Frequency graph extracted from a large Gowers U3 norm

If $\Vert f{\Vert }_{U^3}^8\ge c$ and $\Vert f{\Vert }_{\infty }\le 1$, select for many $a$ a frequency $\varphi (a)$ at which $\widehat{{\partial }_{a}f}$ is large. A box-norm inequality shows that the graph $\{(a,\varphi (a))\}$ has at least $(c/2{)}^8|G{|}^3$ additive quadruples. Additive-combinatorial structure in this graph is the first step toward recovering a quadratic phase.