\leq\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\bigcup_{\alpha\in J:\|\alpha\|_{v}\leq
M^{k}}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)\,$

and furthermore that,

$\displaystyle\mu^{{\mathbb{Q}}_{p}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J:\|\alpha\|_{v}\leq
M^{k}}\Delta\left(R_{\alpha},\rho\left(M^{k+1}\right)p^{\lambda_{0}}\right)\right)$
$\displaystyle\hskip
28.45274pt\leq\sum\limits_{\|\textbf{a}_{0}\|_{(v_{1},\dots,v_{m})}\leq
M^{k}}\,\underset{\Delta(R_{\textbf{a}},\rho(M^{k+1})p^{\lambda_{0}})\cap
B\neq\emptyset}{\sum\limits_{\|(a_{1},\dots,a_{n})\|\leq
M^{k}:}}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\Delta(R_{\textbf{a}},\rho(M^{k+1})p^{\lambda_{0}}\right).$

Now,

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(A_{2}\right)\leq\left(\prod_{i=1}^{m}M^{kv_{i}}\right)\left(2M^{k}r+1\right)^{n}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(B\cap\Delta\left(R_{\textbf{a}},\rho(M^{k+1})p^{\lambda_{0}}\right)\right)\\\
$

by Lemma 5 setting $U=M^{k}$, $V=M^{k+1}$, $\lambda=0$ and
$\lambda_{1}=\lambda_{0}$. Since $\lambda_{0}$ is fixed it is clear that
$\rho_{i}(M^{k+1})p^{\lambda_{0}}<r$.

Then inputting the previous upper bound on the measure of such thickened
resonant sets we have

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{2})\leq
M^{k\sum_{i=1}^{m}v_{i}}\left(2M^{k}r+1\right)^{n}p^{n\lambda_{0}-n}r^{n(m-1)}M^{-(k+1)(m+n)}.$

Since $k$ is chosen sufficiently large such that $M^{k}r>1$ we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times n}(A_{2})\leq
3^{n}r^{nm}p^{n\lambda_{0}-n}M^{-(m+n)}\leq
3^{n}p^{n(\lambda_{0}-1)}M^{-(n+m)}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)$

Thus, by (4.4) we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(A_{2})\leq\frac{1}{4}\mu^{{\mathbb{Q}}_{p}}_{m\times n}(B)\,.$

Hence, by considering (4.1) and the above two calculations we have completed
the proof.

###  4.2. Proof of Theorem 7

We will on occasion use the following estimate

(4.13)
$\\#J_{k}=\\#\left\\{\textbf{a}\in{\mathbb{Z}}^{n+m}:M^{k}<\|\textbf{a}\|_{v}\leq
M^{k+1}\right\\}\leq cM^{k(n+m)}$

for constant $c=2(n+m)3^{n+m-1}\left(M^{n+m}-M^{n+m-\min_{i}v_{i}}\right)$
independent of $k$. To see this observe that

$\displaystyle\sum\limits_{M^{k}<\|\textbf{a}\|_{v}\leq M^{k+1}}1$
$\displaystyle=\sum\limits_{j=1}^{n+m}\underset{|a_{t}|^{1/v_{t}}\leq
M^{k+1}\,,\,1\leq t\leq n+m\,\,t\neq
j}{\sum\limits_{M^{k}<|a_{j}|^{1/v_{j}}\leq M^{k+1}}}1$
$\displaystyle\leq\sum\limits_{j=1}^{n+m}3^{n+m-1}M^{(k+1)\sum\limits_{t=1\,\,t\neq
j}^{n+m}v_{t}}2\left(M^{(k+1)v_{j}}-M^{kv_{j}}\right)$
$\displaystyle=3^{n+m-1}2\sum\limits_{j=1}^{n+m}M^{(k+1)\sum\limits_{t=1\,\,t\neq
j}^{n+m}v_{t}}M^{kv_{j}}\left(M^{v_{j}}-1\right)$ $\displaystyle\leq
3^{n+m-1}2(n+m)\max_{1\leq j\leq n+m}M^{k(n+m)}M^{\sum\limits_{t=1\,\,t\neq
j}^{n+m}v_{t}}\left(M^{v_{j}}-1\right)$
$\displaystyle=3^{n+m-1}2(n+m)\left(M^{n+m}-M^{n+m-\min_{1\leq j\leq
n+m}v_{j}}\right)M^{k(n+m)}.$

Recall

$\displaystyle W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)$ $\displaystyle:=\left\\{X\in
I_{p}^{m\times
n}:\begin{array}[]{c}|\textbf{a}_{0}X_{i}-a_{i}|_{p}<\frac{\psi_{i}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}\quad(1\leq
i\leq n)\\\ \text{ for infinitely many
}\textbf{a}=(\textbf{a}_{0},a_{1},\dots,a_{n})\in{\mathbb{Z}}^{m+n}\end{array}\right\\}$
$\displaystyle=\limsup_{\textbf{a}\in{\mathbb{Z}}^{m+n}}\left\\{X\in
I_{p}^{m\times
n}:|\textbf{a}_{0}X_{i}-a_{i}|_{p}<\frac{\psi_{i}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}\quad(1\leq
i\leq n)\right\\}.$

For ease of notation, we will write for all
$\mathbf{a}=(\mathbf{a}_{0},a_{1},\dots,a_{n})\in{\mathbb{Z}}^{m+n}$

$\mathcal{A}_{\textbf{a}}(\Psi)=\left\\{X\in I_{p}^{m\times
n}:|\textbf{a}_{0}X_{i}-a_{i}|_{p}<\frac{\psi_{i}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}\quad(1\leq
i\leq n)\right\\}.$

For completeness, we prove the convergence case of Theorem 7.

By the Borel-Cantelli convergence Lemma (Lemma 1), we have that

$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\right)=0\quad\text{\rm if
}\quad\sum\limits_{\textbf{a}\in{\mathbb{Z}}^{n+m}}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(\mathcal{A}_{\textbf{a}}(\Psi)\right)<\infty.$

Observe that

$\displaystyle\sum\limits_{\textbf{a}\in{\mathbb{Z}}^{n+m}}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(\mathcal{A}_{\textbf{a}}(\Psi))$
$\displaystyle=\sum\limits_{k=1}^{\infty}\sum_{\begin{subarray}{c}\textbf{a}\in{\mathbb{Z}}^{m+n}:\\\
M^{k}<\|\textbf{a}\|_{v}\leq
M^{k+1}\end{subarray}}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(\mathcal{A}_{\textbf{a}}(\Psi))$
$\displaystyle\asymp_{p,n}\sum\limits_{k=1}^{\infty}\sum_{\begin{subarray}{c}\textbf{a}\in{\mathbb{Z}}^{m+n}:\\\
M^{k}<\|\textbf{a}\|_{v}\leq
M^{k+1}\end{subarray}}\prod_{i=1}^{n}\frac{\psi_{i}(\|\textbf{a}\|_{v})}{\|\textbf{a}\|_{v}}$
$\displaystyle\leq\sum\limits_{k=1}^{\infty}\sum_{\begin{subarray}{c}\textbf{a}\in{\mathbb{Z}}^{m+n}:\\\
M^{k}<\|\textbf{a}\|_{v}\leq
M^{k+1}\end{subarray}}\prod_{i=1}^{n}\frac{\psi_{i}(M^{k})}{M^{k}}$
$\displaystyle\leq\sum\limits_{k=1}^{\infty}cM^{k(n+m)}\prod_{i=1}^{n}\frac{\psi_{i}(M^{k})}{M^{k}}$
$\displaystyle\leq
c\sum\limits_{k=1}^{\infty}M^{k}\left(M^{k(m-1)}\prod_{i=1}^{n}\psi_{i}(M^{k})\right).$

Note that we can trivially assume

$\prod_{i=1}^{n}\psi_{i}(r)<p^{-n}r^{-m}$

for all $r\in{\mathbb{N}}$, otherwise by Lemma 4 we have
$W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)={\mathbb{Z}}_{p}^{m\times n}$. Hence we can
assume

$M^{k(m-1)}\prod_{i=1}^{n}\psi_{i}(M^{k})$

is decreasing in $k\in{\mathbb{N}}$, and so by Cauchy condensation we have
that

$\displaystyle\sum\limits_{\textbf{a}\in{\mathbb{Z}}^{n+m}}\mu^{{\mathbb{Q}}_{p}}_{m\times
n}(\mathcal{A}_{\textbf{a}}(\Psi))\leq
Mc\sum\limits_{r=1}^{\infty}r^{m-1}\prod_{i=1}^{n}\psi_{i}(r)\,.$

So the convergence case follows by the Borel-Cantelli Lemma and convergence
assumption on $\sum\limits_{r=1}^{\infty}r^{m-1}\prod_{i=1}^{n}\psi_{i}(r)$.

To prove the divergence case of Theorem 7 we use Proposition 2 and Theorem 5.
We need to find an $n$-tuple of functions $\rho=(\rho_{1},\dots,\rho_{n})$
such that (4.2) and the conditions (I)-(III) are satisfied for any chosen
$\Psi$. Importantly observe that our $n$-tuple of functions $\Psi$ are of the
form $\frac{\psi_{i}(r)}{r}$ in each coordinate axis $1\leq i\leq n$, and so
we are asking that:

  1. I(${\mathbb{Q}}_{p}$).

For each $1\leq i\leq n$, the function $\frac{\psi_{i}(r)}{r}$ is montonic
decreasing as $r\to\infty$,

  2. II(${\mathbb{Q}}_{p}$).

For each $1\leq i\leq n$, we have $\frac{\psi_{i}(r)}{r}\leq\rho_{i}(r)$ and
$\rho_{i}(r)\to 0$ as $r\to\infty$,

  3. III(${\mathbb{Q}}_{p}$).

For each $1\leq i\leq n$, the function $\frac{\psi_{i}(r)}{r}$ is $c$-regular
on $(M^{k})_{k\geq 1}$ for some fixed constant $0<c<1$.

Firstly, by assumption, each $\psi_{i}$ is monotonic so I(${\mathbb{Q}}_{p}$)
is satisfied. Secondly, note that $\Psi$ is $M^{-1}$-regular on
$(M^{k})_{k\geq 1}$ since in each coordinate

$\frac{\psi_{i}(M^{k+1})}{M^{k+1}}\leq\frac{\psi_{i}(M^{k})}{M^{k+1}}\leq
M^{-1}\frac{\psi_{i}(M^{k})}{M^{k}},$

where the first inequality follows due to the monotonicity of each $\psi_{i}$,
so III(${\mathbb{Q}}_{p}$) is satisfied. Thus it remains to choose functions
$\rho_{i}$ such that II(${\mathbb{Q}}_{p}$) and (4.2) are satisfied.

At each $u\in{\mathbb{N}}$ let $l(u)_{1},\dots,l(u)_{n}$ be an ordering of the
digits $1,\dots,n$ such that

$\psi_{l(u)_{1}}(u)\geq\psi_{l(u)_{2}}(u)\geq\dots\geq\psi_{l(u)_{n}}(u).$

For each $u\in{\mathbb{N}}$ there exists unique $0\leq j\leq n-1$ such that,
for $\nu(u,j)$ defined by

$\nu(u,j)=\left(p^{-n}u^{-m}\left(\prod_{i=1}^{j}\psi_{l(u)_{i}}(u)\right)^{-1}\right)^{\frac{1}{n-j}},$

we have that

$\psi_{l(u)_{1}}(u)\geq\dots\geq\psi_{l(u)_{j}}(u)\geq\nu(u,j)>\psi_{l(u)_{j+1}}(u)\geq\dots\geq\psi_{l(u)_{n}}(u).$

We can assume without loss of generality that $\nu(u,n-1)>\psi_{l(u)_{n}}(u)$
for all sufficiently large $u\in{\mathbb{N}}$, and so such $0\leq j\leq n-1$
exists. Otherwise, we would have that for infinitely many $u\in{\mathbb{N}}$

$p^{-n}u^{-(n+m)}<\prod_{i=1}^{n}\frac{\psi_{l(u)_{i}}(u)}{u}$

and so, by applying Lemma 4, we could conclude that
$W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)=I_{p}^{m\times n}$, thus completing the
proof.

For each $u\in{\mathbb{N}}$ set

$\begin{cases}\phi_{l(u)_{i}}(u)=\psi_{l(u)_{i}}(u)\quad\quad(1\leq i\leq
j),\\\ \phi_{l(u)_{i}}(u)=\nu(u,j)\quad\quad\,\,\,\,\,\,\,\,\,(j+1\leq i\leq
n).\end{cases}$

Note that, by definition, $\phi_{i}(u)\geq\psi_{i}(u)$, and that
$\phi_{i}(u)\to 0$ as $u\to\infty$ for each $1\leq i\leq n$ since either

$\begin{cases}\phi_{i}(u)=\psi_{i}(u)\to 0,\\\
\phi_{i}(u)=\nu(u,j)\leq\psi_{j}(u)\to 0\quad\quad\quad(1\leq j\leq n-1),\\\
\phi_{i}(u)=\nu(u,0)=p^{-1}u^{-\frac{m}{n}}\to 0\end{cases}$

as $u\to\infty$. For each $1\leq i\leq n$ define

$\rho_{i}(u)=\frac{\phi_{i}(u)}{u}\geq\frac{\psi_{i}(u)}{u}\,,$

thus II(${\mathbb{Q}}_{p}$) is satisfied. Lastly, observe that for each
$u\in{\mathbb{N}}$

$\displaystyle\prod_{i=1}^{n}\rho_{i}(u)$
$\displaystyle=\prod_{i=1}^{n}\frac{\phi_{l(u)_{i}}(u)}{u}$
$\displaystyle=u^{-(n-j)}\left(p^{-n}u^{-m}\left(\prod_{i=1}^{j}\psi_{l(u)_{i}}(u)\right)^{-1}\right)\times\prod_{i=1}^{j}\frac{\psi_{l(u)_{i}}(u)}{u}$
$\displaystyle=p^{-n}u^{-(n+m)},$

so (4.2) is satisfied.

Hence Theorem 5 and Proposition 2.2 are applicable. Thus
$\mu^{{\mathbb{Q}}_{p}}_{m\times
n}\left(W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\right)=1$ if

$\sum\limits_{k=1}^{\infty}\prod_{i=1}^{n}\left(\frac{\psi_{i}(u_{k})}{u_{k}\rho_{i}(u_{k})}\right)^{m\left(1-\tfrac{m-1}{m}\right)}=\infty.$

Inputting our chosen $\rho$ we have that

$\sum\limits_{k=1}^{\infty}u_{k}^{n+m}\prod_{i=1}^{n}\frac{\psi_{i}(u_{k})}{u_{k}}\asymp\sum\limits_{k=1}^{\infty}u_{k}^{m}\prod_{i=1}^{n}\psi_{i}(u_{k})\asymp\sum\limits_{r=1}^{\infty}r^{m-1}\prod_{i=1}^{n}\psi_{i}(r),$

where the last line follows by Cauchy condensation with our choice of
$(u_{k})_{k\geq 1}=(M^{k})_{k\geq 1}$. This completes the proof of Theorem 7.

###  4.3. Proof of Theorem 8

Recall that

$\Psi(r)=(r^{-\tau_{1}},\dots,r^{-\tau_{n}}),$

and for ease of notation let

$\mathcal{A}_{\textbf{a}}(\tau)=\left\\{X\in{\mathbb{Z}}_{p}^{m\times
n}:|\textbf{a}_{0}X_{i}-a_{i}|_{p}<\|\textbf{a}\|_{v}^{-\tau_{i}}\quad(1\leq
i\leq n)\right\\},$

so that

$W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)=\limsup_{\textbf{a}\in{\mathbb{Z}}^{n+m}}\mathcal{A}_{\textbf{a}}(\tau).$

The upper bound uses a standard covering argument, but for completeness, we
include it here. Note, for any $N\in{\mathbb{N}}$

$W_{n,m}^{{\mathbb{Z}}_{p}}(\tau)\subset\bigcup_{r\geq
N}\bigcup_{\|\textbf{a}\|_{v}=r}\mathcal{A}_{\textbf{a}}(\tau)$

is a cover of $W_{n,m}^{{\mathbb{Z}}_{p}}(\tau)$ by rectangles. Since the
Hausdorff dimension is determined by covers of balls we consider each $1\leq
j\leq n$ and cover the collections of rectangles
$\mathcal{A}_{\textbf{a}}(\tau)$ in the layer $\|\textbf{a}\|_{v}=r$ by balls
of radius $\|\textbf{a}\|_{v}^{-\tau_{j}}$ to obtain our upper bound. Observe
that in each coordinate axis

$\mathcal{A}_{\textbf{a},i}(\tau)=\left\\{X_{i}\in{\mathbb{Z}}_{p}^{m}:|\textbf{a}_{0}X_{i}-a_{i}|<\|\textbf{a}\|_{v}^{-\tau_{i}}\right\\}$

can be covered by

(4.14)
$\asymp\max\left\\{1,\frac{\|\textbf{a}\|_{v}^{-\tau_{i}}}{\|\textbf{a}\|_{v}^{-\tau_{j}}}\right\\}\times\left(\frac{1}{\|\textbf{a}\|_{v}^{-\tau_{j}}}\right)^{m-1}$

balls of size $\|\textbf{a}\|_{v}^{-\tau_{j}}$. So, for any $s>0$,

$\displaystyle\mathcal{H}^{s}\left(W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\right)$
$\displaystyle\ll\sum\limits_{r\geq
N}r^{-s\tau_{j}}r^{n+m-1}r^{n(m-1)\tau_{j}}\prod_{i=1}^{n}\max\\{1,r^{\tau_{j}-\tau_{i}}\\}$
$\displaystyle\ll\sum\limits_{r\geq
N}r^{-\tau_{j}s+n+m-1+n(m-1)\tau_{j}+\sum\limits_{i:\tau_{j}>\tau_{i}}(\tau_{j}-\tau_{i})},$

which converges for all

$s>s_{j}:=n(m-1)+\frac{n+m-\sum\limits_{i:\tau_{j}>\tau_{i}}(\tau_{i}-\tau_{j})}{\tau_{j}}.$

Hence by definition of Hausdorff dimension
$\operatorname{\dim_{H}}W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)\leq s_{j}$. Since we
can do the same calculation as above for each $1\leq j\leq n$ the upper bound
of Theorem 8 follows.

For the lower bound of Theorem 8 we use Theorem 6 combined with Proposition
2.2. Assume without loss of generality that

$\tau_{1}\geq\tau_{2}\geq\dots\geq\tau_{n}>1.$

In Proposition 2 we pick each $\rho_{i}$ to be of the form

$\rho_{i}(h)=\rho(h)^{\ell_{i}}=\left(p^{-1}h^{-1}\right)^{\ell_{i}}$

for

(4.15) $\boldsymbol{\ell}=(\ell_{1},\dots,\ell_{n})\in(1,m-1)^{n}\quad\text{
with }\,\,\sum\limits_{i=1}^{n}\ell_{i}=n+m.$

Observing such a choice of exponent satisfies the requirements of Proposition
2.2. As in the setup of Theorem 6, set each

$t_{i}=\tau_{i}-\ell_{i}\quad(1\leq i\leq n)\,.$

This choice of $\rho$ function includes the constant $"p^{-1}"$ appearing in
Proposition 2, but note this can safely be removed to obtain our result by
observing that for any choice of $\varepsilon>0$ there exists sufficiently
large $h\in{\mathbb{R}}_{+}$ such that

$h^{-\tau_{i}-\varepsilon}\leq\rho(h)^{\ell_{i}+t_{i}}=\left(p^{-1}h^{-1}\right)^{\tau_{i}}\leq
h^{-\tau_{i}}\,\quad(1\leq i\leq n).$

Consider the following cases:

  1. (1)

(Ball to rectangle) $\tau_{i}\geq\frac{n+m}{n}$ for all $1\leq i\leq n$. Then
let each $\ell_{i}=\frac{n+m}{n}$. Note such choice satisfies (4.15). The set
$A$ from Theorem 6 takes the following order:

$\ell_{1}=\dots=\ell_{n}\leq\tau_{n}\leq\dots\leq\tau_{1}.$

So for any $\ell_{i}$ we have

$\mathcal{K}_{1}=\\{1,\dots,n\\},\quad\mathcal{K}_{2}=\emptyset,\quad\mathcal{K}_{3}=\emptyset,$

which trivially leads to a full dimension lower bound in the case
$A=\ell_{i}$.

For each $\tau_{j}$ we have

$\mathcal{K}_{1}=\emptyset,\quad\mathcal{K}_{2}=\\{j,\dots,n\\},\quad\mathcal{K}_{3}=\\{1,\dots,j-1\\}.$

Hence

$\displaystyle\operatorname{\dim_{H}}W_{n,m}^{{\mathbb{Z}}_{p}}(\Psi)$
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{(n-j+1)m+(m-1)(j-1)+\frac{\frac{n+m}{n}(j-1)-\sum\limits_{j\leq
i\leq n}(\tau_{i}-\frac{n+m}{n})}{\tau_{j}}\right\\}$ (4.16)
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{n(m-1)+\frac{(n+m)-\sum\limits_{j\leq i\leq
n}(\tau_{i}-\tau_{j})}{\tau_{j}}\right\\}.$

  2. (2)

(Rectangle to rectangle) $\tau_{j}<\frac{n+m}{n}$ for some $1\leq j\leq n$.
The idea now is to form a ”Dirichlet-exponent” rectangle that contains the
$\tau$-rectangle. To do this we are trying to find $1\leq u\leq n$ that solves

$u\times\widetilde{D}+\sum\limits_{u<i\leq n}\tau_{i}=n+m,$

for some $\widetilde{D}>0$ with $\tau_{i}>\widetilde{D}$ for all $1\leq i\leq
u$. That is, pick $u$ such that

$\tau_{1}\geq\dots\geq\tau_{u}>\frac{n+m-\sum\limits_{u\leq i\leq
n}\tau_{i}}{u}:=\widetilde{D}\geq\tau_{u+1}\geq\dots\geq\tau_{n}.$

Set

$\displaystyle\ell_{i}$ $\displaystyle=\widetilde{D}\quad(1\leq i\leq u),$
$\displaystyle\ell_{i}$ $\displaystyle=\tau_{i}\quad(u+1\leq i\leq n),$

and observe, by definition of $\widetilde{D}$ and the fact that each
$\tau_{i}>1$, that $\boldsymbol{\ell}$ satisfies (4.15). For $\ell_{i}$ with
$1\leq i\leq u$ we have that

$\mathcal{K}_{1}=\\{1,\dots,u\\},\quad\mathcal{K}_{2}=\\{u+1,\dots,n\\},\quad\mathcal{K}_{3}=\emptyset,$

and for $\ell_{i}$ with $u+1\leq i\leq n$

$\mathcal{K}_{1}=\\{1,\dots,i\\},\quad\mathcal{K}_{2}=\\{i+1,\dots,n\\},\quad\mathcal{K}_{3}=\emptyset.$

Thus, for $A=\ell_{i}$ we obtain the trivial full dimension as a lower bound.

For each $\tau_{j}$ with $1\leq j\leq u$ we have that

$\mathcal{K}_{1}=\emptyset,\quad\mathcal{K}_{2}=\\{j,\dots,n\\},\quad\mathcal{K}_{3}=\\{1,\dots,j-1\\},$

thus the same calculation of (4.16) follows to complete the proof.

##  5\. Complex approximation

In this section we obtain a Lebesgue measure dichotomy on certain sets of
systems of complex linear forms. Under an additional assumption, we calculate
the Hausdorff dimension of such sets when they are null.

