Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions

TL;DR

This paper calculates the Hausdorff dimension of certain sets of irrationals with restricted and slowly growing partial quotients in semi-regular continued fractions, extending previous results and solving a conjecture.

Contribution

It generalizes existing dimension results to semi-regular continued fractions with specific restrictions and growth conditions, using non-autonomous iterated function systems and Bowen's formula.

Findings

Hausdorff dimension formulas for restricted semi-regular continued fractions

Extension of previous regular continued fraction results

Solution of Hirst's conjecture in this context

Abstract

We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second author [Proc. Amer. Math. Soc. {\bf 151} (2023), 3645--3653] and the solution of Hirst's conjecture [B.-W. Wang and J. Wu, Bull. London Math. Soc. {\bf 40} (2008), 18--22], both previously obtained for the regular continued fraction. To prove the result, we construct non-autonomous iterated function systems well-adapted to the given restrictions and growth conditions on partial quotients, estimate the associated pressure functions, and then apply Bowen's formula.