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a/math posted by smh 1 year ago

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.