Let $\mathbb{N}=A\cup B$ where $A$ and $B$ are disjoint. Let $S(A)$ be those integers which are the sum of finitely many distinct elements of $A$, and similarly for $S(B)$. If$$\overline{\delta }(X)=\underset{x\to \infty }{\mathrm{limsup}}\frac{1}{\log x}\sum _{\begin{array}{c}n\in X\\ n<x\end{array}}\frac{1}{n}$$is the upper logarithmic density of $X$ then how large must$$\max (\overline{\delta }(S(A)),\overline{\delta }(S(B)))$$be?