Theorem 7.2 (Entropic Polynomial Freiman–Ruzsa). There exists an absolute constant $α$ satisfying the following: Let $G = 𝔽_{2}^{n}$ and let $X, Y$ be $G$ -valued random variables. Then there exists a subsgroup $H$ of $G$ such that

d [X; U_{H}] + d [U_{H}; Y] \leq α d [X; Y]

where $U_{H}$ is the uniform distribution on $H$ .