# Some Dimensional Results of a Class of Homogeneous Moran Sets

Authors: Jingru Zhang; Yanzhe Li; Manli Lou
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we construct a class of special homogeneous Moran sets:mk-quasi-homogeneous perfect sets, and obtain the Hausdorff dimension of the sets under some conditions. We also prove that the upper box dimension and the packing dimension of the sets can get the maximum value of the homogeneous Moran sets under the condition supk≥1mk<∞ and estimate the upper box dimension of the sets in two cases.


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## Body

## 1. Introduction

The fractal dimensions of the Moran sets have been studied by many authors and close connected with many subjects, such as the multifractals (see [1–3]), the quasi-conformal mappings (see [4, 5]), the ergodic theory (see [6]), and the number theory (see [7]). In these applications, the homogeneous Moran sets play an important role. There are many classical results about the fractal dimensions of the homogeneous Moran sets (see [8, 9]), but Wen et al. [8, 9] only showed the maximal values and the minimal values of the Hausdorff dimension, the upper box dimension, and the packing dimension (see [10]) of the family of the homogeneous Moran sets, and we need to get some more exact expressions of the fractal dimensions for some special homogeneous Moran sets. Wen and Wu [11] constructed a class of homogeneous Moran sets by the gaps, which are called homogeneous perfect sets, and obtained the Hausdorff dimension of the homogeneous perfect sets under some conditions. Later, Wang and Wu [12] obtained the upper box dimension and the packing dimension of the homogeneous perfect sets under some conditions. Hu [13, 14] constructed a special kind of homogeneous Moran sets which is called mk-quasi-homogeneous Cantor sets by the connected components and the gaps and obtained its Hausdorff dimension, upper box dimension, and packing dimension.In this paper, a class of special homogeneous Moran sets, which is calledmk-quasi-homogeneous perfect sets, is constructed by the connected components and the gaps, and several results about the fractal dimensions of the sets are obtained, which extend the results in [11, 12, 14] under some conditions.This paper is organized as follows. In Section2, we give the definitions of the homogeneous Moran sets, the homogeneous perfect sets, and the mk-quasi-homogeneous perfect sets. Our main results are stated in Section 3. Section 4 gives the proofs of our theorems.


## 2. Preliminaries

First, we recall the definition of homogeneous Moran sets.Letnkk≥1 be a sequence of positive integers and ckk≥1 be a sequence of positive real numbers, such that nk≥2, 0<ck<1 and nkck<1, for any k≥1. Let Dk=σ1σ2…σk:1≤σj≤nj,1≤j≤k, for any k≥1, and let D0=∅ and D=∪k=0∞Dk. If σ=σ1σ2…σk∈Dk and 1≤j≤nk+1, let σ∗j=σ1σ2…σkj∈Dk+1.Definition 1.
(homogeneous Moran sets, (see [8]). Let I⊂ℝ be a nonempty closed interval and ℱ=Iσ:σ∈D be a collection of closed subintervals of I. We say that ℱ has homogeneous Moran structure if it satisfies(i)
I∅=I(ii)
For anyk≥0, σ∈Dk, Iσ∗1,Iσ∗2,…,Iσ∗nk+1 are subintervals of Iσ, and the interiors of Iσ∗i and Iσ∗j are disjoint for any i≠j(iii)
For anyk≥1, σ∈Dk−1, and 1≤j≤nk, we have Iσ∗j/Iσ=ck, where A denotes the diameter of the set A⊂ℝ
Let(1)E=Eℱ≔∩k≥1∪σ∈DkIσ.
Then, we callE a homogeneous Moran set satisfying I,nk,ck and denote by ℳI,nk,ck the class of all homogeneous Moran sets associated with I, nk, and ck.
Letℱk=Iσ:σ∈Dk; then, ℱ=∪k≥0ℱk and any Iσ in ℱk is called a k-order basic interval of E. Without loss of generality, we set I=0,1 and assume that Iσ∗1,…,Iσ∗nk are located from left to right for any k≥1 and any σ∈Dk−1.
Based on the basic intervals and the gaps, Wen and Wu [11] constructed a special kind of homogeneous Moran sets: homogeneous perfect sets.Definition 2.
(homogeneous perfect sets, (see [11]). Let E∈ℳI,nk,ck, and we say that J=Iσ:σ∈D satisfies homogeneous perfect structure if it satisfies the following: there exists a sequence of real numbers ηk,j:k≥1,0≤j≤nk such that, for any k≥1,0≤j≤nk, we have ηk,j≥0, and for any k≥0, σ∈Dk, and 1≤i≤nk+1−1, we have minIσ∗1−minIσ=ηk+1,0, minIσ∗i+1−maxIσ∗i=ηk+1,i, and maxIσ−maxIσ∗nk+1=ηk+1,nk+1. Then, we call E=EJ=EI,nk,ck,ηk,j:=∩k≥1∪σ∈DkIσ a homogeneous perfect set satisfying I,nk,ck,ηk,j and denote by JI,nk,ck,ηk,j the class of all homogeneous perfect sets associated with I,nk,ck,ηk,j.
Based on the connected components of the basic intervals and their gaps, we can construct a special class of homogeneous Moran sets:mk-quasi-homogeneous perfect sets.Definition 3.
(mk-quasi-homogeneous perfect sets). Let E∈ℳI,nk,ck, if for all k≥1 and σ∈Dk−1, the k-order basic intervals Iσ∗1,Iσ∗2,…,Iσ∗nk arbitrarily connect forming mk1≤mk≤nk connected components (denote by Jσ∗1,Jσ∗2,…,Jσ∗mk, which are called k-order connected components). We say that ℐ=Iσ:σ∈D satisfies homogeneous perfect structure if it satisfies(i)
I∅=J=I(ii)
There exists a sequence of real numbersηk,l∗:k≥1,0≤l≤mk such that, for any k≥1,0≤l≤mk, we have ηk,j∗≥0 and min1≤l≤mk−1ηk,l∗>0, and for any k≥0, σ∈Dk, and 1≤i≤mk+1−1, we have minJσ∗1−minIσ=ηk+1,0∗, minJσ∗i+1−maxJσ∗i=ηk+1,i∗, and maxIσ−maxJσ∗mk+1=ηk+1,mk+1∗
LetE=Eℐ=EI,J,nk,mk,ck,ηk,l∗:=∩k≥1∪σ∈DkIσ; then, we call E a mk-quasi-homogeneous perfect set satisfying I,J,nk,ck,ηk,l∗ and denote by J∗I,J,nk,mk,ck,ηk,l∗ the class of all mk-quasi-homogeneous perfect sets associated with I,J,nk,ck,ηk,l∗. For any σ∈Dk−1, 1≤j≤mk, denote by ♯i:Iσ∗i⊂Jσ∗j the number of the k-order basic intervals contained in the k-order connected components Jσ∗j.Remark 1.
LetE∈J∗I,J,nk,mk,ck,ηk,l∗; then, we have(i)
Ifmk=nk, for any k≥1, then E is a homogeneous perfect set and E∈JI,nk,ck,ηk,l, where ηk,l=ηk,l∗, for any k≥1, 0≤l≤mk(ii)
Ifmk=nk, ηk,0∗=ηk,mk∗=0, and ηk,1∗=⋅⋅⋅=ηk,mk−1∗, for any k≥1, then E is a homogeneous Cantor set [8]


