Hausdorff theory of dual approximation on planar curves
Hausdorff theory of dual approximation on planar curves
复制标题
DOI：
10.1515/crelle-2015-0073
复制
发表时间：
2014-03
期刊：
影响因子：
--
通讯作者：
Jing-Jing Huang-Jing
中科院分区：
文献类型：
--
作者：
Jing-Jing Huang-Jing
十年前，Beresnevich-Dickinson-Velani [Mem. Amer. Math. Soc. 179（2006），no. 846]发起了一个项目，发展了非退化流形上对偶逼近的一般Hausdorff测度理论。特别是，他们建立了基于普遍存在框架的理论的分歧部分。然而，该项目的收敛对应物仍然是开放的，并且是该主题中的一个主要挑战性问题。直到最近，它甚至不知道任何单一的非退化流形。在本文中，我们解决了这个问题的所有曲线在{\mathbb{R}^{2}}，这代表了第一个完整的理论，它的一个一般类的流形。
Ten years ago, Beresnevich–Dickinson–Velani [Mem. Amer. Math. Soc. 179 (2006), no. 846] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in{\mathbb{R}^{2}}, which represents the first complete theory of its kind for a general class of manifolds.