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

TL;DR

This paper proves that tilings of three-dimensional integer lattices with clusters of prime-squared size exhibit weak periodicity, meaning the tiling pattern repeats in a structured, predictable manner.

Contribution

It establishes a new periodicity result for tilings of $\\mathbb Z^3$ by sets of prime-squared size, extending understanding of tiling symmetries.

Findings

Existence of weakly periodic tilings for prime-squared sized clusters

Tilings can be partitioned into finitely many 1-periodic sets

Results apply specifically to $\\mathbb Z^3$ with prime-squared cardinality sets

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.