Overview
This document presents a concise proof concerning Erd\H{o}s's $B+C$ conjecture, specifically addressing sets of natural numbers with positive upper Banach density. The core finding establishes that such sets can be represented as the sum of two infinite sets. The proof is described as simplifying earlier existing proofs of this phenomenon.
Research Context
The research is situated within the field of number theory, focusing on properties of sets of natural numbers. The central topic is Erd\H{o}s's $B+C$ conjecture. This conjecture deals with the decomposition of sets into sums of other sets. The specific condition under examination is the presence of a positive upper Banach density within a set of natural numbers.
Approach
The approach involves constructing a short proof. The methodology focuses on demonstrating that a set of natural numbers with a positive upper Banach density inherently possesses the property of being expressible as the sum of two infinite sets. The description explicitly states that this new approach offers a simplification compared to prior proofs.
Findings
- Every set of natural numbers with a positive upper Banach density contains the sum of two infinite sets.
- The presented proof for this fact is a short proof.
- The approach simplifies earlier existing proofs regarding this conjecture.