Finite products sets and minimally almost periodic groups

Donald Robertson, Vitaly Bergelson, Pavel Zorin-Kranich, Cory Christopherson

Research output: Contribution to journalArticlepeer-review

Abstract

We characterize those locally compact, second countable, amenable groups in which a density version of Hindman's theorem holds and those countable, amenable groups in which a two-sided density version of Hindman's 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.
Original languageEnglish
Pages (from-to)2126-2167
JournalJournal of Functional Analysis
Volume270
Issue number6
Early online date12 Jan 2016
DOIs
Publication statusPublished - Mar 2016

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