# A survey of recent extensions and generalisations of the Mass Transference
Principle

Demi Allen
(University of Exeter)  Edouard Daviaud
(Université Paris-Est Créteil)

###### Abstract

In this article we survey some recent extensions and generalisations of the
celebrated Mass Transference Principle (Beresnevich and Velani, Annals of
Mathematics, 2006).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Hausdorff measures and dimension
    2. 2.2 Dimension of measures
    3. 2.3 Multifractal analysis of measures
    4. 2.4 Iterated function systems and self-similar measures
    5. 2.5 The open set condition
    6. 2.6 Hausdorff dimension of self-similar sets
    7. 2.7 Singular Value Function
  3. 3 The “original” Mass Transference Principle and the Duffin–Schaeffer Conjecture
    1. 3.1 The Duffin–Schaeffer Conjecture
    2. 3.2 The “original” Mass Transference Principle and the Hausdorff measure Duffin–Schaeffer Conjecture
    3. 3.3 A general mass transference principle from “balls to balls”
  4. 4 Precursors to the Mass Transference Principle
  5. 5 Mass Transference Principle for Systems of Linear Forms
  6. 6 Mass Transference Principle with “Local Scaling Property”
  7. 7 Mass Transference Principle from Balls to Rectangles
  8. 8 Mass Transference Principles from Rectangles to Rectangles
    1. 8.1 The setup
    2. 8.2 Ubiquity for rectangles
    3. 8.3 Mass transference principles from rectangles to rectangles
  9. 9 Mass Transference Principles from Balls to Open Sets
  10. 10 Mass Transference Principles for Inhomogeneous Measures
    1. 10.1 Multifractal mass transference principles
    2. 10.2 A mass transference principle subject to a full inhomogeneous measure statement

##  1 Introduction

The Mass Transference Principle established by Beresnevich and Velani in 2006
[12] is a powerful result enabling one to pass between Lebesgue measure and
Hausdorff measure statements for related $\limsup$ sets of balls. This is
surprising since the latter, Hausdorff measure, is generally viewed as being a
refinement of the former, Lebesgue measure. Indeed, for subsets of
${\mathbb{R}}^{k}$, the $k$-dimensional Hausdorff measure is a constant
multiple times the $k$-dimensional Lebesgue measure [23]. The Mass
Transference Principle has cemented itself as a fundamental result in Metric
Number Theory which has been found to have wide-ranging applications.
Moreover, the Mass Transference Principle has now been extended in a number of
directions and further generalisations and applications are constantly
emerging. Indeed, only a few years ago, a survey was written on
generalisations and applications of the Mass Transference Principle up until
that point [4] but, already, this is somewhat outdated. In this survey, we
will focus on collecting together various generalisations of the Mass
Transference Principle, paying particular attention to the advances made in
the last few years since the writing of [4]. There have also been many
newfound applications of the various mass transference principles we discuss
here. However, due to space constraints, we will not delve into any of these
in any detail in this particular survey.

##  2 Preliminaries

In this section, we will establish some notation and gather together some of
the necessary preliminaries that will be used throughout. The typical
formulation of a mass transference principle result is that it allows us to
infer information regarding (usually) Hausdorff measures of limsup sets in
some setting from _a priori_ weaker statements regarding (usually) Lebesgue
measure of some related limsup set. We recall that if we have a sequence, say
$(A_{i})_{i\in{\mathbb{N}}}$, of subsets of some metric space $X$, then the
_limsup_ of this sequence is

$\displaystyle\limsup_{i\to\infty}{A_{i}}$ $\displaystyle=\\{x\in X:x\in
A_{i}\text{ for i.m. }i\in{\mathbb{N}}\\}$
$\displaystyle=\bigcap_{N=1}^{\infty}{\bigcup_{i=N}^{\infty}{A_{i}}}.$

Throughout, the shorthand “i.m.” stands for “infinitely many”.

###  2.1 Hausdorff measures and dimension

We define a _dimension function_ to be any continuous function
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ such that $f(r)\to 0$ as $r\to 0$.
Recall that for a function $f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$, we say
that $f$ is _doubling_ if there exists a constant $\lambda>1$ such that for
all $x>0$ we have

$f(2x)\leq\lambda f(x).$

For a subset $F$ of ${\mathbb{R}}^{k}$ and a real number $\rho>0$, we say that
a countable collection of balls $\\{B_{i}=B(x_{i},r_{i})\\}_{i\in I}$ is a
_$\rho$ -cover_ for $F$ if $r_{i}<\rho$ for every $i\in I$ and
$F\subset\bigcup_{i\in I}{B_{i}}$. For a dimension function $f$ and a real
number $\rho>0$, the _$\rho$ -approximate Hausdorff $f$-measure_ of $F$ is
defined as

${\cal H}_{\rho}^{f}(F)=\inf\left\\{\sum_{i}{f(r_{i})}:\\{B_{i}\\}\text{ is a
$\rho$-cover of }F\right\\}.$

The _Hausdorff $f$-measure_ of $F$ is then defined as

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

Note that this limit exists (although may be infinite) since as $\rho$
decreases, the number of possible $\rho$-covers is restricted and so the value
of ${\cal H}_{\rho}^{f}(F)$ is monotonically increasing as $\rho$ decreases.
When $f(r)=r^{s}$ for some $s\geq 0$, we write ${\cal H}^{s}$ in place of
${\cal H}^{f}$. The Hausdorff dimension of $F$ is

$\operatorname{\dim_{H}}{F}=\inf\\{s\geq 0:{\cal H}^{s}(F)=0\\}.$

###  2.2 Dimension of measures

Let ${\cal B}({\mathbb{R}}^{k})$ denote the set of all Borel subsets of
${\mathbb{R}}^{k}$ and let ${\cal M}({\mathbb{R}}^{k})$ denote the set of all
Borel probability measures on ${\mathbb{R}}^{k}$. For a measure $\mu\in{\cal
M}({\mathbb{R}}^{k})$, we write $\operatorname{supp}(\mu)$ to denote the
_support_ of $\mu$.

Suppose $\mu\in{\cal M}({\mathbb{R}}^{k})$ and let $x\in\mbox{supp}(\mu)$.
Then the _lower_ and _upper local dimensions_ of $\mu$ at $x$ are defined,
respectively, as

$\displaystyle\underline{\dim}(\mu,x)$ $\displaystyle=\liminf_{r\to
0}\frac{\log(\mu(B(x,r)))}{\log(r)}\;,$ $\displaystyle\overline{\dim}(\mu,x)$
$\displaystyle=\limsup_{r\to 0}\frac{\log(\mu(B(x,r)))}{\log(r)}.$

The _lower Hausdorff dimension_ of the measure $\mu$ is given by

$\underline{\dim}(\mu)=\inf\left\\{\operatorname{\dim_{H}}(E):\,E\in\mathcal{B}(\mathbb{R}^{d}),\,\mu(E)>0\right\\}.$

If $\underline{\dim}(\mu,x)=\overline{\dim}(\mu,x)$ for $\mu$-almost every
$x\in\mbox{supp}(\mu)$, then this common value is simply denoted by
$\dim(\mu)$ and $\mu$ is said to be _exact dimensional_. For further details
see, for example, [23] or [24].

More generally, at various points, we will also consider probability measures
on locally compact metric spaces. If $X$ is a locally compact metric space
equipped with a probability measure $\mu$, we say that $\mu$ is _$\delta$
-Ahlfors-David regular_ if there exist constants $c_{1},c_{2},r_{0}>0$ such
that for any $x\in X$ and for any $0<r<r_{0}$, we have

$c_{1}r^{\delta}\leq\mu(B(x,r))\leq c_{2}r^{\delta}.$

If a measure $\mu$ is $\delta$-Ahlfors-David regular for _some_ $\delta$, we
may simply say that $\mu$ is _Ahlfors-David regular_ if the particular value
of $\delta$ is not of consequence.

###  2.3 Multifractal analysis of measures

The multifractal analysis of measures consists of associating a spectrum that
is correlated to the distribution of the possible local dimensions of the
measure in order to finely characterise its behaviour. More precisely, given
$\mu\in{\cal M}({\mathbb{R}}^{k})$ and a real number $h>0$, it usually relies
on estimating the Hausdorff dimension of the following sets:

$\displaystyle E_{\mu}(h)$
$\displaystyle=\left\\{x\in\mbox{supp}(\mu):\underline{\dim}(\mu,x)=h\right\\},$
$\displaystyle\widetilde{E}_{\mu}(h)$
$\displaystyle=\left\\{x\in\mbox{supp}(\mu):\dim(\mu,x)=h\right\\}.$

Let $\mu\in\mathcal{M}(\mathbb{R}^{d})$ and define

$\Theta_{\mu}(q,r)=\inf\left\\{\sum_{i\in I}\mu(B(x_{i},r))^{q}\right\\}.$

where the infimum is taken over all countable collections of balls
$\left\\{B(x_{i},r)\right\\}_{i\in I}$ satisfying the following two
properties:

  1. (i)

$x_{i}\in\operatorname{supp}(\mu)$ for every $i\in I$, and

  2. (ii)

$B(x_{i},r)\cap B(x_{j},r)=\emptyset$ for $i\neq j.$

We then define

$\tau_{\mu}(q)=\liminf_{r\to 0}\frac{\log(\Theta_{\mu}(q,r))}{\log r}.$

One often aims to correlate the dimension of the above sets with the scaling
functions $\tau_{\mu}(q)$.

We say that the measure $\mu$ satisfies the _multifractal formalism_ at $h$
when

$\operatorname{\dim_{H}}(E_{\mu}(h))=\tau_{\mu}^{*}(h),$

where
$\tau_{\mu}^{*}(h):=\inf_{q\in{\mathbb{R}}}\left\\{hq-\tau_{\mu}(q)\right\\}$
denotes the Legendre transform of $\tau_{\mu}$ at $h$.

