Let $K$ be a totally real number field of degree $n > 0$ over the rationals. I'd like to find an algebraic integer $a ∈ \mathcal{O}_K$ such that the embeddings $σ_1 = \mathrm{id}, σ_2, …, σ_n$ of $K$ into the complex pane $ℂ$ satisfy $$|σ_1(a)| > 2^{2n + 2}, 0 < |σ_2(a)| < \frac 12, …, 0 < |σ_n(a)| < \frac 12.$$
I am sure one can use Minkowski's theorem on convex bodies to find such an element. However, I don't know which lattice to choose.
To get a feeling for what is happening I considered the case of $K = ℚ[\sqrt{2}]$, which is a normal over $ℚ$. Using the lattice $ℤ \times \sqrt{2} ℤ \cong \mathcal{O}_K$ I plotted the respective boundaries and could find the algebraic integer $34 + 24 \sqrt{2}$ satisfying my requirements. However, the area where the demanded points lie is not convex; it is the complement of a parallelogram between two straight lines of distance $1$. Am I on the right track here?
Furthermore, is the condition on $K$ being totally real necessary?