Pair correlations and equidistribution

Christoph Aistleitner*, Thomas Lachmann, Florian Pausinger

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)
175 Downloads (Pure)

Abstract

A deterministic sequence of real numbers in the unit interval is called equidistributed if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs xk,xl∈(xn)1≤n≤N which are within distance s/N of each other is asymptotically ∼2sN. A randomly generated sequence has both of these properties, almost surely. There seems to be a vague sense that having Poissonian pair correlations is a "finer" property than being equidistributed. In this note we prove that this really is the case, in a precise mathematical sense: a sequence whose asymptotic distribution of pair correlations is Poissonian must necessarily be equidistributed. Furthermore, for sequences which are not equidistributed we prove that the square-integral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations.

Original languageEnglish
Pages (from-to)206-220
JournalJournal of Number Theory
Volume182
Early online date18 Jul 2017
DOIs
Publication statusEarly online date - 18 Jul 2017
Externally publishedYes

Keywords

  • Equidistribution
  • Fejér kernel
  • Pair correlations
  • Pseudorandomness

ASJC Scopus subject areas

  • Algebra and Number Theory

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