Abstract view
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The Sizes of Rearrangements of Cantor Sets
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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
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.
