Equidistribution Mod $1$ And Normal Numbers

Abstract

Let $\alpha =0.a_1{a}_2{a}_3\dots$ be an irrational number in base $b>1$, where $0\le a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}{a}_{n+2}\dots a_{n+k})$ 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.

Twenty Three Pages. Keywords: Irrational number; Normal number; Uniform distribution; Borel problem