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

TL;DR

This paper proves that Levin's constructed normal number in base 2 has the optimal order of discrepancy, establishing a lower bound that matches the known upper bound, thus confirming the high quality of its normality.

Contribution

The paper demonstrates that Levin's discrepancy estimate for his normal number is asymptotically optimal, providing a matching lower bound.

Findings

Discrepancy of Levin's number is of order (log N)^2 for infinitely many N

The established lower bound matches the known upper bound

Levin's number achieves the best possible discrepancy rate

Abstract

Mordechay Levin has constructed a number $\alpha$ which is normal in base 2, and such that the sequence $\left\{2^n \alpha\right\}_{n=0,1,2,\ldots}$ 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 $N$.