Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593162 | Journal of Number Theory | 2016 | 13 Pages |
Abstract
Let B={bn,n≥1}B={bn,n≥1} be a strictly increasing sequence of natural numbers, let an(x)an(x) and kn(x)kn(x) be the n-th partial quotients of regular and generalized continued fraction of x, respectively. DefineR(B)={x∈(0,1):an(x)∈B∀n≥1,andan(x)→∞asn→∞},G(B)={x∈(0,1):kn(x)∈B∀n≥1,andkn(x)→∞asn→∞},G¨(B)={x∈(0,1):kn+1(x)−kn(x)∈B∀n≥1,andkn(x)→∞asn→∞}.In this paper, we show that: If B has a subsequence {bni{bni, i≥1}i≥1} such that lognilogbni is convergent and ni+1ni is bounded, thendimHG¨(B)=dimHR(B)whenϵ(k)=−k+cforsomeconstantc≥0;dimHG(B)=2dimHR(B)when−kρ≤ϵ(k)≤kforsomeconstantρ<1, where ϵ(k)ϵ(k) is the parameter function of the generalized continued fractions, and dimHdimH denotes the Hausdorff dimension.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Liang Tang, Ting Zhong,