Sharp o-minimality and lattice point counting
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915294593679360 |
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| author | Harrison-Migochi, Andrew McCulloch, Raymond |
| author_facet | Harrison-Migochi, Andrew McCulloch, Raymond |
| contents | Let $Λ\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the number of lattice points in the fibers $Z_T=\{x\in\mathbb{R}^n:(T,x)\in Z\}$. Here we give an effective version of this result for a family definable in a sharply o-minimal structure expanding $\overline{\mathbb{R}}$. We also give an effective version of the Barroero and Widmer statement for certain sets definable in $\mathbb{R}_{\exp}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp o-minimality and lattice point counting Harrison-Migochi, Andrew McCulloch, Raymond Logic Number Theory Primary 11H06, 03C98, 03C64, Secondary 11P21, 28A75, 52C07 Let $Λ\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the number of lattice points in the fibers $Z_T=\{x\in\mathbb{R}^n:(T,x)\in Z\}$. Here we give an effective version of this result for a family definable in a sharply o-minimal structure expanding $\overline{\mathbb{R}}$. We also give an effective version of the Barroero and Widmer statement for certain sets definable in $\mathbb{R}_{\exp}$. |
| title | Sharp o-minimality and lattice point counting |
| topic | Logic Number Theory Primary 11H06, 03C98, 03C64, Secondary 11P21, 28A75, 52C07 |
| url | https://arxiv.org/abs/2503.01731 |