An improved bound on $(A+A)/(A+A)$

We show that, for a finite set $A$ of real numbers, the size of the set $\frac{A+A}{A+A} = \left\{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right \}$ is bounded from below by $\left|\frac{A+A}{A+A} \right| \gg \frac{|A|^{2+1/4}}{|A / A|^{1/8} \log |A|}.$ This improves a result of Roche-Newton.

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