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The Davenport constant of an interval: a proof that $\mathsf{D}=χ$

Benjamin Girard, Alain Plagne

TL;DR

This work resolves the Davenport-constant problem for intervals of integers by proving $\mathsf{D}(\llbracket -m,M \rrbracket)=m+M-\rho(m,M)$, where $\rho(m,M)$ is defined via a coprimality condition. The authors first reformulate the conjecture through $\chi(\llbracket -m,M \rrbracket)$, then establish exact results for the small $\rho$ cases ($0$–$3$) using detailed analyses of minimal zero-sum sequences. They obtain general upper bounds for $\rho(m,M)$ via Jacobsthal-type bounds on coprimality gaps and complements, finally applying Deng–Zeng’s inverse theorem to cover larger $\rho$; the combination yields the claimed equality for all $(m,M)$. The results connect Davenport constants of integer intervals to Jacobsthal function bounds, providing a complete resolution and advancing understanding of zero-sum phenomena in structured subsets of $\mathbb{Z}$.

Abstract

For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\![ -m,M ]\!]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\![ -m,M ]\!]$. We prove the conjecture that it is equal to $m+M- r$ where $r$ is the smallest integer which can be decomposed as a sum of two non-negative integers $t_1$ and $t_2$ ($r=t_1+t_2$) having the property that $\gcd (M-t_1, m-t_2)=1$.

The Davenport constant of an interval: a proof that $\mathsf{D}=χ$

TL;DR

This work resolves the Davenport-constant problem for intervals of integers by proving , where is defined via a coprimality condition. The authors first reformulate the conjecture through , then establish exact results for the small cases (–) using detailed analyses of minimal zero-sum sequences. They obtain general upper bounds for via Jacobsthal-type bounds on coprimality gaps and complements, finally applying Deng–Zeng’s inverse theorem to cover larger ; the combination yields the claimed equality for all . The results connect Davenport constants of integer intervals to Jacobsthal function bounds, providing a complete resolution and advancing understanding of zero-sum phenomena in structured subsets of .

Abstract

For two positive integers and , we study the Davenport constant of the interval of integers , that is the maximal length of a minimal zero-sum sequence composed of elements from . We prove the conjecture that it is equal to where is the smallest integer which can be decomposed as a sum of two non-negative integers and () having the property that .
Paper Structure (9 sections, 22 theorems, 145 equations)

This paper contains 9 sections, 22 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['theorem-reformulation']}
  3. Proof of special cases of inequality \ref{['ub']}
  4. The case $\rho (m,M)=0$
  5. The case $\rho (m,M)=1$
  6. The case $\rho (m,M)=2$
  7. The case $\rho (m,M)=3$
  8. An upper bound for $\rho (m,M)$
  9. Proving Theorem \ref{['theoprincipal']}

Key Result

Theorem 1

Let $m$ and $M$ be two positive integers. Then,

Theorems & Definitions (43)

  • Conjecture 1
  • Theorem 1
  • Proposition 2
  • Lemma 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 33 more