A Periodicity Result for Tilings of ${\mathbb{Z}}^3$ by Clusters of Prime-Squared Cardinality

Abstract

We show that if ${\mathbb{Z}}^3$ can be tiled by translated copies of a set $F\subseteq {\mathbb{Z}}^3$ of cardinality the square of a prime then there is a weakly periodic $F$-tiling of ${\mathbb{Z}}^3$, that is, there is a tiling $T$ of ${\mathbb{Z}}^3$ by translates of $F$ such that $T$ can be partitioned into finitely many $1$-periodic sets.

27 pages; v3: minor expository revisions and corrections; updated the bibliography