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Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality

T0 review · 1 major / 2 minor · reviewed 2026-05-19 · grok-4.3

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Pith's one-line read For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size.

desk verdict This extends Green-Sawhney to general intersective polynomials by showing the arithmetic level-d inequality stays uniform when auxiliaries change during iteration. read the letter →

arxiv 2605.16216 v1 pith:DSHKI22R submitted 2026-05-15 math.NT math.CO

Carlo Francisco E. Adajar , Rishika Agrawal , Mukul Rai Choudhuri , Chian Yeong Chuah , Steve Fan , Swaroop Hegde , Andrew Lott , Krishnamohan Nandakumar
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Nagendar Reddy Ponagandla
This is my paper · ORCID
classification math.NTmath.CO
keywords Furstenberg-Sarkozytheoremintersectivepolynomialsarithmeticlevel-dinequalitydensityincrementexponentialsumsquasipolynomialboundsadditivecombinatorics
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The pith

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

The reading

The paper adapts the arithmetic level-d inequality of Green and Sawhney to prove a quasipolynomial upper bound for the maximal size of a subset of {1,2,...,X} whose differences avoid all nonzero values of a general intersective polynomial h in Z[x]. The central technical step is to run a density-increment iteration in which the underlying polynomial is allowed to change at each stage while still keeping the inequality effective. This produces the strongest quantitative version presently known for the Furstenberg-Sárközy theorem in the polynomial setting. A sympathetic reader cares because the result quantifies how rapidly sets must thin out to avoid structured differences of polynomial type.

What carries the argument

The arithmetic level-d inequality, which supplies a uniform density-increment bound even when the polynomial is updated at each step of the iteration.

What would settle it

An explicit construction of a subset A of {1,...,X} with |A| larger than the claimed quasipolynomial bound that still avoids all nonzero h(n) differences, or an auxiliary polynomial arising in the iteration for which the level-d inequality fails to produce a positive density increment.

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Extended reading notes

Core claim

By showing that the arithmetic level-d inequality remains effective uniformly across the sequence of auxiliary polynomials generated by the density-increment iteration, the authors obtain a quasipolynomial upper bound on the size of the largest subset of {1,2,...,X} whose difference set contains no nonzero element of the form h(n) for an arbitrary intersective polynomial h.

Load-bearing premise

The arithmetic level-d inequality remains effective uniformly across all auxiliary polynomials arising in the iteration.

Editorial extensions

If this is right

  • The quasipolynomial bound now holds for every fixed intersective polynomial rather than only for squares.
  • The iteration proceeds with a changing polynomial without losing the quasipolynomial gain at each step.
  • Smoothly weighted versions of Rice's exponential-sum estimates are available to support the argument.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The uniformity argument may extend to finite families of intersective polynomials or to multidimensional configurations.
  • Numerical verification for small X and low-degree polynomials could test how close the bound comes to being sharp.
  • The same uniformity technique might combine with other increment methods to handle still broader classes of configurations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper adapts Green and Sawhney's arithmetic level-d inequality to general intersective polynomials h ∈ ℤ[x]. It proves a quasipolynomial upper bound on the size of the largest subset of {1,…,X} whose difference set avoids nonzero values of h(n), via a density-increment iteration in which the underlying polynomial changes at each step. A central contribution is a uniformity statement ensuring the level-d inequality remains effective for all auxiliary polynomials generated during the iteration. The authors also establish smoothly weighted versions of Rice's exponential-sum estimates.

Significance. If the uniformity claim holds with explicit parameter dependence, the result supplies the strongest known quantitative bound for sets without intersective polynomial differences and demonstrates that the Green-Sawhney method extends beyond squares. The weighted exponential-sum estimates are likely to be reusable in other arithmetic-progression or polynomial-difference problems.

major comments (1)
  1. [§3] §3 (Uniformity of the arithmetic level-d inequality): the argument must track the dependence of the implied constants on the degree, leading coefficient, and height of the current auxiliary polynomial explicitly. Without such tracking, it is unclear whether the density-increment threshold remains strong enough for the full iteration to close with a quasipolynomial bound rather than a tower-type loss.
minor comments (2)
  1. [Theorem 1.1] The statement of the main theorem should include the precise form of the quasipolynomial (e.g., X / (log log X)^c) rather than the generic phrase 'quasipolynomial upper bound'.
  2. [§5] Notation for the weighted exponential sums in §5 should be aligned with the unweighted versions introduced earlier to avoid reader confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate revisions to strengthen the presentation of the uniformity argument.

read point-by-point responses

  1. Referee: [§3] §3 (Uniformity of the arithmetic level-d inequality): the argument must track the dependence of the implied constants on the degree, leading coefficient, and height of the current auxiliary polynomial explicitly. Without such tracking, it is unclear whether the density-increment threshold remains strong enough for the full iteration to close with a quasipolynomial bound rather than a tower-type loss.

    Authors: We thank the referee for this observation. The uniformity statement in Section 3 is formulated so that the arithmetic level-d inequality holds for each auxiliary polynomial arising in the iteration, with the implied constants depending on its degree, leading coefficient, and height. To make this fully explicit and confirm that the density-increment thresholds yield only quasipolynomial losses overall, we will revise the manuscript to record the precise functional dependence of all constants on these parameters throughout the iteration analysis. This will verify that no tower-type losses are introduced, consistent with the claimed quasipolynomial bound. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; extends external Green-Sawhney result via independent uniformity argument

full rationale

The derivation adapts the arithmetic level-d inequality from Green and Sawhney (distinct external authors) and proves a new uniformity statement across the auxiliary polynomials that arise during density-increment iteration. This uniformity is explicitly flagged as the paper's key contribution and is not obtained by fitting parameters to the target bound or by self-referential definition. No equations reduce the claimed quasipolynomial bound to a tautology or to a self-citation chain; the iteration closes because the constants are tracked independently of the final result. The paper is therefore self-contained against the external benchmark.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the arithmetic level-d inequality from Green and Sawhney together with the standard definition that h is intersective.

assumptions (1)
  • domain assumption h is an intersective polynomial: h(Z) intersects every arithmetic progression.
    This is the class of polynomials to which the theorem applies, stated in the abstract.

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Cite this review

Pith. "Pith review of Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality." pith.science (2026). https://pith.science/paper/DSHKI22R

@misc{pith2026260516216,
  author       = {Pith},
  title        = {Pith review of: Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSHKI22R}},
  note         = {Machine review of arXiv:2605.16216}
}
abstract

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\in\mathbb{Z}[x]$ and obtain an analogous quasipolynomial upper bound for the largest subset of $\{1,2,\dots,X\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\in \mathbb{Z}$. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences. In contrast to the square case, extending the method to general intersective polynomials requires performing a density increment iteration in which the underlying polynomial changes at each step; a key contribution of this paper is to show that the arithmetic level-$d$ inequality remains effective uniformly across all auxiliary polynomials arising in the iteration. We also develop smoothly weighted versions of the exponential sum estimates of Rice.

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Reference graph

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