Frequency graph extracted from a large Gowers U3 norm
= Frequency graph extracted from a large Gowers U3 norm
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If $\|f\|_{U^3}^8\geq c$ and $\|f\|_\infty\leq1$, select for many $a$ a frequency $\phi(a)$ at which $\widehat{\partial_af}$ is large. A box-norm inequality shows that the graph $\{(a,\phi(a))\}$ 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.