Tiling Abelian Groups with a Single Tile

An Abelian group factorizes as the direct sum G = A \oplus B: if the B-translates of the set (tile) A (i.e., the sets b+A for b \in B), are disjoint and their union is G. Call B the tile set. In this talk we consider conditions on another set C \subset G to tile G with the same tile set B. We first answer a question of Sands regarding such tilings of G when A is a finite tile. A condition considered by Tijdeman and Sands will provide necessary and sufficient conditions for C to tile G with tile set B when the set A is infinite. Our approach comes from the ergodic theory of infinite measure preserving transformations: in particular we use the notion of exhaustive weakly wandering sets. This is joint work with S. Eigen.