Uniform recurrence properties for beta-transformation

TL;DR

This paper investigates the uniform recurrence properties of points under the beta-transformation, characterizing the Hausdorff dimension of sets with prescribed recurrence rates and revealing a phase transition at a critical rate.

Contribution

It provides a precise Hausdorff dimension formula for sets of points with specified uniform recurrence rates in beta-transformations, extending understanding of recurrence behavior.

Findings

Hausdorff dimension of recurrence sets for 0 ≤ r ≤ 1 is ((1 - r)/(1 + r))^2

Recurrence sets with rate r > 1 are countable

The dimension formula reveals a phase transition at r=1

Abstract

For any $\beta > 1$, let $T_\beta: [0,1)\rightarrow [0,1)$ be the $\beta$-transformation defined by $T_\beta x=\beta x \mod 1$. We study the uniform recurrence properties of the orbit of a point under the $\beta$-transformation to the point itself. The size of the set of points with prescribed uniform recurrence rate is obtained. More precisely, for any $0\leq \hat{r}\leq +\infty$, the set $$\left\{x \in [0,1): \forall\ N\gg1, \exists\ 1\leq n \leq N, {\rm\ s.t.}\ |T^n_\beta x-x|\leq \beta^{-\hat{r}N}\right\}$$ is of Hausdorff dimension $\left(\frac{1-\hat{r}}{1+\hat{r}}\right)^2$ if $0\leq \hat{r}\leq 1$ and is countable if $\hat{r}>1$.

Equations

$$\left\{x\in [0,1):\forall N\gg 1,\exists 1\le n\le N,s.t.|T_{\beta }^{n}x-x|\le {\beta }^{-\hat{r}N}\right\}$$

$$\underset{n\to \infty }{\mathrm{liminf}}d(T^{n}x,x)=0$$

$$\underset{n\to \infty }{\mathrm{liminf}}{n}^{\frac{1}{\alpha }}d(T^{n}x,x)<\infty$$

$$\underset{n\to \infty }{\mathrm{liminf}}{n}^{\frac{1}{\alpha }}d(T^{n}x,x)=0$$

$$\Vert n\theta \Vert <N^{-1},$$

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

$$r_{\beta }(x):=\sup \{0\le r\le +\infty :|T_{\beta }^{n}x-x|<({\beta }^n{)}^{-r}\mathrm{for}\mathrm{infinitely}\mathrm{many}n\in \mathbb{N}\}$$

$${\hat{r}}_{\beta }(x):=\sup \{0\le \hat{r}\le +\infty :\mathrm{for}\mathrm{all}N\gg 1,\mathrm{there}\mathrm{is}n\in [1,N],s.t.|T_{\beta }^{n}x-x|<({\beta }^N{)}^{-\hat{r}}\}.$$

$${\dim }_{\mathrm{H}}\left\{x\in [0,1):r_{\beta }(x)\ge r\right\}=\frac{1}{1+r},$$

$${\dim }_{\mathrm{H}}\{x\in [0,1):{\hat{r}}_{\beta }(x)\ge \hat{r}\}={\dim }_{\mathrm{H}}\{x\in [0,1):{\hat{r}}_{\beta }(x)=\hat{r}\}={\left(\frac{1-\hat{r}}{1+\hat{r}}\right)}^2.$$

$$R_{\beta }(\hat{r},r):=\left\{x\in [0,1):{\hat{r}}_{\beta }(x)=\hat{r},r_{\beta }(x)=r\right\}.$$

$${\dim }_{\mathrm{H}}{R}_{\beta }(\hat{r},r)=\frac{r-(1+r)\hat{r}}{(1+r)(r-\hat{r})}.$$

$${\dim }_{\mathrm{H}}\left\{x\in [0,1):r_{\beta }(x)=r\right\}=\frac{1}{1+r}.$$

$$x=\frac{{\epsilon }_1(x,\beta )}{\beta }+\cdots +\frac{{\epsilon }_n(x,\beta )}{{\beta }^n}+\cdots ,$$

$${\Sigma }_{\beta }^n=\{({\epsilon }_1,\dots ,{\epsilon }_n)\in {\mathcal{A}}^n:\exists x\in [0,1),s.t.{\epsilon }_j(x,\beta )={\epsilon }_j,\forall 1\le j\le n\}.$$

$${\Sigma }_{\beta }=\{({\epsilon }_1,{\epsilon }_2,\dots )\in {\mathcal{A}}^{\mathbb{N}}:\exists x\in [0,1),s.t.\epsilon (x,\beta )=({\epsilon }_1,{\epsilon }_2,\dots )\}.$$

$$({\omega }_1,{\omega }_2,\dots ){<}_{\mathrm{lex}}({\omega }_1^{\prime },{\omega }_2^{\prime },\dots )$$

$$1=\frac{{\epsilon }_1(1,\beta )}{\beta }+\cdots +\frac{{\epsilon }_n(1,\beta )}{{\beta }^n}+\cdots ,$$

$${\epsilon }^{\ast }(\beta ):=({\epsilon }_1^{\ast }(\beta ),{\epsilon }_2^{\ast }(\beta ),\dots )=({\epsilon }_1(1,\beta ),{\epsilon }_2(1,\beta ),\dots ,{\epsilon }_m(1,\beta )-1{)}^{\infty }$$

$$1=\frac{{\epsilon }_1^{\ast }(\beta )}{\beta }+\cdots +\frac{{\epsilon }_n^{\ast }(\beta )}{{\beta }^n}+\cdots .$$

$$1=\frac{{\epsilon }_1^{\ast }(\beta )}{x}+\cdots +\frac{{\epsilon }_N^{\ast }(\beta )}{x^N}.$$

$${\epsilon }^{\ast }({\beta }_N)=({\epsilon }_1^{\ast }(\beta ),\dots ,{\epsilon }_N^{\ast }(\beta )-1{)}^{\infty }.$$

$$({\omega }_j,\dots ,{\omega }_n){\le }_{\mathrm{lex}}({\epsilon }_1^{\ast },\dots ,{\epsilon }_{n-j}^{\ast }),\forall 1\le j\le n.$$

$${\Sigma }_{{\beta }_1}^n\subseteq {\Sigma }_{{\beta }_2}^n.$$

$${\beta }^n\le ♯{\Sigma }_{\beta }^n\le \frac{{\beta }^{n+1}}{\beta -1}\mathrm{and}\underset{n\to \infty }{\lim }\frac{\log ♯{\Sigma }_{\beta }^n}{n}=\log \beta .$$

$$I_n(\omega ):=I_n(\omega ,\beta )=\{x\in [0,1):{\epsilon }_j(x,\beta )={\omega }_j,\forall 1\le j\le n\}.$$

$${\beta }^{-(n+N)}\le |I_n(\omega ,\beta )|\le {\beta }^{-n}.$$

$$r_{\beta }(x)=\underset{k\to \infty }{\mathrm{limsup}}\frac{m_k-n_k}{n_k}$$

$${\hat{r}}_{\beta }(x)=\underset{k\to \infty }{\mathrm{liminf}}\frac{m_k-n_k}{n_{k+1}}.$$

$$n_1=\min \{n\ge 1:{\epsilon }_{n+1}={\epsilon }_1\},m_1=\max \{n\ge n_1:|T_{\beta }^{n_1}x-x|<{\beta }^{-(n-n_1)}\}.$$

