Additive features of determinant values over p-adic rings

Abstract

This paper uses exponential sum methods to show that if $E\subset M_2(\mathbb{Z}/p^r)$ is a finite set of $2\times 2$ matrices with sufficiently large density and $j$ is any unit in the finite ring $\mathbb{Z}/p^r$, then there exist at least two elements of $E$ whose difference has determinant $j$.