Assouad Dimensions and Lower Dimensions of Some Moran Sets

We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c∗ > 0.


Introduction
Let us begin with the definition of the Assouad dimension and the lower dimension. For r > 0, E ⊆ R d , and N r (E) denotes the smallest number of open sets required for an rcover of a bounded set E. Definition 1. e Assouad dimension of a nonempty set F ⊆ R d is defined by dim A F � inf α ≥ 0.
If the Hausdorff dimension provides fine, but global, geometric information, then the Assouad dimension which was introduced by Assouad [1] provides coarse, but local, geometric information. e Assouad dimension is a fundamental notion of dimension used to study fractal objects in a wide variety of contexts. An important theme in dimension theory is that dimensions often come in pairs. e natural partner of the Assouad dimension is the lower dimension, which was introduced by Larman [2], where it was called the minimal dimensional number.

Definition 2.
e lower dimension of F is defined by dim L F � inf α ≥ 0.
{ ere exists a constant c > 0 such that, for any 0 < r < R ≤ |F|, and e lower dimension is not monotonic, and the modified lower dimension is defined by e Assouad dimension has recently received an enormous interest in the mathematical literature due to its connections with the doubling property. is lead Larman to introduce the dual notion of dimension, namely, the lower Assouad dimension, often simply called the lower dimension. 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. Olsen [3] gave a simple and direct proof that the Assouad dimension of a graph-directed Moran fractal satisfying the open-set condition which is Ahlfors regular coinciding with its Hausdorff and box dimensions. However, in general, it is difficult to obtain the Assouad dimensions of sets which are not Ahlfors regular. Mackay [4] calculated the Assouad dimension of the self-affine carpets of Bedford and McMullen and his main result solved the problem posed by Olsen [3]. For the Moran sets introduced by Wen [5] which are not Ahlfors regular, Li et al. [6] obtained the Assouad dimensions of Moran sets under suitable condition and studied the Assouad dimensions of Cantor-like sets. Jinjun Li [7] also show that the Assouad dimensions of some Moran sets coincide with their packing and upper box dimensions. However, Li [7] did not compute the lower dimension of this class of fractals, the main conclusions of the paper [6,7] must satisfy the condition that the smallest compression ratio c * > 0 and the paper [7] conjecture that the conclusion remains true if the condition c * > 0 is removed (see Remark 1 [7]). In this paper, we prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition that the smallest compression ratio c * > 0, and we give a negative answer to the conjecture in the paper [7].

Lower Dimensions of Some Moran Sets
Firstly, let us recall the definition of Moran sets introduced by Wen [5]. Let n k k ≥ 1 be a sequence of positive integers. (1) For σ ∈ D, J σ is geometrically similar to J, i.e., there exists a similarity S σ : . For convenience, we write J ∅ � J (2) For all k ≥ 0 and σ ∈ D k , J σ * 1 , J σ * 2 , . . . , J σ * n k+1 are subsets of J σ and satisfy that intJ σ * i ∩ int j σ * j � ∅(i ≠ j) (3) For any k ≥ 1 and σ ∈ D k−1 , 1 ≤ j ≤ n k , where |A| denotes the diameter of A. Suppose that F is a collection of closed subsets of J fulfilling the Moran structure, set It is ready to see that E is a nonempty compact set. e set E � E(F) is called the Moran set associated with the collection F.
Let F k � J σ ; σ ∈ D k and F � ∪ k≥0 F k . e elements of F k are called kth-level basic sets of E and the elements of F are called the basic sets of E. Suppose that the set J and the sequences n k and ϕ k are given. We denote by M � M(J, n k , ϕ k ) the class of the Moran sets satisfying the MSC. We call M(J, n k , ϕ k ) the Moran class associated with the triplet (J, n k , ϕ k ).

Remark 1. From the above definition, we see that if the
Moran sets E 1 , E 2 ∈ M(J, n k , ϕ k ) and E 1 ≠ E 2 , then the relative positions of k th -level basic sets of E 1 and those of E 2 may be different, although they satisfy the same MSC.
Under some mild conditions, Hua et al. [8] gave the Hausdorff packing and upper box dimensions of Moran sets. To state their result, we need some notations. Let M � M(J, n k , ϕ k ) be a Moran class. Let c * : infc i,j and c σ � c 1,σ 1 , . . . , c k,σ k for σ � σ 1 , . . . , σ k ∈ D k . Let where s k satisfies the following equation: We can now present the main result of Hua et al. [8].
Theorem 1 (see [8]). Suppose that M � M(J, n k , ϕ k ) is a Moran class satisfying c * > 0. en, for any E ∈ M, Li [7] computed the Assouad dimension of a fairly general (and important) class of Moran fractals.
Theorem 2 (see [7]). Suppose that M � M(J, n k , ϕ k ) is a Moran class satisfying c * > 0. en, for any E ∈ M, e natural partner of the Assouad dimension is the lower dimension, and we prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions.
2 Journal of Mathematics Lemma 1.

ere exists a probability measure v supported by the Moran E such that
for any k ≥ 1 and σ 0 ∈ D k .
Proof. Take a sequence of probability measures v m m ≥ 1 supported by E such that for any σ 0 ∈ D m . More precisely, we can construct v m as follows. First, we distribute the unit mass among the m th -level basic elements according to (11). Inductively, suppose that we have already distributed the mass of proportion v m (J σ ) to a k th -level basic set J σ (σ ∈ D k , k ≥ m); then, we distribute the mass concentrated on J σ evenly to each of its (k + 1)th-level basic subsets, i.e., Repeating the above procedure, we get the desired measure.
Now, fix some m ≥ 1; for any k < m and σ 0 ∈ D k , we obtain Combining it with (11), we have For any σ 1 ∈ D k , by the definitions of E, and thus by (14), is gives Observing that To summarize, we obtain a sequence of probability measures v m m ≥1 supported by E and satisfy (10) for any k ≤ m and σ 0 ∈ D k . Now, Hellys theorem [9] enables us to extract a subsequence v m n n≥1 converging weakly to a limit measure v.
To verify that v fulfills the desired requirements, we fix some k ≥ 0 and σ 0 ∈ D k . en, by the properties of the weak convergence, (20) On the other hand, take an ε > 0 small enough so that the ε-neighborhood J(ε) of J σ 0 is separated from the other m th -level basic set; then, v m n (J(ε)) � v m n (J σ 0 ). By the properties of weak convergence, the following holds: We have for any k ≥ 1 and Finally, for any x which is not in E, since E is a closed set, there exists an open set U containing x and separated from E, and thus, v(U) ≤ lim n ⟶ ∞ v m n (U) � 0, which asserts that v is supported by E. □ Lemma 2 (see [1,2]). If F ⊂ R d is closed, then For σ ∈ D, we denote by σ-the word obtained by deleting the last letter of σ. For c > 0, we define Γ(c) by Γ(c) � σ ∈ D | c σ < c ≤ c σ− . Lemma 3 (see [7], Lemma 3.1). If c * > 0, there exists a constant l 0 such that # τ ∈ Γ(c) | B(x, c) ∩ J τ ≠ ∅ ≤ l 0 for all x ∈ E and c > 0. Remark 2. Some subtly different definitions of the low dimension are given as follows: dim L F � inf α ≥ 0.
It is easy to check this definition and Definition 2 coincides.
Remark 3. By eorem 5 and eorem 6, we give a negative answer to the conjecture in the paper [8, see Remark 1].

Data Availability
e data used to support the study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.