Problem #431: Are there two infinite sets $A$ and $B$ such that $A+B$...

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Progress

A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper [ElHa15], who showed that if $A,B$ are such sets then for all large $x$ we must have\[\frac{x^{1/2}}{\log x\log\log x} \ll \lvert A \cap [1,x]\rvert \ll x^{1/2}\log\log x\]and similarly for $B$.

Elsholtz [El01] has proved there are no sets $A,B,C$ (all of size at least $2$) such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.

Granville [Gr90] proved, conditional on the prime $k$-tuples conjecture, that there are infinite sets $B$ and $C$ such that\[\{ \tfrac{b+c}{2}: b\in B, c\in C\}\]is a subset of the primes. Tao and Ziegler [TaZi23] gave an unconditional proof that there are infinite sets $B=\{b_1<\cdots\}$ and $C=\{c_1<\cdots\}$ such that\[\{ b_i+c_j : b_i\in B, c_j\in C, i<j\}\]is a subset of the primes.

See also [429] and [432].