## Abstract

We investigate the pair correlation function of the sequence of fractional parts of αn^{d}, n = 1, 2, . . . , N, where d ≥ 2 is an integer and α an irrational. We conjecture that for badly approximable α, the normalized spacings between elements of this sequence have Poisson statistics as N → ∞. We show that for almost all α (in the sense of measure theory), the pair correlation of this sequence is Poissonian. In the quadratic case d = 2, this implies a similar result for the energy levels of the "boxed oscillator" in the high-energy limit. This is a simple integrable system in 2 degrees of freedom studied by Berry and Tabor as an example for their conjecture that the energy levels of generic completely integrable systems have Poisson spacing statistics.

Original language | English (US) |
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Pages (from-to) | 61-70 |

Number of pages | 10 |

Journal | Communications In Mathematical Physics |

Volume | 194 |

Issue number | 1 |

DOIs | |

State | Published - 1998 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics