We present a new approach for establishing the recurrence of a set, through measure rigidity of associated action. Recall, that a subset SS of integers (or of another amenable group GG) is recurrent if for every set EE in integers (in GG) of positive density there exists a non-zero ss in SS such that the intersection of EE and E−sE−s has positive density. By use of measure rigidity results of Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set EE of positive density inside traceless square matrices with integer values, there exists k≥1k≥1 such that the set of characteristic polynomials of matrices in E−EE−E contains ALL characteristic polynomials of traceless matrices divisible by kk. As one of the corollaries we obtain that the set of all possible “discriminants” D={xy−z2∣x,y,z∈B}D={xy−z2∣x,y,z∈B} over a Bohr-zero set BB contains a non-trivial subgroup of the integers.

This talk is based on a joint work with M. Bjorklund (Chalmers).


Alexander Fish

Research Area

University of Sydney


Wed, 04/11/2015 - 2:30pm


RC-4082, The Red Centre, UNSW