The Shrinking Target Problem for Nonautonomous Dynamical Systems Corresponding to Cantor Series Expansion over Formal Laurent Series

Abstract

In this paper, we study the shrinking target problem regarding Q-Cantor series expansions of the formal Laurent series field. We provide the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the unit disk I.

1. Introduction

The goal of the Diophantine approximation in dynamical systems is to investigate the quantitative characteristics of the orbital distribution in a dynamical system, particularly the measure and dimension of the dynamically defined limsup set’s size. The traditional Diophantine approximation and the qualitative character of the orbital density serve as inspiration for the quantitative analysis.

As the theory of Diophantine approximation in the real numbers’ field develops, research will be carried out in other fields, like the p-adic field, formal series, and so on.

Let $\mathbb{F}$ have q elements and be a finite field. Indicate the field of fractions by $\mathbb{F}\left(X\right)$ and the ring of polynomials with coefficients in $\mathbb{F}$ by $\mathbb{F}\left[X\right]$. Let $\mathbb{F}\left(\left(X^{-1}\right)\right)$ be the formal Laurent series field, i.e.,

$$\mathbb{F}\left(\left(X^{-1}\right)\right)=\left\{\sum _{n=n_0}^{+\infty }{c}_n{X}^{-n}|c_n\in \mathbb{F},and{n}_0\in \mathbb{Z}\right\},$$

put x = ${\displaystyle \sum _{n=n_0}^{+\infty }}{c}_n{X}^{-n}\in \mathbb{F}\left(\left(X^{-1}\right)\right),$ where the degree of x is denoted by $\mathrm{deg}\left(x\right)$ = −inf $\{n\in \mathbb{Z}|c_n\ne 0\},$ and the formula is $deg\left(0\right)=-\infty .$

The norm of x to the $\parallel x\parallel =q^{\mathrm{deg}\left(x\right)}$ is defined by us and $\parallel 0\parallel =0$. For each $x,y\in \mathbb{F}\left(\left(X^{-1}\right)\right),$ we must obtain the information of the following values:

• $\parallel x\parallel \ge 0;$ moreover, $\parallel x\parallel =0$ if and only if $x=0;$
• $\parallel xy\parallel =\parallel x\parallel ·\parallel y\parallel ;$
• For $\alpha ,\beta \in \mathbb{F},$ $\parallel \alpha x+\beta y\parallel \le \max (\parallel x\parallel ,\parallel y\parallel );$
• For $\alpha ,\beta \in \mathbb{F}$, $\alpha \ne 0,\beta \ne 0,$ if $\parallel x\parallel \ne \parallel y\parallel ;$ thus,

  $$\parallel \alpha x+\beta y\parallel =\max (\parallel x\parallel ,\parallel y\parallel ).$$

In other words, this norm is non-Archimedean on the field $\mathbb{F}\left(\left(X^{-1}\right)\right)$ and it generates the following metric d as

$$d(x,y)=\parallel x-y\parallel .$$

The following properties of the balls are satisfied by the metric space $(\mathbb{F}\left(\left(X^{-1}\right)\right),d).$ It is complete.

(i) A ball’s center might be considered to be any place within the ball.

(ii) When two balls come together, the one with the bigger radius has to contain the other one.

Let I = $\{x\in \mathbb{F}\left(\left(X^{-1}\right)\right)|\parallel x\parallel <1\}$ be a compact abelian group that is isomorphic to ${\displaystyle \prod _{n\ge 1}}\mathbb{F}.$ Consequently, a unique normalized Haar measure $\mu$ on I given by

$$\mu \left(B(a,q^{-r})\right)=q^{-r}$$

exists, where we have a disc of center $a\in \mathbb{F}\left(\left(X^{-1}\right)\right)$ and radius $q^{-r}$ with $r\in \mathbb{Z},$ denoted by $B(a,q^{-r})$ = $\{x\in \mathbb{F}\left(\left(X^{-1}\right)\right)|\parallel x-a\parallel <q^{-r}\}.$ Keep in mind that $\mathcal{B}\left(I\right)$ is the Borel field on I and that $(I,\mathcal{B}(I),\mu )$ is a probability space with $\mu \left(I\right)$ = 1. Each $x\in \mathbb{F}\left(\left(X^{-1}\right)\right)$ has a unique decomposition (see [1]) of the form $x=\left[x\right]+\left\{x\right\},$ where the fractional part $\left\{x\right\}$ belongs to I and the polynomial component $\left[x\right]$ of x belongs to $\mathbb{F}\left[X\right].$ $\mathbb{F}\left[X\right]$, $\mathbb{F}\left(X\right)$ and $\mathbb{F}\left(\left(X^{-1}\right)\right)$, respectively, correspond to the set of integers, rational numbers, and real numbers.

