Motivated by questions asked by Erdős, we prove that any set
A
⊂
N
A\subset \mathbb {N}
with positive upper density contains, for any
k
∈
N
k\in \mathbb {N}
, a sumset
B
1
+
⋯
+
B
k
B_1+\cdots +B_k
, where
B
1
B_1
, …,
B
k
⊂
N
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
k=2
.