Some dimension relations of the Hirst sets in regular and generalized continued fractions

Abstract

Let B={bn,n≥1} be a strictly increasing sequence of natural numbers, let an(x) and 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, i≥1} such that log⁡nilog⁡bni is convergent and ni+1ni is bounded, thendimH⁡G¨(B)=dimH⁡R(B)whenϵ(k)=−k+cforsomeconstantc≥0;dimH⁡G(B)=2dimH⁡R(B)when−kρ≤ϵ(k)≤kforsomeconstantρ<1, where ϵ(k) is the parameter function of the generalized continued fractions, and dimH denotes the Hausdorff dimension.