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 $n\to\infty$?



Irrational rotations are rank 2 by intervals without spacers
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.


