Uniform Diophantine approximation and run-length function in continued fractions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916648637693952 |
|---|---|
| author | Tan, Bo Zhou, Qing-Long |
| author_facet | Tan, Bo Zhou, Qing-Long |
| contents | We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hatν,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hatν)=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hatν}\Big\}$$ for algebraic irrational points $y\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05855 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform Diophantine approximation and run-length function in continued fractions Tan, Bo Zhou, Qing-Long Number Theory 11K55, 28A80, 11J83 We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hatν,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hatν)=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hatν}\Big\}$$ for algebraic irrational points $y\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions. |
| title | Uniform Diophantine approximation and run-length function in continued fractions |
| topic | Number Theory 11K55, 28A80, 11J83 |
| url | https://arxiv.org/abs/2301.05855 |