The minimum size of a linear set

In this paper, we first determine the minimum possible size of an F q-linear set of rank k in PG(1,q n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F q-linear set of rank k in PG(2,q n). In the case k=n, this confirms a conjecture by Sziklai in [9].