For each $z\in{\mathbb{C}}$, let $\llbracket z\rrbracket$ be the distance from
$z$ to its nearest Gaussian integer; that is

$\llbracket z\rrbracket:=\min\left\\{|z-p|:p\in{\mathbb{Z}}[i]\right\\}.$

where $|\cdot|$ is the Euclidean norm in the complex plane. Let
$I_{{\mathbb{C}}}$ be the compact square

$I_{{\mathbb{C}}}:=\left\\{z\in{\mathbb{C}}:-\frac{1}{2}\leq\Re(z)\leq\frac{1}{2}\;\text{
and }\;-\frac{1}{2}\leq\Im(z)\leq\frac{1}{2}\right\\}.$

Let $m,n\in{\mathbb{N}}$ be arbitrary, they will remain fixed for the rest of
this section. For any $n$-tuple of non-increasing positive functions
$\varphi=(\varphi_{1},\ldots,\varphi_{n}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{m}$
with

$\lim_{q\to\infty}\varphi_{j}(q)=0\quad(1\leq j\leq n)\,,$

and any $m$-tuple of non-decreasing positive functions
$\Phi=(\Phi_{1},\ldots,\Phi_{m}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{m}$ such
that

$\lim_{q\to\infty}\Phi_{k}(q)=\infty\quad(1\leq k\leq m),$

let $W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)\subseteq I_{{\mathbb{C}}}^{m\times
n}$ be the collection of $m\times n$ matrices $A$ with entries in
$I_{{\mathbb{C}}}$ which verify the following property:

for infinitely many integers $u\geq 1$ there is a non-zero
$\mathbf{q}\in{\mathbb{Z}}[i]^{1\times m}$ such that

$\displaystyle\left\llbracket\mathbf{q}\,A_{j}\right\rrbracket$
$\displaystyle<\varphi_{j}(u)\quad(1\leq j\leq n),$ $\displaystyle|q_{k}|$
$\displaystyle\leq\Phi_{k}(u)\quad(1\leq k\leq m).$

For $k\in\\{1,\ldots,m\\}$ and $u\in{\mathbb{N}}$, write

$\Phi_{k}^{-1}(u):=\min\\{v\in{\mathbb{N}}:\Phi_{k}(v)\geq u\\}.$

The set $W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)$ is then

$W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)=\left\\{A\in I_{{\mathbb{C}}}^{m\times
n}:\begin{array}[]{l}\left\llbracket\mathbf{q}\,A_{j}\right\rrbracket<\varphi_{j}\left(\max\left\\{\Phi_{1}^{-1}(|q_{1}|),\ldots,\Phi_{m}^{-1}(|q_{m}|)\right\\}\right)\,\,\,(1\leq
j\leq n)\\\\[8.61108pt] \text{ for infinitely many
}\mathbf{q}\in{\mathbb{Z}}[i]^{1\times m}\end{array}\right\\}.$

In what follows, we denote the Lebesgue measure on ${\mathbb{C}}^{m\times n}$
by $\mu_{m\times n}^{{\mathbb{C}}}$.

######  Theorem 9.

If there are some constants $N_{0},M\in{\mathbb{N}}$ with $M\geq 2$, and
$c_{1},c_{2}>1$ such that for every $j\in{\mathbb{N}}_{\geq N_{0}}$ we have

(5.1) $c_{1}\Phi_{k}(M^{j})\leq\Phi_{k}(M^{j+1})\leq
c_{2}\Phi_{k}(M^{j})\quad(k=1,\ldots,m),$

then

$\mu_{m\times
n}^{{\mathbb{C}}}\left(W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)\right)=\begin{cases}0&\text{\rm
if
}\quad\displaystyle\sum\limits_{q=1}^{\infty}\frac{1}{q}\left(\displaystyle\prod_{j=1}^{n}\varphi_{j}(q)\displaystyle\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}<\infty,\\\\[8.61108pt]
1&\text{\rm if
}\quad\displaystyle\sum\limits_{q=1}^{\infty}\frac{1}{q}\left(\displaystyle\prod_{j=1}^{n}\varphi_{j}(q)\displaystyle\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}=\infty.\end{cases}$

The problem for complex numbers is less studied than the previous contexts.

  * •

$n=m=1$. In $1952$, LeVeque [64] obtained Theorem 9 using complex continued
fractions . In $1982$, Sullivan [74] proved Khintchine theorems for real
numbers and imaginary quadratic fields. In fact, Sullivan’s result holds in
the context of approximation of complex numbers by quotients of elements in
the ring of integers of a given imaginary quadratic fields
$\mathbb{R}(i\sqrt{d})$, where $d$ is a square-free natural number. Sullivan
used the theory of Bianchi groups. The case $d=1$ corresponds to the Picard
group and approximation by Gaussian rationals. Sullivan’s result was derived
by Beresnevich, Dickinson, & Velani in [9, Theorem 7] from the ubiquity
framework. When $d=1$, just like the real case, the set is best studied in
terms of Hurwitz Continued Fractions. To this end, the best known results are
in [17].

  * •

$m,n\in{\mathbb{N}}$. Hussain, non-weighted small linear forms [42] .

For any vector
$\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$ satisfying

(5.2) $\min_{1\leq j\leq n}\tau_{j}>1\;\text{ and
}\;\sum\limits_{j=1}^{n}\tau_{j}\geq m+n,$

define the numbers

$s_{j}(\boldsymbol{\tau}):=2n(m-1)+2\,\frac{m+n-\sum\limits_{r\,:\,\tau_{r}<\tau_{j}}(\tau_{r}-\tau_{j})}{\tau_{j}}\quad(1\leq
j\leq n)$

and the set

$W^{{\mathbb{C}}}_{n,m}(\boldsymbol{\tau}):=\left\\{A\in
I_{{\mathbb{C}}}^{m\times
n}:\begin{array}[]{l}\left\llbracket\mathbf{q}A_{j}\right\rrbracket<\frac{1}{\|\mathbf{q}\|^{\tau_{j}-1}}\;(1\leq
j\leq n)\\\\[8.61108pt] \text{ for infinitely manay
}\mathbf{q}\in{\mathbb{Z}}[i]^{m}\end{array}\right\\}.$

######  Theorem 10.

For any vector $\boldsymbol{\tau}$ satisfying (5.2), we have

$\operatorname{\dim_{H}}W^{{\mathbb{C}}}_{n,m}(\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}.$

The only previously known Hausdorff dimension results for $\Psi$ approximable
complex numbers is for $m=n=1$ from [24] that proves that
$\dim_{H}W^{{\mathbb{C}}}_{1,1}(\tau)=\frac{4}{\tau+1}$ and its generalisation
to an arbitrary approximation function by He and Xiong [40].

The core of the proofs is developed in a slightly different context. To be
more precise, for each complex number $z$ define

$|z|_{\infty}:=\max\\{|\Re(z)|,|\Im(z)|\\}\;\text{ and }\;\llbracket
z\rrbracket_{\infty}:=\min\\{|z-p|_{\infty}:p\in{\mathbb{Z}}[i]\\}.$

First, we show Theorem 9 and Theorem 10 when $|\,\cdot\,|$ and
$\llbracket\,\cdot\,\rrbracket_{\infty}$ are replaced by
$|\,\cdot\,|_{\infty}$ and $\llbracket\,\cdot\,\rrbracket_{\infty}$,
respectively. Afterwards, we use the equivalence between
$|\,\cdot\,|_{\infty}$ and $|\,\cdot\,|$ to conclude the results in the
original setting.

###  5.1. A Minkowski-type theorem

We need the following complex version of Minkowski’s theorem for linear forms
(cfr. [41, Theorem 95]).

######  Lemma 6.

Let $\gamma_{1},\ldots,\gamma_{n},\theta_{1},\ldots,\theta_{m}$ be positive
numbers satisfying $\prod_{k=1}^{n}\gamma_{k}\prod_{j=1}^{m}\theta_{j}\geq 1$.
For every matrix $A\in{\mathbb{C}}^{m\times n}$ there exists a vector
$(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}[i]^{1\times(m+n)}$, $\mathbf{q}\neq
0$, such that

$\displaystyle\left|\,\mathbf{q}\;A_{k}-p_{k}\right|_{\infty}$
$\displaystyle<\gamma_{k}\quad(1\leq k\leq n),$ (5.3)
$\displaystyle\left|q_{j}\right|_{\infty}$
$\displaystyle\leq\theta_{j}\quad(1\leq j\leq m).$

###### Proof.

Take any matrix $A=(a_{j,k})_{1\leq j\leq m,1\leq k\leq
n}\in{\mathbb{C}}^{m\times n}$. For $j\in\\{1,\ldots,m\\}$ and
$k\in\\{1,\ldots,n\\}$, write

$a_{j,k}^{1}:=\Re(a_{j,k})\;\text{ and }\;a_{j,k}^{2}:=\Im(a_{j,k}).$

For each $j\in\\{1,\ldots,n\\}$, define the column vectors $B_{2j-1}$,
$B_{2j}$ by

$B_{2j-1}:=\begin{pmatrix}a_{1,j}^{1}\\\ -a_{1,j}^{2}\\\ \vdots\\\
a_{m,j}^{1}\\\ -a_{m,j}^{2}\end{pmatrix}\;\text{ and
}\;B_{2j}:=\begin{pmatrix}a_{1,j}^{2}\\\ a_{1,j}^{1}\\\ \vdots\\\
a_{m,j}^{2}\\\ a_{m,j}^{1}.\end{pmatrix}.$

Call $B\in{\mathbb{Z}}^{2m\times 2n}$ the matrix whose $r$-th column is
$B_{r}$. Then, a vector $(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}[i]^{m+n}$
solves (6) if and only if the vector
$(\mathbf{Q},\mathbf{P})=(Q_{1},\ldots,Q_{2m},P_{1},\ldots,P_{2n})\in{\mathbb{Z}}^{2m+2n}$
given by

$\mathbf{Q}=(\Re(q_{1}),\Im(q_{1}),\ldots,\Re(q_{m}),\Im(q_{m}))\;\text{ and
}\;\mathbf{P}=(\Re(p_{1}),\Im(p_{1}),\ldots,\Re(p_{n}),\Im(p_{n}))$

solves the system

$\displaystyle\left|\,\mathbf{Q}\,B_{2k-1}-P_{2k-1}\right|$
$\displaystyle<\gamma_{k}\quad(1\leq k\leq n),$
$\displaystyle\left|\,\mathbf{Q}\,B_{2k}-P_{2k}\right|$
$\displaystyle<\gamma_{k}\quad(1\leq k\leq n),$
$\displaystyle\left|Q_{2j-1}\right|$ $\displaystyle\leq\theta_{j}\quad(1\leq
j\leq m),$ (5.4) $\displaystyle\left|Q_{2j}\right|$
$\displaystyle\leq\theta_{j}\quad(1\leq j\leq m).$

Call $I_{2n}$ (resp. $I_{2m}$) the identity matrix of size $2n\times 2n$
(resp. $2m\times 2m$) and let $O_{2m\times 2n}$ be the matrix of size
$2m\times 2n$ whose entries are all equal to $0$. Since

$\prod_{k=1}^{n}\gamma_{k}\prod_{j=1}^{m}\theta_{j}\geq 1$

and the determinant of the matrix

$\begin{pmatrix}B&I_{2m}\\\ -I_{2n}&O_{2m\times 2n}\end{pmatrix}$

is $1$, the system (6) has a non-trivial solution by Theorem 4. ∎

###  5.2. Proof of Theorem 9

Given any $k\in{\mathbb{N}}$ and
$\mathbf{z}=(z_{1},\ldots,z_{k})\in{\mathbb{C}}^{k}$, irrespective of
interpreting it as a row or a column, define

$\|\mathbf{z}\|_{\infty}:=\max\left\\{|z_{1}|_{\infty},\ldots,|z_{k}|_{\infty}\right\\}.$

This way, if $\|\mathbf{z}\|_{2}=\sqrt{|z_{1}|^{2}+\cdots+|z_{k}|^{2}}$ is the
Euclidean norm on ${\mathbb{C}}^{k}$, we have

(5.5)
$\frac{1}{\sqrt{2k}}\|\mathbf{z}\|_{2}\leq\|\mathbf{z}\|_{\infty}\leq\|\mathbf{z}\|_{2}.$

Certainly,

$\frac{1}{\sqrt{2k}}\|\mathbf{z}\|_{2}\leq\frac{1}{\sqrt{2}}\,\max_{1\leq
j\leq k}|z_{j}|\leq\max_{1\leq j\leq
k}|z_{j}|_{\infty}=\|\mathbf{z}\|_{\infty}\leq\max_{1\leq j\leq
k}|z_{j}|\leq\|\mathbf{z}\|_{2}.$

Let $M\in{\mathbb{N}}_{\geq 2}$ be sufficiently large (determined below). We
work with the following objects:

  1. i.

