## Abstract

This is an announcement of the proof of the inverse conjecture for the Gowers U^{s+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 that if f: [N] → [-1, 1] is a function with kfkU^{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 p_{1} < p_{2} < · · · < p_{k} ≤ N of primes, for every k ≥ 3.

Original language | English |
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Pages (from-to) | 69-90 |

Number of pages | 22 |

Journal | Electronic Research Announcements in Mathematical Sciences |

Volume | 18 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

## Keywords

- Gowers norms
- Nilsequences

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