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:

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