Result 076, Real and complex analysis

Real ultraflat Littlewood polynomials and unbounded binary merit factors

Constructs polynomials with N consecutive coefficients in {−1,1}\{-1,1\} whose modulus is (1+o(1))N(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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A sequence of plus and minus signs defines a polynomial whose size varies around a circle. The manuscripts claim these variations can become uniformly negligible, connecting nearly constant magnitude with binary sequences having increasingly large merit factors.

What changes?

Real Littlewood polynomials have N consecutive coefficients, each +1 or -1. The unreviewed manuscript reports that, for every epsilon between zero and one, every sufficiently large integer N admits such a polynomial with modulus between (1 minus epsilon) times the square root of N and (1 plus epsilon) times the square root of N everywhere on the complex unit circle, including +1 and -1. The length threshold may depend on epsilon; signs may be chosen separately at each length.

What does that help mathematicians do?

The square root of N is a natural benchmark: the average squared modulus on the circle is N. If the claimed bounds hold, no fixed relative gap separates the best achievable maximum from this benchmark. The manuscripts also report that the largest binary merit factor tends to infinity through all integer lengths, not merely a subsequence. This would rule out a universal upper bound of the kind proposed by Turyn.

Are there practical applications?

The immediate value is foundational: the result would connect uniform polynomial magnitude with binary sequences whose shifted self-correlations have small total squared size relative to length squared. This clarifies what sign sequences can theoretically achieve. The supplied abstracts describe neither an efficient construction nor a practical implementation, so engineering benefits remain unestablished.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Ultraflat real Littlewood polynomials

October 5, 2026 16 pages

For every ε∈(0,1)\varepsilon\in(0,1) and every sufficiently large integer N, there is a polynomial of length N with coefficients in {−1,1}\{-1,1\} whose modulus lies between (1−ε)N(1-\varepsilon)\sqrt N and (1+ε)N(1+\varepsilon)\sqrt N everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Ultraflat-real-Littlewood-polynomials-October-5-2026,
  author = {{OpenAI}},
  title = {{Ultraflat real Littlewood polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf}{OAI:Ultraflat-real-Littlewood-polynomials-October-5-2026}},
  year = {2026}
}

Nearly minimal maxima and positive minima of Littlewood polynomials

October 5, 2026 28 pages

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

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Asymptotically minimal maxima of real Littlewood polynomials

September 23, 2026 26 pages Main result formalized in Lean

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

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/076.md.

Real ultraflat Littlewood polynomials and unbounded binary merit factors

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives real Littlewood polynomials of every sufficiently large length whose maximum modulus on the unit circle is at most (1+η)N(1+\eta)\sqrt N for any fixed η>0\eta>0. Thus the smallest possible maximum is asymptotically minimal through all integer lengths.

A further statement chooses one family of real sign polynomials for all lengths such that, for every fixed finite p>0p>0, the LpL^p mean of ∣∣PN(z)∣/N−1∣\bigl||P_N(z)|/\sqrt N-1\bigr| on the unit circle tends to zero. The results are existential and do not give an effective convergence rate or a signing algorithm.

Comparator links

Result Comparator statement
Asymptotically minimal Littlewood maximum AsymptoticallyMinimalLittlewood.lean
Finite-exponent flatness of one all-length Littlewood family LittlewoodFiniteFlatness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.