Equi-distributed sequences for a doubling map?

Question

Let us consider the doubling map $\tau:X:=[0,1)\to X:=[0,1)$ defined by $$\tau(x):= 2x-\lfloor 2x \rfloor=2x\,\text{mod}\,1 $$ It is well-known that $\tau$ is ergodic and hence for any continuous function $f$ (integrable is enough), the limit $$ \lim_{n\to \infty}\frac{\sum_{k=0}^{n-1} f(\tau^k(x))}{n}=\int_X f(x) dx $$ is valid for a.e. $x$. My question, what are the points that make this limit valid? That is, given $x_0\in [0,1)$, let us define $$x_n:=\tau^n(x_0) \quad \forall\,n\geq 1. $$ When the seqeuence $\{x_n\}$ is equi-distributed? Or, when the following identity is ture $$ \lim_{n\to \infty}\frac{\sum_{k=0}^{n-1} f(x_k)}{n}=\int_X f(x) dx\quad \forall \,f\in C(X). $$ Is $x_0$ irrational enough?

Answer 1

The requirement is that $x_0$ is normal in base $2$. Irrationality is not sufficient.