Simultaneous Accumulation Points to Sets of d-tuples

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1 ISSN print, online International Journal of Nonlinear Science Vol No.2,pp Simultaneous Accumulation Points to Sets of d-tuples Zhaoxin Yin, Meifeng Dai Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University Zhenjiang, Jiangsu, , P.R. China Received 11 December 2009, accepted 22 January 2010 Abstract: In this paper, we consider a class of fractals associated with accumulation points in d-tuples. By constructing some homogeneous Moran subset, we prove that these fractals have full dimension. Keywords: accumulation point; Hausdorff dimension; homogeneous Moran set 1 Introduction Recently, full dimension subsets have attracted much interest in literature. Some subsets of divergence points, self-similar set, finite Graph and symbolic system [1-4] may have full dimension. In this paper, we consider a class of fractals associated with accumulation points and their Hausdorff dimension, we extend the result of [5]. Cantor series expansion is one of the most applied operation of mathematics in representations of real numbers. We first introduce some definitions on Cantor series expansion. Let Q = q be a sequence of integers with q 2. Suppose d x Z [0, q 1] for all, we call the representation x = 1 d x q 1 q 2 q 1.1 the Cantor series of x with respect to Q, if d x = 0 for infinitely many values of. We call the integers d x =1 the digits of this expansion. By [6], if sup q = +, then for almost all x [0, 1], we have sup d x q = 1 and inf d x q = 0. Before we state our main theorem, let us recall some notations. For a d d is a positive integer equal or larger than 1 tuple of x = x 1,, x d [0, 1] d and a positive integer i, let d i x = d i x 1,, d i x d i.e. d i x is the vector consisting of the i-th digits of x j s j = 1,, d. Next, write Σ q = 0, 1,, q 1 d i.e. Σ q is the family of vector i = i 1,, i d with entries i j 0, 1,, q 1 and for a positive integer, write Σ q1 q = Σ q1 Σ q for the family of strings of ω = i 1 i of length whose entries i j Σ qj x = x 1,, x d [0, 1] d and the integer, we let are vectors of Cantor expansion digits. For d i x d i+ 1 x Corresponding author. address: daimf@ujs.edu.cn Copyright c World Academic Press, World Academic Union IJNS /340

2 Zh. Yin and M. Dai: Simultaneous Accumulation Points to Sets of d-tuples 225 denote the string in Σ qi q i+ 1 obtained by concatenating the following vectors: d i x,, d i+ 1 x. Let A d x q =1 denote the set of all accumulation points of d x q =1. For a closed set A [0, 1]d and = 1, 2,, let E A = x [0, 1] d : x = 1 Now, we are in the position to state our main result. d x with d x i Z [0, q 1] and A q 1 q 2 q d x q =1 = A. Theorem 1 If q = +, then dim H E A = d for every closed set A [0, 1] d, where dim H. denote the Hausdorff dimension [7,8]. The technique of this paper is to construct a Moran subset of E A such that this Moran subset has full dimension d, the method is referred from [5]. 2 Proof of theorem 2.1 Homogeneous Moran set Let n 1 be a sequence of positive integers and c 1 a sequence of positive numbers satisfying n 2, 0 < c < 1, n 1 c 1 < η and n c 1 2, where η is a positive number. For any 1, let D = i 1,, i : 1 i j n j, 1 j, D = 0 D and D 0 =. If σ = i 1,, i D, τ = j 1,, j m D m, write σ τ = i 1,, i, j 1,, j m. Suppose that J is a closed unit cube in R d, the collection of subset F = J σ : σ D of J has a homogeneous Moran structure, if 1 J = J; 2 For every 0 and σ D, Jσ 1,, Jσ n + 1 are subset of J σ with their interiors pairwise disjoint; 3 For every 1 and any σ D 1, 1 j n, we have c = J σ j J σ, where J σ denote the diameter of J σ. σ D J σ a high dimensional homogeneous Moran set determined byj, n d, c, re- We call EF 1 lated information can refer to [9]. Let M = MJ, n d, c be the collection of homogeneous Moran set determined by J, n d, c. Lemma 2 [9] Suppose E M = MJ, n d, c, then 2.2 Proof of theorem dim H E inf d log 2 n 1 n log 2 c 1 c c +1 n +1. Our main idea is to construct a homogeneous Moran subset of E A with Hausdorff dimension d. Fixing the sequence q =1, let then we can have ε = minlog 2 q 1 2, [log 2 q 1 q 1 ] 1 2 log 2 q, 2.1 ε = 0, 2.2 since ε log 2 q log 2 q = 0 as. log 2 q 1 2 By the assumption of Theorem 1, we have q = + and q 1 q = +. Thus q ε = IJNS homepage:

