A zero-one law for uniform Diophantine approximation in Euclidean norm

Abstract

We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a result due to Andersen and Duke on measure zero and uncountability of the set of numbers for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of a dynamical Borel-Cantelli lemma of Maucourant produces a zero-one law for improvement of Dirichlet's theorem in Euclidean norm.