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 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.

Assouad dimensions of complementary sets

“…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.…”

“…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…”

“…By our hypothesis and this observation, we can take N large and an appropriate k such that all the ratios s k+1 /s k , ..., s k+N /s k+N −1 are greater than λ, but s k+N +1 /s k+N ≤ λ. Using (11) gives the estimate…”

“…A similar result was discovered by Hare et al . in [14] in the case of the packing measure. In contrast, it is not difficult to see that all the complementary sets in C a have the same lower and upper box dimension (see [8, ch.…”

On packing measures and a theorem of Besicovitch

“…[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].…”