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

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 $\alpha \neq 2$. As a consequence, we establish that for any $0<a<b,$ the following set $E(a,b)=\Big\{(x,t) \in \mathbb{T}^2 \; : \;a \sqrt{N} \leq \Big|\sum_{n=1}^{N}e(n^2t+nx)\Big| \leq b \sqrt{N} \; \textrm{~infinitely~often}\;\Big\},$ has a Lebesgue measure $0$. We further present an alternative proof of Cordoba's theorem based on Paley-Littlewood inequalities.