Some results on the size of sum and product sets of finite sets of real numbers

Abstract

Let $A$ and $B$ be finite subsets of positive real numbers. Solymosi gave the sum-product estimate $max\left(|A+A|,|A\cdot A|\right)\ge {\left(4\lceil log\left|A\right|\rceil \right)}^{-1∕3}|A{|}^{4∕3}$, where $\lceil \rceil$ is the ceiling function. We use a variant of his argument to give the bound

$$max\left(|A+B|,|A\cdot B|\right)\ge {\left(4\lceil log\left|A\right|\rceil \lceil log\left|B\right|\rceil \right)}^{-1∕3}\left|A{|}^{2∕3}\right|B{|}^{2∕3}.$$

(This isn’t quite a generalization since the logarithmic losses are worse here than in Solymosi’s bound.)

Suppose that $A$ is a finite subset of real numbers. We show that there exists an $a\in A$ such that $|aA+A|\ge c|A{|}^{4∕3}$ for some absolute constant $c$.