###  2.4 Iterated function systems and self-similar measures

Let $D$ be a closed subset of ${\mathbb{R}}^{k}$. A function $\varphi:D\to D$
is said to be a _contraction_ if there exists some constant $0<r<1$ such that

$|\varphi(x)-\varphi(y)|\leq r|x-y|$

for all $x,y\in D$. If we have equality above then we say that $\varphi$ is a
_contracting similarity_ and that $r$ is the _contraction ratio_ of $\varphi$.
We call a finite collection $\Phi=\\{\varphi_{i}:D\to D\\}_{i=1}^{n}$ of at
least two contractions an _iterated function system_ or _IFS_ for short. If
all of the maps in the iterated function system $\Phi$ are contracting
similarities, we say that $\Phi$ is a _self-similar IFS_. It is a well-known
result due to Hutchinson [31] that, for any iterated function system
$\Phi=\\{\varphi_{i}:D\to D\\}_{i=1}^{n}$, there exists a unique, non-empty,
compact set $X\subset D$ such that

$X=\bigcup_{i=1}^{n}{\varphi_{i}(X)}.$

We call $X$ the _attractor_ of the iterated function system $\Phi$. If $\Phi$
is a self-similar IFS, then $X$ is a _self-similar set_. In this case, we may
still refer to the attractor of $\Phi$, but we may alternatively refer to $X$
as being the _self-similar set associated to $\Phi$_. Moreover, given any real
numbers $0<p_{1},...,p_{n}<1$ satisfying $p_{1}+...+p_{n}=1,$ it is also known
([31]) that there exists a unique measure $\mu\in\mathcal{M}(\mathbb{R}^{k})$
with $\operatorname{supp}{\mu}=X$ satisfying

$\mu(\cdot)=\sum_{i=1}^{n}p_{i}\mu(\varphi_{i}^{-1}(\cdot)).$

We call such measures _self-similar measures_ and, due to a result of Feng and
Hu [26], we know that these measures are always exact-dimensional.

###  2.5 The open set condition

Suppose that $\Phi=\\{\varphi_{i}:D\to D\\}_{i=1}^{n}$ is an iterated function
system. Let us write $\Lambda=\\{1,\dots,n\\}$ and define

$\Lambda^{*}=\bigcup_{N=0}^{\infty}{\Lambda^{N}}.$

Thus, $\Lambda^{*}$ consists of all possible finite sequences consisting of
entries from $\Lambda$. For
$\mathbf{i}=(i_{1},\dots,i_{|\mathbf{i}|})\in\Lambda^{*}$, we adopt the
shorthand notation

$\varphi_{\mathbf{i}}=\varphi_{i_{1}}\circ\varphi_{i_{2}}\circ\dots\circ\varphi_{i_{|\mathbf{i}|}}.$

We say that $\Phi$ satisfies the _open set condition_ if there exists a non-
empty bounded open set $U\subset D$ such that

$\varphi_{i}(U)\subset U\quad\text{for each }1\leq i\leq
n\quad\text{and}\quad\varphi_{i}(U)\cap\varphi_{j}(U)=\emptyset\quad\text{for
}i\neq j.$

###  2.6 Hausdorff dimension of self-similar sets

Suppose that $\Phi=\\{\varphi_{i}\\}_{i=1}^{n}$ is a self-similar IFS and
that, for each $1\leq i\leq n$, the map $\varphi_{i}$ has contraction ratio
$r_{i}$. The _similarity dimension_ of $\Phi$, denoted by
$\operatorname{\dim_{S}}{\Phi}$, is defined to be the unique real number
satisfying

$\sum_{i=1}^{n}r_{i}^{\operatorname{\dim_{S}}{\Phi}}=1.$ (1)

In many natural situations, the Hausdorff dimension of a self-similar set
turns out to be equal to the similarity dimension of the corresponding
iterated function system. In particular, it is well-known (see, for example,
[23]) that if $\Phi=\\{\varphi_{i}\\}_{i=1}^{n}$ is a self-similar IFS
satisfying the open set condition and $X$ is the associated self-similar set,
then

$\displaystyle\operatorname{\dim_{H}}{X}=\operatorname{\dim_{S}}{\Phi}.$

###  2.7 Singular Value Function

A quantity which arises naturally when estimating the Hausdorff dimension of
fractal sets that are defined by rectangles (such as self-affine sets or
limsup sets generated by rectangles) is the so-called _singular value
function_. Here we follow the definition as given in [23] but specialise to
the case where the sets of interest are rectangles. Suppose
$\mathbf{x}=(x_{1},\dots,x_{k})\in{\mathbb{R}}^{k}$, $r>0$, and
$1=\tau_{1}\leq...\leq\tau_{k}$. Define

$R=\prod_{i=1}^{k}{B(x_{i},r^{\tau_{i}})}$

to be the rectangle centered at $\mathbf{x}$ with side lengths
$r^{\tau_{1}},...,r^{\tau_{k}}$. For $s\geq 0$, the $s$-dimensional _singular
value function_ associated with $R$ is defined as

$\displaystyle\phi_{s}(R)$ $\displaystyle=r^{\max_{1\leq i\leq
k}s\tau_{i}-\sum_{j=1}^{i}\tau_{i}-\tau_{j}}$ (2)
$\displaystyle=r^{s\tau_{m}-m\tau_{m}+\sum_{j=1}^{m}\tau_{j}},$

where $m$ is the integer satisfying $m<s\leq m+1.$

Given a dimension function $f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$, the
$f$-dimensional Hausdorff content is denoted by ${\cal H}^{f}_{\infty}(E)$ and
defined as

${\cal
H}^{f}_{\infty}(E)=\inf\left\\{\sum_{n\in{\mathbb{N}}}{f(r_{n})}:(B(x_{n},r_{n}))_{n\in\mathbb{N}}\text{
is a sequence of closed balls such that
}E\subset\bigcup_{n\in{\mathbb{N}}}B(x_{n},r_{n})\right\\}.$

When $f(r)=r^{s}$, ${\cal H}^{f}_{\infty}$ is denoted ${\cal H}^{s}_{\infty}.$

As is pointed out by Koivusalo and Rams in [38], up to some multiplicative
constant, the $s$-dimensional singular value of a rectangle $R$ is nothing
more than the $s$-dimensional _Hausdorff content_ of $R$.

##  3 The “original” Mass Transference Principle and the Duffin–Schaeffer
Conjecture

###  3.1 The Duffin–Schaeffer Conjecture

The Mass Transference Principle proved by Beresnevich and Velani in [12] was
borne out of a desire to find a Hausdorff measure analogue of the famous
Duffin–Schaeffer Conjecture in Diophantine Approximation. Given a function
$\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, which we will often refer to as
an _approximating function_ , the _$\psi$ -well approximable points_ in
${\mathbb{R}}$ are defined to be

$\displaystyle{\cal
A}(\psi)=\left\\{x\in[0,1]:\left|x-\frac{p}{q}\right|<\frac{\psi(q)}{q}\text{
for i.m. }(p,q)\in{\mathbb{Z}}\times{\mathbb{N}}\right\\}.$

A classical result due to Khintchine [36] (subsequently modernised by others —
see, for example, [11]) gives us the following characterisation of the
Lebesgue measure of sets of $\psi$-well approximable points. We write ${\cal
L}(X)$ to denote the Lebesgue measure of a subset $X\subset{\mathbb{R}}^{k}$.

######  Theorem 1 (Khintchine’s Theorem).

For $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we have

${\cal L}({\cal A}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}{\psi(q)}<\infty\;,\\\\[8.61108pt] 1&\textrm{if
}\sum_{q=1}^{\infty}{\psi(q)}=\infty\quad\textrm{and $\psi$ is
monotonic.}\end{cases}$

The convergence part of Khintchine’s Theorem can be proved easily via a
straightforward application of the First Borel–Cantelli Lemma (for a statement
of the First Borel–Cantelli Lemma, we refer the reader to [28, Lemma 1.2]). In
particular, note that the proof of the convergence part of Khintchine’s
Theorem requires no monotonicity assumptions on the approximating function
$\psi$. On the other hand, it was shown by Duffin and Schaeffer [20] that the
monotonicity assumption cannot be removed from the divergence part of
Khintchine’s Theorem. In the same paper where they exhibited a counterexample
demonstrating this, they also formulated a conjecture for what ought to be
true instead of Khintchine’s Theorem when we do not assume that the
approximating function $\psi$ is monotonic. To this end, given
$\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we define

$\displaystyle{\cal
A}^{\prime}(\psi)=\left\\{x\in[0,1]:\left|x-\frac{p}{q}\right|<\frac{\psi(q)}{q}\text{
for i.m. }(p,q)\in{\mathbb{Z}}\times{\mathbb{N}}\text{ with
}\gcd(p,q)=1\right\\}.$

Let us denote by $\varphi$ the _Euler totient function_ ; that is, for
$n\in{\mathbb{N}}$,

$\varphi(n)=\\#\\{1\leq k\leq n:\gcd(k,n)=1\\}.$

Duffin and Schaeffer predicted the following [20]:

######  Conjecture 1 (Duffin–Schaeffer Conjecture).

For any approximating function $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we
have

${\cal L}({\cal A}^{\prime}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}{\frac{\varphi(q)\psi(q)}{q}}<\infty\;,\\\\[8.61108pt]
1&\textrm{if
}\sum_{q=1}^{\infty}{\frac{\varphi(q)\psi(q)}{q}}=\infty.\end{cases}$

In fact, the convergence part of the above conjecture follows easily from the
First Borel–Cantelli Lemma. Thus, it is the divergence statement which is the
real substance of the above conjecture. This eluded the mathematical community
for almost 80 years, until there was a recent major breakthrough and the
conjecture was proved by Koukoulopoulos and Maynard [39].

######  Theorem 2.

The Duffin–Schaeffer Conjecture is true.

In the intervening years between the formulation of Conjecture 1 by Duffin and
Schaeffer [20] and its eventual resolution by Koukoulopoulos and Maynard [39],
a natural higher-dimensional analogue of the Duffin–Schaeffer Conjecture was
posed by Sprindžuk and proved by Pollington and Vaughan [44].

