Assouad-Type Dimensions of Homogeneous Moran Sets and Cantor-like Sets

Abstract

Assouad-type dimensions primarily describe the local structure of sets and have attracted significant attention in recent years within the field of fractal geometry. In this paper, we derive a formula for the Assouad dimension and establish an upper bound for the lower dimension for homogeneous Moran sets under the condition ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$. We also present formulas for the Assouad spectrum and the lower spectrum of Cantor-like sets. Our results generalize some results in the references.

1. Introduction

Let $(X,d)$ be a metric space. For any set $F\subset (X,d)$, define the Assouad dimension of F (denoted by ${\dim }_{A}F$) as

$$\begin{array}{cc}\hfill {\dim }_{A}F=\inf \{s\ge 0:& \mathrm{there}\mathrm{exist}b,c>0\mathrm{such}\mathrm{that}\mathrm{for}\mathrm{any}0<r<R<b,x\in F,\hfill \\ & N_r\left(B(x,R)\cap F\right)\le c{({\displaystyle \frac{R}{r}})}^s\},\hfill \end{array}$$

(1)

where $N_r\left(E\right)$ denotes the smallest number of balls with radius r needed to cover the set $E\subset X$. The Assouad dimension was first defined by Assouad [1,2], and it is used in the study of embedding theory in metric spaces. Unlike the Hausdorff dimension, the box dimension and the packing dimension, the Assouad dimension mainly describes the local structure of sets (the “density” of the “densest” part of a set), and it plays an effective role in fractal geometry. More details on the properties and applications of the Assouad dimension can be found in [3,4,5,6].

For any subset F in ${\mathbb{R}}^n$, we have ${\dim }_{H}F\le {\dim }_{P}F\le {\dim }_{A}F$ (${\dim }_H,{\dim }_{P}F$ denote the Hausdorff dimension and the packing dimension), and we also have ${\overline{\dim }}_{B}F\le {\dim }_{A}F$ (${\overline{\dim }}_B$ denotes the upper box dimension) for any bounded set F, see [3]. Furthermore, as shown in the literature [6], for any Ahlfors–David s-regular set F, the Assouad dimension of F is s, which coincides with the Hausdorff dimension and the packing dimension of F. It is difficult to calculate the Assouad dimension of sets that are not Ahlfors–David regular sets. Fraser [4] and Mackay [7] discussed the Assouad dimension of some quasi-self-similar and self-affine sets. Scholars have done many works on the fractal dimensions of the Moran-type sets, see [8,9,10]. For the Assouad dimension, Li, Li, Miao and Xi [11] obtained some results for the Assouad dimension of Cantor-like sets and some Moran sets. Chen [12]; Peng, Wang and Wen [13]; Yang and Du [14]; and Dai, Dong and Wei [15] discussed the Assouad dimension for some fractal sets in $\mathbb{R}$, such as Moran cut-out sets under some conditions and some special homogeneous Moran sets. In this paper, we study the Assouad dimension of homogeneous Moran sets satisfying ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, and obtain a formula for the Assouad dimension, which generalizes the results in [11,13,14,15].

As a dual to the Assouad dimension, Larman [16] introduced the lower dimension. Let $(X,d)$ be a metric space and, for any set $F\subset (X,d)$, define the lower dimension of F (denoted by ${\dim }_{L}F$) as

$$\begin{array}{cc}\hfill {\dim }_{L}F=\sup \{s\ge 0:& \mathrm{there}\mathrm{exist}b,c>0\mathrm{such}\mathrm{that}\mathrm{for}\mathrm{any}0<r<R<b,x\in F,\hfill \\ & N_r\left(B(x,R)\cap F\right)\ge c{({\displaystyle \frac{R}{r}})}^s\}.\hfill \end{array}$$

(2)

Similar to the Assouad dimension, the lower dimension mainly describes the local structure of the sets (the “sparsity” of the “sparsest” part of a set), and also plays an effective role in fractal geometry. However, unlike the Assouad dimension, the lower dimension does not have monotonicity and finite stability properties. More details on the properties and applications of the lower dimension can be found in the literature [3].

By [16], we have ${\dim }_{L}F\le {\underline{\dim }}_{B}F\le {\overline{\dim }}_{B}F\le {\dim }_{A}F$ (${\underline{\dim }}_B$ denotes the lower box dimension) for any totally bounded set F. In general, since the lower dimension does not have monotonicity and finite stability properties, it is always difficult to study the lower dimension of fractal sets. Chen, Wu, and Wei [17] discussed the lower dimensions for some Moran sets. Chen [12], Yang, Li and Hu [18] obtained some results of the lower dimension formulas for some Moran cut-out sets and some homogeneous Moran sets. In this paper, we study the lower dimensions for the homogeneous Moran sets which satisfy ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, and obtain an upper bound of the lower dimension.

