Uniform Diophantine approximation on the Hecke group $\mathbf H_4$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913754219806720 |
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| author | Bakhtawar, Ayreena Kim, Dong Han Lee, Seul Bee |
| author_facet | Bakhtawar, Ayreena Kim, Dong Han Lee, Seul Bee |
| contents | Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $α$, we characterize the sequence of $\mathbf H_4$-best approximations of $α$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $α$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Diophantine approximation on the Hecke group $\mathbf H_4$ Bakhtawar, Ayreena Kim, Dong Han Lee, Seul Bee Number Theory 11J04, 11J17, 11J70 Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $α$, we characterize the sequence of $\mathbf H_4$-best approximations of $α$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $α$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants. |
| title | Uniform Diophantine approximation on the Hecke group $\mathbf H_4$ |
| topic | Number Theory 11J04, 11J17, 11J70 |
| url | https://arxiv.org/abs/2503.18517 |