On the Separation Structure of Lalley-Gatzouras Fractals
Li-Feng Xi Jun Jie Miao Ying Xiong

	

Department of Mathematics, Ningbo University, Ningbo,
	Zhejiang, 315211, P. R. China
xilifengningbo@yahoo.com


Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500, Dongchuan Road, Shanghai 200241, P. R. China
jjmiao@math.ecnu.edu.cn



Department of Mathematics, South China University of Technology,
	Guangzhou, 510641, P. R. China
xiongyng@gmail.com

[2000]28A80



Ying Xiong is the corresponding author.

Supported by National Natural Science Foundation of China (Grant Nos
	11201152, 11371329, 11471124), NSF of Zhejiang Province (No. LR13A010001), K. C. Wong Magna Fund in Ningbo University, Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Morningside Center of Mathematics.




	We obtain a necessary and sufficient condition for Lalley-Gatzouras sets to be uniform disconnected. This enable us to find all Lalley-Gatzouras sets which are quasisymmetrically equivalent to the Cantor ternary set. As another application, we also study the limit behavior of the gap sequence of Lalley-Gatzouras sets.


Valley separation via trigonal warping
    Ying Xiong
    December 30, 2023
======================================





§ INTRODUCTION


Although every compact totally disconnected perfect set is homeomorphic to the Cantor ternary set, such sets in a metric space may have quite different geometric structures. So a natural question is to seek a general way to characterize the sets which are roughly like the Cantor ternary set. David and Semmes <cit.> answer this question in the sense of quasisymmetric equivalence. Two metric spaces  are said to be quasisymmetrically equivalent if there is a quasisymmetric mapping from one onto the other. A mapping f from (X,d) to (Y,ρ) is called quasisymmetric if it is an injection and there exists a homeomorphism η[0,∞)→[0,∞) such that for any x,y,z∈ X and t>0,

    d(x,y)≤ td(x,z)ρ(f(x),f(y))≤η(t)ρ(f(x),f(z)).

We refer to <cit.> and the references therein for detailed description of quasisymmetric equivalence and quasisymmetric mappings.


With the notion of uniform disconnectedness, David and Semmes <cit.> proved that every compact, doubling, uniformly disconnected and uniformly perfect set is quasisymmetrically equivalent to the Cantor ternary set (see <cit.>). Note that every subset of Euclidean spaces is doubling, while Xie, Yin and Sun <cit.> prove that every self-affine set is uniformly perfect if it is not singleton. It is therefore of interest to look at what self-affine sets are uniformly disconnected.

As an important object in fractal geometry and related fields, self-affine sets are composed of smaller affine copies of themselves. In the self-affine construction, contraction ratios may assume different values in different directions. This fact causes huge difficulties to obtain results which hold for all self-affine sets (see  <cit.>). Consequently, there are two approaches in the study of self-affine sets: finding results for generic sets (almost all in some sense, see Falconer <cit.>) or for specific sets (see <cit.>). 

In this paper, we study a special kind of self-affine sets on ℝ^2: Lalley-Gatzouras sets, which were first studied by Lalley and Gatzouras <cit.>, as a generalization of McMullen sets <cit.>. We obtain a necessary and sufficient condition for such sets to be uniformly disconnected. This enable us to find all Lalley-Gatzouras sets which are quasisymmetrically equivalent to the Cantor ternary set. 

As another application, we use this result to study the gap sequence of Lalley-Gatzouras sets. Gap sequence first appear in the work of Besicovitch <cit.> on cutting-out sets in the line. Rao, Ruan and Yang <cit.> generalized this concept to sets in higher dimensional spaces. In the follow-up study, the gap sequence is widely used to explore properties of fractals. In particular, it is often related to box dimension, especially for self-similar sets or cutting-out fractals, see <cit.>. Roughly speaking, if a set E has a regular separation structure (e.g., E is a self-similar sets satisfying the strong separation condition), then its gap sequence {α_k}_k and its box dimension E are related by α_k≍ k^-1/ E, i.e., c^-1α_k≤ k^-1/ E≤ cα_k for a constant c. We will prove this result for Lalley-Gatzouras sets by making use of uniform disconnectedness. 

