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

TL;DR

This paper proves Erdős's conjecture that any positive-density subset of natural numbers contains a sumset of two infinite subsets, using decompositions into structured and pseudo-random parts, and extends the result to amenable groups.

Contribution

It provides a proof of Erdős's sumset conjecture for natural numbers and generalizes the approach to countable amenable groups.

Findings

Every positive-density subset of natural numbers contains a sumset of two infinite subsets.

The proof introduces a decomposition of sequences into structured and pseudo-random parts.

The methods are applicable to countable amenable groups.

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 Erd\H{o}s. 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.