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A note on the largest sum-free sets of integers

Yifan Jing, Shukun Wu

Abstract

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+ω(N)$ for some increasing unbounded function $ω$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest.

A note on the largest sum-free sets of integers

Abstract

Given a set of positive integers, an old question in additive combinatorics asks that whether contains a sum-free subset of size at least for some increasing unbounded function . The question is generally attacked in the literature by considering another conjecture, which asserts that as , . This conjecture, if true, would also imply that a similar phenomenon occurs for -sum-free sets for every . In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set , which might be of independent interest.

Paper Structure

This paper contains 4 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. When $A$ is geometric
  3. A density estimate
  4. When $A$ is not geometric

Key Result

Theorem 1.1

For every $k\geq1$, there is a function $\omega(N)=\log N/\log\log N$, such that for every set $A$ of $N$ positive integers, there exists a maximal $(2k,4k)$-sum-free set $\Omega(2k,4k)\subseteq\mathbb{R}/\mathbb{Z}$, and we have As a consequence, there is an absolute constant $c>0$, such that

Theorems & Definitions (20)

  • Conjecture 1: The sum-free conjecture, combinatorial form
  • Conjecture 2: The sum-free conjecture, analytic form
  • Conjecture 3
  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3: Large sum-free subsets in lacunary sets
  • Theorem 2.4: Littlewood--Paley
  • ...and 10 more