Discrepancy results for the Van der Corput sequence

TL;DR

This paper improves bounds on the discrepancy of the Van der Corput sequence, derives an exact formula for its summatory function, and proves invariance under digit reversal, advancing understanding of its distribution properties.

Contribution

It provides new bounds for the count of indices with low discrepancy, an exact formula for the summatory discrepancy, and establishes invariance under digit reversal in base 2.

Findings

Improved bounds for the number of indices with discrepancy below a threshold.

Derived an exact formula involving a 1-periodic function for the summatory discrepancy.

Proved invariance of discrepancy under digit reversal in base 2.

Abstract

Let $d_N=ND_N(\omega)$ 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 $d_N\leq \log N/100$. 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 $2$.