The pair correlation function of fractional parts of polynomials (Q1270329)

The authors investigate the asymptotic behaviour of the pair correlation function of the sequence of fractional parts of \(\theta_n(\alpha)=\alpha n^d\), \(n=1,\dots,n\), where \(\alpha\) is irrational and \(d\) is an integer greater than one. It is shown that for almost all \(\alpha\) (in the sense of Lebesgue measure) the pair correlation function of this sequence is asymptotically Poissonian, more precisely: For \(d\geq 2\) there is a set \(P\subset R^1\) of full Lebesgue measure such that for any \(\alpha\in P\) and any \(s\geq 0\), as \(N\to \infty\): \[ {1\over N}\#\left\{1<j\neq k<N:\bigl\|\theta_j(\alpha)-\theta_k (\alpha)\bigr\|\leq{s\over N}\right\}\to 2s, \] where \(\|\theta \|\) denotes the distance between \(\theta\) and the nearest integer. Further, it is conjectured that for badly approximable \(\alpha\) (in the sense of diophantine approximation) this limit theorem is also valid. On the other hand, higher-order correlation functions of the sequence of fractional parts do not behave like Poisson statistics.