Effective equidistribution of norm one elements in CM-fields

For a number field $K$ let ${\mathcal{S}}_K$ be the maximal subgroup of the multiplicative group $K^{\times }$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group ${\mathcal{S}}_K$ can be seen as an archimedean counterpart to the group of units ${\mathcal{O}}_K^{\times }$ of the ring of integers ${\mathcal{O}}_K$. If $K=\mathbb{Q}({\mathcal{S}}_K)$ is a CM-field then ${\mathcal{S}}_K\mathrm{/}(K^{\times })$ is a free abelian group of infinite rank. If $K=\mathbb{Q}({\mathcal{S}}_K)$ is not a CM-field then ${\mathcal{S}}_K=\{\pm 1\}$. In the former case ${\mathcal{S}}_K$ is the kernel of the relative norm map from $K^{\times }$ to the multiplicative subgroup $k^{\times }$ of the maximal totally real subfield $k$ of $K$.