Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic dynamics on amenable groups and van der Corput sets

Sohail Farhangi, Robin Tucker-Drob

TL;DR

The paper proves that van der Corput sets on countably infinite amenable groups are independent of the chosen left-Følner sequence, answering a question of Bergelson and Lesigne. It develops a converse to the Furstenberg Correspondence Principle via Reiter sequences and congruent tilings to relate vdC properties to ergodic dynamics, and extends many ℕ-based characterizations of vdC sets to general groups. A key technical contribution is a Ruzsa-type result for amenable groups, linking positive definite functions to ergodic representations, and enabling a unified treatment of operatorial recurrence across countable groups. An appendix collects and extends equivalent vdC/recurrence characterizations, including spectral perspectives and abelian cases, broadening the theoretical framework for recurrence in group dynamics.

Abstract

We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups.

Asymptotic dynamics on amenable groups and van der Corput sets

TL;DR

The paper proves that van der Corput sets on countably infinite amenable groups are independent of the chosen left-Følner sequence, answering a question of Bergelson and Lesigne. It develops a converse to the Furstenberg Correspondence Principle via Reiter sequences and congruent tilings to relate vdC properties to ergodic dynamics, and extends many ℕ-based characterizations of vdC sets to general groups. A key technical contribution is a Ruzsa-type result for amenable groups, linking positive definite functions to ergodic representations, and enabling a unified treatment of operatorial recurrence across countable groups. An appendix collects and extends equivalent vdC/recurrence characterizations, including spectral perspectives and abelian cases, broadening the theoretical framework for recurrence in group dynamics.

Abstract

We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups.
Paper Structure (8 sections, 20 theorems, 57 equations)

This paper contains 8 sections, 20 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Amenable groups and tilings
  5. Koopman representations for positive definite functions
  6. The unique invariant mean on the space of weakly almost periodic functions
  7. Asymptotic dynamics on amenable groups
  8. Appendix: Properties of sets of operatorial recurrence

Key Result

Theorem 1.1

Let $G$ be a countably infinite amenable group and let $\mathcal{F} = (F_n)_{n = 1}^\infty$ be a left-Fø lner sequence in $G$. A set $V \subseteq G$ is an $\mathcal{F}$-vdC set if and only if for any measure preserving system $(X,\mathscr{B},\mu,(\tau_g)_{g \in G})$ and any $f:X\rightarrow\mathbb{S}

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: TilingAmenableGroups
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 26 more