The Coven–Meyerowitz tiling conditions for 3 odd prime factors

Izabella Łaba*, Itay Londner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


It is well known that if a finite set A⊂ Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A⊕ B= ZM of a finite cyclic group. We are interested in characterizing all finite sets A⊂ Z that have this property. Coven and Meyerowitz (J Algebra 212:161–174, 1999) 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=(pipjpk)2, where pi, pj, pk are distinct odd primes. The proof also provides a classification of all such tilings.

Original languageEnglish
Pages (from-to)365-470
Number of pages106
JournalInventiones Mathematicae
Issue number1
StatePublished - Apr 2023
Externally publishedYes


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