A weak polynomial Freiman–Ruzsa conjecture

Let $V$ be a $\mathbb{Z}$-module, and let $A\subseteq V$ be a finite set satisfying

$$|A+A|\leq K|A|$$

for some $K\geq 1$.

Weak polynomial Freiman–Ruzsa conjecture. There is an absolute constant $c>0$ such that one can find $A_1\subseteq A$ with

$$|A_1|>|A|/K^c$$

and elements $\xi_1,\dots,\xi_d\in V$ for some $d<c\log 2K$ satisfying

$$A_1\subseteq \mathbb{Z}\xi_1+\dots+\mathbb{Z}\xi_d.$$

This is a weaker version of the polynomial Freiman–Ruzsa conjecture, which seeks quantitatively optimal dependence on the doubling constant $K$ in Freiman's theorem. The surrounding discussion presents it as a central open problem in additive combinatorics with applications to the sum-product problem; no resolution of this weaker formulation is given here.