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Pair correlation for quadratic polynomials mod 1

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

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 [email protected]
Nadav Yesha
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email [email protected]
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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
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 960 - 983
Copyright
© The Authors 2018 

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