On a theorem of Mattila in the p-adic setting
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917917361176576 |
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| author | Xue, Boqing Pham, Thang Hung, Le Q. Ham, Le Q. Phuong, Nguyen D. |
| author_facet | Xue, Boqing Pham, Thang Hung, Le Q. Ham, Le Q. Phuong, Nguyen D. |
| contents | Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05818 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a theorem of Mattila in the p-adic setting Xue, Boqing Pham, Thang Hung, Le Q. Ham, Le Q. Phuong, Nguyen D. Number Theory Classical Analysis and ODEs Combinatorics Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory. |
| title | On a theorem of Mattila in the p-adic setting |
| topic | Number Theory Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2502.05818 |