Extremely Exceptional Sets on Run-Length Function for Reals in Beta-Dynamical System

Abstract

The extremely exceptional set for the run-length function in the beta-dynamical system is investigated in this study. For any real x in $(0,1]$, the run-length function related to x that measures the maximal length of the initial digit sequence of the $\beta$-expansion of x appears consecutively among the first n digits of the $\beta$-expansion of another real number y in $(0,1]$. The extremely exceptional set consists of all real numbers y with run-length exhibiting extreme oscillatory behavior: the limit inferior of the ratio of the run-length function to the logarithm base $\beta$ of n is zero, while the limit superior of this same ratio is infinity. We prove that the Hausdorff dimension of this set is either 0 or 1, determined solely by the asymptotic scaling of the basic intervals containing x. Crucially, for all x belonging to $(0,1]$, the set is residual in $[0,1]$, which implies that its boxing dimension is 1, which generalizes some known results.

1. Introduction

Let $\beta >1$ and $\beta \in \mathbb{R}$. The β-transformation $T_{\beta }:(0,1]\to (0,1]$, first introduced by Rényi (see [1]) in 1957, is defined as

$$T_{\beta }(x)=\beta x-\lceil \beta x\rceil +1,$$

where $\lceil x\rceil$ represents the smallest integer not less than x. As established in [1], we already know that the iteration of $T_{\beta }$ yields that every real number $x\in (0,1]$ admits a unique series expansion

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

Here, for any $n\ge 1$, we have

$${ϵ}_n(x,\beta )=\lceil \beta T_{\beta }^{n-1}(x)\rceil -1.$$

We refer to ${ϵ}_n(x,\beta )$ as the n-th digit of x. The digit sequence

$$ϵ(x,\beta ):=({ϵ}_1(x,\beta ),\dots ,{ϵ}_n(x,\beta ),\dots )$$

is associated with x and is called the β-expansion of x. For convenience, we simply write this as

$$ϵ(x):=({ϵ}_1(x),\dots ,{ϵ}_n(x),\dots )$$

so there is no potential for confusion. It was demonstrated that the Lebesgue measure is equal to the invariant ergodic measure ${\nu }_{\beta }$ of the $\beta$-transformation (see [2]). The $\beta$-dynamical system $((0,1],T_{\beta },{\nu }_{\beta })$ subsequently garnered a great deal of interest; readers seeking more information may consult [1,2,3] and the references cited therein.

Fix $\beta >1$. Given $x\in (0,1]$ and $n\in \mathbb{N}$, we define the run-length function $r_x(y,n)$ as the greatest integer k, such that the first k digits of x’s $\beta$-expansion occur contiguously in the initial n digits of y’s $\beta$-expansion, that is

$$r_x(y,n)=max\{1\le j\le n:{ϵ}_{i+1}(y)={ϵ}_1(x),\cdots ,{ϵ}_{i+j}(y)={ϵ}_j(x)\mathrm{for}\mathrm{some}0\le i\le n-j\}.$$

Set $r_x(y,n)=0$ when such j does not exist. Since $ϵ(0)=(0,0,\cdots )$ for any $\beta >1$, we can see that the function $r_0(y,n)$ is exactly the maximal length of consecutive zeros in the first n digits of the $\beta$-expansion of y. The study of run-length functions in the $\beta$-dynamical systems traces back to the interesting work on binary sequences carried out by Erdö and Rényi [3]. Their law of large numbers for maximal runs established the fundamental connection between pattern recurrence and logarithmic scaling. It was proven that for Lebesgue, almost every $y\in (0,1]$,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{r_0(y,2)}{{log}_{2}n}}=1.$$

Since then, the run-length function has been further developed and generalized in a variety of ways. For instance, Ma et al. [4] considered the Hausdorff dimension of the exceptional set of points where the limit of ${\displaystyle \frac{r_0(y,2)}{{log}_{2}n}}$ does not equal 1. Subsequent extensions to $\beta$-expansions by Tong et al. [5] revealed the universality of the logarithmic law across irrational bases. Hu et al. [6] specially investigated the set relating to the run-length function in the $\beta$-expansion of 1. More generalization and development of the run-length function related to 0 or 1 can be found in [7,8,9] and so on. However, these research works only focused on considering sets of run-length functions with respect to 0 and 1. Lately, Lü and Wu [10] extended the results by replacing 0 and 1 with any real number $x\in (0,1]$. Let $x\in (0,1]$. They proved that for Lebesgue, almost all $y\in (0,1]$, and it holds that

$$\underset{n\to \infty }{lim}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}=1.$$

Additionally, they researched the Hausdorff dimension for the set of points where the limit of ${\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}$ assumes a prescribed constant value; that is, for any real number $0\le \gamma \le +\infty$, let

$$E_x\left(\gamma \right)=\left\{y\in (0,1]:\underset{n\to +\infty }{lim}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}=\gamma \right\}.$$

They proved that for each $\beta >1$ and $x\in (0,1]$, it holds that

(1) ${dim}_{\mathrm{H}}{E}_x\left(0\right)=1$;

