1.1 d-decaying Gauss-like iterated function systems

Let $\{f_n\}_{n\geq 1}$ be a sequence of $C^{1}$ functions with $f_n:[0,1]\rightarrow [0,1]$ satisfying the following.

1. (i) Open set condition: for any $i\neq j\in \mathbb {N}$ , $f_i((0, 1))\cap f_j((0, 1))=\emptyset $ .

2. (ii) Contraction property: there exists $m\in \mathbb {N}$ and a real number $0<\rho <1$ such that for all $(a_1, \ldots , a_m) \in \mathbb {N}^m$ and $x\in [0,1]$ ,

   $$ \begin{align*} 0<|(f_{a_1}\circ\cdots\circ f_{a_m})'(x)|\leq\rho<1. \end{align*} $$

3. (iii) Regular property: there exists a sequence $\boldsymbol {\xi }=\{\xi _n\}_{n\ge 1}$ such that for any $\varepsilon>0$ , we can find constants $C_1=C_1(\varepsilon ),\ C_2=C_2(\varepsilon )>0$ such that for all $i\in \mathbb {N}$ and $x\in [0,1]$ ,

   $$ \begin{align*} \frac{C_1} {\xi_i^{1+\varepsilon}} \leq |f_i^{\prime}(x)| \leq \frac{C_2} {\xi_i^{1-\varepsilon}}. \end{align*} $$

4. (iv) Bounded distortion property (BDP): there exists a constant $\kappa \geq 1$ such that for every $n\in \mathbb {N}$ and $(a_1, \ldots , a_n)\in \mathbb {N}^n$ ,

   $$ \begin{align*} |(f_{a_1}\circ\cdots\circ f_{a_n})'(x)|\leq\kappa|(f_{a_1}\circ\cdots\circ f_{a_n})'(y)| \end{align*} $$
   for all $x,y\in [0,1]$ .

We say such a system $\{f_n\}_{n\ge 1}$ is a $\boldsymbol {\xi }$ -regular iIFS as defined by Cao et al. [Reference Cao, Wang and Wu3]. Particularly, a system $\{f_n\}_{n\ge 1}$ is called d-decaying iIFS if $\xi _i=i^d\ (d>1)$ for each $i\ge 1$ , and is called d-decaying Gauss-like iIFS if the system in addition fulfills:

1. (v) $\bigcup _{i=1}^\infty f_i([0,1]) = [0,1]$ and $f_{i}(x)<f_{j}(x)$ whenever $i>j.$

One is directed to the work of Jordan and Rams [Reference Jordan and Rams15] for further information.

With the above definition, clearly, there is a natural projection $\Pi : \mathbb {N}^{\mathbb {N}}\rightarrow [0,1]$ defined as

$$ \begin{align*} \Pi(\underline{a})=\lim\limits_{n\rightarrow\infty}f_{a_{1}}\circ\cdots\circ f_{a_{n}}(1) \end{align*} $$

for all $\underline {a}=\{a_n\}_{n\geq 1}\in \mathbb {N}^{\mathbb {N}}$ . Then, each $x\in \Pi (\mathbb {N}^{\mathbb {N}})$ that is the attractor of the iIFS $\{f_n\}_{n\geq 1}$ , corresponds to a sequence of integers $\{a_n\}_{n\geq 1}$ in the sense that

$$ \begin{align*} x=\lim\limits_{n\rightarrow\infty}f_{a_{1}}\circ\cdots\circ f_{a_{n}}(1). \end{align*} $$

We call $\{a_n\}_{n\geq 1}$ the digits of x. It should be pointed out that the sequence of digits of one point may not be unique. In fact, the inverse of $\Pi $ gives, for points $x\in [0,1]$ , their sequences of digits in $\mathbb {N}^{\mathbb {N}}.$ The sequence of digits is unique, except for at most countably many points, which may have zero or two sequences of digits. We remove these points from consideration and the removal of them does not affect the Hausdorff dimension of the sets under consideration. Thus, we write $x=(a_1(x),a_2(x),\ldots )$ for the unique symbolic expansion of any point $x\in [0,1]$ . It is known that there are several classical iIFS closely connected with number theory.

