局部集合近似：Mattila-Vuorinen型集，Reifenberg型集和切集
Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets
摘要：
我们调查的局部和渐近几何的集合$A \subseteq \mathbb{R}^n$和几何模型集$\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$，近似$A$局部均匀小尺度之间的相互作用。本文提出的局部集合近似框架统一和扩展了Jones，Mattila和Vuorinen，Reifenberg和Preiss的思想。我们指出了几个应用这个框架中出现的几何测度论和偏微分方程变分问题。例如，我们证明了在$\mathbb{R}^n$（$n\geq 4$）中的一个$(n-1)$维渐近最优加倍测度的支撑的奇异部分最多有$n-4$的上Minkowski维数.
英文摘要：
We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$ ($n\geq 4$) has upper Minkowski dimension at most $n-4$.
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