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

Ben Green¹ ; Terence Tao² ; Tamar Ziegler³

¹ DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England
² Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555
³ Mathematics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel

Annals of mathematics, Tome 176 (2012) no. 2, pp. 1231-1372.

Résumé

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.

Détail

DOI : 10.4007/annals.2012.176.2.11