On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics
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).
The first author is supported by CERN, European Organization for Nuclear Research, Geneva, as part of the Austrian Doctoral Student Programme.
The second author is supported by the Austrian Science Fund (FWF), START-project Y-901.
The third author is supported by the Austrian Science Fund (FWF): Project F5507-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
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