For $k\in{\mathbb{N}}$ and $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we
define the _simultaneously $\psi$-well approximable points_ in
${\mathbb{R}}^{k}$ as:

$\displaystyle{\cal
A}_{k}(\psi)=\left\\{x\in[0,1]^{k}:\|qx-\mathbf{p}\|<\psi(q)\text{ for i.m.
}(\mathbf{p},q)\in{\mathbb{Z}}^{k}\times{\mathbb{N}}\right\\},$

where $\|\cdot\|$ denotes the supremum norm. In this case, a characterisation
of the Lebesgue measure of sets of simultaneously $\psi$-well approximable
points is again given by a result due to Khintchine [37] (and subsequently
modernised, see [11] for details).

######  Theorem 3 (Khintchine’s Theorem in ${\mathbb{R}}^{k}$).

For $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we have

${\cal L}({\cal A}_{k}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}{\psi(q)^{k}}<\infty\;,\\\\[8.61108pt] 1&\textrm{if
}\sum_{q=1}^{\infty}{\psi(q)^{k}}=\infty\quad\textrm{and $\psi$ is
monotonic.}\end{cases}$

######  Remark 1.

It follows from a result of Gallagher [27] that monotonicity of the
approximating function $\psi$ is not actually required in Theorem 3 when
$k\geq 2$.

To state the higher-dimensional version of the Duffin–Schaeffer Conjecture,
for $k\in{\mathbb{N}}$ and given $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$,
we define

$\displaystyle{\cal A}_{k}^{\prime\prime}(\psi)$
$\displaystyle=\left\\{x\in[0,1]:\|qx-\mathbf{p}\|<\psi(q)\text{ for i.m.
}(\mathbf{p},q)=(p_{1},\dots,p_{k},q)\in{\mathbb{Z}}^{k}\times{\mathbb{N}}\right.$
$\displaystyle\phantom{====================}\left.\text{ with
}\gcd(p_{i},q)=1\text{ for each }1\leq i\leq k\right\\}.$

Sprindžuk postulated the following conjecture regarding the Lebesgue measure
of the sets ${\cal A}_{m}^{\prime\prime}(\psi)$ when $\psi$ is not assumed to
be monotonic.

######  Conjecture 2.

For any approximating function $\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we
have

${\cal L}({\cal A}_{k}^{\prime\prime}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}{\left(\frac{\varphi(q)\psi(q)}{q}\right)^{k}}<\infty\;,\\\\[8.61108pt]
1&\textrm{if
}\sum_{q=1}^{\infty}{\left(\frac{\varphi(q)\psi(q)}{q}\right)^{k}}=\infty.\end{cases}$

Notice that when $k=1$, Conjecture 2 matches Conjecture 1 and so in this sense
is a natural generalisation of it. In the cases when $k\geq 2$, Conjecture 2
has been verified by Pollington and Vaughan [44]. Of course, putting this
together now with Theorem 2 gives a full resolution of Conjecture 2.

######  Theorem 4.

Conjecture 2 is true.

###  3.2 The “original” Mass Transference Principle and the Hausdorff measure
Duffin–Schaeffer Conjecture

In [12], Beresnevich and Velani were aiming to present natural Hausdorff
measure generalisations of Conjectures 1 and 2. The natural, all-encompassing,
statement presented in [12] was the following:

######  Conjecture 3.

Let $k\in{\mathbb{N}}$ and let $f$ be a dimension function such that
$r^{-k}f(r)$ is monotonic. For any approximating function
$\psi:{\mathbb{N}}\to{\mathbb{R}}_{\geq 0}$, we have

${\cal H}^{f}({\cal A}_{k}^{\prime\prime}(\psi))=\begin{cases}0&\textrm{if
}\sum_{q=1}^{\infty}{f\left(\frac{\psi(q)}{q}\right)\varphi(q)^{k}}<\infty\;,\\\\[8.61108pt]
{\cal H}^{f}([0,1]^{k})&\textrm{if
}\sum_{q=1}^{\infty}{f\left(\frac{\psi(q)}{q}\right)\varphi(q)^{k}}=\infty.\end{cases}$

We remark that the convergence part of Conjecture 3 can be proved via a
relatively straightforward covering argument and so the real essence of the
conjecture is the divergence part. Moreover, when $\psi$ is monotonic,
Conjecture 3 is a classical result due to Jarník [35]. Thus, the really
interesting case is when $\psi$ is not assumed to be monotonic.

In the case when $f(r)=r^{k}$, i.e. essentially Lebesgue measure, Conjecture 3
is equivalent to Conjecture 2. Therefore, on the face of it, it would seem
that Conjecture 3 really is more general than Conjectures 1 and 2. Moreover, a
proof of Conjecture 3 would immediately imply the truth of the other two
conjectures. Rather surprisingly, the Mass Transference Principle shows that,
in fact, the converse of this is also true; that a proof of Conjecture 2 would
imply the seemingly more general Conjecture 3.

Given $k\in{\mathbb{N}}$, a dimension function
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$, and a ball $B=B(x,r)$ in
${\mathbb{R}}^{k}$, we define an associated ball

$B^{f}=B(x,f(r))^{\frac{1}{k}}.$

When $f(r)=r^{s}$ for some $s\geq 0$, we write $B^{s}=B^{f}$. Also note that
with this notational convention, we have $B^{k}=B$. The following Mass
Transference Principle was established by Beresnevich and Velani in [12,
Theorem 2].

######  Theorem 5 (The Mass Transference Principle).

Let $\\{B_{i}=B(x_{i},r_{i})\\}_{i\in{\mathbb{N}}}$ be a sequence of balls in
${\mathbb{R}}^{k}$ with $r_{i}\to 0$ as $i\to\infty$. Let
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a dimension function such that
$r^{-k}f(r)$ is monotonic and suppose that for any ball $B$ in
${\mathbb{R}}^{k}$, we have

${\cal H}^{k}\left(B\cap\limsup_{i\to\infty}{B_{i}^{f}}\right)={\cal
H}^{k}(B).$

Then, for any ball $B$ in ${\mathbb{R}}^{k}$, we have

${\cal H}^{f}\left(B\cap\limsup_{i\to\infty}{B_{i}}\right)={\cal H}^{f}(B).$

It was shown already in [12] that a consequence of Theorem 5 was that a proof
of Conjecture 2 would imply Conjecture 3. Since we now have a full resolution
of Conjecture 2, with the cases when $k\geq 2$ due to Pollington and Vaughan
[44], and the case when $k=1$ due to Koukoulopoulos and Maynard [39], we also
have:

######  Theorem 6.

Conjecture 3 is true.

We refer the reader to, for example, [12] for further details on how Theorem 6
follows from Theorem 4. Similar arguments also yield further surprising
consequences of the Mass Transference Principle. For example, one can deduce
Jarník’s Theorem [35] from Khintchine’s Theorem and the Jarník–Besicovitch
Theorem ([34], [14]) can be inferred from Dirichlet’s Theorem. We will not
spend time elaborating on these particular applications here as our primary
focus is to survey some of the more recent developments surrounding the Mass
Transference Principle, but we instead refer the interested reader to the
likes of [12] and [11] for further details.

###  3.3 A general mass transference principle from “balls to balls”

We conclude our section on the “original” Mass Transference Principle by
briefly mentioning a more general version of Theorem 5 also recorded by
Beresnevich and Velani in [12]. Let $(X,d)$ be a locally compact metric space
and let $g:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a doubling dimension
function. Further suppose that there exist constants $0<c_{1}<c_{2}<\infty$
and $r_{0}>0$ such that for any ball $B=B(x,r)$ with $x\in X$ and $r>0$, we
have

$\displaystyle c_{1}g(r)\leq{\cal H}^{g}(B)\leq c_{2}g(r).$ (3)

Given a dimension function $f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ and a
ball $B(x,r)$ in $X$, define

$B^{f}=B(x,g^{-1}f(r)).$

Note that $B^{g}=B$.

In this more general setting, Beresnevich and Velani state the following
general version of the Mass Transference Principle in [12, Theorem 3].

######  Theorem 7.

Let $(X,d)$ and $g$ be as above. Let
$\\{B_{i}=B(x_{i},r_{i})\\}_{i\in{\mathbb{N}}}$ be a sequence of balls in $X$
with $r_{i}\to 0$ as $i\to\infty$. Let
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a dimension function such that
$\frac{f(r)}{g(r)}$ is monotonic. Suppose that for any ball $B$ in $X$, we
have

${\cal H}^{g}\left(B\cap\limsup_{i\to\infty}{B_{i}^{f}}\right)={\cal
H}^{g}(B).$

Then, for any ball $B$ in $X$, we have

${\cal H}^{f}\left(B\cap\limsup_{i\to\infty}{B_{i}^{g}}\right)={\cal
H}^{f}(B)$

In the case that $X={\mathbb{R}}^{k}$ and $g(x)=x^{k}$, Theorem 7 reduces to
Theorem 5. The more general version given in Theorem 7, though, has found
applications in, for example, Diophantine Approximation on fractals, where
Theorem 5 cannot be applied directly. See, for example, [40] and [48].

##  4 Precursors to the Mass Transference Principle

The idea of studying points that are approximable at certain rates using
geometric properties of a “good” measure takes its roots in the 1990’s with
work of Dodson, Rynne and Vickers [19] and work of Dodson, Melian, Pestana and
Velani [18]. More precisely, the notion of ubiquitous systems with respect to
a radius function and a measure was introduced in [19] and refined in [18].
Subsequently, the theory of ubiquitous systems has been further developed and
this theory has become an important topic in the context of Diophantine
approximation. We refer the reader to [8] and references therein for further
information. For the moment, we will focus on the definition of a ubiquitous
system as presented in [18].

######  Definition 1.

Fix a subset $K\subset\mathbb{R}^{k}$ and let $\mu$ be an Ahlfors-David
regular measure on $K$. Let ${\cal R}=\\{x_{\alpha}\\}_{\alpha\in J}$ be a
countable collection of points in $K$ and let $\beta:J\to{\mathbb{R}}_{>0}$.
For each $N\in\mathbb{N}$ define

$J(N)=\left\\{\alpha:\ \beta(\alpha)\leq N\right\\}.$

Suppose that there exists a positive decreasing function
$\rho:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ such that

$\mu\left(K\setminus\bigcup_{\alpha\in
J(N)}\frac{1}{3}B(x_{\alpha},\rho(N))\right)\to 0\quad\text{as }N\to\infty.$
(4)

Then, the system $(\mathcal{R},\beta)$ is said to be _$\mu$ -ubiquitous with
respect to $\rho$_.

