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 S k at infinity is determined by U if there is an affine ( k + 1 ) -dimensional subspace T k + 1 through S k such that U ∩ T k + 1 spans T k + 1 . We examine the extremal case | U | = q n − 1 , and classify point sets not determining every k -subspace in certain cases.