Codimension formulae for the intersection of fractal subsets of Cantor spaces

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.

Accepted version, Proc. Amer. Math. Soc