Browse › Combinatorics › 182 Power savings for intersective polynomial differences and prime arguments
A Power Saving for Polynomial Differences at Prime Arguments
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Abstract
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.
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Notes
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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 of the companion paper, with absolute constants and uniform in $N\ge1$; it lists no formalization of the prime-argument theorem or of the zero-free half-plane input this paper relies on. [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). Shows that the van der Corput property cannot give a power saving for square differences, relevant to the paper's remark that the case $h(x)=(x-1)^2$ implies a power saving for square-difference-free sets.
- Ben Krause, Hamed Mousavi, Terence Tao, Joni Teräväinen, Quantitative bounds for sets lacking polynomial progressions with shifted prime difference (arXiv:2608.19525, 20 Aug 2026). Proves quantitative bounds for polynomial progressions of length at least three with shifted-prime parameters, a multi-term relative of the two-term prime-argument problem treated here.
Digests and expositions
- Gradient Damage, Circulant Hadamard Conjecture, Cycle–Clique Ramsey Numbers and 7 More in Combinatorics (video compilation, 15 October 2026). The chapter beginning at 2:26 illustrates square-difference avoidance, while its accompanying text summarizes the family's intersective-polynomial and prime-argument claims. [family-level]
Elsewhere
- Emergent Mind: family 182
Independent sites; their content is not reviewed here.
Cite
@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}
}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 intersective polynomial differences with an exponent depending only on the degree · October 5, 2026
- A power saving for square-difference-free sets · September 24, 2026