In [18], the following ubiquity theorem was obtained.

######  Theorem 8.

Let $K$ be a subset of $\mathbb{R}^{k}$ and let $\mu$ be a $\delta$-Ahlfors-
David regular measure on $K$. Suppose $(\mathcal{R},\beta)$ is a
$\mu$-ubiquitous system with respect to a function
$\rho:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$. Then, for every $t\geq 1$,

$\operatorname{\dim_{H}}\left\\{y:y\in
B(x_{\alpha},\rho(\beta(\alpha))^{t})\text{ for infinitely many }\alpha\in
J\right\\}\geq\frac{\delta}{t}.$

Note that in (4), the balls all have the same radius $\rho(N).$ In particular,
when $\mathcal{R}$ is a sequence of points $(x_{n})_{n\in\mathbb{N}}$ and
$(\rho(n))_{n\in\mathbb{N}}$ is a sequence of positive numbers satisfying
$\rho(n)\to 0$, defining $\beta:\mathbb{N}\to\mathbb{N}$ by $\beta(n)=n$ and
asking $(\mathcal{R},\beta)$ to be $\mu$-ubiquitous with respect to
$(\rho(n))_{n\in\mathbb{N}}$ is _a priori_ a stronger requirement than asking
that

$\mu\left(\limsup_{n\to\infty}B(x_{n},\rho(n))\right)=1.$ (5)

However, it was only in 2000, with the work of Jaffard [32, 33], that the
above results were refined to obtain results assuming only (5). More
explicitly, Jaffard [32, 33] showed the following.

######  Theorem 9.

Let $K$ be a subset of ${\mathbb{R}}^{k}$, let $\delta\geq 0$, and suppose
that $\mu$ is a $\delta$-Ahlfors-David regular measure on $K$. Let
$(B_{n}:=B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in $K$
such that $r_{n}\to 0$ as $n\to\infty$. If

$\mu\left(\limsup_{n\to\infty}B_{n}\right)=1,$

then for every $t\geq 1$, there exists a dimension function
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ such that

$\lim_{r\to 0}\frac{\log f(r)}{\log
r}=\frac{\delta}{t}\quad\text{and}\quad{\cal
H}^{f}\left(\limsup_{n\to\infty}B(x_{n},r_{n}^{t})\right)>0.$

In particular

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}B(x_{n},r_{n}^{t})\right)\geq\frac{\delta}{t}.$

##  5 Mass Transference Principle for Systems of Linear Forms

Following their discovery of the original Mass Transference Principle (Theorem
5), Beresnevich and Velani subsequently extended the idea and proved a mass
transference principle for systems of linear forms [13] by combining Theorem 5
with a “slicing” result from geometric measure theory. The proof method
employed resulted in some undesirable technical restrictions being imposed on
the result obtained by Beresnevich and Velani in this direction (see Remark
2). These technical restrictions were conjectured to be unnecessary in [7,
Conjecture E] and were later removed in [2]. We state here the mass
transference principle for systems of linear forms given in [2].

Let $k,m\geq 1$ and $\ell\geq 0$ be integers such that $k=m+\ell$. Let ${\cal
R}=(R_{j})_{j\in{\mathbb{N}}}$ be a sequence of $\ell$-dimensional planes in
${\mathbb{R}}^{k}$. Let $\|\cdot\|$ be any fixed norm on ${\mathbb{R}}^{k}$
and, for $j\in{\mathbb{N}}$ and $\delta\geq 0$, define

$\Delta(R_{j},\delta)=\\{\mathbf{x}\in{\mathbb{R}}^{k}:\operatorname{dist}(\mathbf{x},R_{j})<\delta\\},$

where

$\operatorname{dist}(\mathbf{x},R_{j})=\inf\\{\|\mathbf{x}-\mathbf{y}\|:\mathbf{y}\in
R_{j}\\}.$

Let $\Upsilon=(\Upsilon_{j})_{j\in{\mathbb{N}}}$ be a sequence of non-negative
real numbers such that $\Upsilon_{j}\to 0$ as $j\to\infty$. Consider the set

$\Lambda(\Upsilon)=\left\\{\mathbf{x}\in{\mathbb{R}}^{k}:\mathbf{x}\in\Delta(R_{j},\Upsilon_{j})\text{
for infinitely many }j\in{\mathbb{N}}\right\\}.$

######  Theorem 10.

Let ${\cal R}$ and $\Upsilon$ be as defined above. Suppose that
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ is a dimension function such that
$r^{-k}f(r)$ is monotonic and the function
$g:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ defined by $g(r)=r^{-\ell}f(r)$ is
also a dimension function. Let $\Omega$ be a ball in ${\mathbb{R}}^{k}$ and
suppose that for any ball $B$ in $\Omega$, we have

${\cal
H}^{k}\left(B\cap\Lambda\left(g(\Upsilon)^{\frac{1}{m}}\right)\right)={\cal
H}^{k}(B).$

Then, for any ball $B$ in $\Omega$, we have

${\cal H}^{f}(B\cap\Lambda(\Upsilon))={\cal H}^{f}(B).$

######  Remark 2.

In [13, Theorem 4.1], Beresnevich and Velani draw the same conclusion as in
Theorem 10 but subject to the additional condition that there exists an
$m$-dimensional linear subspace $V$ of ${\mathbb{R}}^{k}$ satisfying:

  1. (i)

$V\cap R_{j}\neq\emptyset$ for all $j\in{\mathbb{N}}$, and

  2. (ii)

$\sup_{j\in{\mathbb{N}}}\textrm{diam}(V\cap\Delta(R_{j},1))<\infty$.

Initially, the extra conditions required by Beresnevich and Velani may not
seem too restrictive. However, as well as giving a cleaner statement, Theorem
10 also gives rise to a number of interesting and very general applications in
Diophantine Approximation and it is plausible that some of these may have been
out of reach when trying to use [13, Theorem 4.1]. Applications of Theorem 10
are discussed at some length in [2] and [4]. Further recent applications of
Theorem 10 have also come to light in [3, 29, 51].

##  6 Mass Transference Principle with “Local Scaling Property”

In [1], a general mass transference principle was established which
emcompasses the aforementioned Theorems 5, 7, and 10 and further extends these
results in several directions. In particular, the general mass transference
principle established in [1] holds in locally compact metric spaces, enables
the transference of Hausdorff $g$-measure statements to Hausdorff $f$-measure
statements, and is applicable to a wider class of sets than are covered by the
previously mentioned theorems. For example, the result established in [1]
(Theorem 11 below) holds for limsup sets defined by smooth manifolds in
${\mathbb{R}}^{k}$ as well as self-similar sets arising from iterated function
systems satisfying the open set condition.

While Theorems 5 and 7 apply to limsup sets of balls and Theorem 10 deals with
essentially limsup sets of neighbourhoods of planes, the result in [1] is
applicable to limsup sets defined by neighbourhoods of sets satisfying a
certain _local scaling property_. Essentially, this local scaling property
that is introduced ensures that the sets under consideration behave locally
like affine subspaces. This property appears to be the key feature enabling
one all-encompassing statement which includes and extends several of the
aforementioned results.

Let $(X,d)$ be a locally compact metric space and let $g$ be a doubling
dimension function which furthermore satisfies (3). Given a set $F\subset X$
and a real number $\delta\geq 0$, we define the $\delta$-neighbourhood of $F$
to be

$\Delta(F,\delta)=\\{x\in X:d(x,F)<\delta\\},$

where $d(x,F)=\min\\{d(x,y):y\in F\\}$.

######  Definition 2 (Local Scaling Property).

Let ${\cal F}=(F_{j})_{j\in{\mathbb{N}}}$ be a sequence of sets in $X$ and
suppose $0\leq\kappa<1$ is a real number. We say that ${\cal F}$ satisfies the
_local scaling property_ (henceforth abbreviated to _LSP_) with respect to
$\kappa$ if there exist constants $c_{3},c_{4}>0$ and $r_{0}>0$ such that, for
any real numbers $r$ and $\delta$ with $0<\delta<r<r_{0}$, and for any
$j\in{\mathbb{N}}$ and $x\in F_{j}$, we have that

$\displaystyle c_{3}g(\delta)^{1-\kappa}g(r)^{\kappa}\leq{\cal
H}^{g}(B(x,r)\cap\Delta(F_{j},\delta))\leq
c_{4}g(\delta)^{1-\kappa}g(r)^{\kappa}.$ (6)

If (6) is satisfied for one particular set $F\subset X$, we will also say that
$F$ satisfies the LSP with respect to $\kappa$. It should be clear from
context when we are referring to a single set satisfying the LSP and when we
are referring to a collection of sets satisfying the LSP.

The following examples of sets satisfying the LSP are given in [1]. We merely
state the examples here and, in the case of Examples 3 and 4, refer the reader
to [1] for more detailed arguments for why such sets satisfy the LSP.

######  Example 1.

Suppose ${\cal F}$ is a sequence of points in some locally compact metric
space $(X,d)$. Then ${\cal F}$ satisfies the LSP with respect to $\kappa=0$.

######  Example 2.

Suppose that $F\subset{\mathbb{R}}^{k}$ is any set satisfying the LSP with
respect to some $0\leq\kappa<1$ and suppose that ${\cal
F}=(F_{j})_{j\in{\mathbb{N}}}$ is a sequence of sets where, for each
$j\in{\mathbb{N}}$, $F_{j}$ is the image of $F$ under some isometry. Then
${\cal F}$ satisfies the LSP with respect to $\kappa$.

######  Example 3.

Let ${\cal M}\subset{\mathbb{R}}^{k}$ be a smooth compact $\ell$-dimensional
manifold. Then ${\cal M}$ satisfies the LSP with respect to
$\kappa=\frac{\ell}{k}$.

######  Example 4.

Suppose $N\in{\mathbb{N}}$ and that
$\Phi=\\{\varphi_{j}:{\mathbb{R}}^{k}\to{\mathbb{R}}^{k}\\}_{j=1}^{n}$ is a
self-similar iterated function system. Denote by $K$ the self-similar set
associated with $\Phi$. Furthermore, suppose that $\Phi$ satisfies the open
set condition. Then $K$ satisfies the LSP with respect to
$\kappa=\frac{\operatorname{\dim_{H}}{K}}{k}$.