Full text

Uniform recurrence properties for beta-transformation

00footnotetext: 2010 AMS Subject Classifications: 11K55, 28A80

Lixuan Zheng and Min Wu

LAMA

Université Paris-Est Créteil

61 av Général de Gaulle

94010

Créteil

France

Department of Mathematics

South China University of Technology

Guangzhou 510640

P.R. China

E-mail: lixuan.zheng@u-pec.fr

wumin@scut.edu.cn

(4th June,2018)

Abstract. For any $\beta>1$, let $T_{\beta}:[0,1)\rightarrow[0,1)$ be the $\beta$-transformation defined by $T_{\beta}x=\beta x\mod 1$. We study the uniform recurrence properties of the orbit of a point under the $\beta$-transformation to the point itself. The size of the set of points with prescribed uniform recurrence rate is obtained. More precisely, for any $0\leq\hat{r}\leq+\infty$, the set

[TABLE]

is of Hausdorff dimension $\left(\frac{1-\hat{r}}{1+\hat{r}}\right)^{2}$ if $0\leq\hat{r}\leq 1$ and is countable if $\hat{r}>1$.

Key words and phrases beta-transformation; Hausdorff dimension; uniform recurrence property

1 Introduction

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving dynamical system with a finite Borel measure $\mu$. Let $d$ be a metric on $X$. The well-known Poincaré Recurrence Theorem shows that typically the orbit of a point asymptotically approaches to the point itself. More precisely,

[TABLE]

for $\mu$-almost all $x\in X$. Boshernitzan [3] described the speed of such asymptotic recurrence. In fact, he proved that if there is some $\alpha>0$ such that the $\alpha$-dimensional Hausdorff measure $\mathcal{H}^{\alpha}$ is $\sigma$-finite on $X$ (i.e., $X$ can be written as a countable union of subsets $X_{i}$ with $\mathcal{H}^{\alpha}(X_{i})<\infty$ for all $i=1,2,\ldots$), then

[TABLE]

for $\mu$-almost all $x\in X$, and that if $\mathcal{H}^{\alpha}(X)=0$, then

[TABLE]

for $\mu$-almost all $x\in X$. There are also many other studies on the asymptotic behavior of the orbits motivated by Poincaré Recurrence Theorem including the first return time [2], dynamical Borel-Cantelli Lemma [8], waiting time [11], shrinking target problems [12, 13, 19] and so on.

Different to the asymptotic way of approximation, the famous Dirichlet Theorem provides another point of view of the study on the approximation of the orbits: a uniform way. The Dirichlet Theorem states that for any positive irrational real number $\theta$, for all real number $N\geq 1$, there is an integer $n$ with $1\leq n\leq N$ satisfying

[TABLE]

where $\|\cdot\|$ denotes the distance to the nearest integer. The uniformity lies in that (1.1) has an integer solution for any sufficiently large $N$. Note that $\|n\theta\|=\|T_{\theta}^{n}x-x\|$ where $T_{\theta}:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{R}/\mathbb{Z}$ is defined by $T_{\theta}x=x+\theta$. Thus, the Dirichlet Theorem can be explained as that under the dynamical system $(\mathbb{R}/\mathbb{Z},T_{\theta})$, all points $x$ uniformly return to the point itself with the speed $\frac{1}{N}$. Motivated by the Dirichlet Theorem, some results of the uniform approximation properties have already appeared in [4, 5, 6, 14, 15].

In our paper, we want to investigate the uniform recurrence property of a point to itself IN the beta-dynamical system. We aims at giving the sizes (Lebesgue measure and Huasdorff dimension) of the sets of points with prescribed uniform recurrence rate.

For any real number $\beta>1$, the $\beta$-transformation $T_{\beta}$ on $[0,1)$ is defined by

[TABLE]

We consider the following two exponents of recurrence, one is for asymptotic recurrence, and the other is for uniform recurrence.

Definition 1.1

Let $\beta>1$. For all $x\in[0,1)$, define

[TABLE]

and

[TABLE]

The exponents $r_{\beta}(x)$ and $\hat{r}_{\beta}(x)$ are analogous to the exponents introduced in [1], see also [4, 5]. By the definitions of $r_{\beta}(x)$ and $\hat{r}_{\beta}(x)$, it can be checked that $\hat{r}_{\beta}(x)\leq r_{\beta}(x)$ for all $x\in[0,1)$. Actually, applying Philipp’s result [17], we can deduce that the set $\{x:r_{\beta}(x)=0\}$ is of full Lebesgue measure (see Section 3.1). The asymptotic exponent $r_{\beta}(x)$ has been studied by Tan and Wang [20] who showed that for all $0\leq r\leq+\infty$,

[TABLE]

where $\dim_{\rm H}$ denotes the Hausdorff dimension. We refer the readers to Falconer [9] for more properties of Hausdorrf dimension. Our main result is as follows.

Theorem 1.1

Let $\beta>1$. The set $\{x\in[0,1):\hat{r}_{\beta}(x)=0\}$ is of full Lebesgue measure. When $\hat{r}>1$, the set $\{x\in[0,1):\hat{r}_{\beta}(x)\geq\hat{r}\}$ is countable. When $0\leq\hat{r}\leq 1$, we have

[TABLE]

Actually, Theorem 1.1 follows from the following more general result which gives the Hausdorff dimension of the set of points whose exponents $r_{\beta}(x)$ and $\hat{r}_{\beta}(x)$ are both prescribed. For all $0\leq r\leq+\infty,\ 0\leq\hat{r}\leq+\infty$, let

[TABLE]

Theorem 1.2

Let $\beta>1$. The set $R_{\beta}(0,0)$ is of full Lebesgue measure. When $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$, the set $R_{\beta}(\hat{r},r)$ is countable. When $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r\leq+\infty$, we have

[TABLE]

By Theorem 1.2, the following new result related to the asymptotic exponent $r_{\beta}(x)$ is immediate.

Corollary 1.3

Let $\beta>1$. For all $0\leq r\leq+\infty$, we have

[TABLE]

Our paper is organized as follows. Section 2 contains a brief summary of some classical results on the $\beta$-transformation without proofs. Section 3 is devoted to the proof of Theorem 1.2. The proof of Theorem 1.1 will be given in the last section.

2 $\beta$-transformation

In this section, we will provide some notations and properties on $\beta$-transformation. For more information on $\beta$-transformation, see [7, 10, 16, 18] and the references therein.

