A General Mass Transference Principle

TL;DR

This paper introduces a broad generalization of the Mass Transference Principle, enabling the transfer of measure statements for a wide class of sets including self-similar sets and manifolds, in various metric spaces.

Contribution

It extends the Mass Transference Principle to more general settings and types of sets, broadening its applicability in geometric measure theory.

Findings

Proves a general form of the Mass Transference Principle for $ ext{limsup}$ sets.

Applicable to self-similar sets and smooth manifolds in $ extbf{R}^n$.

Extends previous results to locally compact metric spaces.

Abstract

In this paper we prove a general form of the Mass Transference Principle for $\limsup$ sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in $\mathbb{R}^n$. Our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff $g$-measure statements to Hausdorff $f$-measure statements. This work extends previous results of this type in several distinct directions.

Equations

$$\underset{j\to \infty }{\mathrm{limsup}}{E}_j$$

$$\displaystyle=\big{\{}x:x\in E_{j}\textrm{ for infinitely many }j\in\mathbb{N}\big{\}}.$$

$${\cal H}^{n}\big{(}\/B\cap\limsup_{j\to\infty}B^{f}_{j}{}\,\big{)}={\cal H}^{n}(B)\ .$$

$${\cal H}^{f}\big{(}\/B\cap\limsup_{j\to\infty}B^{n}_{j}\,\big{)}={\cal H}^{f}(B)\ .$$

$$f(2x)<\lambda f(x)$$

$$c_{1}g(r)\le {\mathcal{H}}^g(B(x,r))\le c_{2}g(r)$$

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

$${\mathcal{H}}^g(B\cap \underset{j\to \infty }{\mathrm{limsup}}{B}_j^{f,g})={\mathcal{H}}^g(B).$$

$${\mathcal{H}}^f(B\cap \underset{j\to \infty }{\mathrm{limsup}}{B}_j)={\mathcal{H}}^f(B).$$

$$\Delta (R_j,\delta ):=\{\mathbf{x}\in {\mathbb{R}}^n:\mathrm{dist}(\mathbf{x},R_j)<\delta \},$$

$$\Lambda ({\rm Y}):=\{\mathbf{x}\in {\mathbb{R}}^n:\mathbf{x}\in \Delta (R_j,{{\rm Y}}_j)\text{ for infinitely many }j\in \mathbb{N}\}.$$

$${\mathcal{H}}^n\left(B\cap \Lambda \left(g({\rm Y}{)}^{\frac{1}{m}}\right)\right)={\mathcal{H}}^n(B).$$

$${\mathcal{H}}^f(B\cap \Lambda ({\rm Y}))={\mathcal{H}}^f(B).$$

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

$$c_{3}g(\delta {)}^{1-\kappa }\cdot g(r{)}^{\kappa }\le {\mathcal{H}}^g(B(x,r)\cap \Delta (F_j,\delta ))\le c_{4}g(\delta {)}^{1-\kappa }\cdot g(r{)}^{\kappa }.$$

$$c_3{\delta }^{n-\kappa n}\cdot r^{\kappa n}\le {\mathcal{H}}^n(B(x,r)\cap \Delta (F_j,\delta ))\le c_4{\delta }^{n-\kappa n}\cdot r^{\kappa n}.$$

$$\Lambda ({\rm Y}):=\{x\in X:x\in \Delta (F_j,{{\rm Y}}_j)\text{ for infinitely many }j\in \mathbb{N}\}.$$

$${\mathcal{H}}^g\left(B\cap \Lambda \left(g^{-1}\left({\left(\frac{f({\rm Y})}{g({\rm Y}{)}^{\kappa }}\right)}^{\frac{1}{1-\kappa }}\right)\right)\right)={\mathcal{H}}^g(B).$$

$${\mathcal{H}}^f(B\cap \Lambda ({\rm Y}))={\mathcal{H}}^f(B).$$

$${\mathcal{H}}^n(B\cap \Lambda ({{\rm Y}}^{\frac{s-\kappa n}{(1-\kappa )n}}))={\mathcal{H}}^n(B).$$

$${\mathcal{H}}^s(B\cap \Lambda ({\rm Y}))={\mathcal{H}}^s(B).$$

$${\underline{\dim }}_{\mathrm{B}}(F):=n-\underset{\delta \to 0}{\mathrm{limsup}}\frac{\log {\mathcal{H}}^n(\Delta (F,\delta ))}{\log \delta }$$

$${\overline{\dim }}_{\mathrm{B}}(F):=n-\underset{\delta \to 0}{\mathrm{liminf}}\frac{\log {\mathcal{H}}^n(\Delta (F,\delta ))}{\log \delta }.$$

$$\underline{M}(F):=\underset{\delta \to 0}{\mathrm{liminf}}{\delta }^{n-\mathrm{di}{\mathrm{m}}_{\mathrm{B}}(F)}\cdot {\mathcal{L}}^n(\Delta (F,\delta ))$$

$$\overline{M}(F):=\underset{\delta \to 0}{\mathrm{limsup}}{\delta }^{n-\mathrm{di}{\mathrm{m}}_{\mathrm{B}}(F)}\cdot {\mathcal{L}}^n(\Delta (F,\delta )).$$

