A dichotomy phenomenon for bad minus normed Dirichlet

Abstract

Given a norm ν on $R^2$, the set of ν-Dirichlet improvable numbers ${\mathrm{DI}}_{\nu }$ 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, ${\mathrm{DI}}_{\nu }=\mathrm{BA}\cup Q$, where $\mathrm{BA}$ is the set of badly approximable numbers. Each of the sets ${\mathrm{DI}}_{\nu }$, like $\mathrm{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, $\mathrm{BA}\cap {\mathrm{DI}}_{\nu }$ is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either $\mathrm{BA}\subset {\mathrm{DI}}_{\nu }$ or else $\mathrm{BA}\setminus {\mathrm{DI}}_{\nu }$ has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of ν intersects a precompact $g_t$-orbit, where $\left\{g_t\right\}$ is the one-parameter diagonal subgroup of ${SL}_2\left(R\right)$ acting on the space X of unimodular lattices in $R^2$. Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice $\Lambda \in X$, either $g_R\Lambda$ is unbounded (and then any precompact $g_{R_{>0}}$-orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices in X whose $g_{R_{>0}}$-trajectories are precompact and contain Λ in their closure has full Hausdorff dimension.