# Characteristics of numbers that satisfy inequality related to diophantine approximation with perfect squares

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.

• Do you know any examples? I think such number does not exist. Apr 19 at 19:43
• @DenisShatrov thank you for your comment. I do not actually know Apr 19 at 19:44
• the usual thing is $m^2 - x n^2$ rather than the $m - x n^2$ that you typed Apr 19 at 19:49

Indeed, Weyl's equidsitribution theorem guarantees that for $$x$$ irrational, the $$xn^2$$ mod $$1$$ are equidistributed in $$\mathbb R/\mathbb Z$$ and in particular can be arbitrarily close to $$0$$, which makes $$xn^2$$ arbitrarily close to an integer and thus allows us to make $$m - xn^2$$ arbitrarily small.
On the other hand for rational $$x$$ we can choose $$n$$ so that $$xn^2$$ is equal to an integer by clearing denominators.