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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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