(2) when $0<\gamma <+\infty$, for all x with $\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{-{log}_{\beta }\left|I_n\left(x\right)\right|}{n}}>1$, we have

$$E_x\left(\gamma \right)=⌀.$$

Otherwise, it follows that

$${dim}_{\mathrm{H}}{E}_x\left(\gamma \right)=1;$$

(3) For any x satisfying $\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }\left(-{log}_{\beta }\left|I_n\left(x\right)\right|\right)}{n}}=0$, we deduce that

$${dim}_{\mathrm{H}}{E}_x\left(+\infty \right)=1.$$

If $\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }\left(-{log}_{\beta }\left|I_n\left(x\right)\right|\right)}{n}}>0$, then

$${dim}_{\mathrm{H}}{E}_x\left(+\infty \right)=0.$$

Here, ${dim}_{\mathrm{H}}$ denotes the Hausdorff dimension. In the rest of this paper, ${dim}_{\mathrm{B}}$ means the boxing dimension. The basic definition and related properties of the Hausdorff and boxing dimension can be found in [11]. The investigation of exceptional sets of points violating this universal behavior emerged as a natural direction. For all $0\le a\le b\le +\infty$, Zheng and Wu [12] considered the following exceptional set

$$E_x(a,b)=\left\{y\in (0,1]:\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}=b,\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}=a\right\}$$

and generalized Lü and Wu’s result by giving the Hausdorff dimension of $E_x(a,b)$. In particular, we consider the extremely exceptional set of points with the “worst” divergence corresponding to $a=0$ and $b=+\infty$ and provide the Hausdorff dimension of this set. We employ a more effective method to construct a Cantor subset of $E_x(0,+\infty )$, which is instrumental in our investigation of its residual property.

Although certain irregular sets have zero Lebesgue measure or zero local entropy, they may simultaneously be residual, indicating that they are large in the topological view. The residual property of irregular sets is therefore an attractive topic in number theory for studying their size from a topological perspective. Typical cases include residual sets of irregular points in integer expansions [13]; Baek and Olsen’s demonstration of residual extremely non-normal points in self-similar sets [14]; and the residual property of the set of non-normal numbers in Markov partitions [15]; more examples of such sets can be found in [16,17,18]. Li and Wu [19,20] established the residual property of the set $E_0(0,+\infty )$ for the binary case. Zheng, Wu, and Li [21] subsequently extended the result by showing that the set $E_0(0,+\infty )$ is residual for any $\beta >1$. It is natural to consider the size of the set $E_x(0,+\infty )$ in the same manner. Motivated by this, we show that the set $E_x(0,+\infty )$ is residual in $[0,1]$. The residual set construction (Section 4) leverages the density of full intervals using a $G_{\delta }$-generic approach inspired by [14]. The construction of a $G_{\delta }$ subset of $E_x(0,+\infty )$ is based on the construction of the Cantor subset of $E_x(0,+\infty )$.

We emphasize that the set $E_x(0,+\infty )$ considered here is special. We prove that the Hausdorff dimension of $E_x(0,+\infty )$ differs depending on x, but for every $x\in [0,1]$, it is always residual and thus its boxing dimension is always 1 for any $x\in (0,1]$.

In summary, we characterize the extreme exceptional sets $E_x(0,+\infty )$ in two ways, as follows:

• Establishing a sharp Hausdorff dimension phase transition;
• Proving topological ubiquity, regardless of x.

We conclude this section by outlining the paper’s structure: Section 2 provides essential background on $\beta$-expansions and $\beta$-transformation without proofs. The Hausdorff dimension of the set $E_x(0,+\infty )$ is proved in Section 3, while Section 4 contains the proof of its residual property. We finally present the conclusions of this paper and outline potential future research directions in Section 5.

2. Fundamental Results of Beta Expansions

This section establishes fundamental notation and terminology, while reviewing essential properties of $\beta$-expansions. Comprehensive treatments appear in [1,2,3,22,23,24,25,26,27,28] and associated references.

A representative and widely used $\beta$-transformation is defined by

$$T_{\beta }^{*}(x):=\beta x-\lfloor \beta x\rfloor ,0\le x<1,$$

where $\lfloor x\rfloor$ denotes the greatest integer not exceeding x. We use the transformation $T_{\beta }(x)$ to guarantee that the $\beta$-expansion of any real number $x\in (0,1]$ is infinite; this means that for infinitely many $n\in \mathbb{N}$, we have ${ϵ}_n(x,\beta )>0$ due to $T_{\beta }(x)>0$. Moreover, $\beta$-expansion of all real numbers in $(0,1]$ coincides under both transformations, except when $T_{\beta }^{*}$ produces finite expansions.

The alphabet $\mathcal{A}$ is defined as $\{0,\cdots ,\lceil \beta \rceil -1\}$. Let

$${\mathcal{A}}^{*}=\bigcup _{n=1}^{\infty }{\mathcal{A}}^n.$$

It is clear from the definition of $\beta$-expansions that the n-th digit ${ϵ}_n(x,\beta )$ belongs to the set $\mathcal{A}$ for any integer $n\ge 1$. For any $n,m\in {\mathbb{N}}^{+}$, let $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_n)$ and ${\epsilon }^{\prime }=({\epsilon }_1^{\prime },\dots ,{\epsilon }_m^{\prime })$ be elements of ${\mathcal{A}}^{*}$. We define their concatenation as

$$\epsilon {\epsilon }^{\prime }=({\epsilon }_1,\dots ,{\epsilon }_n,{\epsilon }_1^{\prime },\dots ,{\epsilon }_m^{\prime }).$$

In particular, let $⌀\epsilon =\epsilon$ and $\epsilon ⌀=\epsilon$, where ⌀ denotes the empty word. Let ${\epsilon }^{\mathsf{\ell }}=(\underset{\mathsf{\ell }}{\underbrace{\epsilon ,\dots ,\epsilon }})$ for each $\mathsf{\ell }\in \mathbb{N}$, with ${\epsilon }^0=⌀.$ Additionally, for two arbitrary words $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_n),{\epsilon }^{\prime }=({\epsilon }_1^{\prime },\dots ,{\epsilon }_n^{\prime })\in {\mathcal{A}}^n$, the equality $\epsilon ={\epsilon }^{\prime }$ represents that ${\epsilon }_k={\epsilon }_k^{\prime }$ for each $1\le k\le n$.

