Multi - Lattice Tiles

Let Λ0, . . . , Λn ⊂ R be a finite collection of lattices of the same volume. When does there exist a set Ω ⊂ R that is a fundamental domain for all Λi, i = 0, . . . , n? The main result of this paper is that when the (group-theoretic) sum Λ0 + · · · + Λn of the dual lattices is direct (this means that the equation x0 + · · · + xn = 0, with xi ∈ Λi , has no nontrivial solution), then a Borel measurableΩ ⊂ R exists which is almost a fundamental domain for all the Λi; that is, it covers almost all cosets mod each Λi exactly once. The set Ω, which is generally unbounded, can also be viewed as a “tile” which tiles R by translation with any one of the lattices Λ0, . . . , Λn.