We investigate the interplay between the local and asymptotic geometry of aset $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S}\subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly onsmall scales. The framework for local set approximation developed in this paperunifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, andPreiss. We indicate several applications of this framework to variationalproblems that arise in geometric measure theory and partial differentialequations. 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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机译：我们研究了集合$ A \ subseteq \ mathbb {R} ^ n $的局部和渐近几何与模型集$ \ mathcal {S} \ subset \ mathcal {P}（\ mathbb {R} ^ n）$，在小范围内局部均值$ A $。本文开发的局部集近似框架统一并扩展了Jones，Matila和Vuorinen，Reifenberg和Preiss的思想。我们指出了该框架在几何测度理论和偏微分方程中出现的变分问题的几种应用。例如，我们证明，在$ \ mathbb {R} ^ n $（$ n \ geq 4 $）中，$（n-1）$维渐近最优加倍测度的奇异部分的上端Minkowski维数为大多数$ n-4 $。
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