Simultaneous popular polynomial differences over finite fields

David Conlon; Dingding Dong; Guo-Dong Hong

REVIEW 3 major objections 2 minor 33 references

Simultaneous popular polynomial differences over finite fields

T0 review · 3 major / 2 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Linearly independent polynomials over finite fields share a single popular difference that works for every subconfiguration at once.

desk verdict Clean simultaneous upgrade of Green's popular-difference theorem to polynomials over large F_p, plus a sharp high-dimensional counterexample; abstract-only so proofs unchecked. read the letter →

arxiv 2607.10051 v1 pith:KZAORBM4 submitted 2026-07-11 math.NT math.CO

David Conlon , Dingding Dong , Guo-Dong Hong This is my paper · ORCID

classification math.NTmath.CO MSC 11B3011T0605D10

keywords populardifferencespolynomialconfigurationsfinitefieldssimultaneousarithmeticprogressionsGreen'stheoremlinearindependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Green's popular difference theorem guarantees a single nonzero d that makes the three-term arithmetic progression almost as common as random chance would predict inside any dense set over a large prime field. This paper strengthens that guarantee to whole families of polynomial configurations: for any fixed collection of linearly independent polynomials with zero constant terms, a single d works simultaneously for every subproduct of the corresponding indicators. The density lower bound is the natural random-set value for each subconfiguration, up to an arbitrarily small error. The same paper shows that the simultaneous phenomenon cannot be taken for granted: over vector spaces F_p^n with p fixed and n large, no such d can make both the ordinary progression and the progression with common difference 2d popular at the same time, for a set of density roughly one half. The positive result therefore relies on the polynomial family being independent and on the ambient field being a large prime field rather than a high-dimensional vector space over a fixed field.

What carries the argument

The simultaneous popular-difference statement for a linearly independent family of zero-constant-term polynomials: a single nonzero d that realises the random-set lower bound for every 0-1 weighting of the product of indicator functions.

What would settle it

Exhibit a fixed linearly independent zero-constant family for which, in arbitrarily large prime fields, every dense set admits some nonzero d that fails the random-set lower bound for at least one subconfiguration by a fixed positive amount.

Watch

Extended reading notes

Core claim

For every fixed collection of linearly independent integer polynomials with zero constant terms, every ε>0, all large enough primes p, and every set A of density α in F_p, there exists a single nonzero d such that the density of every subconfiguration formed by a subset of those polynomials is at least α to the power of one plus the number of polynomials used, minus ε. The same simultaneous guarantee fails for the pair of three-term progressions with differences d and 2d over F_p^n.

Load-bearing premise

The polynomials must be linearly independent over the integers and must all have zero constant term; without those structural hypotheses the existence of a single popular d for every subconfiguration is not claimed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know is that this paper upgrades Green's popular-difference theorem to a simultaneous polynomial version over large prime fields and then shows the phenomenon has sharp limits. For any fixed linearly independent family of zero-constant polynomials P1,...,Pk, one nonzero d works at once for every subconfiguration: the density lower bound α^{1+∑ω_i}-ε holds simultaneously for all 2^k choices of which factors to include. They also give a density-1/2 set in F_p^n (n large) where no d makes both d and 2d popular for ordinary 3-APs, so the simultaneous strengthening fails once freeness is lost.

That combination is new and useful. Green's original result is classical; the simultaneous control over all subconfigurations, plus the matching negative example, is the actual contribution. The statements are precise, the hypotheses (linear independence over Z, vanishing constants) are stated up front, and the counterexample is calibrated against the natural 1/8 benchmark. Within additive combinatorics and finite-field ergodic theory this is the right next question, and they answer both the positive and negative sides cleanly.

The soft spots are proportional to the fact that we only have the abstract. The counting or Fourier/Gowers arguments that produce the common d cannot be checked, nor can the explicit construction of the density-1/2 set. Linear independence and zero constants are load-bearing—exactly as the paper's own negative result illustrates—so the claim is tightly conditioned, but that is not a hidden flaw; it is the reason the hypotheses appear. Quantitative dependence on p and the polynomials is also invisible here.

