Let $K$ be a number field. Let $\mathcal{O}_K^{*}$ be the units in the ring of integers of this field.

I am interested in knowing how many units $u,v \in \mathcal{O}_K^{*}$ exist such that $u + v$ is also a unit. This is equivalent to knowing units $u$ such that $1+u$ is also a unit.

In particular, I'm interested when we have cyclotomic number fields as $K$. Some easy examples can be constructed here by using elements of the form $1+\zeta_n^m$.

Are there any results of this type, perhaps about density of such units $u$? Where is the starting point for this?

exceptionalif $1-u$ is also a unit, so that gives you a search term. There is a fair bit of literature about them. Lenstra studied them, he might be responsible for the terminology. $\endgroup$