On the finite pair correlation function of van der Corput sequences

Abstract

In this note we derive an explicit formula for the finite empiric pair correlation function $F_N(s)$ of the van der Corput sequence in base $2$ for all $N\in \mathbb{N}$ and $s\ge 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. Moreover, it can be immediately read off that ${\lim }_{N\to \infty }{F}_N(s)$ exists only for $0\le s\le 1/2$.