From the definitions above, it can be seen that the Assouad dimension and the lower dimension are defined by an infimum and a supremum depending on the two independent scales r and R, but there is no information available as to how the scales approach these values. To see how the infimum and supremum depend on the scales, Fraser and Han [19] restricted the relationship between r and R with ${\displaystyle \frac{\log R}{\log r}}=\theta (0<\theta <1)$ and gave the definitions of the Assouad spectrum and the lower spectrum. Let $(X,d)$ be a metric space, for any set $F\subset (X,d)$ and any $0<\theta <1$, and define the Assouad spectrum of F (denoted by ${\dim }_A^{\theta }F$) as

$$\begin{array}{cc}\hfill {\dim }_A^{\theta }F=\inf \{s\ge 0:& \mathrm{there}\mathrm{exist}b,c>0\mathrm{such}\mathrm{that}\mathrm{for}\mathrm{any}0<R<b,x\in F,\hfill \\ & N_{R^{{\displaystyle \frac{1}{\theta }}}}\left(B(x,R)\cap F\right)\le c{({\displaystyle \frac{R}{R^{\frac{1}{\theta }}}})}^s\},\hfill \end{array}$$

(3)

and define the lower spectrum of F (denoted by ${\dim }_L^{\theta }F$) as

$$\begin{array}{cc}\hfill {\dim }_L^{\theta }F=\sup \{s\ge 0:& \mathrm{there}\mathrm{exist}b,c>0\mathrm{such}\mathrm{that}\mathrm{for}\mathrm{any}0<R<b,x\in F,\hfill \\ & N(B(x,R)\cap F,R^{{\displaystyle \frac{1}{\theta }}})\ge c{({\displaystyle \frac{R}{R^{\frac{1}{\theta }}}})}^s\}.\hfill \end{array}$$

(4)

If F is a totally bounded set, it is clear that ${\dim }_A^{\theta }F$ and ${\dim }_L^{\theta }F$ are continuous functions for any $0<\theta <1$, and we have

$${\overline{\dim }}_{B}F\le {\dim }_A^{\theta }F\le \min \left\{{\displaystyle \frac{{\overline{\dim }}_{B}F}{1-\theta }},{\dim }_{A}F\right\}$$

and

$${\dim }_{L}F\le {\dim }_L^{\theta }F\le {\underline{\dim }}_{B}F.$$

More details for the properties and the applications of the Assouad spectrum and the lower spectrum can be found in [3,19].

Fraser [4] studied the Assouad spectrums and the lower spectrums for some classical self- similar sets and self-conformal sets. Chen [12] obtained some results of the Assouad spectrums and the lower spectrums for some special Moran cut-out sets. Yang and Du [14] obtained some results of the Assouad spectrums. Yang, Li and Hu [18] obtained some results of the lower spectrums for a class of special homogeneous Moran sets which are called homogeneous perfect sets under some conditions. Chen, Wu, and Chang [20] studied some properties of the lower spectrums for uniformly perfect sets. In this paper, we study the Assouad spectrums and the lower spectrums for the Cantor-like sets, and obtain the Assouad spectrum and the lower spectrum formulas for the Cantor-like sets, which generalizes the results in [14,18].

The paper is organized as follows. In Section 2, since the homogeneous Moran sets and the Cantor-like sets are the main research objects of the paper, we recall their definitions and give some remarks. In Section 3, we research the Assouad dimensions and the lower dimensions of the homogeneous Moran sets, and we state our main results and give their proofs. In Section 4, we research the Assouad spectrums and the lower spectrums of the Cantor-like sets; we first give some background and then state our main results and give their proofs.

2. Preliminaries

We recall the definition of the homogeneous Moran sets. Let the sequences ${\left\{n_k\right\}}_{k\ge 1}\subset {\mathbb{Z}}^{+}$ and ${\left\{c_k\right\}}_{k\ge 1}\subset {\mathbb{R}}^{+}$ with $n_k\ge 2$, $n_k{c}_k<1$ for any $k\ge 1$. Set ${\Omega }_0=\{\varnothing \}$, ${\Omega }_k=\{{\sigma }_1{\sigma }_2\cdots {\sigma }_k:1\le {\sigma }_j\le n_j,1\le j\le k\}$ for any $k\ge 1$, and define $\Omega ={\bigcup }_{k=0}^{\infty }{\Omega }_k$. For any $k\ge 1$, $m\ge 1$, if $\sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_k\in {\Omega }_k$,$\tau ={\tau }_1{\tau }_2\cdots {\tau }_m(1\le {\tau }_j\le n_{k+j},1\le j\le m)$, then write $\sigma \ast \tau ={\sigma }_1\cdots {\sigma }_k{\tau }_1\cdots {\tau }_m\in {\Omega }_{k+m}$.

