Dimensions of certain sets of continued fractions with non-decreasing partial quotients

2022/01/31 by Fang, Lulu, Ma, Jihua, Song, Kunkun +1
#11K50 #28A80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2201.12974

Abstract

Let [a₁(x),a₂(x),a₃(x),⋯] be the continued fraction expansion of x∈ (0,1). This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set \x∈(0,1): a₁(x)≤ a₂(x)≤ ⋯, \limsup_n→∞(log a_n(x))/(ψ(n))=1\ for any ψ:ℕ→ℝ⁺ satisfying ψ(n)→∞ as n→∞.

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