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

TL;DR

This paper explores the connection between pair correlation statistics of sequences, quasi-energy spectra in quantum systems, and additive combinatorics, highlighting recent advances and open problems in the distribution of sequence correlations.

Contribution

It introduces the link between pair correlations of sequences and additive combinatorics, and improves a recent result by Jean Bourgain.

Findings

Connection established between sequence pair correlations and additive combinatorics

Reviewed known results and open problems in the field

Slight improvement on Bourgain's result

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.