Length of local cohomology of powers of ideals

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb Z$, $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n)_{\geq -\alpha n})}{n^d}<\infty.$ Combining this with recent strong vanishing results gives that $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}{R/I^n})}{n^d}<\infty$ in many situations. We also establish that the actual limit exists and is rational for certain classes of monomial ideals $I$ such that the lengths of local cohomology of $I^n$ are eventually finite. Our proofs use Gr\"obner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that $\liminf_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n))}{n^d}>0$ when $R/I$ has "nice" singularities in both zero and positive characteristics.