# Multiplicative energy and Cauchy-Schwartz

Let $$A$$ be a finite set in a ring, and define $$E(A) =\left|\left\{(a, b, c, d) \in {A}^{4}: c a=d b\right\}\right|.$$ A number of papers (e.g. here) quote the lower bound $$E({A}) \geq \frac{|{A}|^{4}}{|{A} {A}|}$$ where $$AA = \{ab:a,b\in A\}$$, and say it follows from Cauchy-Schwartz. However, I cannot find a proof, so: how does the above inequality follow from an application of Cauchy-Schwartz?

Let $$|AA|=m$$ and write $$AA=\{e_1,\ldots,e_m\}$$.
Let $$a_i$$ be the number of ways to write $$e_i$$ as the product of two elements of $$A$$. We have $$a_1+\cdots+a_m=|A|^2$$.
But also, $$E(A)=a_1^2+\cdots+a_m^2$$.
Let $$u=(a_1,\ldots,a_m)$$ and $$v=(1/\sqrt{m},\ldots,1/\sqrt{m})$$. Apply Cauchy-Schwarz to these vectors.