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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics