Self-similar sets of zero Hausdorff measure and positive packing measure |
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Authors: | Yuval Peres Károly Simon Boris Solomyak |
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Institution: | (1) Department of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel;(2) Department of Statistics, University of California, 94720 Berkeley, CA, USA;(3) Institute of Mathematics, University of Miskolc, H-315 Miskolc-Egyetemváros, Hungary;(4) Department of Mathematics, University of Washington, Box 354350, 98195 Seattle, WA, USA |
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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 ∑
0
8
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.
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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Keywords: | |
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