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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2409.18635 |
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| _version_ | 1866916412558147584 |
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| author | Li, Bing Li, Ruofan Wu, Yufeng |
| author_facet | Li, Bing Li, Ruofan Wu, Yufeng |
| contents | Let $\{a_n\}_{n\in\mathbb{N}}$, $\{b_n\}_{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $ψ:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all points $(x,y)\in [0,1]^2$ which satisfy $\|a_nx\|\|b_ny\|<ψ(n)$ infinitely often, and the set of all $x\in [0,1]$ satisfying $\|a_nx\|\|b_nx\|<ψ(n)$ infinitely often. This is based on establishing general convergence results for Hausdorff measures of these two sets. We also obtain some results on the set of all $x\in [0,1]$ such that $\max\{\|a_nx\|, \|b_nx\|\}<ψ(n)$ infinitely often. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicative Diophantine approximation with restricted denominators Li, Bing Li, Ruofan Wu, Yufeng Number Theory 11K60, 28A80 Let $\{a_n\}_{n\in\mathbb{N}}$, $\{b_n\}_{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $ψ:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all points $(x,y)\in [0,1]^2$ which satisfy $\|a_nx\|\|b_ny\|<ψ(n)$ infinitely often, and the set of all $x\in [0,1]$ satisfying $\|a_nx\|\|b_nx\|<ψ(n)$ infinitely often. This is based on establishing general convergence results for Hausdorff measures of these two sets. We also obtain some results on the set of all $x\in [0,1]$ such that $\max\{\|a_nx\|, \|b_nx\|\}<ψ(n)$ infinitely often. |
| title | Multiplicative Diophantine approximation with restricted denominators |
| topic | Number Theory 11K60, 28A80 |
| url | https://arxiv.org/abs/2409.18635 |