TILING ABELIAN GROUPS WITH A SINGLE TILE

Abstract. Suppose G is an infinite Abelian group that factorizes as the direct sum G = A⊕B: i.e., the B-translates of the single tile A evenly tile the group G (B is called the tile set). In this note, we consider conditions for another set C ⊂ G to tile G with the same tile set B. In an earlier paper, we answered a question of Sands regarding such tilings of G when A is a finite tile. We now consider extensions of Sands’s question when A is infinite. We offer two approaches to this question. The first approach involves a combinatorial condition used by Tijdeman and Sands. This condition completely characterizes when a set C can tile G with the tile set B; the condition is applied to simplify the proofs and extend some of Sands’s results [8]. The second approach is measure theoretic and follows Eigen, Hajian, and Ito’s work on exhaustive weakly wandering sets for ergodic infinite measure preserving transformations. 1. Introduction. Two subsets A and B of an Abelian group G induce a tiling (or factorization) of G if every element of G has a unique representation as a sum a+b, with a ∈ A and b ∈ B; thus we write G = A ⊕ B. Note that this is the same a