A generalization of Littlewood's $L^{\alpha }$ flat theorem, $\alpha >0$

Abstract

We establish a generalization of Littlewood's criterion on $L^{\alpha }$-flatness by proving that there is no $L^{\alpha }$-flat polynomials, $\alpha >0$, within the class of analytic polynomials on the unit circle of the form $P_n(z)={\sum }_{m=1}^n{c}_m{z}^m,n\in {\mathbb{N}}^{\ast },$ satisfying $$\sum _{m=1}^n|c_m{|}^2\le \frac{K}{n^2}\sum _{m=1}^n{m}^2|c_m{|}^2,$$ where $K$ is an absolutely constant. As a consequence, we confirm the $L^{\alpha }$-Littlewood conjecture, and thereby the $L^1$-Newman and $L^{\infty }$-Erdös conjectures. Our approach combines the $L^{\alpha }$ Littlewood theorem with the generalized Clarkson's second inequality for $L^{\alpha }(X,\mathcal{A},m;B)$, with $B$ a Banach spaces and $1<\alpha \le 2.$ It follows that there are only finitely many Barker sequences, and we further present several applications in number theory and the spectral theory of dynamical systems. Finally, we construct Gauss-Fresnel polynomials that are Mahler-flat, providing a new proof of the Beller-Newman theorem.

27 pages. The results of this paper were announced and presented at the international conference "33èmes Journées Arithmétiques," held at the University of Luxembourg from June 30 to July 4, 2025