$\alpha$ is an irrational number. Is $\liminf_{n\rightarrow\infty}n\{ n\alpha\}$ always positive?

Question

We know that $$\liminf_{n\rightarrow\infty}\ n\{ n\alpha\}=L\left(\alpha\right)$$ exists and $L\left(\alpha\right)\geq 0$.
Then, is the limit always greater than $0,\ \forall\alpha\in\mathbb{R}\setminus\mathbb{Q}$?
$\{n\alpha\}$ is the fractional part of $n\alpha$, i.e. $n\alpha-\left[n\alpha\right]$.
I know that there are infinitely many rational numbers $\frac{p}{q}$ satisfies $\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}$, but if $\frac{1}{q^2}$ is modified as $\frac{C}{q^k}$($C$ is a given constant), does there still exist infinitely many such rational numbers?
Moreover, what about the special cases $\alpha=\sqrt{2}$ or $\alpha=2^\frac{1}{4}$?