Geodesic
The Sizes of Rearrangements of Cantor Sets
Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 354-365

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DOI

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.
DOI : 10.4153/CMB-2011-167-7
Mots-clés : 28A78, 28A80, Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set 
Hare, Kathryn E.; Mendivil, Franklin; Zuberman, Leandro. The Sizes of Rearrangements of Cantor Sets. Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 354-365. doi: 10.4153/CMB-2011-167-7
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     author = {Hare, Kathryn E. and Mendivil, Franklin and Zuberman, Leandro},
     title = {The {Sizes} of {Rearrangements} of {Cantor} {Sets}},
     journal = {Canadian mathematical bulletin},
     pages = {354--365},
     year = {2013},
     volume = {56},
     number = {2},
     doi = {10.4153/CMB-2011-167-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-167-7/}
}
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