Green–Tao nilsequence inverse conjecture for the Gowers norms

Let $s\geq 0$ be an integer and let $0<\delta\leq 1$. For each such $s$ and $\delta$, there is a finite collection $\mathcal M_{s,\delta}$ of $s$-step nilmanifolds $G/\Gamma$, each equipped with a smooth Riemannian metric $d_{G/\Gamma}$, and constants $C(s,\delta),c(s,\delta)>0$. If $N\geq 1$ and $f:[N]\to\mathbb C$ is $1$-bounded with $\|f\|_{U^{s+1}[N]}\geq\delta$, then there are $G/\Gamma\in\mathcal M_{s,\delta}$, $g\in G$, and a function $F:G/\Gamma\to\mathbb C$ of magnitude at most $1$ and Lipschitz constant at most $C(s,\delta)$ with respect to $d_{G/\Gamma}$ such that

$$\left|\mathbb E_{n\in[N]}f(n)\overline{F(g^n x)}\right|\geq c(s,\delta).$$

Inverse conjecture for the Gowers norms. The stated correlation with a bounded Lipschitz nilsequence should hold for every $s$, $\delta$, $N$, and $f$ satisfying the hypotheses.

The conjecture identifies nilsequences as the global obstructions to Gowers uniformity. Its converse is known by repeated applications of the Cauchy–Schwarz inequality, and the case $s=2$ was proved, but the general formulation in the source is presented as a conjecture.