Sumset and inverse sumset theorems for Shannon entropy

TL;DR

This paper extends sumset and inverse sumset theorems from finite sets to discrete random variables using Shannon entropy, classifying variables with small doubling and establishing entropy bounds.

Contribution

It introduces entropy analogues of classical sumset results, characterizing random variables with small doubling as sums of structured components and noise.

Findings

Classifies random variables with small doubling as sums of structured and noise components.

Establishes a sharp lower bound on the entropy of sum of a variable with itself in torsion-free groups.

Provides entropy-based inverse sumset theorems analogous to classical combinatorial results.

Abstract

Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others characterises those finite sets $A$ for which $A+A$ is small. In this paper we establish analogous results in which the finite set $A \subset G$ is replaced by a discrete random variable $X$ taking values in $G$, and the cardinality $|A|$ is replaced by the Shannon entropy $\mathrm{Ent}(X)$. In particular, we classify the random variable $X$ which have small doubling in the sense that $\mathrm{Ent}(X_1+X_2) = \mathrm{Ent}(X)+O(1)$ when $X_1,X_2$ are independent copies of $X$, by showing that they factorise as $X = U+Z$ where $U$ is uniformly distributed on a coset progression of bounded rank, and $\mathrm{Ent}(Z) = O(1)$. When $G$ is torsion-free, we also establish the sharp lower bound $\mathrm{Ent}(X+X) \geq \mathrm{Ent}(X) + {1/2} \log 2 - o(1)$, where $o(1)$ goes to zero as $\mathrm{Ent}(X) \to \infty$.