Nathanson heights in finite vector spaces Original Research Article

Let p be a prime, and let image denote the field of integers modulo p. The Nathanson height of a point image is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace image 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 image of codimension one, the Nathanson height function can only take values about image. We show this by proving a similar result for the coheight on subsets of image, where the coheight of image 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.