Rational points near manifolds and Khintchine theorem
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913816326963200 |
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| author | Beresnevich, Victor Datta, Shreyasi |
| author_facet | Beresnevich, Victor Datta, Shreyasi |
| contents | In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in [Ann. of Math. 175 (2012), 187-235]. Furthermore, we obtain a Jarník-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational points near manifolds and Khintchine theorem Beresnevich, Victor Datta, Shreyasi Number Theory Dynamical Systems 11J83, 11J13, 11K60, 11K55 In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in [Ann. of Math. 175 (2012), 187-235]. Furthermore, we obtain a Jarník-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds. |
| title | Rational points near manifolds and Khintchine theorem |
| topic | Number Theory Dynamical Systems 11J83, 11J13, 11K60, 11K55 |
| url | https://arxiv.org/abs/2505.01227 |