An equivalent definition of packing dimension and its application (Q1015076)

For a set \(E\subseteq \mathbb{R}^d\) and \(\delta>0\), a \(\delta\)-packing of \(E\) is usually defined as a collection of at most countable number of disjoint balls of radii at most \(\delta\) with centers in \(E\); fixing \(s\geq 0\), the \(s\)-dimensional packing measure of \(E\) is defined by \(P^s(E)=\inf \{\sum_iP^s_0(E_i):E\subseteq \bigcup_i^\infty E_i\}\), where \(P_0^s(E_i) =\lim_{\delta\to 0^+}\sup\sum_j|B_j|^s\) and the sup is taken over all \(\delta\)-packings \(\{B_j\}\) of \(E_i\). The packing dimension of \(E\) is then given by \(\dim_pE=\sup\{s:P^s(E)=\infty\}=\inf\{s:P^s (E)=0\}\). The author wants to avoid the inconvenience for evaluating the packing dimension of \(E\), adjacent with the above definition -- in fact the packing used for \(P_0^s(E_i)\) are restricted to the disjoint balls centered at \(E_i\). The author takes the help of Moran sets in his definition only to find out an alternative way to relax the restriction on a packing and at the same time to induce the same value for the packing dimension as introduced earlier. As a direct application, the author has the packing dimension of a class of subsets with prescribed relative group frequencies.