#### Vol. 2, No. 5, 2009

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

### Derrick Hart and Alexander Niziolek

Vol. 2 (2009), No. 5, 603–609
##### Abstract

Let $A$ and $B$ be finite subsets of positive real numbers. Solymosi gave the sum-product estimate $max\left(|A+A|,|A\cdot A|\right)\ge {\left(4⌈log|A|⌉\right)}^{-1∕3}|A{|}^{4∕3}$, where $⌈\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}⌉$ is the ceiling function. We use a variant of his argument to give the bound

$max\left(|A+B|,|A\cdot B|\right)\ge {\left(4⌈log|A|⌉⌈log|B|⌉\right)}^{-1∕3}\phantom{\rule{0.3em}{0ex}}|A{|}^{2∕3}\phantom{\rule{0.3em}{0ex}}|B{|}^{2∕3}.$

(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 $c$.

##### Keywords
sum-product estimate, multiplicative energy, Solymosi bound
##### Mathematical Subject Classification 2000
Primary: 11B13, 11B75