On the star discrepancy of sequences in the unit interval

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