Some properties of the * operation on families of special subsets of R. Algebraic sum of a perfect set and large subsets of R

In the first part of this talk I will present the following lemma. Assume CH. Given a non trivial ideal F of subsets of R which is Borel supported, translation invariant, with cov(F) =c/the continuum/ and a set A which does not belong to F, there is an X such that for every Y from F , X+Y is not equal to R but X+A = R. This lemma leads to the following theorem of G.Horbaczewska and S.Lindner: Suppose that CH holds. Then M=SN* and N=SM*, where M denotes meager sets, N stands for null sets, SN is the family of strongly measure zero sets, and SM is the class of strongly meager sets. In the second part I will discuss two problems related to the algebraic sum of a perfect set and large subsets of R.