Hausdorff dimension of some sets in the theory of continued beta-fractions and its generalized continued fractions

Abstract

Let Ψ={1=ψ1<ψ2<⋯} be a sequence in R with sup{ψi+1−ψi}⩽1 and ψi→∞ (i→∞). We introduce the continued Ψ-fraction which is a generalization of the regular continued fraction and the continued β-fraction with β>1 as a real number. There are already abundant results about Hausdorff dimension of some typical sets for regular continued fractions. We achieve the migration of several important results: when Ψ (or β) satisfies certain condition, the set of real numbers in [0,1) whose continued Ψ-fraction (or continued β-fraction) expansion [a1,a2,⋯] satisfies limsupn→∞an=∞, limn→∞⁡an=∞, limsupn→∞ann=∞, limn→∞⁡ann=∞, limsupn→∞log⁡ann=∞, limn→∞⁡log⁡ann=∞ has Hausdorff dimension 1, 1/2, 1/2, 1/2, 0, 0, respectively.