An update on the sum-product problem

Abstract

AbstractWe improve the best known sum-product estimates over the reals. We 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 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.