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PreprintCC BY-NC-ND 4.0arXiv

Translational tiles without spectra in finite abelian p-groups

  • Shilei Fan
  • Mamateli Kadir
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Abstract

We construct explicit translational tiles without spectra in three finite abelian p-groups. The first is a $64$-point subset of Z44×Z22. The other two are a $512$-point subset of F213 and a $2187$-point subset of F39. Consequently, the tile-to-spectral implication fails for finite abelian p-groups, and it already fails within the class of elementary abelian groups for both p=2 and p=3. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber--clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.

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Cite

BibTeXRIS
BibTeX
@misc{fan2026translational,
  title = {{Translational tiles without spectra in finite abelian p-groups}},
  author = {Shilei Fan and Mamateli Kadir},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00087},
  eprint = {2609.00087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00087},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Translational tiles without spectra in finite abelian p-groups
AU  - Shilei Fan
AU  - Mamateli Kadir
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00087
UR  - https://arxiv.org/abs/2609.00087
AB  - We construct explicit translational tiles without spectra in three finite abelian p-groups. The first is a $64$-point subset of Z_4^4×Z_2^2. The other two are a $512$-point subset of F_2^13 and a $2187$-point subset of F_3^9. Consequently, the tile-to-spectral implication fails for finite abelian p-groups, and it already fails within the class of elementary abelian groups for both p=2 and p=3. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber--clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.
ER  -