Characteristic Polynomial Patterns in Difference Sets of Matrices

TL;DR

This paper demonstrates that in dense subsets of integer trace-zero matrices, the difference set contains all characteristic polynomials of matrices with entries divisible by some integer, linking algebraic properties with measure rigidity results.

Contribution

It establishes a new connection between difference sets of matrices and characteristic polynomial patterns using measure rigidity theory.

Findings

Existence of an integer k such that difference sets contain all characteristic polynomials with entries divisible by k

Link between algebraic matrix properties and measure rigidity results

Extension of polynomial pattern results to matrices with zero trace

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 $k \geq 1$ 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.