Badly approximable vectors on rational quadratic varieties

Approximation in this paper is of vectors on the unit $d$-cube by the projection of integer lattice points onto the same cube. We define badly approximable vectors on a rational quadratic variety and show that sets of these vectors, which are (naturally) indexed by $m \in \QQ$, are winning and strong winning in the sense of Schmidt games. From the winning property, it follows that these sets have full Hausdorff dimension and, moreover, so does their intersection. In most cases, these sets are known to be null sets.