This paper is organized as follows. We recall some basic definitions in Section <ref>; and present main results in Section <ref>. In Section <ref>, we study the uniform disconnectedness and prove Theorem <ref>. Finally, we discuss gap sequence and prove Theorem <ref> in Section <ref>.




 §.§ Basic definitions


We recall the definitions of Lalley-Gatzouras sets, uniform disconnectedness and gap sequence in this subsection.



  §.§.§ Lalley-Gatzouras sets 


Given integers m≥2 and n_i≥0 with i=1,2,…,m. Let b_i>0 be real numbers for i=1,2,…,m such that ∑_i=1^m b_i=1. Write d_1=0 and d_i=b_1+b_2+…+b_i-1 for i=2,3,…,m. Let

    𝒟={(i,j) 1≤ i≤ m for n_i>0 and 1≤ j≤ n_i}.

Let {a_ij}_(i,j)∈𝒟 and {c_ij}_(i,j)∈𝒟 be two sequences of nonnegative numbers such that 1>b_i> a_ij>0 for each (i,j)∈𝒟, ∑_j=1^n_ia_ij≤ 1 for each i, c_i(j+1)-c_ij≥ a_ij for each (i,j)∈𝒟 and 1-c_in_i≥ a_in_i for each i.

For each (i,j)∈𝒟, we define a self-affine transformation on ℝ^2 by

    S_ij(x,y)^T=[ a_ij    0;    0  b_i ]xy+ c_ijd_i,

then the family {S_ij}_(i,j)∈𝒟 forms a self-affine iterated function system. According to Hutchinson <cit.>, there exists an attractor E, called a Lalley-Gatzouras set <cit.>, such that

    E=⋃_(i,j)∈𝒟S_ij(E).


Note that the hypotheses on {a_ij}, {b_i} and {c_ij} guarantee that the open rectangles {S_ij((0,1)^2)}_(i,j)∈𝒟 are pairwise disjoint subsets of (0,1)^2 with edges parallel to the x- and y-axes, are arranged in rows of height b_i, and have height greater than width. See Figure <ref> present the first level structure of a Lalley-Gatzouras set.



As stated in <cit.>, let s_1∈ℝ be the unique real such that

    ∑_i=1, n_i 0^m b^s_1_i=1.

Then the box dimension of E is E=s, where s is given by

    ∑_i=1, n_i 0^m ∑_j=1^n_ib^s_1_ia_ij^s-s_1=1.




  §.§.§ Uniform disconnectedness 


Let E be a subset of a metric space (X,ρ). E is called uniformly disconnected if there is a constant C>1 so that, for each ξ∈ E and r>0, we can find a subset A of E such that B(ξ,r/C)∩ E⊂ A⊂ B(ξ,r) and (A,E∖ A)≥ r/C.



  §.§.§ The gap sequence


The gap sequence of cutting-out sets in the line was first studied by Besicovitch <cit.>. Rao, Ruan and Yang <cit.> generalized this concept to sets in higher dimensional spaces.

Let A be a compact subset of ℝ^2. For distinct x,y∈ A, we say x and y are δ-equivalent if there exists a sequence of points a_0=x, a_1,…, a_k=y of A such that |a_i+1-a_i|≤δ for i=0,1,…,k-1. Let (δ) be the cardinality of the set of δ-equivalent classes of A. Clearly the mapping ℝ→ℕ is non-increasing. We write (δ^-)=lim_h→ 0^+(δ-h). We say a sequence {α_k}_k≥1 is the gap sequence of A if the elements of {α_k}_k≥1 are made of the jump points of the function δ as δ decreases with multiplicity (δ^-)-(δ),  that is, the value of δ is in the gap sequence {α_k}_k if and only if (δ)<(δ^-) and the multiplicity

    card{k: α_k=δ}=(δ^-)-(δ).

