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)