Empty simplices of large width

TL;DR

This paper introduces new constructions of empty lattice simplices with larger width than their dimension, including exhaustive computational results in dimension 10 and a method for arbitrary dimensions with asymptotically linear width growth.

Contribution

It presents the first known examples of empty simplices with width exceeding their dimension, using cyclotomic and circulant matrix constructions.

Findings

Identified five empty cyclotomic simplices of width 11 in dimension 10.

Constructed infinite families of empty simplices with width proportional to dimension.

Demonstrated asymptotic linear growth of width with respect to dimension.

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