Poisson nearest-neighbor spacings on almost every flat torus
For $\alpha = (\alpha_1, \alpha_2, \alpha_3) \in \mathbb{R}^3$ with $\alpha_1 > 0$ and $4\alpha_1\alpha_3 > \alpha_2^2$ let $q_\alpha(m, n) = \alpha_1 m^2 + \alpha_2 mn + \alpha_3 n^2$, and let $\lambda_1 \le \lambda_2 \le \cdots$ be the numbers $$\frac{\pi\, q_\alpha(m, n)}{\sqrt{4\alpha_1\alpha_3 - \alpha_2^2}}, \qquad (m, n) \in \mathbb{Z}^2,\ m > 0 \text{ or } (m = 0,\ n > 0),$$ in increasing order, counted with multiplicity. Up to a constant factor these are the nonzero eigenvalues of the Laplacian on a flat torus, with the multiplicity coming from $(m, n) \mapsto (-m, -n)$ removed, normalized to mean spacing one. The Berry–Tabor conjecture predicts Poisson local statistics for generic integrable systems. Sarnak proved that for almost every $\alpha$ the pair correlation of $(\lambda_j)$ is Poissonian. Aistleitner, Blomer and Radziwiłł evaluated the triple correlation averaged over $\alpha$, and deduced that for almost every $\alpha$, $\limsup_{j \to \infty} (\lambda_{j+1} - \lambda_j) \ge 2.006$; there are sequences with Poissonian pair correlation whose gaps are all at most $2$.
Problem. For Lebesgue-almost every such $\alpha$, is it true, simultaneously for every $s \ge 0$, that $$\lim_{N \to \infty} \frac{1}{N} \#\{j \le N : \lambda_{j+1} - \lambda_j \le s\} = 1 - e^{-s}\,?$$
Source: Berry and Tabor (1977), for generic integrable systems; in this form, Aistleitner, Blomer and Radziwiłł (2024), §1.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NT — Number Theory
- Secondary
- MP, SP
- Keywords
- AIM Open Applied Problems, binary quadratic forms, flat tori, level spacings
References
@article{berry1977,
author = {Berry, M. V. and Tabor, M.},
title = {Level clustering in the regular spectrum},
journal = {Proc. R. Soc. Lond. A},
volume = {356},
number = {1686},
year = {1977},
pages = {375--394},
doi = {10.1098/rspa.1977.0140},
}
@article{aistleitner2024,
author = {Aistleitner, Christoph and Blomer, Valentin and Radziwi{\l}{\l}, Maksym},
title = {Triple correlation and long gaps in the spectrum of flat tori},
journal = {J. Eur. Math. Soc.},
volume = {26},
number = {1},
year = {2024},
pages = {41--74},
doi = {10.4171/JEMS/1356},
eprint = {1809.07881},
archivePrefix = {arXiv},
}
@incollection{sarnak1997,
author = {Sarnak, Peter},
title = {Values at integers of binary quadratic forms},
booktitle = {Harmonic Analysis and Number Theory (Montreal, PQ, 1996)},
series = {CMS Conf. Proc.},
volume = {21},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1997},
pages = {181--203},
}
@misc{aim-berry-tabor-flat-tori,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Poisson nearest-neighbor spacings on almost every flat torus}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/212-berry-tabor-flat-tori.md},
note = {Entry 212 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00408, "Poisson nearest-neighbor spacings on almost every flat torus", clopen.solutions.