Research proof index

Let $T_1,T_2,T_3$ be measurable random variables in a finite abelian group with $T_1+T_2+T_3=0$, and set $\delta=I[T_1:T_2]+I[T_1:T_3]+I[T_2:T_3]$. For any measurable $Y_1,\ldots,Y_n$ and any $\alpha>0$, there is a measurable random variable $U$ such that $d[U;U]+\alpha\sum_{i=1}^n d[Y_i;U]\le\left(2+\frac{\alpha n}{2}\right)\delta+\alpha\sum_{i=1}^n d[Y_i;T_2].$

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 $|c|<mK^{256m^3+1}$.