  The entropic doubling $\sigma_{\operatorname{ent}}[X]$ of a random variable
$X$ taking values in an abelian group $G$ is a variant of the notion of the
doubling constant $\sigma[A]$ of a finite subset $A$ of $G$, but it enjoys
somewhat better properties; for instance, it contracts upon applying a
homomorphism.
  In this paper we develop further the theory of entropic doubling and give
various applications, including:
  (1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew
dimension'' of subsets of $\mathbf{Z}^D$ with small doubling;
  (2) A new proof, and an improvement, of a result of the second author on the
dimension of subsets of $\mathbf{Z}^D$ with small doubling;
  (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$
implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.
