Remarks on sets with small differences and large sums

We use a surprising example of a subset of group Z12 to construct the set X⊂R with int(X−X)=∅ and int(X+X)≠∅. We also build an ideal J on a compact abelian topological group G such that J has the closed (1,1)-Steinhaus property and does not have the closed (1,−1)-Steinhaus property.