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 by degree
The best known ratio at each degree (log scale). The question is whether this frontier reaches 1 as the degree grows. Browse all 2371 polynomials.
Submit a polynomial
Submitting requires an account, so records can be attributed. Log in with GitHub or ORCID to submit.
API
Download the database (JSON) ↓ — every polynomial's metadata (degree, ρ, submitter, status, …) with links to its sign string and canonical blob.
- GET /database.json — the whole database as JSON
- GET /api/polynomials?limit=50&max_degree=… — polynomial list as JSON
- GET /api/polynomial/:id — one polynomial's metadata
- GET /api/polynomial/:id/coefficients — full sign string
- GET /api/polynomial/:id/blob — canonical bit-packed encoding
- POST /api/submit — body {"signs": "+-+…", "note": "…"} or {"generator": {"type":"fekete","p":127,"shift":37}, …} (generators: fekete, rudin-shapiro). Requires Authorization: Bearer luf_… — create a token on your profile.