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

Abstract: 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^{17/10-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)}.]