Hausdorff dimensions of some irregular sets associated with β-expansions

作     者：LI Jin Jun LI Bing 

作者机构：School of Mathematics and Statistics Minnan Normal University School of Mathematics South China University of Technology 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2016年第59卷第3期

页      面：445-458页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：National Natural Science Foundation of China (Grant Nos. 11301473, 11411130372, 11201155 and 11371148) the Natural Science Foundation of Fujian Province (Grant No. 2014J05008) the Education Committee of Fujian Province (Grant No. JA13203) the Program for New Century Excellent Talents in Minnan Normal University (Grant No. MX13002) the Science and Technology Development Fund of Macao (Grant No. 069/2011/A) 

主　　题：β expansion irregular sets Hausdorff dimension 

摘      要：The Hausdorff dimensions of some refined irregular sets associated with β-expansions are determined for any β 1. More precisely, Hausdorff dimensions of the sets {x ∈ [0, 1) :lim inf(n→∞) S_n(x, β)/n= α_1, lim sup (n→∞) S_n(x, β)/n= α_2}, α_1, α_2≥0 are obtained completely, where S_n(x, β) =sum ε_k(x, β) from k=1 to n denotes the sum of the first n digits of the β-expansion of x. As an application, we present another concise proof of that the set of points x ∈ [0, 1) satisfying lim_(n→∞) S_n(x,β)/n does not exist is of full Hausdorff dimension.