New Obstacles to Multiple Recurrence
Ryan Alweiss
TL;DR
The paper tackles a higher-order version of Katznelson's question by constructing a set S that is Nil_d-Bohr recurrent for all d but not a set of topological 2-recurrence, thus giving a negative answer to the higher-order question. The author builds an ℓ^1-based obstruction and a curvature-driven coloring to block 3-term progressions with differences in S, then projects the construction to ℤ while preserving Bohr-set intersection properties via equidistribution results for generalized polynomials. The key insight is that nil-Bohr recurrence captures higher-order structure not constrained by the usual polynomial obstructions, allowing S to interact with nilmanifold dynamics while evading simple degree-2 or higher obstacles. The work resolves the higher-order Katznelson question negatively and motivates further exploration of the relationships between Bohr, topological, and nil-Bohr recurrence, including finite-field analogues and related complexity questions.
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.