Dichotomy Law for a Modified Shrinking Target Problem in Beta Dynamical System

Abstract

Let $\phi :\mathbb{N}\to (0,1]$ be a positive function. We consider the size of the set ${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}$, where “$i.o.n$” stands for “infinitely often”, and $f:(1,\infty )\to [0,1]$ is a Lipschitz function. For any $x\in (0,1]$, it is proved that the Hausdorff measure of ${\mathbb{E}}_f\left(\phi \right)$ fulfill a dichotomy law according to $\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log\phi \left(n\right)}{n}}=-\infty$ or not, where $T_{\beta }$ is the $\beta$-transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the $\beta$ dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the $\beta$ transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function $\phi \left(n\right)$ and the structural properties of the set $E_f\left(\phi \right)$. Specifically, we show that: The Hausdorff dimension of $E_f\left(\phi \right)$ either vanishes or is positive based on the behavior of $\phi \left(n\right)$ as n approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory.

1. Introduction

The study of Diophantine properties of orbits in a dynamical system is generally understood as the analysis of the distributions of the orbits, which can be seen as a quantitative version of Birkhoff’s classic ergodic theorem. This analysis is also relevant to classical Diophantine approximation, which studies the distributions of rational numbers.

Consider a measure-preserving dynamical system $(X,B,\mu ,T)$ with a consistent metric d, where T is ergodic with respect to the measure $\mu$. Birkhoff’s ergodic theorem implies that for any $x_0\in X$,

$$\underset{n\to \infty }{lim\; inf}d(T^{n}x,x_0)=0,\mathrm{a}.\mathrm{s}.$$

However, since this result does not involve the rate of convergence, a natural problem arises: how fast does the above lim inf converge to zero? More specifically, if $\phi :\mathbb{N}\to (0,1)$ is a positive function, what are the metric properties of the set

$$E_{x_0}\left(\phi \right):=\{x\in X:d(T^{n}x,x_0)\le \phi \left(n\right)i.o.n\}$$

(1)

in terms of measure and dimension? The points in $E_{x_0}\left(\phi \right)$ can be thought of as trajectories that hit a shrinking target infinitely often, which is known as the “shrinking target problem” as described by Hill and Velani [1].

Additionally, we can connect this to the “Dichotomy Law”, which states that the behavior of the set $E_{x_0}\left(\phi \right)$ might exhibit two radically different outcomes based on the growth rate of the function $\phi$, leading to scenarios where either the set has full measure or is negligible, depending on whether

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log\phi \left(n\right)}{n}}=-\infty$$

or not.

Another type of shrinking target problem, which aims to quantitatively study Poincar’s recurrence theorem, can be formulated as

$$R\left(\phi \right):=\{x\in X:d(T^{n}x,x)\le \phi \left(n\right)i.o.n\}.$$

The set $R\left(\phi \right)$ can be interpreted as the collection of points whose orbits return to a shrinking neighborhood of the initial point infinitely often.

In this paper, we examine a modified version of the shrinking target problem. To illustrate our motivation, let us start with an example. Consider the rotation map $R_{\alpha }$: $x\to x+\alpha$ on the unit circle. The set studied in classical inhomogeneous Diophantine approximation can be expressed as

$$\begin{array}{c}\hfill \{\alpha \in {\mathbb{Q}}^c:||R_{\alpha }^{n}x-x_0||<r_n,i.o.n\}.\end{array}$$

(2)

where $\left|\right|t\left|\right|,t\in \mathbf{R}$ is the distant to its nearest integer. (For example: Let $x=x_0=0$, the continued fraction expansion of $\alpha$ is $[a_0;a_1,\cdots ,a_n,\cdots ]$, let ${\displaystyle \frac{p_n}{q_n}}=[a_0;a_1,\cdots ,a_n]$ is the approximation factor of $\alpha$, it can be inferred from the properties of continued fractions that

$$|\alpha -{\displaystyle \frac{p_n}{q_n}}|<{\displaystyle \frac{1}{q_n^2}}.$$

(3)

Let ${\left\{q_n\right\}}_{n\ge 1}$ be a subsequence of natural number, from the Equation (3), it follows that sequence $\left\{\right||q_n{\alpha \left|\right|\}}_{n\ge 1}$ converges to 0 at the rate ${\displaystyle \frac{1}{q_n}}$). Bugeaud [2], Levesley [3], Bugeaud and Chevallier [4], among others, have studied the size of this set in terms of Hausdorff measure and Hausdorff dimension. Unlike the shrinking target problem $E_{x_0}\left(\phi \right)$, which considers Diophantine properties in one given system, the set above pertains to the properties of the orbit in a family of dynamical systems. It is the set of parameters $\alpha$ such that $R_{\alpha }$ shares some common property. Building on this idea, our paper examines the same setting as above in the dynamical systems $([0,1],T_{\beta })$ of $\beta$-expansions, where $\beta$ varies in the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$ and the fixed point varies with respect to the function $f\left(\beta \right)$. For more literature on shrinking target problems in $\beta$ dynamical systems, please refer to [5,6,7,8]

Let $\beta >1$. Define $T_{\beta }$ as the $\beta$-transformation:

$$T_{\beta }\left(x\right)=\beta x-\lfloor \beta x\rfloor ,$$

where $\lfloor \xi \rfloor$ denotes the integer part of $\xi$. Rényi [9] was the first to use this kind of mapping in 1957 to represent real number in a non-integer base $\beta >1$. For any $x\in [0,1]$ and $n\ge 1$, define ${\epsilon }_n(x,\beta )=\lfloor \beta T_{\beta }^{n-1}x\rfloor$, which is called the n-th digit of x with respect to the base $\beta$. Consequently, every x in the unite interval $[0,1]$ can be uniquely represented as a finite or infinite series

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

(4)

