The packing measure of the generalized Sierpiński sponge (Q2774183)

Given an increasing function \(\varphi: [0, \infty)\to {\mathbf R}, \varphi(0)=0\), for any subset \(E\) of a metric space, let NEWLINE\[NEWLINE \begin{aligned} \widetilde {P}_\varphi (E)=\lim_{\varepsilon\to 0 } \{\sup\sum_{j=1}^\infty \varphi (\text{ diam } B_j) : & \{B_j\}_{j=1}^\infty \;\text{ is a collection of disjoint closed balls } \\ &\text{of diameters at most } \varepsilon \text{ with centers in } E\}. \end{aligned} NEWLINE\]NEWLINE Define NEWLINE\[NEWLINEP_\varphi(E)=\inf \left \{\sum_{i=1}^\infty \widetilde P_\varphi(E_i): \;E\subset \bigcup_{i=1}^\infty E_i \right \}, NEWLINE\]NEWLINE which is called the packing measure of \(E\) in the measurable function \(\varphi\). In this paper the authors discuss a generalized Sierpiński sponge in \({\mathbf R}^3\) and give the conditions that the packing measures of the generalized Sierpiński sponge are finite or infinite for \(\varphi(t)=t^\theta, \varphi(t)=\frac{t^\theta}{|\log t|}\) and so on.