Fuglede's conjecture holds in 𝔽_2^7

Abstract

We prove that a nonempty subset of the binary vector space 𝔽₂⁷ is spectral if and only if it tiles 𝔽₂⁷ by translations. This is the smallest binary dimension left open by Ferguson and Sothanaphan. For the implication from tiling to spectrality we show that every eight-point subset of 𝔽₂^d admits 2^d−3 frequencies whose pairwise differences avoid the zeros of its Fourier transform; consequently every tile of 𝔽₂^d with at least 2^d−3 elements is spectral. For the converse implication, we use the Smith normal form of the Walsh matrix to exclude every spectral subset of 𝔽₂⁷, and of 𝔽₂⁸, whose size is not a power of two. The sizes $16$ and $32$ are reduced, by a projection argument, to the tiling of point sets attached to binary log-Hadamard matrices of small rank. Using the published classifications of Hadamard matrices of orders $16$ and $32$, we find that exactly sixteen order-$32$ classes up to transposition have dephased binary rank at most seven, and we give explicit tilings for both orientations of each.

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BibTeX RIS

BibTeX

@misc{kadowaki2026fugledes,
  title = {{Fuglede's conjecture holds in $\mathbbF_2^7$}},
  author = {Atsushi Kadowaki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24022},
  eprint = {2610.24022},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24022},
}

RIS

TY  - UNPB
TI  - Fuglede's conjecture holds in 𝔽_2^7
AU  - Atsushi Kadowaki
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24022
UR  - https://arxiv.org/abs/2610.24022
AB  - We prove that a nonempty subset of the binary vector space 𝔽_2^7 is spectral if and only if it tiles 𝔽_2^7 by translations. This is the smallest binary dimension left open by Ferguson and Sothanaphan. For the implication from tiling to spectrality we show that every eight-point subset of 𝔽_2^d admits 2^d−3 frequencies whose pairwise differences avoid the zeros of its Fourier transform; consequently every tile of 𝔽_2^d with at least 2^d−3 elements is spectral. For the converse implication, we use the Smith normal form of the Walsh matrix to exclude every spectral subset of 𝔽_2^7, and of 𝔽_2^8, whose size is not a power of two. The sizes $16$ and $32$ are reduced, by a projection argument, to the tiling of point sets attached to binary log-Hadamard matrices of small rank. Using the published classifications of Hadamard matrices of orders $16$ and $32$, we find that exactly sixteen order-$32$ classes up to transposition have dephased binary rank at most seven, and we give explicit tilings for both orientations of each.
ER  -

Fuglede's conjecture holds in 𝔽₂⁷

Abstract

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