• • Continued fraction system:

  $$ \begin{align*} f_n(x)=\frac{1}{x+n}, \quad x\in[0,1],\ n\in\mathbb{N}. \end{align*} $$
  The sequence of digits $\{a_n(x)\}_{n\geq 1}$ is just the partial quotients of x in its continued fraction expansion.

• • Lüroth system:

  $$ \begin{align*} f_n(x)=\frac{x}{n(n+1)}+\frac{1}{n+1},\quad x\in[0,1],\ n\in\mathbb{N}. \end{align*} $$
  Then, $\{a_n(x)\}_{n\geq 1}$ is just the sequence of digits in the Lüroth series expansion of x.

• • Quadratic Gauss system:

  $$ \begin{align*} f_n(x)=\frac{1}{(x+n)^2},\quad x\in[0,1],\ n\in\mathbb{N}. \end{align*} $$
  Here, the sequence $\{a_n(x)\}_{n\geq 1}$ is just the digits in the series expansion of x induced by the quadratic Gauss map, which is a particular case of f-expansion defined by Rényi [Reference Rényi23]. For more details on the quadratic Gauss map, see [Reference González Robert, Hussain, Shulga and Takahasi9, Section 1].

Both the continued fraction system and the Lüroth system are examples of 2-decaying Gauss-like iIFS, and the quadratic Gauss system is an example of 3-decaying Gauss-like iIFS.

For any $(a_{1},\ldots ,a_{n})\in \mathbb {N}^{n}$ ,

$$ \begin{align*} I_{n}(a_{1},\ldots,a_{n})=f_{a_{1}}\circ\cdots\circ f_{a_{n}}([0,1]) \end{align*} $$

is called an nth-level cylinder denoting the set of points in [0,1] whose symbolic expansions begin with $a_{1},\ldots ,a_{n}$ . Notice that Conditions (ii) and (iii) in the definition of the $\boldsymbol {\xi }$ -regular iIFS can be used to get the upper and lower bounds for the measure of these cylinders. Note that the measures of these cylinders depend on the arbitrarily preassigned parameter $\varepsilon>0$ in Condition (iii). Thus, this makes the upper and lower bounds of the Hausdorff dimension correspondingly dependent on $\varepsilon $ . However, the arbitrariness of $\varepsilon>0$ ensures that the dimension result is independent of it. Hence, we assume without loss of generality that $\varepsilon =0$ in Condition (iii) for simplicity. Thus,

(1-1) $$ \begin{align} C_{1}^{n}\prod\limits_{i=1}^{n}\xi_{a_i}^{-1}\leq|I_{n}(a_{1},\ldots, a_{n})|\leq C_{2}^{n}\prod\limits_{i=1}^{n}\xi_{a_i}^{-1}. \end{align} $$

Determining the fractal dimensions of the symbolic expansions that concern the properties of digits has been an important subject in the study of infinite iterated function systems; see [Reference Cao, Wang and Wu3, Reference González Robert, Hussain, Shulga and Takahasi9, Reference Jordan and Rams15, Reference Liao and Rams17, Reference Zhang26, Reference Zhang27] and references therein. Among them, Jordan and Rams [Reference Jordan and Rams15] considered the dimension of the sets of points with strictly increasing digits in general d-decaying iIFS and obtained the following theorem.

Theorem 1.1 [Reference Jordan and Rams15, Theorem 1.1]

Let $\Phi :\mathbb {N}\rightarrow \mathbb {R}$ be a function such that ${n\leq \Phi (n)\leq \beta n}$ for some $\beta \geq 1$ , then

$$ \begin{align*} \dim_{\mathrm{H}}\Pi\{\underline{a}: a_{n+1}>\Phi(a_n) \text{ for all}\ n\in\mathbb{N}\}=1/d, \end{align*} $$

where we use $\dim _{\mathrm {H}}$ to denote the Hausdorff dimension.

