The Coven-Meyerowitz tiling conditions for 3 odd prime factors

TL;DR

This paper proves that the Coven-Meyerowitz tiling condition (T2) applies to all integer tilings with period equal to the square of the product of three distinct odd primes, and classifies these tilings.

Contribution

It extends the validity of the (T2) tiling condition to new cases involving three odd prime factors and provides a classification of these tilings.

Findings

(T2) holds for tilings with period (p_ip_jp_k)^2

Classification of all such tilings is provided

Supports conjecture that (T2) applies broadly to tilings with multiple prime factors

Abstract

It is well known that if a finite set $A\subset\mathbb{Z}$ tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. We are interested in characterizing all finite sets $A\subset\mathbb{Z}$ that have this property. Coven and Meyerowitz (1998) proposed conditions (T1), (T2) that are sufficient for $A$ to tile, and necessary when the cardinality of $A$ has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of $|A|$. We prove that the Coven-Meyerowitz tiling condition (T2) holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct odd primes. The proof also provides a classification of all such tilings.