Infinite sumsets in sets with positive density

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 .