For convenience, both the series (4) and the digit sequence ${\epsilon }_1(x,\beta ),{\epsilon }_2(x,\beta )\cdots$ are referred to the $\beta$-expansion of x. We denote the latter by $\epsilon (x,\beta )$. For any point $x\in (0,1]$, the distribution of its orbit under $\beta$-transformations may vary depending on the value of $\beta$. when $x=1$, Blanchard [2] classified the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$ according to the distribution of orbits ${\mathcal{O}}_{\beta }:=\{T_{\beta }1:n\ge 1\}$. The classification results are as follows:

• Class C₁: ${\mathcal{O}}_{\beta }$ is ultimately zero.
• Class C₂: ${\mathcal{O}}_{\beta }$ is ultimately non-zero periodic.
• Class C₃: ${\mathcal{O}}_{\beta }$ is an infinite set, but 0 is not an accumulation point of ${\mathcal{O}}_{\beta }$.
• Class C₄: 0 is an accumulation point of $\mathcal{O}$, but $\mathcal{O}$ is not dense in [0, 1].
• Class C₅: ${\mathcal{O}}_{\beta }$ is dense in $[0,1]$.

According to Schmeling’s results [10], Class $C_5$ has full Lebesgue measure. This dense property of ${\mathcal{O}}_{\beta }$ implies a hitting property for L-almost all $\beta >1$, which means that for any $x_0\in [0,1]$ and L-almost all $\beta >1$,

$$\underset{n\to \infty }{lim\; inf}|T_{\beta }^{n}1-x_0|=0.$$

Moreover, Schmeling proved that for any initial point $x\in (0,1]$, its orbit under $\beta$-transformation is dense in $[0,1]$ for L-almost all $\beta >1$ (see Proposition 13.1 in [11]). In other words, for any $x\in (0,1]$ and $x_0\in [0,1]$, we have:

$$\underset{n\to \infty }{lim\; inf}|T_{\beta }^{n}1-x_0|=0for\mathcal{L}-a.e.\beta >1.$$

For the general case, Let $\left\{x_n\right\}\subset [0,1]$ and ${\left\{{\ell }_n\right\}}_{n\ge 1}\subset [0,+\infty )$ be two sequences of real numbers. For any $x\in (0,1]$, Lü and Wu [11] showed that

$${dim}_H{E}_x(\left\{x_n\right\},\phi )=\left\{\begin{array}{cc}\hfill & 0,if\lambda \left(\phi \right)=-\infty ;\hfill \\ \hfill & 1,if\lambda \left(\phi \right)>-\infty .\hfill \end{array}\right.$$

(5)

For more about the properties and conclusions of $\beta$-expansions, please refer to [10,12,13,14]. This paper focuses on the convergence speed in (1). Let $\phi :\mathbb{N}\to (0,1]$ be a positive function. We consider the size of the set

$${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

where $f:(1,\infty )\to [0,1]$ is a Lipschitz function. We prove the following result.

Theorem 1.

Let $\phi :\mathbb{N}\to (0,1]$ be a positive function. we consider the size of the set

$${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

where $f:(1,\infty )\to [0,1]$ is a Lipschitz function. Then for any $x\in (0,1]$, we have

$${dim}_H{E}_f\left(\phi \right)=\left\{\begin{array}{cc}\hfill & 0,if\lambda \left(\phi \right)=-\infty ;\hfill \\ \hfill & 1,if\lambda \left(\phi \right)>-\infty .\hfill \end{array}\right.$$

2. Preliminaries

Now, let’s review some basic properties of $\beta$-expansion for a fixed base $\beta$.

Definition 1.

A finite or infinite sequence ${\epsilon }_1\cdots {\epsilon }_n\cdots$ is called β-admissible if there exists an $x\in [0,1]$ such that the β-expansion of x begins with ${\epsilon }_1\cdots {\epsilon }_n\cdots$.

Let ${\Sigma }_{\beta }^n$ be the set of all $\beta$-admissible sequences with lengths n and ${\Sigma }_{\beta }$ be the set of all infinite $\beta$-admissible sequences.

The lexicographical order ≺ on ${\{0,\cdots ,\lfloor \beta \rfloor \}}^{\mathbb{N}}$ is defined as follows:

$$w=({\epsilon }_1{\epsilon }_2\cdots {\epsilon }_n\cdots )\prec w^{\prime }=({\epsilon }_1^{\prime }{\epsilon }_2^{\prime }\cdots {\epsilon }_n^{\prime }\cdots ),$$

if there exists an integer $k\ge 1$ such that, for all $1\le j<k,{\epsilon }_j={\epsilon }_j^{\prime }$ and ${\epsilon }_k<{\epsilon }_k^{\prime }.$ The notation $w⪯w^{\prime }$ means that $w\prec w^{\prime }$ or $w=w^{\prime }$. This ordering can be extended to finite blocks by identifying a finite block $({\epsilon }_1\cdots {\epsilon }_n)$ with the sequence $({\epsilon }_1\cdots {\epsilon }_{n}00\cdots )$.

Proposition 1

([9,15]). Let $\beta >1$ and $x\in [0,1]$, the following hold.

(1) Let σ represent the shift operator. A word w is in ${\Sigma }_{\beta }$ if and only if for every $k\ge 0$, the condition ${\sigma }^{k}w\prec {\epsilon }^{*}(1,\beta )$ holds, where ${\epsilon }^{*}(1,\beta )$ is defined as follows:

$${\epsilon }^{*}(1,\beta )=\left\{\begin{array}{cc}\epsilon (1,\beta ),\hfill & \mathit{if}\epsilon (1,\beta )\mathit{is}\mathit{infinite};\hfill \\ {({\epsilon }_1(1,\beta ),\dots ,{\epsilon }_{n-1}(1,\beta ),({\epsilon }_n(1,\beta )-1))}^{\infty },\hfill & \mathit{if}\epsilon (1,\beta )\mathit{is}\mathit{finite}.\hfill \end{array}\right.$$

(2) For all integers $i\ge 1$, it holds that ${\sigma }^i\left({\epsilon }^{*}(1,\beta )\right)⪯{\epsilon }^{*}(1,\beta )$.

