Talk:Dirichlet's approximation theorem

Hi, I'm a new member and i'm interested in the following closely related problem. Suppose you are given a real number $a$ and a bound $c>0$. What does the set of all natural (or integral) numbers $m$ satisfying $\vert m\cdot a-b\vert \leq c$ for some integer $b$ look like? Has this set any structure makes it easy to deal with it? Does anybody know if there are any good upper or lower bounds for the cardinality of the intersection of this set with a fixed interval $[i_{1},i_{2}]$? Would be great if anybody could help! Thanks in advance!