On sums and products of integers

Abstract

Erdös and Szemerédi proved that if A A is a set of k k positive integers, then there must be at least c k 1 + δ ck^{1+\delta } integers that can be written as the sum or product of two elements of A A , where c c is a constant and δ > 0 \delta >0 . Nathanson proved that the result holds for δ = 1 31 \delta =\frac 1{31} . In this paper it is proved that the result holds for δ = 1 5 \delta =\frac 15 and c = 1 20 c=\frac 1{20} .