## 3. Main Results

Let(2)s∗=liminfk⟶∞logn1n2,…,nk−logc1c2,…,ck,s∗=limsupk⟶∞logn1n2,…,nk−logc1c2,…,ck,t∗=limsupk⟶∞logn1n2,…,nk+1−logc1c2,…,ck+lognk+1.Our main results are stated as follows.Theorem 1.
LetE∈J∗I,J,nk,mk,ck,ηk,l∗ with ηk,0∗=ηk,mk∗=0. For any k≥1, σ∈Dk−1, and 1≤j≤mk, there exists a positive integer α such that ♯i:Iσ∗i⊂Jσ∗j≤α for the k-order connected component Jσ∗j. If there exist positive constants c1˜,c2˜c2˜≥1 such that at least one of the following two conditions is satisfied for any k≥1:(A)
max1≤l≤mk−1ηk,l∗≤c1˜⋅c1c2,…,ck.(B)
max1≤l≤mk−1ηk,l∗≤c2˜⋅min1≤l≤mk−1ηk,l∗; then,(3)dimHE=liminfk⟶∞logn1n2,…,nk−1mk−logc1c2,…,cknk/mk=s∗.Example 1.
LetE be a mk-quasi-homogeneous perfect set satisfying the conditions of Theorem 1, then we have dimHE=s∗=liminfk⟶∞logn1n2,…,nk−1nk/−logc1c2,…,ck, which is equivalent to Theorem 1.2 (1) in [11]. Notice that if E∈JI,nk,ck,ηk,l0≤l≤nk is a homogeneous perfect set, then E∈E∈J∗I,J,nk,mk,ck,ηk,l∗1≤mk≤nk,0≤l≤mk and satisfies ηk,0∗=ηk,mk∗=0 (for any k≥1) if ηk,0=ηk,nk=0. Thus, Theorem 1 generalizes Theorem 1.2 (A) and (B) in [11] under the condition ηk,0=ηk,nk=0 (for anyk≥1).Theorem 2.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗. If supk≥1mk<∞, then(4)dim¯BE=dimpE=s∗.Example 2.
LetE be a mk-quasi-homogeneous Cantor set (refer to [14]); then, E∈J∗I,J,nk,mk,ck,ηk,l∗ and satisfies ηk,0∗=ηk,mk∗=0 and ηk,1∗=⋯=ηk,mk−1∗, for any k≥1. Since the conclusion of Theorem 2 holds under the condition supk≥1mk<∞, Theorem 2 generalizes Theorem 2.3 in [14].Theorem 3.
LetE∈J∗I,J,nk,mk,ck,ηk,l∗ with ηk,0∗=ηk,mk∗=0 and ak=nk/mk, for any k≥1. If there exists a positive constant c3˜ such that, for any k≥1, there exists a sequence of nonnegative integers Nkk≥1 with Nk<ak and supk≥1Nk<∞, satisfying that, for any k-order connected component Jσ∗jσ∈Dk−1,1≤j≤mk,♯i:Iσ∗i⊂Jσ∗j is one of ak−Nk,…,ak,…,ak+Nk, and the following condition is satisfied:(5)max1≤l≤mk−1ηk,l∗≤c3˜⋅c1c2,…,ck.
Then,(6)dim¯BE=limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Remark 2.
The conditions ofNk and ♯i:Iσ∗i⊂Jσ∗j in Theorem 3 imply that there exist 0<φ1,φ2<∞, such that, for any k≥1,σ∈Dk−1,1≤i,j≤mk, we have φ1≤Jσ∗i/Jσ∗j≤φ2.Theorem 4.
LetE∈J∗I,J,nk,mk,ck,ηk,l∗ with ηk,0∗=ηk,mk∗=0, for any k≥1. Suppose that there exist positive constants c4˜,c5˜,c6˜,γc6˜≥1,γ≥1 such that, for any k≥1, at least one of the following three conditions is satisfied:(C)
max1≤l≤mk−1ηk,l∗≤c4˜⋅c1c2,…,ck, and nk/mk≤γ.(D)
mk⋅min1≤l≤mk−1ηk,l∗≥c5˜⋅c1c2,…,ck−1.(E)
max1≤l≤mk−1ηk,l∗≤c6˜⋅min1≤l≤mk−1ηk,l∗ and nk/mk≤γ; then,(7)dim¯BE=limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Example 3.
We obtain the following conclusions about the relationships between theorems in this paper and Theorem 1.4 in [12]:(i)
If Theorem 1.4 (A) in [12] holds for E which satisfies ηk,0=ηk,nk=0 (for any k≥1) and there exists γ≥1 such that nk/mk≤γ(for any k≥1) or Theorem 1.4 (B) or (C) holds for Eit meansnk/mk=1 which satisfies ηk,0=ηk,nk=0, then we have E∈J∗I,J,nk,mk,ck,ηk,l∗1≤mk≤nk,0≤l≤mk and satisfies the conditions of Theorem 4, then (7) holds for E. Since ηk,0=ηk,nk=0 and mk≤nk≤γmk, (7) is equivalent to (1.4) in [12] for the upper box dimension.(ii)
If Theorem 1.4 (D) in [12] holds for E which satisfies ηk,0=ηk,nk=0(for any k≥1), then supk≥1mk<supk≥1nk<∞. Thus, E∈J∗I,J,nk,mk,ck,ηk,l∗1≤mk≤nk,0≤l≤mkand satisfies the conditions of Theorem 2; then, (4) holds for E. Since supk≥1nk<∞, we have s∗=t∗ and (4) is equivalent to (1.4) in [12].
By the argument above, Theorems2 and 4 generalize Theorem 1.4 in [12] for the upper box dimension under the conditions ηk,0=ηk,nk=0 and nk/mk≤γ(for any k≥1) (notice that if E satisfies (B) or (C) or (D) of Theorem 1.4 in [12], we only need the condition ηk,0=ηk,nk=0).Remark 3.
The conditions of Theorem3 and the conditions of Theorem 4 (C) do not contain each other.
IfE satisfies the conditions of Theorem 3, such that mk≡2, Nk≡1, and supk≥1nk=∞, then ak=nk/mk⟶∞ as nk⟶∞, and E does not satisfy the conditions of Theorem 4 (C).
IfE satisfies the conditions of Theorem 4 (C), such that mk=2k and nk=3k for any k≥1(it means nk/mk≡3/2) and there exists a positive integer k0 such that, for any k≥k0 and any σ∈Dk−1, the numbers of the k-order basic intervals contained in Jσ∗1,Jσ∗2,…,Jσ∗mk are k+1,1,1,…,1, then E does not satisfy the conditions of Theorem 3.Remark 4.
The upper box dimension in Theorems3 or 4 is different from s∗ and t∗.
IfE satisfies the conditions of Theorem 3, such that n1=m1=2, nk=2n1n2,…,nk−1(for any k≥2), mk=2nk−1(for any k≥2), a1=n1/m1=1, ak=nk/mk=2nk−1n1n2,…,nk−2−1(for any k≥2), Nk≡0, and ck=2−nk(for any k≥1) or if E satisfies the conditions of Theorem 4 (D) such that n1=m1=2, nk=2n1n2,…,nk−1(for any k≥2), mk=2nk−1(for any k≥2), and ck=2−nk(for any k≥1), then(8)dim¯BE=limsupk⟶∞1+n1+n1n2+⋯+n1,…,nk−2+nk−1n1+n2+⋯+nk−1+nk−1=12,s∗=limsupk⟶∞1+n1+n1n2+⋯+n1,…,nk−2+n1,…,nk−2nk−1n1+n2+⋯+nk−1+nk=0,t∗=limsupk⟶∞1+n1+n1n2+⋯+n1,…,nk−2+n1,…,nk−1+n1,…,nkn1+n2+⋯+nk−1+nk+n1,…,nk=1.
It can be concluded thatdim¯BE is different from s∗ and t∗.


