An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916055191912448 |
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| author | Peng, Zixuan Wang, Siyuan Wang, Ethan |
| author_facet | Peng, Zixuan Wang, Siyuan Wang, Ethan |
| contents | We study the continuous part of the Dirichlet spectrum $\mathbb{D}$ and improve the best previously published upper bound for the ray-origin constant $δ$. Building on and refining V. A. Ivanov's approach, we introduce a Cantor-type set $F_4^*$ defined by certain restrictions on partial quotients. For its thickness, we prove $τ(\log(F_4^*))>1$, and apply sum-set results for Cantor sets to prove that the set $F_4^* \cdot F_4^*$ is an interval. Finally, we establish a new upper bound $δ\le \frac{111(397+\sqrt{26565})}{65522}\approx0.94866$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21275 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation Peng, Zixuan Wang, Siyuan Wang, Ethan Number Theory We study the continuous part of the Dirichlet spectrum $\mathbb{D}$ and improve the best previously published upper bound for the ray-origin constant $δ$. Building on and refining V. A. Ivanov's approach, we introduce a Cantor-type set $F_4^*$ defined by certain restrictions on partial quotients. For its thickness, we prove $τ(\log(F_4^*))>1$, and apply sum-set results for Cantor sets to prove that the set $F_4^* \cdot F_4^*$ is an interval. Finally, we establish a new upper bound $δ\le \frac{111(397+\sqrt{26565})}{65522}\approx0.94866$. |
| title | An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.21275 |