OpenAI · 182-03 · 2026-09-24 · Combinatorics
A power saving for square-difference-free sets
preprints/A-power-saving-for-square-difference-free-sets-September-24-2026
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 of degree with positive leading coefficient, proves that a subset of avoiding nonzero values as differences has size , with depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when has a unit root modulo every modulus, with exponent allowed to depend on .
We prove that there are absolute constants c > 0 and C < ∞ such that every set with no nonzero square difference satisfies . This answers the fixed-power question posed by Green and Sawhney.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 182: Power savings for intersective polynomial differences and prime arguments. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.