Badly approximable systems of affine forms and incompressibility on fractals

We explore and refine techniques for estimating the Hausdorff dimension of Diophantine exceptional sets and their diffeomorphic images. This work is directly motivated by a recent advance in geometric measure theory, which facilitates the use of games in bounding the dimension of a set's intersection with a sufficiently regular fractal. Specifically, we use a variant of Schmidt's game to deduce the strong C¹ incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms.