A proof of a sumset conjecture of Erdős

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.

54 pages. Corrected proof of Theorem 3.22 and added Example 3.27 Keywords: sum sets, almost periodic functions, ultrafilters