Un crible minorant effectif pour les entiers friables

Let $\mathcal {A}$ be a finite set of positive integers and let $y\geq 1$. We give an effective lower bound of the cardinality of the set $\{n\in \mathcal {A}:p\,|\, n\Rightarrow p\leq y\}$ under the condition of a good knowledge of the level of distribution of $\mathcal {A}$. Some consequences are studied: an application to the friable values of irreducible polynomials or binary forms with integer coefficients, and an application to shifted friable numbers.