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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.

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

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Such numbers do not exist, fulfilling an expectation of Denis Shatrov.

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.

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