  We prove that the countable intersection of $C^1$-diffeomorphic images of
certain Diophantine sets has full Hausdorff dimension. For example, we show
this for the set of badly approximable vectors in $\mathbb{R}^d$, improving
earlier results of Schmidt and Dani. To prove this, inspired by ideas of
McMullen, we define a new variant of Schmidt's $(\alpha, \beta)$-game and show
that our sets are hyperplane absolute winning (HAW), which in particular
implies winning in the original game. The HAW property passes automatically to
games played on certain fractals, thus our sets intersect a large class of
fractals in a set of positive dimension. This extends earlier results of
Fishman to a more general set-up, with simpler proofs.
