A dichotomy phenomenon for Bad minus normed Dirichlet

Given a norm $\nu$ on $\mathbb{R}^2$, the set of $\nu$-Dirichlet improvable numbers $\mathbf{DI}_\nu$ was defined and studied in the papers of Andersen-Duke (Acta Arith. 2021) and Kleinbock-Rao (Internat. Math. Res. Notices 2022). When $\nu$ is the supremum norm, $\mathbf{DI}_\nu = \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is the set of badly approximable numbers. Each of the sets $\mathbf{DI}_\nu$, like $\mathbf{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm $\nu$, $\mathbf{BA} \cap \mathbf{DI}_\nu$ is winning and thus has full Hausdorff dimension. In the present article we prove the following dichotomy phenomenon: either $\mathbf{BA} \subset \mathbf{DI}_\nu$ or else $\mathbf{BA} \smallsetminus \mathbf{DI}_\nu$ has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of $\nu$ intersects a precompact $g_t$-orbit, where $\{g_t\}$ is the one-parameter diagonal subgroup of $\operatorname{SL}_2(\mathbb{R})$ acting on the space $X$ of unimodular lattices in $\mathbb{R}^2$. Thus the aforementioned dichotomy follows from the following dynamical statement: for a lattice $\Lambda\in X$, either $g_\mathbb{R} \Lambda$ is unbounded (and then any precompact $g_{\mathbb{R}_{>0}}$-orbit must eventually avoid a neighborhood of $\Lambda$), or not, in which case the set of lattices in $X$ whose $g_{\mathbb{R}_{>0}}$-trajectories are precompact and contain $\Lambda$ in their closure has full Hausdorff dimension.