Hausdorff dimension of differences of badly approximable sets

The set of badly approximable numbers, Bad, is known to be winning for Schmidt’s game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the set difference also has full dimension using a variant of the Schmidt game, which we call the rapid game, played on the space of unimodular grids.