Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series

TL;DR

This paper proves that any set of integers contains a large sum-free subset of size at least one-third of the original set plus a logarithmic correction, solving a longstanding problem in additive combinatorics.

Contribution

It establishes a new lower bound on the size of sum-free subsets in integer sets, improving previous results by incorporating $L^1$-estimates for trigonometric series.

Findings

Existence of sum-free subset of size at least n/3 + c log log n

Answers a longstanding problem posed by Erdős

Uses $L^1$-estimates for trigonometric series in the proof

Abstract

A set $B$ is said to be \emph{sum-free} if there are no $x,y,z\in B$ with $x+y=z$. We show that there exists a constant $c>0$ such that any set $A$ of $n$ integers contains a sum-free subset $A'$ of size $|A'|\geqslant n/3+c\log \log n$. This answers a longstanding problem in additive combinatorics, originally due to Erd\H{o}s.