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arXiv · 1604.02413

Small gaps in the spectrum of the rectangular billiard

Valentin Blomer·Jean Bourgain·Maksym Radziwiłł·Zeev Rudnick
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Abstract

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.

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Valentin Blomer, Jean Bourgain, Maksym Radziwiłł, Zeev Rudnick. 2016-04-08. Small gaps in the spectrum of the rectangular billiard. https://arxiv.org/abs/1604.02413

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