Difference Sets Containing Squares

Difference Sets Containing Squares

OpenAI's claim

Polynomial differences at prime arguments: For a fixed integer polynomial with positive leading term and a unit root modulo every modulus, the paper claims a fixed power saving for sets avoiding its nonzero values at prime inputs as differences; both the exponent and constant may depend on the polynomial.

Degree-uniform intersective differences: For an intersective polynomial with a positive leading term and degree $k\ge2$, the claimed bound for difference-avoiding subsets is $O_h(N^{1-c_k})$; the exponent depends only on $k$, although the implicit constant depends on the polynomial.

Square-difference-free power saving: The manuscript claims absolute positive constants $c,C$ with $|A|\le C N^{1-c}$ whenever $A\subseteq[1,N]$ has no nonzero square difference; the improvement is a fixed power of $N$.

Mathematical statement of the claimed result

Mathematical claim 1.

An integer polynomial $h$ is prime-intersective if each positive modulus $q$ has a root $r$ with $\gcd(r,q)=1$ and $h(r)\equiv0\pmod q$. If $\deg h\ge2$ and its leading coefficient is positive, then the claim supplies $c_h>0,C_h\ge1$ such that \[(A-A)\cap\{h(p):p\text{ prime}\}\subseteq\{0\}\quad\Longrightarrow\quad|A|\le C_hN^{1-c_h}\] for every integer $N\ge1$ and $A\subseteq\{1,\ldots,N\}$. Both constants may depend on the polynomial.

Mathematical claim 2.

For each degree $k\ge2$ there is $0<c_k<1$ such that every intersective $h\in\mathbb Z[x]$ of degree $k$ with positive leading coefficient has constants $C_h\ge1,N_h$ satisfying \[(A-A)\cap h(\mathbb N_{>0})\subseteq\{0\},\quad A\subseteq\{1,\ldots,N\},\quad N\ge N_h\quad\Longrightarrow\quad|A|\le C_hN^{1-c_k}.\] Intersective means having a root modulo every positive integer. The saving exponent depends only on the degree; the prefactor and threshold may depend on all coefficients.

Mathematical claim 3.

The claimed square-difference bound gives absolute $c>0,C<\infty$ with \[|A|\le CN^{1-c}\] for every $N\ge1$ and $A\subseteq\{1,\ldots,N\}$ having no distinct $a,b$ with $|a-b|=m^2$ for any positive integer $m$. Equivalently every larger subset contains a nonzero square difference.

Outline of the argument

Polynomial differences at prime arguments: The progression-compatible polynomial-difference method is adapted to a distribution supported on prime arguments. Root normalizations retain the required coprimality through successive arithmetic progressions. Prime-distribution estimates from the companion Dirichlet zero-free result supply the local comparison needed to contract the density-dominating multilinear quantity.

Degree-uniform intersective differences: Local roots are normalized until their first nonzero Taylor coefficients are units, without changing their multiplicities. Auxiliary polynomials are chosen to remain compatible under passage to progressions. A multilinear density lower bound and a distribution-comparison argument then produce contraction with numerical choices controlled by the degree rather than the coefficients.

Square-difference-free power saving: A probability law on tuples of residues over many primes defines a nonnegative multilinear quantity dominating density. A rational Fourier lift places the indicator into this residue model. Comparison with shorter intervals works for every residue choice, and a positive fraction of negligible choices makes the quantity contract by a fixed power.

Formal verification

Family 182 describes Lean coverage for the square-difference power saving; the general polynomial and prime-input estimates are separate.

Polynomial differences at prime arguments: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Degree-uniform intersective differences: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Square-difference-free power saving: The manuscript links SquareDifference.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, A Power Saving for Polynomial Differences at Prime Arguments, 2026-10-05; citation key OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, A power saving for intersective polynomial differences with an exponent depending only on the degree, 2026-10-05; citation key OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, A power saving for square-difference-free sets, 2026-09-24; citation key OAI:A-power-saving-for-square-difference-free-sets-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 182: Lean scope.
• Comparator SquareDifference.json.
• Comparator SquareDifference.lean.
• OpenAI catalogue entry.

A Square in a Dense Difference Set

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For a finite set $A\subseteq[N]$, $A-A=\{a-b:a,b\in A\}$. A nonzero square is $u^2$ with $u\in{\mathbb{N}}$. The asymptotic assertion seeks an absolute $c>0$ so that sets at the indicated power-size threshold contain $a,b$ with $a-b=u^2>0$. Cardinality thresholds are rounded to integers; the chosen $c$ must not depend on $N$. [S1] [S2]

Statement recorded in the supplied source.

Is there $c > 0$ such that whenever $A \subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square? [S1]

Residual task reported by the archive.

The embedded review $2026-08-17$ records: Prove any fixed polynomial saving for square-difference-free subsets of [N], or construct examples ruling it out. [S1]

Short English statement

Does a fixed power-saving density threshold already force two selected integers to differ by a nonzero perfect square?

Research attempt

Provenance and scope.

