Maximal sets with no solution to x+y=3z

TL;DR

This paper proves a conjecture about the maximum size of subsets within [0,1] that contain no solutions to x+y=3z, providing an optimal measure bound and characterizing the extremal sets.

Contribution

It establishes the full conjecture by deriving an optimal measure bound for 3-sum-free sets and characterizes the sets that attain this maximum.

Findings

Proved the conjecture with an optimal Lebesgue measure bound.

Characterized the sets with maximal measure avoiding solutions to x+y=3z.

Provided a complete solution to the inverse problem.

Abstract

In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture first made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by Matolcsi and Ruzsa, who made a first significant step towards it. Here, we prove the full conjecture by giving an optimal upper bound for the Lebesgue measure of a 3-sum-free subset A of [0,1], that is, a set containing no solution to the equation x+y=3z where x,y and z are restricted to belong to A. We then address the inverse problem and characterize precisely, among all sets with that property, those attaining the maximal possible measure.