A pair correlation problem, and counting lattice points with the zeta function

Christoph Aistleitner, Marc Alexandre Munsch, Daniel El-Baz

Research output: Contribution to journalArticlepeer-review


The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form (anα)n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and (an)n≥1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number α, in terms of the additive energy of the integer sequence (an)n≥1. In the present paper we develop a similar framework for the case when (an)n≥1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number θ> 1 , the sequence (nθα)n≥1 has Poissonian pair correlation for almost all α∈ R.

Original languageEnglish
Pages (from-to)483-512
Number of pages30
JournalGeometric and Functional Analysis
Issue number3
Publication statusPublished - Jun 2021


  • Diophantine inequality
  • Lattice points
  • Pair correlation
  • Riemann zeta function

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

Fields of Expertise

  • Information, Communication & Computing


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