The $\beta$-transformation is first introduced by Rényi [18]. By the iteration of $T_{\beta}$ defined by (1.2), every real number $x\in[0,1)$ can be uniquely expanded as:

[TABLE]

where $\varepsilon_{n}(x,\beta)=\lfloor\beta T_{\beta}^{n-1}(x)\rfloor$ is the integer part of $\beta T_{\beta}^{n-1}(x)$ for all $n\in\mathbb{N}$. The integer $\varepsilon_{n}(x,\beta)$ is called the $n$-th digit of $x$. We call the sequence $\varepsilon(x,\beta):=(\varepsilon_{1}(x,\beta),\ldots,\varepsilon_{n}(x,\beta),\ldots)$ the $\beta$-expansion of $x$.

We can see that every $n$-th digit $\varepsilon_{n}(x,\beta)$ belongs to $\mathcal{A}:=\{0,1,\cdots,\lceil\beta\rceil-1\}$ where $\lceil x\rceil$ means the smallest integer larger than $x$. A word $(\varepsilon_{1},\ldots,\varepsilon_{n})\in\mathcal{A}^{n}$ is said to be $\beta$-admissible if there is a real number $x\in[0,1)$ such that $(\varepsilon_{1}(x,\beta),\ldots,\varepsilon_{n}(x,\beta))=(\varepsilon_{1},\ldots,\varepsilon_{n})$. Similarly, an infinite sequence $(\varepsilon_{1},\ldots,\varepsilon_{n},\ldots)$ is called $\beta$-admissible if there exists an $x\in[0,1)$ such that $\varepsilon(x,\beta)=(\varepsilon_{1},\ldots,\varepsilon_{n},\ldots)$. Denote by $\Sigma_{\beta}^{n}$ the family of all $\beta$-admissible words of length $n$, that is,

[TABLE]

Write $\Sigma_{\beta}^{\ast}=\bigcup\limits_{n=1}^{\infty}\Sigma_{\beta}^{n}$ the set of all $\beta$-admissible words with finite length. Denote by $\Sigma_{\beta}$ the set of all $\beta$-admissible sequences, that is,

[TABLE]

The lexicographical order $<_{\rm{lex}}$ in the space $\mathcal{A}^{\mathbb{N}}$ is defined as follows:

[TABLE]

if $\omega_{1}<\omega^{\prime}_{1}$ or there exists $j>1$, such that for all $1\leq k\leq j-1$, we have $\omega_{k}=\omega^{\prime}_{k}$ but $\omega_{j}<\omega^{\prime}_{j}$. The symbol $\leq_{\rm{lex}}$ stands for $=$ or $<_{\rm{lex}}$.

We now extend the definition of the $\beta$-transformation to $x=1$. Let $T_{\beta}(1)=\beta-\lfloor\beta\rfloor$. We have

[TABLE]

where $\varepsilon_{n}(1,\beta)=\lfloor\beta T_{\beta}^{n}(1)\rfloor.$ Specially, if the $\beta$-expansion of $1$ is finite, that is, there is an integer $m\geq 1$ such that $\varepsilon_{m}(1,\beta)>0$ and $\varepsilon_{k}(1,\beta)=0$ for all $k>m$, $\beta$ is called a simple Parry number. In this case, set

[TABLE]

where $\omega^{\infty}=(\omega,\omega,\ldots)$. If the $\beta$-expansion of $1$ is not finite, set $\varepsilon^{\ast}(\beta)=\varepsilon(1,\beta)$. In both cases, we have

[TABLE]

The sequence $\varepsilon^{\ast}(\beta)$ is consequently called the infinite $\beta$-expansion of $1$. For any $N$ with $\varepsilon_{N}^{\ast}(\beta)>0$, let $\beta_{N}>1$ be the unique solution of the equation

[TABLE]

Immediately, the infinite $\beta_{N}$-expansion of $1$ is

[TABLE]

We have $0<\beta_{N}<\beta$ and $\beta_{N}\rightarrow\beta$ as $N\rightarrow+\infty$. The real number $\beta_{N}$ is therefore called an approximation of $\beta$.

The following theorem is a characterization of the $\beta$-admissible words established by Parry [16]. It indicates that the $\beta$-dynamical system is totally determined by the infinite $\beta$-expansion of $1$.

Theorem 2.1** (Parry [16])**

Let $\beta>1$.

(1) For any $n\in\mathbb{N}$, $\omega=(\omega_{1},\ldots,\omega_{n})\in\Sigma_{\beta}^{n}$, if and only if,

[TABLE]

(2) If $1<\beta_{1}<\beta_{2}$, then $\varepsilon^{\ast}(\beta_{1})<_{\rm{lex}}\varepsilon^{\ast}(\beta_{2})$. For every $n\geq 1$, it holds that

[TABLE]

The following theorem due to Rényi [18] shows that the dynamical system $([0,1),T_{\beta})$ has topological entropy $\log_{\beta}$. Here and subsequently, we denote by $\sharp$ the cardinality of a finite set.

Theorem 2.2** (Rényi [18])**

For any $n\geq 1$, we have

[TABLE]

For any $\beta$-admissible word $\omega=(\omega_{1},\ldots,\omega_{n})$, let

[TABLE]

The set $I_{n}(\omega)$ is called the cylinder of order $n$ associated to the $\beta$-admissible word $\omega$. It can be checked that the cylinder $I_{n}(\omega)$ is a left-closed and right-open interval (see [10]). Denote by $|I_{n}(\omega)|$ the length of $I_{n}(\omega)$. Then $|I_{n}(\omega)|\leq\beta^{-n}$. Let $I_{n}(x,\beta)$ be the cylinder of order $n$ which contains $x\in[0,1)$. To shorten notation, we write $I_{n}(x)$ instead of $I_{n}(x,\beta)$ and denote by $|I_{n}(x)|$ its length. The cylinder of order $n$ is called full if $|I_{n}(\omega)|=\beta^{-n}$. The corresponding word of the full cylinder is also said to be full.

The full word plays an important role in constructing a Cantor set for the aim of estimating the lower bound of $\dim_{\rm H}R_{\beta}(\hat{r},r)$. A characterization of full words was given by Fan and Wang [10] as follows.

Theorem 2.3** (Fan and Wang [10])**

For any $n\in\mathbb{N}$, the word $\omega=(\omega_{1},\ldots,\omega_{n})$ is full if and only if for all $m\in\mathbb{N}$ and $\omega^{\prime}=(\omega^{\prime}_{1},\ldots,\omega^{\prime}_{m})\in\Sigma_{\beta}^{m}$, the concatenation $\omega\ast\omega^{\prime}=(\omega_{1},\ldots,\omega_{n},\omega^{\prime}_{1},\ldots,\omega^{\prime}_{m})$ is still $\beta$-admissible. Moreover, if $(\omega_{1},\ldots,\omega_{n-1},\omega^{\prime}_{n})$ with $\omega^{\prime}_{n}>0$ is $\beta$-admissible, then $(\omega_{1},\ldots,\omega_{n-1},\omega_{n})$ is full for any $0\leq\omega_{n}<\omega^{\prime}_{n}$.

