Discrepancy results for the Van der Corput sequence

Let $d_N=N{D}_N(\omega )$ 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 $d_N\le \log N\mathrm{/}100$. 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 $2$.