## Abstract

A sequence (x
_{n}) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖x
_{i}−x
_{j}‖≤s/N}=2sN(1+o(1)) for all reals s>0, as N→∞. It is known that, if (x
_{n}) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x
_{1},…,x
_{n}} cannot be bounded along any index subsequence (n
_{t}). First, we improve this by showing that, if (x
_{n}) has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of {x
_{1},…,x
_{n}} is o(n), as n→∞. Furthermore, we show that for every function f:N
^{+}→N
^{+} with lim
_{n}f(n)=∞ there exists a sequence (x
_{n}) with Poissonian pair correlations such that g(n)≤f(n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.

Original language | English |
---|---|

Article number | 112555 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 344 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2021 |

## Keywords

- Distinct gap lengths
- Equidistribution
- Poissonian pair correlations

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

## Fields of Expertise

- Information, Communication & Computing