#### Abstract

We compute the Assouad dimensions and the lower dimensions of a class of homogeneous Cantor sets without the condition that the smallest compression ratio and find that the lower dimension of a homogeneous Cantor set may be any a number in the interval and the Assouad dimension of may be any a number in the interval .

#### 1. Introduction

Let us begin with the definition of the Assouad dimension and the lower dimension. The Assouad dimension of a nonempty set is defined by

The lower dimension of is defined by

is the smallest number of open sets with a diameter less than needed to cover .

The Assouad dimension which is introduced by Assouad [1] has recently received an enormous interest in the mathematical literature due to its connections with the doubling property. The dual notion of the Assouad dimension is the lower dimension which was introduced by Larman [2]. Just like the Assouad dimension, the lower dimension has also received an enormous interest in the mathematical literature due to its connections with the uniform property of metric spaces. As a result of this, a large number of papers have investigated the Assouad dimension and the lower dimension of different classes of fractal sets. Fraser [3] has a detailed discussion of the Assouad dimension, the lower dimension, and its use in fractal geometry. Olsen [4] computed the Assouad dimension of a graph-directed Moran fractal satisfying the open-set conditions which are Ahlfors regular. However, in general, it is difficult to obtain the Assouad dimensions of sets which are not Ahlfors regular. Mackay [5] calculated the Assouad dimension of the self-affine carpets of Bedford and McMullen, and his main result solved the problem posed by Olsen [4]. For the Moran sets introduced by Wen [6] which are not Ahlfors regular, Li et al. [7] obtained the Assouad dimensions of Moran sets under suitable condition. Li [8] also proved that the Assouad dimensions of some Moran sets coincide with their packing and upper box dimensions under the condition that the smallest compression ratio and therefore gave a conjecture that the conclusion remains true if the condition is removed. Xiao [9] proved that the lower dimensions of a class of Moran sets coincide with their Hausdorff dimensions under the condition that the compression sequence and the compression ratio sequence are smoothly changing and found some homogeneous Cantor sets whose Assouad dimension is not equal to their upper box dimensions and packing dimensions and therefore gave a negative answer to the conjecture in the paper [8]. In this paper, we compute the Assouad dimensions and the lower dimensions of a class of homogeneous Cantor sets without the condition that the smallest compression ratio and find that the lower dimension of a homogeneous Cantor set may be any a number in the interval and the Assouad dimension of may be any a number in the interval .

#### 2. Homogeneous Cantor Set

Firstly, let us recall the definition of Moran sets introduced by Wen [5]. Let be a sequence of positive integer (we assume ). Define , and for any , set , and . If and , we define .

Definition 1. Suppose that is a compact set with . Let be a sequence of positive real vectors with , . We say the collection of closed subsets of possesses the Moran structure if it satisfies the following Moran structure conditions (MSC):(1)For , is geometrically similar to ; that is, there exists a similarity such that . For convenience, we write (2)For all and are subsets of and satisfy that whenever (3)For any and ,where denotes the diameter of .Suppose that is a collection of closed subsets of possessing the Moran structure, setIt is ready to see that is a nonempty compact set. The set is called the Moran set associated with the collection .
Let , and . The elements of are called kth-level basic sets of , and the elements of are called the basic sets of . Suppose the set and the sequences are given. We denote by the class of the Moran sets satisfying the MSC. We call the Moran class associated with the triplet .

Definition 2. Suppose is the interval [0,1] and for any in Definition 1. For all , and the left endpoint of is the left endpoint of and the right endpoint of is the right endpoint of . The set is called the homogeneous Cantor set. Write .

Remark 1. Let . In the present paper, the author obtains formulas for the Assouad dimension and the lower dimension of sets belonging to this class of Moran fractals without assuming the condition .

Theorem 1 (see [10]). Suppose the set is a homogeneous Cantor set. Then,where , , and denote the Hausdorff, packing, and upper box dimensions of , respectively.

#### 3. Statement and Proof of Results

Now, consider four conditions that and satisfy.

Condition 1. and there are two strictly increasing sequence and of positive integers with such that(a)(b)The following limitexists (note that is defined as the value of the limit in (6))(c)For each , there is an (only depending on ) such thatfor each pair of positive integers and with and .

Condition 2. .

Condition 3. and there are two strictly increasing sequence and of positive integers with such that(a)(b)The following limitexists (note that is defined as the value of the limit in (8))(c)For each , there is an (only depending on ) such thatfor each pair of positive integers and with and .

Condition 4.

Theorem 2. Suppose the set is a homogeneous Cantor set. Then,

Proof . of Theorem 2. For two positive numbers and with , there exist and such thatFor , we consider the relationship between and . Notice thatwhere is the length of the basic elements of order of the homogeneous Cantor set , and is the length of the interval among the basic elements of order of the homogeneous Cantor set . By a simple calculation, we obtain the relationship between and . The results are shown in Table 1. where .
Secondly, let , we consider the relationship between and . Denote by . By a simple calculation, we obtain the relationship between and . The results are shown in Table 2. where .
Next, we consider the relationship between and . Notice thatBy a simple calculation, we obtain the relationship between and . The results are shown in Tables 3 and 4.
Using (13) and Tables 13, we obtainFor entry 1 in Table 3, by a simple calculation, we obtainfor and .
In order to simplify expression, the notation is introduced. If there exist two constant positive numbers such that we remark . Denote by .
Using (13) and Tables 1 and 2, one getsfor and . It follows that where the constants and be chosen uniformly in and . By a simple calculation that is similar to entry 1 in Table 3, we obtain Table 4.
Notice that is decreasing for and is decreasing for , using (16) and Table 4, we obtain thatIf Condition 1 is satisfied, we prove . Using (c) in Condition 1 and (17) and notice that , it follows that for all small , there exists such thatfor . If , notice that ,By combining (18) and (19), we obtain that . To prove that , it suffices to find a constant and a sequence of points and scales such that and for all for all small . Recall that there exists a sequence whose limit is . Take and , there exists such thatfor all . The proof of is finished.
If Condition 2 is satisfied, there exists a subsequence such that . Let be the left endpoint of some basic elements of order , take and . It is obvious thatNotice that when ; therefore, .
Next, we prove . Using (c) in Condition 3 and (17), and notice that , it follows that for all small , there exists such thatfor . If . Notice that ,By combining (23) and (24), we obtain that . To prove that , it suffices to find a constant and a sequence of points and scales such that and for all for all small . Recall that there exists a sequence whose limit is . Take and , there exists such thatfor all . The proof of is finished.
Finally, we prove if . There exist a subsequence such that . Let be the left endpoint of some basic elements of order , take and . It is obvious thatNotice that when ; therefore, .

Corollary 1. Give a number and , for any number which satisfies that there exists a homogeneous Cantor set such that

Proof . of Corollary 1.

Case 1. .
Let be a sequence of integers such thatDefine the family of parameters as follows: ,By Theorem 1 and Theorem 2, it is obvious that .

Case 2. .
Let be a sequence of integers such thatDefine the family of parameters as follows: ,By Theorem 1 and Theorem 2, it is obvious that

Corollary 2. Give a number and , for any number which satisfies that there exists a homogeneous Cantor set such that

Proof . of Corollary 2.

Case 3. .
Let be a sequence of integers such thatDefine the family of parameters as follows: ,By Theorem 1 and Theorem 2, it is obvious that .

Case 4. .
For Case 2 in Corollary 1, notice that , it is obvious that .

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the scientific research project of the Hubei Provincial Department of Education (B2018358).