Abstract

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point v in Zp^{n is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace V of Zp}n is the least Nathanson height of any of its nonzero points. In this talk, I will investigate the range of the Nathanson height function using a variety of techniques from additive combinatorics. In particular, I will show that on subspaces of Zp^n of codimension one, the Nathanson height function can only take values about p, p/2, p/3, ..... affirmatively answering a question of Nathanson. I prove this by showing a similar result for the coheight on subsets of Zp, where the coheight of a subset A of Zp is the minimum number of times A must be added to itself so that the sum contains 0. I will also present some open questions and conjectures related to the Nathanson height and coheight, and indicate a few possible directions for future research.