Additionally, Jordan and Rams also showed that the dimension does drop below $1/d$ when the digits grow with exponential rates in the d-decaying Gauss-like iIFS. After that, Cao et al. [Reference Cao, Wang and Wu3] investigated the digits of points, which are further restricted to an infinite subset of positive integers in $\boldsymbol {\xi }$ -regular iIFS, including d-decaying Gauss-like systems. They also generalized Łuczak’s work [Reference Łuczak18] by considering the dimensions of sets of points satisfying the doubly exponential growth rate of the digits in Gauss-like systems. The Hausdorff dimension of sets of points with restricted slowly growing digits in $\boldsymbol {\xi }$ -regular iIFS was studied by González-Robert et al. [Reference González Robert, Hussain, Shulga and Takahasi9]. Recently, Liao and Rams [Reference Liao and Rams17] investigated the increasing rate of Birkhoff sums in such systems and calculated the Hausdorff dimensions of sets of points whose Birkhoff sums share the same increasing rate for different unbounded potential functions. Later, Zhang [Reference Zhang26] studied the critical cases for the growth rate functions, which are not discussed in [Reference Liao and Rams17] and pointed out that the corresponding Hausdorff dimension spectrum is right continuous.

Before proceeding, we state the fact that there exists a conformal measure (in our case, it is the one-dimensional Lebesgue measure), which is equivalent to the unique ergodic invariant measure in d-decaying Gauss-like iIFS. To this end, in the following, we recall the classical definitions of conformal iterated function systems and some relevant conclusions coming from [Reference Hanus, Mauldin and Urbański12, Reference Mauldin19, Reference Mauldin and Urbański20].

1.2 Conformal iterated function systems and relevant conclusions

For convenience, we give some notation.

• • $\phi _{\omega }:=\phi _{\omega _1}\circ \cdots \circ \phi _{\omega _n}$ for $\omega =(\omega _1,\ldots ,\omega _n)\in \mathbb {N}^{n},\ n\geq 1$ .

• • $\|\phi _{\omega }^{'}\|:=\sup _{x\in X}|\phi _{\omega }^{'}(x)|$ for $\omega \in \bigcup _{n\geq 1}\mathbb {N}^{n}$ .

• • $C(X)$ denotes the space of continuous functions on X.

• • IntX and $\partial X$ denote the interior and the boundary of X, respectively.

• • The one-dimensional Lebesgue measure is denoted by $\mathcal {L}$ and the n-dimensional Lebesgue measure by $\mathcal {L}^{n}$ .

An iterated function system $S=\{\phi _i\}_{i\in \mathbb {N}}$ is called a conformal iterated function system (c.i.f.s) with seed set X, where X is a nonempty compact connected subset of $\mathbb {R}^n$ , if the following conditions are satisfied:

1. (1) for each $i\in \mathbb {N}$ , $\phi _i$ is an injective map of X into X;

2. (2) the system S is uniformly contractive on X, namely, there exists $0<s<1$ , such that

   $$ \begin{align*} |\phi_i(x)-\phi_i(y)|\leq s|x-y|; \end{align*} $$

3. (3) the open set condition is satisfied for IntX:

   $$ \begin{align*} \phi_i(\text{Int}X)\subseteq\text{Int}X \quad \text{and}\quad \phi_i(\text{Int}X)\cap\phi_j(\text{Int}X)=\emptyset,\quad i,j\in\mathbb{N},i\neq j; \end{align*} $$

4. (4) there exists an open connected set V with $X\subseteq V\subseteq \mathbb {R}^n$ such that each $\phi _i$ , $i\in \mathbb {N}$ , extends to a $C^1$ conformal diffeomorphism of V into V;

5. (5) cone condition: there exists $\gamma ,l$ such that for every $x\in \partial X\subseteq \mathbb {R}^n$ , there is an open cone Con $(x,\gamma ,l)\subset $ IntX with vertex x, central angle of Lebesgue measure $\gamma $ , and altitude l;

6. (6) the BDP is satisfied for S: there is a $K\geq 1$ such that

   $$ \begin{align*} |\phi_{\omega}^{'}(y)|\leq K|\phi_{\omega}^{'}(x)| \end{align*} $$
   for every $\omega \in \bigcup _{n\geq 1}\mathbb {N}^n$ and every pair of points $x,y\in V$ .

