Akhunzhanov and Shatskov (Mosc. J. Comb. Number Theory 3:3-4 (2013),
5–23) defined the two-dimensional Dirichlet spectrum with respect to
Euclidean norm. We consider an analogous definition for arbitrary norms on
and
prove that, for each such norm, the set of Dirichlet-improvable pairs contains the set
of badly approximable pairs, and 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 Hajós–Minkowski-type result for the
critical locus of a cylinder. As a corollary, using a recent result of Kleinbock
and Mirzadeh (arXiv:2010.14065 (2020)), we conclude that for any norm on
the
top of the Dirichlet spectrum is not an isolated point.
Keywords
Dirichlet's theorem, geometry of numbers, critical lattices
