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 in [1].

Some negative results related to Poissonian pair correlation problems

“…Recently, the result of Bourgain has been further extended, see [1,16,17,18]. The result given in [17] is an easy consequence of our Theorem 1 stated below and will be shown in Section 2.…”

“…for example [1] and the references cited therein for detailed information on that topic. Rudnik and Sarnak first studied this concept from a purely mathematical point of view and over the years the topic has attracted wide attention, see e.g., [13,21,24,25,26].…”

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

“…The concept of Poissonian pair correlations has its origin in quantum mechanics, where the spacings of energy levels of integrable systems were studied. See for example [1] and the references cited therein for detailed information on that topic. Rudnick and Sarnak first studied this concept from a purely mathematical point of view and over the years the topic has attracted wide attention, see * The author is supported by the Austrian Science Fund (FWF) Project F5509-N26, which is a part of the Special Research Program "Quasi-Monte Carlo Methods: Theory and Applications" † The author is supported by the Austrian Science Fund (FWF), Project F5507-N26, which is a part of the Special Research Program "Quasi-Monte Carlo Methods: Theory and Applications".…”

“…The notion Poissonian pair correlation has attracted renewed interest in the last few years, due to its connection to several mathematical fields, such as Diophantine approxmation, additive combinatorics and uniform distribution (see e.g., [1,3,4,9,11,19]). The link between the concept of uniform distribution modulo 1 and Poissonian pair correlation has been studied in the one-dimensional case.…”

Pair correlation of sequences with maximal additive energy

“…In the sequel, {•} denotes the fractional part of a real number. The study of Poissonian pair correlations of sequences, especially of sequences of the form ({a n α}) n∈N , where α is irrational, and (a n ) n∈N is a sequence of distinct positive integers, is primarily motivated by certain questions in quantum physics, especially in connection with the Berry-Tabor conjecture in quantum mechanics, cf., [1,12]. The investigation of Poissonian pair correlations was started by Rudnick, Sarnak and Zaharescu, cf., [13,14,15], and was continued by many authors in the subsequent, cf., [2] and the references given there.…”