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

In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture rst 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 rst signicant 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 =3 z 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.