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Given two metric spaces E,F, it is well known that, dimHE +dimHF ≤ dimH(E × F) ≤dimHE +dimPF, dimHE +dimPF ≤ dimP(E × F) ≤dimPE +dimPF, where dimHE, dimPE denote,… Click to show full abstract
Given two metric spaces E,F, it is well known that, dimHE +dimHF ≤ dimH(E × F) ≤dimHE +dimPF, dimHE +dimPF ≤ dimP(E × F) ≤dimPE +dimPF, where dimHE, dimPE denote, respectively, the Hausdorff and packing dimension of E. In this paper, we show that, for any 0 ≤ s,t ≤ 1, there exist E,F ⊂ ℝ such that the following equalities hold simultaneously: dimH(E × F) −dimHE −dimHF = s, dimPE +dimPF −dimP(E × F) = t. This complete the related results of Wei et al. [C. Wei, S. Y. Wen and Z. X. Wen, Remarks on dimensions of Cartesian product sets, Fractals 24(3) (2016) 1650031].
Journal Title: Fractals
Year Published: 2017
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