## Codimension formulae for the intersection of fractal subsets of Cantor spaces

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- by Casey Donoven and Kenneth Falconer PDF
- Proc. Amer. Math. Soc.
**144**(2016), 651-663 Request permission

## 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.## References

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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
- MR Author ID: 65025
- Email: kjf@st-andrews.ac.uk
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 651-663 - MSC (2010): Primary 28A80; Secondary 20E08, 60G57
- DOI: https://doi.org/10.1090/proc12730
- MathSciNet review: 3430842