Approximating rationals with other rationals in way similar to Dirichlet theorem

Question

Suppose we are given a rational $q$. Is it possible that there infinitely many integer solutions $(h, k)$ to following system of inequalities: $0 < |q-h/k| < 1/2k^2$? I think that it is easy to see that for $q=1/2$ (and probably for any rational $a/b$ with $2|b$ and $gcd(a,b)$=1) it is impossible (correct me if I am wrong), but what about other rationals? What about changing constant in the right inequality (making it stronger or weaker)? Maybe it is very easy problem, but I can't any good approach.

Answer 1

Suppose $q = \frac{a}{b}$ is rational, and $\frac{h}{k} \neq q$. Then we have

$$\biggl\lvert \frac{a}{b} - \frac{h}{k}\biggr\rvert = \biggl\lvert \frac{ak - bh}{bk}\biggr\rvert \geqslant \frac{1}{\lvert bk\rvert}.$$

Thus, for every $c > 0$ there can be only finitely many pairs $(h,k)$ such that

$$0 < \biggl\lvert q - \frac{h}{k}\biggr\rvert < \frac{c}{k^2}.$$