It follows from Theorem 2.1(2) that $\Sigma_{\beta_{N}}^{n}\subseteq\Sigma_{\beta}^{n}$ for all $n\geq 1$. For any $\omega\in\Sigma_{\beta_{N}}^{\ast}$, by Theorem 2.3, the word $(\omega,0^{N})$ is full when $\omega$ is regarded as an element of $\Sigma_{\beta}^{n}$. As a result,

[TABLE]

Furthermore, the following theorem due to Bugeaud and Wang [7] implies that the full cylinders are well distributed in the unit interval $[0,1)$.

Theorem 2.4** (Bugeaud and Wang [7])**

There is at least one full cylinder for all $n+1$ consecutive cylinders of order $n$.

We end this section by giving the modified mass distribution principle shown by Bugeaud and Wang [7] which is an important tool to estimate the lower bound of the Hausdorff dimension in $\beta$-dynamical system. For convenience, for all $x\in[0,1)$, we write $I_{n}(x)$ as $I_{n}$ without any ambiguity.

Theorem 2.5** (Bugeaud and Wang [7])**

Let $\mu$ be a Borel measure supported on $E$. If there exists a constant $c>0$ such that for all integer $n$ large enough , the inequality $\mu(I_{n})\leq c|I_{n}|^{s}$ holds for all cylinders $I_{n}$. Then $\dim_{\rm H}E\geq s.$

3 Proof of Theorems 1.2

Before our proof, we will give some useful lemmas.

Lemma 3.1

For any $x\in[0,1)$ whose $\beta$-expansion is not periodic and $r_{\beta}(x)>0$, there exist two sequences $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ such that

[TABLE]

and

[TABLE]

Proof. Suppose $\varepsilon(x,\beta)=(\varepsilon_{1},\varepsilon_{2},\ldots)$. Let

[TABLE]

Suppose that for all $k\geq 1$, $n_{k}$ and $m_{k}$ have been defined. Set

[TABLE]

Note that $r_{\beta}(x)>0$. We always can find the position $n_{k}$ such that $\varepsilon_{n_{k}+1}$ returns to $\varepsilon_{1}$ which means $n_{k}$ is well defined. Since $\beta^{-n}$ is decreasing to [math] as $n$ goes to infinity and $\varepsilon(x,\beta)$ is not periodic, $m_{k}$ is well defined. By the definitions of $n_{k}$ and $m_{k}$, for all $k\geq 1$, we have

[TABLE]

Now we choose two subsequences $\{n_{i_{k}}\}_{k=1}^{\infty}$ and $\{m_{i_{k}}\}_{k=1}^{\infty}$ of $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ such that $\{m_{i_{k}}-n_{i_{k}}\}_{k=1}^{\infty}$ is not decreasing. Let $i_{1}=1$. Assume that $i_{k}$ has been defined. Let

[TABLE]

Since $r_{\beta}(x)>0$, it follows that $m_{k}-n_{k}$ goes to infinity as $k\rightarrow+\infty$. So $i_{k+1}$ is well defined. Then we have the sequence $\{m_{i_{k}}-n_{i_{k}}\}_{k=1}^{\infty}$ is not decreasing. Without causing any confusion, we still use the same symbols $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ to substitute the subsequences $\{n_{i_{k}}\}_{k=1}^{\infty}$ and $\{m_{i_{k}}\}_{k=1}^{\infty}$. We claim that

[TABLE]

In fact, assume

[TABLE]

On the one hand, there is a subsequence $\{j_{k}\}_{k=1}^{\infty}$ such that

[TABLE]

This implies that for all $\delta>0$, there is an integer $k_{0}$, for each $k\geq k_{0}$, $m_{j_{k}}-n_{j_{k}}\geq(c-\delta)n_{j_{k}}$. So

[TABLE]

Consequently, $r_{\beta}(x)\geq c-\delta$ for all $\delta\geq 0$. On the other hand, there is an integer $k_{0}$, for any $k\geq k_{0}$, we have $m_{k}-n_{k}\leq(c+\delta)n_{k}$. So for all $n\geq n_{k_{0}}$, there is an integer $k$ such that $n_{k}\leq n<n_{k+1}$. This means

[TABLE]

Hence, $r_{\beta}(x)<c+\delta$ for any $\delta>0$. Immediately, $r_{\beta}(x)=c$. The same argument can deduce the equality $(\ref{inf})$, we leave it to the readers. $\Box$

Lemma 3.2

For all $x\in[0,1)$ whose $\beta$-expansion is not periodic and $r_{\beta}(x)>0$, let $\{m_{k}\}_{k=1}^{\infty}$ and $\{n_{k}\}_{k=1}^{\infty}$ be defined in Lemma 3.1. Then there is a sequence $\{t_{k}\}_{k=1}^{\infty}$ such that when $t_{k}\geq m_{k}$, we have

[TABLE]

When $t_{k}<m_{k}$, we have

[TABLE]

or

[TABLE]

Proof. Let

[TABLE]

That is, the position $t_{k}$ is chosen to make sure that $(\varepsilon_{n_{k}+1},\ldots,\varepsilon_{t_{k}})$ is the maximal block after position $n_{k}$ which returns to $(\varepsilon_{1},\ldots,\varepsilon_{t_{k}-n_{k}})$. Since $\varepsilon(x,\beta)$ is not periodic, $t_{k}$ is well defined. By the definition of $t_{k}$, it holds that

[TABLE]

Then $T_{\beta}^{n_{k}}x$ and $x$ belong to the interval $I_{t_{k}-n_{k}}(\varepsilon_{1},\ldots,\varepsilon_{t_{k}-n_{k}})$ which implies

[TABLE]

Thus, $t_{k}\leq m_{k}+1$.

When $t_{k}\geq m_{k}$, it follows from (3.11) that

[TABLE]

When $t_{k}<m_{k}$ and $|\varepsilon_{t_{k}+1}-\varepsilon_{t_{k}-n_{k}+1}|\geq 2,$ by (3.11), there is a full cylinder of order $t_{k}-n_{k}+1$ between $x$ and $T_{\beta}^{n_{k}}$. As a consequence,

[TABLE]

which contradicts with the definition of $m_{k}$. Hence, $|\varepsilon_{t_{k}+1}-\varepsilon_{t_{k}-n_{k}+1}|=1$.

