Sharpening Vahlen's result in Diophantine approximation
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910921774858240 |
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| author | Bakhtawar, Ayreena Kraaikamp, Cor |
| author_facet | Bakhtawar, Ayreena Kraaikamp, Cor |
| contents | n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharpening Vahlen's result in Diophantine approximation Bakhtawar, Ayreena Kraaikamp, Cor Dynamical Systems Number Theory 11A55, 11K50, 11K60 n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant. |
| title | Sharpening Vahlen's result in Diophantine approximation |
| topic | Dynamical Systems Number Theory 11A55, 11K50, 11K60 |
| url | https://arxiv.org/abs/2504.20640 |