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

TL;DR

This work connects distributional entropy with additive structure in finite abelian groups by showing that if a finite subset $A$ has its sum distribution $a+b$ (with $a,b$ chosen uniformly from $A$) close to uniform on $A$, then $A$ is close in symmetric difference to a coset of a finite subgroup $H$, quantified up to a logarithmic factor. The authors deploy an entropic Marton-type black box from Green, Manners, and Tao to relate the entropy gap $ΔH(A)$ to a Rusza-type distance to a subgroup, and then translate this into a concrete coset approximation. This yields a sharp distributional analogue of Freiman-type structure theorems for abelian groups, bridging information theory and additive combinatorics with explicit, computable bounds. The results advance understanding of when near-uniform sum distributions force algebraic structure, with potential implications for polynomial Freiman–Ruzsa-type theories and coding-theoretic applications in finite 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.

Theorems & Definitions (16)

• Theorem 1.1: GGMT2, part of Theorem 1.1
• Example 1.2
• Definition 1.3
• Example 1.4
• proof
• Theorem 1.5
• Lemma 2.1
• proof
• Proposition 2.2: GGMT2, Theorem 1.3
• Proposition 2.3: GMT, Proposition 1.3