# The Shrinking Target Problem for Matrix Transformations
of Tori: revisiting the standard problem

Bing Li
(SCUT)  Lingmin Liao
(Wuhan)  Sanju Velani
(York)  Evgeniy Zorin
(York)

(_“You’ve always had the power my dear,
you just had to learn it yourself”_)

###### Abstract

Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a
self-map of the $d$-dimensional torus
${\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$. Let
$\\{E_{n}\\}_{n\in\mathbb{N}}$ be a sequence of subsets of ${\mathbb{T}}^{d}$
and let $W(T,\\{E_{n}\\})$ be the set of points
$\mathbf{x}\in{\mathbb{T}}^{d}$ such that $T^{n}(\mathbf{x})\in E_{n}$ for
infinitely many $n\in{\mathbb{N}}$. For a large class of subsets (namely,
those satisfying the so called bounded property $({\boldsymbol{\rm B}})$ which
includes balls, rectangles, and hyperboloids) we show that the $d$-dimensional
Lebesgue measure of the shrinking target set $W(T,\\{E_{n}\\})$ is zero (resp.
one) if a natural volume sum converges (resp. diverges). In fact, we prove a
quantitative form of this zero-one criteria that describes the asymptotic
behaviour of the counting function $R(x,N):=\\#\big{\\{}1\leq n\leq
N:T^{n}(x)\in E_{n}\\}$. The counting result makes use of a general
quantitative statement that holds for a large class measure-preserving
dynamical systems (namely, those satisfying the so called summable-mixing
property). We next turn our attention to the Hausdorff dimension of
$W(T,\\{E_{n}\\})$. In the case the subsets $E_{n}$ are balls, rectangles or
hyperboloids we obtain precise formulae for the dimension. These shapes
correspond, respectively, to the simultaneous, weighted and multiplicative
theories of classical Diophantine approximation. The dimension results for
balls generalises those obtained in [27] for integer matrices to real
matrices. In the final section, we discuss various problems that stem from the
results proved in the paper.

##  1 Introduction

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving dynamical system. Recall,
that by definition $\mu$ is a probability measure. Now let
$\\{E_{n}\\}_{n\in\mathbb{N}}$ be a sequence of subsets in $\mathcal{B}$ and
let

$\displaystyle W\big{(}T,\\{E_{n}\\}\big{)}$ $\displaystyle:=$
$\displaystyle\limsup_{n\to\infty}T^{-n}(E_{n})$ $\displaystyle=$
$\displaystyle\\{x\in X:T^{n}(x)\in E_{n}\ \hbox{ for infinitely many
}n\in\mathbb{N}\\}\,.$

For obvious reasons the sets $E_{n}$ can be thought of as targets that the
orbit under $T$ of points in $X$ have to hit. The interesting situation is
usually, when working within a metric space, the diameters of $E_{n}$ tend to
zero as $n$ increases. It is thus natural to refer to
$W\big{(}T,\\{E_{n}\\}\big{)}$ as the corresponding shrinking target set
associated with the given dynamical system and target sets. Since $T$ is
measure-preserving $\mu(T^{-n}(E_{n}))=\mu(E_{n})$, and a straightforward
consequence of the (convergent) Borel-Cantelli Lemma is that

$\mu\big{(}W(T,\\{E_{n}\\})\big{)}=0\qquad{\rm if\
}\qquad\sum_{n=1}^{\infty}\mu(E_{n})\,<\,\infty\,.$ (1)

Now two natural questions arise. Both fall under the umbrella of the
“shrinking target problem” formulated in [26].

  *     * (P1)

What is the $\mu$-measure of $W(T,\\{E_{n}\\})$ if the measure sum in (1)
diverges?

    * (P2)

What is the Hausdorff dimension of $W(T,\\{E_{n}\\})$ if the measure sum
converges and so $\mu\big{(}W(T,\\{E_{n}\\})\big{)}=0$?

To be precise, the target sets $E_{n}$ in the original formulation in [26] are
restricted to balls $B_{n}$. The more general setup naturally incorporates a
larger class of problems. For example, within the context of simultaneous
Diophantine approximation, it enables us to address problems associated with
the weighted (the target sets are rectangular) and multiplicative (the target
sets are hyperbola) theories – see Remark 4 in §1.2 below.

In this paper we revisit the shrinking target problem investigated in [27] in
which $T$ is a matrix transformation of the $d$-dimensional torus
$X={\mathbb{T}}^{d}:={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$. There are several
reasons for doing this. Firstly, for integer matrix transformations a solution
to (P1) was announced in [27]; namely, under some regularity condition on the
rate at which the diameters of the balls $B_{n}$ tend to zero, we have that

$m_{d}(W(T,\\{B_{n}\\}))=1\qquad{\rm if\
}\qquad\sum_{n=1}^{\infty}m_{d}(B_{n})\,=\,\infty\,$ (2)

where $m_{d}$ is $d$-dimensional Lebesgue measure. However, the intended paper
establishing this divergent analogue of (1) was never completed111The author
SV would like to take this opportunity to apologise for making an announcement
and then not delivering the goods! and to the best of our knowledge such a
result has not to date appeared in print elsewhere. In this paper not only do
we rectify the situation but we consider the set up in which $T$ is a real
(rather than just integer) matrix transformation and the ‘target’ sets are
general sets rather than just balls. Furthermore, our results are
significantly stronger than statements such as (2). In a nutshell, our
solution to (P1) consists of full measure statements that are quantitative in
nature. Next, turning our attention to (P2), Theorem 2 in [27] provides a
precise formula for the Hausdorff dimension of $W(T,\\{B_{n}\\})$ when $T$ is
an integer matrix transformation diagonalizable over the rationals. In this
paper we investigate the more general situation in which $T$ is real and by
making use of technology that was not available at the time of [27], we show
(for instance) that the aforementioned formula for the Hausdorff dimension
holds for a large class of real diagonal matrix transformations.

At the heart of our solution to (P1) for matrix transformations of tori, is a
result that holds for a large class of measure-preserving dynamical systems.
We start with describing this broader result and then move onto formally
stating our theorems for matrix transformations.

###  1.1 A quantitative full measure result for $\Sigma$-mixing dynamical
systems

Given a measure-preserving dynamical system $(X,\mathcal{B},\mu,T)$ and a
sequence $\\{E_{n}\\}_{n\in\mathbb{N}}$ of subsets in $\mathcal{B}$, we will
show that if $\mu$ is exponentially mixing and the measure sum in (1) diverges
then the associated $\limsup$ set $W\big{(}T,\\{E_{n}\\}\big{)}$ is of full
measure. However, it turns out that a lot more is true. We can establish a
quantitative full measure statement and at the same time work with the
potentially weaker notion of $\Sigma$-mixing.

######  Definition 1.

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving dynamical system and
$\mathcal{C}$ be a collection of measurable subsets of $X$. For
$n\in{\mathbb{N}}$, let

$\phi(n):=\sup\left\\{\;\left|\frac{\mu(E\cap
T^{-n}F)}{\mu(F)}-\mu(E)\right|\,:\,E\in\mathcal{C},F\in\mathcal{C}\right\\}.$
(3)

We say that $\mu$ is $\Sigma$-mixing (short for summable-mixing) with respect
to $(T,\mathcal{C})$ if the series $\sum_{n=1}^{\infty}\phi(n)$ converges.

Recall, that the above of mixing is stronger than that of $\phi$-mixing which
simply requires that $\phi(n)\to 0$ as $n\to\infty$. Also recall, that $\mu$
is _exponentially mixing_ with respect to $(T,\mathcal{C})$ if there exists a
constant $0<\gamma<1$ such that

$\mu(E\cap T^{-n}(F))=\mu(E)\mu(F)+O(\gamma^{n})\mu(F),$ (4)

for any $n\geq 1$ and $E,F\in\mathcal{C}$ – the implied constant in the big O
does not depend on the sets $E$ and $F$. In other words, and not surprisingly,
exponentially mixing and $\Sigma$-mixing coincide whenever $\phi(n)$ converges
to zero exponentially fast. It is worth mentioning that in the standard
definition, condition (4) is required to hold for any $F\in\mathcal{B}$ rather
than just in $\mathcal{C}$. We refer the reader to the survey paper [7] for
further details including “other” variants of the notion of exponentially
mixing. Also, see §1.1.1 below.

As above, let $\\{E_{n}\\}_{n\in\mathbb{N}}$ be a sequence of measurable
subsets of $X$. Then, given $N\in{\mathbb{N}}$ and $x\in X$, consider the
counting function

$R(x,N)=R(x,N;T,\\{E_{n}\\}):=\\#\big{\\{}1\leq n\leq N:T^{n}(x)\in E_{n}\\}.$
(5)

As alluded to in the definition, we will often simply write $R(x,N)$ for
$R(x,N;T,\\{E_{n}\\})$ since the other dependencies will be clear from the
context and are usually fixed. It is easily seen that the convergent statement
(1) is equivalent to saying that if the measure sum converges, then
$\lim_{N\to\infty}R(x,N)$ is finite for $\mu$–almost all $x\in X$. The
following result implies that for a large class of dynamical systems, if the
measure sum diverges then $\mu$–almost all $x\in X$ ‘hit’ the target sets
$E_{n}$ the ‘expected’ number of times.

######  Theorem 1.

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving dynamical system and
$\mathcal{C}$ be a collection of subsets of $X$. Suppose that $\mu$ is
$\Sigma$-mixing with respect to $(T,\mathcal{C})$ and let
$\\{E_{n}\\}_{n\in\mathbb{N}}$ be a sequence of subsets in $\mathcal{C}$.
Then, for any given $\varepsilon>0$, we have that

$R(x,N)=\Phi(N)+O\left(\Phi^{1/2}(N)\ (\log\Phi(N))^{3/2+\varepsilon}\right)$
(6)

for $\mu$-almost all $x\in X$, where

$\Phi(N):=\sum_{n=1}^{N}\mu(E_{n})\,.$

A simple consequence of Theorem 1 is that $\lim_{N\to\infty}R(x,N)=\infty$ for
$\mu$–almost all $x\in X$ if the measure sum diverges and so together with (1)
we obtain the following zero-full measure criterion.

######  Corollary 1.

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving dynamical system and
$\mathcal{C}$ be a collection of subsets of $X$. Suppose that $\mu$ is
$\Sigma$-mixing with respect to $(T,\mathcal{C})$ and let
$\\{E_{n}\\}_{n\in\mathbb{N}}$ be a sequence of subsets in $\mathcal{C}$. Then

$\displaystyle\mu\big{(}W(T,\\{E_{n}\\})\big{)}=\begin{cases}0&\text{if}\ \
\sum_{n=1}^{\infty}\mu\big{(}E_{n}\big{)}<\infty\\\\[8.61108pt] 1&\text{if}\ \
\sum_{n=1}^{\infty}\mu\big{(}E_{n}\big{)}=\infty.\end{cases}$ (7)

Before moving onto considering the specific situation in which $T$ is a matrix
transformation of the torus we discuss previous related works.

####  1.1.1 Connection to other works

We will focus on two previous works that are related to the framework
presented above; i.e. the notion of $\Sigma$-mixing and its consequences. In
an interesting paper [21], Fernández, Melián $\&$ Pestana introduced the
notion of a transformation $T$ being _uniformly mixing at a point $x_{0}\in
X$_. Their notion coincides with our Definition 1 if we restrict the
collection $\mathcal{C}$ to balls $B$ centered at $x_{0}$. The upshot [21,
Theorem 1] is that given a decreasing sequence of balls
$B_{n}:=B(x_{0},r_{n})$, if $T$ is uniformly mixing at $x_{0}$ and
$\sum_{n=1}^{\infty}\mu(B_{n})=\infty$ then

$\lim_{N\to\infty}\frac{R(x,N)}{\Phi(N)}=\lim_{N\to\infty}\frac{\\#\big{\\{}1\leq
n\leq N:T^{n}(x)\in B_{n}\big{\\}}}{\sum_{n=1}^{N}\mu(B_{n})}=1\,.$ (8)

Clearly our Theorem 1 not only implies this asymptotic statement but it also
provides a reasonably sharp estimate for the error term. As a consequence, the
various applications of (8) considered in [21] can be strengthened
accordingly. Indeed, their main motivating application to inner functions [21,
Theorem 2] can be improved to the following statement.

######  Theorem FMP+  .

Let $f:{\mathbb{D}}\to{\mathbb{D}}$ be an inner function with $f(0)=0$, but
not a rotation. Let $\xi_{0}$ be a point in $\partial{\mathbb{D}}$ and let
$\\{r_{n}\\}$ be a decreasing sequence of positive numbers. If
$\sum_{n=1}^{\infty}r_{n}=\infty$, then for any given $\varepsilon>0$, we have
that

$\\#\big{\\{}1\leq n\leq
N:d\big{(}(f^{*})^{n}(\xi),\xi_{0}\big{)}<r_{n}\big{\\}}=\Phi(N)+O\left(\Phi^{1/2}(N)\
(\log\Phi(N))^{3/2+\varepsilon}\right)$

for $\mu$-almost all $x\in X$, where $\Phi(N):=\sum_{n=1}^{N}r_{n}$,
$f^{*}(\xi)=\lim_{r\to 1_{-}}f(r\xi)$ and $d$ is the angular distance in
$\partial{\mathbb{D}}$.

In the later stages of preparing this manuscript, we discovered that our
Theorem 1 overlaps with a result of Philipp [47, Theorem 3] dating back to
1967. Indeed, in his theorem the condition imposed on the sequence
$\\{E_{n}\\}_{n\in\mathbb{N}}$ of measurable sets is in effect equivalent to
our notion of $\Sigma$-mixing with $\mathcal{C}=\mathcal{B}$. It appears that
[47, Theorem 3] has been either entirely overlooked, or at least not fully
exploited in previous works. For the sake of completeness we have decided to
include the proof of Theorem 1 in §2. Moreover, our proof is pretty short and
unlike Philipp’s approach it exploits a rather general tool (Lemma 1 in §2)
for establishing sharp counting statements. To the best of our knowledge, a
slightly weaker version of the tool, which suffices to establish Theorem 1,
first appears in Sprindžuk’s book [51, Lemma 10] which was some ten years
after Philipp’s paper. Furthermore, we have decided to include a self
contained proof of the Corollary 1 since it is rather nifty and some readers
may only be interested in the zero-full measure criterion rather than its
stronger quantitative form.

###  1.2 Quantitative full measure results for matrix transformations

Let $T$ be a $d\times d$ non-singular matrix with real coefficients. Then, $T$
determines a self-map of the $d$-dimensional torus
$X={\mathbb{T}}^{d}:={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$; namely, it sends
$\mathbf{x}\in{\mathbb{T}}^{d}$ to $T\mathbf{x}$ modulo one. In what follows,
$T$ will denote both the matrix and the transformation. It should be obvious
from the context what is meant. Furthermore, for $n\in{\mathbb{N}}$, by
$T^{n}$ we will always mean the $n$-th iteration of the transformation $T$
rather than the matrix multiplied $n$ times. With reference to the general
setup of §1.1, we now describe a broad collection $\mathcal{C}$ of ‘target’
sets contained in ${\mathbb{T}}^{d}$ so that for any sequence
$\\{E_{n}\\}_{n\in\mathbb{N}}$ of subsets in $\mathcal{C}$ we are able to
address the shrinking target ‘measure’ problem (P1) for the associated
$\limsup$ set $W\big{(}T,\\{E_{n}\\}\big{)}$. In order to do this, we require
the notion of the Minkowski content of a set in ${\mathbb{R}}^{d}$. We start
by recalling this basic notion from geometric measure theory.

Let $0\leq s\leq d$ be two positive integers and let $A$ be a subset of
${\mathbb{R}}^{d}$. Let $m_{d}$ denote the $d$-dimensional Lebesgue measure
and $\alpha(d)$ denote the volume of the $d$-dimensional unit Euclidean ball
$\\{\mathbf{x}\in{\mathbb{R}}^{d}:|\mathbf{x}|<1\\}$. By convention, we define
$\alpha(0):=1$. For $0<\epsilon<\infty$, we let $A(\epsilon)$ denote the
$\epsilon$-neighbourhood of $A$; that is

$A(\epsilon):=\\{\mathbf{x}\in{\mathbb{R}}^{d}:{\rm
dist}(\mathbf{x},A)<\epsilon\\}\,.$

Then, following the classical text of Federer [20, Section 3.2.37], the
$s$-dimensional upper and lower Minkowski content of $A$ are defined,
respectively as

$M^{*s}(A):=\limsup_{\epsilon\to
0^{+}}\frac{m_{d}(A(\epsilon))}{\alpha(d-s)\,\epsilon^{d-s}}\qquad{\rm
and}\qquad M^{s}_{*}(A):=\liminf_{\epsilon\to
0^{+}}\frac{m_{d}(A(\epsilon))}{\alpha(d-s)\epsilon^{d-s}}\,.$

If these upper and lower Minkowski contents are equal, then their common value
is called the $s$-dimensional Minkowski content of $A$ and is denoted by
$M^{s}(A)$. In general, the set functions $M^{*s}$ and $M_{*}^{s}$ are not
measures. However, for nice sets it turns out that both equal a constant
multiple of the Lebesgue measure $m_{s}$. In particular, a result of Federer
[20, Theorem 3.2.39] states that if $A$ is a closed $s$-rectifiable subset of
$\mathbb{R}^{d}$ (i.e. the image of a bounded set from $\mathbb{R}^{s}$ under
a Lipschitz function), then the $s$-dimensional Minkowski content of $A$
exists, and is equal to the $s$-dimensional Hausdorff measure of $A$. Recall
that for integer $s$ the latter is a constant multiple of $s$-dimensional
Lebesgue measure. Also, for the sake of completeness it is worth mentioning
that the Minkowski content is intimately related to the Minkowski dimension
which, nowadays is more commonly referred to as the box dimension. When
considering this fractal dimension, $s$ need not be an integer and we put
$\alpha(d-s)=1$ in the above definitions of upper and lower Minkowski
contents. For further details see [16, Section 3.1], [20, Sections 3.2.37-44],
[39, Chapter 5] and references within.

The following proposition identifies the collection $\mathcal{C}$ of ‘target’
sets alluded to above as subsets $E$ of ${\mathbb{T}}^{d}$ for which the
boundary $\partial E$ has bounded $(d-1)$-dimensional upper Minkowski content.
It makes use of the work initiated by Keller [31, 32] on the existence and
properties of absolutely continuous invariant measures for piecewise expanding
maps, and subsequently developed by the likes of Góra $\&$ Boyarsky [24],
Buzzi [10, 11, 12], Buzzi $\&$ Maume-Deschamps [13], Saussol [48] and Tsujii
[53, 54].

######  Proposition 1.

Let $T$ be a real, non-singular matrix transformation of the torus
${\mathbb{T}}^{d}$. Suppose that all eigenvalues of $T$ are of modulus
strictly larger than $1$. Then

  * (i)

there exists an absolutely continuous (with respect to Lebesgue measure
$m_{d}$) invariant probability measure (acim) $\mu$,

  * (ii)

the support $A\subseteq{\mathbb{T}}^{d}$ of any acim $\mu$ can be decomposed
into finitely many disjoint measurable sets $A_{1},\dots,A_{s}$ such that for
each $1\leq i\leq s$ the restriction $\mu|_{A_{i}}$ of $\mu$ to $A_{i}$ is
ergodic and is equivalent to the restriction $m_{d}|_{A_{i}}$ of Lebesgue
measure $m_{d}$ to $A_{i}$,

  * (iii)

each ergodic component $A_{i}$ in (ii) can in turn be decomposed into finitely
many disjoint measurable sets $A_{i1},\dots,A_{ip_{i}}$ such that for each
$1\leq j\leq p_{i}$ the restriction $\mu|_{A_{ij}}$ is mixing with respect to
$T^{p_{i}}$,

  * (iv)

on each mixing component $A_{ij}$ in (iii), the restriction $\mu|_{A_{ij}}$ is
exponentially mixing with respect to $(T^{p_{i}},\mathcal{C})$ for any
collection $\mathcal{C}$ of subsets $E$ of $A_{ij}$ satisfying the bounded
property

$({\boldsymbol{\rm B}}):\quad\sup_{E\in\mathcal{C}}\;M^{*(d-1)}(\partial
E)\,<\,\infty\,.$

######  Remark 1.

By definition, the restriction $\mu|_{A}$ of a probability measure $\mu$ to a
set $A$ with $\mu(A)>0$ is normalized so that it too is a probability measure.
In other words, for an arbitrary measurable set $E$

$\mu|_{A}(E):={\tiny\frac{1}{\mu(A)}}\,\mu(E)\,.$

For each $1\leq i\leq s$, the sets $A_{ij}$ ($1\leq j\leq p_{i}$) appearing in
part (iii) are referred to as the mixing components of $A_{i}$
$(=\bigcup_{j=1}^{p_{i}}A_{ij})$ and the positive integers $p_{i}$ are the
period of the mixing components. These mixing components satisfy the property
that

$T(A_{ij})=A_{ij+1}\quad(1\leq j\leq p_{i}-1)\qquad{\rm and}\qquad
T(A_{ip_{i}})=A_{i1}\,.$

Also, for the sake of clarity, completeness and convenience, recall that if
$\mu$ and $\nu$ are two measures on the same measurable space, then $\mu$ is
_absolutely continuous_ with respect to $\nu$ (written $\mu\ll\nu$) if
$\mu(E)=0$ for every measurable set $E$ for which $\nu(E)=0$. Moreover, the
measures $\mu$ and $\nu$ are _equivalent_ if $\mu\ll\nu$ and $\nu\ll\mu$ and
are said to be _strongly equivalent_ or _comparable_ if there exists a
constant $C\geq 1$ such that for every measurable set $E$

${C^{-1}}\nu(E)\leq\mu(E)\leq C\nu(E).$

Given a measure-preserving dynamical system $(X,\mathcal{B},\mu,T)$, the
invariant measure $\mu$ is ergodic if for every set $E\in\mathcal{B}$ with
$T^{-1}E=E$ we have either $\mu(E)=0$ or $\mu(E)=1$. Moreover, $\mu$ is said
to be mixing with respect to $T$ (often referred to strong-mixing) if for
every $E,F\in\mathcal{B}$

$\lim_{n\to\infty}\mu(E\cap T^{-n}F)=\mu(E)\mu(F).$

Clearly, exponentially mixing tells us that the implied error term in the
above limit decays exponentially. Also, if we put $F=E$ we immediately see
that mixing implies ergodic.

