Characteristic polynomial patterns in difference sets of matrices

We show that for every subset E of positive density in the set of integer square-matrices with zero traces, there exists an integer k >= 1 such that the set of characteristic polynomials of matrices in E - E contains the set of all characteristic polynomials of integer matrices with zero traces and entries divisible by k. Our theorem is derived from results by Benoist-Quint on measure rigidity for actions on homogeneous spaces.