  Let $X$ be a subset of $\N^t$ or $\Z^t$. We can associate with $X$ a function
${\cal G}_X:\N^t\longrightarrow\N$ which returns, for every $(n_1, ..., n_t)\in
\N^t$, the number ${\cal G}_X(n_1, ..., n_t)$ of all vectors $x\in X$ such
that, for every $i=1,..., t, |x_{i}| \leq n_{i}$. This function is called the
{\em growth function} of $X$. The main result of this paper is that the growth
function of a semi-linear set of $\N^t$ or $\Z^t$ is a box spline. By using
this result and some theorems on semi-linear sets, we give a new proof of
combinatorial flavour of a well-known theorem by Dahmen and Micchelli on the
counting function of a system of Diophantine linear equations.