(3) For any x in the interval $(0,1]$, the function mapping β to $\epsilon (x,\beta )$ is monotonically increasing for $\beta >1$. This means that if $1<{\beta }_1<{\beta }_2$, then $\epsilon (x,{\beta }_1)\prec \epsilon (x,{\beta }_2)$ holds true.

Lemma 1

([15]). For any $\beta >1$,

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

where ♯ denotes the cardinality of a finite set.

Cylinder sets in beta-expansion. For any $({\epsilon }_1,\cdots ,{\epsilon }_n)\in {\sum }_{\beta }^n$, we denote

$$I_{n,\beta }({\epsilon }_1,\cdots ,{\epsilon }_n)=\{x\in [0,1]:{\epsilon }_j(x,\beta )={\epsilon }_j,1\le j\le n\}$$

as an n-th order cylinder (with respect to the base $\beta$), to simplify notations, we also use $I_{n,\beta }$ to denote the n-th order cylinder (with respect to the base $\beta$).

The following simple calculation is used several times in the sequel, we state it in the follows.

Proposition 2.

Let $1<{\beta }_0<{\beta }_1$ and $0\le {\epsilon }_i\le {\beta }_0$ for all $i\ge 1$. If ${\epsilon }_1\cdots {\epsilon }_n\ne 0^n$, Let $1\le n_0\le n$ be a integer such that ${\epsilon }_0\ne 0$, Then, for every $n\ge 1$, we have

$$\left({\displaystyle \frac{{\epsilon }_1}{{\beta }_0}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{{\beta }_0^n}}\right)-\left({\displaystyle \frac{{\epsilon }_1}{{\beta }_1}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{{\beta }_1^n}}\right)\ge {\displaystyle \frac{n_0{\epsilon }_{n_0}}{{\beta }_1^{n_0+1}}}({\beta }_1-{\beta }_0).$$

(6)

Cylinder sets in the parameter space. In the following, we consider the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$ instead of a fixed $\beta >1$. From now to the end of this paper, x is a fixed point in $(0,1]$.

Definition 2.

A finite or infinite sequence w of non-negative integers is called admissible if there exists a $\beta >1$ such that the β-expansion of x begins with w.

For all $n\ge 1$, we define ${\Omega }_n$ as the set of all possible prefixes of length n of the $\beta$-expansion of x in some base $\beta >1$, i.e.,

$${\Omega }_n=\{{\epsilon }_1(x,\beta )\cdots {\epsilon }_n(x,\beta ):\beta >1\}.$$

For any $n\in \mathbb{N}$ and $w\in {\Omega }_n$, we define $\underline{\beta }\left(w\right)$ as follows: if $w=0^n$, then $\underline{\beta }\left(w\right)=1$; Otherwise, we define $\underline{\beta }\left(w\right)\ge 1$ as the unique positive solution of the equation $x={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{w_i}{{\beta }^n}}$. For any $\beta >1$, since the first digit of the expansion $\epsilon (1,\beta )=\lfloor \beta \rfloor \ge 1$. We have

$$\underline{\beta }\left(w\right)=1\iff “x\in (0,1)andw=0^n”orx=1andw={10}^{n-1}.$$

We denote the lexicographically largest word in ${\Omega }_{n+k}$ with w as a prefix by $w^{\left(k\right)}$, where k is a natural number.

Lemma 2

([11]). Let $w\in {\Omega }_n$, where $n\in \mathbb{N}$. The following statements are true:

(1) If $\underline{\beta }\left(w\right)>1$, then $\epsilon (x,\underline{\beta }\left(w\right))=w{0}^{\infty }$;

(2) The sequence ${\left\{\underline{\beta }\left(w^{\left(k\right)}\right)\right\}}_{k\ge 1}$ converges. For any $n\in \mathbb{N}$ and $w\in {\Omega }_n$, we define $I\left(w\right)$ as

$$I\left(w\right)=\left\{\beta >1:{\epsilon }_1(x,\beta )\cdots {\epsilon }_n(x,\beta )=w\right\},$$

which we call a cylinder set of order n in the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$.

Lemma 3

([11]). Let $w\in {\Omega }_n$ where $n\in \mathbb{N}$.

• If $\underline{\beta }\left(w\right)>1$, the cylinder $I\left(w\right)$ is represented by the half-open interval $[\underline{\beta }\left(w\right),\overline{\beta }\left(w\right))$;
• If $\underline{\beta }\left(w\right)=1$, the cylinder $I\left(w\right)$ is represented by the half-open interval $[1,\overline{\beta }\left(w\right))$.

We define a function $f_w:(1,+\infty )\to [0,+\infty )$ for any $n\in \mathbb{N}$ and $w\in {\Omega }_n$ as follows:

$$f_w\left(\beta \right)={\beta }^n(x-\sum _{i=1}^n{\displaystyle \frac{w_i}{{\beta }^i}}),where\beta \in (1,+\infty ).$$

On the interval $I\left(w\right)$, $f_w\left(\beta \right)$ is equivalent to $T_{\beta }^{n}x$ when viewed as a function of $\beta$. Due to

$$f_w^{\prime }\left(\beta \right)={\beta }^{n-1}(nx-\sum _{i=1}^{n-1}{\displaystyle \frac{(n-i)w_i}{{\beta }^i}})\ge x{\beta }^{n-1}\ge 0,$$

$f_w$ is continuous and strictly increasing function on the interval $I\left(w\right)$. We define $J\left(w\right)$ as $f_w\left(I\left(w\right)\right)$, which is a subinterval of $[0,1)$ because $I\left(w\right)$ is an interval. By Lemma 3, we observe that:

• if $\underline{\beta }\left(w\right)>1$, then $J\left(w\right)=[0,t)$ for some $t\in (0,1]$ with $f_w\left(\overline{\beta }\left(w\right)\right)=t;$
• if $\underline{\beta }\left(w\right)=1$, then $J\left(w\right)=(x,t)$ or $(0,t)$ for some $x\in (0,1)$ or $x=1$, respectively.