This individual attempt was generated by GPT-6 Astra Ultra on 1 October 2026. The method was to derive explicit partial results and test the first proposed extension adversarially. The original statement and sources below are retained. No claim of mathematical novelty or complete solution is made.

Formal target and primary-source context.

A square-difference-free set $A\subset[N]$ has no two elements with positive difference $u^2$, $u\geq1$. The question asks for a fixed power saving in its maximum cardinality. The inspected Green--Sawhney manuscript proves a bound of shape $Ne^{-c\sqrt{\log N}}$. That remains larger than $N^{1-\delta}$ for every fixed $\delta>0$ at sufficiently large $N$, since $\sqrt{\log N}=o(\log N)$. We construct a simple digit family and derive an exact finite-field upper bound, then check why reducing modulo a prime fails to connect the two at the requested scale.

A square-free difference construction in base five.

For $k\geq1$, let \[ A_k=\left\{\sum_{i=0}^{k-1}\varepsilon_i5^i:\varepsilon_i\in\{0,2\}\right\}. \] It has $2^k$ elements and maximum $(5^k-1)/2$. Take distinct $a,b\in A_k$, and let $j$ be their lowest differing digit. Then \[ a-b=5^j u,\qquad u\equiv2\text{ or }-2\pmod5. \] There is no ambiguity from higher digits: after factoring out $5^j$, they are multiples of five, while the first nonzero digit difference is $\pm2$. If $a-b$ were a nonzero integer square, its five-adic valuation $j$ would be even and its unit part $u$ would be a nonzero square modulo five. But $2$ and $3=-2$ are both nonsquares modulo five. This contradiction proves that $A_k$ has no nonzero square difference of either sign. Translating by one puts it in a positive interval without changing differences.

This yields examples of cardinality order $N^{\log_5 2}$ along an explicit sequence of lengths. Therefore a proposed universal power exponent cannot exceed $1-\log_5 2$ by this construction alone. This is not asserted to be the strongest known construction; it is a transparent example with a complete valuation proof.

An exact quadratic-character bound over prime fields.

Let $p\equiv1\pmod4$ be prime and $B\subset{\mathbb{F}}_p$ have all nonzero differences quadratic nonresidues. Let $m=|B|$ and let $\chi$ be the quadratic character, with $\chi(0)=0$. Then \[ \sum_{a,b\in B}\chi(a-b)=-m(m-1). \] For a nonzero additive frequency $r$, the Fourier transform of $\chi$ has modulus $\sqrt p$, by the usual Gauss-sum identity; its zero coefficient is zero. A short proof of the relevant modulus follows from $\sum_x\chi(x)\chi(x+h)=-1$ for $h\ne0$. To check this correlation, divide by $h$, complete the square, and count solutions to $v^2=u(u+1)$: multiplying by four converts it to $(2u+1)^2-(2v)^2=1$, whose factor pairs number $p-1$. Thus the correlation is $-1$, and Fourier inversion gives modulus $\sqrt p$ at every nonzero frequency.

The convolution operator with kernel $\chi$ therefore has operator norm $\sqrt p$ on mean-zero functions. Apply it to $1_B-m/p$; its squared norm is $m-m^2/p$ in counting measure. We obtain \[ m(m-1)\leq\sqrt p\,(m-m^2/p), \qquad m\leq\sqrt p. \] For $m>0$, the final inequality follows by division and rearrangement; the empty set is immediate. This is a strong power bound in a finite-field model.

The attempted modular transfer breaks.

An integer square-difference-free set need not have only quadratic-nonresidue differences modulo $p$. A positive integer can be a square modulo $p$ without being an integer square. For example, two integers differing by six have no square difference, but six is a square modulo five. Thus the finite-field character condition is strictly stronger than the original one. Choosing a prime comparable to $N$ does not repair this implication. A modular proof would need extra information forcing an actual integer square among the differences, not merely a quadratic-residue class.

Verification and remaining gap.

The companion script checks every pair in the digit construction for $k=1,\ldots,5$, obtaining sizes $2,4,8,16,32$ and no square differences. The prime-field argument explicitly uses $p\equiv1\pmod4$, ensuring the undirected difference condition is consistent. Its normalisations and diagonal zero terms are retained. The work proves an explicit infinite lower family, a finite-field power bound, and the failure of the direct reduction between them. The unresolved task is an integer upper bound with any fixed exponent saving; neither the current subexponential literature bound nor the stronger modular hypothesis supplies it.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 126 (GREEN-038), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Ben Green and Mehtaab Sawhney, New bounds for the Furstenberg-Sarkozy theorem, arXiv:2411.17448 (2025 version). https://arxiv.org/abs/2411.17448. Bibliographic information from the supplied record.

• S3 Ben Green, 100 Open Problems, current Problem 65. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The author-hosted document was inspected on 27 September 2026; the archive’s local code is not a problem-number locator in this paper.

• E1 B. Green and M. Sawhney, New bounds for the Furstenberg--Sarkozy theorem, current manuscript. https://arxiv.org/abs/2411.17448. Inspected 1 October 2026.