How the dimension of some GCF_ϵ sets change with proper choice of the parameter function ϵ(k)

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• 作者：Xi Wu^a ; Li Yan^b ; Ting Zhong^a ; ^{zhongtingjsu@aliyun.com}
• 关键词：11K50 ; 28A80
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：1-13
• 全文大小：305 K
• 卷排序：174

文摘

For a parameter function ϵ(k)ϵ(k) satisfying the condition ϵ(k)+k+1>0ϵ(k)+k+1>0, let x=[k1(x),k2(x),⋯]ϵx=[k1(x),k2(x),⋯]ϵ denote the GCFϵ expansion of x. In this paper, we consider the fractional set asEϵ(a,b)={x∈(0,1):kn(x)≥abnfor infinitely manyn∈N} and obtain that:dimH⁡Eϵ(a,b)={11+b,when ϵ(k)=−k;1b,when −kρ≤ϵ(k)≤k and ρ<1;1b−β+1,when ϵ(k)∼kβ and b≥β≥1;1,when ϵ(k)∼kβ and b≤β, where real a,b>1a,b>1, and dimHdimH denotes the Hausdorff dimension.It is interesting that when we choose ϵ(k)=−kϵ(k)=−k, Eϵ(a,b)Eϵ(a,b) has the same size with the similar set of regular continued fractions; and when −kρ≤ϵ(k)≤k−kρ≤ϵ(k)≤k and ρ<1ρ<1, Eϵ(a,b)Eϵ(a,b) has the same size with the similar set of Engel series.