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Linear equations and chromatic thresholds in $B_h$ sets

T0 review · 0 major / 3 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read A B_h set without pairwise distinct solutions to a linear equation with more than 2h variables must be smaller by a constant factor than the largest known B_h sets.

desk verdict Extends the Sidon-set Roth analogs to general B_h sets by proving extremal ones are Fourier pseudorandom, plus some new chromatic constructions for non-invariant equations. read the letter →

arxiv 2606.30767 v2 pith:TTJT425C submitted 2026-06-29 math.CO math.NT

Nathan Tung This is my paper · ORCID
classification math.COmath.NT
keywords B_hsetsSidonlinearequationschromaticnumberFourierpseudorandomnessRoththeoremadditivecombinatorics
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

The paper establishes that B_h sets of near-maximum size must contain solutions to linear equations with more than 2h variables, extending Roth-type theorems to this sparse setting. A key step shows that the largest B_h sets are Fourier pseudorandom, which transfers dense results to the sparse case. For Sidon sets, avoiding non-translation-invariant equations with zero-sum subcollections of five or more coefficients forces either a very small set or a Cayley graph of bounded chromatic number. Constructions demonstrate large Sidon sets that avoid equations with four-coefficient zero-sums yet produce graphs of unbounded chromatic number.

What carries the argument

Fourier pseudorandomness of extremal B_h sets, which serves as the input that transfers dense Roth-type results into the sparse B_h setting.

What would settle it

An explicit construction of a B_h set that reaches within a constant factor of the maximum size while avoiding all pairwise distinct solutions to some linear equation with more than 2h variables.

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Extended reading notes

Core claim

If a B_h set is free of pairwise distinct solutions to a linear equation with more than 2h variables then it must be a constant factor smaller than the best-known upper bound on the size of any B_h set. Extremal B_h sets are Fourier pseudorandom. If the forbidden equation has a certain subdivision structure, an asymptotic saving is obtained. For Sidon sets, forbidding a non-translation-invariant equation with a zero-sum subcollection of at least five coefficients implies the set is either very small or generates a Cayley graph with bounded chromatic number.

Load-bearing premise

Extremal B_h sets are Fourier pseudorandom.

Editorial extensions

If this is right

  • Large B_h sets must contain solutions to many linear equations with more than 2h variables.
  • When the equation has a subdivision structure, the size bound improves to an asymptotic saving.
  • Sidon sets avoiding equations with zero-sum subcollections of five or more coefficients are either small or generate bounded-chromatic-number Cayley graphs.
  • Sidon sets avoiding only equations with four-coefficient zero-sums can still be large and generate unbounded-chromatic-number Cayley graphs.

Reading between the lines

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

  • The pseudorandomness property may allow similar transfers for other additive-combinatorial statements beyond linear equations.
  • Chromatic-threshold characterizations could extend from Sidon sets to general B_h sets.
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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

0 major / 3 minor

Summary. The paper derives sparse analogs of Roth-type results in near-maximal B_h sets. It shows that a B_h set without pairwise distinct solutions to a linear equation with more than 2h variables must be smaller by a constant factor than the best-known upper bound on |B_h| sets. A key input is the proof that extremal B_h sets are Fourier pseudorandom. For Sidon sets (h=2), it further shows that forbidding a non-translation-invariant equation E with a zero-sum subcollection of at least five coefficients forces either small size or bounded chromatic number in the generated Cayley graph, while constructing large Sidon sets free of equations with four-coefficient zero-sums that generate graphs of unbounded chromatic number. This extends prior Sidon-set work of Conlon-Fox-Sudakov-Zhao and Prendiville, and provides a sparse analog of Liu-Wu-Yang-Zhang on chromatic thresholds.

Significance. If the Fourier pseudorandomness of extremal B_h sets holds, the results give a structural dichotomy for equation-free subsets of near-maximal B_h sets and supply new examples of sparse sets with controlled chromatic thresholds. The pseudorandomness input is a reusable tool that strengthens the sparse additive combinatorics toolkit, and the constructions for Sidon sets separate the behavior of different zero-sum lengths in a manner parallel to the dense case.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise definition of 'pairwise distinct solutions' used in the main theorem (likely in §2 or §3) to avoid ambiguity with the standard B_h definition.
  2. Notation for the linear equation E and its subdivision structure (mentioned for the asymptotic saving case) should be introduced with a numbered display equation early in the paper.
  3. The reference list should include the full citation details for Conlon-Fox-Sudakov-Zhao and Prendiville to facilitate comparison with the dense analogs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, for highlighting the significance of the Fourier pseudorandomness result as a reusable tool, and for recommending minor revision. The report correctly identifies the extensions of prior work by Conlon-Fox-Sudakov-Zhao, Prendiville, and Liu-Wu-Yang-Zhang.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The derivation establishes Fourier pseudorandomness of extremal B_h sets as an explicit key input and then derives constant-factor size reductions for subsets free of certain linear equations, extending prior results on Sidon sets by other authors (Conlon-Fox-Sudakov-Zhao, Prendiville, Liu-Wu-Yang-Zhang). No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; all supporting results are external, and the argument is presented as a direct sparse analog without internal reduction to its own inputs. The paper is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard mathematical foundations and domain-specific assumptions in additive combinatorics and Fourier analysis; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

assumptions (2)
  • standard math Standard axioms of set theory and arithmetic
    Background for all combinatorial arguments.
  • domain assumption Fourier analysis on abelian groups applies to pseudorandomness of extremal B_h sets
    Explicitly invoked as key input in the abstract.

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Cite this review

Pith. "Pith review of Linear equations and chromatic thresholds in $B_h$ sets." pith.science (2026). https://pith.science/paper/TTJT425C

@misc{pith2026260630767,
  author       = {Pith},
  title        = {Pith review of: Linear equations and chromatic thresholds in $B_h$ sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTJT425C}},
  note         = {Machine review of arXiv:2606.30767}
}
abstract

We derive sparse analogs of several Roth-type results, showing that they hold in $B_h$ sets of near-maximum size. It is shown that if a $B_h$ set is free of pairwise distinct solutions to a linear equation with more than $2h$ variables then it must be a constant factor smaller than the best-known upper bound on the size of any $B_h$ set. As a key input, it is established that extremal $B_h$ sets are Fourier pseudorandom. If the forbidden equation has a certain subdivision structure, an asymptotic saving is obtained. The case of Sidon sets ($h=2$) was previously studied by Conlon, Fox, Sudakov, and Zhao as well as Prendiville. When forbidding a non-translation-invariant equation $E$ from a Sidon set, it is shown that if $E$ has a zero-sum subcollection of at least five coefficients then the Sidon set must either be very small or generate a Cayley graph with bounded chromatic number. On the other hand, large Sidon sets are constructed that generate Cayley graphs with unbounded chromatic number and are also free of multiple equations with zero-sum subcollections of four coefficients. This can be viewed as a sparse analog of a result of Liu, Wu, Yang, and Zhang characterizing linear equations with vanishing chromatic threshold.

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Reference graph

Works this paper leans on

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