# Units with small log-Minkowski norm

Let $\mathbb{K}$ be a number field, let $r_1, r_2$ be the numbers of real resp. pairs of complex conjugate embeddings, and let $\sigma_i; \,i = 1,2, \ldots, r_1$ count the real embeddings, while $(\sigma_{r_1+i}, \sigma_{r_1+r_2+1}); \,i = 1, 2, \ldots, r_2$ are the pairs of complex conjugate embeddings. The norm indices $n_i = 1$ for $1 \leq i \leq r_1$ and $n_i = 2$ for $r_1 < i \leq r_1 + r_2$.

The log-Minkowski map $\ell : \mathbb{K}^{\times} \rightarrow \mathbb{R}^{r_1+r_2}$ is given by \begin{eqnarray*} x \mapsto \left( \log( | \sigma_i(x) |^{n_i} \right)_{i=1}^{r_1+r_2} . \end{eqnarray*}

Let $E \subset \mathcal{O}^{\times}(\mathbb{K})$ be a $\mathbb{Z}$-free submodule of the units such that $\mathcal{O}^{\times}(\mathbb{K}) = E \cdot W$, with $W \subset \mathbb{K}$ the roots of unity. Let $\upsilon_{\mathbb{K}} = \inf\{ | \ell( \varepsilon ) | : \varepsilon \in E \}$ be the smallest euclidean length of an $\ell$-image of some unit in $E$.

What lower bound for $\upsilon_{\mathbb{K}}$ can one prove? The case when the field is neither totally real nore CM is of particular interest.

• If $\varepsilon$ is a root of unity then $|\ell(\varepsilon)|=0$. You are presumably ruling out this case? And what do you want the implied constants to depend on in the big-O? For a given number field the min is the min, it's an invariant of the number field. – Kevin Buzzard Jan 18 '17 at 23:30
• Kevin - thank you, of course I meant non-torsion units! And the constants can depend on things like the extension degree $n$ or the embedding numbers $r_i, i = 1, 2$ ... not on themselves, of course. Meanwhile I was informed that the problem is addressed and there are some papers on the extremal value of inidividual embeddings, which is a quite related problem. If $\K$ is in addition galois, then Minkowski units can be very well approximated, since they have equal length in the log-embedding. How can one descend from the Minkowski-units lattice to the one of units? – Preda Jan 20 '17 at 7:48
• I am still confused by this question. If the implied constant in $O(\sqrt{r_1+r_2})$ can depend on $r_1$ and $r_2$ then there's no point having the $\sqrt{r_1+r_2}$ in the bracket as you can just put it into the constant. Note that I'm extremely unlikely to actually be able to answer this question, I am just trying to understand it. Perhaps to an expert what I'm asking is obvious but I don't really have a feeling for this sort of thing. – Kevin Buzzard Jan 20 '17 at 8:35
• Kevin, I do not know what a realistic order of magnitude would be! I just mentioned that I can find in some cases a proof of that order, but I cannot in the general case. So please forget the example. The question is "what lower bound can one prove when $\mathbb{K}$ is neither real nore CM". One expects the bound to be a function of $n$, or maybe individually of $r1,r2$ (?!) , but I made no spefic suggestion, so I think trying to infer from the example is not in the spirit of my question. Therefore, I remove the example! – Preda Jan 21 '17 at 11:38