Characteristic polynomial patterns in difference sets of matrices

M. Bjorklund, A. Fish

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 $k⩾1$ 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.

Keywords: Ergodic Ramsey Theory, Measure rigidity.

AMS Subject Classification: Primary: 37A45; Secondary: 11P99, 11C99.