Constructing universally small subsets of a given packing index in Polish groups

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 $|A_0\cap g{A}_0|=\mathfrak{c}$ 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 $\pack^\sharp(A_κ)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $\kappa$.

6 pages