Some results on the size of sum and product sets of finite sets of real numbers

Abstract

Let A and B be finite subsets of positive real numbers. Solymosi gave the sumproduct estimate max.jAC Aj;jA Aj/ .4dlogjAje/ 1=3 jAj 4=3 , whered e is the ceiling function. We use a variant of his argument to give the bound max.jAC Bj;jA Bj/ .4dlogjAjedlogjBje/ 1=3 jAj 2=3 jBj 2=3 : (This isn’t quite a generalization since the logarithmic losses are worse here than in Solymosi’s bound.) Suppose that A is a finite subset of real numbers. We show that there exists an a2 A such thatja AC Aj cjAj 4=3 for some absolute constant c.