Multiplicative energy
= Multiplicative energy
{title2=$E(A,B)$}
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 $E(A,B)\geq |A|^2|B|^2/|A\cdot B|$.