A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet another) variation on Kronecker's theorem on units. In turn, this implies that on any Oeljeklaus-Toma manifold associated to a number field with
![$2t$](img41.png)
complex embeddings and
![$s$](img42.png)
real embeddings with
![$1<s\leq t$](img43.png)
there is no LCK metric.