TITLE: Square-free diophantine approximation
QUESTION [9 upvotes]: Given an irrational algebraic number $\alpha$ (and maybe I want to add: of degree greater than $2$?), do there exist infinitely many relatively prime and square-free $p$,$q$ with
$$|\alpha - p/q | < \frac{1}{q^2}\ .$$
I would guess "yes" based on a combination of Dirichlet's approximation theorem and the positive density of square-free integers among all the integers, but I can't even think of one example where I can prove this.

REPLY [12 votes]: I believe this is an open problem. Heath-Brown (1984) proved that there are infinitely many solutions to $|\alpha-p/q| < q^{-5/3+\epsilon}$ in square-free numbers. Harman (1984) proved that for almost all $\alpha$ there are infinitely many solutions to $|\alpha-p/q| < q^{-2+\epsilon}$ in square-free numbers. See paper1 and paper2.