  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.
