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Codimension formulae for the intersection of fractal subsets of Cantor spaces (1409.8070v2)
Published 29 Sep 2014 in math.MG
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.