Additive features of determinant values over p-adic rings

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 \( j \).