A word $({ϵ}_1,\dots ,{ϵ}_n)$ is β-admissible if it constitutes the initial n symbols of the $\beta$-expansion for some x; that is, ${ϵ}_1(x,\beta )={ϵ}_1,\cdots ,{ϵ}_n(x,\beta )={ϵ}_n.$ Similarly, we call an infinite sequence $({ϵ}_1,\dots ,{ϵ}_n,\dots )$ admissible if there is an $x\in (0,1]$ such that ${ϵ}_k(x,\beta )={ϵ}_k$ for all $k\ge 1$, meaning that the $\beta$-expansion of x is exactly $({ϵ}_1,\dots ,{ϵ}_n,\dots )$. Denote ${\Sigma }_{\beta }^n$ as the family of all $\beta$-admissible words of length n, that is,

$${\Sigma }_{\beta }^n=\left\{(w_1,\dots ,w_n)\in {\mathcal{A}}^n:\exists x\in (0,1],\mathrm{such}\mathrm{that}{ϵ}_k(x,\beta )=w_k,\mathrm{for}\mathrm{all}1\le k\le n\right\}.$$

Define ${\Sigma }_{\beta }^n$ as the set of length-n $\beta$-admissible words, that is,

$${\Sigma }_{\beta }^{*}=\bigcup _{n=1}^{\infty }{\Sigma }_{\beta }^n.$$

Let ${\Sigma }_{\beta }$ represent the collection of all infinite $\beta$-admissible sequences; those realizable as $\beta$-expansions of some $x\in (0,1]$.

As is shown in [2,26], the $\beta$-expansion of the number 1 is fundamental for analyzing the orbit dynamics of 1 and characterizing $\beta$-admissible words. We therefore define

$$ϵ(1,\beta )=({ϵ}_1^{*},\cdots ,{ϵ}_n^{*},\cdots )$$

as the $\beta$-expansion of 1. For any integer $n\in {\mathbb{N}}^{+}$, let

$${\Gamma }_n={\Gamma }_n(\beta ):=\underset{1\le k\le n}{max}\underset{j}{max}\{j\ge 0:{ϵ}_{k+1}^{*}=\cdots ={ϵ}_{k+j}^{*}=0\}.$$

(1)

Let ${\mathcal{A}}^{\mathbb{N}}$ be the symbolic space of $\mathcal{A}$, that is,

$${\mathcal{A}}^{\mathbb{N}}=\{({\epsilon }_1,{\epsilon }_2,\dots ):{\epsilon }_j\in \mathcal{A},j\ge 0\}.$$

The space ${\mathcal{A}}^{\mathbb{N}}$ is endowed with the lexicographical order, denote ${<}_{\mathrm{lex}}$ and be defined as follows:

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

if and only if there is an integer $i\in {\mathbb{N}}^{+}$ that satisfies, for each $1\le j<i$, ${\epsilon }_j={\epsilon }_j^{\prime }$ but ${\epsilon }_i<{\epsilon }_i^{\prime }$. Here, ${\le }_{\mathrm{lex}}$ stands for = or ${<}_{\mathrm{lex}}$.

The admissibility of sequences is characterized in Parry’s theorem [2], with the $\beta$-expansion of 1 playing an essential role:

Lemma 1

(Parry [2]). Let $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots )$. For any $\beta >1$ and $n\in {\mathbb{N}}^{+}$, it follows that $({\epsilon }_1,\dots ,{\epsilon }_n)\in {\Sigma }_{\beta }^n$ if and only if ${\sigma }^j\epsilon {\le }_{\mathrm{lex}}({ϵ}_1^{*},\dots ,{ϵ}_{n-j}^{*}),\forall j\ge 1,$ where σ is the shift operator satisfying $\sigma \epsilon =({\epsilon }_2,{\epsilon }_3,\dots ).$

This lemma immediately leads to the following result:

Remark 1.

For any $\beta >1$ and admissible $\omega \in {\Sigma }_{\beta }^n$, $\omega 0^k$ is admissible for every $k\ge 1$, where $0^k$ represents the word $(\underset{k}{\underbrace{0,0,\dots ,0}})$.

Rényi [1] provided an estimate of the cardinality of the set ${\Sigma }_{\beta }^n$. In the remainder of this paper, we will denote the cardinality of a finite set using the symbol ♯.

Lemma 2

(Rényi [1]). For all $n\in {\mathbb{N}}^{+}$, it follows that

$${\beta }^n\le ♯{\Sigma }_{\beta }^n\le {\displaystyle \frac{{\beta }^{n+1}}{\beta -1}}.$$

For an admissible word $ϵ=({ϵ}_1,\dots ,{ϵ}_n)$, we refer to the n-th order basic interval relative to $\beta$ as

$$I_n(ϵ):=I_n(ϵ,\beta )=\{x\in (0,1]:{ϵ}_k(x,\beta )={ϵ}_k,\mathrm{for}\mathrm{all}1\le k\le n\}.$$

As established in [25], $I_n(ϵ)$ is fundamentally a left-open, right-closed interval with left endpoint ${\sum }_{k=1}^n{\displaystyle \frac{{ϵ}_k}{{\beta }^k}}$. Li and Wu [26] showed that $|I_n(ϵ)|\le {\beta }^{-n}$, where $|I_n(ϵ)|$ means the length of the interval $I_n(ϵ).$ For any $x\in (0,1]$, write

$$I_n(x,\beta )=I_n({ϵ}_1(x),\cdots ,{ϵ}_n(x))=\{y\in (0,1]:{ϵ}_k(y,\beta )={ϵ}_k(x,\beta ),\forall 1\le k\le n\}.$$

We adopt the abbreviation $I_n(x):=I_n(x,\beta )$ for the remainder of this work, provided no confusion arises. A basic interval $I_n(ϵ)$ is called full if it satisfies $|I_n(ϵ)|={\beta }^{-n}.$ In addition, the corresponding word of a full basic interval is said to be full.

