Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.)
Is there any known characterization for the pairs of positive irrationals $\alpha$ and $\beta$ for which $S(\alpha)\cap S(\beta)=\varnothing$?
Not a complete answer: these sequences are called Beatty sequences. It's a nice exercise to show that $S(\alpha)$ and $S(\beta)$ partition the positive integers when $\frac{1}{\alpha} + \frac{1}{\beta} = 1$, hence for $\alpha, \beta$ satisfying this condition we can take $S(n \alpha), S(m \beta)$ where $n, m \in \mathbb{N}$.
You are right about $S(\alpha)\cap S(\beta)=\emptyset\iff \dfrac{n}{\alpha}+\dfrac{m}{\beta}=1$ for some $n,m\in\mathbb Z^+$ (see Theorem 8 of the cited paper).
The implication $\Longleftarrow$ is easy so let's assume that $\frac{n}{\alpha}+\frac{m}{\beta}\neq 1$ for all $n,m\in\mathbb Z^+$. Then one of the following is true.
It is enough to show that any one of items 1, 2 or 3 implies that $S(\alpha)\cap S(\beta)\neq\emptyset$. Elementary proofs of these can be found in this paper (Theorem 5, Theorem 6 and Theorem 7 respectively).
I suggest looking in Joe Roberts amazing book I thought I remembered a partition into 3 such sequences but actually it is $\lfloor \tau \lfloor \tau n \rfloor \rfloor$,$\lfloor \tau \lfloor \tau^2 n \rfloor \rfloor$,$\lfloor \tau^2 n \rfloor $ So not exactly what was requested. Maybe that is not so deep.
I put in this answer partly to get the link to the book in.