For a fixed $\beta$, in accordance with the $\beta$-expansion’s full cylinders sets, we can define the same concept in the parameter space. If $J\left(w\right)=[0,1)$, we term $I\left(w\right)$ a full cylinder of order n in the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$. We define ${\Lambda }_n=\{w\in {\Omega }_n:J\left(w\right)=[0,1)\}$, which is the set of all $w\in {\Lambda }_n$ such that $I\left(w\right)$ is a full cylinder set of order n in the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$. It is important to note that $\beta \left(w\right)>1$ for all $w\in {\Lambda }_n$, as mentioned earlier.

Lemma 4

([11]). (1) For any $n\in \mathbb{N}$ and $w\in {\Omega }_n$, if $\underline{\beta }\left(w\right)>1$, then $w\in {\Lambda }_n$ if and only if $f_w\left(\overline{\beta }\left(w\right)\right)=1,$ in other words, we can express w as

$$x={\displaystyle \frac{w_1}{\overline{\beta }\left(w\right)}}+{\displaystyle \frac{w_2}{\overline{\beta }{\left(w\right)}^2}}+\cdots +{\displaystyle \frac{w_n}{\overline{\beta }{\left(w\right)}^n}}+{\displaystyle \frac{1}{\overline{\beta }{\left(w\right)}^n}}.$$

(2) For any $n\in \mathbb{N}$, if we consider every consecutive set of $n+1$ cylinders in order n within the parameter space $\{\beta \in \mathbb{R}:\beta >1\}$, there will be at least one complete cylinder present.

For more descriptions and discussions about the full cylinder set, please refer to reference [11].

3. Proof of Main Theorem

Theorem 2

(Mass transference principle [16]). Let X be an interval in $\mathbb{R}$ and ${\left\{B(y_j,r_j)\right\}}_{j\ge 1}$ be a sequence of balls in X with $B(y_j,r_j)=(y_j-r_j,y_j+r_j)$ and $r_j\to 0$ as $j\to \infty$. Let g be a dimension function such that ${\displaystyle \frac{g\left(x\right)}{x}}$ is monotonic and suppose that for any B in X,

$$L(B\cap \underset{j\to \infty }{lim\; sup}{B}_j^g)=L\left(B\right),where{B}_j^g=B(y_j,g\left(r_j\right)).$$

Then, for any ball B in X,

$${\mathcal{H}}^g(B\cap \underset{j\to \infty }{lim\; sup}{B}_j)={\mathcal{H}}^g\left(B\right).$$

Lemma 5.

For any $n\ge 1$ and $w_1\cdots w_n\in {\Omega }_n$, if $\underline{\beta }\left(w\right)>1$, then the set $I\left(w\right)$ is a half open interval $[\underline{\beta }\left(w\right),\overline{\beta }\left(w\right))$, and $\left|I\left(w\right)\right|\le {\displaystyle \frac{1}{x\overline{\beta }{\left(w\right)}^{n-1}}}$, especially, when $w\in {\Lambda }_n$, we have

$$\left|I\left(w\right)\right|\ge {\displaystyle \frac{1}{n\overline{\beta }{\left(w\right)}^{n-1}x}}$$

Proof. 

By ([11] Lemma 3), we known that the set $I\left(w\right)$ is a half open interval $[\underline{\beta }\left(w\right),\overline{\beta }\left(w\right))$ and $\left|I\left(w\right)\right|\le {\displaystyle \frac{1}{x\overline{\beta }{\left(w\right)}^{n-1}}}$. Next, when $w\in {\Lambda }_n$, we will show $\left|I\left(w\right)\right|\ge {\displaystyle \frac{1}{n\overline{\beta }{\left(w\right)}^{n-1}x}}$.

Give $n\ge 1$ and $w\in {\Omega }_n$, we consider the function $f_w\left(\beta \right)=T_{\beta }^{n}x={\beta }^{n}x-({\beta }^{n-1}{w}_1+{\beta }^{n-2}{w}_2+\cdots +w_n)$, this function is continuous and strictly increasing on $I\left(w\right)$, since $w\in {\Lambda }_n$, by Lemma 4, we have

$$f_w\left(\overline{\beta }\left(w\right)\right)={\left(\overline{\beta }\left(w\right)\right)}^{n}x-({\left(\overline{\beta }\left(w\right)\right)}^{n-1}{w}_1+{\left(\overline{\beta }\left(w\right)\right)}^{n-2}{w}_2+\cdots +w_n)=1.$$

(7)

Since $w\in {\Lambda }_n$ we have $\underline{\beta }\left(w\right)>1$. By the definition of $\underline{\beta }\left(w\right)$, we have

$$f_w\left(\underline{\beta }\left(w\right)\right)={\left(\underline{\beta }\left(w\right)\right)}^{n}x-({\left(\underline{\beta }\left(w\right)\right)}^{n-1}{w}_1+{\left(\underline{\beta }\left(w\right)\right)}^{n-2}{w}_2+\cdots +w_n)=0.$$

(8)

By (7)and (8), we have

$$f_w\left(\overline{\beta }\left(w\right)\right)-f_w\left(\underline{\beta }\left(w\right)\right)=1.$$

Since

$$f_w^{\prime }\left(\beta \right)=n{\beta }^{n-1}x-((n-1){\beta }^{n-2}{w}_1+\cdots +w_{n-1}),$$

(9)

by simple calculation, we have

$${\beta }^{n-1}x\le f_w^{\prime }\left(\beta \right)\le n{\beta }^{n-1}x.$$

Then we have

$$\overline{\beta }\left(w\right)-\underline{\beta }\left(w\right)={\displaystyle \frac{1}{f_w^{\prime }\left(\beta \right)}}\ge {\displaystyle \frac{1}{n\overline{\beta }{\left(w\right)}^{n-1}x}}.$$

□

Lemma 6.

For any $n\ge 1$ and $w=w_1\cdots w_n\in {\Lambda }_n$, then there exists a point $y^{*}$ in the closure of $I\left(w\right)$ such that

$$T_{y^{*}}^{n}x=f\left(y^{*}\right).$$

Proof. 

