Constructing universally small subsets of a given packing index in Polish groups
Volume 125 / 2011
Colloquium Mathematicum 125 (2011), 213-220-
MSC: 03E15, 03E50, 22A05, 54H05, 54H11.
DOI: 10.4064/cm125-2-6
Abstract
A subset of a Polish space $X$ is called universally small if it belongs to each ccc $\sigma$-ideal with Borel base on $X$. Under CH in each uncountable Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$ such that $|A_0\cap gA_0|=\mathfrak c$ for each $g\in G$. For each cardinal number $\kappa\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small subset $A$ of $G$ with sharp packing index ${\rm pack}^\sharp(A_\kappa)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $\kappa$.
