Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 11 / 3 / 1 / Solution

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.