3 226 International Journal of NonlinearScience,Vol.92010,No.2,pp Given δ [0, 1], we define a sequence I δ, of intervals as follows [ δq q 1 ε I δ, = 2, δq 1 ] [, if δ [ 1 2, 1]; δq, δq + q 1 ε + 1 ], if δ [0, Lemma 3 [5] Suppose q = +, and ε, I δ, are defined as above. There exists some integer 0 > 0 such that for any δ [0, 1] and any 0, we have Let a = a,1,, a,d, b = b,1,, b,d. We choose a denumerable dense subset = a 1, a 1, a 2, a 1, a 2, a 3, a 1, a 2, a 3, a 4, I δ, [0, q 1] and I δ, Z q 1 ε a 1,, a,. i.e. for any positive integer of A and let b n n 1 denote the sequence for any j N [1, ]. Suppose 0 is the integer mentioned in Lemma 2. Let Then we have M A = Lemma 4 M A E A. b 1 2 +j = a j 2.6 x [0, 1] d : x = d x with d x i Z [0, q 1] I b,i for all 0. q 1 q 2 q 1 Proof. It suffices to prove that for any x = x 1,, x d M A, A d x q =1 Given x M A, by 2.4 the following estimation holds for each 1: d x, b = d x 1 b,1 q q d x d b,d q 2 dq 1 = A q ε. 2.8 For any a l, we can select a subsequence n l 1 such that b l n = a l. Then by 2.8, the following holds Because is a dense subset of A, we have A d x q On the other hand, if x = x 1,, x d A d l n x 1 = a l,1,, q =1 d l n x d q = a l,d. A 2.9 d x q =1, then there exist a subsequence m such that d m x 1 = x d m x d q 1,, = x q d. Since A is a closed set and d m x q, b m dq 1 + q ε 0 as, IJNS for contribution: editor@nonlinearscience.org.u

4 Zh. Yin and M. Dai: Simultaneous Accumulation Points to Sets of d-tuples 227 we deduce x = d m x 1 d m x d,, q q = b m,1,, A. b m,d 2.10 which implies A d x q =1 A By 2.9 and 2.11, we complete the proof. Lemma 5 dim H M A = d Proof. The set M A is a homogeneous Moran set with c = q 1. By Lemma 2, for any 0 and b,i [0, 1], And n = q 2 for any < 0. By 2.1, one can easily see n = Z [0, q 1] I b,i q 1 ε Using Lemma 1, 2.2, 2.12 and 2.13, we have ε log 2 q = log 2 q 1 q 1 dim H M A inf inf = = = d d log 2 n 1 n log 2 c 1 c c +1 n +1 d[log 2 q 1 q 0 + log 2 q 1 ε log 2 [ 1 1 q 1 q +1 q 1 ε ] d log 2 q 1 q log 2 q 1 q + ε +1 log 2 q +1 d 1 + ε +1 log 2 q +1 log 2 q 1 q q 1 ε ] Proof of Theorem 1. The conclusion of Theorem 1 follows immediately by Lemma 3 and Lemma 4. Acnowledgements Research is supported by the National Science Foundation of China and the Education Foundation of Jiangsu Province08KJB References [1] L. Olsen. Applications of multifractal divergence points to sets of d-tuples of numbers defined by their N-adic expansion. Bull. Sci. Math., 128:2004, [2] Y. Jiang and M. F. Dai. Properties of Distribution Class and Spectral Class for a self-similar Set. International Journal of Nonlinear Science., 43:2007, IJNS homepage:

5 228 International Journal of NonlinearScience,Vol.92010,No.2,pp [3] Q. Wang, M. jin and L. F. Xi. Fitness of Graph Based on Fractal Dimension. International Journal of Nonlinear Science., 42:2007, [4] Q. L. Guo. Hausdorff Dimension of Level set Related to Symbolic system. International Journal of Nonlinear Science., 31:2007, [5] Y. Wang, Z. X. Wen and L. F. Xi. Some fractals associated with Cantor expansions. J. Math. Anal., 354:2009, [6] J. Galambos. Representations of Real Numbers by Infinite Series, Lecture Notes in Math. Springer-Verlag, Berlin/Heidelberg [7] K. J. Falconer. Techniques in Fractal Geometry. John Wiley Sons, Ltd., Chichester.,1997. [8] K. J. Falconer. Fractal Geometry: Mathematic Foundation and Applications. John Wiley Sons, Ltd., Chichester.,1990. [9] P. Yan. Dimensions of a class of high-dimensional homogeneous Moran sets and Moran classes. Progress in Natural Science., 129:2002, IJNS for contribution: editor@nonlinearscience.org.u