Proof. By the definition of d-decaying Gauss-like iIFS, Conditions (1)–(4), (6) are obviously satisfied for $S=\{f_i\}_{i\in \mathbb {N}}$ if we take $X=[0,1]$ and $V=(-1,2)$ . As for Condition (5), the cone condition can be replaced with a weaker condition (see [Reference Graf, Mauldin and Williams11, page 72] and [Reference Mauldin and Urbański20, Section 2]):

(1-2) $$ \begin{align} \inf\limits_{x\in\partial X}\inf\limits_{0<r<1}\mathcal{L}^{n}(B(x,r)\cap\text{Int}X)/\mathcal{L}^{n}(B(x,r))>0, \end{align} $$

where $B(x,r)$ denotes the open ball with center x and radius r. In our setting, Condition (1-2) is clearly satisfied when we let $X=[0,1]$ ; thus, $\mathcal {L}^{n}=\mathcal {L}$ , Int $X=(0,1)$ and ${\partial X=\{0,1\}}$ .

The topological pressure function P for the c.i.f.s $S=\{\phi _i\}_{i\in \mathbb {N}}$ is defined as

$$ \begin{align*} P(t):=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\log\sum\limits_{\omega\in\mathbb{N}^n}\|\phi_{\omega}^{'}\|^t. \end{align*} $$

Detailed properties of this pressure function can be found in [Reference Mauldin19–Reference Mauldin and Urbański21]. As shown in [Reference Mauldin19], there are two disjoint classes of c.i.f.s, regular and irregular. A system is called regular if there exists $t\geq 0$ such that $P(t)=0$ , otherwise irregular.

Let J be the attractor of the c.i.f.s $S=\{\phi _i\}_{i\in \mathbb {N}}$ , that is, $J=\bigcup _{i\in \mathbb {N}}\phi _i(J)$ . A Borel probability measure m is said to be t-conformal if it is supported on the set J, and for every Borel set $A\subseteq X$ and $i,j\in \mathbb {N},\ i\neq j$ ,

$$ \begin{align*} m(\phi_i(A))=\int_{A}|\phi_i^{'}|^t\,dm \quad \text{and}\quad m(\phi_i(X)\cap\phi_j(X))=0. \end{align*} $$

It is shown in [Reference Mauldin19] that the c.i.f.s is regular if and only if there exists a t-conformal measure (t is such that $P(t)=0$ ), and then $t=\dim _{\mathrm {{H}}}J$ .

The following lemma provides a way to determine the conformal measure.

For the regular system, there are at least two ways for deriving an ergodic invariant measure $m^{\ast }$ , which is equivalent to the t-conformal measure. One of the methods relies on using Banach limits (see [Reference Mauldin19, Theorems 8.1 and 8.2] for more details) and the other depends on applying the Frobenius–Perron operator $L=L_t:\ C(X)\rightarrow C(X)$ defined as

$$ \begin{align*} L(f)(x)=\sum\limits_{i\in\mathbb{N}}|\phi_i^{'}(x)|^tf(\phi_i(x)). \end{align*} $$

Lemma 1.4 [Reference Mauldin19, Theorem 8.3]

For m-almost every (a.e.) $x\in X$ , $\lim _{n\rightarrow \infty }L^n(1)(x)=g(x)$ exists and $g=dm^{\ast }/dm$ . In particular, for m-a.e. $x\in X$ ,

$$ \begin{align*} g(x)=L(g)(x)=\sum\limits_{i\in\mathbb{N}}|\phi_i^{'}(x)|^tg(\phi_i(x)). \end{align*} $$

Based on Lemmas 1.3 and 1.4, we immediately obtain the following result.

1.3 Statements of results

Much attention has been paid to the convergence exponent for kinds of sequences (see Pólya and Szegő [Reference Pólya and Szegő22, page 26]) in number theory and multifractal analysis of various dynamical systems in the past decades. For more details, we refer to [Reference Besicovitch and Taylor2, Reference Dodson4, Reference Fan, Liao, Ma and Wang6, Reference Wang and Wu25] and references therein. Following the work [Reference Fang, Ma, Song and Wu7, Reference Fang, Ma, Song and Yang8] in continued fraction systems, it is natural to ask whether the Hausdorff dimension of the level sets defined by the convergence exponent is closely related to polynomial decay of the derivative in d-decaying Gauss-like iIFS. In the following, we confirm that this is indeed the case. The convergence exponent of the sequence of digits $\{a_n(x)\}_{n\geq 1}$ for each $x\in (0,1)$ in d-decaying Gauss-like iIFS is defined by

