Self-similar sets of zero Hausdorff measure and positive packing measure
The authors prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension. For instance, if \(1/5< r< 1/3\) then the set \({\mathcal K}^r_u\) of all sums \(\sum^\infty_{n=0} a_nr^n\) with \(a_n\in \{0, 1,u\}\) has this property for almost every \(u\) from a certain nonempty interval, notably \(u\in [3,6]\) for \(r= 1/4\). It is, however, an unsolved problem to exhibit specific parameters \(r\), \(u\) for which the conclusion holds. Note that the set \({\mathcal K}^r_u\) can be identified with the orthogonal projection of the \(s\)-dimensional Sierpiński gasket on the line \(y= ux\). The authors also describe the families of projections of self-similar sets such that the projected sets have zero Hausdorff measure. The Hausdorff measure result is established using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand's projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. To prove the result on packing measure, the authors derive bounds that also determine which kernels assign positive capacity to typical projections.
scientific article; zbMATH DE number 1455701
scientific article