An extension of the direction problem

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