Communications in Mathematical Physics

, Volume 194, Issue 1, pp 61–70

The Pair Correlation Function of Fractional Parts of Polynomials

  • Zeév Rudnick
  • Peter Sarnak

DOI: 10.1007/s002200050348

Cite this article as:
Rudnick, Z. & Sarnak, P. Comm Math Phys (1998) 194: 61. doi:10.1007/s002200050348

Abstract:

We investigate the pair correlation function of the sequence of fractional parts of αnd, 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Zeév Rudnick
    • 1
  • Peter Sarnak
    • 2
  1. 1.Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, IsraelIL
  2. 2.Department of Mathematics, Princeton University, Princeton NJ 08544, USAUS