Empty simplices of large width

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URI: https://hdl.handle.net/10902/35891
ISSN: 2050-5094
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Autoría
Doolittle, Joseph; Katthän, Lukas; Nill, Benjamin; Santos, FranciscoAutoridad Unican
Fecha
2025-02
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© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Publicado en
Forum of Mathematics, Sigma, 2025, 13, e21
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Cambridge University Press
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Resumen/Abstract
An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: - We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension 10 and volume up to 2³¹. Among them, we find five empty ones of width 11 and none of larger width. - Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as d/arcsinh(1) ~ 1.1346 d.
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© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.Excepto si se señala otra cosa, la licencia del ítem se describe como © The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.