Normality and equidistribution under integer multiplication ($x\text{ normal in base }K⟺(\{K^{n}x\}{)}_{n\ge 0}\text{ equidistributed}$)

A real number $x$ is a normal number in base $K$ exactly when its iterates under the integer multiplication map on the circle, $T_K(x)=\{Kx\}$, form an equidistributed sequence. Braces denote the fractional part. Multiplication by $K$ shifts the positional notation one digit to the left:A length-$r$ word with base-$K$ integer value $a$ specifies the half-open interval $[a/K^r,(a+1)/K^r)$, whose length is $K^{-r}$. Visiting that interval is exactly encountering that word at the next digit position. These digit intervals form arbitrarily fine grids, so their correct limiting visiting frequencies imply equidistribution on every interval. This converts a question about digits into one about a dynamical system. Bailey and Crandall develop this connection using equidistribution of the fractional parts of $K^{n}x$. ^[ref]

In base ten the word $12$ corresponds to $[0.12,0.13)$. For Champernowne constant, counting the visits $T_{10}^n(x)$ for $0\le n\le 99998$ gives precisely the $2088$ occurrences of $12$ in its first $100000$ digits. For $x=1/3$, the base-two integer multiplication map on the circle instead alternates between $1/3$ and $2/3$; it never visits $[0,1/4)$, the interval for $00$. This two-point orbit is not an equidistributed sequence.