Abstract We show that for a finite, nonempty subset A of a group, the quotient set A −1 A ≔ a 1 −1 a 2: a 1, a 2 ∈ A has size A^-1 A 5 3 A ∣ A − 1 A ∣≥ 5 3 ∣ A ∣, unless A is densely contained in a coset, or in a union of two cosets of a finite subgroup.
Vsevolod F. Lev (Sun,) studied this question.