(1-3) $$ \begin{align} \tau_1(x):=\inf \bigg\{s\geq0: \sum\limits_{n\geq1}a_n^{-s}(x)<\infty\bigg\}. \end{align} $$

Applying the Birkhoff ergodic theorem to d-decaying Gauss-like iIFS, we deduce from (1-1) and Theorem 1.5 that for $\mathcal {L}$ -a.e. $x \in (0,1)$ ,

(1-4) $$ \begin{align} \nonumber\lim\limits_{n\to\infty}\sqrt[n]{a_1(x)\cdots a_n(x)}&=e^{\int_{0}^{1}\log a_1(x)\,dm^\ast}\\ &\asymp e^{\sum_{k\geq1}\int_{I_1(k)}\log a_1(x)\,dx}\asymp e^{\sum_{k\geq1}({\log k}/{k^d})}. \end{align} $$

Here, we write $a\asymp b$ for $1/C\leq a/b\leq C$ , where C is an absolute constant. Then, it is immediate that infinitely many of $a_n(x)$ are less than some positive constant for $\mathcal {L}$ -a.e $x \in (0,1)$ . This shows that $\tau _1(x)=\infty $ for $\mathcal {L}$ -a.e. $x \in (0,1)$ . Therefore, it is natural to investigate the sizes of such Lebesgue null sets from the viewpoint of multifractal analysis, that is, the Hausdorff dimension of

$$ \begin{align*} E(\alpha):= \{x\in(0,1): \tau_1(x)=\alpha\}, \quad 0\leq\alpha<\infty. \end{align*} $$

Theorem 1.6. For any $0\leq \alpha <\infty $ ,

$$ \begin{align*} \dim_{\mathrm{H}}E(\alpha)=1/d. \end{align*} $$

As we know, the multifractal properties of sets relevant to the growth rate of the product of two consecutive digits have become an emerging and vital subject in view of the pioneering work of Kleinbock and Wadleigh [Reference Kleinbock and Wadleigh16], who considered the improvability of Dirichlet’s theorem in metric Diophantine approximation. In recent years, there have been abundant relevant results in continued fraction systems; see [Reference Bakhtawar and Feng1, Reference Huang and Wu13, Reference Hussain, Kleinbock, Wadleigh and Wang14, Reference Song and Zhang24] for more details. Motivated by this, we also wonder about the multifractal properties of the convergence exponent, which are relevant to the growth rate of weighted products of distinct digits with finite numbers in d-decaying Gauss-like iIFS. To be precise, we first consider the convergence exponent defined as

(1-5) $$ \begin{align} \tau_2(x):=\inf \bigg\{s\geq0: \sum\limits_{n\geq1}(a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x))^{-s}< \infty \bigg\}, \end{align} $$

where $m\geq 1$ and the weights $\{t_i\}_{0\leq i\leq m}$ are a sequence of nonnegative real numbers. Without loss of generality, we assume that $t_0\neq 0$ and at least one $t_i\neq 0\ (1\leq i\leq m)$ here and in the following. Notice that the assumption is valid. Indeed, for the case when the weights $\{t_i\}_{0\leq i\leq m}$ are a sequence of positive real numbers, the convergence exponent $\tau _2(x)$ is defined by the weighted products of consecutive digits, while for some $t_i=0\ (1\leq i\leq m)$ , it is defined by the weighted products of digits that are missing at some positions. It follows from (1-4) that $\tau _2(x)=\infty $ for $\mathcal {L}$ -a.e $x \in (0,1)$ . Then, we are also interested in the Hausdorff dimension of Lebesgue null sets

$$ \begin{align*} E(\alpha,\{t_i\}_{0\leq i\leq m}):= \{x\in(0,1): \tau_2(x)=\alpha\}, \quad 0\leq\alpha<\infty. \end{align*} $$

Theorem 1.7. For any $0\leq \alpha <\infty $ ,

$$ \begin{align*} \dim_{\mathrm{H}}E(\alpha,\{t_i\}_{0\leq i\leq m})=1/d. \end{align*} $$

