等分布模$1$与正态数
Equidistribution Mod $1$ And Normal Numbers
摘要：
设$α=0.a_1a_2a_3\ldots$是基$b>1$中的无理数，其中$0\leq a_i<b$。如果$k$数字的每个块$(a_{n+1}a_{n+2}\ldots a_{n+k})$以概率$1/b^k$出现，则数字$α\in (0,1)$是\textit{normal number}。本说明中给出了实数$\sqrt{2}$在基$10$中正态性的证明。基于不同的方法给出了三种不同的证明：一个条件证明和两个无条件证明。
英文摘要：
Let $α=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $α\in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A proof of the normality of the real number $\sqrt{2}$ in base $10$ is presented in this note. Three different proofs based on different methods are given: a conditional proof, and two unconditional proofs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
