  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.