## 4. Proofs of the Theorems

We shall use the following lemmas to prove the theorems.Lemma 1.
(mass distribution principle, (see [10]). Let E be a Borel set and μ be a Borel measure supported on E such that 0<μE<∞. Suppose that, for some s, there are numbers c>0 and δ>0 such that μU≤c|U|s, for all sets U with U≤δ. Then, dimHE≥s.Lemma 2.
(see [13]). Let E∈J∗I,J,nk,mk,ck,ηk,l∗; then,(9)dimHE≤liminfk⟶∞logn1n2,…,nk−1mk−logc1c2,…,cknk/mk.Lemma 3.
(see [8]). Let E∈ℳI,nk,ck, then(10)s∗≤dimpE≤dim¯BE≤t∗.Lemma 4.
(see [15]) Let E∈J∗I,J,nk,mk,ck,ηk,l∗; then,(11)dim¯BE=infs>0:∑k=1∞n1n2⋯nk∑i=0mk+1ηk+1,i∗s<∞.


### 4.1. Proof of Theorem 1

Since the conditions of Theorem1 imply that nk/mk≤α, for all k≥1, we have liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk=liminfk⟶∞logn1n2,…,nk−1nk/−logc1c2,…,ck=s∗. We only need to prove dimHE=liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk.By Lemma2, it is suffices to prove that if E satisfies the conditions of Theorem 1, then dimHE=liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk.For convenience, let(12)δk=c1c2,…,ck,α¯k=max1≤l≤mk−1ηk,l∗,α¯k=min1≤l≤mk−1ηk,l∗,for anyk≥1.For any0<l<liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk, there exists a positive integer K such that, for any k≥K, we have(13)n1n2,…,nk−1mkδkαl≥n1n2,…,nk−1mkδknkmkl≥1.Letμ be a Borel probability measure supported on E, such that, for each k-order basic intervals Iσ∗ik≥1,σ∈Dk−1,1≤i≤nk, μIσ∗i=n1n2,…,nk−1nk−1. Then, for each k-order connected component Jσ∗j1≤j≤mk, μJσ∗j≤α/n1n2,…,nk−1nk≤α/n1n2,…,nk−1mk.LetU be an interval with 0<U=δ<δK; then, there exists a positive integer k>K such that αδk≤δ<αδk−1. Thus, the number of the k−1-order basic intervals that intersect U is at most 2α≤2α2. Denote by g the number of the k-order connected components that intersect U. In order to estimate μU, we divide our proof into the following two cases:Case (i):αδk≤δ<α¯k.It is obvious thatg≤2. Then,(14)μU≤2αn1n2,…,nk−1mk≤2αδkαl≤2αδl=2αUl.Case (ii):maxαδk,α¯k≤δ<αδk−1We discuss separately according to the two conditions (A) and (B) of Theorem1.Condition (A):α¯k≤c1˜c1c2,…,ck. Then, we have(15)δk−1=∑i=1mk−1ηk,i∗+nkδk≤mk−1α¯k+nkδk≤c1˜mk−1δk+nkδk≤c1˜mkδk+nkδk≤c1˜+1nkδk.On the one hand, since the number of thek−1-order basic intervals that intersect U is at most 2α2, we have g≤2α2mk. On the other hand, the number of the k-order connected components that intersect U is at most δ/δk+2≤2δ/δk+1≤4δ/δk. Then, by (13) and (15), we have(16)μU≤αn1n2,…,nk−1mk⋅min2α2mk,4δδk≤αn1n2,…,nk−1mk2α2mk1−l4δδkl≤8αδl1n1n2,…,nk−1mkmk1−l1δklα2−2l≤8αδl1n1n2,…,nk−1mkαlnk1nkδklα2−l≤8α3δl1n1n2,…,nk−1αlnkmkc1˜+1l1δk−1l≤8α4δl1n1n2,…,nk−2δk−1αl1nk−1c1˜+1l≤8α4c1˜+1lδlmk−1nk−1≤8α4c1˜+1lδl≤8α4c1˜+1δl=8α4c1˜+1Ul.Condition (B):α¯k≤c2˜α¯k. In this case, we have(17)δk−1≤mk−1α¯k+nkδk≤c2˜mkα¯k+αmkδk=mkc2˜α¯k+αδk≤mkc2˜α¯k+αδk≤2c2˜mkmaxαδk,α¯k.On the one hand,g≤2α2mk. On the other hand, by the definition of g, we have(18)g−2δk≤δ,g−1α¯k≤δ.Then,(19)g≤δmaxδk,α¯k+2≤4δmaxδk,α¯k≤4δαmaxαδk,α¯k.Combining (13), (17), and (19), we have(20)μU≤αn1n2,…,nk−1mk⋅min2α2mk,4δαmaxαδk,α¯k≤αn1n2,…,nk−1mk2α2mk1−l4δαmaxαδk,α¯kl≤8δlαn1n2,…,nk−1mkα2−2lmk1−l2c2˜mkδk−1lαl≤16c2˜α3δl1n1n2,…,nk−2nk−1δk−1αl≤16c2˜α3δl1n1n2,…,nk−2mk−1δk−1αl≤16c2˜α3δl=16c2˜α3Ul.By (14), (16), (20), and Lemma 1, we have dimHE≥l. By the arbitrariness of l, we obtain dimHE≥liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk and complete the proof of Theorem 1.


