The exact order of discrepancy for Levin's normal number in base 2

Mordechay B. Levin in [4] has constructed a number alpha which is normal in base 2, and such that the sequence {2n alpha}n=0,1,2,... has very small discrepancy DN. Indeed we have N center dot DN = O((log N)2). That means, that alpha is normal of extremely high quality. In this paper we show that this estimate is best possible, i.e., N center dot DN >= c center dot (log N)2 for infinitely many N.