The following constitutes our most general measure theoretic result for the
shrinking target problem for matrix transformations of tori. As we shall see
the “divergent” part, which is the hard part, is essentially an immediate
consequence of combining Theorem 1 and Proposition 1.

######  Theorem 2.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that all eigenvalues of $T$ are of modulus strictly
larger than $1$ and let $\mathcal{C}$ be any collection of subsets $E$ of
$\mathbb{T}^{d}$ satisfying the bounded property $({\boldsymbol{\rm B}})$.
Furthermore, let $\mu$ be an acim and suppose it has support
${\mathbb{T}}^{d}$ and is mixing with respect to $T$. Then for any sequence
$\\{E_{n}\\}_{n\in\mathbb{N}}$ of subsets in $\mathcal{C}$ and
$\varepsilon>0$, we have that

$R(\mathbf{x},N)=\Phi(N)+O\left(\Phi^{1/2}(N)\
(\log\Phi(N))^{3/2+\varepsilon}\right)$ (9)

for $\mu$-almost all (equivalently $m_{d}$-almost all)
$\mathbf{x}\in\mathbb{T}^{d}$, where $\Phi(N):=\sum_{n=1}^{N}\mu(E_{n})$. In
particular,

$\displaystyle
m_{d}\big{(}W(T,\\{E_{n}\\})\big{)}=\mu\big{(}W(T,\\{E_{n}\\})\big{)}=\begin{cases}0&\text{if}\
\ \sum_{n=1}^{\infty}\mu\big{(}E_{n}\big{)}<\infty\\\\[8.61108pt] 1&\text{if}\
\ \sum_{n=1}^{\infty}\mu\big{(}E_{n}\big{)}=\infty.\end{cases}$ (10)

We note that the existence of the acim measure $\mu$ in Theorem 2 is
guaranteed by part (i) of Proposition 1 and that the assumptions imposed on
it, namely that the support of $\mu$ is the whole space ${\mathbb{T}}^{d}$ and
that $\mu$ is mixing with respect to $T$, are often satisfied. Indeed, this is
the situation when the eigenvalues of $T$ are large in modulus or the
coefficients of $T$ are integers. Regarding the former we have the following
precise statement. We will come to the integer situation shortly (see Theorem
5 below).

######  Theorem 3.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that all eigenvalues of $T$ are of modulus strictly
larger than $1+\sqrt{d}$. Let $\mathcal{C}$ be any collection of subsets of
$\mathbb{T}^{d}$ satisfying the bounded property $({\boldsymbol{\rm B}})$.
Then there is a unique acim $\mu$, such that for any sequence
$\\{E_{n}\\}_{n\in\mathbb{N}}$ in $\mathcal{C}$ and $\varepsilon>0$, the
counting formula (9) holds for $\mu$-almost all (equivalently $m_{d}$-almost
all) $\mathbf{x}\in\mathbb{T}^{d}$, where $\Phi(N):=\sum_{n=1}^{N}\mu(E_{n})$.
In particular, the zero-full measure criteria (10) holds.

######  Remark 2.

The fact that the acim $\mu$ appearing in Theorem 3 is unique is a trivial
consequence of the fact that any acim satisfying the hypotheses of Theorem 2
has to be unique. Indeed, to see that this is the case, suppose there exist
two such measures. Then by part (ii) of Proposition 1, both are equivalent to
$m_{d}$. By assumption, both are mixing with respect to $T$ and hence ergodic.
It thus follows (see [57, Theorem 6.10]) that the two measures are equal.

######  Remark 3.

By using the full force of Proposition 1, the assumptions on $\mu$ in Theorem
2 can be completely dropped if we restrict our attention to the shrinking
target set $W(T^{p},\\{E_{n}\\})\cap A$. Here $A\subseteq{\mathbb{T}}^{d}$ is
the support of the acim $\mu$ (guaranteed by part (i) of Proposition 1) and
$p:=p_{1}p_{2}\ldots p_{s}$ where the integers $p_{i}$ are the periods of the
mixing components associated with part (iii) of Proposition 1. Establishing
Theorem 4 below is an illustration of precisely this remark in action. In
short, the point of making the assumptions on $\mu$ in Theorem 2 is to obtain
a simple statement for the size of $W(T,\\{E_{n}\\})$ in terms of the
probability measure $m_{d}$ supported on ${\mathbb{T}}^{d}$.

######  Remark 4.

We consider two special families of target sets that correspond to “natural”
setups within the classical theory of Diophantine approximation. Let
$\psi:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ be a real positive function and fix
some point $\mathbf{a}:=(a_{1},\ldots,a_{d})\in{\mathbb{T}}^{d}$. For
$n\in{\mathbb{N}}$, let

$B_{n}=B(\mathbf{a},\psi(n)):=\Big{\\{}\mathbf{x}\in\mathbb{T}^{d}:\max_{1\leq
i\leq d}\|x_{i}-a_{i}\|\leq\psi(n)\Big{\\}}$

and

$H_{n}=H(\mathbf{a},\psi(n)):=\Big{\\{}\mathbf{x}\in\mathbb{T}^{d}:\prod_{1\leq
i\leq d}\|x_{i}-a_{i}\|\leq\psi(n)\Big{\\}}\,,$ (11)

where $\|\,.\,\|$ denotes the distance to the nearest integer. Clearly,
$B_{n}$ is a ball with respect to the maximum norm and $H_{n}$ is a
hyperboloid – both are centred at the fixed point $\mathbf{a}$. In turn, let

$W(T,\psi,\mathbf{a}):=\\{\mathbf{x}\in\mathbb{T}^{d}:T^{n}(\mathbf{x})\in
B(\mathbf{a},\psi(n))\ \ \text{for infinitely many}\ n\in\mathbb{N}\\},$

and

$W^{\times}(T,\psi,\mathbf{a}):=\\{\mathbf{x}\in\mathbb{T}^{d}:T^{n}(\mathbf{x})\in
H(\mathbf{a},\psi(n))\ \ \text{for infinitely many}\ n\in\mathbb{N}\\},$

denote the corresponding shrinking target sets. The former is intimately
related to sets studied within the classical simultaneous theory of
Diophantine approximation and the latter to the multiplicative theory. To see
this explicitly, suppose that $T$ is an integer, diagonal matrix. In fact,
suppose that

$T={\rm diag}\,(t_{1},\ldots,t_{d})\quad{\rm with}\quad t_{i}\geq 2\,$

and for convenience suppose $\mathbf{a}$ is the origin 0. Then, on using the
fact that $T$ is integer, it is easily seen that for any given
$\mathbf{x}:=(x_{1},\ldots,x_{d})\in[0,1)^{d}$ we have

$T^{n}(\mathbf{x})\in
B(\textbf{0},\psi(n))\quad\Longleftrightarrow\quad\max_{1\leq i\leq
d}\|t_{i}^{n}x_{i}\|\leq\psi(n)$

and

$T^{n}(\mathbf{x})\in
H(\textbf{0},\psi(n))\quad\Longleftrightarrow\quad\prod_{1\leq i\leq
d}\|t_{i}^{n}x_{i}\|\leq\psi(n)\,.$

It is evident that both the families of target sets $\\{B_{n}\\}_{n\geq 1}$
and $\\{H_{n}\\}_{n\geq 1}$ satisfy the bounded property $({\boldsymbol{\rm
B}})$. Thus, at the very least, our theorems incorporate both the simultaneous
and multiplicative aspects of the classical theory of Diophantine
approximation in which the denominators of the rational approximates are
restricted to lacunary sequences. For the explicit statements see Corollaries
2 $\&$ 3 below. In fact, our bounded property $({\boldsymbol{\rm B}})$
condition is far more general than the so called property $({\boldsymbol{\rm
P}})$ condition (see §5.1) imposed by Gallagher in his elegant and influential
paper [22]. We reiterate that our results hold for any family of target sets
$\\{E_{n}\\}_{n\geq 1}$ whose boundaries are rectifiable and are of uniformly
bounded $(d-1)$-dimensional Lebesgue measure.

We now investigate natural situations in which the measure $\mu$ associated
with Theorem 2 is strongly equivalent to Lebesgue measure $m_{d}$ on
${\mathbb{T}}^{d}$. For such situations we can replace $\mu$ by $m_{d}$ in the
finite sum $\Phi(N)$ and the righthand side of (10) and thus obtain statements
entirely in terms of Lebesgue measure. To start with, let us stick with real,
non-singular matrices and suppose that $T$ is diagonal with all eigenvalues
(or equivalently diagonal entries) $\beta_{1},\beta_{2},\dots,\beta_{d}$ of
modulus strictly larger than $1$. Now with this in mind, let
$\beta\in\mathbb{R}$ such that $|\beta|>1$ and let $\mu_{\beta}$ be
corresponding Parry measure for positive $\beta$ or the Yrrap measure for
negative $\beta$ – see $\S\ref{secmetricresult}$ for background and further
details. Also, let $K(\beta)$ denote the support of $\mu_{\beta}$. Then (see
Proposition 3 below),

$K(\beta)=[0,1]\qquad{\rm if}\qquad\beta\in(-\infty,-g]\,\cup\,(1,+\infty)\,,$
(12)

and $K(\beta)$ is a finite union of closed intervals contained in $[0,1]$ if
$\beta\in(-g,-1)$. Here and throughout,

$g:=(\sqrt{5}+1)/2$

is the golden ratio. Now returning to the transformation $T$ of the torus
${\mathbb{T}}^{d}$, we consider the product measure $\nu$ of the corresponding
one-dimensional Parry-Yrrap measures $\mu_{\beta_{i}}$; that is

$\nu:=\mu_{\beta_{1}}\times\mu_{\beta_{2}}\times\cdots\times\mu_{\beta_{d}}\,.$
(13)

Then by definition, the support of $\nu$ is

$K:=\prod_{i=1}^{d}K(\beta_{i}),$

and in view of (12) we have that $K={\mathbb{T}}^{d}$ if all $\beta_{i}$ are
in $(-\infty,-g]\,\cup\,(1,+\infty)$. On exploiting the properties of the
Parry-Yrrap measures $\mu_{\beta_{i}}$ and using the full force of Proposition
1 (see Remark 3) we are able to show that $\nu$ is exponentially mixing with
respect to $(T,\mathcal{C})$ for any collection $\mathcal{C}$ of subsets $E$
of $K$ satisfying the bounded property $({\boldsymbol{\rm B}})$. The details
of this are given in §3.3 and is at the heart of establishing the following
statement for real, diagonal matrix transformations.

######  Theorem 4.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal and all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ are of modulus strictly larger than $1$.
Let $\nu$ be the product measure given by (13) with support
$K\subseteq\mathbb{T}^{d}$. Let $\mathcal{C}$ be any collection of subsets $E$
of $K$ satisfying the bounded property $({\boldsymbol{\rm B}})$. Then for any
sequence $\\{E_{n}\\}_{n\in\mathbb{N}}$ in $\mathcal{C}$ and $\varepsilon>0$,
the counting formula (9) holds for $\nu$-almost all (equivalently
$m_{d}|_{K}$-almost all) $\mathbf{x}\in\mathbb{T}^{d}$, where
$\Phi(N):=\sum_{n=1}^{N}\nu(E_{n})$. In particular,

$\displaystyle
m_{d}|_{K}\big{(}W(T,\\{E_{n}\\})\big{)}=\nu\big{(}W(T,\\{E_{n}\\})\big{)}=\begin{cases}0&\text{if}\
\ \sum_{n=1}^{\infty}\nu\big{(}E_{n}\big{)}<\infty\\\\[8.61108pt] 1&\text{if}\
\ \sum_{n=1}^{\infty}\nu\big{(}E_{n}\big{)}=\infty.\end{cases}$

Furthermore, if all the eigenvalues of $T$ are in
$(-\infty,-g]\cup(1,+\infty)$ then $K=\mathbb{T}^{d}$ and we can replace $\nu$
by $m_{d}$ in the above; i.e. the counting formula (9) holds for
$m_{d}$-almost all $\mathbf{x}\in\mathbb{T}^{d}$, where
$\Phi(N):=\sum_{n=1}^{N}m_{d}(E_{n})$ and in particular

$\displaystyle m_{d}\big{(}W(T,\\{E_{n}\\})\big{)}=\begin{cases}0&\text{if}\ \
\sum_{n=1}^{\infty}m_{d}\big{(}E_{n}\big{)}<\infty\\\\[8.61108pt] 1&\text{if}\
\ \sum_{n=1}^{\infty}m_{d}\big{(}E_{n}\big{)}=\infty.\end{cases}$

In the case the collection $\mathcal{C}$ of subsets of $K$ is restricted to
rectangles with sides parallel to the axes (they clearly satisfy the bounded
property $({\boldsymbol{\rm B}})$) we can avoid using Proposition 1 and give a
self-contained and reasonably elementary proof of the above theorem (see
§3.3.1). In particular, it is more than enough to establish the following
corollary for balls (cubes); i.e., when we take $E_{n}=B_{n}$ (see Remark 4)
in the above theorem.

######  Corollary 2.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal with all eigenvalues in
$(-\infty,-g]\cup(1,+\infty)$. Let $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be
real positive function and $\mathbf{a}\in{\mathbb{T}}^{d}$. Then for any
$\varepsilon>0$, we have that

$\\#\big{\\{}1\leq n\leq N:T^{n}(\mathbf{x})\in
B(\mathbf{a},\psi(n))\big{\\}}=\Phi(N)+O\left(\Phi^{1/2}(N)\
(\log\Phi(N))^{3/2+\varepsilon}\right)$

for $m_{d}$-almost all $\mathbf{x}\in\mathbb{T}^{d}$, where
$\Phi(N):=\sum_{n=1}^{N}(2\psi(n))^{d}$. In particular,

$\displaystyle
m_{d}\big{(}W(T,\psi,\mathbf{a})\big{)}=\begin{cases}0&\text{if}\
\sum_{n=1}^{\infty}\psi(n)^{d}<\infty\\\\[8.61108pt] 1&\text{if}\
\sum_{n=1}^{\infty}\psi(n)^{d}=\infty.\end{cases}$

In fact, if we assume that $\psi(n)\to 0$ as $n\to\infty$, we are able to
appropriately extend Corollary 2 to the situation in which the eigenvalues are
in $(-\infty,-g]\,\cup\,[-1,+\infty)$. In other words, we can incorporate the
interval [-1,1] into the allowed range of the eigenvalues. This is the subject
of §3.3.2 below.

In another direction, if $T$ is an integer matrix transformation we are able
to use a nifty “reduction” argument to relax the condition that $T$ is
diagonal in Theorem 4 to $T$ is diagonalizable over ${\mathbb{Z}}$. This
reduction argument is the subject of §3.3.3 below. In fact, for integer
matrices far more is true.

######  Theorem 5.

Let $T$ be an integer, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that all eigenvalues of $T$ are of modulus strictly
larger than $1$ and and let $\mathcal{C}$ be any collection of subsets $E$ of
$\mathbb{T}^{d}$ satisfying the bounded property $({\boldsymbol{\rm B}})$.
Then, for any sequence $\\{E_{n}\\}_{n\in\mathbb{N}}$ of subsets in
$\mathcal{C}$ and $\varepsilon>0$, the counting formula (9) holds for
$m_{d}$-almost all $\mathbf{x}\in\mathbb{T}^{d}$, where
$\Phi(N):=\sum_{n=1}^{N}m_{d}(E_{n})$. In particular,

$\displaystyle m_{d}\big{(}W(T,\\{E_{n}\\})\big{)}=\begin{cases}0&\text{if}\ \
\sum_{n=1}^{\infty}m_{d}\big{(}E_{n}\big{)}<\infty\\\\[8.61108pt] 1&\text{if}\
\ \sum_{n=1}^{\infty}m_{d}\big{(}E_{n}\big{)}=\infty.\end{cases}$

To end with, we illustrate natural “number theoretic” consequences of our
results. Let $t_{1},\ldots,t_{d}~{}\geq~{}2$ be integers and let $T={\rm
diag}\,(t_{1},\ldots,t_{d})$. Then with reference to Remark 4, it follows that

$W(T,\psi,\mathbf{a})=\\{\mathbf{x}\in[0,1)^{d}:\max_{1\leq i\leq
d}\|t_{i}^{n}x_{i}-a_{i}\|\leq\psi(n)\ \ \text{for infinitely many}\
n\in\mathbb{N}\\}$

and

$W^{\times}(T,\psi,\mathbf{a})=\\{\mathbf{x}\in[0,1)^{d}:\prod_{1\leq i\leq
d}\|t_{i}^{n}x_{i}-a_{i}\|\leq\psi(n)\ \ \text{for infinitely many}\
n\in\mathbb{N}\\}\,.$

Thus, Theorem 5 implies the following statement for multiplicative Diophantine
approximation. In fact, since $T$ is diagonal, it is also covered by the
“furthermore part” of Theorem 4.

######  Corollary 3.

Let $t_{1},\ldots,t_{d}\geq 2$ be integers and let $T={\rm
diag}\,(t_{1},\ldots,t_{d})$. Let $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be
real positive function such that $\psi(x)<2^{-d}$ and
$\mathbf{a}=(a_{1},\ldots,a_{d})\in{\mathbb{T}}^{d}$. Then

$\\#\Big{\\{}1\leq n\leq N:\prod_{1\leq i\leq
d}\|t_{i}^{n}x_{i}-a_{i}\|\leq\psi(n)\Big{\\}}=\Phi(N)+O\left(\Phi^{1/2}(N)\
(\log\Phi(N))^{3/2+\varepsilon}\right)$

for $m_{d}$-almost all $\mathbf{x}=(x_{1},\ldots,x_{d})\in\mathbb{T}^{d}$,
where

$\Phi(N)=\sum_{n=1}^{N}2^{d}\psi(n)\left(\sum_{s=0}^{d-1}\frac{1}{s!}\left(\log\frac{1}{2^{d}\psi(n)}\right)^{s}\right)$

In particular,

$\displaystyle
m_{d}\big{(}W^{\times}(T,\psi,\mathbf{a})\big{)}=\begin{cases}0&\text{if}\
\sum_{n=1}^{\infty}\psi(n)\,\big{(}\log\frac{1}{\psi(n)}\big{)}^{d-1}<\infty\\\\[12.91663pt]
1&\text{if}\
\sum_{n=1}^{\infty}\psi(n)\,\big{(}\log\frac{1}{\psi(n)}\big{)}^{d-1}=\infty.\end{cases}$

The analogous statement for the simultaneous set $W(T,\psi,\mathbf{a})$ is
clearly covered by Corollary 2 above. The condition that $\psi(x)<2^{-d}$ is
only required for the counting statement.

Corollary 3 is probably most familiar to number theorists within the context
of when $t_{1}=\ldots=t_{d}$. This corresponds to approximating arbitrary
points $\mathbf{x}\in[0,1]^{d}$ by “shifted” rational points
$((p_{1}+a_{1})/q,\ldots,(p_{d}+a_{d})/q)$ with denominators $q$ restricted to
an integer lacunary sequence. In this setup, the zero-full measure criterion
within the corollary can just as easily be deduced from the elegant work of
Gallagher [22] mentioned in Remark 4. Also, under the same setup and the
assumption that $\psi$ is non-increasing, the corresponding quantitative
version (with a slightly worse error term) can be deduced from [25, Theorem
4.6].

######  Remark 5.

For $n\in{\mathbb{N}}$, let $H_{n}=H(\mathbf{a},\psi(n))$ be the hyperboloid
region given by (11). Clearly, Corollary 3 follows directly from Theorem 5 on
letting $E_{n}=H_{n}$ and on showing that

$m_{d}(H_{n})=2^{d}\psi(n)\left(\sum_{s=0}^{d-1}\frac{1}{s!}\left(\log\frac{1}{2^{d}\psi(n)}\right)^{s}\right)\,.$
(14)

For the sake of completeness we will provide the details of this measure
calculation in §3.2.1.