We also write (δ, A) to emphasize the dependence on the set A.

For an example, the gap sequence of the Cantor ternary set is 

    1/3,1/9,1/9,…,1/3^n,…,1/3^n_2^n-1,….




 §.§ Main results


Our first result concerns the uniform disconnectedness of Lalley-Gatzouras sets.

	A Lalley-Gatzouras set E is uniformly disconnected if and only if E is totally disconnected and there exists an i∈{1,2,…,m} such that n_i=0.


David and Semmes <cit.> proved that every compact, doubling, uniformly disconnected and uniformly perfect set is quasisymmetrically equivalent to the Cantor ternary set. Since self-affine sets are always doubling and uniformly perfect (see  <cit.>), Theorem <ref> implies

	A Lalley-Gatzouras set E is quasisymmetrically equivalent to the Cantor ternary set if and only if E is totally disconnected and there exists an i∈{1,2,…,m} such that n_i=0.

Geometrically, the condition n_i=0 for some i∈{1,2,…,m} means that there are some horizontal line segments in [0,1]^2 do not intersect with the Lalley-Gatzouras set. Corollary <ref> implies that the Lalley-Gatzouras sets that satisfy this condition are essentially different from that do not satisfy this condition.

As another application of Theorem <ref>, we obtain the relationship between the gap sequence of Lalley-Gatzouras set and its box dimension. 

	Let E be a totally disconnected Lalley-Gatzouras set with n_i=0 for some i∈{1,2,…,m} and {α_k}_k the gap sequence of E. Then
	
    α_k≍ k^-1/ E.


Here the symbol α_k≍β_k means that there exists a constant c>1 such that c^-1β_k≤α_k≤ cβ_k for all k.



§ UNIFORM DISCONNECTEDNESS OF LALLEY-GATZOURAS SETS


We shall prove Theorem <ref> in this section. Let E be a Lally-Gatzouras set with the IFS {S_ij}_(i,j)∈𝒟 defined by (<ref>). 



 §.§ Proof of necessity
 The following equivalent condition for uniform disconnectedness is useful (see <cit.>). E is uniformly disconnected if and only if there is ϵ_0>0 such that no pair of distinct points ξ,ξ'∈ E can be connected by an ϵ_0-chain in E, where an ϵ_0-chain connecting ξ and ξ' in E is a sequence of points ξ=ξ_0, ξ_1, …, ξ_n=ξ' in E satisfying |ξ_i-ξ_i+1|≤ϵ_0|ξ-ξ'| for all i=0,1,…,n-1.

We prove the necessary part of Theorem <ref> by using the above equivalent condition. Suppose that n_i0 for all i=1,2,…,m. It suffices to show that, for any ϵ_0∈(0,1), there are two points ξ,ξ' can be connected by an ϵ_0-chain in E.

For this, observe that the condition n_i0 for all i implies that the projection of E to y-axis is [0,1]. Hence we can find a sequence {(x_k,k/n)}_0≤ k≤ n⊂ E, where n is a positive integer with 1/n<ϵ_0/2. Pick an affine mapping T=S_i_1j_1∘ S_i_2j_2∘…∘ S_i_ℓ j_ℓ, where S_i_kj_k defined by (<ref>), such that 

    a_i_1j_1a_i_2j_2⋯ a_i_ℓ j_ℓ/b_i_1b_i_2⋯ b_i_ℓ≤ϵ_0/2.


