We show that for a finite, nonempty subset A of a group, the quotient set A⁻¹A:=₁⁻¹a₂ a₁,a₂∈ A\ has size |A⁻¹A|≥53\,|A|, unless A is densely contained in a coset, or in a union of two cosets of a finite subgroup.
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Vsevolod F. Lev (2024) studied this question.
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