Abstract
It is an open question whether the fractional parts of nonlinear polynomials at integers have the same finescale 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 2, 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 BerryTabor conjecture in the theory of quantum chaos.
Original language  English 

Pages (fromto)  960983 
Number of pages  24 
Journal  Compositio Mathematica 
Volume  154 
Issue number  5 
Early online date  20 Mar 2018 
DOIs  
Publication status  Published  May 2018 
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Profiles

Professor Jens Marklof
 Science Faculty Office  Dean of the Faculty of Science and Professor of Mathematical Physics
 Probability, Analysis and Dynamics
 Pure Mathematics
 Ergodic theory and dynamical systems
Person: Academic , Member