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Triple correlation and long gaps in the spectrum of flat tori

Aistleitner, Christoph and Blomer, Valentin and Radziwiłł, Maksym (2018) Triple correlation and long gaps in the spectrum of flat tori. . (Unpublished)

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We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequence we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most (resp. at least) Poissonian, and that (b) almost all flat tori contain infinitely many gaps in their spectrum that are at least 2.006 times longer than the average gap. The significance of the constant 2.006 lies in the fact that there exist sequences with Poissonian pair correlation and with consecutive spacings bounded uniformly from above by 2, as we also prove in this paper. Thus our result goes beyond what can be deduced solely from the Poissonian behavior of the pair correlation.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:The first author is supported by the Austrian Science Fund FWF, projects F-5512 and Y-901. The second author was partially supported by a SNF-DFG lead agency grant BL 915/2-2. The third author acknowledges support of a Sloan fellowship.
Funding AgencyGrant Number
FWF Der WissenschaftsfondsF-5512
FWF Der WissenschaftsfondsY-901
Deutsche Forschungsgemeinschaft (DFG)BL 915/2-2
Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:billard, flat torus, long gaps, spectrum, Berry-Tabor conjecture, Poisson statistics, pair correlation, triple correlation, Diophantine inequalities
Classification Code:2010 Mathematics Subject Classification. 35P20, 11D75, 11K36, 11E16, 11L07
Record Number:CaltechAUTHORS:20210825-184506223
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110512
Deposited By: George Porter
Deposited On:26 Aug 2021 22:29
Last Modified:26 Aug 2021 22:29

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