###  1.3 Dimension results for matrix transformations

We address the shrinking target ‘dimension’ problem (P2) in the case $T$ is a
self-map of the $d$-dimensional torus ${\mathbb{T}}^{d}$ and the target sets
are a sequence $\\{B_{n}\\}_{n\in\mathbb{N}}$ of balls as in the original
formulation of the problem. The following two theorems constitute our main
dimension results. It turns out that these statements for balls can be
exploited to determine the dimension of shrinking targets sets in the case the
targets are a sequence $\\{H_{n}\\}_{n\in\mathbb{N}}$ of hyperboloids.
Throughout, given a real positive function
$\psi:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ we let $\lambda=\lambda(\psi)$
denote its lower order at infinity; that is

$\lambda=\lambda(\psi):=\liminf_{n\to\infty}\frac{-\log\psi(n)}{n}.$

######  Theorem 6.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal with all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ strictly larger than $1$. Assume that
$1<\beta_{1}\leq\beta_{2}\leq\cdots\leq\beta_{d}$. Let
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a real positive function and
$\mathbf{a}\in\mathbb{T}^{d}$. Then

$\dim_{\rm H}W(T,\psi,\mathbf{a})=\min_{1\leq i\leq d}\theta_{i}(\lambda),$

where

$\theta_{i}(\lambda):=\frac{i\log\beta_{i}-\sum\limits_{k:\beta_{k}>\beta_{i}e^{\lambda}}(\log\beta_{j}-\log\beta_{i}-\lambda)+\sum\limits_{k>i}\log\beta_{j}}{\lambda+\log\beta_{i}}\,.$

######  Remark 6.

We will in fact deduce the above theorem from a more general statement
concerning rectangular target sets – see Theorem 12 in §4.2.

In the case $d=1$, the above result corresponds to the main result in [49]. It
turns out that while we are currently unable to prove in full generality the
analogue of Theorem 6 that incorporates negative eigenvalues, we can do so in
the one dimensional case. Thus, the following statement for $\beta<-1$ is new
and extends the work of Shen $\&$ Wang [49] from positive to arbitrary
$\beta$-transformations $T_{\beta}$.

######  Theorem 7.

Let $\beta$ be a real number with $|\beta|>1$ and $K(\beta)$ be the support of
the associated Parry-Yrrap measure. Let $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$
be a real positive decreasing function and $a\in K(\beta)$. Then

$\dim_{\rm H}W(T_{\beta},\psi,a)=\frac{\log|\beta|}{\lambda+\log|\beta|}\,.$

The proof of Theorem 7 makes use of a general approximation technique for any
piecewise linear map of the unit interval with constant slope. The associated
result (Proposition 7 in §7) may prove to be useful for other problems.

We now mention two consequences of our main dimension theorems. The first is
that if $T$ is an integer matrix transformation, then in Theorem 6 we can
replace the condition that $T$ is diagonal by $T$ is diagonalizable over
${\mathbb{Z}}$.

######  Theorem 8.

Let $T$ be an integer, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonalizable over ${\mathbb{Z}}$ with
all eigenvalues $\beta_{1},\beta_{2},\dots,\beta_{d}$ strictly larger than
$1$. Assume that $1<\beta_{1}\leq\beta_{2}\leq\cdots\leq\beta_{d}$. Let
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a real positive function and
$\mathbf{a}\in\mathbb{T}^{d}$. Then

$\dim_{\rm H}W(T,\psi,\mathbf{a})=\min_{1\leq i\leq d}\theta_{i}(\lambda)\,.$

The theorem follows from Theorem 6 by using a “reduction” argument – see
§4.2.2. The second is that the above theorems for balls enables us to
establish the dimension of the multiplicative set
$W^{\times}(T,\psi,\mathbf{a})$. In fact, we only require the $d=1$ statement
and so we are able to utilize the more general Theorem 7.

######  Theorem 9.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal and all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ are of modulus strictly larger than $1$.
Assume that $1<|\beta_{1}|\leq|\beta_{2}|\leq\cdots\leq|\beta_{d}|$. Let
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a real positive decreasing function
and $\mathbf{a}\in\mathbb{T}^{d}$ with $a_{d}\in K(\beta_{d})$. Then

$\dim_{\rm
H}W^{\times}(T,\psi,\mathbf{a})=d-1+\frac{\log|\beta_{d}|}{\lambda+\log|\beta_{d}|}\,.$

This consequence of Theorem 7 was pointed out to us by Baowei Wang. We thank
him for sharing his insight and indeed for providing the details of the proof
which forms the appendix. We stress that making use of Theorem 7, rather than
the previously known $d=1$ case of Theorem 6 due to Shen $\&$ Wang [49] is
crucial. The latter requires that all the eigenvalues are positive and
strictly larger than one and would thus yield a weaker version of Theorem 9.

##  2 Establishing Theorem 1 and Corollary 1

The following statement [25, Lemma 1.5] represents an important tool in the
theory of metric Diophantine approximation for establishing counting
statements. It has its bases in the familiar variance method of probability
theory and can be viewed as the quantitative form of the (diveregnce) Borel-
Cantelli Lemma [2, Lemma 2.2].

######  Lemma 1.

Let $(X,\cal{B},\mu)$ be a probability space, let
$(f_{n}(x))_{n\in{\mathbb{N}}}$ be a sequence of non-negative $\mu$-measurable
functions defined on $X$, and $(f_{n})_{n\in{\mathbb{N}}},\
(\phi_{n})_{n\in{\mathbb{N}}}$ be sequences of real numbers such that

$0\leq f_{n}\leq\phi_{n}\hskip 19.91692pt(n=1,2,\ldots).$

Suppose that for arbitrary $a,b\in{\mathbb{N}}$ with $a<b$, we have

$\int_{X}\left(\sum_{n=a}^{b}\big{(}f_{n}(x)-f_{n}\big{)}\right)^{2}\mathrm{d}\mu(x)\,\leq\,C\\!\sum_{n=a}^{b}\phi_{n}$
(15)

for an absolute constant $C>0$. Then, for any given $\varepsilon>0$, we have

$\sum_{n=1}^{N}f_{n}(x)\,=\,\sum_{n=1}^{N}f_{n}\,+\,O\left(\Phi(N)^{1/2}\log^{\frac{3}{2}+\varepsilon}\Phi(N)+\max_{1\leq
k\leq N}f_{k}\right)$ (16)

for $\mu$-almost all $x\in X$, where $\Phi(N):=\sum\limits_{n=1}^{N}\phi_{n}$.

Note that in statistical terms, if the sequence $f_{n}$ is the mean of
$f_{n}(x)$; i.e.

$f_{n}=\int_{X}f_{n}(x)\mathrm{d}\mu(x)\,,$

then the l.h.s. of (15) is simply the variance ${\rm Var}(Z_{a,b})$ of the
random variable

$Z_{a,b}=Z_{a,b}(x):=\sum_{n=a}^{b}f_{n}(x)\,.$

In particular,

${\rm Var}(Z_{a,b})=\mathbb{E}(Z^{2}_{a,b})-\mathbb{E}(Z_{a,b})^{2}$

where

$\mathbb{E}(Z_{a,b})=\int_{X}Z_{a,b}(x)\mathrm{d}\mu(x)\,.$

The following extremely useful classical inequality that bounds the
probability that a random variable is small, in terms of its expectation and
second moment, is a well know consequence of the Cauchy-Schwarz inequality.

######  Lemma 2 (Paley-Zygmund Inequality).

Let $(X,\cal{B},\mu)$ be a probability space and $Z$ be a non-negative random
variable. Then for any $0<\lambda<1$, we have that

$\mu\Big{(}\\{x\in
X:Z(x)>\lambda\mathbb{E}(Z)\\}\Big{)}\,\geq\,(1-\lambda)^{2}\
\frac{\mathbb{E}(Z)^{2}}{\mathbb{E}(Z^{2})}\,.$

We shall see that a straightforward application of Lemma 2 leads to a direct
proof of Corollary 1. Deducing Theorem 1 from Lemma 1 is also pretty
straightforward.

###### Proof of Theorem 1.

Given a sequence $\\{E_{n}\\}_{n\in\mathbb{N}}$ of subsets in $\mathcal{C}$,
we consider Lemma 1 with

$f_{n}(x):={\chi}_{{T^{-n}(E_{n})}}(x)={\Large\chi}_{{}_{E_{n}}}(T^{n}(x))\quad{\rm
and}\quad f_{n}:=\phi_{n}:=\mu(E_{n})\,,$ (17)

where $\chi_{{}_{E}}$ is the characteristic function of the set $E\subset X$.
Then, clearly for any $x\in X$ and $N\in{\mathbb{N}}$, we have that the

${\rm l.h.s.\ of\ }\eqref{ebc_conclusion}\ =\ \\#\big{\\{}1\leq n\leq
N:T^{n}(x)\in E_{n}\\}:=\ R(x,N)\,$

and so (16) and (6) coincide. Thus, to complete the proof of Theorem 1 we need
to verify that (15) is satisfied. Note that by definition, $f_{n}$ is the mean
of $f_{n}(x)$ and so

${\rm l.h.s.\ of\ }\eqref{ebc_condition1}\ =\ {\rm Var}(Z_{a,b})\ =\
\mathbb{E}(Z_{a,b}^{2})-\mathbb{E}(Z_{a,b})^{2}\,$ (18)

where

  * $\bullet$

${\rm Var}(Z_{a,b})$ is variance of the random variable

$Z_{a,b}=Z_{a,b}(x)=\sum\limits_{n=a}^{b}\chi_{E_{n}}(T^{n}(x))=\sum\limits_{n=a}^{b}\chi_{T^{-n}E_{n}}(x),$
(19)

  * $\bullet$

the expectation

$\mathbb{E}(Z_{a,b})=\sum\limits_{n=a}^{b}\mu(E_{n})\,,$ (20)

  * $\bullet$

and the second moment

$\displaystyle\mathbb{E}(Z_{a,b}^{2})$ $\displaystyle=\sum_{a\leq m,n\leq
b}\mu\left(T^{-m}(E_{m})\cap T^{-n}(E_{n})\right)$ (21)
$\displaystyle=\sum_{a\leq n\leq b}\mu(E_{n})+2\sum_{a\leq m<n\leq
b}\mu\left(E_{m}\cap T^{-(n-m)}(E_{n})\right).$

By making use of the $\Sigma$-mixing property (3), it follows that

$\displaystyle\sum_{a\leq m<n\leq b}\mu\left(E_{m}\cap
T^{-(n-m)}(E_{n})\right)$ $\displaystyle\leq$ $\displaystyle\sum_{a\leq
m<n\leq b}\\!\\!\mu(E_{m})\mu(E_{n})\ +\sum_{a\leq m<n\leq
b}\\!\\!\\!\phi(n-m)\ \mu(E_{n})$ $\displaystyle\leq$
$\displaystyle\sum_{a\leq m<n\leq b}\\!\\!\mu(E_{m})\mu(E_{n})\ +\sum_{a\leq
n\leq b}\left(\sum_{a\leq m<n}\\!\\!\\!\phi(n-m)\right)\mu(E_{n})$
$\displaystyle\leq$ $\displaystyle\sum_{a\leq m<n\leq
b}\\!\\!\mu(E_{m})\mu(E_{n})\ +\ \kappa\sum_{a\leq n\leq b}\mu(E_{n})$

where $\kappa:=\sum_{n=1}^{\infty}\phi(n)<\infty$. This together with (21)
implies that

$\displaystyle\mathbb{E}(Z_{a,b}^{2})$ $\displaystyle\leq$
$\displaystyle(2\kappa+1)\sum_{a\leq n\leq b}\mu(E_{n})\ +\ 2\,\sum_{a\leq
m<n\leq b}\\!\\!\mu(E_{m})\mu(E_{n})$ (22) $\displaystyle\leq$
$\displaystyle(2\kappa+1)\sum_{a\leq n\leq b}\mu(E_{n})\ +\ \left(\sum_{a\leq
n\leq b}\\!\\!\mu(E_{n})\right)^{2}\,.$

The upshot of (18), (20) and (22) is that

${\rm Var}(Z_{a,b})\ \leq\ (2\kappa+1)\sum_{a\leq n\leq b}\mu(E_{n})\,.$

This verifies (15) with $C=2\kappa+1$ and thereby completes the proof of
Theorem 1. ∎

###### Proof of Corollary 1.

In view of (1), we assume that the sum in (7) diverges. With the same notation
as in the proof of Theorem 1, we start with the observation that for any
$\lambda>0$

$\mu\Big{(}W(T,\\{E_{n}\\})\Big{)}\ \geq\
\mu\Big{(}\limsup_{b\to\infty}(Z_{1,b}>\lambda\mathbb{E}(Z_{1,b}))\Big{)}\
\geq\
\limsup_{b\to\infty}\mu\Big{(}Z_{1,b}>\lambda\mathbb{E}(Z_{1,b})\Big{)}\,.$
(23)

To estimate the measure on the r.h.s. we use the Paley-Zygmund inequality
(Lemma 2) and the estimates (20) and (22). With this in mind, for any
$0<\lambda<1$, it follows that

$\displaystyle\mu\Big{(}Z_{1,b}>\lambda\mathbb{E}(Z_{1,b})\Big{)}$
$\displaystyle\geq$ $\displaystyle(1-\lambda)^{2}\ \
\frac{\mathbb{E}(Z_{1,b})^{2}}{\mathbb{E}(Z_{1,b}^{2})}$ $\displaystyle\geq$
$\displaystyle(1-\lambda)^{2}\ \frac{\Big{(}\sum\limits_{1\leq n\leq
b}\mu(E_{n})\Big{)}^{2}}{\Big{(}\sum\limits_{1\leq n\leq
b}\mu(E_{n})\Big{)}^{2}+(2\kappa+1)\sum\limits_{1\leq n\leq b}\mu(E_{n})}\,.$

By the divergent sum hypothesis, on letting $b\to\infty$ and $\lambda\to 0$,
we obtain that

$\limsup_{b\to\infty}\mu\Big{(}Z_{1,b}>\lambda\mathbb{E}(Z_{1,b})\Big{)}=1\,.$

which together with (23) completes the proof of the corollary. ∎

##  3 Establishing measure results for matrix transformations

###  3.1 Proof of Proposition 1

Let $X\subset\mathbb{R}^{d}$ be a compact set. The proof of Proposition 1
makes essential use of the work of Saussol [48] for general piecewise
expanding maps $T:X\longrightarrow X$ on $X$. Clearly, our particular case in
which $X=[0,1]^{d}$ and the real, non-singular matrix $T$ (with the modulus of
all eigenvalues strictly larger than one) that sends $\mathbf{x}$ to
$T\mathbf{x}\ {\rm mod}\ 1$ defines a piecewise expanding map on $X$. In what
follows will state and apply Saussol’s results to our setup. So with this in
mind, the first three parts, apart for the equivalence of the restricted
measures $\mu|_{A_{i}}$ and $m_{d}|_{A_{i}}$ in part (ii), follow from [48,
Theorem 5.1]. To prove the equivalence of the restricted measures we note that
by [48, Proposition 5.1], the Randon-Nykodym derivative $f$ of $\mu|_{A_{i}}$
with respect to $m_{d}|_{A_{i}}$ is $m_{d}$-almost surely strictly positive.
Hence, for any measurable subset $E$, if $m_{d}|_{A_{i}}(E)>0$, then

$\mu|_{A_{i}}(E)=\int_{E}f(\mathbf{x})m_{d}|_{A_{i}}(d\mathbf{x})>0$

and so $m_{d}|_{A_{i}}\ll\mu|_{A_{i}}$. The other direction follows directly
from part (i).

It remains to prove part (iv) of the proposition. Without loss of generality,
we will assume that $A_{ij}$ is $A$ and thus $\mu|_{A_{ij}}=\mu$ and also
$\mu$ is mixing with respect to $T$. The key to proving part (iv) is to use
the fact that the acim $\mu$ satisfies the property of exponential decay of
correlations. With this mind, for a set $E\subset\mathbb{R}^{d}$, define the
oscillation of $\varphi\in L^{1}(m_{d})$ over $E$ as

${\rm osc}(\varphi,E):=\underset{E}{\textrm{ess-sup}}(\varphi)-\underset{\
E}{\textrm{ess-inf}}(\varphi).$

For given real numbers $0<\alpha\leq 1$ and $0<\epsilon_{0}<1$, define the
following $\alpha$-seminorm

$|\varphi|_{\alpha}:=\underset{0<\epsilon\leq\epsilon_{0}}{\sup\
}\epsilon^{-\alpha}\int_{\mathbb{R}^{d}}{\rm osc}(\varphi,B(x,\epsilon))dx.$

Let $V_{\alpha}$ be the space of $L^{1}(m_{d})-$functions such that
$|\varphi|_{\alpha}<\infty$ endowed with the norm

$\|\varphi\|_{\alpha}:=\|\varphi\|_{L^{1}(m_{d})}+|\varphi|_{\alpha}.$

Then $(V_{\alpha},\|\cdot\|_{\alpha})$ is a Banach space which does not depend
on the choice of $\epsilon_{0}$ and $V_{\alpha}\subset L^{\infty}(m_{d})$ (see
[48, Section 3]). Since the acim $\mu$ is mixing with respect to $T$, it
follows from [48, Theorem 6.1] that there exist constants $C>0$ and
$0\leq\gamma<1$ such that for all $\psi\in V_{\alpha}$, for all $\phi\in
L^{1}(\mu)$ and for all $n\in\mathbb{N}$, we have

$\displaystyle\left|\int_{[0,1]^{d}}\\!\\!\\!\\!\\!\psi.\phi\circ T^{n}\,d\mu\
-\ \int_{[0,1]^{d}}\\!\\!\\!\\!\\!\\!\\!\psi\
d\mu\int_{[0,1]^{d}}\\!\\!\\!\\!\\!\\!\\!\phi\ d\mu\right|\ \leq\ C\
\|\psi\|_{\alpha}\ \|\phi\|_{1}\ \gamma^{n}.$ (24)

Now, let $\mathcal{C}$ be any collection of subsets of $A$ satisfying the
bounded property $({\boldsymbol{\rm B}})$. Take $\psi=\chi_{E}$ and
$\phi=\chi_{F}$ with $E,F\in\mathcal{C}$ and assume for the moment that

$\displaystyle\sup_{E\in\mathcal{C}}\|\chi_{E}\|_{\alpha}<\infty.$ (25)

Note that (25) implies that $\psi=\chi_{E}\in V_{\alpha}$ and thus together
with (24), we obtain that

$\displaystyle\left|\mu(E\cap T^{-n}F)-\mu(E)\mu(F)\right|\leq
C\cdot\|\chi_{E}\|_{\alpha}\cdot\mu(F)\cdot\gamma^{n}\leq
C\cdot\Big{(}\sup_{E\in\mathcal{C}}\|\chi_{E}\|_{\alpha}\Big{)}\cdot\gamma^{n}\cdot\mu(F).$

Hence the exponentially mixing property (4) is satisfied for the collection
$\mathcal{C}$. This proves part (iv) modulo (25).

We now prove (25). To start with, observe that for the characteristic function
$\chi_{E}$, the oscillation can only be non-zero on the boundary $\partial E$
of $E$. It can be verified that for any $\epsilon>0$,

${\rm osc}(\chi_{E},B(x,\epsilon))\,\leq\,\chi_{(\partial E)(\epsilon)}(x),$

where $(\partial E)(\epsilon)$ is the $\epsilon$-neighborhood of $\partial E$.
Thus,

$\epsilon^{-\alpha}\int_{\mathbb{R}^{d}}{\rm
osc}(\chi_{E},B(x,\epsilon))dx\leq\epsilon^{-\alpha}\cdot m_{d}((\partial
E)(\epsilon)).$

By the bounded property $({\boldsymbol{\rm B}})$ imposed on $\mathcal{C}$,
there exists a constant $C_{1}$ such that the $(d-1)$-dimensional upper
Minkowski content of $\partial E$

$M^{*(d-1)}(\partial E)\leq C_{1},$

for all $E\in\mathcal{C}$. Hence, by the definition of $M^{*(d-1)}$, there
exists a constant $C_{0}$ such that for all $E\in\mathcal{C}$

$m_{d}((\partial E)(\epsilon))\,<\,C_{0}\epsilon.$

Consequently, for all $E\in\mathcal{C}$,

$|\chi_{E}|_{\alpha}=\underset{0<\epsilon\leq\epsilon_{0}}{\sup\
}\epsilon^{-\alpha}\int_{\mathbb{R}^{d}}{\rm
osc}(\chi_{E},B(x,\epsilon))dx\,\leq\,C_{0}\epsilon_{0}^{1-\alpha}.$

On the other hand, for $E\subset[0,1]^{d}$, we have that

$\|\chi_{E}\|_{L^{1}(m_{d})}=m_{d}(E)\leq 1.$

Thus, for all $E\in\mathcal{C}$, it follows that

$\|\chi_{E}\|_{\alpha}\leq 1+C_{0}\epsilon_{0}^{1-\alpha}.$

Therefore, (25) holds and this completes the proof of the proposition.
$\square$

######  Remark 7.

For the sake of completeness, we mention that in the case that $T$ is an
integer, non-singular, matrix transformation of the torus $\mathbb{T}^{d}$
with all eigenvalues in modulus strictly larger than one, Fan [17] proved the
exponential decay of correlation formula (24). Also in the case $d=1$, the
first three parts of Proposition 1 coincide with the Main Result of Wagner
[56]

###  3.2 Proof of Theorems 2, 3 and 5

###### Proof of Theorem 2.

In view of parts (i) to (iii) of Proposition 1, if the acim $\mu$ has
${\mathbb{T}}^{d}$ as its support and is mixing with respect to $T$, then
$\mu$ has one mixing component (namely the whole space $\mathbb{T}^{d}$) of
period one. Furthermore, by part (iv) of Proposition 1, on this unique mixing
component, $\mu$ is exponentially mixing with respect to $(T,\mathcal{C})$ for
any collection $\mathcal{C}$ of subsets of $\mathbb{T}^{d}$ satisfying the
bounded property $({\boldsymbol{\rm B}})$. The desired counting part (9) of
the theorem and the zero-full measure criteria (10) with respect to the
measure $\mu$ now immediately follow on applying Theorem 1 and Corollary 1
respectively. To complete the proof of Theorem 2, it remains to prove that the
measures $\mu$ and $m_{d}$ share the same zero and full measure sets (note
that we have already shown above that $\mu\big{(}W(T,\\{E_{n}\\})\big{)}$ is
either 0 or 1). This follows directly from part (ii) of Proposition 1 since it
implies that $\mu$ is equivalent to $m_{d}$ and it is easily seen that
equivalent measures share the same zero and full measure sets. ∎

The following lemma will be used in establishing Theorems 3 and 5.

