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 everyu ∈ [3, 6], the set of all sums ∑ 80 a n 4−n a n 4−n with digits witha 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.
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Research of Y. Peres was partially supported by NSF grant #DMS-9803597.
Research of K. Simon was supported in part by the OTKA foundation grant F019099.
Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.
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Peres, Y., Simon, K. & Solomyak, B. Self-similar sets of zero Hausdorff measure and positive packing measure. Isr. J. Math. 117, 353–379 (2000). https://doi.org/10.1007/BF02773577
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DOI: https://doi.org/10.1007/BF02773577