normal number

Simply Normal Numbers

Let $x\in\mathbb{R}$ and $2\le b\in\mathbb{N}$ . Consider the sequence of digits of $x$ in base $b$ : $$x=x_1x_2\cdots x_m.x_{m+1}\cdots x_n \cdots,\mbox{ where }x_i\in\{0,\ldots,b-1\}.$$ We are interested in finding out how often a given digit $s$ shows up in the above representation of $x$ . If we denote by $N(s,n)$ the number of occurrences of $s$ in the first $n$ digits of $x$ , we can calculate the ratio $$\frac{N(s,n)}{n}.$$ As $n$ approaches $\infty$ , this ratio may converge to a limit, called the frequency of $s$ in $x$ . The frequency of $s$ in $x$ is necessarily between 0 and 1. If all base-$b$ digits are equally frequent, i.e. if the frequency of each digit $s$ , $0\le s<b$ , is $1/b$ , then we say that $x$ is simply normal in base $b$ . For example, in base $5$ , the number $$0.1234012340123401234....$$ is simply normal.

Some real numbers have two digital representations in base $b$ . Given such a number $x$ , there is no ambiguity as to whether $x$ is simply normal or not, for it is easy to see that 0 occurs in $x$ with frequency 1 in one representation, and $b-1$ occurs in $x$ with frequency 1 in the other. Hence such a number $x$ is not simply normal.

It is still an open question if the transcendental constants $\pi$ , $e$ , and $\ln(2)$ are simply normal, although empirical evidence tends to support this claim. Actually, we don't even know which digits occur infinitely often in the base 10 expansion of $\pi$ (there must be at least two). The following table shows the number of occurrences of each digit as they appear in the decimal representation of $\pi$ , evaluated to just over 10,000 places.