Some Additive Combinatorics Problems in Matrix Rings

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers

A quantitative version of the non-abelian idempotent theorem

Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the non-abelian idempotent theorem.Comment: 82 pp. Changed the title from `Indicator functions in the Fourier-Eymard algebra'. Corrected the proof of Lemma 19.1. Expanded the introduction. Corrected typo