TY - JOUR

T1 - The Coven–Meyerowitz tiling conditions for 3 odd prime factors

AU - Łaba, Izabella

AU - Londner, Itay

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2023/4

Y1 - 2023/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85142375061&partnerID=8YFLogxK

U2 - 10.1007/s00222-022-01169-y

DO - 10.1007/s00222-022-01169-y

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AN - SCOPUS:85142375061

SN - 0020-9910

VL - 232

SP - 365

EP - 470

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -