Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

Abstract

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta >1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \hbox { for infinitely many}, \, n\in \mathbb{N }\,\Big \}$ is determined, where $x_0$ is a given point in $[0,1]$ and $\{\ell _n\}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\rightarrow \infty$. For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of $\beta$ with a common prefix in the expansion of 1) in the parameter space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$.

1 Introduction

The study of Diophantine properties of the orbits in a dynamical system has recently received much attention. This study contributes to a better understanding of the distribution of the orbits in a dynamical system. Let $(X,\mathcal{B },\mu , T)$ be a measure-preserving dynamical system with a consistent metric $d$. If $T$ is ergodic with respect to the measure $\mu$, then Birkhoff’s ergodic theorem yields the following hitting property, namely, for any $x_0\in X$ and $\mu$-almost all $x\in X$,

$$\begin{aligned} \liminf _{n\rightarrow \infty } d(T^n(x),x_0)=0. \end{aligned}$$

(1.1)

One can then ask, what are the quantitative properties of the convergence speed in (1.1)? More precisely, for a given sequence of balls $B(x_0,r_n)$ with center $x_0\in X$ and shrinking radius $\{r_n\}$, what are the metric properties of the set

$$\begin{aligned} F(x_0,\{r_n\}):=\Big \{x\in X: d(T^nx, x_0)< r_n\, {\text {for infinitely many}}\, n\in \mathbb{N }\Big \} \end{aligned}$$

in the sense of measure and in the sense of dimension? More generally, let $\{B_n\}_{n\ge 1}$ be a sequence of measurable sets with $\mu (B_n)$ decreasing to 0 as $n\rightarrow \infty$. The study of the metric properties of the set

$$\begin{aligned} \Big \{\, x\in X: T^nx\in B_n\, {\text {for infinitely many}}\, n\in \mathbb{N } \,\Big \} \end{aligned}$$

(1.2)

is called the dynamical Borel–Cantelli Lemma [6] or the shrinking target problem [12].

In this paper, we consider a modified shrinking target problem. Let us begin with an example to illustrate the motivation. Let $R_\alpha :x \mapsto x+\alpha$ be a rotation map on the unit circle. Then the set studied in classical inhomogeneous Diophantine approximation can be written as

$$\begin{aligned} \Big \{\, \alpha \in \mathbb{Q }^c: |R_{\alpha }^n0-x_0|<r_n, \, {\text {for infinitely many}}\, n\in \mathbb{N } \,\Big \}, \end{aligned}$$

(1.3)

where $|x-y|$ means the distance between $x,y\in \mathbb{R }$. The size of the set (1.3) in the sense of Hausdorff measure and Hausdorff dimension was studied by Bugeaud [3], Levesley [15], Bugeaud and Chevallier [4] etc. Compared with the shrinking target problem (1.2), instead of considering the Diophantine properties in one given system, the set (1.3) concerns the properties of the orbit of some given point (the orbit of 0) in a family of dynamical systems. It is the set of parameters $\alpha$ such that $R_\alpha$ share some common properties.

Following this idea, in this paper, we consider the same setting as (1.3) in the dynamical systems $([0,1], T_{\beta })$ of $\beta$-transformations with $\beta$ varying in the parameter space $\{\, \beta \in \mathbb{R }: \beta >1 \,\}$.

It is well-known that $\beta$-transformations are typical examples of one-dimensional expanding systems, whose properties are reflected by the orbit of some critical point. In the case of $\beta$-transformations, this critical point is the unit 1. This is because the $\beta$-expansion of 1 (or the orbit of 1 under $T_\beta$) can completely characterise all admissible sequences in the $\beta$-shift space (see [17]), the lengths and the distribution of cylinders induced by $T_\beta$ [8], etc. Upon this, in this current work, we study the Diophantine properties of $\{T^n_{\beta }1\}_{n\ge 1}$, the orbit of 1, as $\beta$ varies in the parameter space $\{\, \beta \in \mathbb{R }: \beta >1\,\}$.

Blanchard [1] gave a kind of classification of the parameters in the space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$ according to the distribution of $\fancyscript{O}_{\beta }:=\{T^n_{\beta }1\}_{n\ge 1}$: (i) ultimately zero; (ii) ultimately non-zero periodic; (iii) 0 is not an accumulation point of $\fancyscript{O}_\beta$ (exclude those $\beta$ in classes (i,ii)); (iv) non-dense in $[0,1]$ (exclude $\beta$’s in classes (i,ii,iii)); and (v) dense in $[0,1]$. It was shown by Schmeling [21] that the class (v) is of full Lebesgue measure (the results in [21] give more, that for almost all $\beta$, all allowed words appear in the expansion of 1 with regular frequencies). This dense property of $\fancyscript{O}_{\beta }$ for almost all $\beta$ gives us a type of hitting property, i.e., for any $x_0\in [0,1]$,

$$\begin{aligned} \liminf _{n\rightarrow \infty } |T^n_{\beta }1-x_0|=0, \quad \text {for } \mathcal{L }{\text {-a.e.}} \, \beta >1, \end{aligned}$$

(1.4)

where $\fancyscript{L}$ is the Lebesgue measure on $\mathbb{R }$. Similarly as for (1.1), we would like to investigate the possible convergence speed in (1.4).

Fix a point $x_0\in [0,1]$ and a sequence of positive integers $\{\ell _n\}_{n\ge 1}$. Consider the set of $\beta >1$ for which $x_0$ can be well approximated by the orbit of 1 under the $\beta$-transformations with given shrinking speed, namely the set

$$\begin{aligned} E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\, \beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \, {\text {for infinitely many}}\, n\in \mathbb{N } \,\Big \}. \end{aligned}$$

(1.5)

This can be viewed as a kind of shrinking target problem in the parameter space.

When $x_0=0$ and $\ell _n=\alpha n \, (\alpha > 0)$, Persson and Schmeling [18] proved that

$$\begin{aligned} \dim _\mathsf H E(\{\alpha n\}_{n\ge 1}, 0)=\frac{1}{1+\alpha }, \end{aligned}$$

where $\dim _\mathsf H$ denotes the Hausdorff dimension. For a general $x_0\in [0, 1]$ and a sequence $\{\ell _n\}$, we have the following.

Theorem 1.1

Let $x_0\in [0,1]$ and let $\{\ell _n\}_{n\ge 1}$ be a sequence of positive integers such that $\ell _n\rightarrow \infty$ as $n\rightarrow \infty$. Then

$$\begin{aligned} \dim _\mathsf H E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\frac{1}{1+\alpha }, \quad {\text {where}}\, \alpha =\liminf _{n\rightarrow \infty }\frac{\ell _n}{n}. \end{aligned}$$

In other words, the set in (1.5) consists of the points in the parameter space $\{\,\beta >1: \beta \in \mathbb{R }\,\}$ for which the orbit $\{\,T^n_{\beta }1: n\ge 1\,\}$ is close to the same point $x(\beta )=x_0$ for infinitely many moments in time. What can be said if the point $x(\beta )$ is also allowed to vary continuously with $\beta >1$? Let $x=x(\beta )$ be a function on $(1, +\infty )$, taking values on $[0, 1]$. The setting (1.5) changes to

$$\begin{aligned} \widetilde{E}\big (\{\ell _n\}_{n\ge 1}, x\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x(\beta )|<\beta ^{-\ell _n}, \, {\text {for infinitely many}}\, n\in \mathbb{N }\,\Big \}. \end{aligned}$$

(1.6)

As will become apparent, the proof of Theorem 1.1 also works for this general case $x=x(\beta )$ after some minor adjustments, and we can therefore state the following theorem.

Theorem 1.2

Let $x=x(\beta ): (1, +\infty )\rightarrow [0, 1]$ be a Lipschtiz continuous function and $\{\ell _n\}_{n\ge 1}$ be a sequence of positive integers such that $\ell _n\rightarrow \infty$ as $n\rightarrow \infty$. Then

$$\begin{aligned} \dim _\mathsf H \widetilde{E}\big (\{\ell _n\}_{n\ge 1}, x\big )=\frac{1}{1+\alpha }, \quad {\text {where}}\, \alpha =\liminf _{n\rightarrow \infty }\frac{\ell _n}{n}. \end{aligned}$$

Theorems 1.1 (as well as Theorem 1.2) can be viewed as a generalisation of the result of Persson and Schmeling [18]. But there are essential differences between the three cases when the target $x_0=0,\,x_0\in (0,1)$ and $x_0=1$. The following three remarks serve as an outline of the differences.

Remark 1

The generality of $\{\ell _n\}_{n\ge 1}$ gives no extra difficulty compared with the special sequence $\{\ell _n = \alpha n\}_{n\ge 1}$. However, there are some essential difficulties when generalizing $x_0$ from zero to non-zero. The idea used in [18], to construct a suitable Cantor subset of $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )$ to get the lower bound of $\dim _\mathsf H E(\{\ell _n\}_{n\ge 1}, x_0)$, is not applicable for $x_0\ne 0$. For any $\beta >1$, let

$$\begin{aligned} \varepsilon _1(x,\beta ),\varepsilon _2(x,\beta ),\ldots \end{aligned}$$

be the digit sequence of the $\beta$-expansion of $x$. To guarantee that the two points $T^n_{\beta }1$ and $x_0$ are close enough, a natural idea is to require that

$$\begin{aligned} \varepsilon _{n+1}(1,\beta )=\varepsilon _1(x_0, \beta ),\, \ldots ,\, \varepsilon _{n+\ell }(1,\beta )=\varepsilon _{\ell }(x_0, \beta ) \end{aligned}$$

(1.7)

for some $\ell \in \mathbb{N }$ sufficiently large. When $x_0=0$, the $\beta$-expansions of $x_0$ are the same (all digits are 0) no matter what $\beta$ is. Thus to fulfill (1.7), one needs only to consider those $\beta$ for which a long string of zeros follows $\varepsilon _n(1,\beta )$ in the $\beta$-expansion of 1. But when $x_0\ne 0$, the $\beta$-expansions of $x_0$ under different $\beta$ are different. Furthermore, the expansion of $x_0$ is not known to us, since $\beta$ has not been determined yet. This difference constitutes a main difficulty in constructing points $\beta$ fulfilling the conditions in the definition of $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )$.

To overcome this difficulty, a better understanding of the parameter space seems necessary. In Sect. 3, we analyse the length and the distribution of a cylinder in the parameter space which relies heavily on a new notion that we call the recurrence time of a word.

Remark 2

When $x_0\ne 1$, the set $E(\{\ell _n\}_{n\ge 1}, x_0)$ can be regarded as a type of shrinking target problem with fixed target. While when $x_0=1$, the set $E(\{\ell _n\}_{n\ge 1}, x_0)$ is the set of $\beta$ for which the orbit of 1 returns to a shrinking neighbourhood of itself infinitely often. In this case, we have a so-called recurrence problem. There are some differences between these two cases. Therefore, their proofs for the lower bounds of $\dim _\mathsf H E(\{\ell _n\}_{n\ge 1}, x_0)$ are given separately in Sects. 5 and 6.

Remark 3

If $x(\beta )$, when developed in base $\beta$, is the same for all $\beta \in (\beta _0, \beta _1)$, then with an argument based on Theorem 15 in [18], one can give the dimension of $\widetilde{E}(\{\ell _n\}_{n\ge 1},x(\beta ))$. However as far as a general function $x(\beta )$ is concerned, the idea used in proving Theorem 1.1 can be applied to give a complete solution of the dimension of $\widetilde{E}(\{\ell _n\}_{n\ge 1}, x(\beta ))$.

For more dimensional results related to the $\beta$-transformations, the readers are referred to [10, 19, 21, 25, 26] and references therein. For more dimensional results concerning the shrinking target problems, see [2, 5, 9, 11–13, 22–24, 27] and references therein.

2 Preliminary

This section is devoted to recalling some basic properties of $\beta$-transformations and fixing some notation. For more information on $\beta$-transformations, see [1, 14, 17, 20] and references therein.

The $\beta$-expansion of real numbers was first introduced by Rényi [20], which is given by the following algorithm. For any $\beta >1$, let

$$\begin{aligned} T_{\beta }(0):=0, \quad T_{\beta }(x)=\beta x-\lfloor \, \beta x\rfloor , \, x\in (0,1), \end{aligned}$$

(2.1)

where $\lfloor \xi \rfloor$ is the integer part of $\xi \in \mathbb{R }$. By taking

$$\begin{aligned} \varepsilon _n(x,\beta )=\lfloor \, \beta T_{\beta }^{n-1}x\rfloor \in \mathbb{N } \end{aligned}$$

recursively for each $n\ge 1$, every $x\in [0,1)$ can be uniquely expanded into a finite or an infinite sum

$$\begin{aligned} x=\frac{\varepsilon _1(x,\beta )}{\beta } + \cdots + \frac{\varepsilon _n(x,\beta )}{\beta ^n} + \cdots , \end{aligned}$$

(2.2)

which is called the $\beta$-expansion of $x$ and the sequence $\{\varepsilon _n(x,\beta )\}_{n\ge 1}$ is called the digit sequence of $x$. We also write (2.2) as $\varepsilon (x,\beta ) = (\varepsilon _1(x,\beta ), \ldots , \varepsilon _n(x,\beta ),\ldots )$. The system $([0,1), T_{\beta })$ is called a $\beta$-transformation, $\beta$-dynamical system or a $\beta$-system.

Definition 2.1

A finite or an infinite sequence $(w_1, w_2, \ldots )$ is said to be admissible (with respect to the base $\beta$), if there exists an $x\in [0,1)$ such that the digit sequence (in the $\beta$-expansion) of $x$ begins with $(w_1, w_2, \ldots )$.

Denote by $\Sigma _{\beta }^n$ the collection of all $\beta$-admissible sequences of length $n$ and by $\Sigma _{\beta }$ that of all infinite admissible sequences. Write $\mathcal{A }=\{0,1,\ldots ,\beta -1\}$ when $\beta$ is an integer and otherwise, $\mathcal{A }=\{0,1,\ldots ,\lfloor \beta \rfloor \}$. Let $S_\beta$ be the closure of $\Sigma _\beta$ under the product topology on $\mathcal{A }^{\mathbb{N }}$. Then $(S_\beta ,\sigma |_{S_\beta })$ is a subshift of the symbolic space $(\mathcal{A }^{\mathbb{N }},\sigma )$, where $\sigma$ is the shift map on $\mathcal{A }^{\mathbb{N }}$.

Let us now turn to the infinite $\beta$-expansion of 1, which plays an important role in the study of $\beta$-expansions. At first, apply the algorithm (2.1) to the number $x=1$. Then the number 1 can also be expanded into a series, denoted by

$$\begin{aligned} 1 = \frac{\varepsilon _1(1,\beta )}{\beta } + \cdots + \frac{\varepsilon _n(1,\beta )}{\beta ^n} + \cdots . \end{aligned}$$

If the above series is finite, i.e. there exists $m\ge 1$ such that $\varepsilon _m(1,\beta )\ne 0$ but $\varepsilon _n(1,\beta )=0$ for all $n>m$, then $\beta$ is called a simple Parry number. In this case, the digit sequence of 1 is defined by

$$\begin{aligned} \varepsilon ^*(1,\beta ) := (\varepsilon _1^*(\beta ), \varepsilon _2^*(\beta ), \ldots ) = (\varepsilon _1(1,\beta ), \ldots , \varepsilon _{m-1}(1,\beta ), \varepsilon _m(1,\beta )-1)^\infty , \end{aligned}$$

where $(w)^\infty$ denotes the periodic sequence $(w,w,w,\ldots )$. If $\beta$ is not a simple Parry number, the digit sequence of 1 is defined by

$$\begin{aligned} \varepsilon ^*(1,\beta ) := (\varepsilon _1^*(\beta ), \varepsilon _2^*(\beta ), \ldots ) = (\varepsilon _1(1, \beta ),\varepsilon _2(1, \beta ),\ldots ). \end{aligned}$$

In both cases, the sequence $(\varepsilon _1^*(\beta ),\varepsilon _2^*(\beta ),\ldots )$ is called the infinite $\beta$-expansion of 1 and we always have that

$$\begin{aligned} 1 = \frac{\varepsilon _1^*(\beta )}{\beta } + \cdots + \frac{\varepsilon _n^*(\beta )}{\beta ^n}+\cdots . \end{aligned}$$

(2.3)

The lexicographical order $\prec$ between two infinite sequences is defined as follows:

$$\begin{aligned} w=(w_1, w_2, \ldots , w_n,\ldots )\prec w'=(w_1', w_2', \ldots , w_n',\ldots ) \end{aligned}$$

if there exists $k \ge 1$ such that $w_j =w_j'$ for $1\le j<k$, while $w_k<w_k'$. The notation $w\preceq w'$ means that $w\prec w'$ or $w=w'$. This ordering can be extended to finite blocks by identifying a finite block $(w_1,\ldots ,w_n)$ with the sequence $(w_1,\ldots ,w_n,0,0,\ldots )$.

The following result due to Parry [17] is a criterion for the admissibility of a sequence which relies heavily on the infinite $\beta$-expansion of 1.

Theorem 2.2

(Parry [17])

• (1) Let $\beta > 1$. For each $n\ge 1$, a block of non-negative integers $w=(w_1,\ldots , w_n)$ belongs to $\Sigma _{\beta }^n$ if and only if

  $$\begin{aligned} \sigma ^i w \preceq \varepsilon _1^*(1,\beta ),\ldots , \varepsilon _{n-i}^*(1,\beta )\quad \hbox { for all } \, 0\le i<n. \end{aligned}$$

• (2) The function $\beta \mapsto \varepsilon ^*(1,\beta )$ is increasing with respect to the variable $\beta >1$. Therefore, if $1<\beta _1<\beta _2$, then

  $$\begin{aligned} \Sigma _{\beta _1}\subset \Sigma _{\beta _2}, \quad \Sigma _{\beta _1}^n\subset \Sigma _{\beta _2}^n \quad (\text{ for } \text{ all } \, n\ge 1). \end{aligned}$$

At the same time, Parry also presented a characterisation of when a sequence of integers is the infinite expansion of 1 for some $\beta >1$. First, we introduce the notion of a self-admissible word.

Definition 2.3

A word $w = (\varepsilon _1, \ldots , \varepsilon _n)$ is called self-admissible if for all $1\le i <n$

$$\begin{aligned} \sigma ^i(\varepsilon _1,\ldots ,\varepsilon _n)\preceq \varepsilon _1,\ldots ,\varepsilon _{n-i}. \end{aligned}$$

An infinite digit sequence $w=(\varepsilon _1,\varepsilon _2, \ldots )$ is said to be self-admissible if for all $i\ge 1,\,\sigma ^iw\preceq w$.

Theorem 2.4

(Parry [17]) A digit sequence $(\varepsilon _1, \varepsilon _2, \ldots )$ with $\varepsilon _1\ge 1$ is the infinite expansion of 1 for some $\beta >1$ if and only if it is self-admissible.

The following result of Rényi implies that the dynamical system $([0,1), T_{\beta })$ admits $\log \beta$ as its topological entropy. Here and hereafter $\sharp$ denotes the cardinality of a finite set.

Theorem 2.5

(Rényi [20]) Let $\beta >1$. For any $n\ge 1$,

$$\begin{aligned} \beta ^n\le \sharp \Sigma _{\beta }^n\le {\beta ^{n+1}}/({\beta -1}). \end{aligned}$$

3 Distribution of regular cylinders in parameter space

From this section on, we turn to the parameter space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$, instead of considering a fixed $\beta >1$. We will address the length of a cylinder in the parameter space, which is closely related to the notion of recurrence time.

Definition 3.1

Let $(\varepsilon _1, \ldots , \varepsilon _n)$ be self-admissible. A cylinder in the parameter space is defined as

$$\begin{aligned} I_n^P(\varepsilon _1,\ldots ,\varepsilon _n)=\Big \{\, \beta >1: \varepsilon _1(1,\beta )= \varepsilon _1,\ldots ,\varepsilon _n(1,\beta )=\varepsilon _n \,\Big \}, \end{aligned}$$

i.e., the set of $\beta$ for which the $\beta$-expansion of 1 begins with the common prefix $\varepsilon _1, \ldots , \varepsilon _n$. Denote by $C_n^P$ the collection of cylinders of order $n$ in the parameter space.

When $(\varepsilon _1,\ldots ,\varepsilon _n)$ is a self-admissible word, we will sometimes talk about “the cylinder $(\varepsilon _1,\ldots ,\varepsilon _n)$”. When we do so, we mean the cylinder $I_n^P(\varepsilon _1,\ldots ,\varepsilon _n)$.

3.1 Recurrence time of words

Definition 3.2

Let $w=(\varepsilon _1, \ldots , \varepsilon _n)$ be a word of length $n$. The recurrence time $\tau (w)$ of $w$ is defined as

$$\begin{aligned} \tau (w):=\inf \big \{\, k\ge 1: \sigma ^k(\varepsilon _1,\ldots ,\varepsilon _n) = (\varepsilon _1,\ldots ,\varepsilon _{n-k}) \,\big \}. \end{aligned}$$

If such an integer $k$ does not exist, then $\tau (w)$ is defined to be $n$ and $w$ is said to be a non-recurrent word.

From the definition of the recurrence time $\tau (\cdot )$, it is clear that if $w=(\varepsilon _1, \ldots , \varepsilon _n)$ is recurrent with $\tau (w)=k<n$, then

$$\begin{aligned} (\varepsilon _1, \ldots , \varepsilon _n) = (\varepsilon _1, \ldots , \varepsilon _k)^{\lfloor n/k\rfloor } \varepsilon _1, \ldots , \varepsilon _{n-k\lfloor n/k\rfloor }, \end{aligned}$$

where $\lfloor \xi \rfloor$ denotes the integer part of $\xi$.

Applying the definition of recurrence time and the criterion of self-admissibility of a sequence, we obtain the following.

Lemma 3.3

Let $w=(\varepsilon _1,\ldots , \varepsilon _n)$ be self-admissible with the recurrence time $\tau (w)=k$. Then for each $1\le i<k$,

$$\begin{aligned} \varepsilon _{i+1},\ldots , \varepsilon _k \prec \varepsilon _1,\ldots ,\varepsilon _{k-i}. \end{aligned}$$

(3.1)

Proof

The self-admissibility of $w$ ensures that

$$\begin{aligned} \varepsilon _{i+1},\ldots ,\varepsilon _k, \varepsilon _{k+1},\ldots , \varepsilon _n\preceq \varepsilon _1,\ldots , \varepsilon _{k-i}, \varepsilon _{k-i+1},\ldots , \varepsilon _{n-i}. \end{aligned}$$

The recurrence time $\tau (w)=k$ of $w$ implies that for $1\le i<k$,

$$\begin{aligned} \varepsilon _{i+1},\ldots ,\varepsilon _k, \varepsilon _{k+1},\ldots , \varepsilon _n\ne \varepsilon _1,\ldots , \varepsilon _{k-i}, \varepsilon _{k-i+1},\ldots , \varepsilon _{n-i}. \end{aligned}$$

Combining the above two facts, we arrive at

$$\begin{aligned} \varepsilon _{i+1},\ldots ,\varepsilon _k, \varepsilon _{k+1},\ldots , \varepsilon _n \prec \varepsilon _1,\ldots , \varepsilon _{k-i}, \varepsilon _{k-i+1},\ldots , \varepsilon _{n-i}. \end{aligned}$$

(3.2)

When $k=n,\,(\varepsilon _{k+1},\ldots ,\varepsilon _n)$ is an empty word. Then the result follows directly by (3.2). Now we assume $k<n$ and compare the suffixes of the two words in (3.2). By the definition of $\tau (w)$, the left one ends with

$$\begin{aligned} \varepsilon _{k+1}, \ldots , \varepsilon _n=\varepsilon _1, \ldots ,\varepsilon _{n-k}, \end{aligned}$$

while the right one ends with

$$\begin{aligned} \varepsilon _{k-i+1}, \ldots , \varepsilon _{n-i}. \end{aligned}$$

By the self-admissibility of $\varepsilon _1,\ldots ,\varepsilon _n$, we get

$$\begin{aligned} \varepsilon _{k+1}, \ldots , \varepsilon _{n} = \varepsilon _1, \ldots , \varepsilon _{n-k}\succeq \varepsilon _{k-i+1}, \ldots , \varepsilon _{n-i}. \end{aligned}$$

(3.3)

Then the formulae (3.2) and (3.3) enable us to conclude the result. $\square$

We give a sufficient condition to ensure that a word is non-recurrent.

Lemma 3.4

Assume that $(\varepsilon _1, \ldots , \varepsilon _{m-1},\varepsilon _m)$ and $(\varepsilon _1, \!\ldots \!, \varepsilon _{m-1}, \overline{\varepsilon }_m)$ are both self-admissible and $0\le \varepsilon _m<\overline{\varepsilon }_m$. Then

$$\begin{aligned} \tau (\varepsilon _1, \ldots , \varepsilon _m) = m. \end{aligned}$$

Proof

Let $\tau (\varepsilon _1, \ldots , \varepsilon _m)=k.$ Suppose that $k<m$. We show that this will lead to a contradiction. Write $m=tk+i$ with $0<i\le k$. By the definition of the recurrence time $\tau$, we have

$$\begin{aligned} \sigma ^{tk}(\varepsilon _1, \ldots , \varepsilon _m) = (\varepsilon _{tk+1}, \ldots , \varepsilon _{m}) = (\varepsilon _1, \ldots , \varepsilon _i). \end{aligned}$$

(3.4)

From the self-admissibility of the other sequence $(\varepsilon _1, \ldots , \varepsilon _{m-1},\overline{\varepsilon }_m)$, we know

$$\begin{aligned} \sigma ^{tk}(\varepsilon _1, \ldots , \varepsilon _{m-1}, \overline{\varepsilon }_m) = (\varepsilon _{tk+1}, \ldots , \overline{\varepsilon }_{m})\preceq (\varepsilon _1, \ldots , \varepsilon _i). \end{aligned}$$

(3.5)

The assumption $\varepsilon _m<\overline{\varepsilon }_m$ implies that

$$\begin{aligned} (\varepsilon _{tk+1}, \ldots , \varepsilon _{m}) \prec (\varepsilon _{tk+1}, \ldots , \overline{\varepsilon }_{m}). \end{aligned}$$

Combining this with (3.4) and (3.5), we arrive at the contradiction $(\varepsilon _1, \ldots , \varepsilon _i) \prec (\varepsilon _1, \ldots , \varepsilon _i)$. $\square$

3.2 Maximal admissible sequences in parameter space

Now we recall a result of Schmeling [21] concerning the length of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$.

Lemma 3.5

([21]) The cylinder $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ is a half-open interval $[\beta _0, \beta _1)$. The left endpoint $\beta _0$ is given as the only solution in $(1, \infty )$ of the equation

$$\begin{aligned} 1=\frac{\varepsilon _1}{\beta }+\cdots +\frac{\varepsilon _n}{\beta ^n}. \end{aligned}$$

The right endpoint $\beta _1$ is the limit of the unique solutions $\{\beta _N\}_{N\ge n}$ in $(1,\infty )$ of the equations

$$\begin{aligned} 1 = \frac{\varepsilon _1}{\beta } + \cdots + \frac{\varepsilon _n}{\beta ^n} + \frac{\varepsilon _{n+1}}{\beta ^{n+1}} + \cdots + \frac{\varepsilon _N}{\beta ^N}, \quad N\ge n \end{aligned}$$

where $(\varepsilon _1, \ldots , \varepsilon _n, \varepsilon _{n+1}, \ldots , \varepsilon _N)$ is the maximal self-admissible sequence of length $n+N$ beginning with $\varepsilon _1, \ldots , \varepsilon _n$ in the lexicographical order. Moreover,

$$\begin{aligned} \big |I_n^P(\varepsilon _1,\dots ,\varepsilon _n)\big |\le {\beta _1^{-n+1}}. \end{aligned}$$

Therefore, to give an accurate estimate on the length of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$, we are led to determine the maximal self-admissible sequences beginning with a given self-admissible word $\varepsilon _1, \ldots , \varepsilon _n$.

Lemma 3.6

Let $w=(\varepsilon _1, \ldots , \varepsilon _n)$ be self-admissible with $\tau (w)=k$. Then for each $m\ge 1$ and $0\le \ell <k$ with $km+\ell \ge n$, the periodic sequence

$$\begin{aligned} (\varepsilon _1, \ldots , \varepsilon _k)^{m}\varepsilon _1, \ldots , \varepsilon _{\ell }, \end{aligned}$$

is the maximal self-admissible sequence of length $km+\ell$ beginning with $\varepsilon _1, \ldots , \varepsilon _n$. Consequently, if we denote by $\beta _1$ the right endpoint of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$, then the $\beta _1$-expansion of 1 and the infinite $\beta _1$-expansion of 1 are given respectively as

$$\begin{aligned} \varepsilon (1, \beta _1)=(\varepsilon _1, \ldots , \varepsilon _k+1),\quad \varepsilon ^*(1, \beta _1)=(\varepsilon _1, \ldots , \varepsilon _k)^{\infty }. \end{aligned}$$

Proof

By Lemma 3.3, we get for all $1\le i<k$

$$\begin{aligned} \varepsilon _{i+1},\ldots ,\varepsilon _k\prec \varepsilon _1,\ldots ,\varepsilon _{k-i}. \end{aligned}$$

(3.6)

For each $m\in \mathbb{N }$ and $0\le \ell <k$ with $km+\ell \ge n$, we check that

$$\begin{aligned} w_0 = (\varepsilon _1, \ldots , \varepsilon _k)^{m}\varepsilon _1, \ldots , \varepsilon _{\ell } \end{aligned}$$

is the maximal self-admissible sequence beginning with $\varepsilon _1, \ldots , \varepsilon _{n}$ of length $mk+\ell$.

The admissibility of $w_0$ follows directly from (3.6). Now we show that $w_0$ is maximal. Let

$$\begin{aligned} w = (\varepsilon _1, \ldots , \varepsilon _k)^tw_1, \ldots , w_k, \ldots , w_{(m-t-1)k+1}, \ldots , w_{(m-t)k}, w_{(m-t)k+1}, \ldots , w_{(m-t)k+\ell } \end{aligned}$$

be a self-admissible word different from $w_0$, where $t\ge 1$ is the maximal integer such that $w$ begins with $(\varepsilon _1, \ldots , \varepsilon _k)^t$. We distinguish two cases according to $t<m$ or $t=m$. We consider only the case $t<m$, since the other case can be treated similarly.

If $t<m$, then

$$\begin{aligned} w_{1}, \ldots , w_{k}\ne \varepsilon _1, \ldots , \varepsilon _k. \end{aligned}$$

The self-admissibility of $w$ ensures that

$$\begin{aligned} w_{1}, \ldots , w_{k} \preceq \varepsilon _1, \ldots , \varepsilon _k. \end{aligned}$$

Hence, we arrive at

$$\begin{aligned} w_{1}, \ldots , w_{k}\prec \varepsilon _1, \ldots , \varepsilon _k. \end{aligned}$$

(3.7)

This shows $w\prec w_0$. $\square$

The following fact is just the self-admissibility of $w_0$ proven in Lemma 3.6. We state it as a corollary for later use.

Corollary 3.7

Assume that $(\varepsilon _1, \ldots , \varepsilon _k)$ is a non-recurrent word. Then for any integer $m\ge 1$ and $0\le \ell <k$, the word

$$\begin{aligned} (\varepsilon _1, \ldots , \varepsilon _k)^{m}, \varepsilon _1, \ldots , \varepsilon _{\ell } \end{aligned}$$

is self-admissible.

The following simple calculation will be used several times in the sequel, so we state it in advance.

Lemma 3.8

Let $1<\beta _0<\beta _1$ and $0\le \varepsilon _k< \beta _0$ for all $k\ge 1$. Then for every $n\ge 1$,

$$\begin{aligned} \left( \frac{\varepsilon _1}{\beta _0}+\cdots +\frac{\varepsilon _n}{\beta _0^n}\right) - \left( \frac{\varepsilon _1}{\beta _1}+\cdots +\frac{\varepsilon _n}{\beta _1^n}\right) \le \frac{\beta _0}{(\beta _0-1)^2}(\beta _1-\beta _0). \end{aligned}$$

Now we apply Lemma 3.6 to give a lower bound on the length of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$.

Theorem 3.9

Let $w = (\varepsilon _1, \ldots , \varepsilon _n)$ be self-admissible with $\tau (w)=k$. Let $\beta _0$ and $\beta _1$ be the left and right endpoints of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$. Then we have

$$\begin{aligned} \big |I_n^P(\varepsilon _1, \ldots , \varepsilon _n)\big | \ge \left\{ \begin{array}{l@{\quad }l} C\beta _1^{-n}, &{} \mathrm{when } \, k=n;\\ C\frac{1}{\beta _1^{n}}\left( \frac{\varepsilon _{t+1}}{\beta _1} + \cdots + \frac{\varepsilon _k+1}{\beta _1^{k-t}}\right) , &{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

(3.8)

where $C:=(\beta _0-1)^2/\beta _0$ is a constant depending on $\beta _0$; the integers $t$ and $\ell$ are given by $\ell k <n\le (\ell +1)k$ and $t=n-\ell k$.

Proof

When $k=n$, the endpoints $\beta _0$ and $\beta _1$ of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ are given respectively as the solutions to

$$\begin{aligned} 1=\frac{\varepsilon _1}{\beta _0} + \cdots + \frac{\varepsilon _n}{\beta _0^n}, \quad {\text {and}}\quad 1=\frac{\varepsilon _1}{\beta _1} + \cdots + \frac{\varepsilon _n+1}{\beta _1^n}. \end{aligned}$$

(3.9)

Thus,

$$\begin{aligned}&\frac{1}{\beta _1^{n}} = \left( \frac{\varepsilon _1}{\beta _0} + \cdots + \frac{\varepsilon _n}{\beta _0^n}\right) - \left( \frac{\varepsilon _1}{\beta _1}+\cdots +\frac{\varepsilon _n}{\beta _1^n}\right) \le C^{-1}(\beta _1-\beta _0). \end{aligned}$$

Then $|I_n^P(\varepsilon _1,\dots ,\varepsilon _n)|=\beta _1 - \beta _0\ge C\beta _1^{-n}$.

When $k<n$, the endpoints $\beta _0$ and $\beta _1$ of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ are given respectively as the solutions to

$$\begin{aligned} 1 = \frac{\varepsilon _1}{\beta _0} + \cdots + \frac{\varepsilon _n}{\beta _0^n}, \quad {\text {and}} \quad 1 = \frac{\varepsilon _1}{\beta _1} + \cdots + \frac{\varepsilon _n}{\beta _1^n} + \frac{\varepsilon _{t+1}}{\beta _1^{n+1}} + \cdots + \frac{\varepsilon _k+1}{\beta _1^{(\ell +1)k}}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\varepsilon _{t+1}}{\beta _1^{n+1}} + \cdots + \frac{\varepsilon _k+1}{\beta _1^{(\ell +1)k}} =\left( \frac{\varepsilon _1}{\beta _0} + \cdots + \frac{\varepsilon _n}{\beta _0^n}\right) -\left( \frac{\varepsilon _1}{\beta _1} +\cdots +\frac{\varepsilon _n}{\beta _1^n}\right) \le C^{-1}(\beta _1-\beta _0), \end{aligned}$$

and we obtain the desired result. $\square$

Combining Lemma 3.5 and Theorem 3.9, we know that when $(\varepsilon _1, \ldots , \varepsilon _n)$ is a non-recurrent word, the length of $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ satisfies

$$\begin{aligned} C\beta _1^{-n}\le |I_n^P(\varepsilon _1, \ldots , \varepsilon _n)| \le \beta _1^{-n}. \end{aligned}$$

In this case, $I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ is called a regular cylinder.

The following corollary of Theorem 3.9 indicates that if the digit 1 appears regularly in a self-admissible sequence $w$, then we can have a good lower bound for the length of the cylinder generated by $w$. This will be applied in constructing a Cantor subset of $E(\{\ell _n\}_{n\ge 1}, x_0)$.

Corollary 3.10

Let $w=(\varepsilon _1, \ldots , \varepsilon _n)$ be self-admissible and $d$ an integer such that for every $0\le i\le n-d$, the word $(w_{i+1}, \ldots , w_{i+d})$ is nonzero. Then we have

$$\begin{aligned} |I_n(w)|\ge C \beta _1^{-n-d}, \end{aligned}$$

where $C$ and $\beta _1$ are as those in Theorem 3.9.

Proof

Let $\tau (w)=k$. When $n$ is a multiple of $k$, the maximal self-admissible sequence beginning with $w$ is just the periodic sequence $(w, w, w,\ldots )$. Then the desired result follows with the same argument as that for the first inequality in (3.8).

When $n$ is not a multiple of $k$, we argue as follows. Keep the notation as in Theorem 3.9. If $k-t\ge d$, then $(\varepsilon _{t+1}, \ldots , \varepsilon _{t+d})$ is nonzero. Thus by the second inequality in (3.8), we have $|I_n(w)|\ge C\beta _1^{-(n+d)}$. If $k-t<d$, then still by the second inequality in (3.8), we have

$$\begin{aligned} |I_n(w)|\ge C\beta _1^{-n}\cdot \beta _1^{-(k-t)}\ge C\beta _1^{-(n+d)}. \end{aligned}$$

$\square$

3.3 Distribution of regular cylinders

The following result presents a relationship between the recurrence time of two consecutive cylinders in the parameter space.

Proposition 3.11

Let $w_1,w_2$ be two self-admissible words of length $n$. Assume that $w_2\prec w_1$ and $w_2$ is next to $w_1$ in the lexicographic order. If $\tau (w_1)<n$, then

$$\begin{aligned} \tau (w_2)>\tau (w_1). \end{aligned}$$

Proof

Since $\tau (w_1):=k_1<n,\,w_1$ can be written as

$$\begin{aligned} w_1 = (\varepsilon _1, \ldots , \varepsilon _{k_1})^{t}, \varepsilon _1, \ldots , \varepsilon _{\ell }, \quad {\text {for some integers }}\, t\ge 1\, {\text {and}}\, 1\le \ell \le k_1. \end{aligned}$$

It is clear that $\varepsilon _1\ge 1$ which ensures the self-admissibility of the sequence

$$\begin{aligned} w = (\varepsilon _1, \ldots , \varepsilon _{k_1})^{t}, \underbrace{0, \ldots , 0}_{\ell }. \end{aligned}$$

Since $w_2$ is less than $w_1$ and is next to $w_1$, we have

$$\begin{aligned} w\preceq w_2\prec w_1. \end{aligned}$$

This implies that $w_1$ and $w_2$ have common prefixes up to at least $k_1 \cdot t$ terms. Then $w_2$ can be expressed as

$$\begin{aligned} w_2 = (\varepsilon _1, \ldots , \varepsilon _{k_1})^{t}, \varepsilon '_1, \ldots , \varepsilon '_{\ell }. \end{aligned}$$

First, we claim that $\tau (w_2):=k_2\ne k_1$. Otherwise, by the definition of $\tau (w_2)$, we obtain

$$\begin{aligned} \varepsilon '_1, \ldots , \varepsilon '_{\ell } = \varepsilon _1, \ldots , \varepsilon _{\ell }, \end{aligned}$$

which indicates that $w_1=w_2$.

Second, we show that $k_2$ cannot be strictly smaller than $k_1$. Otherwise, consider the prefix $\varepsilon _1, \ldots , \varepsilon _{k_1}$ which is also the prefix of $w_1$. If $k_2<k_1$, we have

$$\begin{aligned} \varepsilon _{k_2+1}, \ldots , \varepsilon _{k_1} = \varepsilon _1, \ldots , \varepsilon _{k_1-k_2}, \end{aligned}$$

which contradicts Lemma 3.3 by applying to $w_1$.

Therefore, $\tau (w_2)>\tau (w_1)$ holds. $\square$

The following corollary indicates that cylinders with regular length (equivalent with $\beta _1^{-n}$) are well distributed among the parameter space. This result was found for the first time by Persson and Schmeling [18].

Corollary 3.12

Among any $n$ consecutive cylinders in $C_n^P$, there is at least one with regular length.

Proof

Let $w_1\succ w_2\succ \cdots \succ w_n$ be $n$ consecutive cylinders in $C_n^P$. By Theorem 3.9, it suffices to show that there is at least one word $w$ whose recurrence time is equal to $n$. If this is not the case, then by Proposition 3.11, we have

$$\begin{aligned} 1\le \tau (w_1)<\tau (w_2)<\cdots <\tau (w_n)<n, \end{aligned}$$

i.e. there would be $n$ different integers in $\{1, 2, \ldots , n-1\}$. This is impossible. This completes the proof. $\square$

4 Proof of Theorem 1.1: upper bound

The proof of the upper bound of $\dim _\mathsf H E(\{\ell _n\}_{n\ge 1}, x_0)$ is given in a unified way no matter whether $x_0=1$ or not. Before providing an upper bound of $\dim _\mathsf H E(\{\ell _n\}_{n\ge 1}, x_0)$, we begin with a lemma.

Lemma 4.1

Let $(\varepsilon _1, \ldots , \varepsilon _n)$ be self-admissible. Then the set

$$\begin{aligned} \Big \{\, T^n_{\beta }1: \beta \in I_n^P(\varepsilon _1, \ldots , \varepsilon _n) \,\Big \} \end{aligned}$$

(4.1)

is a half-open interval $[0,a)$ for some $a\le 1$. Moreover, $T^n_{\beta }1$ is continuous and increasing on $\beta \in I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$.