We claim that ξ=T(x_0,0) and ξ'=T(x_n,1) are two points connected by the ϵ_0-chain {T(x_k,k/n)}_0≤ k≤ n in E. Indeed,

    |T(x_k,k/n)   -T(x_k+1,(k+1)/n)|≤ a_i_1j_1a_i_2j_2⋯ a_i_ℓ j_ℓ|x_k-x_k+1|+b_i_1b_i_2⋯ b_i_ℓ/n
       ≤ b_i_1b_i_2⋯ b_i_ℓ(ϵ_0|x_k-x_k+1|/2+1/n)
    ≤ b_i_1b_i_2⋯ b_i_ℓ(ϵ_0/2+ϵ_0/2)
       =ϵ_0 b_i_1b_i_2⋯ b_i_ℓ≤ϵ_0 |T(x_0,0)-T(x_n,1)|=ϵ_0|ξ-ξ'|.

Thus, the claim is true and the proof of necessary part is completed.



 §.§ Proof of sufficiency

Suppose that E is totally disconnected and there exists an i∈{1,2,…,m} such that n_i=0, we will show that E is uniformly disconnected. The proof is heavily dependent on the special geometrical structure of Lalley-Gatzouras sets. Roughly speaking, the condition n_i=0 for some i means that the set E is well separated in the vertical direction. Thus, the focus is on the analysis of separation in the horizontal direction. 

We divide the proof into three steps. Step 1 studies the projection of E on the y-axis and the fiber of E on each horizontal line. Step 2 deals with the separation structure of the projection and the fibers. Step 3 gives the proof of sufficiency.



  §.§.§ Step 1
 In this step, we determine the projection of E on the y-axis and the fiber of E on each horizontal line.

Recall that the IFS of E is defined by (<ref>). Let

    ℐ={i∈{1,2,…,m} n_i0}.

For i∈ℐ, set ϕ_i(y)=b_iy+d_i. The projection of E to y-axis, denoted by F, is the self-similar set generated by the IFS Φ={ϕ_i i∈ℐ}, i.e.,

    F=⋃_i∈ℐϕ_iF.


Now we study the fiber E_y of E for y∈ F, where E_y={x(x,y)∈ E}. Clearly, (x,y)∈ E if and only if there are infinite words i_1i_2…∈ℐ^ℕ and j_1j_2… with j_k∈{1,2,…,n_i_k} such that

    (x,y)∈⋂_k≥1S_i_1j_1∘ S_i_2j_2…∘ S_i_kj_k([0,1]^2).

Since S_ij(x,y)↦(a_ijx+c_ij,b_iy+d_i), this is equivalent to that 

    y∈⋂_k≥1ϕ_i_1ϕ_i_2∘…∘ϕ_i_k([0,1])

and that

    x∈⋂_k≥1ψ_i_1j_1ψ_i_2j_2∘…∘ψ_i_kj_k([0,1]),  where ψ_ij(x)=a_ijx+c_ij.


For i∈ℐ, let

    Ψ_i={ψ_ij j∈{1,2,…,n_i}}.

For a finite word i_1i_2… i_k∈ℐ^k, let

    K_i_1… i_k=⋃_f_1∈Ψ_i_1,…,f_k∈Ψ_i_k
    f_1∘…∘ f_k[0,1].

For an infinite word 𝐢=i_1i_2…∈ℐ^ℕ, let

    K_𝐢=⋂_k≥1 K_i_1i_2… i_k.

Then (<ref>) is equivalent to x∈ K_𝐢. Given y∈ F, an infinite word 𝐢=i_1i_2…∈ℐ^ℕ is called the coding of y if (<ref>) holds. Note that each point y∈ F belongs to at most two different set ϕ_i(F) and ϕ_j(F), where i,j∈ℐ. Hence every y∈ F has at most two different codings. 
Consequently, we have

	For y∈ F, the fiber E_y has the form
	
    E_y=
    		
    		K_𝐢,   if y has only one coding 𝐢;
    
    		K_𝐢∪ K_𝐣,   if y has two codings 𝐢 and 𝐣.



Although the fibers E_y's are different for different y's, the set K_𝐢 is close to K_𝐣 in the Hausdorff metric if the coding 𝐢 is close to 𝐣. Recall that, for any two nonempty compact sets A and B, the Hausdorff metric  is defined by

    (A,B)=max(max_x∈ A(x,B),max_x∈ B(x,A)).

