Hausdorff dimensions of some exceptional sets in Engel expansions
Hausdorff dimensions of some exceptional sets in Engel expansions
复制标题
DOI：
10.1016/j.jnt.2017.09.015
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Jia Liu
中科院分区：
文献类型：
--
作者：
lv meiying;Jia Liu
Given any real number x ∈ (0, 1], denote its Engel expansion.by ∞n=1.1 d1(x)···dn(x) , where {dj (x), j ≥ 1} is a sequence of.positive integers satisfying d1(x) ≥ 2 and dj+1(x) ≥ dj (x) (j ≥ 1). Suppose φ : N → R+ is a function satisfying φ(n +.1) − φ(n) → ∞ as n → ∞. In this paper, we consider the set.E(φ) = x ∈ (0, 1] : limn→∞.log dn(x) φ(n) = 1,.and we quantify the size of E(φ) in the sense of Hausdorff.dimension. As applications, we get the Hausdorff dimensions.of the sets x ∈ (0, 1] : limn→∞.log dn(x) nβ = γ and x ∈ (0, 1] :.limn→∞.log dn(x) τ n = η, where β > 1, γ > 0 and τ > 1, η > 0.
