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. (Lean)

Manuscripts (3)

Ultraflat real Littlewood polynomials

October 5, 2026 · source · cite as \cite{OAI:Ultraflat-real-Littlewood-polynomials-October-5-2026}

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.

Nearly minimal maxima and positive minima of Littlewood polynomials

October 5, 2026 · source · cite as \cite{OAI:Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026}

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.

Asymptotically minimal maxima of real Littlewood polynomials

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026}

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.

Result 076 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.