  We show that for every subset $E$ of positive density in the set of integer
square-matrices with zero traces, there exists an integer $k \geq 1$ such that
the set of characteristic polynomials of matrices in $E-E$ contains the set of
\emph{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.
