## Abstract

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∥U^{4} ≥ 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.

Original language | American English |
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Pages (from-to) | 1-50 |

Number of pages | 50 |

Journal | Glasgow Mathematical Journal |

Volume | 53 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

Externally published | Yes |

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