# Self-similar sets of zero Hausdorff measure and positive packing measure

- Received:

DOI: 10.1007/BF02773577

- Cite this article as:
- Peres, Y., Simon, K. & Solomyak, B. Isr. J. Math. (2000) 117: 353. doi:10.1007/BF02773577

## Abstract

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost every*u* ∈ [3, 6], the set of all sums ∑_{0}^{8}*a*_{n}4^{−n}*a*_{n}4^{−n} with digits with*a*_{n} ∈ {0, 1,*u*} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result 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.