Hausdorff dimension of sets with restricted, slowly growing partial quotients

2021/11/04 by Takahasi, Hiroki
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.02694

Abstract

I. J. Good (1941) showed that the set of irrational numbers in (0,1) whose partial quotients a_n tend to infinity is of Hausdorff dimension 1/2. A number of related results impose restrictions of the type a_n∈ B or a_n≥ f(n), where B is an infinite subset of \mathbb N and f is a rapidly growing function with n. We show that, for an arbitrary B and an arbitrary f with values in [min B,∞) and tending to infinity, the set of irrational numbers in (0,1) such that a_n∈ B, a_n≤ f(n) for all n∈\mathbb N, and a_n→∞ as n→∞ is of Hausdorff dimension τ(B)/2, where τ(B) is the exponent of convergence of B.

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