### 4.2. Proof of Theorem 2

By Lemma3, it is suffices to prove dim¯BE≤s∗. The conclusion is obvious if s∗=1. Now, we assume s∗<1. For any 1>t>s∗=limsupk⟶∞logn1n2,…,nk/−logc1c2,…,ck, there exist ε>0, k1>0 such that, for any k≥k1,(21)t>logn1n2,…,nk−logc1c2,…,ck+ε.Letbk=t−logn1n2,…,nk/−logc1c2,…,ck>ε; then, n1n2,…,nkδkt=δkbk. By the definitions of nk and ck, we have δk=c1c2,…,ck≤2−k. Since mk≥1, for any k≥1, we have mk+1≥2 and mk+1≤2mk. By the convexity of xs0<s<1, we obtain that(22)∑i=0mk+1ηk+1,i∗t≤mk+1+1∑i=0mk+1ηk+1,i∗mk+1+1t=mk+1+11−ck+1nk+1δkmk+1+1t.Letm=supk≥1mk<∞; by (21) and (22), we have(23)∑k=k1∞n1n2,…,nk∑i=0mk+1ηk+1,i∗t≤∑k=k1∞n1n2,…,nkmk+1+11−ck+1nk+1δkmk+1+1t≤2m∑k=k1∞n1n2,…,nkδkmk+1+1t=2m∑k=k1∞n1n2,…,nkδktmk+1+1−t=2m∑k=k1∞δkbkmk+1+1−t≤2m∑k=k1∞δkbk2−t≤2m∑k=k1∞2−kε⋅1<∞.By Lemma4, we have dim¯BE≤t. By the arbitrariness of t, we have dim¯BE≤s∗ and complete the proof of Theorem 1.


### 4.3. Proof of Theorem 3

First, we use Lemma4 to estimate the upper bound for dim¯BE.Proposition 1.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗; then,(24)dim¯BE≤limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
For any1>p>limsupk⟶∞logn1n2,…,nkmk+1/−logc1c2,…,ck/mk+1, there exist ξ>0 and k2>0 such that, for any k≥k2,(25)p>logn1n2,…,nkmk+1−logc1c2,…,ck/mk+1+ξ,and it follows that(26)ξ−plogc1c2,…,ckmk+1>logn1n2,…,nkmk+1,which implies(27)n1n2,…,nkmk+1<c1c2,…,ckmk+1ξ−p.
Byδk=c1c2,…,ck≤2−k, for any k≥0, and the convexity of xs0<s<1, we have(28)∑i=0mk+1ηk+1,i∗p≤mk+1+1∑i=0mk+1ηk+1,i∗mk+1+1p=mk+1+11−ck+1nk+1δkmk+1+1p.
Sincemk≥1for anyk≥1, we have mk+1≤2mk for any k≥1. Then, by (27) and (28), we have(29)∑k=k2∞n1n2,…,nk∑i=0mk+1ηk+1,i∗p≤∑k=k2∞n1n2,…,nkmk+1+11−ck+1mk+1δkmk+1+1p≤2∑k=k2∞n1n2,…,nkmk+1δkmk+1+1p≤2∑k=k2∞δkmk+1ξ−pδkmk+1p=2∑k=k2∞δkmk+1ξ≤2∑k=k2∞2−kξ<∞.
By Lemma4, we have dim¯BE≤p. By the arbitrariness of p, we have dim¯BE≤limsupk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,ck−1/mk.
Now, we estimate the lower bound fordim¯BE.Proposition 2.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗ satisfies the conditions of Theorem 3; then,(30)dim¯BE≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
Letr1=1/m1 and rk=c1c2,…,ck−1/mk (for any k≥2). We estimate logNrkE/−logrk for a fixed k large enough.
By the condition of Theorem3, we have α¯k≤c3˜⋅c1c2,…,ck. Let Mk be the maximum number of the k-order connected components intersecting a ball of diameter rk; then, we have(31)Mk−2ak−Nkc1c2,…,ck≤rk=c1c2,…,ck−1mk≤nkc1c2,…,ck+mk−1α¯kmk≤nkc1c2,…,ck+mk−1c3˜⋅c1c2,…,ckmk≤nkc1c2,…,ck+nk−1c3˜⋅c1c2,…,ckmk≤c1c2,…,cknk+nkc3˜mk,which implies(32)Mk−2≤nk/mk1+c3˜ak−Nk.
LetN=supk≥1Nk, then we have nk/mk/ak−Nk≤ak+Nk/ak−Nk=1+2Nk/ak−Nk≤1+2Nk≤2N+1. Then, Mk≤2+1+c3˜2N+1. Thus,(33)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+1+c3˜2N+1−logc1c2,…,ck−1/mk.
Sincerk⟶0 as k⟶∞, then, by (33), we have(34)dim¯BE=limsupk⟶∞logNεE−logε≥limsupk⟶∞logNrkE−logrk≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.
From Propositions1 and 2, we have Theorem 3.