Some properties and characterizations of full words were shown in [2,25], as follows:

Lemma 3

(Parry [2], Fan and Wang [25] and Zheng and Wu [12]). Let $\beta >1$. The following properties hold:

(1) For any $h\in {\mathbb{N}}^{+}$ and any $ϵ0^h\in {\Sigma }_{\beta }^{*}$, the word $ϵ\in {\Sigma }_{\beta }^n$.

(2) For each $h\in {\mathbb{N}}^{+}$, the word $ϵ0^h$ is full if ϵ is full.

(3) A word ϵ is full if and only if $ϵ{ϵ}^{\prime }\in {\Sigma }_{\beta }^{*}$ for all ${ϵ}^{\prime }\in {\Sigma }_{\beta }^{*}$.

(4) If $({ϵ}_1,\cdots ,{ϵ}_{n-1},{ϵ}_n^{\prime })({ϵ}_n^{\prime }>0)$ is admissible, then the word $({ϵ}_1,\cdots ,{ϵ}_{n-1},{ϵ}_n)$ is full for all integers ${ϵ}_n$ satisfying $0\le {ϵ}_n<{ϵ}_n^{\prime }$.

(5) If both words ϵ and ${ϵ}^{\prime }$ are full, then the concatenated word $ϵ{ϵ}^{\prime }$ is also full.

(6) For any $n\in {\mathbb{N}}^{+}$, the interval $I_{n+{\Gamma }_n+1}({ϵ}_1,\cdots ,{ϵ}_n,0^{{\Gamma }_n+1})$ is full.

(7) For all $ϵ\in {\sum }_{\beta }^n$ and $m\ge n$, if $|I_n(ϵ)|\ge {\beta }^{-m}$, then $ϵ0^{m-n}$ is full.

Assuming $\beta >1$. For any $x\in (0,1]$ and $n\in {\mathbb{N}}^{+}$, let

$${\mathsf{\ell }}_n(x)=min\{h\ge n:({ϵ}_1(x),\cdots ,{ϵ}_{h-1}(x),1)\in {\Sigma }_{\beta }^h\}.$$

(2)

Since the $\beta$-expansion of every $x\in (0,1]$ is infinite, it follows that $n\le {\mathsf{\ell }}_n(x)<+\infty$ for each $n\in {\mathbb{N}}^{+}$. Moreover, the function $n\to {\mathsf{\ell }}_n$ is non-decreasing. In addition, Lemma 3 (3) implies that the word $({ϵ}_1(x),\cdots ,{ϵ}_{{\mathsf{\ell }}_n(x)-1}(x),0)$ is full for all $n\in {\mathbb{N}}^{+}$. This leads to the following result provided by Lü and Wu [10]:

Lemma 4

(Lü and Wu [10]). Fix $\beta >1$. For any $x\in (0,1]$, it holds that

$$\underset{n\to \infty }{lim}{\displaystyle \frac{-{log}_{\beta }\left|I_n\left(x\right)\right|}{{\mathsf{\ell }}_n\left(x\right)}}=1$$

Bugeaud and Wang established the following distribution property for full words [22]:

Lemma 5

(Bugeaud and Wang [22]). Let $\beta >1$. For each $n\in {\mathbb{N}}^{+}$, any $n+1$ consecutive β-admissible words (in the natural ordering of their intervals) include at least one full word.

We end this section by offering the modified mass distribution principle given by Bugeaud and Wang [22], which provides a method to establish lower bounds on ${dim}_{\mathrm{H}}{E}_x(0,+\infty )$.

Lemma 6

(Bugeaud and Wang [22]). Given $\beta >1$, consider a measurable set $U\subset (0,1]$ and a Borel measure μ supported on U. Suppose there exist constants $c>0$ and $N\in {\mathbb{N}}^{+}$ such that for all $n\ge N$ and every basic interval $I_n(x)$ ($x\in (0,1]$),

$$\mu (I_n(x))\le c{\left|I_n(x)\right|}^s$$

holds. It follows that

$${dim}_{\mathrm{H}}E\ge s.$$

3. Hausdorff Dimension

As mentioned in Section 1, we modified the construction of the Cantor subset of $E_x(0,+\infty )$ in [12], retaining the main idea in [12]. We restate the following theorem and provide a brief proof sketch for constructing a Cantor subset of $E_x(0,+\infty )$.

Theorem 1.

Fix $\beta >1$. For any $x\in (0,1]$, it holds that

$${dim}_{\mathrm{H}}{E}_x(0,+\infty )=\left\{\begin{array}{ccc}\hfill 1& ,\hfill & \hfill \mathrm{if}\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }(-{log}_{\beta }|I_n(x)|)}{n}}=0;\\ \hfill 0& ,\hfill & \hfill \mathrm{if}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{{log}_{\beta }(-{log}_{\beta }|I_n(x)|)}{n}}>0.\end{array}\right.$$

Proof. 

