An update on the sum-product problem

Abstract: We improve the best known sum-product estimates over the reals. We prove that [ \max(|A+A|,|AA|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,, ] for a finite $A\subset \mathbb R$, following a streamlining of the arguments of Solymosi, Konyagin and Shkredov. We include several new observations to our techniques. Furthermore, [ |AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,. ] Besides, for a convex set $A$ we show that [ |A+A|\geq |A|^{\frac{30}{19}-o(1)}\,. ] This paper is largely self-contained.