Intrinsic Diophantine approximation for overlapping iterated function systems

Diophantine Approximation is the study of approximations of vectors in \({\mathbb {R}}^d\) by elements of \({\mathbb {Q}}^d\). Given a set \(X\subset {\mathbb {R}}^d\), it is natural to wonder how well elements of X can be approximated by elements of \({\mathbb {Q}}^d\) contained within X. Similarly, it is natural to wonder how well elements of X can be approximated by elements of \({\mathbb {Q}}^d\) lying outside of X. These two questions are the motivation behind the topics of intrinsic Diophantine Approximation and extrinsic Diophantine Approximation respectively. Often the set X is taken to be a smooth manifold or a fractal set. A tremendous amount of work has been done on these two topics when X is taken to be such a set. For further details we refer the reader to the papers [7, 9,10,11, 13, 14, 16,17,18,19, 26, 27, 34, 35, 37, 39, 40] and the references therein. In this paper we study intrinsic Diophantine Approximation when the set X is a self-similar set. We will provide a more thorough introduction to this topic in Sect. 2. The main result of this paper is a general theorem on the metric properties of a family of limsup sets defined using iterated function systems. As we will see, this theorem implies a number of results in intrinsic Diophantine Approximation.

In what remains of this introductory section we will provide the relevant background from Fractal Geometry and state Theorem 1.1, which is our main result. In Sect. 2 we will show how Theorem 1.1 can be used to obtain a number of results for intrinsic Diophantine Approximation on self-similar sets. In Sect. 3 we will prove Theorem 1.1. In Sect. 4 we will apply the mass transference principle of Beresnevich and Velani together with Theorem 1.1 to deduce further results on the Hausdorff measure of certain limsup sets.

1.1 Background from Fractal Geometry

We call a map \(\phi :{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^d\) a similarity if there exists \(r\in (0,1)\) such that \(\Vert \phi (x)-\phi (y)\Vert =r\Vert x-y\Vert \) for all \(x,y\in {\mathbb {R}}^d\). We call a finite set of similarities an iterated function system or IFS for short. An important result due to Hutchinson [25] states that for any IFS \(\Phi =\{\phi _{a}\}_{a\in {\mathcal {A}}},\) there exists a unique non-empty compact set X satisfying

X is called the self-similar set of \(\Phi \). When the elements of \(\Phi \) all have the same contraction ratio, i.e. \(r_{a}=r_{a'}\) for all \(a,a'\in {\mathcal {A}}\), then we say that an IFS is equicontractive. Importantly we can view X as the image of \({\mathcal {A}}^{{\mathbb {N}}}\) under an appropriate projection map: Let \(\pi :{\mathcal {A}}^{{\mathbb {N}}}\rightarrow X\) be given by

Here 0 can be replaced with any other vector in \({\mathbb {R}}^d\). Importantly the map \(\pi \) is surjective and continuous (when \({\mathcal {A}}^{{\mathbb {N}}}\) is equipped with the product topology). Given an IFS \(\Phi =\{\phi _{a}\}_{a\in {\mathcal {A}}}\) we define the similarity dimension of \(\Phi \) to be the unique solution to the equation

We denote the similarity dimension of an IFS \(\Phi \) by \(\dim _{S}\left( \Phi \right) \). Notice that if \(\Phi \) is equicontractive then \(\dim _{S}\left( \Phi \right) =\frac{\log \#{\mathcal {A}}}{-\log r}\) where r is the common contraction ratio. It is well known that the Hausdorff dimension of a self-similar set X always satisfies the following upper bound:

For many iterated function systems this inequality is in fact an equality, see [15, 22, 23, 33]. We say that \(\Phi \) satisfies the strong separation condition if \(\phi _{a}(X)\cap \phi _{a'}(X)=\emptyset \) for all \(a,a'\in {\mathcal {A}}\) such that \(a\ne a'\). An IFS \(\Phi \) is said to satisfy the open set condition if there exists a bounded open set O such that \(\phi _{a}(O)\subset O\) for all \(a\in {\mathcal {A}}\), and \(\phi _{a}(O)\cap \phi _{a'}(O)=\emptyset \) whenever \(a\ne a'\). It is known that the strong separation condition implies the open set condition, and that under either of these assumptions we have equality in (1.1).

To prove equality in (1.1) in the overlapping case one often uses self-similar measures. These are defined as follows: Given an IFS \(\{\phi _{a}\}_{a\in A}\) and a probability vector \({\textbf{p}}=\left( p_{a}\right) _{a\in {\mathcal {A}}},\) then there exists a unique Borel probability measure \(\mu _{{\textbf{p}}}\) satisfying

We call \(\mu _{{\textbf{p}}}\) the self-similar measure corresponding to \(\Phi \) and \({\textbf{p}}\). Given \({\textbf{p}}\), if we let \({\mathfrak {m}}_{{\textbf{p}}}\) denote the corresponding Bernoulli measure on \({\mathcal {A}}^{{\mathbb {N}}}\) then it is also the case that \(\mu _{{\textbf{p}}}=\pi {\mathfrak {m}}_{{\textbf{p}}}\). For our purposes we will only need to focus on one particular self-similar measure, namely the one corresponding to the probability vector \(\left( r_{a}^{\dim _{S}\left( \Phi \right) }\right) _{a\in {\mathcal {A}}}.\) This self-similar measure is distinguished amongst the family of self-similar measures. Studying its properties often allows one to prove equality in (1.1). For an IFS \(\Phi \), we will denote the self-similar measure corresponding to \(\left( r_{a}^{\dim _{S}\left( \Phi \right) }\right) _{a\in {\mathcal {A}}}\) by \(\mu _{\Phi },\) or simply \(\mu \) if the choice of \(\Phi \) is implicit. Similarly, we will denote \(\left( r_{a}^{\dim _{S}\left( \Phi \right) }\right) _{a\in {\mathcal {A}}}\) by \({\textbf{p}}_{\Phi }\) or simply \({\textbf{p}}\), and the corresponding Bernoulli measure on \({\mathcal {A}}^{{\mathbb {N}}}\) by \({\mathfrak {m}}_{\Phi }\) or \({\mathfrak {m}}\). For a probability vector \({\textbf{p}}\) we denote the entropy of \({\textbf{p}}\) by

Suppose now that in addition to \({\textbf{p}}\) we are also given an IFS \(\Phi \), we then define the Lyapnuov exponent of \(\Phi \) and \({\textbf{p}}\) to be

We conclude this overview of the relevant topics from Fractal Geometry by introducing some notation. In what follows, we denote an element of \(\cup _{n=1}^{\infty }A^n\) or \({\mathcal {A}}^{{\mathbb {N}}}\) by \({\textbf{a}}\) or \({\textbf{b}}\). Given an IFS \(\Phi =\left\{ \phi _{a}\right\} _{a\in {\mathcal {A}}}\) and a word \({\textbf{a}}=(a_1,\ldots ,a_n),\) we let \(\phi _{{\textbf{a}}}:=\phi _{a_1}\circ \cdots \circ \phi _{a_n}\) and \(r_{{\textbf{a}}}:=\prod _{l=1}^{n}r_{a_l}.\) Given a word \({\textbf{a}}\) we let \(X_{{\textbf{a}}}=\phi _{{\textbf{a}}}\left( X\right) \). Given a finite word \({\textbf{a}}\) and a finite word or infinite sequence \({\textbf{b}},\) we let \({\textbf{a}}{\textbf{b}}\) denote the concatenation of \({\textbf{a}}\) and \({\textbf{b}}\). For a finite word \({\textbf{a}}\) we let \({\textbf{a}}^k\) denote the k-fold concatenation of \({\textbf{a}}\) with itself. Similarly \({\textbf{a}}^{\infty }\) denotes the periodic element of \({\mathcal {A}}^{{\mathbb {N}}}\) obtained by concatenating \({\textbf{a}}\) with itself indefinitely. We denote the length of a finite word \({\textbf{a}}\) by \(|{\textbf{a}}|\). Finally, given a finite word \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^{n}\) we let

We will often refer to \([{\textbf{a}}]\) as the cylinder set corresponding to \({\textbf{a}}\).

1.2 Statement of Theorem 1.1

The family of limsup sets that will be the main focus of this paper are defined as follows: Given an IFS \(\Phi \) and a function \(\Psi :\cup _{n=1}^{\infty } {\mathcal {A}}^{n}\rightarrow [0,\infty ),\) we let

Alternatively, \(W_{\Phi }(\Psi )\) is the set of \(x\in {\mathbb {R}}^d\) such that for infinitely many n, there exists \({\textbf{a}}\in {\mathcal {A}}^n\) and \(0\le l\le n-1\) such that

The connection between \(W_{\Phi }(\Psi )\) and intrinsic Diophantine Approximation will be made clear in Sect. 2. Our main result demonstrates that for certain choices of \(\Psi ,\) the measure of \(W_{\Phi }(\Psi )\) is determined by naturally occurring volume sums. One cannot expect such a behaviour to occur for all choices of \(\Psi \). Indeed Example 2.1 from [2] shows that for a related family of limsup sets, if we want the measure of these limsup sets to be determined by volume sums, then the underlying \(\Psi \) should reflect the different rates of scaling within the IFS. As such we will often restrict ourselves to \(\Psi \) of the form

where \(g:{\mathbb {N}}\rightarrow [0,\infty )\). A similar restriction was also adopted in [1, 2, 5]. Functions of the form

for some \(g:(0,\infty )\rightarrow (0,\infty )\) were considered in a similar situation in [21] and later in [8, Section 12.4]. Note that if \(\Phi \) is equicontractive, then the set of \(\Psi \) of the form (1.2) coincides with the set of \(\Psi \) of the form (1.3). Moreover, both of these sets coincide with the set of \(\Psi \) such that \(\Psi ({\textbf{a}})\) only depends upon the length of \({\textbf{a}}\).

Our main result is the following statement.

Theorem 1.1

Let \(\Phi =\left\{ \phi _{a}\right\} _{a\in {\mathcal {A}}}\) be an IFS and \(\Psi :\cup _{n=1}^{\infty }{\mathcal {A}}^n\rightarrow [0,\infty ).\) Then the following statements are true:

1. 1.

   For any \(s\ge 0\), suppose that

   $$\begin{aligned} \sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^n}n \cdot \Psi ({\textbf{a}})^{s}<\infty . \end{aligned}$$

   Then \({\mathcal {H}}^{s}\left( W_{\Phi }(\Psi )\right) =0.\)

2. 2.

   Assume that

   $$\begin{aligned} h_{{\textbf{p}}}<-2\log \sum _{a\in {\mathcal {A}}}p_{a}^{2} \end{aligned}$$

   (1.4)

   and \(\Psi \) is of the form \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|)\) for some non-increasing \(g:{\mathbb {N}}\rightarrow [0,\infty ).\) If

   $$\begin{aligned} \sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^n}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty \end{aligned}$$

   then \(\mu \left( W_{\Phi }(\Psi )\right) =1.\)

3. 3.

   Assume that \(\Phi \) is equicontractive and \(\Psi \) is of the form \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|)\) for some \(g:{\mathbb {N}}\rightarrow [0,\infty ).\) If

   $$\begin{aligned} \sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^n}n \cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty \end{aligned}$$

   then \(\mu \left( W_{\Phi }(\Psi )\right) =1.\)

We conclude this section with some remarks on Theorem 1.1.

Remark 1.2

Statement 3 of Theorem 1.1 was proved for the IFS \(\left\{ \phi _{1}(x)=\frac{x}{3},\phi _{2}(x)\right. \)\(\left. =\frac{x+2}{3}\right\} \) by Tan, Wang, and Wu in [37]. Note that this IFS has the middle third Cantor set as its self-similar set. In a recent talk Wang [38] commented that the methods used in [37] could be generalised to prove Statement 3 of Theorem 1.1 for equicontractive IFSs acting on \({\mathbb {R}}\) that satisfy the strong separation condition. During this talk Wang posed the question as to what happens for IFSs that are not equicontractive. This paper was in part motivated by this question and Statement 2 of Theorem 1.1 provides a partial answer. Importantly, as well as providing information in the non-equicontractive case, Theorem 1.1 also applies in arbitrary dimensions and requires no separation assumptions on the IFS. The techniques of [37] do not apply in this generality. That being said, our method of proof largely follows the same overall strategy as [37]. The major differences being that we require additional arguments to control the different rates of scaling within our potentially non-equicontractive IFS, and we also require a new argument to address the potential overlaps that may be present within the IFS. The latter argument uses ideas from [5, Section 7] where the author reinterpreted their problem in terms of a limsup set on the sequence space \({\mathcal {A}}^{{\mathbb {N}}}\). This reinterpretation addresses the problem of potential overlaps.

Remark 1.3

If \(\Phi \) satisfies the open set condition then it is known that \(\mu \) is equivalent to the restriction of the \({\mathcal {H}}^{\dim _{S}\left( \Phi \right) }\)-dimensional Hausdorff measure on X. As such, under the open set condition, Statements 1, 2, and 3 of Theorem 1.1 provide a nearly complete description of the \(\mu \) measure of \(W_{\Phi }(\Psi )\) for \(\Psi \) of the form \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|)\). Moreover, if we assume that \(\Phi \) is equicontractive and satisfies the open set condition, then Statements 1 and 3 do provide a complete description. In the overlapping case, i.e. when the open set condition is not satisfied, then Statements 2 and 3 can be used to deduce a number of corollaries on the Hausdorff dimension of \(W_{\Phi }(\Psi )\). For if \(\dim \mu =\min \{\dim _{S}\left( \Phi \right) ,d\}\) and \(\mu (W_{\Phi }(\Psi ))=1,\) then we must have \(\dim _{H}(W_{\Phi }(\Psi ))\ge \min \{\dim _{S}\left( \Phi \right) ,d\}.\) Moreover because \(W_{\Phi }(\Psi )\) is a subset of the self similar set X, and X satisfies (1.1), we must then have \(\dim _{H}(W_{\Phi }(X))=\min \{\dim _{S}\left( \Phi \right) ,d\}.\) The important part in this argument is determining when we have \(\dim \mu =\min \{\dim _{S}\left( \Phi \right) ,d\}.\) A number of significant breakthroughs on this topic have been made in recent years, see [22, 23, 33]. These papers provide general sufficient conditions which guarantee \(\dim \mu =\min \{\dim _{S}\left( \Phi \right) ,d\}.\) We won’t state the results of these papers in their full generality here. Instead we will focus on one particular consequence that is relevant to our purposes. Suppose that \(\Phi =\{\phi _{a}(x)=r_{a}x+t_{a}\}\) is an IFS acting on \({\mathbb {R}}\) and that each \(r_{a}\) is algebraic, then it follows from the results of [33] that if \(\Phi \) does not contain an exact overlap then \(\dim \mu =\min \{\dim _{S}\left( \Phi \right) ,d\}.\) We recall that an IFS is said to contain an exact overlap if there exists \({\textbf{a}},{\textbf{b}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^n\) such that \(\phi _{{\textbf{a}}}=\phi _{{\textbf{b}}}\) and \({\textbf{a}}\ne {\textbf{b}}\).

