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The Sizes of Rearrangements of Cantor Sets

2011; Cambridge University Press; Volume: 56; Issue: 2 Linguagem: Inglês

10.4153/cmb-2011-167-7

1496-4287

Kathryn E. Hare, Franklin Mendivil, Leandro Zuberman,

Functional Equations Stability Results

Abstract A linear Cantor setC 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.