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The Sizes of Rearrangements of Cantor Sets

Published online by Cambridge University Press:  20 November 2018

Kathryn E. Hare ,
Franklin Mendivil  and
Leandro Zuberman
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON e-mail: kehare@uwaterloo.ca
Franklin Mendivil
Affiliation:
Department of Mathematics and Statistics, Acadia University, Wolfville, NS e-mail: franklin.mendivil@acadiau.ca
Leandro Zuberman
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON e-mail: kehare@uwaterloo.ca
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Abstract

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A linear Cantor set $C$ with zero Lebesgue measure is associated with the countable collection of the bounded complementary open intervals. A rearrangment of $C$ has the same lengths of its complementary intervals, but with different locations. We study the Hausdorff and packing $h$-measures and dimensional properties of the set of all rearrangments of some given $C$ for general dimension functions $h$. For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure.

Keywords

28A7828A80Hausdorff dimensionpacking dimensiondimension functionsCantor setscut-out set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 354 - 365
Copyright
Copyright © Canadian Mathematical Society 2013

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