Distinguishing sets of strong recurrence from van der Corput Sets

TL;DR

This paper constructs a set of strong recurrence that is not a van der Corput set, clarifying the relationship between these classes and answering open questions in the field.

Contribution

It demonstrates that the class of strong recurrence sets properly contains some sets outside van der Corput sets, and vice versa, refining the understanding of their relationship.

Findings

Constructed a strong recurrence set not being a van der Corput set

Proved the class of strong recurrence is not a subset of van der Corput sets

Clarified the proper subclass relationship between these sets

Abstract

We construct a set of strong recurrence which is not a van der Corput set. This shows that the class of enhanced van der Corput sets is a proper subclass of sets of strong recurrence. In addition, we derive that the class of sets of strong recurrence is not a subclass of van der Corput sets. This answers some questions asked by Bergelson and Lesigne.