On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups

We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another description: if a finite subset A of an abelian group G is such that the distribution of the sums $a+b$ with (a,b) ∈ A × A is only slightly more spread out than the uniform distribution on A, then A has small symmetric difference with some finite coset of G. The resulting bounds are necessarily sharp up to a logarithmic factor.