Linear equations and chromatic thresholds in $B_h$ sets

Nathan Tung

arxiv: 2606.30767 · v2 · pith:TTJT425Cnew · submitted 2026-06-29 · 🧮 math.CO · math.NT

Linear equations and chromatic thresholds in B_h sets

Nathan Tung This is my paper

Pith reviewed 2026-07-01 01:47 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords B_h setsSidon setslinear equationschromatic numberFourier pseudorandomnessadditive combinatoricsRoth-type theorems

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The pith

B_h sets avoiding pairwise distinct solutions to linear equations with more than 2h variables must be smaller than the maximum size by a constant factor.

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

The paper establishes that in a B_h set free of such solutions, the size drops by a fixed positive proportion below the best known upper bound for unrestricted B_h sets. This holds because near-maximal B_h sets are Fourier pseudorandom, allowing Roth-type density increment arguments to transfer to the sparse setting. For the special case of Sidon sets and non-translation-invariant equations whose coefficients contain a zero-sum subset of size at least five, the set is either small or the Cayley graph it generates has bounded chromatic number. Large Sidon sets that avoid equations with zero-sum subsets of size four can still produce Cayley graphs of unbounded chromatic number.

Core claim

If a B_h set contains no pairwise distinct solutions to a linear equation with more than 2h variables, then its cardinality is at most a constant factor smaller than the largest known B_h sets. Extremal B_h sets are Fourier pseudorandom. When the forbidden equation has a subdivision structure an asymptotic improvement follows. In the Sidon case, forbidding an equation whose coefficients admit a zero-sum subcollection of five or more terms forces either small size or bounded chromatic number on the associated Cayley graph, while constructions show that four-term zero-sum subcollections permit large sets with unbounded chromatic number.

What carries the argument

Fourier pseudorandomness of extremal B_h sets, which transfers density-increment arguments from the dense setting to control the size of equation-free subsets.

If this is right

For equations with a certain subdivision structure the size reduction improves from constant-factor to asymptotic.
In Sidon sets, the presence of a zero-sum subcollection of five or more coefficients forces either very small size or bounded chromatic number on the generated Cayley graph.
Sidon sets avoiding all equations whose coefficients have zero-sum subcollections of size four can still generate Cayley graphs of unbounded chromatic number while remaining large.
The Fourier pseudorandomness property supplies the main technical step that lets the sparse Roth-type statements hold inside near-maximal B_h sets.

Where Pith is reading between the lines

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

The results indicate that the extremal size of B_h sets is attained only when many linear equations are represented.
The transition between bounded and unbounded chromatic number at five versus four coefficients may mark a structural threshold that persists for higher h.
Similar size reductions might hold for other families of sets with controlled additive energy, such as sum-free sets or bases of higher order.

Load-bearing premise

Extremal B_h sets are Fourier pseudorandom.

What would settle it

An explicit construction of a B_h set whose size meets or exceeds the current upper bound minus only o(1) times that bound, yet contains no pairwise distinct solutions to some fixed linear equation with more than 2h variables, would falsify the size-reduction claim.

read the original 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.

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.

Desk Editor's Note private letter to a colleague

The paper proves extremal B_h sets are Fourier pseudorandom and uses that to get constant-factor size drops for those avoiding linear equations with more than 2h variables, while extending the Sidon chromatic threshold results.

read the letter

The main point is that large B_h sets without solutions to those linear equations end up smaller by a constant factor than the best known upper bounds, and this follows from first showing the extremal ones are Fourier pseudorandom. The work also gives new constructions for Sidon sets that control the chromatic number of the associated Cayley graphs depending on the zero-sum structure of the forbidden equation.

What is actually new is the extension from the h=2 Sidon case to arbitrary h, along with the zero-sum coefficient constructions that produce both bounded and unbounded chromatic number examples. It handles the non-translation-invariant equations cleanly and recovers the sparse analog of the Liu-Wu-Yang-Zhang characterization. The pseudorandomness step is presented as something they establish rather than assume, which removes the main potential weakness in the size bound.

The soft spots are minor. The constant-factor saving is the expected level of result in this area, and the subdivision structure case for asymptotic savings is stated but would need the full proofs to check edge cases. No circularity appears in the citations to the earlier Sidon papers, and the argument structure matches the stress-test description.

This is for additive combinatorialists already working on B_h sets, sparse Roth analogs, or chromatic thresholds in Cayley graphs. Someone following the Conlon-Fox-Sudakov-Zhao and Prendiville lines will see the direct extensions and get value from the new constructions.

I would send it to a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes that extremal B_h sets are Fourier pseudorandom. It then shows that any B_h set free of 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 the size of B_h sets. For the special case h=2 (Sidon sets), the paper proves that forbidding a non-translation-invariant equation E with a zero-sum subcollection of at least five coefficients forces the set to be small or to generate a Cayley graph of bounded chromatic number, while also constructing large Sidon sets free of certain four-coefficient zero-sum equations that generate Cayley graphs of unbounded chromatic number.

Significance. The Fourier pseudorandomness result for near-maximal B_h sets is a substantive technical contribution that enables the transfer of Roth-type density-increment arguments to sparse additive bases. The constant-factor size reduction supplies a quantitative sparse analog of classical results, while the chromatic-threshold statements for Sidon sets extend the translation-invariant work of Liu-Wu-Yang-Zhang to a non-invariant setting with explicit zero-sum conditions. These advances are likely to be useful in further sparse extremal problems in additive combinatorics.

minor comments (2)

[Abstract] Abstract: the phrase 'pairwise distinct solutions' is used without a parenthetical gloss; a brief clarification of the precise notion of solution (e.g., whether variables are required to be distinct from each other and from the equation coefficients) would aid readers who consult only the abstract.
[Introduction] The transition between the B_h results and the Sidon-set chromatic-number results would benefit from an explicit sentence stating how the subdivision-structure hypothesis in the general case specializes to the zero-sum-subcollection condition used for h=2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no point-by-point responses to provide at this stage. We will address any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation establishes Fourier pseudorandomness of extremal B_h sets as an internal result, then applies it to obtain the constant-factor size reduction for equation-free sets. This is presented as a derived input rather than a fitted parameter or self-definition. Prior Sidon results are cited to external authors (Conlon-Fox-Sudakov-Zhao, Prendiville), not as load-bearing self-citations. No equations reduce by construction to inputs, no ansatz is smuggled, and no uniqueness theorem is imported from the same authors. The central claim retains independent content from the pseudorandomness step and external benchmarks on B_h set sizes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract mentions no explicit free parameters or invented entities; relies on standard tools from additive combinatorics.

axioms (1)

standard math Standard results from additive combinatorics and Fourier analysis
Invoked to establish pseudorandomness of extremal B_h sets and size bounds.

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