Characteristic Polynomial Patterns in Difference Sets of Matrices

Abstract

We show that for every subset $E$ of positive density in the set of integer square-matrices with zero traces, there exists an integer $k\ge 1$ 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.

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