Codimension formulae for the intersection of fractal subsets of Cantor spaces

Casey Donoven; Kenneth John Falconer

doi:10.1090/proc12730

Codimension formulae for the intersection of fractal subsets of Cantor spaces

Casey Donoven, Kenneth John Falconer

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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
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	https://doi.org/10.1090/proc12730
Publication status	Published - Feb 2016

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© 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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Cite this

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Codimension formulae for the intersection of fractal subsets of Cantor spaces

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