oai-math-revised beta

Browse › Real and complex analysis › 076 Real ultraflat Littlewood polynomials and unbounded binary merit factors

Asymptotically minimal maxima of real Littlewood polynomials

OpenAI · September 23, 2026

Lean

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

We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in $\{-1,1\}$ is $(1+o(1))\sqrt N$, 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.

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.

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, links the ultraflat companion PDF and the comparator Lean file for this paper's Theorem 1.1, and notes that the 16-page PDF "appears to heavily rely on two other related PDFs". [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, one of the contrary nonflatness claims examined in this paper's Appendix A, and el Abdalaoui replies.

Formalization

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026,
  author = {{OpenAI}},
  title = {{Asymptotically minimal maxima of real Littlewood polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf}{OAI:Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026}},
  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: