Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

Abstract

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 Hajós--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.

16 pages, 3 figures; the new version has a stronger version of the main theorem with a simpler proof

References in corpus (1)

• Nondense orbits on homogeneous spaces and applications to geometry and number theory

Cited by in corpus (1)

• Constructing Displacement Vectors