On a theorem of Mattila in the p-adic setting

Let $A,B$ be subsets of $(\mathbb{Z}\mathrm{/}{p}^r\mathbb{Z}{)}^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $\mathrm{\mid }gA-B\mathrm{\mid }\gg p^{2r}$ for a positive proportion of $g\in S{O}_2(\mathbb{Z}\mathrm{/}{p}^r\mathbb{Z})$. 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.