### 4.4. Proof of Theorem 4

It is obvious that the conclusion of Proposition1 also holds if E satisfies the conditions of Theorem 4, then it is suffices to estimate the lower bound of dim¯BE under the conditions of Theorem 4.Proposition 3.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗ satisfies the conditions of Theorem 4; then,(35)dim¯BE≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
We discuss separately according to the three conditions (C), (D), and (E) of Theorem4 to estimate logNrkE/−logrk for a fixed k large enough, and we use the definitions of rk and Mk in the proof of Theorem 3.Condition (C): in this case,α¯k≤c4˜⋅c1c2,…,ck and nk/mk≤γ. Then, we have(36)Mk−2c1c2,…,ck≤rk=c1c2,…,ck−1mk≤nkc1c2,…,ck+mk−1α¯kmk≤nkc1c2,…,ck+mk−1c4˜⋅c1c2,…,ckmk≤nkc1c2,…,ck+nk−1c4˜⋅c1c2,…,ckmk≤c1c2,…,cknk+nkc4˜mk,it follows(37)Mk−2≤nk+nkc4˜mk≤γ1+c4˜,which impliesMk≤2+γ1+c4˜. Thus,(38)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+γ1+c4˜−logc1c2,…,ck−1/mk.Condition (D): in this case,mk⋅α¯k≥c5˜⋅c1c2,…,ck−1. Then,(39)α¯k≥c5˜c1c2,…,ck−1mk=c5˜rk,which implies that the number of thek-order connected components intersecting a ball of diameter c5˜rk is at most 2; then, the number of the k-order connected components intersecting a ball of diameter rk is at most 21/c5˜+1:=Tk. Thus,(40)logNrkE−logrk≥log1/Tkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log21/c5˜+1logc1c2,…,ck−1/mk.Condition (E): in this case,α¯k≤c6˜⋅α¯k. We distinguish two cases.Case 1:α¯k<c1c2,…,ck; then, α¯k≤c6˜⋅c1c2,…,ck. Similar to the discussions in condition (C), we have(41)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+γ1+c6˜−logc1c2,…,ck−1/mk.Case 2:α¯k≥c1c2,…,ck. In this case, we have(42)c1c2,…,ck−1≤nkc1c2,…,ck+mk−1α¯k≤nkc1c2,…,ck+mk−1c6˜α¯k≤nkc1c2,…,ck+nkc6˜α¯k≤nkα¯k+nkc6˜α¯k,and it follows that(43)α¯k≥c1c2,…,ck−1nk1+c6˜≥c1c2,…,ck−1γmk1+c6˜=1γ1+c6˜⋅rk.
Similarly to the discussions in Condition (D), we have(44)logNrkE−logrk≥logn1n2,…,nk−1mk−log2γc6˜+1+1logc1c2,…,ck−1/mk.
Notice thatrk⟶0 as k⟶∞. Then, by (38), (40), (41), and (44), we have(45)dim¯BE=limsupε⟶0logNεE−logε≥limsupk⟶∞logNrkE−logrk≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.
From Propositions1 and 3, we have Theorem 4.


## 4.1. Proof of Theorem 1

Since the conditions of Theorem1 imply that nk/mk≤α, for all k≥1, we have liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk=liminfk⟶∞logn1n2,…,nk−1nk/−logc1c2,…,ck=s∗. We only need to prove dimHE=liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk.By Lemma2, it is suffices to prove that if E satisfies the conditions of Theorem 1, then dimHE=liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk.For convenience, let(12)δk=c1c2,…,ck,α¯k=max1≤l≤mk−1ηk,l∗,α¯k=min1≤l≤mk−1ηk,l∗,for anyk≥1.For any0<l<liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk, there exists a positive integer K such that, for any k≥K, we have(13)n1n2,…,nk−1mkδkαl≥n1n2,…,nk−1mkδknkmkl≥1.Letμ be a Borel probability measure supported on E, such that, for each k-order basic intervals Iσ∗ik≥1,σ∈Dk−1,1≤i≤nk, μIσ∗i=n1n2,…,nk−1nk−1. Then, for each k-order connected component Jσ∗j1≤j≤mk, μJσ∗j≤α/n1n2,…,nk−1nk≤α/n1n2,…,nk−1mk.LetU be an interval with 0<U=δ<δK; then, there exists a positive integer k>K such that αδk≤δ<αδk−1. Thus, the number of the k−1-order basic intervals that intersect U is at most 2α≤2α2. Denote by g the number of the k-order connected components that intersect U. In order to estimate μU, we divide our proof into the following two cases:Case (i):αδk≤δ<α¯k.It is obvious thatg≤2. Then,(14)μU≤2αn1n2,…,nk−1mk≤2αδkαl≤2αδl=2αUl.Case (ii):maxαδk,α¯k≤δ<αδk−1We discuss separately according to the two conditions (A) and (B) of Theorem1.Condition (A):α¯k≤c1˜c1c2,…,ck. Then, we have(15)δk−1=∑i=1mk−1ηk,i∗+nkδk≤mk−1α¯k+nkδk≤c1˜mk−1δk+nkδk≤c1˜mkδk+nkδk≤c1˜+1nkδk.On the one hand, since the number of thek−1-order basic intervals that intersect U is at most 2α2, we have g≤2α2mk. On the other hand, the number of the k-order connected components that intersect U is at most δ/δk+2≤2δ/δk+1≤4δ/δk. Then, by (13) and (15), we have(16)μU≤αn1n2,…,nk−1mk⋅min2α2mk,4δδk≤αn1n2,…,nk−1mk2α2mk1−l4δδkl≤8αδl1n1n2,…,nk−1mkmk1−l1δklα2−2l≤8αδl1n1n2,…,nk−1mkαlnk1nkδklα2−l≤8α3δl1n1n2,…,nk−1αlnkmkc1˜+1l1δk−1l≤8α4δl1n1n2,…,nk−2δk−1αl1nk−1c1˜+1l≤8α4c1˜+1lδlmk−1nk−1≤8α4c1˜+1lδl≤8α4c1˜+1δl=8α4c1˜+1Ul.Condition (B):α¯k≤c2˜α¯k. In this case, we have(17)δk−1≤mk−1α¯k+nkδk≤c2˜mkα¯k+αmkδk=mkc2˜α¯k+αδk≤mkc2˜α¯k+αδk≤2c2˜mkmaxαδk,α¯k.On the one hand,g≤2α2mk. On the other hand, by the definition of g, we have(18)g−2δk≤δ,g−1α¯k≤δ.Then,(19)g≤δmaxδk,α¯k+2≤4δmaxδk,α¯k≤4δαmaxαδk,α¯k.Combining (13), (17), and (19), we have(20)μU≤αn1n2,…,nk−1mk⋅min2α2mk,4δαmaxαδk,α¯k≤αn1n2,…,nk−1mk2α2mk1−l4δαmaxαδk,α¯kl≤8δlαn1n2,…,nk−1mkα2−2lmk1−l2c2˜mkδk−1lαl≤16c2˜α3δl1n1n2,…,nk−2nk−1δk−1αl≤16c2˜α3δl1n1n2,…,nk−2mk−1δk−1αl≤16c2˜α3δl=16c2˜α3Ul.By (14), (16), (20), and Lemma 1, we have dimHE≥l. By the arbitrariness of l, we obtain dimHE≥liminfk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,cknk/mk and complete the proof of Theorem 1.


