We shall give a gentle overview of the Falconer distance problem and then recast it in the language of complexity. We shall then describe a natural generalization of the distance problem in the context of finite point configurations. In the second half of the talk, we shall introduce the Vapnik-Chervonenkis dimension as the measure of complexity for families of indicator functions of spheres and show how this point of view leads to a variety of natural results pertaining to large point configurations in sets of a given size in vector spaces over finite fields, and sets of given Hausdorff dimension in Euclidean space.