Empty simplices of large width

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 $d/\operatorname{arcsinh}(1) \sim 1.1346\,d$.