An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

It is known that there is a constant c > 0 such that for every sequence x₁, x₂, . . . in [0, 1) we have for the star discrepancy DN*$D_N^* $ of the first N elements of the sequence that NDN*≥c⋅log⁡N$ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c^∗ be the supremum of all such c with this property. We show c^∗ > 0.065664679 . . . , thereby slightly improving the estimates known until now.