PFR

If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then

If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then

For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $S:=Y_1+Y_2+Y_3+Y_4$, $T_1:=Y_1+Y_2$, $T_2:=Y_1+Y_3$. Then

For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $T_1:=Y_1+Y_2, T_2:=Y_1+Y_3, T_3:=Y_2+Y_3$ and $S:=Y_1+Y_2+Y_3+Y_4$. Then

If $X_1, X_2$ is a $\phi$-minimizer, then $d[X_1;X_2] = 0$.

For η ≤ 1/8, there exist phi-minimizers X₁, X₂ at zero Rusza distance. For η < 1/8, all minimizers are fine, by dist_of_min_eq_zero. For η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.

For any random variables $Y_1,Y_2$, there exist a subgroup $H$ such that $2\rho(U_H) \leq \rho(Y_1) + \rho(Y_2) + 8 d[Y_1;Y_2].$

If $|A+A| \leq K|A|$, then there exists a subgroup $H$ and $t\in G$ such that $|A \cap (H+t)| \geq K^{-4} \sqrt{|A||V|}$, and $|H|/|A|\in[K^{-8},K^8]$.

If $A \subset {\bf F}_2^n$ is finite non-empty with $|A+A| \leq K|A|$, then there exists a subgroup $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$.

Corollary of better_PFR_conjecture in which the ambient group is not required to be finite (but) then $H$ and $c$ are finite.

$d[X_1+\tilde X_1; X_2+\tilde X_2] \geq k - \frac{\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$

$I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$

A pair of measures minimizing $\tau$ exists.

A pair of random variables minimizing $τ$ exists.

For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one can lower bound $d[X'_1|Z;X'_2|W]$ by $k - \eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ).$