On the number of gaps of sequences with Poissonian pair correlations

Christoph Aistleitner, Thomas Lachmann, Paolo Minelli, Paolo Leonetti

Publikation: Beitrag in einer FachzeitschriftArtikelBegutachtung

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.

Originalspracheenglisch
Aufsatznummer112555
Seitenumfang13
FachzeitschriftDiscrete Mathematics
Jahrgang344
Ausgabenummer11
DOIs
PublikationsstatusVeröffentlicht - Nov. 2021

ASJC Scopus subject areas

  • Theoretische Informatik
  • Diskrete Mathematik und Kombinatorik

Fields of Expertise

  • Information, Communication & Computing

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