## Abstract

We prove the inverse conjecture for the Gowers U^{s+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 ||f||_{Us+1[N]} ≥ δ, then there is a bounded complexitys-step nilsequence F(g(n)Γ) that 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.

Original language | American English |
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Pages (from-to) | 1231-1372 |

Number of pages | 142 |

Journal | Annals of Mathematics |

Volume | 176 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2012 |

Externally published | Yes |

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