The Sizes of Rearrangements of Cantor Sets

Hare, Kathryn E. ; Mendivil, Franklin ; Zuberman, Leandro

Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 354-365

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-167-7

Mots-clés : 28A78, 28A80, Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set

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