The sum-product problem for integers with few prime factors

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, 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 our paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial (structural results for sets with small multiplicative doubling), analytical (backwards martingales) and number theoretical (the subspace theorem).
Passcode: a six digit number $N=(4!)^2+(p-5)^2$ where p is the smallest prime such that p>600.