Abstract
Moscow J. Comb. Number Th. 11 (2022) 97-114 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 $\mathbb{R}^2$ and prove that, for each such
norm, the set of Dirichlet improvable pairs contains the set of badly
approximable pairs, hence is hyperplane absolute winning. To prove this we make
a careful study of some classical results in the geometry of numbers due to
Chalk--Rogers and Mahler to establish a Haj\'{o}s--Minkowski type result for
the critical locus of a cylinder. As a corollary, using a recent result of the
first named author with Mirzadeh, we conclude that for any norm on
$\mathbb{R}^2$ the top of the Dirichlet spectrum is not an isolated point.