When $\varepsilon_{t_{k}-n_{k}+1}=\varepsilon_{t_{k}+1}+1$. By the definition of $t_{k}$, we have

[TABLE]

Noting that $|T^{n_{k}}_{\beta}x-x|=x-T^{n_{k}}_{\beta}x<\beta^{-(m_{k}-n_{k})}$, we have

[TABLE]

Note that

[TABLE]

Thus,

[TABLE]

Then,

[TABLE]

When $\varepsilon_{t_{k}+1}=\varepsilon_{t_{k}-n_{k}+1}+1$, we have

[TABLE]

Noticing that $|T^{n_{k}}_{\beta}x-x|=T^{n_{k}}_{\beta}x-x<\beta^{-(m_{k}-n_{k})}$, we have

[TABLE]

This implies

[TABLE]

$\Box$

Remark 1

The two sequences $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ chosen here are exactly the same sequences in Bugeaud and Liao [5]. However, the property that $n_{k+1}>m_{k}=t_{k}>n_{k}$ which always holds in [5] fails in our case. Then uncertainty of the relationship between $m_{k}$, $t_{k}$ and $n_{k+1}$ makes the construction of $\varepsilon(x,\beta)$ of all $x\in R_{\beta}(r,\hat{r})$ be much more complicated.

Now we will investigate the relationship between $m_{k}$ and $n_{k}$ for the case $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$.

Lemma 3.3

For each $x\in R_{\beta}(\hat{r},r)$ with $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$, let $m_{k}$ and $n_{k}$ be defined in Lemma 3.1. Then there is an integer $k^{\prime}$ such that $n_{k+1}<m_{k}$ for all $k\geq k^{\prime}$.

Proof. Since $\hat{r}>\frac{r}{1+r}$, by (3.5) and (3.6), there is a positive number $\varepsilon>0$ and an integer $k^{\prime\prime}$, such that for any $k\geq k^{\prime\prime}$, we have

[TABLE]

By (3.5), we obtain

[TABLE]

Consequently, there is an integer $k^{\prime}\geq k^{\prime\prime}$, such that for all $k\geq k^{\prime}$,

[TABLE]

which implies $n_{k+1}\leq(1-\varepsilon)m_{k}.$ $\Box$

The following lemma gives the sequences we will use to construct the covering of $R_{\beta}(r,\hat{r})$ when proving the upper bound of $\dim_{\rm H}R_{\beta}(r,\hat{r})$ for the case $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r\leq+\infty$.

Lemma 3.4

For all $x\in R_{\beta}(\hat{r},r)$ with $0\leq\hat{r}\leq 1,\ \frac{\hat{r}}{1-\hat{r}}\leq r<+\infty$, there are two sequences $\{n_{k}^{\prime}\}_{k=1}^{\infty}$ and $\{m_{k}^{\prime}\}_{k=1}^{\infty}$ such that for any large enough $k$, there is a positive real number $C$ satisfying

[TABLE]

Proof. For all $x\in R_{\beta}(\hat{r},r)\left(0\leq\hat{r}\leq 1,\ \frac{\hat{r}}{1-\hat{r}}\leq r<+\infty\right)$, the $\beta$-expansion of $x$ is not periodic and $r_{\beta}(x)>0$. Let $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ be the sequences defined in Lemma 3.1.

By (3.6), for any $0<\delta_{1}<\frac{\hat{r}_{\beta}(x)}{2}$, for sufficiently large $k$, we have

[TABLE]

Note that $\hat{r}_{\beta}(x)\leq\frac{r_{\beta}(x)}{1+r_{\beta}(x)}$. It follows from (3.5) and (3.6) that

[TABLE]

Consequently,

[TABLE]

We claim that, for all $\delta>0$, there are infinitely many $k$, such that

[TABLE]

In fact, notice that $m_{k}-n_{k}>0,\ n_{k+1}m_{k}>0$ for all $k\geq 1$. When

[TABLE]

by (3.12), we have

[TABLE]

When

[TABLE]

then for all $0<\delta_{2}\leq\frac{(\hat{r}_{\beta}(x)-\delta_{1})(\hat{r}_{\beta}(x)-2\delta_{1})}{2(1+\hat{r}_{\beta}(x)-\delta_{1})},$ there are infinitely many $k\geq k_{0}$ such that

[TABLE]

where the last inequality follows from the first part of (3.12).

Therefore, by (3.12) and the choice of $\delta_{2}$,

[TABLE]

Letting $\delta=\frac{\delta_{2}}{\hat{r}_{\beta}(x)-\delta_{1}}$, we obtain the claim.

Now we choose the subsequence $\{n_{j_{k}}\}_{k=1}^{\infty}$ and $\{m_{j_{k}}\}_{k=1}^{\infty}$ of $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ satisfying (3.13). For simplicity, let $n^{\prime}_{k}=n_{j_{k}}$ and $m^{\prime}_{k}=m_{j_{k}}$. Then $\{n^{\prime}_{k}\}_{k=1}^{\infty}$ increases at least exponentially since

[TABLE]

In conclusion, the new sequences satisfy that, there is a large enough $k_{0}$ and a positive real number $C$ such that for all $k\geq k_{0}$, we have $k\leq C\log n^{\prime}_{k}.$ $\Box$

We now give an estimation of the numbers of the sum of all the lengths of the blocks which are “fixed” in the prefix of length $m_{k}$ ($m_{k}$ is defined in Lemma 3.4) of the infinite sequence $\varepsilon(x,\beta)$ where $x\in R_{\beta}(\hat{r},r)\ \left(0\leq\hat{r}\leq 1,\ \frac{\hat{r}}{1-\hat{r}}\leq r<+\infty\right)$ for all sufficiently large $k$.

Lemma 3.5

For all $x\in R_{\beta}(\hat{r},r)$ with $0\leq\hat{r}\leq 1,\ \frac{\hat{r}}{1-\hat{r}}\leq r<+\infty$, let $\{m^{\prime}_{k}\}_{k=1}^{\infty}$ and $\{n^{\prime}_{k}\}_{k=1}^{\infty}$ be the sequences defined in Lemma 3.4. Then for all large enough integer $k$, we have

[TABLE]

Proof. By Lemma 3.4, the sequences $\{n^{\prime}_{k}\}_{k=1}^{\infty}$ and $\{m^{\prime}_{k}\}_{k=1}^{\infty}$ are the subsequences of$\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$, the equalities (3.5) and (3.6) become:

[TABLE]

and

[TABLE]

By (3.15) and (3.16), for any real number $0<\epsilon<\frac{\hat{r}_{\beta}(x)}{2}$, there exits a large enough $k_{1}$, such that for all $k\geq k_{1}$,

[TABLE]

and

[TABLE]

By (3.17) and (3.18), we have

[TABLE]

Hence,

[TABLE]

Let

[TABLE]

Note that $\frac{\hat{r}_{\beta}(x)-\epsilon}{r_{\beta}(x)+\epsilon}<1$. There exists $k^{\prime}\geq k$ such that for all $\ell\geq k^{\prime}$, we have

[TABLE]

Thus,

[TABLE]

This implies

[TABLE]

By (3.19), for any small enough real number $\epsilon^{\prime}>0$, there is a sufficiently large integer $k\geq\{k^{\prime},k_{1}\}$ such that

[TABLE]

$\Box$

We will prove Theorem 1.2 by dividing into three cases: $r=\hat{r}=0$, $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$ and $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r\leq+\infty$.

