Consider the Fibonacci words $B\_n$:


* $B\_1 = 1$
* $B\_2 = 10$
* $B\_3 = 101$
* $B\_4 = 10110$
* $B\_5 = 10110101$


(start with $B\_1=1$, and go from $B\_n$ to $B\_{n+1}$ by replacing every occurence of $1$ in $B\_n$ with $10$ and every occurence of $0$ with $1$).


The word $B\_n$ has $f\_n$ digits, the $n$-th Fibonacci number. 


For a given integer $\alpha \geq 2$ and $0 \leq j < \alpha$, denote by 
$$
R\_n(\alpha,j) = \#\bigl\{1 \leq i \leq f\_n \mid B\_n(i)=1 \,\text{ and }\, i\!\!\!\!\pmod\alpha=j \bigr\}
$$
the number of integers $i$ in $\{1, \ldots, f\_n\}$ such that $i$ is a "rabbit integer" (I mean $B\_n(i)=1$) and $i$ is congruent to $j$ modulo $\alpha$. 


Does one know an asymptotic estimate of $R\_n(\alpha,j)$ when $n \to \infty$ ? 



Estimate of the number of rabbit integers with a given congruence
You are asking about subsequences of the infinite Fibonacci word [A005614](https://oeis.org/A005614). 


One way to describe this sequence is the indices where $\lfloor (n+1) \phi \rfloor \gt \lfloor n \phi \rfloor$, or equivalently when $n \phi \mod 1$ is in a particular interval of length $1/\phi$, namely $[1-1/\phi,1)$. Your subsequences with indices in an arithmetic progression correspond to when $(\alpha n + j) \phi = n (\alpha \phi) + j \phi$ is in that interval mod $1$, or equivalently when $n(\alpha \phi)$ is in a translated interval. For any positive integer $\alpha,$ $\alpha \phi$ is irrational, so the multiples are equidistributed and they hit the interval with density $1/\phi$. 


This tells you that $$\lim\_{n\to \infty} \frac{R\_n(\alpha,j)}{|B\_n|} = \frac{1}{\alpha \phi}$$ but it might be possible to get good estimates on $|B\_n| - \alpha \phi R\_n(\alpha,j)$.


