Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions

TL;DR

This work determines the exact Hausdorff dimensions of two families of fractal sets defined via arithmetic progressions in the strictly increasing partial quotients of continued fractions. By combining covering arguments with fundamental intervals and the Gauss map for upper bounds and a mass-distribution framework with Cantor-type measures for lower bounds, the authors derive explicit formulas: for α=liminf ν_n/n, dim_H F((ν_n)) = 1/[2(1+α)] when α<∞ (and 0 if α=∞); and for β=lim (σ_{n+1}-σ_n)/n, dim_H G((σ_n)) = (β-1)/(2β) when β<∞ (and 1/2 when β=∞). The results reveal sharp transitions in dimension controlled by α and β, and identify precise no-dimension-drop regimes (α=0 for F, β=∞ for G). These findings provide a quantitative characterization of how prescribed AP-constraints in continued-fraction digits impact fractal size, enriching the understanding of thin sets arising from number-theoretic expansions.

Abstract

Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: IKKhi64NT25
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Fal97Fal14
• Lemma 2.5
• proof