\Lambda{\backslash\left\{0\right\}}}r^{m} \gamma f(r\beta {\mathbf{x}}) = (V^{-1}\zeta(m))\int_{W} f({\mathbf{x}}) \mbox{d} {\mathbf{x}},$$ by the definition of Riemann integral. Exchanging the limit and the sum is justified by dominated convergence, given the condition on $f$.

If $\Lambda=\mathbb{Z}^n$ and $\phi_p$ is the reduction modulo $p$, we obtain mod-$p$ lattices as in [@Loeliger]. It is clear that $\Lambda_p = p\mathbb{Z}^n$ satisfies the hypothesis of Theorem \[thm:AverageEnsembleGeneral\], with $m=n$ and $\alpha = k/n$. This implies Theorem 1 of [@Loeliger].

A sequence of surjective homomorphisms $(\phi_j)_{j=1}^{\infty}    $, $\phi_j:\Lambda \to \mathbb{F}_{p_j}^n$ is said to be *non-degenerate* if $$\lambda(\Lambda_{p_j}) \geq c p_{j}^{\frac{n}{m}},$$ for some constants $c > 0$. Similarly, the sequence of associated ensembles (Equation \[eq:ensemble\]) are said to be non-degenerate.

It follows that if the reductions are non-degenerate, the associated ensemble admits the Minkowski-Hlawka theorem.

If $m=n$, the natural reduction to $\mathbb{F}_p^n$ is as follows. Given a basis ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_n$ for $\Lambda$, take $\phi_p$ to be the linear map defined by $\phi_p({\mathbf{x}}_i) = \bm{e}_i \in \mathbb{F}_p^n$, where $\mathbf{e}_i$ the $i$-th canonical vector $(0,\ldots,0,1,0,\ldots,0)$. It is clear that $\phi$ is surjective and $\ker \phi_p = p \Lambda$, therefore the associated sequence of reductions is non-degenerate. This provides a systematic way of constructing good sublattices of a given lattice. \[ex:naturalReduction\]

Taking $f(x)$ 