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

TL;DR

This paper establishes a new lower bound on the size of the set formed by ratios of sums of elements from a finite real set, improving previous bounds by Roche-Newton.

Contribution

It provides an improved lower bound on the ratio set size involving sums and ratios, advancing understanding in additive combinatorics.

Findings

Lower bound on |(A+A)/(A+A)| involving |A| and |A/A|

Improves previous bounds by Roche-Newton

Enhances understanding of sum and ratio set interactions

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.