Openai 2026 power saving intersective polynomial differences

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theorem_1_1: The manuscript's main claim: for each degree k >= 2 an exponent c_k > 0, depending only on k, such that subsets of {1, ..., N} with no difference equal to a nonzero value h(n), n >= 1, of an intersective h of degree k with positive leading coefficient have at most C_h N^(1-c_k) elements. Claims checked only.

OpenAI, A power saving for intersective polynomial differences with an exponent depending only on the degree, OpenAI Math Release preprint, October 5, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026; the held PDF, power-saving-intersective-polynomial-differences.pdf in the release, is retained as openai_2026_power_saving_intersective_polynomial_differences.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026,
  author = {{OpenAI}},
  title = {{A power saving for intersective polynomial differences with an exponent depending only on the degree}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf}{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026}},
  year = {2026}
}

The held PDF has 47 pages with a text layer; its title page is dated October 5, 2026 and its metadata records a creation date of 6 October 2026. The TeX source is build/manuscript/main.tex (75 lines), which inputs seven section files under sections/ (introduction.tex, local.tex, tuples.tex, lifts.tex, kernel.tex, transfer.tex and induction.tex), the third of which inputs one figure file, figures/list-regrouping.tex; that is 3,566 TeX lines in all, with a 14-entry bibliography references/references.bib. main.tex also inputs a glyphtounicode-cmex file that the bundle does not include. The bundle is the text read here.

Attestation. The release README states that the manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model", each using on average "three hours of ChatGPT Pro thinking compute"; the exceptions it names (a zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties) concern other families. The manuscript's own README carries only the title, author ("OpenAI"), date and citation block and adds no sentence about authorship or assistance. The title page names OpenAI as the sole author, and the PDF has no acknowledgment, affiliation or funding note. These are the source's own statements, recorded as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here, and nothing on this card is independently reviewed.

Formalization. The release lists no formalization of this manuscript. Its family's Lean page, lean/docs/182.md, carries the family's title but names as its accompanying paper only the square-difference-free companion below; its one comparator, lean/ComparatorChallenges/SquareDifference.lean (theorem OAI.SquareDifference.power_saving), states, in words, that there are reals $c>0$ and $C$ such that every finite set of integers in $[1,N]$, $N\ge1$, with no difference $a-b$ equal to $m^2$ for an integer $m\ge1$, has at most $CN^{1-c}$ elements. That is the square case only, not Theorem 1.1 here. The release's catalog lean/formalization.yaml lists no entry for either manuscript at this revision, and its status.main_results names no comparator of this family. All of this is read statically from the release; nothing was built, replayed or audited in this repository.

Companions. The release groups this manuscript in its family, "Power savings for intersective polynomial differences and prime arguments", with two others. A power saving for square-difference-free sets (September 24, 2026), filed at the square-difference card, claims absolute constants $c>0$ and $C$ with $|A|\le CN^{1-c}$ for every $A\subseteq\{1,\ldots,N\}$ with no nonzero square difference; this manuscript cites it (its Theorem 1.1) as the immediate predecessor of its method and says it adapts that paper's reflection-positive tuple laws and signed multilinear functional, with all required proofs included. An observation of this card, not a claim of either manuscript: the case $h(x)=x^2$ of Theorem 1.1 here has exactly that hypothesis, and with the trivial bound $|A|\le N$ for $N<N_h$ it gives a bound of that form for every $N\ge1$. A Power Saving for Polynomial Differences at Prime Arguments (October 5, 2026), filed at the prime-argument card, claims in its abstract, which the release's manuscript map CONTENTS.md repeats, a power saving $|A|\le C_hN^{1-c_h}$ for sets whose differences avoid the nonzero values $h(p)$ at primes $p$, for $h$ of degree at least two with positive leading coefficient and a unit root modulo every positive integer, with $c_h$ and $C_h$ depending on $h$, obtained through a companion zero-free half-plane for Dirichlet $L$-functions. Its introduction (p. 2) calls this manuscript its "direct methodological predecessor" and says it adapts this manuscript's tuple laws, Fourier lifts, signed kernels and contracting induction to prime arguments; this manuscript does not cite it. The release holds no earlier dated version of this manuscript. The companions' arguments were not read for this card.

