Year: 2026
Type: preprint
Abstract: 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$$k\, n^2 \sum_{m=1}^{n} a_m^2 \;\leq\; \sum_{m=1}^{n} m^2 a_m^2,$$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.
Source: arXiv (Cornell University)
Author: Abdalaoui, el Houcein el
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223315899