In the following, we use |A| to denote the diameter of the set A.

	For 𝐢=i_1i_2… i_ki_k+1… and 𝐣=i_1i_2… i_kj_k+1…, where i_k+1 j_k+1, we have
	
    (K_𝐢,K_𝐣)≤max_f_1∈Ψ_i_1,…,f_k∈Ψ_i_k|f_1∘…∘ f_k[0,1]|=
    		a^*_i_1a^*_i_2⋯ a^*_i_k,

	where Ψ_i's are defined by (<ref>) and a^*_i=max_ja_ij for i∈ℐ.


	By symmetry, it suffices to show that
	
    (x,K_𝐣)≤ a^*_i_1a^*_i_2⋯ a^*_i_k  for x∈ K_𝐢.

	Fix an x∈ K_𝐢. By the definition of K_𝐢, there are f_1∈Ψ_i_1,…,f_k∈Ψ_i_k such that x∈ f_1∘…∘ f_k[0,1]. Since 𝐣=i_1i_2… i_kj_k+1…, there is an x'∈ K_𝐣∩ f_1∘…∘ f_k[0,1]. Hence
	
    (x,K_𝐣)≤|x-x'|≤|f_1∘…∘ f_k[0,1]|≤ a^*_i_1a^*_i_2⋯ a^*_i_k.





  §.§.§ Step 2


This step studies the separation structure of the projection F and the sets K_𝐢's. We begin with the projection F. 

To avoid confusion with the Hausdorff metric, we use  to denote the Euclidean distance and (A_1,A_2):=inf{|x-y| x∈ A_1,y∈ A_2} for nonempty sets A_1,A_2. For δ>0, let

    ℐ_δ={i_1i_2… i_k∈ℐ^k k≥1,
    b_i_1b_i_2⋯ b_i_k≤δ<b_i_1b_i_2⋯ b_i_k-1}.

We say w,w'∈ℐ_δ are δ-connected if there exist
w_0(=w),w_1,…,w_n(=w')∈ℐ_δ such that
(ϕ_w_k(F),ϕ_w_k+1(F))≤δ for
k=0,1,…,n-1. Here ϕ_w=ϕ_i_1∘ϕ_i_2∘…∘ϕ_i_ℓ for w=i_1i_2… i_ℓ, where ϕ_i's are defined in (<ref>). For w∈ℐ_δ, let

    ℐ_δ(w)={v∈ℐ_δv and w are δ-connected}.


The following lemma describes the separation structure of F.

	Suppose that there exists i∈{1,2,…,m} with n_i=0, then
	
    L:=sup_δ,w#ℐ_δ(w)<∞.



	Since n_i=0 for some i∈{1,2,…,m}, there exists an interval I such that I⊂[0,1]∖ F. Let η=|I| be the length of I. Write b_*=min_1≤ i≤ mb_i.
	
	If δ≥η b_*, we have
	
    #ℐ_δ(w)≤#ℐ_δ≤δ^-1≤η^-1b_*^-1  for all w∈ℐ_δ.

	
	If δ<η b_*, fix a w∈ℐ_δ. Write δ'=δ/(η b_*). We claim that there are at most two different u∈ℐ_δ' such that
	
    ϕ_u(F)∩⋃_v∈ℐ_δ(w)ϕ_v(F)∅.

	It follows from the claim that
	
    ⋃_v∈ℐ_δ(w)ϕ_v(F)⊂ϕ_u_1(F)∪ϕ_u_2(F)  for some u_1,u_2∈ℐ_δ'.

	(Here we allow that u_1=u_2. In this case, u_1 is the only one word in ℐ_δ' satisfying (<ref>)). Consequently,
	
    #ℐ_δ(w)≤|ϕ_u_1(F)|+|ϕ_u_2(F)|/min_v∈ℐ_δ(w)|ϕ_v(F)|≤2δ'|F|/δ b_*|F|=2η^-1b_*^-2.