Read status: claims checked for Theorem 1.1, read clause by clause in the release's TeX source together with the definition of an intersective polynomial and the notation it uses (sections/introduction.tex, label thm:main at lines 14--26, definitions at lines 3--12, remarks at lines 28--31); also read clause by clause were the statements of Condition (U) and Lemmas 2.1 and 2.2 (sections/local.tex, labels local:unit-condition at lines 57--61, local:normalization at lines 66--84 and local:family at lines 147--172), Corollary 4.6 (sections/lifts.tex, lift:density at lines 601--612) and Propositions 7.1 and 7.2 (sections/induction.tex, induction:recurrence at lines 23--44 and induction:decay at lines 168--182). The statements of the other numbered results named under Contents were read in their TeX files; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The numbering below is the PDF's (results are numbered within sections in one sequence); the title page, abstract and table of contents occupy pp. 1--2.

• Section 1, Introduction (pp. 2--5). Calls $h\in\mathbb Z[x]$ intersective when it has a root modulo every integer $q\ge1$. The condition is necessary, since without a root modulo $q$ the multiples of $q$ form a set of positive density whose differences all miss $h(\mathbb N_+)$, and it does not require a rational root. States Theorem 1.1 (p. 2) and remarks that its constants may depend on every coefficient of $h$ while the exponent depends only on the degree. The history cites Furstenberg and Sárközy for square differences, Kamae and Mendès France for intersective polynomials, Pintz, Steiger and Szemerédi, Bloom and Maynard ($N(\log N)^{-c\log\log\log N}$ for squares), Lucier, Rice, Arala (the Bloom--Maynard bound for every intersective polynomial of degree at least two), Green and Sawhney ($N\exp(-c\sqrt{\log N})$ for squares, their Theorem 1.1) and Adajar et al. (arXiv:2605.16216v2, Theorem 1.1: $N\exp(-c_{h,\mu}(\log N)^\mu)$ for every fixed intersective $h$ and $0<\mu<1/2$), and the release's square-difference manuscript as the immediate predecessor, with Hatami and Conlon--Lee for related reflection arguments. The overview (pp. 3--5) names the five steps that the later sections carry out, and the notation subsection (p. 5) fixes the uniformity conventions: a bound written $O_k$ is uniform over all data of degree $k$ satisfying the unit condition of Section 2.
• Section 2, auxiliary polynomials and local estimates (pp. 5--12). Fixes a compatible root $\mathfrak z_p\in\mathbb Z_p$ of $h$ at every prime, with multiplicity $m_p$ and first nonzero Taylor coefficient $b_{m_p,p}$ there, a unit for all but finitely many $p$; Condition (U) asks that it be a unit at every prime. Lemma 2.1 (p. 6) normalizes: $g(x)=h(r+D_0x)/M$ is an intersective integer polynomial of degree $k$ with positive leading coefficient satisfying (U), with $Mg(n)=h(r+D_0n)$ and $r+D_0n\ge1$ for $n\ge1$. Following Lucier's auxiliary-polynomial construction (Section 8 of his paper), it defines the completely multiplicative $\lambda$ with $\lambda(p)=p^{m_p}$ and $h_l(x)=h(r_l+lx)/\lambda(l)$; Lemma 2.2 (p. 7) says each $h_l$ is an intersective integer polynomial of degree $k$ with the same root multiplicities, keeps (U) when $h$ has it, and satisfies $h_l(Dx-i)=\lambda(D)h_{lD}(x)$ for some $0\le i<D$, with $D\mid\lambda(D)\le D^k$, so a set in a progression of step $\lambda(D)$ avoiding $h_l$-differences becomes, in progression coordinates, a set avoiding $h_{lD}$-differences. With $\delta=1/(100k2^k)$ and $\gamma=\delta/10$, Lemma 2.3 (p. 9) bounds complete sums of $h_l$ modulo $p^j$ by $p^{-8\delta j}$, and over regular arguments ($h_l'\ne0$ modulo $p$) by $p^{-4\delta}$, for $p\ge P_1$; Lemma 2.4 (p. 