Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $\mathbb{R}^2$ prove that, each such norm, set improvable pairs contains badly approximable pairs, hence is hyperplane absolute winning. To this we make careful study some classical results in geometry numbers due Chalk--Rogers Mahler establish Haj\'{o}s--Minkowski type result critical locus cylinder. As corollary, using first named author Mirzadeh, conclude that any top not isolated point.