# American Institute of Mathematical Sciences

June  2006, 16(2): 361-365. doi: 10.3934/dcds.2006.16.361

## Tiling Abelian groups with a single tile

 1 Northeastern University, Department of Mathematics, Boston, MA 02115 2 University of Massachusetts Lowell, Department of Mathematics, One University Avenue, Lowell, MA 01854, United States

Received  January 2005 Revised  June 2005 Published  July 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.
Citation: S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
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