Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 182-01. A power saving for intersective polynomial differences with an exponent depending only on the degree
Manuscript map · lines 5715–5724
OpenAI@2026-10-07#rem:oai-p-182-01
Authors: OpenAI. Family 182.
Title: A power saving for intersective polynomial differences with an exponent depending only on the degree.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
Nearby in this section
- section: Manuscript map
- remark: Remark: 001-01. Milne's rationality conjecture for abelian varieties
- remark: Remark: 002-01. Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank
- remark: Remark: 002-02. The Selmer converse for elliptic curves at every prime
- remark: Remark: 002-03. The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
- remark: Remark: 003-01. The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane s 7/8
- remark: Remark: 003-02. The Quasi-Riemann Hypothesis (alternate 11/12 proof)
- remark: Remark: 003-03. Uniform exclusion of Landau–Siegel zeros