ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

Abstract

We derive an explicit formula for the N-point correlation $F_N(s)$ 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. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$ .