  For an m by n real matrix A, we investigate the set of badly approximable
targets for A as a subset of the m-torus. It is well known that this set is
large in the sense that it is dense and has full Hausdorff dimension. We
investigate the relationship between its measure and Diophantine properties of
A. On the one hand, we give the first examples of a non-singular matrix A such
that the set of badly approximable targets has full measure with respect to
some non-trivial 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 A that slightly strengthens non-singularity, and show that under the
assumption that A satisfies this condition, the set of badly approximable
targets is a null-set with respect to any non-trivial algebraic measure on the
torus. For this we use naive homogeneous dynamics, harmonic analysis, and a
novel concept we refer to as mixing convergence of measures.
