On the Minimum Size of Linear Sets

Recently, a lower bound on the size of linear sets in projective spaces intersecting a hyperplane in a canonical subgeometry was established. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $\pi$ in a canonical subgeometry. We obtain a tight lower bound on the size of any ${\mathbb F}_{q}$-linear set spanning $\mathrm{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $\pi$ is a hyperplane, and in the case that $\pi$ is a lower dimensional subspace.