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On the Hausdorff
h-measure of Cantor sets
Carlos Cabrelli, Franklin Mendivil, Ursula M. Molter and Ronald Shonkwiler |
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Vol. 217 (2004), No. 1, 45–59
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Abstract |
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We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set. It is well-known that not every Cantor set on the line is an s-set for some 0 ≤ s ≤ 1. However, if the sequence associated to the Cantor set C is nonincreasing, we show that C is an h-set for some continuous, concave dimension function h. We construct the function h from the sequence associated to the set C. |
Milestones
Received: 28 May 2003
Published: 1 November 2004
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Authors |
| Carlos Cabrelli | |
| Departamento de Matemática Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria, Pabellón I 1428 Capital Federal Argentina |
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| Franklin Mendivil | |
| Department of Mathematics and
Statistics Acadia University Wolfville Nova Scotia B4P 2R6 Canada |
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| Ursula M. Molter | |
| Ronald Shonkwiler | |
| School of Mathematics Georgia Institute of Technology Atlanta GA 30332 |
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