Let $\alpha$, $\beta$ be two irrational numbers.
Is there a good way to find some integer $n,m$ that $|n\alpha-\beta-m|$ is sufficiently small?
For example, if $\beta=0$, we know that there exists $n,m$ that $|n\alpha-m|<\frac{1}{n}$ and this can be easily found by using continued fractions of $\alpha$.
Since $n\alpha$'s are equally distributed, I'm pretty sure that for 'most' $\alpha,\beta$ there exist a $C_{\alpha,\beta}$ such that there exist infinitely many $n,m$ satisfying $|n\alpha-\beta-m|<\frac{C_{\alpha,\beta}}{n}$
Is there a good way to prove existence of such $C$ or some good heuristic algorithm on obtaining $n,m$?
