Pompeiu's problem on discrete space

Let [unk] be a finite family of finite subsets of the n-dimensional lattice Zⁿ, and let τ denote the group of all translations of Zⁿ. We shall here consider the Pompeiu problem for the family [unk]—namely, to determine when the only function f:Zⁿ → C such that [Formula: see text] is the zero function.