$J:=\\{\alpha=(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}[i]^{m+n}:\|\mathbf{p}\|_{\infty}\leq
2m\|\mathbf{q}\|_{\infty}\\}$,

  2. ii.

$\beta:J\to{\mathbb{R}}_{+}$,
$\alpha=(\mathbf{q},\mathbf{p})\mapsto\beta_{\alpha}=\max\left\\{\Phi_{1}^{-1}\left(|q_{1}|_{\infty}\right),\ldots,\Phi_{m}^{-1}\left(|q_{m}|_{\infty}\right)\right\\}$,

  3. iii.

$(u_{j})_{j\geq 1}$ given by $u_{j}=M^{j}$ for all $j\in{\mathbb{N}}$,

  4. iv.

For each $\alpha=(\mathbf{q},\mathbf{p})\in J$, define
$R_{\alpha,j}:=\left\\{A_{j}\in I_{{\mathbb{C}}}^{m\times
1}:\mathbf{q}\,A_{j}=p_{j}\right\\}$ for $j\in\\{1,\ldots,n\\}$ and the
resonant set $R_{\alpha}$ is $R_{\alpha}:=\prod_{j=1}^{n}R_{\alpha,j}$,

  5. v.

In this setting, we have $\kappa_{j}=1-\frac{1}{m}=\frac{2m-2}{2m}$ and
$\delta_{j}=2m$ for $j\in\\{1,\ldots,n\\}$.

Let us expand on the $\kappa$-scaling property. If $V$ is a finite dimensional
complex vector space and if $W$ is a subspace of $V$, then both $V$ and $W$
are real vector spaces and their dimension as real vector spaces is twice
their dimension as complex vector spaces. We then rely upon [2, Section 2.2]
to obtain $\kappa_{j}$.

We also consider the positive functions
$\rho:(\rho_{1},\ldots,\rho_{n}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{n}$ given by

$\rho_{j}(u):=\sqrt{2}\,M\frac{\varphi_{j}(u)}{\|\Phi(u)\|_{\infty}}\left(\prod_{s=1}^{n}\varphi_{s}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{-1/n}\quad(1\leq
j\leq n,\,u\in{\mathbb{N}})$

and $\Psi=(\psi_{1},\ldots,\psi_{n}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{n}$
given by

$\psi_{j}(u):=\frac{\varphi_{j}(u)}{2m\|\Phi(u)\|_{\infty}}\quad(1\leq j\leq
n,\,u\in{\mathbb{N}}).$

Note that

(5.6)
$\prod_{j=1}^{n}\frac{\psi_{j}(u)}{\rho_{j}(u)}=\left(2^{3/2}mM\right)^{-n}\prod_{j=1}^{n}\varphi_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)\quad(u\in{\mathbb{N}}).$

When there is a strictly increasing sequence of natural numbers
$(r_{s})_{s\geq 1}$ satisfying

$\prod_{j=1}^{n}\varphi_{s}(r_{s})\prod_{k=1}^{m}\Phi_{k}(r_{s})>1\quad(s\in{\mathbb{N}}),$

then, by Lemma 6, we have $\mu_{m\times
n}^{{\mathbb{C}}}\left(W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)\right)=\mu_{m\times
n}^{{\mathbb{C}}}\left(I_{{\mathbb{C}}}^{m\times n}\right)$. Thus, we assume

(5.7) $\prod_{j=1}^{n}\varphi_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)\leq
1\quad(u\in{\mathbb{N}}).$

######  Lemma 7.

The system $((R_{\alpha})_{\alpha\in J},\beta)$ is a weighted ubiquitous
system with respect to the function $\rho$.

In what follows, given a vector $\mathbf{z}\in{\mathbb{C}}^{m\times 1}$ (resp.
$\mathbf{z}\in{\mathbb{C}}^{1\times m}$) , we denote by
$\overline{\mathbf{z}}$ the vector in ${\mathbb{C}}^{m\times 1}$ (resp. in
${\mathbb{C}}^{1\times m}$) whose $r$-th coordinate is the complex conjugate
of the $r$-th coordinate of $\mathbf{z}$.

######  Proposition 3.

Let $f_{j}:{\mathbb{N}}\to{\mathbb{R}}_{+}$, $1\leq j\leq n$, be positive
functions such that $f_{j}(u)\to 0$ as $u\to\infty$ and

$\prod_{j=1}^{n}f_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)=1\quad(u\in{\mathbb{N}}).$

For all large $u\in{\mathbb{N}}$ and all $A$ in the interior of
$I_{{\mathbb{C}}}^{m\times n}$, there is some
$\alpha=(\mathbf{q},\mathbf{p})\in J$ satisfying $\beta_{\alpha}\leq u$ and

(5.8)
$A\in\prod_{j=1}^{n}\Delta\left(R_{\alpha,j};\sqrt{2}\,\frac{f_{j}(u)}{\|\mathbf{q}\|_{\infty}}\right).$

###### Proof.

Take $A$ as in the statement. By Lemma 6, there is some non-zero
$(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}[i]^{m+n}$ such that

$\displaystyle\left|\mathbf{q}\,\overline{A_{j}}-p_{j}\right|_{\infty}$
$\displaystyle<f_{j}(u)\quad(1\leq j\leq n),$
$\displaystyle\left|q_{k}\right|_{\infty}$
$\displaystyle\leq\Phi_{k}(u)\quad(1\leq k\leq m).$

The vector $\alpha=(\mathbf{q},\mathbf{p})$ satisfies $\beta_{\alpha}\leq u$
and

$\displaystyle\left|\mathbf{q}\,\overline{A_{j}}-p_{j}\right|$
$\displaystyle<\sqrt{2}\,f_{j}(u)\,\quad\,(1\leq j\leq n),$
$\displaystyle\left|q_{k}\right|$
$\displaystyle\leq\sqrt{2}\,\Phi_{k}(u)\quad(1\leq k\leq m).$

Let us verify (5.8). If $\mathbf{q}^{\top}$ is the transpose of $\mathbf{q}$,
then
$\mathbf{v}=\overline{A_{j}}-\frac{\overline{\mathbf{q}}\,\overline{A_{j}}}{\|\mathbf{q}\|^{2}}\mathbf{q}^{\top}$
is orthogonal to $\mathbf{q}^{\top}$ and

$\overline{A_{j}}=\frac{\overline{\mathbf{q}}\,\overline{A_{j}}}{\|\mathbf{q}\|_{2}^{2}}\mathbf{q}^{\top}+\mathbf{v}.$

For any $\mathbf{a}$ such that $\overline{\mathbf{a}}\in R_{\alpha,j}$ write
$\mathbf{w}=\mathbf{a}-\frac{\overline{p_{j}}}{\|\mathbf{q}\|^{2}}\mathbf{q}^{\top}$,
then $\mathbf{w}$ is also orthogonal to $\mathbf{q}^{\top}$ and, the
Pythagoras theorem,

$\displaystyle\left\|A_{j}-\overline{\mathbf{a}}\right\|_{2}^{2}$
$\displaystyle=\left\|\overline{A_{j}}-\mathbf{a}\right\|_{2}^{2}$
$\displaystyle=\left\|\left(\frac{\overline{\mathbf{q}}\,A_{j}}{\|\mathbf{q}\|^{2}}-\frac{\overline{p_{j}}}{\|\mathbf{q}\|^{2}}\right)\mathbf{q}^{\top}\right\|_{2}^{2}+\left\|\mathbf{v}-\mathbf{w}\right\|_{2}^{2}\geq\frac{|\overline{\mathbf{q}}\,A_{j}-\overline{p_{j}}|^{2}}{\|\mathbf{q}\|_{2}^{2}}=\frac{|\mathbf{q}\,\overline{A_{j}}-p_{j}|^{2}}{\|\mathbf{q}\|_{2}^{2}}.$

For $\mathbf{w}=\mathbf{v}$, the triangle inequality along with $f_{j}(u)\to
0$ as $u\to\infty$ lead us to the conclusion $\mathbf{a}\in R_{\alpha,j}$ when
$u$ is large. Hence,

$\min\left\\{\|A_{j}-\mathbf{a}\|_{\infty}:\mathbf{a}\in
R_{\alpha,j}\right\\}\leq\min\left\\{\|A_{j}-\mathbf{a}\|_{2}:\mathbf{a}\in
R_{\alpha,j}\right\\}=\frac{|\mathbf{q}\,\overline{A_{j}}-p_{j}|}{\|\mathbf{q}\|_{2}}\leq\frac{\sqrt{2}\,f_{j}(u)}{\|\mathbf{q}\|_{\infty}}.$

∎

Define the functions $f_{j}:{\mathbb{N}}\to{\mathbb{R}}$,
$j\in\\{1,\ldots,N\\}$, by

$f_{j}(u):=\varphi_{j}(u)\left(\prod_{s=1}^{n}\varphi_{s}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{-1/n}\quad\quad(u\in{\mathbb{N}}).$

For each $s\in{\mathbb{N}}$, define the set

$\widetilde{J}_{s}:=\left\\{\alpha=(\mathbf{q},\mathbf{p})\in
J:\frac{\Phi_{k}(u_{s})}{M}\leq|q_{k}|_{\infty}\leq\Phi_{k}(u_{s})\quad(1\leq
k\leq m)\right\\}.$

Observe that $\beta_{\alpha}\leq u_{s}$ when
$\alpha=(\mathbf{q},\mathbf{p})\in\widetilde{J}_{s}$. Then, since
$\beta_{\alpha}\to\infty$ as $\alpha\to\infty$, we may choose an adequate
$l_{j}$ ensuring $\widetilde{J}_{j}\subseteq J_{j}$. For
$k\in\\{1,\ldots,m\\}$, write

$J_{s,k}:=\left\\{\alpha\in
J:\left|q_{k}\right|_{\infty}\leq\frac{\Phi_{k}(u_{s})}{M}\text{ and
}|q_{j}|_{\infty}\leq\Phi_{j}(u_{s})\quad(j\in\\{1,\ldots
m\\}\setminus\\{k\\})\right\\}.$

Let $B=\prod_{k=1}^{n}B(X_{k};r)$ be an arbitrary whose closure is contained
in the interior of $I_{{\mathbb{C}}}^{m\times n}$. In view of Proposition 3,
for large $s\in{\mathbb{N}}$ we have

$\displaystyle B$ $\displaystyle=B\cap\bigcup_{\alpha:\beta_{\alpha}\leq
u_{s}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\sqrt{2}\,\frac{f_{j}(u_{s})}{\beta_{\alpha}}\right)$
$\displaystyle=\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{s}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\sqrt{2}\,\frac{f_{j}(u_{s})}{\beta_{\alpha}}\right)\right)\,\cup\,\left(B\cap\bigcup_{h=1}^{m}\bigcup_{\alpha\in
J_{s,h}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\sqrt{2}\,\frac{f_{j}(u_{s})}{\beta_{\alpha}}\right)\right).$

######  Proposition 4.

There is some $N(r)\in{\mathbb{N}}$ such that for all $j\in\\{1,\ldots,n\\}$
and $\mathbf{q}\in{\mathbb{Z}}[i]^{1\times m}$, every $s\in{\mathbb{N}}_{\geq
N(r)}$ satisfies

