On thin-complete ideals of subsets of groups - Banakh, T. O.; Lyaskovska, N. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if$xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $τ(\mathcal{F})$.If $τ(\mathcal{F}) = \mathcal{F}$, then $\mathcal{F}$ is called thin-complete. The thin-completion $τ*(\mathcal{F})$ of $\mathcal{F}$ is the smallest thin-complete subfamily of $\mathcal{P}_G$ that contains $\mathcal{F}$.Answering questions of Lutsenko and Protasov, we prove that a set $A \subset G$ belongs to $τ*(G)$ if and only if for any sequence $(g_n)_{n\in ω}$ of non-zero elements of G there is $n\in ω$ such that$$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$Also we prove that for an additive family $\mathcal{F} \subset \mathcal{P}_G$ its thin-completion $τ*(\mathcal{F})$ is additive. If the group $G$ is countable and torsion-free, then the completion $τ*(\mathcal{F}_G)$ of the ideal $\mathcal{F}_G$ of finite subsets of $G$ is coanalytic and not Borel in the power-set $\mathcal{P}_G$ endowed with the natural compact metrizable topology.