It follows from Lemma 4 that we only need to prove that for any $x\in (0,1]$, we can obtain

$${dim}_{\mathrm{H}}{E}_x(0,+\infty )=\left\{\begin{array}{ccc}\hfill 1& ,\hfill & \hfill \mathrm{if}\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }{\mathsf{\ell }}_n(x)}{n}}=0;\\ \hfill 0& ,\hfill & \hfill \mathrm{if}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{{log}_{\beta }{\mathsf{\ell }}_n(x)}{n}}>0.\end{array}\right.$$

In fact, for the case with

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }{\mathsf{\ell }}_n(x)}{n}}>0,$$

let

$$F=\bigcup _{N=1}\bigcap _{n=N}^{\infty }\left\{y\in (0,1]:r_x(y,n)\ge ⌊{\displaystyle \frac{2{log}_{\beta }n}{ϵ}}⌋+1\right\}.$$

Then, it follows that $E_x\left(a,+\infty \right)\subset F$ and we therefore obtain

$${dim}_{\mathrm{H}}{E}_x(0,+\infty )=0$$

since it was shown in [10] (Proposition 6) that

$${dim}_{\mathrm{H}}F=0.$$

For the case with

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{log}_{\beta }{\mathsf{\ell }}_n(x)}{n}}=0.$$

(3)

For any $x\in (0,1]$, choose $N_0\in {\mathbb{N}}^{+}$ sufficiently large so that for all $N\ge N_0$, we have

$${\beta }^N\ge 2{(N+1)}^2.$$

For all $N\ge N_0$, set

$$G_N(x)=\{\omega \in {\Sigma }_{\beta }^N:\omega \ne ({\epsilon }_{k+1}(x),\cdots ,{\epsilon }_{k+N}(x)),\forall 0\le i\le N-1,\mathrm{and}\omega \mathrm{is}\mathrm{full}\}$$

(4)

Then, Lemmas 2 and 5 yield

$$♯G_N(x)\ge {\displaystyle \frac{♯{\Sigma }_{\beta }^N}{N+1}}-N-1\ge {\displaystyle \frac{{\beta }^N}{2(N+1)}}.$$

(5)

For simplicity, write $l_i(x)$ by $l_i$ for any $i\ge 0$, where $l_i(x)$ is defined in (2). Let ${\mathcal{E}}_0=\{⌀\}$, $a_0=0$ and $a_n={\sum }_{i=1}^n{\mathsf{\ell }}_i{\mathsf{\ell }}_{i+1}$. Set $n_0\ge N_0$. Assume that $n_{k-1}$ and ${\mathcal{E}}_{k-1}$ has been defined. Let

$$n_k=min\left\{n:a_n>{\beta }^{k{\mathsf{\ell }}_{n_{k-1}}},a_n\ge k{a}_{n_{k-1}}\right\}\mathrm{and}{t}_k=⌊{\displaystyle \frac{a_{n_k}-a_{n_{k-1}}}{{\mathsf{\ell }}_{n_{k-1}}}}⌋.$$

(6)

Now, define

$${\mathcal{E}}_k=\left\{u{\epsilon }_1(x)\cdots {\epsilon }_{{\mathsf{\ell }}_{n_{k-1}}-1}(x)0{\omega }_k^{(1)}\cdots {\omega }_k^{(t_k)}:u\in {\mathcal{E}}_{k-1}\right\},$$

where

$${\omega }_k^{(i)}=\left\{\begin{array}{ccc}\hfill u_k^{(i)}\in G_{{\mathsf{\ell }}_{n_{k-1}}}(x)& ,\hfill & \hfill \mathrm{when}1\le i\le t_k-1;\\ \hfill 0^{a_{n_k}-a_{n_{k-1}}-t_k{\mathsf{\ell }}_{n_{k-1}}}& ,\hfill & \hfill \mathrm{when}i=t_k.\end{array}\right.$$

It follows from Lemma 3 (2) (4) (5) that ${\mathcal{E}}_k\subset {\Sigma }_{\beta }^{a_{n_k}}$ and any word in ${\mathcal{E}}_k$ is full for each $k\in N$. Finally, let

$$E_k=\bigcup _{\omega \in {\mathcal{E}}_k}I(\omega ),E=\bigcap _{k=1}^{\infty }{E}_k.$$

We now show that $E\subset E_x(0,+\infty ).$ In fact, for any $y\in E$, the construct of E shows that when $a_{n_{k-1}}+{\mathsf{\ell }}_{n_{k-1}}\le n\le a_{n_k}$. We have

$${\mathsf{\ell }}_{n_{k-1}}-1\le r_x(y,n)\le 2{\mathsf{\ell }}_{n_{k-1}}.$$

Note that ${\left\{{\mathsf{\ell }}_n\right\}}_{n\ge 1}$ is non-decreasing. We have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}& \ge \underset{k\to \infty }{lim}{\displaystyle \frac{r_x(y,a_{n_{k-1}}+{\mathsf{\ell }}_{n_{k-1}})}{{log}_{\beta }(a_{n_{k-1}}+{\mathsf{\ell }}_{n_{k-1}})}}\hfill \\ \hfill & \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\mathsf{\ell }}_{n_{k-1}}-1}{{log}_{\beta }({\sum }_{i=1}^{n_{k-1}}{\mathsf{\ell }}_i{\mathsf{\ell }}_{i+1}+{\mathsf{\ell }}_{n_{k-1}})}}\hfill \\ \hfill & \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\mathsf{\ell }}_{n_{k-1}}-1}{{log}_{\beta }((n_{k-1}+1){\mathsf{\ell }}_{n_{k-1}+1}^2)}}\hfill \\ \hfill & =+\infty \hfill \end{array}$$

(7)

where (3) and ${\mathsf{\ell }}_n\ge n$ imply the last equality. Moreover,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}& \le \underset{k\to \infty }{lim}{\displaystyle \frac{r_x(y,a_{n_k})}{{log}_{\beta }{a}_{n_k}}}\hfill \\ \hfill & \le \underset{k\to \infty }{lim}{\displaystyle \frac{2{\mathsf{\ell }}_{n_{k-1}}}{{log}_{\beta }{\beta }^{k{\mathsf{\ell }}_{n_{k-1}}}}}\hfill \\ \hfill & =0.\hfill \end{array}$$

(8)

Combination of (7) and (8) gives $E\subset E_x(0,+\infty )$.

