On the size of the set $AA+A$

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