Israel Journal of Mathematics

, Volume 117, Issue 1, pp 353–379

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


DOI: 10.1007/BF02773577

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


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 everyu ∈ [3, 6], the set of all sums ∑08an4nan4n with digits withan ∈ {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.

Copyright information

© Hebrew University 2000

Authors and Affiliations

  1. 1.Department of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  3. 3.Institute of MathematicsUniversity of MiskolcMiskolc-EgyetemvárosHungary
  4. 4.Department of MathematicsUniversity of WashingtonSeattleUSA