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

Abstract

AbstractThe 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 $$(a_n \alpha )_{n \ge 1}$$ ( a n α ) n ≥ 1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here $$\alpha $$ α is a real parameter, and $$(a_n)_{n \ge 1}$$ ( a n ) 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 $$\alpha $$ α , in terms of the additive energy of the integer sequence $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 . In the present paper we develop a similar framework for the case when $$(a_n)_{n \ge 1}$$ ( a n ) 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 $$\theta >1$$ θ > 1 , the sequence $$(n^\theta \alpha )_{n \ge 1}$$ ( n θ α ) n ≥ 1 has Poissonian pair correlation for almost all $$\alpha \in {\mathbb {R}}$$ α ∈ R .