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.

arXiv comment25 pages; intro changed, and typos fixed

References in corpus (1)

• Invariants for critical dimension groups and permutation-Hermite equivalence