Reduction of integer tiles via CRT and base-p digits

Abstract

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

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BibTeX RIS

BibTeX

@misc{li2026reduction,
  title = {{Reduction of integer tiles via CRT and base-p digits}},
  author = {Chunlin Li and Jie Wu and Wenchuan Hu and Chao Wang and Erxiao Wang and Fuhai Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14026},
  eprint = {2609.14026},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14026},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Reduction of integer tiles via CRT and base-p digits
AU  - Chunlin Li
AU  - Jie Wu
AU  - Wenchuan Hu
AU  - Chao Wang
AU  - Erxiao Wang
AU  - Fuhai Zhu
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14026
UR  - https://arxiv.org/abs/2609.14026
AB  - Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).
ER  -

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