Let $x \in [0,1]$ be an irrational algebraic number, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that $$ kx \equiv y \mod 1? $$ This can be thought of a question about circle rotations from dynamics: marking a point at $0$, if we rotate the circle by an angle that is an irrational algebraic number, does there exist some iteration of the rotation where the marked point equals some target value in $\mathbb{R}/\mathbb{Z}$?
For example, when $x = \sqrt{2}/4$, and $y = \sqrt{2}/2$, we can find that there exists a $k=2$ solving the problem. But it is unclear if this is the case in general.