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

TL;DR

This paper establishes an entropy inequality related to Marton’s conjecture, showing that sets with nearly uniform sum distributions in abelian groups are close to cosets, with bounds tight up to a logarithmic factor.

Contribution

It introduces a new entropy inequality that links sum distribution spread to structural proximity to cosets in abelian groups.

Findings

Sets with nearly uniform sum distributions are close to cosets.

Bounds are sharp up to a logarithmic factor.

Provides a black box for related combinatorial inequalities.

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.