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 $s \geq 0$. 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 $0 \leq s \leq 1/2$.