Now we turn to stating the main result of [1]. Suppose that ${\cal
F}=(F_{j})_{j\in{\mathbb{N}}}$ is a sequence of sets in $X$ and
$\Upsilon=(\Upsilon_{j})_{j\in{\mathbb{N}}}$ is a sequence of non-negative
real numbers with $\Upsilon_{j}\to 0$ as $j\to\infty$. We define the limsup
set

$\displaystyle\Lambda(\Upsilon)$ $\displaystyle=\\{x\in
X:x\in\Delta(F_{j},\Upsilon_{j})\quad\text{for infinitely many
}j\in{\mathbb{N}}\\}.$

The following general mass transference principle was established in [1,
Theorem 1].

######  Theorem 11.

Let $(X,d)$ be a locally compact metric space and let $g$ be a doubling
dimension function which also satisfies (3). Let ${\cal
F}=(F_{j})_{j\in{\mathbb{N}}}$ be a sequence of sets in $X$ satifying the LSP
with respect to some $0\leq\kappa<1$, and let
$\Upsilon=(\Upsilon_{j})_{j\in{\mathbb{N}}}$ be a sequence of non-negative
real numbers such that $\Upsilon_{j}\to 0$ as $j\to\infty$. Suppose that $f$
is a dimension function such that $f/g$ is monotonic and $f/g^{\kappa}$ is
also a dimension function.

Suppose that for any ball $B$ in $X$ we have

$\displaystyle{\cal
H}^{g}\left(B\cap\Lambda\left(g^{-1}\left(\left(\frac{f(\Upsilon)}{g(\Upsilon)^{\kappa}}\right)^{\frac{1}{1-\kappa}}\right)\right)\right)={\cal
H}^{g}(B).$

Then, for any ball $B$ in $X$, we have

${\cal H}^{f}(B\cap\Lambda(\Upsilon))={\cal H}^{f}(B).$

The strategy used for proving Theorem 11 mirrors the ideas used in [2] for
proving Theorem 10. In turn, the ideas there are inspired by those used by
Beresnevich and Velani in [12] for proving the “original” Mass Transference
Principle (Theorem 5). If we take ${\cal F}$ to be a sequence of points in a
locally compact metric space and $\kappa=0$, then Theorem 11 matches Theorem
7. If our underlying metric space is ${\mathbb{R}}^{k}$, $g(r)=r^{k}$, ${\cal
F}$ is a sequence of points in ${\mathbb{R}}^{k}$, and $\kappa=0$, then
Theorem 11 reduces to Theorem 5. We can also derive Theorem 10 from Theorem 11
by taking our metric space to be a suitable ball $\Omega$ in
${\mathbb{R}}^{k}$, $g(r)=r^{j}$, ${\cal F}$ to be a sequence of
$\ell$-dimensional planes in ${\mathbb{R}}^{k}$, and $\kappa=\frac{\ell}{k}$.
Thus, Theorem 11 incorporates several of the previously discovered mass
transference principles. However, as can be seen from Examples 3 and 4 above,
Theorem 11 is also more widely applicable to much more general sets.

##  7 Mass Transference Principle from Balls to Rectangles

The variants of the mass transference principle we exhibit in this section and
the next allow one to derive Hausdorff measure results for limsup sets arising
from rectangles. These results are particularly relevant to the topic of
weighted approximation in various contexts and have already found several
applications in this direction [5, 9, 10, 30, 45, 46, 50].

In this section we will present the mass transference principle from “balls to
rectangles” proved by Wang, Wu, and Xu in [49]. In the next section we will
discuss mass transference principles from “rectangles to rectangles”.

Let $k\in{\mathbb{N}}$ and suppose that $(x_{n})_{n\in{\mathbb{N}}}$ is a
sequence of points in the $k$-dimensional unit cube $[0,1]^{k}$. Let
$(r_{n})_{n\in{\mathbb{N}}}$ be a sequence of non-negative real numbers such
that $r_{n}\to 0$ as $n\to\infty$. We define

$\displaystyle W_{0}$
$\displaystyle=[0,1]^{k}\cap\limsup_{n\to\infty}{B(x_{n},r_{n})}$
$\displaystyle=\left\\{x\in[0,1]^{k}:x\in B(x_{n},r_{n})\text{ for i.m.
}n\in{\mathbb{N}}\right\\}.$

For $\mathbf{a}=(a_{1},a_{2},\dots,a_{k})\in{\mathbb{R}}^{k}_{\geq 0}$,
$x=(x_{1},\dots,x_{k})\in[0,1]^{k}$, and $r>0$, define

$B^{\mathbf{a}}(x,r)=\prod_{i=1}^{k}{B(x_{i},r^{a_{i}})}.$

In other words, $B^{\mathbf{a}}(x,r)$ is the rectangle with centre $x$ and
side-lengths $(r^{a_{1}},\dots,r^{a_{k}})$. Given $(x_{n})_{n\in{\mathbb{N}}}$
and $(r_{n})_{n\in{\mathbb{N}}}$ as above and
$\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}^{k}_{\geq 0}$ with $1\leq
a_{1}\leq a_{2}\leq\dots\leq a_{k}$, we define

$\displaystyle W_{\mathbf{a}}$
$\displaystyle=[0,1]^{k}\cap\limsup_{n\to\infty}{B^{\mathbf{a}}(x_{n},r_{n})}$
$\displaystyle=\left\\{x\in[0,1]^{k}:x\in B^{\mathbf{a}}(x_{n},r_{n})\text{
for i.m. }n\in{\mathbb{N}}\right\\}.$

In [49, Theorem 1.2], Wang, Wu, and Xu proved the following result regarding
Hausdorff dimension.

######  Theorem 12.

Let $(x_{n})_{n\in{\mathbb{N}}}$ be a sequence of points in the
$k$-dimensional unit cube $[0,1]^{k}$, let $(r_{n})_{n\in{\mathbb{N}}}$ be a
sequence of non-negative real numbers such that $r_{n}\to 0$ as $n\to\infty$,
and let $\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}^{k}_{\geq 0}$ be such
that $1\leq a_{1}\leq a_{2}\leq\dots\leq a_{k}$. Suppose that ${\cal
L}(W_{0})=1$. Then

$\operatorname{\dim_{H}}{W_{\mathbf{a}}}\geq\min_{1\leq j\leq
k}\left\\{\frac{k+ja_{j}-\sum_{i=1}^{j}{a_{i}}}{a_{j}}\right\\}.$

Subject to the additional constraint that $a_{k}>1$, Wang, Wu and Xu proved
the following result regarding Hausdorff measure in [49, Theorem 1.3].

######  Theorem 13.

Let $(x_{n})_{n\in{\mathbb{N}}}$ be a sequence of points in the
$k$-dimensional unit cube $[0,1]^{k}$, let $(r_{n})_{n\in{\mathbb{N}}}$ be a
sequence of non-negative real numbers such that $r_{n}\to 0$ as $n\to\infty$,
and let $\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}^{k}_{\geq 0}$ be such
that $1\leq a_{1}\leq a_{2}\leq\dots\leq a_{k}$ and $a_{k}>1$. Suppose that
${\cal L}(W_{0})=1$ and write

$s=\min_{1\leq j\leq
k}\left\\{\frac{k+ja_{j}-\sum_{i=1}^{j}{a_{i}}}{a_{j}}\right\\}.$

Then ${\cal H}^{s}{(W_{\mathbf{a}})}=\infty$.

##  8 Mass Transference Principles from Rectangles to Rectangles

The results from the previous section, giving us mass transference principles
from “balls to rectangles” have been further generalised recently by Wang and
Wu in [47] allowing for mass transference principles from “rectangles to
rectangles”.

###  8.1 The setup

Fix $k\in{\mathbb{N}}$ and, for each $1\leq i\leq k$, let
$(X_{i},|\cdot|_{i},m_{i})$ be a bounded locally compact metric measure space.
Further suppose, for each $1\leq i\leq k$, that $m_{i}$ is a
$\delta_{i}$-Ahlfors-David regular probability measure for some
$\delta_{i}>0$. In [47], Wang and Wu establish mass transference principles
from rectangles to rectangles in the product space $(X,|\cdot|,m)$, where

$\displaystyle X=\prod_{i=1}^{k}{X_{i}},\qquad
m=\prod_{i=1}^{k}{m_{i}},\qquad\text{and}\quad|\cdot|=\max_{1\leq i\leq
k}{|\cdot|_{i}}.$

We note that a ball $B(x,r)$ in $X$, where $x=(x_{1},\dots,x_{k})\in X$ and
$r>0$, is a product of balls from the spaces $\\{X_{i}\\}_{i=1}^{k}$, i.e.

$B(x,r)=\prod_{i=1}^{k}{B(x_{i},r)}.$

Next, let $J$ be an infinite countable indexing set and let
$\beta:J\to{\mathbb{R}}_{>0}$ be such that for any real number $M>1$ we have

$\\#\\{\alpha\in J:\beta(\alpha)<M\\}<\infty.$

Let $(\ell_{n})_{n\in{\mathbb{N}}}$ and $(u_{n})_{n\in{\mathbb{N}}}$ be
sequences of integers such that $u_{n}\geq\ell_{n}$ for all $n\in{\mathbb{N}}$
and $\ell_{n}\to\infty$ as $n\to\infty$. One implication of these assumptions
will be that without loss of generality we may assume that $\ell_{n}>0$ and
$u_{n}>0$ for all $n\in{\mathbb{N}}$. For $n\in{\mathbb{N}}$, define

$J_{n}=\\{\alpha\in J:\ell_{n}\leq\beta(\alpha)\leq u_{n}\\}.$

For each $1\leq i\leq k$, let $({\cal R}_{\alpha,i})_{\alpha\in J}$ be a
collection of subsets in $X_{i}$. We define the corresponding sequence of
_resonant sets_ in $X$ by setting

${\cal R}_{\alpha}=\prod_{i=1}^{k}{{\cal R}_{\alpha,i}}.$

Let $\rho:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a non-increasing function
such that $\rho(u)\to 0$ as $u\to\infty$. For
$\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}_{\geq 0}^{k}$ and
$\rho\in{\mathbb{R}}_{>0}$, define

$\Delta({\cal R}_{\alpha},\rho^{\mathbf{a}})=\prod_{i=1}^{k}{\Delta({\cal
R}_{\alpha,i},\rho^{a_{i}})},$

where $\Delta({\cal R}_{\alpha,i},\rho^{a_{i}})$ denotes the
$\rho^{a_{i}}$-neighbourhood of ${\cal R}_{\alpha,i}$ in $X_{i}$. Note that
$\Delta({\cal R}_{\alpha},\rho^{\mathbf{a}})$ defines a “rectangle” in $X$. To
establish the mass transference principles given in [47], Wang and Wu further
require that, for each $1\leq i\leq k$, the collections $({\cal
R}_{\alpha,i})_{\alpha\in J}$ satisfy the local scaling property (see
Definition 2) with respect to some $0\leq\kappa<1$ and where
$g(r)=r^{\delta_{i}}$.

