[论文解读] The Davenport constant of an interval: a proof that $\mathsf{D}=χ$

Benjamin Girard, Alain Plagne|arXiv (Cornell University)|Jan 12, 2026

Limits and Structures in Graph Theory被引用 0

一句话总结

作者证明区间 ⟪-m,M⟫ 的 Davenport 常数等于 m+M−ρ(m,M)，通过对极小零和序列和函数 ρ(m,M) 的详细分析确认了猜想恒等 D(⟪-m,M⟫)=χ(⟪-m,M⟫)。

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 $\gcd (M-t_1, m-t_2)=1$.

研究动机与目标

• Motivate the study of Davenport constants for intervals of integers and the search for a closed form.
• Define the Davenport constant for the interval ⟪-m,M⟫ and relate it to the Jacobsthal-like function ρ(m,M).
• Prove that D(⟪-m,M⟫)=m+M−ρ(m,M) and thus establish D=χ for these intervals.
• Develop the technical framework to bound D(⟪-m,M⟫) via structural analysis of minimal zero-sum sequences.
• Describe the behavior of ρ(m,M) and its role in attaining the conjectured formula.

提出的方法

• Reformulate the conjecture using the function ρ(m,M) and show that proving D(⟪-m,M⟫) ≤ m+M−ρ(m,M) suffices.
• Prove equality for key special cases ρ(m,M)=0,1,2,3 through detailed combinatorial constructions and inverse theorems.
• Use permutations of sequence elements to bound partial sums within shrinking intervals, ensuring minimal zero-sum sequences cannot be too long.
• Employ lemmas on gcd structure and minimal zero-sum sequences of the form M^α(−m)^β and related variants.
• Leverage results from Deng and Zeng to handle the case ρ(m,M)≥4 and complete the proof of Theorem 1.

实验结果

研究问题

• RQ1 What is the exact value of the Davenport constant for the interval ⟪-m,M⟫?
• RQ2 Is D(⟪-m,M⟫) equal to χ(⟪-m,M⟫) for all positive integers m,M?
• RQ3 How does the coprimality structure of m, M and nearby integers affect the maximal length of a minimal zero-sum sequence?
• RQ4 How can the function ρ(m,M) be used to obtain sharp upper bounds for D(⟪-m,M⟫)?
• RQ5 What are the special cases (ρ(m,M)=0,1,2,3) where an inverse result can be obtained?

主要发现

• D(⟪-m,M⟫) = m+M − ρ(m,M) for all positive integers m,M.
• χ(⟪-m,M⟫) = m+M − ρ(m,M), yielding D=χ as a corollary.
• The case ρ(m,M)=0 occurs when gcd(m,M)=1 and yields D(⟪-m,M⟫)=m+M.
• For ρ(m,M)=1 or 2, D(⟪-m,M⟫) equals m+M−1 or m+M−2 respectively, with inverse results in some cases.
• The analysis uses permutation-based control of partial sums, gcd-based bounds, and structured minimal zero-sum sequences such as M^α(−m)^β.
• The approach connects to Deng and Zeng’s inequality framework to finalize the proof when ρ(m,M)≥4.