Littlewood Ultraflat

Do L¹-ultraflat Littlewood polynomials exist?

A Littlewood polynomial has every coefficient equal to +1 or −1: P(z) = ε₀ + ε₁z + ⋯ + εₙzⁿ, εⱼ ∈ {±1}. On the unit circle its L² norm is exactly √(n+1), and its L¹ norm is at most that. The flatness ratio

ρ(P) = ‖P‖L¹(𝕊¹) / √(n+1) < 1

measures how close |P| comes to constant magnitude in the mean. A sequence of Littlewood polynomials is L¹-ultraflat if its ratios tend to 1. Whether any such sequence exists is an open problem; by a theorem of Guenais, a positive answer yields an ergodic measure-preserving system with simple spectrum having a Lebesgue component, answering a form of a question of Banach.

This site tracks the largest ratio attained at each degree. Random signs give ρ ≈ 0.8862 (→ √π/2); the question is how far above that — and how close to 1 — the records climb as the degree grows.

The official score is the mean of |P| over M equispaced points of the circle, with M doubled until two successive values agree to 1e-10. Scores shown are accurate to the displayed digits unless marked otherwise.

Records

Top ratios over all degrees, one entry per symmetry class (negation, reversal and z → −z give the same ratio).

No records yet — be the first.

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