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The pair correlation is a localized statistic for sequences in the unitinterval. Pseudo-random behavior with respect to this statistic is calledPoissonian behavior. The metric theory of pair correlations of sequences of theform $(a_n \alpha)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak andZaharescu. Here $\alpha$ is a real parameter, and $(a_n)_{n \geq 1}$ is aninteger sequence, often of arithmetic origin. Recently, a general framework wasdeveloped which gives criteria for Poissonian pair correlation of suchsequences for almost every real number $\alpha$, in terms of the additiveenergy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper wedevelop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequenceof reals rather than integers, thereby pursuing a line of research which wasrecently initiated by Rudnick and Technau. As an application of our method, weprove that for every real number $\theta>1$, the sequence $(n^\theta \alpha)_{n\geq 1}$ has Poissonian pair correlation for almost all $\alpha \in\mathbb{R}$.

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