  We study the problem of finding a function $f$ with ``small support'' that
simultaneously tiles with finitely many lattices $\Lambda_1, \ldots, \Lambda_N$
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.
