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.

Authors

Ben Green

DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England

Terence Tao

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555

Tamar Ziegler

Mathematics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel