I was recently asked this question by a student, and I don't know a nice, elegant way to solve it (actually, I'm not sure I know how to solve it at all).
Let $S(\alpha)=\lbrace \lfloor n\alpha\rfloor\ |\ n\in\mathbb{Z}^+\rbrace$. Show that $S(\sqrt{2})$ and $S(2+\sqrt{2})$ are disjoint.
I remember some tricks involved here, like replacing $\sqrt{2}$ by $1+(\sqrt{2}-1)$, and showing that $\lfloor n\sqrt{2}\rfloor=\lfloor m(2+\sqrt{2})\rfloor$ implied some inequalities on $n$, or $m$. So my question is: what is an elegant way to show this disjointness?
I should mention the student who asked me has just finished a calculus sequence (!), so I would prefer to avoid anything advanced.
Thanks.