Counting rational points near manifolds: a refined estimate, a conjecture and a variant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915697952555008 |
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| author | Hickman, Jonathan Srivastava, Rajula Wright, James |
| author_facet | Hickman, Jonathan Srivastava, Rajula Wright, James |
| contents | Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$ case we formulate a conjecture concerning this count. The conjecture is motivated in part by interpreting certain codimension $2$ submanifolds of $\mathbb{R}^{2m+2}$ as complex hypersurfaces in $\mathbb{C}^{m+1}$ and using the complex structure to provide a natural reformulation of the curvature condition. Finally, we provide further evidence for the conjecture by proving a natural variant for $n \geq 2$ in which rationals are replaced with Gaussian rationals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting rational points near manifolds: a refined estimate, a conjecture and a variant Hickman, Jonathan Srivastava, Rajula Wright, James Number Theory 11D75, 11J25, 11K60, 42B20 Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$ case we formulate a conjecture concerning this count. The conjecture is motivated in part by interpreting certain codimension $2$ submanifolds of $\mathbb{R}^{2m+2}$ as complex hypersurfaces in $\mathbb{C}^{m+1}$ and using the complex structure to provide a natural reformulation of the curvature condition. Finally, we provide further evidence for the conjecture by proving a natural variant for $n \geq 2$ in which rationals are replaced with Gaussian rationals. |
| title | Counting rational points near manifolds: a refined estimate, a conjecture and a variant |
| topic | Number Theory 11D75, 11J25, 11K60, 42B20 |
| url | https://arxiv.org/abs/2512.23204 |