Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring, and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in $R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show that $$ \lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d} =\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d} $$ always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals $I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth.

We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.