THE PAIR CORRELATION-FUNCTION OF FRACTIONAL-PARTS OF POLYNOMIALS

We investigate the pair correlation function of the sequence of fracti onal parts of alpha n(d), n = 1,2,..., N, where d greater than or equa l to 2 is an integer and alpha an irrational. We conjecture that for b adly approximable alpha, the normalized spacings between elements of t his sequence have Poisson statistics as N --> infinity. We show that f or almost all alpha (in the sense of measure theory), the pair correla tion of this sequence is Poissonian. In the quadratic case d = 2, this implies a similar result for the energy levels of the ''boxed oscilla tor'' in the high-energy limit. This is a simple integrable system in 2 degrees of freedom studied by Ferry and Tabor as an example for thei r conjecture that the energy levels of generic completely integrable s ystems have Poisson spacing statistics.