On the size of the set $AA+A$

Abstract

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A|\gg |A{|}^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A\subset \mathbb{R}$ such that $|AA+A|=o(|A{|}^2)$, disproving a conjecture of Balog.

References in corpus (1)

• Some remarks on the asymmetric sum--product phenomenon