On Katznelson's Question for skew product systems

TL;DR

This paper addresses Katznelson's Question in topological dynamics, providing positive results for specific skew product systems and linking recurrence properties to combinatorial structures in number sets.

Contribution

It offers a positive answer for certain skew product extensions of equicontinuous systems and identifies key frequency conditions for recurrence.

Findings

Recurrence is guaranteed in certain skew product systems.

Identifies frequency conditions necessary for recurrence.

Derives combinatorial results on difference sets of syndetic subsets.

Abstract

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew product extensions of equicontinuous systems, including systems of the form $(x,t) \mapsto (x + \alpha, t + h(x))$. We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.