The sum-product problem for integers with few prime factors

Abstract. It was asked by E. Szemer´edi if, for a finite set A ⊂ Z, one can improve estimates for max{|A + A|, |A · A|}, under the constraint that all integers involved have a bounded number of prime factors – that is, each a ∈ A satisfies ω(a) ≤ k. In this paper, answer Szemer´edi’s question in the affirmative by showing that this maximum is of order|A|^{5/3 −o(1)} provided k ≤ (log |A|) ^1−ε for some ε > 0. In fact, this will follow from an estimatefor additive energy which is best possible up to factors of size |A| o⁽¹⁾.