The Q-Cantor series expansion of a real number was first introduced by Cantor [2] in 1896. It is a generalization of the binary expansion by taking $q_n$ = b for all $n.$ Since then, research on the Q-Cantor series has developed rapidly. For instance, Erdős and Rényi [3,4] studied normal numbers and various statistical properties of real numbers with respect to large classes of Cantor series expansions. Ref. [5] observed uniformly distributed sequences mod 1 and Cantor’s series representation. Han and Ma [6] investigated uniform Diophantine approximation in the nonautonomous dynamic system generated by the Cantor series expansions. Readers can refer to [7,8,9,10,11,12,13] for more information on the Q-Cantor series. We define the Q-Cantor expansions of formal Laurent series in this study.

Given a series of polynomials $Q=\left\{q_i\right|i\in \mathbb{N}\},$ such that, for every $k\ge 1,$ $\parallel q_i\parallel >1,$ we define the Q-Cantor transformation $T_{Q,n}\left(x\right)$ on I for each positive integer n as follows:

$$T_{Q,n}\left(x\right)=q_{n}x-\left[q_{n}x\right],$$

and the transformation $T_Q^n\left(x\right)$ on I by

$$T_Q^n\left(x\right)=q_1{q}_2\cdots q_{n}x-[q_1{q}_2\cdots q_{n}x].$$

After that, each $x\in I$ can be represented by

$$x={\displaystyle \frac{{\epsilon }_1\left(x\right)}{q_1}}+{\displaystyle \frac{{\epsilon }_2\left(x\right)}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n\left(x\right)}{q_1{q}_2\cdots q_n}}+\cdots .$$

(1)

If we let ${\epsilon }_1\left(x\right)=\left[q_{1}x\right]$ and ${\epsilon }_n\left(x\right)=\left[q_n{T}_Q^{n-1}\left(x\right)\right]$ for all $n\ge 2,$ we call the form (1) the Q-Cantor expansion of x in base $Q,$ for simplicity, denoted by

$$({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_n\left(x\right),\cdots ).$$

Since ${\epsilon }_1\left(x\right)$ = $\left[q_{1}x\right],$ ${\epsilon }_n\left(x\right)$ = $\left[q_n{T}_Q^{n-1}\left(x\right)\right],$ and $\parallel T_Q^{n-1}\left(x\right)\parallel <1$, then $\parallel {\epsilon }_n\left(x\right)\parallel <\parallel q_n\left(x\right)\parallel$ (i.e., deg$\left({\epsilon }_n\left(x\right)\right)<\mathrm{deg}\left(q_n\right)$), for all $x\in I$ and $n\ge 1$; moreover, we have

$$P_n=\{\epsilon \in \mathbb{F}\left[X\right]|\parallel \epsilon \parallel <\parallel q_n\parallel \}.$$

Noting that, $\#P_n=\parallel q_n\parallel ,$ or the number of all possible digits is $\parallel q_n\parallel .$ We can show that for any given sequence ${\left\{{\epsilon }_n\right\}}_{n\ge 1}$ with ${\epsilon }_n\in P_n,$ there exist a unique $x\in I$ such that ${\epsilon }_n\left(x\right)={\epsilon }_n,$ for all $n\ge 1.$ The dynamical system corresponding to the Cantor series expansion is said to be nonautonomous because, at different stages of the iteration process, the action is different. $T_{Q,n}$ is used in the n-th step to run the iteration. The continued fraction expansions studied by Besicovitch [14] and Jarník [15] served as the model for the research of the Cantor expansions. Refs. [16,17] examined the metrical and ergodic theory of the continued fraction expansion on $I.$ Regarding the shrinking target problem, Wang, Fan and Zhang examined the hitting sets and quantitative recurrence of the $\beta$-transformation $T_{\beta }$ on the unit disk I of formal Laurent series field in [18]. The shrinking target problem for matrix transformations of tori was studied by Li, Liao, Velani and Zorin in [19]. To gain further insight into the diminishing target issue, readers may consult references [20,21,22,23,24]. In contrast to every research listed above, we operate within the framework of a nonautonomous dynamical system; in this nonautonomous dynamical system, the shrinking problem can be expressed in the following way: Let $\phi :\mathbb{N}\to (0,1)$ be a positive function and $y\in [0,1];$ moreover; put

$$E_y\left(\phi \right):=\{x\in [0,1]||T_Q^n\left(x\right)-y|<\phi \left(n\right)i.o.n\}.$$

In [25], Fishman, Mance, Simmons and Urbański obtained the Hausdorff dimension of $E_y\left(\phi \right)$ when $y=0.$ In [26], Sun and Cao, given the complete answer on the size of $E_y\left(\phi \right),$ obtained not only the dimension but also the f- Hausdorff measure for the general dimension function $f,$ which includes the case of the Lebesgue measure of $E_y\left(\phi \right).$ In this paper, we obtain the Hausdorff dimension of $E_y\left(\phi \right)$ over the field $\mathbb{F}\left(\left(X^{-1}\right)\right)$ when $y=0.$ More precisely, we obtained the Hausdorff dimension of the following set:

$$D_{\infty }^Q\left(\varphi \left(n\right)\right)=\{x\in I|\parallel T_Q^n\left(x\right)\parallel <q^{-\varphi \left(n\right)}i.o.\},$$

where $\varphi \left(n\right)$ is any positive function defined on $\mathbb{N}$ with $\varphi \left(n\right)\to \infty ,$ $n\to \infty .$ Since it is difficult to find the technique to solve the case when $Q=\left\{q_n\right\},$ $n\ge 1$ increases too fast, we always assume that the degree of $Q=\left\{q_n\right\}$, $n\ge 1$ is bounded, and we completely answer this question from the viewpoint of a Hausdorff dimension by using the following theorem.

