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

Figures (2)

• Figure 2: Illustration of parts (1) and (2) of Lemma \ref{['lemma:empty']}. Left: the simplex $m\,A$ and its decomposition into $d+1$ dilated unimodular simplices $m\,A_i$. Center: our circulant simplex $S(d,m)$, contained in the Minkowski sum $\Delta + m\,A$. Right: the simplices $e_{i-1} + m A_{i}$ contain all the lattice points in $\Delta + m\,A$. In the picture we have $m=2.6$, which is bigger than the emptyness threshold in dimension two, $m_0(2)=1$.
• Figure 3: Illustration of part (3) of Lemma \ref{['lemma:empty']}. In this picture $m=0.8$, below the threshold $m_0(2)=1$. The simplex $e_{d} + m A_{0}$ lies completely in the open half-space $\{u <0\}$, except for its vertex $e_{d} + m a_{d}$, common to $S(d,m)$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2: partially Codenotti and Santos CodenottiSantos2020
• proof
• Theorem 1.3
• Conjecture 1.8
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof