论文标题
不良减去规范的二分法现象
A dichotomy phenomenon for Bad minus normed Dirichlet
论文作者
论文摘要
给定$ \ mathbb {r}^2 $上的规范$ν$，在Andersen-Duke（ActaArith。2021）和Kleinbock-rao（kleinbock-rao（intericate notterices）中，定义并研究了$ν$ -Dirichlet的一组$ν$ -Dirichlet Rivebable Number $ \ Mathbf {di}_ν$。当$ν$是最高规范时，$ \ mathbf {di}_ν= \ mathbf {ba} \ cup \ mathbb {q} $，其中$ \ mathbf {ba} $是差不多的数字。每个集合$ \ mathbf {di}_ν$，例如$ \ mathbf {ba} $，均为零，满足Schmidt的获胜属性。因此，对于每一个规范$ν$，$ \ mathbf {ba} \ cap \ mathbf {di}_ν$都在获胜，因此具有完整的Hausdorff Dimension。在本文中，我们证明了以下二分法现象：要么$ \ Mathbf {ba} \ subset \ mathbf {di}_ν$或其他$ \ Mathbf {ba} \ smallSetMinus \ Mathbf {di}_ν$具有Full hausdorff ditemension。我们为两种情况中的每一个提供了几个示例。二分法是基于$ν$的关键基因座是否与预交$ g_t $ -orbit相交，其中$ \ {g_t \} $是$ \ propatatorName {sl} _2 _2 _2（\ nathbb {r}）$ $ x $ x $ x $ x $ x $ x y的单参数的对角亚组$ \ mathbb {r}^2 $。 Thus the aforementioned dichotomy follows from the following dynamical statement: for a lattice $Λ\in X$, either $g_\mathbb{R} Λ$ is unbounded (and then any precompact $g_{\mathbb{R}_{>0}}$-orbit must eventually avoid a neighborhood of $Λ$), or not, in which case the set of lattices in $ x $的$ g _ {\ mathbb {r} _ {> 0}} $ - 轨迹是预发的，并且在其关闭中包含$λ$具有完整的Hausdorff尺寸。
Given a norm $ν$ on $\mathbb{R}^2$, the set of $ν$-Dirichlet improvable numbers $\mathbf{DI}_ν$ was defined and studied in the papers of Andersen-Duke (Acta Arith. 2021) and Kleinbock-Rao (Internat. Math. Res. Notices 2022). When $ν$ is the supremum norm, $\mathbf{DI}_ν= \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is the set of badly approximable numbers. Each of the sets $\mathbf{DI}_ν$, like $\mathbf{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm $ν$, $\mathbf{BA} \cap \mathbf{DI}_ν$ is winning and thus has full Hausdorff dimension. In the present article we prove the following dichotomy phenomenon: either $\mathbf{BA} \subset \mathbf{DI}_ν$ or else $\mathbf{BA} \smallsetminus \mathbf{DI}_ν$ 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 $\{g_t\}$ 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 $Λ\in X$, either $g_\mathbb{R} Λ$ is unbounded (and then any precompact $g_{\mathbb{R}_{>0}}$-orbit must eventually avoid a neighborhood of $Λ$), or not, in which case the set of lattices in $X$ whose $g_{\mathbb{R}_{>0}}$-trajectories are precompact and contain $Λ$ in their closure has full Hausdorff dimension.
