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)