	Combining this with the case δ≥η b_*, we have
	
    L=sup_δ,w#ℐ_δ(w)≤ 2η^-1b_*^-2.

	
	It remains to verify the claim. Indeed, if otherwise, suppose that there are distinct u_1,u_2,u_3∈ℐ_δ' satisfying (<ref>). Without loss of generality, suppose that ϕ_u_1(F) is on the left side of ϕ_u_2(F) and ϕ_u_3(F) is on the right side of ϕ_u_2(F). Since I⊂[0,1]∖ F, we have
	
    ⋃_v∈ℐ_δ(w)ϕ_v(F)=A∪ B,

	where A contains all the points which are on the left side of ϕ_u_2(I) and B contains all the points which are on the right side of ϕ_u_2(I). Clearly, A and B are nonempty and (A,B)≥|ϕ_u_2(I)|>δ'η b_*=δ. This contradicts the definition of ℐ_δ(w).


Now we turn to the separation structure of K_𝐢. 

	There exists a constant λ>0 such that, for each open interval
	I and each infinite word 𝐢, we can find an open interval
	J⊂ I satisfying |J|≥λ|I| and J∩ K_𝐢=∅.


Fix an open interval I and an infinite word 𝐢=i_1i_2…. Divide I into three intervals of equal length, let I' be the middle open
interval. If I'∩ K_𝐢=∅, we can take J=I' and
λ=1/3. 

Otherwise, by (<ref>), we can find a set A of the form

    A=ψ_i_1j_1∘ψ_i_2j_2∘…∘ψ_i_kj_k[0,1]

such that A∩ I'∅ and A⊂ I. We further require that A is the set of largest length satisfying the above conditions. We have

    |A|≥3^-1a_*|I|,  where a_*=min_i,ja_ij.

To see this, suppose otherwise |A|<3^-1a_*|I| and let

    A^*=ψ_i_1j_1∘ψ_i_2j_2∘…∘ψ_i_k-1j_k-1[0,1].

Then |A^*|<a_i_kj_k|A|<3^-1I and A^*∩ I'⊃ A∩ I'∅. Since I'⊂ I is the middle open interval of length |I|/3, we have A^*⊂ I. Therefore A^* also satisfies the conditions and |A^*|>|A|. This contradicts the fact that |A| is largest.

Since E is totally disconnected, for each i∈ℐ, we can find an open interval
I_i⊂[0,1]∖⋃_j=1^n_iψ_ij[0,1]. Let

    J=ψ_i_1j_1∘ψ_i_2j_2∘…∘ψ_i_kj_k I_i_k+1,

then J⊂ A⊂ I, J∩ K_𝐢=∅ and
|J|=|A|·|I_i_k+1|≥3^-1a_*|I_i_k+1|·|I| by (<ref>). Thus, J is desired and we can take

    λ=3^-1a_*·min_i∈ℐ|I_i|.





  §.§.§ Step 3

We can prove the sufficiency of Theorem <ref> now. Let (x_0,y_0)∈ E and δ>0 sufficiently small. Pick w∈ℐ_δ such that y_0∈ϕ_wF. Let

    B=⋃_v∈ℐ_δ(w)ϕ_vF.

By the definition of ℐ_δ(w) (see (<ref>)), we have

    (y_0-δ,y_0+δ)∩ F⊂ B  and  (B,F∖ B)>δ.

By (<ref>),

    |B|≤ 2Lδ.


For each v∈ℐ_δ(w), pick an infinite word 𝐢_v which starts with the finite word v. Let

    G=⋃_v∈ℐ_δ(w)K_𝐢_v.

Let

    h_δ=max_i_1i_2… i_k∈ℐ_δ
    
    i_ℓ j_ℓ∈𝒟 for 1≤ℓ≤ ka_i_1j_2a_i_2j_2⋯ a_i_kj_k/b_i_1b_i_2⋯
    b_i_k,
 
note that h_δ→0 as δ→0. By Lemma <ref> and <ref>, for all y∈ B, we have

    E_y⊂ G_δ h_δ:={x(x,G)≤δ h_δ}.

