Constructing universally small subsets of a given packing index in Polish groups
Banakh, Taras, 1968- ; Lyaskovska, Nadia
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 ▫$\text{pack}^\sharp(A_\kappa) = \sup \{|\mathcal{D}|^+ : \mathcal{D} \subset \{gA\}_{g\in G}$▫ is disjoint▫$\}$▫ equal to ▫$\kappa$▫.
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