OpenAI · 076-03 · 2026-09-23 · Real and complex analysis
Asymptotically minimal maxima of real Littlewood polynomials
preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026
Family 076. Ultraflat real Littlewood polynomials.
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
Constructs polynomials with consecutive coefficients in whose modulus is uniformly on the entire unit circle, for every sufficiently large integer length . Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in is , as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 076: Real ultraflat Littlewood polynomials and unbounded binary merit factors. 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.