Functions Tiling Simultaneously with Two Arithmetic Progressions

We consider measurable functions f$f$ on R$\mathbb {R}$ that tile simultaneously by two arithmetic progressions alpha Z$\alpha \mathbb {Z}$ and beta Z$\beta \mathbb {Z}$ at respective tiling levels p$p$ and q$q$. We are interested in two main questions: what are the possible values of the tiling levels p,q$p,q$, and what is the least possible measure of the support of f$f$? We obtain sharp results which show that the answers depend on arithmetic properties of alpha,beta$\alpha , \beta$ and p,q$p,q$, and in particular, on whether the numbers alpha,beta$\alpha , \beta$ are rationally independent or not.