  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 $|[1..n]. [1..n]| = o(n^2)$.