3.1 Case for $r=\hat{r}=0$

Note that $\hat{r}_{\beta}(x)\leq r_{\beta}(x)$ for all $x\in[0,1)$. Then

[TABLE]

So if we prove that $r_{\beta}(x)=0$ for $\mathcal{L}$-almost all $x\in[0,1)$, we have $\hat{r}_{\beta}(x)=0$ for $\mathcal{L}$-almost all $x\in[0,1)$.

Hence, we only need to prove

[TABLE]

Since $\sum\limits_{n=1}^{\infty}\beta^{-\frac{1}{k}n}<\infty$ for all $k\geq 1$, it follows from Philipp [17](Theorem 2A) that

[TABLE]

Moreover, we have

[TABLE]

and

[TABLE]

Thus,

[TABLE]

3.2 Case for $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$

When $\varepsilon(x,\beta)$ is periodic, we have $\hat{r}=r=+\infty$ and the set with such $\varepsilon(x,\beta)$ is countable.

When $\varepsilon(x,\beta)$ is not periodic, let $m_{k}$ and $n_{k}$ be defined in Lemma 3.1. We will construct a countable set $D$ such that $\varepsilon(x,\beta)\in D$ for all $x\in R_{\beta}(\hat{r},r)$ with $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$. By Lemma 3.3, there is an integer $k^{\prime}$, such that for all $k\geq k^{\prime}$, we have $n_{k+1}<m_{k}$. Let $t_{k}$ be defined by (3.10). Suppose $t_{k}=an_{k}+p_{k}$ where $1\leq a\leq\lfloor r\rfloor+1$ and $1\leq p_{k}<n_{k}$. Then, by (3.11),

[TABLE]

where $\omega^{k}=(\underbrace{\omega,\ldots,\omega}_{k})$ for any $\omega\in\Sigma^{\ast}_{\beta},\ k\in\mathbb{N}$. Therefore, when $n_{k+1}\leq t_{k}\leq m_{k}+1$, by the definitions of $\{n_{k}\}_{k=1}^{\infty}$ and $\{t_{k}\}_{k=1}^{\infty}$, we have

[TABLE]

By comparing the equalities (3.11), (3.20) and (3.21), we can check that $n_{k+1}\neq a^{\prime}n_{k}$ for all integer $1\leq a^{\prime}\leq a$. Suppose that there are two integers $1\leq a^{\prime}\leq a$ and $1\leq j_{k}<n_{k}$ such that $n_{{k+1}}=a^{\prime}n_{k}+j_{k}$. Then

[TABLE]

When $t_{k}<n_{k+1}<m_{k}$, by (3.8), (3.9) and (3.20), it follows that

[TABLE]

or

[TABLE]

Now we will construct the countable set $D$. For all $\omega=(\omega_{1},\ldots,\omega_{n})\in\Sigma_{\beta}^{n}$, define

[TABLE]

If such $i$ does not exist, let $s_{1}(\omega)=n$. If $s_{1}(\omega)<n$, let

[TABLE]

Suppose that $s_{k}(\omega)$ and $t_{k}(\omega)$ have been defined, let

[TABLE]

If this $i$ does not exist, let $s_{k+1}(\omega)=n$. If $s_{k+1}(\omega)<n$, let

[TABLE]

Let $k(\omega)=\max\{k:1\leq s_{k}(\omega)<n\}$ and $k(\omega)=0$ if such $k$ does not exist.

For all $\omega=(\omega_{1},\ldots,\omega_{n})\in\Sigma_{\beta}^{n}$, when $k(\omega)>0$, let

[TABLE]

where $\left(\omega^{a},u,\omega_{t_{k}(\omega)+1}-1,\varepsilon_{1}^{\ast}(\beta),\ldots,\varepsilon_{j}^{\ast}(\beta)\right)=\left(\omega^{a},u,\omega_{t_{k}(\omega)+1}-1\right)$ when $j=0$, and

[TABLE]

where $\left(\omega^{a},u,\omega_{t_{k}(\omega)+1}+1,0^{j}\right)=\left(\omega^{a},u,\omega_{t_{k}(\omega)+1}+1\right)$ when $j=0$. When $k(\omega)=0$, let

[TABLE]

where $\left(\omega^{a},\omega_{1}-1,\varepsilon_{1}^{\ast}(\beta),\ldots,\varepsilon_{j}^{\ast}(\beta)\right)=\left(\omega^{a},\omega_{1}-1\right)$ when $j=0$, and

[TABLE]

where $\left(\omega^{a},\omega_{1}+1,0^{j}\right)=\left(\omega^{a},\omega_{1}+1\right)$ when $j=0$.

Now fix $\omega=(\omega_{1},\ldots,\omega_{n})\in\Sigma_{\beta}^{n}$ and let $M(\omega)=M^{\prime}(\omega)\bigcup M^{\prime\prime}(\omega)\bigcup\left\{\omega\right\}$. Set $M_{1}(\omega)=M(\omega)\bigcup\{(\omega^{2})\}$. Suppose that $M_{k}(\omega)$ has been defined. Let

[TABLE]

Then we have $\sharp M_{k}(\omega)<\infty$ for all $k\geq 1$ and we also have $M_{k}(\omega)\subset\mathcal{A}^{\ast}$. Let

[TABLE]

By the former analysis in this section, for any $x\in R_{\beta}(\hat{r},r)$ with $0\leq\frac{r}{1+r}<\hat{r}\leq+\infty$, we have $\varepsilon(x,\beta)\in D$ and the set $D$ is countable. As a consequence, the set $R_{\beta}(\hat{r},r)\left(0\leq\frac{r}{1+r}<\hat{r}\leq+\infty\right)$ is countable.

3.3 Case for $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r\leq+\infty$

Note that $\{x\in[0,1):r_{\beta}(x)=+\infty\}\subseteq\{x\in[0,1):r_{\beta}(x)\geq r\}$ for all $0\leq r\leq+\infty$. When $0\leq\hat{r}\leq\frac{r}{1+r},\ r=+\infty$, by (1.3), it holds that

[TABLE]

Consequently, letting $r\rightarrow+\infty$, we conclude that

[TABLE]

By the fact that $R_{\beta}(r,\hat{r})\subseteq\{x\in[0,1):r_{\beta}(x)=+\infty\}$, we have $\dim R_{\beta}(r,\hat{r})=0.$

When $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r<+\infty$. Our proof is divided into two parts.