10) bounds the derivative valuation at some small argument at each small prime. The admissible residues modulo $p^{E_p}$ are then defined (p. 11) as values of $h_l$ at arguments whose derivative valuation is within that bound when $p<P_1$, and at regular arguments when $p\ge P_1$; Lemma 2.5 (p. 11) is the uniform lifting lemma that explains this modulus; Lemma 2.6 (p. 11) gives $L$ admissible residues summing to zero, for one cycle length $L$. Under (U), $P_1$ and $L$ depend only on $k$, and each exponent $E_p$ only on $k$ and $p$. Rice's revised version (Sections 2.4--2.5 and 4) is cited for similar restrictions.
• Section 3, weighted reflection-positive tuple laws (pp. 12--22). Takes slots $V=\Pi^w$, words of $w=2t+1$ blocks each a permutation of $[L]$, so $d=|V|=(L!)^{2t+1}$ is even, with reflection cuts that swap two symbols in one block; a law on $\mathbb F_p^V$ is reflection positive when every cut matrix is positive semidefinite. The edge weight counts the regular arguments $x$ with $h_l(x)=y$ in $\mathbb F_p$, divided by $k$. Lemma 3.1 (p. 13) is a graph-counting and list-mixing bound; measures indexed by sets of marked blocks are compared in Lemmas 3.2--3.4 (pp. 15--16). Proposition 3.5 (p. 17): for large $p$, uniformly in $l$, a mixture $\mu_{l,p}$ is a reflection-positive, translation-invariant probability law with uniform single-slot marginals, and outside an exceptional part of mass at most $p^{-4}$ every increment along a designated $L$-cycle of slots is a value of $h_l$ at a regular argument. Lemma 3.6 (p. 18) is the covariance: the map $z\mapsto\lambda(D)^{-1}(z-a')$ carries the law for $l$ to the law for $lD$ when $p\nmid D$. Lemma 3.7 (p. 19) is conditional mixing with saving $p^{-\gamma}$, and Proposition 3.8 (p. 20) shows that the prime threshold $P$ and the small-prime modulus $M_0=\prod_{p<P}p^{E_p}$ depend only on $k$ under (U). The section adapts Lemmas 2.1--2.2 and Section 3 of the square manuscript.
• Section 4, multilinear estimates and Fourier lifts (pp. 22--30). Proposition 4.1 (p. 23) is a signed Hölder inequality for the product laws over a finite set of primes at least $P$, proved by repeated folding across cuts after the chessboard method of Fröhlich, Israel, Lieb and Simon (their Theorem 4.1); Lemma 4.2 (p. 24) makes its $d$th root a seminorm that dominates the mean, contracts under coordinate averaging and is carried exactly from $l$ to $lD$ by the covariance map. Lemma 4.3 (p. 24) gives decay $R^{-\gamma}$ when one slot has no Fourier mass at denominators at most $R$. Proposition 4.4 (p. 26) bounds the $L^{2r}$ norm of the Fourier lift of a function $f$ on an interval of length at least $10Q^6$ (its Fourier coefficients at the frequencies whose squarefree denominators, at most $Q$, are built from the chosen primes, moved to the product of residue fields) by $C_{r,\epsilon}\|f\|_\infty Q^\epsilon$, and Lemma 4.5 (p. 28) prices the fixing of coordinates. Section 4.4 (p. 29) fixes $\eta=1/(100k)$, $\zeta=\eta/(20k)$, $\nu=\eta/(10\cdot2^k)$, $\beta=\nu/100$, $\tau=\beta\gamma/(1000k(d+1))$ and $\sigma=\gamma\tau/4$, the cutoff $Q=N^\beta$ and the density quantity $Y_l(N,A)$, the multilinear form of the lift of $1_A$; Corollary 4.6 (p. 29) gives $(|A|/N)^d\le Y_l(N,A)\le C_\epsilon N^\epsilon$. The section adapts the square manuscript's Proposition 4.1 and Lemmas 4.2--4.3 and its two lift lemmas, which this manuscript cites as Lemmas 5.1--5.2; in the released square manuscript they are Lemmas 5.2--5.3.
