Sums of two units in number fields

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$. Let ${\mathcal{N}}_K$ be the set of positive integers $n$ such that there exist units $\epsilon ,\delta \in {\mathcal{O}}_K^{\times }$ satisfying $\epsilon +\delta =n$. We show that ${\mathcal{N}}_K$ is a finite set if $K$ does not contain any real quadratic subfield. In the case where $K$ is a cubic field, we also explicitly classify all solutions to the unit equation $\epsilon +\delta =n$ when $K$ is either cyclic or has negative discriminant.