Definition 1

(Homogeneous Moran sets [21]). Let $I\ne \varnothing$ be a closed interval in $\mathbb{R}$. We say the collection of the closed intervals $\mathcal{F}=\{I_{\sigma }\subset I:\sigma \in \Omega \}$ fulfills the homogeneous Moran structure if it satisfies:

(i)

$I_{\varnothing }=I$;

(ii)

For any $k\ge 1$ and $\sigma \in {\Omega }_{k-1}$, $I_{\sigma \ast 1,}$$I_{\sigma \ast 2,}\cdots$, $I_{\sigma \ast n_k}$ are closed intervals with ${\bigcup }_{i=1}^{n_k}{I}_{\sigma \ast i}\subset I_{\sigma }$ and $\mathit{int}\left(I_{\sigma \ast i_1}\right)\cap \mathit{int}\left(I_{\sigma \ast i_2}\right)=\varnothing$ for any $1\le i_1<i_2\le n_k$, where $\mathit{int}\left(A\right)$ denotes the interiors of the set $A\subset \mathbb{R}$;

(iii)

For any $k\ge 1$, $\sigma \in {\Omega }_{k-1}$, and $1\le i\le n_k$,

$${\displaystyle \frac{|I_{\sigma \ast i}|}{|I_{\sigma }|}}=c_k,$$

where $\left|A\right|$ denotes the diameter of the set $A\subset \mathbb{R}$.

If the collection $\mathcal{F}=\{I_{\sigma }:\sigma \in \Omega \}$ fulfills the homogeneous Moran structure, then we call

$$E=E\left(\mathcal{F}\right):=\bigcap _{k\ge 1}\bigcup _{\sigma \in {\Omega }_k}{I}_{\sigma }$$

(5)

• a homogeneous Moran set defined by $\mathcal{F}$. For any $k\ge 1$, if $\sigma \in {\Omega }_k$, then $I_{\sigma }\in \mathcal{F}$ is called a k-level basic interval of E. Denote by $\mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$ the class of all homogeneous Moran sets which are associated with $I,\left\{n_k\right\}$ and $\left\{c_k\right\}$.

Remark 1.

Without loss of generality, we assume that $I=[0,1]$.

Let

$$N_k=\prod _{i=1}^k{n}_i,{\delta }_0=1,{\delta }_k=\prod _{i=1}^k{c}_i$$

for any $k\ge 1$; then, $N_k$ is the total number of the k-level basic intervals of E and ${\delta }_k$ is the length of any k-level basic interval of E.

Remark 2.

Note that $n_k\ge 2$, $n_k{c}_k<1$ for any $k\ge 1$ implies that $c_k<{\displaystyle \frac{1}{2}}$ for any $k\ge 1$. Then we have ${\lim }_{k\to +\infty }{\delta }_k=0$.

Remark 3.

Any uniform Cantor set (see [13]) is a homogeneous Moran set, any homogeneous perfect set (see in [14]) is a homogeneous Moran set, and any parameterized homogeneous Moran set (see in [15]) is a homogeneous Moran set.

More details on the homogeneous Moran sets can be found in [8,9,21,22,23].

The definition of the Cantor-like sets is introduced by Li, Li, Miao and Xi in [11] as follows.

Definition 2

(Cantor-like sets [11]). Let the sequence ${\left\{a_k\right\}}_{k\ge 1}\subset {\mathbb{R}}^{+}$ satisfy ${\sum }_{k=1}^{\infty }{a}_k<+\infty$. Let ${\left\{n_k\right\}}_{k\ge 1}\subset {\mathbb{Z}}^{+}$ and ${\left\{c_k\right\}}_{k\ge 1}\subset {\mathbb{R}}^{+}$ be the sequences with $n_k\ge 2$, $0<c_k<1$ and $c_{\ast }={\inf }_k\left\{c_k\right\}>0$ for any $k\ge 1$. Set ${\Omega }_0=\{\varnothing \}$, ${\Omega }_k=\{{\sigma }_1{\sigma }_2\cdots {\sigma }_k:1\le {\sigma }_j\le n_j,1\le j\le k\}$ for any $k\ge 1$, and define $\Omega ={\bigcup }_{k=0}^{\infty }{\Omega }_k$. For any $k\ge 1$, $m\ge 1$, if $\sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_k\in {\Omega }_k$,$\tau ={\tau }_1{\tau }_2\cdots {\tau }_m(1\le {\tau }_j\le n_{k+j},1\le j\le m)$; then, write $\sigma \ast \tau ={\sigma }_1\cdots {\sigma }_k{\tau }_1\cdots {\tau }_m\in {\Omega }_{k+m}$.

Let $J\subset {\mathbb{R}}^n(n\ge 1)$ be a closed set with $\mathit{int}\left(J\right)\ne \varnothing$. For any $k\ge 1$ and $\sigma \in {\Omega }_{k-1}$, if

$$J_{\sigma \ast 1},J_{\sigma \ast 2},\cdots ,J_{\sigma \ast n_k}\subset J_{\sigma }$$

are geometrically similar to $J_{\sigma }$ with

$$c_k(1-a_k)\le {\displaystyle \frac{|J_{\sigma \ast j}|}{|J_{\sigma }|}}\le c_k(1+a_k)$$

for any $1\le j\le n_k$, and $\mathit{int}\left(J_{\sigma \ast i_1}\right)\cap \mathit{int}\left(J_{\sigma \ast i_2}\right)=\varnothing$ for any $1\le i_1<i_2\le n_k$, then

$$E=\bigcap _{k\ge 1}\bigcup _{\sigma \in {\Omega }_k}{J}_{\sigma }$$

(6)

is called a Cantor-like set. Denote by $\mathcal{C}(J,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$ the class of all Cantor-like sets associated with $J,\left\{n_k\right\},\left\{c_k\right\}$ and $\left\{a_k\right\}$.

