It was asked by E. Szemer\'edi 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, answer Szemer\'edi's question in the affirmative by showing that this maximum is of order $|A|^{ \frac{ 5}{ 3}-o(1)}$ provided $k\leq (\log|A|)^{ 1-\epsilon}$ for some $\epsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{ o(1)}$.
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