Dimension estimates for sets of uniformly badly approximable systems of linear forms

The set of badly approximable $m \times n$ matrices is known to have Hausdorff dimension $mn$. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed $c$. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions.