Small gaps in the spectrum of the rectangular billiard

We study the size of the minimal gap between the firstN eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio, in comparison to the corresponding quantity for a Poissonian sequence. Ifis a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1=N, which is essentially consistent with Poisson statistics. We also give related results for a set ofÕs of full measure. However, on a fine scale we show that Poisson statistics is violated for all. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.