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

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 (anα)n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and (an)n≥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 α, in terms of the additive energy of the integer sequence (an)n≥1. In the present paper we develop a similar framework for the case when (an)n≥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 θ> 1 , the sequence (nθα)n≥1 has Poissonian pair correlation for almost all α∈ R.