I was wondering how random unit lattices in number fields are. To make this more precise:

If $K$ is a number field with embeddings $\sigma_1, \dots, \sigma_n, \overline{\sigma_{r+1}}, \dots, \overline{\sigma_n} \to \mathbb{C}$ (so we have $r$ real embeddings and $2 (n - r)$ complex embeddings), let $\mathcal{O}_K$ be the ring of integers and $\Lambda_K := \{ (\log |\sigma_1(\varepsilon)|^{d_1}, \dots, \log |\sigma_n(\varepsilon)|^{d_n}) \mid \varepsilon \in \mathcal{O}_K^\ast \}$ be the unit lattice, where $d_i = 1$ if $\sigma_i(K) \subseteq \mathbb{R}$ and $d_i = 2$ otherwise.

Then $\Lambda_K$ is always contained in $H := \{ (x_1, \dots, x_n) \in \mathbb{R}^n \mid \sum_{i=1}^n x_i = 0 \}$, and $\det \Lambda_K$ is the regulator $R_K$ of $K$. Let us normalize $\Lambda_K$ by $\hat{\Lambda}_K := \frac{1}{\sqrt[n]{R_K}} \Lambda_K$; then $\det \hat{\Lambda}_K = 1$.

Now my question is: can we say something on how random the lattices $\hat{\Lambda}_K$ are among all lattices in $H$ of determinant 1? (For example, for fixed signature $(r, n-r)$ of $K$.)

Since these lattices are not completely random (they consist of vectors of logarithms of algebraic numbers), it is maybe better to ask something like this:

Given $\varepsilon > 0$ and a lattice $\Lambda \subseteq H$ with determinant 1, does there exists a number field $K$ of signature $(r, n - r)$ such that there is a basis $(v_i)_i$ of $\hat{\Lambda}_K$ and a basis $(w_i)_i$ of $\Lambda$ such that $\|v_i - w_i\| < \varepsilon$ for all $i$?

And if this exists, can one bound the discriminant of $K$ (or any other invariant of $K$) in terms of $\varepsilon$?

(Of course, this question is only interesting when $n > 2$.)

I assume that this is a very hard problem, so I'd be happy about any hint on whether something about this is known, whether someone is working on this, how one could proof such things, etc.

1more comment