Sárközy's conjecture on additive irreducibility of smooth numbers

Fix $0<\epsilon<1$. An integer is $y$-smooth if all of its prime factors are at most $y$, and let $\mathcal{S}_{n^{\epsilon}}$ denote the set of integers $n$ with no prime factor greater than $n^{\epsilon}$. A set is asymptotically additively irreducible if it has no asymptotic additive decomposition into two sets each containing at least two elements. Sárközy's conjecture. The set $\mathcal{S}_{n^{\epsilon}}$ is asymptotically additively irreducible. This is the smooth-number analogue of Ostmann's problem. The paper presents it as an open problem posed by Sárközy; the methods that solved the ternary prime problem do not directly transfer to this setting.