######  Lemma 3.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that (i) all eigenvalues of $T$ are of modulus
strictly larger than $1+\sqrt{d}$ or that (ii) $T$ is integer and all
eigenvalues of $T$ are of modulus strictly larger than $1$. Then there is a
unique acim $\mu$. Furthermore, such an acim $\mu$ has ${\mathbb{T}}^{d}$ as
its support and is of maximal entropy.

###### Proof.

We are given that all eigenvalues of $T$ are of modulus strictly larger than
$1$. Thus, by part (i) of Proposition 1, there exists an acim $\mu$. By [10,
Proposition 1], if all eigenvalues of $T$ have modulus strictly larger than
$1+\sqrt{d}$ or if $T$ is integers, then the dynamical system
$(\mathbb{T}^{d},T)$ is topological transitive. Hence, by [10, Theorem 3 and
its Corollary] the acim $\mu$ has the whole space ${\mathbb{T}}^{d}$ as its
support and is the unique maximal entropy measure. ∎

We now prove Theorems 3 and 5.

###### Proof of Theorem 3.

By [10, Lemma 5], if the eigenvalues of $T$ are all of modulus strictly larger
than $1+\sqrt{d}$, then $T$ is locally eventually onto. Thus the unique
maximal entropy acim $\mu$ coming from Lemma 3 which has ${\mathbb{T}}^{d}$ as
its support, is exact (see for example [43, Theorem 5.2.12]) and hence mixing
with respect to $T$ (see [42, Proposition 12.2]). In other words, this measure
$\mu$ satisfies the hypotheses of Theorem 2. On applying Theorem 2, we obtain
Theorem 3. ∎

###### Proof of Theorem 5.

Observe that if $T$ is integer, then $T$ is an endomorphism of the torus
${\mathbb{T}}^{d}$ (see for example, [57, Theorem 0.15]). By [57, Corollary
1.10.1 and Theorem 1.28], the Lebesgue measure $m_{d}$ is mixing with respect
to $T$. Furthermore, by [57, Theorem 8.15], $m_{d}$ is of maximal entropy.
Thus $m_{d}$ is nothing but the unique maximal entropy acim $\mu$ in Lemma 3.
Hence, Theorem 5 follows on applying Theorem 2 with $\mu=m_{d}$. ∎

####  3.2.1 Proof of Corollary 3

As noted in Remark 5, the corollary follows directly from Theorem 5 on showing
that the $d$-dimensional Lebesgue measure $m_{d}$ of the hyperboloid region
$H(\mathbf{a},\psi(n))$ satisfies (14). It is easily versified that
$m_{d}(H(\mathbf{a},\psi(n)))$ is independent of the ‘shift’
$\mathbf{a}\in{\mathbb{T}}^{d}$. So with this in mind, it suffices to prove
the following statement.

######  Lemma 4.

Given $d\in{\mathbb{N}}$ and $\delta>0$, let

$H_{d}(\delta):=\big{\\{}(x_{1},\dots,x_{d})\in[0,1)^{d}:\|x_{1}\|\dots\|x_{d}\|<\delta\big{\\}}\,.$
(26)

Then

$m_{d}(H_{d}(\delta))=\begin{cases}1&\text{ if }\ \delta\geq
2^{-d}\\\\[8.61108pt]
2^{d}\delta\left(\sum_{t=0}^{d-1}\frac{1}{t!}\left(\log\frac{1}{2^{d}\delta}\right)^{t}\right)&\text{
if }\ \delta<2^{-d}\,.\end{cases}$ (27)

###### Proof.

To simplify computations, first note that the measure of $H_{d}(\delta)$ is
equal to $2^{d}$ times the measure of $[0,1/2]^{d}\cap H_{d}(\delta)$.
Furthermore, it is technically simpler to work with points restricted to
$[0,1/2]^{d}$ since the inequality under consideration is equivalent to

$x_{1}\dots x_{d}\leq\delta.$ (28)

So, from now on we will focus on computing the measure of the set

$V_{d}(\delta):=\big{\\{}(x_{1},\dots,x_{d})\in[0,1/2]^{d}\,:\,{x_{1}\dots
x_{d}}\leq\delta\big{\\}}\,$ (29)

and recall that

$2^{d}\,m_{d}\big{(}V_{d}(\delta)\big{)}\ =\
m_{d}\big{(}H_{d}(\delta)\big{)}\,.$ (30)

_Case (a): if $\delta\geq 2^{-d}$._ Then it is a easily versified that
$V_{d}(\delta)=[0,1/2]^{d}$ and this together with (30) implies (27). So,
without loss of generality we can assume that $\delta<2^{-d}$.

_Case (b): if $\delta<2^{-d}$._ In view of (30), to establish (27) we need to
show that for any $d\in{\mathbb{N}}$ and $0<\delta<2^{-d}$

$m_{d}\big{(}V_{d}(\delta)\big{)}\,=\,\delta\
\sum_{t=0}^{d-1}\frac{1}{t!}\left(\log\frac{1}{2^{d}\delta}\right)^{t}\,.$
(31)

This we now do by induction on $d$. For $d=1$, we have that

$m_{1}\big{(}V_{1}(\delta)\big{)}=m_{1}\big{(}\left\\{x\in[0,1/2]\,:\,x\leq\delta\right\\}\big{)}\,=\,\delta\,$
(32)

and this coincides with (31). Now let $d\geq 2$ and observe that we can
rewrite (28) as

$x_{1}\dots x_{d-1}\leq\delta/x_{d}.$ (33)

Note that since $(x_{1},\dots,x_{d})\in[0,1/2]^{d}$ , the left hand side of
(33) is not bigger than $(1/2)^{d-1}$. Hence, it follows that for any
$0<x_{d}\leq 2^{d-1}\delta$, inequality (33) is satisfied for all $0\leq
x_{1},\dots,x_{d-1}\leq 1/2$. The $m_{d}$-measure of the set of such points
$(x_{1},\dots,x_{d})$ is thus equal to $2^{-d+1}\times 2^{d-1}\delta=\delta$.
On the other hand, for any fixed value of $x_{d}\in(2^{d-1}\delta,1/2]$, the
$m_{d-1}$-measure of the set of points $(x_{1}\dots x_{d-1})\in[0,1/2]^{d-1}$
satisfying (33) is by definition equal to
$m_{d-1}\big{(}V_{d-1}(\delta/x_{d})\big{)}$. The upshot is that for any
$d\geq 2$ and $0<\delta<2^{-d}$

$m_{d}\big{(}V_{d}(\delta)\big{)}\,=\,\delta\,+\,\int_{2^{d-1}\delta}^{1/2}m_{d-1}\big{(}V_{k-1}(\delta/x_{d})\big{)}\;\mathrm{d}x_{d}.$
(34)

Now assume that (31) holds with $d-1$ in place of $d$. Then, it follows via
(34) that

$\displaystyle m_{d}\big{(}V_{d}(\delta)\big{)}$ $\displaystyle=$
$\displaystyle\delta\ +\ \int_{2^{d-1}\delta}^{1/2}\
\frac{\delta}{x_{d}}\left(\sum_{t=0}^{d-2}\frac{1}{t!}\left(\log\frac{x_{d}}{2^{d-1}\delta}\right)^{t}\right)\,\mathrm{d}x_{d}$
$\displaystyle=$ $\displaystyle\delta\ +\
\delta\,\int_{1}^{\frac{1}{2^{d}\delta}}\
\left(\sum_{t=0}^{d-2}\frac{1}{t!}\frac{(\log y)^{t}}{y}\right)\,\mathrm{d}y$
$\displaystyle=$ $\displaystyle\delta\ +\ \delta\
\sum_{t=0}^{d-2}\frac{1}{t!}\ \int_{1}^{\frac{1}{2^{d}\delta}}\ \frac{(\log
y)^{t}}{y}\ \mathrm{d}y$ $\displaystyle=$ $\displaystyle\delta\ +\ \delta\
\sum_{t=0}^{d-2}\frac{1}{(t+1)!}\left(\log\frac{1}{2^{d}\delta}\right)^{t+1}$
$\displaystyle=$ $\displaystyle\delta\ +\ \delta\
\sum_{t=1}^{d-1}\frac{1}{t!}\left(\log\frac{1}{2^{d}\delta}\right)^{t}\,=\,\delta\
\sum_{t=0}^{d-1}\frac{1}{t!}\left(\log\frac{1}{2^{d}\delta}\right)^{t}\,.$

This completes the induction step and so establishes (31) for any $d\geq 1$.

∎

###  3.3 Proof of Theorem 4

We start by summarising various basic facts concerning $\beta$-transformations
that will be utilized in proving Theorem 4. So, with this in mind, let $\beta$
be a real number such that $|\beta|>1$ and let $T_{\beta}:[0,1)\to[0,1)$ be
the associated $\beta$-transformation given by

$T_{\beta}(x)=\beta x\ (\text{mod}\ 1).$

For obvious reasons, when $\beta<-1$ the corresponding transformation is
refereed to as the negative $\beta$-transformation.

For $\beta>1$, Rényi [44, Theorem 1] proved that there exists a unique
$T_{\beta}$-invariant measure $\mu_{\beta}$ (the so called Parry measure) that
is strongly equivalent to (one-dimensional) Lebesgue measure $m_{1}$ on the
unit interval. Clearly, this implies that $\mu_{\beta}$ is absolutely
continuous with respect to Lebesgue measure. For the negative
$\beta$-transformation, Ito and Sadahiro [28] proved that there is a unique
$T_{\beta}$-invariant measure $\mu_{\beta}$ (the so called Yrrap measure)
which is absolutely continuous with respect to Lebesgue measure $m_{1}$. The
following proposition implies that $\mu_{\beta}$ is in fact strongly
equivalent to $m_{1}$ when $\beta\leq-g$. Note that in view of [28, Theorem
16], for the negative $\beta$-transformation the corresponding density
function of $\mu_{\beta}$ with respect to $m_{1}$ is given by

$h_{\beta}(x):=\frac{1}{F(\beta)}\sum_{n\geq 0\atop T_{\beta}^{n}1\geq
x}\frac{1}{\beta^{n}},$

where

$F(\beta):=\int_{0}^{1}\sum_{n\geq 0\atop T_{\beta}^{n}1\geq
x}\frac{1}{\beta^{n}}dx$

is the normalising function.

######  Proposition 2.

Let $\beta\leq-g$. Then the Yrrap measure $\mu_{\beta}$ is strongly equivalent
to the Lebesgue measure $m_{1}$ on the unit interval. More precisely, there
exists a constant $C(\beta)>0$ such that

$C(\beta)^{-1}\leq h_{\beta}(x)\leq C(\beta)\qquad\forall\quad x\in(0,1)\,.$

###### Proof.

For $\beta=-g$, it is easily verified that

$h_{\beta}(x)=\begin{cases}\frac{1}{3+\beta}\ \ &\text{if}\quad 0<x\leq
2+\beta\\\\[8.61108pt] \frac{-\beta}{3+\beta}\ \ &\text{if}\quad
2+\beta<x<1.\end{cases}$

Hence, we can choose $C(\beta)=\frac{1}{3+\beta}$. Without loss of generality,
assume $\beta<-g$ and note that

$1+\sum_{n=0}^{\infty}\frac{1}{\beta^{2n+1}}\ =\
\frac{\beta^{2}+\beta-1}{\beta^{2}-1}\ \leq\ F(\beta)\ \leq\
\sum_{n=0}^{\infty}\frac{1}{\beta^{2n}}\ =\ \frac{\beta^{2}}{\beta^{2}-1}\,.$

It then immediately follows that

$h_{\beta}(x)\ \leq\
\frac{1}{F(\beta)}\sum_{n=0}^{\infty}\frac{1}{\beta^{2n}}\ =\
\frac{\beta^{2}}{\beta^{2}+\beta-1}$

and

$h_{\beta}(x)\ \geq\
\frac{1}{F(\beta)}\left(1+\sum_{n=0}^{\infty}\frac{1}{\beta^{2n+1}}\right)\ =\
\frac{\beta^{2}+\beta-1}{\beta^{2}}.$

Hence, we can choose $C(\beta)=\frac{\beta^{2}}{\beta^{2}+\beta-1}\,$ . ∎

The following result identifies the nature of the support of the
$T_{\beta}$-invariant measure $\mu_{\beta}$.

######  Proposition 3.

Let $\beta$ be a real number with $|\beta|>1$, $\mu_{\beta}$ be the associated
Parry-Yrrap measure and let $K(\beta)$ denote the support of $\mu_{\beta}$.
Then

$K(\beta)=[0,1]\qquad{\rm if}\quad\beta\in(-\infty,-g]\,\cup\,(1,+\infty)\,,$

and $K(\beta)$ is a finite union of closed intervals contained in $[0,1]$ if
$\beta\in(-g,-1)$. Furthermore, $\mu_{\beta}$ is mixing with respect to
$T_{\beta}$ and is equivalent to the measure $m|_{K(\beta)}$; i.e. the one-
dimensional Lebesgue measure restricted to $K(\beta)$.

From this point onwards, given $\beta$ with $|\beta|>1$, $\mu_{\beta}$ will
always denote the associated Parry-Yrrap measure and $K(\beta)$ will denote
the support of $\mu_{\beta}$.

###### Proof.

If $\beta\in(-\infty,-g]\,\cup\,(1,+\infty)$, the result immediately follows
from the fact that $\mu_{\beta}$ is strongly equivalent to the Lebesgue
measure $m_{1}$ on $[0,1]$ – this is Proposition 2 for $\beta\leq-g$ and as
already mentioned established by Rényi [44, Theorem 1] for $\beta>1$. In
general, Keller [30] proved for any $|\beta|>1$ the support of $\mu_{\beta}$
is a finite union of closed intervals. The precise description of the closed
intervals for the non-trivial case when $\beta\in(-g,-1)$ was given by Liao
$\&$ Steiner [37, Theorem 2.1].

For the furthermore part, it follows via Rokhlin [45, Section 4.5] for
$\beta>1$ and Liao $\&$ Steiner [37, Corollary 2.3] for $\beta<-1$, that the
$T_{\beta}$-invariant measure $\mu_{\beta}$ is exact and hence mixing with
respect to $T_{\beta}$ ([42, Proposition 12.2]). In turn, by the the Main
Result in [56] it follows that $\mu_{\beta}$ is equivalent to $m_{1}$
restricted to $K(\beta)$. ∎

Of course, as indicated in above proof of the proposition, for
$\beta\in(-\infty,-g]\,\cup\,(1,+\infty)$ we have that $\mu_{\beta}$ is
strongly equivalent to the Lebesgue measure $m_{1}$ on $[0,1]$ rather than
simply equivalent. The next statement is a straight forward consequence of
Proposition 3 and basic properties of product measures.

######  Lemma 5.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal and all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ are of modulus strictly larger than $1$.
Then the product measure

$\nu:=\mu_{\beta_{1}}\times\mu_{\beta_{2}}\times\cdots\times\mu_{\beta_{d}}$

has support $K:=\prod_{i=1}^{d}K(\beta_{i})$ and is a $T$-invariant mixing
measure that is equivalent to $m_{d}|_{K}$; i.e. the $d$-dimensional Lebesgue
measure restricted to $K$.

###### Proof.

By definition, the product measure $\nu$ has support
$K=\prod_{i=1}^{d}K(\beta_{i})$ and in view of Proposition 3 it is
$T$-invariant and equivalent to the measure $m_{d}|_{K}$. Furthermore, since
each $\mu_{\beta_{i}}$ is mixing with respect to $T_{\beta_{i}}$, on following
the proof of [57, Theorem 1.24] it is easily verified that $\nu$ is mixing
with respect to $T$. ∎

We now show that Theorem 4 is an easy consequence of Lemma 5 together with
Proposition 1 and Theorem 1. In the next section we will provide a self-
contained and essentially elementary proof of Theorem 4 in the case the
collection $\mathcal{C}$ of subsets of $K$ is restricted to rectangles with
sides parallel to the axes. This is an important class of “target sets” that
clearly satisfy the bounded property $({\boldsymbol{\rm B}})$ and the proof
will avoid appealing to Proposition 1.

###### Proof of Theorem 4 using Proposition 1.

Lemma 5 implies that product measure $\nu$ is a $T$-invariant mixing measure
equivalent to $m_{d}|_{K}$. Hence by Proposition 1, $\nu$ is exponentially
mixing with respect to $(T,\mathcal{C})$ where $\mathcal{C}$ is any collection
of subsets $E$ of $K$ satisfying the bounded property $({\boldsymbol{\rm
B}})$. Then the main counting part of Theorem 4 immediately follows from
Theorem 1. For the “furthermore” part we first recall (as we have done so
several times) that by Rényi [44, Theorem 2] and Proposition 2, for any
$\beta\in(-\infty,-g]\cup(1,+\infty)$ the Parry-Yrrap measure $\mu_{\beta}$ is
strongly equivalent to the Lebesgue measure $m_{1}$ on $[0,1]$. It thus
follows that the product measure $\nu$ is strongly equivalent to $m_{d}$
restricted to $K={\mathbb{T}}^{d}$. The upshot of this is that we can replace
$\nu$ by $m_{d}$ in the first part of the theorem and thereby completes the
proof of Theorem 4. ∎

####  3.3.1 Theorem 4 for rectangles: a self contained and direct proof

In the proof of Theorem 4 given above, we make use of Proposition 1 to deduce
that the product measure $\nu$ is exponentially mixing with respect to
$(T,\mathcal{C})$ where $\mathcal{C}$ is any collection of subsets $E$ of $K$
satisfying the bounded property $({\boldsymbol{\rm B}})$. The following result
enables us to bypass the proposition in the case $\mathcal{C}$ is restricted
to rectangles with sides parallel to the axes.

######  Lemma 6.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal and all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ are of modulus strictly larger than $1$.
Let
$\nu:=\mu_{\beta_{1}}\times\mu_{\beta_{2}}\times\cdots\times\mu_{\beta_{d}}$
be the product measure and $K:=\prod_{i=1}^{d}K(\beta_{i})$ be its support.
Then $\nu$ is exponentially mixing with respect to $(T,\mathcal{R})$ for any
collection $\mathcal{R}$ of rectangles of $K$ with sides parallel to the axes.

It is easily seen that by appealing to Lemma 6 instead of Proposition 1 in the
“Proof of Theorem 4 using Proposition 1” given in the previous section, we
obtain the special case of Theorem 4 in which $\mathcal{C}$ is any collection
$\mathcal{R}$ of rectangles of $K$ with sides parallel to the axes. In other
words, it enables us to provide a self-contained and direct proof of Theorem 4
for rectangular target sets.

###### Proof of Lemma 6.

First, we assert that for any $\beta\in\mathbb{R}$ with $|\beta|>1$,
$\mu_{\beta}$ is exponentially mixing with respect to
$(T_{\beta},\mathcal{C})$ where $\mathcal{C}$ is any collection of intervals
of $K(\beta)$. When $\beta>1$, this is the classical result of Gel’fond [23,
Formula (12)] and Philipp [47, Lemma 7]. When $\beta<-1$, the assertion
follows from a general result of Baladi [1, Theorem 3.4] for piecewise
monotone expanding interval maps.

We now verify that the product $\nu$ satisfies the desired exponentially
mixing property. With this in mind, let $E=B(z_{1},r_{1})\times\cdots\times
B(z_{d},r_{d})$ and $F=B(z^{\prime}_{1},r^{\prime}_{1})\times\cdots\times
B(z^{\prime}_{d},r^{\prime}_{d})$ be any two rectangles in $\mathcal{R}$. Then

$\displaystyle\nu(E\cap
T^{-n}F)=\int\chi_{E}(x_{1},\dots,x_{d})\,\chi_{F}(T_{\beta_{1}}^{n}x_{1},\dots,T_{\beta_{d}}^{n}x_{d})\
d\mu_{\beta_{1}}(x_{1})\cdots\,d\mu_{\beta_{d}}(x_{d})\,,$

and by the property of the product measure the right hand side equals

$\displaystyle\int\chi_{B(z_{1},r_{1})}(x_{1})\chi_{B(z_{1}^{\prime},r_{1}^{\prime})}(T_{\beta_{1}}^{n}x_{1})d\mu_{\beta_{1}}(x_{1})\cdots\int\chi_{B(z_{d},r_{d})}(x_{d})\chi_{B(z_{d}^{\prime},r_{d}^{\prime})}(T_{\beta_{d}}^{n}x_{d})d\mu_{\beta_{d}}(x_{d}).$

It then follows by the exponentially mixing property of $T_{\beta_{i}}$, that

$\displaystyle\nu(E\cap
T^{-n}F)=\prod_{i=1}^{d}\Big{(}\,\mu_{\beta_{i}}(B(z_{i},r_{i}))\,\mu_{\beta_{i}}(B(z_{i}^{\prime},r_{i}^{\prime}))+O(\gamma_{i}^{n})\,\mu_{\beta_{i}}(B(z_{i}^{\prime},r_{i}^{\prime}))\,\Big{)}$

where $0<\gamma_{i}<1$. This together with the fact that

$\nu(E)=\mu_{\beta_{1}}(B(z_{1},r))\cdots\mu_{\beta_{d}}(B(z_{d},r))\,,\quad\nu(F)=\mu_{\beta_{1}}(B(z_{1}^{\prime},r^{\prime}))\cdots\mu_{\beta_{d}}(B(z_{d}^{\prime},r^{\prime}))$

and $\mu_{\beta_{i}}(B(z_{i},r))\leq 1$ $(1\leq i\leq d)$ implies that

$\displaystyle\nu(E\cap T^{-n}F)=\nu(E)\nu(F)+O(\gamma^{n})\mu(F)\quad{\rm
where}\quad\gamma=\max\\{\gamma_{1},\dots,\gamma_{d}\\}\,.$

In other words, (4) holds for rectangles in $\mathcal{R}$ and we are done. ∎

####  3.3.2 Extending Corollary 2 to incorporate eigenvalues in $[-1,1]$

We show that on assuming $\psi(n)\to 0$ as $n\to\infty$, we can naturally
extend Corollary 2 to the situation that all the eigenvalues of $T$ are in
$(-\infty,-g]\,\cup\,[-1,+\infty)$; that is to say, we can incorporate the
interval $[-1,1]$. In short, assuming that $T$ has eigenvalues in $[-1,1]$, we
do this in most cases by reformulating the shrinking target set
$W(T,\psi,\mathbf{a})$ in terms of related “lower dimensional’ sets
$W(T_{*},\psi,\mathbf{a}_{*})$ for which Corollary 2 in its current form is
applicable. In other words, all the eigenvalues of the related transformation
$T_{*}$ are in $(-\infty,-g]\,\cup\,(1,+\infty)$. Let $T$ be a real, non-
singular, diagonal matrix transformation of the torus $\mathbb{T}^{d}$ with
eigenvalues $\beta_{1},\beta_{2},\dots,\beta_{d}$ in
$(-\infty,-g]\cup[-1,+\infty)$. Without loss of generality, assume that there
is at least one eigenvalue in $[-1,1]$. In fact, let us assume that there is
only one such eigenvalue, say $\beta_{1}$. It should be self-evident how to
deal with the situation in which that are multiple eigenvalues in $[-1,1]$. We
consider the three separate situations depending on whether $|\beta_{1}|<1$,
$\beta_{1}=1$ or $\beta_{1}=-1$. Note that
$T=\text{diag}(\beta_{1},\cdots,\beta_{d})$ and so for any
$\mathbf{x}=(x_{1},\dots,x_{d})\in{\mathbb{T}}^{d}$

$T^{n}(\mathbf{x})=(T_{\beta_{1}}^{n}(x_{1}),T_{\beta_{2}}^{n}(x_{2}),\cdots,T_{\beta_{d}}^{n}(x_{d})).$

(i) _We assume $|\beta_{1}|<1$._ We distinguish between three subcases:

  * •

Case 1: $\beta_{1}=0$. Then it is easily verified that

$\displaystyle W(T,\psi,\mathbf{a})=\begin{cases}\emptyset&\text{if}\
a_{1}\neq 0\\\\[4.30554pt] \mathbb{T}\times
W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ a_{1}=0,\end{cases}$

where

$T_{*}:=\text{diag}(\beta_{2},\cdots,\beta_{d})\quad{\rm\ and\
}\quad\mathbf{a}_{*}:=(a_{2},\cdots,a_{d})\in\mathbb{T}^{d-1}\,.$ (35)

  * •

Case 2: $0<\beta_{1}<1$. Then $T_{\beta_{1}}^{n}(x_{1})=\beta_{1}^{n}x_{1}\to
0$ as $n\to\infty$ for any $x_{1}\in\mathbb{T}$. Note that zero is the unique
fixed point of $T_{\beta_{1}}$. Thus,

$\big{\\{}x_{1}\in[0,1):T_{\beta_{1}}^{n}(x_{1})<\psi(n)\ \hbox{ for
infinitely many }n\in\mathbb{N}\big{\\}}\,=\,I_{*}\ \text{ or }\
\overline{I_{*}},$

where $I_{*}:=[0,\min\\{1,\tau\\})$, $\overline{I_{*}}$ is the closure of the
set $I_{*}$ and

$\tau:=\limsup\limits_{n\to\infty}\,\psi(n)\,|\beta_{1}|^{-n}.$ (36)

Indeed, the set under consideration is $I_{*}$ if
$\psi(n)\,|\beta_{1}|^{-n}\leq\tau$ for infinitely many $n\in\mathbb{N}$ and
$\overline{I_{*}}$ otherwise. Hence, with $T_{*}$ and $\mathbf{a}_{*}$ as in
(35), it follows that

$\displaystyle W(T,\psi,\mathbf{a})=\begin{cases}\emptyset&\text{if}\ \
a_{1}\neq 0\\\\[4.30554pt] \\{0\\}\times
W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ \ a_{1}=0\ \text{and}\
\tau=0,\\\\[4.30554pt] I_{*}\times W(T_{*},\psi,\mathbf{a}_{*})\text{ \ \ or \
\ }\overline{I_{*}}\times W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ \ a_{1}=0\
\text{and}\ \tau>0.\end{cases}$

  * •

Case 3: $-1<\beta_{1}<0$. Let $F$ be the countable set of all preimages of
zero; that is,

$F:=\big{\\{}x_{1}\in\mathbb{T}:T_{\beta_{1}}^{n}(x_{1})=0\ \text{for some}\
n\geq 0\big{\\}}.$

If $x_{1}\notin F$, then $T_{\beta_{1}}x_{1}=\beta_{1}x_{1}+1$ and

$T_{\beta_{1}}^{n}x_{1}=\beta_{1}^{n}x_{1}+\beta_{1}^{n-1}+\beta_{1}^{n-2}+\cdots+\beta_{1}+1\to\frac{1}{1-\beta_{1}}$

as $n\to\infty$. Note that zero and $\frac{1}{1-\beta_{1}}$ are the fixed
points of $T_{\beta_{1}}$. Thus,

$\big{\\{}x_{1}\in[0,1)\setminus F:T_{\beta_{1}}^{n}(x_{1})<\psi(n)\ \
\text{for infinitely many}\ n\in\mathbb{N}\big{\\}}=J_{*}\setminus F\ \text{
or }\ \overline{J_{*}}\setminus F,$

where
$J_{*}:=\left((\frac{1}{1-\beta_{1}}-\tau,\frac{1}{1-\beta_{1}}+\tau)\cap[0,1)\right)$,
$\overline{J_{*}}$ is the closure of the set $J_{*}$ and $\tau$ is given by
(36). Indeed, the set under consideration is $J_{*}\setminus F$ if
$\psi(n)\,|\beta_{1}|^{-n}\leq\tau$ for infinitely many $n\in\mathbb{N}$ and
$\overline{J_{*}}\setminus F$ otherwise. Hence, with $T_{*}$ and
$\mathbf{a}_{*}$ as in (35), it follows that

$\displaystyle W(T,\psi,\mathbf{a})=\begin{cases}F\times
W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ \ a_{1}=0\\\\[4.30554pt]
\emptyset&\text{if}\ \ a_{1}\neq\frac{1}{1-\beta_{1}},a_{1}\neq
0\\\\[4.30554pt] \\{\frac{1}{1-\beta_{1}}\\}\times
W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ \ a_{1}=\frac{1}{1-\beta_{1}}\
\text{and}\ \tau=0,\\\\[4.30554pt] J_{*}\times
W(T_{*},\psi,\mathbf{a}_{*})\text{ \ \ or \ \ }\overline{J_{*}}\times
W(T_{*},\psi,\mathbf{a}_{*})&\text{if}\ \ a_{1}=\frac{1}{1-\beta_{1}}\
\text{and}\ \tau>0.\end{cases}$

The upshot is that in order to determine the size of $W(T,\psi,\mathbf{a})$ or
the behaviour of the associated counting function we need to investigate the
shrinking target set
$W(T_{*},\psi,\mathbf{a}_{*})\subseteq{\mathbb{T}}^{d-1}$. Recall, that for
ease of discussion we are assuming that $\beta_{1}$ is the only eigenvalue of
$T$ in $[-1,1]$. Thus, all the eigenvalues of the related transformation
$T_{*}$ are in $(-\infty,-g]\,\cup\,(1,+\infty)$ and so Corollary 2 is
applicable with $T$ replaced by $T_{*}$, $\mathbf{a}$ replaced by
$\mathbf{a}_{*}$ and $d$ replaced by $d-1$.

(ii) _We assume $\beta_{1}=1$._ For any
$\mathbf{x}=(x_{1},\dots,x_{d})\in\mathbb{T}^{d}$ and
$\mathbf{a}=(a_{1},\cdots,a_{d})\in\mathbb{T}^{d}$, the condition
$T^{n}(\mathbf{x})\in B(\mathbf{a},\psi(n))$ implies that

$\|x_{1}-a_{1}\|<\psi(n).$

Now, since we are assuming that $\psi(n)\to 0$ and $n\to\infty$, it follows
that for any $\mathbf{x}\in W(T,\psi,\mathbf{a})$ we must have that
$x_{1}=a_{1}$. Hence, with $T_{*}$ and $\mathbf{a}_{*}$ as in (35), it follows
that

$W(T,\psi,\mathbf{a})=\\{a_{1}\\}\times W(T_{*},\psi,\mathbf{a}_{*})\,.$

As in (i), the upshot is that we need to investigate the shrinking target set
$W(T_{*},\psi,\mathbf{a}_{*})\subseteq{\mathbb{T}}^{d-1}$ and that for this
setup Corollary 2 is applicable with $T$ replaced by $T_{*}$, $\mathbf{a}$
replaced by $\mathbf{a}_{*}$ and $d$ replaced by $d-1$.

(iii) _We assume $\beta_{1}=-1$._ Then, for $x_{1}\neq 0$

$\displaystyle T_{\beta_{1}}^{n}(x_{1})=\begin{cases}x_{1}&\text{if}\ \ n\
\text{is even}\\\\[4.30554pt] -x_{1}+1&\text{if}\ \ n\ \text{is
odd}.\end{cases}$

If $a_{1}=0$, then the same reasoning as in (ii) shows that
$W(T,\psi,\mathbf{a})=\\{0\\}\times W(T_{*},\psi,\mathbf{a}_{*})$ and so we
can apply Corollary 2 with $T$ replaced by $T_{*}$, $\mathbf{a}$ replaced by
$\mathbf{a}_{*}$ and $d$ replaced by $d-1$. If $a_{1}\neq 0$, it follows that
for any $\mathbf{x}\in W(T,\psi,\mathbf{a})$ we must have that $x_{1}=a_{1}$
or $x_{1}=-a_{1}+1$. Thus, with $T_{*}$ and $\mathbf{a}_{*}$ as in (35), we
have that

$W(T,\psi,\mathbf{a})=\\{a_{1}\\}\times W^{\prime}(T_{*},\psi,\mathbf{a}_{*})\
\bigcup\ \\{1-a_{1}\\}\times W^{\prime\prime}(T_{*},\psi,\mathbf{a}_{*}),$

where

$W^{\prime}(T_{*},\psi,\mathbf{a}_{*}):=\\{\mathbf{x}\in\mathbb{T}^{d-1}:T_{*}^{n}(\mathbf{x})\in
B(\mathbf{a}_{*},\psi(n))\ \ \text{for infinitely many even}\
n\in\mathbb{N}\\},$
$W^{\prime\prime}(T_{*},\psi,\mathbf{a}_{*}):=\\{\mathbf{x}\in\mathbb{T}^{d-1}:T_{*}^{n}(\mathbf{x})\in
B(\mathbf{a}_{*},\psi(n))\ \ \text{for infinitely many odd}\
n\in\mathbb{N}\\}.$

Now observe that $W^{\prime}(T_{*},\psi,\mathbf{a}_{*})$ and
$W^{\prime\prime}(T_{*},\psi,\mathbf{a}_{*})$ are shrinking target sets with
respect to the transformation $T_{*}\circ T_{*}$ of the torus
$\mathbb{T}^{d-1}$. Indeed,

$W^{\prime}(T_{*},\psi,\mathbf{a}_{*})=\\{\mathbf{x}\in\mathbb{T}^{d-1}:(T_{*}\circ
T_{*})^{n}(x)\in B(\mathbf{a}_{*},\psi(2n))\ \ \text{for infinitely many}\
n\in\mathbb{N}\\},$

and

$W^{\prime\prime}(T_{*},\psi,\mathbf{a}_{*})=\\{\mathbf{x}\in\mathbb{T}^{d-1}:(T_{*}\circ
T_{*})^{n}(x)\in T_{*}^{-1}B(\mathbf{a}_{*},\psi(2n+1))\ \ \text{for
infinitely many}\ n\in\mathbb{N}\\}.$

Now in essence, the above procedure removes the presence of the problematic
eigenvalue $\beta_{1}=-1$ but still none of the results we have established
for matrix transformations are applicable to the above shrinking targets sets.
The reason for this is simple. The composition map $T_{*}\circ T_{*}$ is not a
matrix transformation of the torus $\mathbb{T}^{d-1}$. The upshot is that we
need to appeal to Theorem 1 and its corollary directly. In view of the
argument set out in §3.3.1 that provides a self-contained proof of Theorem 4
for rectangular target sets, it is easily seen that the desired counting and
measure statements for $W^{\prime}(T_{*},\psi,\mathbf{a}_{*})$ and
$W^{\prime\prime}(T_{*},\psi,\mathbf{a}_{*})$ would follow on showing that the
product measure $\nu$ is exponentially mixing with respect to $(T_{*}\circ
T_{*},\mathcal{R})$. To establish the latter, we first recall (see the start
of the proof of Lemma 6) that for any $\beta\in\mathbb{R}$ with $|\beta|>1$,
$\mu_{\beta}$ is exponentially mixing with respect to
$(T_{\beta},\mathcal{C})$ where $\mathcal{C}$ is any collection of intervals
of $K(\beta)$. Hence by definition, there exists a constant $0<\gamma<1$ such
that

$\mu_{\beta}\big{(}E\cap
T^{-2n}F\big{)}=\mu_{\beta}(E)\mu_{\beta}(F)+O(\gamma^{2n})\,\mu_{\beta}(F)$

for any $E,F\in\mathcal{C}$. In other words,

$\mu_{\beta}\big{(}E\cap(T\circ
T)^{-n}F\big{)}=\mu_{\beta}(E)\mu_{\beta}(F)+O\big{(}(\gamma^{2})^{n}\big{)}\,\mu_{\beta}(F).$

and so $\mu_{\beta}$ is exponentially mixing with respect to $(T_{\beta}\circ
T_{\beta},\mathcal{C})$. Then, on mimicking the proof of Lemma 6 with $T$
replaced by $T_{*}\circ T_{*}$ and $d$ replaced by $d-1$, we conclude that
$\nu$ is exponentially mixing with respect to $(T_{*}\circ T_{*},\mathcal{R})$
for any collection $\mathcal{R}$ of rectangles of $K$ with sides parallel to
the axes.

####  3.3.3 A nifty “reduction” argument

In this section we show that when $T$ is an integer matrix transformation, the
diagonal assumption in Theorem 4 can be relaxed to $T$ is diagonalizable over
${\mathbb{Z}}$. So, suppose $T$ is diagonalizable over $\mathbb{Z}$. Then by
definition, there exist a nonsingular integer matrix $P$ and a diagonal
integer matrix $D$ such that product matrix relationship $P\cdot T=D\cdot P$
holds. In turn, there exists an invertible mapping

$\phi:\mathbb{T}^{d}\to\mathbb{T}^{d}\quad\hbox{\ such that \ }\quad\phi\circ
T=D\circ\phi\,.$ (37)

Obviously, the diagonal entries of $D$ are the eigenvalues of $T$ and we
assume these integers are of modulus strictly larger than $1$.

Now recall that for diagonal transformations such as $D$, the “Proof of
Theorem 4 using Proposition 1” makes key use of the fact that the product
measure $\nu$ is $D$-invariant and is exponentially mixing with respect to
$(D,\mathcal{C})$ where $\mathcal{C}$ is any collection of subsets $E$ of $K$
satisfying the bounded property $({\boldsymbol{\rm B}})$. We claim that the
image measure $\nu\circ\phi$ is $T$-invariant and exponentially mixing with
respect to $(D,\mathcal{C})$. The proof of Theorem 4 can then be modified in
the obvious manner to deal with the more general (integer) situation in which
$T$ is diagonalizable over $\mathbb{Z}$. To establish the claim, first note
that for any measurable set $A\subset\mathbb{T}^{d}$, on using the fact that
$\nu$ is $D$-invariant it follows that

$\nu\circ\phi(T^{-1}A)=\nu(\phi(T^{-1}A))=\nu(D^{-1}(\phi(A)))=\nu(\phi(A))=\nu\circ\phi(A).$

Thus, $\nu\circ\phi$ is $T$-invariant. Next, since $\phi$ is linear the
(inverse) image of any collection $\mathcal{C}$ of subsets $E$ of $K$
satisfying the bounded property $({\boldsymbol{\rm B}})$ also satisfies the
bounded property $({\boldsymbol{\rm B}})$. Hence, for any $E,F\in\mathcal{C}$,
noting that $\phi(E\cap T^{-n}F)=\phi(E)\cap\phi(T^{-n}F)$, it follows that
there exists a constant $0<\gamma<1$ such that

$\displaystyle\nu\circ\phi(E\cap T^{-n}F)$ $\displaystyle=$
$\displaystyle\nu\big{(}\phi(E\cap
T^{-n}F)\big{)}=\nu\big{(}\phi(E)\cap\phi(T^{-n}F)\big{)}$ $\displaystyle=$
$\displaystyle\nu\big{(}\phi(E)\cap D^{-n}\circ\phi(F)\big{)}$
$\displaystyle=$
$\displaystyle\nu\big{(}\phi(E)\big{)}\,\nu\big{(}\phi(F)\big{)}\,+\,O\big{(}\gamma^{n}\big{)}\
\nu\big{(}\phi(F)\big{)}$ $\displaystyle=$ $\displaystyle\nu\circ\phi(E)\
\nu\circ\phi(F)\,+\,O(\gamma^{n})\ \nu\circ\phi(F)\,$

as desired. Note that the third displayed line uses the fact that $\nu$ is
exponentially mixing with respect to $(D,\mathcal{C})$.

######  Remark 8.

We remark that for real non-integer matrices, we cannot in general use the
above argument to extend Theorem 4 to the situation that $T$ is diagonalizable
over ${\mathbb{Z}}$. In short, the commutative property $\phi\circ
T=D\circ\phi$ may not be true since the product matrix relationship $P\cdot
T=D\cdot P$ does not guarantee that

$P(T\mathbf{x}\ {\rm mod}\ 1)\ {\rm mod}\ 1=T(P\mathbf{x}\ {\rm mod}\ 1)\ {\rm
mod}\ 1.$

##  4 Establishing dimension results for matrix transformations

We begin with a brief account in which we bring together various statements
concerning Hausdorff measure and dimension that we will utilise in the course
of establishing Theorems 6 and 7.

###  4.1 Preliminaries

We start by defining Hausdorff measure and dimension for completeness and for
establishing some notation. Let $X$ be a subset of ${\mathbb{R}}^{d}$. For
$\rho>0$, a countable collection $\left\\{B_{i}\right\\}$ of Euclidean balls
in ${\mathbb{R}}^{d}$ of diameter $d_{i}\leq\rho$ for each $i$ such that
$X\subset\bigcup_{i}B_{i}$ is called a $\rho$-cover for $X$. Let $s$ be a non-
negative number and define

${\cal H}^{s}_{\rho}(X)\;=\;\inf\left\\{\sum_{i}d_{i}^{s}\ :\\{B_{i}\\}{\rm\
is\ a\ }\rho{\rm-cover\ of\ }X\right\\}\;,$

where the infimum is taken over all possible $\rho$-covers of $X$. The
s-dimensional Hausdorff measure ${\cal H}^{s}(X)$ of $X$ is defined by

${\cal H}^{s}(X)=\lim_{\rho\rightarrow 0}{\cal
H}^{s}_{\rho}(X)=\sup_{\rho>0}{\cal H}^{s}_{\rho}(X)$

and the Hausdorff dimension dim $X$ of $X$ by

$\dim_{\rm H}\,X=\inf\left\\{s:{\cal H}^{s}(X)=0\right\\}=\sup\left\\{s:{\cal
H}^{s}(X)=\infty\right\\}\,.$

Further details and alternative definitions of Hausdorff measure and dimension
can be found in [16, 39]. It is easily verified (see [16, Corollary 2.4]) that
the Hausdorff dimension of a set is invariant under bi-Lipschitz maps.

######  Lemma 7.

Let $X$ be a subset of ${\mathbb{R}}^{d}$ and $f:X\to{\mathbb{R}}^{d}$ be a
bi-Lipschitz map; i.e.

$c_{1}\,|x-y|\leq|f(x)-f(y)|\leq c_{2}\,|x-y|\qquad(x,y\in X)$

where $0<c_{1}\leq c_{2}<\infty$, then $\dim_{\rm H}f(X)=\dim_{\rm H}X\,$.

We now describe a deep and powerful mechanism for obtaining lower bounds for
the Hausdorff dimension of a large class of “rectangular’ $\limsup$ sets.

####  4.1.1 Mass Transference Principle for Rectangles

The discussion below is tailored to the application we have in mind. It is far
from the most general and powerful setup of the Mass Transference Principle.
We begin by describing the original ‘balls to balls’ principle which is all
that is required for directly proving Theorem 6 and Theorem 7. However, we
will deduce Theorem 6 from a more general statement concerning rectangular
target sets and for this we will require the more versatile ‘rectangle to
rectangle’ principle.

To set the scene, let $X$ be a locally compact subset of ${\mathbb{R}}^{d}$
equipped with a non-atomic probability measure $\mu$. Suppose there exist
constants $\delta>0$, $0<a\leq 1\leq b<\infty$ and $r_{0}>0$ such that

$a\,r^{\delta}\ \leq\ \mu(B)\ \leq\ b\,r^{\delta}$ (38)

for any ball $B=B(x,r)$ with $x\in X$ and radius $r\leq r_{0}$. Such a measure
is said to be _$\delta$ -Ahlfors regular_. It is well known that if $X$
supports a $\delta$-Ahlfors regular measure $\mu$, then $\dim_{\rm H}X=\delta$
and moreover that $\mu$ is strongly equivalent to $\delta$-dimensional
Hausdorff measure ${\cal H}^{\delta}$ – see [16, 39] for the details. The
latter implies that (38) is valid with $\mu$ replaced by $\mathcal{H}^{s}$.
Next, given $s>0$ and a ball $B=B(x,r)$ we define the scaled ball

$B^{s}:=B\big{(}x,r^{\frac{s}{\delta}}\big{)}\,.$

and so by definition $B^{\delta}=B$. The following Mass Transference Principle
[3] allows us to transfer $\mathcal{H}^{\delta}$-measure theoretic statements
for $\limsup$ subsets of $X$ to general $\mathcal{H}^{s}$-measure theoretic
statements.

######  Theorem 10 (MTP: balls to balls).

Let $X$ be a locally compact subset of ${\mathbb{R}}^{d}$ equipped with a
$\delta$-Ahlfors regular measure $\mu$. Let $\\{B_{n}\\}_{n\in{\mathbb{N}}}$
be a sequence of balls in $X$ with radius $r(B_{n})\to 0$ as $n\to\infty$. Let
$s\geq 0$ and suppose that

$\mathcal{H}^{\delta}\big{(}\limsup_{n\to\infty}B_{n}^{s}\big{)}=\mathcal{H}^{\delta}(X).$

Then,

$\mathcal{H}^{s}\big{(}\limsup_{n\to\infty}B_{n}\big{)}=\mathcal{H}^{s}(X).$

Note that by the definition of Hausdorff dimension, Theorem 10 implies that

$\dim_{\rm H}\big{(}\limsup_{n\to\infty}B_{n}\big{)}\geq s\,,$ (39)

and moreover that $\mathcal{H}^{s}(\limsup_{n\to\infty}B_{n})=\infty$ if
$s<\delta$. We now describe a recent result due to Wang $\&$ Wu [58] that
gives a lower bound for the Hausdorff dimension of $\limsup$ sets defined via
rectangles rather than just balls. So with this in mind, fix an integer $p\geq
1$ and for $1\leq i\leq p$, let $X_{i}$ be a subset of ${\mathbb{R}}^{d_{i}}$.
Obviously, if $B_{i}$ is a ball in $X_{i}$ then $\prod_{i=1}^{p}B_{i}$ is in
general a rectangle in the product space $\prod_{i=1}^{p}X_{i}$. The following
statement for $\limsup$ sets arising from sequences of such rectangles is a
much simplified version of [58, Theorem 3.4]. As we shall see, it is more than
adequate for our purpose.

######  Theorem 11 (MTP: rectangles to rectangles).

For each $1\leq i\leq p$, let $X_{i}$ be a locally compact subset of
${\mathbb{R}}^{d_{i}}$ equipped with a $\delta_{i}$-Ahlfors regular measure
$\mu_{i}$. Let $\\{B_{i,n}\\}_{n\in{\mathbb{N}}}$ be a sequence of balls in
$X_{i}$ with radius $r(B_{i,n})\to 0$ as $n\to\infty$ for each $1\leq i\leq p$
and assume that there exist
${\mathbf{v}}=(v_{1},\dots,v_{p})\in(\mathbb{R}^{+})^{p}$ and a sequence
$\\{r_{n}\\}_{n\in{\mathbb{N}}}$ of positive real numbers such that

$\displaystyle r(B_{i,n})=r_{n}^{v_{i}}\quad\text{for all \ }1\leq i\leq p.$
(40)

Suppose that there exists
$(s_{1},\dots,s_{p})\in\prod_{i=1}^{p}(0,\delta_{i})$ such that

$\mu_{1}\times\cdots\times\mu_{p}\Big{(}\limsup_{n\to\infty}{\textstyle\prod_{i=1}^{p}}B_{i,n}^{s_{i}}\Big{)}=\mu_{1}\times\cdots\times\mu_{p}\Big{(}{\textstyle\prod_{i=1}^{p}}X_{i}\Big{)}.$
(41)

Then, we have that

$\dim_{\rm
H}\big{(}\limsup_{n\to\infty}{\textstyle\prod_{i=1}^{p}}B_{i,n}\big{)}\geq\min_{1\leq
i\leq p}s({\mathbf{u}},{\mathbf{v}},i),\,$

where ${\mathbf{u}}=(u_{1},\dots,u_{p})$ with $u_{i}=s_{i}v_{i}/\delta_{i}$
for $1\leq i\leq p$, and

$s({\mathbf{u}},{\mathbf{v}},i):=\sum_{k\in\mathcal{K}_{1}(i)}\delta_{k}+\sum_{k\in\mathcal{K}_{2}(i)}\delta_{k}\big{(}1-\frac{v_{k}-u_{k}}{v_{i}}\big{)}+\sum_{k\in\mathcal{K}_{3}(i)}\frac{u_{k}\delta_{k}}{v_{i}}\,,$

with the sets

$\mathcal{K}_{1}(i):=\\{1\leq k\leq p:u_{k}\geq
v_{i}\\},\qquad\mathcal{K}_{2}(i):=\\{1\leq k\leq p:v_{k}\leq v_{i}\\},$

and

$\mathcal{K}_{3}(i):=\\{1,\dots,p\\}\setminus\left(\mathcal{K}_{1}(i)\cup\mathcal{K}_{2}(i)\right)$

forming a partition of $\\{1,\dots,p\\}$.

Note that since the radius $r(B_{i,n})\to 0$ as $n\to\infty$ for each $1\leq
i\leq p$ we automatically have that $\lim_{n\to\infty}r_{n}=0$. Also note that
if (40) holds for some
${\mathbf{v}}=(v_{1},\dots,v_{p})\in(\mathbb{R}^{+})^{p}$, then it holds for
$c{\mathbf{v}}=(cv_{1},\dots,cv_{p})$ where $c>0$ is a constant. Thus, the
choice of $v$ and therefore $u$ is not unique. However, it is easily seen that
the “dimension number” $s({\mathbf{u}},{\mathbf{v}},i)$ is not effected; i.e.

$s({\mathbf{u}},{\mathbf{v}},i)=s(c{\mathbf{u}},c{\mathbf{v}},i)\,.$

For the sake of convenience, we refer to a collection of rectangles
$\\{R_{n}:={\textstyle\prod_{i=1}^{p}}B_{i,n}\\}_{n\in\mathbb{N}}$ with
sidelengths satisfying (40) as a collection of _rectangles with exponent
$\mathbf{v}$._ Thus, Theorem 11 can be regarded as the dimension analogue of
the original Mass Transference Principle for $\limsup$ sets arising from
rectangles with exponent $\mathbf{v}$.

######  Remark 9.

It is worth mentioning that assumption (40) can be easily weakened to the
following statement: for any $1\leq i,j\leq p$,

$\lim_{n\to\infty}{\log r(B_{i,n})\over\log r(B_{j,n})}\ \ \text{ \ \ exists
and is finite.}$

To see this, let $r_{n}=r(B_{1,n})$. Then, there exists
${\mathbf{v}}=(v_{1},\dots,v_{p})\in(\mathbb{R}^{+})^{p}$ and sequences
$\\{v_{i,n}\\}_{n\in\mathbb{N}}$ for each $1\leq i\leq p$, such that

$r(B_{i,n})=r_{n}^{v_{i,n}}\quad\text{with}\ \lim_{n\to\infty}v_{i,n}=v_{i}\
.$

Now, given $\epsilon>0$ consider the associated $\limsup$ set
$R^{*}_{\epsilon}$ obtained by replacing the balls $B_{i,n}$ by balls
$B^{*}_{i,n}$ with the same centre but radius
$r(B^{*}_{i,n})=r_{n}^{v_{i}+\epsilon}$ $(1\leq i\leq p)$. Then,

$R^{*}_{\epsilon}:=\limsup_{n\to\infty}{\textstyle\prod_{i=1}^{p}}B^{*}_{i,n}\subset\limsup_{n\to\infty}{\textstyle\prod_{i=1}^{p}}B_{i,n}$

and on applying Theorem 11 we obtain a lower bound for
$\dim_{\mathrm{H}}R^{*}_{\epsilon}$ which converges to the desired dimensional
number as $\epsilon\to 0$.

######  Remark 10.

In the above setup the rectangles arise as products of balls $B_{i}$ in
$X_{i}$. Balls of radius $\rho$ are of course $\rho$-neighbourhoods of special
points; namely their centres. The general form of the Mass Transference
Principle of Wang $\&$ Wu is based on the framework of ubiquitous systems.
This allows them to naturally consider the situation in which the balls
$B_{i}$ are replaced by $\rho$-neighbourhoods of special sets called resonant
sets. It is easily verified that the $\kappa$-scaling property for resonant
sets within the general dimension statement [58, Theorem 3.4] is satisfied
with $\kappa=0$ when the resonant sets are points. In turn, this together with
[58, Proposition 3.1] directly yields Theorem 11. It is also worth mentioning
that if we replace the full measure condition (41) by the stronger ‘local
ubiquity system for rectangles’ condition [58, Definition 3.2] and also assume
that the radii of the balls in the given sequence are non-increasing, then we
are able to conclude [58, Theorem 3.2] that

$\mathcal{H}^{s}\big{(}\limsup_{n\to\infty}{\textstyle\prod_{i=1}^{p}}B_{i,n}\big{)}=\mathcal{H}^{s}\big{(}{\textstyle\prod_{i=1}^{p}}X_{i}\big{)}\quad{\rm
with}\quad s=\min_{1\leq i\leq p}s({\mathbf{u}},{\mathbf{v}},i).$

In other words, we obtain a complete ‘rectangle to rectangle’ analogue of the
original Mass Transference Principle.

It is easily seen that in the case of balls, Theorem 11 coincides with the
dimension statement (39). Indeed, when $p=1$ we have that that
$\mathcal{K}_{1}(1)=\mathcal{K}_{3}(1)=\emptyset$ and
$\mathcal{K}_{2}(1)=\\{1\\}$. Hence, it follows that

$s({\mathbf{u}},{\mathbf{v}},1)=\delta_{1}(1-\frac{v_{1}-u_{1}}{v_{1}})=\frac{\delta_{1}u_{1}}{v_{1}}=s_{1}\,,$

and so Theorem 11 implies that

$\dim_{\rm H}\limsup_{n\to\infty}B_{1,n}\geq s_{1}$

as claimed.

###  4.2 Proof of Theorem 6 via a dimension theorem for rectangular targets

As mentioned in Remark 6, straight after the statement of Theorem 6, we deduce
the theorem from a more general statement concerning rectangular target sets.
This we now describe and prove. For $1\leq i\leq d$, let
$\psi_{i}:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ be a real positive function.
For convenience, let $\Psi:=(\psi_{1},\dots,\psi_{d})$ and for
$n\in{\mathbb{N}}$ let $\Psi(n):=(\psi_{1}(n),\dots,\psi_{d}(n))$. Fix some
point $\mathbf{a}:=(a_{1},\ldots,a_{d})\in{\mathbb{T}}^{d}$ and for
$n\in{\mathbb{N}}$, let

$R\big{(}\mathbf{a},\Psi(n)\big{)}:=\Big{\\{}\mathbf{x}\in\mathbb{T}^{d}:\|x_{i}-a_{i}\|\leq\psi_{i}(n)\
(1\leq i\leq d)\Big{\\}}.$

Clearly, $R\big{(}\mathbf{a},\Psi(n)\big{)}$ is a rectangle centred at the
fixed point $\mathbf{a}$. In turn, let

$W(T,\Psi,\mathbf{a}):=\big{\\{}\mathbf{x}\in\mathbb{T}^{d}:T^{n}(\mathbf{x})\in
R\big{(}\mathbf{a},\Psi(n)\big{)}\ \ \text{for infinitely many}\
n\in\mathbb{N}\big{\\}}\ .$

It is evident that the family of rectangular target sets
$\\{R\big{(}\mathbf{a},\Psi(n)\big{)}\\}_{n\geq 1}$ satisfy the bounded
property $({\boldsymbol{\rm B}})$ and indeed the stronger property
$({\boldsymbol{\rm P}_{\\!\mathbf{a}}})$ based on Gallagher’s property
$({\boldsymbol{\rm P}})$ as described in §5.1. Note that when
$\psi:=\psi_{1}=\dots=\psi_{d}$, the rectangles are squares and so

$W(T,\Psi,\mathbf{a})=W(T,\psi,\mathbf{a})\,.$

Also, it is easily verified that if

$T={\rm diag}\,(\beta_{1},\ldots,\beta_{d})\qquad(\beta_{i}\in{\mathbb{R}})\,$

then

$W(T,\Psi,\mathbf{a})=\big{\\{}\mathbf{x}\in\mathbb{T}^{d}:|T_{\beta_{i}}^{n}x_{i}-a_{i}|\leq\psi_{i}(n)\
(1\leq i\leq d)\ \ \text{for infinitely many}\ n\in\mathbb{N}\big{\\}}\,,$
(42)

where $T_{\beta_{i}}$ is the standard $\beta$-transformation with
$\beta=\beta_{i}$.

It turns out that the Hausdorff dimension of the shrinking target set
$W(T,\Psi,\mathbf{a})$ is dependent on the set $\mathcal{U}(\Psi)$ of
accumulation points $\mathbf{t}=(t_{1},t_{2},\ldots,t_{d})$ of the sequence
$\big{\\{}(-\frac{\log\psi_{1}(n)}{n},\cdots,-\frac{\log\psi_{d}(n)}{n})\big{\\}}_{n\geq
1}$.

######  Theorem 12.

Let $T$ be a real, non-singular matrix transformation of the torus
$\mathbb{T}^{d}$. Suppose that $T$ is diagonal with all eigenvalues
$\beta_{1},\beta_{2},\dots,\beta_{d}$ strictly larger than $1$. Assume that
$1<\beta_{1}\leq\beta_{2}\leq\cdots\leq\beta_{d}$. For $1\leq i\leq d$, let
$\psi_{i}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a real positive function and
$\mathbf{a}\in\mathbb{T}^{d}$. Assume that $\mathcal{U}(\Psi)$ is bounded.
Then

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})=\sup_{\mathbf{t}\in\mathcal{U}(\Psi)}\min_{1\leq i\leq
d}\big{\\{}\theta_{i}(\mathbf{t})\big{\\}},$

where

$\theta_{i}(\mathbf{t}):=\sum_{k\in\mathcal{K}_{1}(i)}1+\sum_{k\in\mathcal{K}_{2}(i)}\left(1-\frac{t_{k}}{\log\beta_{i}+t_{i}}\right)+\sum_{k\in\mathcal{K}_{3}(i)}\frac{\log\beta_{k}}{\log\beta_{i}+t_{i}}$

and, in turn

$\mathcal{K}_{1}(i):=\\{1\leq k\leq d:\log\beta_{k}>\log\beta_{i}+t_{i}\\},\
\mathcal{K}_{2}(i):=\\{1\leq k\leq
d:\log\beta_{k}+t_{k}\leq\log\beta_{i}+t_{i}\\},\ $

and

$\mathcal{K}_{3}(i):=\\{1,\dots,d\\}\setminus(\mathcal{K}_{1}(i)\cup\mathcal{K}_{2}(i)).$

######  Remark 11.

It is easily seen the value of $\theta_{i}(\mathbf{t})$ remains unchanged if
replace $>$ to $\geq$ in $\mathcal{K}_{1}(i)$, and/or replace $\leq$ by $<$ in
$\mathcal{K}_{2}(i)$.

We now deduce Theorem 6 from Theorem 12.

