What is the frequency of $w$ appearing in the binary expansion of almost every $x \in (0,1)\setminus \{ \frac p{2^n}: 0<p<2^n; n\geq 1\}$

Question

Fix a word $w$ of length $k$, $w \in \{0,1\}^k$. What is the frequency of $w$ appearing in the binary expansion of almost every $x \in (0,1)\setminus \{ \frac p{2^n}: 0<p<2^n; n\geq 1\}$

My attempt:

I was thinking to apply doubling map. Here $w=w_1....w_k$ and taking $f=\chi_{( \frac{w_1}2+....+ \frac{w_k}{2^k}, \frac{w_1}2+...+ \frac{w_k}{2^k}+ \frac1{2^k})}$

Then applying Birkhoff ergodic theorem but I am not getting the desired result. I am bit confused.

Answer 1

Here you are almost done,

from Birkhoff ergodic theorem you know that for a.e $x \in (0,1)$

$ \frac 1n \sum_{j=0}^\infty f(T^jx) \to \int_X f d\mu= \frac1 {2^k}=\frac 1n \{$ the number of times $w$ appeears in $a_1,..,a_n\}$