The Pair Correlation Function of Fractional Parts of Polynomials

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.