Abstract : Let $k > 2$ be a real number. We inquire into the following question : what is the maximal size (inner Lebesque measure) and the form of a set avoiding solutions to the linear equation $x + y = kz$ ? This problem was used for $k$ an integer larger than 4 to guess the density and the form of a corresponding maximal set of positive integers less than $N$. Nevertheless, in the case $k = 3$, the discrete and the continuous setting happen to be very different. Although the structure of maximal sets in the continuous setting is quite easy to describe for $k$ far enough from 2, it is more difficult to handle as $k$ comes closer to 2. Joint work with Alain Plagne.