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arXiv · 2502.05818

On a theorem of Mattila in the p-adic setting

Boqing Xue·Thang Pham·Le Q. Hung·Le Q. Ham·Nguyen D. Phuong
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Abstract

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.

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Boqing Xue, Thang Pham, Le Q. Hung, Le Q. Ham, Nguyen D. Phuong. 2025-02-09. On a theorem of Mattila in the p-adic setting. https://arxiv.org/abs/2502.05818

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