2. Main Results

Theorem 1.

Let $\varphi \left(n\right)$ be a positive function defined on $\mathbb{N}$ with $\varphi \left(n\right)\to \infty$, $n\to \infty$, where we denote

$$D_{\infty }^Q\left(\varphi \left(n\right)\right)=\{x\in I|\parallel T_Q^n\left(x\right)\parallel <q^{-\varphi \left(n\right)}i.o.\}.$$

Then,

$${\mathit{dim}}_H\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)=\underset{n\to \infty }{\mathit{limsup}}{\displaystyle \frac{\mathit{log}\parallel q_1{q}_2\cdots q_n\parallel }{\mathit{log}\parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$$

To prove this theorem, we introduce the following:

Definition 1.

For any given block $({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)$ with ${\epsilon }_i\in P_i$ $(1\le i\le n),$

$$J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)=\{x\in I|{\epsilon }_1\left(x\right)={\epsilon }_1,{\epsilon }_2\left(x\right)={\epsilon }_2,\cdots ,{\epsilon }_n\left(x\right)={\epsilon }_n\},$$

is called the nth cylinder of the Q-Cantor expansion.

Lemma 1 

(See [27]). Let $x,$ y$\in \mathbb{F}\left(\left(X^{-1}\right)\right),$ if $\parallel x-y\parallel <1$; then, $\left[x\right]=\left[y\right].$

Lemma 2.

For any cylinder $J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n),$ we have

$$J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)=B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}).$$

As a consequence, $\mu \left(J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\right)={\parallel q_1{q}_2\cdots q_n\parallel }^{-1}.$

Proof. 

For any $x\in J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n),$ we have

$$x={\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}}+{\displaystyle \frac{{\epsilon }_{n+1}\left(x\right)}{q_1{q}_2\cdots q_n{q}_{n+1}}}+\cdots .$$

Since $\parallel T_Q^n\left(x\right)\parallel <1$ for any $n\ge 1,$ we have

$$\parallel x-({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}})\parallel =\parallel {\displaystyle \frac{{\epsilon }_{n+1}\left(x\right)}{q_1{q}_2\cdots q_n{q}_{n+1}}}+{\displaystyle \frac{{\epsilon }_{n+2}\left(x\right)}{q_1{q}_2\cdots q_{n+1}{q}_{n+2}}}+\cdots +\cdots \parallel .$$

Since $\parallel {\epsilon }_{n+1}\left(x\right)\parallel <\parallel q_{n+1}\parallel ,$ we obtain

$$\parallel {\displaystyle \frac{{\epsilon }_{n+1}\left(x\right)}{q_1{q}_2\cdots q_n{q}_{n+1}}}+{\displaystyle \frac{{\epsilon }_{n+2}\left(x\right)}{q_1{q}_2\cdots q_{n+1}{q}_{n+2}}}+\cdots +\parallel <\parallel {\displaystyle \frac{1}{q_1{q}_2\cdots q_n}}\parallel ={\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}.$$

Thus, $x\in B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }})$, i.e., $J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\subset B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots$$+{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}).$

On the other hand, for any $x\in B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}),$ it follows that

$$\parallel x-({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}})\parallel <{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}.$$

(2)

Multiply $\parallel q_1\parallel$ on both sides of (2); thus, we can get

$$\parallel q_{1}x-({\epsilon }_1+{\displaystyle \frac{{\epsilon }_2}{q_2}}+{\displaystyle \frac{{\epsilon }_3}{q_2{q}_3}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_2{q}_3\cdots q_n}})\parallel <{\displaystyle \frac{1}{\parallel q_2{q}_3\cdots q_n\parallel }}.$$

Thus,

$${\epsilon }_1\left(x\right)=\left[q_{1}x\right]=[{\epsilon }_1+{\displaystyle \frac{{\epsilon }_2}{q_2}}+{\displaystyle \frac{{\epsilon }_3}{q_2{q}_3}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_2{q}_3\cdots q_n}}]={\epsilon }_1.$$

From the algorithm of Q-Cantor expansions, for all $1\le i\le n,$ we have

$$\parallel T_Q^{i-1}\left(x\right)-({\displaystyle \frac{{\epsilon }_i}{q_i}}+{\displaystyle \frac{{\epsilon }_{i+1}}{q_i{q}_{i+1}}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_i\cdots q_n}})\parallel <{\displaystyle \frac{1}{\parallel q_i\cdots q_n\parallel }}<1.$$

So, ${\epsilon }_i\left(x\right)$ = ${\epsilon }_i$ for all 1 ≤ i ≤ $n.$ Therefore,

$$B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }})\subset J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n).$$

Thus,

$$J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)=B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }}).$$

Since $\mu \left(B({\displaystyle \frac{{\epsilon }_1}{q_1}}+{\displaystyle \frac{{\epsilon }_2}{q_1{q}_2}}+\cdots +{\displaystyle \frac{{\epsilon }_n}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }})\right)={\parallel q_1{q}_2\cdots q_n\parallel }^{-1}$, we have

$$\mu \left(J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\right)={\parallel q_1{q}_2\cdots q_n\parallel }^{-1}.$$

□

From this proposition, we know that every cylinder is a ball. Conversely, we have the following:

Proposition 1.

