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Published 2017 | Submitted
Journal Article Open

Small gaps in the spectrum of the rectangular billiard


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 α, in comparison to the corresponding quantity for a Poissonian sequence. If α 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 α's of full measure. However, on a fine scale we show that Poisson statistics is violated for all α. 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.

Additional Information

© 2017 Société Mathématique de France. V.B. was supported in part by the Volkswagen Foundation and NSF grant 1128155 while enjoying the hospitality of the Institute for Advanced Study. J.B. was partially supported by NSF grant DMS-1301619. M.R. is funded by the National Science and Engineering Research Council of Canada. Z.R. was supported by the Friends of the Institute for Advanced Study, and by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 320755. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

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August 19, 2023
October 18, 2023