Asymptotic hollowness of lattice simplices

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.