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arXiv · 2209.08318

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

Yuto Nakajima·Hiroki Takahasi
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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.

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Yuto Nakajima, Hiroki Takahasi. 2024-07-17. Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions. https://arxiv.org/abs/2209.08318

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