# Density of the set of numbers that are "good approximators" to a given real in the sense of Dirichlet's approximation theorem

Let $$\mathbb{N}$$ be the set of positive integers. Given a set $$A\subseteq \mathbb{N}$$ we let the (upper) density of $$A$$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$

If $$\alpha\in\mathbb{R}$$, we say $$q\in\mathbb{N}$$ is good for approximating $$\alpha$$ if there is $$p\in\mathbb{Z}$$ such that $$|\alpha - \frac{p}{q}|< \frac{1}{q^2},$$

and denote the set of those positive integers by $$G_\alpha$$. The approximation theorem of Dirichlet states that $$G_\alpha$$ is infinite for any $$\alpha\in\mathbb{R}$$.

Question. Given $$\delta\in[0,1]$$, is there $$\alpha\in\mathbb{R}$$ such that $$\mu^+(G_\alpha) = \delta$$?

If $$\alpha$$ is irrational, then the sequence $$\alpha,2\alpha,\ldots$$ is equidistributed modulo 1 (Weyl's theorem). Thus the inequality $$|q\alpha-p|<1/q$$ holds for $$q$$ of density $$0$$. If $$\alpha=a/b$$ ($$a,b$$ are coprime) is rational, then the density of your numbers equals $$1/b$$.