Wednesday 19 August 2015 from 11:00–12:00 in Carslaw 535A

Recurrence, measure rigidity and characteristic polynomial patterns in difference sets of matrices

Sasha Fish (Sydney)

Abstract

We present a new approach for establishing the recurrence of a set, through measure rigidity of associated action. Recall, that a subset $S$ of integers (or of another amenable group $G$) is recurrent if for every set $E$ in integers (in $G$) of positive density the sets $S$ and $E-E$ intersect non-trivially. By use of measure rigidity results of Benoist-Quint for algebraic actions on homogeneous spaces and our method, we prove that for every set $E$ of positive density inside traceless square matrices with integer values, there exists $k\ge 1$ such that the set of characteristic polynomials of matrices in $E-E$ contains ALL characteristic polynomials of traceless matrices divisible by $k$. This talk is based on a joint work with M. Bjorklund (Chalmers)