The sum-product problem for integers with few prime factors

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbb{Z}$, one can improve estimates for $\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }A\cdot A\mathrm{\mid }\}$, under the constraint that all integers involved have a bounded number of prime factors – that is, each $a\in A$ satisfies $\omega (a)\le k$. In this paper, answer Szemerédi’s question in the affirmative by showing that this maximum is of order $\mathrm{\mid }A{\mathrm{\mid }}^{\frac{5}{3}-o(1)}$ provided $k\le (\log \mathrm{\mid }A\mathrm{\mid }{)}^{1-\epsilon }$ for some $\epsilon >0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $\mathrm{\mid }A{\mathrm{\mid }}^{o(1)}$.