to be the indicator function of a ball in part (ii) of Theorem \[thm:AverageEnsembleGeneral\], we recover Theorem \[thm:2\]. Another function of interest is $f(x)= e^{-\tau \left\|x\right\|^2}$ for $\tau > 0$, yielding the *theta series* $$\Theta_{\Lambda}(\tau) = \sum_{x \in \Lambda} e^{-\tau \left\|x\right\|^2}.$$ A corollary of part (i) of the theorem is the following:

The average theta series of a sequence of non-degenerate ensemble satisfies $$\lim_{p_j \to \infty} E_{\mathbb{L}_{p_j}}\left[\Theta_{\Lambda}(\tau)\right] = V^{-1} \left(\frac{\pi}{\tau}\right)^{m/2} + 1.$$ \[cor:thetaSeries\]

Corollary \[cor:thetaSeries\] can, for instance, be applied to the construction of sufficiently flat Gaussian measures for secure communications (cf [@LLBS_12]).

The condition for non-degeneracy can be re-written as $$\liminf_{p_j\to\infty} \gamma(\Lambda_{p_j})  > 0,$$ where $\gamma(\Lambda) = \lambda(\Lambda)/V(\Lambda)^{1/m}$ is the Hermite parameter of $\Lambda$. In other words, non-degeneracy is equivalent to non-vanishing Hermite parameter of the sequence of kernel lattices.

We close this section with another consequence of Theorem \[thm:AverageEnsembleGeneral\]. We shall refer to each ratio $$\Delta_i(\Lambda) = \frac{\mbox{vol } \mathcal{B}_{\lambda_i/2}}{V(\Lambda)}, i = 1, \ldots, m,
\label{eq:successiveDensities}$$ as the $i$-th successive density of a lattice $\Lambda$. For a sequence of non-degenerate ensemble, put $\mathbb{L} = \bigcup_{j=1}^{\infty}\mathbb{L}_{p_j}$.

For any $\varepsilon > 0$, there exists $\Lambda_{p_j} \in \mathbb{L}$ such that $$\prod_{i=1}^m \Delta_i(\Lambda)^{1/m} \geq \frac{2 m \zeta(m)(1-\varepsilon)}{e(1-e^{-m})}.$$

The proof follows from a 