Heden's bound on the tail of a vector space partition

A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion of $q^r$-divisible sets of $k$-subspaces in $\mathbb{F}_q^v$. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden.