  We study the size of the minimal gap between the first N eigenvalues of the
Laplacian on a rectangular billiard having irrational squared aspect ratio
$\alpha$, in comparison to the corresponding quantity for a Poissonian
sequence. If $\alpha$ is 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 of $\alpha$'s of full measure. However, on a fine
scale we show that Poisson statistics is violated for all $\alpha$. 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.
