Coven-Meyerowitz T2 necessity through coprime stripe collapse
Abstract
We prove that every finite subset of the integers which tiles by translations satisfies the Coven-Meyerowitz condition T2, with no restriction on the number of prime factors or their exponents. Together with the necessity of T1 and the sufficiency of T1 and T2 proved by Coven and Meyerowitz, this gives their proposed characterization of finite integer tiles. The proof uses strong induction on a cyclic tiling period. Character identities produce periodic Boolean product stripes; integral descent to a coprime quotient and the Frobenius identity force a common orientation. Independent phase shifts then give smaller-period tilings from which the mixed cyclotomic zeros of the original factors can be recovered. A companion Lean formalization verifies the unrestricted T2 necessity statement.
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BibTeX
@misc{prajapati2026covenmeyerowitz,
title = {{Coven-Meyerowitz T2 necessity through coprime stripe collapse}},
author = {Jitendra Prajapati},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22280},
eprint = {2609.22280},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22280},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coven-Meyerowitz T2 necessity through coprime stripe collapse
AU - Jitendra Prajapati
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.22280
UR - https://arxiv.org/abs/2609.22280
AB - We prove that every finite subset of the integers which tiles by translations satisfies the Coven-Meyerowitz condition T2, with no restriction on the number of prime factors or their exponents. Together with the necessity of T1 and the sufficiency of T1 and T2 proved by Coven and Meyerowitz, this gives their proposed characterization of finite integer tiles. The proof uses strong induction on a cyclic tiling period. Character identities produce periodic Boolean product stripes; integral descent to a coprime quotient and the Frobenius identity force a common orientation. Independent phase shifts then give smaller-period tilings from which the mixed cyclotomic zeros of the original factors can be recovered. A companion Lean formalization verifies the unrestricted T2 necessity statement.
ER -