Set generated by irrational number in quotient group $\mathbb{R}/\mathbb{Z}$

Question

I'm thinking of the set $S_r = \{nr \mod 1|n\in \mathbb{Z^+}\}$ for a fixed irrational number $r\in[0,1)$.

I just found that from an algebraic perspective, $S_r$ is equivalent to the set generated by $r$ in $\mathbb{R}/\mathbb{Z}$ (or may thinking about unit circle group).

There are many good questions about the set of rational numbers in [0,1) (even it forms a group), but I cannot find any good properties about irrational numbers for $S_r$.

I noticed the following facts:

1. $S_r$ is dense.
2. $q \notin S_r$ where $q$ is a rational number.
3. If r is an algebraic number, every transcendental number in $[0,1)$ is not in $S_r$.
4. If r is a transcendental number, every algebraic number in $[0,1)$ is not in $S_r$.

But still cannot cover all the numbers. Is there a named set or known keywords about $S_r$?