Let $x$ be an irrational number. Is the set $\ \{nx \mod 1: n \in \mathbb{N} \}\ $ dense in $[0,1]\ ?$
Obviously if $x$ were rational, the set would be finite and therefore would have no limit points, so my question follows naturally.
Certainly, the set must have at least one limit point in $[0,1],\ $ e.g. see my answer here.
If the answer is no, how do we find the set of limit points in $[0,1]\ ?$
I had a look at this question, but I'm not sure it's the same question and I'm not very knowledgeable on continued fractions.
