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A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets)

Theorem 1.1: absolute $c>0$ , $C$ with $|A|\le CN^{1-c}$ for every square-difference-free $A\subseteq[N]$ ; consequently the square-difference graph on $[N]$ has chromatic number at least $C^{-1}N^c$ . Companions: for an intersective $h$ of degree $k\ge2$ , a power saving $O_h(N^{1-c_k})$ with exponent depending only on $k$ ; and for $h$ with a unit root modulo every modulus, a power saving $C_hN^{1-c_h}$ for differences $h(p)$ at primes. The exponent is tiny and not computed, far from Ruzsa's lower-bound exponent $0.733$ ; the true exponent is not addressed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets) · Power saving for square-difference-free sets

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

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Named evidence

VibeMathed record: A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Green and Sawhney's question as cited. Lean: Comparator challenge SquareDifference, declaration OAI.SquareDifference.power_saving. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: there exist reals $c>0$ and $C$ such that for every $N\ge1$ every finite set of integers $A\subseteq[1,N]$ with $a-b\ne m^2$ for all $a,b\in A$ and $m\ge1$ has $|A|\le CN^{1-c}$ . That is exactly the headline. Not rebuilt here. The paper says the exponent is extremely small and not computed. The two 5 October companions (intersective polynomials, prime arguments) are not formalized; the prime-argument one depends on a companion zero-free half-plane theorem for Dirichlet L-functions.

Source ↗

A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets)

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean Comparator challenge SquareDifference (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Power saving for intersective polynomial differences, exponent depending only on the degree (companion)

paper · pending

paper link supplied by the VibeMathed registry.

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A Power Saving for Polynomial Differences at Prime Arguments (companion)

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Green and Sawhney, New bounds for the Furstenberg-Sarkozy theorem (arXiv:2411.17448)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge SquareDifference (OpenAI math release)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

A Power Saving for Polynomial Differences at Prime Arguments (companion) evidence for this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets) evidence for this event

A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets) parent of this event

Lean Comparator challenge SquareDifference (OpenAI math release) evidence for this event

A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets) evidence for this event

Power saving for intersective polynomial differences, exponent depending only on the degree (companion) evidence for this event

Green and Sawhney, New bounds for the Furstenberg-Sarkozy theorem (arXiv:2411.17448) evidence for this event

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