Let $B(x,r)\subset I$ be a ball. Then, there exists $n_0\in \mathbb{N}$ such that

$$J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0+1}\left(x\right))\subset B(x,r)\subset J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0}\left(x\right)).$$

Proof. 

There exists $n_1\in \mathbb{N}$, such that $q^{-(n_1+1)}<r\le q^{-n_1}$; therefore,

$$B(x,r)=B(x,q^{-n_1}).$$

Since d is a discrete metric, we choose $n_0\in \mathbb{N},$ such that

$$\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0}\right)\le n_1<\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0+1}\right).$$

Thus, we can get

$$q^{-n_1}<q^{-(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0}\right))}={\parallel q_1{q}_2\cdots q_{n_0}\parallel }^{-1}.$$

Therefore,

$$B(x,q^{-n_1})\subset B(x,{\parallel q_1{q}_2\cdots q_{n_0}\parallel }^{-1})=J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0}\left(x\right)).$$

On the other hand, for any $x\in J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0+1}\left(x\right)),$ since

$$q^{-n_1}>q^{-(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0+1}\right))}.$$

We have

$$J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0+1}\left(x\right))=B(x,q^{-(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0+1}\right))})\subset B(x,q^{-n_1}).$$

Therefore,

$$x\in B(x,q^{-(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)+\cdots +\mathrm{deg}\left(q_{n_0+1}\right))})\subset B(x,q^{-n_1}).$$

Thus,

$$J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0+1}\left(x\right))\subset B(x,q^{-n_1})=B(x,r)\subset J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{n_0}\left(x\right)).$$

□

Proposition 2.

Let $x_0\in I$ and $n_0\in \mathbb{Z}$; if P = $\{x\in F\left[X\right]|\parallel x\parallel <\parallel q_1{q}_2\cdots q_n\parallel \},$ we have

$$T_Q^{-n}\left(B(x_0,q^{-n_0})\right)=\bigcup _{p\in P}B({\displaystyle \frac{p+x_0}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{n_0}}}).$$

Proof. 

For any x$\in T_Q^{-n}\left(B(x_0,q^{-n_0})\right)$, i.e., $T_Q^n\left(x\right)\in B(x_0,q^{-n_0}),$ we have

$$\parallel q_1{q}_2\cdots q_{n}x-[q_1{q}_2\cdots q_{n}x]-x_0\parallel <q^{-n_0}.$$

Thus, we can get

$$\parallel x-{\displaystyle \frac{[q_1{q}_2\cdots q_{n}x]+x_0}{q_1{q}_2\cdots q_n}}\parallel <{\displaystyle \frac{q^{-n_0}}{\parallel q_1{q}_2\cdots q_n\parallel }}.$$

Therefore, $x\in B({\displaystyle \frac{[q_1{q}_2\cdots q_{n}x]+x_0}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{q^{-n_0}}{\parallel q_1{q}_2\cdots q_n\parallel }})\subset \bigcup _{p\in P}B({\displaystyle \frac{p+x_0}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{n_0}}}).$

On the other hand, if $x\in {\displaystyle \bigcup _{p\in P}}B({\displaystyle \frac{p+x_0}{q_1\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1\cdots q_n\parallel q^{n_0}}}),$ then there exist $p_1\in P$ and $x\in B({\displaystyle \frac{p_1+x_0}{q_1\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1\cdots q_n\parallel q^{n_0}}}),$ such that

$$\parallel x-{\displaystyle \frac{p_1+x_0}{q_1{q}_2\cdots q_n}}\parallel <{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{n_0}}},,i.e.,\parallel q_1{q}_2\cdots q_{n}x-p_1-x_0\parallel <q^{-n_0}<1.$$

With Lemma 1, we can get $[q_1{q}_2\cdots q_{n}x]=[p_1+x_0]=\left[p_1\right]$; thus,

$$\parallel q_1{q}_2\cdots q_{n}x-p_1-x_0\parallel =\parallel q_1{q}_2\cdots q_{n}x-[q_1{q}_2\cdots q_{n}x]-x_0\parallel =\parallel T_Q^n\left(x\right)-x_0\parallel <q^{-n_0}.$$

Therefore, $T_Q^n\left(x\right)\in B(x_0,q^{-n_0}),$, i.e., $x\in T_Q^{-n}\left(B(x_0,q^{-n_0})\right).$

Therefore, we have

$$T_Q^{-n}\left(B(x_0,q^{-n_0})\right)=\bigcup _{p\in P}B({\displaystyle \frac{p+x_0}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{n_0}}}).$$