3.3.1 The upper bound of $\dim_{\rm H}R_{\beta}(\hat{r},r)$

We now construct a covering of the set $R_{\beta}(\hat{r},r)$ with $0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r<+\infty$. Let $\{n^{\prime}_{k}\}_{k=1}^{\infty}$ and $\{m^{\prime}_{k}\}_{k=1}^{\infty}$ be the sequences such that

[TABLE]

Given $k\geq 1$, we collect all of the points $x$ with $\hat{r}_{\beta}(x)\geq\hat{r}$ and $r_{\beta}(x)=r$. We first calculate the possible choices of digits among the $m^{\prime}_{k}$ prefix of $(\omega_{1},\omega_{2},\ldots)$. For the “free” blocks in the prefix of length $m^{\prime}_{k}$ of the infinite sequence $\varepsilon(x,\beta)$. Write their lengths as $d_{1},\cdots,d_{k}$. It follows immediately that $d_{i}=0$ when $n^{\prime}_{i}<m^{\prime}_{i-1}$ and $d_{i}=n^{\prime}_{i}-m^{\prime}_{i-1}$ when $n^{\prime}_{i}>m^{\prime}_{i-1}$ for all $2\leq i\leq k$. Moreover, $d_{1}=n_{1}$. Let $m_{0}=0$. By (3.14) and the choice of $m_{k}$ and $n_{k}$, it follows from Lemma 3.4 that, for any sufficiently large $k\geq\max\{k_{0},k_{1},k^{\prime}\}$,

[TABLE]

Applying Lemma 3.4, we have

[TABLE]

where $n_{0}=\delta\sum\limits_{n^{\prime}_{i}<m^{\prime}_{i-1},k_{0}\leq i\leq k}(m^{\prime}_{i-1}-n^{\prime}_{i})<+\infty$. By (3.17) and Lemma 3.4, we have

[TABLE]

By (3.14), it holds that

[TABLE]

Combining (3.23), (3.24) and (3.25), we obtain

[TABLE]

where $\epsilon^{\prime\prime}$ is a small enough real number. By Theorem 2.2, we deduce that, for every blocks with length $d_{i}$, there are no more than

[TABLE]

ways of the words can be chosen. Thus, there are at most

[TABLE]

choices of the “free” blocks in total. Notice that there are at most $n_{k}$ possible choices for the first index of the $k$ blocks. This indicates that there are at most ${n^{\prime}_{k}}^{k}$ possible choices for the position of the “free” blocks. For the “fixed” block, it follows from (3.7), (3.8), (3.9) that the block $(\varepsilon_{1},\ldots,\varepsilon_{m^{\prime}_{k}-n^{\prime}_{k}})$ has at most $2\lfloor\beta\rfloor(m^{\prime}_{k}-n^{\prime}_{k}+1)\leq 2\beta(m^{\prime}_{k}-n^{\prime}_{k}+1)$ choices which means there are at most $\left(2\beta(m^{\prime}_{k}-n^{\prime}_{k}+1)\right)^{k}$ choices of the “fixed” blocks in total. By Lemma 3.4 and (3.22), for all sufficient large $k$, the set of all real number belonging to $R_{\beta}(\hat{r},r)$ is contained in a union of no more than

[TABLE]

cylinders of order $m^{\prime}_{k}$ whose length is at most

[TABLE]

where the last inequalities follows from (3.22). Denote

[TABLE]

Then for any $s>s_{0}$, we have

[TABLE]

Letting $\epsilon^{\prime\prime}\rightarrow 0$, we conclude that

[TABLE]

3.3.2 Construction of Cantor Set

We construct a Cantor subset of $R_{\beta}(\hat{r},r)\left(0\leq\hat{r}\leq\frac{r}{1+r},\ 0<r<+\infty\right)$ as follows. Fix $\delta>0$. Let $\beta_{N}$ be the approximation of $\beta$ which is defined in Section 2. Notice that $\beta_{N}\rightarrow\beta$ as $N\rightarrow\infty$, we can choose sufficiently large integer $N$ with $\varepsilon_{N}^{\ast}(\beta)>0$ and $M$ large enough such that

[TABLE]

Now we choose two sequences $\{n_{k}\}_{k=1}^{\infty}$ and $\{m_{k}\}_{k=1}^{\infty}$ such that $n_{k}<m_{k}<n_{k+1}$ with $n_{1}>2M$. The sequence $\{m_{k}-n_{k}\}_{k=1}^{\infty}$ is non-decreasing. In addition,

[TABLE]

and

[TABLE]

Actually, we can choose the following sequences.

(1) When $\hat{r}=0$, $0<r<+\infty$, let

[TABLE]

By a small adjustment, we can obtain the required sequences.

(2) When $0<\hat{r}\leq\frac{r}{1+r},\ 0<r<+\infty$, let

[TABLE]

Note that $r<\hat{r}$. Both of the sequences $\{n^{\prime}_{k}\}_{k=1}^{\infty}$ and $\{m^{\prime}_{k}\}_{k=1}^{\infty}$ increase to infinity as $k$ increases. We can adjust these sequences to make sure that they satisfy the required properties. Without any ambiguity, the statement that $\omega\in\Sigma_{\beta_{N}}^{\ast}$ is full means that $\omega$ is full when regarding it as an element of $\Sigma_{\beta}^{\ast}$.

For all $k\in\mathbb{N}$, let $m_{k}=\ell_{k}n_{k}+p_{k}$ with $0\leq p_{k}<n_{k}$ and $n_{k+1}-m_{k}=t_{k}M+q_{k}$ with $0\leq q_{k}<M$. For any integer $n>N$ and $\omega\in\Sigma_{\beta_{N}}^{n}$, define the function $a_{p_{k}}$ of $\omega$ by

[TABLE]

where $\omega|_{i}=(\omega_{1},\ldots,\omega_{i})$. Then $a_{p_{k}}(\omega)$ is full when regarding it as an element of $\Sigma_{\beta}^{\ast}$. For convenience, denote

[TABLE]

for all $1\leq i<n$ and $\omega^{n}=\omega$. Let

[TABLE]

Set

[TABLE]

Fix $v_{1}=\left((u)^{\lfloor\frac{n_{1}}{M}\rfloor},0^{n-\lfloor\frac{n_{1}}{M}\rfloor M}\right)\in\mathcal{D}_{1}$. Define

[TABLE]

Suppose that $\mathcal{D}_{k-1}$ and $\mathcal{G}_{k-1}$ are well defined. Fix $u_{k-1}\in\mathcal{G}_{k-1}$. Let

[TABLE]

Define

[TABLE]

We can see that all of the words in $\mathcal{M}$ and $\mathcal{D}_{k}$ are full when regarding them as an element of $\Sigma_{\beta}^{\ast}$. Hence, by Theorem 2.3, $\mathcal{G}_{k}$ is well defined.

Now define

[TABLE]

We claim that $E_{N}$ is a subset $R_{\beta}(\hat{r},r)$.

