  Every set $A$ of positive integers with upper Banach density 1 contains an
infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that
$B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I}
B_i \subseteq A$ for every nonempty finite set $I$ of positive integers.
