Weighted uniform Diophantine approximation of systems of linear forms

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