An extension of the direction problem

Let UU be a point set in the nn-dimensional affine space AG(n,q) over the finite field of qq elements and 0≤k≤n−20≤k≤n−2. In this paper we extend the definition of directions determined by  UU: a kk-dimensional subspace SkSk at infinity is determined   by UU if there is an affine (k+1)(k+1)-dimensional subspace Tk+1Tk+1 through SkSk such that U∩Tk+1U∩Tk+1 spans Tk+1Tk+1. We examine the extremal case |U|=qn−1|U|=qn−1, and classify point sets not   determining every kk-subspace in certain cases.