  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.
