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. {\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.

13 pages, 1 figure. Former title: The dimension theory of semi-regular continued fractions