**Abstract:** In this paper we show that every set \(A \subset \mathbb{N}\) with positive density contains \(B + C\) for some pair \(B, C\) of infinite subsets of \(\mathbb{N}\), settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.