Nathanson heights in finite vector spaces

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point v∈Zpn is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace V⊆Zpn is the least Nathanson height of any of its nonzero points. In this paper, we resolve a quantitative conjecture of Nathanson [M.B. Nathanson, Heights on the finite projective line, Int. J. Number Theory, in press], showing that on subspaces of Zpn 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 Zp, where the coheight of A⊆Zp 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.