## 4.2. Proof of Theorem 2

By Lemma3, it is suffices to prove dim¯BE≤s∗. The conclusion is obvious if s∗=1. Now, we assume s∗<1. For any 1>t>s∗=limsupk⟶∞logn1n2,…,nk/−logc1c2,…,ck, there exist ε>0, k1>0 such that, for any k≥k1,(21)t>logn1n2,…,nk−logc1c2,…,ck+ε.Letbk=t−logn1n2,…,nk/−logc1c2,…,ck>ε; then, n1n2,…,nkδkt=δkbk. By the definitions of nk and ck, we have δk=c1c2,…,ck≤2−k. Since mk≥1, for any k≥1, we have mk+1≥2 and mk+1≤2mk. By the convexity of xs0<s<1, we obtain that(22)∑i=0mk+1ηk+1,i∗t≤mk+1+1∑i=0mk+1ηk+1,i∗mk+1+1t=mk+1+11−ck+1nk+1δkmk+1+1t.Letm=supk≥1mk<∞; by (21) and (22), we have(23)∑k=k1∞n1n2,…,nk∑i=0mk+1ηk+1,i∗t≤∑k=k1∞n1n2,…,nkmk+1+11−ck+1nk+1δkmk+1+1t≤2m∑k=k1∞n1n2,…,nkδkmk+1+1t=2m∑k=k1∞n1n2,…,nkδktmk+1+1−t=2m∑k=k1∞δkbkmk+1+1−t≤2m∑k=k1∞δkbk2−t≤2m∑k=k1∞2−kε⋅1<∞.By Lemma4, we have dim¯BE≤t. By the arbitrariness of t, we have dim¯BE≤s∗ and complete the proof of Theorem 1.


