Fuglede's conjecture holds in 𝔽_p^d for p=2, d=8 and p=11, d=3

Abstract

Fuglede's conjecture in 𝔽_p^d says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in 𝔽₂⁸ and 𝔽₁₁³. Over 𝔽₂ we show more generally that the Fourier support of a nonconstant indicator of density different from one half contains a plane. Consequently every spectral set of density one quarter has a subspace tiling complement. This supplies one of the dimension-eight reductions left by Kadowaki; the others are completed using classified Hadamard matrices and perfect binary codes. In 𝔽₂⁹ we prove that every tile is spectral and that every spectral set has power-of-two cardinality. All spectral cardinalities except $64$ are shown to tile, and we give further reductions of the remaining case. We also construct a spectral set of density one eighth whose Fourier support contains no three-dimensional subspace. For 𝔽₁₁³ we extend the geometric method of Fallon, Mayeli, and Villano by using moment identities, polynomial plane profiles, weighted incidence counts, and exact finite computations. The computations and their verification programs are described in the appendices.

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BibTeX RIS

BibTeX

@misc{ferguson2026fugledes,
  title = {{Fuglede's conjecture holds in $\mathbbF_p^d$ for $p=2,\,d=8$ and $p=11,\,d=3$}},
  author = {Samuel J. Ferguson and Joshua Iosevich},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30712},
  eprint = {2610.30712},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30712},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Fuglede's conjecture holds in 𝔽_p^d for p=2, d=8 and p=11, d=3
AU  - Samuel J. Ferguson
AU  - Joshua Iosevich
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30712
UR  - https://arxiv.org/abs/2610.30712
AB  - Fuglede's conjecture in 𝔽_p^d says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in 𝔽_2^8 and 𝔽_11^3. Over 𝔽_2 we show more generally that the Fourier support of a nonconstant indicator of density different from one half contains a plane. Consequently every spectral set of density one quarter has a subspace tiling complement. This supplies one of the dimension-eight reductions left by Kadowaki; the others are completed using classified Hadamard matrices and perfect binary codes. In 𝔽_2^9 we prove that every tile is spectral and that every spectral set has power-of-two cardinality. All spectral cardinalities except $64$ are shown to tile, and we give further reductions of the remaining case. We also construct a spectral set of density one eighth whose Fourier support contains no three-dimensional subspace. For 𝔽_11^3 we extend the geometric method of Fallon, Mayeli, and Villano by using moment identities, polynomial plane profiles, weighted incidence counts, and exact finite computations. The computations and their verification programs are described in the appendices.
ER  -

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