Asymptotic hollowness of lattice simplices

TL;DR

This paper studies the asymptotic behavior of lattice simplices defined by certain integer tuples, characterizing when these simplices are hollow and identifying all such cases in four dimensions.

Contribution

It provides a complete characterization of nontrivial asymptotically hollow tuples, including an effective criterion and explicit classification in four dimensions.

Findings

Nontrivial asymptotically hollow tuples are characterized by modular inequalities.

Existence of a computable constant C determining the hollow property for large N.

Complete classification of nontrivial asymptotically hollow triples in four dimensions.

Abstract

An $(n-1)$-tuple $a = (a(1), \dots, a(n-1))$ consisting of positive integers is said to be asymptotically hollow if there exist infinitely many positive integers $N$ such that the convex hull, $K(a(n))$, in $n$-dimensional Euclidean space of $\{ 0,e_1, \dots, e_{n-1}, \alpha(N)^T\}$ is hollow (has no lattice points in its interior), where $e_i$ run over all but the last standard basis elements, and $\alpha(N) $ is the row $(a(1), \dots, a(N-1), N)$. The tuple is trivial if $\min a(i) = 1$. Nontrivial asymptotically hollow tuples are characterized in terms of modular inequalities, and turn out to be rare. We show that for a tuple $a$, there exists an effectively computable constant $C$ (depending on $a$) such that if for some $N > C$, $K(\alpha(N))$ is (not) hollow, then for all $M > C$, $K(\alpha(M))$ is (not) hollow (respectively). When $n = 4$, the nontrivial asymptotically hollow triples are completely determined; there are eleven of them, together with a one-parameter family.