Ben Green, Terence Tao and Tamar Ziegler. An inverse theorem for the Gowers Us+1[N]-norm. Announcement.
In this note we announce the proof of the inverse conjecture for the Gowers Us+1[N]-norm for all s ≥ 3; this is new for s ≥ 4, the cases s = 1,2,3 having been previously established. More precisely we outline a proof (details of which will appear in a forthcoming paper) that if f : [N] -> [-1,1] is a function with || f ||U^s+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. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p1 < p2 < ... < pk ≤ N of primes, for every k ≥ 3.