Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1}  \sim \ell_n/3$. There's also a word $B_\infty$ since $B_{n+1}$ is an extension of $B_n$. 

While dealing with my own research, I discovered the following conjecture. Let $S \subset \mathbb{N}$ be $S = j\mathbb{Z} + k$ (density $d(S)=1/j$) or $S=\bigl\{n \mid \{n\gamma\} \in [a,b)\bigr\}$ (density $d(S)=b-a$, where $0 \leq a < b \leq 1$ and $\gamma \in (0,1)$ is irrational). Then:
$$
\frac{\#\{i \mid 1 \leq i \leq N, i \in S, B_\infty(i)=1 \}}{N} \underset{N \to \infty}{\longrightarrow} \frac{1}{3}d(S).
$$
Do you know a proof of this result ? 

### Update 

Let me develop how I got this conjecture. According to Theorem 1 of [this paper by Hanson & Pledger](http://link.springer.com/article/10.1007%2FBF00537020), the conjecture is equivalent to 
$$
(\ast)\colon \quad \frac{1}{\ell_n} \sum_{\substack{k=0 \\ B_n(k+1)=1}}^{\ell_n-1} f(T^k\cdot) \overset{L^2}{\longrightarrow} \frac{1}{3}E(f \mid {\cal I}) 
$$
for every invertible *mpt* $T$, every $f \in L^2$, and where ${\cal I}$ is the $T$-invariant $\sigma$-field. 

In fact, I have almost proved $(\ast)$ (I have only proved it for ergodic $T$ - I would have entirely proved it if I knew that the answer to  [to this question](http://mathoverflow.net/questions/233201/invariant-sigma-field-of-a-product-with-a-weakly-mixing-transformation) were yes). 

Note that $(\ast)$ is very close to an application of the ergodic theorem. 
By the ergodic theorem applied to $T \times C$ where $C$ is the Chacon map and $T$ is ergodic (hence $T \times C$ is always ergodic), we have 
$$
(\#)\colon\quad \frac{1}{\ell_n} \sum_{k=0}^{\ell_n-1} f(T^kx){\boldsymbol 1}_{C^k u \in [2/3,1[} \longrightarrow \frac{1}{3}E(f) \quad \text{for almost all $(x,u)$},
$$
and for $u=0$ one has 
$$
\frac{1}{\ell_n} \sum_{k=0}^{\ell_n-1} f(T^kx){\boldsymbol 1}_{C^k u \in [2/3,1[} = 
\frac{1}{\ell_n} \sum_{\substack{k=0 \\ B_n(k+1)=1}}^{\ell_n-1} f(T^kx).
$$
But I don't know if something allows us to apply $(\#)$ for $u=0$. Anyway I wonder whether there is a proof of the conjecture without using Hanson & Pledger's theorem.