  Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the
commensurability growth function assigns to each $n$ the cardinality of the set
of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap
\Delta] = n$. This growth function gives a new setting where methods of F.
Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent
groups" apply to study arithmetic lattices in an algebraic group. In
particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with
arithmetic lattice $\Gamma$, the Dirichlet function associated to the
commensurability growth function satisfies an Euler decomposition. Moreover,
the local parts are rational functions in $p^{-s}$, where the degrees of the
numerator and denominator are independent of $p$. This gives regularity results
for the set of arithmetic lattices in $G$.
