On a theorem of Mattila in the p-adic setting

Abstract: Let $A, B$ be subsets of $(\mathbb{Z}/p<sup>r\mathbb{Z})<sup>2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p<sup>{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p<sup>r\mathbb{Z})$. 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.