The Coven-Meyerowitz tiling conditions for 3 odd prime factors

DOI10.1007/S00222-022-01169-YarXiv2106.14044MaRDI QIDQ6371320

Publication date: 26 June 2021

Abstract: It is well known that if a finite set

A s u b s e t m a t h b b Z

tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization

A o p l u s B = m a t h b b Z_{M}

of a finite cyclic group. We are interested in characterizing all finite sets

A s u b s e t m a t h b b 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_{i} p_{j} p_{k})^{2}

, where

p_{i}, p_{j}, p_{k}

are distinct odd primes. The proof also provides a classification of all such tilings.

Mathematics Subject Classification ID

Other combinatorial number theory (11B75) Finite abelian groups (20K01) Combinatorial aspects of tessellation and tiling problems (05B45)

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