论文标题
levin在基本2中的正常数的确切差异顺序
The exact order of discrepancy for Levin's normal number in base 2
论文作者
论文摘要
Mordechay Levin构建了一个数字$α$，这在基本2中是正常的，因此序列$ \ weft \ {2^nα\ right \} _ {n = 0,1,2，\ ldots} $具有很小的差异$ d_n $。实际上，我们有$ n \ cdot d_n = \ mathcal {o} \ left（\ left（\ log n \ right）^2 \ right）$。这意味着，$α$的质量非常高。在本文中，我们表明该估计是最好的，即，$ n \ cdot d_n \ geq c \ cdot \ left（\ log n \ right）^2 $对于无限的许多$ n $。
Mordechay Levin has constructed a number $α$ which is normal in base 2, and such that the sequence $\left\{2^n α\right\}_{n=0,1,2,\ldots}$ 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 $α$ 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 $N$.
