Additive decompositions of sets with restricted prime factors

TL;DR

This paper proves that certain sets with restricted prime factors, like smooth numbers, cannot be expressed as a sum of three sets, confirming a conjecture and refining results on prime sumset decompositions.

Contribution

It establishes that smooth numbers cannot be decomposed into ternary sumsets, confirming Sárközy's conjecture, and improves understanding of prime number sumset structures.

Findings

Smooth numbers cannot be written as ternary sumsets

Confirmed Sárközy's conjecture on sumset decompositions

Sharpened results on sumset decompositions of primes

Abstract

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.