  Mordechay Levin has constructed a number $\alpha$ which is normal in base 2,
and such that the sequence $\left\{2^n \alpha\right\}_{n=0,1,2,\ldots}$ has
very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O}
\left(\left(\log N\right)^2\right)$. That means, that $\alpha$ is normal of
extremely high quality. In this paper we show that this estimate is best
possible, i.e., $N\cdot D_N \geq c \cdot \left(\log N\right)^2$ for infinitely
many $N$.
