New Obstacles to Multiple Recurrence
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 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 N without three-term arithmetic progressions with common differences in S, but so that S lacks the usual polynomial obstacles to arithmetic progressions.
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