Ben Green, Terence Tao and Tamar Ziegler. An inverse theorem for the Gowers U4-norm. To appear Glasgow Mathematical Journal.
We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] &rarr C is a function with |f(n)| ≤ 1 for all n and || f ||U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n) Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes.