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arXiv · 1409.8070

Codimension formulae for the intersection of fractal subsets of Cantor spaces

Casey Donoven·Kenneth Falconer
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Abstract

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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Casey Donoven, Kenneth Falconer. 2014-09-29. Codimension formulae for the intersection of fractal subsets of Cantor spaces. https://arxiv.org/abs/1409.8070

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