Sum-product conjecture

Let $A$ be a finite subset of real numbers. Define the sumset and productset by

$$A+A=\\{a+b:a,b\in A\\},\qquad A\cdot A=\\{ab:a,b\in A\\}.$$

Sum-product conjecture. For arbitrarily small positive $\epsilon$,

$$\max(|A+A|,|A\cdot A|)\geqslant\mathcal{O}(|A|^{2-\epsilon}).$$

This is a central conjecture in additive combinatorics concerning simultaneous additive and multiplicative structure; the supplied text does not state its resolution status.