I am trying to figure out, when is $\lfloor ax \rfloor = a\lfloor x \rfloor$ for $x,a \in \mathbb{R}$, where $\lfloor x \rfloor$ is floor function, and I'm completely stuck.
Is there a general rule, for which pairs of $a$ and $x$ is the equation above true? It's obvious that $a \in \mathbb{Q}$ but I couldn't get any further than that. Clearly, for fixed $a={b\over c}$ we have at least one $x$, namely $x=c$ and every multiple of $c$. But how to find all such $a$ and $x$?
First note that if $a \lfloor x \rfloor \notin \Bbb{Z}$, then $\lfloor ax \rfloor \in \Bbb{Z}$ so it can't be equal. Now let's suppose that $a \lfloor x \rfloor \in \Bbb{Z}$.
Now let's mark $r = x \mod 1$. Then $x = \lfloor x \rfloor + r$. And we have:$$ \lfloor ax \rfloor = a \lfloor x \rfloor\\ \lfloor a\lfloor x\rfloor + ar \rfloor = a \lfloor x \rfloor $$ And because $a\lfloor x\rfloor\in\Bbb{Z}$:$$ a\lfloor x\rfloor + \lfloor ar \rfloor = a \lfloor x \rfloor\\ \lfloor ar \rfloor = 0 $$ And this holds only for $ar \in [0;1)$.