This is for people who already care about popular differences, polynomial Szemerédi, or Gowers norms over finite fields. A serious referee should see the full proofs; the results are important enough inside the subfield and look carefully formulated. I would send it out for review rather than desk-reject.

Referee Report

3 major / 2 minor

Summary. The manuscript claims a simultaneous popular-difference theorem for polynomial configurations over prime fields: for any fixed collection of linearly independent polynomials P_1,...,P_k in Z[t] with zero constant terms, every ε>0, all large primes p, and every A⊆F_p of density α, there exists nonzero d such that the configuration density E_x 1_A(x)∏ 1_A(x+P_i(d))^{ω_i} is at least α^{1+∑ω_i}-ε simultaneously for every ω∈{0,1}^k. A complementary negative result asserts that over F_p^n (p fixed, n→∞) simultaneous popularity of both d and 2d for three-term APs fails: there exist sets of density 1/2+o(1) for which the min of the two configuration densities is at most 1/8-c for some c>0 independent of n.

Significance. If the positive theorem holds as stated, it is a clean and natural strengthening of Green's popular-difference theorem to a simultaneous multi-configuration setting for polynomial progressions, of clear interest in additive combinatorics over finite fields. The negative result supplies a sharp limitation by exhibiting an explicit conflict between two linearly related configurations, showing that freeness hypotheses cannot be dropped casually. The claims are stated with external density benchmarks (α^3, α^{1+∑ω_i}, 1/8) and are in principle falsifiable; the simultaneous formulation and the density-1/2 counterexample construction are the main contributions.

major comments (3)

Only the abstract is available for review, so the derivations, error estimates, counting lemmas, and the explicit construction of the density-1/2 counterexample over F_p^n cannot be checked. The central positive claim and the sharpness statement therefore remain unverified; a full assessment of soundness is impossible on the present material.
The positive theorem (as stated in the abstract) takes linear independence of the P_i over Z and vanishing constant terms as hypotheses. These are load-bearing: the paper's own negative result already shows that simultaneous popularity fails once freeness is lost (d versus 2d). The manuscript must make explicit where independence and P_i(0)=0 enter the argument (e.g., equidistribution of d↦(P_1(d),...,P_k(d)), Gowers-norm control, or Fourier analysis) and should indicate whether either hypothesis can be relaxed.
The negative result asserts a uniform gap c>0 below 1/8 for the min of the two 3-AP densities, for sets of density 1/2+o_n(1). Without the construction or the quantitative estimates, it is unclear whether the o_n(1) and the constant c are robust, or whether the same obstruction appears already in F_p (rather than only in high-dimensional F_p^n). This gap is essential to the claim that the simultaneous strengthening of Green is false in that regime.

minor comments (2)

The abstract is clearly written and the statements are easy to parse; once the full text is available, ensure that the main theorems are numbered and that the dependence of the 'sufficiently large p' threshold on ε, k and the degrees of the P_i is recorded explicitly.
A brief comparison with existing popular-difference or popular-polynomial results (beyond Green) would help place the simultaneous bound and the 1/8-c obstruction in context.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; abstract-only claims are existence theorems and a counterexample against external density benchmarks.

full rationale

Only the abstract is available. It states Green’s popular-difference theorem as background, then asserts a simultaneous lower bound α^{1+∑ω_i}-ε for every subconfiguration ω∈{0,1}^k under the explicit structural hypotheses that the fixed polynomials are linearly independent over Z and have zero constant terms, together with a matching negative construction over F_p^n showing that simultaneous popularity of d and 2d fails for density 1/2+o(1). No derivation steps, fitted parameters, uniqueness theorems, or load-bearing self-citations appear in the supplied text. The target densities are classical combinatorial benchmarks (α^3, α^{1+∑ω_i}, 1/8), not quantities defined from the paper’s own inputs. Linear independence and vanishing constants are openly declared hypotheses, not smuggled conclusions. Consequently the abstract exhibits no self-definitional loop, no fitted-input-called-prediction, and no circular self-citation chain. Score 0 is the honest finding for an abstract-only review of this form.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only review: free parameters are not visible; the work is asymptotic existence over large primes and a construction in high dimension. Background axioms are standard finite-field and additive-combinatorics assumptions (density, expectation over F_p or F_p^n, linear independence of polynomials). No new physical entities are introduced.

