# Minimal diameter of a class in a number field

Let $\mathbb{K}$ be a number field of degree $n$, let $A = \mathcal{O}(\mathbb{K})$ be the ring of integers and consider the Minkowski embedding $\mathbb{K} \rightarrow \mathbb{R}^n$ given by $x \mapsto (\sigma_i(x))$, with $\sigma_i$ the various embeddings of $\mathbb{K}$.

For a full lattice $L \subset \mathbb{R}^n$, we define the orthonormality defect by $o(L) = \frac{\delta(L)}{2 \det(L)^{1/n}}$, where $\delta(L)$ is the diameter of the fundamental paralellepiped of the lattice.

To an ideal $\mathfrak{a} \subset A$ we associate the lattice $L_{\mathfrak{a}} \subset A$ consisting of the images of algebraic integers of $\mathfrak{a}$ in $\mathbb{R}^n$. To an ideal class $\mathfrak{K}$ we associate the radius $\rho(\mathfrak{K}) := \inf_{\mathfrak{a} \in \mathfrak{K}} ( o(L_{\mathfrak{a}} ))$. The radius is thus the smallest orthonomrality defect of a lattice in the ideal. What useful upper bounds can one find for $\rho(\mathfrak{K})$?