$${\mathcal{H}}^l({\pi }_l(B(y,r)\cap M))\asymp r^l.$$

$${\mathcal{H}}^{n-l}(\{z\in \Delta (M,\delta ):\pi (z)=\pi (y)\})\asymp {\delta }^{n-l}.$$

$${\mathcal{H}}^n(B(y,r)\cap \Delta (M,\delta ))\asymp r^l{\delta }^{n-l}.$$

$$|{\varphi }_i(x)-{\varphi }_i(y)|=r_i|x-y|\text{for all }x,y\in {\mathbb{R}}^n,$$

$$K=\bigcup _{i=1}^l{\varphi }_i(K).$$

Full text

2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11K60.

Keywords and phrases: Mass Transference Principle, Hausdorff measures, $\limsup$ sets, Diophantine Approximation.

1 Introduction

1.1 Background

In Diophantine Approximation, Dynamical Systems, and Probability Theory, many sets of interest can be characterised as $\limsup$ sets. Recall that, given a countable collection of sets $(E_{j})_{j\in{\mathbb{N}}},$ we define the corresponding $\limsup$ set to be

[TABLE]

Often we are interested in determining the metric properties of $\limsup E_{j}.$ When the sequence $(E_{j})_{j\in{\mathbb{N}}}$ is a collection of balls a powerful tool in determining the metric properties of $\limsup E_{j}$ is the Mass Transference Principle [5], which allows us to infer Hausdorff measure statements from seemingly less general Lebesgue measure statements.

Given a ball $B:=B(x,r)$ in ${\mathbb{R}}^{n}$ and a dimension function $f:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ (see Section 3 for definitions), define another corresponding ball $B^{f}:=B(x,f(r)^{\frac{1}{n}})$. Throughout, ${\mathbb{R}}^{+}:=[0,\infty)$. When $f(r)=r^{s}$ for some real number $s>0$, we write $B^{s}$ in place of $B^{f}$. In particular, $B^{n}=B$ for $n\in{\mathbb{N}}$. The following was established by Beresnevich and Velani in [5].

Mass Transference Principle**.**

Let $(B_{j})_{j\in{\mathbb{N}}}$ be a sequence of balls in ${\mathbb{R}}^{n}$ with $r(B_{j})\to 0$ as $j\to\infty$. Let $f$ be a dimension function such that $x^{-n}f(x)$ is monotonic and suppose that, for any ball $B$ in ${\mathbb{R}}^{n}$,

[TABLE]

Then, for any ball $B$ in ${\mathbb{R}}^{n}$,

[TABLE]

We denote by ${\cal H}^{f}(X)$ the Hausdorff $f$-measure of a set $X\subset{\mathbb{R}}^{n}$. For $s\geq 0$, ${\cal H}^{s}(X)$ denotes the standard Hausdorff $s$-measure of $X$. These notions will be formally introduced in Section 3. It is worth noting at this point though that if $X$ is a Borel subset of ${\mathbb{R}}^{n}$, then ${\cal H}^{n}(X)$ is a constant multiple times the $n$-dimensional Lebesgue measure of $X$ (see [9] for further details). Thus, as discussed previously, the Mass Transference Principle genuinely does enable us to transfer Lebesgue measure statements to Hausdorff measure ones.

A generalisation of the Mass Transference Principle, which is applicable to $\limsup$ sets of balls in locally compact metric spaces and not restricted to $\limsup$ sets in ${\mathbb{R}}^{n}$, was also given by Beresnevich and Velani in [5]. Furthermore, this generalisation allows for the transference of Hausdorff $g$-measure (not just Lebesgue measure) statements to Hausdorff $f$-measure statements, where $g$ and $f$ are dimension functions subject to some mild conditions. Before stating this result formally, we require some preliminaries.

We say that a function $f:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ is doubling if there exists a constant $\lambda>1$ such that

[TABLE]

for all $x>0$.

Let $(X,d)$ be a locally compact metric space and let $g$ be a doubling dimension function. Moreover, suppose that there exist constants $0<c_{1}<1<c_{2}<\infty$ and $r_{0}>0$ such that

[TABLE]

for all $x\in X$ and $0<r<r_{0}$. Given another dimension function $f$ and a ball $B:=B(x,r)$ in $X$ we define

[TABLE]

The following theorem was established in [5].

Theorem MTP***.**

Let $(X,d)$ and $g$ be as above and let $(B_{j})_{j\in{\mathbb{N}}}$ be a sequence of balls in $X$ with $r(B_{j})\to 0$ as $j\to\infty$. Let $f$ be a dimension function such that $f/g$ is monotonic and suppose that for any ball $B$ in $X$ we have

[TABLE]

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

[TABLE]

Compared with the Mass Transference Principle, this general theorem applies to $\limsup$ sets of balls in more general metric spaces and deals with a larger class of (Hausdorff) measures. As an example, Theorem MTP* is applicable when $X$ is, say, the middle-third Cantor set. In this case, Theorem MTP* has been utilised in [17] to solve a problem posed by Mahler regarding the existence of very well approximable points in the middle-third Cantor set.

