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Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements — arXiv:2610.42005

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.42005
License (as archived): CC0-1.0
Archived: 2026-11-02T08:00:33.820Z
SHA-256 of the archived text: 797d6cc5f431e82bd3ca2d34b6d1bef8d3380dfed5265ea77070aa9b548cdd27
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How to cite

Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements — arXiv:2610.42005 — arXiv (en), https://arxiv.org/abs/2610.42005 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_c8ee79ea985d (Metraipolis archive dc_c8ee79ea985d, sha256 797d6cc5f431e82bd3ca2d34b6d1bef8d3380dfed5265ea77070aa9b548cdd27)

BibTeX

@misc{dcc8ee79ea985d,
  title = {Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements — arXiv:2610.42005},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.42005},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_c8ee79ea985d; sha256 797d6cc5f431e82bd3ca2d34b6d1bef8d3380dfed5265ea77070aa9b548cdd27},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.42005. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct an explicit 64-point subset of $\mathbb F_2^9$ which tiles by translation but has no linear-subspace tiling complement. Together with our earlier dimension-eight theorem and a product argument, this shows that every nonempty tile in $\mathbb F_2^d$ has a linear-subspace complement precisely when $d\le8$. The example is spectral, so it is not a counterexample to Fuglede's conjecture. Its construction uses six three-dimensional subspaces of $\mathbb F_2^6$, specified by a small table. We classify all its tiling complements and spectra: there are exactly 512 complements and 64 spectra, with one translation class of each. Every spectrum has normalized Wiener norm $15/8$. An appendix proves that Goyal's related construction is affinely inequivalent but has the same tiling complements and spectra. The construction and both classifications are proved from first principles. We also give elementary proofs of the positive assertion in dimensions at most seven and for all tile sizes other than sixteen in dimension eight. The remaining case uses our previously established theorem based on the classification of perfect binary codes of length fifteen.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO) ; Classical Analysis and ODEs (math.CA).
Pagina del lavoro: https://arxiv.org/abs/2610.42005 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T07:42:45.750Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).