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PreprintCC BY 4.0arXiv

Weak Tiling by Unions of Non-overlapping Unit Cubes

  • Tianyu Chen
  • Shilei Fan
  • Mihail N. Kolountzakis
  • Chun-Kit Lai
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Abstract

We study weak tiling by finite unions of pairwise non-overlapping unit cubes in ℝd that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in ℝ2 as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026weak,
  title = {{Weak Tiling by Unions of Non-overlapping Unit Cubes}},
  author = {Tianyu Chen and Shilei Fan and Mihail N. Kolountzakis and Chun-Kit Lai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34852},
  eprint = {2609.34852},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34852},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Weak Tiling by Unions of Non-overlapping Unit Cubes
AU  - Tianyu Chen
AU  - Shilei Fan
AU  - Mihail N. Kolountzakis
AU  - Chun-Kit Lai
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34852
UR  - https://arxiv.org/abs/2609.34852
AB  - We study weak tiling by finite unions of pairwise non-overlapping unit cubes in ℝ^d that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in ℝ^2 as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.
ER  -