A note on the largest sum-free sets of integers

TL;DR

This paper proves that large sum-free subsets of positive integers of size exceeding a third of the set exist, using a structural analysis approach that advances understanding in additive combinatorics.

Contribution

It provides a direct proof of the existence of large sum-free subsets, extending previous conjectures and introducing a novel structural analysis method.

Findings

Proves the existence of large sum-free subsets in positive integers.

Establishes a connection between sum-free sets and a conjecture involving exponential sums.

Introduces a new structural analysis technique for additive combinatorics.

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 $N/3+\omega(N)$ for some increasing unbounded function $\omega$. 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.