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# The Sizes of Rearrangements of Cantor Sets

Published:2011-08-31
Printed: Jun 2013
• Kathryn E. Hare,
Department of Pure Mathematics, University of Waterloo, Waterloo, ON
• Franklin Mendivil,
Department of Mathematics and Statistics, Acadia University, Wolfville, NS
• Leandro Zuberman,
Departamento de Matemática, FCEN-UBA, Buenos Aires, Argentina
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## Abstract

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: Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set
 MSC Classifications: 28A78 - Hausdorff and packing measures 28A80 - Fractals [See also 37Fxx]