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

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 $$\sum _{m=1}^n\mathrm{\mid }{c}_m{\mathrm{\mid }}^2\le \frac{K}{n^2}\sum _{m=1}^n{m}^2\mathrm{\mid }{c}_m{\mathrm{\mid }}^2,$$ where $K$ is an absolutely constant. As a consequence, we confirm the $L^{\alpha }$-Littlewood conjecture, and thereby the $L^1$-Newman and $L^{\mathrm{\infty }}$-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.