  Following the development of weighted asymptotic approximation properties of
matrices, we introduce the analogous uniform approximation properties (that is,
study the improvability of Dirichlet's Theorem). An added feature is the use of
general norms, rather than the supremum norm, to quantify the approximation. In
terms of homogeneous dynamics, the approximation properties of an $m \times n$
matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb
R})/\mathrm{SL}_{m+n}({\mathbb Z})$ avoiding a compact subset of the space of
lattices called the critical locus defined with respect to the corresponding
norm. The trajectory is formed by the action of a one-parameter diagonal
subgroup corresponding to the weights. We first state a very precise form of
Dirichlet's theorem and prove it for some norms. Secondly we show, for these
same norms, that the set of Dirichlet-improvable matrices has full Hausdorff
dimension. Though the techniques used vary greatly depending on the chosen
norm, we expect these results to hold in general.
