bibkey: girardplagne2026boxes
authors: Benjamin Girard and Alain Plagne
year: 2026
title: The Davenport constant of low-dimensional boxes
doi: 10.48550/arXiv.2610.16562
url: https://arxiv.org/abs/2610.16562v1
claim: "Section 2 asks whether the fixed-support box Davenport ratio has a limit in every positive dimension; Theorem 5 gives the three-dimensional asymptotic constant 16."
strata_touched:
  - D5/S3/Arith/Lattices/CofactorKernel
  - D5/S3/Arith/Lattices/CubeSimplexDeterminant
  - D5/S3/Arith/Lattices/PrimitivePencil
  - D5/S3/Arith/Lattices/MinimalKernelPacking
  - D5/S3/Arith/Lattices/GirardPlagneBoxLimit
license: citation-only
triage: anchor

Fixed-support Davenport constants of integer boxes

Verified locator

DOI: https://doi.org/10.48550/arXiv.2610.16562

Source: https://arxiv.org/abs/2610.16562v1

The version is arXiv:2610.16562v1. Section 2 is titled “New results”. Theorem 5 and the question below appear on printed page 4.

Source statements

Section 2, printed page 2, defines the quantity:

we study the quantity D^(d+1)(⟦−m, m⟧^d) which is defined as the maximal length of a minimal zero-sum sequence over ⟦−m, m⟧^d having a support of size d + 1.

Section 2, printed page 4, asks:

It appears tempting to conjecture that for every integer d ≥ 1, the ratio D^(d+1)(⟦−m, m⟧^d)/m^d has a limit when m tends to infinity. This is clear when d = 1, and it follows from Theorems 1 and 5 when d ∈ {2, 3}. However, this problem remains open when d ≥ 4.

Section 1, printed page 1, defines minimality:

Moreover, if S is a zero-sum sequence such that ∑ᵢ∈ᴵ xᵢ ≠ 0 for every ∅ ≠ I ⊊ ⟦1, ℓ⟧, it will be called a minimal zero-sum sequence.

The sum of every nonempty proper subsequence is required to be nonzero. Multiplicity functions with values in the natural numbers represent sequences; their pointwise order represents subsequence inclusion. A support has one element per distinct vector.

Mathematical correspondence

For dimension d and radius m, the box consists of integer vectors v satisfying ∀ i, |v i| ≤ m. The length set records all nonzero minimal zero-sum multiplicity functions with support cardinality exactly d + 1. Its natural-number supremum is the source’s fixed-support Davenport constant. The formal claim includes nonemptiness and boundedness for every m ≥ 1.

The repository-derived strengthening identifies the limit with delta d, the finite maximum of the absolute determinant after adjoining a top row of ones to a d by d + 1 sign matrix. Full rank gives a primitive cofactor kernel and its exact length formula. A cube vertex estimate bounds the determinant, and a packing estimate makes deficient rank lower order. Smith form supplies primitive integer pencils approaching an extremal simplex.

Theorem 1’s two-dimensional formula and Theorem 5’s three-dimensional asymptotic agree with this limit. The source’s unrestricted box Davenport constant questions and exact higher-dimensional values are separate questions. No general evaluation of the augmented maximal determinant is asserted.

Numerical evidence and literature boundary

The independent experiment is docs/reports/girard-plagne-2026-box-davenport-limit/check.py in the-omega-institute/trureturing-experiments, commit 73c308cec1a907b063b6e7a614083b5f71b0b0d8, SHA-256 94291f19d14913886f357a5e8d364f36d1ac02fd602cb34bbae46d37030c2fea. The supplied reading is python3 check.py, exit 0, ALL_OK. These computations do not replace the general proof.

The supplied literature screen covers arXiv:2510.20412 (v1 and v2), arXiv:2601.07950, Girard–Zotova arXiv:2510.23543, Plagne–Tringali arXiv:1405.4363, and arXiv:2610.14546. No settlement was found in that screen. Sturmfels’s monograph, Chapters 6–7, was not obtained. Literature completeness is ASSUMED-UNVERIFIED; no exhaustive priority claim is made.