Gauges for the Cookie-cutter Sets

Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Al...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2009-12, Vol.25 (12), p.2119-2126
Main Authors: Dai, Yu Xia, Wen, Sheng You
Format: Article
Language:English
Subjects:
Cookie
Gauges
Hausdorff
Mathematics
Mathematics and Statistics
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Summary:Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-009-7455-6