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

$$\frac{A+A}{A+A} = \left\{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right \}$$

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 (2016).