Sumsets with restricted number of prime factors

Abstract

In 1934, Erdős and Turan proved that if a1, a2, . . . , an are distinct positive integers such that the product Π1≤i<j≤n(ai + aj) has at most k distinct prime factors, then n < 3 × 2k − 1. In this paper, we introduce the notion of the p-graph of a set of positive integers and improve the bound to n ≤ 2k. In particular, we prove that the bound 2k is optimal for k = 2. Beyond these, we also obtain some related results and pose several problems for further research.