Suppose that $\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}_{\geq 0}^{k}$ is
fixed. Given $\mathbf{t}=(t_{1},\dots,t_{k})\in{\mathbb{R}}_{\geq 0}^{k}$,
define the limsup set

$W(\mathbf{t})=\left\\{x\in X:x\in\Delta({\cal
R}_{\alpha},\rho(\beta(\alpha))^{\mathbf{a}+\mathbf{t}})\quad\text{for i.m.
}\alpha\in J\right\\}.$

This is essentially the limsup set obtained by “shrinking” the rectangles
$\Delta({\cal R}_{\alpha},\rho(\beta(\alpha))^{\mathbf{a}})$. In [47], Wang
and Wu also consider variations on the set $W(\mathbf{t})$ with more general
approximating functions. However, for simplicity, here we will restrict our
attention to the setting outlined above.

In [47], Wang and Wu essentially provide two “types” of mass transference
principle from rectangles to rectangles. The first type requires the original
sequence of “unshrunk” rectangles to satisfy certain _ubiquity_ conditions, as
defined in the following section. The second type is more reflective of the
typical statements of mass transference principles seen in the previous
sections. More precisely, the second type of mass transference principle from
rectangles to rectangles exhibited by Wang and Wu in [47] allows us to infer
information about the Hausdorff dimension of sets of the form $W(\mathbf{t})$
(as defined above) subject to knowing that the limsup set of “orginal”
rectangles is of full measure.

###  8.2 Ubiquity for rectangles

Inspired by the notions of local and global ubiquitous systems defined by
Beresnevich, Dickinson, and Velani in [8], Wang and Wu defined the natural
corresponding notions of _local ubiquity_ and _uniform local ubiquity_ for
rectangles in [47]. Throughout we assume the same setting and use the same
notation as described in the previous subsection.

######  Definition 3 (Local ubiquity for rectangles).

We say that $(\\{{\cal R}_{\alpha}\\}_{\alpha\in J},\beta)$ is a _locally
ubiquitous system of rectangles_ with respect to $(\rho,\mathbf{a})$ if there
exists a constant $c>0$ such that for any ball $B$ in $X$ we have

$\limsup_{n\to\infty}{m\left(B\cap\bigcup_{\alpha\in J_{n}}{\Delta({\cal
R}_{\alpha},\rho(u_{n})^{\mathbf{a}})}\right)}\geq cm(B).$

######  Definition 4 (Uniform local ubiquity for rectangles).

We say that $(\\{{\cal R}_{\alpha}\\}_{\alpha\in J},\beta)$ is a _uniformly
locally ubiquitous system of rectangles_ with respect to $(\rho,\mathbf{a})$
if there exists a constant $c>0$ such that for any ball $B$ in $X$, there is
some $n_{0}(B)\in{\mathbb{N}}$ such that

$m\left(B\cap\bigcup_{\alpha\in J_{n}}{\Delta({\cal
R}_{\alpha},\rho(u_{n})^{\mathbf{a}})}\right)\geq cm(B)$

for all $n\geq n_{0}(B)$.

###  8.3 Mass transference principles from rectangles to rectangles

Throughout this subsection, let us again assume the setting described in
Section 8.1. Recall that for each $1\leq i\leq k$, we assume that the measure
$m_{i}$ is $\delta_{i}$-Ahlfors-David regular for some $\delta_{i}>0$. Also
recall that we assume that for each $1\leq i\leq k$, the collections $({\cal
R}_{\alpha,i})_{\alpha\in J}$ satisfy the local scaling property (Definition
2) with respect to some $0\leq\kappa<1$ with $g(r)=r^{\delta_{i}}$. Assuming
local ubiquity for rectangles, Wang and Wu first establish the following mass
transference principle from rectangles to rectangles in [47].

######  Theorem 14.

Assume the setting as described above and further suppose that $(\\{{\cal
R}_{\alpha}\\}_{\alpha\in J},\beta)$ is a locally ubiquitous system of
rectangles with respect to $(\rho,\mathbf{a})$. Then

$\operatorname{\dim_{H}}{W(\mathbf{t})}\geq\min_{A\in{\cal
A}}\left\\{\sum_{j\in{\cal K}_{1}}{\delta_{j}}+\sum_{j\in{\cal
K}_{2}}{\delta_{j}}+\kappa\sum_{j\in{\cal
K}_{3}}{\delta_{j}}+(1-\kappa)\frac{\sum_{j\in{\cal
K}_{3}}{a_{j}\delta_{j}}-\sum_{j\in{\cal
K}_{2}}{t_{j}\delta_{j}}}{A}\right\\}$

where

${\cal A}=\\{a_{i},a_{i}+t_{i}:1\leq i\leq k\\}$

and, for each $A\in{\cal A}$, the sets ${\cal K}_{1},{\cal K}_{2},{\cal
K}_{3}$ are defined by

$\displaystyle{\cal K}_{1}$ $\displaystyle=\\{1\leq j\leq k:a_{j}\geq A\\},$
$\displaystyle{\cal K}_{2}$ $\displaystyle=\\{1\leq j\leq k:a_{j}+t_{j}\leq
A\\}\setminus{\cal K}_{1},$ $\displaystyle{\cal K}_{3}$
$\displaystyle=\\{1,\dots,k\\}\setminus({\cal K}_{1}\cup{\cal K}_{2}).$

Note that the sets ${\cal K}_{1},{\cal K}_{2},{\cal K}_{3}$ give a partition
of $\\{1,\dots,k\\}$.

Furthermore, Wang and Wu also obtain the following Hausdorff measure result in
[47].

######  Theorem 15.

Under the same assumptions as in Theorem 14, for any ball $B$ in $X$, we have

${\cal H}^{s}(B\cap W(\mathbf{t}))={\cal H}^{s}(B)$

for

$s=\min_{A\in{\cal A}}\left\\{\sum_{j\in{\cal
K}_{1}}{\delta_{j}}+\sum_{j\in{\cal K}_{2}}{\delta_{j}}+\kappa\sum_{j\in{\cal
K}_{3}}{\delta_{j}}+(1-\kappa)\frac{\sum_{j\in{\cal
K}_{3}}{a_{j}\delta_{j}}-\sum_{j\in{\cal
K}_{2}}{t_{j}\delta_{j}}}{A}\right\\}.$

Assuming uniform local ubiquity for rectangles, Wang and Wu generalise
Theorems 14 and 15 in [47] to analogous results with more general
approximating functions. However, we shall not elaborate further on these more
general statements here.

We conclude this section by stating the “full measure” mass transference
principle from rectangles to rectangles also established by Wang and Wu in
[47].

######  Theorem 16.

Assume the setting as described above and further suppose that

$m\left(\limsup_{\begin{subarray}{c}\alpha\in J\\\
\beta(\alpha)\to\infty\end{subarray}}{\Delta({\cal
R}_{\alpha},\rho(\beta(\alpha))^{\mathbf{a}})}\right)=m(X).$

Then,

$\operatorname{\dim_{H}}{W(\mathbf{t})}\geq\min_{A\in{\cal
A}}\left\\{\sum_{j\in{\cal K}_{1}}{\delta_{j}}+\sum_{j\in{\cal
K}_{2}}{\delta_{j}}+\kappa\sum_{j\in{\cal
K}_{3}}{\delta_{j}}+(1-\kappa)\frac{\sum_{j\in{\cal
K}_{3}}{a_{j}\delta_{j}}-\sum_{j\in{\cal
K}_{2}}{t_{j}\delta_{j}}}{A}\right\\}$

where

${\cal A}=\\{a_{i},a_{i}+t_{i}:1\leq i\leq k\\}$

and, for each $A\in{\cal A}$, the sets ${\cal K}_{1},{\cal K}_{2},{\cal
K}_{3}$ are defined by

$\displaystyle{\cal K}_{1}$ $\displaystyle=\\{1\leq j\leq k:a_{j}\geq A\\},$
$\displaystyle{\cal K}_{2}$ $\displaystyle=\\{1\leq j\leq k:a_{j}+t_{j}\leq
A\\}\setminus{\cal K}_{1},$ $\displaystyle{\cal K}_{3}$
$\displaystyle=\\{1,\dots,k\\}\setminus({\cal K}_{1}\cup{\cal K}_{2}).$

Note that the conclusion in Theorem 16 is the same as that in Theorem 14.

##  9 Mass Transference Principles from Balls to Open Sets

In this section we present mass transference principles which further
generalise some of the theorems seen in earlier sections. In particular, we
present results which allow us to pass from Lebesgue measure statements for
limsup sets defined by sequences of balls to Hausdorff measure and dimension
statements for related limsup sets generated by arbitrary open sets.

