View source for Power-saving bounds for square-difference-free sets

'''Power-saving bounds for square-difference-free sets''' are the subject of three manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). Write ''s''(''N'') for the largest size of a set ''A'' ⊆ {1, …, ''N''} in which no two elements differ by a nonzero perfect square. Furstenberg and Sárközy independently proved ''s''(''N'') = ''o''(''N''), and whether a bound ''s''(''N'') ≤ ''CN''<sup>1−''c''</sup> holds was an open question. The manuscripts claim such a bound, together with extensions to intersective polynomials and to polynomial values at primes. A human paper first posted on the same day as the repository proves a weaker bound by a different method, and its second version comments on the OpenAI draft. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''Claim (squares).''' There are absolute constants ''c'' > 0 and ''C'' < ∞ such that every square-difference-free ''A'' ⊆ {1, …, ''N''} has |''A''| ≤ ''CN''<sup>1−''c''</sup>, for every ''N'' ≥ 1 (Theorem 1.1). The manuscript states that this answers the fixed-power question posed by Green and Sawhney. It also says that the exponent is "extremely small" and that the argument "does not yield a useful numerical value". Source: the manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026 A power saving for square-difference-free sets], dated 24 September 2026. Its outline (§1.2) describes the method: a seminorm built from a reflection-positive probability law over prime fields dominates a fixed power of the density; the seminorm is applied to a signed Fourier lift of ''A''; and a comparison along square-step progressions, together with a signed kernel supported on actual square shifts, makes this quantity decay by a power of ''N''.
* '''Claim (intersective polynomials).''' For each degree ''k'' ≥ 2 there is ''c''<sub>''k''</sub> > 0 such that, if ''h'' is an intersective integer polynomial of degree ''k'' with positive leading coefficient (one with a root modulo every positive integer), every ''A'' ⊆ {1, …, ''N''} whose differences avoid the nonzero values ''h''(1), ''h''(2), … has |''A''| = ''O''<sub>''h''</sub>(''N''<sup>1−''c''<sub>''k''</sub></sup>). The exponent depends only on the degree. Source: the manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026 A power saving for intersective polynomial differences with an exponent depending only on the degree], dated 5 October 2026.
* '''Claim (prime arguments).''' If ''h'' has degree at least two, a positive leading coefficient and a unit root modulo every positive integer, then every ''A'' ⊆ {1, …, ''N''} whose differences avoid the nonzero values ''h''(''p'') at primes ''p'' has |''A''| ≤ ''C''<sub>''h''</sub>''N''<sup>1−''c''<sub>''h''</sub></sup>. Source: the manuscript [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026 A Power Saving for Polynomial Differences at Prime Arguments], dated 5 October 2026. Its prime-distribution input (Theorem 5.1) is the zero-free half-plane ℜ''s'' > 7/8 for all Dirichlet ''L''-functions, taken from the companion manuscript in the same collection (see [[The quasi-Riemann hypothesis]]).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/182.md lean/docs/182.md] lists only the square-difference manuscript. It states that the formalization proves the fixed power saving for sets with no nonzero square difference, uniformly for every ''N'' ≥ 1. The Comparator statement is [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SquareDifference.lean <code>SquareDifference.lean</code>]. The intersective-polynomial and prime-argument results are not covered by that scope note.
* '''Dates.''' The manuscripts carry the dates given above. The repository's initial public commit, [https://github.com/openai/math/commit/adc7f1241b42e322a6451854ab7e4b4c146bf78a <code>adc7f12</code>], which contains all three, is timestamped 6 October 2026, 21:58:50 UTC.

== Earlier published results ==
* '''Upper bounds.''' Furstenberg (1977) and Sárközy (1978) independently proved ''s''(''N'') = ''o''(''N''). Later bounds of the form ''N''/(log ''N'')<sup>''c'' log log log log ''N''</sup> (Pintz, Steiger and Szemerédi, 1988) and ''N''/(log ''N'')<sup>''c'' log log log ''N''</sup> (Bloom and Maynard, ''Compositio Mathematica'') followed. Green and Sawhney ([https://arxiv.org/abs/2411.17448 arXiv:2411.17448], ''New bounds for the Furstenberg–Sárközy theorem'') prove ''s''(''N'') ≪ ''N'' exp{−''c''(log ''N'')<sup>1/2</sup>} and ask whether a fixed power saving holds. For intersective polynomials of degree at least two, Adajar et al. ([https://arxiv.org/abs/2605.16216 arXiv:2605.16216]) obtain savings exp{−''c''(log ''N'')<sup>''μ''</sup>} for every fixed ''μ'' < 1/2.
* '''Lower bounds.''' Ruzsa (1984) constructed square-difference-free sets of size ≫ ''N''<sup>0.733077…</sup>. Beigel and Gasarch ([https://arxiv.org/abs/0804.4892 arXiv:0804.4892]) and Lewko (2015) improved the exponent to 0.7334117…, and Krachun ([https://arxiv.org/abs/2608.01325 arXiv:2608.01325], ''Square-Difference-Free Sets beyond the Three-Quarter Barrier'') to 0.7527964…. After the release, E. Naslund posted a computer-assisted construction with exponent 0.7580758, formalized in Lean ([https://hexagonmath.org/2610.00151v1 Hexagon 2610.00151v1], 8 October 2026). That note concerns lower bounds and does not refer to the OpenAI manuscripts.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-06 (v1), 2026-10-12 (v2) || [https://arxiv.org/abs/2610.07765 arXiv:2610.07765] (v1, v2), S. Fan and A. Lott; v1 titled ''A new bound for the Furstenberg–Sárközy theorem using the van der Corput property'', v2 retitled ''The optimal van der Corput property for squares'' || Parallel work (weaker bound, different method); remark and critique (v2) || Proves ''s''(''N'') ≤ ''N'' exp{−''c''(log ''N'' log log ''N'')<sup>1/2</sup>}, improving Green and Sawhney, by constructing nonnegative trigonometric polynomials whose frequencies are squares (a quantitative van der Corput property with signed coefficients). Both versions also prove a matching lower bound for the constant coefficient of every such polynomial; the v2 abstract concludes that "the van der Corput property cannot be used to obtain a power-saving bound". A remark in v2, linking to the repository's initial commit and to its Lean scope note, refers to "a draft claiming the power-saving bound". It states that v1 was submitted to arXiv on 6 October 2026 at 05:02:35 UTC, 16 hours and 56 minutes before the repository's initial commit, describes the draft as appearing "AI-written and largely unreadable", and notes that the repository claims a Lean certificate. The paper also announces a forthcoming power saving for cube differences by the same method. Its AI disclosure states that the authors used a language model during the work and verified the manuscript themselves. || arXiv v2.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial", which MathVet defines as "only a special case, or only one of several co-equal headline claims". Its note says that only the square case ''h''(''x'') = ''x''<sup>2</sup> is stated in Lean and that the intersective-polynomial and prime-argument results are not formalized. It records a sandboxed Comparator pass for the challenge <code>SquareDifference</code> (8 October 2026). || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]