Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by $\sigma_{s+1},\overline{\sigma_{s+1}},\dots,\sigma_{s+t},\overline{\sigma_{s+t}}$. set $n=s+2t$. Now we define a map $\Sigma\colon K \to \mathbb{R}^{n}$ by $$ \alpha\in K \mapsto (\sigma_1(\alpha),\ldots,\sigma_{s}(\alpha), \Re \sigma_{s+1}(\alpha), \Im \sigma_{s+1}(\alpha), \ldots, \Re \sigma_{s+t}(\alpha), \Im \sigma_{s+t}(\alpha))\in\mathbb{R}^n. $$ We get a lattice $\Lambda = \Sigma(O)$.

What upper bounds are known for the covering radius of the lattice $\Lambda$?

This is equivalent to the following question.

What upper bounds are known for the diameter of the Voronoi cell corresponding to the lattice $\Lambda$?

(diameter of the Voronoi cell means smallest radius of a ball containing the Voronoi cell)


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