Hausdorff dimensions of some exceptional sets in Engel expansions

JOURNAL OF NUMBER THEORY

卷:185

Article

Lu, Meiying¹  Liu, Jia² 

[1] Chongqing Normal Univ, Sch Math Sci, Chongqing 401331, Peoples R China

[2] Anhui Univ Finance & Econ, Inst Stat & Appl Math, Benbu 233030, Peoples R China

关键词: Engel expansions;    Exceptional sets;    Hausdorff dimension;   

DOI  :  10.1016/j.jnt.2017.09.015

来源: Elsevier

【 摘 要 】

Given any real number x is an element of(0,1], denote its Engel expansion by Sigma(infinity)(n=1) 1/d(1)(x)...d(n)(x), where {d(j)(x),j >= 1} is a sequence of positive integers satisfying d(1)(x) >= 2 and d(j+1)(x) >= d(j) (x) (j >= 1). Suppose phi : N -> R+ is a function satisfying phi(n + 1) - phi(n) -> infinity as n -> 4 infinity. In this paper, we consider the set E(phi) = {x is an element of(0,1] : lim(n ->infinity) logd(n)(x)/phi(n) =1}, and we quantify the size of E(phi) in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets {x is an element of(0,1] : lim(n ->infinity) log d(n)(x)/n(beta)=gamma} and {x is an element of(0,1] : lim(n ->infinity) logd(n)(x)/tau(n) = eta}, where beta > 1, gamma > 0 and tau > 1, eta > 0. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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