OpenAI · 182-02 · 2026-10-05 · Combinatorics

A Power Saving for Polynomial Differences at Prime Arguments

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026

Family 182. Power savings for polynomial-difference-free sets.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

For every fixed intersective integer polynomial $h$ of degree $k \geq 2$ with positive leading coefficient, proves that a subset of ${1, \dots, N}$ avoiding nonzero values $h (1), h (2), \dots$ as differences has size $O_{h} (N^{1 - c_{k}})$ , with $c_{k} > 0$ 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 $h$ .

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 $A \subseteq {1, \dots, N}$ whose differences avoid all nonzero values $h (p)$ at primes satisfies $∣ A ∣ \leq C_{h} N^{1 - c_{h}}$ , where $c_{h} > 0$ and $C_{h} \geq 1$ 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.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

A Lean scope note is stored for family 182, but it does not name this preprint. It names 182-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.