1. CMB 2011 (vol 56 pp. 354)
|The Sizes of Rearrangements of Cantor Sets|
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.
Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set
2. CMB 2000 (vol 43 pp. 330)
|Maximal Operators and Cantor Sets |
We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on $L^2$ if the Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set