Remark 1.4

The inequality (1.4) and the non-increasing assumption on g in Statement 2 are both technical assumptions and are believed to be non-optimal. Note that in Statement 3 there are no monotonicity conditions imposed on g. We expect that both of these assumptions can be removed. For the purposes of our exposition, we highlight that in the case of an IFS consisting of two similarities \(\{\phi _{1}(x)=r_{1}x+t_{1},\phi _{2}(x)=r_{2}x+t_2\},\) then (1.4) is satisfied if \(r_{1}^{\dim _{S}\left( \Phi \right) }\) satisfies

As such we see that (1.4) is satisfied by a significant proportion of those IFSs consisting of two similarities. The inequalities above can be derived by considering the functions \(f:(0,1)\rightarrow {\mathbb {R}}\) and \(g:(0,1)\rightarrow {\mathbb {R}}\) given by \(f(x)=-x\log x -(1-x)\log (1-x)\) and \(g(x)=-2\log (x^2+(1-x)^2),\) and using a computer to determine those x for which \(f(x)<g(x)\). We remark that when \(\Phi \) is an equicontractive IFS then we have \(h_{{\textbf{p}}}=-\log \sum _{a\in {\mathcal {A}}}p_{a}^{2},\) and so (1.4) is automatically satisfied in this case. It follows from this observation and the fact that the quantities on each side of (1.4) depend continuously on \({\textbf{p}}\), that if we fix the number of maps within our IFS to be N for some \(N\in {\mathbb {N}}\) and identify the space of contraction ratios with \((0,1)^{N}\), then for a non-empty open set of contraction ratios the inequality (1.4) is satisfied. We will see another explicit example where the inequality (1.4) is satisfied in Fig. 1.

Remark 1.5

It is a simple exercise to show that a function \(g:{\mathbb {N}}\rightarrow [0,\infty )\) satisfies

if and only if

It will on occasion be more convenient to use this latter divergence condition.

Notation. In this paper we will adopt the following notational convention. Given a set S and two functions \(f,g:S\rightarrow {\mathbb {R}}\) we write \(f\ll g\) if there exists \(C>0\) such that \(|f(x)|\le C|g(x)|\) for all \(x\in S\). We write \(f\asymp g\) if \(f\ll g\) and \(g \ll f\).

2.1 Background

The study of intrinsic Diophantine Approximation for self-similar sets has its origins in a question of Mahler [30]. He asked how well can elements of the middle third Cantor set C be approximated by rational numbers lying within C. To the best of the author’s knowledge, the first significant progress in this direction was the work of Levesley, Salp, and Velani [29]. They considered rational approximations of the from \(p/3^n\). Or equivalently, rational approximations provided by the end points of the sets of the form \(\phi _{{\textbf{a}}}(C)\). They proved a general Khintchine type result for approximations of this type, see [29, Theorem 1]. Using this theorem, they were able to prove that there exist well-approximable numbers in the middle third Cantor set that are not Liouville. This was an unproved assertion attributed to Mahler. Bugeaud also proved this assertion using a different method in [13]. He in fact provided explicit examples of elements of the middle third Cantor set with any irrationality exponent. In [14] Bugeaud and Durand posed a conjecture on the value of the Hausdorff dimension of the set of points in the middle third Cantor set whose irrationality exponent exceeds a given parameter. Interestingly this conjecture suggests that a phase transition should occur for the value of the Hausdorff dimension of this set. The main result of [14] shows that a version of this conjecture holds almost surely for a particular random model of C. The following intrinsic analogue of Dirichlet’s theorem for C was proved by Broderick, Fishman, and Reich [11].

Theorem 2.1

For any \(x\in C\) and \(Q>1\), there exists \(p/q\in C\) with \(1\le q\le Q\) such that

Theorem 2.1 was shown to be optimal by Fishman and Simmons [19], and Fishman, Merrill, and Simmons [17].

The study of intrinsic Diophantine approximation for self-similar sets naturally leads one to study limsup sets that are in a sense built using the underlying iterated function system. A number of papers have appeared which study such sets, see [1,2,3,4,5,6, 31, 32]. Despite being a problem that was originally motivated by number theoretic considerations, the study of these limsup sets is connected to topics from Ergodic Theory and Fractal Geometry. Interestingly the metric properties of these limsup sets can be related to the absolute continuity of self-similar measures, see [5].

2.2 Applications

For the rest of this section we restrict our attention to iterated function systems of the form

Where for all \(a\in {\mathcal {A}}\) we have \({\mathfrak {p}}_{a}\in {\mathbb {Z}}^{d},\) \(q_{a}\in {\mathbb {Z}},\) and \(q_{a}\) also satisfies \(|q_{a}|\ge 2.\) For this IFS, the projection map \(\pi \) takes the following simplified form:

For such an IFS, the following lemma connects those x contained in \(X\cap {\mathbb {Q}}^d\) with eventually periodic sequences. It is a particular case of a more general result due to Schleischitz [34]. For this reason we omit its proof.

Lemma 2.2

Let \(\Phi \) be an IFS of the form \(\Phi =\left\{ \phi _{a}(x)=\frac{x+{\mathfrak {p}}_{a}}{q_{a}}\right\} \). Then for any \(x\in X\) we have that \(x\in {\mathbb {Q}}^{d}\) if and only if there exists \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^{n}\) and \(0\le l \le |{\textbf{a}}|-1\) such that

If \(x\in {\mathbb {Q}}^d\) and therefore by Lemma 2.2 is equal to \(\pi \left( a_{1}\ldots a_{l}\left( a_{l+1}\ldots a_{|{\textbf{a}}|}\right) ^{\infty }\right) \) for some \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^{n}\) and \(0\le l \le |{\textbf{a}}|-1\), we can use properties of geometric series and (2.1) to conclude that there exists \({\mathfrak {p}}_{x}\in {\mathbb {Z}}^d\) such that

One of the major difficulties in understanding the properties of the rational numbers within a self-similar set is not knowing if any cancellation occurs between the entries in the vector \({\mathfrak {p}}_{x}\) and the \(\prod _{j=1}^{l}q_{a_j}\cdot \left( \prod _{j=l+1}^{n}q_{a_j}-1\right) \) term. It is possible that these two terms contain many common factors and as such x could be written in a significantly reduced form. This makes studying intrinsic Diophantine Approximation for self-similar sets challenging. This difficulty is avoided in this paper by considering the quantity \(q_{\text {int}}\) defined below. For ease of exposition we split what remains of this section into two cases, when \(\Phi \) is equicontractive and the general case. Our applications are much simpler to state in the equicontractive case.

2.2.1 The equicontractive case

In this section we assume that \(\Phi \) is equicontractive, i.e. there exists \(q_{\Phi }\in {\mathbb {Z}}\) such that \(q_{a}=q_{\Phi }\) for all \(a\in A\). By the above we know that if \({\mathfrak {p}}/q\in X\) then there exists \({\mathfrak {p}}' \in {\mathbb {Z}}^d,\) \(n\in {\mathbb {N}}\), and \(0\le l\le n-1\) such that

We define the intrinsic denominator of \({\mathfrak {p}}/q\in X\) to be

This is a generalisation of the notion of intrinsic denominator defined in [19] for \(\Phi \) satisfying the strong separation condition. Given a function \(\Psi :{\mathbb {N}}\rightarrow [0,\infty )\) we may then define a limsup set as follows:

Fishman and Simmons in [19] proved a version of Khintchine’s theorem for limsup sets of the form \(W_{\Phi }^{*}(\Psi )\) when the underlying \(\Phi \) is equicontractive, acting on \({\mathbb {R}},\) and satisfies the strong separation condition. Importantly this result did not provide a complete metric description for the sets \(W_{\Phi }^{*}(\Psi ),\) even in the restricted case of equicontractive IFSs acting on \({\mathbb {R}}\) which satisfy the strong separation condition. The divergence condition they needed for a full measure statement was not optimal. This issue was addressed in a recent paper by Tan, Wang, and Wu [37] who established a complete analogue of Khintchine’s theorem for the set \(W_{\Phi }^{*}(\Psi )\) for the IFS \(\left\{ \phi _{1}(x)=\frac{x}{3}, \phi _{2}(x)=\frac{x+2}{3}\right\} .\) Note that this IFS has the middle third Cantor set as its self-similar set.

Theorem 2.3

[37, Theorem 1.4] Let \(\Phi =\left\{ \phi _{1}(x)=\frac{x}{3}, \phi _{2}(x)=\frac{x+2}{3}\right\} \) and \(\Psi :{\mathbb {N}}\rightarrow [0,\infty )\) be a non-increasing function. Then

The proof given in [37] can be generalised to prove an analogue of Theorem 2.3 for any equicontractive IFS acting on \({\mathbb {R}}\) satisfying the strong separation condition. Our main result in this direction is the following statement. It generalises Theorem 2.3 to arbitrary dimensions and requires no separation conditions for \(\Phi \).

