If $(A+A)/(A+A)$ is small then the ratio set is large

TL;DR

This paper establishes a new lower bound on the size of the ratio set derived from sumsets, showing that if the ratio set is small, then the original set must have a large ratio set, advancing sum-product problem understanding.

Contribution

The paper provides an improved lower bound for the ratio set size in terms of the set size and ratio set, refining previous results and addressing related conjectures.

Findings

If the ratio set is small, the original set's ratio set must be large.

The new bound improves previous results for subquadratic ratio sets.

Answers a conjecture related to sum-product estimates in additive combinatorics.

Abstract

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