Irrational rotation $\{n\alpha (\mod 1)\}$ is dense in $S^1$,
so every point $x\in S^1$ can be approximated by a subsequence. However,
the question is: how large is the set of points in $S^1$ which can be
approximated by $\{n_i \alpha\} (\mod 1)$ with a given approximation
speed, that is, the set
$$
E_\phi = \{x\in S^1; x\in (n\alpha - \phi(n), n\alpha + \phi(n)) (\mod
1) i. o. \}.
$$
One usually considers nonincreasing $\phi$. This question was considered
by many mathematicians, in particular Khintchine, Bernik and Dodson,
Bugeaud, Schmeling and Troubetskoy, Fan and Wu, Xu. I also worked on
this problem (with Lingmin Liao). In this talk I plan to give the final
answer. This is a joint work with Bao-wei Wang.