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Nearly minimal maxima and positive minima of Littlewood polynomials

OpenAI · October 5, 2026

Lean

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Abstract

For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in $\{-1,1\}$ whose modulus lies between $\sqrt N/16$ and $(1+\eta)\sqrt N$ everywhere on the unit circle.

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Notes

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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.

Comments and discussion

  • Thomas Bloom, Erdős problems in the OpenAI proofs (erdosproblems.com blog, 8 October 2026). Lists Erdős Problem #1150 among the release's claimed results via the ultraflat companion PDF, which it says "appears to heavily rely on two other related PDFs" of 28 and 26 pages (this paper and the September 23 one). [family-level]
  • Erdős Problems forum, Problem #1150 discussion thread (erdosproblems.com, comments February–August 2026). Predates the release; T. Bloom and T. Tao comment on gaps in el Abdalaoui's arXiv:2504.21499, a contrary nonflatness claim that this paper cites, and el Abdalaoui replies.

Formalization

  • OpenAI math release, Lean scope notes for family 076 (release Lean docs, October 2026). Covers only the upper bound $(1+\eta)\sqrt N$ of the September 23 predecessor and an $L^p$ flatness statement; the lower bound $\sqrt N/16$ of this paper is not part of the stated scope. [family-level]

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026,
  author = {{OpenAI}},
  title = {{Nearly minimal maxima and positive minima of Littlewood polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf}{OAI:Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/076.md. The Lean statements have not been checked by people against the paper's claims.

Family 076: Real ultraflat Littlewood polynomials and unbounded binary merit factors

Constructs polynomials with N consecutive coefficients in $\{-1,1\}$ whose modulus is $(1+o(1))\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length N. 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.

Other papers in this family: