## Abstract

We examine the dimensions of the intersection of a subset E of an m-ary Cantor space 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 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.

Original language | English |
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Pages (from-to) | 651-663 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 144 |

Issue number | 2 |

Early online date | 26 Jun 2015 |

DOIs | |

Publication status | Published - Feb 2016 |