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Codimension formulae for the intersection of fractal subsets of Cantor spaces


Authors: Casey Donoven and Kenneth Falconer
Journal: Proc. Amer. Math. Soc. 144 (2016), 651-663
MSC (2010): Primary 28A80; Secondary 20E08, 60G57
DOI: https://doi.org/10.1090/proc12730
Published electronically: June 26, 2015
MathSciNet review: 3430842
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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.


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Additional Information

Casey Donoven
Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
Email: cd65@st-andrews.ac.uk

Kenneth Falconer
Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom
Email: kjf@st-andrews.ac.uk

DOI: https://doi.org/10.1090/proc12730
Received by editor(s): September 30, 2014
Received by editor(s) in revised form: January 15, 2015, and January 16, 2015
Published electronically: June 26, 2015
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society