A generalization of Littlewood's $L^α$ flat theorem, $α>0$

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}}^*,$ satisfying $$ \sum_{m=1}^{n}|c_m|^2 \leq \frac{K}{n^2} \sum_{m=1}^{n}m^2 |c_m|^2, $$ where $K$ is an absolutely constant. As a consequence, we confirm the $L^\alpha$-Littlewood conjecture, and thereby the $L^1$-Newman and $L^\infty$-Erd\"os 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 $1 < \alpha \leq 2.$ 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.