Power savings for polynomial differences

Source and verification

This article explains family 182 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Forbidden polynomial differences

For $A\subseteq [N]=\{1,\dots ,N\}$, write $A-A=\{a-b:a,b\in A\}$. A square-difference-free set contains no pair whose positive difference is a nonzero square. More generally, one can forbid nonzero values of an integer polynomial $h$ among these differences.

A polynomial is intersective if it has a root modulo every positive integer. This is a necessary local condition for forcing its values as differences in every positive-density set. If $h$ has no root modulo $q$, the multiples of $q$ avoid all its values as differences while retaining density $1/q$.

For prime arguments, the relevant condition is stronger: for every $q$, there must be a root $r$ of $h$ modulo $q$ with $\gcd (r,q)=1$. Call such a polynomial prime-intersective. For example, $(x-1{)}^2$ is prime-intersective because $r=1$ works at every modulus. The polynomial $x^2$ is intersective but not prime-intersective.

🏷️ Three power-saving statements

Manuscript claims

Square-difference-free subsets satisfy $|A|\le C{N}^{1-c}$ for some absolute $c>0$. The exponent is not optimized or supplied as a useful numerical value. (OpenAI, 2026)

For every degree $k\ge 2$, there is $c_k>0$ such that every fixed intersective polynomial $h$ of degree $k$ with positive leading coefficient satisfies

$$\begin{array}{c}(A-A)\cap h({\mathbb{N}}_{+})\subseteq \{0\}\\ ⟹|A|\le C_h{N}^{1-c_k}.\end{array}$$

for sufficiently large $N$ depending on $h$. The exponent depends only on the degree; the constant and threshold may depend on all coefficients. (OpenAI, 2026a)

For a fixed prime-intersective polynomial of degree at least two and positive leading coefficient, avoiding all nonzero values $h(p)$ at primes gives $|A|\le C_h{N}^{1-c_h}$ for every $N\ge 1$. Here the exponent may depend on the polynomial. (OpenAI, 2026b)

Zero values impose no restriction. The statements concern differences in the integers, not a cyclic group. The prime-argument paper uses the claimed uniform Dirichlet zero-free half-plane from family 003 as an essential companion input; it is stronger than a classical near-one zero-free region.

🏷️ An elementary coloring consequence

If an $r$-coloring of $[N]$ avoids a monochromatic square difference, each color class has size at most $C{N}^{1-c}$. Summing gives $N\le rC{N}^{1-c}$, hence $N\le (Cr{)}^{1/c}$.

Thus any longer interval forces two same-colored integers $a,a+d^2$. The integer $d$ need not have that color. Equivalently, the graph joining integers at square distance has chromatic number at least $C^{-1}{N}^c$.

This deduction illustrates the force of a fixed power saving. A density bound tending to zero without a specified rate would not give a polynomial bound in the number of colors.

🏷️ A multilinear quantity dominating density

The square-difference argument replaces the density by a nonnegative multilinear quantity $Y(N,A)$ satisfying

$$(|A|/N{)}^d\le Y(N,A)$$

for a fixed even number of slots $d$. A Fourier lift retains selected rational Fourier coefficients of $1_A$ on a product of finite residue spaces. The lift has mean $|A|/N$ but can take negative values.

The tuple law is reflection positive: splitting its slots across certain reflections gives a positive semidefinite coupling matrix. Repeated Cauchy–Schwarz yields a mixed inequality valid for signed inputs and a seminorm dominating the mean. A version valid only for nonnegative functions would not justify using the signed Fourier lift.

Apart from a small exceptional mass, a designated cycle of slots has square-valued increments at each prime. Conditional mixing and moment bounds make high-denominator Fourier terms negligible. Crucially, the prime-by-prime exceptional masses remain summable after the changes of scale required by the induction.

🏷️ Cancellation and contracting scales

Fixing residue coordinates and then splitting into progressions of step $D^2$ preserves square-difference avoidance after rescaling. The seminorm compares the original functional with the same functional on shorter intervals, with a factor tending to one rather than a fixed loss repeated at every scale.

