Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908930487091200 |
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| author | Rusu, Andreas Sebe, Gabriela Ileana |
| author_facet | Rusu, Andreas Sebe, Gabriela Ileana |
| contents | We investigate the Hausdorff dimension of level sets defined by digit growth rates in $θ$-expansions, a generalization of regular continued fractions. For any $α\geq 0$, we prove that the set \[ E_θ(α) = \left\{ x \in [0, θ] \setminus \mathbb{Q} : \lim_{n \to {+}\infty} \frac{L_{n,θ}(x) \log n \log \log n}{S_{n,θ}(x) - L_{n,θ}(x)} = α\right\} \] has full Hausdorff dimension. This extends previous work of Zhang and {Lü} (2016) on regular continued fractions to the broader framework of $θ$-expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through Hölder-continuous mappings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions Rusu, Andreas Sebe, Gabriela Ileana Dynamical Systems Number Theory We investigate the Hausdorff dimension of level sets defined by digit growth rates in $θ$-expansions, a generalization of regular continued fractions. For any $α\geq 0$, we prove that the set \[ E_θ(α) = \left\{ x \in [0, θ] \setminus \mathbb{Q} : \lim_{n \to {+}\infty} \frac{L_{n,θ}(x) \log n \log \log n}{S_{n,θ}(x) - L_{n,θ}(x)} = α\right\} \] has full Hausdorff dimension. This extends previous work of Zhang and {Lü} (2016) on regular continued fractions to the broader framework of $θ$-expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through Hölder-continuous mappings. |
| title | Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2604.00587 |