The original inverse Gowers conjecture

Let $p\geq 2$ be a fixed prime, let $d\geq 1$, and let $F:\mathbb{F}_p^n\to\mathbb{C}$ be a bounded function with $\|F\|_{\infty}\leq 1$. If its Gowers norm satisfies

$$\|F\|_{U^{d+1}}\geq\varepsilon,$$

then there exists a degree-$d$ polynomial $P:\mathbb{F}_p^n\to\mathbb{F}_p$ such that

$$\left|\left\langle F,e^{2\pi iP/p}\right\rangle\right|\geq\delta(p,d,\varepsilon).$$

Original inverse Gowers conjecture. For every such $F$, a lower bound on the $U^{d+1}$ norm implies correlation with a phase polynomial of degree $d$, with correlation bounded below by a positive quantity depending only on $p$, $d$, and $\varepsilon$. This is the inverse problem for the Gowers norm: large uniformity norm should reveal low-degree polynomial structure. The precise validity and dependence of the bound are not established here.