Sumset and inverse sumset theorems for Shannon entropy

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 $\mathrm{\mid }A\mathrm{\mid }$ is replaced by the Shannon entropy $\mathrm{E}\mathrm{n}\mathrm{t}(X)$. In particular, we classify the random variable $X$ which have small doubling in the sense that $\mathrm{E}\mathrm{n}\mathrm{t}(X_1+X_2)=\mathrm{E}\mathrm{n}\mathrm{t}(X)+O(1)$ 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 $\mathrm{E}\mathrm{n}\mathrm{t}(Z)=O(1)$. When $G$ is torsion-free, we also establish the sharp lower bound $\mathrm{E}\mathrm{n}\mathrm{t}(X+X)\ge \mathrm{E}\mathrm{n}\mathrm{t}(X)+1\mathrm{/}2\log 2-o(1)$, where $o(1)$ goes to zero as $\mathrm{E}\mathrm{n}\mathrm{t}(X)\to \mathrm{\infty }$.