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We study the size of the minimal gap between the first N eigenvalues of theLaplacian on a rectangular billiard having irrational squared aspect ratio$\alpha$, in comparison to the corresponding quantity for a Poissoniansequence. If $\alpha$ is a quadratic irrationality of certain type, such as thesquare root of a rational number, we show that the minimal gap is roughly ofsize 1/N, which is essentially consistent with Poisson statistics. We also giverelated results for a set of $\alpha$'s of full measure. However, on a finescale we show that Poisson statistics is violated for all $\alpha$. The proofsuse a variety of ideas of an arithmetical nature, involving Diophantineapproximation, the theory of continued fractions, and results in analyticnumber theory.

Small gaps in the spectrum of the rectangular billiard