It is shown that the L α-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 6norm 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 $\alpha \geq 2$. As a consequence, we establish that for any $a,b>0$ , the following set $$E(a,b)=\Big\{(x,t) \in \T^2 \; : \;a \sqrt{N} \leq \Big|\sum_{n=1}^{N}e(n^2t+nx)\Big| \leq b \sqrt{N} \;\textrm{~infinitely~often}\;\Big\},$$has a Lebesgue measure $0$. We further present an alternative proof of Cordoba's theorem based on Paley-Littlewood inequalities.