On a theorem of Davenport-Schmidt on Dirichlet improvable pairs

Abstract :  In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers.