2013
DOI: 10.4153/cmb-2011-167-7
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The Sizes of Rearrangements of Cantor Sets

Abstract: Abstract.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 Hausdorf… Show more

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Cited by 8 publications
(8 citation statements)
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“…This investigation began with Besicovitch and Taylor in [3] where they proved that the Cantor set associated with a has the maximal Hausdorff dimension of all sets in C a . Moreover, they showed that given any s ∈ [0, dim H C a ], there was some E ∈ C a with dim H E = s. The analogous result was subsequently shown in [13] for packing dimension.…”
Section: Dimensional Properties Of Complementary Setsmentioning
confidence: 58%
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“…This investigation began with Besicovitch and Taylor in [3] where they proved that the Cantor set associated with a has the maximal Hausdorff dimension of all sets in C a . Moreover, they showed that given any s ∈ [0, dim H C a ], there was some E ∈ C a with dim H E = s. The analogous result was subsequently shown in [13] for packing dimension.…”
Section: Dimensional Properties Of Complementary Setsmentioning
confidence: 58%
“…By hypothesis, there is an index j with k + n − N 0 < j ≤ k + n and s j+1 /s j < 1/2 − δ. This ensures that (11) 2δs j ≤ s j − 2s j+1 = 2 −j (a…”
Section: Maximal Assouad Dimensionmentioning
confidence: 99%
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“…[3] and [7] for two deep recent examples. Packing measures have also been recently applied to the study Cantor sets defined in terms of their gaps [2] and their rearrangements [6].…”
Section: Introduction and Statements Of Resultsmentioning
confidence: 99%