Pair correlation of sequences: metric results and a modified additive energy.

The uniform distribution of a sequence $\{x_n\}_{n\geq 1}$ measures the pseudo-random behavior at a global scale. At a more localized scale, we can study the pair correlation 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 $(a_n \alpha)_{n \geq 1}$  has been pioneered by Rudnick, Sarnak and Zaharescu. Recently, a general framework was developed which gives a criterion for Poissonian pair correlation of such sequences for almost $\alpha \in (0,1)$, in terms of the additive energy of the integer sequence $\{a_n\}_{n \geq 1}$. In the present talk we will discuss a similar framework in the more delicate case where $\{a_n\}_{n \geq 1}$ is a sequence of reals. We give a criterion involving a modified version of the additive energy expressed via a diophantine inequality. We give several concrete applications of our method and present some open problems.
This is joint work with Christoph Aistleitner and Daniel EL-Baz.

Zoom Meeting ID: 919 6497 4060
For password see the email or contact Pieter Moree (moree@mpim...).