Functions tiling simultaneously with two arithmetic progressions

We consider measurable functions (Formula presented.) on (Formula presented.) that tile simultaneously by two arithmetic progressions (Formula presented.) and (Formula presented.) at respective tiling levels (Formula presented.) and (Formula presented.). We are interested in two main questions: what are the possible values of the tiling levels (Formula presented.), and what is the least possible measure of the support of (Formula presented.) ? We obtain sharp results which show that the answers depend on arithmetic properties of (Formula presented.) and (Formula presented.), and in particular, on whether the numbers (Formula presented.) are rationally independent or not.