Functions tiling with several lattices

We study the problem of finding a function $f$ with ``small support’’ that simultaneously tiles with finitely many lattices ${\mathrm{\Lambda }}_1,\dots ,{\mathrm{\Lambda }}_N$ in $d$-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.