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Fatness and thinness of uniform Cantor sets for doubling measures

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Abstract

Let E = E({n k }, {C k })) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if Σ(n k c k )p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that Σ(n k c k )q < ∞ and Σ(n k c k )p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510641, China

    FengJi Peng

  2. Department of Mathematics, Hubei University, Wuhan, 430062, China

    ShengYou Wen

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  1. FengJi Peng
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  2. ShengYou Wen
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Correspondence to ShengYou Wen.

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Peng, F., Wen, S. Fatness and thinness of uniform Cantor sets for doubling measures. Sci. China Math. 54, 75–81 (2011). https://doi.org/10.1007/s11425-010-4148-7

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  • DOI: https://doi.org/10.1007/s11425-010-4148-7

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