PFR

We have I[Z_1 : Z_2 | W], I[Z_2 : Z_3 | W], I[Z_1 : Z_3 | W] ≤ 4m^2 η k.

We have $\bbH[W] \leq (2m-1)k + \frac1m \sum_{i=1}^m \bbH[X_i]$.

We have $\bbH[Z_2] \leq (8m^2-16m+1) k + \frac{1}{m} \sum_{i=1}^m \bbH[X_i]$.

We have $\bbI[W : Z_2] \leq 2(m-1) k$.

We have $\sum_{i=1}^m d[X_i;Z_2|W] \leq 4(m^3-m^2) k$.

Let $G$ be an abelian group, let $(T_1,T_2,T_3)$ be a $G^3$-valued random variable such that $T_1+T_2+T_3=0$ holds identically, and write [ \delta := \bbI[T_1 : T_2] + \bbI[T_1 : T_3] + \bbI[T_2 : T_3]. ] Let $Y_1,\dots,Y_n$ be some further $G$-valued random variables and let $\alpha>0$ be a constant. Then there exists a random variable $U$ such that

We have $k = 0$.

Suppose that $G$ is a finite abelian group of torsion $m$. Suppose that $X$ is a $G$-valued random variable. Then there exists a subgroup $H \leq G$ such that [ d[X;U_H] \leq 64 m^3 d[X;X].].

A uniform distribution on a set with doubling constant K has self Rusza distance at most log K.

Every subgroup H of a finite m-torsion abelian group G contains a subgroup H' of order between k and mk, if 0 < k < |H|.

Suppose that $G$ is a finite abelian group of torsion $m$. If $A \subset G$ is non-empty and $|A+A| \leq K|A|$, then $A$ can be covered by most $mK^{64m^3+1}$ translates of a subspace $H$ of $G$ with $|H| \leq |A|$.

Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset.

If G is torsion-free and X, Y are G-valued random variables then d[X; 2Y] ≤ 5d[X; Y].

Let $G=\mathbb{F}_2^n$ and X, Y be G-valued random variables such that [\mathbb{H}(X)+\mathbb{H}(Y)> (20/\alpha) d[X;Y],] for some $\alpha > 0$. There is a non-trivial subgroup $H\leq G$ such that [\log \lvert H\rvert <(1+\alpha)/2 (\mathbb{H}(X)+\mathbb{H}(Y))] and [\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))< \alpha (\mathbb{H}(X)+\mathbb{H}(Y))] where $\psi:G\to G/H$ is the natural projection homomorphism.