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

Abstract

We show that, for a finite set $A$ of real numbers, the size of the set $$\frac{A+A}{A+A}=\left\{\frac{a+b}{c+d}:a,b,c,d\in A,c+d\ne 0\right\}$$ 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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