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

Abstract

Mordechay Levin has constructed a number $\alpha$ which is normal in base 2, and such that the sequence ${\left\{2^n\alpha \right\}}_{n=0,1,2,\dots }$ 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., $N\cdot D_N\ge c\cdot {\left(\log N\right)}^2$ for infinitely many $N$.

arXiv commentThere was a gap in the proof in a previous version. Here now is the corrected version