Nathanson heights in finite vector spaces

Abstract

Let $p$ be a prime, and let $\mathbb{Z}_p$ denote the field of integers modulo $p$. The \emph{Nathanson height} of a point $v \in \mathbb{Z}_p^n$ is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace $V \subseteq \mathbb{Z}_p^n$ is the least Nathanson height of any of its nonzero points. In this paper, we resolve a conjecture of Nathanson [M. B. Nathanson, Heights on the finite projective line, International Journal of Number Theory, to appear], showing that on subspaces of $\mathbb{Z}_p^n$ of codimension one, the Nathanson height function can only take values about $p, p/2, p/3, ....$ We show this by proving a similar result for the coheight on subsets of $\mathbb{Z}_p$, where the \emph{coheight} of $A \subseteq \mathbb{Z}_p$ is the minimum number of times $A$ must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture.