In this talk I shall describe new results regarding generic properties of certain sets on fractals. Questions regarding these (often exceptional) sets, arising from number theory, dynamics and Diophantine approximation theory, have been extensively studied in recent years utilizing Schmidt´s game and properties of the class of friendly measures. In order to highlight the main ideas in many of these proofs, I shall reprove a slight modification of Schmidt´s original result regarding badly approximable numbers, pointing out where generalizations have been made using modern ideas and techniques.

I wish to emphasize that the talk is accessible to anyone interested, as I will try and make it as self contained as possible.