Ultraflat real Littlewood polynomials
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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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\,…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
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…