An inverse theorem for the Gowers Us+1[N]-norm

We prove the inverse conjecture for the Gowers Us+1[N]-norm for all s >= 1; this is new for s >= 4. More precisely, we establish that if f : [N] -> [-1, 1] is a function with parallel to f parallel to(Us+1) ([N]) >= 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.