□

3. Upper Bounds of Dimension

The definition of a Hausdorff measure on I is the same as that on $R_n$ (see [28]). Given $s>0$ and a subset E of I, the s-Hausdorff measure is given by

$${\displaystyle {\mathcal{H}}^s\left(E\right)=\underset{\sigma \to 0}{lim}\left\{\inf \sum _{\mathrm{j}}{\left(\mathrm{diam}\left({\mathrm{D}}_{\mathrm{j}}\right)\right)}^{\mathrm{s}}\right\},}$$

where the infimum takes over all covers of E with disks $D_j$ with a diameter of at most $\sigma$, and diam denotes the diameter of a set. The Hausdorff dimension of the set E is defined by

$${\dim }_{\mathrm{H}}\left(E\right)=\inf \{\mathrm{s}\ge 0|{\mathcal{H}}^{\mathrm{s}}\left(\mathrm{E}\right)=0\}.$$

Proposition 3.

Let $\varphi \left(n\right)$ be a positive function defined on $\mathbb{N}$ with $\varphi \left(n\right)\to \infty ,$ $n\to \infty .$ Thus,

$${\dim }_{\mathrm{H}}\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\le \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$$

Proof. 

Since $P=\{x\in \mathbb{F}\left[x\right]|\parallel x\parallel <\parallel q_1{q}_2\cdots q_n\parallel \},$ we have

$$\begin{array}{cc}\hfill D_{\infty }^Q\left(\varphi \left(n\right)\right)& =\{x\in I|\bigcup _{p\in P}B({\displaystyle \frac{p}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{\varphi \left(n\right)}}})i.o.\}\hfill \\ \hfill & =\bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }\{x\in I|\bigcup _{p\in P}B({\displaystyle \frac{p}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{\varphi \left(n\right)}}})\}.\hfill \end{array}$$

Thus, the union of ball $B({\displaystyle \frac{p}{q_1{q}_2\cdots q_n}},{\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{\varphi \left(n\right)}}})$ is a cover of the set $D_{\infty }^Q\left(\varphi \left(n\right)\right).$

Denote $s=\underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}$; for any $\epsilon >0,$ by the definition of a Hausdorff measure and Lemma 2, we have

$${\mathcal{H}}^{s+\epsilon }\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\le \sum _{n=N}^{\infty }\sum _{p\in P}\left({\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel q^{\varphi \left(n\right)}}}\right){)}^{s+\epsilon }.$$

Since $\#P$ = $\parallel q_1{q}_2\cdots q_n\parallel$, we have

$$\begin{array}{cc}\hfill {\mathcal{H}}^{s+\epsilon }\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)& \le \sum _{n=N}^{\infty }\parallel q_1{q}_2\cdots q_n\parallel {\displaystyle \frac{1}{(\parallel q_1{q}_2\cdots q_n{\parallel )}^{s+\epsilon }{q}^{\varphi \left(n\right)(s+\epsilon )}}}\hfill \\ \hfill & =\sum _{n=N}^{\infty }(\parallel q_1{q}_2\cdots q_n{\parallel )}^{(1-(s+\epsilon \left)\right)}{q}^{-\varphi \left(n\right)(s+\epsilon )},\hfill \end{array}$$

(3)

Note that $(s+\epsilon )>\underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}$; therefore, there exists ${\epsilon }^{\prime }>0,$ for any $n>>1,$ $(s+\epsilon )>(1+{\epsilon }^{\prime }){\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}$, i.e., $\parallel q_1{q}_2\cdots q_n{\parallel }^{1-(s+\epsilon )}{q}^{-\varphi \left(n\right)(s+\epsilon )}<{\parallel q_1{q}_2\cdots q_n\parallel }^{-{\epsilon }^{\prime }}.$

Thus, the right series of (3) is convergent, which implies that ${\mathcal{H}}^{s+\epsilon }\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)<+\infty .$ Therefore,

$${\dim }_{\mathrm{H}}\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\le \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}+\epsilon .$$

By letting $\epsilon \to 0^{+},$ we can get

$${\dim }_{\mathrm{H}}\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\le \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$$

□

4. Lower Bounds of Dimension

The method for constructing Cantor-like subsets of $D_{\infty }^Q\left(\varphi \left(n\right)\right)$ is to estimate the lower bounds of the Hausdorff dimensions of the sets $D_{\infty }^Q\left(\varphi \left(n\right)\right).$

Proposition 4.

Let $\varphi \left(n\right)$ be a positive function defined on $\mathbb{N}$ with $\varphi \left(n\right)\to \infty ,$ $n\to \infty .$ Thus,

$${\dim }_{\mathrm{H}}\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\ge \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$$

Firstly, we state the mass distribution principle over the field $\mathbb{F}\left(\left(X^{-1}\right)\right)$ that will be used later.

Lemma 3.

Let $E\subset I$ be a Borel set and ν be a probability measure with $\nu \left(E\right)>0.$ If the constants $C>0$ and $\sigma >0$ exist, such that

$$\begin{array}{c}\hfill \nu \left(D\right)\le C{\left(\mathrm{diam}\left(\mathrm{D}\right)\right)}^s\end{array}$$

(4)

for all disks D with $\mathrm{diam}\left(\mathrm{D}\right)\le \sigma$, then

$${\dim }_{\mathrm{H}}\left(E\right)\ge s.$$

Proof of Proposition 4. 

