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The Gowers U3 norm derivative identity isLet $\kappa =\Vert F{\Vert }_{U^3(G)}^8$. Since $\Vert I{\Vert }_{U^3(G)}\ge \mathbb{E}I=N/M\ge 1/16$, by repeated Cauchy-Schwarz, the interval hypothesis gives $\kappa \ge 16^{-8}{\delta }^8$. Also $\Vert {\partial }_{h}F{\Vert }_{U^2}^4\le 1$. Therefore the sethas density at least $\kappa /2$ in $G$.

Use normalized Fourier coefficients on a finite abelian group $\hat{g}(k)={\mathbb{E}}_{x}g(x)e(-kx/M)$. The Gowers U2 norm and Parseval identity giveFor $g={\partial }_{h}F$, the final mean is at most one. Thus each $h\in H$ has a frequency $\theta (h)=k_h/M$ withThe derivative is identically zero unless $h$ is represented by an integer in $[-N+1,N-1]$, so $H$ lies in the requested interval of shifts. Its size is $\gg {\delta }^{8}M\gg {\delta }^{8}N$. Keep one such frequency for each shift; these same choices will satisfy the energy conclusion below.

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

With normalized averages on the finite abelian group $G$, the Gowers U2 norm is defined byThe Gowers U3 norm is defined bywhere $\mathcal{C}$ denotes complex conjugation.