On thin-complete ideals of subsets of groups - Banakh, T. O.; Lyaskovska, N. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let F P_G be a left-invariant lower family of subsets of a group G. A subset A G is called F-thin if xA yA F for any distinct elements x, y G. The family of all F-thin subsets of G is denoted by τ(F). If τ(F) = F, then F is called thin-complete. The thin-completion τ*(F) of F is the smallest thin-complete subfamily of P_G that contains F. Answering questions of Lutsenko and Protasov, we prove that a set A G belongs to τ*(G) if and only if for any sequence (g_n)_nω of non-zero elements of G there is nω such that _i_0,...,i_n {0, 1 g_0^i_0...g_n^i_n A F. Also we prove that for an additive family F P_G its thin-completion τ*(F) is additive. If the group G is countable and torsion-free, then the completion τ*(F_G) of the ideal F_G of finite subsets of G is coanalytic and not Borel in the power-set P_G endowed with the natural compact metrizable topology.