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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.23320 |
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| _version_ | 1866915519280447488 |
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| author | Cao, Yang Huang, Zhizhong Zhang, Runlin |
| author_facet | Cao, Yang Huang, Zhizhong Zhang, Runlin |
| contents | Let $k$ be a number field. Let $q(x_1,\cdots,x_n)$ be a non-degenerate integral quadratic form in $n\geq 3$ variables with coefficients in $k$ and $m\in k^\times$. Let $X$ be the affine quadric defined by $q=m$ in $\mathbb{A}^n_k$. Based on results on effective equidistribution of $S$-integral points in symmetric spaces, we establish the following:
(i) The arithmetic purity of strong approximation off any single place of $k$ for $X$;
(ii) The geometric sieve for $p_0$-integral points on $X$ when $k=\mathbb{Q}$ and $p_0$ is a prime number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23320 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics Cao, Yang Huang, Zhizhong Zhang, Runlin Number Theory Dynamical Systems 14G05, 37A17, 11N35 Let $k$ be a number field. Let $q(x_1,\cdots,x_n)$ be a non-degenerate integral quadratic form in $n\geq 3$ variables with coefficients in $k$ and $m\in k^\times$. Let $X$ be the affine quadric defined by $q=m$ in $\mathbb{A}^n_k$. Based on results on effective equidistribution of $S$-integral points in symmetric spaces, we establish the following: (i) The arithmetic purity of strong approximation off any single place of $k$ for $X$; (ii) The geometric sieve for $p_0$-integral points on $X$ when $k=\mathbb{Q}$ and $p_0$ is a prime number. |
| title | Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics |
| topic | Number Theory Dynamical Systems 14G05, 37A17, 11N35 |
| url | https://arxiv.org/abs/2509.23320 |