The Mass Transference Principle was originally motivated by a desire to establish a Hausdorff measure analogue of the famous Duffin–Schaeffer Conjecture in Metric Number Theory. Since their initial announcement, the Mass Transference Principle and Theorem MTP* have been shown to have applications in many distinct areas of mathematics. In particular, the fields of Number Theory, Dynamical Systems, and Fractal Geometry have all benefited significantly from these results. For further applications of the Mass Transference Principle and Theorem MTP* see [2, 4, 5, 15, 23].

In the Euclidean setting, Beresnevich and Velani extended the Mass Transference Principle in another direction to allow for the transference of Lebesgue measure statements to Hausdorff measure statements for $\limsup$ sets defined via neighbourhoods of “approximating planes” [6]. The result they obtained in this case, [6, Theorem 3], was subject to extra technical conditions arising from their particular proof strategy. Recently, the first author and Beresnevich have removed these additional constraints in [1], thus unlocking a number of previously inaccessible applications in Diophantine Approximation. We include here a statement of [1, Theorem 1] to enable ease of comparison of this mass transference principle for planes with the statement of the main result of the present article, namely Theorem 1 below.

Let $n,m\geq 1$ and $l\geq 0$ be integers such that $n=m+l$. Let ${\cal R}:=(R_{j})_{j\in{\mathbb{N}}}$ be a family of $l$-dimensional planes in ${\mathbb{R}}^{n}$. For every $j\in{\mathbb{N}}$ and $\delta\geq 0$, define

[TABLE]

where $\operatorname{dist}({\mathbf{x}},R_{j})=\inf\{\|{\mathbf{x}}-{\mathbf{y}}\|:{\mathbf{y}}\in R_{j}\}$ and $\|\cdot\|$ is any fixed norm on ${\mathbb{R}}^{n}$.

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

[TABLE]

The following is shown in [1].

Theorem AB**.**

Let ${\cal R}$ and $\Upsilon$ be as given above. Let $f$ and $g:r\to g(r):=r^{-l}f(r)$ be dimension functions such that $r^{-n}f(r)$ is monotonic and let $\Omega$ be a ball in ${\mathbb{R}}^{n}$. Suppose that, for any ball $B$ in $\Omega$,

[TABLE]

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

[TABLE]

While we will be concerned here with generalising the aforementioned variations of the Mass Transference Principle, we remark here, for completeness, that progress towards mass transference principles has also been made in some other settings. For example, progress towards proving a mass transference principle for rectangles in the Euclidean setting has been made by Wang, Wu and Xu in [24] and an implicit multifractal mass transference principle is given by Fan, Schmeling and Troubetzkoy in [10].

In this paper, we extend the results of [1, 5, 6] by proving a general version of the Mass Transference Principle that applies to $\limsup$ sets in a locally compact metric space which are defined in terms of neighbourhoods of sets satisfying a certain local scaling property (see Section 1.2). Our main result, Theorem 1, extends the known mass transference principles of [1, 5, 6] in several manners. First of all, while it incorporates mass transference principles for balls and planes, Theorem 1 is also applicable to more exotic sets. For example, we are able to consider $\limsup$ sets generated by sequences of neighbourhoods of smooth compact manifolds or self-similar fractals satisfying the open set condition. Furthermore, unlike in previously known variants of the Mass Transference Principle, as long as the local scaling property is satisfied, the sets generating the $\limsup$ sets in Theorem 1 need not all be of the same type (e.g. all balls or all planes). Secondly, we deal with $\limsup$ sets in a locally compact metric space $(X,d)$ and are not confined to the Euclidean setting. Finally, the result we derive allows us to transfer Hausdorff $g$-measure statements to Hausdorff $f$-measure statements, where $f$ and $g$ are dimension functions subject to some mild conditions.

Compared with its predecessors, the greater generality of Theorem 1 opens up a number of new possible applications to explore. We include one such application in Section 6 where we use Theorem 1 to deduce Hausdorff measure and dimension results for a family of random $\limsup$ sets. Within Diophantine Approximation, it is reasonable to expect that Theorem 1 will enable the establishment of further Hausdorff measure statements relating to approximation on manifolds (see [3, 4] and the references therein for more on this problem). Within Dynamical Systems, it is also reasonable to expect that Theorem 1 will allow one to study a wider class of shrinking target problems, in particular when our target is allowed to have a more exotic structure (see [20, 22, 23] and the references therein for more on this problem). We hope to return to these topics in a later work.

1.2 The Main Result

Let $(X,d)$ be a locally compact metric space and let $g$ be a doubling dimension function satisfying (2). Given $F\subset X$ and $\delta\geq 0$, we define the $\delta$-neighbourhood of $F$ to be

[TABLE]

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

The following local scaling property appears to be the key which enables us to prove a “unifying” mass transference principle which incorporates and extends the results presented in Section 1.1.

