Multi-recurrence and van der Waerden systems

TL;DR

This paper investigates recurrence properties in dynamical systems related to combinatorial theorems like van der Waerden, establishing new connections and providing measure-theoretic and dynamical proofs for key combinatorial results.

Contribution

It introduces a measure-theoretical analog of multi-transitivity and offers a dynamical proof of the existence of a zero Banach density C-set, advancing the understanding of recurrence in dynamics.

Findings

Established relations between recurrence properties and combinatorial problems.

Presented a measure-theoretical analog of Glasner's multi-transitivity result.

Provided a dynamical proof for the existence of a C-set with zero Banach density.

Abstract

We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem. We describe relations between these properties, study their consequences for dynamics, and demonstrate connections to combinatorial problems. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity in topological weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.