PFR

If $G=\mathbb{F}_2^d$ and X, Y are G-valued random variables and $\alpha < 1$ then there is a subgroup $H\leq \mathbb{F}_2^d$ such that [\log \lvert H\rvert \leq (1 + α) / (2 * (1 - α)) * (\mathbb{H}(X)+\mathbb{H}(Y))] and if $\psi:G \to G/H$ is the natural projection then [\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq 20/\alpha * d[\psi(X);\psi(Y)].]

If $G=\mathbb{F}_2^d$ and X, Y are G-valued random variables then there is a subgroup $H\leq \mathbb{F}_2^d$ such that [\log \lvert H\rvert \leq 2 * (\mathbb{H}(X)+\mathbb{H}(Y))] and if $\psi:G \to G/H$ is the natural projection then [\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq 34 * d[\psi(X);\psi(Y)].]

Let $\phi : G\to H$ be a homomorphism and $A,B\subseteq G$ be finite subsets. If $x,y\in H$ then let $A_x=A\cap \phi^{-1}(x)$ and $B_y=B\cap \phi^{-1}(y)$. There exist $x,y\in H$ such that $A_x,B_y$ are both non-empty and [d[\phi(U_A);\phi(U_B)]\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert} \leq (\mathbb{H}(\phi(U_A))+\mathbb{H}(\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).]

Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality $\log\frac{|A| |B|}{|A_x| |B_y|} ≤ 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$ and one has the dimension bound $n \log 2 ≤ \log |G/N| + 40 d[U_A; U_B]$.

In fact one has equality here, but this is trickier to prove and not needed for the argument.

If $A,B\subseteq \mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\subseteq A$ and $B'\subseteq B$ such that [\log\frac{\lvert A\rvert\lvert B\rvert}{\lvert A'\rvert\lvert B'\rvert}\leq 34 d[U_A;U_B]] such that $\max(\dim A',\dim B')\leq \frac{40}{\log 2} d[U_A;U_B]$.

If $A\subseteq \mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\leq \log K$ then there exists a non-empty $A'\subseteq A$ such that $\lvert A'\rvert\geq K^{-17}\lvert A\rvert$ and $\dim A'\leq \frac{40}{\log 2} \log K$.

Let $A\subseteq \mathbb{Z}^d$ and $\lvert A-A\rvert\leq K\lvert A\rvert$. There exists $A'\subseteq A$ such that $\lvert A'\rvert \geq K^{-17}\lvert A\rvert$ and $\dim A' \leq \frac{40}{\log 2} \log K$.