For any $k\ge 1$, we also write

$$N_k=\prod _{i=1}^k{n}_i,{\delta }_0=1,{\delta }_k=\prod _{i=1}^k{c}_i.$$

Remark 4.

Note that for any $k\ge 1$, we have $n_k\ge 2$, and for any $\sigma \in {\Omega }_{k-1}$, we have $\mathit{int}\left(J_{\sigma \ast i_1}\right)\cap \mathit{int}\left(J_{\sigma \ast i_2}\right)=\varnothing$ for any $1\le i_1<i_2\le n_k$. Combining ${\sum }_{k=1}^{\infty }{a}_k<+\infty$, we have ${\lim }_{k\to +\infty }{\delta }_k=0$.

Remark 5.

If $E\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$ is a homogeneous Moran set with $c_{\ast }={\inf }_k\left\{c_k\right\}>0$, then E is a Cantor-like set with $E\in \mathcal{C}(I,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$ and $a_k=0$ for all $k\ge 1$. Cantor-like sets need not be Moran sets (see [21]).

3. The Assouad Dimensions and the Lower Dimensions of the Homogeneous Moran Sets

3.1. Main Results

In this section, we research the Assouad dimensions and the lower dimensions of the homogeneous Moran sets, our main results are stated as follows.

Theorem 1.

Let $E\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$. If ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, then

$${\dim }_{A}E=\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}.$$

Remark 6.

Any homogeneous Cantor set $E\in \mathcal{C}(\left\{n_k\right\},\left\{c_k\right\})$ (the definition of the homogeneous Cantor set can be seen in [13]), any homogeneous perfect set $F\in \mathcal{J}(I,\left\{n_k\right\},\left\{c_k\right\},\left\{{\eta }_{k,j}\right\})$ (the definition of the homogeneous perfect set can be seen in [14]) and any parameterized homogeneous Moran set $G\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\},\left\{{\epsilon }_k\left(t\right)\right\})$ (the definition of the parameterized homogeneous Moran set can be seen in [15]) are homogeneous Moran sets, where $E,F,G\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$. If $c_{\ast }={\inf }_k\left\{c_k\right\}>0$, then ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$. Thus Theorem 1 generalizes Theorem 2.1 of [13] and Theorem 1 of [15] under the condition ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$; it also generalizes Theorem 1(2) of [14] and Corollary 1 of [11].

Theorem 2.

Let $E\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$. If ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, then

$${\dim }_{L}E\le \underset{l\to +\infty }{\lim \mathrm{lnf}}\underset{k\ge 1}{\inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}.$$

3.2. The Equivalent Definitions of the Assouad Dimension and the Lower Dimension

To prove Theorems 1 and 2, we require the following equivalent definitions of the Assouad dimension and the lower dimension (see [12]).

Lemma 1

(Equivalent definitions [12]). Let F be a compact set in a metric space $(X,d)$; then,

$$\begin{array}{cc}\hfill {\dim }_{A}F=\inf \{s\ge 0:& \mathit{there}\mathit{exist}b,c>0\mathit{such}\mathit{that}\mathit{for}\mathit{any}0<r<b,\mathit{any}\hfill \\ & r<R<\left|F\right|,\mathit{and}\mathit{any}x\in F,N_r\left(B(x,R)\cap F\right)\le c{({\displaystyle \frac{R}{r}})}^s\},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\dim }_{L}F=\sup \{s\ge 0:& \mathit{there}\mathit{exist}b,c>0\mathit{such}\mathit{that}\mathit{for}\mathit{any}0<r<b,\mathit{any}\hfill \\ & r<R<\left|F\right|,\mathit{and}\mathit{any}x\in F,N_r\left(B(x,R)\cap F\right)\ge c{({\displaystyle \frac{R}{r}})}^s\}.\hfill \end{array}$$

(8)

3.3. Proofs of Theorems 1 and 2

We start to study the Assouad dimensions and the lower dimensions of homogeneous Moran sets under the condition ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$. Let $M={\sup }_{k\ge 1}\left\{n_k\right\}$.

Proof of Theorem 1.

To prove Theorem 1, we first prove that

$${\dim }_{A}E\ge \underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}.$$

If $s>{\dim }_{A}E$, by (7), there exist constants $b,C>0$, such that for any $0<r<b$, any $r<R<\left|E\right|$, and any $x\in E$,

$$N_r\left(B(x,R)\cap E\right)\le C{({\displaystyle \frac{R}{r}})}^s.$$

Then there exist $L_1>0$ and $C_1>0$, such that for any $k\ge 1$, $l\ge L_1$, $R={\delta }_k$ and $r={\delta }_{k+l}$,

$$N_r\left(B(x,R)\cap E\right)\le C_1{({\displaystyle \frac{R}{r}})}^s=C_1{(c_{k+1}···c_{k+l})}^{-s}$$

(9)

for any $x\in E$.

