How the dimension of some GCF epsilon, sets change with proper choice of the parameter function epsilon(k)
How the dimension of some GCF epsilon, sets change with proper choice of the parameter function epsilon(k)
复制标题
某些 GCF epsilon 集合的维数如何随着参数函数 epsilon(k) 的正确选择而变化
DOI：
10.1016/j.jnt.2016.10.013
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发表时间：
2017
影响因子：
0.7
通讯作者：
Zhong Ting
中科院分区：
文献类型：
--
作者：
Wu Xi;Yan Li;Zhong Ting
对于满足条件的参数函数λ (k)+ k+ 1>，设x=[k1 (x), k2 (x)，⋯]λ表示x的GCF λ展开式。在本文中，我们考虑分数集E λ (a, b)={x∈(0,1)：对于无限多个n∈n}， k n (x)≥a b n，并得到：当λ (k)= - k时，dim H λ E λ (a, b)={11 1+ b；b，当−k ρ≤λ (k)≤k且ρ< 1时；1当λ (k) ~ k β和λ≥β≥1时，b−β+ 1；1，当ε (k) ~ k β和β≤β时，其中实a， b> 1，暗H表示Hausdorff维数。有趣的是，当我们选择ε (k)=−k时，ε (a, b)与相似的正则连分数集具有相同的大小；当−k ρ≤ε (k)≤k且ρ< 1时，ε (a, b)与相似的Engel级数集具有相同的大小。
For a parameter function ϵ (k) satisfying the condition ϵ (k)+ k+ 1> 0, let x=[k 1 (x), k 2 (x),⋯] ϵ denote the GCF ϵ expansion of x. In this paper, we consider the fractional set as E ϵ (a, b)={x∈(0, 1): k n (x)≥ a b n for infinitely many n∈ N} and obtain that: dim H E ϵ (a, b)={1 1+ b, when ϵ (k)=− k; 1 b, when− k ρ≤ ϵ (k)≤ k and ρ< 1; 1 b− β+ 1, when ϵ (k)∼ k β and b≥ β≥ 1; 1, when ϵ (k)∼ k β and b≤ β, where real a, b> 1, and dim H denotes the Hausdorff dimension. It is interesting that when we choose ϵ (k)=− k, E ϵ (a, b) has the same size with the similar set of regular continued fractions; and when− k ρ≤ ϵ (k)≤ k and ρ< 1, E ϵ (a, b) has the same size with the similar set of Engel series.