We now define a Borel measure $\mu$ supported on E, initialized by $\mu (I(⌀))=\mu ((0,1])=1$. For each $k\in \mathbb{N}$ and every cylinder set in ${\mathcal{E}}_{k+1}$, set

$$\mu (I_{a_{n_{k+1}}}(\omega ))={\displaystyle \frac{\mu (I_{a_{n_k}}(u))}{♯{\mathcal{E}}_{k+1}}}={\displaystyle \frac{\mu (I_{a_{n_k}}(u))}{{(♯G_{{\mathsf{\ell }}_{n_k}}(x))}^{t_{k+1}-1}}},$$

where $u\in {\mathcal{E}}_k$ is the prefix of $\omega$. For $\omega ⊈{\mathcal{E}}_{k+1}(k\ge 1)$ with $k\ge 1$, we assign $\mu (I_{a_{n_{k+1}}}(\omega ))=0$. We simplify the notation by writing $I_n$ instead of $I_n(y)$ for any $y\in (0,1]$, without causing confusion. Whenever $I_n\cap E\ne ⌀$ for a basic interval $I_n$, assign k as the unique integer fulfilling $a_{n_k}<n\le a_{n_{k+1}}$, set

$$\mu (I_n)=\sum _{I_{a_{n_{k+1}}}\subset I_n}\mu (I_{a_{n_{k+1}}}),$$

where the sum is taken over all $I_{a_{n_{k+1}}}$ related to $\omega \in {\mathcal{E}}_{k+1}$ contained in $I_n$. Standard measure-theoretic arguments establish that $\mu$ is a well-defined finite pre-measure, uniquely extendable to a Borel probability measure on $(0,1]$.

As the typical technique for establishing a lower bound on ${dim}_{\mathrm{H}}{E}_x(0,+\infty )$, ${dim}_{\mathrm{H}}{E}_x(0,+\infty )$. Our next step is giving the local dimension

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{log\mu (I_n)}{log|I_n|}}$$

for all intervals which satisfy that $I_n\bigcap E\ne ⌀$. For any $k\ge 1$, for all $\omega =({\omega }_1,\cdots ,{\omega }_{a_{n_k}})\in {\mathcal{E}}_k$, the definitions of $\mu$ and (5) imply

$$\begin{array}{c}\hfill \mu (I_{a_{n_k}}(\omega ))={\displaystyle \frac{\mu \left(I_{a_{n_{k-1}}}({\omega }_1,\cdots ,{\omega }_{a_{n_{k-1}}})\right)}{♯{\mathcal{E}}_{k+1}}}={\displaystyle \frac{1}{{\prod }_{i=1}^k{(♯G_{{\mathsf{\ell }}_{n_{i-1}}}(x))}^{t_i-1}}}\le \prod _{i=1}^k{\displaystyle \frac{2^{t_i-1}{({\mathsf{\ell }}_{n_{i-1}}+1)}^{t_i-1}}{{\beta }^{{\mathsf{\ell }}_{n_{i-1}}(t_i-1)}}}.\end{array}$$

For each $n\in {\mathbb{N}}^{+}$, let $k\in \mathbb{N}$ be the unique integer such that $a_{n_k}\le n<a_{n_{k+1}}$. To estimate $\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{log\mu (I_n)}{log|I_n|}}$, it is divided into three cases.

Case 1 When $a_{n_k}\le n\le a_{n_k}+{\mathsf{\ell }}_{n_k}$. Then,

$$\begin{array}{c}\hfill \mu (I_n)=\mu (I_{a_{n_k}})\le \prod _{i=1}^k{\displaystyle \frac{2^{t_i-1}{({\mathsf{\ell }}_{n_{i-1}}+1)}^{t_i-1}}{{\beta }^{{\mathsf{\ell }}_{n_{i-1}}(t_i-1)}}}.\end{array}$$

Note that

$$\left|I_n\right|\ge \left|I_{a_{n_k}+{\mathsf{\ell }}_{n_k}}\right|={\displaystyle \frac{1}{{\beta }^{a_{n_k}+{\mathsf{\ell }}_{n_k}}}}.$$

By (6), we deduce that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{log\mu (I_n)}{log|I_n|}}\ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\displaystyle \sum _{i=1}^k}{\mathsf{\ell }}_{n_{i-1}}(t_i-1)-{\displaystyle \sum _{i=1}^k}(t_i-1){log}_{\beta }(2({\mathsf{\ell }}_{n_{i-1}}+1))}{a_{n_k}+{\mathsf{\ell }}_{n_k}}}=1.\end{array}$$

Case 2 When $a_{n_k}+{\mathsf{\ell }}_{n_k}<n\le a_{n_k}+t_{k+1}{\mathsf{\ell }}_{n_k}$, we can rewrite n as $a_{n_k}+t{\mathsf{\ell }}_{n_k}+q$ with $1\le t\le t_{k+1}$ and $0\le q<{\mathsf{\ell }}_{n_k}$. For this case, it holds that

$$\begin{array}{c}\hfill \mu (I_n)\le \mu (I_{a_{n_k}})·{\displaystyle \frac{1}{{(♯G_{{\mathsf{\ell }}_{n_k}(x)})}^t}}\le \prod _{i=1}^k{\displaystyle \frac{2^{t_i-1}{({\mathsf{\ell }}_{n_{i-1}}+1)}^{t_i-1}}{{\beta }^{{\mathsf{\ell }}_{n_{i-1}}(t_i-1)}}}·{\displaystyle \frac{2^t{({\mathsf{\ell }}_{n_k}+1)}^t}{{\beta }^{t{\mathsf{\ell }}_{n_k}}}}.\end{array}$$