Local Scaling Property (LSP): Given a sequence of sets ${\cal F}:=(F_{j})_{j\in{\mathbb{N}}}$ in $X$ and $0\leq\kappa<1$, we say that ${\cal F}$ satisfies the local scaling property (LSP) with respect to $\kappa$ if there exist constants $c_{3},c_{4},r_{1}>0$ such that, for any $0<r<r_{1}$, $\delta<r$, $j\in{\mathbb{N}}$ and $x\in F_{j},$ we have

[TABLE]

If (3) is satisfied for one specific set $F$, say, we will also say that $F$ satisfies the local scaling property with respect to $\kappa$. It should be clear from context when we are referring to an individual set and when we are referring to a sequence of sets.

If we restrict ourselves to the case where $X=\mathbb{R}^{n}$ and ${\cal H}^{g}={\cal H}^{n}$ (that is, essentially, Lebesgue measure), then (3) takes the form

[TABLE]

Many familiar subsets of Euclidean space satisfy (4) for an appropriate choice of $\kappa$. For example, in the next section we show that any smooth compact manifold embedded in $\mathbb{R}^{n}$ satisfies (4). In addition, we show that self-similar sets satisfying the open set condition also satisfy the LSP. In these examples we will see that $\kappa$ is related to the box counting dimension of the sets we are considering and we offer some discussion as to why this should be the case. In the meantime, we present here two trivial examples of sets satisfying the LSP.

Example 1*.*

If ${\cal F}$ were a sequence of points, then it follows from (2) that ${\cal F}$ satisfies the LSP with respect to $\kappa=0$.

Example 2*.*

Suppose $F\subset\mathbb{R}^{n}$ is a specific set satisfying (4). One can then define a sequence ${\cal F}:=(F_{j})_{j\in{\mathbb{N}}}$ by defining each $F_{j}$ to be the image of $F$ under some isometry. Clearly ${\cal F}$ provides us with a sequence of sets satisfying the LSP.

Next, suppose $(\Upsilon_{j})_{j\in{\mathbb{N}}}$ is a sequence of non-negative reals such that $\Upsilon_{j}\to 0$ as $j\to\infty.$ Consider the $\limsup$ set

[TABLE]

Our main result is the following theorem.

Theorem 1**.**

Let $(X,d)$ and $g$ be as above. Let ${\cal F}:=(F_{j})_{j\in{\mathbb{N}}}$ be a sequence of sets satisfying 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 reals such that $\Upsilon_{j}\to 0$ as $j\to\infty$. Let $f$ be a dimension function such that $f/g$ is monotonic and $f/g^{\kappa}$ is a dimension function. Suppose that for any ball $B$ in $X$ we have

[TABLE]

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

[TABLE]

Taking $(F_{j})_{j\in{\mathbb{N}}}$ to be a sequence of points in $X$ and $\kappa=0$, as in Example 1, Theorem 1 coincides with Theorem MTP* given above. If, further, we insist that $X={\mathbb{R}}^{n}$ and $g(r)=r^{n}$, we recover the usual Mass Transference Principle. Theorem AB can also be deduced as a special case of Theorem 1 by taking $X=\Omega$ to be a suitable ball in ${\mathbb{R}}^{n}$, $(F_{j})_{j\in{\mathbb{N}}}$ to be a sequence of $l$-dimensional planes $(R_{j})_{j\in{\mathbb{N}}}$ in ${\mathbb{R}}^{n}$, $\kappa=\frac{l}{n}$, and $g(r)=r^{n}$.

Restricting our attention to Euclidean space and $s$-dimensional Hausdorff measures, Theorem 1 takes the following simpler form.

Corollary 1**.**

Let ${\cal F}:=(F_{j})_{j\in{\mathbb{N}}}$ be a sequence of subsets of $\mathbb{R}^{n}$ satisfying 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 reals such that $\Upsilon_{j}\to 0$ as $j\to\infty$. Let $s>\kappa n$ and suppose that for any ball $B$ in $\mathbb{R}^{n}$ we have

[TABLE]

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

[TABLE]

1.3 Structure of the paper

The rest of the paper is arranged as follows. Some examples of sets satisfying the LSP are outlined in Section 2. In Section 3 we recall some useful preliminaries from geometric measure theory. In Section 4 we prove some key technical lemmas which will be required throughout the proof of Theorem 1, which is presented in full in Section 5. Finally, in Section 6, we use Theorem 1 to deduce the Hausdorff dimension and measure of some random $\limsup$ sets.

2 Some sets satisfying the local scaling property

In this section we provide some discussion on connections between the LSP and box counting dimension and Minkowski content. Thereafter, we briefly detail two collections of sets that satisfy the LSP for appropriate choices of parameters.

2.1 Box counting dimension, Minkowski content and the LSP

In this section, we highlight the connection between (4) and the Minkowski/box counting dimension of a set and why these quantities are likely to be related to $\kappa$ for sets satisfying the LSP.

