Erdős Problem #1211 — Logarithmic density of subset sums in a two-colouring

Partition $N=A⊔B$, and let $S(A)$ and $S(B)$ be the sets of finite sums of distinct elements of $A$ and $B$. If $\overline δ_{\log}(X)=\limsup_{x→∞}(1/\log x)\sum_{n<x,n∈X}1/n$, how small can $\max\{\overline δ_{\log}(S(A)),\overline δ_{\log}(S(B))\}$ be?