A disproof of $L^{\alpha }$ polynomials Rudin conjecture, $2\le \alpha <4.$

Abstract

It is shown that the $L^{\alpha }$-norms polynomials Rudin conjecture fails. Our counterexample is inspired by Bourgain's work on NLS. Precisely, his study of the Strichartz's inequality of the $L^6$-norm of the periodic solutions given by the two dimension Weyl sums. We gives also a lower bound of the $L^{\alpha }$-norm of such solutions for $\alpha \ne 2$. As a consequence, we establish that for any $0<a<b,$ the following set $E(a,b)=\{(x,t)\in {\mathbb{T}}^2:a\sqrt{N}\le |{\sum }_{n=1}^{N}e(n^{2}t+nx)|\le b\sqrt{N}\text{ infinitely often}\},$ has a Lebesgue measure $0$. We further present an alternative proof of Cordoba's theorem based on Paley-Littlewood inequalities.

17 pages. Scientific comments and criticism are welcome