For any $y_0>1$ and $k\ge 1$. Let $y_k$ and $y_{k+1}$ be the unique solution of the following equations respectively

$${\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}+{\displaystyle \frac{f\left(y_{k-1}\right)}{y_k^n}}=x$$

(10)

$${\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}}+{\displaystyle \frac{f\left(y_k\right)}{y_{k+1}^n}}=x.$$

(11)

Next, we will divide two cases to proof the above Lemma.

Case 1: If $f\left(y_k\right)\le f\left(y_{k-1}\right)$,

By (10) and (11), we have $y_{k+1}\le y_k$,

$${\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}\le {\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}}$$

and

$${\displaystyle \frac{f\left(y_{k-1}\right)}{y_k^n}}\ge {\displaystyle \frac{f\left(y_k\right)}{y_{k+1}^n}},$$

by a simple calculation, we have

$$0\le y_{k+1}^{n}f\left(y_{k-1}\right)-y_k^{n}f\left(y_k\right)\le y_{k+1}^{n}f\left(y_{k-1}\right)-y_{k+1}^{n}f\left(y_k\right)=y_{k+1}^n(f\left(y_{k-1}\right)-f\left(y_k\right)).$$

(12)

Let $h\left(x\right)={\displaystyle \frac{w_1}{x}}+\cdots +{\displaystyle \frac{w_n}{x^n}},n_0:=\begin{array}{c}\hfill min\end{array}\{i:w_i\ne 0\}$, by Proposition 2, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}-({\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}})& \ge {\displaystyle \frac{n_0{w}_{n_0}}{y_k^{n_0+1}}}(y_k-y_{k+1})\hfill \end{array}$$

(13)

By (12) and (13), we have

$$\begin{array}{cc}\hfill 0\le {\displaystyle \frac{n_0{w}_{n_0}}{y_k^{n_0+1}}}(y_k-y_{k+1})\le & {\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}}-\left({\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}\right)\hfill \\ \hfill & \le {\displaystyle \frac{f\left(y_{k-1}\right)}{y_k^n}}-{\displaystyle \frac{f\left(y_k\right)}{y_{k+1}^n}}\hfill \\ \hfill & \le {\displaystyle \frac{y_{k+1}^n(f\left(y_{k-1}\right)-f\left(y_k\right))}{y_k^n{y}_{k+1}^n}}\hfill \\ \hfill & \le {\displaystyle \frac{\left|L\right(y_k-y_{k-1}\left)\right|}{y_k^n}}\hfill \end{array}$$

Therefore,

$$|y_{k+1}-y_k|\le {\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}|y_k-y_{k-1}|.$$

Case 2: If $f\left(y_k\right)>f\left(y_{k-1}\right)$,

By (10) and (11), we have $y_{k+1}>y_k$,

$${\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}>{\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}}$$

and

$${\displaystyle \frac{f\left(y_{k-1}\right)}{y_k^n}}<{\displaystyle \frac{f\left(y_k\right)}{y_{k+1}^n}},$$

by a simple calculation, we have

$$0\le y_k^{n}f\left(y_k\right)-y_{k+1}^{n}f\left(y_{k-1}\right)\le y_k^{n}f\left(y_k\right)-y_k^{n}f\left(y_{k-1}\right)=y_k^n(f\left(y_k\right)-f\left(y_{k-1}\right)).$$

(14)

Let $h\left(x\right)={\displaystyle \frac{w_1}{x}}+\cdots +{\displaystyle \frac{w_n}{x^n}},n_0:=min\{i:w_i\ne 0\}$, by Proposition 3, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}-({\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}})& \ge {\displaystyle \frac{n_0{w}_{n_0}}{y_{k+1}^{n_0+1}}}(y_{k+1}-y_k)\hfill \end{array}$$

(15)

By (14), (15), we have

$$\begin{array}{cc}\hfill 0\le {\displaystyle \frac{n_0{w}_{n_0}}{y_{k+1}^{n_0+1}}}(y_{k+1}-y_k)\le & {\displaystyle \frac{w_1}{y_k}}+\cdots +{\displaystyle \frac{w_n}{y_k^n}}-\left({\displaystyle \frac{w_1}{y_{k+1}}}+\cdots +{\displaystyle \frac{w_n}{y_{k+1}^n}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{f\left(y_k\right)}{y_{k+1}^n}}-{\displaystyle \frac{f\left(y_{k-1}\right)}{y_k^n}}\hfill \\ \hfill \le & {\displaystyle \frac{y_k^n(f\left(y_k\right)-f\left(y_{k-1}\right))}{y_k^n{y}_{k+1}^n}}\hfill \\ \hfill \le & {\displaystyle \frac{\left|L\right(y_k-y_{k-1}\left)\right|}{y_{k+1}^n}}\hfill \end{array}$$

Therefore,

$$|y_{k+1}-y_k|\le {\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}|y_k-y_{k-1}|.$$

If $n_0\ge {\displaystyle \frac{1}{2}}n$, then ${\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}\le {\displaystyle \frac{\left|L\right|{\beta }_2}{\frac{1}{2}n}}$. When n large enough, we have

$${\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}\le {\displaystyle \frac{\left|L\right|{\beta }_2}{\frac{1}{2}n}}<1.$$

If $n_0<{\displaystyle \frac{1}{2}}n$, then ${\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}\le {\displaystyle \frac{\left|L\right|}{y_{k+1}^{\frac{1}{2}n-1}}}\le {\displaystyle \frac{\left|L\right|}{{\beta }_1^{\frac{1}{2}n-1}}}$, when n large enough, we have

$${\displaystyle \frac{\left|L\right|y_k^{n_0+1}}{n_0{w}_{n_0}{y}_k^n}}\le {\displaystyle \frac{\left|L\right|}{y_{k+1}^{\frac{1}{2}n-1}}}\le {\displaystyle \frac{\left|L\right|}{{\beta }_1^{\frac{1}{2}n-1}}}<1.$$

Thus, ${\left\{y_k\right\}}_{k\ge 1}$ is a Cauchy sequence, and its limit, denoted by $y^{*}\in [0,1]$, satisfies

$${\displaystyle \frac{w_1}{y^{*}}}+\cdots +{\displaystyle \frac{w_n}{{y^{*}}^n}}+{\displaystyle \frac{f\left(y_{*}\right)}{{y^{*}}^n}}=x.$$

• When $0\le f\left(y^{*}\right)<1$, one has $x\in I\left(w\right)$, and $T_{y^{*}}^{n}x=f\left(y^{*}\right).$
• When $f\left(y^{*}\right)=1$, by Lemma 4, we have

  $$y^{*}=\overline{\beta }\left(w\right)and{T}_{y^{*}}^n=f\left(y^{*}\right)=1.$$

□

Proof of Theorem 1.