It follows from the fact that every word $\omega \in G_{{\mathsf{\ell }}_{n_k}(x)}$ is full that

$$\left|I_n\right|\ge \left|I_{a_{n_k}+(t+1){\mathsf{\ell }}_{n_k}}\right|={\displaystyle \frac{1}{{\beta }^{a_{n_k}+(t+1){\mathsf{\ell }}_{n_k}}}}.$$

This means that, for all $a_{n_k}+{\mathsf{\ell }}_{n_k}<n\le a_{n_k}+t_{k+1}{\mathsf{\ell }}_{n_k}$, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}& {\displaystyle \frac{log\mu (I_n)}{log|I_n|}}\hfill \\ \hfill & \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\displaystyle \sum _{i=1}^k}\left({\mathsf{\ell }}_{n_{i-1}}(t_i-1)+t{\mathsf{\ell }}_{n_k}-(t_i-1){log}_{\beta }(2({\mathsf{\ell }}_{n_{i-1}}+1))\right)-t{log}_{\beta }2({\mathsf{\ell }}_{n_k}+1)}{a_{n_k}+(t+1){\mathsf{\ell }}_{n_k}}}\hfill \\ \hfill & =1.\hfill \end{array}$$

Case 3 When $a_{n_k}+t_{k+1}{\mathsf{\ell }}_{n_k}<n<a_{n_{k+1}}$. The construction of E gives

$$\mu (I_n)=\mu (I_{a_{n_{k+1}}})\le \prod _{i=1}^{k+1}{\displaystyle \frac{2^{t_i-1}{({\mathsf{\ell }}_{n_i}+1)}^{t_i-1}}{{\beta }^{{\mathsf{\ell }}_{n_{i-1}}(t_i-1)}}},$$

and

$$\left|I_n\right|={\displaystyle \frac{1}{{\beta }^n}}\ge {\displaystyle \frac{1}{{\beta }^{a_{n_{k+1}}}}}.$$

We consequently obtain that for all $a_{n_k}+t_{k+1}{\mathsf{\ell }}_{n_k}<n<a_{n_{k+1}}$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{log\mu (I_n)}{log|I_n|}}& \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\displaystyle \sum _{i=1}^{k+1}}{\mathsf{\ell }}_{n_{i-1}}(t_i-1)-{\displaystyle \sum _{i=1}^{k+1}}(t_i-1){log}_{\beta }(2({\mathsf{\ell }}_{n_{i-1}}+1))}{a_{n_{k+1}}}}\hfill \\ \hfill & =1.\hfill \end{array}$$

As a result, by Lemma 6, we have ${dim}_{\mathrm{H}}E\ge 1$, that is ${dim}_{\mathrm{H}}E=1$. This yields

$${dim}_{\mathrm{H}}{E}_x(0,+\infty )=1.$$

We complete our proof. □

4. Residual Property

It is sufficient to prove this by constructing a subset V of $E_x(0,+\infty )$ that is both a $G_{\delta }$ set and dense in $[0,1]$.

For each $k\in \mathbb{N}$, let ${\Gamma }_k$ be defined as (1). For convenience, let

$$u_k=k+{\Gamma }_k+1,v_k=⌊{\beta }^{k{\mathsf{\ell }}_{u_k}}⌋,s_k=⌊{\displaystyle \frac{v_k}{{\mathsf{\ell }}_{u_k}}}⌋,\mathrm{and}{p}_k=v_k-(s_k+1){\mathsf{\ell }}_{u_k}-u_k.$$

Recall the definition of $G_n(x)$ as (4). For any $n\ge 1$, choose a fix word ${\omega }_n(x)\in G_n(x)$. Now define

$$V:=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }\bigcup _{({\epsilon }_1,\cdots ,{\epsilon }_k)\in {\Sigma }_{\beta }^k}\mathrm{int}\left(I_{v_k}({\epsilon }_1,\cdots ,{\epsilon }_k,0^{{\Gamma }_k+1},{\epsilon }_1(x),\cdots ,{\epsilon }_{{\mathsf{\ell }}_{u_k-1}}(x),0,{\omega }_{{\mathsf{\ell }}_{u_k}}^{s_k},0^{p_k}\right).$$

Remark 2.

For any $({\epsilon }_1,\cdots ,{\epsilon }_k)\in {\Sigma }_{\beta }^k$, we can obtain that $\left({\epsilon }_1,\cdots ,{\epsilon }_k,0^{{\Gamma }_k+1}\right)$ is full by Lemma 3 (6). We can also obtain that the words $\left({\epsilon }_1(x),\cdots ,{\epsilon }_{{\mathsf{\ell }}_{u_k-1}}(x),0\right)$, ${\omega }_{{\mathsf{\ell }}_{u_k}}$, and $0^{p_k}$ are full words. This consequently gives that V is well defined.

Note that the set $\mathrm{int}(I_{\left|\epsilon \right|}(\epsilon ))$ is open, V is consequently a $G_{\delta }$ set. We only need to show that $V\subset E_x(0,+\infty )$ and V is dense in $[0,1]$, as in the following two lemmas:

Lemma 7.

$V\subset E_x(0,+\infty )$.

Proof. 

For all $y\in V$, the construction of V implies the existence of infinitely many k satisfying

$$\epsilon (y,\beta )=\left({\epsilon }_1,\cdots ,{\epsilon }_k,0^{{\Gamma }_k+1},{\epsilon }_1(x),\cdots ,{\epsilon }_{{\mathsf{\ell }}_{u_k-1}}(x),0,{\omega }_{{\mathsf{\ell }}_{u_k}}^{s_k},0^{p_k}\right),$$

for some $({\epsilon }_1,\cdots ,{\epsilon }_k)\in {\Sigma }_{\beta }^k$.

