I read from the book "Which Way did the Bicycle Go" that it is unknown whether for every $c>0$ there are infinitely many integers $n$ such that $|n\sin n|<c.$
Let $\mathbb{Q}_{m}$ be the set of rationals where the absolute value of the denominator is at most $m$ when the rational number has been written in its simplest form.
What can we say about the following weakened problem?
For every $c>0$ there is an integer $m$ and infinitely many rationals $q\in\mathbb{Q}_m$ such that $|q\sin q|<c.$
Has this been solved for some fixed $m?$