Multiplicative energy and Cauchy-Schwartz

Question

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?

Answer 1

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.