This question asked for bounds greater than $\omicron\left(n\right)$ on the error $$ E_n=|T\cap\{1,2,\ldots\}|-\ell n. $$ where $$ \ell=\lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n} \qquad\text{ and }\qquad T=\{k: k\alpha-\lfloor k\alpha\rfloor \in I\}$$ and $\alpha$ is an arbitrary irrational number. The question was answered in negative for such an arbitrary $\alpha$, but in positive under the additional hypothesis that the number has irrationality measure equals to $2$.
I would be very interested in knowing if there are similar results for numbers with irrationality measure $\le m$ where $m$ is greater than $2$.