Nearest Neighbor Distances on a Circle: Multidimensional Case

Abstract

We study the distances, called spacings, between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E+1, and study how the spacings between these levels change for various choices of E, particularly when E goes to infinity. Primarily, we study the case in which the spring constant is a badly approximable vector. We first give the proof by Boshernitzan-Dyson that the number of distinct spacings has a uniform bound independent of E. Then, if the spring constant has components forming a basis of an algebraic number field, we show that, when normalized up to a unit, the spacings are from a finite set. Moreover, in the specific case that the field has one fundamental unit, the probability distribution of these spacings behaves quasiperiodically in log E. We conclude by studying the spacings in the case that the spring constant is not badly approximable, providing examples for which the number of distinct spacings is unbounded.Comment: Version 2 is updated to include more discussion of previous works. 17 pages with five figures. To appear in the Journal of Statistical Physic

Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

“…It was conjectured by Erdös (Geelen and Simpson 1993) that the latter equation should hold whenever 1, α and β are Q-linearly independent. This conjecture was disproved in Bleher et al (2012), where it was established that the set of (α, β) for which sup N ∈N G (α,β) (N , N ) < +∞, although of zero Lebesgue measure, has full Hausdorff dimension. It is an open problem to determine whether there exists a pair (α, β) such that sup N ∈N G (α,β) …”

Gaps problems and frequencies of patches in cut and project sets

“…It is worth mentioning that, in the special case when d = 1, our proofs of Theorems 1.1 and 1.2 can be simplified and, regardless of Diophantine properties, we can obtain uniform bounds for #ξ 1 (r) and #ξ 2 (r). On the other hand we will prove, using a modification of an argument from [2], that for any k and d with d ≥ 2, there is a large collection of subspaces for which the upper bounds in our previous theorems cannot be obtained.…”

“…They also give us a convenient mathematical model for producing patterns which occur naturally in physical materials known as quasicrystals [16,17]. The particular problems which we are studying in this paper have direct consequences to understanding the distribution of distances between molecules and patterns in quasicrystals [19], as well as to related problems of understanding the distribution of energy levels in sums of forced harmonic oscillators [2].…”

Higher dimensional Steinhaus and Slater problems via homogeneous dynamics

“…In this paper we are firstly interested in a higher dimensional version of the Steinhaus problem, which was previously studied by Geelen and Simpson [17], Fraenkel and Holzman [15], Chevallier [8], Boshernitzan [5,6], Dyson [12], and Bleher, Homma, Ji, Roeder, and Shen [4]. For this problem our goal is twofold: to demonstrate the close connection between the multi-dimensional Steinhaus problem and the Littlewood conjecture, and to show how well known results from ergodic theory on the space of unimodular lattices in R d can be used to shed new light on a question of Erdős as stated by Geelen and Simpson [17,Section 4].…”