Rational integers as sums of units -- the quadratic case

TL;DR

This paper investigates the asymptotic behavior of natural numbers below a large threshold that can be expressed as sums of units in quadratic number fields, solving a problem posed in 2007.

Contribution

It provides the asymptotic count of such numbers in quadratic fields, addressing a previously open problem by Jarden and Narkiewicz.

Findings

Asymptotic formulas for the count of numbers as sums of units

Resolution of Jarden and Narkiewicz's 2007 problem for quadratic fields

Enhanced understanding of units in quadratic number rings

Abstract

How many natural numbers below $X$ can be written as a sum of $k$ units of the ring of integers of a given number field? We give the asymptotics as $X$ gets large for quadratic number fields. This solves a problem of Jarden and Narkiewicz from 2007 for quadratic number fields.