Discrepancy Results for The Van Der Corput Sequence

Let d_N = ND_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 ≤ 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 give a new proof of the fact that d_N is invariant under digit reversal in base 2.