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arxiv: 2605.23050 · v1 · pith:FLAK4WKVnew · submitted 2026-05-21 · 🧮 math.DS · math.CO· math.NT

Sets of large values of polynomial multi-correlation functions

Vitaly Bergelson , Rigoberto Zelada This is my paper

Pith reviewed 2026-05-25 05:02 UTC · model grok-4.3

classification 🧮 math.DS math.COmath.NT
keywords polynomial multi-correlationssyndetic setslinear independenceA-IP* propertymultiple recurrenceergodic theoryupper Banach density
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The pith

Sets of large polynomial multi-correlations are syndetic precisely when the polynomials are linearly independent.

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

The paper examines sets R_ε^{p1,...,pL}(A) consisting of those n in Z^d where the measure of the intersection A ∩ T1^{-p1(n)}A ∩ ⋯ ∩ TL^{-pL(n)}A exceeds μ(A)^{L+1} minus ε. It proves these sets are syndetic for every positive-measure A and every ε if and only if the polynomials are linearly independent, and that linear independence further forces the stronger A-IP* property. The same linear-independence condition yields an A-IP* combinatorial statement for sets of positive upper Banach density in Z^D. These results answer a question posed by Frantzikinakis and Kuca and connect to polynomial Szemerédi theorems.

Core claim

For non-constant polynomials p1,...,pL in Z[x1,...,xd] with zero constant term and commuting invertible measure-preserving transformations T_j, the set R_ε^{p1,...,pL}(A) is syndetic if and only if the polynomials are linearly independent; under the same independence condition the sets are A-IP*. In the combinatorial setting, when D > L > 1 the corresponding S_ε sets in Z^D with positive upper Banach density are A-IP* but this property is sharp and cannot be strengthened to IP*.

What carries the argument

The sets R_ε^{p1,...,pL}(A) that collect those n where the multi-correlation measure exceeds the product threshold minus ε, under polynomial shifts by commuting transformations.

If this is right

  • Linear independence of the polynomials forces every R_ε set to be syndetic and A-IP*.
  • The same independence yields A-IP* for the combinatorial S_ε sets of positive upper Banach density when D > L > 1.
  • The A-IP* property cannot be upgraded to IP* in the combinatorial case when D > L > 1.
  • An amplified version of the IP-polynomial Szemerédi theorem follows from the density polynomial Hales-Jewett conjecture via the techniques developed.

Where Pith is reading between the lines

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

  • The syndeticity characterization may allow one to decide recurrence questions for specific polynomial families by checking linear independence alone.
  • The sharpness result for D > L > 1 suggests that further strengthenings of multiple-recurrence statements will require additional assumptions beyond linear independence.
  • The A-IP* conclusion might transfer to other notions of largeness, such as piecewise syndeticity, under the same independence hypothesis.

Load-bearing premise

The polynomials are non-constant with zero constant term and act through commuting invertible measure-preserving transformations on a probability space.

What would settle it

An explicit example of linearly dependent polynomials for which some R_ε set fails to be syndetic, or of linearly independent polynomials for which some R_ε set fails to be A-IP*.

Figures

Figures reproduced from arXiv: 2605.23050 by Rigoberto Zelada, Vitaly Bergelson.

