Normal number

Definition

Let $b \ge 2$ be an integer, and let $x$ be a real number. Write the fractional part $x - \floor{x}$ in base $b$ as $0.d_1 d_2 d_3 \ldots$ with digits $d_i \in \set{0, 1, \dots, b-1}$. When there are two such expansions, take the one that does not end in an infinite run of the digit $b - 1$. For a finite string $w$ of $k$ digits and an integer $n \ge 1$, let $N(w, n)$ be the number of indices $i$ with $1 \le i \le n - k + 1$ such that $d_i d_{i+1} \cdots d_{i+k-1} = w$. The number $x$ is normal in base $b$ if $$\lim_{n \to \infty} \frac{N(w, n)}{n} = b^{-k}$$ for every $k \ge 1$ and every string $w$ of $k$ digits. The number $x$ is absolutely normal if it is normal in base $b$ for every integer $b \ge 2$.