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 1; this is new for s 4.More precisely, we establish that if f :then there is a boundedcomplexity s-step nilsequence F (g(n)Γ) that correlates with f , where the bounds on the complexity and correlation depend only on s and δ.From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.

Locations

• Annals of Mathematics - View - PDF
• arXiv (Cornell University) - View - PDF
• Annals of Mathematics - View - PDF
• arXiv (Cornell University) - View - PDF