= Normal number
{title2=$\lim_{N\to\infty}F_N(w)=K^{-r}$}
{wiki}
A <real number> is normal in an <integer> base $K\geq2$ when every finite <word> of digits occurs in its <positional notation> with the same limiting <digit block frequency> as every other <word> of the same length. Write its <fractional part> as $x=0.d_1d_2\ldots$ in base $K$, using the expansion that is not eventually all $K-1$ digits when two expansions are possible. For every length $r\geq1$ and every <word> $w\in\{0,\ldots,K-1\}^r$, the requirement is
$$
\lim_{N\to\infty}\frac{\left|\{1\leq j\leq N-r+1:(d_j,\ldots,d_{j+r-1})=w\}\right|}{N-r+1}=K^{-r}.
$$
Occurrences overlap: the <word> $11$ occurs twice in $111$. There are $K^r$ possible <words> of length $r$, so $K^{-r}$ is the equal share for each. In a <decimal representation>, this means frequency $1/10$ for each digit, $1/100$ for each pair, and $1/1000$ for each triple, continuing through every finite length. The definition agrees with that used by Copeland and Erdős in their work on explicit <normal numbers>. https://users.renyi.hu/~p_erdos/1946-01.pdf[Arthur H. Copeland and Paul Erdős, Note on Normal Numbers, 1946, opening definition]{ref}
The appeal is that a precise, deterministic <real number> can have the same limiting <digit block frequencies> as repeated draws from a <uniform distribution on a finite set>, with <independence> between draws. <Champernowne constant> is an explicit example. Normality is useful for studying digit distribution, but balanced digits alone establish only a <simply normal number>, and a finite sample cannot establish any of these infinite <limits>.
Émile Borel introduced <normal numbers> and established their prevalence in his 1909 paper applying <probability> to <number theory>; the stronger all-base statement is discussed under <absolutely normal number>. https://doi.org/10.1007/BF03019651[Émile Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rendiconti del Circolo Matematico di Palermo 27, 247–271, 1909]{ref}
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