BY-MO 2004 Problem 5

Suppose that $A$ and $B$ are sets of real numbers such that $$A\subset B+\alpha \mathbb{Z} \text{and} B\subset A+\alpha\mathbb{Z} \text{for all} \alpha>0$$ (where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$ (a) Does it follow that $A=B$ (b) The same question, with the assumption that $B$ is bounded