Pair correlation of real-valued vector sequences

Abstract

In this article, we investigate the fine-scale statistics of real-valued arithmetic sequences. In particular, we focus on real-valued vector sequences, generalizing previous works of Boca et al. and the first author on the local statistics of integer-valued and rational-valued vector sequences, respectively. As the main results, we prove the Poissonian behavior of the pair correlation function for certain classes of real-valued vector sequences. This is achieved by extrapolating conditions on the number of solutions of Diophantine inequalities using twisted moments of the Riemann zeta function. Later, we give concrete examples of sequences in this set-up where these conditions are satisfied.