Abstract: We will discuss an infinite game which produces a class of full-dimension sets that is closed under countable intersection and under taking images by C^1 diffeomorphisms. In particular, the countable intersection of C^1 diffeomorphic images of such a set meets any sufficiently regular fractal in a set of full Hausdorff dimension. We present joint work with L. Fishman and D. Simmons, where we have shown that, for any surjective endomorphism of a torus, the set of nondense orbits belongs to the above-mentioned class. We also show that the set of badly approximable systems of linear forms has this property. We will then discuss recent work with D. Kleinbock which uses the game, along with techniques from homogeneous dynamics, to produce complements to the above results concerning `uniform’ versions of the sets.