## 4.3. Proof of Theorem 3

First, we use Lemma4 to estimate the upper bound for dim¯BE.Proposition 1.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗; then,(24)dim¯BE≤limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
For any1>p>limsupk⟶∞logn1n2,…,nkmk+1/−logc1c2,…,ck/mk+1, there exist ξ>0 and k2>0 such that, for any k≥k2,(25)p>logn1n2,…,nkmk+1−logc1c2,…,ck/mk+1+ξ,and it follows that(26)ξ−plogc1c2,…,ckmk+1>logn1n2,…,nkmk+1,which implies(27)n1n2,…,nkmk+1<c1c2,…,ckmk+1ξ−p.
Byδk=c1c2,…,ck≤2−k, for any k≥0, and the convexity of xs0<s<1, we have(28)∑i=0mk+1ηk+1,i∗p≤mk+1+1∑i=0mk+1ηk+1,i∗mk+1+1p=mk+1+11−ck+1nk+1δkmk+1+1p.
Sincemk≥1for anyk≥1, we have mk+1≤2mk for any k≥1. Then, by (27) and (28), we have(29)∑k=k2∞n1n2,…,nk∑i=0mk+1ηk+1,i∗p≤∑k=k2∞n1n2,…,nkmk+1+11−ck+1mk+1δkmk+1+1p≤2∑k=k2∞n1n2,…,nkmk+1δkmk+1+1p≤2∑k=k2∞δkmk+1ξ−pδkmk+1p=2∑k=k2∞δkmk+1ξ≤2∑k=k2∞2−kξ<∞.
By Lemma4, we have dim¯BE≤p. By the arbitrariness of p, we have dim¯BE≤limsupk⟶∞logn1n2,…,nk−1mk/−logc1c2,…,ck−1/mk.
Now, we estimate the lower bound fordim¯BE.Proposition 2.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗ satisfies the conditions of Theorem 3; then,(30)dim¯BE≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
Letr1=1/m1 and rk=c1c2,…,ck−1/mk (for any k≥2). We estimate logNrkE/−logrk for a fixed k large enough.
By the condition of Theorem3, we have α¯k≤c3˜⋅c1c2,…,ck. Let Mk be the maximum number of the k-order connected components intersecting a ball of diameter rk; then, we have(31)Mk−2ak−Nkc1c2,…,ck≤rk=c1c2,…,ck−1mk≤nkc1c2,…,ck+mk−1α¯kmk≤nkc1c2,…,ck+mk−1c3˜⋅c1c2,…,ckmk≤nkc1c2,…,ck+nk−1c3˜⋅c1c2,…,ckmk≤c1c2,…,cknk+nkc3˜mk,which implies(32)Mk−2≤nk/mk1+c3˜ak−Nk.
LetN=supk≥1Nk, then we have nk/mk/ak−Nk≤ak+Nk/ak−Nk=1+2Nk/ak−Nk≤1+2Nk≤2N+1. Then, Mk≤2+1+c3˜2N+1. Thus,(33)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+1+c3˜2N+1−logc1c2,…,ck−1/mk.
Sincerk⟶0 as k⟶∞, then, by (33), we have(34)dim¯BE=limsupk⟶∞logNεE−logε≥limsupk⟶∞logNrkE−logrk≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.
From Propositions1 and 2, we have Theorem 3.


## 4.4. Proof of Theorem 4

It is obvious that the conclusion of Proposition1 also holds if E satisfies the conditions of Theorem 4, then it is suffices to estimate the lower bound of dim¯BE under the conditions of Theorem 4.Proposition 3.
SupposeE∈J∗I,J,nk,mk,ck,ηk,l∗ satisfies the conditions of Theorem 4; then,(35)dim¯BE≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.Proof.
We discuss separately according to the three conditions (C), (D), and (E) of Theorem4 to estimate logNrkE/−logrk for a fixed k large enough, and we use the definitions of rk and Mk in the proof of Theorem 3.Condition (C): in this case,α¯k≤c4˜⋅c1c2,…,ck and nk/mk≤γ. Then, we have(36)Mk−2c1c2,…,ck≤rk=c1c2,…,ck−1mk≤nkc1c2,…,ck+mk−1α¯kmk≤nkc1c2,…,ck+mk−1c4˜⋅c1c2,…,ckmk≤nkc1c2,…,ck+nk−1c4˜⋅c1c2,…,ckmk≤c1c2,…,cknk+nkc4˜mk,it follows(37)Mk−2≤nk+nkc4˜mk≤γ1+c4˜,which impliesMk≤2+γ1+c4˜. Thus,(38)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+γ1+c4˜−logc1c2,…,ck−1/mk.Condition (D): in this case,mk⋅α¯k≥c5˜⋅c1c2,…,ck−1. Then,(39)α¯k≥c5˜c1c2,…,ck−1mk=c5˜rk,which implies that the number of thek-order connected components intersecting a ball of diameter c5˜rk is at most 2; then, the number of the k-order connected components intersecting a ball of diameter rk is at most 21/c5˜+1:=Tk. Thus,(40)logNrkE−logrk≥log1/Tkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log21/c5˜+1logc1c2,…,ck−1/mk.Condition (E): in this case,α¯k≤c6˜⋅α¯k. We distinguish two cases.Case 1:α¯k<c1c2,…,ck; then, α¯k≤c6˜⋅c1c2,…,ck. Similar to the discussions in condition (C), we have(41)logNrkE−logrk≥log1/Mkn1n2,…,nk−1mk−logrk≥logn1n2,…,nk−1mk−log2+γ1+c6˜−logc1c2,…,ck−1/mk.Case 2:α¯k≥c1c2,…,ck. In this case, we have(42)c1c2,…,ck−1≤nkc1c2,…,ck+mk−1α¯k≤nkc1c2,…,ck+mk−1c6˜α¯k≤nkc1c2,…,ck+nkc6˜α¯k≤nkα¯k+nkc6˜α¯k,and it follows that(43)α¯k≥c1c2,…,ck−1nk1+c6˜≥c1c2,…,ck−1γmk1+c6˜=1γ1+c6˜⋅rk.
Similarly to the discussions in Condition (D), we have(44)logNrkE−logrk≥logn1n2,…,nk−1mk−log2γc6˜+1+1logc1c2,…,ck−1/mk.
Notice thatrk⟶0 as k⟶∞. Then, by (38), (40), (41), and (44), we have(45)dim¯BE=limsupε⟶0logNεE−logε≥limsupk⟶∞logNrkE−logrk≥limsupk⟶∞logn1n2,…,nk−1mk−logc1c2,…,ck−1/mk.
From Propositions1 and 3, we have Theorem 4.



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