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

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., 151 (2023), 3645–3653] and the solution of Hirst's conjecture [B.-W. Wang and J. Wu, Bull. London Math. Soc., 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.