An inverse theorem for the Gowers U⁴-norm

We prove the so-called inverse conjecture for the Gowers U ^s+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → 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)) 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 the main result of the present paper with several previous results 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.