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 ?