Ben Green, Terence Tao and Tamar Ziegler. An inverse theorem for the Gowers Us+1[N]-norm.
We prove the \emph{inverse conjecture for the Gowers Us+1[N]-norm for all s ≥ 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||Us+1[N] ≥ δ then there is a bounded-complexity s-step nilsequence F(g(n)Γ) which 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.