  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.