Note that for any $x\in E$, x is contained in a $(k+1)$-level basic interval of E and the diameter of any $(k+1)$-level basic interval of E is ${\delta }_{k+1}<{\delta }_k=R={\displaystyle \frac{1}{2}}\left|B(x,R)\right|$. Then, $B(x,R)$ contains at least one $(k+1)$-level basic interval of E. Note that any $x\in E$, x is also contained in a $(k+l)$-level basic interval of E and the diameter of any $(k+l)$-level basic interval of E is ${\delta }_{k+l}=r={\displaystyle \frac{1}{2}}\left|B(x,r)\right|$ (then $2{\delta }_{k+l}=\left|B(x,r)\right|$). Considering that the interiors of any two basic intervals of E are disjoint, we have that $B(x,r)$ meets at most four $(k+l)$-level basic intervals of E (otherwise $B(x,r)$ may meet at least five $(k+l)$-level basic intervals of E, which implies that $\left|B(x,r)\right|\ge 3{\delta }_{k+l}$, which contradicts $2{\delta }_{k+l}=\left|B(x,r)\right|$). Thus

$$\begin{array}{cc}\hfill N_r\left(B(x,R)\cap E\right)& \ge {\displaystyle \frac{n_{k+2}···n_{k+l}}{4}}\hfill \\ & \ge {\displaystyle \frac{1}{4M}}(n_{k+1}···n_{k+l})\hfill \end{array}$$

(10)

for any $x\in E$.

Combining (9) and (10), we have

$${\displaystyle \frac{1}{4M}}(n_{k+1}···n_{k+l})\le C_1{(c_{k+1}···c_{k+l})}^{-s}$$

(11)

for any $k\ge 1$ and $l\ge L_1$. This implies that there is a constant $C_2$, such that

$${\displaystyle \frac{\log n_{k+1}···n_{k+l}}{-\log c_{k+1}···c_{k+l}}}\le s+{\displaystyle \frac{C_2}{\log c_{k+1}···c_{k+l}}}$$

for any $k\ge 1$ and $l\ge L_1$.

Let $l\to +\infty$. We have

$$\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}\le s.$$

By the arbitrariness of $s>{\dim }_{A}E$, we have

$${\dim }_{A}E\ge \underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}.$$

Now we prove

$${\dim }_{A}E\le \underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}.$$

For any $s>\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$, there is a positive integer $L_2$, such that for any $l\ge L_2$ and $k\ge 1$, we have

$$s>{\displaystyle \frac{\log n_{k+1}···n_{k+l}}{-\log c_{k+1}···c_{k+l}}}.$$

Otherwise for any $\overline{L}>1$, there exist $\overline{L_1}>\overline{L}$ and $\overline{k}\ge 1$, such that ${\displaystyle \frac{\log n_{\overline{k}+1}···n_{\overline{k}+\overline{L_1}}}{-\log c_{\overline{k}+1}···c_{k+\overline{L_1}}}}\ge s$, which contradicts $s>\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$. Then we have

$$n_{k+1}···n_{k+l}\le {(c_{k+1}···c_{k+l})}^{-s}.$$

(12)

For any $0<R<{\delta }_1$, there exists a positive integer $k\ge 2$, such that

$${\delta }_k<R\le {\delta }_{k-1}.$$

(13)

For any $0<r<{\delta }_{k+L_2}$, there exists a positive integer $l\ge L_2$, such that

$${\delta }_{k+l+1}<r\le {\delta }_{k+l}.$$

(14)

Then for any $x\in E$, $B(x,R)$ meets at most four $(k-1)$-level basic intervals of E, and $B(x,r)$ contains at least one $(k+l+1)$-level basic interval of E. Note that $l\ge L_2$, and by (12)–(14), we have

$$\begin{array}{cc}\hfill N_r\left(B(x,R)\cap E\right)& \le 4n_k···n_{k+l+1}\hfill \\ & \le 4M^2(n_{k+1}···n_{k+l})\hfill \\ & \le 4M^2{(c_{k+1}···c_{k+l})}^{-s}\hfill \\ & \le 4M^2{({\displaystyle \frac{R}{r}})}^s\hfill \end{array}$$

(15)

for any $x\in E$.

For any ${\delta }_{k+L_2}\le r<R<{\delta }_1$, $B(x,r)$ contains at least one $(k+L_2+1)$-level basic interval of E. Then

$$\begin{array}{cc}\hfill N_r\left(B(x,R)\cap E\right)& \le 4n_k\cdots n_{k+L_2+1}\hfill \\ \hfill & \le 4M^{L_2+2}\hfill \\ \hfill & \le 4M^{L_2+2}{({\displaystyle \frac{R}{r}})}^s\hfill \end{array}$$

(16)

for any $x\in E$.

Combine (15) and (16), we have ${\dim }_{A}E\le s$. By the arbitrariness of $s>\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$, we obtain

$${\dim }_{A}E\le \underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}},$$

which completes the proof of Theorem 1. □

Proof of Theorem 2.

