A proof of a sumset conjecture of Erd\H{o}s

@article{Moreira2018APO,
  title={A proof of a sumset conjecture of Erd\H\{o\}s},
  author={Joel Moreira and Florian Karl Richter and D. Robertson},
  journal={arXiv: Combinatorics},
  year={2018}
}
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. 
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