• Section 5, a signed kernel for polynomial differences (pp. 30--37). The pair multiplier $\Gamma$ of a cycle edge satisfies $|\Gamma(\xi)|\le q(\xi)^{-\gamma}$. Proposition 5.1 (p. 30): for $l\le N^\zeta$, a modulus $D=M_0\prod_{p\in B}p\le N^\beta$ and two functions on $[N]$ bounded by one, supported in residue classes modulo $D$ with admissible difference, with the first support entirely below the second and no difference between them in $h_l(\mathbb N_+)$, the $\Gamma$-weighted pair sum over squarefree denominators up to $H=N^{\beta/2}$ coprime to $D$, times $D$, is $\ll H^{-\gamma/2}$. The proof builds a signed kernel supported on positive values of $h_l$, with derivative weights and a truncated divisor expansion, whose Fourier transform matches a model kernel uniformly (Lemma 5.2, p. 32): Weyl differencing on the minor arcs (Lemma 5.3, p. 33) and a comparison of local coefficients on the major arcs (Lemma 5.4, p. 35); its bilinear form vanishes on avoiding pairs (Section 5.5, p. 36). It adapts Section 6 of the square manuscript (Proposition 6.1).
• Section 6, comparison and cancellation (pp. 37--42). Defines the maximal functional $\mathcal Y_l(N)$, the largest $Y_l(N,A)$ over $A\subseteq[N]$ avoiding nonzero $h_l$-differences. Lemma 6.1 (p. 37): fixing prime coordinates of a lift restricts to a residue class. Proposition 6.2 (p. 38) bounds, uniformly in $l$, the multilinear form of lifts with the prime coordinates in $B$ fixed, for $q_B\le N^\tau$, by $\Lambda_B^d\mathcal Y_{lD}(n_B)+O(N^{-\sigma})$ at the shorter scale $n_B=\lceil N/\lambda(D)\rceil$, $D=M_0q_B$, with $1\le\Lambda_B\le1+M_0^kN^{k\tau-1}$. Lemma 6.3 (p. 39): the residue assignments modulo $M_0$ whose cycle increments are admissible have proportion at least $\rho_*=M_0^{1-L}$. Proposition 6.4 (p. 40): for these and $l\le N^\zeta$, the integral against the nonexceptional laws is $\ll N^{-\sigma}$, by locating an ordered cycle edge and applying Proposition 5.1. The section adapts Sections 7--8 of the square manuscript, whose comparison and contraction results this manuscript cites as Propositions 7.2 and 8.1; in the released square manuscript they are Propositions 7.2 and 8.2.
• Section 7, exceptional terms and the power-saving induction (pp. 42--46). Proposition 7.1 (p. 42) is a recurrence bounding $\mathcal Y_l(N)$, for $l\le N^\zeta$, by $C_1N^{-\sigma}$ plus $1+u_N$ times a sum, with weights $1-\rho_l$ and $\rho_l\eta_{l,B}$, of values $\mathcal Y_{lD_B}(n_B)$ at shorter scales, where $u_N\to0$ and the exceptional masses $\eta_{l,B}$ sum to at most $1/4$. Proposition 7.2 (p. 44) gives $\mathcal Y_l(N)\le C_0l^{b_0}N^{-a_0}$ for all positive $l,N$, with $a_0$, $b_0$ explicit in $k,\zeta,\sigma,M_0,\rho_*$ and depending only on $k$ under (U). The proof of Theorem 1.1 (pp. 45--46) sets $c_k=a_0/d$. The section adapts the square manuscript's recurrence and decay results, which this manuscript cites as Proposition 9.2 and Theorem 9.3; in the released square manuscript they are Proposition 9.1 and Theorem 9.2.
• References (p. 47): fourteen entries, eleven with DOIs and three as arXiv versions or a release citation (Green--Sawhney, Adajar et al., the square manuscript).

The manuscript flags nothing as numerical, computer-assisted or conditional, and it states that all estimates it needs from the reflection-positive method are proved in the text. All structural parameters are explicit functions of $k$ except the prime thresholds $P_1$ and $P$, fixed only as large enough in terms of $k$; no numerical value of $c_k$ is given. The release folder for this manuscript holds only the PDF, the TeX build directory and the README; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. The problem pages found by searching for square and polynomial differences concern other objects: Problems 438 and 439 ask about squares among sums of two elements, and Problem 894 about differences lying in a lacunary sequence; Theorem 1.1 says nothing about either kind of set.