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 $\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{12/7-o(1)}$ and, for any $m\geq 2$, $\max(\lvert mA\rvert, \lvert A^{(m)}\rvert) \geq \lvert A\rvert^{\frac{2}{3}m+\frac{1}{3}-o(1)}.$ 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 $\lvert A+AA\rvert \geq \lvert A\rvert^{2-o(1)}$.

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