A signed kernel supported on actual positive squares has zero bilinear contribution on ordered subsets of an avoiding set. Its Fourier transform approximates the pair correlations of the tuple law. This converts arithmetic avoidance into cancellation on a fixed positive proportion of small-prime residue assignments.

The other assignments are controlled by the universal scale comparison. The combined recurrence yields $Y(N,A)\ll N^{-a}$ for some fixed $a>0$, and hence $|A|/N\ll N^{-a/d}$. The power saving emerges from a strict contraction, not merely from finding one large Fourier coefficient.

🏷️ Tracking the polynomial through progressions

For a general intersective polynomial, rescaling changes the forbidden polynomial. The source constructs an auxiliary family $h_l$ and a multiplicative scaling function $\lambda$ with

$$\begin{array}{c}{h}_l(Dx-i)=\lambda (D)h_{lD}(x),\\ D\mid \lambda (D),\lambda (D)\le D^{\mathrm{deg}h}.\end{array}$$

This gives exact covariance of the avoidance condition along progressions. Local normalization controls the structural parameters that determine the exponent using only the degree, while constants for the original polynomial remain coefficient-dependent.

The pair distribution supplied by a useful tuple law need not equal the natural distribution of polynomial values. A separately constructed signed kernel realizes its local multipliers on actual positive polynomial values. Its support ensures cancellation even though its coefficients can have either sign.

The induction keeps the growing auxiliary parameter explicit, proving a bound of the form $Y_l(N,A)\le C{l}^b{N}^{-a}$. Omitting the factor $l^b$ would demand unjustified uniformity when the auxiliary polynomial has grown large relative to the interval.

🏷️ The additional prime input

Restricting arguments to primes requires tuple distributions supported on unit residue classes, and a kernel supported on values $h(p)$. Major-arc estimates use the companion zero-free theorem to control primes in progressions; minor arcs use prime-sum decompositions and differencing. These inputs match the local unit distributions before the contraction argument is applied.

The prime result is therefore not obtained simply by discarding composite arguments from an integer-argument proof. The kernel approximation and its prime-distribution input must be rebuilt. The family scope page documents only the square-difference statement, not these two broader companions.

🏷️ Key steps and proof outlines

Expand exceptional measures before estimating a signed integral

The square-difference argument packages density in a nonnegative lifted quantity $Y(N)$, bounded below by a fixed power of the density. Its Fourier representation nevertheless has a signed integrand. This distinction determines how local exceptions must be removed. For each relevant prime $p$, let ${\mu }_p$ be the local probability measure and ${ϵ}_p$ its exceptional submeasure, of mass ${\eta }_p\le p^{-4}$. Then

$$\underset{p}{⨂}{\mu }_p-\underset{p}{⨂}({\mu }_p-{ϵ}_p)=\sum _{\varnothing \ne B}(-1{)}^{|B|+1}{\kappa }_B,{\kappa }_B=\underset{p\in B}{⨂}{ϵ}_p\otimes \underset{p\notin B}{⨂}{\mu }_p.$$

Each ${\kappa }_B$ is positive and has mass ${\eta }_B={\prod }_{p\in B}{\eta }_p$. One can estimate its integral and then use the triangle inequality. A pointwise comparison of the signed Fourier integrand against a positive integrand would not give the required conclusion. (OpenAI, 2026c)

Write $q_B={\prod }_{p\in B}p$. After putting sufficiently small primes into a fixed modulus $M_0$, the local decay gives

$$\sum _{B\ne \varnothing }{q}_B{\eta }_B=\prod _p(1+p{\eta }_p)-1\le \frac{1}{4}.$$

This weighted estimate pays for the shorter intervals created by conditioning. For $q_B>N^{\tau }$, translation invariance gives uniform single-coordinate marginals under ${\kappa }_B/{\eta }_B$; Hölder’s inequality and the lifted moment estimate bound the entire large-modulus tail by $C{N}^{-\tau +\epsilon }$. Taking $\epsilon =\tau /2$ makes this a decaying error. For the remaining subsets, the primes outside $B$ retain their original local laws, so the scale-transfer lemma still applies.