Proof

Note that for any $\beta \in I_n^P (\varepsilon _1, \ldots , \varepsilon _n)$, we have

$$\begin{aligned} 1=\frac{\varepsilon _1}{\beta }+\cdots +\frac{\varepsilon _n +T^n_{\beta }1}{\beta ^n}. \end{aligned}$$

Thus

$$\begin{aligned} T^n_{\beta }1 = \beta ^n - \beta ^n \left( \frac{\varepsilon _1}{\beta }+\cdots +\frac{\varepsilon _n}{\beta ^n}\right) . \end{aligned}$$

Denote

$$\begin{aligned} f(\beta )=\beta ^n-\Big (\varepsilon _1 \beta ^{n-1}+\varepsilon _2 \beta ^{n-2}+\cdots +\varepsilon _n\Big ). \end{aligned}$$

(4.2)

Then the set in (4.1) is just the set

$$\begin{aligned} \{\, f(\beta ): \beta \in I_n^P (\varepsilon _1, \ldots , \varepsilon _n) \,\}. \end{aligned}$$

To show the monotonicity of $\beta \mapsto T^n_{\beta }1$, it suffices to show that the derivative $f'(\beta )$ is positive. In fact,

$$\begin{aligned} f'(\beta )&= n\beta ^{n-1}-\Big ( (n-1)\varepsilon _1\beta ^{n-2} + (n-2) \varepsilon _2\beta ^{n-3} + \cdots +\varepsilon _{n-1} \Big )\\&\ge n\beta ^{n-1}-(n-1)\beta ^{n-1}\left( \frac{\varepsilon _1}{\beta }+ \cdots +\frac{\varepsilon _{n-1}}{\beta ^{n-1}}\right) \\&\ge n\beta ^{n-1}-(n-1)\beta ^{n-1} >0. \end{aligned}$$

Since $f$ is continuous and $I_n^P (\varepsilon _1, \ldots , \varepsilon _n)$ is an interval with the left endpoint $\beta _0$ given as the solution to the equation

$$\begin{aligned} 1=\frac{\varepsilon _1}{\beta }+\cdots +\frac{\varepsilon _n}{\beta ^n}, \end{aligned}$$

the set (4.1) is an interval with 0 being its left endpoint and some right endpoint $a\le 1$. $\square$

Now we give an upper bound of $\dim _\mathsf H E \big (\{\ell _n\}_{n\ge 1}, x_0\big )$. For any $1<\beta _0<\beta _1$, denote

$$\begin{aligned} E(\beta _0,\beta _1) = \big \{\, \beta _0<\beta \le \beta _1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \, {\text {i.o.}}\, n\in \mathbb{N } \,\big \}. \end{aligned}$$

For any $\delta >0$, we partition the parameter space $(1,\infty )$ into $\{(a_i, a_{i+1}]: i\ge 1\}$ with $\frac{\log a_{i+1}}{\log a_i} < 1+\delta$ for all $i\ge 1$. Then

$$\begin{aligned} E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\cup _{i=1}^{\infty }E(a_i,a_{i+1}). \end{aligned}$$

By the $\sigma$-stability of the Hausdorff dimension, it suffices to give an upper bound on $\dim _\mathsf H E (\beta _0,\beta _1)$ for any $1<\beta _0<\beta _1$ with $\frac{\log \beta _1}{\log \beta _0} < 1+\delta$.

Proposition 4.2

For any $1<\beta _0<\beta _1$, we have

$$\begin{aligned} \dim _\mathsf H E(\beta _0,\beta _1)\le \frac{1}{1+\alpha }\frac{\log \beta _1}{\log \beta _0}. \end{aligned}$$

(4.3)

Proof

Let $B(x,r)$ be a ball with center $x\in [0, 1]$ and radius $r$. By using the simple inclusion $B(x_0, \beta ^{-\ell _n})\subset B(x_0, \beta _0^{-\ell _n})$ for any $\beta >\beta _0$, we have

$$\begin{aligned} E(\beta _0, \beta _1)&= \bigcap _{N=1}^{\infty } \bigcup _{n=N}^{\infty } \left\{ \, \beta \in (\beta _0, \beta _1]: T_\beta ^n1\in B(x_0, \beta ^{-\ell _n}) \,\right\} \\&\subset \bigcap _{N=1}^{\infty } \bigcup _{n=N}^{\infty }\left\{ \, \beta \in (\beta _0, \beta _1] : T_\beta ^n1\in B(x_0, \beta _0^{-\ell _n}) \,\right\} \\&= \bigcap _{N=1}^{\infty } \bigcup _{n=N}^{\infty }\bigcup _{(i_1,\ldots , i_n) \in \Sigma _{\beta _0,\beta _1}^{P,n}} I_n^P(i_1, \ldots , i_n;\beta _0^{-\ell _n}), \end{aligned}$$

where $\Sigma _{\beta _0,\beta _1}^{P,n}$ denotes the set of self-admissible words of length $n$ between $(\varepsilon _1^*(\beta _0),\dots ,\varepsilon _n^*(\beta _0))$ and $(\varepsilon _1^*(\beta _1),\dots ,\varepsilon _n^*(\beta _1))$ in the lexicographic order, and

$$\begin{aligned} I_n^P(i_1,\ldots , i_n;\beta _0^{-\ell _n}) := \{\, \beta \in (\beta _0, \beta _1]: \beta \in I_n^P(i_1, \ldots , i_n), \, T_\beta ^n1\in B(x_0, \beta _0^{-\ell _n}) \,\}. \end{aligned}$$

By Lemma 4.1, we know that the set $I_n^P(i_1, \ldots , i_n;\beta _0^{-\ell _n})$ is an interval. In case it is non-empty we denote its left and right endpoints by $\beta _0'$ and $\beta _1'$ respectively. Thus

$$\begin{aligned} \beta _1'\le i_1+\frac{i_2}{\beta _1'}+\cdots +\frac{i_n}{\beta _1'^{n-1}}+\frac{x_0+\beta _0^{-\ell _n}}{\beta _1'^{n-1}} \end{aligned}$$

and

$$\begin{aligned} \beta _0'\ge i_1 + \frac{i_2}{\beta _0'} + \cdots +\frac{i_n}{\beta _0'^{n-1}} + \frac{x_0-\beta _0^{-\ell _n}}{\beta _0'^{n-1}}\ge i_1 + \frac{i_2}{\beta _1'}+\cdots +\frac{i_n}{\beta _1'^{n-1}} + \frac{x_0-\beta _0^{-\ell _n}}{\beta _1'^{n-1}}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\beta _1' - \beta _0' \\&\quad \le \left( i_1+\frac{i_2}{\beta _1'}+\cdots +\frac{i_n}{\beta _1'^{n-1}}+\frac{x_0+ \beta _0^{-\ell _n}}{\beta _1'^{n-1}}\right) - \left( i_1+\frac{i_2}{\beta _1'}+\cdots +\frac{i_n}{\beta _1'^{n-1}}+ \frac{x_0-\beta _0^{-\ell _n}}{\beta _1'^{n-1}}\right) \\&\quad = \frac{2\beta _0^{-\ell _n}}{\beta _1'^{n-1}}\le \frac{2\beta _0^{-\ell _n}}{\beta _0^{n-1}}=2\beta _0^{-(\ell _n+n-1)}. \end{aligned}$$

By the monotonicity of $\varepsilon (1,\beta )$ with respect to $\beta$ (Theorem 2.2 (2)), we have $\varepsilon (1,\beta )\in \Sigma _{\beta _1}$ for any $\beta <\beta _1$. Therefore

$$\begin{aligned} \sharp \Sigma _{\beta _0,\beta _1}^{P,n}\le \sharp \Sigma _{\beta _1}^n\le \frac{\beta _1^{n+1}}{\beta _1-1}, \end{aligned}$$

where the last inequality follows from Theorem 2.5. It is clear that the family

$$\begin{aligned} \Big \{\, I_n^P(i_1, \ldots , i_n,\beta _0^{-\ell _n}) : (i_1, \ldots , i_n) \in \Sigma _{\beta _0, \beta _1}^{P,n}, n\ge N\,\Big \} \end{aligned}$$

is a cover of the set $E(\beta _0,\beta _1)$. Recall that $\alpha = \liminf \limits _{n\rightarrow \infty }{\ell _n/n}$. Thus for any $s> \frac{1}{1+\alpha } \frac{\log \beta _1}{\log \beta _0}$, we have

$$\begin{aligned} \mathcal{H }^s(E(\beta _0,\beta _1))&\le \liminf _{N\rightarrow \infty } \sum _{n\ge N} \sum _{(i_1, \ldots , i_n) \in \Sigma _{\beta _0,\beta _1}^{P,n}} \big | I_n^P (i_1, \ldots , i_n,\beta _0^{-\ell _n}) \big |^s \\&\le \liminf _{N\rightarrow \infty } \sum _{n\ge N} \frac{\beta _1^{n+1}}{\beta _1-1} \cdot 2^s\cdot \beta _0^{-(\ell _n+n-1)s}<\infty . \end{aligned}$$

This gives the estimate (4.3). $\square$

5 Lower bound of $E(\{\ell _n\}_{n\ge 1}, x_0): x_0\ne 1$

The proof of the lower bound of $\dim _\mathsf H E\big (\{\ell _n\}_{n\ge 1}, x_0\big )$, when $x_0\ne 1$, is done by using a classic method: first construct a Cantor subset $\mathcal{F }$, then define a measure $\mu$ supported on $\mathcal{F }$, and estimate the Hölder exponent of the measure $\mu$. At last, conclude the result by applying the following mass distribution principle [7, Proposition 4.4].

Proposition 5.1

(Falconer [7]) Let $E$ be a Borel subset of $\mathbb{R }^d$ and $\mu$ be a Borel measure with $\mu (E)>0$. Assume that, for any $x\in E$

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{\log \mu (B(x,r))}{\log r}\ge s. \end{aligned}$$

Then $\dim _\mathsf H E\ge s.$

Instead of dealing with $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )$ directly, we give some technical modifications by considering the following set

$$\begin{aligned} E=\big \{\, \beta >1: |T^n_{\beta }1-x_0|<4(n+\ell _n)\beta ^{-\ell _n}, \, {\text {i.o.}}\, n\in \mathbb{N } \,\big \}. \end{aligned}$$

It is clear that if we replace $\beta ^{-\ell _n}$ by $\beta ^{-(\ell _n+n \epsilon )}$ for any $\epsilon >0$ in defining $E$ above, the set $E$ will be a subset of $E(\{\ell _n\}_{n\ge 1}, x_0)$. Therefore, once we show the dimension of $E$ is bounded from below by $1/(1+\alpha )$, so is $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )$. We always assume in the following that $\alpha >0$, if not, just replace $\ell _n$ by $\ell _n+n\epsilon$. In the remaining part of this section, we are going to prove that

$$\begin{aligned} \dim _\mathsf H E \ge \frac{1}{1+\alpha }, \quad {\text {with} \quad \alpha >0}. \end{aligned}$$

5.1 Cantor subset

Let $x_0$ be a real number in $[0,1)$. Let $\beta _0>1$ be such that its expansion $\varepsilon (1,\beta _0)$ of 1 is infinite, i.e. there are infinitely many nonzero terms in $\varepsilon (1,\beta _0)$. The infinity of the digit sequence $\varepsilon (1,\beta _0)$ implies that for each $n\ge 1$, the number $\beta _0$ is not the right endpoint of the cylinder $I_n^P(\beta _0)$ containing $\beta _0$ by Lemma 3.6. Hence we can choose another $\beta _1>\beta _0$ such that the $\beta _1$-expansion $\varepsilon (1,\beta _1)$ of 1 is infinite and has a sufficiently long common prefix with $\varepsilon (1,\beta _0)$ so that

$$\begin{aligned} \frac{\beta _1(\beta _1-\beta _0)}{(\beta _0 - 1)^2} \le \frac{1-x_0}{2}. \end{aligned}$$

(5.1)

Let

$$\begin{aligned} M = \min \{\, n\ge 1: \varepsilon _n(1,\beta _0)\ne \varepsilon _n(1,\beta _1) \,\}, \end{aligned}$$

that is, $\varepsilon _i(1,\beta _0)=\varepsilon _i(1,\beta _1)$ for all $1\le i<M$ and $\varepsilon _M(1,\beta _0)\ne \varepsilon _M(1,\beta _1)$. Let $\beta _2$ be the maximal element beginning with $w(\beta _0):=(\varepsilon _1(1,\beta _0),\ldots , \varepsilon _{M}(1,\beta _0))$ in its infinite expansion of 1, that is, $\beta _2$ is the right endpoint of $I_M^P(w(\beta _0))$. Then it follows that $\beta _0<\beta _2<\beta _1$. Note that the word

$$\begin{aligned} (\varepsilon _1(1,\beta _0), \ldots , \varepsilon _{M-1}(1,\beta _0), \varepsilon _{M}(1,\beta _1)) = (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M-1}(1,\beta _1), \varepsilon _{M}(1,\beta _1)) \end{aligned}$$

is self-admissible and $\varepsilon _M(1,\beta _0)< \varepsilon _M(1,\beta _1)$. So by Lemma 3.4, we know that $\tau (w(\beta _0))=M$. As a result, Lemma 3.6 compels that the infinite $\beta _2$-expansion of 1 is

$$\begin{aligned} \varepsilon ^*(1,\beta _2) = (\varepsilon _1(1,\beta _0), \ldots , \varepsilon _{M}(1,\beta _0))^{\infty }. \end{aligned}$$

(5.2)

Since the following fact will be used frequently, we highlight it here:

$$\begin{aligned} \varepsilon _1^*(1,\beta _2), \ldots , \varepsilon _{M}^*(1,\beta _2)\prec \varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1). \end{aligned}$$

(5.3)

Lemma 5.2

For any $w\in S_{\beta _2}$, the sequence

$$\begin{aligned} \varepsilon =\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1), 0^M, w \end{aligned}$$

is self-admissible.

Proof

This will be checked by using properties of the recurrence time and the fact (5.3). Denote $\tau (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1)) = k$. Then $\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1)$ is periodic with a period $k$. Thus $\varepsilon$ can be rewritten as

$$\begin{aligned} (\varepsilon _1, \ldots , \varepsilon _k)^{t_0} \varepsilon _1, \ldots , \varepsilon _s, 0^M, w \end{aligned}$$

for some $t_0\in \mathbb{N }$ and $0\le s<k$. We will compare $\sigma ^i\varepsilon$ and $\varepsilon$ for all $i\ge 1$. The proof is divided into three steps according to $i\le M,\,M<i<2M$ or $i\ge 2M$.

