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

Ida Aichinger (), Christoph Aistleitner () and Gerhard Larcher ()
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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

A chapter in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018, pp 1-16 from Springer

Abstract: 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 ).

Date: 2018
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DOI: 10.1007/978-3-319-72456-0_1

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