Theorem 2.4

Let \(\Phi \) be an equicontractive IFS of the form \(\Phi =\left\{ \phi _{a}(x)=\frac{x+{\mathfrak {p}}_{a}}{q_{\Phi }}\right\} .\) Let \(\Psi :{\mathbb {N}}\rightarrow [0,\infty )\) be a non-increasing function. Then the following statements are true:

1. 1.

   For any \(s\ge 0\), suppose that \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^n)^{s}<\infty .\) Then \({\mathcal {H}}^{s}(W_{\Phi }^*(\Psi ))=0.\)

2. 2.

   If \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^n)^{\frac{\log \#{\mathcal {A}}}{\log q_{\Phi }}}=\infty \) then \(\mu (W_{\Phi }^*(\Psi ))=1.\)

Proof

We begin by remarking that for any word \({\textbf{a}}\) and \(0\le l\le |{\textbf{a}}|-1\) we have

Proof of the convergence case. To prove the convergence case let \(\Psi ':\cup _{n=1}^{\infty }{\mathcal {A}}^{n}\rightarrow [0,\infty )\) be given by

Now notice that for any \({\mathfrak {p}}/q\in X\), there must exist \({\textbf{a}}\) and \(0\le l\le |{\textbf{a}}|-1\) such that

Therefore, using the fact \(\Psi \) is non-increasing together with (2.2) we have

Therefore \(W_{\Phi }^{*}(\Psi )\subseteq W_{\Phi }(\Psi ').\) By Theorem 1.1 it will follow that \({\mathcal {H}}^{s}(W_{\Phi }^{*}(\Psi ))=0\) if \(\sum _{n=1}^{\infty } \sum _{{\textbf{a}}\in {\mathcal {A}}^n}n\cdot \Psi '({\textbf{a}})^{s}<\infty .\) This latter inequality is equivalent to \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^{n-1})^{s}<\infty .\) However, this inequality is implied by our assumption \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^n)^{s}<\infty .\)

Proof of the divergence case. Let \(g:{\mathbb {N}}\rightarrow [0,\infty )\) be given by

Now define \(\Psi ':\cup _{n=1}^{\infty }{\mathcal {A}}^{n}\rightarrow [0,\infty )\) by \(\Psi '({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})\cdot g(|{\textbf{a}}|)\). For any \({\textbf{a}}\in {\mathcal {A}}^n\) and \(0\le l\le |{\textbf{a}}|-1\) there exists \({\mathfrak {p}}/q\in X\) such that

By the non-increasing assumption on \(\Psi \), the definition of intrinsic denominator, and (2.2), it follows that

Therefore \(W_{\Phi }(\Psi ')\subseteq W_{\Phi }^{*}(\Psi )\). By Theorem 1.1, it will follow that \(\mu (W_{\Phi }^{*}(\Psi ))=1\) if we can show that

By the definition of g this is equivalent to our assumption \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^n)^{\frac{\log \#{\mathcal {A}}}{\log q_{\Phi }}}=\infty .\) Therefore our result follows. \(\square \)

If a rational vector \({\mathfrak {p}}/q\in X\) is in its reduced form then we must have \(q\le q_{\text {int}}({\mathfrak {p}}/q)\). This observation together with Theorem 2.4 implies the following statement for traditional rational approximations, where the neighbourhood is defined in terms of the denominator of \({\mathfrak {p}}/q\) rather than \(q_{\text {int}}({\mathfrak {p}}/q)\).

Corollary 2.5

Let \(\Phi \) be an equicontractive IFS of the form \(\Phi =\{\phi _{{\textbf{a}}}(x)=\frac{x+{\mathfrak {p}}_{a}}{q_{\Phi }}\}.\) Let \(\Psi :{\mathbb {N}}\rightarrow [0,\infty )\) be a non-increasing function. If \(\sum _{n=1}^{\infty }n\cdot \#{\mathcal {A}}^{n}\cdot \Psi (q_{\Phi }^n)^{\frac{\log \#{\mathcal {A}}}{\log q_{\Phi }}}=\infty \) then

See Fig. 2 for an example of two IFSs to which Theorem 2.4 can be applied.

2.2.2 The general case

In this section we no longer assume that \(\Phi \) is equicontractive. We formulate two statements, one when \(\Phi \) is potentially overlapping, and one when \(\Phi \) satisfies the strong separation condition.

Given \(g:{\mathbb {N}}\rightarrow [0,\infty )\) and \({\mathfrak {p}}/q\in X\) we define

To each \(g:{\mathbb {N}}\rightarrow [0,\infty )\) we associate the set

The following statement is essentially Theorem 1.1 rephrased in terms of rational approximations.

Theorem 2.6

Let \(\Phi \) be an IFS of the form \(\Phi =\left\{ \phi _{{\textbf{a}}}(x)=\frac{x+{\mathfrak {p}}_{a}}{q_{a}}\right\} \) and \(g:{\mathbb {N}}\rightarrow [0,\infty )\). Then the following statements are true

1. 1.

   For any \(s\ge 0\), suppose that \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{s}<\infty .\) Then \({\mathcal {H}}^{s}(W_{\Phi }^{**}(g))=0.\)

2. 2.

   If g is non-increasing, \(h_{{\textbf{p}}}<-2\log \sum _{a\in {\mathcal {A}}}p_{a}^{2},\) and \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty \) then \(\mu (W_{\Phi }^{**}(g))=1.\)

Proof

This result follows from Theorem 1.1 together with the observation that if \(x\notin {\mathbb {Q}}^d\) then \(x\in W_{\Phi }^{**}(\Psi _{g})\) if and only if \(x\in W_{\Phi }(\Psi )\) for \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|).\) \(\square \)

For what remains of this section we will always assume that \(\Phi \) satisfies the strong separation condition. Because of the strong separation condition, for any \({\mathfrak {p}}/q\in X\) there exists a unique sequence \((a_n)\in {\mathcal {A}}^{{\mathbb {N}}}\) satisfying \(\pi ((a_n))={\mathfrak {p}}/q\). We emphasise that \((a_n)\) must be eventually periodic. Given \({\mathfrak {p}}/q\in X\) we define the intrinsic denominator of \({\mathfrak {p}}/q\) to be

We denote this quantity by \(q_{\text {int}}({\mathfrak {p}}/q).\) For any \({\mathfrak {p}}/q\in X\) we define

Similarly, we let

It can be shown that \(n({\mathfrak {p}}/q)\) and \(l({\mathfrak {p}}/q)\) are the unique parameters satisfying

and

Given \(g:{\mathbb {N}}\rightarrow [0,\infty )\) we define a limsup set as follows, let

Theorem 2.7

Let \(\Phi \) be an IFS of the form \(\Phi =\left\{ \phi _{a}(x)=\frac{x+{\mathfrak {p}}_a}{q_a}\right\} \) satisfying the strong separation condition and let \(g:{\mathbb {N}}\rightarrow [0,\infty )\). Then the following statements are true:

1. 1.

   For any \(s\ge 0\), suppose that \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{s}<\infty .\) Then \({\mathcal {H}}^{s}(W_{\Phi }^{***}(g))=0\).

2. 2.

