Hostname: page-component-6766d58669-r8qmj Total loading time: 0 Render date: 2026-05-22T20:49:20.545Z Has data issue: false hasContentIssue false
  • English
  • Français

Pair correlation for quadratic polynomials mod 1

Part of: Dynamical systems with hyperbolic behavior Ergodic theory Discontinuous groups and automorphic forms General mathematical topics and methods in quantum theory Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  20 March 2018

Jens Marklof  and
Nadav Yesha
Jens Marklof
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email j.marklof@bristol.ac.uk
Nadav Yesha
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email nadav.yesha@kcl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

It is an open question whether the fractional parts of non-linear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure convergence in the space of polynomials of a given degree. We will here provide explicit Diophantine conditions on the coefficients of polynomials of degree two, under which the convergence of an averaged pair correlation density can be established. The limit is consistent with the Poisson distribution. Since quadratic polynomials at integers represent the energy levels of a class of integrable quantum systems, our findings provide further evidence for the Berry–Tabor conjecture in the theory of quantum chaos.

Keywords

pair correlationdistribution mod 1Poisson statisticsBerry–Tabor conjecture

MSC classification

Primary: 11J71: Distribution modulo one
Secondary: 11F27: Theta series; Weil representation; theta correspondences 37A17: Homogeneous flows 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 81Q50: Quantum chaos

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 960 - 983
Copyright
© The Authors 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Berry, M. V. and Tabor, M., Level clustering in the regular spectrum , Proc. R. Soc. Lond. Ser. A 356 (1977), 375–394.Google Scholar
Bleher, P. M., The energy level spacing for two harmonic oscillators with generic ratio of frequencies , J. Stat. Phys. 63 (1991), 261–283.Google Scholar
Boca, F. P. and Zaharescu, A., Pair correlation of values of rational functions (mod p) , Duke Math. J. 105 (2000), 267–307.CrossRefGoogle Scholar
Cellarosi, F. and Marklof, J., Quadratic Weyl sums, automorphic functions, and invariance principles , Proc. Lond. Math. Soc. (3) 113 (2016), 775–828.Google Scholar
De Bièvre, S., Degli Esposti, M. and Giachetti, R., Quantization of a class of piecewise affine transformations on the torus , Comm. Math. Phys. 176 (1996), 73–94.CrossRefGoogle Scholar
Eskin, A., Margulis, G. and Mozes, S., Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2-tori , Ann. of Math. (2) 161 (2005), 679–725.CrossRefGoogle Scholar
Greenman, C. D., The generic spacing distribution of the two-dimensional harmonic oscillator , J. Phys. A 29 (1996), 4065–4081.Google Scholar
Heath-Brown, D. R., Pair correlation for fractional parts of 𝛼n 2 , Math. Proc. Cambridge Philos. Soc. 148 (2010), 385–407.Google Scholar
Kurlberg, P. and Rudnick, Z., The distribution of spacings between quadratic residues , Duke Math. J. 100 (1999), 211–242.CrossRefGoogle Scholar
Margulis, G. and Mohammadi, A., Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms , Duke Math. J. 158 (2011), 121–160.CrossRefGoogle Scholar
Marklof, J., The Berry–Tabor conjecture , in European Congress of Mathematics, Vol. II (Barcelona, 2000), Progress in Mathematics, vol. 202 (Birkhäuser, Basel, 2001), 421–427.Google Scholar
Marklof, J., Pair correlation densities of inhomogeneous quadratic forms , Ann. of Math. (2) 158 (2003), 419–471.Google Scholar
Marklof, J. and Strömbergsson, A., Equidistribution of Kronecker sequences along closed horocycles , Geom. Funct. Anal. 13 (2003), 1239–1280.Google Scholar
Pandey, A., Bohigas, O. and Giannoni, M. J., Level repulsion in the spectrum of two-dimensional harmonic oscillators , J. Phys. A 22 (1989), 4083–4088.Google Scholar
Pellegrinotti, A., Evidence for the poisson distribution for quasi-energies in the quantum kicked-rotator model , J. Stat. Phys. 53 (1988), 1327–1336.Google Scholar
Rudnick, Z. and Sarnak, P., The pair correlation function of fractional parts of polynomials , Comm. Math. Phys. 194 (1998), 61–70.Google Scholar
Rudnick, Z., Sarnak, P. and Zaharescu, A., The distribution of spacings between the fractional parts of n 2𝛼 , Invent. Math. 145 (2001), 37–57.Google Scholar
Sinai, Ya. G., The absence of the Poisson distribution for spacings between quasi-energies in the quantum kicked-rotator model , Phys. D 33 (1988), 314–316.Google Scholar
Slater, N., Gaps and steps for the sequence n𝜃 mod 1 , Proc. Cambridge Philos. Soc. 63 (1967), 1115–1123.Google Scholar
Strömbergsson, A., An effective Ratner equidistribution result for SL(2, ℝ) ⋉ℝ 2 , Duke Math. J. 164 (2015), 843–902.Google Scholar
Truelsen, J. L., Divisor problems and the pair correlation for the fractional parts of n 2𝛼 , Int. Math. Res. Not. IMRN (2010), 3144–3183.Google Scholar
Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins , Math. Ann. 77 (1916), 313–352.Google Scholar
Zelditch, S., Level spacings for integrable quantum maps in genus zero , Comm. Math. Phys. 196 (1998), 289–318; Addendum: ‘Level spacings for integrable quantum maps in genus zero’, Comm. Math. Phys. 196 (1998), 319–329.Google Scholar
Zelditch, S. and Zworski, M., Spacing between phase shifts in a simple scattering problem , Comm. Math. Phys. 204 (1999), 709–729.Google Scholar