Figure 1
Figure 1. Figure 1: Here are the explanatory remarks concerning the families of sets appearing in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial consequences of the Density Polynomial Hales-Jewett conjecture (DPHJ) naturally lead to the study of sets of large returns which are defined as $$ R_\epsilon^{p_1,...,p_L}(A):=\{n\in Z^d\,|\,\mu(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A)>\mu^{L+1}(A)-\epsilon\}, $$ where the $T_j$'s are commuting and invertible $\mu$-preserving transformations, $A$ is measurable, and $\epsilon>0$. We obtain new results dealing with the sets of the form $R_\epsilon^{p_1,...,p_L}(A)$. Among other things, we show that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ is syndetic if and only if $p_1,...,p_L$ are linearly independent, answering a question asked by Frantzikinakis-Kuca. Moreover, the linear independence of $p_1,...,p_L$ implies that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ has the A-IP$^*$ property (="almost" IP$^*$ property), which is stronger than syndeticity. The following is one of the new combinatorial results obtained in this paper. Suppose that $p_1,...,p_L$ are linearly independent. For any set $E\subseteq Z^D$ with upper Banach density $d^*(E)>0$, any non-zero $v_1,..., v_L\in Z^D$, and any $\epsilon>0$, the set $$ S_\epsilon^{p_1,...,p_L}(E):=\{ n\in Z^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L))>(d^*(E))^{L+1}-\epsilon\} $$ is A-IP$^*$. Furthermore, we prove that when $D>L>1$, this result is sharp: the A-IP$^*$ property cannot be upgraded to IP$^*$. The techniques developed in this paper lead to some additional applications. For example, we show that an amplified form of the IP-polynomial Szemeredi theorem conjectured by Bergelson- McCutcheon follows from the DPHJ.

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 / 2 minor

Summary. The paper studies sets R_ε^{p1,...,pL}(A) of large polynomial multi-correlations for commuting invertible measure-preserving transformations T_j and measurable A. It proves that such sets are syndetic if and only if the non-constant zero-constant-term polynomials p1,...,pL ∈ Z[x1,...,xd] are linearly independent, and that linear independence implies the stronger A-IP* property. This answers a question of Frantzikinakis-Kuca. Combinatorial consequences include that the analogous sets S_ε^{p1,...,pL}(E) are A-IP* for positive upper Banach density E ⊆ Z^D when the polynomials are linearly independent, with sharpness shown for D > L > 1 (A-IP* cannot be strengthened to IP*). The techniques also yield an amplified IP-polynomial Szemerédi theorem from the Density Polynomial Hales-Jewett conjecture.

Significance. If the proofs hold, the iff characterization of syndeticity (and the A-IP* implication) supplies a precise structural result for the return sets that arise in ergodic proofs of polynomial Szemerédi theorems. The sharpness statement for the combinatorial version when D > L > 1 is a concrete contribution. Deriving the amplified IP-polynomial Szemerédi theorem from DPHJ is a useful application. The work strengthens the link between linear independence of polynomials and recurrence properties in multiple dimensions.

minor comments (2)
  1. [Definition of R_ε sets] In the definition of R_ε^{p1,...,pL}(A), the phrase 'linearly independent' is used without an explicit statement of the coefficient field (Q or R); this should be clarified at first use in §1 or the introduction.
  2. [Combinatorial applications] The combinatorial sharpness claim (D > L > 1) is stated in the abstract but the precise counter-example construction or reference to the relevant theorem number is not indicated in the provided abstract; ensure the section containing the counter-example is cross-referenced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the iff characterization of syndeticity, the A-IP* implication, the sharpness result for the combinatorial version, and the application to the amplified IP-polynomial Szemerédi theorem. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The central claims (iff syndeticity of R_ε^{p1,...,pL}(A) precisely when the polynomials are linearly independent, plus the A-IP* implication and the combinatorial sharpness for D>L>1) are stated as new theorems proved via ergodic theory. The setup (non-constant zero-constant-term polynomials, commuting invertible mpt's) is explicitly part of the definition of the return sets rather than derived from them. The result answers an external question of Frantzikinakis-Kuca and invokes a conjecture of Bergelson-McCutcheon only as an application, not as a load-bearing premise. No self-definitional equations, fitted inputs renamed as predictions, or uniqueness theorems imported from the authors' prior work appear in the abstract or stated claims. The derivation is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; ledger populated from definitions appearing in the abstract. Full list of background ergodic-theory axioms cannot be audited.

axioms (2)
  • domain assumption T_j are commuting invertible μ-preserving transformations on a probability space
    Invoked in the definition of R_ε^{p1,...,pL}(A)
  • domain assumption p1,...,pL are non-constant polynomials in Z[x1,...,xd] with zero constant term
    Stated at the beginning of the abstract

pith-pipeline@v0.9.0 · 6040 in / 1343 out tokens · 23895 ms · 2026-05-25T05:02:11.652983+00:00 · methodology

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