Instead of the usual box counting definition of lower and upper box counting dimension (see, for example, [9]), there is the following equivalent notion that is defined using the volume of a $\delta$-neighbourhood of a set. Given a bounded set $F\subset\mathbb{R}^{n}$, the lower and upper box counting dimension of $F$ are defined to be, respectively,

[TABLE]

and

[TABLE]

When these limits coincide we call the common value the box counting dimension of $F$ and denote it by $\operatorname{dim_{B}}(F)$. It follows from this definition that if $F$ were a set whose box counting dimension exists, and we were interested in whether (4) held for some $\kappa$, then the natural candidate for $\kappa$ would be $\dim_{B}(F)\cdot n^{-1}.$

Two other useful quantities that describe how the volume of a $\delta$-neighbourhood of a set $F$ scales are the lower and upper Minkowski content. Given a bounded set $F$ contained in $\mathbb{R}^{n}$ whose box counting dimension exists, we define the lower and upper Minkowski content to be, respectively,

[TABLE]

and

[TABLE]

Here ${\cal L}^{n}$ is the $n$-dimensional Lebesgue measure. When $\underline{M}(F)=\overline{M}(F)$ we call the common value the Minkowski content and denote it by $M(F).$ Determining conditions under which a set $F$ has both $\underline{M}(F)$ and $\overline{M}(F)$ positive and finite is a well studied problem, see [16]. Equation (4) does not follow directly from the positivity and finiteness of both $\underline{M}(F)$ and $\overline{M}(F).$ However, if $F$ were reasonably homogeneous in small neighbourhoods, we would expect this property to be sufficient to deduce (4).

2.2 Smooth compact manifolds

Let $M\subset\mathbb{R}^{n}$ be a smooth compact manifold of dimension $l$. Let us start by remarking that when $M$ is a smooth compact manifold, the tangent space map sending $x\to T_{x}M$ is a continuous map from $M$ into the Grassmanian of $l$-dimensional subspaces of $\mathbb{R}^{n}$. Fixing $x\in M$, and applying a rotation if necessary, we can identify $T_{x}M$ with $\mathbb{R}^{l}\times\{0^{n-l}\}.$ Since the tangent space map is continuous we know that for any $y$ sufficiently close to $x$ its tangent space $T_{y}M$ is approximately $\mathbb{R}^{l}\times\{0^{n-l}\}.$ As such, we can assert that there exists $R_{x}>0$ such that the following two properties hold:

• •

For any $y\in B(x,R_{x})\cap M$ and $r>0$ such that $B(y,r)\subset B(x,R_{x}),$ we have

[TABLE]

Here $\pi_{l}$ is the projection map from $\mathbb{R}^{n}$ to $\mathbb{R}^{l}$ sending $(x_{1},\ldots,x_{n})$ to $(x_{1},\ldots,x_{l})$.

• •

For any $y\in B(x,R_{x})\cap M$ and $\delta<R_{x}$ we have

[TABLE]

The implied constants in (6) and (7) depend only upon $M$ and $x$. Via an application of (6), (7), and Fubini’s Theorem, it can be shown that for any $y\in B(x,R_{x}/2),$ $r\leq R_{x}/2,$ and $\delta\leq r$ we have

[TABLE]

Again, the implied constants in (8) depend only upon $M$ and $x$. Since $\{B(x,R_{x}/2)\}_{x\in M}$ covers $M,$ it follows by a compactness argument that there exists $R>0$ such that for any $y\in M,$ $r\leq R$ and $\delta\leq r$ we have

[TABLE]

Thus, we conclude that $M$ satisfies the LSP with respect to $\kappa=\frac{l}{n}$.

2.3 Self-similar sets satisfying the open set condition

Let $\Phi=\{\phi_{1},\ldots,\phi_{l}\}$ be a collection of contracting similarities acting on ${\mathbb{R}}^{n}$; that is, $\Phi$ is a collection of maps such that

[TABLE]

and $0<r_{i}<1$ for each $1\leq i\leq l$. It is well-known (see, for example, [9]) that there exists a unique non-empty compact set $K\subset\mathbb{R}^{n}$ such that

[TABLE]

We say that $\Phi$ satisfies the open set condition if there exists an open set $O\subset\mathbb{R}^{n}$ such that $\phi_{i}(O)\subset O$ for each $1\leq i\leq l$ and $\phi_{i}(O)\cap\phi_{j}(O)=\emptyset$ whenever $i\neq j$. In [12] it is shown that if $\Phi$ satisfies the open set condition then there exist constants $b_{1},b_{2}>0$ such that for all $\delta$ sufficiently small

[TABLE]

where $d$ is the box counting dimension of $K$. In [12], (9) was proved for the $n$-dimensional Lebesgue measure. The statement given above follows since ${\cal H}^{n}$ is equal to a scalar multiple of the $n$-dimensional Lebesgue measure.

We now prove that the local scaling property holds for $K$ with $\kappa=\frac{d}{n}$. To this end, fix $x\in K$ and some small number $r>0$. Then there exists a sequence $(a_{i})_{i\in{\mathbb{N}}}\in\{1,\ldots,l\}^{\mathbb{N}}$ such that

[TABLE]

Next, note that there exists $N\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

For this value of $N$ and $\delta<r/2$, we have

[TABLE]

The last line follows since $r_{a_{1}}\cdots r_{a_{N}}\asymp r$ by our choice of $N$. Thus, we have proved that the lower bound in the LSP holds in this case. It remains to prove the upper bound.