Comparing the results of Theorems 1.6 and 1.7, we know that the Hausdorff dimensions of the corresponding level sets do not vary by changing the number of digits and weights. Thus, it is natural to consider the cases when the Hausdorff dimensions of the above level sets rely on the weights $\{t_i\}_{0\leq i\leq m}$ . Note that if the sequence of digits $\{a_n(x)\}_{n\geq 1}$ is nondecreasing, then the sequence $\{a^{t_0}_n(x)\cdots a^{t_m}_{n+m}(x)\}_{n\geq 1}$ is also nondecreasing. By a result of [Reference Wang and Wu25, Lemma 2.1], the convergence exponent defined by (1-5) can be written as

(1-6) $$ \begin{align} \tau_{2}(x)=\limsup\limits_{n\to\infty}\frac{\log n}{\log(a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x))}. \end{align} $$

This shows that the convergence exponent can be determined by the growth rate of digits. In this case, the multifractal spectrum of $\tau _{2}(x)$ is closely related to the Hausdorff dimension of the set of real numbers for which their digits tend to infinity with certain growth rates in some sense. In the following, we continue to study the multifractal spectrum of $\tau _{2}(x)$ , that is, the Hausdorff dimension of the intersection of sets $E(\beta ,\{t_i\}_{0\leq i\leq m})$ and $\Lambda $ for any $0\leq \beta \leq \infty $ , where

$$ \begin{align*} \Lambda=\{x\in(0,1): a_{n}(x)\leq a_{n+1}(x)\, \text{ for all }\ n\geq1\}. \end{align*} $$

It can be concluded by Theorem 1.1 that $\dim _{\mathrm {H}}\Lambda =1/d$ . Let $\alpha =1/\beta $ . Then, we deduce from (1-6) that

$$ \begin{align*} E(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})&:=E(\beta,\{t_i\}_{0\leq i\leq m})\cap\Lambda\\ &=\bigg\{x\in\Lambda: \liminf\limits_{n\to\infty}\frac{\log(a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x))}{\log n}=\alpha\bigg\}. \end{align*} $$

In the following, we show that the Hausdorff dimension of the level sets $E(\Lambda ,\alpha ,\{t_i\}_{0\leq i\leq m})$ is indeed dependent on the level $\alpha $ and the weights $\{t_i\}_{0\leq i\leq m}$ . Now, we are in a position to state the results.

Theorem 1.8. Let $\Sigma _t=\sum _{0\leq i\leq m}t_i>0$ . Then, for any $0\leq \alpha \leq \infty $ ,

$$ \begin{align*} \dim_{\mathrm{H}}E(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})= \begin{cases} 0,& 0\leq\alpha<\Sigma_t, \\ \dfrac{\alpha-\Sigma_t}{d\alpha}, & \Sigma_t\leq\alpha<\infty, \\[8pt] \dfrac{1}{d}, & \alpha=\infty. \end{cases} \end{align*} $$

Replacing the lower limit by the limit or upper limit in the set $E(\Lambda ,\alpha ,\{t_i\}_{0\leq i\leq m})$ , we are also interested in the Hausdorff dimension of the sets

$$ \begin{align*} F(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})=\bigg\{x\in\Lambda: \lim\limits_{n\to\infty} \frac{\log(a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x))}{\log n}=\alpha\bigg\} \end{align*} $$

and

$$ \begin{align*} G(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})=\bigg\{x\in\Lambda: \limsup\limits_{n\to\infty} \frac{\log(a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x))}{\log n}=\alpha\bigg\}. \end{align*} $$

Theorem 1.9. For any $0\leq \alpha \leq \infty $ ,

$$ \begin{align*} \dim_{\mathrm{H}}F(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})=\dim_{\mathrm{H}}G(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m})=\dim_{\mathrm{H}}E(\Lambda,\alpha,\{t_i\}_{0\leq i\leq m}). \end{align*} $$

Throughout this paper, we use $\mathcal {H}^{s}$ to denote the s-dimensional Hausdorff measure of a set, $\lfloor x\rfloor $ the largest integer not exceeding x, and $\sharp $ the cardinality of a set. The paper is organized as follows. In Section 3, we collect and establish some dimensional results for later use. Section 3 is devoted to the proofs of the main results.