Distinct Matroid Base Weights and Additive Theory

Let M be a matroid on a set E and let w : E −→ G be a weight function, where G is a cyclic group. Assuming that w(E) satisfies the Pollard’s Condition (i.e. Every non-zero element of w(E)−w(E) generates G), we obtain a formulae for the number of distinct base weights. If |G| is a prime, our result coincides with a result Schrijver and Seymour. We also describe Equality cases in this formulae. In the prime case, our result generalizes Vosper’s Theorem.