###### Proof of Theorem 6 modulo Theorem 12.

For the real positive function $\psi$ in Theorem 6, we first suppose that its
lower order at infinity is bounded; that is

$\lambda=\lambda(\psi):=\liminf_{n\to\infty}\frac{-\log\psi(n)}{n}<+\infty.$

With this in mind, put $\psi_{1}=\psi_{2}=\cdots=\psi_{d}=\psi$ in the
statement of Theorem 12 and note that any $\mathbf{t}$ in $\mathcal{U}(\Psi)$
is of the form $\mathbf{t}=(t,t,\dots,t)$ where $t\in\mathcal{U}(\psi)$ – the
set of accumulation points of the sequence
$\big{\\{}-\frac{\log\psi(n)}{n}\big{\\}}_{n\geq 1}$. We remark that
$\lambda<+\infty$ means that $\mathcal{U}(\Psi)$ is bounded. Hence, for any
$1\leq i\leq d$ we have that

$\theta_{i}(t):=\theta_{i}(\mathbf{t})=\sum_{k:\beta_{k}>\beta_{i}e^{t}}1+\sum_{k:\beta_{k}\leq\beta_{i}}\left(1-\frac{t}{\log\beta_{i}+t}\right)+\sum_{k:\beta_{i}e^{t}\geq\beta_{k}>\beta_{i}}\frac{\log\beta_{k}}{\log\beta_{i}+t}.$

Now let

$k_{1}:=\max\\{1\leq k\leq d:\beta_{k}\leq\beta_{i}\\},$

and

$k_{2}:=\max\\{1\leq k\leq d:\beta_{k}\leq\beta_{i}e^{t}\\}.$

Then, the above expression for $\theta_{i}(t)$ becomes

$\theta_{i}(t)=\sum_{k=k_{2}+1}^{d}1+\sum_{k=1}^{k_{1}}\frac{\log\beta_{i}}{\log\beta_{i}+t}+\sum_{k=k_{1}+1}^{k_{2}}\frac{\log\beta_{k}}{\log\beta_{i}+t}$
(43)

and noting that $\beta_{k}=\beta_{i}$ for $i+1\leq k\leq k_{1}$ whenever
$k_{1}>i$, it follows that

$\displaystyle\theta_{i}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{k=k_{2}+1}^{d}(\log\beta_{i}+t)+i\log\beta_{i}+\sum_{k=i+1}^{k_{2}}\log\beta_{k}}{\log\beta_{i}+t}$
$\displaystyle=$
$\displaystyle\frac{i\log\beta_{i}-\sum_{k=k_{2}+1}^{d}(\log\beta_{k}-\log\beta_{i}-t)+\sum_{k=i+1}^{d}\log\beta_{k}}{\log\beta_{i}+t}.$

The upshot of this together with Theorem 12 is that

$\dim_{\rm H}W(T,\psi,\mathbf{a})=\sup_{t\in\mathcal{U}(\psi)}\min_{1\leq
i\leq d}\theta_{i}(t).$

Now observe that since $\theta_{i}(t)$ is a decreasing function in $t$, the
above right hand side is equal to $\min_{1\leq i\leq d}\theta_{i}(\lambda)$
where $\lambda=\lambda(\psi)$ is the lower order at infinity of the function
$\psi$. Thus, under the assumption that $\lambda$ is bounded, we have that

$\dim_{\rm H}W(T,\psi,\mathbf{a})=\min_{1\leq i\leq d}\theta_{i}(\lambda)$

as desired. To deal with the case that $\lambda=\lambda(\psi)=+\infty$, given
any real number $M>0$ consider the function
$\psi_{M}:\mathbb{R}^{+}\to\mathbb{R}^{+}:x\to e^{-xM}$. Then, by definition
$\mathcal{U}(\psi_{M})=\\{M\\}$ and for $M$ sufficiently large

$W(T,\psi,\mathbf{a})\subset W(T,{\psi_{M}},\mathbf{a})\,$

and so it follows that

$0\leq\dim_{\rm H}W(T,\psi,\mathbf{a})\leq\dim_{\rm
H}W(T,{\psi_{M}},\mathbf{a})\leq\min_{1\leq i\leq d}\theta_{i}(M).$ (44)

Now with reference to (43), we have that $t=M$ and so for $M$ sufficiently
large: $k_{2}=d$ for $1\leq i\leq d$. Hence, for any $1\leq i\leq d$

$\lim_{M\to\infty}\theta_{i}(M)=\lim_{M\to\infty}\left(\sum_{k=1}^{i}\frac{\log\beta_{i}}{\log\beta_{i}+M}+\sum_{k=i+1}^{d}\frac{\log\beta_{k}}{\log\beta_{i}+M}\right)=0.$

This together with (44) implies that

$\dim_{\rm H}W(T,\psi,\mathbf{a})=0.$

∎

####  4.2.1 Proof of Theorem 12

We start with a brief discussion that sums up various fundamental notions and
statements that we will require during the course of establishing Theorem 12.
The statements are concerned with the distribution of the preimages of a fixed
ball under a given $\beta$-transformation $T_{\beta}$. As usual, let
$\beta\in\mathbb{R}$ such that $|\beta|>1$ and let

$\mathcal{Q}=\left\\{\Big{[}0,{1\over|\beta|}\Big{)},\dots,\Big{[}{k\over|\beta|},{k+1\over|\beta|}\Big{)},\dots,\Big{[}{\lfloor|\beta|\rfloor\over|\beta|},1\Big{)}\right\\}$

be the natural partition of $[0,1)$. The $n$-th refinement of $\mathcal{Q}$ is
defined as

$\mathcal{Q}^{n}:=\left\\{Q_{i_{0}}\cap
T_{\beta}^{-1}(Q_{i_{1}})\cap\cdots\cap T_{\beta}^{-(n-1)}(Q_{i_{n-1}}):\quad
Q_{i_{j}}\in\mathcal{Q}\ \text{for }0\leq j\leq n-1\right\\}.$

The elements in $\mathcal{Q}^{n}$ are called _cylinders of order $n$_.
Evidently, the cylinders are disjoint and the restriction of $T_{\beta}^{n}$
on each cylinder is continuous linear of slop $\beta^{n}$. Now given a point
$a\in{\mathbb{T}}$, consider the preimage of the ball $B(a,r)$ under
$T_{\beta}^{n}$. It can be verified that this preimage consists of disjoint
intervals whose lengths are bounded above by $2r|\beta|^{-n}$. Indeed, we can
write

$T_{\beta}^{-n}\big{(}B(a,r)\big{)}=\bigcup_{j=1}^{N_{n}}I_{n,j},$ (45)

where each $I_{n,j}$ is an interval lying in some cylinder of order $n$ and
$N_{n}$ is the number of such intervals.

For $\beta>1$, via the work of Rényi [44], it follows that the (total) number
of cylinders of order $n$ of $T_{\beta}$ is bounded from above by
$\beta^{n+1}/(\beta-1)$. Hence,

$N_{n}\leq\beta^{n+1}/(\beta-1).$ (46)

######  Remark 12.

It is worth mentioning that on exploiting the well-known fact that the
topological entropy of $T_{\beta}$ is $\log|\beta|$ for any $\beta$ with
$|\beta|>1$, we obtain the weaker bound

$N_{n}\leq|\beta|^{n(1+\epsilon)}$ (47)

for any $\epsilon>0$ and $n$ sufficiently large. This suffices for not only
proving Lemma 8 below but more importantly for establishing the upper bound
for the dimension within the context of Theorems 7 and 9 in which $\beta$ is
allowed to be negative.

Now, suppose $\beta>1$. Recall that a cylinder $I$ of order $n$ is said to be
_full_ for $T_{\beta}$ if $T_{\beta}^{n}(I)={\mathbb{T}}$. With this in mind,
Bugeaud $\&$ Wang [9, Theorem 1.2] proved that every $(n+1)$ consecutive
cylinders of order $n$ contains at least one full cylinder. This gives rise to
the following useful fact that we will make use of on multiple occasions.

Fact BW:  _The distance between any two consecutive full cylinders is less
than $(n+1)\beta^{-n}$. Furthermore, since any full cylinder intersects
$T_{\beta}^{-n}\big{(}B(a,r)\big{)}$ it follows that the distance between any
two consecutive intervals $I_{n,j}$ and $I_{n,j+1}$ is less than
$(n+3)\beta^{-n}$; i.e.,_

${\rm dist}(I_{n,j},I_{n,j+1})\leq(n+3)\beta^{-n}.$

We now move onto the task of proving Theorem 12. This will be done by
establishing the upper and lower bounds for $\dim W(T,\Psi,\mathbf{a})$
separately.

######  Proposition 4.

Under the setting of Theorem 12, we have that

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})\leq\sup_{\mathbf{t}\in\mathcal{U}(\Psi)}\min_{1\leq
i\leq d}\\{\theta_{i}(\mathbf{t})\\}.$

We first establish the proposition in the special case that
$\mathcal{U}(\Psi)$ consists of a single point.

######  Lemma 8.

Under the setting of Theorem 12, assume in addition that there exists
$\mathbf{t}=(t_{1},\dots,t_{d})\in(\mathbb{R}^{+})^{d}$ such that

$\lim_{n\to\infty}\frac{-\log\psi_{i}(n)}{n}=t_{i}\quad{\rm for\ all}\ \ 1\leq
i\leq d.$

Then

$\dim_{\rm H}W(T,\Psi,\mathbf{a})\leq\min_{1\leq i\leq
d}\\{\theta_{i}(\mathbf{t})\\}.$

###### Proof.

Observe that we can re-write (42) as

$W(T,\Psi,\mathbf{a})=\limsup_{n\to\infty}\
T_{\beta_{1}}^{-n}\big{(}B(a_{1},\psi_{1}(n))\big{)}\times\cdots\times
T_{\beta_{d}}^{-n}\big{(}B(a_{d},\psi_{d}(n))\big{)},$ (48)

where $T_{\beta_{i}}$ is the standard $\beta$-transformation with
$\beta=\beta_{i}$. As usual, we do not distinguish between
$\beta$-transformations acting on the unit interval $[0,1)$ or the torus
${\mathbb{T}}$. The proof of Lemma 8 relies on finding an “efficient” covering
by balls of the $\limsup$ set (48). So with this in mind, for any $1\leq i\leq
d$, by (45) we have that

$T_{\beta_{i}}^{-n}\big{(}B(a_{i},\psi_{i}(n))\big{)}=\bigcup_{j=1}^{N_{i,n}}I_{n,j}^{(i)}\,,$
(49)

where each $I_{n,j}^{(i)}$ is an interval lying in some cylinder of order $n$
and $N_{i,n}$ is the number of such intervals. Now for $n\in{\mathbb{N}}$, let

$J_{n}:=\big{\\{}\mathbf{j}=(j_{1},\ldots,j_{d}):1\leq j_{i}\leq N_{i,n}\
(1\leq i\leq d)\big{\\}}$

and for $\mathbf{j}\in J_{n}$, let

$R_{n,\mathbf{j}}:=I_{n,j_{1}}^{(1)}\times\cdots\times I_{n,j_{d}}^{(d)}\,.$

In turn, let

$R_{n}=\bigcup_{\mathbf{j}\in J_{n}}R_{n,\mathbf{j}}\qquad{\rm and\
}\qquad\mathcal{R}_{n}:=\big{\\{}R_{n,\mathbf{j}}\,:\,\mathbf{j}\in
J_{n}\big{\\}}\,.$

Then in view of (49), we can re-write (48) as

$W(T,\Psi,\mathbf{a})=\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}R_{n}\,$

and it follows that for any $N\geq 1$,

$W(T,\Psi,\mathbf{a})\,\subset\,\bigcup_{n=N}^{\infty}R_{n}\,.$

In other words, the collection $\\{\mathcal{R}_{n}:n=N,N+1,\ldots\\}$ of
rectangles $R_{n,\mathbf{j}}$ form a cover for the set $W(T,\Psi,\mathbf{a})$.
Now, observe that along the direction of the $i$-th axis $(1\leq i\leq d)$, by
construction for each $1\leq j<N_{i,n}$ the sides $I_{n,j}^{(i)}$ and
$I_{n,j+1}^{(i)}$ are disjoint and thus the rectangles in $\mathcal{R}_{n}$
are disjoint. On the other hand, by Fact BW

$\text{dist}(I_{n,j}^{(i)},I_{n,j+1}^{(i)})\leq(n+3)\beta_{i}^{-n}$

and so along the direction of the $i$-th axis, the distance between
consecutive rectangles in $\mathcal{R}_{n}$ is at most $(n+3)\beta_{i}^{-n}$.

We now estimate the number of balls $B_{i,n}$ of diameter
$2\psi_{i}(n)\beta_{i}^{-n}$ (the sidelength of the rectangles in
$\mathcal{R}_{n}$ along the direction of the $i$-th axis) needed to cover the
set $R_{n}$. We start by covering a fixed generic rectangle
$R=R_{n,\mathbf{j}}\in\mathcal{R}_{n}$. It is easily verified that we can find
a collection $\mathcal{B}_{i,n}(R)$ of balls $B_{i,n}$ that covers $R$ with

$\\#\mathcal{B}_{i,n}(R)\ \leq\ 2^{d}\prod_{1\leq k\leq d\
:\atop\psi_{k}(n)\beta_{k}^{-n}\geq\psi_{i}(n)\beta_{i}^{-n}}\frac{\psi_{k}(n)\beta_{k}^{-n}}{\psi_{i}(n)\beta_{i}^{-n}}\,.$
(50)

Indeed, we can simply take the natural cover in which we split $R$ into closed
balls $B_{i,n}$ which are disjoint apart from at the boundary. Now observe
that the collection $\mathcal{B}_{i,n}(R)$ will also cover other rectangles in
$\mathcal{R}_{n}$ along the direction of the $k$-th axis ($1\leq k\leq d$) if
the separation in that direction is small compared to the diameter of the
balls $B_{i,n}$; that is, in view of Fact BW if

$\ \ (n+3)\beta_{k}^{-n}<2\psi_{i}(n)\beta_{i}^{-n}\,.$

In particular, this leads to the following lower bound for the number
$M_{i,n}(R)$ of rectangles covered by $\mathcal{B}_{i,n}(R)$ :

$\displaystyle M_{i,n}{(R)}$ $\displaystyle:=$
$\displaystyle\\#\Big{\\{}R_{n,\mathbf{j}}\in\mathcal{R}_{n}\,:\,R_{n,\mathbf{j}}\subseteq\bigcup_{B_{i,n}\in\mathcal{B}_{n,i}(R)}\\!\\!B_{i,n}\Big{\\}}$
(51) $\displaystyle\geq$ $\displaystyle\\!\\!\\!\\!\prod_{1\leq k\leq d\
:\atop(n+3)\beta_{k}^{-n}<\psi_{i}(n)\beta_{i}^{-n}}\\!\\!\frac{2\psi_{i}(n)\beta_{i}^{-n}}{(n+3)\beta_{k}^{-n}}\,.$

The upshot is that there is a collection $\mathcal{B}_{n,i}$ of balls of
$B_{i,n}$ that cover the set $R_{n}$ with

$\\#\mathcal{B}_{n,i}\ \leq\ \frac{\\#\mathcal{R}_{n}}{M_{i,n}{(R)}}\ \
\\#\mathcal{B}_{n,i}(R)\ \ =\ \ \prod_{j=1}^{d}N_{j,n}\ \cdot\
\frac{\\#\mathcal{B}_{n,i}(R)}{M_{i,n}{(R)}}\,.$

This together with (46), (50) and (51) implies that

$\displaystyle\\#\mathcal{B}_{n,i}\ $ $\displaystyle\leq$
$\displaystyle{\prod_{j=1}^{d}\frac{\beta_{j}^{n+1}}{\beta_{j}-1}}\
\cdot\prod_{1\leq k\leq d\
:\atop(n+3)\beta_{k}^{-n}<\psi_{i}(n)\beta_{i}^{-n}}\\!\\!\\!\\!\frac{(n+3)\beta_{k}^{-n}}{2\psi_{i}(n)\beta_{i}^{-n}}\
\ \cdot\ \ 2^{d}\\!\\!\\!\\!\prod_{1\leq k\leq d\
:\atop\psi_{k}(n)\beta_{k}^{-n}\geq\psi_{i}(n)\beta_{i}^{-n}}\\!\\!\\!\\!\frac{\psi_{k}(n)\beta_{k}^{-n}}{\psi_{i}(n)\beta_{i}^{-n}}$
$\displaystyle=$ $\displaystyle
2^{d}\cdot{\prod_{j=1}^{d}\frac{\beta_{j}^{n+1}}{\beta_{j}-1}}\ \
\prod_{k\in\mathcal{K}_{n,1}(i)}\frac{(n+3)\beta_{k}^{-n}}{2\psi_{i}(n)\beta_{i}^{-n}}\
\
\prod_{k\in\mathcal{K}_{n,2}(i)}\frac{\psi_{k}(n)\beta_{k}^{-n}}{\psi_{i}(n)\beta_{i}^{-n}},$

where

$\displaystyle\mathcal{K}_{n,1}(i)$ $\displaystyle:=$
$\displaystyle\left\\{1\leq k\leq
d:(n+3)\beta_{k}^{-n}<\psi_{i}(n)\beta_{i}^{-n}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{1\leq k\leq d:-{\log(n+3)\over
n}+\log\beta_{k}>-{\log\psi_{i}(n)\over n}+\log\beta_{i}\right\\},$

and

$\displaystyle\mathcal{K}_{n,2}(i)$ $\displaystyle:=$
$\displaystyle\left\\{1\leq k\leq
d:\psi_{k}(n)\beta_{k}^{-n}\geq\psi_{i}(n)\beta_{i}^{-n}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{1\leq k\leq d:-{\log\psi_{k}(n)\over
n}+\log\beta_{k}\leq-{\log\psi_{i}(n)\over n}+\log\beta_{i}\right\\}.$

Thus, given $\rho>0$ and on choosing $N$ sufficiently large so that
$2\psi_{i}(n)\beta_{i}^{-n}<\rho$ for any $n\geq N$, it follows from the
definition of $s$-dimensional Hausdorff measure that for any $s>0$

$\displaystyle\mathcal{H}_{\rho}^{s}(W(T,\Psi,\mathbf{a}))\\!\\!$
$\displaystyle\leq$ $\displaystyle\\!\\!\sum_{n=N}^{\infty}\
\\#\mathcal{B}_{n,i}\ \left(2\psi_{i}(n)\beta_{i}^{-n}\right)^{s}$ (52)
$\displaystyle\leq$
$\displaystyle\sum_{n=N}^{\infty}2^{d}\cdot{\prod_{j=1}^{d}\frac{\beta_{j}^{n+1}}{\beta_{j}-1}}\prod_{k\in\mathcal{K}_{n,1}(i)}\frac{(n+3)\beta_{k}^{-n}}{2\psi_{i}(n)\beta_{i}^{-n}}\prod_{k\in\mathcal{K}_{n,2}(i)}\frac{\psi_{k}(n)\beta_{k}^{-n}}{\psi_{i}(n)\beta_{i}^{-n}}\cdot\left(2\psi_{i}(n)\beta_{i}^{-n}\right)^{s}$
$\displaystyle=$ $\displaystyle
C\sum_{n=N}^{\infty}\exp\left\\{-n\cdot\ell_{n}\right\\},$

where $C:=2^{s+d}\prod_{j=1}^{d}\frac{\beta_{j}}{\beta_{j}-1}\ $ is a constant
and

$\displaystyle\ell_{n}=\ell_{n}(i)\ :=\ -\sum_{j=1}^{d}\log\beta_{j}\ \
\\!\\!\ $
$\displaystyle-\sum\limits_{k\in\mathcal{K}_{n,1}(i)}\left({\log(n+3)\over
n}-\log\beta_{k}-{\log\psi_{i}(n)\over n}+\log\beta_{i}\right)$
$\displaystyle-\sum\limits_{k\in\mathcal{K}_{n,2}(i)}\left(\frac{\log\psi_{k}(n)}{n}-\log\beta_{k}-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}\right)$
$\displaystyle+\ \ s\left(-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}\right).$

Now note that $\sum_{n=1}^{\infty}\exp\left\\{-n\cdot\ell_{n}\right\\}$
converges as long as

$\limsup_{n\to\infty}\ell_{n}>0\,,$

and that this is equivalent to the condition that $s$ is strictly larger than
the upper limit of

$\displaystyle h_{n}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!h_{n}(i)\ :=\
\frac{\sum\limits_{j=1}^{d}\log\beta_{j}+\sum\limits_{k\in\mathcal{K}_{n,1}(i)}\left({\log(n+3)\over
n}-\log\beta_{k}-{\log\psi_{i}(n)\over
n}+\log\beta_{i}\right)}{-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \qquad+\ \ \
\frac{\sum\limits_{k\in\mathcal{K}_{n,2}(i)}\left(\frac{\log\psi_{k}(n)}{n}-\log\beta_{k}-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}\right)}{-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}}\
$ $\displaystyle=$
$\displaystyle\\!\\!\sum_{k\in\mathcal{K}_{n,1}(i)}1+\sum_{k\in\mathcal{K}_{n,2}(i)}\left(1-{-\frac{\log\psi_{k}(n)}{n}\over-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}}\right)+\sum_{k\in\mathcal{K}_{n,3}(i)}{\log\beta_{k}\over-\frac{\log\psi_{i}(n)}{n}+\log\beta_{i}},$

where

$\displaystyle\mathcal{K}_{n,3}(i)\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\{1,\dots,d\\}\setminus(\mathcal{K}_{n,1}(i)\cup\mathcal{K}_{n,2}(i))$
$\displaystyle=$ $\displaystyle\\!\\!\\!\left\\{1\leq k\leq d:-{\log(n+3)\over
n}+\log\beta_{k}\leq-{\log\psi_{i}(n)\over
n}+\log\beta_{i}<-{\log\psi_{k}(n)\over n}+\log\beta_{k}\right\\}\,.$

So, by the additional assumption imposed in the lemma, it follows that

$\limsup_{n\to\infty}h_{n}=\lim_{n\to\infty}h_{n}=\sum_{k\in\mathcal{K}_{1}(i)}1+\sum_{k\in\mathcal{K}_{2}(i)}\left(1-\frac{t_{k}}{\log\beta_{i}+t_{i}}\right)+\sum_{k\in\mathcal{K}_{3}(i)}\frac{\log\beta_{k}}{\log\beta_{i}+t_{i}}=\theta_{i}(\mathbf{t}).$

The upshot of the above is that for any $1\leq i\leq d$ and
$s>\theta_{i}(\mathbf{t})$, we have that

$\sum_{n=1}^{\infty}\exp\left\\{-n\cdot\ell_{n}\right\\}<\infty$

and hence together with (52) we obtain that

$0\leq\mathcal{H}^{s}(W(T,\Psi,\mathbf{a}))=\lim\limits_{\rho\to
0}\mathcal{H}_{\rho}^{s}(W(T,\Psi,\mathbf{a}))\leq\lim\limits_{N\to\infty}C\sum_{n=N}^{\infty}\exp\left\\{-n\cdot\ell_{n}\right\\}=0\,.$

In turn, it follows from the definition of Hausdorff dimension that $\dim_{\rm
H}W(T,\Psi,\mathbf{a})\leq\theta_{i}(\mathbf{t})\,.$ This upper bound estimate
is true for any $1\leq i\leq d$, and so it implies that

$\dim_{\rm H}W(T,\Psi,\mathbf{a})\leq\min_{1\leq i\leq
d}\theta_{i}(\mathbf{t})$

as desired. ∎

Armed with Lemma 8, it is relatively straightforward to prove the general
upper bound statement for the Hausdorff dimension of $W(T,\Psi,\mathbf{a})$.

