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

Wooyeon Kim, Jens Marklof, Matthew Welsh

TL;DR

This work verifies the Berry-Tabor conjecture for the two-point spectrum of a quantum particle in a 3D box under Diophantine side-length ratios by translating spectral counts into averages of Siegel theta functions and analyzing unipotent flows on products of SL(2,R). The authors develop a uniform variant of the quantitative Oppenheim conjecture for a special signature $(3,3)$ form, employing a two-regime analysis (major/minor arcs) and a precise shrinking-interval scaling with $L\asymp M^{d-2}$, to extract both the bulk and diagonal contributions to the pair correlations. A key technical advance is a new bound for the Margulis function in \\Gamma^3\\backslash G^3 under Diophantine conditions, coupled with avoidance estimates and a global contraction framework to control escape of mass in cusps. The combination of theta-operator methods, Ratner-type equidistribution, and Diophantine geometry yields a rigorous Poisson (independent) two-point statistics result, extending the Eskin–Margulis–Mozes program to dimension three and providing explicit convergence rates in the special diagonal form setting. These results sharpen our understanding of local spectral statistics for integrable quantum systems and highlight the role of shrinking-interval counting in capturing universal Poisson behavior.

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.

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

TL;DR

This work verifies the Berry-Tabor conjecture for the two-point spectrum of a quantum particle in a 3D box under Diophantine side-length ratios by translating spectral counts into averages of Siegel theta functions and analyzing unipotent flows on products of SL(2,R). The authors develop a uniform variant of the quantitative Oppenheim conjecture for a special signature form, employing a two-regime analysis (major/minor arcs) and a precise shrinking-interval scaling with , to extract both the bulk and diagonal contributions to the pair correlations. A key technical advance is a new bound for the Margulis function in \\Gamma^3\\backslash G^3 under Diophantine conditions, coupled with avoidance estimates and a global contraction framework to control escape of mass in cusps. The combination of theta-operator methods, Ratner-type equidistribution, and Diophantine geometry yields a rigorous Poisson (independent) two-point statistics result, extending the Eskin–Margulis–Mozes program to dimension three and providing explicit convergence rates in the special diagonal form setting. These results sharpen our understanding of local spectral statistics for integrable quantum systems and highlight the role of shrinking-interval counting in capturing universal Poisson behavior.

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 . 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 . 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 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 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.
Paper Structure (20 sections, 29 theorems, 167 equations)

This paper contains 20 sections, 29 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. The Berry-Tabor conjecture
  3. Uniform variant of the quantitative Oppenheim conjecture
  4. Escape of mass and Margulis functions
  5. Preliminaries
  6. Theta functions and Shale-Weil representation
  7. Automorphy of theta functions
  8. Margulis functions
  9. Diophantine property
  10. Asymptotics of $\mathcal{N}_{M,L}(F,\psi)$
  11. Small $\xi$
  12. Remaining $\xi$
  13. Escape of mass
  14. Local contraction
  15. Global contraction
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(\xi_j)_j$ be the deterministic sequence given by the unfolded spectrum unfold of the Dirichlet Laplacian for a box in $\mathbb{R}^3$ with edge lengths $\ell_1,\ell_2,\ell_3$. If the ratios $\ell_i^2/\ell_j^2$ are $\kappa$-Diophantine for all $i \neq j$ with $\kappa$ sufficiently close to 1, th

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • ...and 39 more