A pair correlation problem, and counting lattice points with the zeta function

Abstract

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form (a_n alpha )_{n ge 1} has been pioneered by Rudnick, Sarnak and Zaharescu. Here alpha is a real parameter, and (a_n)_{n ge 1} is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number alpha , in terms of the additive energy of the integer sequence (a_n)_{n ge 1}. In the present paper we develop a similar framework for the case when (a_n)_{n ge 1} is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number theta >1, the sequence (n^theta alpha )_{n ge 1} has Poissonian pair correlation for almost all alpha in {mathbb {R}}.