Growth of multiplicities of graded families of ideals

Let (R,m) be a Noetherian local ring of dimension d>0. Let I
- ={In}n∈N be a graded family of m-primary ideals in R. We examine how far off from a polynomial can the length function ℓR(R/In) be asymptotically. More specifically, we show that there exists a constant γ>0 such that for all n≥0,ℓR(R/In+1)−ℓR(R/In)<γnd−1.