Given $r>0$ let

[TABLE]

Importantly $\{(\phi_{a_{1}}\circ\cdots\circ\phi_{a_{k}})(K)\}_{(a_{1},\ldots,a_{k})\in I_{r}}$ forms a cover of $K$. In [9] it is shown that when the open set condition holds, for any $x\in K$ and $r>0$ we have

[TABLE]

for some $C>0$ independent of $r$.

Fixing $x\in K$ and $r>0$ and using (9) and (11), for $\delta<\frac{r}{2}$ we obtain

[TABLE]

Combining (2.3) and (2.3), we see that $K$ satisfies (4) for $\delta<r/2$. Equation (4) trivially holds for $\delta\in[r/2,r)$ since, in that case, $B(x,r)\cap\Delta(K,\delta)$ contains a ball of radius $r/2$ and is contained in a ball of radius $r$. Therefore (4) holds for all $\delta<r$ and $K$ satisfies the LSP with respect to $\kappa=\frac{d}{n}$ as claimed.

3 Preliminaries

In this section we state some definitions and recall some well known facts from geometric measure theory. Throughout this paper we will say that $f:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ is a dimension function if $f$ is a left continuous, non-decreasing function such that $f(r)\to 0$ as $r\to 0\,$. Given a ball $B:=B(x,r)$ in $X$ and a dimension function $f$, we define

[TABLE]

The Hausdorff $f$-measure with respect to the dimension function $f$ is defined as follows. Suppose $F\subset X$, let $f$ be a dimension function and let $\rho>0$. A $\rho$-cover for $F$ is any countable collection of balls $\{B_{i}\}_{i\in{\mathbb{N}}}$ with $r(B_{i})<\rho$ for every $i\in{\mathbb{N}}$ and $F\subset\bigcup_{i\in{\mathbb{N}}}{B_{i}}$. We define

[TABLE]

The Hausdorff $f$-measure of $F$ with respect to the dimension function $f$ is then defined as

[TABLE]

A simple consequence of the definition of ${\cal H}^{f}$ is the following useful fact (see, for example, [9]).

Lemma 1**.**

Let $(X,d)$ be as above. Suppose $f$ and $g$ are dimension functions such that the ratio $f(r)/g(r)\to 0$ as $r\to 0$, then ${\cal H}^{f}(F)=0$ whenever ${\cal H}^{g}(F)<\infty$.

When $f(r)=r^{s}$ ($s\geq 0$), the measure ${\cal H}^{f}$ is the familiar $s$-dimensional Hausdorff measure, which we denote by ${\cal H}^{s}$. The Hausdorff dimension, $\operatorname{dim_{H}}F$, of a set $F$ is defined as

[TABLE]

When calculating the Hausdorff dimension of a set, a usual strategy is to obtain upper and lower bounds separately. It is often the case that calculating an upper bound is relatively straightforward while determining a lower bound is much more difficult. Nevertheless, a standard tool which can frequently be employed in obtaining lower bounds for Hausdorff dimension is the following Mass Distribution Principle.

Lemma 2** (Mass Distribution Principle).**

Let $\mu$ be a probability measure supported on a subset $F$ of $X$. Suppose there are positive constants $c$ and $r_{0}$ such that

[TABLE]

for any ball $B$ with radius $r\leq r_{0}\,$. If $E$ is a subset of $F$ with $\mu(E)=\lambda>0$ then ${\cal H}^{f}(E)\geq\lambda/c$.

For this precise statement of the Mass Distribution Principle, see [5, Section 2]. For further general information regarding Hausdorff measures and dimension we refer the reader to [9, 18].

Let $B:=B(x,r)$ be a ball in $(X,d)$. For any $\alpha>0$, we denote by $\alpha B$ the ball $B$ scaled by a factor $\alpha$; i.e. $\alpha B(x,r):=B(x,\alpha r)$. A useful covering lemma which we will use throughout is the following (see [18]).

Lemma 3** (The $5r$-covering lemma).**

Let $(X,d)$ be a metric space. Every family ${\cal F}$ of balls of uniformly bounded diameter in $X$ contains a disjoint subfamily ${\cal G}$ such that

[TABLE]

We will also make use of the following adaptation of [7, Lemma 4].

Lemma 4**.**

Let $B$ be a ball in the locally compact metric space $(X,d)$ and let $g$ be a doubling dimension function. Let $(S_{i})_{i\in{\mathbb{N}}}$ be a sequence of subsets in $B$ and let $(\delta_{i})_{i\in{\mathbb{N}}}$ be a sequence of positive numbers such that $\delta_{i}\to 0$ as $i\to\infty$. Let

[TABLE]

Then, for any real number $C>1$,

[TABLE]

Note that [7, Lemma 4] is stated in the setting of Euclidean space. Going through the steps in the proof of this lemma one can verify the above analogue holds in our setting. To prove this analogue we require a notion of the Lebesgue Density Theorem that holds for our metric space $(X,d)$ equipped with the measure ${\cal H}^{g}$. Such an analogue is known to exist when $g$ satisfies our doubling bound (1), see for example [21].

