On the Sum of Two Sets of Integers

Abstract

In his beautiful paper: proof of the fundamental theorem on the density of sums of sets of positive integers' Mr. Mann succeeded in proving the (a, A)hypothesis and a generalization of it that had been conjectured for more than ten years. We found that his method can be simplified considerably and even yields some stronger results. Let A, B respectively be sets of nonnegative integers a, b. Let C = A + B be the set of all integers of the form a + b. Let A(x), B(x), C(x) denote the number of positive integers of the sets <x.2 Mr. Mann proved the following theorem: If 0 C A and O C B, and if C(n) < n, then