On a combinatorial method for counting smooth numbers in sets of integers

Abstract

In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums a + b , a ∈ A , b ∈ B . The proof of this result is completely combinatorial and elementary.