Close a recurrence with a genuine contraction

The square substitution produces a shorter length

$$D_B=M_0{q}_B,n_B=\lceil \frac{N}{D_B^2}\rceil .$$

The ceiling is not discarded: its normalization factor is at most $1+M_0^2{N}^{2\tau -1}$. There is a fixed positive probability $\rho$ of a favorable residue configuration. The favorable nonexceptional contribution cancels, while the remaining contributions transfer to shorter scales. Suppressing only a factor $1+u_N$ with $u_N\to 0$, the recurrence has the form

$$Y(N)\le K{N}^{-{\sigma }_0}+(1+u_N)\left((1-\rho )Y(n_{\varnothing })+\rho \sum _{B\ne \varnothing }{\eta }_{B}Y(n_B)\right).$$

Why a small power saving follows

Try the inductive bound $Y(n)\le C{n}^{-a}$. If $2a\le 1$, the exceptional scale loss is affordable because ${\sum }_B{\eta }_B{q}_B^{2a}\le 1/4$. Choose $a>0$ so small that

$$M_0^{2a}(1-3\rho /4)<1.$$

The recurrence then returns strictly less than $C{N}^{-a}$, leaving room for the analytic error and the ceiling factor. Choose the large-scale threshold first and enlarge $C$ afterward to cover the finite initial range.

This is the proof’s quantitative engine: positivity of the lifted density quantity, summable exceptional masses, and strict contraction after rescaling. Finally, if $Y(N)\ge {\delta }^d$, where $\delta$ is density and $d$ is the number of lift coordinates, then $Y(N)\ll N^{-a}$ yields $\delta \ll N^{-a/d}$.

Preserve the auxiliary parameter through polynomial rescaling

For general intersective polynomials, rescaling changes the auxiliary polynomial as well as the interval. Denote its parameter by $\ell$ and its lifted quantity by $Y_{\ell }(N)$. The useful induction is

$$Y_{\ell }(N)\le C{\ell }^b{N}^{-a}\text{for every }\ell ,N,$$

rather than only in the analytic range $\ell \le N^{\zeta }$. The successor parameter is $\ell D_B$, so an otherwise valid induction could leave that range. Set $b=2a/\zeta$. Outside it, the factor ${\ell }^b{N}^{-a}\ge N^a$ absorbs the elementary subpower estimate. Inside it, the analytic descent applies. (OpenAI, 2026a)

For degree $k$, the shorter scale satisfies $n_B\ge N/D_B^k$. The combined loss is therefore $D_B^{ka+b}$. Choosing $ka+b\le 1$ makes the same weighted exceptional-mass estimate applicable, and choosing the exponent still smaller ensures strict contraction. The degree-uniform paper fixes this exponent through degree-dependent parameters before absorbing the original polynomial’s coefficients into constants and initial thresholds. A residue-class normalization then transfers the auxiliary difference back to a nonzero difference of the original polynomial.

The prime-argument companion uses the same parameter-aware descent, but the construction of its prime-weighted kernels depends on the stated uniform zero-free input. Its arithmetic input and exponent dependence must therefore be retained when interpreting the conclusion; the recurrence alone does not establish the required distribution of prime arguments. These mechanisms are developed in the square paper’s “Induction,” the degree-uniform paper’s induction section, and the prime-argument paper’s “Descent.” (OpenAI, 2026b)

See Also

• Arithmetic progressions concerns linear patterns of arbitrary fixed length, with a different quantitative iteration.
• Zero-free half-planes supplies the explicitly identified analytic input to the prime-argument companion.

🏷️ Manuscripts and verification scope

• A power saving for intersective polynomial differences with an exponent depending only on the degree — pinned manuscript and companion source.
• A Power Saving for Polynomial Differences at Prime Arguments — pinned manuscript and companion source.
• A power saving for square-difference-free sets — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects only the uniform square-difference theorem. It does not document the general intersective or prime-argument companions. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A power saving for intersective polynomial differences with an exponent depending only on the degree.

🐻  OpenAI 2026b. A Power Saving for Polynomial Differences at Prime Arguments.

🐻  OpenAI 2026c. A power saving for square-difference-free sets.