Cantor空间分形子集交的余维数公式
Codimension formulae for the intersection of fractal subsets of Cantor spaces
摘要：
我们研究了$m$元Cantor空间$\mathcal{C}^m$的子集$E$与子集$F$的像在关于自然度量的随机等距下的交集的维数。我们得到几乎必然的上界的Hausdorff和上计盒维数的交集，和一个下界的本质上确界的豪斯多夫维数。交集的维数通常是$\max\{\dim E +\dim F -\dim \mathcal{C}^m, 0\}$，类似于其他余维定理。上限估计来自覆盖的预期大小，而下限估计则更加复杂，使用鞅来定义交集上的随机测度，以方便潜在的理论论证。
英文摘要：
We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\mathcal{C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\max\{\dim E +\dim F -\dim \mathcal{C}^m, 0\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.
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