   If g is non-increasing, \(h_{{\textbf{p}}}<-2\log \sum _{a\in {\mathcal {A}}}p_{a}^{2},\) and \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty ,\) then \(\mu (W_{\Phi }^{***}(g))=1.\)

Proof

We begin our proof by remarking that there exist constants \(C_{1},C_{2}>0\) such that for any word \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^{n}\) and \(0\le l\le |{\textbf{a}}|-1\) we have

The convergence case. It follows from (2.3) that for any \({\mathfrak {p}}/q\in X\) we have

Therefore \(W_{\Phi }^{***}(g)\subseteq W_{\Phi }(\Psi )\) for \(\Psi ({\textbf{a}})=\frac{g(|{\textbf{a}}|){\text {Diam}}(X_{{\textbf{a}}})}{C_{1}}.\) In which case \({\mathcal {H}}^{s}(W_{\Phi }^{***}(g))=0\) if \({\mathcal {H}}^{s}(W_{\Phi }(\Psi ))=0.\) However this last inequality follows from Theorem 1.1 and our assumption \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot \left( {\text {Diam}}(X_{{\textbf{a}}})g(n)\right) ^{s}<\infty .\)

The divergence case. For any \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^n\) and \(0\le l\le |{\textbf{a}}|-1\) there exists \({\mathfrak {p}}/q\in X\) such that \({\mathfrak {p}}/q=\pi \left( a_{1}\ldots a_{l}\left( a_{l+1}\ldots a_{|{\textbf{a}}|}\right) ^{\infty }\right) .\) Therefore

It now follows from the fact g is non-increasing together with (2.3) that

Therefore \(W_{\Phi }(\Psi )\subset W_{\Phi }^{***}(g)\) for \(\Psi \) given by \(\Psi ({\textbf{a}})=\frac{g(|{\textbf{a}}|){\text {Diam}}(X_{{\textbf{a}}})}{C_{2}}\), and our result follows if \(\mu (W_{\Phi }(\Psi ))=1.\) However \(\mu (W_{\Phi }(\Psi ))=1\) follows from Theorem 1.1 together with our assumption \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty .\) \(\square \)

As remarked upon in the equicontractive case, if \({\mathfrak {p}}/q\in X\) is in its reduced from then we must have \(q\le q_{\text {int}}({\mathfrak {p}}/q)\). This observation together with Theorem 2.7 implies the following corollary.

Corollary 2.8

Let \(\Phi \) be an IFS of the form \(\Phi =\left\{ \phi _{a}(x)=\frac{x+{\mathfrak {p}}_a}{q_a}\right\} \) satisfying the strong separation condition and \(h_{{\textbf{p}}}<-2\log \sum _{a\in {\mathcal {A}}}p_{a}^{2}\). Let \(g:{\mathbb {N}}\rightarrow [0,\infty )\) be a non-increasing function. If \(\sum _{n=1}^{\infty }\sum _{{\textbf{a}}\in {\mathcal {A}}^{n}}n\cdot ({\text {Diam}}(X_{{\textbf{a}}})g(n))^{\dim _{S}\left( \Phi \right) }=\infty \) then

In this section we prove Theorem 1.1. Statement 1 of this theorem is proved by a standard covering argument and as such is omitted. We will only prove Statement 2 of Theorem 1.1 in full. Statement 3 is proved via an almost identical method. Where appropriate we will indicate in the footnotes where the proofs differ and why Statement 3 does not require the assumption g is non-increasing.

3.1 Technical preliminaries

Given a word \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^n\) we let \(C_{t}({\textbf{a}})\) denote the number of distinct words of length t appearing within \({\textbf{a}}\). Given a probability vector \({\textbf{p}}\) and \(n\in {\mathbb {N}}\) we let

We also let

When the choice of \({\textbf{p}}\) is implicit, we simply denote \(k_{n,{\textbf{p}}}\) by \(k_{n}\) and \({\mathcal {F}}_{n,{\textbf{p}}}\) by \({\mathcal {F}}_{n}.\)

The following lemma is a suitable adaptation of Lemma 4.1. from [37].

Lemma 3.1

Let \({\textbf{p}}\) be a probability vector and \({\mathfrak {m}}\) be the corresponding Bernoulli measure on \({\mathcal {A}}^{{\mathbb {N}}}\). Then for n sufficiently large, we have

Proof

Given \({\textbf{a}}\in {\mathcal {A}}^{n}\) and \({\textbf{b}}\in {\mathcal {A}}^{k_n},\) let

It is convenient to express our proof using the language of probability theory. As such let \((Z_l)_{l=1}^n\) be a sequence of independent and identically distributed random variables taking values in \({\mathcal {A}}\) such that \({\mathbb {P}}(Z_l=a)=p_{a}\) for all \(a\in {\mathcal {A}}\). Let \(\Sigma _{k_n}^{n}\) be the real valued random variable given by

We start by bounding the expectation of \(\Sigma _{k_n}^{n}\):

With (3.1) in mind, we now bound \({\mathbb {P}}(Z_{l+1}\ldots Z_{l+k_n}=Z_{j+1}\ldots Z_{j+k_{n}})\) from above. If \(l=j\) then clearly \({\mathbb {P}}(Z_{l+1}\ldots Z_{l+k_n}=Z_{j+1}\ldots Z_{j+k_{n}})=1.\) We remark that for any parameters \(l,j,k\in {\mathbb {N}}\), if \(l+k<j+1\) then by independence we have

We will use the fact that if \(l+k<j+1\) then

throughout our proof.

We now proceed via a case analysis. If \(l+k_{n}<j+1\) then (3.2) immediately implies

Now suppose that \(l+\frac{k_n}{4}\le j+1\le l+k_{n}\). Observe that \(Z_{l+1}\ldots Z_{l+k_n}=Z_{j+1}\ldots Z_{j+k_{n}}\) implies that \(Z_{2l-j+k_n+1}\ldots Z_{l+k_n}=Z_{l+k_{n}+1}\ldots Z_{j+k_{n}}.\) Therefore by (3.2) we have

Let us now suppose that \(l+1<j+1< l+\frac{k_{n}}{4}\) and \(Z_{l+1}\ldots Z_{l+k_n}=Z_{j+1}\ldots Z_{j+k_{n}}.\) Notice that \(Z_{l+1}\ldots Z_{l+k_n}=Z_{j+1}\ldots Z_{j+k_{n}}\) implies that \(Z_{l+i+1}\ldots Z_{l+i+\lfloor \frac{k_n}{4}\rfloor }=Z_{j+i+1}\ldots Z_{j+i+\lfloor \frac{k_n}{4}\rfloor }\) for any \(0\le i<k_{n}-\lfloor \frac{k_n}{4}\rfloor .\) Repeatedly applying this identity, it follows that

for any d such that \(l+d(j-l)+1<k_{n}-\lfloor \frac{k_n}{4}\rfloor .\) Since \(j-l<\frac{k_n}{4}\), it follows that we can pick d such that \(l+\lfloor \frac{k_n}{4}\rfloor <l+d(j-l)+1\le k_{n}-\lfloor \frac{k_n}{4}\rfloor .\) Taking such a d, it then follows from (3.2) that

Recalling (3.1), we have

Applying the bounds provided by (3.3), (3.4), (3.5) we have

By the definition of \(k_{n}\) we know that

Moreover, as \(k_{n}\) grows logarithmically in n, and \(\left( \sum _{a\in {\mathcal {A}}}p_{a}^2\right) ^{\lfloor \frac{k_n}{4}\rfloor }\) decays to zero polynomially fast, we know that

for all n sufficiently large. Therefore for n sufficiently large we have

Let B be the event that

We now bound the probability of B from above. It follows from (3.6) that for n sufficiently large we have

In the penultimate line we used that for n sufficiently large we have \(n-k_{n}+1\ge \frac{4n}{5}\). This is because \(k_n\) grows logarithmically in n. Therefore

This means that \({\mathbb {P}}(B^c)\ge 7/32\). Since

this completes our proof. \(\square \)

Let us now suppose that we are given a probability vector \({\textbf{p}}\) and an IFS \(\Phi \). Recall that the entropy of \({\textbf{p}}\) and the Lyapunov exponent of \({\textbf{p}}\) are defined to be

respectively. Notice that when \({\textbf{p}}=\left( r_{a}^{\dim _{S}\left( \Phi \right) }\right) _{a\in {\mathcal {A}}}\) then \(\dim _{S}(\Phi )=\frac{h_{{\textbf{p}}}}{\chi _{\Phi ,{\textbf{p}}}}\). When the choice of \({\textbf{p}}\) and \(\Phi \) is implicit, we simply denote \(h_{{\textbf{p}}}\) by h and \(\chi _{\Phi ,{\textbf{p}}}\) by \(\chi \). Given a word \({\textbf{a}}\in {\mathcal {A}}^n\) and \(\epsilon >0\) we let