$\\#\left\\{p\in{\mathbb{Z}}[i]:B(X_{j},r)\cap\Delta\left(R_{\alpha,j},\sqrt{2}\,\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\neq\varnothing\right\\}\leq(8rm\|\mathbf{q}\|_{\infty}+2)^{2}.$

###### Proof.

Let $j\in\\{1,\ldots,n\\}$ and $0\neq\mathbf{q}\in{\mathbb{Z}}[i]^{m}$ be
arbitrary. Take $p\in{\mathbb{Z}}[i]$ be such that for some $A_{j},Y_{j}\in
I_{{\mathbb{C}}}^{m\times 1}$ we have

$\|X_{j}-Y_{j}\|_{\infty}<r,\quad\mathbf{q}A_{j}=p,\;\text{ and
}\;\|A_{j}-Y_{j}\|_{\infty}<\sqrt{2}\,\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}.$

That is, the $\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}$-thickened
hyperplane $\mathbf{q}A_{j}=p$ has non-empty intersection with $B(X_{j},r)$.
By Cauchy’s inequality and (5.5), we have

$|\mathbf{q}X_{j}-p|_{\infty}\leq|\mathbf{q}(X_{j}-A_{j})|\leq\|\mathbf{q}\|_{2}\|X_{j}-A_{j}\|_{2}\leq
2m\|\mathbf{q}\|_{\infty}\left(\|X_{j}-Y_{j}\|_{\infty}+\|Y_{j}-A_{j}\|_{\infty}\right)$

and hence

$|\mathbf{q}X_{j}-p|_{\infty}<2m\|\mathbf{q}\|_{\infty}\left(r+\sqrt{2}\,\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right).$

As a consequence, for every large $s\in{\mathbb{N}}$ (depending on $r$) we
have

$|\mathbf{q}X_{j}-p|_{\infty}\leq 4mr\|\mathbf{q}\|_{\infty}.$

The proposition now follows from the next elementary estimate:

(5.9) $\\#\\{z\in{\mathbb{Z}}[i]:\max\\{|\Re(z)|,|\Im(z)|\\}\leq
R\\}\leq(2R+2)^{2}\quad(R>0).$

∎

######  Proposition 5.

If $M>2^{5n}3^{m}m^{n+1}$ and writing $\alpha=(\mathbf{q},\mathbf{p})$, every
large $s\in{\mathbb{N}}$ satisfies

(5.10) $\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{k=1}^{m}\bigcup_{\alpha\in
J_{s,k}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\sqrt{2}\,\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right)\leq\frac{1}{2}\mu_{m\times
n}^{{\mathbb{C}}}(B).$

###### Proof.

Take $s\in{\mathbb{N}}$. For each $k\in\\{1,\ldots,m\\}$, write

$J_{s,k}^{\prime}:=\left\\{\mathbf{q}\in{\mathbb{Z}}[i]^{1\times
m}:\alpha=(\mathbf{q},\mathbf{p})\in J_{s,k}\right\\}\,,$

so, by (5.9), every large $s\in{\mathbb{N}}$ verifies

$\\#J_{n,k}^{\prime}\leq
3^{2m}\left(\frac{1}{M}\prod_{j=1}^{m}\Phi_{j}(u_{n})\right)^{2}.$

Let $G_{{\mathbb{C}}}$ be the intersection in (5.10). By the $\kappa$-scaling
property and Proposition 4,

$\mu_{m\times
n}^{{\mathbb{C}}}(G_{{\mathbb{C}}})\leq\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}(8mr\|\mathbf{q}\|_{\infty}+2)^{2n}\left(\prod_{j=1}^{n}\sqrt{2}\,\frac{f_{j}(u_{s})r^{m-1}}{\|\mathbf{q}\|_{\infty}}\right)^{2}.$

Recall that $(a+b)^{2n}\leq(2a)^{2n}+(2b)^{2n}$ for all $a,b\geq 0$, then

(5.11) $\mu_{m\times n}^{{\mathbb{C}}}(G_{{\mathbb{C}}})\leq
2^{9n}m^{2n}r^{2mn}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}\prod_{j=1}^{n}f_{j}^{2}(u_{s})+2^{5n}r^{2n(m-1)}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}\prod_{j=1}^{n}\frac{f_{j}^{2}(u_{s})}{\|\mathbf{q}\|_{\infty}^{2}}.$

We can bound the the first sum in (5.11) using $M>2^{5n}3^{m}m^{n+1}$ as
follows:

$\displaystyle
2^{9n}m^{2n}r^{2mn}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,j}^{\prime}}\prod_{k=1}^{n}f_{k}^{2}(u_{s})$
$\displaystyle=\frac{2^{9n}m^{2n}r^{2mn}}{\Phi_{1}^{2}(u_{s})\cdots\Phi_{m}^{2}(u_{s})}\sum\limits_{k=1}^{m}\\#J_{s,j}^{\prime}$
$\displaystyle\leq\frac{2^{9n}3^{2m}m^{2n+1}}{M^{2}}r^{2nm}$
$\displaystyle<\frac{1}{4}\mu_{m\times n}^{{\mathbb{C}}}(B).$

The second sum in (5.11) tends to $0$ as $s$ tends to $\infty$. Indeed, since
$\\#\\{q\in{\mathbb{Z}}[i]:|q|_{\infty}=s\\}=8s$, we have

$\displaystyle\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}\prod_{j=1}^{n}\frac{f_{j}^{2}(u_{s})}{\|\mathbf{q}\|^{2}}$
$\displaystyle=\frac{1}{\Phi_{1}^{2}(u_{s})\cdots\Phi_{m}^{2}(u_{s})}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}\frac{1}{\|\mathbf{q}\|^{2n}}$
$\displaystyle\leq\frac{1}{\Phi_{1}^{2}(u_{s})\cdots\Phi_{m}^{2}(u_{s})}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}\frac{1}{\|\mathbf{q}\|^{2}}$
$\displaystyle\leq\frac{1}{\Phi_{1}^{2}(u_{s})\cdots\Phi_{m}^{2}(u_{s})}\sum\limits_{k=1}^{m}\sum\limits_{q_{k}=1}^{\Phi_{k}(u_{s})/M}\frac{8q_{k}}{q_{k}^{2}}\left(\frac{\Phi_{1}(u_{s})\cdots\Phi_{m}(u_{s})}{\Phi_{k}(u_{s})}\right)^{2}$
$\displaystyle\leq\frac{8}{\Phi_{1}^{2}(u_{s})\cdots\Phi_{m}^{2}(u_{s})}\sum\limits_{k=1}^{m}\left(\frac{\Phi_{1}(u_{s})\cdots\Phi_{m}(u_{s})}{\Phi_{k}(u_{s})}\right)^{2}\left(\log\Phi_{k}(u_{s})-\log
M\right)$ $\displaystyle\leq
8\sum\limits_{k=1}^{m}\frac{\log\Phi_{k}(u_{s})}{\Phi_{k}(u_{s})^{2}}\to
0\text{ as }s\to\infty.$

∎

###### Proof of Lemma 7.

It only remains to show the ubiquity condition with respect to $\rho$. For any
$s\in{\mathbb{N}}$, the definition of $\widetilde{J}_{s}$ tells us that

$\|\mathbf{q}\|_{\infty}\geq\frac{1}{M}\left\|\left(\Phi_{1}(u_{s}),\ldots,\Phi_{m}(u_{s})\right)\right\|_{\infty}\quad\left(\alpha=(\mathbf{q},\mathbf{p})\in\widetilde{J}_{s}\right),$

then

$\displaystyle\mu_{m\times n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\alpha\in
J_{s}}\Delta\left(R_{\alpha},\rho\right)\right)$ $\displaystyle=\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\alpha\in
J_{s}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},\sqrt{2}M\,\frac{f_{k}(u_{s})}{\|\Phi(u_{s})\|_{\infty}}\right)\right)$
$\displaystyle\geq\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{s}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},\sqrt{2}M\,\frac{f_{k}(u_{s})}{\|\Phi(u_{s})\|_{\infty}}\right)\right)$
$\displaystyle\geq\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{s}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},\sqrt{2}\,\frac{f_{k}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right)$
$\displaystyle\geq\frac{1}{2}\mu_{m\times n}^{{\mathbb{C}}}(B).$

∎

Approximating by finitely many balls whose closure is contained in the
interior of $I_{{\mathbb{C}}}^{m\times n}$, we see that previous estimates
also hold for any ball $B$ in $I_{{\mathbb{C}}}^{m\times n}$.

Lemma 8 below allows us to impose an additional assumption on $\psi$ and
$\Phi$ without losing any generality. We omit its proof for it is shown as
Lemma 6.1 in [57]. The only difference is that in the last step we deal with
the series $\sum_{t}(c_{1}^{-2(d-\varepsilon)})^{t}<\infty$. Call
$W_{\infty}(\varphi,\Phi)$ the set of matrices $A\in I_{{\mathbb{C}}}^{m\times
n}$ such that

$\displaystyle\left\llbracket\mathbf{q}A_{j}\right\rrbracket_{\infty}$
$\displaystyle<\phi_{j}(u)\quad(1\leq j\leq n),$
$\displaystyle|q_{k}|_{\infty}$ $\displaystyle\leq\Phi_{k}(u)\quad(1\leq k\leq
m).$

######  Lemma 8.

Under the assumptions of Theorem 9, there are functions
$\widetilde{\varphi}=(\widetilde{\varphi}_{1},\ldots,\widetilde{\varphi}_{n})$
such that $W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)\subseteq
W_{n,m}^{{\mathbb{C}}}(\widetilde{\varphi},\Phi)$, $\mu_{m\times
n}^{{\mathbb{C}}}\left(W_{n,m}^{{\mathbb{C}}}(\varphi,\Phi)\right)=\mu_{m\times
n}^{{\mathbb{C}}}\left(W_{n,m}^{{\mathbb{C}}}(\tilde{\varphi},\Phi)\right)$,
and

$\lim_{u\to\infty}\|\Phi(u)\|_{\infty}^{n}\prod_{j=1}^{m}\Phi_{k}(u)\prod_{j=1}^{n}\varphi_{j}(u)=\infty\quad(1\leq
j\leq n).$

######  Lemma 9.

Theorem 9 holds if we replace $|\,\cdot\,|$ and
$\llbracket\,\cdot\,\rrbracket$ with $|\,\cdot\,|_{\infty}$ and
$\llbracket\,\cdot\,\rrbracket_{\infty}$, respectively.

###### Proof.

First, we verify the hypotheses of Theorem 5.

  1. 1.

The system $((R_{\alpha})_{\alpha\in J},\beta)$ is ubiquitous with respect to
$\rho$ and the sequence $(l_{j})_{j\geq 1}$, $(u_{j})_{j\geq 1}$ defined as
above.

  2. 2.

The function $\Psi=(\psi_{1},\ldots,\psi_{n})$ is $c$-regular. Indeed, by
(5.1) and the monotonicity of each $\varphi_{j}$ and each $\Phi_{k}$, we have

$\psi_{j}(u_{s+1})=\frac{\varphi_{j}(u_{s+1})}{2m\|\Phi(u_{s+1})\|_{\infty}}\leq\frac{\varphi_{j}(u_{s})}{c_{1}2m\|\Phi(u_{s})\|_{\infty}}=\frac{1}{c_{1}}\psi_{j}(u_{s})\quad(s\in{\mathbb{N}}).$

  3. 3.

