A Combinatorial Method for Counting Smooth Numbers in Sets of Integers

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In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and we apply our result to give an easy proof of a result which says that dense subsets A and B of 1,2,...,x always produce asymptotically the expected number of xʳ - smooth sums a+b, where a in A and b in B. Recall that a number n is said to be y-smooth if all its prime divisors are at most y.