On sets with small additive doubling in product sets

TL;DR

This paper proves that sets with polynomial growth cannot have large subsets with small additive doubling within their product sets, extending classical results and showing the additive energy is asymptotically small.

Contribution

It extends the classical Multiplication Table theorem to sets of small doubling and polynomial growth, revealing limitations on the structure of their product sets.

Findings

Product sets of polynomial growth cannot contain large small-doubling subsets.

Additive energy of such product sets is asymptotically smaller than |B|^6.

Generalizes Erdős's Multiplication Table theorem to broader classes of 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 Erd\H{o}s saying that $|[1..n]. [1..n]| = o(n^2)$.