The sum-product problem for integers with few prime factors

Abstract It was asked by E. Szemerédi if, for a finite set $A\subset {\mathbb {Z}}$ , one can improve estimates for $\max \{|A+A|,|A\cdot A|\}$ , under the constraint that all integers involved have a bounded number of prime factors, that is, each $a\in A$ satisfies $\omega (a)\leq k$ . In this paper we show that this maximum is at least of order $|A|^{\frac {5}{3}-o_\epsilon (1)}$ provided $k\leq (\log |A|)^{1-\varepsilon }$ for any $\varepsilon \gt 0$ . In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$ .