Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914873470877696 |
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| author | Nakajima, Yuto Takahasi, Hiroki |
| author_facet | Nakajima, Yuto Takahasi, Hiroki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08318 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions Nakajima, Yuto Takahasi, Hiroki Dynamical Systems Number Theory 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. |
| title | Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2209.08318 |