Let $\epsilon >0$ be an arbitrary real number. Let $\left\{N_k\right\}$ and $\left\{n_k\right\}$ be two sequences of natural numbers with

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{\varphi \left(n_k\right)}{\log \parallel q_1{q}_2\cdots q_{n_k}\parallel }}=\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{\varphi \left(n\right)}{\log \parallel q_1{q}_2\cdots q_n\parallel }},\end{array}$$

(5)

with every $N_i$ of $\left\{N_k\right\}$ being the maximum number that is satisfied, as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\parallel q_{n_i+1}{q}_{n_i+2}\cdots q_{n_i+N_i}\parallel }}\ge q^{-\varphi \left(n_i\right)}.\end{array}$$

(6)

We choose a subsequence ${\left\{n_{k_i}\right\}}_{i\ge 1}$ of the sequence ${\left\{n_k\right\}}_{i\ge 1}$ (for simplicity, we still denote the subsequence by ${\left\{n_k\right\}}_{i\ge 1}$) such that

$$\begin{array}{c}\hfill n_1>1,n_{k+1}>n_k+N_k+1,\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{k}{\log \parallel q_1{q}_2\cdots q_{n_k}\parallel }}=0,\sum _{j=1}^{k-1}\varphi \left(n_j\right)\le {\displaystyle \frac{\epsilon }{2s}}\log \parallel q_1{q}_2\cdots q_{n_k}\parallel ,\end{array}$$

(8)

where $s=\underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$ For any $k\ge 1$, denote the sets of integers

$$S_k=\{n_k+j|1\le j\le N_k+1\},S=\bigcup _{k=1}^{\infty }{S}_k.$$

Let

$$E_{\varphi }^{\left\{n_k\right\}}=\{x\in I|{\epsilon }_{n_k+1}={\epsilon }_{n_k+2}=\cdots ={\epsilon }_{n_k+N_k+1}=0,\forall k\ge 1\}.$$

Then,

$$E_{\varphi }^{\left\{n_k\right\}}\subset D_{\infty }^Q\left(\varphi \left(n\right)\right).$$

We can also describe the set $E_{\varphi }^{\left\{n_k\right\}}$ according to the following structure.

Let $D_n$ be the collection of n-cylinders $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)$ satisfying ${\epsilon }_i\in P_i$ for $1\le i\le n,$ ${\epsilon }_i=0$ if $i\in S_k$ for some $k.$ Thus,

$$E_{\varphi }^{\left\{n_k\right\}}=\bigcap _{n=1}^{\infty }\bigcup _{J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\in D_n}J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n).$$

Thus, the set $E_{\varphi }^{\left\{n_k\right\}}$ is a Cantor-like subset of $D_{\infty }^Q\left(\varphi \left(n\right)\right)$.

Using the mass distribution principle, we will now provide a lower bound on the Hausdorff dimension of the set $E_{\varphi }^{\left\{n_k\right\}}.$ First, we create a measure $\nu$ supported on $E_{\varphi }^{\left\{n_k\right\}},$ or a mass distribution on $E_{\varphi }^{\left\{n_k\right\}}.$

We define the measure $\nu$ on the cylinders firstly. Let $\nu \left(I\right)=1,$ and $\nu \left(J\left({\epsilon }_1\right)\right)={\displaystyle \frac{1}{\parallel q_1\parallel }},$ for any ${\epsilon }_1\in P_1.$ Suppose the measure of the $(n-1)$ cylinder $\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_{n-1})\right)$ is well defined. Next, we define $\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)$ as the following:

(1)

If $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\notin D_n,$ then $\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)=0.$

(2)

If $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\in D_n,$ then

$$\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)=\left\{\begin{array}{ccc}\hfill \nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_{n-1})\right)& & n\in S,\hfill \\ \hfill \parallel q_n{\parallel }^{-1}\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_{n-1})\right)& & n\notin S.\hfill \end{array}\right.$$

The measure $\nu$ is well defined on all cylinders because we can verify that

$$\sum _{{\epsilon }_{n+1}\in P_{n+1}}\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_{n+1})\right)=\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right),$$

and

$$\sum _{{\epsilon }_i\in P_i}\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)=1.$$

The measure $\nu$ can be defined on the measurable space $(I,\mathcal{B}(I\left)\right).$

We assert that the requirement (4) is satisfied with measure $\nu .$ That is, two constants, $C>0$ and $\sigma >0$, exist for each $\epsilon >0,$ such that

$$\begin{array}{c}\hfill \nu \left(B(x,r)\right)\le C{r}^{s-\epsilon }\end{array}$$

(9)

for any ball $B(x,r)$ with $r<\sigma ,$ where

$$s=\underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}.$$

Next, we can apply the mass distribution principle to the set $E_{\varphi }^{\left\{n_k\right\}}$; thus, we obtain

$${\dim }_{\mathrm{H}}\left(E_{\varphi }^{\left\{n_k\right\}}\right)\ge \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}-\epsilon .$$

Note that $E_{\varphi }^{\left\{n_k\right\}}\subset D_{\infty }^Q\left(\varphi \left(n\right)\right)$; thus,

$${\dim }_{\mathrm{H}}\left(D_{\infty }^Q\left(\varphi \left(n\right)\right)\right)\ge \underset{n\to \infty }{\mathrm{limsup}}{\displaystyle \frac{\log \parallel q_1{q}_2\cdots q_n\parallel }{\log \parallel q_1{q}_2\cdots q_n\parallel +\varphi \left(n\right)}}-\epsilon .$$

By letting $\epsilon \to 0,$ we complete the proof.

