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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 $\frac{1}{\alpha} + \frac{1}{\beta} = 1$, so the problem reduces to determining when $S(\alpha) \subseteq S(\beta)$. Certainly we can take $\alpha = n \beta$, and plausibly this is all we can do.