By observing V, we have

$$r_x(y,k+{\Gamma }_k+{\mathsf{\ell }}_{u_k})\ge {\mathsf{\ell }}_{u_k}-1,$$

which means

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}& \ge \underset{k\to \infty }{lim}{\displaystyle \frac{r_x\left(y,k+{\Gamma }_k+{\mathsf{\ell }}_{u_k}+1\right)}{{log}_{\beta }\left(k+{\Gamma }_k+{\mathsf{\ell }}_{u_k}+1\right)}}\hfill \\ \hfill & \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\mathsf{\ell }}_{u_k}-1}{{log}_{\beta }\left(k+{\Gamma }_k+{\mathsf{\ell }}_{u_k}+1\right)}}\hfill \\ \hfill & \ge \underset{k\to \infty }{lim}{\displaystyle \frac{{\mathsf{\ell }}_{u_k}-1}{{log}_{\beta }(2{\mathsf{\ell }}_{u_k})}}=+\infty ,\hfill \end{array}$$

where the last inequality follows that ${\mathsf{\ell }}_{u_k}\ge u_k=k+{\Gamma }_k+1$.

Furthermore,

$$r_x(y,v_k-p_k)\le 3{\mathsf{\ell }}_{u_k},$$

which implies

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{r_x(y,n)}{{log}_{\beta }n}}& \le \underset{k\to \infty }{lim}{\displaystyle \frac{3{\mathsf{\ell }}_{u_k}}{{log}_{\beta }(v_k-p_k)}}\hfill \\ \hfill & \le \underset{k\to \infty }{lim}{\displaystyle \frac{3{\mathsf{\ell }}_{u_k}}{{log}_{\beta }\left(\lfloor {\beta }^{k{\mathsf{\ell }}_{u_k}}\rfloor -{\mathsf{\ell }}_{u_k}\right)}}=0.\hfill \end{array}$$

The proof is finished. □

Lemma 8.

V is dense in $[0,1]$.

Proof. 

For any $z\in [0,1]$ and $r>0$, we need to search for a real number $z^{\prime }\in V$ with $|z-z^{\prime }|\le r$. Suppose that $\epsilon (z,\beta )=({\epsilon }_1(z),{\epsilon }_2(z),\cdots )$. Let q be a sufficiently large integer such that ${\beta }^{-q}\le r$. Then, $({\epsilon }_1(z),\cdots ,{\epsilon }_q(z))\in {\Sigma }_{\beta }^q$. Now, let

$$z^{\prime }\in \mathrm{int}\left(I_{v_q}\left({\epsilon }_1,\cdots ,{\epsilon }_q,0^{{\Gamma }_q+1},{\epsilon }_1(x),\cdots ,{\epsilon }_{{\mathsf{\ell }}_{u_q-1}}(x),0,{\omega }_{{\mathsf{\ell }}_{u_q}}^{s_q},0^{p_q}\right)\right),$$

where ${\omega }_{{\mathsf{\ell }}_{u_q}}\in {\mathcal{M}}_{{\mathsf{\ell }}_{u_q}}(x)$ is chosen in the construction of V. We then have

$$|z-z^{\prime }|\le {\beta }^{-q}\le r,$$

from which it follows that the $\beta$-expansions of z and $z^{\prime }$ have the prefix $({\epsilon }_1(z),\cdots ,{\epsilon }_q(z))$. As a conclusion, the set V is dense in $[0,1]$. □

Theorem 2.

For every $\beta >1$ and $x\in (0,1]$, $E_x(0,+\infty )$ is residual in $[0,1]$

Proof. 

The basic category theorem implies that V is residual in $[0,1]$. It therefore follows from Lemmas 7 and 8 that the set $E_x(0,+\infty )$ is also residual in $[0,1]$. □

Furthermore, Theorem 2 implies that the following corollary holds:

Corollary 1.

For every $\beta >1$ and $x\in (0,1]$, it holds that

$${dim}_{\mathrm{B}}{E}_x(0,+\infty )=1.$$

Proof. 

It can be deduced from Theorem 2 that the set $E_x(0,+\infty )$ is dense in $[0,1]$, which means

$$\overline{E_x(0,+\infty )}=[0,1],$$

where $\overline{E_x(0,+\infty )}$ is the closure of $E_x(0,+\infty )$. Consequently,

$${dim}_{\mathrm{B}}{E}_x(0,+\infty )={dim}_{\mathrm{B}}\overline{E_x(0,+\infty )}=1,$$

which implies ${dim}_{\mathrm{B}}{E}_x(0,+\infty )=1$. □

5. Conclusions

We proposed and analyzed the Hausdorff dimension and the residual properties of the extremely exceptional set on the run-length function with respect to $x\in [0,1)$ in this paper. The method of constructing a Cantor subset of the extremely exceptional set can be applied to various problems relating to fractals of the run-length function. In addition, this approach to the construction of a subset of the extremely exceptional set, which is both a $G_{\delta }$ set and dense in $[0,1]$, is useful for showing the residue of an irregular set. What should be pointed out here is that the set we considered here has different Hausdorff dimension when x differs, but it is always residual, and so its boxing dimension will always be 1 for any $x\in [0,1)$. This gives an example to distinguish the Hausdorff dimension and the boxing dimension.

In our future work, we will focus on addressing the other fractal dimensions such as the Pacing, Assouad, or Minkowski dimensions of the extremely exceptional set. Inspired by [29], we also aim to generalize the extremely exceptional set by changing the function ${log}_{\beta }n$ to be a increasing function $\phi$ with $\phi (n)\to +\infty$ as $n\to +\infty$, and we will further investigate the size of the level set

$$F_x(\alpha ,\beta )=\left\{x|\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{r_x(y,n)}{\phi (n)}}=\alpha \mathrm{and}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{r_x(y,n)}{\phi (n)}}=\beta \right\}.$$