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

Speaker: 

Alexander Fish

Affiliation: 

University of Sydney

Date: 

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

Venue: 

RC-4082, The Red Centre, UNSW

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 there exists a non-zero $s$ in $S$ such that the intersection of $E$ and $E - s$ has positive density. By use of measure rigidity results of Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set $E$ 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 $k$. As one of the corollaries we obtain that the set of all possible “discriminants” $D = \left\{xy-z^2 \mid x,y,z \in B\right\}$ over a Bohr-zero set $B$ contains a non-trivial subgroup of the integers.

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

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