In what follows we use the Vinogradov notation, writing $A\ll B$ if $A\leq cB$ for some positive constant $c$ and $A\gg B$ if $A\geq c^{\prime}B$ for some positive constant $c^{\prime}$. If $A\ll B$ and $A\gg B$ we write $A\asymp B$ and say that $A$ and $B$ are comparable.

4 The $K_{G,B}$-Lemma

Before proving Theorem 1, we formulate suitable analogues of [1, Lemma 4] and [1, Lemma 5] which will be required in the present setting. Let

[TABLE]

Given a ball $B$ in $X$ and $j\in{\mathbb{N}},$ we define

[TABLE]

The following is the analogue of [1, Lemma 4] or [5, Lemma 5], the so-called $K_{G,B}$-Lemma, we obtain in the setting currently under consideration.

Lemma 5**.**

Let $(X,d)$, ${\cal F}$, $\Upsilon$, $g$ and $f$ be as given in Theorem 1 and assume that the hypotheses of Theorem 1 hold. Then, for any ball $B$ in $X$ and any $G\in{\mathbb{N}}$, there exists a finite collection

[TABLE]

satisfying the following properties:

1. (i)

if $(A;j)\in K_{G,B}$ then $3A\subset B$; 2. (ii)

if $(A;j),(A^{\prime},j^{\prime})\in K_{G,B}$ are distinct then $3A\cap 3A^{\prime}\neq\emptyset$; and 3. (iii)

there exists a constant $c_{5}\geq 0$ independent of our choice of ball $B$ such that

[TABLE]

Similarly to [1, Lemma 4], the collection $K_{G,B}$ here is a collection of balls drawn from the families $\Phi_{j}(B)$. These balls correspond to the $\limsup$ set $\Lambda(\tilde{\Upsilon})$. From each of these balls what we are actually interested in is extracting a suitable collection of balls corresponding to the $\limsup$ set $\Lambda(\Upsilon)$. We adopt the notation from [1] and write $(A;j)$ for a generic ball from $K_{G,B}$ to “remember” the index $j$ of the family $\Phi_{j}(B)$ that the ball $A$ comes from, but just write $A$ if we are referring only to the ball $A$ (as opposed to the pair $(A;j)$). Making such a distinction is necessary for us to be able to choose the “right” collection of balls within $A$ that at the same time lie in an $\Upsilon_{j}$-neighbourhood of the relevant $F_{j}$. Indeed, for $j\neq j^{\prime}$ we could have $A=A^{\prime}$ for some $A\in\Phi_{j}(B)$ and $A^{\prime}\in\Phi_{j^{\prime}}(B)$.

Proof of Lemma 5.

For $j\in{\mathbb{N}}$ and a fixed ball $B$ in $X$, consider the set of balls

[TABLE]

It follows from our assumption (5) that for any $G\geq 1$ we have

[TABLE]

Observe that $\tilde{\Upsilon}_{j}\to 0$ as $j\to\infty$ because $g$ and $f/g^{\kappa}$ are dimension functions. Therefore for $j\in{\mathbb{N}}$ sufficiently large,

[TABLE]

Therefore for any sufficiently large $G\in{\mathbb{N}}$, we have

[TABLE]

Suppose $G^{\prime}\in{\mathbb{N}}$ is large enough that the above inequality holds for any $G\geq G^{\prime}$. Clearly for any $G<G^{\prime}$ we also have

[TABLE]

Therefore for any $G\in{\mathbb{N}}$ it follows that

[TABLE]

Next, by the $5r$-covering Lemma (Lemma 3), there exists a disjoint subcollection ${\cal G}\subset\{(L;j):j\geq G,~{}L\in\Phi_{j}^{3}(B)\}$ satisfying

[TABLE]

Let ${\cal G}^{\prime}:=\{(\tfrac{1}{3}L;j):(L;j)\in{\cal G}\}$ be the balls from the collection ${\cal G}$ all scaled by a factor of $1/3$. Note that the balls in ${\cal G}^{\prime}$ are still disjoint when scaled by 3. By the above, we have that

[TABLE]

It follows from (1), (2), and the disjointness of the balls in ${\cal G}^{\prime}$ that

[TABLE]

Now also using (13) and (14), we see that

[TABLE]

Thus, there exists a constant $c^{\prime}>0$ such that

[TABLE]

Since the balls in ${\cal G}^{\prime}$ are disjoint and contained in $B$, it follows that

[TABLE]

Consequently, there must exist $N_{0}\in{\mathbb{N}}$ such that

[TABLE]

We define $K_{G,B}$ to be the subcollection of $(A;j)\in{\cal G}^{\prime}$ with $G\leq j<N_{0}.$ By the above we see that $K_{G,B}$ is a finite collection of balls while still satisfying the required properties (i)–(iii) with $c_{5}=\frac{c^{\prime}}{2}$. ∎

As mentioned previously, from each of the balls in $K_{G,B}$ we wish to extract a collection of balls corresponding to $\Lambda(\Upsilon)$. The desired properties and existence of such collections are summarised in the following lemma, which constitutes the required analogue of [1, Lemma 5] in this setting.

