On units with complex Galois conjugates of equal absolute value

Résumé

We investigate the following question:

 Given a number field K with s real embeddings and 2t complex ones has a group of units  such that all elements in U have all its complex conjugates of same absolute value, does it follow that t = 1?

This fact has an interesting implication in complex hermitian geometry, namely it describes all Oeljeklaus–Toma manifolds carrying locally conformally Kähler structures. We prove that the stated question has an affirmative answer under a (relatively mild) condition on K, namely that for some finitely may extensions L of it, L has finitely many units lying on some specific circle.