  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.
