Special Dynamical Systems Seminar
Vidhu Prasad
University of Massachusetts at Lowell
Tiling Abelian Groups with a Single Tile
An Abelian group factorizes as the direct sum G = A \oplus B: if the
Btranslates 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.
