Research proof index

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

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 $\log|A|+\log|B|-\log|A_x|-\log|B_y|\le34\bigl(d[U_A;U_B]-d[U_{A_x};U_{B_y}]\bigr).$