By (5.7), for $j\in\\{1,\ldots,n\\}$ we have $\rho_{j}(u)\geq\psi_{j}(u)$ for
all $u\in{\mathbb{N}}$ and Lemma 8 implies

$\lim_{u\to\infty}\rho_{j}(u)=0.$

Combining (5.6), $\delta_{j}(1-\kappa_{j})=2$ for all $j\in\\{1,\ldots,n\\}$,
and Theorem 5, the set $W_{\infty}(\varphi,\Phi)$ is either of full measure or
null depending on the divergence or convergence of the series

(5.12)
$\sum\limits_{s=0}^{\infty}\left(\prod_{j=1}^{n}\varphi_{j}(u_{s})\prod_{k=1}^{m}\Phi_{k}(u_{s})\right)^{2}.$

Take any $s\in{\mathbb{N}}$. Since each $\varphi_{j}$ is non-increasing, it
follows that

$\displaystyle\sum\limits_{u_{s-1}\leq
q<u_{s}}\,\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}$
$\displaystyle\leq\sum\limits_{u_{s-1}\leq
q<u_{s}}\,\frac{1}{u_{s-1}}\left(\prod_{j=1}^{n}\varphi_{j}(u_{s-1})\prod_{k=1}^{m}c_{2}\Phi_{k}(u_{s-1})\right)^{2}$
$\displaystyle\text{ (by }\eqref{EQ:LinearForms:C})$
$\displaystyle=\left(\frac{u_{s}}{u_{s-1}}-1\right)c_{2}^{2m}\,\left(\prod_{j=1}^{n}\varphi_{j}(u_{s-1})\prod_{k=1}^{m}\Phi_{k}(u_{s-1})\right)^{2}$
$\displaystyle=\left(M-1\right)c_{2}^{2m}\,\left(\prod_{j=1}^{n}\varphi_{j}(u_{s-1})\prod_{k=1}^{m}\Phi_{k}(u_{s-1})\right)^{2};$

as a consequence,

$\displaystyle\sum\limits_{q=1}^{\infty}\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}$
$\displaystyle=\sum\limits_{s=1}^{\infty}\sum\limits_{u_{s-1}\leq
q<u_{n}}\,\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}$
$\displaystyle\leq\left(M-1\right)c_{2}^{2m}\,\sum\limits_{s=1}^{\infty}\left(\prod_{j=1}^{n}\varphi_{j}(u_{s-1})\prod_{k=1}^{m}\Phi_{k}(u_{s-1})\right)^{2}.$

Similarly, we may see that

$\displaystyle\sum\limits_{u_{s-1}\leq
q<u_{s}}\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{n}\Phi_{k}(q)\right)^{2}$
$\displaystyle\geq\sum\limits_{u_{s-1}\leq
q<u_{s}}\frac{1}{u_{s}}\left(\prod_{j=1}^{n}\varphi_{j}(u_{s})\prod_{k=1}^{n}\Phi_{k}(u_{s-1})\right)^{2}$
$\displaystyle\geq\frac{1}{c_{2}^{2m}}\left(1-\frac{1}{M}\right)\left(\prod_{j=1}^{n}\varphi_{j}(u_{s})\prod_{k=1}^{n}\Phi_{k}(u_{s})\right)^{2}.$

Therefore, the convergence of the series in (5.12) is equivalent to that of

(5.13)
$\sum\limits_{q=1}^{\infty}\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}.$

The divergence part of Theorem 9 follows from Theorem 5. In order to prove the
convergence part, for each $u\in{\mathbb{N}}$, let
$\mathcal{A}(u;\varphi,\Phi)$ be the set of matrices $A\in
I_{{\mathbb{C}}}^{m\times n}$ for which there is some non-zero
$\mathbf{q}\in{\mathbb{Z}}[i]^{m}$ satisfying

$\displaystyle\llbracket\mathbf{q}A_{j}\rrbracket_{\infty}<\varphi_{j}(u)$
$\displaystyle\quad(1\leq j\leq n),$
$\displaystyle|q_{k}|_{\infty}\leq\Phi_{k}(u)$ $\displaystyle\quad(1\leq k\leq
m).$

If $\mathbf{q}\in{\mathbb{Z}}[i]^{m}$, $\mathbf{q}\neq 0$, verifies
$|q_{k}|_{\infty}\leq\Phi_{k}(u)$ for $k\in\\{1,\ldots,m\\}$, define

$\mathcal{A}_{\mathbf{q}}(u;\varphi,\Phi):=\left\\{A\in
I_{{\mathbb{C}}}^{m\times
n}:\llbracket\mathbf{q}A_{j}\rrbracket_{\infty}<\varphi_{j}(u)\quad(1\leq
j\leq n)\right\\}$

and, for all $\mathbf{p}\in{\mathbb{Z}}[i]^{n}$,

$\mathcal{A}_{\mathbf{q},\mathbf{p}}(u;\varphi,\Phi):=\left\\{A\in
I_{{\mathbb{C}}}^{m\times
n}:|\mathbf{q}A_{j}-p_{j}|_{\infty}<\varphi_{j}(u)\quad(1\leq j\leq
n)\right\\}.$

The following estimates hold for some constants depending on $m$ and $n$:

$\displaystyle\\#\left\\{\mathbf{q}\in{\mathbb{Z}}[i]^{m}:|q_{k}|_{\infty}\leq\Phi_{k}(u)\quad(1\leq
k\leq m)\right\\}$
$\displaystyle\ll_{m,n}\left(\prod_{k=1}^{m}\Phi_{k}(u)\right)^{2},$
$\displaystyle\\#\left\\{\mathbf{p}\in{\mathbb{Z}}[i]^{n}:\mathcal{A}_{\mathbf{q},\mathbf{p}}(u;\varphi,\Phi)\neq\varnothing\right\\}$
$\displaystyle\ll_{m,n}\|\mathbf{q}\|_{\infty}^{2n}$
$\displaystyle\mu_{m\times
n}^{{\mathbb{C}}}(\mathcal{A}_{\mathbf{q},\mathbf{p}}(u;\varphi,\Phi))$
$\displaystyle\ll_{m,n}\frac{1}{\|\mathbf{q}\|_{\infty}^{2n}}\left(\prod_{j=1}^{n}\varphi_{j}(u)\right)^{2}.$

Therefore, we have

$\mu_{m\times
n}^{{\mathbb{C}}}(\mathcal{A}(u;\varphi,\Phi))\ll_{m,n}\left(\prod_{j=1}^{n}\varphi_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{2}.$

Choose $s\in{\mathbb{N}}$ such that $u_{s-1}\leq u<u_{s}$, so
$\varphi(u_{s})\leq\varphi(u)\leq\varphi(u_{s-1})$ and
$\Phi(u_{s-1})\leq\Phi(u)\leq\Phi(u_{s})$. Hence, by (5.1),

$\mathcal{A}(u;\varphi,\Phi)\subseteq\mathcal{A}(u_{s-1};\varphi,c_{2}\Phi),$

which implies

$\limsup_{u\to\infty}\mathcal{A}(u;\varphi,\Phi)\subseteq\limsup_{s\to\infty}\mathcal{A}(u_{s};\varphi,c_{2}\Phi).$

Finally, since the series in (5.12) converges if and only if the series in
(5.13) converges, the result follows from the Borel-Cantelli lemma. ∎

###### Proof of Theorem 9.

Since $|z|_{\infty}\leq|z|\leq\sqrt{2}|z|_{\infty}$ for all
$z\in{\mathbb{C}}$, we have

$W_{\infty}(\varphi,\Phi)\subseteq W(\varphi,\Phi)\subseteq
W_{\infty}(\sqrt{2}\,\varphi,\sqrt{2}\,\Phi).$

When the series in (5.13) converges, the set
$W_{\infty}(\sqrt{2}\,\varphi,\sqrt{2}\,\Phi)$ is null and $W(\varphi,\Phi)$
is also null. The divergence of the series implies the full measure of
$W_{\infty}(\varphi,\Phi)$ and, hence, the full measure of $W(\varphi,\Phi)$.
∎

###  5.3. Proof of Theorem 10

Consider again the norm $\|\cdot\|_{\infty}$. Pick
$\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$ satisfying
(5.2). Let $\Phi=(\Phi_{1},\ldots,\Phi_{m})$ be determined by

$\Phi_{k}(q)=q\quad(1\leq k\leq m,\;q\in{\mathbb{N}})$

and let $\varphi=(\varphi_{1},\ldots,\varphi_{n})$ be given by

$\varphi_{j}(q)=\frac{1}{q^{\tau_{j}-1}}\quad(1\leq j\leq
n,\;q\in{\mathbb{N}}).$

If $\eta=\frac{1}{2}(\tau_{1}+\cdots+\tau_{n}-n-m)>0$, we have for all
$q\in{\mathbb{N}}$

$\frac{1}{q}\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}=\frac{1}{q^{1+\eta}},$

and Theorem 9 implies $\mu_{m\times n}^{{\mathbb{C}}}(W(\varphi,\Phi))=0$. It
is trivial to choose $\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})$ such that
for some $j\in\\{1,\ldots,n\\}$ we have

$\tau_{1}+\cdots+\tau_{n}>n\tau_{j}+n+m;$

(for example, $\tau_{1}=\cdots=\tau_{n-1}=3(n+m)$ and $\tau_{n}=2$). For such
$\boldsymbol{\tau}$ and $j$, the function $\rho_{j}$ defined as in the
previous section does not converge to $0$ when its argument tends to $\infty$.
However, we may find a suitable $n$-tuple of positive functions
$\tilde{\rho}=(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{n})$ and a sequence
$(\tilde{l}_{s})_{s\geq 1}$ such that, for a sufficiently large $M$, the
system $((R_{\alpha})_{\alpha\in J},\beta)$ is ubiquitous with respect to
$\tilde{\rho}$ and $(\tilde{l}_{s})_{s\geq 1}$, $(u_{s})_{s\geq 1}$.

Let $M\in{\mathbb{N}}$ large (we determine how large $M$ should be below) and
take $J$, $R_{\alpha}$ for $\alpha\in J$, $\beta$, and $(u_{s})_{s\geq 1}$ as
in Lemma 7. Call $(\tilde{l}_{s})_{s\geq 1}$ the sequence given by
$l_{s}=M^{s-1}$ for all $s\in{\mathbb{N}}$. In this context, we have

$\beta_{\alpha}=\|\mathbf{q}\|_{\infty}\quad\left(\alpha=(\mathbf{q},\mathbf{p})\in
J\right).$

Given any $\textbf{a}=(a_{1},\ldots,a_{n})\in{\mathbb{R}}^{n}$ such that

(5.14) $\min_{1\leq j\leq n}a_{j}\geq 1\;\text{ and
}\;\sum\limits_{j=1}^{n}a_{j}=m+n,$

define
$\tilde{\rho}=(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{n}):{\mathbb{N}}\to{\mathbb{R}}^{n}$
by

$\tilde{\rho}_{j}(u)=\sqrt{2}\,\frac{1}{u^{a_{j}-1}}\quad(1\leq j\leq
n,\;u\in{\mathbb{N}}).$

######  Lemma 10.

For any $\mathbf{a}=(a_{1},\ldots,a_{n})\in{\mathbb{R}}^{n}$ satisfying
(5.14), the system $(\\{R_{\alpha}\\}_{\alpha\in J},\beta)$ is ubiquitous with
respect to $\tilde{\rho}$ and $(\tilde{l}_{s})_{s\geq 1}$, $(u_{s})_{s\geq
1}$.

We omit several steps in the forthcoming proof, for it resembles that of Lemma
7.

