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Gauges for the cookie-cutter sets

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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 \( \tfrac{{g(t)}} {{t^s }} \)< ∞.

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Authors and Affiliations

  1. Department of Mathematics, Hubei University, Wuhan, 430062, P. R. China

    Yu Xia Dai & Sheng You Wen

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  1. Yu Xia Dai
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  2. Sheng You Wen
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Correspondence to Sheng You Wen.

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Supported by NSFC (Grant Nos. 10571063, 10771164) and HuBei JiaoYuTing (Grant No. D20061001) 1) Corresponding author

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Dai, Y.X., Wen, S.Y. Gauges for the cookie-cutter sets. Acta. Math. Sin.-English Ser. 25, 2119–2126 (2009). https://doi.org/10.1007/s10114-009-7455-6

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  • DOI: https://doi.org/10.1007/s10114-009-7455-6

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