A product theorem in free groups

If $A$ is a finite subset of a free group with at least two noncommuting elements, then $|A\cdot A\cdot A|\geqslant\frac{|A|^2}{(\log|A|)^{O(1)}}$. More generally, the same conclusion holds in an arbitrary virtually free group, unless AA generates a virtually cyclic subgroup. The central part of the proof of this result is carried on by estimating the number of collisions in multiple products $A_1\cdot\ldots\cdot A_k$. We include a few simple observations showing that in this “statistical” context the analogue of the fundamental Plünnecke–Ruzsa theory looks particularly simple and appealing.