Borel sets without perfectly many overlapping translations, III

We expand the results of Rosłanowski and Shelah [11] , [10] to all perfect Abelian Polish groups ( H , + ) . In particular, we show that if α < ω 1 and 4 ≤ k < ω , then there is a ccc forcing notion adding a Σ 2 0 set B ⊆ H which has ℵ α many pairwise k –overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, X − X ⊆ Y − Y imply that a translation of X or − X is included in Y .