The Coven-Meyerowitz tiling conditions for 3 prime factors: the even case

TL;DR

This paper proves that a specific tiling characterization by Coven and Meyerowitz applies to all integer tilings with period equal to the square of the product of three distinct primes, extending previous results to the even case.

Contribution

It extends the Coven-Meyerowitz tiling characterization to the case where the period is the square of a product of three distinct primes, including even cases, and improves previous proofs.

Findings

Proved the Coven-Meyerowitz conjecture (T2) for the even case with three prime factors.

Extended previous results from odd to even periods of the form (p_i p_j p_k)^2.

Split the original lengthy paper into two parts for clarity and further results.

Abstract

We consider finite sets $A\subset\mathbb{Z}$ tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. Building on por previous work, we prove that a tentative characterization of finite tiles proposed by Coven and Meyerowitz holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct primes. This extends the main result of [15] (Invent. Math. 2023), where we assumed that $M$ is odd. We also improve parts of the argument from [15]. We have split the earlier (70-page) version into two papers. The current version (49 pages) is the first of the two. The main result is the same as in the previous version: we prove (T2) in the 3-prime even case. The second paper will be posted shortly as a new submission. It will have a new main result where we prove (T2) for a new class of tilings (proved very recently, not included in v1 of this paper). Splitting-related results from the earlier 70-page version of this paper have been moved there.