The family of central Cantor sets with packing dimension zero

Abstract

As in the recent article of M. Balcerzak, T. Filipczak and P. Nowakowski [Georgian Math Journal 26 (2019)], we identify the family $\mathcal{CS}$ of central Cantor subsets of [0,1] with the Polish space $X:=(0,1)^{\mathbb{N}}$ equipped with the probabilty product measure $\mu$. We investigate the size of the family $\mathcal{P}_0$ of sets in $\mathcal{CS}$ with packing dimension zero. We show that $\mathcal{P}_0$ is meager and of $\mu$ measure zero while it is treated as the corresponding subset of $X$. We also check possible inclusions between $\mathcal{P}_0$ and other subfamilies of $\mathcal{CS}$ consisting of small sets.