We then define

Lemma 3.2

Let \(\Phi \) be an IFS, \({\textbf{p}}\) be a probability vector, and \({\mathfrak {m}}\) be the Bernoulli measure corresponding to \({\textbf{p}}\). For any \(\epsilon >0,\) there exists \(\gamma \in (0,1)\) such that

Proof

As in the proof of the previous lemma, it is useful to express this proof in terms of random variables. Let \((Z_{l})_{l=1}^n\) be a sequence of independent and identically distributed random variables taking values in \({\mathcal {A}}\) such that \({\mathbb {P}}(Z_l=a)=p_{a},\) and let \(\epsilon >0\) be arbitrary. We start our proof by bounding from above the expectation

By the linearity of expectation we have

By Hoeffding’s inequality for large deviations [24], there exists \(\gamma _1:=\gamma _1(\epsilon ,{\textbf{p}},\Phi )\in (0,1)\) such that

Therefore

Now by Markov’s inequality, we have

By an analogous argument, it can be shown that there exists \(\gamma _{2}\in (0,1)\) such that

Now let \(\gamma =\max \{\gamma _{1},\gamma _{2}\}\). Clearly if \((Z_l)_{l=1}^n\) is such that either

for \(\lfloor \frac{n}{20}\rfloor \) values of l,  it must satisfy

for at least \(\frac{1}{2}\lfloor \frac{n}{20}\rfloor \) values of l or

for at least \(\frac{1}{2}\lfloor \frac{n}{20}\rfloor \) values of l. As such we may conclude that

This completes our proof. \(\square \)

Combining Lemmas 3.1 and 3.2, we see that the following statement holds.

Lemma 3.3

Let \(\Phi \) be an IFS, \({\textbf{p}}\) be a probability vector, and \({\mathfrak {m}}\) be the Bernoulli measure corresponding to \({\textbf{p}}\). For any \(\epsilon >0\), it is the case that for all n sufficiently large there exists a set \(Good(n,\epsilon )\subset {\mathcal {A}}^n\) satisfying:

1. 1.

   \({\mathfrak {m}}\left( \bigcup _{{\textbf{a}}\in Good(n,\epsilon )}[{\textbf{a}}]\right) \ge 7/64.\)

2. 2.

   For each \({\textbf{a}}\in Good(n,\epsilon )\) there exists a set \(W_{{\textbf{a}}}\subset \{0,\ldots , n-k_{n}\}\) satisfying:

   1. (a)

      \(\#W_{{\textbf{a}}}\ge \Big \lfloor \frac{n}{20}\Big \rfloor \).

   2. (b)

      If \(l,l'\in W_{{\textbf{a}}}\) and \(l\ne l'\) then \(a_{l+1}\ldots a_{l+k_{n}}\ne a_{l'+1}\ldots a_{l'+k_{n}}\).

   3. (c)

      For each \(l\in W_{{\textbf{a}}}\) we have

      $$\begin{aligned} \prod _{i=1}^{k_n}r_{a_{l+i}}\in \left[ e^{k_n\left( -\chi -\epsilon \right) },e^{k_n\left( -\chi +\epsilon \right) }\right] \end{aligned}$$

      and

      $$\begin{aligned} \prod _{i=1}^{k_n}p_{a_{l+i}}\in \left[ e^{k_n\left( -h-\epsilon \right) },e^{k_n\left( -h+\epsilon \right) }\right] . \end{aligned}$$

Lemma 3.3 is the main technical result in this section and will play an important part in our proof of Theorem 1.1.^{Footnote 1}

3.2 Proof of Statement 2 from Theorem 1.1

Before moving on to our proof of Statement 2,  it is useful to record for later reference a number of technical results. We start by recalling a well known lemma.

Lemma 3.4

Let \((X,A,{\mathfrak {m}})\) be a finite measure space and \(E_n\in A\) be a sequence of sets such that \(\sum _{n=1}^{\infty }{\mathfrak {m}}(E_n)=\infty .\) Then

Lemma 3.4 is due to Kochen and Stone [28]. For a proof of this lemma see either [20, Lemma 2.3] or [36, Lemma 5]. The following density lemma follows from [8, Lemma 6]. It has been phrased for our purposes.

Lemma 3.5

Let \({\mathfrak {m}}\) be a Bernoulli measure on \({\mathcal {A}}^{{\mathbb {N}}}\) and \(E\subset {\mathcal {A}}^{{\mathbb {N}}}\). Suppose that there exists \(c>0\) such that for each finite word \({\textbf{c}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^{n}\) we have

Then \({\mathfrak {m}}(E)=1\).

For the rest of this section we fix an IFS \(\Phi \) satisfying

We emphasise that throughout this section \({\textbf{p}}\) will always be the probability vector corresponding to \((r_{a}^{\dim _{S}\left( \Phi \right) }).\) It follows from this inequality that we can pick \(\epsilon >0\) sufficiently small such that

and

For the rest of this section we fix an \(\epsilon >0\) such that (3.7) and (3.8) are both satisfied. These are the only conditions \(\epsilon \) will need to satisfy. Let \(\Psi :\cup _{n=1}^{\infty }{\mathcal {A}}^n\rightarrow [0,\infty )\) be an arbitrary function of the form \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}}) g(|{\textbf{a}}|)\) for some non-increasing \(g:{\mathbb {N}}\rightarrow [0,\infty ).\) We also assume that g is such that

As previously remarked, this divergence condition is equivalent to

The following two lemmas allow us to replace g with a function whose decaying behaviour we know more about.

Lemma 3.6

Assume that g is a non-increasing function satisfying (3.9). Let \(g_{1}:{\mathbb {N}}\rightarrow [0,\infty )\) be given by

Then

Proof

Our argument is an adaptation of the proof of Lemma 6.3 from [12]. Since g is non-increasing, and so is the sequence \(\left( n^{-2/\dim _{S}\left( \Phi \right) }\right) \), we see that the function \(g_{1}\) is also non-increasing. Let us now suppose that

Since \(g_{1}\) is non-increasing, for any m sufficiently large we have

Equation (3.10) then implies that

Combining the two equations above, we may conclude that

for all m sufficiently large. This means that \(g_{1}(n)=g(n)\) for n sufficiently large. Therefore by (3.10) we must have

This contradicts our initial assumption that g satisfies (3.9). Therefore we must have.^{Footnote 2}

\(\square \)

Lemma 3.7

Assume that g is a non-increasing function satisfying (3.9) and let \(g_1\) be as in Lemma 3.6. Let \(g_{2}:{\mathbb {N}}\rightarrow [0,\infty )\) be given by

Then

Proof

This follows from Lemma 3.6 and the fact \(\sum _{n=1}^{\infty }n\cdot \left( \frac{1}{n^{4/\dim _{S}\left( \Phi \right) }}\right) ^{\dim _{S}\left( \Phi \right) }<\infty .\) \(\square \)

Let \(g_{2}\) be as in Lemma 3.7. We now define \(\Psi _{2}:\cup _{n=1}^{\infty }{\mathcal {A}}^n\rightarrow [0,\infty )\) by \(\Psi _{2}({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g_{2}(|{\textbf{a}}|).\) Since \(g_{2}(n)\le g(n)\) for all n it follows that \(W_{\Phi }(\Psi _{2})\subset W_{\Phi }(\Psi )\). Therefore to prove Statement 2 of Theorem 1.1 it is sufficient to show that \(\mu (W_{\Phi }(\Psi _{2}))=1.\)

Let \({\textbf{c}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^n\) be arbitrary and fixed. We will show that

for some c independent of \({\textbf{c}}\). Lemma 3.5 then implies that \({\mathfrak {m}}\left( \pi ^{-1}(W_{\Phi }(\Psi _{2})\right) =1\). Since \(\mu =\pi {\mathfrak {m}}\) this implies that \(\mu \left( W_{\Phi }(\Psi _{2})\right) =1.\) Therefore to complete our proof it suffices to show that (3.11) holds.

