Additive decompositions of sets with restricted prime factors

We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by S\'{a}rk\"ozy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.