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Published August 26, 2021 | Submitted
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Triple correlation and long gaps in the spectrum of flat tori


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.

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.

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