Un crible minorant effectif pour les entiers friables

Let $\mathcal{A}$ be a finite set of positive integers and $y\geq 1$. We give an effective lower bound of the cardinality of the set $\{n\in\mathcal{A};\,p|n\Longrightarrow p\leq y\}$ under the condition of a good knowledge of the level of distribution of the set $\mathcal{A}$. 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.