On sets with small additive doubling in product sets

Journal article, 2015

Text: Following the sum-product paradigm, we prove that for a set B of polynomial growth, the product set B.B cannot contain large subsets with small doubling and size of order |B|2. It follows that the additive energy of B.B is asymptotically o(|B|6). In particular, we extend to sets with small doubling and of polynomial growth the classical Multiplication Table theorem of Erdos which says that |[1. . .n].[1. . .n]|=o(n2).