Erdős sum-product conjecture

Prove that for every finite set A of real numbers and every ε>0, max(|A+A|,|AA|)≫ε |A|^(2−ε), equivalently that the optimal sum-product exponent is c=2.

Background

The sum-product problem asks how large at least one of the additive sumset A+A and multiplicative product set AA must be for a finite set of real numbers. The paper formulates the problem through the largest exponent c for which max(|A+A|,|AA|)≫ε |A|^c−ε.

Known results give exponents only slightly above 4/3, while the conjectured value c=2 would express the strongest possible incompatibility between additive and multiplicative structure. The paper improves the best known exponent marginally but leaves the conjecture unresolved.