Irrational rotations are rank 2 by intervals without spacers

Question

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$.

S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. 1, 35--65. MR1436950] states (Thm. 5): Every irrational rotation is of rank at most two by intervals (without spaces). For a proof he refers to a more general result (Thm. 7) stating that An ergodic exchange of $s$ intervals is of rank at most $s$ by intervals, without spacers.

As far as I understand, it means that I can find a nested family of intervals $J_1\supset J_2\supset \ldots$ such that $J_n=F_n\cup G_n$, where $F_n=[a_n,b_n)$ and $G_n= [b_n,c_n)$ are left-closed, right-open intervals on the unit circle (interpreted here as $[0,1)$), and for each $n=1,2,\ldots$ there are positive integers $h_n^F$ and $h^G_n$ such that $$ [0,1)=\bigcup_{j=0}^{h_n^F-1}R_{\alpha}^j(F_n) \cup \bigcup_{j=0}^{h_n^G-1}R_{\alpha}^j(G_n) $$ is a disjoint union, and hence $$ \{R_{\alpha}^j(F_n):j=0,1,\ldots,h_n^F-1\} \cup \{R_{\alpha}^j(G_n):j=0,1,\ldots,h_n^G-1\} $$ is a partition refining the previous one $$ \{R_{\alpha}^j(F_{n-1}):j=0,1,\ldots,h_{n-1}^F-1\} \cup \{R_{\alpha}^j(G_{n-1}):j=0,1,\ldots,h_{n-1}^G-1\}. $$ So here are my questions:

1. Is the above interpretation correct? (Hope it is, but due to a certain vaguesness in defining ``rank by intervals'' I am not 100% sure.)
2. What can be said about $h_n^F$ and $h^G_n$ and their relation to each other as $n\to\infty$?

Answer 1

As was pointed out, the answer is related to the continued fraction expansion. If $\alpha=\frac{1}{c_1+\frac1{c_2}...}$, we fix rational approximations $\frac{p_k}{q_k}=\frac{1}{c_1+\frac1{c_2+...\frac1{c_k}}}$. $q_k+q_{k-1}$ iterated preimages ( or images) of a point decompose the unit circle in the wished intervals that form two towers of heights $q_k$ and $q_{k-1}$. For example, $T^i(0)$, $i=0..(q_k+q_{k-1}-1)$, decompose unit circle into interval

• $[T^i(0),T^{i+q_{k-1}}(0))$, $i=0..q_k-1$,
• $[T^{i+q_k}(0),T^{i}(0))$, $i=0..q_{k-1}-1$.

The orientation switches with the parity of $k$. But it is likely not important for you. So, the heights are $q_k$ and $q_{k-1}$, possibly the best sequence related with $\alpha$ you can play with. Let me recall, that $q_{k+1}=c_{k+1}q_k+q_{k-1}$. The ratio can be as close to 1 as possible, along the subsequence, or unbouded in other case, or constant e.g. for Fibonnacci.