Result 182, Combinatorics

Power savings for intersective polynomial differences and prime arguments

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $\{1,\ldots,N\}$ avoiding nonzero values $h(1),h(2),\ldots$ as differences has size $O_h(N^{1-c_k})$ , with $c_k\gt 0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.

Lean formalization New or sharp bound

The bigger picture

Why it matters

How large can a collection of integers be if no two differ by a square or another prescribed polynomial value? The unreviewed manuscripts report power-saving bounds, showing that such avoidance forces the collection to become quantitatively sparse.

What changes?

Fix an integer polynomial h of degree k at least two with positive leading coefficient. Intersective means every positive integer divides some value of h. Subsets of 1,...,N avoiding nonzero h(n) differences for positive integers n have size at most C_h times N to the power 1-c_k. The positive c_k depends only on k; C_h may depend on h. For prime inputs, each modulus must admit a root sharing no factor greater than one with it; both constants may depend on h.

What does that help mathematicians do?

The bounds give a threshold above which a forbidden difference must occur, not merely a statement that avoiding sets eventually have vanishing density. For squares, the reported consequence is a bound with absolute positive constants. More generally, fixing the degree fixes the saving exponent across all qualifying polynomials, although their multiplicative constants can differ. Researchers can therefore compare these avoidance problems using a common quantitative scale.

Are there practical applications?

The immediate value is foundational: these results quantify how divisibility conditions on a polynomial constrain the size of integer sets avoiding its values as differences. The prime-input manuscript also connects this question to prime distribution, using a companion zero-free half-plane theorem for Dirichlet L-functions to supply the needed estimates. This is a theoretical connection, not a demonstrated practical algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

A power saving for intersective polynomial differences with an exponent depending only on the degree

October 5, 2026 47 pages

An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent $c_k\gt 0$ such that every set $A\subseteq\{1,\ldots,N\}$ whose differences avoid all nonzero values $h(1),h(2),\ldots$ , where h is an intersective polynomial of degree k with positive leading coefficient, satisfies $|A|=O_h(N^{1-c_k})$ . The implied constant may depend on h, but the power-saving exponent depends only on its degree.

PDF LaTeX source

Cite (BibTeX)

@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}
}

A Power Saving for Polynomial Differences at Prime Arguments

October 5, 2026 42 pages

Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $A\subseteq\{1,\ldots,N\}$ whose differences avoid all nonzero values $h(p)$ at primes satisfies $|A|\le C_hN^{1-c_h}$ , where $c_h\gt 0$ and $C_h\ge1$ depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026,
  author = {{OpenAI}},
  title = {{A Power Saving for Polynomial Differences at Prime Arguments}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf}{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026}},
  year = {2026}
}

A power saving for square-difference-free sets

September 24, 2026 48 pages

We prove that there are absolute constants c > 0 and C < ∞ such that every set $A\subseteq\{1,\ldots,N\}$ with no nonzero square difference satisfies $|A|\le C N^{1-c}$ . This answers the fixed-power question posed by Green and Sawhney.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026,
  author = {{OpenAI}},
  title = {{A power saving for square-difference-free sets}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf}{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/182.md.

Power savings for intersective polynomial differences and prime arguments

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

A power saving for square-difference-free sets

Scope

The formalization proves a fixed power saving for sets with no nonzero square difference. There are absolute constants $c>0$ and $C$ such that every $A\subseteq\{1,\ldots,N\}$ satisfying $a-b\ne m^2$ for all $a,b\in A$ and integers $m\ge1$ has $|A|\le C N^{1-c}$ . The bound is uniform for every integer $N\ge1$ .

Comparator links

Result	Comparator statement
Power saving for square-difference-free sets	SquareDifference.lean

At a glance

Subject: Combinatorics
Manuscripts: 3
Pages: 137
Dated: Sep 24 to Oct 5, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Fixed-power question posed by Green and SawhneyProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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