Endgame for the Torsion PFR theorem #
Z_1+Z_2+Z_3= 0
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 $\sum_{i=1}^m d[X_i;Z_2|W] \leq 8(m^3-m^2) k$.
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 $$ d[U;U] + \alpha \sum_{i=1}^n d[Y_i;U] \leq \Bigl(2 + \frac{\alpha n}{2} \Bigr) \delta + \alpha \sum_{i=1}^n d[Y_i;T_2]. $$
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].]
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 at most $K ^ {(64m^3+2)/2}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of $G$ with $|H|/|A| \in [K^{-64m^3}, K^{64m^3}]$
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 $|H| \leq |A|$.