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 \(k\geqslant 1\) 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.