Fix $k\in{\mathbb{N}}$, let $(x_{n})_{n\in{\mathbb{N}}}$ be a sequence of
points in $[0,1]^{k}$ and let $(r_{n})_{n\in{\mathbb{N}}}$ be a sequence of
positive real numbers such that $r_{n}\to 0$ as $n\to\infty$. Consider the
sequence of balls $(B_{n}:=B(x_{n},r_{n}))_{n\in\mathbb{N}}$ in $[0,1]^{k}$
and suppose that

$\mathcal{L}\left(\limsup_{n\to\infty}{B_{n}}\right)=1.$

In this situation, it follows from the mass transference principles presented
earlier in Sections 3 and 7 that we can estimate

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}{B(x_{n},r_{n}^{\delta})}\right)$

where $\delta>0$ and

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}(B^{\mathbf{a}}(x_{n},r_{n}))\right)$

where $\mathbf{a}=(a_{1},\dots,a_{k})\in{\mathbb{R}}^{k}_{\geq 0}$ and $1\leq
a_{1}\leq...\leq a_{k}$. The notation $B^{\mathbf{a}}(x,r)$ here is as defined
in Section 7. It is natural to ask whether one can derive a more general mass
transference principle when one shrinks each ball $B_{n}$ into a set
$U_{n}\subset B_{n}$ subject to some appropriate geometric constraints. It is
easy to see that if the sets $U_{n}$ have empty interior, then not much can be
said in general so we will consider the case where the sets $U_{n}$ are open.
In this case Koivusalo and Rams [38] obtained a lower bound for

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}{U_{n}}\right)$

in terms of the Hausdorff content of the sets $U_{n}$. More precisely, they
established the following result in [38].

######  Theorem 17.

Let $(B_{n}:=B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in
$[0,1]^{k}$ with $r_{n}\to 0$ as $n\to\infty$ and suppose that

${\cal L}\left(\limsup_{n\to\infty}{B_{n}}\right)=1.$

Let $(U_{n})_{n\in{\mathbb{N}}}$ be sequence of open subsets of $[0,1]^{k}$
such that for every $n\in{\mathbb{N}}$, we have $U_{n}\subset B_{n}$. Suppose
that there exists a real number $0<s<k$ such that for every $n\in\mathbb{N}$
we have

$\mathcal{H}^{s}_{\infty}(U_{n})\geq\mathcal{L}(B_{n}).$ (7)

Then

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}U_{n}\right)\geq s.$

Koivusalo and Rams in fact prove a different statement in [38] from which they
deduce Theorem 17 as an immediate corollary. The statement which they prove is
formulated in terms of a generalised version of the singular value function
(defined by (2)). The advantage of the generalised singular value function
introduced in [38] is that it applies to any set rather than being limited to
rectangles or ellipsoids. To be more precise, let $E\subset{\mathbb{R}}^{k}$
be a Borel set and let $s\geq 0$ be a real number. We define the _generalised
singular value function_ of $E$ as

$\phi_{s}(E)=\sup_{\mu\in\mathcal{M}(E)}\inf_{x\in
E}\inf_{r>0}\frac{r^{s}}{\mu(B(x,r))}.$

With this definition, Koivusalo and Rams were able to prove the following
result in [38].

######  Theorem 18.

Let $(B_{n}:=B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in
$[0,1]^{k}$ with $r_{n}\to 0$ as $n\to\infty$ and suppose that

${\cal L}\left(\limsup_{n\to\infty}{B_{n}}\right)=1.$

Let $(U_{n})_{n\in{\mathbb{N}}}$ be sequence of open subsets of $[0,1]^{k}$
such that for every $n\in{\mathbb{N}}$, we have $U_{n}\subset B_{n}$. Suppose
that there exists a real number $s\geq 0$ such that for every $n\in\mathbb{N}$
we have

$\phi_{s}(U_{n})\geq{\cal L}(B_{n}).$ (8)

Then

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}U_{n}\right)\geq s.$

That Theorem 17 is a corollary of Theorem 18 follows from the observation of
Koivusalo and Rams in [38, Proposition 2.1] that Hausdorff content and the
singular value function agree up to a multiplicative constant.

Notice that Theorems 17 and 18 only relate to the Hausdorff dimension of
$\limsup_{n\to\infty}U_{n}$. A natural refinement of Theorem 18, determining
the Hausdorff $f$-measures of $\limsup_{n\to\infty}U_{n}$, was established by
Zhong in [52]. To obtain Hausdorff $f$-measure results for general dimension
functions, Zhong first further extended the definition of the generalised
singular value function in the following way. Let
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a dimension function, let
$E\subset{\mathbb{R}}^{k}$ and let $s\geq 0$. We define the _generalised
$f$-singular value function_ of $E$ as

$\phi_{f}(E)=\sup_{\mu\in\mathcal{M}(E)}\inf_{x\in
E}\inf_{r>0}\frac{f(r)}{\mu(B(x,r))}.$

Zhong proved the following in [52].

######  Theorem 19.

Let $(B_{n}:=B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in
$[0,1]^{k}$ with $r_{n}\to 0$ as $n\to\infty$ and suppose that

${\cal L}\left(\limsup_{n\to\infty}{B_{n}}\right)=1.$

Let $(U_{n})_{n\in{\mathbb{N}}}$ be sequence of open subsets of $[0,1]^{k}$
such that for every $n\in{\mathbb{N}}$, we have

$U_{n}\subset B_{n}\qquad\text{and}\qquad\lim_{n\to\infty}{\frac{{\cal
L}(B_{n})}{{\cal L}(U_{n})}}=\infty.$

Let $f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ be a dimension function such
that $r^{-k}f(r)$ is monotonically decreasing. Suppose that for every
$n\in\mathbb{N}$ we have

$\phi_{f}(U_{n})\geq{\cal L}(B_{n}).$ (9)

Then, for any ball $B$ in $[0,1]^{k}$, we have

${\cal H}^{f}\left(B\cap\limsup_{n\to\infty}{U_{n}}\right)={\cal H}^{f}(B).$

We add to this section that a comparable result was also obtained by Persson
in [42]. In order to state this result, let us define the _Riesz energy_ of a
set $E$.

######  Definition 5.

Let $E\subset\mathbb{R}^{k}$ be a Borel set and let $0\leq s\leq k$. We define
the _$s$ -dimensional Riesz energy_ of $E$ as

$I_{s}(E)=\int_{E}\int_{E}\frac{1}{|x-y|^{s}}dxdy.$

We also need to define the _class of large intersection of dimension $s$_.

######  Definition 6.

Let $0\leq s\leq k$ be a real number. Then the _$s$ -dimensional class of
large intersection of $[0,1]^{k}$_, denoted by ${\cal G}_{s}([0,1]^{k})$, is
defined to be the set of all $G_{\delta}$-sets (countable intersections of
dense open sets) $E$ such that

${\cal H}^{t}_{\infty}(E\cap B(x,r))=r^{t}$

for every $t<s$ and for every ball $B(x,r)\subset[0,1]^{k}$.

Persson proved the following result in [42].

######  Theorem 20.

Let $(B_{n}:=B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in
$[0,1]^{k}$ with $r_{n}\to 0$ as $n\to\infty$ and suppose that

${\cal L}\left(\limsup_{n\to\infty}B_{n}\right)=1.$

Let $(U_{n})_{n\in\mathbb{N}}$ be a sequence of open sets in $[0,1]^{k}$ such
that $U_{n}\subset B_{n}$ for every $n\in{\mathbb{N}}$. Let

$s=\sup\left\\{t>0:\sup_{n\in{\mathbb{N}}}{\frac{I_{t}(U_{n}){\cal
L}(B_{n})}{{\cal L}(U_{n})^{2}}}<\infty\right\\}.$

Then

$\operatorname{\dim_{H}}\limsup_{n\to\infty}U_{n}\geq s,$

and

$\limsup_{n\to\infty}U_{n}\in\mathcal{G}_{s}([0,1]^{k}).$

It is worth noting that the bound provided by Theorem 20 does not improve upon
Theorems 17 and 18 in terms of dimension, see [42] for further discussion
regarding this point. However, Theorem 20 implies that the set
$\limsup_{n\to\infty}U_{n}$ belongs to $\mathcal{G}_{s}([0,1]^{k})$, which is
not established by Theorem 17.

In the setting of general metric spaces, the following mass transference
principle from balls to arbitrary open sets has recently been established by
Eriksson-Bique in [22].

######  Theorem 21.

Let $(X,d)$ be a complete metric space, let $s\geq 0$ be a real number and let
$(B(x_{n},r_{n}))_{n\in{\mathbb{N}}}$ be a sequence of balls in $X$ with
$r_{n}\to 0$ as $n\to\infty$. Suppose that, for every ball $B=B(x,r)$ in $X$,
we have

$\mathcal{H}^{s}_{\infty}\left(B\cap\limsup_{n\to\infty}B(x_{n},r_{n})\right)\geq
Cr^{s}.$ (10)

Suppose that $(U_{n})_{n\in{\mathbb{N}}}$ is an arbitrary sequence of open
sets in $X$ such that $U_{n}\subset B(x_{n},r_{n})$ for every
$n\in{\mathbb{N}}$. Furthermore, suppose that there exists some $0\leq t\leq
s$ such that for every $n\in{\mathbb{N}}$ we have

${\cal H}^{t}_{\infty}(U_{n})\geq r_{n}^{s}.$ (11)

Then, there exists a constant $\kappa>0$ such that for every ball $B$ in $X$,
we have

$\mathcal{H}^{t}_{\infty}\Big{(}B\cap\limsup_{n\to\infty}U_{n}\Big{)}\geq\kappa\mathcal{H}^{t}_{\infty}(B).$

In particular, $\operatorname{\dim_{H}}\limsup_{n\to\infty}U_{n}\geq t.$

Note that, by (10), the hypothesis on the sequence $(B_{n})_{n\in\mathbb{N}}$
is comparable to belonging to the $s$-dimensional class of large intersection.
Moreover, by (11), in the setting of Theorem 21, the set
$\limsup_{n\to\infty}U_{n}$ also satisfies a property that is comparable to
belonging to the $t$-dimensional class of large intersection. For the sake of
simplicity, Theorem 21 is stated only for traditional Hausdorff $s$-measure
but we remark for completeness that Eriksson-Bique obtains results for more
general dimension functions in [22].

##  10 Mass Transference Principles for Inhomogeneous Measures

In this section we discuss mass transference principles where the measures
involved are not necessarily Ahlfors-David regular. We will refer to measures
which are not Ahlfors-David regular as _inhomogeneous measures_.

