More on the sum-product problem for integers with few prime factors

We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then $$\max (\mid A+A\mid ,\mid AA\mid )\ge \mid A{\mid }^{12\mathrm{/}7-o(1)}$$ and, for any $m\ge 2$, $$\max (\mid mA\mid ,\mid A^{(m)}\mid )\ge \mid A{\mid }^{\frac{2}{3}m+\frac{1}{3}-o(1)}\mathrm{.}$$ Finally, we show that if $A\subset \mathbb{Q}$ is a finite set of rationals in which the numerator and denominator of every $x\in A$ is divisible by $O(1)$ many primes then $\mid A+AA\mid \ge \mid A{\mid }^{2-o(1)}$.