The Sizes of Rearrangements of Cantor Sets

Canadian mathematical bulletin

Franklin Mendivil²  Leandro Zuberman¹  Kathryn E. Hare³ 

[1] Departamento de Matemática, FCEN-UBA, Buenos Aires, Argentina;Department of Mathematics and Statistics, Acadia University, Wolfville, NS;Department of Pure Mathematics, University of Waterloo, Waterloo, ON

关键词: Hausdorff dimension;    packing dimension;    dimension functions;    Cantor sets;    cut-out set;   

DOI  :  10.4153/CMB-2011-167-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

A linear Cantor set $C$ with zero Lebesgue measure is associated withthe countable collection of the bounded complementary open intervals. Arearrangment of $C$ has the same lengths of its complementaryintervals, but with different locations. We study the Hausdorff and packing $h$-measures and dimensional properties of the set of all rearrangments ofsome given $C$ for general dimension functions $h$. For each set ofcomplementary lengths, we construct a Cantor set rearrangement which has themaximal 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.

【 授权许可】

Unknown