###### Proof.

We may show–as we did with (5.8)–that for every $A$ in the interior of
$I_{{\mathbb{C}}}^{m\times n}$ and any large $s\in{\mathbb{N}}$ there is some
$\alpha=(\mathbf{q},\mathbf{p})\in J$ such that
$1\leq\|\mathbf{q}\|_{\infty}\leq u_{s}$ and

$A\in\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\frac{\sqrt{2}}{\|\mathbf{q}\|_{\infty}u_{s}^{a_{j}-1}}\right).$

Similar to (5.10), we may choose a sufficiently large $M$ for which any ball
$B=\prod_{j=1}^{n}B_{j}\subseteq X$ and any large $s\in{\mathbb{N}}$
(depending on the radius of $B$) satisfy

$\displaystyle\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\begin{subarray}{c}\alpha=(\mathbf{q},\mathbf{p})\in
J\\\
\beta_{\alpha}<\tilde{l}_{s}\end{subarray}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\frac{\sqrt{2}}{\|\mathbf{q}\|_{\infty}u_{s}^{a_{j}-1}}\right)\right)$
$\displaystyle=\mu_{m\times
n}^{{\mathbb{C}}}\left(\bigcup_{q=1}^{\tilde{l}_{s}-1}\bigcup_{\begin{subarray}{c}\alpha\in
J\\\
\beta_{\alpha}=q\end{subarray}}\prod_{j=1}^{n}B_{j}\cap\Delta\left(R_{\alpha,j},\frac{\sqrt{2}}{\|\mathbf{q}\|_{\infty}u_{s}^{a_{j}-1}}\right)\right)$
$\displaystyle\leq\frac{1}{2}\mu_{m\times n}^{{\mathbb{C}}}(B),$

which implies

$\mu_{m\times
n}^{{\mathbb{C}}}\left(B\cap\bigcup_{\begin{subarray}{c}\alpha=(\mathbf{q},\mathbf{p})\in
J\\\ \tilde{l}_{s}\leq\beta_{\alpha}\leq
u_{s}\end{subarray}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\frac{\sqrt{2}}{u_{s}^{a_{j}-1}}\right)\right)\geq\frac{1}{2}\mu_{m\times
n}^{{\mathbb{C}}}(B).$

∎

For any $\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$
satisfying (5.2), define the sets

$W^{{\mathbb{C}}}_{\infty}(\boldsymbol{\tau}):=\left\\{A\in
I_{{\mathbb{C}}}^{m\times
n}:\left\llbracket\mathbf{q}A_{j}\right\rrbracket_{\infty}<\frac{1}{\|\mathbf{q}\|_{\infty}^{\tau_{j}-1}}\quad(1\leq
j\leq n)\text{ for i. m. }\mathbf{q}\in{\mathbb{Z}}[i]^{m}\right\\}$

and

$\widetilde{W}^{{\mathbb{C}}}_{\infty}(\gamma,\boldsymbol{\tau}):=\bigcap_{Q\in{\mathbb{N}}}\bigcup_{\begin{subarray}{c}\mathbf{q}\in{\mathbb{Z}}[i]^{m}\\\
\|\mathbf{q}\|_{\infty}=Q\end{subarray}}\bigcup_{\begin{subarray}{c}\mathbf{p}\in{\mathbb{Z}}[i]^{n}\\\
\alpha=(\mathbf{q},\mathbf{p})\in
J\end{subarray}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\frac{\gamma}{\|\mathbf{q}\|_{\infty}^{\tau_{j}}}\right)\quad(\gamma>0).$

Clearly, when we consider $\|\,\cdot\,\|_{\infty}$ and $|\,\cdot\,|_{\infty}$
and the function $\Psi=(\psi_{1},\ldots,\psi_{n})$ given by
$\psi_{j}(u)=\gamma u^{-\tau_{j}}$, the set
$\widetilde{W}^{{\mathbb{C}}}_{\infty}(\gamma,\boldsymbol{\tau})$ is precisely
$W^{I_{{\mathbb{C}}}^{m\times n}}(\Psi)$ as defined in Section 2.

######  Lemma 11.

If $\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}_{+}$
satisfies (5.2) and $\gamma>0$, then

$\operatorname{\dim_{H}}\widetilde{W}_{\infty}(\gamma,\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}.$

###### Proof.

Let us further assume that $\gamma=1$.

Upper bound. First, for all $Q\in{\mathbb{N}}$ we have

$\\#\\{\mathbf{q}\in{\mathbb{Z}}[i]^{m}:\|\mathbf{q}\|_{\infty}=Q\\}\asymp_{m}Q^{2m-1},$

because

$\displaystyle\\#\left\\{\mathbf{q}\in{\mathbb{Z}}[i]^{m}:\|\mathbf{q}\|_{\infty}=Q\right\\}$
$\displaystyle=\\#\left\\{\mathbf{q}\in{\mathbb{Z}}^{2m}:\|\mathbf{q}\|_{\infty}=Q\right\\}$
$\displaystyle=\\#\left\\{\mathbf{q}\in{\mathbb{Z}}^{2m}:\|\mathbf{q}\|_{\infty}\leq
Q\right\\}-\\#\left\\{\mathbf{q}\in{\mathbb{Z}}^{2m}:\|\mathbf{q}\|_{\infty}\leq
Q-1\right\\}$ $\displaystyle=(2Q+1)^{2m}-(2Q-1)^{2m}$
$\displaystyle=2\sum\limits_{j=0}^{2m-1}(2Q+1)^{2m-1-j}(2Q-1)^{j},$

which is a polynomial of degree $2m-1$ in $Q$. For all
$j,l\in\\{1,\ldots,m\\}$ the number of balls of radius
$\|\mathbf{q}\|^{-\tau_{j}}$ required to cover
$\triangle(R_{\alpha,l},\|\mathbf{q}\|_{\infty}^{-\tau_{l}})$ is
asymptotically equivalent to

$\left(\max\left\\{1,\frac{\|\mathbf{q}\|_{\infty}^{-\tau_{l}}}{\|\mathbf{q}\|_{\infty}^{-\tau_{j}}}\right\\}\|\mathbf{q}\|_{\infty}^{\tau_{j}(m-1)}\right)^{2}.$

Hence, for all $s>0$, we have

$\displaystyle\mathcal{H}^{s}\left(\widetilde{W}_{\infty}(1,\boldsymbol{\tau})\right)$
$\displaystyle\ll\liminf_{N\to\infty}\sum\limits_{Q\geq
N}Q^{2m-1}Q^{2n}Q^{-s\tau_{j}}Q^{2\tau_{j}n(m-1)}Q^{2\sum\limits_{\tau_{l}<\tau_{j}}\tau_{j}-\tau_{l}}$
$\displaystyle=\liminf_{N\to\infty}\sum\limits_{Q\geq
N}Q^{2m+2n-1+2\tau_{j}n(m-1)+2\sum\limits_{\tau_{l}<\tau_{j}}\tau_{j}-\tau_{l}}Q^{-s\tau_{j}}.$

The last series above converges if and only if the exponent of $Q$ is strictly
less than $-1$, or equivalently if $s>s_{j}(\boldsymbol{\tau})$, so
$\operatorname{\dim_{H}}W_{\infty}(1,\boldsymbol{\tau})\leq s_{j}$.

Lower bound. Let $J$, $(R_{\alpha})_{\alpha\in J}$, $\beta$,
$(\tilde{l}_{s})_{s\geq 1}$ and $(u_{s})_{s\geq 1}$ as in Lemma 10. We only
establish the setup, the computations are quite similar to the $p$-adic case.
Suppose without loss of generality that
$\tau_{1}\geq\tau_{2}\geq\ldots\geq\tau_{n}>1$ and recall that $\delta_{k}=2m$
for $k\in\\{1,\ldots,n\\}$. First, assume that $\tau_{n}\geq\frac{n+m}{n}$.
For $j\in\\{1,\ldots,n\\}$ define

$a_{j}:=\frac{n+m}{n}\;\text{ and }\;t_{j}:=\tau_{j}-a_{j}.$

Then, the order of $\mathcal{A}$ is

$a_{1}+t_{1}\geq a_{2}+t_{2}\geq\ldots\geq a_{n}+t_{n}\geq
a_{1}=\ldots=a_{n}.$

Suppose that $\tau_{n}<\frac{n+m}{n}$ and let $K\in\\{1,\ldots,n\\}$ be the
largest integer such that

$\tau_{K}>\frac{m+n-(\tau_{K+1}+\cdots+\tau_{n})}{K}.$

For each $j\in\\{1,\ldots,n\\}$, write

$a_{j}:=\begin{cases}\tau_{j},&\quad\quad(K+1\leq j\leq n),\\\\[8.61108pt]
\displaystyle\frac{m+n-(\tau_{K+1}+\cdots+\tau_{d})}{K},&\quad\quad(1\leq
j\leq K).\end{cases}$

Then, $\mathcal{A}$ is ordered as follows:

$a_{1}+t_{1}\geq a_{2}+t_{2}\geq\ldots\geq
a_{K}+t_{K}>a_{1}=\ldots=a_{K}>a_{K+1}=a_{K+1}+t_{K+1}\geq\ldots\geq
a_{n}=a_{n}+t_{n}.$

By Theorem 6,
$\operatorname{\dim_{H}}\widetilde{W}_{\infty}(1,\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}$.

Now assume that $\gamma>0$ is arbitrary and take $0<\varepsilon<\max_{1\leq
j\leq n}\tau_{j}-1$. Since $\|\mathbf{q}\|_{\infty}^{\varepsilon}>\gamma$ for
all but finitely many $\mathbf{q}\in{\mathbb{Z}}[i]^{m}$, we have

$\widetilde{W}_{\infty}(1,\tau_{1}+\varepsilon,\ldots,\tau_{d}+\varepsilon)\subseteq\widetilde{W}_{\infty}(\gamma,\boldsymbol{\tau})\subseteq\widetilde{W}_{\infty}(1,\tau_{1}-\varepsilon,\ldots,\tau_{d}-\varepsilon).$

The lemma follows by letting $\varepsilon\to 0$, because
$\tau\mapsto\operatorname{\dim_{H}}\widetilde{W}_{\infty}(1,\boldsymbol{\tau})$
is continuous. ∎

######  Lemma 12.

If $\boldsymbol{\tau}$ satisfies (5.2), then

$\operatorname{\dim_{H}}W_{\infty}(\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}.$

###### Proof.

Arguing as in the proof of (5.8) and using (5.5), we obtain

$\widetilde{W}_{\infty}\left((2m)^{-1},\tau\right)\subseteq
W_{\infty}(\boldsymbol{\tau})\subseteq\widetilde{W}_{\infty}\left(\sqrt{2},\tau\right).$

The result now follows from Lemma 11. ∎

In a similar fashion, we may show that the corresponding set obtained from the
usual complex absolute value $|\,\cdot\,|$ has the same Hausdorff dimension:

(5.15)
$\operatorname{\dim_{H}}W^{{\mathbb{C}}}_{n,m}(\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}.$

##  6\. Quaternion approximation

In this section we study sets of linear forms in quaternion space. We refer
the reader to [38, Section 20] for the elementary algebraic aspects of
quaternions and Hurwitz integers, to [22] for a beautiful account on metrical
Diophantine approximation aspects of quaternions, and to [19] for an overview
of the theory. It is worth stressing that no metrical results whatsoever are
known in the higher (quaternion) dimensions, other than what is presented
below.

Fix two natural numbers $m$ and $n$. Let $i,j$ be two symbols and define the
operations

(6.1) $i^{2}=-1,\;j^{2}=-1,\;ij=-ji$

We write $k:=ij$. The ring of quaternions $\mathbb{H}$ is the skew field whose
elements are the objects $a+bi+cj+dk$ with $a,b,c,d\in{\mathbb{R}}$ along with
the sum defined coordinate-wise and the product obtained by following the
usual rules and (6.1). Given any $\xi=a+bi+cj+dk\in\mathbb{H}$, its conjugate
$\overline{\xi}$ is

$\overline{\xi}=a-bi-cj-dk$

and its norm $|\xi|$ is

$|\xi|=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}},$

so $|\xi|=\sqrt{\xi\overline{\xi}}$. The definition of the product implies
that for any $\xi,\zeta\in\mathbb{H}$ we have
$\overline{\xi\zeta}=\overline{\zeta}\,\overline{\xi}$ and, hence,
$|\xi\zeta|=|\xi||\zeta|$.

