### Difference sets in abelian groups

Let A be a finite non-empty subset of an abelian group G, and let D:=A-A be the difference set.

Suppose that any dÎ D has strictly more than |A|/2 representations of the form d=a'-a'' with a',a''Î A. I claim that then D is a subgroup of G: indeed, by the pigeonhole principle for any d1, d2Î D there exist a pair of representations d1=a1'-a1'', d2=a2'-a2'' such that a1''=a2'', and it follows that d1-d2=a1'-a2'Î D.

Assume now that any dÎ D is only guaranteed to have at least |A|/2 representations as d=a'-a'' with a',a''Î A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:=HÈ (g+H), where H<G is a finite subgroup and gÎ G is chosen so that the order of g in the factor-group G/H is at least five. Then D=(-g+H)È HÈ (g+H) is not a subgroup, but rather a union of three cosets. At the same time, it is easily seen that any dÎ D has at least |H|=|A|/2 representations of the form d=a'-a''.

The question is whether the counterexample above is unique. In other words, given that any dÎ D has at least |A|/2 representations as d=a'-a'', is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 boundary.

Problem. Suppose that any dÎ D:=A-A has more than |A|/3 representations of the form d=a'-a'' with a',a''Î A. Is it necessarily true that D is either a subgroup or a union of three cosets?