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

TL;DR

This paper investigates the fractal structure of irrational numbers with strictly increasing continued fraction partial quotients that contain arbitrarily long arithmetic progressions, linking number theory and fractal geometry.

Contribution

It introduces a detailed analysis of the Hausdorff dimension of sets of irrationals with specific arithmetic progression properties in their continued fraction expansions.

Findings

Determines the Hausdorff dimension of these special sets.

Shows the prevalence of such numbers within the space of irrationals.

Provides new insights into the structure of continued fractions with arithmetic progressions.

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.