This question came up when thinking about an [older question](https://mathoverflow.net/questions/348541/density-of-the-set-of-numbers-that-are-good-approximators-to-a-given-real-in-t) 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](https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) 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 $\alpha\in \mathbb{Z}$?



On the set of good approximators in the sense of Dirichlet's theorem
The answer is yes. For an irrational number $\alpha$, the inequality says that the fractional part $\{q\alpha\}$ lies in $(0,1/q)\cup(1-1/q,1)$. In particular, $\{q\alpha\}\not\in[1/3,2/3]$ when $q\geq 3$. However, it is easy to show by Dirichlet's approximation theorem that the fractional parts $\{q\alpha\}$ are dense in $(0,1)$, and even uniformly distributed by Weyl's theorem. Hence $G\_\alpha=\mathbb{N}$ implies that $\alpha$ is a rational number, say $\alpha=r/s$ with $s\in\mathbb{N}$ coprime to $r\in\mathbb{Z}$. Then, for every $q\in\mathbb{N}$, there is $p\in\mathbb{Z}$ such that
$$\left|\frac{r}{s}-\frac{p}{q}\right|<\frac{1}{q^2},\qquad\text{that is}\qquad |rq-ps|<\frac{s}{q}.$$
For $q>s$ this forces $rq-ps=0$, hence also $s\mid q$. In particular, taking $q=s+1$, we conclude that $s\mid s+1$, hence $s=1$. So $\alpha=r$ is an integer.


**P.S.** After writing the above answer, I realize that the key step is already present in Fedor Petrov's response to the [OP's earlier question](https://mathoverflow.net/questions/348541/density-of-the-set-of-numbers-that-are-good-approximators-to-a-given-real-in-t).


