## 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.

: Primary: 37A45; Secondary: 11P99, 11C99.

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 Tuesday, August 4, 2015