is to apply to Theorem \[thm:AverageEnsembleGeneral\], rather than indicator function of a ball, to a bounded-support function that allows us to analyze the generalized densities of the ensemble. After ensuring the existence of a lattice with good generalized densities, it is possible to apply standard linear transformations (see e.g. [@Vance Thm. 2.2]) in order to ensure the existence of a lattice with good *packing density*.

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*Proof of Corollary \[cor:betterCyclotomic\]:* . For $\Lambda_0 \subset \mathcal{O}_K^t$ let the $i$-th successive minima of $\Lambda_0$ (over $K$) be the smallest $i$-such that the ball $B_r^{nt}$ contains the canonical embedding of $i$ linearly independent vectors (over $K$). More formally $$\begin{split}
\lambda_i^K(\Lambda_0) = \min \left\{r > 0: \dim \mbox{span}_{K} \left(\sigma^{-1} \left(\sigma(\Lambda_0) \cap \mathcal{B}_r \right)\right)=i \right\}.
\end{split}$$ Notice that $\lambda_1^K(\Lambda_0) = \lambda_1(\sigma(\Lambda_0))$ and, in general $\lambda_i^K(\Lambda_0) \geq \lambda_i(\sigma(\Lambda_0))$. Also, if ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_t$ are linearly independent over $K$ and achieve the sucessive minima of $\Lambda_0$, then the embeddings $\sigma({\mathbf{x}}_1),\ldots, \sigma({\mathbf{x}}_t)$ are linearly independent and primitive in $\sigma(\Lambda_0) \subset \mathbb{R}^{nt}$. Now let $f:\mathbb{R}^{nt} \to \mathbb{R}$ be the following function with limited support: $$f({\mathbf{y}})=\left\{ \begin{array}{cc} 
1/n & \mbox{if }\left\|{\mathbf{y}}\right\| \leq re^{(1-t)/tn} \\
\frac{1}{nt} - \log\left(\frac{\left\|{\mathbf{y}}\right\|}{r}\right) & \mbox{if } re^{(1-t)/tn}\leq \left\|{\mathbf{y}}\right\| \leq re^{1/tn}\\
0 & \mbox{otherwise.}\end{array}\right.$$ We have $$\int_{\mathbb{R}^{nt}} f({\mathbf{y}}) \mbox{d}{\mathbf{y}}= 