A dichotomy phenomenon for bad minus normed Dirichlet

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. Notices 2022 (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 the critical locus of ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the space X of 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 in X whose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension.