  It is known that there is a constant $c > 0$ such that for every sequence
$x_1, x_2, \ldots$ in $[0,1)$ we have for the star discrepancy $D_N^*$ of the
first $N$ elements of the sequence that $N D_N^* \ge c \cdot \log N$ holds for
infinitely many $N$. Let $c^*$ be the supremum of all such $c$ with this
property. We show $c^* > 0.0646363$, thereby improving the until now known
estimates.
