Inverse theorem for the Gowers U3 norm over a finite field ($U^3$ inverse theorem)

For fixed prime $p$, a one-bounded function on ${\mathbb{F}}_p^n$ with $U^3$ norm at least $\delta$ has correlation bounded below in terms of $p$ and $\delta$ with a quadratic phase, with the usual nonclassical formulation in small characteristic.

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.

If a dense set has too few four-term arithmetic progressions, its balanced function has large $U^3$ norm. A Frequency graph extracted from a large Gowers U3 norm, followed by the Balog-Szemerédi-Gowers theorem and a Freiman theorem, yields correlation with a quadratic phase. Restricting to a suitable level set gives a density increment, and iteration proves the Finite-field Szemerédi theorem for four-term arithmetic progressions.