Lemma 6**.**

Let $(X,d)$, ${\cal F}$, $\Upsilon$, $f$, $g$, and $B$ be as in Lemma 5 and assume that the hypotheses of Theorem 1 hold. Furthermore, assume that $f(r)/g(r)\to\infty$ as $r\to 0$. Let $K_{G,B}$ be as in Lemma 5. Then, provided that $G$ is sufficiently large, for any $(A;j)\in K_{G,B}$ there exists a collection ${\cal C}(A;j)$ of balls satisfying the following properties:

1. (i)

each ball in ${\cal C}(A;j)$ is of radius $\Upsilon_{j}$ and is centred on $F_{j};$ 2. (ii)

if $L\in{\cal C}(A;j)$ then $3L\subset A;$ 3. (iii)

if $L,M\in{\cal C}(A;j)$ are distinct then $3L\cap 3M=\emptyset;$ 4. (iv)

\displaystyle{\cal H}^{g}\big{(}\Delta(F_{j},\Upsilon_{j})\cap\tfrac{1}{2}A\big{)}~{}\ll~{}{\cal H}^{g}\left(\bigcup_{L\in{\cal C}(A;j)}{L}\right)~{}\leq~{}\,{\cal H}^{g}\big{(}\Delta(F_{j},\Upsilon_{j})\cap A\big{)}; and 5. (v)

there exist some constants $d_{1},d_{2}>0$ such that

[TABLE]

Proof.

We begin by showing that

[TABLE]

To this end, suppose that $N\in{\mathbb{N}}$. We aim to show that for all sufficiently large $j\in{\mathbb{N}}$ we have

[TABLE]

Observe that (17) holds if

[TABLE]

Furthermore, by repeated application of (1), we see that

[TABLE]

where $\lambda$ is the doubling constant. Consequently, (18) holds if

[TABLE]

Rearranging the above we get

[TABLE]

By the assumptions that $\frac{f(r)}{g(r)}\to\infty$ as $r\to 0$ and $\Upsilon_{j}\to 0$ as $j\to\infty$, we see that this inequality holds for sufficiently large $j\in{\mathbb{N}}$, thus verifying (16).

In light of (16) we can assume that $G$ is sufficiently large so that

[TABLE]

Let $x_{1},\dots,x_{t}\in F_{j}\cap\tfrac{1}{2}A$ be a maximal collection of points such that

[TABLE]

Define ${\cal C}(A;j)$ to be the collection of balls

[TABLE]

By construction, property (i) is satisfied by the collection ${\cal C}(A;j)$. Next, recall that $A\in\Phi_{j}(B)$ and so $\tfrac{1}{2}A$ has radius $\tfrac{1}{2}\tilde{\Upsilon}_{j}$. If $L:=B(x_{i},\Upsilon_{j})\in{\cal C}(A;j)$ then any $y\in 3L$ satisfies $d(y,x_{i})<3\Upsilon_{j}$. Supposing $x_{0}$ is the centre of $A$, we also have $d(x_{i},x_{0})\leq\tfrac{1}{2}\tilde{\Upsilon}_{j}$. Combining (19) and the triangle inequality we obtain

[TABLE]

Therefore property (ii) follows. In addition property (iii) follows from (20).

It is a consequence of the maximality of $x_{1},\dots,x_{t}$ that for any $x\in F_{j}\cap\tfrac{1}{2}A,$ there exists an $x_{i}$ from this collection such that $d(x,x_{i})\leq 6\Upsilon_{j}$. Consequently

[TABLE]

Therefore, by (1) and (2),

[TABLE]

However, by property (ii), we have

[TABLE]

This together with the previous inequality proves property (iv).

As a byproduct of (19) we have that $\Upsilon_{j}<\frac{\tilde{\Upsilon}_{j}}{2}$ for all $j\geq G.$ Therefore, by the LSP we have

[TABLE]

Combining this with the doubling property (1) and the LSP we obtain

[TABLE]

By (2) and the disjointness of the balls in ${\cal C}(A;j)$ we have

[TABLE]

Combining the above with (21) and property (iv) we get

[TABLE]

So property (v) holds. ∎

5 Proof of Theorem 1

5.1 Strategy

Fix an arbitrary ball $B_{0}$ in $X$ and suppose the assumptions of Theorem 1 hold. Our goal is to show that

[TABLE]

Since $f/g$ is monotonic, there are three situations to consider:

1. (a)

$\frac{f(r)}{g(r)}\to\infty$ as $r\to 0$; 2. (b)

$\frac{f(r)}{g(r)}\to 0$ as $r\to 0$; and 3. (c)

$\frac{f(r)}{g(r)}\to\ell$ as $r\to 0$, where $0<\ell<\infty$.

If we are in situation (b) it follows from Lemma 1 that ${\cal H}^{f}(B_{0})=0$. Since $B_{0}\cap\Lambda(\Upsilon)\subset B_{0}$ the result follows.

If we are in case (c) it can be shown that $\tilde{\Upsilon}_{j}\asymp\Upsilon_{j}$. It then follows from Lemma 4 that ${\mathcal{H}}^g(B_{\mathrm{0</}}$