###  10.1 Multifractal mass transference principles

In many dynamical settings and in multifractal analysis, limsup sets generated
by balls well distributed according to a multifractal measure arise naturally
and various authors have developed tools similar to the mass transference
principle of Beresnevich and Velani [12] in order to deal with such cases. The
first result of this kind was established in 2007 by Barral and Seuret in [6].
Barral and Seuret obtained dimension estimates similar to those obtained by
Jaffard [32, 33] and Beresnevich and Velani [12], but where the measure
involved is a multinomial measure, a Gibbs measure on the dyadic grid, a
multiplicative cascade or, more generally, a measure which carries enough
self-similarity. The particular case where the measure is a Gibbs measure on
the dyadic grid was also established independently later on in 2013 by Fan,
Shmeling and Troubetzkoy in [25] when studying shrinking targets associated
with the doubling map on $\mathbb{T}^{1}$.

In order to state the results of [6] and [25], let us first recall several
relevant definitions. Let $f:\left\\{0,1\right\\}^{\mathbb{N}}\to\mathbb{R}$
be a Hölder continuous function, let $k\in{\mathbb{N}}$ and let
$\mathbf{i}=(i_{1},...,i_{k})\in\left\\{0,1\right\\}^{k}.$ Denote by $\sigma$
the _left-shift_ on $\left\\{0,1\right\\}^{\mathbb{N}}$; that is,

$\sigma(i_{1},i_{2},i_{3},i_{4},\dots)=(i_{2},i_{3},i_{4},i_{5},\dots)$

for $(i_{1},i_{2},i_{3},i_{4},\dots)\in\\{0,1\\}^{{\mathbb{N}}}$. For finite
words
$\mathbf{i}=(i_{1},\dots,i_{|\mathbf{i}|})\in\bigcup_{N=0}^{\infty}{\\{0,1\\}^{N}}$
we define the _cylinder set_ determined by $\mathbf{i}$ as

$[\mathbf{i}]=\left\\{\mathbf{j}\in\\{0,1\\}^{{\mathbb{N}}}:i_{\ell}=j_{\ell}\quad\text{for
}1\leq\ell\leq|\mathbf{i}|\right\\}.$

Next, define

$\displaystyle
Y_{k,f}(\mathbf{i})=\sup_{\mathbf{j}\in[\mathbf{i}]}e^{\sum_{0\leq i\leq
k-1}f(\sigma^{i}(\mathbf{j}))}$

and

$\displaystyle
Z_{k,f}=\sum_{\mathbf{i}\in\left\\{0,1\right\\}^{k}}Y_{k,f}(\mathbf{i}).$

We define the pressure of $f$ as

$P_{f}=\lim_{k\to\infty}\frac{\log Z_{k,f}}{k}.$ (12)

The _Gibbs measure_ associated with the potential $f$ is defined to be the
unique probability measure on $\Lambda^{\mathbb{N}},$ $\nu_{f}$, such that for
every $k\in\mathbb{N}$, for every $\mathbf{i}\in\left\\{0,1\right\\}^{k}$ and
every $\mathbf{j}\in[\mathbf{i}]$, we have

$C_{1}e^{\sum_{0\leq i\leq
k-1}f(\sigma^{i}(\mathbf{j})-kP_{f}}\leq\nu_{f}([\mathbf{i}])\leq
C_{2}e^{\sum_{0\leq i\leq k-1}f(\sigma^{i}(\mathbf{j}))-kP_{f}}.$ (13)

for some constants $C_{1},C_{2}>0$. It is proved in [15] that such a measure
exists.

For
$\mathbf{i}=(i_{1},i_{2},i_{3},i_{4},\dots)\in\left\\{0,1\right\\}^{\mathbb{N}}$,
the _projection associated with the dyadic grid_
$\pi:\left\\{0,1\right\\}^{\mathbb{N}}\to[0,1]$ is defined as

$\pi(\mathbf{i})=\sum_{n\in{\mathbb{N}}}2^{-n}i_{n}.$

The following result was established by Barral and Seuret in [6] with the
aforementioned special case being proved independently later by Fan, Schmeling
and Troubetzkoy in [25].

######  Theorem 22.

Let $f$ be a potential on $\left\\{0,1\right\\}^{\mathbb{N}}$, let $\nu_{f}$
the Gibbs measure associated with $f$, and let $\mu_{f}=\nu_{f}\circ\pi^{-1}.$
Let $(B_{n}:=B(x_{n},r_{n}))_{n\in\mathbb{N}}$ be any sequence of balls (i.e.
intervals) in $[0,1]$ such that $r_{n}\to 0$ as $n\to\infty$ and

$\mu_{f}(\limsup_{n\to\infty}B_{n})=1.$

Then, for any $\delta\geq 1$, we have

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}B(x_{n},r_{n}^{\delta})\right)\geq\frac{\dim(\mu_{f})}{\delta}.$

Theorem 22 gives rise to the following “multifractal mass transference
principle” as observed in [6, 25].

######  Corollary 1.

Let

$\mathcal{D}=\left\\{\left[\frac{i}{2^{k}},\frac{i+1}{2^{k}}\right]:n\in\mathbb{N},0\leq
i\leq 2^{k}-1\right\\}$

be the set of closed dyadic intervals in $[0,1]$ and let
$\mu\in\mathcal{M}(\mathbb{R})$ a Gibbs measure associated with the dyadic
grid. For

$D=\left[\frac{i}{2^{k_{D}}},\frac{i+1}{2^{k_{D}}}\right]\in\mathcal{D},$

define

$\varepsilon_{D}=\sqrt{\frac{\log\log k_{D}}{\log k_{D}}}.$

For every $\alpha\in\mathbb{R}$ and $\delta\geq 1$, we have

$\operatorname{\dim_{H}}\limsup_{\begin{subarray}{c}D\in\mathcal{D}:\\\
2^{-k_{D}(\alpha-\varepsilon_{D})}\leq\mu(D)\leq
2^{-k_{D}(\alpha-\varepsilon_{D})}\end{subarray}}D^{\delta}=\frac{\tau_{\mu}^{*}(\alpha)}{\delta}.$

As the multifractal analysis of Gibbs measures is naturally correlated with
the multifractal analysis of the Birkhoff average of a given potential,
Theorem 22 turns out to be particularly well suited to the study of shrinking
targets which are dynamically defined. For example, Theorem 22 has been used
to study shrinking targets associated with the doubling map by Fan, Schmeling
and Troubetzkoy in [25] and shrinking targets associated with expanding Markov
maps by Liao and Seuret in [41]. We conclude this section by noting that other
multifractal mass transference principle type statements have been obtained in
the random (rather than deterministic) setting by Ekström and Persson [21] and
Persson and Rams [43].

###  10.2 A mass transference principle subject to a full inhomogeneous
measure statement

When dealing with inhomogeneous measures (i.e, measures which are not Ahlfors-
David regular), one needs to introduce a geometric quantity which plays the
same role as the Hausdorff content does in Theorem 17. The following quantity
was introduced in [16] for this purpose. Let $A\subset\mathbb{R}^{k}$ be a
Borel set and let $\mu\in\mathcal{M}(\mathbb{R}^{k}).$ For $s\geq 0,$ we
define the _$\mu$ -essential Hausdorff content_ of $A$ by setting

$\mathcal{H}^{\mu,s}_{\infty}(A)=\inf\left\\{\mathcal{H}^{s}_{\infty}(E):E\subset
A\text{ and }\mu(E)=\mu(A)\right\\}.$

Using this quantity, the following Theorem was established in [16].

######  Theorem 23.

Let $\mu\in\mathcal{M}(\mathbb{R}^{k})$, and suppose
$(B_{n}:=B(x_{n},r_{n}))_{n\in\mathbb{N}}$ is a sequence of closed balls in
$\mathbb{R}^{k}$ with $r_{n}\to 0$ as $n\to\infty$ and such that

$\mu\left(\limsup_{n\to\infty}\frac{1}{2}B_{n}\right)=1.$

Let $(U_{n})_{n\in\mathbb{N}}$ be a sequence of open sets such that
$U_{n}\subset B_{n}$ for every $n\in\mathbb{N}$. Let $0\leq
s\leq\underline{\dim}(\mu)$ be a real number such that for every
$n\in\mathbb{N}$ we have

$\mathcal{H}^{\mu,s}_{\infty}(U_{n})\geq\mu(B_{n}).$

Then there exists a dimension function
$f:{\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}$ such that

$\lim_{r\to 0}\frac{\log f(r)}{\log
r}=s\quad\text{and}\quad\mathcal{H}^{f}\left(\limsup_{n\to\infty}U_{n}\right)>0.$

In particular,

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}U_{n}\right)\geq s.$

In [16], precise estimates of the $\mu$-essential content are provided. In
particular, the following corollary is established in [16].

######  Corollary 2.

Let $m\geq 2$ be an integer, let
$\Phi=\left\\{\varphi_{1},...,\varphi_{m}\right\\}$ be a self-similar IFS and
let $X$ be the attractor of $\Phi$. Let $\mu\in\mathcal{M}(\mathbb{R}^{k})$ be
a self-similar measure associated with $\Phi$. Recall that
$\operatorname{supp}{\mu}=X$. Let $(B_{n}:=B(x_{n},r_{n}))_{n\in\mathbb{N}}$
be a sequence of balls in ${\mathbb{R}}^{k}$ centered on $X$ such that
$r_{n}\to 0$ and $n\to\infty$ and

$\mu\left(\limsup_{n\to\infty}B_{n}\right)=1.$

Then for every $\delta\geq 1,$

$\operatorname{\dim_{H}}\left(\limsup_{n\to\infty}B(x_{n},r_{n}^{\delta})\right)\geq\frac{\dim(\mu)}{\delta}.$

Note that this corollary holds without any separation assumption on the
iterated function system $\Phi$. Corollary 2 has been used in [17] to study
the dimension of certain shrinking targets associated with some overlapping
self-similar iterated function systems.

#### Acknowledgements.

We thank the organisers of the Fractals and Related Fields IV conference in
Porquerolles in September 2022 for an excellent conference on a beautiful
island.

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