Product set ($A\cdot B$)

The product set of two sets of real numbers is $A\cdot B=\{ab:a\in A,b\in B\}$. It is a multiplicative counterpart of a sumset, not the Cartesian product $A\times B$.

For finite sets of nonzero real numbers, the multiplicative energy is the number of quadruples $(a,b,c,d)\in A\times B\times A\times B$ satisfying $a/b=c/d$. Swapping $b,d$ shows that it also counts quadruples with $ab=cd$. Thus $E(A,B)={\sum }_x{r}_{A\cdot B}(x{)}^2$, where $r_{A\cdot B}(x)$ counts product representations. The Cauchy-Schwarz inequality yields $E(A,B)\ge |A{|}^2|B{|}^2/|A\cdot B|$.

For finite sets of positive real numbers $A,B$ with $|B|\ge 2$, $E(A,B)\le 4\lceil \log |B|\rceil |A+A||B+B|$, with natural or binary logarithm. Partition ray occupancies of $A\times B$ into $\lceil \log |B|\rceil$ classes of multiplicative width $e$, or width $2$ for the binary convention. Injectivity of sums on two distinct rays and disjoint open planar sectors bound each class's sum of squared occupancies by $4|A+A||B+B|$. A singleton class contributes at most $|A||B|$. Combining this with the energy lower bound gives a sum-product inequality.