OpenAI partially disproves Littlewood's 1961 $L^\alpha$-flat theorem

Abstract

We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^\alpha$-flat polynomials. The theorem asserts that no $L^\alpha$-flat polynomials exist within the class of trigonometric polynomials of the form$$\sum_{m=1}^{n} a_m \cos(m\theta+\alpha_m), \qquad n \in \mathbb{N}^*,$$satisfying$$k\, n^2 \sum_{m=1}^{n} a_m^2 \;\leq\; \sum_{m=1}^{n} m^2 a_m^2,$$where $k$ is an absolute constant (alternatively expressed on the right-hand side as $K = 1/k$). We show that OpenAI's counterexample is contained within the class of hyper-lacunary sequences. Consequently, it offers only a limited refutation, as such sequences fail to satisfy an implicit assumption of Littlewood's theorem. We also discuss broader questions surrounding this counterexample and provide a brief historical overview.