Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori

Abstract

Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum particle in a three-dimensional box, subject to a Diophantine condition on the domain's proportions. A permissible choice of width, height and depth is for example $1,2^{1/3},2^{-1/3}$. This extends previous work of Eskin, Margulis and Mozes (Annals of Math., 2005) in dimension two, where the problem reduces to the quantitative Oppenheim conjecture for quadratic forms of signature $(2,2)$. The difficulty in three and higher dimensions is that we need to consider the distribution of indefinite forms in shrinking rather than fixed intervals, which we are able to resolve for special diagonal forms of signature $(3,3)$ in various scalings, including a rate of convergence. A key step of our approach is to represent the relevant counting problem as an average of a theta function on $\mathrm{SL}(2,\mathbb{Z})^3\backslash\mathrm{SL}(2,\mathbb{R})^3$ over an expanding family of one-parameter unipotent orbits. The asymptotic behaviour of these unipotent averages follows from Ratner's measure classification theorem and subtle escape of mass estimates.