TITLE: Numbers with disjoint sets of multiples of integer parts
QUESTION [9 upvotes]: 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 $S(\alpha)\cap S(\beta)=\varnothing$?

REPLY [6 votes]: 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.

The numbers $1,\dfrac{1}{\alpha},\dfrac{1}{\beta}$ are linearly independent over $\mathbb Q$,
there exist some $n,m,k\in\mathbb Z^+$ such that $\left|\dfrac{n}{\alpha}-\dfrac{m}{\beta}\right|=k$,
there exist some $n,m,k\in\mathbb Z^+$ such that $\dfrac{n}{\alpha}+\dfrac{m}{\beta}=k$ with $k>1$, and $\gcd(n,m,k)=1$.

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).