Browse › Combinatorics › 182 Power savings for intersective polynomial differences and prime arguments
A power saving for intersective polynomial differences with an exponent depending only on the degree
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Power savings for intersective polynomial differences and prime arguments: Lean scope (OpenAI Math Release, lean/docs/182.md, 6 October 2026). The family scope note reports a formalization only of the square-difference theorem (the case $h(x)=x^2$ of this paper's result), with absolute constants and uniform in $N\ge1$, and lists no formalization of the general intersective-polynomial theorem. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Steve Fan and Andrew Lott, The optimal van der Corput property for squares (arXiv:2610.07765v2, 12 October 2026). For $h(x)=x^2$, it gives an $N\exp(-c\sqrt{\log N\log\log N})$ upper bound and shows that the van der Corput property cannot give a power saving, a comparison point for the paper's fixed-power claim obtained by a different, reflection-positivity-based method.
- Alexandros Kalogirou, Andrew Lott, Ákos Magyar, Akash Singha Roy, The Furstenberg–Sárközy theorem for sums of an even number of odd powers (arXiv:2609.16595, 15 Sep 2026). Proves power-saving bounds for differences avoiding sums of an even number of odd $k$-th powers, with an exponent depending on $k$ and the number of summands, which can be compared with the paper's degree-only exponent $c_k$.
- Khalid Younis, Sets whose differences avoid a bracket quadratic (arXiv:2608.30078, 30 Aug 2026). Proves a fixed power saving $N^{1-c}$ for differences that avoid a bracket quadratic (generalised polynomial), a nearby extension of polynomial Sárközy-type power savings.
Elsewhere
- Emergent Mind: family 182
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/182.md. The Lean statements have not been checked by people against the paper's claims.
Family 182: 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.
Other papers in this family:
- A Power Saving for Polynomial Differences at Prime Arguments · October 5, 2026
- A power saving for square-difference-free sets · September 24, 2026