On sets with small additive doubling in product sets

TextFollowing the sum-product paradigm, we prove that for a set B   of polynomial growth, the product set B.BB.B cannot contain large subsets with small doubling and size of order |B|2|B|2. It follows that the additive energy of B.BB.B is asymptotically o(|B|6)o(|B|6). In particular, we extend to sets with small doubling and of polynomial growth the classical Multiplication Table theorem of Erdős which says that |[1…n].[1…n]|=o(n2)|[1…n].[1…n]|=o(n2).VideoFor a video summary of this paper, please visit http://youtu.be/GKgpM1OcJVM.