Erdős's sum-product conjecture for positive integers

Let $A$ be a finite set of positive integers, and write

$$A+A=\{x+y:x,y\in A\},\qquad AA=\{xy:x,y\in A\}.$$

Erdős's sum-product conjecture. For every $\epsilon>0$ there is an $n_0$ such that, whenever $|A|\ge n_0$,

$$\big|(A+A)\cup(AA)\big|\ge |A|^{2-\epsilon}.$$

This is the classical sum-product conjecture, originating with Erdős and later substantially advanced by Erdős and Szemerédi. The source presents equivalent formulations using sum-product pairs and notes that the conjecture is not supported by the finite dataset examined; its general status remains open.