Nathanson heights in finite vector spaces
نویسنده
چکیده
Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point v ∈ Z p is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace V ⊆ Z p 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 Z p 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.
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