Sums of two units in number fields
Magdaléna Tinková, Robin Visser, Pavlo Yatsyna
TL;DR
This work studies the set $\\mathcal{N}_K = \\{ n \\in \\mathbb{Z}_{>0} : \\exists \\varepsilon,\\delta \\in \\mathcal{O}_K^{\\times} \,\\text{with} \\varepsilon + \\delta = n \\\}$ of positive integers representable as a sum of two units in the ring of integers of a number field $K$. It proves finiteness of $\\mathcal{N}_K$ for number fields not containing a real quadratic subfield, deriving an explicit bound via Amoroso–Viada bounds on unit equations in three unknowns, and it specializes to cubic fields to give a complete classification of unit-sum solutions in both cyclic and complex cubic fields, separating infinite families from finitely many sporadic cases. For cyclic cubic fields $K_a$ and complex cubic fields $L_a$, the paper provides precise descriptions of all solutions to $\\varepsilon + \\delta = n$, identifies the sporadic solutions, and yields explicit $\\mathcal{N}_K$ sets in many instances, including computational tables for several families. Computational methods based on discriminants, minimal polynomials, and Thue equations are developed to determine $\\mathcal{N}_K$ for individual fields, with concrete results and tables for cyclic and complex cubics. The work ends with open problems on averages, extrema, and algorithmic determination of $\\mathcal{N}_K$ and its generalizations to sums of more units.
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 $\varepsilon, δ\in \mathcal{O}_K^\times$ satisfying $\varepsilon + δ= n$. 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 $\varepsilon + δ= n$ when $K$ is either cyclic or has negative discriminant.