assumptions (3)

domain assumption Linear independence of the fixed polynomials P1,...,Pk over Z and vanishing constant terms
Stated as the hypothesis under which the simultaneous popular-difference theorem is claimed; without it the existence of a single d for all 2^k subconfigurations is not asserted.
standard math Standard density and expectation formalism over finite fields F_p and vector spaces F_p^n
The statements are phrased in the usual language of additive combinatorics; these are background conventions, not ad-hoc inventions of the paper.
domain assumption For every ε>0 the prime p is taken sufficiently large (depending on ε and the fixed polynomial family)
The positive theorem is asymptotic in p; the quantitative dependence is not given in the abstract and is part of the existence claim.

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Cite this review

Pith. "Pith review of Simultaneous popular polynomial differences over finite fields." pith.science (2026). https://pith.science/paper/KZAORBM4

@misc{pith2026260710051,
  author       = {Pith},
  title        = {Pith review of: Simultaneous popular polynomial differences over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZAORBM4}},
  note         = {Machine review of arXiv:2607.10051}
}

abstract

Green's popular difference theorem says that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $\alpha$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq \alpha^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $\alpha$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{\omega_i} \geq \alpha^{1+\sum_i\omega_i}-\varepsilon \] simultaneously for every $\omega=(\omega_1,\dots,\omega_k)\in\{0,1\}^k$. We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime $p$, there is a constant $c>0$ such that, for all sufficiently large $n$, one can find a set $A\subseteq\mathbb F_p^n$ of density $1/2+o_n(1)$ satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both $d$ and $2d$ are simultaneously popular differences for three-term arithmetic progressions is false.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [1]

Fox, Jacob and Pham, Huy Tuan , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2021 , NUMBER =. doi:10.1093/imrn/rny240 , URL =

work page doi:10.1093/imrn/rny240 2021

[2] [2]

Fox, Jacob and Pham, Huy Tuan and Zhao, Yufei , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2023 , NUMBER =. doi:10.4171/jems/1271 , URL =

work page doi:10.4171/jems/1271 2023

[3] [3]

Frantzikinakis, Nikos and Kra, Bryna , TITLE =. J. London Math. Soc. , FJOURNAL =. 2006 , NUMBER =. doi:10.1112/S0024610706023374 , URL =

work page doi:10.1112/s0024610706023374 2006

[4] [4]

2023 , isbn =

Zhao, Yufei , title =. 2023 , isbn =

2023

[5] [5]

Additive combinatorics , SERIES =

Green, Ben , TITLE =. Additive combinatorics , SERIES =. 2007 , ISBN =. doi:10.1090/crmp/043/06 , URL =

work page doi:10.1090/crmp/043/06 2007

[6] [6]

Lyall, Neil and Magyar, \'Akos , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/blms/bdr011 , URL =

work page doi:10.1112/blms/bdr011 2011

[7] [7]

Duke Math

Peluse, Sarah , TITLE =. Duke Math. J. , FJOURNAL =. 2019 , NUMBER =. doi:10.1215/00127094-2018-0051 , URL =

work page doi:10.1215/00127094-2018-0051 2019

[8] [8]

Berger, Aaron and Sah, Ashwin and Sawhney, Mehtaab and Tidor, Jonathan , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/tran/8593 , URL =

work page doi:10.1090/tran/8593 2022