In fact, for any $x\in E_{N}$, suppose $\varepsilon(x,\beta)=(\varepsilon_{1},\varepsilon_{2},\ldots)$. We first prove that for every $n$ with $m_{k}\leq n<n_{k+1}$, we have

[TABLE]

where the last equality follows from the fact that $n_{1}>2M$ and the construction of $E_{N}$. In fact, if it is not true, then there is $m_{k}\leq n<n_{k+1}-2M$ such that

[TABLE]

Assume $n=m_{k}+tM+q\ (0\leq t\leq t_{k}-1)$. Then we have

[TABLE]

This implies

[TABLE]

which is a contradiction. Note that $n\geq 2M$. Suppose

[TABLE]

Without loss of generality, suppose $\varepsilon_{n+j+1}>\varepsilon_{j}+1$. Notice that both $(\varepsilon_{n+1},\ldots,\varepsilon_{n+j+1})$ and $(\varepsilon_{1},\ldots,\varepsilon_{j+1})$ belong to $\Sigma_{\beta_{N}}^{M}$. Then

[TABLE]

Hence, for any $m_{k}\leq n<n_{k+1}$, we have

[TABLE]

We now show that $r_{\beta}(x)=r.$

On the one hand, for any $\delta>0$, notice that

[TABLE]

Then there exits $k_{0}$ large enough such that for all $k\geq k_{0}$, we have $m_{k}-n_{k}+N<(r+\delta)n_{k}$. Consequently, by (2.4), for all $n\geq n_{k_{0}}$, there is $k\geq k_{0}$, such that $n_{k}\leq n<n_{k+1}$. The same argument as (3.29) gives

[TABLE]

By the definition of $r_{\beta}(x)$, it holds that $r_{\beta}(x)<r+\delta$ for all $\delta>0$. So $r_{\beta}(x)\leq r.$

On the other hand, for all $\delta>0$, there is $k_{0}$ such that for any $k\geq k_{0}$. Thus, we have $m_{k}-n_{k}\geq(r-\delta)n_{k}$. When $n=n_{k}$, we obtain

[TABLE]

As a consequence, $r_{\beta}(x)\geq r-\delta$ for any $\delta>0$, which implies $r_{\beta}(x)\geq r.$

The proof of $\hat{r}_{\beta}(x)=\hat{r}$ is similar to the argument of $r_{\beta}(x)=r.$ We leave the details to the readers.

Our Cantor set is therefore constructed.

3.3.3 The lower bound of $\dim_{\rm H}R_{\beta}(\hat{r},r)$

The rest of this section is devoted to estimating the lower bound of Hausdorff dimension of $E_{N}$ by the modified mass distribution principal.

As the classical method of giving the lower bound of $\dim_{\rm H}E_{N}$, we first define a Borel probability measure $\mu$ on $E_{N}$. Set

[TABLE]

For each $k\geq 1,$ if $u_{k+1}=\left((u_{k},v_{k+1})^{\ell_{k+1}},a_{p_{k+1}}(u_{k},v_{k+1})\right)\in\mathcal{G}_{k+1}$, let

[TABLE]

If $u_{k+1}\notin\mathcal{G}_{k+1}$ ($k\geq 1$), let $\mu\left(I_{m_{k+1}}(u_{k+1})\right)=0$. For each $n\in\mathbb{N}$ and each cylinder $I_{n}$ with $I_{n}\cap E_{N}\neq\emptyset$, let $k\geq 1$ be the integer such that $m_{k}<n\leq m_{k+1}$, and set

[TABLE]

where the sum is taken over all the basic cylinders associate to $u_{k+1}\in\mathcal{G}_{k+1}$ contained in $I_{n}$. We can see that $\mu$ satisfies the consistency property which ensures that it can be uniquely extended to a Borel probability measure on $E_{N}$.

Now we will estimate the local dimension $\liminf\limits_{n\rightarrow\infty}\frac{\log\mu(I_{n})}{\log|I_{n}|}$ for all basic cylinder $I_{n}$ with $I_{n}\cap E_{N}\neq\emptyset$. We claim that, there exists $k_{0}$, for all $k\geq k_{0}$, we have

[TABLE]

for all $k\geq k_{0}$. Combining Theorems 2.2, 2.4 and the definition of $\mathcal{M}$, by (3.26), we have

[TABLE]

Immediately, for any $k\geq 1$, the relationship between $\mathcal{M}$ and $\mathcal{D}_{k}$ gives

[TABLE]

where $c=\beta^{-M(1-\delta)}$. As a consequence,

[TABLE]

Therefore, for all $i\geq 2$,

[TABLE]

In gerneral $n\geq 1$, there is an integer $k\geq 1$ such that $m_{k}<n\leq m_{k+1}$. We distinguish two cases to estimate $\mu(I_{n})$.

Case 1. When $m_{k}<n\leq n_{k+1}$, write $n=m_{k}+tM+q$ with $0\leq t\leq t_{k}$ and $0\leq q<M$. then

[TABLE]

Case 2. When $n_{k+1}<n\leq m_{k+1}$, by the construction of $E_{N}$, it follows that

[TABLE]

By (2.4), in both two cases,

[TABLE]

Since

[TABLE]

it holds that

[TABLE]

By (3.27) and (3.28),

[TABLE]

By the Stolz-Cesàro theorem, we deduce that

[TABLE]

Hence,

[TABLE]

Let $\delta\rightarrow 0$. The modified mass distribution principle (Theorem 2.5) gives

[TABLE]

for all sufficient large enough $N$.

4 Proof of Theorem 1.1

When $\hat{r}=0$, note that

[TABLE]

Theorem 1.2 yields that the set $\{x\in[0,1):\hat{r}_{\beta}(x)=0\}$ is of full Lebesgue measure. When $\hat{r}>1$, we have

[TABLE]

for all $0\leq r\leq+\infty$. It follows from Theorem 1.2 that the set $\{x\in[0,1):\hat{r}_{\beta}(x)\geq\hat{r}\}$ is countable. When $0<\hat{r}\leq 1$, applying the same process as Section 3.3.1, we deduce that, for all $\theta>0$,

[TABLE]

Now considering the final part of (4.33) as a function of $r$, we can find that

[TABLE]

for all $r\geq 0$. Note that

[TABLE]

By the $\sigma$-stability of Hausdorff dimension (see [9]), we have

[TABLE]

Letting $n\rightarrow+\infty$, we therefore conclude that

[TABLE]

Finally, we use the maximization method of Bugeaud and Liao [5] for the estimation of the lower bound of $\dim_{\rm{H}}\{x\in[0,1):\hat{r}_{\beta}(x)=\hat{r}\}$. In fact, fix $0<\hat{r}\leq 1$, the function $r\mapsto\dim_{\rm H}R_{\beta}(\hat{r},r)$ is continuous and reaches its maximum at the unique point $r=\frac{2\hat{r}}{1-\hat{r}}$ . By calculation, we conclude that the maximum is exactly equal to $\left(\frac{1-\hat{r}}{1+\hat{r}}\right)^{2}$. Therefore,

[TABLE]

Acknowledgement The authors thank Lingmin Liao for his valuable suggestion. This work was partially supported by NSFC 11771153, Guangzhou Natural Science Foundation 2018B0303110005.