Recall that $\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log\phi \left(n\right)}{n}}$ and

$${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

Next, we divide the proof into two cases:

• When $\mathit{\lambda }\left(\mathit{\phi }\right)=-\infty$.

At first, we express the set ${\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)$ as a form reflecting its limsup nature. Recall that

$${\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2]:=\{\beta \in [{\beta }_1,{\beta }_2):|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

(16)

Let L denote the Lipschitz constant of f, i.e.,

$$\left|f\right(x)-f(y\left)\right|\le L|x-y|,forx,y\in [0,1].$$

then

$$\begin{array}{cc}\hfill {\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)=& \bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }\{\beta \in [{\beta }_1,{\beta }_2]:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)\}.\hfill \\ \hfill =& \bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }\bigcup _{{\Omega }_n({\beta }_1,{\beta }_2)}I(\omega ,\phi ).\hfill \end{array}$$

(17)

where

$${\Omega }_n({\beta }_1,{\beta }_2)=\{w\in {\Omega }_n:I\left(w\right)\cap [{\beta }_1,{\beta }_2]\ne \varnothing \}$$

and

$$I(w,\phi )=\{\beta \in I\left(w\right)\cap [{\beta }_1,{\beta }_2]:|T_{\beta }^n\left(x\right)-f\left(x\right)|<\phi \left(n\right)\}\}.$$

Let ${\Sigma }_n\left({\beta }_2\right)=\{w=w_1\cdots w_n:\exists y\in [0,1]suchthat{\epsilon }_1(y,{\beta }_2\cdots {\epsilon }_n(y,{\beta }_2))=w\}$, then, for any $w\in {\Omega }_n({\beta }_1,{\beta }_2)$, by Lemma 1, we have

$$♯{\Omega }_n({\beta }_1,{\beta }_2)\le ♯{\Sigma }_{{\beta }_2}^n\le {\displaystyle \frac{{\beta }_2^{n+1}}{{\beta }_2-1}}.$$

(18)

For any word $w\in {\Omega }_n({\beta }_1,{\beta }_2)$, by Lemma 6, there exists a ${\beta }_{*}\in I_n\left(w\right)$ such that

$$T_{{\beta }_{*}}^n\left(x\right)=f\left({\beta }_{*}\right).$$

Therefore,

$$\begin{array}{cc}\hfill |T_{\beta }^{n}x-f\left(\beta \right)|=& |T_{\beta }^{n}x-T_{{\beta }_{*}}^n\left(x\right)+f\left({\beta }_{*}\right)-f\left(\beta \right)|\hfill \\ \hfill \le & |T_{\beta }^{n}x-T_{{\beta }_{*}}^n\left(x\right)|+|f\left(\beta \right)-f\left({\beta }_{*}\right)|\hfill \\ \hfill \le & n{\beta }_2^{n-1}x|\beta -{\beta }^{*}|+L|\beta -{\beta }^{*}|\hfill \\ \hfill =& (n{\beta }_2^{n-1}x+L)|\beta -{\beta }^{*}|.\hfill \end{array}$$

(19)

where the point ${\beta }_{*}$ is only decided by f and w. Similarly, when n is large enough, we have

$$|T_{\beta }^{n}x-f\left(\beta \right)|\ge |n{\beta }_1^{n-1}x-L\left|\right|\beta -{\beta }^{*}|.$$

(20)

Note that $\{I_n({\epsilon }_1\cdots {\epsilon }_n):{\epsilon }_1\cdots {\epsilon }_n)\in {\mathcal{D}}_n\}$ is a partition of $[0,1]$, defining

$$\begin{array}{cc}\hfill E_n^{\left(1\right)}:=& \bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}\{\beta \in I\left(w\right)\cap [{\beta }_1,{\beta }_2]:|\beta -{\beta }^{*}\left(w\right)|<{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_2^{n-1}x+L}}\}\hfill \\ \hfill =& \bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}(I\left(w\right)\cap [{\beta }_1,{\beta }_2]\cap ({\beta }^{*}-{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_2^{n-1}x+L}},{\beta }^{*}+{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_2^{n-1}x+L}})),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E_n^{\left(2\right)}:=& \bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}\{x\in I\left(w\right)\cap [{\beta }_1,{\beta }_2]:|\beta -{\beta }^{*}\left(w\right)|<{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}}\}\hfill \\ \hfill =& \bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2]}(I\left(w\right)\cap [{\beta }_1,{\beta }_2]\cap ({\beta }^{*}-{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}},{\beta }^{*}+{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}})).\hfill \end{array}$$

Rewrite $E_n$ as follows:

$$E_n=\bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}\{\beta \in [{\beta }_1,{\beta }_2]:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)\}.$$

(21)

By (19) and (20), we have

$$E_n^{\left(1\right)}\subset E_n\subset E_n^{\left(2\right)}.$$

(22)

Thus, for each $N\ge 1$,

$$\bigcup _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}({\beta }^{*}-{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}},{\beta }^{*}+{\displaystyle \frac{\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}})$$

is a cover of ${\mathbb{E}}_y\left(\phi \right)$. By (18) and $\lambda \left(\phi \right)=\infty$, thus we have the result as follows:

