  Motivated by questions asked by Erdos, we prove that any set
$A\subset{\mathbb N}$ with positive upper density contains, for any
$k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where
$B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof uses ergodic theory
and relies on structural results for measure preserving systems. Our techniques
are new, even for the previously known case of $k=2$.
