Take the 2-minute tour ×  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_{