If $s<{\dim }_{L}E$, by (8), there exist constants $b,C>0$, such that for any $0<r<b$, any $r<R<\left|E\right|$, and any $x\in E$,

$$N_r\left(B(x,R)\cap E\right)\ge C{({\displaystyle \frac{R}{r}})}^s.$$

Then there exist $L_3>0$ and $C_3>0$, such that for any $k\ge 1$, $l\ge L_3$, $R={\delta }_k$ and $r={\delta }_{k+l}$,

$$N_r\left(B(x,R)\cap E\right)\ge C_3{({\displaystyle \frac{R}{r}})}^s=C_3{(c_{k+1}···c_{k+l})}^{-s}$$

(17)

for any $x\in E$.

Note that for any $x\in E$, by the same argument in the proof of Theorem 1, $B(x,R)$ meets at most four k-level basic intervals of E, and $B(x,r)$ contains at least one $(k+l+1)$-level basic interval of E. Thus

$$\begin{array}{cc}\hfill N_r\left(B(x,R)\cap E\right)& \le 4n_{k+1}···n_{k+l+1}\hfill \\ & \le 4M(n_{k+1}···n_{k+l})\hfill \end{array}$$

(18)

for any $x\in E$.

Combine (17) and (18), we have

$$4M(n_{k+1}···n_{k+l})\ge C_3{(c_{k+1}···c_{k+l})}^{-s}$$

(19)

for any $k\ge 1$ and $l\ge L_3$, which implies that there is a constant $C_4$, such that

$${\displaystyle \frac{\log n_{k+1}···n_{k+l}}{-\log c_{k+1}···c_{k+l}}}\ge s+{\displaystyle \frac{C_4}{\log c_{k+1}···c_{k+l}}}$$

for any $k\ge 1$ and $l\ge L_3$.

Let $l\to +\infty$. We have

$$\underset{l\to +\infty }{\lim \inf }\underset{k\ge 1}{\inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}\ge s.$$

By the arbitrariness of $s<{\dim }_{L}E$, we have

$${\dim }_{L}E\le \underset{l\to +\infty }{\lim \inf }\underset{k\ge 1}{\inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}},$$

which completes the proof of Theorem 2. □

Remark 7.

To estimate the lower dimension, we cannot use the dual method of the estimation of the Assouad dimension to estimate the lower bound since $c_{\ast }={\inf }_k\left\{c_k\right\}=0$ implies that ${\displaystyle \frac{R}{r}}$ has no uniformly upper bound when ${\delta }_{k+L_2}\le r<R$. It could be the case that $\underset{l\to +\infty }{\lim \inf }\underset{k\ge 1}{\inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$ is not the supremum of the lower dimension of a homogeneous Moran set $E\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$ with ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, but we have not found any examples to support this.

4. The Assouad Spectrums and the Lower Spectrums of the Cantor-like Sets

4.1. Some Background

Let X be a metric space, $s\ge 0$, and A be a compact set in X. If for any $x\in A$ and $0<r\le \left|A\right|$, there exist $c\ge 1$ and a Borel measure $\mu$ supported on A satisfying

$$c^{-1}{r}^s\le \mu \left(B(x,r)\right)\le c{r}^s,$$

then we call A an Ahlfors s-regular set [24].

The Ahlfors regular sets contain many typical fractal sets, such as the self-similar and the self-conformal sets which satisfy the open set condition, which play important roles in the study of fractal geometry. For any Ahlfors s-regular set F, the Assouad dimension of F is s, which coincides with the Hausdorff dimension and the packing dimension of F, see [6].

Reference [24] showed that for any Ahlfor s-regular set A, there are constants $\lambda \ge 1$, $a\in (0,1)$, and $1<\alpha \le \beta <+\infty$, such that for any $x,x_1,x_2\in A$ and $r\in (0,|A\left|\right)$,

$${\lambda }^{-1}\le {\displaystyle \frac{\mu \left(B(x_1,r)\right)}{\mu \left(B(x_2,r)\right)}}\le \lambda ,$$

(20)

$$\alpha \le {\displaystyle \frac{\mu \left(B(x,r)\right)}{\mu \left(B(x,ar)\right)}}\le \beta ,$$

(21)

which implies that $\mu$ is a doubling measure (the definition of a doubling measure can be seen in [24]) on the set A. For any set E satisfying (20) and (21), Lü, Lou, Wen and Xi [25] showed that ${\dim }_{H}E=\underset{r\to 0}{\lim \inf }h\left(r\right),{\dim }_{P}E=\underset{r\to 0}{\lim \sup }h\left(r\right)$, where $h\left(r\right):(0,ϵ)\to {\mathbb{R}}^{+}$, which is called the scale function of E, which is the function such that

$$0<\underset{r<ϵ}{\inf }h\left(r\right)\le \underset{r<ϵ}{\sup }h\left(r\right)<+\infty$$

(22)

for a positive constant $ϵ$. Furthermore, for any $x\in E$ and $0<r<ϵ$, there exists a constant $c\ge 0$ with

$$|h\left(r\right)\log r-\log \mu \left(B(x,r)\right)|\le c.$$

(23)

For an Ahlfors s-regular set, we can take $h\left(r\right)=s$.

4.2. Main Results

In this section, we research the Assouad spectrums and the lower spectrums of the Cantor-like sets, our main results are stated as follows.

Theorem 3.

Let $E\in \mathcal{C}(J,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$, and then

$${\dim }_A^{\theta }E=\underset{k\to +\infty }{\lim \sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}},$$

where $l(\theta ,k)=\max \{l\in {\mathbb{Z}}^{+}:{\delta }_l\ge {\left({\delta }_k\right)}^{{\displaystyle \frac{1}{\theta }}}\}$ for any $k\ge 1$.

Theorem 4.

Let $E\in \mathcal{C}(J,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$, and then

$${\dim }_L^{\theta }E=\underset{k\to +\infty }{\lim \inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}},$$

where $l(\theta ,k)=\max \{l\in {\mathbb{Z}}^{+}:{\delta }_l\ge {\left({\delta }_k\right)}^{{\displaystyle \frac{1}{\theta }}}\}$ for any $k\ge 1$.

Remark 8.

Any homogeneous perfect set $E\in \mathcal{J}(I,\left\{n_k\right\},\left\{c_k\right\},\left\{{\eta }_{k,j}\right\})$ (the definition of a homogeneous perfect set can be seen in [14]) with $c_{\ast }={\inf }_k\left\{c_k\right\}>0$ is a Cantor-like set, where $E\in \mathcal{C}(I,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$ with $a_k=0$ for all $k\ge 1$. Thus Theorem 3 generalizes Theorem 2 of [14] and Theorem 4 generalizes Theorem 2 of [18].

4.3. Proofs of Theorems 3 and 4

We are ready to prove Theorems 3 and 4. The following propositions provide expressions for the Assouad spectrum and the lower spectrum in terms of the scale function for the sets, satisfying (20) and (21).

Proposition 1

([14]). Let F be a set in a metric space X, and F satisfies (20) and (21) with a scale function $h\left(r\right)$. Then,

$${\dim }_A^{\theta }F=\underset{r\to 0}{\lim \sup }\left|{\displaystyle \frac{h\left(r\right)\log r-h(r^{\frac{1}{\theta }})\log r^{\frac{1}{\theta }}}{(1-\frac{1}{\theta })\log r}}\right|.$$

(24)

Proposition 2

([18]). Let F be a set in a metric space X, and F satisfies (20) and (21) with a scale function $h\left(r\right)$. Then,

$${\dim }_L^{\theta }F=\underset{r\to 0}{\lim \inf }\left|{\displaystyle \frac{h\left(r\right)\log r-h(r^{\frac{1}{\theta }})\log r^{\frac{1}{\theta }}}{(1-\frac{1}{\theta })\log r}}\right|.$$

(25)

The next proposition shows a scale function of a Cantor-like set.

Proposition 3

([11]). Let $E\in \mathcal{C}(J,\left\{n_k\right\},\left\{c_k\right\},\left\{a_k\right\})$, and then F satisfies (20) and (21) with a scale function

$$h\left(r\right)={\displaystyle \frac{\log N_k}{-\log {\delta }_k}}$$

for any $k\ge 1$ and ${\delta }_k\left|J\right|<r\le {\delta }_{k-1}\left|J\right|$.

Now we start the final proof of Theorems 3 and 4.

Let $E\in \mathcal{C}(J,\left\{n_k\right\},\left\{c_k\right\},\left\{a_k\right\})$. For any $r\in (0,\min \{1,\left|J\right|\left\}\right)$, there is $k\ge 1$ satisfying

$${\delta }_k\left|J\right|<r\le {\delta }_{k-1}\left|J\right|.$$

Since $l(\theta ,k)=\max \{l\in {\mathbb{Z}}^{+}:{\delta }_l\ge {\left({\delta }_k\right)}^{{\displaystyle \frac{1}{\theta }}}\}$, we have $l(\theta ,k)\ge k$ and

$${\delta }_{l(\theta ,k)+1}<{\left({\delta }_k\right)}^{{\displaystyle \frac{1}{\theta }}}\le {\delta }_{l(\theta ,k)}.$$

Let $l^{\ast }(\theta ,r)$ be the non-negative integer satisfying

$${\delta }_{l^{\ast }(\theta ,r)+1}<{({\displaystyle \frac{r}{\left|J\right|}})}^{{\displaystyle \frac{1}{\theta }}}\le {\delta }_{l^{\ast }(\theta ,r)}.$$

Note that

$${\delta }_{l(\theta ,k)+1}<{\left({\delta }_k\right)}^{{\displaystyle \frac{1}{\theta }}}<{({\displaystyle \frac{r}{\left|J\right|}})}^{{\displaystyle \frac{1}{\theta }}}$$

and

$${({\displaystyle \frac{r}{\left|J\right|}})}^{{\displaystyle \frac{1}{\theta }}}\le {\left({\delta }_{k-1}\right)}^{{\displaystyle \frac{1}{\theta }}}\le {\delta }_{l(\theta ,(k-1\left)\right)},$$

