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If A is a nite subset of a free group with at least two noncommuting elements then jA ¡ A ¡ Aj ! jAj 2 (log jAj) O(1) . More generally, the same conclusion holds in an arbitrary virtually free group, unless A 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 ¡ : : : ¡ A k . We include a few simple observations showing that in this \statistical" context the analogue of the fundamentaldoi:10.4007/annals.2014.179.2.1 fatcat:h3yfxnbfr5fuvopccnioslbffy