Bad approximation by fractions with quadratic denominators

Question

Are there irrational numbers $\alpha$ satisfaying that

$$ \left|\alpha-\frac{m}{k^2}\right| \geq \frac{\varepsilon}{k^2} \text{ for some } \varepsilon>0 \text{ and all } k\in \mathbb{N}_{\geq 1},m \in \mathbb{Z}?$$

Answer 1

I have found a suitable reference. The answer to my question is "No", see here.