Pair correlation of the fractional parts of

  • Christopher Lutsko

    Rutgers University, Piscataway, USA; Universität Zürich, Zürich, Switzerland
  • Athanasios Sourmelidis

    Graz University of Technology, Graz, Austria
  • Niclas Technau

    University of Wisconsin–Madison, Madison, USA
Pair correlation of the fractional parts of $\alpha n^{\theta}$ cover
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Fix and consider the sequence . Since the seminal work of Rudnick–Sarnak (1998) and due to the Berry–Tabor conjecture in quantum chaos, the fine-scale properties of these dilated monomial sequences have been intensively studied. In this paper, we show that for and , the pair correlation function is Poissonian. While (for a given ) this strong pseudo-randomness property has been proven for almost all values of , there are next-to-no instances where this has been proven for explicit . Our result holds for all and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner–El-Baz–Munsch (2021).

Cite this article

Christopher Lutsko, Athanasios Sourmelidis, Niclas Technau, Pair correlation of the fractional parts of . J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1449