Codimension formulae for the intersection of fractal subsets of Cantor spaces

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 inte...

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Bibliographic Details
Published inarXiv.org
Main Authors Casey Donoven, Falconer, Kenneth
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.01.2015
Subjects
Fractals
Intersections
Lower bounds
Martingales
Upper bounds
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Summary: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.
ISSN:2331-8422