Sums of two units in number fields

Abstract

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 $\epsilon ,\hat{I}´\in {\mathcal{O}}_K^{\times }$ satisfying $\epsilon +\hat{I}´=n$. 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 $\epsilon +\hat{I}´=n$ when $K$ is either cyclic or has negative discriminant.

21 pages; to appear in Mathematische Zeitschrift. Made some minor revisions, and added Proposition 2.2 and Proposition 6.5