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

Yuval Peres, Károly Simon, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

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 an4-n with digits with an ∈ {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.

Original languageEnglish
Pages (from-to)353-379
Number of pages27
JournalIsrael Journal of Mathematics
Volume117
DOIs
StatePublished - 2000
Externally publishedYes

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