A very important caveat to the previous lemma is the fact that there must exist an infinite number of primes $p$ such that the construction above is possible. This follows from Chebotarev’s density theorem (e.g. [@Algebraic1][ Cor. 13.6, p. 547]{}), which implies that the natural density $\delta$ of primes that split completely in $K$ is positive (indeed, one has $0 < \delta \leq 1/n!$).

Very similarly, it is possible to prove that the constructions in [@ConsAOggier] are non-degenerate.

Suppose that $K$ contains $r(K)$ roots of unity. Let $\mu$ be a root of unity contained in $\mathcal{O}_K$. It follows that $\left\| \sigma(\mu {\mathbf{x}})\right\| = \left\| \sigma({\mathbf{x}})\right\|$ (e.g. [@Autissier Lem. 3.1]). Therefore, each $\Lambda$ constructed as in Definition \[def:con2\] contains at least $r(K)$ minimal vectors and we automatically obtain the density $r(K)(1-\varepsilon)/2^{nt}$ (this argument was used by Venkatesh [@Venkatesh] to prove that cyclotomic lattices ($\mathbb{Z}[\mu]$-lattices where $\mu$ is a root of unity) achieve density $m(1-\varepsilon)/2^{2\phi(m)}$). For $t$-dimensional $\mathcal{O}_K$ lattices, however, there is a loss of a linear factor of $t$ in the enumerator. Nevertheless, we can improve the following density up to that of Corollary \[cor:betterCyclotomic\], using a method by Rogers [@Rogers], recently employed in [@Vance] to quaternionic lattices. The basic idea 