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On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • Ida Aichinger

    (European Organization for Nuclear Research, CERN)

  • Christoph Aistleitner

    (Institute for Analysis and Number Theory, TU Graz)

  • Gerhard Larcher

    (Institute for Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz)

Abstract

The investigation of the pair correlation statistics of sequences was initially motivated by questions concerning quasi-energy-spectra of quantum systems. However, the subject has been developed far beyond its roots in mathematical physics, and many challenging number-theoretic questions on the distribution of the pair correlations of certain sequences are still open. We give a short introduction into the subject, recall some known results and open problems, and in particular explain the recently established connection between the distribution of pair correlations of sequences on the torus and certain concepts from additive combinatorics. Furthermore, we slightly improve a result recently given by Jean Bourgain in Aistleitner et al. (Isr. J. Math., to appear. Available at https://arxiv.org/abs/1606.03591 ).

Suggested Citation

  • Ida Aichinger & Christoph Aistleitner & Gerhard Larcher, 2018. "On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 1-16, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_1
    DOI: 10.1007/978-3-319-72456-0_1
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