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 Cm 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 Cm, 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.
Citation
Donoven , C & Falconer , K J 2016 , ' Codimension formulae for the intersection of fractal subsets of Cantor spaces ' , Proceedings of the American Mathematical Society , vol. 144 , no. 2 , pp. 651-663 . https://doi.org/10.1090/proc12730
Publication
Proceedings of the American Mathematical Society
Status
Peer reviewed
ISSN
0002-9939Type
Journal article
Rights
© 2015. American Mathematical Society. First published in Proceedings of the American Mathematical Society in 2015, published by the American Mathematical Institute. . The final published version of this work is available at www.ams.org / https://dx.doi.org/10.1090/proc12730
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