Browse › Real and complex analysis › 076 Real ultraflat Littlewood polynomials and unbounded binary merit factors
Ultraflat real Littlewood polynomials
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Abstract
For every $\varepsilon\in(0,1)$ and every sufficiently large integer N, there is a polynomial of length N with coefficients in $\{-1,1\}$ whose modulus lies between $(1-\varepsilon)\sqrt N$ and $(1+\varepsilon)\sqrt N$ 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.
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Notes
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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). Links this paper's PDF for Erdős Problem #1150, summarizing it as "there exist 'ultraflat' polynomials", and notes it is 16 pages but "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, a contrary nonflatness claim that this paper's footnote 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 two-sided ultraflat statement of this paper is not part of the stated scope. [family-level]
Related prior literature mentioned by others
- Andrew Odlyzko, Search for ultraflat polynomials with plus and minus one coefficients (in Connections in Discrete Mathematics, Cambridge University Press, 2018). Flagged by the erdosproblemaday.com review of family 076 as uncited; its extensive computations suggested that ultraflat ±1 polynomials do not exist, the opposite of this paper's Theorem 1. [AI-generated per source]
- Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba, Flat Littlewood polynomials exist (Annals of Mathematics 192, 2020). The AI Proved What? exposition identifies this constant-factor two-sided flatness theorem as prior context for the family's stronger asymptotic claims. [family-level]
Digests and expositions
- Orply / Perplexity, Every Sufficiently Large Degree Admits an Ultraflat Real Littlewood Polynomial (9 October 2026). Explains this manuscript's continuous-wave construction, endpoint curvature, Fourier-tail control and defect-sensitive rounding, attributing the ultraflatness claim to the manuscript. [AI-generated per source]
- Latent Seminar, Ultraflat Littlewood polynomials and autocorrelation (6 October 2026). Explains the three claims, the merit-factor implication, stationary phase, packing, discrepancy rounding and lower-bound refinements, explicitly limiting its review to statements and proof mechanisms. [family-level]
- AI Proved What?, Flat with signs only? Ultraflat Littlewood polynomials and merit factors (Episode 076, October 2026). Video and transcript explain the progression of the three manuscripts, quadratic-phase waves, rounding and the documented Lean scope. [AI-generated per source] [family-level]
- Patrick White with Claude, Erdős #1150: OpenAI ultraflat Littlewood, refereed (erdosproblemaday.com, 12 October 2026). Referee-style reading of the three family-076 papers by seven Claude "referee seats", which report no gap found in this two-sided paper; lists it as "paper only", without formalization. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 076
Independent sites; their content is not reviewed here.
Cite
@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}
}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:
- Nearly minimal maxima and positive minima of Littlewood polynomials · October 5, 2026
- Asymptotically minimal maxima of real Littlewood polynomials · September 23, 2026