This question came up when thinking about an older question that hasn't been answered as of now.

Let $\mathbb{N}$ be the set of positive integers. 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 $$\left|\alpha - \frac{p}{q}\right|< \frac{1}{q^2},$$

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

Clearly, if $\alpha\in\mathbb{Z}$, the set $G_\alpha$ is as large as it can get, that is $G_\alpha = \mathbb{N}$.

**Question.** Does $G_\alpha = \mathbb{N}$ imply $\alpha\in \mathbb{Z}$?