论文标题
一个新的Hausdorff内容绑定到Limsup集
A New Hausdorff Content Bound for Limsup Sets
论文作者
论文摘要
我们为Limsup集提供了一个新的Hausdorff内容，这与Falconer的大交叉点有关。法尔康纳（Falconer）的一组大交叉路口满足了一个空间中所有球的内容。相比之下，我们的主要定理仅假定对构成相关极限集合的球的比例不变结合。 我们给出了这些思想的四个应用和我们的主要定理：与毒液近似相关的质量转移原理的新证明和概括，这是对随机LIMSUP集的相关结果，Federer对一组有限周围表征的新证明以及有关通用路径的陈述和衡量理论边界的表征。新的一般质量转移原理将一个球集合的内容传递给另一组集合的内容 - 但是，该内容绑定必须保留在空间中的所有球上。我们方法的好处是极大地简化了论证以及估计Hausdorff内容的新工具。 新方法使我们能够在先前的工作中分配许多假设。具体而言，我们的一般质量转移原理和随机LIMSUP集的边界不假定AHLFORS的规律性。此外，它们适用于任何完整的度量空间。我们的一般Hausdorff内容估计适用于任何完整的度量空间中的LIMSUP集，使这种通用性成为可能。
We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only assumes a scale-invariant bound for the balls forming the limit superior set in question. We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer's characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle transfers a content bound of one collection of balls, to the content bound of another collection of sets -- however, this content bound must hold on all balls in the space. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content. The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.
