We say that a number $x\in\mathbb{R}$ is $\varepsilon$-approximated by a fraction $p/q$, with $(p,q) = 1$, if $$\Big| x-\frac{p}{q}\Big| < \frac{\varepsilon}{q^2}.$$ I'm curious about the density of fraction that approximate a number, that is, about an estimate of the quantity $$d_\varepsilon(N) := \frac{1}{N}\lvert\{1\le q\le N\mid \lvert x-p/q\rvert < \varepsilon/q^2, \textrm{ for some } p\textrm{ comprime with } q\}\rvert$$ Almost every number $x\in\mathbb{R}$ is $\varepsilon$-approximated by infinitely many fractions—see Khinchine Theorem—but I don't know how many fractions approximate a number.
If $x$ is $\varepsilon$-approximated only by finitely many fractions, then $d_\varepsilon(N)\le c_{x,\varepsilon}/N$ when $N$ is large enough; e.g. if $x$ is rational. I wonder if $d_\varepsilon(N)$ can be expanded as an effective asymptotic series, say $d_\varepsilon(N) = c_{x,\varepsilon}\log N/N + \mathcal{O}(1/N^a)$ if $x$ satisfies such and such properties, and the values of $c_{x,\varepsilon}$ and $a$ are ... .