Step One. Let $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)$ be any n-th cylinder.

Case (1). $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\notin D_n$; thus,

$$\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)=0.$$

We utilize the construction of $\nu$, which evidently induces Inequality (9).

Case (2). (a) $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\in D_n,n\notin S.$

That is, $n_k+N_k+1<n\le n_{k+1}$ for some $k\in \mathbb{N}.$ By the construction of the measure $\nu$, we have

$$\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)={\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_{n_1}\parallel }}·{\displaystyle \frac{1}{\parallel q_{n_1+N_1+2}{q}_{n_1+N_1+3}\cdots q_{n_2}\parallel }}\cdots {\displaystyle \frac{1}{\parallel q_{n_k+N_k+2}{q}_{n_k+N_k+3}\cdots q_n\parallel }}.$$

Since $\mathrm{diam}$ $\left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)$ = ${\displaystyle \frac{1}{\parallel q_1{q}_2\cdots q_n\parallel }},$ we know that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)}{{\left(\mathrm{diam}\left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)\right)}^{s-\epsilon }}}& ={\displaystyle \frac{\parallel q_1{q}_2\cdots q_n{\parallel }^{s-\epsilon }}{\parallel q_1\cdots q_{n_1}\parallel \parallel q_{n_1+N_1+2}\cdots q_{n_2}\parallel \cdots \parallel q_{n_k+N_k+2}\cdots q_n\parallel }}\hfill \\ \hfill & =\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{\parallel q_{n_1+1}\cdots q_{n_1+N_1+1}\parallel }^{s-\epsilon }\hfill \\ & \cdots \parallel q_{n_1+N_1+2}\cdots q_{n_2}{\parallel }^{s-1-\epsilon }\cdots {\parallel q_{n_k+N_k+2}\cdots q_n\parallel }^{s-1-\epsilon }\hfill \\ & \le \parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\varphi \left(n_1\right)+\varphi \left(n_2\right)\cdots +\varphi \left(n_k\right))(s-1-\epsilon )}\parallel q_{n_1+1}\cdots q_{n_1+N_1+1}\parallel \hfill \\ & \parallel q_{n_2+1}{q}_{n_2+2}\cdots q_{n_2+N_2+1}\parallel \cdots \parallel q_{n_k+1}{q}_{n_k+2}\cdots q_{n_k+N_k+1}\parallel \hfill \\ \hfill & \le \parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\varphi \left(n_1\right)+\varphi \left(n_2\right)\cdots +\varphi \left(n_k\right))(s-1-\epsilon )+k·(N+1)·M}\hfill \\ \hfill & =\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots \mathrm{deg}\left(q_{n_k}\right)t_k}\hfill \end{array}$$

where $t_k=[{\displaystyle \frac{{\sum }_{j=1}^{k-1}\varphi \left(n_j\right)}{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right)}}+{\displaystyle \frac{\varphi \left(n_k\right)}{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right)}}](s-1-\epsilon )+{\displaystyle \frac{k·(N+1)·M}{\mathrm{deg}\left(q_1\right)+\cdots +\mathrm{deg}\left(q_{n_k}\right)}},$ the third inequality holds because of (6) and $s-1-\epsilon <0,$ the fourth inequality holds because N is the maximum number of ${\left\{N_i\right\}}_{1\le i\le k}$, and M is the maximum degree of $\parallel q_i{\parallel }_{n_1+1\le i\le n_1+N_1+1,\cdots ,n_k+1\le i\le n_k+N_k+1}.$ Combining the definitions of s and (8), we have

$$\underset{k\to \infty }{\mathrm{limsup}}{t}_k=-{\displaystyle \frac{1}{2s}}\epsilon .$$

Therefore, there exists a constant $C,$ such that

$$\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right))t_k}<C.$$

Thus, (9) holds for such cylinder $J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)$ and such constant $C.$

(b). $J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\in D_n,n\in S.$

That is, $n_k<n\le n_k+N_k+1$ for some $k\in \mathbb{N}.$ By the construction of measure $\nu$, we have

$$\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)={\displaystyle \frac{1}{\parallel q_1\cdots q_{n_1}\parallel }}·{\displaystyle \frac{1}{\parallel q_{n_1+N_1+2}\cdots q_{n_2}\parallel }}\cdots {\displaystyle \frac{1}{\parallel q_{n_{k-1}+N_{k-1}+2}{q}_{n_{k-1}+N_{k-1}+3}\cdots q_{n_k}\parallel }}.$$

