A note on homogeneous Cantor set (Q2749021)

Let \(\mathbf c\) be the collection of homogeneous Cantor sets on the unit interval \([0,1]\) generated by the sequence of positive integers \(\{n_j\}_{\geq 1}\) and \(\{c_j\}_{\geq 1}\) the sequence of contracting ratios with \(0<c_j<1\) for any \(j\). The authors prove that if \(E\in {\mathbf c}\) NEWLINE\[NEWLINE\frac{{\mathbb H}^s(E)}{{\mathbb R}_s(E)}=\frac{1}{2}.NEWLINE\]NEWLINE where \(s\) is the Hausdorff dimension of the set \(E\), \({\mathbb{H}}^s\) is the Hausdorff \(s\)-dimensional measure and \({\mathbb R}_s(E)=\liminf\prod_{j=1}^kn_jc_j^s\).