An inverse theorem for the Gowers $U^{s+1}[N]$-norm

Abstract

We prove the inverse conjecture for the Gowers $U^{s+1}[N]$-norm for all $s \geq 1$; this is new for $s \geq 4$. More precisely, we establish that if $f : [N] \rightarrow [-1,1]$ is a function with $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$, then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ that correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.