If $(A+A)\mathrm{/}(A+A)$ is small then the ratio set is large

In this paper, we consider the sum-product problem of obtaining lower bounds for the size of the set $$\frac{A+A}{A+A}:=\{\frac{a+b}{c+d}:a,b,c,d\in A,c+d\mathrm{\ne }0\},$$ for an arbitrary finite set $A$ of real numbers. The main result is the bound $$\mid \frac{A+A}{A+A}\mid \gg \frac{\mathrm{\mid }A{\mathrm{\mid }}^{2+\frac{2}{25}}}{\mathrm{\mid }A:A{\mathrm{\mid }}^{\frac{1}{25}}\log \mathrm{\mid }A\mathrm{\mid }},$$ where $A:A$ denotes the ratio set of $A$. This improves on a result of Balog and the author (arXiv:1402.5775), provided that the size of the ratio set is subquadratic in $\mathrm{\mid }A\mathrm{\mid }$. That is, we establish that the inequality $$\mid \frac{A+A}{A+A}\mid \ll \mathrm{\mid }A{\mathrm{\mid }}^2\Rightarrow \mathrm{\mid }A:A\mathrm{\mid }\gg \frac{\mathrm{\mid }A{\mathrm{\mid }}^2}{{\log }^{25}\mathrm{\mid }A\mathrm{\mid }}\mathrm{.}$$ This extremal result answers a question similar to some conjectures in a recent paper of the author and Zhelezov (arXiv:1410.1156).