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Abstract Given a norm ν on , the set of ν‐Dirichlet improvable numbers was defined and studied in the papers (Andersen and Duke,Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao,Internat. Math. Res. Notices2022 (2022) 5617–5657). When ν is the supremum norm, , where is the set of badly approximable numbers. Each of the sets , like , is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either or else has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether thecritical locusof ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the spaceXof unimodular lattices in . Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice , either is unbounded (and then any precompact ‐orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices inXwhose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension. 

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 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.