Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements
Abstract
We construct an explicit 64-point subset of 𝔽29 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 𝔽2d has a linear-subspace complement precisely when d≤8. The example is spectral, so it is not a counterexample to Fuglede's conjecture. Its construction uses six three-dimensional subspaces of 𝔽26, 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.
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BibTeX
@misc{ferguson2026tiling,
title = {{Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements}},
author = {Samuel J. Ferguson and Joshua Iosevich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42005},
eprint = {2610.42005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42005},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements
AU - Samuel J. Ferguson
AU - Joshua Iosevich
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42005
UR - https://arxiv.org/abs/2610.42005
AB - We construct an explicit 64-point subset of 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 𝔽_2^d has a linear-subspace complement precisely when d≤8. The example is spectral, so it is not a counterexample to Fuglede's conjecture. Its construction uses six three-dimensional subspaces of 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.
ER -