Hausdorff measures for a class of homogeneous Cantor sets (Q1945971)

Let \(E\) be a ``homogeneous'' Cantor set determined by \((n_k)_{k\in\mathbb{N}}\), \((c_k)_{k\in\mathbb{N}}\), which means, in particular, that any \(E\)-determining basic interval of level \(k-1\) contains \(n_k\) intervals of level \(k\) which are of equal length \(\lambda_k=c_k \lambda_{k-1}\) and equally spaced. Let \(y_k\) be the length of the gap between two consecutive basic intervals of level \(k\) contained in the same basic interval of level \(k-1\). Under the assumption that \((y_k)\) is decreasing \textit{C. Q. Qu, H. Rao} and \textit{W. Y. Su} [Acta Math. Sin., Engl. Ser. 17, No. 1, 15--20 (2001; Zbl 0990.28003)] proved that, if \(s\) is the Hausdorff dimension of \(E\), the \(s\)-dimensional Hausdorff measure of \(E\) is \(\liminf \prod_{j=1}^k n_jc_j^s\). In the present paper, the author proves the same conclusion relaxing the monotony condition of \((y_k)\).