Upper and lower bounds for the packing measure in relation to the Hausdorff measure
The packing measure \(p^ h(E)\) of a set E in an Euclidean space as considered here is defined by \[ p^ h(E)=\inf \{\sum_{i}P^ h(E_ i);\quad E\subseteq \cup_{i}E_ i\}, \] where h is a Hausdorff function and \(P^ h(E_ i)\) denotes the upper limit of all sums \(\sum_{j}h(2r_ j)\) of packings of \(E_ i\) by pairwise disjoint closed balls \(B(x_ j,r_ j)\) centred at \(E_ i\). If further \(m^ h\) denotes the Hausdorff measure and \(\underline d^ h_ m\) the lower Hausdorff density function defined as follows \[ \underline d^ h_ m(x)=\liminf_{r\to 0} [m^ h(E\cap B(x,r))/h(2r)] \] then for a partition of a \(p^ h\)-measurable set E into \(p^ h\)-measurable sets \(E_ n\) satisfying \(\underline d^ h_ m(x)<a_ n<1\forall x\in E\) a.e. \(m^ h\) the author obtains \[ p^ h(E)\geq \sum_{n}a_ n^{- 1}m^ h(E_ n). \] A similar estimate for an upper bound is found. In the second part symmetric Cantor sets of the real line are considered. Provided that Hausdorff and packing dimension coincide the author gives an upper bound for the lower Hausdorff density function. The estimations concerning endpoints of symmetric sets are convincing, but for limit points which are not endpoints the reviewer could only follow the proof under the additional assumption that the length of the removed middle interval is larger than the two which are left.
scientific article; zbMATH DE number 4204723