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A vector space partition of $\mathbb{F}_q^v$ is a collection of subspacessuch that every non-zero vector is contained in a unique element. We improve alower bound of Heden, in a subcase, on the number of elements of the smallestoccurring dimension in a vector space partition. To this end, we introduce thenotion of $q^r$-divisible sets of $k$-subspaces in $\mathbb{F}_q^v$. Bygeometric arguments we obtain non-existence results for these objects, whichthen imply the improved result of Heden.

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