If $A\subseteq \mathbb{N}$ is a multiset of integers such that$$|A\cap \{1,\dots ,N\}|\gg N$$for all $N$ then must $A$ be subcomplete? That is, must$$P(A)=\left\{\sum _{n\in B}n:B\subseteq A\text{ finite }\right\}$$contain an infinite arithmetic progression?