Multi-recurrence and van der Waerden systems

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.

29 pages. Small changes during the proofing stage

References in corpus (2)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Invariant scrambled sets, uniform rigidity and weak mixing

Cited by in corpus (5)

• Positive topological entropy and $\Delta$-weakly mixing sets
• Two remarks on the set of recurrent vectors
• Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems
• On chaotic minimal center of attraction of a Lagrange stable motion for topological semi flows
• An extension of Furstenberg's structure theorem for Noetherian modules and multiple recurrence theorems III