### Abstract

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemerédi's Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on ℓ^{∞} analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.

Original language | English (US) |
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Pages (from-to) | 219-276 |

Number of pages | 58 |

Journal | Journal d'Analyse Mathematique |

Volume | 108 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2009 |

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

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## Cite this

*Journal d'Analyse Mathematique*,

*108*(1), 219-276. https://doi.org/10.1007/s11854-009-0024-1