An update on the sum-product problem

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.