Thus, we know

$$\begin{array}{cc}\hfill {\displaystyle \frac{\nu \left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)}{{\left(\mathrm{diam}\left(J({\epsilon }_1,{\epsilon }_2\cdots ,{\epsilon }_n)\right)\right)}^{s-\epsilon }}}& ={\displaystyle \frac{\parallel q_1{q}_2\cdots q_n{\parallel }^{s-\epsilon }}{\parallel q_1{q}_2\cdots q_{n_1}\parallel \parallel q_{n_1+N_1+2}{q}_{n_1+N_1+3}\cdots q_{n_2}\parallel \cdots \parallel q_{n_{k-1}+N_{k-1}+2}\cdots q_{n_k}\parallel }}\hfill \\ \hfill & =\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{\parallel q_{n_1+1}\cdots q_{n_1+N_1+1}\parallel }^{s-\epsilon }\hfill \\ \hfill & \cdots \parallel q_{n_{k-1}+N_{k-1}+2}{q}_{n_{k-1}+N_{k-1}+3}\cdots q_{n_k}{\parallel }^{s-1-\epsilon }{\parallel q_{n_k+1}{q}_{n_k+2}\cdots q_n\parallel }^{s-\epsilon }\hfill \\ & \le \parallel q_1\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\varphi \left(n_1\right)+\varphi \left(n_2\right)\cdots +\varphi \left(n_{k-1}\right))(s-1-\epsilon )}\parallel q_{n_1+1}\cdots q_{n_1+N_1+1}\parallel \hfill \\ & \parallel q_{n_2+1}{q}_{n_2+2}\cdots q_{n_2+N_2+1}\parallel \cdots \parallel q_{n_k+1}{q}_{n_k+2}\cdots q_n{\parallel }^{s-\epsilon }\hfill \\ \hfill & \le \parallel q_1\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\varphi \left(n_1\right)+\varphi \left(n_2\right)\cdots +\varphi \left(n_{k-1}\right))(s-1-\epsilon )}\parallel q_{n_1+1}\cdots q_{n_1+N_1+1}\parallel \hfill \\ & \parallel q_{n_2+1}{q}_{n_2+2}\cdots q_{n_2+N_2+1}\parallel \cdots \parallel q_{n_k+1}{q}_{n_k+2}\cdots q_{n_k+N_k+1}\parallel \hfill \\ \hfill & \le \parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\varphi \left(n_1\right)+\varphi \left(n_2\right)\cdots +\varphi \left(n_{k-1}\right))(s-1-\epsilon )+k·(N+1)·M}\hfill \\ \hfill & =\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{(\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right))t_k^{\prime }}\hfill \end{array}$$

where $t_k^{\prime }={\displaystyle \frac{{\sum }_{j=1}^{k-1}\varphi \left(n_j\right)}{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right)}}(s-1-\epsilon )+{\displaystyle \frac{k·(N+1)·M}{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right)}},$ the third inequality holds because of (6) and $s-1-\epsilon <0,$ the fourth inequality is because $n\le n_k+N_k+1$ and $s<1,$ the fifth inequality is because N is the maximum number of ${\left\{N_i\right\}}_{1\le i\le k}$, and M is the maximum degree of $\parallel q_i{\parallel }_{n_1+1\le i\le n_1+N_1+1,\cdots ,n_k+1\le i\le n_k+N_k+1}.$ Combining the definitions of s and (8), we have

$$\underset{k\to \infty }{\mathrm{limsup}}{t}_k^{\prime }=-{\displaystyle \frac{1}{2s}}\epsilon .$$

Therefore, there exists a constant $C^{\prime },$ such that

$$\parallel q_1{q}_2\cdots q_{n_1}{\parallel }^{s-1-\epsilon }{q}^{\mathrm{deg}\left(q_1\right)+\mathrm{deg}\left(q_2\right)\cdots +\mathrm{deg}\left(q_{n_k}\right)t_k^{\prime }}<C^{\prime }.$$

Thus, (9) holds for such cylinder $J({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)$ and such constant $C^{\prime }.$

Step two: For any ball $B(x,r)$, by Proposition 2.5, there exists an integer $m\in \mathbb{N}$ such that

$$J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{m+1})\subset B(x,r)\subset J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_m).$$

Thus, we have

$$\begin{array}{cc}\hfill \nu \left(B\right(x,r\left)\right)& \le \nu \left(J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_m)\right)\hfill \\ \hfill & \le C{\left(diam\left(J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_m)\right)\right)}^s\hfill \\ \hfill & =C(\parallel q_{m+1}{\parallel diam\left(J({\epsilon }_1\left(x\right),{\epsilon }_2\left(x\right),\cdots ,{\epsilon }_{m+1})\right))}^s\hfill \\ \hfill & \le C(\parallel q_{m+1}{\parallel )}^s{r}^s.\hfill \end{array}$$

The outcome of step one justifies the second inequality. As a result, (9) is true for any ball, and we complete the proof of Proposition 4. □

Proof of Theorem 1. 

By combining Proposition 3 and Proposition 4, we know that Theorem 1 holds. □