Playing a little bit with Groessencharacters a stumbled on the following question:
Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$ and assume that $r\geq 1$. Let $\epsilon_1,\ldots,\epsilon_{r}$ be a system of $\mathbf{Z}$-linearly independent (in the multiplicative notation) units of $\mathcal{O}_K$. For $x\in K$ and $j\in\{1,2,\ldots,r_2\}$ let $x^{(j)}$ be the image of $x$ under the $j$-th complex embedding (where the $r_2$ choices of complex embeddings are made so that no complex embedding are related by complex conjugation).
Q: Is it possible to choose $K$ and $\{\epsilon_1,\ldots,\epsilon_{r}\}$ so that the above assumptions are satisfied and such that there exists an $(r+r_2)$-tuple $((m_1,\ldots,m_r);(n_1,\ldots,n_{r_2}))\in\mathbf{Z}^{r+r_2}\backslash\{0\}$ such that $$ 2\pi m_j-\sum_{k=1}^{r_2} n_k\cdot\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)=0, $$ for all $j\in\{1,\ldots,r\}$ ?
This looks like a delicate question about the $\mathbf{Q}$-linear independence of the real numbers $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)$.
added The non-CM assumption is essential since for a CM unit $\epsilon_j$, we have $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)\in\pi\mathbf{Q}$. Also, if $\epsilon_j\in\mathcal{O}_K^{\times}$ is chosen to be totally real then $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)\in 2\pi\mathbf{Z}$. So may be one could come up with an example that would combine CM units and totally real units...
added following Kevin's comment. Let us look at the infinite part of Groesscharacter $\chi$ (so by character I mean continuous and unitary), which we denote by $\chi_{\infty}$. Then it is a continuous group homomorphism $\chi_{\infty}:(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\rightarrow S^1$ which is invariant under a finite index subgroup of $U\leq\mathcal{O}_K^{\times}$. If we take $U$ to be the group generated by the $\epsilon_j$'s then such characters are "essentially" parametrized by
(1) an $(r+r_2)$-tuples $(m;n)\in\mathbf{Z}^{r+r_2}$,
(2) a "sign character", i.e., a character which is trivial on the connected component of the identity of $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times})$, and
(3) a real number $t$.
The question I raised is essentially equivalent to ask if the pair $(m;n)$ is uniquely determined by $\chi_{\infty}$ ? For example if $K$ is totally real then it is easy to see that (this is the calculation that was performed by Hecke in the first paper where he introduces such characters) two distinct $r$-tuples $m$ and $m'$ give rise to two distinct characters $\chi_{\infty}$ and $\chi_{\infty}'$. In general, this uniqueness already fails for quartic CM fields (and also for those of larger degrees). So it looked to me like a natural question to ask for what number fields does this "uniqueness" fail ?
