TY - JOUR

T1 - Self-similar sets of zero Hausdorff measure and positive packing measure

AU - Peres, Yuval

AU - Simon, Károly

AU - Solomyak, Boris

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0002548704&partnerID=8YFLogxK

U2 - 10.1007/bf02773577

DO - 10.1007/bf02773577

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AN - SCOPUS:0002548704

SN - 0021-2172

VL - 117

SP - 353

EP - 379

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -