# Normal number

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2010 Mathematics Subject Classification: Primary: 11K16 [MSN][ZBL]

A real number $\alpha$, $0 < \alpha < 1$, having the following property: For every natural number $s$, any given $s$-tuple $\delta = (\delta_1,\ldots,\delta_s)$ consisting of the symbols $0,1,\ldots,g-1$ appears with asymptotic frequency $1/g^s$ in the sequence $$\label{eq:a1} \alpha_1,\ldots,\alpha_n,\ldots$$ obtained from the expansion of $\alpha$ as an infinite fraction in base $g$, $$\alpha = \frac{\alpha_1}{g} + \cdots + \frac{\alpha_n}{g^n} + \cdots \ .$$

In more detail, let $g>1$ be a natural number and let $$\label{eq:a2} (\alpha_1,\ldots,\alpha_s), (\alpha_2,\ldots,\alpha_{s+1}), \ldots$$ be the infinite sequence of $s$-tuples corresponding to \ref{eq:a1}. Let $N(n,\delta)$ denote the number of occurrences of the tuple $\delta = (\delta_1,\ldots,\delta_s)$ among the first $n$ tuples of \ref{eq:a2}. The number $$\alpha = \frac{\alpha_1}{g} + \cdots + \frac{\alpha_n}{g^n} + \cdots \ .$$ is said to be normal if for any number $s$ and any given $s$-tuple $\delta$ consisting of the symbols $0,\ldots,g-1$, $$\lim_{n \rightarrow \infty} \frac{N(n,s)}{ n } = \frac{1}{g^s} \ .$$

The concept of a normal number was introduced for $g=10$ by E. Borel (see [B], [B2], p. 197). He called a real number $\alpha$ weakly normal to the base $g$ if $$\lim_{n \rightarrow \infty} \frac{N(n,\delta)}{n} = \frac{1}{g}$$ where $N(n,\delta)$ is the number of occurrences of $\delta$, $0 \le \delta \le g-1$, among the first $n$ terms of the sequences $\alpha_1,\alpha_2,\ldots$ and normal if $\alpha, g\alpha, g^2\alpha, \ldots$ are weakly normal to the bases $g, g^2, \ldots$. He also showed that for a normal number $$\lim_{n \rightarrow \infty} \frac{N(n,s)}{ n } = \frac{1}{g^s}$$ for any $s$ and any given $s$-tuple $\delta = (\delta_1,\ldots,\delta_s)$. Later it was proved (see [Pi], [NZ], and also [Po]) that the last relation is equivalent to Borel's definition of a normal number.

A number $\alpha$ is called absolutely normal if it is normal with respect to every base $g$. The existence of normal and absolutely-normal numbers was established by Borel on the basis of measure theory. The construction of normal numbers in an explicit form was first achieved in [C]. Earlier (see [S], [L]) an effective procedure for constructing normal numbers was indicated. For other methods for constructing normal numbers and for connections between the concepts of normality and randomness see [Po].

Uniform distribution of the fractional parts $\{ \alpha g^x \}$, $x = 0, 1, \ldots$ on the interval $[0,1]$ is equivalent to $\alpha$ being normal.