A density question

Question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?

Answer 1

Yes. In fact, for any irrational number $\alpha$ and and real number $\beta$, the set $\{\alpha n+\beta \bmod 1\colon n\in\Bbb N\}$ is dense in $[0,1]$. (A proof follows from showing that every interval of the form $[0,\varepsilon)$ contains some multiple of $\alpha\bmod1$.) The answer to your question follows from taking $\alpha=2\theta$ and $\beta=\theta$.