Let , and let G be the difference set.
D:=A-A
Suppose that any strictly more than representations of the form |A|/2 with d=a'-a''. I claim that then a',a''Î A is a subgroup of D: indeed, by the pigeonhole principle for any G there exist a pair of representations d_{1}, d_{2}Î D such that d_{1}=a_{1}'-a_{1}'', d_{2}=a_{2}'-a_{2}'''', and it follows that a_{1}''=a_{2}.
d_{1}-d_{2}=a_{1}'-a_{2}'Î D
Assume now that any at least representations as |A|/2 with d=a'-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',a''Î A, where A:=HÈ (g+H) is a finite subgroup and H<G is chosen so that the order of gÎ G in the factor-group g is at least five. Then G/H is not a subgroup, but rather a union of three cosets. At the same time, it is easily seen that any D=(-g+H)È HÈ (g+H) has at least dÎ D representations of the form |H|=|A|/2.
d=a'-a''
The question is whether the counterexample above is unique. In other words, given that any representations as |A|/2, is it necessarily true that d=a'-a'' is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the D boundary.
|A|/2
Problem.
Suppose that any representations of the form |A|/3 with d=a'-a''. Is it necessarily true that a',a''Î A is either a subgroup or a union of three cosets?
D |