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Rational integers as sums of units -- the quadratic case

Christopher Frei, Martin Widmer, Volker Ziegler

TL;DR

The paper solves the Jarden–Narkiewicz problem for quadratic fields by proving a sharp asymptotic for the count of integers representable as sums of at most $k$ units in a real quadratic field $L$, with the main term governed by the fundamental unit $\eta$ via $\rho=\lfloor k/2\rfloor$ and constants depending on parity. The authors reduce the problem to counting traces of units with no vanishing subsums, using a finite, structured reduction to unit-trace sums and controlling non-uniqueness via unit-equation finiteness arguments. They establish a precise asymptotic for the trace-sum counts $T_{L,\ell}^{\mathbf{c}}(X) = (2\log X/\log\eta)^{\ell} + O((\log X)^{\ell-1})$ and show that collisions among representations contribute only $O((\log X)^{\rho-1})$, allowing the dominant term to be extracted. Consequently, they obtain $\operatorname{N}_{L,k}(X)= c_k (2\log X/\log\eta)^{\rho} + O((\log X)^{\rho-1})$ with $c_k = 1/\rho!$ for even $k$ and $c_k = 3/\rho!$ for odd $k$, and a parallel result for the exact-sum variant. Additionally, the paper analyzes local solubility of the associated fibre equations, showing when fibers admit points locally, which complements the global counting picture and connects unit equations with Diophantine geometry in a concrete, quantitative way.

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.

Rational integers as sums of units -- the quadratic case

TL;DR

The paper solves the Jarden–Narkiewicz problem for quadratic fields by proving a sharp asymptotic for the count of integers representable as sums of at most units in a real quadratic field , with the main term governed by the fundamental unit via and constants depending on parity. The authors reduce the problem to counting traces of units with no vanishing subsums, using a finite, structured reduction to unit-trace sums and controlling non-uniqueness via unit-equation finiteness arguments. They establish a precise asymptotic for the trace-sum counts and show that collisions among representations contribute only , allowing the dominant term to be extracted. Consequently, they obtain with for even and for odd , and a parallel result for the exact-sum variant. Additionally, the paper analyzes local solubility of the associated fibre equations, showing when fibers admit points locally, which complements the global counting picture and connects unit equations with Diophantine geometry in a concrete, quantitative way.

Abstract

How many natural numbers below can be written as a sum of units of the ring of integers of a given number field? We give the asymptotics as gets large for quadratic number fields. This solves a problem of Jarden and Narkiewicz from 2007 for quadratic number fields.
Paper Structure (9 sections, 10 theorems, 78 equations)

This paper contains 9 sections, 10 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Reduction to unit trace sums
  3. Proof of Proposition \ref{['prop:trace_sum_reduction']}
  4. Counting unit trace sums
  5. Proof of Proposition \ref{['prop:Counting-Tracesums']}
  6. Upper bounds for non-unique tuples
  7. Proof of Proposition \ref{['prop:non_unique_bound']}
  8. Proof of Theorem \ref{['thm:main']}
  9. Local solubility

Key Result

Theorem 1

Let $\eta>1$ be the fundamental unit of the real quadratic field $L$, let $k\in\mathbb{N}$ and define $\rho:=\lfloor k/2\rfloor$. Then, for $X\geq 2$, where

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Remark 1
  • Corollary 2
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • Lemma 2
  • ...and 7 more