On the size of the set $AA+A$

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