This question has come up in my algorithms and physics research. I apologize if this is very basic, but I am new to number theory and it seems this is a number-theoretic question. What can we say about positive numbers $x >0$ for which there exists a constant $C(x)>0$ depending on $x$, such that the inequality $$ \left\lvert m - x n^2 \right\rvert \geq C $$ is satisfied for all natural numbers $m, n \in \mathbb{N}$? What can we say about the set of these numbers $x$? This seems related to definitions of diophantine numbers, but I am a complete beginner in number theory and thus could not find any answers. Can we say something about the cardinality or measure of the set of such numbers $x$? Any and all help would be appreciated.