On the inverse Goldbach problem

We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq \mathbb{N}$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$\frac{\sqrt{x}}{\log x}\ll A(x),B(x)\ll \sqrt{x}\mathrm{.}$$ Second, we show that, if $A,B,C\subseteq \mathbb{N}$ have at least two elements each, then $$((A+B+C)△\mathbb{P})(x)\gg \frac{\sqrt{x}}{(\log x{)}^2}\mathrm{.}$$ An analogous estimate is obtained for sumsets of $k\ge 3$ sets. The proofs use Montgomery’s large sieve, Gallagher’s larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.