Discrepancy results for the Van der Corput sequence

Lukas Spiegelhofer

paper

Discrepancy results for the Van der Corput sequence

arXiv:1710.01560

Abstract

Let $d_{N} = N D_{N} (ω)$ be the discrepancy of the Van der Corput sequence in base $2$ . We improve on the known bounds for the number of indices $N$ such that $d_{N} \leq lo g N /100$ . Moreover, we show that the summatory function of $d_{N}$ satisfies an exact formula involving a $1$ -periodic, continuous function. Finally, we show that $d_{N}$ is invariant under digit reversal in base $2$ .

arXiv comment9 pages