Abstract:We study the zero-threshold lifting problem for polynomial multiple correlation sequences with respect to the measure-theoretic pro-nilfactor. The structure theory for polynomial multiple averages implies that, at every positive threshold, positivity on the pro-nilfactor lifts to positivity in the original system, except on a set of zero upper Banach density. We demonstrate that this lifting property does not hold at the zero threshold.
Specifically, we construct an ergodic system and two sets of positive measure for which the pro-nilfactor correlation is positive along a set of times with positive upper density, while the corresponding exact correlation vanishes on this set. This provides a negative answer to a question posed by Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. %\cite{GKLMMRR}. Additionally, we prove a corresponding rigidity property. For any ergodic system, any essentially distinct family of integer polynomials vanishing at the origin, and any tuple of non-negative bounded functions, the zero-threshold discrepancy set is not piecewise syndetic.
From: Kangbo Ouyang [view email]
[v1]
Wed, 24 Jun 2026 08:54:39 UTC (16 KB)
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