• (1) $i\le M$. When $i=tk$ for some $t\in \mathbb{N }$, then $\sigma ^i(\varepsilon )$ and $\varepsilon$ have a common prefix up to the $(M-tk)$th digit. Following this prefix, the next $k$ digits in $\sigma ^i(\varepsilon )$ are $0^k$, while they are $(\varepsilon _{s+1}, \ldots , \varepsilon _k,\varepsilon _{1}, \ldots , \varepsilon _s)$ in $\varepsilon$, which implies that $\sigma ^i\varepsilon \prec \varepsilon$. When $i=tk+\ell$ for some $0<\ell <k$, then $\sigma ^i(\varepsilon )$ begins with $(\varepsilon _{\ell +1}, \ldots , \varepsilon _s,0^{k-s})$ if $t=t_0$ and begins with $(\varepsilon _{\ell +1}, \ldots , \varepsilon _k)$ if $t<t_0$. By Lemma 3.3, we know that

  $$\begin{aligned} \varepsilon _{\ell +1}, \ldots ,\varepsilon _s,0^{k-s} \preceq \varepsilon _{\ell +1}, \ldots ,\varepsilon _k\prec \varepsilon _1, \ldots ,\varepsilon _{k-\ell }. \end{aligned}$$

  Thus $\sigma ^i(\varepsilon )\prec \varepsilon .$

• (2) $M<i<2M$. For this case, it is trivial because $\sigma ^i\varepsilon$ begins with 0.

• (3) $i=2M+\ell$ for some $\ell \ge 0$. Then the sequence $\sigma ^i(\varepsilon )$ begins with the subword $(w_{\ell +1}, \ldots , w_{\ell +M})$ of $w$. Since $w\in S_{\beta _2}$, we have

  $$\begin{aligned} w_{\ell +1}, \ldots , w_{\ell +M}\preceq \varepsilon _1^*(1,\beta _2), \ldots , \varepsilon _{M}^*(1,\beta _2) \prec \varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1). \end{aligned}$$

  where the last inequality follows from (5.3). Therefore, $\sigma ^i(\varepsilon )\prec \varepsilon$.

$\square$

Now we use Lemmas 4.1, 5.2 and a suitable choice of the self-admissible sequence to show that the interval defined in (4.1) can be large enough. Fix $q\ge M$ such that

$$\begin{aligned} 0^q\prec \varepsilon _{M+1}(1,\beta _1), \ldots , \varepsilon _{M+q}(1,\beta _1), \end{aligned}$$

that is, find a position $M+q$ in $\varepsilon (1,\beta _1)$ with nonzero element $\varepsilon _{M+q} (1,\beta _1)$. The choice of the integer $q$ guarantees that the cylinder $I_{M+q}^P (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _M(1,\beta _1),0^q)$ lies on the left hand side of $\beta _1$.

Lemma 5.3

Suppose $\beta _0$ and $\beta _1$ are close enough such that (5.1) holds. For any $w\in \Sigma _{\beta _2}^{n-M-q}$ ending with $M$ zeros, the interval

$$\begin{aligned} \Gamma _n=\Big \{\, T^n_{\beta }1: \beta \in I_n^P(\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1), 0^q, w) \,\Big \} \end{aligned}$$

contains $(x_0+1)/2$.

Proof

Recall $\varepsilon ^*(1,\beta _2) = (\varepsilon _1(1,\beta _0), \ldots , \varepsilon _M(1,\beta _0))^{\infty } := (e_1, \ldots , e_M)^{\infty }$. Since $w$ ends with $M$ zeros, the sequence $(w, (e_1, \ldots , e_M)^{\infty })$ is in $S_{\beta _2}$. Thus, the number $\beta ^*$ for which

$$\begin{aligned} \varepsilon ^*(1,\beta ^*)=\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1), 0^q, w, e_1,e_2, \ldots e_M, e_1,e_2,\ldots \end{aligned}$$

belongs to the closure of $I_n^P(\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1), 0^q, w)$ by Lemma 5.2. Note that $\beta ^*\le \beta _1$ by the choice of $q$. For such a number $\beta ^*$,

$$\begin{aligned} T^n_{\beta ^*}1 = \frac{e_1}{\beta ^*} + \frac{e_2}{\beta ^{*2}} + \cdots \ge \frac{e_1}{\beta _1} + \frac{e_2}{\beta _1^2} + \cdots . \end{aligned}$$

Note also that

$$\begin{aligned} 1=\frac{e_1}{\beta _2}+\frac{e_2}{\beta _2^2}+\cdots . \end{aligned}$$

Thus

$$\begin{aligned} 1-T_{\beta ^*}^n1 \le \left( \frac{e_1}{\beta _2}+\frac{e_2}{\beta _2^2}+\cdots \right) - \left( \frac{e_1}{\beta _1}+\frac{e_2}{\beta _1^2}+\cdots \right) \le \frac{\beta _1(\beta _1-\beta _0)}{(\beta _0 - 1)^2}. \end{aligned}$$

Hence $T^n_{\beta ^*}1 > \frac{x_0 + 1}{2}$ by (5.1). Then we obtain the statement of Lemma 5.3. $\square$

Now we are in the position to construct a Cantor subset $\mathcal{F }$ of $E$. Let $\mathfrak N$ be a subsequence of integers such that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\ell _n}{n}=\lim _{n\in \mathfrak N , \, n\rightarrow \infty }\frac{\ell _{n}}{n}=\alpha >0. \end{aligned}$$

5.1.1 Generation 0 of the Cantor set

Write

$$\begin{aligned} \varepsilon ^{(0)}=(\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1), 0^q), \quad {\text {and}}\, \mathbb{F }_0=\{\varepsilon ^{(0)}\}. \end{aligned}$$

Then the 0th generation of the Cantor set is defined as

$$\begin{aligned} \mathcal{F }_0=\Big \{\, I_{M+q}^P(\varepsilon ^{(0)}): \varepsilon ^{(0)}\in \mathbb{F }_0 \,\Big \}. \end{aligned}$$

5.1.2 Generation 1 of the Cantor set

Recall that $M$ is the integer defined for $\beta _2$ in the beginning of this subsection. Let $N\gg M$. Denote by $U_\ell$ a collection of words in $S_{\beta _2}$:

$$\begin{aligned} U_\ell =\Big \{\, u = (0^M, 1, 0^M, a_1, \ldots ,a_{N}, 0^M, 1,0^M): (a_1, \ldots , a_{N})\in S_{\beta _2} \,\Big \}, \end{aligned}$$

(5.4)

where $\ell =4M+2+N$ is the length of the words in $U_\ell$. Without causing any confusion, in the sequel, the family $\mathbb{F }_0$ of words is also called the 0th generation of the Cantor set $\mathcal{F }$.

Remark 4

We give a remark on the way that the family $U_{\ell }$ is constructed.

• (1) The first $M$-zeros guarantee that for any $\beta _2$-admissible word $v$ and any element $u\in U_{\ell }$, the concatenation $(v, u)$ is still $\beta _2$-admissible.

• (2) With the same reason as for (1), the other three blocks of $0^M$ guarantee that $U_{\ell }\subset \Sigma _{\beta _2}^{\ell }$.

• (3) The two digits 1 are added so that the digit 1 appears regularly in $u\in U_{\ell }$ (recall Corollary 3.10).

Let $m_{0}=M+q$ be the length of $\varepsilon ^{(0)}\in \mathbb{F }_{0}$. Choose an integer $n_1\in \mathfrak N$ such that $n_1\gg m_{0},\,\beta _0^{-{n_1}}\le 2(\beta _0-1)^2/\beta _1$ and $4(n_1+\ell _{n_1})\beta ^{-\ell _{n_1}} < \frac{1-x_0}{2}$ by noting $\alpha >0$. Write

$$\begin{aligned} n_1-m_{0}=t_1\ell +i_1, \quad {\text {for some}}\quad t_1\in \mathbb{N }, \, 0\le i_1<\ell . \end{aligned}$$

First, we collect a family of self-admissible sequences beginning with $\varepsilon ^{(0)}$:

$$\begin{aligned} \mathfrak M (\varepsilon ^{(0)}) = \Big \{\, (\varepsilon ^{(0)}, u_1, \ldots , u_{t_1-1}, u_{t_1},0^{i_1}): u_1, \ldots , u_{t_1}\in U_\ell \,\Big \}. \end{aligned}$$

Here the self-admissibility of the elements in $\mathfrak M (\varepsilon ^{(0)})$ follows from Lemma 5.2.

Second, for each $w\in \mathfrak M (\varepsilon ^{(0)})$, we will extract an element belonging to $\mathbb{F }_1$ (the first generation of $\mathcal{F }$). Let $\Gamma _{n_1}(w):=\{T_\beta ^{n_1}1: \beta \in I_{n_1}^P(w)\}.$ By Lemma 5.3, we have that

$$\begin{aligned} \Gamma _{n_1}=\Gamma _{n_1}(w)\supset B(x_0, 4(n_1+\ell _{n_1})\beta _0^{-\ell _{n_1}}). \end{aligned}$$

(5.5)

Now we consider the set of all possible self-admissible sequences of order $n_1+\ell _{n_1}$ beginning with $w$, denoted by

$$\begin{aligned} \mathbb{A }(w) := \big \{\, (w, \eta _1, \ldots , \eta _{\ell _{n_1}}): (w, \eta _1, \ldots , \eta _{\ell _{n_1}}) \, \text{ is } \text{ self-admissible } \, \big \}. \end{aligned}$$

Then

$$\begin{aligned} \Gamma _{n_1}(w)= \bigcup _{\varepsilon \in \mathbb{A }(w)} \big \{\, T_{\beta }^{n_1}1 : \beta \in I_{n_1+\ell _{n_1}}^P(\varepsilon ) \,\big \}. \end{aligned}$$

(5.6)

We show that for each $\varepsilon \in \mathbb{A }(w)$,

$$\begin{aligned} \big |\big \{\, T_{\beta }^{n_1}1:\beta \in I_{n_1+\ell _{n_1}}^P(\varepsilon ) \,\big \}\big |\le 4\beta _0^{-\ell _{n_1}}. \end{aligned}$$

(5.7)

In fact, for each pair $\beta ,\beta '\in I_{n_1+\ell _{n_1}}^P(\varepsilon )$, we have

$$\begin{aligned} T^{n_1}_{\beta }1 = \frac{\eta _1}{\beta } + \cdots + \frac{\eta _{\ell _{n_1}} + y}{\beta ^{\ell _{n_1}}}, \qquad T^{n_1}_{\beta '}1 = \frac{\eta _1}{\beta '} + \cdots + \frac{\eta _{\ell _{n_1}} + y'}{\beta '^{\ell _{n_1}}} \end{aligned}$$

for some $0\le y, y'\le 1$. Then

$$\begin{aligned} \Big |T^{n_1}_{\beta }1-T^{n_1}_{\beta '}1\Big |&\le \sum _{k=1}^{\ell _{n_1}} \left| \frac{\eta _k}{\beta ^k} - \frac{\eta _k}{\beta '^k}\right| + \frac{1}{\beta ^{\ell _{n_1}}} + \frac{1}{\beta '^{\ell _{n_1}}}\\&\le \frac{\beta _1}{(\beta _0-1)^2} \beta _0^{-n_1 - \ell _{n_1}} + \frac{1}{\beta ^{\ell _{n_1}}} + \frac{1}{\beta '^{\ell _{n_1}}} \le 4\beta _0^{-\ell _{n_1}}. \end{aligned}$$

Now Lemma 4.1, together with the estimate (5.7), enables us to conclude the following simple facts:

• for each $\varepsilon \in \mathbb{A }(w),\,\big \{\, T_{\beta }^{n_1}1:\beta \in I^P_{n_1+\ell _{n_1}}(\varepsilon ) \,\big \}$ is an interval, since $I^P_{n_1+\ell _{n_1}}(\varepsilon )$ is an interval;

• for every pair $\varepsilon , \varepsilon '\in \mathbb{A }(w)$, if $\varepsilon \prec \varepsilon '$, then by the monotonicity of $T_{\beta }^{n_1}1$ with respect to $\beta$ we have that $\big \{\, T_{\beta }^{n_1}1:\beta \in I^P_{n_1+\ell _{n_1}}(\varepsilon ) \,\big \}$ lies on the left hand side of $\big \{T_{\beta }^{n_1}1:\beta \in I^P_{n_1+\ell _{n_1}}(\varepsilon ')\big \}$. Therefore, the intervals in the union in (5.6) are arranged in $[0,1]$ consecutively;

• moreover, there are no gaps between adjoint intervals in the union in (5.6), since $\Gamma _{n_1}(w)$ is an interval;

• the length of the interval $\big \{\, T_{\beta }^{n_1}1:\beta \in I^P_{n_1+\ell _{n_1}}(\varepsilon ) \,\big \}$ is less than $4\beta _0^{-\ell _{n_1}}$.

By these four facts, we conclude that there are at least $(n_1+\ell _{n_1})$ consecutive cylinders $I_{n_1+\ell _{n_1}}^P(\varepsilon )$ with $\varepsilon \in \mathbb{A }(w)$ such that $\big \{\, T_{\beta }^{n_1}1:\beta \in I^P_{n_1+\ell _{n_1}}(\varepsilon ) \,\big \}$ is contained in the ball $B(x_0, 4(n_1+\ell _{n_1})\beta _0^{-\ell _{n_1}})$. Thus by Corollary 3.12, there exists a cylinder, denoted by

$$\begin{aligned} I_{n_1+\ell _{n_1}}^P(w, w_1^{(1)}, \ldots , w_{\ell _{n_1}}^{(1)}) \end{aligned}$$

satisfying that

• The word $(w, w_1^{(1)}, \ldots , w_{\ell _{n_1}}^{(1)})$ is non-recurrent;

• The set $\big \{\, T_{\beta }^{n_1}1:\beta \in I_{n_1+\ell _{n_1}}^P (w, w_1^{(1)}, \ldots , w_{\ell _{n_1}}^{(1)}) \,\big \}$ is contained in the ball $B(x_0, 4(n_1+\ell _{n_1})\beta _0^{-\ell _{n_1}})$. Thus, for any $\beta \in I_{n_1+\ell _{n_1}}^P(w, w_1^{(1)},\ldots , w_{\ell _{n_1}}^{(1)})$,

  $$\begin{aligned} \Big |T^{n_1}_{\beta }1-x_0\Big | < 4(n_1+\ell _{n_1})\beta _0^{-\ell _{n_1}}. \end{aligned}$$

  (5.8)

This is the cylinder corresponding to $w\in \mathfrak M (\varepsilon ^{(0)})$ that we are looking for in composing the first generation of the Cantor set.

