On sets with small additive doubling in product sets

Abstract

Following the sum-product paradigm, we prove that for a set $B$ with polynomial growth, the product set $B.B$ cannot contain large subsets with size of order $|B{|}^2$ with small doubling. It follows that the additive energy of $B.B$ is asymptotically $o(|B{|}^6)$. In particular, we extend to sets of small doubling and polynomial growth the classical Multiplication Table theorem of Erdős saying that $|[\mathrm{1..}n].[\mathrm{1..}n]|=o(n^2)$.

References in corpus (2)

• Energies and structure of additive sets
• Using the subspace theorem to bound unit distances