The Dicks–Ivanov problem and the Hamidoune problem

We report on what we call the Hamidoune problem, inspired by a problem by Dicks and Ivanov. The problem asks if the inequality | A | + | B | − 1 2 | A B | − 1 2 | A ⋅ 2 B | ⩽ max { 2 , | g H | : H ⩽ G , g ∈ G , g H ⊆ A ⋅ 2 B } holds when A and B are finite subsets of a group G , each one with at least two elements, and A ⋅ 2 B denotes the set of elements which can be written in at least two different ways as a product of one element in A and one in B .