Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 147 / 1 / iv

Remove the quadratic phase by settingThe phase around the parallelogram isso the hypothesis says exactly that $\Vert g{\Vert }_{U^2}^4\ge c$, with harmless negative choices of the increments. The Fourier identity for the Gowers U2 norm and the Parseval identity now givewhere the last inequality uses $\Vert f{\Vert }_{\infty }\le 1$. Thus the Quadratic phase detection by the Gowers U2 norm supplies an $r\in {\mathbb{Z}}_N$ such that