  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.
