New Obstacles to Multiple Recurrence

TL;DR

The paper constructs a set that is recurrent for nil-Bohr sets but not for multiple recurrence, providing a negative answer to a higher-order Katznelson question about Bohr and topological recurrence.

Contribution

It introduces a set that separates recurrence for nil-Bohr sets from multiple recurrence, addressing a key open problem in recurrence theory.

Findings

Constructed a set not of multiple recurrence but of recurrence for nil-Bohr sets.

Provided a counterexample to a higher-order Katznelson question.

Showed the set lacks polynomial obstacles to arithmetic progressions.

Abstract

We show that there is a set which is not a set of multiple recurrence despite being a set of recurrence for nil-Bohr sets. This answers Huang, Shao, and Ye's \enquote{higher-order} version of Katznelson's Question on Bohr recurrence and topological recurrence in the negative. Equivalently, we construct a set $S$ so that there is a finite coloring of $\mathbb{N}$ without three-term arithmetic progressions with common differences in $S$, but so that $S$ lacks the usual polynomial obstacles to arithmetic progressions.