Tiling Abelian groups with a single tile
Stanley Eigen, V. S. Prasad · Discrete and Continuous Dynamical Systems · 2006
Suppose $G$ is an infinite Abelian group that factorizes as the direct sum $G = A \oplus B$: 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 $C \subset G$ 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.