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

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