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$$\sum _{m=1}^n{a}_m\cos (m\theta +{\alpha }_m),n\in {\mathbb{N}}^{\ast },$$satisfying$$k{n}^2\sum _{m=1}^n{a}_m^2\le \sum _{m=1}^n{m}^2{a}_m^2,$$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.

arXiv comment9 pages. 4 quotations. 14 references. Comments, feedback and collaboration are welcome