An extension of the direction problem

AbstractLet 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