| We prove that the countable intersection of C 1-diffeomorphic images of certain sets arising in dynamics and Diophantine approximation has full Hausdorff dimension. For example, we prove this for the set of badly approximable vectors, improving earlier results of Schmidt and Dani. To prove this we define two new variants of Schmidt's (alpha,beta) game, and show that each of our sets is winning in one of these new games. These properties both pass automatically to games played on fractals, thus the above countable intersection must intersect a large class of fractals in a set of positive dimension. This extends results of Fishman to a more general class of fractals, with simpler proofs. |
