Erdős problem 354

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is $$\{ \lfloor \alpha\rfloor,\lfloor \gamma\alpha\rfloor,\lfloor \gamma^2\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor \gamma\beta\rfloor,\lfloor \gamma^2\beta\rfloor,\ldots\}$$ complete?