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

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.

16 pages. Dedicated to Ian H. Sloan on the occasion of his 80th birthday. This paper will appear in the volume: "Contemporary Computational Mathematics - a celebration of the 80th birthday of Ian Sloan". (Arxiv version 2: corrected an error in the proof of Lemma 2. Thanks to Andreas Weingartner for pointing out this error to us.)