## 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.

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 language | English |
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Pages (from-to) | 2126-2167 |

Journal | Journal of Functional Analysis |

Volume | 270 |

Issue number | 6 |

Early online date | 12 Jan 2016 |

DOIs | |

Publication status | Published - Mar 2016 |