Triple correlation and long gaps in the spectrum of flat tori

TL;DR

This paper studies the eigenvalue spectrum of flat tori, showing that their triple correlations tend to Poissonian limits and that most contain infinitely many long spectral gaps, revealing complex spectral structures beyond pair correlation behavior.

Contribution

It introduces new results on the triple correlation of eigenvalues of flat tori and demonstrates the existence of arbitrarily long spectral gaps, extending understanding beyond pair correlation.

Findings

Triple correlation limits are almost surely Poissonian.

Most flat tori have infinitely many spectral gaps at least 2.006 times longer than average.

Sequences with Poissonian pair correlation can have bounded spacings up to 2.

Abstract

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.