Effective equidistribution

I have an irrational number $\alpha$ (in this case, $\alpha=1/(2\pi)$, but hopefully answers will be more general) and I am interested in finding bounds on the size of $$ T=\{k: k\alpha-\lfloor k\alpha\rfloor \in I\} $$

for some interval $I\in[0,1)$. (Actually, I think it would be more natural to take intervals of $S^1$, or finite unions of same, but this is sufficient for my purposes.)

By the equidistribution theorem we know that $$ \lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n}=\ell $$ where $\ell$ is the length of the interval $I$. But I would like to know more about the error term $$ E_n=|T\cap\{1,2,\ldots\}|-\ell n. $$

It is not possible to put a bound better than $o(n)$ for arbitrary irrational $\alpha$. More precisely, if $f\colon\mathbb{N}\to\mathbb{R}^+$ is any function with $\liminf_{n\to\infty}f(n)/n=0$ then there exists uncountably many irrational $\alpha$ with $\limsup_{n\to\infty}\lvert E_n\rvert/f(n)=\infty$. See my answer on mathoverflow.

However, for specific $\alpha$ it is possible to do better. As shown in the answer linked above, we have $E_n=o(n^x)$ for any $x > 1/2$ so long as $\alpha$ has irrationality measure 2, and almost every real number has irrationality measure 2. It is an unsolved problem as to whether $\pi$ (and, equivalently, $1/(2\pi)$) has irrationality measure 2, but it is expected that it does.