It follows that

    {(x,y)∈ E y∈ B}⊂ G_δ h_δ× B.


Combining (<ref>) and #ℐ_δ(w)≤ L, using Lemma <ref> at most L times, we have, for each open interval I, there exists an open interval J⊂ I satisfying J∩ G=∅ and
|J|≥λ^L|I|. Let I'=(x_0-2δ,x_0-δ) and
I”=(x_0+δ,x_0+2δ), suppose J'=(a',b')⊂ I' and
J”=(a”,b”)⊂ I” are such subintervals. Let G'=G∩[b',a”]
and G”=G∖ G', then (G',G”)≥λ^Lδ.

Finally, let δ be so small that δ
h_δ<3^-1λ^Lδ, then

    G_δ h_δ=G'_δ h_δ∪ G”_δ h_δ  and  (G'_δ h_δ,G”_δ h_δ)
    ≥3^-1λ^Lδ,

where G'_δ h_δ={x(x,G')≤δ h_δ} and G”_δ h_δ={x(x,G”)≤δ h_δ}. We also have

    (x_0-δ,x_0+δ)⊂ G'_δ h_δ  and  
    |G'_δ h_δ|≤4δ.

Combining this with (<ref>), (<ref>), (<ref>)
and (<ref>), we have

    (x_0-δ,x_0+δ)×(y_0-δ,y_0+δ)∩ E
    ⊂(G'_δ h_δ× B)∩ E,  
    |E∩(G'_δ h_δ× B)|≤8Lδ

and

    (E∩(G'_δ h_δ× B),
    E∖(G'_δ h_δ× B))≥3^-1λ^Lδ.

This means that the set E is uniformly disconnected and the proof of sufficient part is completed.



§ GAP SEQUENCE OF LALLEY-GATZOURAS SETS


This section is devoted to the proof of Theorem <ref>. We divide the proof into three lemmas.


	Let {α_k}_k≥1 be the gap sequence of a compact set A. Then
	
    α_k≍ k^-1/γ(δ, A)≍δ^-γ.

	Here γ>0.



	If A⊂ℝ^d is a uniformly disconnected compact set, then
	
    (δ, A)≍ N_δ(A)  for δ>0.
 
	Here N_δ(A) denotes the smallest number of sets of diameter δ that cover A. 


	Recall that (δ,A) is the number of δ-equivalent classes of A. Fix δ>0, for x∈ A, let A_x be the set of all points δ-equivalent to x.
	
	By the definition of uniform disconnectedness, we have |A_x|≤ cδ for all x∈ A and a constant c. Therefore,
	
    (δ,A)≥ N_cδ(A).

	
	By the definition of δ-equivalence, we have (A_x,A_y)>δ if A_x A_y. Therefore,
	
    (δ,A)≤ N_δ/2(A).

	
	Now notice that in Euclidean spaces, we always have N_cδ(A)≍ N_δ(A)≍ N_δ/2(A). Together with (<ref>) and (<ref>), this completes the proof.




	Let E be a Lalley-Gatzouras set, then 
	
    N_δ(E)≍δ^- E.



	This lemma is proved by Lalley and Gatzouras <cit.>, although they didn't give an explicit statement regarding this. See the process of proofs of Lemma 2.1, Lemma 2.4 and Theorem 2.4 in <cit.>.


Clearly, Theorem <ref> follows from Theorem <ref> and the three lemmas above.



 §.§ An unsolved problem


We end this paper with the following unsolved problem:
What is the limit behavior of gap sequence of Lalley-Gatzouras set when n_i0 for all i?


The authors <cit.> solved this problem for McMullen sets, but the argument is not applicable for Lalley-Gatzouras sets.


10
	
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