齐性空间上拟射线的有界轨道与带权函数的丢番图逼近
Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions
浏览论文内容
中文总结 AI 辅助
研究齐性空间上拟射线有界轨道，通过施氏等分布定理描述\(G\)中一类子集\(F\)，使其有界轨道点集有全豪斯多夫维数，还证明了拟乘法权函数丢番图逼近中不良逼近矩阵集有全豪斯多夫维数。
中文摘要 AI 辅助
设\(G\)为连通半单实李群，\(\Gamma\)为\(G\)中的不可约格，\(X = G/\Gamma\)。设\(F = \{g_t: t\geq0\}\)是\(G\)的非拟幂单单参数子半群。已知\(X\)中具有有界\(F\)轨道的点集具有全豪斯多夫维数。若\(U\)是相对于\(g_1\)的扩张水平球面子群，对任意\(x\in X\)，使得\(ux\)的\(F\)轨道有界的\(u\in U\)的集合也具有全豪斯多夫维数。本文取\(U\)为\(G\)的水平球面子群，应用施氏关于相对于\(U\)的扩张锥元素的等分布定理，描述了\(G\)中一类子集\(F\)，对于这类子集上述全豪斯多夫维数结论也成立。作为应用，证明了在拟乘法权函数的丢番图逼近设定下，不良逼近矩阵集具有全豪斯多夫维数。
英文摘要
Let $G$ be a connected semisimple real Lie group, $Γ$ an irreducible lattice in $G$ and $X = G/Γ$. Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$. Then it is known that the set of points in $X$ with bounded $F$-trajectories has full Hausdorff dimension. In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$, then for any $x \in X$ the set of points $u \in U$ such that the $F$-trajectory of $ux$ is bounded has full Hausdorff dimension. In this paper we take $U$ to be a horospherical subgroup of $G$ and apply Shi's equidistribution theorem for elements of the expanding cone with respect to $U$ to describe a class of subsets $F$ in $G$, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.
