1. Gauges for the cookie-cutter sets.
- Author
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Yu Xia Dai and Sheng You Wen
- Subjects
- *
HAUSDORFF measures , *MATHEMATICAL functions , *EQUATIONS , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
Let E be a cookie-cutter set with dim H 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 < lim inf t→0 $$ \tfrac{{g(t)}} {{t^s }} $$ < ∞. 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 sup t→0 $$ \tfrac{{g(t)}} {{t^s }} $$ < ∞. [ABSTRACT FROM AUTHOR]
- Published
- 2009
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