Let us fix an N sufficiently large so that Lemma 3.3 applies for all \(n\ge N\) for our choice of \(\epsilon .\) We may also assume that N is sufficiently large so that

for \(n\ge N\), and so that there exists \(\gamma \in (1,2)\) for which

for \(n\ge N\). The existence of \(\gamma ,\) and the fact that (3.12) and (3.13) are satisfied for n sufficiently large, follows from (3.7), (3.8) and the fact \(\dim _{S}\left( \Phi \right) =\frac{h}{\chi }\).

Let \(n\ge N\), for each \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}\) we consider the ball

If \(g_{2}(|{\textbf{c}}|+n)\ne 0\) then there exists \(h_{{\textbf{a}},l}\in {\mathbb {N}}\) and \(l+1\le j_{{\textbf{a}},l}\le n\) such that

and

It follows from the fact \({\textbf{p}}=(r_{{\textbf{a}}}^{\dim _{S}\left( \Phi \right) })\) and \({\mathfrak {m}}\) is the Bernoulli measure on \({\mathcal {A}}^{{\mathbb {N}}}\) corresponding to \({\textbf{p}}\), that for any word \({\textbf{a}}\in \cup _{n=1}^{\infty }{\mathcal {A}}^n\) we have

Combining (3.15) together with (3.16) we can deduce that

We will use the cylinder sets \(\left[ {\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}\right] \) to prove that (3.11) holds. Before doing that it is useful to prove some properties of the parameters \(h_{{\textbf{a}},l}\) and \(j_{{\textbf{a}},l}\).

Lemma 3.8

Let \(n\ge N\) be such that \(g_{2}(|{\textbf{c}}|+n)\ne 0,\) and let \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}.\) For \(h_{{\textbf{a}},l}\) and \(j_{{\textbf{a}},l}\) as defined above, if \(h_{{\textbf{a}},l}=1\) then \(j_{{\textbf{a}},l}>l+k_{n}.\)

Proof

If \(h_{{\textbf{a}},l}=1\) then

implies

By Lemma 3.6 and Lemma 3.7 we know that \(g_{2}(|{\textbf{c}}|+n)\le n^{-2/\dim _{S}\left( \Phi \right) }\). Therefore

Importantly, by (2c) from Lemma 3.3 we know that

Equation (3.12) states that

for \(n\ge N.\) It follows therefore from (3.18) and (3.19) that

Therefore we must have \(j_{{\textbf{a}},l}> l+k_{n}\). \(\square \)

If we combine (2b). from Lemma 3.3 together with Lemma 3.8, we may conclude the following lemma.

Lemma 3.9

Assume that \(n\ge N\) is such that \(g_{2}(|{\textbf{c}}|+n)\ne 0\) and let \({\textbf{a}}\in Good(n,\epsilon )\). If \(l,l'\in W_{{\textbf{a}}}\) and \(l\ne l'\) then

Lemma 3.10

Let \(n\ge N\) be such that \(g_{2}(|{\textbf{c}}|+n)\ne 0,\) and let \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}.\) There exists \(C=C({\textbf{c}})\) such that for \(h_{{\textbf{a}},l}\) and \(j_{{\textbf{a}},l}\) as defined above, we have

Proof

If \(g_{2}(|{\textbf{c}}|+n)\ne 0\) then by Lemma 3.7 we know that it must satisfy \(g_{2}(|{\textbf{c}}|+n)\ge \frac{1}{(|{\textbf{c}}|+n)^{4/\dim _{S}\left( \Phi \right) }}.\) Equation (3.15) then implies that if \(g_{2}(|{\textbf{c}}|+n)\ne 0\) then

This in turn implies that

Taking logarithms and then manipulating the resulting expression, one can show that the above implies that there exists \(C=C({\textbf{c}})\) such that

\(\square \)

For each \(n\ge N\) such that \(g_{2}(|{\textbf{c}}|+n)\ne 0\) we let

Lemma 3.9 tells us that any pair of cylinder sets in this union are disjoint. If \(n\ge N\) is such that \(g_{2}(|{\textbf{c}}|+n)= 0\) then set \(E_{n}=\emptyset \). Importantly (3.14) implies that

Therefore to prove (3.11) it is sufficient to show that

We will prove that (3.20) holds using Lemma 3.4. Before that it is necessary to check that the hypothesis of this lemma are satisfied.

Lemma 3.11

For \(n\ge N\) we have \({\mathfrak {m}}(E_n)\asymp m([{\textbf{c}}])\cdot n\cdot g_{2}(|{\textbf{c}}|+n)^{\dim _{S}\left( \Phi \right) }.\)

Proof

This lemma is obviously true if n is such that \(g_{2}(|{\textbf{c}}|+n)=0\). As such we restrict our attention to those \(n\ge N\) for which \(g_{2}(|{\textbf{c}}|+n)\ne 0\). Recall that by Lemma 3.9, for distinct \(l,l'\in W_{{\textbf{a}}}\) we have \({\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}\ne {\textbf{c}}a_{1}\ldots a_{l'}(a_{l'+1}\ldots a_{n})^{h_{{\textbf{a}},l'}}a_{l'+1}\ldots a_{j_{{\textbf{a}},l'}}\). Therefore we have

\(\square \)

It follows from Lemma 3.7 and Lemma 3.11 that \(\sum _{n=1}^{\infty }{\mathfrak {m}}(E_n)=\infty \). So our sequence of sets \((E_n)\) satisfies the hypothesis of Lemma 3.4. To complete our proof we need to get good upper bounds for \({\mathfrak {m}}(E_n\cap E_m)\). We restrict our attention to those n and m satisfying \(n<m,\) \(g_2(|{\textbf{c}}|+n)\ne 0,\) and \(g_{2}(|{\textbf{c}}|+m)\ne 0\). For these n and m we see that

The following proposition gives good upper bounds for the terms in this sum. The parameter C in the statement of this proposition is the same C as in Lemma 3.10.

Proposition 3.12

Let \(n,m\ge N\) be such that \(n<m\), \(g_2(|{\textbf{c}}|+n)\ne 0,\) and \(g_{2}(|{\textbf{c}}|+m)\ne 0.\) Then for \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}\) the following holds:

1. I.

   If \(n<m\le n+C\log n\) then

   $$\begin{aligned}&{\mathfrak {m}}\left( \left[ {\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}\right] \cap E_{m}\right) \\&\quad \ll \,{\mathfrak {m}}([{\textbf{c}}{\textbf{a}}])e^{(h+\epsilon ) k_{m}}g_{2}(|{\textbf{c}}|+n)^{\dim _{S}\left( \Phi \right) } g_{2}(|{\textbf{c}}|+m)^{\dim _{S}\left( \Phi \right) }\\&\qquad + {\mathfrak {m}}([{\textbf{c}}{\textbf{a}}])\left( \max _{a\in {\mathcal {A}}}p_{a}\right) ^{m-n}g_{2}(|{\textbf{c}}|+m)^{\dim _{S}\left( \Phi \right) }\\&\qquad +\,m\cdot {\mathfrak {m}}([{\textbf{c}}{\textbf{a}}]) g_{2}(|{\textbf{c}}|+m)^{\dim _{S}\left( \Phi \right) }g_{2}(|{\textbf{c}}|+n)^{\dim _{S}\left( \Phi \right) }. \end{aligned}$$
2. II.

   If \(m> n+C\log n\) then

   $$\begin{aligned}{} & {} {\mathfrak {m}}\left( \left[ {\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}\right] \cap E_{m}\right) \\{} & {} \quad \ll m\cdot {\mathfrak {m}}([{\textbf{c}}{\textbf{a}}]) g_{2}(|{\textbf{c}}|+m)^{\dim _{S}\left( \Phi \right) }g_{2}(|{\textbf{c}}|+n)^{\dim _{S}\left( \Phi \right) }. \end{aligned}$$

Proof

We prove each statement separately.

