Donald Robertson

Triangles in Cartesian Squares of Quasirandom Groups

We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.

Finite products sets and minimally almost periodic groups

Abstract We characterize those locally compact, second countable, amenable groups in which a density version of Hindmans theorem holds and those countable, amenable groups in which a two-sided density version of Hindmans theorem holds. In both cases the possible failure can be attributed to an abundance of finite-dimensional unitary representations, which allows us to construct sets with large density that do not contain any shift of a set of measurable recurrence, let alone a shift of a finite products set. The possible success is connected to the ergodic–theoretic phenomenon of weak mixing via a two-sided version of the Furstenberg correspondence principle. We also construct subsets with large density that are not piecewise syndetic in arbitrary non-compact amenable groups. For countably infinite amenable groups, the symbolic systems associated to such sets admit invariant probability measures that are not concentrated on their minimal subsystems.