Finally the first generation of the Cantor set is defined as

$$\begin{aligned} \mathbb{F }_1 = \Big \{\, \varepsilon ^{(1)}=(w, w_1^{(1)}, \ldots , w_{\ell _{n_1}}^{(1)}) : w\in \mathfrak M (\varepsilon ^{(0)}) \, \Big \}, \qquad \mathcal{F }_1 = \bigcup _{\varepsilon ^{(1)}\in \mathbb{F }_1}I_{n_1+\ell _{n_1}}^P(\varepsilon ^{(1)}), \end{aligned}$$

where $w_1^{(1)}, \ldots , w_{\ell _{n_1}}^{(1)}$ depend on $w\in \mathfrak M (\varepsilon ^{(0)})$, but for simplicity we do not emphasize this dependence in our notation. Let $m_1=n_1+\ell _{n_1}.$

5.1.3 From generation $k-1$ to generation $k$ of the Cantor set $\mathcal{F }$

Assume that the $(k-1)$th generation $\mathbb{F }_{k-1}$ has been well defined, and is composed by a collection of non-recurrent words.

To repeat the process of the construction of the Cantor set, we present similar results as Lemmas 5.2 and 5.3.

Lemma 5.4

Let $\varepsilon ^{(k-1)}\in \mathbb{F }_{k-1}$. Then for any $u\in S_{\beta _2}$ ending with $M$ zeros, the sequence

$$\begin{aligned} (\varepsilon ^{(k-1)}, u) \end{aligned}$$

is self-admissible.

Proof

Let $1\le i<m_{k-1}$, where $m_{k-1}$ is the order of $\varepsilon ^{(k-1)}$. Since $\varepsilon ^{(k-1)}$ is non-recurrent, an application of Lemma 3.3 yields that

$$\begin{aligned} \sigma ^i(\varepsilon ^{(k-1)}, u)\prec \varepsilon ^{(k-1)}. \end{aligned}$$

Moreover, combining the assumption of $u\in S_{\beta _2}$ and (5.3), we obtain that any block of $M$ consecutive digits in $u$ is strictly less than the prefix of $\varepsilon ^{(k-1)}$. In other words, when $m_{k-1}\le i\le m_{k-1}+|u|-M$, we have $\sigma ^i(\varepsilon ^{(k-1)}, u)\prec \varepsilon ^{(k-1)}.$

At last, since $u$ ends with $M$ zeros, clearly when $i\ge m_{k-1}+|u|-M$, we have $\sigma ^i(\varepsilon ^{(k-1)}, u)\prec \varepsilon ^{(k-1)}$. $\square$

Lemma 5.5

For any $\varepsilon ^{(k-1)}\in \mathbb{F }_{k-1}$ and $u\in S_{\beta _2}$ ending with $M$ zeros, write $n=|\varepsilon ^{(k-1)}|+|u|$. Then

$$\begin{aligned} \Gamma _n=\Big \{\, T^n_{\beta }1: \beta \in I_n^P(\varepsilon ^{(k-1)}, u) \,\Big \} \end{aligned}$$

contains $(x_0+1)/2$.

Proof

With the same argument as Lemma 5.4, we can prove that the sequence $(\varepsilon ^{(k-1)}, u, (e_1,\ldots ,e_M)^{\infty })$ is self-admissible. Then with the same argument as that in Lemma 5.3, we can conclude the assertion. $\square$

Let $\varepsilon ^{(k-1)}\in \mathbb{F }_{k-1}$ be a word of length $m_{k-1}$. Choose an integer $n_k\in \mathfrak N$ such that $n_k\gg m_{k-1}$. Write

$$\begin{aligned} n_k-m_{k-1}=t_k\ell +i_k, \quad {\text {for some}}\quad 0\le i_k<\ell . \end{aligned}$$

We collect a family of self-admissible sequences beginning with $\varepsilon ^{(k-1)}$:

$$\begin{aligned} \mathfrak M (\varepsilon ^{(k-1)})= \Big \{\, \varepsilon ^{(k-1)}, u_1,\ldots ,u_{t_k-1}, u_{t_k},0^{i_k}: u_1,\ldots ,u_{t_k}\in U_\ell \,\Big \}. \end{aligned}$$

(5.9)

Here the self-admissibility of the elements in $\mathfrak M (\varepsilon ^{(k-1)})$ follows from Lemma 5.4.

Then in the light of Lemma 5.5, the remaining argument for the construction of $\mathbb{F }_k$ (the $k$th generation of $\mathcal{F }$) is absolutely the same as that for $\mathbb{F }_1$.

For each $w\in \mathfrak M (\varepsilon ^{(k-1)})$, we can extract a non-recurrent word of length $n_k+\ell _{n_k}$ belonging to $\mathbb{F }_k$, denoted by

$$\begin{aligned} (w, w^{(k)}_1,\ldots ,w^{(k)}_{\ell _{n_k}}). \end{aligned}$$

Then the $k$th generation $\mathbb{F }_k$ is defined as

$$\begin{aligned} \mathbb{F }_k=\Big \{\, \varepsilon ^{(k)}=(w, w^{(k)}_1,\ldots ,w^{(k)}_{\ell _{n_k}}): w\in \mathfrak M (\varepsilon ^{(k-1)}),\, \varepsilon ^{(k-1)}\in \mathbb{F }_{k-1} \, \Big \}, \end{aligned}$$

(5.10)

and

$$\begin{aligned} \mathcal{F }_k=\bigcup _{\varepsilon ^{(k)}\in \mathbb{F }_{k}}I^P_{n_k+\ell _{n_k}}(\varepsilon ^{(k)}). \end{aligned}$$

Note also that $w^{(k)}_1, \ldots , w^{(k)}_{\ell _{n_k}}$ depend on $w$ for each $w\in \mathfrak M (\varepsilon ^{(k-1)})$.

Continuing this procedure, we get a nested sequence $\{\mathcal{F }_k\}_{k\ge 1}$ consisting of cylinders. Finally, the desired Cantor set is defined as

$$\begin{aligned} \mathcal{F }=\bigcap _{k=1}^{\infty }\bigcup _{\varepsilon ^{(k)}\in \mathbb{F }_k}I_{|\varepsilon ^{(k)}|}^P(\varepsilon ^{(k)}) = \bigcap _{k=1}^{\infty }\bigcup _{\varepsilon ^{(k)}\in \mathbb{F }_k}I_{n_k+\ell _{n_k}}^P(\varepsilon ^{(k)}). \end{aligned}$$

Lemma 5.6

$\mathcal{F }\subset E$.

Proof

This is clear by (5.8). $\square$

5.2 Measure supported on $\mathcal{F }$

Though $\mathcal{F }$ can only be viewed as a locally homogeneous Cantor set, we define a measure uniformly distributed on $\mathcal{F }$. This measure is defined along the cylinders with non-empty intersection with $\mathcal{F }$. For any $\beta \in \mathcal{F }$, let $\{I_n^P(\beta )\}_{n\ge 1}$ be the cylinders containing $\beta$ and write

$$\begin{aligned} \varepsilon (1,\beta )=(\varepsilon ^{(k-1)}, u_1,\ldots , u_{t_k}, 0^{i_k}, w_1^{(k)},\ldots , w_{\ell _{n_k}}^{(k)}, \ldots ). \end{aligned}$$

To simplify the notation, we still use $u_{t_k}$ to denote $(u_{t_k}, 0^{i_k})$ in the above formula. Then the $\beta$-expansion of 1 will be read as

$$\begin{aligned} \varepsilon (1,\beta )=(\varepsilon ^{(k-1)}, u_1,\ldots , u_{t_k}, w_1^{(k)},\ldots , w_{\ell _{n_k}}^{(k)}, \ldots ). \end{aligned}$$

Note that the order of $\varepsilon ^{(k-1)}$ is $n_{k-1}+\ell _{n_{k-1}}$.

Now define

$$\begin{aligned} \mu \big (I_{M+q}^P(\varepsilon ^{(0)})\big )=1, \end{aligned}$$

and let

$$\begin{aligned} \mu \big (I_{n_1}^P(\varepsilon ^{(0)}, u_1,\ldots ,u_{t_1})\big )=\left( \frac{1}{\sharp \Sigma _{\beta _2}^N}\right) ^{t_1}. \end{aligned}$$

In other words, the measure is uniformly distributed among the offsprings of the cylinder $I_{M+q}^P(\varepsilon ^{(0)})$ with nonempty intersection with $\mathcal{F }$.

Next for each $n_1<n\le n_1+\ell _{n_1}$, let

$$\begin{aligned} \mu \big (I_{n}^P(\beta )\big )=\mu \big (I_{n_1}^P(\beta )\big ). \end{aligned}$$

Assume that $\mu \big (I_{n_{k-1}+\ell _{n_{k-1}}}^P(\beta )\big )$, i.e. $\mu \big (I_{n_{k-1}+\ell _{n_{k-1}}}^P(\varepsilon ^{(k-1)})\big )$ has been defined.

• (1) Define

  $$\begin{aligned} \mu \big (I_{n_k}^P(\varepsilon ^{(k-1)},u_1,\ldots ,u_{t_k})\big ) := \left( \frac{1}{\sharp \Sigma _{\beta _2}^{N}}\right) ^{t_k} \mu \big (I_{|\varepsilon ^{(k-1)}|}^P(\varepsilon ^{(k-1)})\big ) = \left( \prod _{j=1}^k \big (\sharp \Sigma _{\beta _2}^{N}\big )^{t_j} \right) ^{-1}.\nonumber \\ \end{aligned}$$

  (5.11)

• (2) When $n_{k-1}+\ell _{n_{k-1}}<n<n_k$, let

  $$\begin{aligned} \mu \big (I_n^P(\beta )\big )=\sum _{I^P_{n_k}(w)\in \mathcal{F }_k: I^P_{n_k}(w)\cap I_n^P(\beta )\ne \emptyset }\mu \big (I_{n_k}^P(w)\big ). \end{aligned}$$

  More precisely, when $n=n_{k-1}+\ell _{n_{k-1}}+t\ell$,

  $$\begin{aligned} \mu \big (I_n^P(\beta )\big )=\prod _{j=1}^{k-1}\left( \frac{1}{\sharp \Sigma _{\beta _2}^{N}}\right) ^{t_j}\cdot \left( \frac{1}{\sharp \Sigma _{\beta _2}^{N}}\right) ^{t}, \end{aligned}$$

  (5.12)

  and when $n=n_{k-1}+\ell _{n_{k-1}}+t\ell +i$ for some $i\ne 0$, we have

  $$\begin{aligned} \mu \big (I_{n_{k-1}+\ell _{n_{k-1}}+t\ell }^P(\beta )\big )\ge \mu \big (I_n^P(\beta )\big )\ge \max \Big \{\, \mu \big (I_{n_{k-1}+\ell _{n_{k-1}}+(t+1)\ell }^P(\beta )\big ), \, \mu \big (I_{n_k}^P(\beta )\big ) \,\Big \}.\nonumber \\ \end{aligned}$$

  (5.13)

• (3) When $n_k<n\le n_k+\ell _{n_k}$, take

  $$\begin{aligned} \mu \big (I_{n}^P(\beta )\big )=\mu \big (I_{n_k}^P(\beta )\big ). \end{aligned}$$

  (5.14)

5.3 Lengths of cylinders

Now we estimate the lengths of cylinders with non-empty intersection with $\mathcal{F }$.

Let $(\varepsilon _1,\ldots ,\varepsilon _n)$ be self-admissible such that $I_n^P := I_n^P(\varepsilon _1, \ldots , \varepsilon _n)$ has non-empty intersection with $\mathcal{F }$. Thus there exists $\beta \in \mathcal{F }$ such that $I_n^P$ is just the cylinder containing $\beta$. Let $n_k\le n<n_{k+1}$ for some $k\ge 1$. The estimate of the length of $I_n^P$ is divided into two cases according to the range of $n$.

• (1) When $n_k\le n<n_{k}+\ell _{n_k}$. The length of $I_n^P$ is bounded from below by the length of the cylinder containing $\beta$ with order $n_{k}+\ell _{n_k}+M$. By the construction of $\mathcal{F }_k$, we know that $\varepsilon (1,\beta )$ can be expressed as

  $$\begin{aligned} \varepsilon (1,\beta )=(\varepsilon ^{(k)}, 0^M, 1,\ldots ), \end{aligned}$$

  which implies the self-admissibility of $(\varepsilon ^{(k)}, 0^M, 1)$. Then clearly $(\varepsilon ^{(k)}, 0^M, 0)$ is self-admissible as well. Then by Lemma 3.4, we know that $(\varepsilon ^{(k)}, 0^M, 0)$ is non-recurrent. Thus,

  $$\begin{aligned} \big |I_n^P\big |&\ge \big |I_{n_k+\ell _{n_k}+M}^P(\beta )\big |\ge \big |I_{n_k+\ell _{n_k}+M+1}^P(\varepsilon ^{(k)}, 0^{M},0)\big |\nonumber \\&\ge C\beta _1^{-(n_k+\ell _{n_k}+M+1)}:=C_1 \beta _1^{-(n_k+\ell _{n_k})}. \end{aligned}$$

  (5.15)

• (2) When $n_{k}+\ell _{n_k}\le n<n_{k+1}$. Let $t=n-n_k-\ell _{n_k}$. Write $\varepsilon (1,\beta )$ as

  $$\begin{aligned} \varepsilon (1,\beta ) = (\varepsilon ^{(k)}, \eta _1,\ldots , \eta _{t},\ldots ) \end{aligned}$$

  for some $(\eta _1,\ldots ,\eta _{t})\in \Sigma _{\beta _2}^{t}$. Lemma 5.4 tells us that

  $$\begin{aligned} (\varepsilon ^{(k)}, \eta _1,\ldots ,\eta _{t}, 0^M, 1, 0^M) \end{aligned}$$

  is self-admissible. Then with the same argument as case (1), we obtain

  $$\begin{aligned} \big |I_n^P\big |\ge \big |I_{n+M+1}^P(\varepsilon ^{(k)},\eta _1, \ldots , \eta _{t}, 0^M, 0)\big |\ge C\beta _1^{-(n+M+1)}:=C_1\beta _1^{-n}. \end{aligned}$$

  (5.16)

5.4 Measure of balls

We estimate the measure of arbitrary balls $B(\beta ,r)$ with $\beta \in \mathcal{F }$ and $r$ small enough.

