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arXiv · 2103.14925

Empty simplices of large width

Joseph Doolittle·Lukas Katthän·Benjamin Nill·Francisco Santos
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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 $2^{31}$. 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 $d/\operatorname{arcsinh}(1) \sim 1.1346\,d$.

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Joseph Doolittle, Lukas Katthän, Benjamin Nill, Francisco Santos. 2021-03-27. Empty simplices of large width. https://doi.org/10.1017/fms.2024.131

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