### Abstract

Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system (X,χμT) and a set A ∈ χ of positive measure, the set of integers n such that μ (A ∩ T^{n} A ∩ T^{2n} ... ∪ T ^{kn} A) > μ (A)^{k+1} - ∈ is syndetic. The size of this set, surprisingly enough, depends on the length (k + 1) of the arithmetic progression under consideration. In an ergodic system, for k = 2 and k = 3, this set is syndetic, while for k ≥ 4 it is not. The main tool is a decomposition result for the multicorrelation sequence ∫ f(x)f(T^{n}x) f(T ^{2n}x) ... f(T^{kn}x) dμ (x), where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d * (E) > 0 and for all ∈ ;> 0, the set {n ∈ ℤ: d* (E ∪ ( E + n) ∪ (E + 2n) ∪ (E + 3n)) > d* (E)^{4} - ∈} is syndetic.

Original language | English |
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Pages (from-to) | 261-303 |

Number of pages | 43 |

Journal | Inventiones Mathematicae |

Volume | 160 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2005 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Inventiones Mathematicae*,

*160*(2), 261-303. https://doi.org/10.1007/s00222-004-0428-6