Together with the $\mu$-measure and the lengths of cylinders with non-empty intersection with $\mathcal{F }$ given in the last two subsections, it follows directly that

Corollary 5.7

For any $\beta \in \mathbb{F }$,

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \mu \big (I_{n}^P(\beta )\big )}{\log |I_{n+1}^P(\beta )|}\ge \frac{1}{1+\alpha }\frac{\log \beta _2}{\log \beta _1}\frac{N}{\ell }, \end{aligned}$$

(5.17)

where $N$ and $\ell$ are the integers in the definition of $U_{\ell }$ (see (5.4)).

Now we refine the cylinders containing some $\beta \in \mathcal{F }$ as follows. For each $\beta \in \mathcal{F }$ and $n\ge 1$, define

$$\begin{aligned} J_n(\beta )=\left\{ \begin{array}{l@{\quad }l} I^P_{n_k+\ell _{n_k}}(\beta ), &{} \hbox {when } n_k\le n<n_k+\ell _{n_k} \hbox { for some } k\ge 1; \\ I_n^P(\beta ), &{} \hbox {when } n_k+\ell _{n_k}\le n<n_{k+1} \hbox { for some } k\ge 1. \end{array} \right. \end{aligned}$$

(5.18)

and call $J_n(\beta )$ the basic interval of order $n$ containing $\beta$.

Fix a ball $B(\beta ,r)$ with $\beta \in \mathcal{F }$ and $r$ small. Let $n$ be the integer such that

$$\begin{aligned} \big |J_{n+1}(\beta )\big |\le r<\big |J_n(\beta )\big |. \end{aligned}$$

Let $k$ be the integer such that $n_k\le n<n_{k+1}$. The difference of the lengths of $J_{n+1}(\beta )$ and $J_n(\beta )$ (i.e., $|J_{n+1}(\beta )|<|J_n(\beta )|$) yields that

$$\begin{aligned} n_{k}+\ell _{n_k}\le n<n_{k+1}. \end{aligned}$$

Recall the definition of $\mu$. It should be noticed that

$$\begin{aligned} \mu \big (J_n(\beta )\big )=\mu \big (I_n^P(\beta )\big ), \quad \text{ for } \text{ all } \, n\in \mathbb{N }. \end{aligned}$$

Then all basic intervals $J$ with the same order are of equal $\mu$-measure. So, to bound the measure of the ball $B(\beta , r)$ from above, it suffices to estimate the number of basic intervals with non-empty intersection with the ball $B(\beta , r)$. We denote this number by $\mathcal{N }$. Note that by (5.16) and (5.18), when $n_{k}+\ell _{n_k}\le n<n_{k+1}$, all basic intervals of order $n$ are of length no less than $C_1\beta _1^{-n}$ . Since $r\le |J_n(\beta )|\le \beta _0^{-n}$, we have

$$\begin{aligned} \mathcal{N }\le 2r/(C_1\beta _1^{-n})+2\le 2\beta _0^{-n}/(C_1\beta _1^{-n})+2\le C_2\beta _0^{-n}\beta _1^{n}. \end{aligned}$$

It follows that

$$\begin{aligned} \mu \big (B(\beta ,r)\big )\le C_2\beta _0^{-n}\beta _1^{n} \cdot \mu \big (I_n^P(\beta )\big ). \end{aligned}$$

(5.19)

Now we give a lower bound for $r$. When $n<n_{k+1}-1$, we have

$$\begin{aligned} r\ge \big |J_{n+1}(\beta )\big |=\big |I^P_{n+1}(\beta )\big |\ge C_1\beta _1^{-n-1}. \end{aligned}$$

(5.20)

When $n=n_{k+1}-1$, we have

$$\begin{aligned} r\ge \big |J_{n+1}(\beta )\big |\ge C_1\beta _1^{-n_{k+1}-\ell _{n_{k+1}}} \end{aligned}$$

(5.21)

Thus, by the formulae (5.19) (5.20) (5.21) and Corollary 5.7, we have

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{\log \mu (B(\beta , r))}{\log r}\ge \left( \frac{\log \beta _0-\log \beta _1}{\log \beta _1}+\frac{\log \beta _2}{\log \beta _1}\frac{N}{\ell }\right) \frac{1}{1+\alpha }. \end{aligned}$$

Applying the mass distribution principle (Proposition 5.1), we obtain

$$\begin{aligned} \dim _\mathsf H E \ge \left( \frac{\log \beta _0-\log \beta _1}{\log \beta _1}+\frac{\log \beta _2}{\log \beta _1}\frac{N}{\ell }\right) \frac{1}{1+\alpha }. \end{aligned}$$

Letting $N\rightarrow \infty$ and then $\beta _1\rightarrow \beta _0$, we arrive at

$$\begin{aligned} \dim _\mathsf H E \ge \frac{1}{1+\alpha }. \end{aligned}$$

6 Lower bound of $E(\{\ell _n\}_{n\ge 1}, x_0): x_0=1$

We still use the classic strategy to estimate the dimension of $E(\{\ell _n\}_{n\ge 1}, 1)$ from below. In fact, we will show a little stronger result: for any $\beta _0<\beta _1$, the Hausdorff dimension of the set $E(\{\ell _n\}_{n\ge 1}, 1)\cap (\beta _0, \beta _1)$ is $1/(1+\alpha )$.

The first step is devoted to constructing a Cantor subset $\mathcal{F }$ of $E(\{\ell _n\}_{n\ge 1}, 1)$. We begin with some notation.

As in the beginning of Sect. 5.1, we can require that $\beta _0$ and $\beta _1$ are sufficiently close such that the common prefix

$$\begin{aligned} (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M-1}(1,\beta _1)) \end{aligned}$$

of $\varepsilon (1,\beta _0)$ and $\varepsilon (1,\beta _1)$ contains at least four nonzero terms. Assume that $\varepsilon (1,\beta _1)$ begins with the word ${o}=(a_1, 0^{r_1-1}, a_2,0^{r_2-1}, a_3, 0^{r_3-1}, a_4)$ with $a_i\ne 0$. Let

$$\begin{aligned} \overline{o}=(0^{r_1}, 1, 0^{r_2}, 1, 0^{r_3}), \quad \overline{O}=(0^{r_1}, 1, 0^{r_2+1}). \end{aligned}$$

By the self-admissibility of ${o}$, it follows that if $a_1=1$, then $\min \{r_2, r_3\}\ge r_1$. So it is direct to check that for any $i\ge 0$, we have

$$\begin{aligned} \sigma ^i(\overline{o})\prec \varepsilon _1(1,\beta _1), \ldots , \varepsilon _{(r_1+r_2+r_3+2)-i}(1,\beta _1). \end{aligned}$$

(6.1)

Recall that $\beta _2$ is given in (5.2). Fix an integer $\ell \gg M$. Define the collection

$$\begin{aligned} U_\ell =\big \{u=(\overline{o}, \varepsilon _1, \ldots , \varepsilon _{\ell -r_1-r_2-r_3-2-M}, 0^M) \in \Sigma _{\beta _2}^{\ell }\big \}. \end{aligned}$$

Following the same argument as the case (3) in proving Lemma 5.2 and then by (5.3), we have for any $u\in U_{\ell }$ and $i\ge r_1+r_2+r_3+2$,

$$\begin{aligned} \sigma ^i(u) \prec (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1)). \end{aligned}$$

(6.2)

Combining (6.1) and (6.2), we get for any $u\in U_{\ell }$ and $i\ge 0$,

$$\begin{aligned} \sigma ^i(u) \prec (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _{M}(1,\beta _1)). \end{aligned}$$

(6.3)

Recall that $q$ is the integer such that

$$\begin{aligned} (\varepsilon _{M+1}(1,\beta _1), \ldots , \varepsilon _{M+q}(1,\beta _1))\ne 0^q. \end{aligned}$$

With the help of (6.3), we present a result with the same role as that of Lemma 5.2.

Lemma 6.1

Let $k\in \mathbb{N }$. For any $u_1, \ldots ,u_k\in U_\ell$, the word

$$\begin{aligned} \varepsilon =(\varepsilon _1(1,\beta _1), \ldots , \varepsilon _M(1,\beta _1), 0^q, u_1, u_2, \ldots , u_k) \end{aligned}$$

is non-recurrent.

Proof

We check that $\sigma ^i(\varepsilon )\prec \varepsilon$ for all $i\ge 1$. When $i<M+q$, the argument is absolutely the same as that for $i<M+q$ in Lemma 5.2. When $i\ge M+q$, it follows by (6.3). $\square$

6.1 Construction of the Cantor subset

Now we return to the set

$$\begin{aligned} E_0 := \big \{\, \beta _0<\beta <\beta _1: \big |T^n_{\beta }1-1| < \beta ^{-\ell _n}, \, {\text {i.o.}}\, n\in \mathbb{N }\,\big \}. \end{aligned}$$

We will use the following strategy to construct a Cantor subset of $E_0$.

• Strategy: If the $\beta$-expansion of 1 has a long periodic prefix with period $n$, then $T^n_{\beta }1$ and 1 will be close enough.

Let $\{n_k\}_{k\ge 1}$ be a subsequence of integers such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\ell _{n_k}}{n_k}=\liminf _{n\rightarrow \infty }\frac{\ell _n}{n}=\alpha , \quad {\text {and}} \quad n_{k+1}\gg n_k, \, \text{ for } \text{ all } \, k\ge 1. \end{aligned}$$

6.1.1 First generation $\mathcal{F }_1$ of the Cantor set $\mathcal{F }$

Let $\varepsilon ^{(0)} = (\varepsilon _1(1,\beta _1), \ldots , \varepsilon _M(1,\beta _1), 0^q)$ and $m_0=M+q$. Write $n_1=m_0+t_1\ell +i_1$ for some $t_1\in \mathbb{N }$ and $0\le i_1<\ell$. Now consider the collection of self-admissible words of length $n_1$

$$\begin{aligned} \mathfrak M (\varepsilon ^{(0)}) = \big \{\, (\varepsilon ^{(0)}, u_1, \ldots , u_{t_1}, 0^{i_1}): u_1, \ldots , u_{t_1}\in U_\ell \,\big \}. \end{aligned}$$

Lemma 6.1 says that all the elements in $\mathfrak M (\varepsilon ^{(0)})$ are non-recurrent words.

Enlarging $\ell _{n_1}$ by at most $m_0+\ell$ if necessary, the number $\ell _{n_1}$ can be written as

$$\begin{aligned} \ell _{n_1}=z_1n_1+m_0+j_1\ell , \quad {\text {with}}\, z_1\in \mathbb{N }, \, 0\le j_1<t_1. \end{aligned}$$

(6.4)

Corollary 3.7 convinces us that for any $(\varepsilon _1, \ldots , \varepsilon _{n_1})\in \mathfrak M (\varepsilon ^{(0)})$, the word

$$\begin{aligned} \varepsilon :=\Big (\big (\varepsilon _1, \ldots , \varepsilon _{n_1}\big ), \big (\varepsilon _1, \ldots , \varepsilon _{n_1}\big )^{z_1}, \big (\varepsilon ^{(0)}, u_1, \ldots , u_{j_1}\big )\Big ) \end{aligned}$$

(6.5)

is self-admissible. In other words, $\varepsilon$ is a periodic self-admissible word with length $n_1+\ell _{n_1}$. We remark that the suffix $\big (\varepsilon ^{(0)}, u_1, \ldots , u_{j_1}\big )$ is the prefix of $(\varepsilon _1, \ldots , \varepsilon _{n_1})$ but not chosen freely.

Now consider the cylinder

$$\begin{aligned} I_{n_1+\ell _{n_1}}^P := I^P_{n_1+\ell _{n_1}} \Big (\big (\varepsilon _1, \ldots , \varepsilon _{n_1}\big )^{z_1+1}, \big (\varepsilon ^{(0)}, u_1, \ldots , u_{j_1}\big )\Big ). \end{aligned}$$

It is clear that for each $\beta \in I_{n_1+\ell _{n_1}}^P$, the $\beta$-expansion of $T^{n_1}_{\beta }1$ and that of 1 coincide for the first $\ell _{n_1}$ terms. So, we conclude that for any $\beta \in I^P_{n_1+\ell _{n_1}}$,

$$\begin{aligned} \big |T_{\beta }^{n_1}1-1\big |<\beta ^{-\ell _{n_1}}. \end{aligned}$$

(6.6)

Now we prolong the word in (6.5) to a non-recurrent word. Still by Corollary 3.7, we know that $(\varepsilon , u_{j_1+1})$ is self-admissible, which implies the admissibility of the word

$$\begin{aligned} (\varepsilon , 0^{r_1}, 1, 0^{r_2}, 1). \end{aligned}$$

So, by Lemma 3.4, we obtain that the word $(\varepsilon , \overline{O})$ is non-recurrent. Then finally, the first generation $\mathcal{F }_1$ of the Cantor set $\mathcal{F }$ is defined as

$$\begin{aligned} \mathcal{F }_1&= \Big \{\,I^P_{(n_1+\ell _{n_1}+r_1+r_2+2)} \Big (\big (\varepsilon _1, \ldots , \varepsilon _{n_1}\big )^{z_1+1}, \big (\varepsilon ^{(0)}, u_1, \ldots , u_{j_1}, \overline{O}\big )\Big ): (\varepsilon _1, \ldots ,\varepsilon _{n_1})\\&\quad \in \mathfrak M (\varepsilon ^{(0)}) \,\Big \}. \end{aligned}$$

6.1.2 Second generation $\mathcal{F }_2$ of the Cantor set $\mathcal{F }$

Let $m_1=n_1+\ell _{n_1}+r_1+r_2+2$ and write

$$\begin{aligned} n_2=m_1+t_2\ell +i_2\quad {\text { for some}}\quad t_2\in \mathbb{N }, \quad 0\le i_2<\ell . \end{aligned}$$

For each $\varepsilon ^{(1)}\in \mathcal{F }_1$, consider the collection of self-admissible words of length $n_2$

$$\begin{aligned} \mathfrak M (\varepsilon ^{(1)}) = \big \{\, (\varepsilon ^{(1)}, u_1, \ldots , u_{t_2}, 0^{i_2}): u_1, \ldots , u_{t_2}\in U_\ell \,\big \}. \end{aligned}$$

By noting that $\varepsilon ^{(1)}$ is non-recurrent and by the formula (6.3), we know that all elements in $\mathfrak M (\varepsilon ^{(1)})$ are non-recurrent words.

Similar to the modification on $\ell _{n_1}$, by enlarging $\ell _{n_2}$ by at most $m_1+\ell$ if necessary, the number $\ell _{n_2}$ can be written as

$$\begin{aligned} \ell _{n_2}=z_2n_2+m_1+j_2\ell , \quad {\text {with}}\quad z_2\in \mathbb{N }, \quad 0\le j_2<t_2. \end{aligned}$$

(6.7)

Then, following the same line as for the construction of the first generation, we get the second generation $\mathcal{F }_2$, defined by

$$\begin{aligned} \mathcal{F }_2&= \Big \{\, I^P_{(n_2+\ell _{n_2}+r_1+r_2+1)} \Big (\big (\varepsilon _1, \ldots ,\varepsilon _{n_2}\big )^{z_2+1}, \big (\varepsilon ^{(1)}, u_1, \ldots , u_{j_2}, \overline{O}\big )\Big ): (\varepsilon _1, \ldots ,\varepsilon _{n_2})\\&\quad \in \mathfrak M (\varepsilon ^{(1)}) \,\Big \}. \end{aligned}$$

We remark that the suffix $\big (\varepsilon ^{(1)}, u_1, \ldots , u_{j_2}\big )$ is the prefix of $(\varepsilon _1,\ldots ,\varepsilon _{n_2})$ but not chosen freely. Then let $m_2=n_2+\ell _{n_2}+r_1+r_2+2$.

Then, proceeding along the same line, we get a nested sequence $\mathcal{F }_k$ consisting of a family of cylinders. The desired Cantor set is defined as

$$\begin{aligned} \mathcal{F }=\bigcap _{k\ge 1}\mathcal{F }_k. \end{aligned}$$

Noting (6.6), we know that $\mathcal{F } \subset E_0.$

6.2 Estimate on the supported measure

The remaining argument for the dimension of $\mathcal{F }$ is almost the same as what we did in Sect. 5: constructing an evenly distributed measure supported on $\mathcal{F }$ and then applying the mass distribution principle. Thus, we will not repeat it here.

7 Proof of Theorem 1.2

The proof of Theorem 1.2 can be established with almost the same argument as that for Theorem 1.1. Therefore only differences of the proof are marked below.

7.1 Proof of the upper bound

For each self-admissible sequence $(i_1,\ldots ,i_n)$, denote

$$\begin{aligned} J_n (i_1, \ldots , i_n):=\Big \{\, \beta \in I_n^P (i_1, \ldots , i_n): |T_\beta ^n1-x(\beta )|< \beta _0^{-\ell _n} \,\Big \}. \end{aligned}$$

These sets correspond to the sets $I_n^P (i_1, \ldots , i_n; \beta _0^{-\ell _n})$ studied in the proof of Proposition 4.2, where the upper bound for the case of constant $x_0$ was obtained. We have that

$$\begin{aligned} \left( \widetilde{E}\big (\{\ell _n\}_{n\ge 1}, x\big )\cap (\beta _0,\beta _1)\right) \subset \bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }\quad \bigcup _{(i_1, \ldots , i_n) \, {\text {self-admissible}}} J_n (i_1, \ldots , i_n). \end{aligned}$$

What remains is to estimate the diameter of $J_n (i_1, \ldots , i_n)$ for any self-admissible sequence $(i_1, \ldots , i_n)$. If we can get a good estimate of the diameter, then we can do as in the proof of Proposition 4.2 to get an upper bound of the dimension of $\widetilde{E}\big (\{\ell _n\}_{n\ge 1}, x\big )\cap (\beta _0,\beta _1)$.

Suppose $J_n$ is non-empty, and let $\beta _2 < \beta _3$ denote the infimum and supremum of $J_n$. Let $L$ be such that $\beta \mapsto x (\beta )$ is Lipschitz continuous, with constant $L$. Denote by $\psi$ the map $\beta \mapsto T_\beta ^n (1)$, and note that $\psi$ satisfies

$$\begin{aligned} |\psi (\beta _3) - \psi (\beta _2)| \ge \beta _0^n\cdot |\beta _3 - \beta _2|. \end{aligned}$$

Clearly, $\beta _2$ and $\beta _3$ must satisfy

$$\begin{aligned} |\psi (\beta _3) - \psi (\beta _2)| - |x (\beta _3) - x(\beta _2)| < 2 \beta _0^{-\ell _n}, \end{aligned}$$

and hence, we must have

$$\begin{aligned} \beta _0^n \cdot |\beta _3 - \beta _2| - L\cdot |\beta _3 - \beta _2| < 2 \beta _0^{-\ell _n}. \end{aligned}$$

(7.1)

Take $K > 2$. Then we must have $|\beta _3 - \beta _2| \le K \beta _0^{-\ell _n - n}$ for sufficiently large $n$, otherwise (7.1) will not be satisfied.

Thus, we have proved that $|J_n (i_1,\ldots ,i_n)| \le K \beta _0^{- \ell _n - n}$ for some constant $K$. This is all what is needed to make the proof of Proposition 4.2 work also for the case of non-constant $x_0$.

7.2 Proof of the lower bound

• Case 1. If $x(\beta )=1$ for all $\beta \in [\beta _0, \beta _1]$, this falls into the proof of Theorem 1.1.

• Case 2. Otherwise, we can find a subinterval of $(\beta _0,\beta _1)$ such that the supremum of $x(\beta )$ on this subinterval is strictly less than 1. We denote by $0\le x_0<1$ the supremum of $x(\beta )$ on this subinterval. We note that with this definition of $x_0$, Lemma 5.3 still holds.

Now that we have Lemma 5.3, we can get a lower bound in the same way as in Sect. 5, i.e. we construct a Cantor set with desired properties. The proof is more or less unchanged, but some minor changes are nessesary, as we will describe below.

The sets $\mathbb{F }_0$ and $\mathfrak M (\varepsilon ^{(0)})$ are defined as before, and we consider a $w \in \mathfrak M (\varepsilon ^{(0)})$. On the interval $I_{n_1}^P (w)$ we define $\psi :\beta \mapsto T_\beta ^{n_1} (1)$, and we observe that there are constants $c_1$ and $c_2$ such that

$$\begin{aligned} c_1 \beta _0^{n_1} \le \psi ' (\beta ) \le c_2 \beta _0^{n_1}, \end{aligned}$$

holds for all $\beta \in I_{n_1}^P (w)$. As in the proof of the upper bound, we let $L$ denote the Lipschitz constant of the function $\beta \mapsto x(\beta )$.

We need to estimate the size of the set

$$\begin{aligned} J = \{\, \beta \in I_{n_1}^P (w) : \psi (\beta ) \in B (x_0(\beta ), C (n_1 + \ell _{n_1}) \beta _0^{-\ell _{n_1}}) \,\}. \end{aligned}$$

The constant $C$ appearing in the definition of $J$ above, was equal to 4 in Sect. 5. We remark that the value of $C$ has no influence on the result of the proof, so we may choose it more freely, as will be done here.

Lemma 5.3 implies that there is a $\beta _a \in J$ such that $\psi (\beta _a) = x (\beta _a)$. Suppose $\beta _b \in I_{n_1}^P (w)$ is such that $|\beta _a - \beta _b| < 4 (n_1 + \ell _{n_1}) \beta _0^{-n_1 - \ell _{n_1}}$. We can choose $C$ so large that we have

$$\begin{aligned} |\psi (\beta _b) - x (\beta _b)|&\le |\psi (\beta _a) - \psi (\beta _b)| + |x (\beta _a) - x (\beta _b)| \\&\le c_2 4 (n_1 + \ell _{n_1}) \beta _0^{-\ell _{n_1}} + L\cdot 4 (n_1 + \ell _{n_1}) \beta _0^{- n_1 - \ell _{n_1}} < C (n_1 + \ell _{n_1}) \beta _0^{-\ell _{n_1}}. \end{aligned}$$

This proves that $\beta _b$ is in $J$, and hence, $J$ contains an interval of length at least $4 (n_1 + \ell _{n_1}) \beta _0^{-n_1 - \ell _{n_1}}$.

Analogous to the estimate in (5.7), we have that $|I_{n_1+\ell _{n_1}}^P (\varepsilon )| \le 4 \beta _0^{-n_1 - \ell _{n_1}}$. This implies that there are at least $(n_1 + \ell _{n_1})$ consequtive cylinders $I_{n_1 + \ell _{n_1}}^P (\varepsilon )$ with the desired hitting property, where $\varepsilon \in \mathbb{A } (w)$.

With the changes indicated above, the proof then continues just as in Sect. 5.

8 Application

This section is devoted to an application of Theorem 1.1. For each $n\ge 1$, denote by $\ell _n(\beta )$ the length of the longest string of zeros just after the $n$th digit in the $\beta$-expansion of 1, namely,

$$\begin{aligned} \ell _n(\beta ) := \max \{\, k\ge 0: \varepsilon _{n+1}^*(\beta ) = \cdots = \varepsilon _{n+k}^*(\beta )=0 \,\}. \end{aligned}$$

Let

$$\begin{aligned} \ell (\beta )=\limsup _{n\rightarrow \infty }\frac{\ell _n(\beta )}{n}. \end{aligned}$$

Li and Wu [16] gave a kind of classification of betas according to the growth of $\{\ell _n\}_{n\ge 1}$ as follows:

$$\begin{aligned}&A_0=\Big \{\,\beta >1: \{\ell _n(\beta )\}\, \text {is bounded}\,\Big \};\\&A_1=\Big \{\,\beta >1: \{\ell _n(\beta )\}\, \text{ is } \text{ unbounded } \text{ and }\quad \ell (\beta )=0 \,\Big \};\\&A_2=\Big \{\,\beta >1: \ell (\beta )>0\,\Big \}. \end{aligned}$$

We will use the dimensional result of $E(\{\ell _n\}_{n\ge 1}, x_0)$ to determine the size of $A_1, A_2$ and $A_3$ in the sense of Lebesgue measure $\mathcal{L }$ and Hausdorff dimension. In the argument below only the dimension of $E(\{\ell _n\}_{n\ge 1}, x_0)$ when $x_0=0$ is used. In other words, the result in [18] by Persson and Schmeling is already sufficient for the following conclusions.

Proposition 8.1

(Size of $A_0$) $\mathcal{L }(A_0)=0$ and $\dim _\mathsf H (A_0)=1$.

Proof

The set $A_0$ is nothing but the collections of $\beta$ with specification properties. Then this proposition is just Theorem A in [21]. $\square$

Proposition 8.2

(Size of $A_2$) $\mathcal{L }(A_2)=0$ and $\dim _\mathsf H (A_2)=1$.

Proof

For any $\alpha >0$, let

$$\begin{aligned} F(\alpha ) = \left\{ \, \beta >1: \ell (\beta )\ge \alpha \,\right\} . \end{aligned}$$

Then $A_2=\bigcup _{\alpha >0}F(\alpha )$. Since $F(\alpha )$ is increasing with respect to $\alpha$, the above union can be expressed as a countable union. Now we show that for each $\alpha >0$

$$\begin{aligned} \dim _\mathsf H F(\alpha )=\frac{1}{1+\alpha }, \end{aligned}$$

which is sufficient for the desired result.

Recall the algorithm of $T_{\beta }$. Since for each $\beta \in A_2$, the $\beta$-expansion of 1 is infinite, then for each $n\ge 1$, we have

$$\begin{aligned} T^n_{\beta }1 = \frac{\varepsilon _{n+1}^*(\beta )}{\beta } + \frac{\varepsilon _{n+2}^*(\beta )}{\beta ^2} + \cdots . \end{aligned}$$

Then by the definition of $\ell _n(\beta )$, it follows that

$$\begin{aligned} \beta ^{-(\ell _n(\beta )+1)} \le T_\beta ^n1 \le (\beta +1)\beta ^{-(\ell _n(\beta )+1)}. \end{aligned}$$

(8.1)

As a consequence, for any $\delta >0$,

$$\begin{aligned} F(\alpha )\subset \{\,\beta >1: T_\beta ^n1<(\beta +1)\beta ^{-n(\alpha -\delta )-1}\, {\text {for infinitely many}}\, n\in \mathbb{N }\,\}. \end{aligned}$$

(8.2)

On the other hand, it is clear that

$$\begin{aligned} \{\,\beta >1: T_\beta ^n1<\beta ^{-n\alpha }\, {\text {for infinitely many}}\, n\in \mathbb{N }\,\}\subset F(\alpha ). \end{aligned}$$

(8.3)

Applying Theorem 1.1 to (8.2) and (8.3), we get that

$$\begin{aligned} \dim _\mathsf H F(\alpha )= \frac{1}{1+\alpha }. \end{aligned}$$

$\square$

Since $A_1=(1,\infty ){\setminus } (A_0\cup A_2)$, it follows directly that

Proposition 8.3

(Size of $A_1$) The set $A_1$ is of full Lebesgue measure.