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 1st-named author and Wadleigh [ 17], 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 the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) 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 the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an m × n matrix are governed by a trajectory in SL m+n (R)/ SL m+n (Z) avoiding a compact subset of the space of lattices called the critical locus defined with respect to the corresponding norm. The trajectory is formed by the action of a oneparameter diagonal subgroup corresponding to the weights. We first state a very precise form of Dirichlet's theorem and prove it for some norms. Secondly we show, for these same norms, that the set of Dirichlet-improvable matrices has full Hausdorff dimension. Though the techniques used vary greatly depending on the chosen norm, we expect these results to hold in general.

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We state a norm sensitive Diophantine approximation problem arising form the work of the first-named author and Wadleigh [KWa], and extend the set-up by replacing the supremum norm with an arbitrary norm, following the approach of Andersen and Duke [AD]. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices. In particular, the choice of the Euclidean norm on R 2 corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of the dynamical Borel-Cantelli lemma of Maucourant [Mau] produces, given an approximation function ψ, a zero-one law for the set of α ∈ R for which the inequality αq−p ψ(t) 2 + q t 2 < 2 √ 3 has non-trivial integer solutions for all large enough t.