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A short proof of Erdős's $B+C$ conjecture

Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson

Abstract

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.

A short proof of Erdős's $B+C$ conjecture

Abstract

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.

Paper Structure

This paper contains 4 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from ergodic theory
  3. Reformulation from combinatorics to ergodic theory
  4. Proof of main dynamical result

Key Result

Theorem 1.1

For any $A\subset\mathbb{N}$ with positive upper Banach density, there exist infinite sets $B,C\subset \mathbb{N}$ such that $B+C=\{b+c:b\in B,~c\in C\}\subset A$.

Theorems & Definitions (12)

  • Theorem 1.1: Erdős's $2^{\mathrm{nd}}$ sumset conjecture
  • Theorem 1.2: Erdős's $1^{\mathrm{st}}$ sumset conjecture
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5: Correspondence principle
  • Theorem 3.1: Main dynamical result
  • proof : Proof that \ref{['thm_main_dynamical']} implies \ref{['thm_mrr']}
  • proof : Proof of \ref{['thm_main_dynamical']}
  • ...and 2 more