Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$