Quadratic phase detection by the Gowers U2 norm

Let $|f|\le 1$ on ${\mathbb{Z}}_N$ and put $g(x)=f(x){\omega }^{-x^2}$. If $\Vert g{\Vert }_{U^2}^4\ge c$, then the Parseval identity givesConsequently $f$ correlates with a quadratic phase: for some $r\in {\mathbb{Z}}_N$,