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.