Length, multiplicity, and multiplier ideals

Let $(R,\mathfrak{m})$ be an $n$-dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ of $R$, the relationship between the singularities of the scheme defined by $\mathfrak{a}$ and those defined by the multiplier ideals $\mathcal{J}(\mathfrak{a}^c)$, with $c$ varying in $\mathbb{Q}_+$, are quantified in this paper by showing that the Samuel multiplicity of $\mathfrak{a}$ satisfies $e(\mathfrak{a}) \ge (n+k)^n/c^n$ whenever $\mathcal{J}(\mathfrak{a}^c) \subseteq \mathfrak{m}^{k+1}$. This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.