可解群的不变测度与丢番图逼近
Invariant measures for solvable groups and diophantine approximation
摘要：
我们表明，如果$\mathcal{L}$是一条线在平面上包含一个糟糕的逼近向量，那么几乎每一个点在$\mathcal{L}$不承认狄利克雷定理的改进。我们的证明依赖于一个措施的分类结果，一定的措施不变下的非阿贝尔二维群的齐性空间$\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$。利用测度分类定理，我们重新证明了Shah关于平面非退化曲线（不一定是解析的）的一个结果，并证明了带权的丢番图逼近框架下的类似结果.我们还表明，有线段在$\mathbb{R}^3$，其中包含严重的逼近点，并承认所有的点在狄利克雷定理的改进。
英文摘要：
We show that if $\mathcal{L}$ is a line in the plane containing a badly approximable vector, then almost every point in $\mathcal{L}$ does not admit an improvement in Dirichlet's theorem. Our proof relies on a measure classification result for certain measures invariant under a non-abelian two dimensional group on the homogeneous space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. Using the measure classification theorem, we reprove a result of Shah about planar nondegenerate curves (which are not necessarily analytic), and prove analogous results for the framework of Diophantine approximation with weights. We also show that there are line segments in $\mathbb{R}^3$, which do contain badly approximable points, and for which all points do admit an improvement in Dirichlet's theorem.
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