Tiling and weak tiling in $({\mathbb{Z}}_p{)}^d$

Abstract

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $({\mathbb{Z}}_p{)}^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1,2$, and prove some partial results for $d=3$.