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

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