When regarded as a real vector space, $\mathbb{H}$ is isomorphic to
${\mathbb{R}}^{4}$ and an isomorphism is determined by

$1\mapsto(1,0,0,0),\;i\mapsto(0,1,0,0),\;j\mapsto(0,0,1,0),\;k\mapsto(0,0,0,1).$

Under this identification between $\mathbb{H}$ and $\mathbb{R}^{4}$, the real
bi-linear map
$\langle\,\cdot\,,\,\cdot\,\rangle:\mathbb{H}\times\mathbb{H}\to\mathbb{R}$
given by

$\langle\zeta,\xi\rangle:=\frac{1}{2}\left(\overline{\zeta}\xi-\zeta\overline{\xi}\right)\quad(\zeta,\xi\in\mathbb{H})$

is the usual inner product. As a consequence, the function from
$\mathbb{H}^{n}\times\mathbb{H}^{n}$ to $\mathbb{R}$ given by

$\left\langle(\xi_{1},\ldots,\xi_{n}),(\zeta_{1},\ldots,\zeta_{n})\right\rangle_{\mathbb{H}^{n}}\mapsto\sum_{j=1}^{n}\langle\xi_{j},\zeta_{j}\rangle_{\mathbb{H}}$

for all $(\xi_{1},\ldots,\xi_{n})$,
$(\zeta_{1},\ldots,\zeta_{n})\in\mathbb{H}^{n}$ is the usual inner product on
$\mathbb{R}^{4n}$.

The set of Lipschitz integers, given by

$\\{a+ib+cj+dk:a,b,c,d\in{\mathbb{Z}}\\},$

is the obvious choice for integers in $\mathbb{H}$. However, it is customary
to work instead with the Hurwitz integers ${\mathbb{Z}}_{\mathbb{H}}$, defined
as

${\mathbb{Z}}_{\mathbb{H}}:=\left\\{\frac{a}{2}+\frac{b}{2}i+\frac{c}{2}j+\frac{d}{2}k:a,b,c,d\in{\mathbb{Z}}\text{
and }a\equiv b\equiv c\equiv d\pmod{2}\right\\}.$

Clearly, the Lipschitz integers are contained in ${\mathbb{Z}}_{\mathbb{H}}$.
The reason Hurwitz integers are preferred over Lipschitz integers is of an
algebraic kind: Hurwitz integers are a Euclidean domain while Lipschitz
integers are not [19, Chapter 5]. Hurwitz integers have $24$ invertible
elements or units:

$\pm 1,\;\pm i,\;\pm j,\;\pm k,\;\frac{1}{2}\left(\pm 1+\pm i+\pm j+\pm
k\right).$

As noted in [22], ${\mathbb{Z}}_{\mathbb{H}}$ is the additive subgroup of
$\mathbb{H}$ generated by $i$, $j$, $k$, $\frac{1+i+j+k}{2}$ and, as a sub-
lattice of ${\mathbb{R}}^{4}$, the determinant of ${\mathbb{Z}}_{\mathbb{H}}$
is $\frac{1}{2}$. Let $I_{\mathbb{H}}$ be the Voronoi region for
${\mathbb{Z}}_{\mathbb{H}}$ containing $0$, that is

$I_{\mathbb{H}}:=\\{\xi\in\mathbb{H}:|\xi|\leq|\xi-\zeta|\text{ for all
}\zeta\in{\mathbb{Z}}_{\mathbb{H}}\\}.$

If $\llbracket\xi\rrbracket_{\mathbb{H}}$ denotes the distance between
$\xi\in\mathbb{H}$ and its nearest Hurwitz integer, then $I_{\mathbb{H}}$ is
precisely the set of quaternions $\xi$ for which
$\llbracket\xi\rrbracket_{\mathbb{H}}=|\xi|$. Observe that $I_{\mathbb{H}}$ is
the convex hull of
$\\{\pm\frac{1}{2},\pm\frac{1}{2}i,\pm\frac{1}{2}j,\pm\frac{1}{2}k\\}$ and
that its Lebesgue measure is $\frac{1}{2}$. Call $I_{\mathbb{H}}^{m\times n}$
the set of $m\times n$ matrices $A$ with entries in $I_{\mathbb{H}}$.

Given an $n$-tuple of non-increasing positive functions
$\varphi=(\varphi_{1},\ldots,\varphi_{n}):{\mathbb{N}}\to{\mathbb{R}}^{n}$
such that

$\lim_{q\to\infty}\phi_{j}(q)=0\quad(1\leq j\leq n)$

and an $m$-tuple of non-decreasing positive functions
$\Phi=(\Phi_{1},\ldots,\Phi_{m}):{\mathbb{N}}\to{\mathbb{R}}^{m}$ such that

$\lim_{q\to\infty}\Phi_{k}(q)=\infty\quad(1\leq k\leq m),$

we call $W_{n,m}^{\mathbb{H}}(\varphi,\Phi)\subseteq I_{\mathbb{H}}^{m\times
n}$ the set of $m\times n$ matrices $A$ with the following property:

there are infinitely many integers $u\geq 1$ such that the system

$\displaystyle\left\llbracket\mathbf{q}\,A_{j}\right\rrbracket$
$\displaystyle<\varphi_{j}(u)\quad(1\leq j\leq n)$ $\displaystyle|q_{k}|$
$\displaystyle\leq\Phi_{k}(u)\quad(1\leq k\leq m)$

has a non-zero solution $\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}^{1\times m}$.

We denote by $\mu_{m\times n}^{\mathbb{H}}$ the Lebesgue measure on
$\mathbb{H}^{m\times n}$.

######  Theorem 11.

If there are some constants $N_{0},M\in{\mathbb{N}}$, $M\geq 2$, and
$c_{1},c_{2}>1$ such that for every $j\in{\mathbb{N}}_{\geq N_{0}}$ we have

(6.2) $c_{1}\Phi_{k}(M^{j})\leq\Phi_{k}(M^{j+1})\leq
c_{2}\Phi_{k}(M^{j})\quad(1\leq k\leq m),$

then,

$\mu_{m\times
n}^{\mathbb{H}}\left(W_{n,m}^{\mathbb{H}}(\varphi,\Phi)\right)=\begin{cases}0,&\text{\rm
if
}\quad\displaystyle\sum_{q=1}^{\infty}\frac{1}{q}\left(\displaystyle\prod_{j=1}^{n}\varphi_{j}(q)\displaystyle\prod_{k=1}^{m}\Phi_{k}(q)\right)^{4}<\infty,\\\
\mu_{m\times n}^{\mathbb{H}}(I_{\mathbb{H}}^{m\times n}),&\text{\rm if
}\quad\displaystyle\sum_{q=1}^{\infty}\frac{1}{q}\left(\displaystyle\prod_{j=1}^{n}\varphi_{j}(q)\displaystyle\prod_{k=1}^{m}\Phi_{k}(q)\right)^{4}=\infty.\end{cases}$

In [22], Dodson and Everitt solved the one dimensional case using the theory
of ubiquitous systems of balls as introduced in [9]. More precisely, given a
non-increasing function $\psi:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that
$\psi(t)=\psi(\lfloor t\rfloor)$ for all $t>0$, they proved that the set

$V(\psi):=\left\\{\xi\in I_{\mathbb{H}}:|\xi q-p|<|q|\psi(|q|)\text{ for i.m.
}p,q\in{\mathbb{Z}}_{\mathbb{H}}\right\\}$

is of either zero or full measure according to the convergence or divergence
of

$\sum_{q=1}^{\infty}q^{7}\psi(q)^{4}.$

When $m=n=1$, the function $\Phi_{1}$ is the identity map and $\varphi$ is
decreasing, Theorem 11 tells us that the set

$\left\\{\xi\in I_{\mathbb{H}}:|q\xi-p|<\varphi(\lfloor|q|\rfloor)\text{ for
i.m. }p,q\in{\mathbb{Z}}_{\mathbb{H}}\right\\}$

is of zero or full measure according to the convergence or divergence of

$\sum_{q=1}^{\infty}q^{3}\varphi(q)^{4}.$

Although quaternions are not commutative, the proof of Theorem 11 also allows
us to conclude the 0-1 dichotomy for the set

$\left\\{\xi\in I_{\mathbb{H}}:|\xi q-p|<\varphi(\lfloor|q|\rfloor)\text{ for
i.m. }p,q\in{\mathbb{Z}}_{\mathbb{H}}\right\\}.$

The result of Dodson and Everitt now follows.

Let $\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$ be any
vector for which (5.2) holds. Define

$s_{j}(\boldsymbol{\tau}):=4n(m-1)+4\,\frac{m+n-\sum\limits_{r\,:\,\tau_{r}<\tau_{j}}(\tau_{r}-\tau_{j})}{\tau_{j}}\quad(1\leq
j\leq n)$

and

$W_{n,m}^{\mathbb{H}}(\boldsymbol{\tau}):=\left\\{A\in I_{\mathbb{H}}^{m\times
n}:\left\llbracket\mathbf{q}A_{j}\right\rrbracket<\frac{1}{\|\mathbf{q}\|^{\tau_{j}-1}}\;(1\leq
j\leq n)\text{ for i. m.
}\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}^{m}\right\\}.$

######  Theorem 12.

If $\tau\in{\mathbb{R}}^{n}$ satisfies (5.2), then

$\operatorname{\dim_{H}}W_{n,m}^{\mathbb{H}}(\boldsymbol{\tau})=\min\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\\}.$

Moreover,
$\mathcal{H}^{s}\left(W_{n,m}^{\mathbb{H}}(\boldsymbol{\tau})\right)=\infty$.

Theorems 11 and 12 can be shown as their complex counterpart. We thus omit a
considerable amount of detail in their proofs. First, as before, we replace
$|\,\cdot\,|$ with the more manageable norm $|\,\cdot\,|_{\infty}$ given by

$|\xi|_{\infty}:=\max\\{|a|,|b|,|c|,|d|\\}\quad\left(\xi=a+bi+cj+dk\in\mathbb{H}\right)$

and $\llbracket\,\cdot\,\rrbracket$ with the function
$\llbracket\,\cdot\,\rrbracket_{\infty}:\mathbb{H}\to{\mathbb{R}}_{+}$ given
by

$\llbracket\xi\rrbracket_{\infty}:=\min\\{|\xi-\zeta|_{\infty}:\zeta\in{\mathbb{Z}}_{\mathbb{H}}\\}\quad(\xi\in\mathbb{H}).$

The quaternion version of Minkowski’s theorem for linear forms, Lemma 13
below, follows from Theorem 4 and
$\det({\mathbb{Z}}_{\mathbb{H}})=\frac{1}{2}$.

######  Lemma 13.

Let $\gamma_{1},\ldots,\gamma_{n},\theta_{1},\ldots,\theta_{m}$ be positive
numbers satisfying
$\prod_{j=1}^{n}\gamma_{j}\prod_{k=1}^{m}\theta_{k}\geq\frac{1}{2^{m+n}}$. For
every matrix $A\in\mathbb{H}^{m\times n}$ there exists a vector
$(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}_{\mathbb{H}}^{1\times(m+n)}$ with
$\mathbf{q}\neq 0$ such that

$\displaystyle\left|\,\mathbf{q}\;A_{t}-p_{t}\right|_{\infty}$
$\displaystyle<\gamma_{t}\quad(1\leq t\leq n),$ (6.3)
$\displaystyle\left|q_{r}\right|_{\infty}$
$\displaystyle\leq\theta_{r}\quad(1\leq r\leq m).$

###  6.1. Ubiquity for quaternions

Given $k\in{\mathbb{N}}$, for any
$\xi=(\xi_{1},\ldots,\xi_{k})\in\mathbb{H}^{k}$ define

$\|\xi\|_{\infty}:=\max\left\\{|z_{1}|_{\infty},\ldots,|z_{k}|_{\infty}\right\\}.$

Hence, if $\|\xi\|_{2}=\sqrt{|\xi_{1}|^{2}+\cdots+|\xi_{k}|^{2}}$, we have

(6.4) $\frac{1}{2\sqrt{k}}\|\xi\|_{2}\leq\|\xi\|_{\infty}\leq\|\xi\|_{2}.$

Consider a sufficiently large $M$ and the next objects:

  1. i.