###### Proof of Proposition 4.

To prove the proposition, we first cover the accumulation set
$\mathcal{U}(\Psi)$. For any $\varepsilon>0$, since $\mathcal{U}(\Psi)$ is
bounded, we can find a family $\mathcal{B}_{\epsilon}$ of finitely many balls
of the form

$B=\prod_{i=1}^{d}[b^{(i)}_{B},b^{(i)}_{B}+\varepsilon]\qquad(b^{(i)}_{B}\geq
0)$

that cover $\mathcal{U}(\Psi)$. For $B\in\mathcal{B}_{\epsilon}$, let

$\mathcal{N}(B)=\left\\{n\in\mathbb{N}:\left(\frac{-\log\psi_{1}(n)}{n},\cdots,\frac{-\log\psi_{d}(n)}{n}\right)\
\in\ \prod_{i=1}^{d}[b^{(i)}_{B},b^{(i)}_{B}+\varepsilon]\right\\}.$

Without loss of generality, we assume that $\\#\mathcal{N}(B)=\infty$ since,
otherwise, there is no accumulation point in the ball $B$. We claim that
$W(T,\Psi,\mathbf{a})$ is a subset of

$\bigcup_{B\in\mathcal{B}}\Big{\\{}\mathbf{x}\in\mathbb{T}^{d}:\|T_{\beta_{i}}^{n}x_{i}-a_{i}\|\leq
e^{-nb^{(i)}_{B}}\ \ (1\leq i\leq d)\ \ \text{for infinitely many}\
n\in\mathcal{N}(B)\Big{\\}}.$

Indeed, for any $\mathbf{x}\in W(T,\Psi,\mathbf{a})$, there exists a sequence
$\\{n_{j}\\}_{j\in{\mathbb{N}}}$ depending on $\mathbf{x}$ such that for any
$1\leq i\leq d$

${\|}T_{\beta_{i}}^{n_{j}}x_{i}-a_{i}{\|}\leq\psi_{i}(n_{j})\quad\ \forall\ \
j\geq 1.$

Since there are only finitely many balls $B\in\mathcal{B}_{\epsilon}$ which
cover $\mathcal{U}(\Psi)$, there exists some $B\in\mathcal{B}_{\epsilon}$ that
contains infinitely many points of

$\Big{\\{}\big{(}\frac{-\log\psi_{1}(n_{j})}{n_{j}},\cdots,\frac{-\log\psi_{d}(n_{j})}{n_{j}}\big{)}\Big{\\}}_{j\in{\mathbb{N}}}\,.$

Thus, for these infinitely many $j$’s, we have that for any $1\leq i\leq d$

${\|}T_{\beta_{i}}^{n_{j}}x_{i}-a_{i}{\|}\,\leq\,\psi_{i}(n_{j})\ \leq\
e^{-n_{j}b^{(i)}_{B}}\,.$

This establishes the claim and by the countable stability property of
Hausdorff dimension, it follows that $\dim_{\rm H}W(T,\Psi,\mathbf{a})$ is
less than or equal to

$\max_{B\in\mathcal{B}_{\epsilon}}\ \dim_{\rm
H}\Big{\\{}\mathbf{x}\in\mathbb{T}^{d}:{\|}T_{\beta_{i}}^{n}x_{i}-a_{i}{\|}\leq
e^{-nb^{(i)}_{B}}\ (1\leq i\leq d)\ \ \text{for infinitely many}\
n\in\mathbb{N}\Big{\\}}.$

Now observe that

$\lim_{n\to\infty}\frac{-\log e^{-nb_{B}^{(i)}}}{n}\quad{\rm for\ all}\ \
1\leq i\leq d\,,$

and so on applying Lemma 8 we deduce that

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})\leq\max_{B\in\mathcal{B}_{\epsilon}}\min_{1\leq i\leq
d}\theta_{i}\big{(}(b^{(1)}_{B},\dots,b^{(d)}_{B})\big{)}.$

Then on letting $\varepsilon\to 0$, by the continuity of
$\theta_{i}(\mathbf{t})$ with respect to $\mathbf{t}$, we conclude that

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})\leq\sup_{\mathbf{t}\in\mathcal{U}(\Psi)}\min_{1\leq
i\leq d}\\{\theta_{i}(\mathbf{t})\\}.$

This completes the proof of Proposition 4.

∎

We now turn out attention to establishing the lower bound for the Hausdorff
dimension of $W(T,\Psi,\mathbf{a})$.

######  Proposition 5.

Under the setting of Theorem 12, we have that

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})\geq\sup_{\mathbf{t}\in\mathcal{U}(\Psi)}\min_{1\leq
i\leq d}\\{\theta_{i}(\mathbf{t})\\}.$

###### Proof.

The proof of Proposition 5 relies on constructing a suitable $\limsup$ type
subset of $W(T,\Psi,\mathbf{a})$ which enables us to exploit the ‘rectangles
to rectangles’ Mass Transference Principle (Theorem 11). With this in mind, by
(48) and (49), we know that $W(T,\Psi,\mathbf{a})$ is a limsup set of
rectangles with sides given by the intervals $I_{n,j}^{(i)}$ ($1\leq i\leq
d$). Recall, that the sides correspond to the intersection of
$T_{\beta_{i}}^{-n}\big{(}B(a_{i},\psi_{i}(n))\big{)}$ and a cylinder of order
$n$ and as in the proof of Proposition 4, we do not distinguish between
$\beta$-transformations acting on the unit interval or the torus. Thus, if for
each $1\leq i\leq d$, we select only those intervals which are intersections
of $T_{\beta_{i}}^{-n}\big{(}B(a_{i},\psi_{i}(n))\big{)}$ and full cylinders
of order $n$, we will obtain a $\limsup$ type subset of
$W(T,\Psi,\mathbf{a})$; that is

$W(T,\Psi,\mathbf{a})\supset\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\bigcup_{j_{1}=1}^{M_{1,n}}\cdots\bigcup_{j_{d}=1}^{M_{d,n}}B(x_{n,j_{1}}^{{(1)}},\beta_{1}^{-n}\psi_{1}(n))\times\cdots\times
B(x_{n,j_{d}}^{(d)},\beta_{d}^{-n}\psi_{d}(n))\,,$ (53)

where $\\{x_{n,j_{i}}^{(i)},1\leq j_{i}\leq M_{i,n}\\}$ are the preimages of
$a_{i}$ under $T_{\beta_{i}}^{n}$ that fall within full cylinders of order $n$
for $T_{\beta_{i}}$ and $M_{i,n}$ is the number of such full cylinders. Now
with (53) and Fact BW in mind, it follows that for each $1\leq i\leq d$ the
enlarged collection of balls or rather intervals
$\big{\\{}B(x_{n,j_{i}}^{(i)},(n+3)\beta_{i}^{-n}):1\leq j_{i}\leq
M_{i,n}\big{\\}}$ covers ${\mathbb{T}}$, that is

${\mathbb{T}}\ =\
\bigcup_{j_{i}=1}^{M_{i,n}}B\big{(}x_{n,j_{i}}^{(i)},(n+3)\beta_{i}^{-n}\big{)}\,.$
(54)

Now fix a point $\mathbf{t}=(t_{1},\dots,t_{d})\in\mathcal{U}(\Psi)$. Then by
definition and the fact that $\mathcal{U}(\Psi)$ is bounded, there exists a
subsequence $\\{n_{l}\\}_{l\in{\mathbb{N}}}$ such that

$\lim\limits_{l\to\infty}\frac{-\log\psi_{i}(n_{l})}{n_{l}}=t_{i}\quad{\rm
for\ all}\ \ 1\leq i\leq d.$

It is easily verified that for any $0<\varepsilon<1$, there exists
$N=N(\varepsilon)>0$ such that

$\frac{(1-\varepsilon)\log\beta_{i}}{(1-\varepsilon)\log\beta_{i}+t_{i}}\
\leq\
\frac{-\frac{\log(n_{l}+3)}{n_{l}}+\log\beta_{i}}{\log\beta_{i}+\frac{-\log\psi_{i}(n_{l})}{n_{l}}}$
(55)

for all $l\geq N$ and $1\leq i\leq d$. Let

$s_{i}:=\frac{(1-\varepsilon)\log\beta_{i}}{(1-\varepsilon)\log\beta_{i}+t_{i}}\
\quad(1\leq i\leq d).$

Then, (55) is equivalent to

$\big{(}\beta_{i}^{-n_{l}}\psi_{i}(n_{l})\big{)}^{s_{i}}\geq(n_{l}+3)\beta_{i}^{-n_{l}},$

which together with (54) implies that for any $l\geq N$

${\mathbb{T}}=\bigcup_{j_{i}=1}^{M_{i,n_{l}}}B\big{(}x_{n_{l},j_{i}}^{(i)},\big{(}\beta_{i}^{-n_{l}}\psi_{i}(n_{l})\big{)}^{s_{i}}\big{)}.$

In turn, it follows that for any $l\geq N$

${\mathbb{T}}^{d}\ =\
\bigcup_{j_{1}=1}^{M_{1,n_{l}}}\cdots\bigcup_{j_{d}=1}^{M_{d,n_{l}}}B\big{(}x_{n_{l},j_{1}}^{(1)},(\beta_{1}^{-n_{l}}\psi_{1}(n_{l}))^{s_{1}}\big{)}\times\cdots\times
B\big{(}x_{n_{l},j_{d}}^{(d)},(\beta_{d}^{-n_{l}}\psi_{d}(n_{l}))^{s_{d}}\big{)}$

and so

${\mathbb{T}}^{d}\ =\
\limsup_{n\to\infty}\bigcup_{j_{1}=1}^{M_{1,n}}\cdots\bigcup_{j_{d}=1}^{M_{d,n}}B\big{(}x_{n,j_{1}}^{(1)},(\beta_{1}^{-n}\psi_{1}(n))^{s_{1}}\big{)}\times\cdots\times
B\big{(}x_{n,j_{d}}^{(d)},(\beta_{d}^{-n}\psi_{d}(n))^{s_{d}}\big{)}.$ (56)

The upshot is that given the $\limsup$ set of rectangles appearing on the
right hand side of (53), the corresponding $\limsup$ set of
‘$(s_{1},\ldots,s_{d})$-scaled up’ rectangles satisfies (41) with $p=d$,
$X_{i}={\mathbb{T}}$, $\delta_{i}=1$ and $\mu_{i}=m_{1}$ (one-dimensional
Lebesgue measure) for each $1\leq i\leq d$. Thus on applying Theorem 11 with
$u_{i}=(1-\varepsilon)\log\beta_{i}$ and
$v_{i}=(1-\varepsilon)\log\beta_{i}+t_{i}$ ($1\leq i\leq d$), we obtain the
lower bound

$\dim_{\rm H}W(T,\Psi,\mathbf{a})\geq\min_{1\leq i\leq d}s(i,\varepsilon)$

where

$s(i,\varepsilon):=\sum_{k\in\mathcal{K}_{1}(i,\varepsilon)}1+\sum_{k\in\mathcal{K}_{2}(i,\varepsilon)}\left(1-\frac{t_{k}}{(1-\varepsilon)\log\beta_{i}+t_{i}}\right)+\sum_{k\in\mathcal{K}_{3}(i,\varepsilon)}\frac{(1-\varepsilon)\log\beta_{k}}{(1-\varepsilon)\log\beta_{i}+t_{i}}$

and where $\mathcal{K}_{1}(i,\varepsilon)$, $\mathcal{K}_{2}(i,\varepsilon)$,
$\mathcal{K}_{3}(i,\varepsilon)$ is the partition of $\\{1,\dots,d\\}$ given
by

$\displaystyle\mathcal{K}_{1}(i,\varepsilon)$ $\displaystyle:=$
$\displaystyle\big{\\{}k:(1-\varepsilon)\log\beta_{k}\geq(1-\varepsilon)\log\beta_{i}+t_{i}\big{\\}}$
$\displaystyle\mathcal{K}_{2}(i,\varepsilon)$ $\displaystyle:=$
$\displaystyle\big{\\{}k:(1-\varepsilon)\log\beta_{k}+t_{k}\leq(1-\varepsilon)\log\beta_{i}+t_{i}\big{\\}},$
$\displaystyle\mathcal{K}_{3}(i,\varepsilon)$ $\displaystyle:=$
$\displaystyle\big{\\{}1,\dots,d\big{\\}}\setminus\big{(}\mathcal{K}_{1}(i,\varepsilon)\cup\mathcal{K}_{2}(i,\varepsilon)\big{)}.$

Fix $1\leq i\leq d$. On letting $\varepsilon\to 0$, we find that
$\mathcal{K}_{1}(i,\varepsilon)\to\\{k:\log\beta_{k}>\log\beta_{i}+t_{i}\\}=\mathcal{K}_{1}(i),$
$\mathcal{K}_{2}(i,\varepsilon)\to\mathcal{K}_{2}(i)$ and
$\mathcal{K}_{3}(i,\varepsilon)\to\mathcal{K}_{3}(i)$. Thus

$\lim\limits_{\varepsilon\to
0}s(i,\varepsilon)=\theta_{i}(\mathbf{t})\quad{\rm and}\quad\dim_{\rm
H}W(T,\Psi,\mathbf{a})\geq\min\limits_{1\leq i\leq d}\theta_{i}(\mathbf{t}).$

Moreover, since this is valid for any $\mathbf{t}\in\mathcal{U}(\Psi)$ it
follows that

$\dim_{\rm
H}W(T,\Psi,\mathbf{a})\geq\sup_{\mathbf{t}\in\mathcal{U}(\Psi)}\min_{1\leq
i\leq d}\\{\theta_{i}(\mathbf{t})\\}$

and we are done. ∎

####  4.2.2 Proof of Theorem 8

We show that when $T$ is an integer matrix transformation, the diagonal
assumption in Theorem 6 can be relaxed to $T$ is diagonalizable over
${\mathbb{Z}}$. This thereby proves Theorem 8. So, suppose $T$ is
diagonalizable over $\mathbb{Z}$. Then by definition, there exist a diagonal
integer matrix $D$ and an invertible mapping $\phi$ satisfying (37). It is
easily versified that $T^{n}(\mathbf{x})\in B(\mathbf{a},\psi(n))$ if and only
if $D^{n}(\phi(\mathbf{x}))\in\phi\big{(}B(\mathbf{a},\psi(n))\big{)}$. Since
$\phi$ is a bi-Lipschitz map, we can find two positive constants $0<c_{1}\leq
c_{2}<\infty$ such that

$B\big{(}\phi(\mathbf{a}),c_{1}\psi(n)\big{)}\subset\phi\big{(}B(\mathbf{a},\psi(n)\big{)}\subset
B\big{(}\phi(\mathbf{a}),c_{2}\psi(n)\big{)}.$

In turn, Lemma 7 implies that the Hausdorff dimension of

$W(T,\psi,\mathbf{a}):=\\{\mathbf{x}\in\mathbb{T}^{d}:T^{n}(\mathbf{x})\in
B(\mathbf{a},\psi(n))\ \ \text{for infinitely many}\ n\in\mathbb{N}\\}$

is the same as that of

$\big{\\{}x\in\mathbb{T}^{d}:D^{n}(\mathbf{x})\in B(\phi(\mathbf{a}),\psi(n))\
\ \text{for infinitely many}\ n\in\mathbb{N}\big{\\}}.$

Thus, without loss of generality, we only need to prove the desired dimension
result in the case that $T$ is diagonal.

###  4.3 Proof of Theorem 7

The proof of Theorem 7 will make use of a general statement (namely,
Proposition 7 below) concerning Markov subsystems which may be of independent
interest. In short, these systems provide a “nice” approximation to one-
dimensional piecewise linear dynamical systems. To start with, let us recall
the notion of a Markov system for a one-dimensional expanding dynamical system
$(X,T)$. With this in mind, let $X$ be a compact set in $\mathbb{R}$ and
$T:X\to X$ be an expanding map. Furthermore, let $\Lambda$ be a subset of $X$.
A partition $\mathcal{P}_{\Lambda}$ of $\Lambda$ into finite or countable
collection of sets $P(k)$ is called a _Markov partition_ if
$\Lambda:=\bigcap_{n=0}^{\infty}T^{-n}\big{(}\cup P(k)\big{)}\,$ and

  * (i)

the interior of $P(j)$ and $P(k)$ are disjoint if $j\neq k$,

  * (ii)

$T$ restricted on each $P(j)$ is one to one,

  * (iii)

if $T(P(j))$ intersects the interior of $P(k)$ for some $j$ and $k$ then
$P(k)\subseteq\overline{T(P(j))}.$

In turn, the system $(\Lambda,T|_{\Lambda},\mathcal{P}_{\Lambda})$ is called a
_Markov subsystem of $(X,T)$_. In the case $\Lambda=X$, we simply write
$(X,T,\mathcal{P})$ and referred to it as a _Markov system_.

An important property regarding Markov subsystems that we shall utilise is
given by the following statement. It is a direct consequence of [38, Theorems
4.2.9 $\&$ 4.2.11].

######  Proposition 6.

Let $X$ be a compact set in $\mathbb{R}$ and $T:X\to X$ be an expanding map.
Let $(\Lambda,T|_{\Lambda},\mathcal{P}_{\Lambda})$ be a Markov subsystem of
$(X,T)$ with finite partition $\mathcal{P}_{\Lambda}=\\{P(i)\\}_{1\leq i\leq
N}$ whose incidence matrix is primitive. Suppose that for any $1\leq k\leq N$,
$T|_{P(k)}$ is $C^{1+\alpha}$ for some $\alpha>0$. Then the measure
$\mathcal{H}^{\delta}|_{\Lambda}$ is $\delta$-Ahlfors regular where
$\delta:=\dim_{\rm H}\Lambda$.

The following statement provides a lower bound for $\dim_{\rm H}\Lambda$ in
the case $T$ is piecewise linear. Throughout, we suppose that the absolute
value of the slope of such a map $T$ is constant and will be denoted by
${\beta(T)}$.

######  Proposition 7.

Let $T$ be a piecewise linear map on $[0,1]$ and assume that ${\beta(T)}>8$.
Then there exists a Markov subsystem
$(\Lambda,T|_{\Lambda},\mathcal{P}_{\Lambda})$ of $([0,1],T)$ with a finite
partition $\mathcal{P}_{\Lambda}=\\{P(i)\\}_{1\leq i\leq m}$ where each $P(i)$
is an interval and $T|_{P(i)}$ is linear, such that

$\dim_{\rm H}\Lambda\geq 1-\frac{\log 8}{\log{\beta(T)}}.$

###### Proof.

Let $\tilde{\mathcal{P}}=\\{\tilde{P}(i)\\}_{i=1}^{m}$ be a partition of
$[0,1]$ such that for each $1\leq i\leq m$ the set $\tilde{P}(i)$ is an
interval and $T|_{\tilde{P}(i)}$ is linear. Without loss of generality, we can
assume that

$\max\\{|\tilde{P}(i)|:\tilde{P}(i)\in\tilde{\mathcal{P}}\\}\leq
2\kappa\quad{\rm with\
}\quad\kappa:=\min\\{|\tilde{P}(i)|:\tilde{P}(i)\in\tilde{\mathcal{P}}\\}\,.$

Indeed, if $|\tilde{P}(i)|>2\kappa$ for some $1\leq i\leq m$, then there
exists $\ell\in\mathbb{N}$ such that

$2^{\ell}\kappa<|\tilde{P}(i)|\leq 2^{\ell+1}\kappa.$

Hence, we can subdivide $\tilde{P}(i)$ into $2^{\ell}$ equal pieces and take
these subintervals as part of partition rather than $\tilde{P}(i)$. The map
$T$ restricted to each piece of the new partition is still linear and by
construction the length of each piece if bounded above by $2\kappa$.

For any interval $\tilde{P}\in\tilde{\mathcal{P}}$, let

$P:=\tilde{P}\cap T^{-1}\Big{(}\ \overline{\bigcup_{1\leq i\leq
m}\\{\tilde{P}(i)\in\tilde{\mathcal{P}}:\tilde{P}(i)\subset T(\tilde{P})\\}}\
\Big{)}.$

Now since $T|_{\tilde{P}(i)}$ is linear, the intervals $\tilde{P}(i)$
contained in $T(\tilde{P})$ are adjacent intervals in the partition
$\tilde{\mathcal{P}}$. Hence, $P$ is a subinterval of $\tilde{P}$.
Furthermore, since $T|_{\tilde{P}}$ is linear with slope $\pm{\beta(T)}$ we
have that $|T(\tilde{P})|={\beta(T)}\cdot|\tilde{P}|\geq{\beta(T)}\kappa$. So
the number of $\tilde{P}(i)\in\tilde{\mathcal{P}}$ that intersect
$T(\tilde{P})$ is at least the integer part of
${\beta(T)}\kappa/2\kappa=[{\beta(T)}/2]$. Here we use the fact that
$|\tilde{P}(i)|\leq 2\kappa$ for all intervals in the partition. Thus, on
using the fact that ${\beta(T)}>8$, we have that

$\\#\big{\\{}\tilde{P}(i)\in\tilde{\mathcal{P}}:\tilde{P}(i)\subset
T(\tilde{P})\big{\\}}\geq[{\beta(T)}/2]-2\geq 1\,.$ (57)

The upshot of this is that

$P\neq\emptyset\,.$