On linear combinations of units with bounded coefficients and double-base digit expansions

TL;DR

This paper generalizes a classical result on expressing algebraic integers as sums of units and explores applications to signed double-base digit expansions, providing new insights into number field representations.

Contribution

It extends Belcher's 1970s result on algebraic integers as sums of units and applies this to develop signed double-base digit expansions.

Findings

Generalized unit sum representations for algebraic integers

Established conditions for signed double-base digit expansions

Provided applications to number field representations

Abstract

Let $\ord$ be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in $\ord$ is the sum of pairwise distinct units, if the unit equation $u+v=2$ has a non-trivial solution $u,v\in\ord^*$. We generalize this result and give applications to signed double-base digit expansions.