A proof of a sumset conjecture of Erdős


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 Erd\H os. 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.


Joel Moreira

Northwestern University, Evanston, IL

Florian K. Richter

Northwestern University, Evanston, IL

Donald Robertson

The University of Utah, Salt Lake City, UT