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

Abstract: 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 $(a,b) \in A \times A$ 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.