The Sizes of Rearrangements of Cantor Sets
Printed: Jun 2013
Kathryn E. Hare,
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.
Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set
28A78 - Hausdorff and packing measures
28A80 - Fractals [See also 37Fxx]