• For the Hausdorff measure

$$\begin{array}{cc}\hfill {\mathcal{H}}^s({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2))\le & \underset{N\to \infty }{lim\; inf}\sum _{n=N}^{\infty }\sum _{w\in {\Omega }_n({\beta }_1,{\beta }_2)}{\left|I(\omega ,\phi )\right|}^s\hfill \\ \hfill \le & \underset{N\to \infty }{lim\; inf}\sum _{n=N}^{\infty }{\displaystyle \frac{{\beta }_2^{n+1}}{{\beta }_2-1}}{({\displaystyle \frac{2\phi \left(n\right)}{n{\beta }_1^{n-1}x-L}})}^s<\infty .\hfill \end{array}$$

This shows that ${dim}_H\left({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\right)\le s$, then we have ${dim}_H\left({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\right)=0$, thus

$$\begin{array}{c}\hfill {dim}_H\left({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\right)=\underset{n\in \mathbb{N}}{sup}{dim}_H{E}_f^{\phi }(1+{\displaystyle \frac{1}{n}},n)=0.\end{array}$$

• When $\mathit{\lambda }\left(\mathit{\phi }\right)>-\infty$.

For any $-\infty <\gamma <\lambda \left(\phi \right)$, choose a subsequence ${\left\{m_k\right\}}_{k\ge 1}$ of positive integers such that

$${\displaystyle \frac{\lambda +\lambda \left(\phi \right)}{2}}<{\displaystyle \frac{log\phi \left(m_k\right)}{m_k}}.$$

For any $n\ge 1$ and $w\in {\Omega }_n({\beta }_1,{\beta }_2)$, we have $\left|I\left(w\right)\right|\le {\displaystyle \frac{1}{x{\left(\overline{\beta }\left(w\right)\right)}^{n-1}}}<{\displaystyle \frac{1}{x{\beta }_1^{n-1}}}$ By (Lemma 5), therefore, we have ${\displaystyle \frac{♯{\Omega }_n({\beta }_1,{\beta }_2)}{n}}\to \infty$. By Lemma 4, for any $k\ge 1$, there exist ${\ell }_k$ full cylinders $I\left(w\left(1\right)\right),I\left(w\left(2\right)\right),\cdots I\left(w\left({\ell }_k\right)\right)$ of order $m_k$ in the parameter space satisfy

1.

$w\left(1\right)\prec w\left(2\right)\prec \cdots \prec w\left({\ell }_k\right)$;

2.

$I\left(w\left(i\right)\right)\subset [{\beta }_1,{\beta }_2]$ for all $1\le i\le l_k$;

3.

$[{\beta }_1,{\beta }_2]\subset {\bigcup }_{i=1}^{l_k}B(x_i,{\displaystyle \frac{m_k+2}{x{\beta }_1^{m_k-1}}}),\forall x_i\in I\left(w\left(i\right)\right),1\le i\le {\ell }_k$, where the notation $B(x_i,{\displaystyle \frac{m_k+2}{x{\beta }_1^{m_k-1}}})$ denotes the open ball centered at $x_i$ with radius ${\displaystyle \frac{m_k+2}{x{\beta }_1^{m_k-1}}}$.

For any $k\ge 1$ and $1\le i\le l_k$, by (22) the set $I\left(w\right(i\left)\right)$ contains an interval $J_i\subset I\left(w\left(i\right)\right)$ of the length ${\displaystyle \frac{\phi \left(m_k\right)}{2(m_k{\beta }_2^{m_k-1}x+L)}}$. We choose $y_j$ to be the center of $J_j$ and $r_j$ to be the radius of the interval $J_j$. One can check that

1.

$r_j\to 0asj\to \infty$;

2.

$[{\beta }_1,{\beta }_2]\subset \underset{j\to \infty }{lim\; sup}B(y_j,r_j^{{\displaystyle \frac{1}{1-\gamma /log{\beta }_2}}{\displaystyle \frac{log{\beta }_1}{log{\beta }_2}}})$.

Therefore, by the mass transference principle, we have

$$\begin{array}{cc}\hfill {\mathcal{H}}^s\left({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\right)\le & {\mathcal{H}}^{{\displaystyle \frac{1}{1-\gamma /log{\beta }_2}}{\displaystyle \frac{log{\beta }_1}{log{\beta }_2}}}([{\beta }_1,{\beta }_2]\cap \underset{j\to \infty }{lim\; sup}B(y_j,r_j))\hfill \\ \hfill \le & {\mathcal{H}}^{{\displaystyle \frac{1}{1-\gamma /log{\beta }_2}}{\displaystyle \frac{log{\beta }_1}{log{\beta }_2}}}\left([{\beta }_1,{\beta }_2]\right)=\infty .\hfill \end{array}$$

(23)

where the last inequality follows from the assumption that $\lambda \left(\phi \right)=-\infty$. Then we have ${dim}_H{\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\le s$. Therefore, ${dim}_H{\mathbb{E}}_f^{\phi }\ge \underset{N\to \infty }{lim\; sup}{dim}_H{\mathbb{E}}_f^{\phi }(N,N+1)=1.$ □

4. Further Discussion

Rewrite ${\mathbb{E}}_f\left(\phi \right)$ as follows:

$${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

Corollary 1.

Let $\phi :\mathbb{N}\to (0,1)$ be a positive function and $f:(1,\infty )\to [0,1]$ is a Lipschitz function. Then for any $x\in (0,1]$,

$$L\left({\mathbb{E}}_f\left(\phi \right)\right)=0,$$

where L denotes the Lebesgue measure.

Proof. 

