Distinct Matroid Base Weights and Additive Theory

Let $M$ be a matroid on a set $E$ and let $w:E\longrightarrow G$ 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.