Growth of multiplicities of graded families of ideals

Abstract: Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = {I_n}{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$, $$\ell_R(R/I{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$