Balog's conjecture on the size of $AA+A$

Let $A e othing$ be a finite set of positive real numbers, and define

$$AA+A:=\{ab+c:a,b,c\in A\}.$$

Balog's conjecture. For every such set $A$,

$$|AA+A|\geq |A|^2.$$

The conjecture was proposed as an analogue of the Erdős–Szemerédi sum-product conjecture. It is refuted by the construction in this paper, which gives arbitrarily large finite sets of rational numbers for which $|AA+A|$ is smaller than $|A|^2$ by a factor involving a power of $\log\log|A|$.