A dichotomy phenomenon for Bad minus normed Dirichlet

TL;DR

This paper establishes a dichotomy in the behavior of badly approximable numbers relative to Dirichlet improvability, linking it to the intersection properties of the critical locus of a norm with certain dynamical orbits in lattice space.

Contribution

It proves a new dichotomy phenomenon for Bad minus normed Dirichlet sets, connecting Diophantine approximation with dynamical properties of lattice orbits.

Findings

Either Bad numbers are fully contained in Dirichlet improvable sets or their difference has full Hausdorff dimension.

The dichotomy depends on whether the critical locus intersects a precompact orbit.

The result links Diophantine approximation properties to dynamical behavior of lattices.

Abstract

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.