Badly approximable grids and k-divergent lattices

Let (Formula presented.) be a matrix. In this paper, we investigate the set (Formula presented.) of badly approximable targets for (Formula presented.), where (Formula presented.) is the (Formula presented.) -torus. It is well known that (Formula presented.) is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of (Formula presented.) and Diophantine properties of (Formula presented.). On the one hand, we give the first examples of a nonsingular (Formula presented.) such that (Formula presented.) has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on (Formula presented.) that slightly strengthens nonsingularity, and show that under the assumption that (Formula presented.) satisfies this condition, (Formula presented.) is a null-set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.