An update on the sum-product problem

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