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arxiv: 2606.25581 · v1 · pith:OVQVSKINnew · submitted 2026-06-24 · 🧮 math.DS

Zero-Threshold Discrepancies for Multiple Correlation Sequences

Kangbo Ouyang , Jiahao Qiu , Xiangdong Ye This is my paper

Pith reviewed 2026-06-25 19:31 UTC · model grok-4.3

classification 🧮 math.DS
keywords zero-thresholdpro-nilfactormultiple correlationsergodic systemslifting propertydiscrepancy setspiecewise syndetic
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The pith

The lifting property for positivity of multiple correlation sequences fails at the zero threshold for pro-nilfactors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether positivity of correlations on the measure-theoretic pro-nilfactor lifts to the full ergodic system at the zero threshold. For any positive threshold, known structure theory ensures lifting except on zero upper Banach density sets. The authors construct a counterexample: an ergodic system and positive measure sets where the pro-nilfactor correlation stays positive on a positive upper density set of times, but the exact correlation is zero there. This negates a question from prior work. They also prove a rigidity result that the zero-threshold discrepancy set is never piecewise syndetic.

Core claim

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. Additionally, we prove that 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.

What carries the argument

The zero-threshold discrepancy between the pro-nilfactor multiple correlation sequence and the exact correlation sequence in an ergodic system.

If this is right

  • Positivity on the pro-nilfactor does not imply positivity in the system at zero threshold, unlike for positive thresholds.
  • There exist ergodic systems and positive-measure sets where pro-nilfactor correlations are positive on positive upper density times but exact correlations vanish.
  • The zero-threshold discrepancy set is never piecewise syndetic for any ergodic system and any essentially distinct polynomial family vanishing at the origin.
  • This provides a negative answer to the question of whether the lifting property extends to the zero threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Zero-threshold phenomena in multiple recurrence may require tools distinct from those used for positive thresholds.
  • The non-piecewise-syndetic rigidity could interact with other notions of largeness in sets of return times.
  • Similar discrepancies might appear when replacing the pro-nilfactor with other characteristic factors.

Load-bearing premise

The structure theory for polynomial multiple averages guarantees lifting of positivity above any positive threshold but does not force the same at exactly zero.

What would settle it

Finding that the exact correlation remains positive on the positive-density set in the constructed system would falsify the claim that the lifting property fails at zero threshold.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript studies the zero-threshold lifting problem for polynomial multiple correlation sequences with respect to the measure-theoretic pro-nilfactor. It shows that unlike the positive-threshold case (where positivity lifts except on zero upper Banach density sets, by structure theory), this fails at zero threshold. Specifically, the authors construct an ergodic system (X, μ, T) together with sets A, B of positive measure such that the pro-nilfactor correlation is positive along a positive upper-density set of times, while the exact correlation vanishes identically on that set. This gives a negative answer to a question of Glasscock et al. The paper also proves a rigidity result: 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 never piecewise syndetic.

Significance. If the explicit construction holds, the result is significant because it isolates a sharp distinction at the zero threshold that is invisible in the positive-threshold theory. The concrete counterexample (rather than an abstract non-existence argument) and the accompanying rigidity theorem on discrepancy sets are both contributions that could influence work on multiple recurrence, nilfactors, and related questions in ergodic theory and combinatorial number theory.

minor comments (1)
  1. [Abstract] Abstract: the citation appears as a commented-out LaTeX command (%\cite{GKLMMRR}); ensure the full reference list includes the Glasscock et al. paper and that the in-text citation is active.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive recommendation to accept the manuscript and for recognizing the significance of the explicit counterexample and the rigidity result on discrepancy sets. We appreciate the careful summary of the main contributions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result is a negative answer via explicit construction of an ergodic system and sets A, B where pro-nilfactor correlation is positive on a positive-density set of times but exact correlation vanishes identically. This contrasts with (but does not derive from) known positive-threshold lifting from structure theory. No equations reduce to self-definitions, no fitted parameters are relabeled as predictions, and the cited question comes from an independent group. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no free parameters or invented entities are described. The main background assumption is the known positive-threshold lifting from structure theory.

axioms (1)
  • domain assumption Structure theory for polynomial multiple averages implies positivity lifting at every positive threshold except on zero upper Banach density sets
    Explicitly invoked in the abstract to set up the contrast with the zero-threshold case.

pith-pipeline@v0.9.1-grok · 5704 in / 1246 out tokens · 49340 ms · 2026-06-25T19:31:20.811624+00:00 · methodology

discussion (0)

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Reference graph

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19 extracted references · 1 linked inside Pith

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