Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or
closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is
irrational. Consider in $\mathbb{S}^{1}$ the sequence $z\_{n}=e^{in\alpha}$.
Then this sequence is dense in $\mathbb{S}^{1}$ by Kronecker's Theorem or by
ergodicity. Let's associate with the arc $J$ its "indicator sequence"
$s(J)={s\_{n}\}$ of zeroes and ones defined as follows:


$s\_{n}=1$ if $z\_{n}\in J$ and $s\_{n}=0$ if $z\_{n}\notin J$.


So, we get something like 
0 0 1 1 1 0 0 0 1 1 0 0 1. . .
Suppose that we are given such a sequence $s(J)$ for some $J$ and some
$\alpha$. By the Ergodic Theorem one gets the measure of arc $J$ as the limit


$\mathtt{meas}(J)=2\pi\underset{n\rightarrow\infty}{\lim}\frac{\sigma\_{n}}{n}$
where $\sigma\_{n}$ is the number of 1's in ${s\_{1},s\_{2},...,s\_{n}}$.


OK, but is it possible to detect the "frequency" $\alpha$ only by the 0-1 data
contained in the sequence $s\_{n}$? More precisely, my question is:


Let $\{s\_{n}\}$ be a sequence of 0's and 1's and we know that it is an
"indicator sequence" for some arc $J\subset\mathbb{S}^{1}$ and some angle
$\alpha$. Is it then possible to get $\alpha$ by some formula similar to the
above one for the measure of $J$? This would be something like a "rotation
number" of sequence $\{s\_{n}\}$.


Similar question may be posed for the torus $\mathbb{T}^{n\text{ }}$and an
open set $J\subset\mathbb{T}^{n\text{ }}$. Then we should detect not only the
frequencies $\alpha\_{1},\alpha\_{2},...$ but also the "dimension" $n$ of the
sequence. Here $\alpha\_{1},...,\alpha\_{n},\pi$ have to be independent over
$\mathbb{Z}$.


[I know that the "indicator sequence" is a standard construction in symbolic
dynamics, but I am not very involved in the topic, so references are welcome.]


P.S. Curly brackets {} are not displayed in math mode. How to fix the problem?



How to detect frequency?
It seems likely to me that $\alpha$ can be computed by calculating the frequencies of subwords of the coding sequence, but in a manner which depends on certain parameters. For example, if $\alpha<\min\{|J|,2\pi-|J|\}$ then the interval $J \setminus J +\alpha$ has length precisely $\alpha$, and it follows easily that $\alpha$ equals the frequency of the subword 01. On the other hand if $|J|$ is very small and $\alpha, 2\pi-\alpha$ are both larger than $|J|$, then the frequency of the subwords 01 and 10 are both $|J|$, while the subword 00 has frequency $1-2|J|$, and we cannot gain anything by considering words of length 1 or 2. So the frequencies of words of arbitrary length probably need to be considered.


The articles "Coding rotations on intervals" by Berstel and Vuillon, and "Three-distance theorems and combinatorics on words" by Alessandri and Berthé appear to be relevant (especially Lemma 1 in the latter) but do not seem to yield a complete answer.


