  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 $N D^{*}_N\geq c\cdot \log N$ holds for
infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this
property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the
estimates known until now.
