An improved bound for the size of the set $A/A+A$

TL;DR

This paper proves a new lower bound on the size of the set formed by the sum of the quotient set and the original set for positive real numbers, improving previous bounds with a specific exponent and logarithmic factor.

Contribution

It introduces an improved lower bound for the set size of A/A+A, advancing the understanding of sum-product type problems in additive combinatorics.

Findings

Established that |A/A+A| is at least on the order of |A|^{3/2 + 1/26} divided by the square root of log |A|.

Provides a tighter bound compared to previous results in the sum-product problem.

Enhances the theoretical framework for analyzing sum and quotient sets of finite positive real sets.

Abstract

It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$