Rewrite ${\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)$ as follows:

$${\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2):=\{\beta \in [{\beta }_1,{\beta }_2]:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}.$$

Through the same process as in the proof of Part I of Theorem 1, we have ${\mathcal{H}}^s\left({\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\right)<\infty$ for any $s>{\displaystyle \frac{log{\beta }_1}{log{\beta }_2-\lambda \left(\phi \right)}}$, which implies that

$${dim}_H{\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)\le {\displaystyle \frac{log{\beta }_1}{log{\beta }_2-\lambda \left(\phi \right)}}.$$

(24)

Let ${\beta }_1=1+e^{-n\lambda \left(\phi \right)}$ and ${\beta }_2=1+e^{-(n+1)\lambda \left(\phi \right)}$, for any $n\in \mathbb{N}$, since $\lambda \left(\phi \right)<0$, then we have

$${dim}_H{\mathbb{E}}_f^{\phi }({\beta }_1,{\beta }_2)<1$$

By (24), we have

$${dim}_H{\mathbb{E}}_f^{\phi }(1+e^{-n\lambda \left(\phi \right)},1+e^{-(n+1)\lambda \left(\phi \right)})\le 1.$$

Therefore

$$L\left({\mathbb{E}}_f^{\phi }(1+e^{-n\lambda \left(\phi \right)},1+e^{-(n+1)\lambda \left(\phi \right)})\right)=0.$$

Then, we have

$$L\left({\mathbb{E}}_f\left(\phi \right)\right)\le \sum _{n\in \mathbb{Z}}L({\mathbb{E}}_f\left(\phi \right)(1+e^{-n\lambda \left(\phi \right)}),1+e^{-(n+1)\lambda \left(\phi \right)})=0.$$

□

Diophantine Analysis

For any $x_0\in [0,1]$ and a sequence of positive integers ${\left\{{\ell }_n\right\}}_{n\ge 1}$ such that ${\ell }_n\to \infty$ as $n\to \infty$. Persson and Schmeling [17] and Li et al. [18] studied a set $E(x_0,\left\{{\ell }_n\right\})$ which is defined as follows:

$$E(x_0,\left\{{\ell }_n\right\})=\{\beta >1:|T_{\beta }^{n}1-x_0|<{\beta }^{-{\ell }_n}forinfinitelymanyn\in \mathbb{N}\}.$$

When $x_0=0$ and ${\ell }_n=\alpha n(\alpha >0)$, Persson and Schmeling [17] proved that the Hausdorff dimension of $E(x_0,\left\{{\ell }_n\right\})$ is equal to ${\displaystyle \frac{1}{1+\alpha }}.$ Li et al. [18] studied the generalized case and obtain the following result

$${dim}_{H}E(x_0,\left\{{\ell }_n\right\})={\displaystyle \frac{1}{1+\alpha }}\mathrm{with}\alpha =\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{{\ell }_n}{n}}.$$

Let ${\left\{x_n\right\}}_{n\ge 1}\subset [0,1]$ and ${\left\{{\ell }_n\right\}}_{n\ge 1}\subset [0,+\infty )$ be two sequences of real numbers. Then, for any $x\in (0,1]$, Lü and Wu [11] show that

$${dim}_H{E}_x(\left\{x_n\right\},\left\{{\ell }_n\right\})\cap [{\beta }_1,{\beta }_2]={\displaystyle \frac{1}{1+\alpha }},$$

for any $1\le {\beta }_1\le {\beta }_2$. Where $\alpha =\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{{\ell }_n}{n}}$ and

$$E_x(\left\{x_n\right\},\left\{{\ell }_n\right\})=\{\beta >1:|T_{\beta }^{n}x-x_n|\le {\beta }^{-{\ell }_n}.$$

Using the same argument as in the proof of Theorem 1, we obtain the following result:

Theorem 3.

Let ${\left\{x_n\right\}}_{n\ge 1}\subset [0,1]$ and ${\left\{{\ell }_n\right\}}_{n\ge 1}\subset [0,+\infty )$ be two sequences of real numbers, and $f:(1,\infty )\to [0,1]$ is a Lipschitz function. Then for any $x\in (0,1]$, the following holds: 

$${dim}_H\left(E_f^{{\ell }_n}\cap [{\beta }_1,{\beta }_2]\right)={\displaystyle \frac{1}{1+\alpha }}\mathrm{for}\mathrm{any}1<{\beta }_1<{\beta }_2.$$

 where $\alpha =\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{{\ell }_n}{n}}$ and $E_f^{{\ell }_n}=\{\beta >1:|T_{\beta }^{n}x-f\left(\beta \right)|<{\beta }^{-{\ell }_n}fori.o.n\}$.

$$L\left({\mathbb{E}}_f\left(\phi \right)\right)=0,$$

Theorem 1 provides profound insights into the structure of the set $E_f\left(\varphi \right)=\{\beta >1:|T_n^{\beta }\left(x\right)-f\left(\beta \right)|<\varphi \left(n\right)\mathrm{i}.\mathrm{o}.\},$ where $\varphi :\mathbb{N}\to (0,1]$ is a positive function, and $f:(1,\infty )\to [0,1]$ is a Lipschitz function. The result addresses the Hausdorff dimension of this set for any fixed $x\in (0,1]$, revealing two distinct scenarios based on the behavior of the summability of the function $\varphi \left(n\right)$.

5. Conclusions

Firstly, if $\lambda \left(\phi \right)=-\infty$, this implies that the function $\phi$ decays sufficiently rapidly as n increases, leading to negligible contributions in terms of the density of the set $E_f\left(\varphi \right)$. Consequently, the Hausdorff dimension of this set is zero, which indicates that $E_f\left(\phi \right)$ can be considered to be “small” in a measure-theoretic sense essentially, it occupies no space within the interval $(1,\infty )$. This result underscores the fact that when the deviations from the Lipschitz behavior of $f\left(\beta \right)$ are sufficiently controlled by $\phi \left(n\right)$, the set of $\beta$ values that violate this control becomes insignificant.

On the other hand, if $\lambda \left(\phi \right)>-\infty$, it indicates that $\phi \left(n\right)$ does not decay too rapidly, allowing for a denser collection of $\beta$ values where the inequality $|T_n^{\beta }\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)$ holds infinitely often. In this case, the Hausdorff dimension of the set $E_f\left(\phi \right)$ is one, suggesting that this set occupies a “larger” space within $(1,\infty )$. This finding signifies the robustness of the Lipschitz function’s behavior as there exist many $\beta$ values for which the approximations given by $T_n^{\beta }\left(x\right)$ remain close to $f\left(\beta \right)$ in a sufficiently frequent manner.