Then we have

$$l(\theta ,(k-1))\le l^{\ast }(\theta ,r)\le l(\theta ,k).$$

Let $l^{\ast \ast }(\theta ,r)$ be a non-negative integer satisfying

$${\delta }_{l^{\ast \ast }(\theta ,r)+1}\left|J\right|<r^{{\displaystyle \frac{1}{\theta }}}\le {\delta }_{l^{\ast \ast }(\theta ,r)}\left|J\right|.$$

Since $0<c_{\ast }<c^{\ast }={\sup }_k\left\{c_k\right\}<1$ and ${\sum }_{k=1}^{+\infty }{a}_k<+\infty$, we have ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, ${\sup }_{k\ge 1}\{l(\theta ,k)-l(\theta ,(k-1))\}<+\infty$ and ${\sup }_{0<r<\min \{1,|J\left|\right\}}\left\{\right|l^{\ast \ast }(\theta ,r)-l^{\ast }(\theta ,r)\left|\right\}<+\infty$, which together with Propositions 1–3 yields

$$\begin{array}{cc}\hfill {\dim }_A^{\theta }E& =\underset{r\to 0}{\lim \sup }\left|{\displaystyle \frac{h\left(r\right)\log r-h(r^{\frac{1}{\theta }})\log r^{\frac{1}{\theta }}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{r\to 0}{\lim \sup }\left|{\displaystyle \frac{h\left(r\right)\log r-\frac{\log N_{l^{\ast \ast }(\theta ,r)+1}}{-\log {\delta }_{l^{\ast \ast }(\theta ,r)+1}}\log {\delta }_{l^{\ast \ast }(\theta ,r)+1}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{r\to 0}{\lim \sup }\left|{\displaystyle \frac{h\left(r\right)\log r-\frac{\log N_{l^{\ast }(\theta ,r)+1}}{-\log {\delta }_{l^{\ast }(\theta ,r)+1}}\log {\delta }_{l^{\ast }(\theta ,r)+1}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{k\to +\infty }{\lim \sup }\left|{\displaystyle \frac{\frac{\log N_k}{-\log {\delta }_k}\log {\delta }_k-\frac{\log N_{l(\theta ,k)}}{-\log {\delta }_{l(\theta ,k)}}\log {\delta }_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}\right|\hfill \\ & =\underset{k\to +\infty }{\lim \sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\dim }_L^{\theta }E& =\underset{r\to 0}{\lim \inf }\left|{\displaystyle \frac{h\left(r\right)\log r-h(r^{\frac{1}{\theta }})\log r^{\frac{1}{\theta }}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{r\to 0}{\lim \inf }\left|{\displaystyle \frac{h\left(r\right)\log r-\frac{\log N_{l^{\ast \ast }(\theta ,r)+1}}{-\log {\delta }_{l^{\ast \ast }(\theta ,r)+1}}\log {\delta }_{l^{\ast \ast }(\theta ,r)+1}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{r\to 0}{\lim \inf }\left|{\displaystyle \frac{h\left(r\right)\log r-\frac{\log N_{l^{\ast }(\theta ,r)+1}}{-\log {\delta }_{l^{\ast }(\theta ,r)+1}}\log {\delta }_{l^{\ast }(\theta ,r)+1}}{(1-\frac{1}{\theta })\log r}}\right|\hfill \\ & =\underset{k\to +\infty }{\lim \inf }\left|{\displaystyle \frac{\frac{\log N_k}{-\log {\delta }_k}\log {\delta }_k-\frac{\log N_{l(\theta ,k)}}{-\log {\delta }_{l(\theta ,k)}}\log {\delta }_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}\right|\hfill \\ & =\underset{k\to +\infty }{\lim \inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}.\hfill \end{array}$$

5. Conclusions

In this paper, we prove that if $E\in \mathcal{M}(I,\left\{n_k\right\},\left\{c_k\right\})$ is a homogeneous Moran set with ${\sup }_{k\ge 1}\left\{n_k\right\}<+\infty$, then the Assouad dimension of E is $\underset{l\to +\infty }{\lim \sup }\underset{k\ge 1}{\sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$, and an upper bound of the lower dimension of E is $\underset{l\to +\infty }{\lim \inf }\underset{k\ge 1}{\inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{k+l}}{-\log c_{k+1}\cdots c_{k+l}}}$. We also prove that if $E\in \mathcal{C}(J,\left\{c_k\right\},\left\{n_k\right\},\left\{a_k\right\})$ is a Cantor-like set, then the Assouad spectrum of E is ${\dim }_A^{\theta }E=\underset{k\to +\infty }{\lim \sup }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}$, and the lower spectrum of E is ${\dim }_L^{\theta }E=\underset{k\to +\infty }{\lim \inf }{\displaystyle \frac{\log n_{k+1}\cdots n_{l(\theta ,k)}}{(1-\frac{1}{\theta })\log {\delta }_k}}$.