A proof of a sumset conjecture of Erdős

By Joel Moreira, F. Richter, D. Robertson. Annals of Mathematics, 2018

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

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