According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ for any initial value $u_0=x$.

This observation is just the motivation of the following question. I don't need to recall the sequence $(u_n)$ in order to state the question.

Because of this observation, one could expect that for almost all $x$,

$$
\lim_{N \to \infty} \sum_{k=0}^{\infty} a_{N,k} {\boldsymbol 1}_{T^kx \in I} = |I|
$$
where $T$ is an irrational rotation, and $a_{N,k} = \dfrac{\#\{i | 0 \leq i \leq N-1, K_i = k\}}{N}$
where $K_N=\sum_{i = 0}^{N-1} \epsilon_i2^i$ and ${(\epsilon_n)}_{n \geq 0}$ is a sequence of independent $0$-$1$ symmetric Bernoulli random variables (thus $a_{N,k}=0$ for $k > 2^N-1$ and $\sum_{k=0}^{2^N-1} a_{N,k}=1)$.

Observe that the $a_{N,k}$ are random, so I should also say that the above limit holds for almost all sequences ${(\epsilon_n)}_{n \geq 0}$.

Thus the expectation is that the following limit holds for almost every realisation of the $a_{N,k}$: $$ \sum_{k=0}^{\infty} a_{N,k} f(T^k x) \overset{a.s.}{\to} E(f) $$ for every $f \in L^1$. Is it true ?

I suspect this is true when $T$ has discrete spectrum and $T^{2^n}$ is ergodic for every $n \geq 0$ (*i.e.* the product of $T$ with the dyadic odometer is ergodic). By the way my simulations yield the same observation when $T$ is an ergodic product of two rotations.

Let me rephrase the problem in terms of an algorithm. The input is a trajectory `y(0), y(1), ...`

of an ergodic stationary process and the output is `x(0), x(1), ...`

:

```
SET x(0)=y(0)
SET i=0
REPEAT
SET k=2^i
SIMULATE epsilon = 0 or 1
IF epsilon=0 SET x(i+1)=x(i) ELSE SET x(i+1)=y(k)
SET i=i+1
```

Then the claim is that the empirical distribution given by `x(0), x(1), ...`

is the invariant measure of the stationary process.