Proof of Statement I. Assume that \(n<m\le n+C\log n\). Let \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}\). If \(m\le l +h_{{\textbf{a}},l}(n-l)+(j_{{\textbf{a}},l}-l)\) then at most one \({\textbf{b}}\in Good(m,\epsilon )\) is such that \([{\textbf{c}}{\textbf{b}}]\) has a non-empty intersection with \(\left[ {\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}\right] .\) Let us assume that such a \({\textbf{b}}\) exists. Otherwise \({\mathfrak {m}}([{\textbf{c}}a_{1}\ldots a_{l}(a_{l+1}\ldots a_{n})^{h_{{\textbf{a}},l}}a_{l+1}\ldots a_{j_{{\textbf{a}},l}}]\cap E_{m})=0\) and our upper bound holds trivially. In this case we see that

Lemma 3.8 implies that if

then we must have

By Lemma 3.3 we know that for each \(l'\in W_{{\textbf{b}}}\) the cylinder set \(\left[ {\textbf{c}}b_{1}\ldots b_{m}b_{l'+1}\ldots b_{l'+k_m}\right] \) satisfies

Therefore, by (3.17) and a measure argument we have

Applying the above observations we see that

Because \({\textbf{b}}\) must have \({\textbf{a}}\) as a prefix we see that

Substituting (3.22) into the last line in the above, we have shown that if \(m\le l +h_{{\textbf{a}},l}(n-l)+(j_{{\textbf{a}},l}-l)\) then^{Footnote 3}

Now suppose that \(m> l +h_{{\textbf{a}},l}(n-l)+(j_{{\textbf{a}},l}-l).\) In this case

Adding together the upper bounds obtained in (3.23) and (3.24) we obtain the desired upper bound which holds for all m satisfying \(n<m\le n+C\log n\).

Proof of Statement II. Assume \(m>n+C\log n\). Let \({\textbf{a}}\in Good(n,\epsilon )\) and \(l\in W_{{\textbf{a}}}\). If \(m>n+C\log n\) then by Lemma 3.10 we must have \(m>l +h_{{\textbf{a}},l}(n-l)+(j_{{\textbf{a}},l}-l)\). In which case the same argument as is used in the proof of the second part of Statement I applies and we have the desired bound

\(\square \)

Equipped with Proposition 3.12 we will now prove the following statement.

Proposition 3.13

There exists a constant \(C_{1}=C_{1}({\textbf{c}})\) such that

Proof

We start our proof by rewriting \(\sum _{n,m=N}^{Q}{\mathfrak {m}}(E_{n}\cap E_m):\)

We will focus on the three terms A, B, and C individually. By Lemma 3.11 we have the following bound for term A:

Now focusing on the term B,  if we apply Statement I from Proposition 3.12 we have

Focusing on the term B1 in the above, we know by (3.13) that

for some \(\gamma \in (1,2)\). Using this inequality we have

It follows from Lemma 3.6 and Lemma 3.7 that \(g_{2}(|{\textbf{c}}|+m)^{\dim _{S}\left( \Phi \right) }\le m^{-2}\) for all \(m\in {\mathbb {N}}\). Therefore we have

In the line marked M.V.T. we use the mean value theorem. In the penultimate line in the above we used that \(g_{2}(|{\textbf{c}}|+n)^{\dim _{S}\left( \Phi \right) }\le n^{-2}\). Because \(\gamma \in (1,2)\) we know that \(\sum _{n=1}^{\infty }\frac{C\log n}{n^{3-\gamma }}<\infty \). Therefore we can assert that there exists a constant \(C_{1}=C_{1}({\textbf{c}})\) so that the term B1 satisfies

Turning our attention to the term B2 we see that

Now using the fact that \(\sum _{n=1}^{m-1}n\cdot \left( \max _{a\in {\mathcal {A}}}p_{a}\right) ^{m-n}\ll m\) we see that

So our term B2 must satisfy

Now focusing on the term B3, we have

Combining (3.26), (3.27), and (3.28) we have the following bound for the term B

By an analogous argument to that used to bound B3, by applying Statement II from Proposition 3.12 it can also be shown that the term C satisfies

Combining (3.25), (3.29), and (3.30) we may conclude our desired bound

\(\square \)

Combining Proposition 3.13 together with Lemma 3.4 and Lemma 3.11 we may conclude that

Thus (3.20) holds and our proof of Statement 2 from Theorem 1.1 is complete. We emphasise that in the penultimate line in the above we used the fact that the constant \(C_{1}\) does not affect the limit. This is important because the implicit constant in (3.20) needs to be independent of \({\textbf{c}}\) if we want to apply Lemma 3.5.

The mass transference principle of Beresnevich and Velani [10] is a powerful tool that allows one to derive information on the Hausdorff measure of a limsup set. We do not state it in its full generality, but instead content ourselves with the following which is better suited for our purposes.

Let \(X\subset {\mathbb {R}}^d\). Then X is said to be Ahlfors regular if there exists \(C_{1},C_{2}>0\) such that

for all \(x\in X\) and r sufficiently small. Given an Ahlfors regular set X, a ball B(x, r) in X, and \(s>0\), we let \(B^{s}=B\left( x,r^{s/\dim _{H}(X)}\right) \). The following theorem is a simplified version of Theorem 3 from [10].

Theorem 4.1

Let X be Ahlfors regular and \((B_l)\) be a sequence of balls in X with radii tending to zero. Let \(s>0\) and suppose that for any ball B in X we have

Then, for any ball B in X

It is a well known fact that if an IFS \(\Phi \) satisfies the open set condition then the corresponding self-similar set is Ahlfors regular. It is also well known that if \(\Phi \) satisfies the open set condition then \(\mu \) is equivalent to the restriction of \({\mathcal {H}}^{\dim _{S}\left( \Phi \right) }\) on X. Combining these facts together with Theorem 1.1 and Theorem 4.1, we may deduce the following statement.

Theorem 4.2

Let \(\Phi =\left\{ \phi _a\right\} _{a\in {\mathcal {A}}}\) be an IFS which satisfies the open set condition. Let \(\Psi :\cup _{n=1}^{\infty }{\mathcal {A}}^{n}\rightarrow [0,\infty )\) be given by \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|)\) for some function \(g:{\mathbb {N}}\rightarrow [0,\infty )\) satisfying

For \(t\ge 1\) let \(\Psi ^{t}:\cup _{n=1}^{\infty }{\mathcal {A}}^n\rightarrow [0,\infty )\) be given by \(\Psi ^{t}({\textbf{a}})=\Psi ({\textbf{a}})^t.\) Then the following statements are true:

1. 1.

   Assume that

   $$\begin{aligned} h_{{\textbf{p}}}<-2\log \sum _{a\in {\mathcal {A}}}p_{a}^{2} \end{aligned}$$

   and that g is non-increasing. Then for any \(t\ge 1\) we have \({\mathcal {H}}^{\dim _{H}(X)/t}(W_{\Phi }(\Psi ^{t}))={\mathcal {H}}^{\dim _{H}/t}(X)\).

2. 2.

   If \(\Phi \) is equicontractive then for any \(t\ge 1\) we have \({\mathcal {H}}^{\dim _{H}(X)/t}(W_{\Phi }(\Psi ^{t}))={\mathcal {H}}^{\dim _{H}/t}(X)\).

We emphasise that if \(\Phi \) is not equicontractive and \(\Psi \) is of the form \(\Psi ({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g(|{\textbf{a}}|),\) then for \(t>1\) there exists no \(g_{t}:{\mathbb {N}}\rightarrow [0,\infty )\) such that \(\Psi ^{t}\) satisfies \(\Psi ^{t}({\textbf{a}})={\text {Diam}}(X_{{\textbf{a}}})g_{t}(|{\textbf{a}}|)\) for all \({\textbf{a}}\in {\mathcal {A}}\). As such Theorem 4.2 allows us to make positive measure statements about a new class of functions.

We conclude this section by mentioning that by following the arguments used in Sect. 2, one can use Theorem 4.2 to prove a number of statements on the Hausdorff measure of certain limsup sets arising from the study of intrinsic Diophantine Approximation on self-similar sets. We leave the details to the interested reader.