Research proof index

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

Polynomial Freiman-Ruzsa theorem for bounded-torsion groups. Let $G$ be a finite abelian group in which $mx=0$ for every $x$, with $m\ge2$. If $A\subseteq G$ is nonempty and $|A+A|\le K|A|$, then there are a subgroup $H\le G$ and a set $c$ such that $A\subseteq c+H$, $|H|\le|A|$, and $|c|<mK^{256m^3+1}$.

An asymmetric weak-PFR estimate. Let $A,B$ be nonempty finite subsets of a rank-$n$ free $\mathbb Z$-module $G$. There are a subgroup $N\le G$, cosets $x,y\in G/N$, and nonempty fibers $A_x=\{a\in A:a+N=x\}$ and $B_y=\{b\in B:b+N=y\}$ such that $n\log2\le\log|G/N|+40d[U_A;U_B]$ and $\log|A|+\log|B|-\log|A_x|-\log|B_y|\le34\bigl(d[U_A;U_B]-d[U_{A_x};U_{B_y}]\bigr).$