{"id":"openai-math/Ultraflat-real-Littlewood-polynomials","pdf":"https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf","url":"https://github.com/openai/math/tree/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026","tags":["analysis","polynomials","littlewood-polynomials"],"score":0,"title":"Ultraflat real Littlewood polynomials","domain":"Real and complex analysis","family":"076","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":2,"published":"2026-10-05","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=uniformly%20on%20the%20entire%20unit%20circle%2C%20answering%20that%20stronger%20question","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"uniformly on the entire unit circle, answering that stronger question. First-passage percolation. In Analysis of Boolean Functions week 5 and 6 (2013), I discussed passage-time fluctuations and the challenge of improving variance bounds in first-passage percolation. The release announces absence…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://arxiv.org/abs/2610.30797","date":"2026-10-22","kind":"web","site":"arXiv.org","text":"We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^α$-flat polynomials. The theorem asserts that no $L^α$-flat polynomials exist within the class of trigonometric polynomials of the form$$\\sum_{m=1}^{n} a_m \\cos(mθ+α_m), \\qquad n \\in \\mathbb{N}^*,$$satisfying$$k\\,…","added":"2026-10-24","title":"OpenAI partially disproves Littlewood's 1961 $L^\\alpha$-flat theorem","author":"el Houcein el Abdalaoui","record":"arxiv/2610.30797"}],"formalized":false,"family_title":"Real ultraflat Littlewood polynomials and unbounded binary merit factors"}