## Abstract

Let (an)n be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy E(A _{N} ) of the cut-offs AN={an:n≤N}, and (an)n possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of “second order”. Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of α∈ [0 , 1] satisfying that (〈αan〉)n with E(A _{N} ) = Ω (N ^{3} ) does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.

Original language | English |
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Pages (from-to) | 137-156 |

Number of pages | 20 |

Journal | Monatshefte fur Mathematik |

Volume | 189 |

Issue number | 1 |

DOIs | |

State | Published - 1 May 2019 |

Externally published | Yes |

## Keywords

- Additive energy
- Diophantine approximation
- Metric number theory
- Poissonian pair correlations